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https://dsp.stackexchange.com/questions/51198/design-of-fir-filters-with-arbitrary-magnitude-and-phase-responses | # Design of FIR Filters with Arbitrary Magnitude and Phase Responses
I found many information in this thesis "Algorithms for the Constrained Design of Digital Filters with Arbitrary Magnitude and Phase Responses",i want to understand it.
This question is a continuation of a previous question(Complex Least Squares Approximation).
$$H(e^{j\omega})=\sum_{n=0}^{N-1}h[n]e^{-jn\omega}=\mathbf{c}^H(\omega)\cdot \mathbf{h}\tag{1}$$
I want to approximate (1) by the following desired frequency response:
$$D(\omega)=M(\omega)e^{jP(\omega)}\tag{2}$$
My problem with this specifications :
• Frequency domain specification is not symmetrical (linear phase filter with not symmetric impulse response),with order of filter and constant group delay, constant weighting function .
1-Before to start the design filter, must be know the problem is linear or no,does this have relation by 'h'?
In the case of FIR filters, we formulate the design problem using the complex least squares .The passband error function is
$$E(\mathbf{h})=\int_{\omega_1}^{\omega_2}| H(e^{j\omega})-D(e^{j\omega})|^2d\omega \tag{3}$$
How can i solve the integral in this case?or what is the steps which allow me to formulate an equation by least squares?
In complex leastsquares there is continuous and discrete approximation .However, often these integrals cannot be solved analytically.In this case one either has to resort to numerical integration or formulate the problem a priori as a discrete approximation problem. What is the better for (3)?use the continuous or discrete approximation?
For the sake of clarity, let me point out that you do not approximate $(1)$ by $(2)$, but the other way around: you approximate a desired frequency response, i.e., your specification, by the filter's frequency response, and this is done by choosing the coefficients in the vector $\mathbf{h}$ such that some error measure (e.g., least squares) is minimized.
Since the frequency response $(1)$ is linear in $\mathbf{h}$, the problem is a linear least squares problem, which can be solved by solving a system of linear equations. This is true for the continuous error measure $(3)$ as well as for the discrete error measure
$$E(\mathbf{h})=\sum_{k=1}^K|H(e^{j\omega_k})-D(e^{j\omega_k})|^2\tag{1}$$
where $K$ is the number of discrete frequency points $\omega_k$. In general we require $K\gg N$, where $N$ is the number of FIR filter coefficients in the vector $\mathbf{h}$.
In the general case it is more convenient to use the discrete error measure $(1)$. There are only a few special cases in which the integral of the continuous error measure (Eq. $(3)$ in your question) can be solved analytically, otherwise you have to resort to numerical approximations anyway. And even if an analytical solution is available, the difference when minimizing the discrete error measure is negligible if the number of frequency points $K$ is sufficiently large.
Minimizing the discrete squared error $(1)$ is equivalent to solving the following overdetermined linear system of equations in a least squares sense:
$$H(e^{j\omega_k})\stackrel{!}{=}D(e^{j\omega_k}),\qquad k=1,2,\ldots,K\tag{2}$$
Formulation $(2)$ is convenient because many software packages such as Matlab have syntactically elegant and computationally efficient methods to solve such systems.
I've implemented a complex FIR filter design based on solving $(2)$ in the (extremely simple) Matlab/Octave function cfirls.m.
• L :thank you for your answer, i have a question ,how can i find the least-squares error from the function 'cfirls.m'? Aug 13 '18 at 13:48
• @K.n90: The function doesn't return the error. You can compute it by evaluating the expression given by Eq. (1) in my answer. Aug 13 '18 at 14:05
• L:what is the better to plot the function ,use the plot or fvtool? Aug 13 '18 at 16:25
• @K.n90: Just a matter of preference, I always use 'plot'. Try both and see what you like better. Aug 13 '18 at 16:26
• L:i have a problem in design the filter,if i used 'h=cfirls(N,f,d,w)' , frequency response is H=hexp(-jpi*f*(0:(N-1)))', i don't get a best design ,how can i solve the problem ? Aug 14 '18 at 10:43 | 2021-09-20 09:07:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8008400201797485, "perplexity": 429.0286401187458}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00237.warc.gz"} |
https://deepai.org/publication/on-the-expressive-power-of-deep-polynomial-neural-networks | On the Expressive Power of Deep Polynomial Neural Networks
We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreducible algebraic variety upon taking closure. This paper proposes the dimension of this variety as a precise measure of the expressive power of polynomial neural networks. We obtain several theoretical results regarding this dimension as a function of architecture, including an exact formula for high activation degrees, as well as upper and lower bounds on layer widths in order for deep polynomials networks to fill the ambient functional space. We also present computational evidence that it is profitable in terms of expressiveness for layer widths to increase monotonically and then decrease monotonically. Finally, we link our study to favorable optimization properties when training weights, and we draw intriguing connections with tensor and polynomial decompositions.
Authors
• 15 publications
• 11 publications
• 81 publications
11/30/2019
Counting invariant subspaces and decompositions of additive polynomials
The functional (de)composition of polynomials is a topic in pure and com...
12/21/2021
NN2Poly: A polynomial representation for deep feed-forward artificial neural networks
Interpretability of neural networks and their underlying theoretical beh...
04/22/2018
Torus polynomials: an algebraic approach to ACC lower bounds
We propose an algebraic approach to proving circuit lower bounds for ACC...
03/18/2021
Neural tensor contractions and the expressive power of deep neural quantum states
We establish a direct connection between general tensor networks and dee...
02/07/2021
Towards a mathematical framework to inform Neural Network modelling via Polynomial Regression
Even when neural networks are widely used in a large number of applicati...
04/26/2013
An Algorithm for Training Polynomial Networks
We consider deep neural networks, in which the output of each node is a ...
11/24/2016
Survey of Expressivity in Deep Neural Networks
We survey results on neural network expressivity described in "On the Ex...
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1 Introduction
A fundamental problem in the theory of deep learning is to study the
functional space of deep neural networks. A network can be modeled as a composition of elementary maps, however the family of all functions that can be obtained in this way is extremely complex. Many recent papers paint an accurate picture for the case of shallow networks (e.g., using mean field theory chizat_global_2018 ; mei_mean_2018 ) and of deep linear networks arora_convergence_2018 ; arora_optimization_2018 ; kawaguchi_deep_2016 , however a similar investigation of deep nonlinear networks appears to be significantly more challenging, and require very different tools.
In this paper, we consider a general model for deep polynomial neural networks
, where the activation function is a polynomial (
-th power) exponentiation. The advantage of this framework is that the functional space associated with a network architecture is algebraic, so we can use tools from algebraic geometry harris_algebraic_1995 for a precise investigation of deep neural networks. Indeed, for a fixed activation degree and architecture (expressed as a sequence of widths), the family of all networks with varying weights can be identified with an algebraic variety , embedded in a finite-dimensional Euclidean space. In this setting, an algebraic variety can be thought of as a manifold that may have singularities.
In this paper, our main object of study is the dimension of as a variety (in practice, as a manifold), which may be regarded as a precise measure of the architecture’s expressiveness. Specifically, we prove that this dimension stabilizes when activations are high degree, and we provide an exact dimension formula for this case (Theorem 14). We also investigate conditions under which fills its ambient space. This question is important from the vantage point of optimization, since an architecture is “filling” if and only if it corresponds to a convex functional space (Proposition 6). In this direction, we prove a bottleneck property, that if a width is not sufficiently large, the network can never fill the ambient space regardless of the size of other layers (Theorem 19).
In a broader sense, our work introduces a powerful language and suite of mathematical tools for studying the geometry of network architectures. Although this setting requires polynomial activations, it may be used as a testing ground for more general situations and, e.g., to verify rules of thumb rigorously. Finally, our results show that polynomial neural networks are intimately related to the theory of tensor decompositions Landsberg-book . In fact, representing a polynomial as a deep network corresponds to a type of decomposition of tensors which may be viewed as a composition of decompositions of a recently introduced sort LORS-2019 . Using this connection, we establish general non-trivial upper bounds on filling widths (Theorem 10). We believe that our work can serve as a step towards many interesting research challenges in developing the theoretical underpinnings of deep learning.
1.1 Related work
The study of the expressive power of neural networks dates back to seminal work on the universality of networks as function approximators cybenko_approximation_1989 ; hornik_multilayer_1989 . More recently, there has been research supporting the hypothesis of “depth efficiency”, i.e., the fact that deep networks can approximate functions more efficiently than shallow networks delalleau2011shallow ; martens_expressive_2014 ; cohen_expressive_2016 ; cohen_convolutional_2016 . Our paper differs from this line of work, in that we do not emphasize approximation properties, but rather the study of the functions that can be expressed exactly using a network.
Most of the aforementioned studies make strong hypotheses on the network architecture. In particular, delalleau2011shallow ; martens_expressive_2014 focus on arithmetic circuits, or sum-product networks poon_sum-product_2012 . These are networks composed of units that compute either the product or a weighted sum of their inputs. In cohen_expressive_2016 , the authors introduce a model of convolutional arithmetic circuits. This is a particular class of arithmetic circuits that includes networks with layers of 1D convolutions and product pooling. This model does not allow for non-linear activations (beside the product pooling), although the follow-up paper cohen_convolutional_2016
extends some results to ReLU activations with sum pooling. Interestingly, these networks are related to Hierarchical Tucker (HT) decomposition of tensors.
The polynomial networks studied in this paper are not arithmetic circuits, but feedforward deep networks with polynomial -th power activations. This is a vast generalization of a setting considered in several recent papers venturi2018a ; du_power_2018 ; soltanolkotabi_theoretical_2018 , that study shallow (two layer) networks with quadratic activations (). These papers show that if the width of the intermediate layer is at least twice the input dimension, then the quadratic loss has no “bad” local minima. This result in line with our Proposition 5, which explains in this case the functional space is convex and fills
the ambient space. We also point out that polynomial activations are required for the functional space of the network to span a finite dimensional vector space
leshno_multilayer_1993 ; venturi2018a .
The polynomial networks considered in this paper do not correspond to HT tensor decompositions as in cohen_expressive_2016 ; cohen_convolutional_2016 , rather they are related to a different polynomial/tensor decomposition attracting very recent interest FOS-PNAS ; LORS-2019 . These generalize usual decompositions, however their algorithmic and theoretical understanding are, mostly, wide open. Neural networks motivate several questions in this vein.
Main contributions.
Our main contributions can be summarized as follows.
• We give a precise formulation of the expressiveness of polynomial networks in terms of the algebraic dimension of the functional space as an algebraic variety.
• We spell out the close, two-way relationship between polynomial networks and a particular family of decompositions of tensors.
• We prove several theoretical results on the functional space of polynomial networks. Notably, we give a formula for the dimension that holds for sufficiently high activation degrees (Theorem 14) and we prove a tight lower bound on the width of the layers for the network to be “filling” in the functional space (Theorem 19).
Notation.
We use to denote the space of homogeneous polynomials of degree in variables with coefficients in . This set is a vector space over of dimension , spanned by all monomials of degree in variables. In practice, is isomorphic to , and our networks will correspond to points in this high dimensional space. The notation expresses the fact that a polynomial of degree in variables can always be identified with a symmetric tensor in that collects all of its coefficients.
2 Basic setup
A polynomial network is a function of the form
pθ(x)=WhρrWh−1ρr…ρrW1x,Wi∈Rdi×di−1, (1)
where the activation raises all elements of to the -th power (). The parameters (with ) are the network’s weights, and the network’s architecture is encoded by the sequence (specifying the depth and widths ). Clearly, is a homogeneous polynomial mapping of degree , i.e., .
For fixed degree and architecture , there exists an algebraic map
Φd,r:θ↦pθ=⎡⎢ ⎢⎣pθ1⋮pθdh+1⎤⎥ ⎥⎦, (2)
where each is a polynomial in variables. The image of is a set of vectors of polynomials, i.e., a subset of , and it is the functional space represented by the network. In this paper, we consider the “Zariski closure” of the functional space.111The Zariski closure of a set is the smallest set containing that can be described by polynomial equations. We refer to as functional variety of the network architecture, as it is in fact an irreducible algebraic variety. In particular, can be studied using powerful machinery from algebraic geometry.
Remark 1.
The functional variety may be significantly larger than the actual functional space , since the Zariski closure is typically larger than the closure with respect to the standard the Euclidean topology. On the other hand, the dimensions of the spaces and agree, and the set is usually “nicer” (it can be described by polynomial equations, whereas an exact implicit description of may require inequalities).
2.1 Examples
We present some examples that describe the functional variety in simple cases.
Example 2.
A linear network is a polynomial network with . In this case, the network map is simply matrix multiplication:
θ=(Wh,Wh−1,…,W1)↦pθ=WhWh−1…W1x. (3)
The functional space is the set of matrices with rank at most . This set is already characterized by polynomial equations, as the common zero set of all minors, so in this case. The dimension of is .
Example 3.
Consider and . The input variables are , and the parameters are the weights
W1=[w111w112w121w122],W2=⎡⎢⎣w211w212w221w222w231w232⎤⎥⎦. (4)
The network map is a triple of quadratic polynomials in , that can be written as
W2ρ2W1x=⎡⎢ ⎢⎣w211(w111x1+w112x2)2+w212(w121x1+w122x2)2w221(w111x1+w112x2)2+w222(w121x1+w122x2)2w231(w111x1+w112x2)2+w232(w121x1+w122x2)2⎤⎥ ⎥⎦. (5)
The map in (2) takes (that have parameters) to the three quadratics in displayed above. The quadratics have a total of coefficients, however these coefficients are not arbitrary, i.e., not all possible triples of polynomials occur in the functional space. Writing for the coefficient of in in (5) (with ) then it is a simple exercise to show that
det⎡⎢ ⎢ ⎢⎣c(1)11c(1)12c(1)22c(2)11c(2)12c(2)22c(3)11c(3)12c(3)22⎤⎥ ⎥ ⎥⎦=0. (6)
This cubic equation describes the functional variety , which is in this case an eight-dimensional subset (hypersurface) of .
2.2 Objectives
The main goal of this paper is to study the dimension of as the network’s architecture and the activation degree vary. This dimension may be considered a precise and intrinsic measure of the polynomial network’s expressivity
, quantifying degrees of freedom of the functional space. For example, the dimension reflects the number of input/output pairs the network can interpolate, as each sample imposes one linear constraint on the variety
.
In general, the variety lives in the ambient space , which in turn only depends on the activation degree , network depth , and the input/output dimensions and . We are thus interested in the role of the intermediate widths in the dimension of .
Definition 4.
A network architecture has a filling functional variety for the activation degree if .
It is important to note that if the functional variety is filling, then actual functional space (before taking closure) is in general only thick, i.e., it has positive Lebesgue measure in (see Remark 1). On the other hand, given an architecture with a thick functional space, we can find another architecture whose functional space is the whole ambient space.
Proposition 5 (Filling functional space).
Fix and suppose has a filling functional variety . Then the architecture has a filling functional space, i.e., .
In summary, while an architecture with a filling functional variety may not necessarily have a filling functional space, it is sufficient to double all the intermediate widths for this stronger condition to hold. As argued below, we expect architectures with thick/filling functional spaces to have more favorable properties in terms of optimization and training. On the other hand, non-filling architectures may lead to interesting functional spaces for capturing patterns in data. In fact, we show in Section 3.2 that non-filling architectures generalize families of low-rank tensors.
2.3 Connection to optimization
The following two results illustrate that thick/filling functional spaces are helpful for optimization.
Proposition 6.
If the closure of a set is not convex, then there exists a convex function on whose restriction to has arbitrarily “bad” local minima (that is, there exist local minima whose value is arbitrarily larger than that of a global minimum).
Proposition 7.
If a functional space is not thick, then it is not convex.
These two facts show that if the functional space is not thick, we can always find a convex loss function and a data distribution that lead to a landscape with arbitrarily bad local minima. There is also an obvious weak converse, namely that if the functional space is filling
, then any convex loss function will have a unique global minimum (although there may be “spurious” critical points that arise from the non-convex parameterization).
3 Architecture dimensions
In this section, we begin our study of the dimension of . We describe the connection between polynomial networks and tensor decompositions for both shallow (Section 3.1) and deep (Section 3.2) networks, and we present some computational examples (Section 3.3).
3.1 Shallow networks and tensors
Polynomial networks with are closely related to CP tensor decomposition Landsberg-book . Indeed in the shallow case, we can verify the network map sends to:
W2ρrW1x=(d1∑i=1W2(:,i)⊗W1(i,:)⊗r)⋅x⊗r=:Φ(W2,W1)⋅x⊗r. (7)
Here is a partially symmetric tensor, expressed as a sum of partially symmetric rank terms, and denotes contraction of the last indices. Thus the functional space is the set of rank partially symmetric tensors. Algorithms for low-rank CP decomposition could be applied to to recover and . In particular, when , we obtain a symmetric tensor. For this case, we have the following.
Lemma 8.
A shallow architecture is filling for the activation degree if and only if every symmetric tensor has rank at most .
Furthermore, the celebrated Alexander-Hirschowitz Theorem alexander1995 from algebraic geometry provides the dimension of for all shallow, single-output architectures.
Theorem 9 (Alexander-Hirschowitz).
If , the dimension of is given by , except for the following cases:
• , ,
• , , ,
• , , ,
• , , ,
• , , .
3.2 Deep networks and tensors
Deep polynomial networks also relate to a certain iterated tensor decomposition. We first note the map may be expressed via the so-called Khatri-Rao product from multilinear algebra. Indeed maps to:
SymRowWh((Wh−1…(W2(W∙r1))∙r…)∙r). (8)
Here the Khatri-Rao product operates on rows: for , the power replaces each row, , by its vectorized -fold outer product, . Also in (8), SymRow denotes symmetrization of rows, regarded as points in a certain linear operator.
Another viewpoint comes from using polynomials and inspecting the layers in reverse order. Writing for the output polynomials at depth , the top output at depth is:
wh11prθ1+wh12prθ2+…+wh1dh−1prθdh−1. (9)
This expresses a polynomial as a weighted sum of -th powers of other (nonlinear) polynomials. Recently, a study of such decompositions has been initiated in the algebra community LORS-2019 . Such expressions extend usual tensor decompositions, since weighted sums of powers of homogeneous linear forms correspond to CP symmetric decompositions. Accounting for earlier layers, our neural network expresses each in (9) as -th powers of lower-degree polynomials at depth , so forth. Iterating the main result in FOS-PNAS on decompositions of type (9), we obtain the following bound on filling intermediate widths.
Theorem 10 (Bound on filling widths).
Suppose and satisfy
dh−i≥min(dh⋅rid0,(rh−i+d0−1rh−i)) (10)
for each . Then the functional variety is filling.
3.3 Computational investigation of dimensions
We have written code222Available at https://github.com/mtrager/polynomial_networks. in the mathematical software SageMath sagemath that computes the dimension of for a general architecture and activation degree . Our approach is based on randomly selecting parameters and computing the rank of the Jacobian of in (2). This method is based on the following lemma, coming from the fact that the map is algebraic.
Lemma 11.
For all , the rank of the Jacobian matrix is at most the dimension of the variety . Furthermore, there is equality for almost all (i.e., for a non-empty Zariski-open subset of ).
Thus if is full rank at any , this witnesses a mathematical proof is filling. On the other hand if the Jacobian is rank-deficient at random
, this indicates with “probability 1" that
is not filling. We have implemented two variations of this strategy, by leveraging backpropagation:
1. Backpropagation over a polynomial ring. We defined a network class over a ring , taking as input a vector variables . Performing automatic differentiation (backpropagation) of the output function yields polynomials corresponding to , for any entry of a weight matrix . Extracting the coefficients of the monomials in , we recover the entries of the Jacobian of .
2. Backpropagation over a finite field. We defined a network class over a finite field . After performing backpropagation at a sufficient number of random sample points , we can recover the entries of the Jacobian of by solving a linear system (this system is overdetermined, but it will have an exact solution since we use exact finite field arithmetic). The computation over provides the correct dimension over for almost all primes .
The first algorithm is simpler and does not require interpolation, but is generally slower. We present examples of some of our computations in Tables 1 and 2. Table 1 shows minimal architectures that are filling, as the depth varies. Here, “minimal” is with respect to the partial ordering comparing all widths. It is interesting to note that for deeper networks, there is not a unique minimally filling network. Also conspicuous is that minimal filling widths are “unimodal", (weakly) increasing and then (weakly) decreasing. Arguably, this pattern conforms with common wisdom.
Conjecture 12 (Minimal filling widths are unimodal).
Fix , , and . If is a minimal filling architecture, there is such that and .
Table 2 shows examples of computed dimensions, for varying architectures and degrees. Notice that the dimension of an architecture stabilizes as the degree increases.
4 General results
This section presents general results on the dimension of . We begin by pointing out symmetries in the network map , under suitable scaling and permutation.
Lemma 13 (Multi-homogeneity).
For all diagonal matrices and permutation matrices (), the map returns the same output under the replacement:
W1 ←P1D1W1 W2 ←P2D2W2D−r1PT1 W3 ←P3D3W3D−r2PT2 ⋮ Wh ←WhD−rh−1PTh−1.
Thus the dimension of a generic fiber (pre-image) of is at least .
Our next result deduces a general upper bound on the dimension of . Conditional on a standalone conjecture in algebra, we prove that equality in the bound is achieved for all sufficiently high activation degrees . An unconditional result is achieved by varying the activation degrees per layer.
Theorem 14 (Naive bound and equality for high activation degree).
If , then
dimVd,r≤min(dh+h∑i=1(di−1−1)di,dh(d0+rh−1−1rh−1)). (11)
Conditional on Conjecture 16, for fixed satisfying (), there exists such that whenever , we have an equality in (11). Unconditionally, for fixed satisfying (), there exist infinitely many such that the image of has dimension .
Proposition 15.
Given integers , there exists with the following property. Whenever are homogeneous polynomials of the same degree in variables, no two of which are linearly dependent, then are linearly independent if .
Conjecture 16.
In the setting of Proposition 15, may be taken to depend only on and .
Proposition 15 and Conjecture 16 are used in induction on for the equality statements in Theorem 14. Our next result uses the iterative nature of neural networks to provide a recursive bound.
Proposition 17 (Recursive Bound).
For all and , we have:
dimV(d0,…,dh),r≤dimV(d0,…,dk),r+dimV(dk,…,dh),r−dk. (12)
Using the recursive bound, we can prove an interesting bottleneck property for polynomial networks.
Definition 18.
The width in layer is an asymptotic bottleneck (for , and ) if there exists such that for all and all , …, , , then the widths are non-filling.
This expresses our finding that too narrow a layer can “choke" a polynomial network, such that there is no hope of filling the ambient space, regardless of how wide elsewhere or deep the network is.
Theorem 19 (Bottlenecks).
If , then is an asymptotic bottleneck. Moreover conditional on Conjecture 2 in nicklasson-2017 , then is not an asymptotic bottleneck.
Proposition 17 affords a simple proof is an asymptotic bottleneck. However to obtain the full statement of Theorem 19, we seem to need more powerful tools from algebraic geometry.
5 Conclusion
We have studied the functional space of neural networks from a novel perspective. Deep polynomial networks furnish a framework for nonlinear networks, to which the powerful mathematical machinery of algebraic geometry may be applied. In this respect, we believe polynomial networks can help us access a better understanding of deep nonlinear architectures, for which a precise theoretical analysis has been extremely difficult to obtain. Furthermore, polynomials can be used to approximate any continuous activation function over any compact support (Stone?Weierstrass theorem). For these reasons, developing a theory of deep polynomial networks is likely to pay dividends in building understanding of general neural networks.
In this paper, we have focused our attention on the dimension of the functional space of polynomial networks. The dimension is the first and most basic descriptor of an algebraic variety, and in this context it provides an exact measure of the expressive power of an architecture. Our novel theoretical results include a general formula for the dimension of the architecture attained in high degree, as well as a tight lower bound and nontrivial upper bounds on the width of layers in order for the functional variety to be filling. We have also demonstrated intriguing connections with tensor and polynomial decompositions, including some which appear in very recent literature in algebraic geometry.
The tools and concepts introduced in this work for fully connected feedforward polynomial networks can be applied in principle to more general algebraic network architectures. Variations of our algebraic model could include multiple polynomial activations (rather than just single exponentiations) or more complex connectivity patterns of the network (convolutions, skip connections, etc.). The functional varieties of these architectures could be studied in detail and compared. Another possible research direction is a geometric study of the functional varieties, beyond the simple dimension. For example, the degree or the Euclidean distance degree draisma_euclidean_2013 of these varieties could be used to bound the number of critical points of a loss function. Additionally, motivated by Section 3.2, we would like to develop computational methods for constructing a network architecture that represents an assigned polynomial mapping. Such algorithms might lead to “closed form” approaches for learning using polynomial networks (similar to SVD or tensor decomposition), as a provable counterpoint to gradient descent methods. Our research program might also shed light on the practical problem of choosing an appropriate architecture for a given application.
Acknowledgements
We thank Justin Chen, Amit Moscovich, Claudiu Raicu and Steven Sam for their help. JK was partially supported by the Simons Collaboration on Algorithms and Geometry. MT and JB were partially supported by the Alfred P. Sloan Foundation, NSF RI-1816753 and Samsung Electronics.
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Appendix A Technical proofs
See 5
Proof.
We mimic the proof of Theorem 1 in [5]. As is thick, equivalently contains some Euclidean open ball (see Chevalley’s theorem [18]). But given any point , we may write for some and . Thus in the architecture , we may set the “top half" of weights to represent , the “bottom half" to represent , and so scaling appropriately, all together the network represents . ∎
See 6
Proof.
We write for the closure of . Let a line that intersects in (at least) two closed disjoint intervals . Such line always exists because is not convex. It is easy to construct a convex function that is outside of and has (arbitrarily) different minima when restricted to : this amounts to constructing a convex function with assigned minima on disjoint closed intervals. ∎
See 7
Proof.
It is enough to argue that does not lie on a linear subspace (i.e., that its affine hull is the whole ambient space). Indeed, because has zero-measure, this implies that it cannot coincide with its convex hull. To show the claim, we observe that always contains all vectors of polynomials of the form , where is a linear form in variables (this follows by induction on ). The vectors span the ambient space, as any polynomial can be written as a linear combination of powers of linear forms. ∎
See 8
Proof.
This is clear as the network outputs . ∎
See 10
Proof.
It is equivalent to show that the network map with scalars extended to (i.e., allowing complex weights), denoted , has full-measure image. For this, we use induction on . The key input is Theorem 4 of [16], which states generic homogeneous polynomials over of degree in variables can be written as a sum of many -th powers of degree polynomials over , when .
The base case is trivial. Thus assume and that the image has full measure for . If , then for generic , the entries of form a vector space basis of , so the image of is filling. On the other hand if , then the image of is full measure by [16] and the inductive hypothesis. ∎
See 11
Proof.
We note entries of are polynomials in , thus minors of are polynomials in , so has a Zariski-generic rank (the largest size of minor that is a nonzero polynomial), which is also the maximum rank of . By basic algebraic geometry, this is the dimension of (see “generic submersiveness" of algebraic maps in characteristic 0 [18]). ∎
See 13
Proof.
This is from the multi-homogeneity of the -th power activation by substituting. ∎
See 14
Proof.
We know the dimension of equals the dimension of the domain of minus the dimension of a generic fiber of (see generic freeness [15]). Thus by Lemma 13, . At the same time, the dimension of is at most that of its ambient space . Combining produces the bound (10).
For the next statement, we temporarily assume Conjecture 16. We shall prove by induction on the stronger result that for the generic fibers of are precisely as described in Lemma 13 (and no more). The base case is trivial. Thus assume and that for the generic fiber is exactly as in Lemma 13, whenever . For the induction step, we let be a threshold which works in Conjecture 16 for and , and then we set . Now with fixed generic weights , we consider any other weights satisfying
WhρrWh−1…ρrW1x=~Whρr~Wh−1…ρr~W1x (13)
for . Write for the output of the LHS in (13) at depth , and similarly for the RHS. By genericity and , the polynomials are pairwise linearly independent. Comparing the top outputs at depth in (13), we get two decompositions of type (9):
wh11prθ1+…+wh1dh−1prθdh−1=~wh11~prθ1+…+~wh1dh−1~prθdh−1. (14)
Since , by Conjecture 16 there must be two linearly dependent summands in (14). Permuting as necessary we may assume these are the first two terms on both sides. Scaling as necessary we may assume , and then subtract from (14) to get:
(wh11−~wh11)prθ1+…+wh1dh−1prθdh−1=~wh12~prθ2+…+~wh1dh−1~prθdh−1. (15)
Invoking Conjecture 16 again, we may remove another summand from the RHS, so on until the RHS is 0. Then each individual summand in the LHS must be 0 too, by pairwise linear independence and Conjecture 16 once more. We have argued that (up to scales and permutation) it must hold and . Comparing other outputs at depth in (13) gives (up to scales and permutation). Thus by the inductive hypothesis, the fiber through is as in Lemma 13 and no more. This completes the induction.
For the unconditional result with differing degrees per layer, the argument runs closely along similar lines, but it relies on Proposition 15 in place of Conjecture 16. For brevity, details are omitted. ∎
See 15
Proof.
It is shown in [4] (via Wronskian and Vandermonde determinants) that for any particular , no two of which are linearly dependent, there exists such that if . The dependence on particular can be removed as follows.
Let be the set of -tuples, no two entries of which are linearly dependent. So is Zariski-open, described by the non-vanishing of minors. Further let be the subset of -tuples whose -th powers are linearly independent, similarly Zariski-open. Consider the chain of inclusions . By [4], the union of this chain equals . Thus by Noetherianity of affine varieties, there exists with [15]. Now works. ∎
See 17
Proof.
This bound encapsulates the bracketing:
(WhρrWh−1…Wk+1)ρr(WkρrWk−1…W1x). (16)
More formally, the network map factors as:
(17)
by first sending to the pair of bracketed terms in (16) and then the pair to the composite in (16). The closure of the image of the first map in (17) is . On the other hand, the second map in (17) has -dimensional generic fibers, by multiplying with a diagonal matrix . Combining these facts gives the result. ∎
See 19
Proof.
We first point out that Proposition 17 gives an elementary proof is an asymptotic bottleneck. This is because as grows the ambient dimension grows like , while the RHS bound grows like , so if then cannot fill for .
To gain a factor of 2 in the bottleneck bound, we start by writing for the output polynomials at depth , that is, for . Fixing , we consider , a subalgebra of the Veronese ring . The key idea is to compare the Hilbert polynomials of and of [6]. If the Hilbert polynomials differ in any non-constant terms, this means the dimension of the degree piece of minus that of diverges to as goes to . At the same time, however we vary weights (keeping fixed), the output polynomials remain in the algebra . Additionally, for varying and , the possible -vectors of degree polynomials in variables, , comprise a bounded-dimensional variety. The upshot is that if it need always be the case (based on ) that the Hilbert polynomials of and have non-constant difference, then must be an asymptotic bottleneck. Thus it suffices to check the Hilbert polynomial property holds for all if . To this end, we derived the following general result:
Claim.
Given integers and . Then whenever are homogeneous polynomials of the same degree in variables, the algebra and the Veronese algebra have Hilbert polynomials with non-constant difference.
Proof of claim.
First, it suffices to check the claim for generic . Second, the difference in Hilbert polynomials identifies with the Hilbert polynomial of the sheaf [18]. Here () is the projective Veronese variety, the linear projection | 2022-01-22 04:52:35 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8842165470123291, "perplexity": 747.093626838151}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303747.41/warc/CC-MAIN-20220122043216-20220122073216-00352.warc.gz"} |
https://stats.stackexchange.com/questions/401338/how-can-i-perform-all-pairwise-comparisons-of-categorical-levels-in-a-linear-reg?noredirect=1 | # How can I perform all pairwise comparisons of categorical levels in a linear regression with other control variables? [duplicate]
I want to perform a linear regression with some categorical predictor $$z$$ and some other continuous and categorical control variables. I use one-hot dummy encoding for $$z$$, with e.g. $$z_a$$ as the base level and dummies for levels $$z_b, z_c,$$ and $$z_d$$. However, with this regression I can only compare and test the difference between levels $$z_b, z_c,$$ and $$z_d$$ with level $$z_a$$, but not with each other.
To compare $$z_b$$ and $$z_c$$ I could use a new dummy encoding with $$z_b$$ as base level and perform a new regression. For all pair-wise comparisons I could choose each level of $$z$$ as the base level once. Is there an easier way to do this (in Python or R)?
From what I understand, with Tukey's test I can compare all means, but cannot correct for any other variables. | 2020-01-28 08:43:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5467897057533264, "perplexity": 343.3008257565433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251776516.99/warc/CC-MAIN-20200128060946-20200128090946-00461.warc.gz"} |
http://www.math.wpi.edu/Course_Materials/MA1021A97/lab3/lab_template.html | # Differentiation
Note:All example commands given in this write-up and the output from them should be included in your lab report.
## D and diff
Maple knows how to take many derivatives. Its main commands for doing this are D and diff. D is designed to differentiate functions, whereas diff is for differentiating expressions. However, if proper notation is used, diff can also be used with functions. To review the difference between a function and an expression, check the two examples below. The f statement defines a function, the g statement defines an expression.
> f:=x->x^2+7*x+5;
> g:=x^3-5*x+8;
Here are some examples that show how D and diff work. Check the difference between these two commands.
> D(sin);
> D(sin)(x);
Also, check these two.
> D(f);
> D(f)(x);
Carefully consider these.
> diff(g,x);
> diff(f(x),x);
See what happens with these.
> diff(g(x),x);
> diff(f,x);
After the last four examples, you should be convinced that proper notation is very important in doing derivatives in Maple.
Maple can also do higher derivatives. Check these commands.
> diff(g,x$2); > diff(f(x),x$2);
The \$ sign is used to indicate repeated differentiation, the number following it tells how many times the function or expression is to be differentiated. D will also do repeated differentiation.
> (D@@2)(f)(x);
More information on diff and D can be obtained through Maple help screens.
## Composition
Maple can be used to perform composition of functions (Note: functions, not expressions). The symbol for composition is @. Try these commands
> (f@f)(x);
> simplify(");
> h:=x->x^3+x^2-3*x+4;
> p:=x->8*x+6;
> (f@h@p)(x);
> simplify(");
> (p@h@f)(x);
> simplify(");
Are and the same?
## Implicit Differentiation
The implicitdiff command can be used to find derivatives of implicitly defined functions. The syntax is as follows
> implicitdiff(f,y,x);
where f is an expression or equation, y is the dependent variable and x is the independent variable. Thus the command as just stated would compute . If f is given as an expression Maple will assume that the implicit equation is f = 0. Check the results of the following commands.
> f:=x^2*y^2+y^3;
> implicitdiff(f,y,x);
> g:=x^2+y^3=1;
> implicitdiff(g,y,x);
Second derivatives can also be taken with implicitdiff. The following command computes .
> implicitdiff(g,y,x,x);
Maple also has a command for plotting implicitly defined functions. It is in the package plots which must be called before using the command.
> with(plots);
> implicitplot(x^2-y^2=1,x=-3..3,y=-3..3);
## Exercises
1.
Find an equation of the line tangent to the graph of at the point (2,0). (Note that Maple has a sqrt function.)
2.
Find the third derivative of evaluated at .
3.
Consider the function . Find the points at which the graph of this function has a horizontal tangent line. (The numer command could be useful here.) Use plot to graph the function. What geometric property seems to be indicated by a horizontal tangent line?
4.
Consider the function
(a)
In order to take the derivative of the given function by using the Chain Rule, it is necessary to picture the function as the composite of several functions. Open a text section and explain how that is done for this problem.
(b)
Use Maple to take the composite of the functions you found in (a). Make sure this composite really gives the original function.
(c)
Open a text section and explain how the derivative is found by means of the Chain Rule.
(d)
Check your answer in (c) by using Maple to get the derivative.
5.
Consider a function y that is implicitly defined by y5 - 3x2y + x3 = -1.
(a)
Attach a handwritten page on which you find .
(b)
Use Maple to check the accuracy of your work.
(c)
Find an equation of the line tangent to the graph of the function at the point (1,1).
Labs and Projects for Calculus using Maple
This document was generated using the LaTeX2HTML translator Version 97.1 (release) (July 13th, 1997)
Copyright © 1993, 1994, 1995, 1996, 1997, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
The command line arguments were:
latex2html -split 1 lab_template.tex.
The translation was initiated by Christine M Palmer on 9/23/1997
Christine M Palmer
9/23/1997 | 2017-11-23 07:31:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8384678363800049, "perplexity": 1285.3394958757185}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806760.43/warc/CC-MAIN-20171123070158-20171123090158-00120.warc.gz"} |
https://ora.ox.ac.uk/objects/uuid:67fc4c97-a244-4573-8c48-9f60cd6cc0ea | Journal article
### Photoproduction of events with rapidity gaps between jets at HERA
Abstract:
The photoproduction of dijet events, where the two jets with the highest transverse energy are separated by a large gap in pseudorapidity, have been studied with the ZEUS detector using an integrated luminosity of 39 pb -1. Rapidity-gap events are defined in terms of the energy flow between the jets, such that the total summed transverse energy in this region is less than some value ETCUT. The data show a clear excess over the predictions of standard photoproduction models. This is interprete...
### Authors
Journal:
European Physical Journal C
Volume:
50
Issue:
2
Pages:
283-297
Publication date:
2007-04-05
DOI:
EISSN:
1434-6052
ISSN:
1434-6044
URN:
uuid:67fc4c97-a244-4573-8c48-9f60cd6cc0ea
Source identifiers:
109952
Local pid:
pubs:109952
Language:
English | 2021-10-18 15:26:44 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.848565399646759, "perplexity": 3003.712668453506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585203.61/warc/CC-MAIN-20211018124412-20211018154412-00712.warc.gz"} |
https://www.physicsforums.com/threads/continuous-almost-everywhere.230002/ | # Continuous almost everywhere
## Main Question or Discussion Point
what does continuous almost everywhere - alpha means?
I know that the term almost everywhere means that the property holds everywhere on the measurable space except on a subset of measure 0,what I don't really understand is the term almost everywhere-alpha.
Last edited: | 2019-12-12 05:48:29 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9843826293945312, "perplexity": 858.3851360370245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540537212.96/warc/CC-MAIN-20191212051311-20191212075311-00091.warc.gz"} |
https://proofwiki.org/wiki/Mathematician:Jacob_Bernoulli | Mathematician:Jacob Bernoulli
Mathematician
Swiss mathematician best known for his work on probability theory, analytic geometry and development of the calculus.
Also developed the field of calculus of variations.
Developed the technique of Separation of Variables, and in $1696$ solved what is now known as Bernoulli's (Differential) Equation.
Invented polar coordinates.
Elder brother of Johann Bernoulli, with whom he famously quarrelled.
He and Johann, having encountered Leibniz's early papers in his Acta Eruditorum, became his most important students.
Solved the Brachistochrone problem, which had been posed in $1696$ by his brother Johann.
Also investigated the catenary and the logarithmic spiral.
Swiss
History
• Born: 27 Dec 1654 in Basel, Switzerland
• 1687: Became Professor of Mathematics at Basel
• Died: 16 Aug 1705 in Basel, Switzerland
Theorems and Definitions
Results named for Jacob Bernoulli can be found here.
Definitions of concepts named for Jacob Bernoulli can be found here.
Notable Quotes
Invito patre sidera verso (Against my father's will I study the stars)
-- Personal motto, created in memory of his father who opposed his desire to study mathematics and astronomy and tried to force him to study to become a theologian.
Also known as
Jacob Bernoulli is also known as James, Jacques or Jakob.
Sometimes reported as Jakob I, or Jacob I, so as to distinguish him from Jakob II Bernoulli. | 2021-04-11 19:39:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.598389744758606, "perplexity": 5230.027950252321}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038064898.14/warc/CC-MAIN-20210411174053-20210411204053-00409.warc.gz"} |
https://physics.stackexchange.com/questions/121536/is-it-possible-to-have-a-perfectly-black-material/121538 | # Is it possible to have a perfectly black material?
After reading this NASA article about the "blackest material", the following stuck out to me.
The tiny gaps between the tubes absorb 99.5 percent of the light that hits them
Is it possible to create a material that absorbs, not just all visible light, but all electromagnetic radiation?
• In general, it is not possible to have anything "perfect". – jinawee Jun 23 '14 at 22:31
• Do keep in mind that electromagnetic energy comes in 'packets' (photons) and that the energy of a photon is proportional to the frequency (or inversely proportional to the wavelength). So, unless there is a material that can, in principle, absorb a photon of arbitrarily large energy without being destroyed, then the answer is no. – Alfred Centauri Jun 23 '14 at 23:15
• @jinawee: That might be true most of the time, but not all. We do have superconductors. – Gerard Jun 24 '14 at 15:06
• A possibly related answer; note that the material in the article seems to be versatile enough to be applied like paint and to be black from all angles of incidence, constraints which might limit the total absorption you can achieve. – rob Jun 24 '14 at 16:57
We can make it absorb a lot of energy but if you read about black-body radiation effect you will notice that as energy is introduced into the object it will similarly radiate a small amount of the energy back, at room temperature appears black, as most of the energy it radiates is infra-red and cannot be perceived by the human eye. At higher temperatures, black bodies glow with increasing intensity and colors that range from dull red to blindingly brilliant blue-white as the temperature increases.
This means that later even if we did have an $100$ percent absorbing materials of all electro-magnetic radiation the object will radiate some energy due to heating or other similar process. That in mind, it will always leak out some energy.To conclude, we could say it will only absorb radiation for short-time before releasing enough photons which could be detected by an photon detector, therefore "breaking" the truly blackest or absorbing material due to this effect.
As a summary as long as the object absorbs some form of energy it can never be completely black.
Furthermore, an blackhole is very black however it even is not as the power in the Hawking radiation (if it was proven, currently it is hypothesized) from a solar mass $M$ which is equal to $1.98855\pm 0.00025 * 10^{30}$ black hole turns out to be a minuscule 9 × 10−29 watts. It is indeed an extremely good approximation to call such an object 'black'. As $$P =\hbar c^2/15360\pi G^2M^2 = 9.004 * 10^{-29}$$
That being said, even black-holes one of the strongest objects cannot escape being truly black.
I believe user43495's answer to be quite accurate, and right, but I will add something here just as food for thought.
It has been hypothesised that the LHC could create miniature black holes. Even if it can't, there are many in space[citation needed]. These would be able to "absorb" all kinds of electromagnetic radiation. Within a black hole's event horizon, escape velocity is higher than the speed of light. Black holes are black bodies, but even their own radiated photons cannot escape past their event horizon. That should be as dark as you can get.
Stephen Hawkings has some works where he describes a kind of radiation emanated from black holes. Editions in italic: the existence of this Hawkings Radiation has been not been proved true or false yet. But supposing it exists, it is a kind of black body radiation. However, its energy is not proportional to temperature as in most black bodies. Rather, it's inversely proportional to the black hole's mass. I think with sufficient mass, a black hole may be darker than the material quoted in the question.
(Thanks user43495 for reminding me that Hawkings Radiation is still hypothetical.)
• Hawking radiation is still hypothesized and never had been seen in laboratories or even observed (except in Mathematics) therefore I edited your answer to express its hypothesized stage as it could confuse individuals as they could be misled into thinking Hawking Radiation has been detected. – LogicProgrammer Jun 24 '14 at 15:17
• Like to point out that while it is sufficiently black, a black hole's event horizon is not something I would classify as a "material" – Jim Jun 24 '14 at 17:25
• @Jim I mean the whole black hole as a material, not just its event horizon. – Renan Jun 24 '14 at 18:16
• @Jim Im sure Renan was attempting to show the closest thing to an truly dark object – LogicProgrammer Jun 24 '14 at 18:19
• I suppose it depends on your definition of "material". While I was looking for something tangible enough to create something out of (clothing, car, spaceship, synthetic planet, etc.) I do appreciate the offer of "blackest" possible thing. – David Starkey Jun 24 '14 at 18:28 | 2019-10-14 20:22:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5844122767448425, "perplexity": 435.48332407713644}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986655310.17/warc/CC-MAIN-20191014200522-20191014224022-00290.warc.gz"} |
http://www.math.iitb.ac.in/~seminar/geometry/khare-december15-2014.html | Date & Time: Monday, December 15, 2014, 14:30-15:30.
Venue: Ramanujan Hall
Title: : Faces and supports of highest weight modules
Speaker: Apoorva Khare, Stanford University
Abstract: We present three formulas to compute the set of weights of all simple highest weight modules (and others) over a complex semisimple Lie algebra $\mathfrak{g}$. These formulas are direct and do not involve cancellations. Our results extend the notion of the Weyl polytope to general highest weight $\mathfrak{g}$-modules $\mathbb{V}^\mu$.
We further classify and describe the vertices, faces, and their symmetries for a very large class of highest weight modules, including all parabolic Verma modules and their simple quotients. Finally, we completely classify inclusions between faces of arbitrary $\mathbb{V}^\mu$, in the process extending results of Vinberg, Chari, Cellini, and others from finite-dimensional modules to all highest weight modules. | 2017-10-23 18:45:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5137122869491577, "perplexity": 708.4553289716382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826283.88/warc/CC-MAIN-20171023183146-20171023203146-00475.warc.gz"} |
http://www.mammal.cn/CN/10.16829/j.slxb.150206 | • 论文 •
### 草原鼢鼠土丘与不同根系类型植物的关系
1. （1 内蒙古农业大学草原与资源环境学院，呼和浩特 010011）（2 草业与草地资源教育部重点实验室，呼和浩特 010011）（3内蒙古农业大学啮齿动物研究中心，呼和浩特 010011）
• 出版日期:2019-03-30 发布日期:2019-03-26
• 通讯作者: 武晓东 E-mail:wuxiaodong_hgb@163.com
• 基金资助:
国家自然科学基金（31772667，31602003，31560669）；农业部公益性行业项目（201203041）；内蒙古自然科学基金项目（2016MS0351）
### The relationships between the mounds of transbaikal zokor and plants with different root types
JI Yu, YUAN Shuai, FU Heping, YUE Chuang, MAN Duhu, YANG Suwen, WU Xiaodong
1. (1 College of Grassland,Resources and Environment,Inner Mongolia Agriculture University,Hottot 010011,China)
(2 Key Laboratory of Prataculture and Grassland Resource,Ministry of Education,Hohhot 010011,China)
(3 Rodent Research Center,Inner Mongolia Agriculture University,Hottot 010011,China)
• Online:2019-03-30 Published:2019-03-26
Abstract: An experiment with three quadrants（20m×50m）was conducted to monitor the locations of plants and the number of new and old mounds in the habitat of Myospalax aspalax in natural ground in Baiyinkulun pasture, Xinlinguole, Inner Mongolia from 2013 to 2015. The relationships between the number of new and old mounds and plant biomass, plant number and species were analyzed by Partial RDA (Partial Redundancy Analysis). The results showed that in spring, the number of new zokor mounds was positively correlated with the axis-rooted system and creeping-root system which accounted for much of the variation. The number of old mounds of zokor was positively correlated with sparse grove system. The number of old mounds was negatively correlated with the axis-rooted system and creeping-root system, and accounted for much of the variation. In autumn, the number of new mounds of zokor was positively correlated with the sparse grove system, dense clumping system and creeping-root system, and accounted for much of the variation. The number of old zokor mounds was positively correlated with the sparse grove system, dense clumping system and creeping-rooted system, and accounted for much of the variation. These results suggest that there were seasonal differences in zokor’s mound-forming activity and micro-habitat selection, and those activities were affected by food resources. Axis-rooted system and creeping-root system had significant effects on the spring activities of zokor, while the the sparse grove system, dense clumping system and creeping-root system had important effects on zokor's autumn activities. | 2023-02-03 00:19:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19915485382080078, "perplexity": 12909.095837822722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500041.2/warc/CC-MAIN-20230202232251-20230203022251-00614.warc.gz"} |
http://physics.stackexchange.com/tags/atomic-physics/hot | # Tag Info
30
There is an interesting diagram in the wiki article on underground nuclear testing - the picture file is here This shows that the crater you get from a nuclear explosion depends on the depth of burial: I think the most interesting diagrams are the ones labeled (e) and (f) - where the explosion happens at great depth. In that case, you get a "tight ...
18
A lot of different forms, but mostly kinetic energy. A good table is given at Hyperphysics. The energy released from fission of uranium-235 is about 215 MeV. This is divided into: Kinetic energy of fragments (heat): ~168 MeV Assorted gamma rays: ~15-24 MeV Beta particles (electrons/positrons) and their kinetic energy: ~8 MeV Assorted neutrons and their ...
9
Well, we could say, yes, that is simply how quantum mechanics works. But these are not the axioms of quantum mechanics, and the exclusion principle in particular is really only understood in the context of quantum field theory. The electron does not "spiral in" because it doesn't move in the classical sense at all. At the scale of the size of an atom, ...
6
I would think that there is a cavity after the explosion, as well as - Rock is compressible, and the surrounding rock is squeezed together - There is a bulge on the surface, it's just too flat to be noticeable I don't think the rock would be porous since that would allow radiation to leak out of the site. Added: From Wikipedia: When testing went ...
3
The energy that is released when a the absorption of a neutron causes a heavy atom nucleus to fission into two daughter nuclei comes from the tighter binding energy of the two daughter nuclei compared to the weaker (smaller) binding energy of the original nucleus. This extra energy is mostly released in the form of the kinetic energy of the two daughter ...
3
Yes it's possible, and it's called (unsurprisingly) two photon emission. However the probability of two or multi photon emission is generally lower than for single photon emission by several orders of magnitude, so it's hard to observe.
3
The mass of an atom is always less than the sum of the masses of the particles that compose it. The lack of mass (or energy, from E = mc^2) is called binding energy and it is the energy expended by the particles to remain confined inside of the atom. When fission occurs, not more spending of energy to hold together the individual particles. So the energy ...
2
Laser modes are the eigen-modes of a laser resonator: only specific distributions of electro-magnetic field can "resonate" in each particular resonator. Due to the 3D nature of our space, each mode is described by 3 numbers, or indices, $m$, $n$, $q$. The latter is the longitudinal mode number, and is easy to understand: to form a standing wave, the ...
2
1. Why can you only get the ground state? According to the wikipedia article linked to, the variational approach works by taking a wave function with some parameters (for example, a gaussian with mean $\mu$ and standard devation $\sigma$) and minimizing the energy with respect to these parameters. The minimum energy parameters give you a wave function that ...
2
As another side note, if you know the ground state, it IS possible to use the variational method to get the next-lowest energy. You simply restrict your trial wavefunctions to be orthogonal to the ground state, and then minimize as usual. If you know the two lowest ground states, you can use the variational method to find the third-lowest energy, etc. Also, ...
1
There is no such thing as "proving a postulate" theoretically, because your proof will always need to start from somewhere. What you can hope for is to show that a set A of postulates, which seems clunky and ad hoc, actually follows from a simpler set B of postulates. The clearest example of this is when Einstein showed that the Lorentz transformations, ...
1
Why is not really important, how is. If you ask yourself why then the answers can be many, for example Why does gravity make two masses attract each other? The answer is because it does, what is really important is how and for that you have a first theory, Newton's Law of Gravitation, this theory is only true for relatively small masses (or masses with ...
1
This is just an additional remark – the answer by NowIGetToLearnWhatAHeadIs is fine. But you can get all states from the variational method, just not with the trial function method (unless your trial function space contains the exact excited states). While the excited states are usually not extrema of the variation functional, they are stationary points! ...
1
Oh yes, certainly. Of course there is, by your definition, no way to determine this, even in principle. Since these "ghost atoms" are unobservable, they emit no radiation that we can sense. Unlike dark matter, whose existence we have deduced from gravitational effects, they do not affect us gravitationally. They do not collide with regular atoms, and they do ...
1
Apparently, the CANDU reactor can accept a variety of fuels, including what would be considered "waste" from other reactor types, although some amount of reprocessing is involved.
1
Even in the classical model, an infinite amount of levels doesn't necessarily mean that it occupies an infinite amount of space. You can divide any finite distance into infinitely many bits (for instance, $1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$). EDIT: I'd forgotten about the $r\sim N^2$ relation that the OP mentions below, so yes, although ...
1
With such a two-electron spin wave function both electrons cannot be in the 1s state due to the electron parity (the wave function should be antisymmetric relative to exchange of two particles). When both electrons are in the orbital 1s, the only possible spin function reads $$\frac{1}{\sqrt{2}} \left( |\uparrow \downarrow >-| \downarrow \uparrow> ... 1 If we write:$$ \psi_+ \propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta e^{i\phi} $$and:$$ \psi_- \propto \frac{r}{a_o}e^{-r/2a_0}\sin \theta e^{-i\phi} $$then because any superposition of solutions is also a solution we can write the solutions:$$ \psi_{px} = \psi_+ + \psi_- $$and likewise:$$ \psi_{py} = \psi_+ - \psi_- $$which gives us:$$ ...
1
Considering the way matter waves are associated with all moving particles, it seems inconceivable to me that electrons cannot move in other than elliptical orbits. Close examination of the harmonics & resonance effects of phase waves & matter waves, it becomes apparent that they need to move in elliptical orbits in order to harmonize the way they do. ...
1
In the real world transitions are rarely (never?) forbidden because the assumptions we make rarely hold exactly. You are quite correct that the 21 cm transition has $\Delta\ell = 0$ and is therefore forbidden. However it can occur (very slowly) as a magnetic dipole transition. The lifetime of the excited state is around $10^{15}$ seconds, and we can observe ...
1
Distinct spectral lines (rather than an infinite range of energies and hence white light) are a result of only specific transitions being allowed, which implies a set of distinct and unvarying energy levels. Classical models of the atom before quantum mechanics couldn't explain why there should only be certain electron energy levels.
1
This is going to take a bit, so be prepared. Also, some of what I say will not be exactly correct, in order to get the larger point across, but bear with me. First, the Rutherford model really did not speak to the question of emission lines. It simply noted that, on the basis of Rutherford's scattering experiments, all of the protons in an atom had to be ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2015-07-01 16:01:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.737691342830658, "perplexity": 420.44511912704525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094957.74/warc/CC-MAIN-20150627031814-00096-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://nanoelectronics.unibas.ch/research/graphene/ | Ballistic graphene offers a promising platform for electron optical devices. We have developed a versatile technology that allows to suspend graphene and complement it with arbitrary bottom and top-gate structures.1 Using current annealing we demonstrated exceptional high nobilities approaching 102 m2/Vs. These suspended devices are ballistic over micrometer length scales and display intriguing interference patterns in the electrical conductance when different gate potentials and magnetic fields are applied.2 There are great similarities between the propagation of light in a dielectric and electrons in graphene, but also differences. In particular, a negative refractive index is straightforward to realize in graphene, but hard in optics. We have used pn junctions to define an electron waveguide by electrostatics3 and guide electrons in snake-states due to alternating cyclotron motion in a small magnetic field,4 to realized mirrors and momentum filters,2 beam splitters5 and Fabry-Perot-like cavities as well as more complex interferometers.
Bilayer graphene is an exciting material, widely extending the range of phenomena compared to monolayer graphene, due its massive nature and much larger interaction parameter. In bilayer graphene a gap can be opened by applying a potential difference between the two layers, for example. Furthermore, the eight-fold ground-state degeneracy of the zero-energy Landau level provides a large Hilbert space, where interaction is expected to lift the degeneracy resulting in novel composite particles. We have discovered that the ground-state in undoped ultraclean high-mobility bilayer graphene is gapped in the absence of magnetic and electric field.6 The new phase, which spontaneously appears driven by interactions, was latter assigned to an antiferromagnetic one. We have also seen recently that a gap appears, albeit much weaker, for graphene with four layers, but it is absent in graphene with a odd layer number.
In current projects we use high-quality h-BN encapsulated graphene and study superlattice effects induced by lattice of the substrate.7 We further work extensively on superconducting graphene devices with side contacts. We have realized Josephson junctions with sputter superconducting films. We also explore isospin currents due to valley effects using pn junctions and suspended graphene where we aim to control the strain in latter with the goal to realize a valley physics with a pseudomagnetic field.
Funding: NCCR-QSIT, SNI
1. R. Maurand, P. Rickhaus, P. Makk, S. Hess, E. Tovari, C. Handschin, M. Weiss and CS, Carbon 79, 486 (2014).
2. P. Rickhaus, R. Maurand, M. H. Liu, M. Weiss, K. Richter and CS, Nat. Commun. 4, 2342 (2013).
3. P. Rickhaus, M. H. Liu, P. Makk, R. Maurand, S. Hess, S. Zihlmann, M. Weiss, K. Richter and CS, Nano Lett. 15 (9), 5819-5825 (2015).
4. P. Rickhaus, P. Makk, M. H. Liu, E. Tovari, M. Weiss, R. Maurand, K. Richter and CS, Nat. Commun. 6, 6470 (2015).
5. P. Rickhaus, P. Makk, M. H. Liu, K. Richter and CS, Appl. Phys. Lett. 107 (25), 251901 (2015).
6. F. Freitag, J. Trbovic, M. Weiss and CS, Phys. Rev. Lett. 108 (6), 076602 (2012).
7. C. Handschin, P.Makk, P. Rickhaus, M.-H. Liu, K. Watanabe, T. Taniguchi, K. Richter, C. Schönenberger, Nano Lett. 17, 328 (2016).
Relevant papers (keyword: GRAPHENE):
##### 2022
• Phase-dependent microwave response of a graphene Josephson junction
R. Haller, G. Fülöp, D. Indolese, J. Ridderbos, R. Kraft, Luk Yi Cheung, J. H. Ungerer, K. Watanabe, T. Taniguchi, D. Beckmann, R. Danneau, P. Virtanen, and C. Schönenberger.
Phys. Rev. Research 4, 13198 (2022)
[arXiv:2108.00989 ] [ Open Data ] [Abstract ]
Gate-tunable Josephson junctions embedded in a microwave environment provide a promising platform to in-situ engineer and optimize novel superconducting quantum circuits. The key quantity for the circuit design is the phase-dependent complex admittance of the junction, which can be probed by sensing an rf SQUID with a tank circuit. Here, we investigate a graphene-based Josephson junction as a prototype gate-tunable element enclosed in a SQUID loop that is inductively coupled to a superconducting resonator operating at 3 GHz. With a concise circuit model that describes the dispersive and dissipative response of the coupled system, we extract the phase-dependent junction admittance corrected for self-screening of the SQUID loop. We decompose the admittance into the current-phase relation and the phase-dependent loss and as these quantities are dictated by the spectrum and population dynamics of the supercurrent-carrying Andreev bound states, we gain insight to the underlying microscopic transport mechanisms in the junction. We theoretically reproduce the experimental results by considering a short, diffusive junction model that takes into account the interaction between the Andreev spectrum and the electromagnetic environment, from which we deduce a lifetime of ~17 ps for non-equilibrium populations.
• 2D materials shrink superconducting qubits
C. Schönenberger.
Nature Materials (News & Views) 21, 381 (2022)
[Abstract ]
The exceptional quality of hexagonal boron nitride crystals that can be cleaved into few layers provides ultrathin dielectrics, thereby opening a route to ultrasmall capacitors with large capacitances. With such capacitors, the superconducting transmon qubit is scaled down by orders of magnitude. % Associated Content: `Hexagonal boron nitride as a low-loss dielectric for superconducting quantum circuits and qubits’ by J. J.-I. Wang et al. Nature Materials 21, 398–403 (2022)
##### 2021
• Boosting proximity spin orbit coupling in graphene/WSe2 heterostructures via hydrostatic pressure
B. Fülöp, A. Márffy, S. Zihlmann, M. Gmitra, E. Tóvári, B. Szentpéteri, M. Kedves, K. Watanabe, T. Taniguchi, J. Fabian, C. Schönenberger, P. Makk, and S. Csonka.
npj 2D Materials and Applications 5, 82 (2021)
[arXiv:2103.13325 ] [ Open Data ] [Abstract ]
Van der Waals heterostructures composed of multiple few layer crystals allow the engineering of novel materials with predefined properties. As an example, coupling graphene weakly to materials with large spin orbit coupling (SOC) allows to engineer a sizeable SOC in graphene via proximity effects. The strength of the proximity effect depends on the overlap of the atomic orbitals, therefore, changing the interlayer distance via hydrostatic pressure can be utilized to enhance the interlayer coupling between the layers. In this work, we report measurements on a graphene/WSe2 heterostructure exposed to increasing hydrostatic pressure. A clear transition from weak localization to weak anti-localization is visible as the pressure increases, demonstrating the increase of induced SOC in graphene.
• New method of transport measurements on van der Waals heterostructures under pressure
B. Fülöp, A. Márffy, E. Tóvári, M. Kedves, S. Zihlmann, D. Indolese, Z. Kovács-Krausz, K. Watanabe, T. Taniguchi, C. Schönenberger, Kézsmárki I. P. Makk, and S. Csonka.
J. Apl. Phys. 130, 64303 (2021)
[arXiv:2103.14617 ] [ Open Data ] [Abstract ]
The interlayer coupling, which has a strong influence on the properties of van der Waals heterostructures, strongly depends on the interlayer distance. Although considerable theoretical interest has been demonstrated, experiments exploiting a variable interlayer coupling on nanocircuits are scarce due to the experimental difficulties. Here, we demonstrate a novel method to tune the interlayer coupling using hydrostatic pressure by incorporating van der Waals heterostructure based nanocircuits in piston-cylinder hydrostatic pressure cells with a dedicated sample holder design. This technique opens the way to conduct transport measurements on nanodevices under pressure using up to 12 contacts without constraints on the sample at the fabrication level. Using transport measurements, we demonstrate that a hexagonal boron nitride capping layer provides a good protection of van der Waals heterostructures from the influence of the pressure medium, and we show experimental evidence of the influence of pressure on the interlayer coupling using weak localization measurements on a transitional metal dichalcogenide/graphene heterostructure.
• Superconducting contacts to a monolayer semiconductor
M. Ramezani, Correa I. Sampaio, K. Watanabe, T. Taniguchi, C. Schönenberger, and A. Baumgartner.
Nano Letters 21, 5614 (2021)
[arXiv:2102.06227 ] [ Open Data ] [Abstract ]
We demonstrate superconducting vertical interconnect access (VIA) contacts to a mono-layer of molybdenum disulfide (MoS2), a layered semiconductor with highly relevant elec-tronic and optical properties. As a contact material we use MoRe, a superconductor with a high critical magnetic field and high critical temperature. The electron transport is mostly dominated by a single superconductor/normal conductor junction with a clear superconductor gap. In addition, we find MoS2 regions that are strongly coupled to the superconductor, resulting in resonant Andreev tunneling and junction dependent gap characteristics, suggesting a superconducting proximity effect. Magnetoresistance measurements show that the band-structure and the high intrinsic carrier mobility remain intact in the bulk of the MoS2. This type of VIA contact is applicable to a large variety of layered materials and superconducting
• Global strain-induced scalar potential in graphene devices
L. Wang, A. Baumgartner, P. Makk, S. Zihlmann, B. S. Varghese, D. I. Indolese, K. Watanabe, T. Taniguchi, and C. Schönenberger.
Comm. Phys. 4, 147 (2021)
[arXiv:2009.03035 ] [ Open Data ] [Abstract ]
By mechanically distorting a crystal lattice it is possible to engineer the electronic and optical properties of a material. In graphene, one of the major effects of such a distortion is an energy shift of the Dirac point, often described as a scalar potential. We demonstrate how such a scalar potential can be generated systematically over an entire electronic device and how the resulting changes in the graphene work function can be detected in transport experiments. Combined with Raman spectroscopy, we obtain a characteristic scalar potential consistent with recent theoretical estimates. This direct evidence for a scalar potential on a macroscopic scale due to deterministically generated strain in graphene paves the way for engineering the optical and electronic properties of graphene and similar materials by using external strain.
##### 2020
• Out-of-plane corrugations in graphene based van der Waals heterostructures
S. Zihlmann, P. Makk, M. K. Rehmann, L. Wang, M. Kedves, D.Indolese, K. Watanabe, T. Taniguchi, D. M. Zumbühl, and C. Schönenberger.
Phys. Rev. B 102, 195404 (2020)
[arXiv:2004.02690 ] [ Open Data ] [Abstract ]
Two dimensional materials are usually envisioned as flat, truly 2D layers. However out-of-plane corrugations are inevitably present in these materials. In this manuscript, we show that graphene flakes encapsulated between insulating crystals (hBN, WSe2), although having large mobilities, surprisingly contain out-of-plane corrugations. The height fluctuations of these corrugations are revealed using weak localization measurements in the presence of a static in-plane magnetic field. Due to the random out-of-plane corrugations, the in-plane magnetic field results in a random out-of-plane component to the local graphene plane, which leads to a substantial decrease of the phase coherence time. Atomic force microscope measurements also confirm a long range height modulation present in these crystals. Our results suggest that phase coherent transport experiments relying on purely in-plane magnetic fields in van der Waals heterostructures have to be taken with serious care.
• Compact SQUID realized in a double layer graphene heterostructure
D. I. Indolese, P. Karnatak, A. Kononov, R. Delagrange, R. Haller, L. Wang, P. Makk, K. Watanabe, T. Taniguchi, and C. Schönenberger.
Nano Letters 20, 7129–7135 (2020)
[arXiv:2006.05522 ] [ Open Data ] [Abstract ]
Two-dimensional systems that host one-dimensional helical states are exciting from the perspective of scalable topological quantum computation when coupled with a superconductor. Graphene is particularly promising for its high electronic quality, versatility in van der Waals heterostructures and its electron and hole-like degenerate 0$th$ Landau level. Here, we study a compact double layer graphene SQUID (superconducting quantum interference device), where the superconducting loop is reduced to the superconducting contacts, connecting two parallel graphene Josephson junctions. Despite the small size of the SQUID, it is fully tunable by independent gate control of the Fermi energies in both layers. Furthermore, both Josephson junctions show a skewed current phase relationship, indicating the presence of superconducting modes with high transparency. In the quantum Hall regime we measure a well defined conductance plateau of 2$e^2/h$ an indicative of counter propagating edge channels in the two layers. Our work opens a way for engineering topological superconductivity by coupling helical edge states, from graphene’s electron-hole degenerate 0$th$ Landau level via superconducting contacts.
• Experimental demonstration of the suppression of optical phonon splitting in 2D materials by Raman spectroscopy
De M. Luca, X. Cartoixa, D. Indolese, J. Martín-Sánchez, K. Watanabe, T. Taniguchi, C. Schönenberger, R. Trotta, R. Rurali, and I. Zardo.
2D Materials 7 (2020)
[arXiv:2009.07618 ] [Abstract ]
Raman spectroscopy is one of the most extended experimental techniques to investigate thin-layered 2D materials. For a complete understanding and modeling of the Raman spectrum of a novel 2D material, it is often necessary to combine the experimental investigation to density-functional-theory calculations. We provide the experimental proof of the fundamentally different behavior of polar 2D vs 3D systems regarding the effect of the dipole−dipole interactions, which in 2D systems ultimately lead to the absence of optical phonons splitting, otherwise present in 3D materials. We demonstrate that non-analytical corrections (NACs) should not be applied to properly model the Raman spectra of few-layered 2D materials, such as WSe2 and h-BN, corroborating recent theoretical predictions [Nano Lett. 2017, 17 (6), 3758-3763]. Our findings are supported by measurements performed on tilted samples that allow increasing the component of photon momenta in the plane of the flake, thus unambiguously setting the direction of an eventual NAC. We also investigate the influence of the parity of the number of layers and of the type of layer-by-layer stacking on theeffect of NACs on the Raman spectra.
• Mobility enhancement in graphene by in situ reduction of random strain fluctuations
L. Wang, P. Makk, S. Zihlmann, A. Baumgartner, D. I. Indolese, K. Watanabe, T. Taniguchi, and C. Schönenberger.
Phys. Rev. Lett. 124, 157701 (2020)
[arXiv:1909.13484 ] [ Open Data ] [Abstract ]
Microscopic corrugations are ubiquitous in graphene even when placed on atomically flat substrates. These result in random local strain fluctuations limiting the carrier mobility of high quality hBN-supported graphene devices. We present transport measurements in hBN-encapsulated devices where such strain fluctuations can be in situ reduced by increasing the average uniaxial strain. When ∼0.2\% of uniaxial strain is applied to the graphene, an enhancement of the carrier mobility by ∼35\% is observed while the residual doping reduces by ∼39\%. We demonstrate a strong correlation between the mobility and the residual doping, from which we conclude that random local strain fluctuations are the dominant source of disorder limiting the mobility in these devices. Our findings are also supported by Raman spectroscopy measurements.
##### 2019
• In-situ strain tuning in hBN-encapsulated graphene electronic devices
L. Wang, S. Zihlmann, A. Baumgartner, J. Overbeck, K. Watanabe, T. Taniguchi, P. Makk, and C. Schönenberger.
Nano Letters 19, 4097-4102 (2019)
[arXiv:1904.06737 ] [ Open Data ] [Abstract ]
Using a simple setup to bend a flexible substrate, we demonstrate deterministic and reproducible in-situ strain tuning of graphene electronic devices. Central to this method is the full hBN encapsulation of graphene, which preserves the exceptional quality of pristine graphene for transport experiments. In addition, the on-substrate approach allows one to exploit strain effects in the full range of possible sample geometries and at the same time guarantees that changes in the gate capacitance remain negligible during the deformation process. We use Raman spectroscopy to spatially map the strain magnitude in devices with two different geometries and demonstrate the possibility to engineer a strain gradient, which is relevant for accessing the valley degree of freedom with pseudo-magnetic fields. Comparing the transport characteristics of a suspended device with those of an on-substrate device, we demonstrate that our new approach does not suffer from the ambiguities encountered in suspended devices
• GHz nanomechanical resonator in an ultraclean suspended graphene p-n junction
Minkyung Jung, P. Rickhaus, S. Zihlmann, A. Eichler, P. Makk, and C. Schönenberger.
Nanoscale 11, 4355 (2019)
[arXiv:1812.06412 ] [Abstract ]
We demonstrate high-frequency mechanical resonators in ballistic graphene p–n junctions. Fully suspended graphene devices with two bottom gates exhibit ballistic bipolar behavior after current annealing. We determine the graphene mass density and built-in tension for different current annealing steps by comparing the measured mechanical resonant response to a simplified membrane model. We consistently find that after the last annealing step the mass density compares well with the expected density of pure graphene. In a graphene membrane with high built-in tension, but still of macroscopic size with dimensions 3 × 1 micrometer^2, a record resonance frequency of 1.17 GHz is observed after the final current annealing step. We further compare the resonance response measured in the unipolar with the one in the bipolar regime. Remarkably, the resonant signals are strongly enhanced in the bipolar regime. This enhancement is caused in part by the Fabry-Pérot resonances that appear in the bipolar regime and possibly also by the photothermoelectric effect that can be very pronounced in graphene p–n junctions under microwave irradiation.
• New generation of Moiré superlattices in doubly aligned hBN/graphene/hBN heterostructures
L. Wang, S. Zihlmann, Ming-Hao Liu, P. Makk, K. Watanabe, T. Taniguchi, A. Baumgartner, and C. Schönenberger.
Nano Letters 19, 2371-2376 (2019)
[arXiv:1812.10031 ] [Abstract ]
The specific rotational alignment of two-dimensional lattices results in a moiré superlattice with a larger period than the original lattices and allows one to engineer the electronic band structure of such materials. So far, transport signatures of such superlattices have been reported for graphene/hBN and graphene/graphene systems. Here we report moiré superlattices in fully hBN encapsulated graphene with both the top and the bottom hBN aligned to the graphene. In the graphene, two different moiré superlattices form with the top and the bottom hBN, respectively. The overlay of the two superlattices can result in a third superlattice with a period larger than the maximum period (\SI{14}{nm}) in the graphene/hBN system, which we explain in a simple model. This new type of band structure engineering allows one to artificially create an even wider spectrum of electronic properties in two-dimensional materials.
• Non-equilibrium properties of graphene probed by superconducting tunnel spectroscopy
S. Zihlmann, P. Makk, S. Castillas, J. Gramich, K. Thodkar, S. Caneva, R. Wang, S. Hofmann, and C. Schönenberger.
Phys. Rev. B 99, 75419 (2019)
[arXiv:1811.08746 ] [Abstract ]
We report on non-equilibrium properties of graphene probed by superconducting tunnel spectroscopy. A hexagonal boron nitride (hBN) tunnel barrier in combination with a superconducting Pb contact is used to extract the local energy distribution function of the quasiparticles in graphene samples in different transport regimes. In the cases where the energy distribution function resembles a Fermi-Dirac distribution, the local electron temperature can directly be accessed. This allows us to study the cooling mechanisms of hot electrons in graphene. In the case of long samples (device length L much larger than the electron-phonon scattering length le−ph), cooling through acoustic phonons is dominant. We find a cross-over from the dirty limit with a power law T3 at low temperature to the clean limit at higher temperatures with a power law T4 and a deformation potential of 13..3 eV. For shorter samples, where L is smaller than le−ph but larger than the electron-electron scattering length le−e, the well-known cooling through electron out-diffusion is found. Interestingly, we find strong indications of an enhanced Lorenz number in graphene. We also find evidence of a non-Fermi-Dirac distribution function, which is a result of non-interacting quasiparticles in very short samples
##### 2018
• Wideband and on-chip excitation for dynamical spin injection into graphene
D. I. Indolese, S. Zihlmann, P. Makk, C. Jünger, K. Thodkar, and C. Schönenberger.
Phys. Rev. Appl. 10, 44053 (2018)
[arXiv:1806.09356 ] [Abstract ]
Graphene is an ideal material for spin transport as very long spin relaxation times and lengths can be achieved even at room temperature. However, electrical spin injection is challenging due to the conductivity mismatch problem. Spin pumping driven by ferromagnetic resonance is a neat way to circumvent this problem as it produces a pure spin current in the absence of a charge current. Here, we show spin pumping into single layer graphene in micron scale devices. A broadband on-chip RF current line is used to bring micron scale permalloy (Ni80Fe20) pads to ferromagnetic resonance with a magnetic eld tunable resonance condition. At resonance, a spin current is emitted into graphene, which is detected by the inverse spin hall voltage in a close-by platinum electrode. Clear spin current signals are detected down to a power of a few milliwatts over a frequency range of 2 GHz to 8 GHz. This compact device scheme paves the way for more complex device structures and allows the investigation of novel materials.
• Signatures of van Hove singularities probed by the supercurrent in a graphene – hBN superlattice
D. I. Indolese, R. Delagrange, P. Makk, J. R. Wallbank, K. Wanatabe, T. Taniguchi, and C. Schönenberger.
Phys. Rev. Lett. 121, 137701 (2018)
[arXiv:1805.10184 ] [Abstract ]
The bandstructure of graphene can be strongly modified when it is aligned with its Boron Nitride substrate. A moiré superlattice forms, which manifests itself by the appearance of new Dirac points, accompanied by van Hove singularities. In this work, we present supercurrent measurements in a Josephson junction made from such a graphene superlattice in the long and diffusive transport regime, where the supercurrent depends on the Thouless energy. We can then estimate the specific density of states of the graphene superlattice from the combined measurement of the critical current and the normal state resistance. The result matches with theoretical predictions and highlights the strong increase of the density of states at the van Hove singularities. By measuring the magnetic field dependence of the supercurrent, we find the presence of edge currents at these singularities. We explain it by the reduction of the Fermi velocity associated with the flat band at the van Hove singularity, which suppresses the supercurrent in the bulk while the electrons at the edge remain less localized, resulting in an edge supercurrent. We attribute this different behavior of the edges to defects or chemical doping.
• Observation of High Accuracy Resitance Quantization in CVD Graphene
K. Thodkar, C. Schönenberger, M. Calame, F. Lüönd, and Overrney F. B. ad Jeanneret.
2018 Conference on Precision Electromagnetic Measurements (CPEM 2018) pages 1-2 (2018)
[Abstract ]
A prime technique to produce graphene is Chemical Vapor Deposition (CVD). In this paper, the first observation of high accuracy resistance quantization in CVD graphene samples grown on polycrystalline copper foils is shown. High precision measurements performed using a cryogenic current comparator reveal a resistance quantization accuracy of 100 parts in 10^9.
• Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in WSe2/Gr/hBN heterostructures
S. Zihlmann, A. W. Cummings, J. H. Garcia, M. Kedves, K. Watanabe, T. Taniguchi, C. Schönenberger, and P. Makk.
Phys. Rev. B 97, 75434 (2018)
[arXiv:1712.05678 ] [Abstract ]
Large spin-orbital proximity effects have been predicted in graphene interfaced with a transition metal dichalcogenide layer. Whereas clear evidence for an enhanced spin-orbit coupling has been found at large carrier densities, the type of spin-orbit coupling and its relaxation mechanism remained unknown. We show for the first time an increased spin-orbit coupling close to the charge neutrality point in graphene, where topological states are expected to appear. Single layer graphene encapsulated between the transition metal dichalcogenide WSe2 and hBN is found to exhibit exceptional quality with mobilities as high as 100 000 cm2/Vs. At the same time clear weak anti-localization indicates strong spin-orbit coupling and a large spin relaxation anisotropy due to the presence of a dominating symmetric spin-orbit coupling is found. Doping dependent measurements show that the spin relaxation of the in-plane spins is largely dominated by a valley-Zeeman spin-orbit coupling and that the intrinsic spin-orbit coupling plays a minor role in spin relaxation. The strong spin-valley coupling opens new possibilities in exploring spin and valley degree of freedoms in graphene with the realization of new concepts in spin manipulation.
• Charge transport in a single molecule transistor probed by scanning tunneling microscopy
S. Bouvron, R. Maurand, A. Graf, P. Erler, L. Gragnaniello, M. Skripnik, D. Wiedmann, C. Engesser, C. Nef, W. Fu, C. Schönenberger, F. Paulya, and M. Fonin.
Nanoscale 10, 1487-1493 (2018)
[Abstract ]
We report on the scanning tunneling microscopy/spectroscopy (STM/STS) study of cobalt phthalocyanine (CoPc) molecules deposited onto a back-gated graphene device. We observe a clear gate voltage (Vg) dependence of the energy position of the features originating from the molecular states. Based on the analysis of the energy shifts of the molecular features upon tuning Vg, we are able to determine the nature of the electronic states that lead to a gapped differential conductance. Our measurements show that capacitive couplings of comparable strengths exist between the CoPc molecule and the STM tip as well as between CoPc and graphene, thus facilitating electronic transport involving only unoccupied molecular states for both tunneling bias polarities. These findings provide novel information on the interaction between graphene and organic molecules and are of importance for further studies, which envisage the realization of single molecule transistors with non-metallic electrodes.
• Quantum-Confined Stark Effect in a MoS2 Monolayer van der Waals Heterostructure
J. G. Roch, N. Leisgang, G. Froehlicher, P. Makk, K. Watanabe, T. Taniguchi, C. Schönenberger, and R. J. Warburton.
Nano Letters 18, 1070-1074 (2018)
[arXiv:1710.09750 ] [Abstract ]
The optics of dangling-bond-free van der Waals heterostructures containing transition metal dichalcogenides are dominated by excitons. A crucial property of a confined exciton is the quantum confined Stark effect (QCSE). Here, such a heterostructure is used to probe the QCSE by applying a uniform vertical electric field across a molybdenum disulfide (MoS2) monolayer. The photoluminescence emission energies of the neutral and charged excitons shift quadratically with the applied electric field, provided that the electron density remains constant, demonstrating that the exciton can be polarized. Stark shifts corresponding to about half the homogeneous linewidth were achieved. Neutral and charged exciton polarizabilities of (7.8 +/- 1.0) × 10−10 and (6.4 +/- 0.9) × 10−10 D m V−1 at relatively low electron density (~10^12 cm−2) have been extracted, respectively. These values are one order of magnitude lower than the previously reported values but in line with theoretical calculations. The methodology presented here is versatile and can be applied to other semiconducting layered materials.
• Spin transport in two-layer-CVD-hBN/graphene/hBN heterostructures
M. Gurram, S. Omar, S. Zihlmann, P. Makk, Q. C. Li, Y. F. Zhang, C. Schönenberger, and B. J. van Wees.
Phys. Rev. B 97, 45411 (2018)
[arXiv:1712.00815 ] [Abstract ]
We study room-temperature spin transport in graphene devices encapsulated between a layer-by-layer-stacked two-layer-thick chemical vapor deposition (CVD) grown hexagonal boron nitride (hBN) tunnel barrier, and a few-layer-thick exfoliated-hBN substrate. We find mobilities and spin-relaxation times comparable to that of SiO2 substrate-based graphene devices, and we obtain a similar order of magnitude of spin relaxation rates for both the Elliott-Yafet and D’Yakonov-Perel’ mechanisms. The behavior of ferromagnet/two-layer-CVDhBN/ graphene/hBN contacts ranges from transparent to tunneling due to inhomogeneities in the CVD-hBN barriers. Surprisingly, we find both positive and negative spin polarizations for high-resistance two-layer-CVDhBN barrier contacts with respect to the low-resistance contacts. Furthermore, we find that the differential spininjection polarization of the high-resistance contacts can be modulated by dc bias from −0.3 to +0.3 V with no change in its sign, while its magnitude increases at higher negative bias. These features point to the distinctive spin-injection nature of the two-layer-CVD-hBN compared to the bilayer-exfoliated-hBN tunnel barriers.
##### 2017
• Giant Valley-Isospin Conductance Oscillations in Ballistic Graphene
C. Handschin, P.Makk, P. Rickhaus, R. Maurand, K. Watanabe, T. Taniguchi, K. Richter, Ming-Hao Liu, and C. Schönenberger.
Nano Letters 17, 5389-5393 (2017)
[arXiv:1708.09614 ] [Abstract ]
At high magnetic fields the conductance of graphene is governed by the half-integer quantum Hall effect. By local electrostatic gating a p−n junction perpendicular to the graphene edges can be formed, along which quantum Hall channels copropagate. It has been predicted by Tworzidło and co-workers that if only the lowest Landau level is filled on both sides of the junction, the conductance is determined by the valley (isospin) polarization at the edges and by the width of the flake. This effect remained hidden so far due to scattering between the channels copropagating along the p−n interface (equilibration). Here we investigate p−n junctions in encapsulated graphene with a movable p−n interface with which we are able to probe the edge configuration of graphene flakes. We observe large quantum conductance oscillations on the order of e2/h which solely depend on the p−n junction position providing the first signature of isospin-defined conductance. Our experiments are underlined by quantum transport calculations.
• Restoring the Electrical Properties of CVD Graphene via Physisorption of Molecular Adsorbates
K. Thodkar, D-. Thompson, F. Lüönd, L. Moser, F. Overney, L. Marot, C. Schönenberger, B. Jeanneret, and M. Calame.
ACS Appl. Mater. Interfaces 9(29), 25014-25022 (2017)
[Abstract ]
Chemical vapor deposition (CVD) is a powerful technique to produce graphene for large-scale applications. Polymer-assisted wet transfer is commonly used to move the graphene onto silicon substrates, but the resulting devices tend to exhibit p-doping, which decreases the device quality and reproducibility. In an effort to better understand the origin of this effect, we coated graphene with n-methyl-2-pyrrolidone (NMP) and hexamethyldisilazane (HMDS) molecules that exhibit negligible charge transfer to graphene but bind more strongly to graphene than ambient adsorbents. Using Raman spectroscopy, X-ray photoelectron spectroscopy (XPS), electrical transport measurements, and quantum mechanical computer simulations, we show that the molecules help in the removal of p-doping, and our data indicate that the molecules do this by replacing ambient adsorbents (typically O2 and water) on the graphene surface. This very simple method of improving the electronic properties of CVD graphene by passivating its surface with common solvent molecules will accelerate the development of CVD graphene-based devices
• Contactless Microwave Characterization of Encapsulated Graphene p-n Junctions
V. Ranjan, S. Zihlmann, P. Makk, K. Watanabe, T. Taniguchi, and C. Schönenberger.
Phys. Rev. Appl. 7(5), 54015 (2017)
[arXiv:1702.02071 ] [Abstract ]
Accessing intrinsic properties of a graphene device can be hindered by the influence of contact electrodes. Here, we capacitively couple graphene devices to superconducting resonant circuits and observe clear changes in the resonance-frequency and -widths originating from the internal charge dynamics of graphene. This allows us to extract the density of states and charge relaxation resistance in graphene p-n junctions without the need of electrical contacts. The presented characterizations pave a fast, sensitive and non-invasive measurement of graphene nanocircuits.
• Fabry-Pérot Resonances in a Graphene/hBN Moiré Superlattice
C. Handschin, P.Makk, P. Rickhaus, M. -H. Liu, K. Watanabe, T. Taniguchi, K. Richter, and C. Schönenberger.
Nano Letters 17, 328-333 (2017)
[arXiv:1701.09141 ] [Abstract ]
While Fabry-Pérot (FP) resonances and Moiré superlattices are intensively studied in graphene on hexagonal boron nitride (hBN), the two effects have not been discussed in their coexistence. Here we investigate the FP oscillations in a ballistic pnp-junctions in the presence and absence of a Moiré superlattice. First, we address the effect of the smoothness of the confining potential on the visibility of the FP resonances and carefully map the evolution of the FP cavity size as a function of densities inside and outside the cavity in the absence of a superlattice, when the cavity is bound by regular pn-junctions. Using a sample with a Moiré superlattice, we next show that an FP cavity can also be formed by interfaces that mimic a pn-junction but are defined through a satellite Dirac point due to the superlattice. We carefully analyze the FP resonances, which can provide insight into the band-reconstruction due to the superlattice.
##### 2016
• Gate-controlled conductance enhancement from quantum Hall channels along graphene p-n junctions
E. Tovari, P. Makk, Ming-Hao Liu, P. Rickhaus, Z. Kovas-Krausz, C. Schönenberger, and S. Csonka.
Nanoscale 8(47), 19910-19916 (2016)
[arXiv:1606.08007 ] [Abstract ]
The formation of quantum Hall channels inside the bulk of graphene is studied using various contact and gate geometries. p-n junctions are created along the longitudinal direction of samples, and enhanced conductance is observed in the case of bipolar doping due to the new conducting channels formed in the bulk, whose position, propagating direction and, in one geometry, coupling to electrodes are determined by the gate-controlled filling factor across the device. This effect could be exploited to probe the behavior and interaction of quantum Hall channels protected against uncontrolled scattering at the edges.
• Microwave Photodetection in an Ultraclean Suspended Bilayer Graphene p–n Junction
M. Jung, P. Rickhaus, S. Zihlmann, and C. Schönenberger.
Nano Letters 16(11), 6988 (2016)
[Abstract ]
We explore the potential of bilayer graphene as a cryogenic microwave photodetector by studying the microwave absorption in fully suspended clean bilayer graphene p–n junctions in the frequency range of 1–5 GHz at a temperature of 8 K. We observe a distinct photocurrent signal if the device is gated into the p–n regime, while there is almost no signal for unipolar doping in either the n–n or p–p regimes. Most surprisingly, the photocurrent strongly peaks when one side of the junction is gated to the Dirac point (charge-neutrality point CNP), while the other remains in a highly doped state. This is different to previous results where optical radiation was used. We propose a new mechanism based on the phototermal effect explaining the large signal. It requires contact doping and a distinctly different transport mechanism on both sides: one side of graphene is ballistic and the other diffusive. By engineering partially diffusive and partially ballistic devices, the photocurrent can drastically be enhanced.
• Role of hexagonal boron nitride in protecting ferromagnetic anostructures from oxidation
S. Zihlmann, P. Makk, C. A. F. Vaz, and C. Schönenberger.
2D Materials 3(1), 11008 (2016)
[arXiv:1509.03087 ] [Abstract ]
Ferromagnetic contacts are widely used to inject spin polarized currents into non-magnetic materials such as semiconductors or 2-dimensional materials like graphene. In these systems, oxidation of the ferromagnetic materials poses an intrinsic limitation on device performance. Here we investigate the role of ex situ transferred chemical vapour deposited hexagonal boron nitride (hBN) as an oxidation barrier for nanostructured cobalt and permalloy electrodes. The chemical state of the ferromagnets was investigated using x-ray photoemission electron microscopy because of its high sensitivity and lateral resolution. We have compared the oxide thickness formed on ferromagnetic nanostructures covered by hBN to uncovered reference structures. Our results show that hBN reduces the oxidation rate of ferromagnetic nanostructures suggesting that it could be used as an ultra-thin protection layer in future spintronic devices.
• Spin transport in fully hexagonal boron nitride encapsulated graphene
M. Gurram, S. Omar, S. Zihlmann, P. Makk, C. Schönenberger, and B. J. van Wees.
Physical Review B 93(11), 115441 (2016)
[arXiv:1603.04357 ] [Abstract ]
We study fully hexagonal boron nitride (hBN) encapsulated graphene spin valve devices at room temperature. The device consists of a graphene channel encapsulated between two crystalline hBN flakes: thick-hBN flake as a bottom gate dielectric substrate which masks the charge impurities from Si_{O2}/Si substrate and single-layer thin-hBN flake as a tunnel barrier. Full encapsulation prevents the graphene from coming in contact with any polymer/chemical during the lithography and thus gives homogeneous charge and spin transport properties across different regions of the encapsulated graphene. Further, even with the multiple electrodes in-between the injection and the detection electrodes which are in conductivity mismatch regime, we observe spin transport over 12.5 $\mu$m-long distance under the thin-hBN encapsulated graphene channel, demonstrating the clean interface and the pinhole-free nature of the thin hBN as an efficient tunnel barrier.
• Comparative study of single and multi domain CVD graphene using large-area Raman mapping and electrical transport characterization
K. Thodkar, El M. Abbassi, F. Lüönd, F. Overney, C. Schönenberger, B. Jeanneret, and M. Calame.
physica status solidi (RRL) – Rapid Research Letters 10(11), 807 (2016)
[Abstract ]
We systematically investigate the impact of granularity in CVD graphene films by performing Raman mapping and electrical characterization of single (SD) and multi domain (MD) graphene. In order to elucidate the quality of the graphene film, we study its regional variations using large-area Raman mapping and compare the G and 2D peak positions of as-transferred chemical vapor deposited (CVD) graphene on SiO2 substrate. We find a similar upshift in wavenumber in both SD and MD graphene in comparison to freshly exfoliated graphene. In our case, doping could play the dominant role behind the observation of such upshifts rather than the influence due to strain. Interestingly, the impact of the polymer-assisted wet transfer process is the same in both the CVD graphene types. The electrical characterization shows that SD graphene exhibits a substantially higher (a factor 5) field-effect mobility when compared to MD graphene. We attribute the low sheet resistance and mobility enhancement to a decrease in charge carrier scattering thanks to a reduction of the number of grain boundaries and defects in SD graphene.
• Signatures of single quantum dots in graphene nanoribbons within the quantum Hall regime
E. Tovari, P. Makk, P. Rickhaus, C. Schönenberger, and S. Csonka.
Nanoscale 8, 11480 (2016)
[arXiv:1601.01628 ] [Abstract ]
We report on the observation of periodic conductance oscillations near quantum Hall plateaus in suspended graphene nanoribbons. They are attributed to single quantum dots that form in the narrowest part of the ribbon, in the valleys and hills of a disorder potential. In a wide flake with two gates, a double-dot system’s signature has been observed. Electrostatic confinement is enabled in single-layer graphene due to the gaps that form between Landau levels, suggesting a way to create gate-defined quantum dots that can be accessed with quantum Hall edge states.
##### 2015
• Gate tuneable beamsplitter in ballistic graphene
P. Rickhaus, P. Makk, M. -H. Liu, K. Richter, and C. Schönenberger.
Applied Physics Letters 107, 251901 (2015)
[arXiv:1511.03044 ] [Abstract ]
We present a beam splitter in a suspended, ballistic, multiterminal, bilayer graphene device. By using local bottomgates, a p-n interface tilted with respect to the current direction can be formed. We show that the p-n interface acts as a semi-transparent mirror in the bipolar regime and that the reflectance and transmittance of the p-n interface can be tuned by the gate voltages. Moreover, by studying the conductance features appearing in magnetic field, we demonstrate that the position of the p-n interface can be moved by 1μm. The herein presented beamsplitter device can form the basis of electron-optic interferometers in graphene
• Gate tuneable beamsplitter in ballistic graphene
P. Makk.
Nature Physics 11, 894-895 (2015)
• Snake trajectories in ultraclean graphene p–n junctions
P. Rickhaus, P. Makk, Ming-Hao Liu, E. Tovari, M. Weiss, R. Maurand, and C. Schönenberger.
Nature Communications 6, 6470 (2015)
[arXiv:1502.01935 ] [Abstract ]
Snake states are trajectories of charge carriers curving back and forth along an interface. There are two types of snake states, formed by either inverting the magnetic field direction or the charge carrier type at an interface. The former has been demonstrated in GaAs–AlGaAs heterostructures, whereas the latter has become conceivable only with the advance of ballistic graphene where a gap-less p–n interface governed by Klein tunnelling can be formed. Such snake states were hidden in previous experiments due to limited sample quality. Here we report on magneto-conductance oscillations due to snake states in a ballistic suspended graphene p–n junction, which occur already at a very small magnetic field of 20 mT. The visibility of 30 percent is enabled by Klein collimation. Our finding is firmly supported by quantum transport simulations. We demonstrate the high tunability of the device and operate it in different magnetic field regimes.
• Scalable Tight-Binding Model for Graphene
Ming-Hao Liu, P. Rickhaus, P. Makk, E. Tovari, R. Maurand, F. Tkatschenko, M. Weiss, C. Schönenberger, and K. Richter.
Phys. Rev. Lett. 114, 36601 (2015)
[arXiv:1407.5620 ] [Abstract ]
Artificial graphene consisting of honeycomb lattices other than the atomic layer of carbon has been shown to exhibit electronic properties similar to real graphene. Here, we reverse the argument to show that transport properties of real graphene can be captured by simulations using “theoretical artificial graphene.” To prove this, we first derive a simple condition, along with its restrictions, to achieve band structure invariance for a scalable graphene lattice. We then present transport measurements for an ultraclean suspended single-layer graphene pn junction device, where ballistic transport features from complex Fabry-Pérot interference (at zero magnetic field) to the quantum Hall effect (at unusually low field) are observed and are well reproduced by transport simulations based on properly scaled single-particle tight-binding models. Our findings indicate that transport simulations for graphene can be efficiently performed with a strongly reduced number of atomic sites, allowing for reliable predictions for electric properties of complex graphene devices. We demonstrate the capability of the model by applying it to predict so-far unexplored gate-defined conductance quantization in single-layer graphene.
• Point contacts in encapsulated graphene
C. Handschin, B. Fülöp, P. Makk, S. Blanter, M. Weiss, K. Watanabe, T. Taniguchi, S. Csonka, and C. Schönenberger.
Applied Physics Letters 107(18), 183108 (2015)
[arXiv:1509.04137v1.pdf ] [Abstract ]
We present a method to establish inner point contacts on hexagonal boron nitride (hBN) encapsulated graphene heterostructures with dimensions as small as 100 nm by pre-patterning the top-hBN in a separate step prior to dry-stacking. 2 and 4-terminal field effect measurements between different lead combinations are in qualitative agreement with an electrostatic model assuming point-like contacts. The measured contact resistances are 0.5-1.5 k$\Omega$ per contact, which is quite low for such small contacts. By applying a perpendicular magnetic fields, an insulating behaviour in the quantum Hall regime was observed, as expected for inner contacts. The fabricated contacts are compatible with high mobility graphene structures and open up the field for the realization of several electron optical proposals.
• Guiding of Electrons in a Few-Mode Ballistic Graphene Channel
P. Rickhaus, M. -H. Liu, P. Makk, R. Maurand, S. Hess, S. Zihlmann, M. Weiss, K. Richter, and Schönenberger Richter (Uni. C. -. in cooperation with group Regensburg).
Nano Letters 15(5819) (2015)
[arXiv:1509.02653 ] [Abstract ]
In graphene, the extremely fast charge carriers can be controlled by electron-optical elements, such as waveguides, in which the transmissivity is tuned by the wavelength. In this work, charge carriers are guided in a suspended ballistic few-mode graphene channel, defined by electrostatic gating. By depleting the channel, a reduction of mode number and steps in the conductance are observed, until the channel is completely emptied. The measurements are supported by tight-binding transport calculations including the full electrostatics of the sample.
• Graphene spintronics: the European Flagship perspective
S. Roche, J. Akerman, B. Beschoten, J. -C. Charlier, M. Chshiev, S. P. Dash, B. Dlubak, J. Fabian, A. Fert, M. Guimaraes, F. Guinea, I. Grigorieva, C. Schönenberger, P. Seneor, C. Stampfer, S. O.Valenzuela, X. Waintal, and B. van Wees.
2D Materials 2, 30202 (2015)
##### 2014
• CVD Graphene Electrical Quantum Metrology
K. Thodkar, C. Nef, W. Fu, C. Schönenberger, M. Calame, F. Lüönd, F. Overney, B. Jeckelmann, and B. Jeanneret.
IEEE Proceedings of the Conference on Precision Electromagnetic Measurements (CPEM 2014) pages 540-541 (2014)
[Abstract ]
Graphene, a two dimensional material with sp2 hybridized carbon atoms arranged in honey comb lattice, is known for its unique electronic and mechanical properties. Soon after the isolation of 2D graphene crystals Quantum Hall effect (QHE) has been observed in this material at room temperature. The Quantum Hall plateaus in graphene have large spacing between the Landau levels in comparison to other 2DEGs, which makes it an ideal material for a quantum resistance standard defined by the electron charge and Planck s constant. We will present results for graphene by Chemical Vapor Deposition (CVD) and transferred to SiO2/Si using different techniques. The transferred graphene films were patterned into millimeter scale Hall bar geometry and characterized using confocal Raman spectroscopy. First electrical transport measurements will be presented.
• Large-scale fabrication of BN tunnel barriers for graphene spintronics
W. Fu, P. Makk, R. Maurand, M. Bräuninger, and C. Schönenberger.
Journal of Applied Physics 116(20), 74306 (2014)
[arXiv:1407.1439 ] [Abstract ]
We have fabricated graphene spin-valve devices utilizing scalable materials made from chemical vapor deposition (CVD). Both the spin-transporting graphene and the tunnel barrier material are CVD-grown. The tunnel barrier is realized by Hexagonal boron nitride, used either as a monolayer or bilayer and placed over the graphene. Spin transport experiments were performed using ferromagnetic contacts deposited onto the barrier. We find that spin injection is still greatly suppressed in devices with a monolayer tunneling barrier due to resistance mismatch. This is, however, not the case for devices with bilayer barriers. For those devices, a spin relaxation time of ∼260 ps intrinsic to the CVD graphene material is deduced. This time scale is comparable to those reported for exfoliated graphene, suggesting that this CVD approach is promising for spintronic applications which require scalable materials.
• Fabrication of ballistic suspended graphene with local-gating
R. Maurand, P. Rickhaus, P. Makk, S. Hess, E. Tovari, C. Handschin, M. Weiss, and C. Schönenberger.
Carbon 79, 486-492 (2014)
[arXiv:1409.4751 ] [Abstract ]
Herein we discuss the fabrication of ballistic suspended graphene nanostructures supplemented with local gating. Using in-situ current annealing, we show that exceptional high mobilities can be obtained in these devices. A detailed description is given of the fabrication of bottom and different top-gate structures, which enable the realization of complex graphene structures. We have studied the basic building block, the p-n junction in detail, where a striking oscillating pattern was observed, which can be traced back to Fabry-Perot oscillations that are localized in the electronic cavities formed by the local gates. Finally we show some examples how the method can be extended to incorporate multi-terminal junctions or shaped graphene. The structures discussed here enable the access to electron-optics experiments in ballistic graphene.
• High-yield fabrication of nm-sized gaps in monolayer CVD graphene
C. Nef, L. Pósa, P. Makk, W. Fu, A. Halbritter, C. Schönenberger, and M. Calame.
Nanoscale 6, 7249-7254 (2014)
[Abstract ]
Herein we demonstrate the controlled and reproducible fabrication of sub-5 nm wide gaps in single-layer graphene electrodes. The process is implemented for graphene grown via chemical vapor deposition using an electroburning process at room temperature and in vacuum. A yield of over 95 percent for the gap formation is obtained. This approach allows producing single-layer graphene electrodes for molecular electronics at a large scale. Additionally, from Raman spectroscopy and electroburning carried out simultaneously, we can follow the heating process and infer the temperature at which the gap formation happens.
• Electrolyte gate dependent high-frequency measurement of graphene field-effect transistor for sensing applications
W. Fu, El M. Abbassi, T. Hasler, M. Jung, M. Steinacher, M. Calame, C. Schönenberger, G. Puebla-Hellmann, S. Hellmüller, T. Ihn, and A. Wallraff.
Appl. Phys. Lett. 104, 13102 (2014)
[arXiv:1401.0381 ] [Abstract ]
We performed radiofrequency (RF) reflectometry measurements at 2.4 GHz on electrolyte-gated graphene field-effect transistors, utilizing a tunable stub-matching circuit for impedance matching. We demonstrate that the gate voltage dependent RF resistivity of graphene can be deduced, even in the presence of the electrolyte which is in direct contact with the graphene layer. The RF resistivity is found to be consistent with its DC counterpart in the full gate voltage range. Furthermore, in order to access the potential of high-frequency sensing for applications, we demonstrate time-dependent gating in solution with nanosecond time resolution.
• Rendering graphene supports hydrophilic with non-covalent aromatic functionalization for transmission electron microscopy
R. S. Pantelic, W. Fu, C. Schönenberger, and H. Stahlberg.
Appl. Phys. Lett. 104, 134103 (2014)
[arXiv:1403.6976 ] [Abstract ]
Amorphous carbon films have been routinely used to enhance the preparation of frozen-hydrated transmission electron microscopy (TEM) samples, either in retaining protein concentration, providing mechanical stability or dissipating sample charge. However, strong background signal from the amorphous carbon support obstructs that of the sample, and the insulating properties of amorphous carbon films preclude any efficiency in dispersing charge. Graphene addresses the limitations of amorphous carbon. Graphene is a crystalline material with virtually no phase or amplitude contrast and unparalleled, high electrical carrier mobility. However, the hydrophobic properties of graphene have prevented its routine application in Cryo-TEM. This letter reports a method for rendering graphene TEM supports hydrophilic – a convenient approach maintaining graphene’s structural and electrical properties based on non-covalent, aromatic functionalization.
##### 2013
• Hydrogen plasma microlithography of graphene supported on a Si/SiO2 substrate
B. Eren, T. Glatzel, M. Kisiel, W. Fu, R. Pawlak, U. Gysin, C. Nef, L. Marot, M. Calame, C. Schönenberger, and E. Meyer.
Appl. Phys. Lett. 102, 71601 (2013)
[Abstract ]
In this work, a silicon stencil mask with a periodic pattern is used for hydrogen plasma microlithography of single layer graphene supported on a Si/SiO2 substrate. Obtained patterns are imaged with Raman microscopy and Kelvin probe force microscopy, thanks to the changes in the vibrational modes and the contact potential difference (CPD) of graphene after treatment. A decrease of 60 meV in CPD as well as a significant change of the D/G ratio in the Raman spectra can be associated with a local hydrogenation of graphene, while the topography remains invariant to the plasma exposure.
• Ballistic interferences in suspended graphene
P. Rickhaus, R. Maurand, M. Weiss, C. Schönenberger, Ming-Hao Liu, and K. Richter.
Nature communications 4(2342), 1-6 (2013)
[Abstract ]
Graphene is the 2-dimensional (2D) carbon allotrope with the atoms arranged in a honeycomb lattice [1]. The low-energy electronic excitations in this 2D crystal are described by massless Dirac fermions that have a linear dispersion relation similar to photons [2, 3]. Taking advantage of this optics-like electron dynamics, generic optical elements like lenses, beam splitters and wave guides have been proposed for electrons in engineered and ballistic graphene [4, 5]. Tuning of these elements rely on the ability to adjust the carrier concentration in defined areas, including the possibility to create bipolar regions of opposite charge (p-n regions). However, the combination of ballistic transport and complex electrostatic gating remain challenging. Here, we report on the fabrication and characterization of fully suspended graphene p-n junctions. By local electrostatic gating, resonant cavities can be defined, leading to complex Fabry-Perot interference patterns in the unipolar and the bipolar regime. The amplitude of the observed conductance oscillations account for quantum interference of electrons that propagate ballistically over long distances exceeding 1 micrometer. We also demonstrate that the visibility of the interference pattern is enhanced by Klein collimation at the p-n interface [6, 7]. This finding paves the way to more complex gate controlled ballistic graphene devices and brings electron optics in graphene closer to reality.
• Low-bias active control of TeraHertz-waves by coupling large-area CVD-graphene to a TeraHertz-Metamaterial
F. Valmorra, G. Scalari, C. Maissen, W. Fu, C. Schönenberger, J. W. Choi, H. G. Park, Hyung Gyu, M. Beck, and J. Faist.
Nano Lett. 13, 3193-3198 (2013)
[Abstract ]
We propose an hybrid graphene/metamaterial device based on terahertz electronic split-ring resonators directly evaporated on top of a large-area single-layer CVD graphene. Room temperature time-domain spectroscopy measurements in the frequency range from 250 GHz to 2.75 THz show that the presence of the graphene strongly changes the THz metamaterial transmittance on the whole frequency range. The graphene gating allows active control of such interaction, showing a modulation depth of 11.5\% with an applied bias of 10.6 V. Analytical modeling of the device provides a very good qualitative and quantitative agreement with the measured device behavior. The presented system shows potential as a THz modulator and can be relevant for strong light–matter coupling experiments
• Spin Symmetry of the Bilayer Graphene Ground State
F. Freitag, M. Weiss, R. Maurand, J. Trbovic, and C. Schönenberger.
Phys. Rev. B 87, 161402 (2013)
[Abstract ]
We show nonlinear transport experiments on clean, suspended bilayer graphene that reveal a gap in the density of states. Looking at the evolution of the gap in magnetic fields of different orientation, we find that the groundstate is a spin-ordered phase. Of the three possible gapped groundstates that are predicted by theory for equal charge distribution between the layers, we can therefore exclude the quantum anomalous Hall phase, leaving the layer antiferromagnet and the quantum spin Hall phase as the only possible gapped groundstates for bilayer graphene
##### 2012
• Homogeneity of Bilayer Graphene
F. Freitag, M. Weiss, R. Maurand, J. Trbovic, and C. Schönenberger.
Solid State Communications 152, 2053-2057 (2012)
[arXiv:1207.4424 ] [Abstract ]
We present non-linear transport measurements on suspended, current annealed bilayer graphene devices. Using a multi-terminal geometry we demonstrate that devices tend to be inhomogeneous and host two different electronic phases next to each other. Both of these phases show gap-like features of different magnitude in non-linear transport at low charge carrier densities, as already observed in previous studies. Here, we investigate the magnetic field dependence and find that both features grow with increasing field, the smaller one with 0.6meV/T, the larger one with a 5-10 times higher field dependence. We attribute the larger of the two gaps to an interaction induced broken symmetry state and the smaller one to localization in the more disordered parts of the device.
• Spontaneously Gapped Ground State in Suspended Bilayer Graphene
F. Freitag, J. Trbovic, M. Weiss, and C. Schönenberger.
Phys. Rev. Lett. 108, 76602 (2012)
• Quantum Hall Effect in Graphene with Superconducting Electrodes
P. Rickhaus, M. Weiss, L. Marot, and C. Schönenberger.
Nano Letters 12, 1942 (2012)
##### 2011
• Conductance fluctuations in graphene devices with superconducting contacts in different charge density regimes
F. Freitag, J. Trbovic, and C. Schönenberger.
Phys. Status Solidi B (arXiv:1108.4599) 248, 2649 (2011)
##### 2010
• Superconductivity-enhanced conductance fluctuations in few layer graphene
J. Trbovic, N. Minder, F. Freitag, and C. Schönenberger.
Nanotechnology 21, 274005 (2010) | 2023-02-02 21:32:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5500288009643555, "perplexity": 4871.383320260281}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500041.18/warc/CC-MAIN-20230202200542-20230202230542-00168.warc.gz"} |
http://svn.tug.org/pipermail/texhax/2016-July/022375.html | # [texhax] \magstep reduces text area
Rodolfo Medina rodolfo.medina at gmail.com
Tue Jul 19 21:52:11 CEST 2016
David Carlisle <d.p.carlisle at gmail.com> writes:
> plain tex sets hsize and vsize in "true in" units so they are not
> affected by magnification, so if you magnify you have larger fonts
> with the same physical area which appears expressed as same size
> fonts, in a smaller physical area due to the way \mag works.
>
> David
>
>
> On 19 July 2016 at 20:03, Rodolfo Medina <rodolfo.medina at gmail.com> wrote:
>> I noticed that the use of \magstep, e.g.: \magnification=\magstep1 reduces
>> considerably the text area. Why?
So you say that it would be only an eye effect? But I'm sure the text area is
really reduced, right margin is much larger...
Thanks,
Rodolfo | 2018-02-19 16:14:56 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9951283931732178, "perplexity": 13430.740884317105}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812756.57/warc/CC-MAIN-20180219151705-20180219171705-00623.warc.gz"} |
http://openstudy.com/updates/501ad933e4b02742c0b234be | ## katiebugg a1 = 5, an = 3, n = 20 one year ago one year ago
1. amistre64
baking cupcakes are you?
2. katiebugg
no
3. katiebugg
find the sum of the arithmetic sequence having the data given.
4. amistre64
oh, thats doable then, use the formula: $\frac n2(a1+an)$
5. katiebugg
thanks! | 2014-04-19 10:26:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8046977519989014, "perplexity": 9333.64667655972}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00260-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-1-foundations-for-algebra-1-7-the-distributive-property-practice-and-problem-solving-exercises-page-52/90 | ## Algebra 1: Common Core (15th Edition)
Simplify this expression: $-6(3-3x-7y)+2y-x$ Answer: $17x+44y-18$
Simplify this expression: $-6(3-3x-7y)+2y-x$ Use Distributive Property: $(-6)(3) +(-6)(-3x)+(-6)(-7y)+2y-x$ Simplify: $-18+18x+42y+2y-x$ Combine like terms: $-18+18x+42y+2y-x$ Simplify: $17x+44y-18$ Answer: $17x+44y-18$ | 2022-05-25 06:35:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9390144348144531, "perplexity": 3486.1307033419457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662580803.75/warc/CC-MAIN-20220525054507-20220525084507-00785.warc.gz"} |
https://math.stackexchange.com/questions/2405996/positive-integers-d-1-d-2-ldots-d-n-divide-1995-prove-that-for-s | # Positive integers $d_1$, $d_2$, $\ldots$, $d_n$ divide $1995$. Prove that, for some $i,j$, the numerator of $\dfrac{d_i}{d_j}$ is at least $n$.
(Probably Correct Statement) Distinct positive integers $d_1$, $d_2$, $\ldots$, $d_n$ divide $1995$. Prove that, for some indices $i,j\in\{1,2,\ldots,n\}$, the numerator of the reduced fraction $\dfrac{d_i}{d_j}$ is at least $n$.
Well, the 16 divisors of 1995 are 1 3 5 7 15 19 21 35 57 95 105 133 285 399 665 1995. So, everything starting from 19/1(or, if you want quotient to be greater than 1, from 95/3) suits. But how to make it more general?
• What did you try? – Parcly Taxel Aug 25 '17 at 17:35
• I just rolled back an edit because I saw no conceivable justification for the changes that were made. The included image is clear about what the statement of the question is. – David K Aug 25 '17 at 17:40
• Remarks: (1) The notation $d_1\mid d_2\mid \ldots \mid d_n\mid 1995$ means $d_1\mid d_2$, $d_2\mid d_3$, $\ldots$, $d_{n-1}\mid d_n$, and $d_n\mid 1995$. (2) The problem is not clear. If the $d_i$'s are allowed to be the same, then the statement is definitely false. It is perhaps the case that the $d_i$'s must be pairwise distinct (which should have been clearly stated in the problem). – Batominovski Aug 25 '17 at 17:40
• @Famke Your conclusion is true but your assumptions are much stronger than the olympiad problem allowed. You're dealing with a different question. – David K Aug 25 '17 at 17:52
• @Piquito The quotient is less than $2$, but the numerator of the quotient is either $15$ or $19$, both of which are greater than $2$. – Batominovski Aug 25 '17 at 18:55
Claim: If $p_1,p_2,\dots,p_m$ are distinct primes with $p_i\geq 2^{i}-1$, then for any set of distinct divisors $d_1<d_2<\dots<d_n$ of $N=p_1p_2\cdots p_m$, there is some fraction $\frac{d_i}{d_j}$ with reduced numerator $\geq n$.
This gives your result since $1995=3\cdot 5\cdot 7\cdot 19$.
Proof:
We will prove by induction on $m$.
If $m=1$, then all divisors are $1,p_1$. Since $p_1\geq 2$ you are done.
If true for $m$, then take $p_1,\dots,p_{m+1}$ with $p_i\geq 2^{i}-1$.
If none of $d_1,\dots,d_n$ are divisible by $p_{m+1}$, then it reduces to the case of $m$.
If all of the $d_i$ are divisible by $p_{m+1}$, then we can take the set of distinct divisors $\frac{d_1}{p_{m+1}},\dots,\frac{d_n}{p_{m+1}}$ of $p_1\cdot p_m$, and reduce to the case of $m$.
If $d_i$ is divisible by $p_{m+1}$ and $d_j$ is not, then $\frac{d_i}{d_j}$ has $p_{m+1}$ as a factor of the numerator. So that handles the cases when $n\leq p_{m+1}$. But if $n> p_{m+1}\geq 2^{m+1}-1$, then, since there are exactly $2^{m+1}$ divisors of $N$, $n=2^{m+1}$ and $d_1,\dots,d_n$ is all of the divisors. In particular, one of the $d_i=1$ and another $d_j=N$, and since $\{d_1,\dots,d_n\}\subseteq \{1,\dots,N\}$, you have $N\geq n$.
All of the $16$ divisors of $1995$ are of the form $3^{\delta_1}5^{\delta_2}7^{\delta_3}19^{\delta_4}$ where each exponent is equal to $0$ or $1$.
There are $8$ divisors of the form $19\cdot3^{\delta_1}5^{\delta_2}7^{\delta_3}$ and $8$ divisors of the form $3^{\delta_1}5^{\delta_2}7^{\delta_3}$.
►All set of divisors containing $n\ge9$ elements should contain at least one element of each of the two forms in which case we choose as numerator a factor of $19$ and as denominator a divisor coprime with it. This end the proof for $n\ge9$ because $19\gt 16$.
It remains the proof for $n=1,2,3,4,5,6,7,8$.
► For $n=8$ if all the divisors are multiples of $19$ we choose, for example the quotient $\dfrac{19\cdot3\cdot5}{19\cdot7}$ For all other set of $8$ elements containing a multiple of $19$ and a divisor coprime with $19$ we choose as numerator the multiple of $19$ and as denominator the divisor coprime to $19$
►We can now to consider just divisors of the form $3^{\delta_1}5^{\delta_2}7^{\delta_3}$.
For $n=8$ we choose $\dfrac{d_i}{d_j}=\dfrac{7\cdot3\cdot5}{5}$ (there are other examples).
For $n=7$ one of the $8$ involved divisors must lack. If it is the largest we choose $\dfrac{d_i}{d_j}=\dfrac{7\cdot5}{3}$ and if not we choose as numerator the mentioned largest (there are other examples).
For $n=6$ two of the $8$ concerned divisors should lack. If its are the two largest we choose $\dfrac{d_i}{d_j}=\dfrac{7\cdot3}{5}$ and if not we choose $\dfrac{d_i}{d_j}=\dfrac{7\cdot3\cdot5}{5}$(there are other examples).
And so on for $n=5,4,3,2$ and the trivial case $n=1$. | 2020-01-25 03:15:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9716358184814453, "perplexity": 155.4238946009586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250628549.43/warc/CC-MAIN-20200125011232-20200125040232-00020.warc.gz"} |
http://tjsullivan.org.uk/tag/owhadi/ | # Tim Sullivan
Clear Search
### Computing with dense kernel matrices at near-linear cost in MMS
The paper “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity” by Florian Schäfer, Houman Owhadi, and myself has just appeared in print in Multiscale Modeling and Simulation. This paper shows how a surprisingly simple algorithm — the zero fill-in incomplete Cholesky factorisation — with respect to a cleverly-chosen sparsity pattern allows for near-linear complexity compression, inversion, and approximate PCA of square matrices of the form
$$\Theta = \begin{bmatrix} G(x_{1}, x_{1}) & \cdots & G(x_{1}, x_{N}) \\ \vdots & \ddots & \vdots \\ G(x_{N}, x_{1}) & \cdots & G(x_{N}, x_{N}) \end{bmatrix} \in \mathbb{R}^{N \times N} ,$$
where $$\{ x_{1}, \dots, x_{N} \} \subset \mathbb{R}^{d}$$ is a data set and $$G \colon \mathbb{R}^{d} \times \mathbb{R}^{d} \to \mathbb{R}$$ is a covariance kernel function. Such matrices play a key role in, for example, Gaussian process regression and RKHS-based machine learning techniques.
F. Schäfer, T. J. Sullivan, and H. Owhadi. “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity.” Multiscale Modeling & Simulation: A SIAM Interdisciplinary Journal 19(2):688–730, 2021. doi:10.1137/19M129526X
Abstract. Dense kernel matrices $$\Theta \in \mathbb{R}^{N \times N}$$ obtained from point evaluations of a covariance function $$G$$ at locations $$\{ x_{i} \}_{1 \leq i \leq N}$$ arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions of elliptic boundary value problems and homogeneously-distributed sampling points, we show how to identify a subset $$S \subset \{ 1 , \dots , N \}^2$$, with $$\# S = O ( N \log (N) \log^{d} ( N /\varepsilon ) )$$, such that the zero fill-in incomplete Cholesky factorisation of the sparse matrix $$\Theta_{ij} 1_{( i, j ) \in S}$$ is an $$\varepsilon$$-approximation of $$\Theta$$. This factorisation can provably be obtained in complexity $$O ( N \log( N ) \log^{d}( N /\varepsilon) )$$ in space and $$O ( N \log^{2}( N ) \log^{2d}( N /\varepsilon) )$$ in time; we further present numerical evidence that $$d$$ can be taken to be the intrinsic dimension of the data set rather than that of the ambient space. The algorithm only needs to know the spatial configuration of the $$x_{i}$$ and does not require an analytic representation of $$G$$. Furthermore, this factorization straightforwardly provides an approximate sparse PCA with optimal rate of convergence in the operator norm. Hence, by using only subsampling and the incomplete Cholesky factorization, we obtain, at nearly linear complexity, the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky factorization we also obtain a solver for elliptic PDE with complexity $$O ( N \log^{d}( N /\varepsilon) )$$ in space and $$O ( N \log^{2d}( N /\varepsilon) )$$ in time.
Published on Thursday 15 April 2021 at 12:00 UTC #publication #prob-num #schaefer #owhadi
### Computing with dense kernel matrices at near-linear cost
Florian Schäfer, Houman Owhadi, and I have just uploaded a revised and improved version of our preprint “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity” to the arXiv. This paper shows how a surprisingly simple algorithm — the zero fill-in incomplete Cholesky factorisation — with respect to a cleverly-chosen sparsity pattern allows for near-linear complexity compression, inversion, and approximate PCA of square matrices of the form
$$\Theta = \begin{bmatrix} G(x_{1}, x_{1}) & \cdots & G(x_{1}, x_{N}) \\ \vdots & \ddots & \vdots \\ G(x_{N}, x_{1}) & \cdots & G(x_{N}, x_{N}) \end{bmatrix} \in \mathbb{R}^{N \times N} ,$$
where $$\{ x_{1}, \dots, x_{N} \} \subset \mathbb{R}^{d}$$ is a data set and $$G \colon \mathbb{R}^{d} \times \mathbb{R}^{d} \to \mathbb{R}$$ is a covariance kernel function. Such matrices play a key role in, for example, Gaussian process regression and RKHS-based machine learning techniques.
Abstract. Dense kernel matrices $$\Theta \in \mathbb{R}^{N \times N}$$ obtained from point evaluations of a covariance function $$G$$ at locations $$\{ x_{i} \}_{1 \leq i \leq N}$$ arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions of elliptic boundary value problems and homogeneously-distributed sampling points, we show how to identify a subset $$S \subset \{ 1 , \dots , N \}^2$$, with $$\# S = O ( N \log (N) \log^{d} ( N /\varepsilon ) )$$, such that the zero fill-in incomplete Cholesky factorisation of the sparse matrix $$\Theta_{ij} 1_{( i, j ) \in S}$$ is an $$\varepsilon$$-approximation of $$\Theta$$. This factorisation can provably be obtained in complexity $$O ( N \log( N ) \log^{d}( N /\varepsilon) )$$ in space and $$O ( N \log^{2}( N ) \log^{2d}( N /\varepsilon) )$$ in time; we further present numerical evidence that $$d$$ can be taken to be the intrinsic dimension of the data set rather than that of the ambient space. The algorithm only needs to know the spatial configuration of the $$x_{i}$$ and does not require an analytic representation of $$G$$. Furthermore, this factorization straightforwardly provides an approximate sparse PCA with optimal rate of convergence in the operator norm. Hence, by using only subsampling and the incomplete Cholesky factorization, we obtain, at nearly linear complexity, the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky factorization we also obtain a solver for elliptic PDE with complexity $$O ( N \log^{d}( N /\varepsilon) )$$ in space and $$O ( N \log^{2d}( N /\varepsilon) )$$ in time.
Published on Tuesday 26 March 2019 at 12:00 UTC #preprint #prob-num #schaefer #owhadi
### Computing with dense kernel matrices at near-linear cost
Florian Schäfer, Houman Owhadi, and I have just uploaded a preprint of our latest paper, “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity” to the arXiv. This paper builds upon the probabilistic-numerical ideas of “gamblets” (elementary gables upon the solution of a PDE) introduced by Owhadi (2016) to provide near-linear cost $$\varepsilon$$-approximate compression, inversion and principal component analysis of dense kernel matrices, the entries of which come from Green's functions of suitable differential operators.
Abstract. Dense kernel matrices $$\Theta \in \mathbb{R}^{N \times N}$$ obtained from point evaluations of a covariance function $$G$$ at locations $$\{x_{i}\}_{1 \leq i \leq N}$$ arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions elliptic boundary value problems and approximately equally spaced sampling points, we show how to identify a subset $$S \subset \{ 1,\dots, N \} \times \{ 1,\dots,N \}$$, with $$\#S = O(N \log(N)\log^{d}(N/\varepsilon))$$, such that the zero fill-in block-incomplete Cholesky decomposition of $$\Theta_{i,j} 1_{(i,j) \in S}$$ is an $$\varepsilon$$-approximation of $$\Theta$$. This block-factorisation can provably be obtained in $$O(N \log^{2}(N)(\log(1/\varepsilon)+\log^{2}(N))^{4d+1})$$ complexity in time. Numerical evidence further suggests that element-wise Cholesky decomposition with the same ordering constitutes an $$O(N \log^{2}(N) \log^{2d}(N/\varepsilon))$$ solver. The algorithm only needs to know the spatial configuration of the $$x_{i}$$ and does not require an analytic representation of $$G$$. Furthermore, an approximate PCA with optimal rate of convergence in the operator norm can be easily read off from this decomposition. Hence, by using only subsampling and the incomplete Cholesky decomposition, we obtain at nearly linear complexity the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky decomposition we also obtain a near-linear-time solver for elliptic PDEs.
Published on Tuesday 13 June 2017 at 07:00 UTC #preprint #prob-num #schaefer #owhadi
### Bayesian Brittleness in SIAM Review
The 2015 Q4 issue of SIAM Review will carry an article by Houman Owhadi, Clint Scovel, and myself on the brittle dependency of Bayesian posteriors as a function of the prior. This is an abbreviated presentation of results given in full earlier this year in Elec. J. Stat. The PDF is available for free under the terms of the Creative Commons 4.0 licence.
H. Owhadi, C. Scovel, and T. J. Sullivan. “On the brittleness of Bayesian inference.” SIAM Review 57(4):566–582, 2015. doi:10.1137/130938633
Abstract. With the advent of high-performance computing, Bayesian methods are becoming increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods can impact the making of sometimes critical decisions in increasingly complicated contexts, the sensitivity of their posterior conclusions with respect to the underlying models and prior beliefs is a pressing question to which there currently exist positive and negative answers. We report new results suggesting that, although Bayesian methods are robust when the number of possible outcomes is finite or when only a finite number of marginals of the data-generating distribution are unknown, they could be generically brittle when applied to continuous systems (and their discretizations) with finite information on the data-generating distribution. If closeness is defined in terms of the total variation (TV) metric or the matching of a finite system of generalized moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusion. The mechanism causing brittleness/robustness suggests that learning and robustness are antagonistic requirements, which raises the possibility of a missing stability condition when using Bayesian inference in a continuous world under finite information.
Published on Friday 6 November 2015 at 12:00 UTC #publication #siam-review #ouq #bayesian #owhadi #scovel
### Bayesian Brittleness in Elec. J. Stat.
The Electronic Journal of Statistics has published an article by Houman Owhadi, Clint Scovel, and myself on the brittle dependency of Bayesian posteriors as a function of the prior.
H. Owhadi, C. Scovel, and T. J. Sullivan. “Brittleness of Bayesian inference under finite information in a continuous world.” Electronic Journal of Statistics 9(1):1–79, 2015. doi:10.1214/15-EJS989
Abstract. We derive, in the classical framework of Bayesian sensitivity analysis, optimal lower and upper bounds on posterior values obtained from Bayesian models that exactly capture an arbitrarily large number of finite-dimensional marginals of the data-generating distribution and/or that are as close as desired to the data-generating distribution in the Prokhorov or total variation metrics; these bounds show that such models may still make the largest possible prediction error after conditioning on an arbitrarily large number of sample data measured at finite precision. These results are obtained through the development of a reduction calculus for optimization problems over measures on spaces of measures. We use this calculus to investigate the mechanisms that generate brittleness/robustness and, in particular, we observe that learning and robustness are antagonistic properties. It is now well understood that the numerical resolution of PDEs requires the satisfaction of specific stability conditions. Is there a missing stability condition for using Bayesian inference in a continuous world under finite information?
Published on Tuesday 3 February 2015 at 10:00 UTC #publication #elec-j-stat #ouq #bayesian #owhadi #scovel | 2021-07-31 09:47:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7609976530075073, "perplexity": 651.2126340123375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154085.58/warc/CC-MAIN-20210731074335-20210731104335-00384.warc.gz"} |
https://chemistry.stackexchange.com/questions/114414/why-are-diene-structures-in-the-c-ring-of-anthocyanins-singled-out-for-their-s | # Why are “diene” structures in the C-ring of anthocyanins singled out for their singlet-oxygen quenching ability?
Anthocyanins are a class of water-soluble vacuolar pigments. They are sensitive to pH, and can undergo a series of structural changes according to a pH-dependent equilibrium as illustrated in Figure 1. For this question, the flavylium cation (the form predominant at pH 1-4, which is also generally used to represent anthocyanins) and the quinonoidal anhydrobase (the form predominant at a neutral pH) are of particular interest.
Figure 1: The generalised structure of the anthocyanin flavylium cation, indicating the positions and numbering of the A, B and C ring of anthocyanins; the pH-dependent equilibrium between the different forms of anthocyanins.
I was initially under the impression that the antioxidant strength of the various forms of anthocyanins would mainly be dependent on the extent of conjugation (and thus electron delocalization) across the entire molecule, with more extensive delocalization stabilizing the anthocyanin radical formed upon interaction with singlet oxygen or some other reactive oxygen species. By this logic, the flavylium cation would have the highest antioxidant activity not only because it retains its hydroxyl groups for hydrogen donation to radicals, but also because there is conjugation across the entire flavonoid skeleton, even the oxonium function at position 4. So I was surprised when some papers singled out conjugated dienes (as opposed to the extent of conjugation across the entire molecule) to explain singlet-oxygen quenching ability.
In a paper$$^1$$ which argued that the quinonoidal anhydrobase is a better singlet-oxygen quencher:
In our experiments carried out at neutral pH, the anthocyanins exhibited a blue color, indicating that the quinonoidal anhydrobase likely dominated. This form is the most efficient for the quenching of nonradical singlet oxygen because its unsaturated diene conjugation (double-single-double bond arrangement) allows for the addition of singlet oxygen and the formation of endoperoxide at these electron-rich double bonds.
Figure 2: The diene structure (arrows) on the quinonoidal anhydrobase 2 is an excellent substrate for singlet oxygen.
In a follow-up paper$$^2$$ by another group arguing for the efficacy of the flavylium cation as a quencher of singlet-oxygen:
In the case of the flavylium cation form, AM1 semiempirical molecular orbital calculations indicated that 4’-OH substitution in the ring B causes resonance increment with C-ring allowing strong delocalization of the pi electrons and the positive charge producing also a 1,3-diene system in the ring. C [29] (Scheme 2). Therefore, it can be expected that increasing the number of electron-donating groups in the anthocyanin skeleton stabilizes the resonance structure of the diene system form in the flavylium cation favouring a [4+2] type interaction with the electrophilic species $$\ce{^1O2}$$.
Why these dienes? It appears that there are loads of other conjugated “dienes” in the anthocyanin molecules. Groups of double-single-double bonds on the C-ring which include the O at position 4 as one of the 4 atoms are automatically eliminated because a diene is defined as “a molecule which contains two alkene linkages”. But why aren’t the “dienes” on the A-ring of the flavylium cation and the B-ring of both the flavylium cation and the quinonoidal anhydrobase considered as substrates for singlet oxygen? With reference to the 3,5 diene of the quinonoidal base and the 1,3 diene in the resonance structure of the flavylium cation, there are 2 possible key differences I can think of:
1. It is necessary for one of the carbon atoms of a singlet-oxygen quenching diene to be connected to the electron withdrawing oxygen at position 4 (which does this through inductance, being an [![electronegative atom) to reduce the electron electron density of the conjugated pi system. (Is a system that is too “electron-rich” somehow a bad thing?) Wouldn’t this reduce the reactivity of the diene towards the electrophilic singlet oxygen, however? Is there something I am missing here?
2. The dienes that occupy these positions have unpaired electrons, thereby making them more reactive towards singlet oxygen, whereas dienes in other positions do not have unpaired electrons. I can’t give a source, but I remember hearing from somewhere that dienes can have unpaired electrons, depending on the molecular orbital diagram (I have no idea how this works though, if it is relevant an explanation would be appreciated).
In summary: is the reference to the presence/ creation of a “diene” system simply a convenient way of referring to the positions of the anthocyanin which are somehow more likely to react with singlet oxygen (with one of the two scenarios above being a possible explanation), or is this system somehow isolated from the rest of the conjugated pi-system? (The latter seems unlikely to me.) If the former is true, which explanation, if any, is correct?
References
1) Jang, Y. P., Zhou, J., Nakanishi, K., & Sparrow, J. R. (2005). Anthocyanins protect against A2E photooxidation and membrane permeabilization in retinal pigment epithelial cells. Photochemistry and photobiology, 81(3), 529–536. doi:10.1562/2004-12-14-RA-402
2) De Rosso, V. V., Moran Vieyra, F. E., Mercadante, A. Z., & Borsarelli, C. D. (2008). Singlet oxygen quenching by anthocyanin's flavylium cations. Free radical research, 42(10), 885-891. (Accessible via ResearchGate) | 2019-11-12 23:38:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7924761772155762, "perplexity": 4026.393834503386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665809.73/warc/CC-MAIN-20191112230002-20191113014002-00121.warc.gz"} |
https://math.stackexchange.com/questions/4091993/equation-of-rotated-ellipse-semi-major-axis-is-changing | # Equation of Rotated Ellipse - Semi Major Axis is Changing
I am looking at astronomical observations of gas. The gas is orbiting a black hole with circular radius, $$R$$. However, from Earth it appears that the gas is an inclined ellipse. This is because projection effects cause the circle to be rotated once about its $$y$$ axis, creating an ellipse, before being rotated by an angle $$\theta$$ in the $$z$$ axis (out of the page). Perhaps the below image will help visualise this:
Image showing a circle being first rotated about the $$y$$ axis by an undefined amount, before being rotated about the $$z$$ axis by angle $$\theta$$.
I am looking to fit an ellipse to the $$x,y$$ coordinates I have for the gas on a map of the sky. From the general equation of an inclined ellipse, we have that,
$$\frac{(x \cos(\theta) + y\sin(\theta))^2}{a^2} + \frac{(x \sin(\theta) - y\cos(\theta))^2}{b^2} = 1 ,$$
and from how I have defined the rotation by $$\theta$$, we have that $$b = R$$. I can measure the semi-minor axis, $$a$$, on my map, however the gas emission does not extend out to the semi-major axis. I would like to determine $$R$$, hence I figured that if I rearranged the above for $$b = R$$, then I should find that each $$x,y$$ value would return the same value for $$R$$. Rearranging gives the following, $$\frac{(x\sin(\theta)-y\cos(\theta))^2}{1-(\frac{x\cos(\theta)+y\sin(\theta)}{a})^2} = R^2,$$
at which point I should apologise if any errors have crept in at this point - the main point I'm trying to get across is that, to my reckoning, any $$x,y$$ on the gas ellipse will return the same value for $$R$$ (ignoring errors and physics) when plugged into the above equation. However, when I did do this, my values for $$R^2$$ decreased proportionally to $$x^2 + y^2$$. Again, let's totally ignore the physics here and focus on the mathematics. Can I please check if I have done anything wrong up to this point? Am I correct in stating that $$b = R$$ from the original circle, and that the above equation should hold for all points on the ellipse? What I also found odd, was that if I rearranged the first equation for $$y$$ by getting it in a quadratic form, and plotted $$y$$ vs $$x$$ on Desmos for different values of $$\theta$$, $$a$$ and $$R$$, I did not get an inclined ellipse. I would like to do this so I can plot different ellipses on my data and find the best fitting one. This equation for $$y$$ was pretty ghastly so I will spare you seeing it, but I assume my problem is the same as above, and I have made a mathematical error somewhere in my understanding.
Thank you!
I should add that my coordinate system is centered such that $$x_0, y_0 = 0$$
Rotation matrix $$R_\alpha = \begin{bmatrix}\cos(\alpha)&-\sin(\alpha)\\\sin(\alpha)&\cos(\alpha)\end{bmatrix}$$
Assuming the center of the ellipse is at the origin, rotation by the angle θ for $$ax^2+by^2=1$$ would be: $$a(y sinθ+x cosθ)^2+b(y cosθ-x sinθ)^2=1$$.
To derive the formula, watch the video and solve for $$y=\pm \sqrt {\frac {1-ax^2}{b}}$$ | 2022-05-17 00:25:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 34, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8629274964332581, "perplexity": 169.01127753890137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662515466.5/warc/CC-MAIN-20220516235937-20220517025937-00054.warc.gz"} |
https://research.utwente.nl/en/publications/young-inequality-for-surface-convolutions-in-mathbfr3 | Young inequality for surface convolutions in $\mathbf{R}^{3}$
Research output: Book/ReportReportProfessional
Abstract
We prove a Young inequality for convolutions defined on a Lipschitz continuous surface in $\mathbf{R}^{3}$.
Original language Undefined Enschede Numerical Analysis and Computational Mechanics (NACM) 7 0169-2690 Published - 2004
Publication series
Name Memoranda Department of Applied Mathematics, University of Twente 1735 0169-2690
Keywords
• METIS-220064
• EWI-3555
• MSC-26D10
• IR-65919
Cite this
Izsak, F., & van der Vegt, J. J. W. (2004). Young inequality for surface convolutions in $\mathbf{R}^{3}$. (Memoranda; No. 1735). Enschede: Numerical Analysis and Computational Mechanics (NACM). | 2020-03-31 03:02:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.506022036075592, "perplexity": 12478.013085530765}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370499280.44/warc/CC-MAIN-20200331003537-20200331033537-00460.warc.gz"} |
https://www.gamedev.net/forums/topic/552748-image-display-in-pygame-wont-work/ | Public Group
Image display in pygame wont work.
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""" mypygame.py
draw an image on the screen"""
import pygame
pygame.init()
def main():
screen = pygame.display.set_mode((640, 480))
pygame.display.set_caption("Display an image")
background = pygame.Surface(screen.get_size())
background = background.convert()
background.fill((96, 96, 0))
mypygame = mypygame.convert()
clock = pygame.time.Clock()
keepGoing = True
while keepGoing:
clock.tick(30)
for event in pygame.event.get():
if event.type == pygame.QUIT:
keepGoing = False
screen.blit(background, (0, 0))
screen.blit(example, (96, 96))
pygame.display.flip()
if __name__ == "__main__":
main()
When I save and run the module, all I get is a black window. What am I doing wrong? BTW I'm using pygame 1.9.1 the version that's required for 2.5.4. [Edited by - Unknownforest on November 14, 2009 9:15:11 PM]
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Original post by Unknownforestmypygame = pygame.image.load("example.png")mypygame = mypygame.convert()...screen.blit(background, (0, 0))screen.blit(example, (96, 96))...pygame.display.flip()
Here is the relevant portion of yr code where the problem is. Yr line, screen.blit(example, (96, 96)) should be screen.blit(mypygame, (96, 96)). Right?
Check if this works.
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It worked with small PNGs but when I went with a PNG that's 640x400 it didn't work.
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Quote:
Original post by Unknownforest... but when I went with a PNG that's 640x400 it didn't work.
What do you mean by this? Can you please be more specific?
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Quote:
Original post by signal_
Quote:
Original post by Unknownforest... but when I went with a PNG that's 640x400 it didn't work.
What do you mean by this? Can you please be more specific?
The image won't show up in the window.
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Okay. I just downloaded Pygame & Python. I wiped this comp one month ago so I didn't re-install it yet.
Try this:
import pygame, sys, ospygame.init()from pygame.locals import *def GetInput(): key = pygame.key.get_pressed() for event in pygame.event.get(): if event.type == pygame.QUIT or key[K_ESCAPE]: pygame.quit(); sys.exit() def main(): screen = pygame.display.set_mode((640, 480)) pygame.display.set_caption("Display an image") background = pygame.Surface(screen.get_size()) background = background.convert() background.fill((96, 96, 0)) img0 = pygame.image.load("0.png").convert() img1 = pygame.image.load("1.png").convert() clock = pygame.time.Clock() keepGoing = True imgX = 0; imgY = 0; directionX = 1; directionY = 1; while keepGoing: clock.tick(30) GetInput() screen.blit(background, (0, 0)) screen.blit(img0, (0, 0)) screen.blit(img1, (imgX, imgY)) if imgX < 0 or imgX > (screen.get_width() - img1.get_width()): directionX *= -1 if imgY < 0 or imgY > (screen.get_height() - img1.get_height()): directionY *= -1 imgX += directionX imgY += directionY pygame.display.flip()if __name__ == "__main__": main()
Some notes:
1. I used 2 images to make this program: img0 and img1. img0, for me, is 640x480 so it covers the screen. img1 is a 96x96 sprite that bounces around the screen. I added some simple movement and screen-edge collision detection.
2. Use 'source' tags when posting code; they preserve the formatting. You can just copy-n-paste mine....
3. Try this. It should work. If it doesn't something is wrong on yr end (I'm using the same Python and Pygame as you are).
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import pygame, sys, ospygame.init()from pygame.locals import *def GetInput(): key = pygame.key.get_pressed() for event in pygame.event.get(): if event.type == pygame.QUIT or key[K_ESCAPE]: pygame.quit(); sys.exit() def main(): screen = pygame.display.set_mode((640, 480)) pygame.display.set_caption("Display an image") background = pygame.Surface(screen.get_size()) background = background.convert() background.fill((96, 96, 0)) sky6 = pygame.image.load("6.png").convert() sky4 = pygame.image.load("4.png").convert() clock = pygame.time.Clock() keepGoing = True skyX = 0; skyY = 0; directionX = 1; directionY = 1; while keepGoing: clock.tick(30) GetInput() screen.blit(background, (0, 0)) screen.blit(sky6, (0, 0)) screen.blit(sky4, (skyX, skyY)) if skyX < 0 or skyX > (screen.get_width() - sky4.get_width()): directionX *= -1 if skyY < 0 or skyY > (screen.get_height() - sky4.get_height()): directionY *= -1 sky4X += directionX sky4Y += directionY pygame.display.flip()if __name__ == "__main__": main()
I use the approach of using sky6 as the background since it's 0, 0. And then I used sky4 for movement. Both are 96x96 .pngs. But all I get is.
Traceback (most recent call last): File "C:\Documents and Settings\anonymous\Desktop\thatfolder\pygame.py", line 1, in <module> import pygame, sys, os File "C:\Documents and Settings\anonymous\Desktop\thatfolder\pygame.py", line 2, in <module> pygame.init()AttributeError: 'module' object has no attribute 'init'
What wrong here? What did I do wrong in placing the pngs?
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Are you sure you have exactly:
import pygame, sys, os
pygame.init()
at the start of your pygame.py? No weird spaces or anything else?
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Quote:
Original post by mattdAre you sure you have exactly:import pygame, sys, ospygame.init()at the start of your pygame.py? No weird spaces or anything else?
Yeah, OP, make sure you preserve formatting. Why don't you try typing or copy-pasting the code into IDLE. That way you can see any error messages that pop up. Maybe something is wrong with yr setup since the code is running fine on my pc.
So try inputting the code into IDLE line-by-line and post the results. Here is my result from the code that I typed up:
Python 2.5.4 (r254:67916, Dec 23 2008, 15:10:54) [MSC v.1310 32 bit (Intel)] on win32Type "copyright", "credits" or "license()" for more information. **************************************************************** Personal firewall software may warn about the connection IDLE makes to its subprocess using this computer's internal loopback interface. This connection is not visible on any external interface and no data is sent to or received from the Internet. **************************************************************** IDLE 1.2.4 ==== No Subprocess ====>>> import pygame, sys, os>>> >>> from pygame.locals import *>>> pygame.init()(6, 0)>>> screen = pygame.display.set_mode((640, 480))>>> pygame.display.set_caption("Display an image")>>> background = pygame.Surface(screen.get_size())>>> background = background.convert()>>> background.fill((96, 96, 0))<rect(0, 0, 640, 480)>>>> img0 = pygame.image.load("0.png").convert()>>> img1 = pygame.image.load("1.png").convert()>>> imgX = 0;>>> imgY = 0;>>> screen.blit(background, (0, 0))<rect(0, 0, 640, 480)>>>> screen.blit(img0, (0, 0))<rect(0, 0, 640, 480)>>>> screen.blit(img1, (imgX, imgY))<rect(0, 0, 96, 96)>>>> >>> pygame.display.flip()>>> pygame.quit();
Does yrs look like this? Post the result if it doesn't.
This way you can see what each line of the code does. IDLE basically will interpret each line of code you give it, kinda like stepping thru a program with a debugger.
Pay special attention to the line pygame.init() when you type it in to IDLE. If you receive
..., line 2, in <module> pygame.init()AttributeError: 'module' object has no attribute 'init'
then that is troubling. It might mean that things are not setup correctly for you. Also, what operating system are you using? Is yr processor 32 or 64 bit?
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This is what I did real quick. BTW I am running off of a 32 bit OS. Windows XP. OS X should be 64 bit I think. Since whenever the hell I get this up and running I'd like this build to actually run well in modern macs if there really is going to be a crap load of issues with mac os 9.2 and GameSprockets.
>>> import pygame, sys, os>>> >>> from pygame.locals import *>>> pygame.init()(6, 0)>>> screen = pygame.display.set_mode((640, 480))>>> pygame.display.set_caption("Display an image")>>> background = pygame.Surface(screen.get_size())>>> background.fill((96, 96, 0))<rect(0, 0, 640, 480)>>>> sky6 = pygame.image.load("sky6.png").convert()Traceback (most recent call last): File "<pyshell#8>", line 1, in <module> sky6 = pygame.image.load("sky6.png").convert()error: Couldn't open sky6.png>>>
I guess I could give copy and pasting another shot. If this doesn't work I don't know what the hell I'm suppose to do about making a pygame app.
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× | 2018-06-19 05:27:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17687343060970306, "perplexity": 13900.883365942824}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267861899.65/warc/CC-MAIN-20180619041206-20180619061206-00353.warc.gz"} |
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-2-quiz-sections-2-1-2-3-page-71/8 | ## Trigonometry (11th Edition) Clone
Published by Pearson
# Chapter 2 - Quiz (Sections 2.1-2.3) - Page 71: 8
#### Answer
$\theta = 60^{\circ}$ $\theta = 120^{\circ}$
#### Work Step by Step
$sin~\theta = \frac{\sqrt{3}}{2}$ $\frac{y}{r} = \frac{\sqrt{3}}{2}$ We can let $y = \sqrt{3}$ and $r = 2$. Then: $x^2 = r^2-y^2$ $x = \pm \sqrt{r^2-y^2}$ $x = \pm \sqrt{(2)^2-(\sqrt{3})^2}$ $x = 1$ or $x = -1$ If $x = 1$ and $y = \sqrt{3}$, then $\theta$ makes an angle of $60^{\circ}$ above the positive x-axis. Then $\theta = 60^{\circ}$ If $x = -1$ and $y = \sqrt{3}$, then $\theta$ makes an angle of $60^{\circ}$ above the negative x-axis. Then $\theta = 180^{\circ}-60^{\circ} = 120^{\circ}$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2021-03-06 23:30:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8902896046638489, "perplexity": 442.0021239493094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178375529.62/warc/CC-MAIN-20210306223236-20210307013236-00618.warc.gz"} |
http://www.numdam.org/articles/10.1051/ps:2007054/ | Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 258-272.
In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow Royer’s approach (Royer, 1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box ${\left[-n,n\right]}^{d}$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincaré. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.
DOI : https://doi.org/10.1051/ps:2007054
Classification : 82B20, 60K35, 26D10
Mots clés : Ising model, unbounded spins, functional inequalities, Beckner inequalities
@article{PS_2008__12__258_0,
author = {Zitt, Pierre-Andr\'e},
title = {Functional inequalities and uniqueness of the {Gibbs} measure - from {log-Sobolev} to {Poincar\'e}},
journal = {ESAIM: Probability and Statistics},
pages = {258--272},
publisher = {EDP-Sciences},
volume = {12},
year = {2008},
doi = {10.1051/ps:2007054},
zbl = {1187.82033},
mrnumber = {2374641},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007054/}
}
TY - JOUR
AU - Zitt, Pierre-André
TI - Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré
JO - ESAIM: Probability and Statistics
PY - 2008
DA - 2008///
SP - 258
EP - 272
VL - 12
PB - EDP-Sciences
UR - http://www.numdam.org/articles/10.1051/ps:2007054/
UR - https://zbmath.org/?q=an%3A1187.82033
UR - https://www.ams.org/mathscinet-getitem?mr=2374641
UR - https://doi.org/10.1051/ps:2007054
DO - 10.1051/ps:2007054
LA - en
ID - PS_2008__12__258_0
ER -
Zitt, Pierre-André. Functional inequalities and uniqueness of the Gibbs measure - from log-Sobolev to Poincaré. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 258-272. doi : 10.1051/ps:2007054. http://www.numdam.org/articles/10.1051/ps:2007054/
[1] F. Barthe, P. Cattiaux and C. Roberto, Interpolated inequalities between exponential and gaussian, Orlicz hypercontractivity and application to isoperimetry. Revistra Mat. Iberoamericana 22 (2006) 993-1067. | MR 2320410 | Zbl 1118.26014
[2] F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line. Studia Math. 159 (2003) 481-497. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). | MR 2052235 | Zbl 1072.60008
[3] T. Bodineau and B. Helffer, Correlations, spectral gaps and log-Sobolev inequalities for unbounded spins systems, in Differential equations and mathematical physics, Birmingham, International Press (1999) 27-42. | MR 1764741
[4] T. Bodineau and F. Martinelli, Some new results on the kinetic ising model in a pure phase. J. Statist. Phys. 109 (2002) 207-235. | MR 1927919 | Zbl 1027.82028
[5] P. Cattiaux, I. Gentil and A. Guillin, Weak logarithmic Sobolev inequalities and entropic convergence. Prob. Theory Rel. Fields 139 (2007) 563-603. | MR 2322708 | Zbl 1130.26010
[6] P. Cattiaux and A. Guillin, On quadratic transportation cost inequalities. J. Math. Pures Appl. 86 (2006) 342-361. | MR 2257848 | Zbl 1118.58017
[7] R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis, Lect. Notes Math. Springer, Berlin 1745 (2000) 147-168. | MR 1796718 | Zbl 0986.60017
[8] M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, in Séminaire de Probabilités, XXXV, Lect. Notes Math. Springer, Berlin 1755 (2001) 167-194. | Numdam | MR 1837286 | Zbl 0979.60096
[9] S.L. Lu and H.-T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 (1993) 399-433. | MR 1233852 | Zbl 0779.60078
[10] L. Miclo, An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | MR 1710983 | Zbl 0942.60081
[11] G. Royer, Une initiation aux inégalités de Sobolev logarithmiques. Number 5 in Cours spécialisés. SMF (1999). | MR 1704288 | Zbl 0927.60006
[12] D.W. Stroock and B. Zegarliński, The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992) 175-193. | MR 1182416 | Zbl 0758.60070
[13] D.W. Stroock and B. Zegarliński, On the ergodic properties of Glauber dynamics. J. Stat. Phys. 81(5/6) (1995). | MR 1361304 | Zbl 1081.60562
[14] N. Yoshida, The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Annales de l'Institut H. Poincaré 37 (2001) 223-243. | Numdam | MR 1819124 | Zbl 0992.60089
[15] B. Zegarliński. The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Comm. Math. Phys. 175 (1996) 401-432. | MR 1370101 | Zbl 0844.46050
[16] P.-A. Zitt, Applications d'inégalités fonctionnelles à la mécanique statistique et au recuit simulé. PhD thesis, University of Paris X, Nanterre (2006). http://tel.archives-ouvertes.fr/tel-00114033.
Cité par Sources : | 2022-05-18 13:00:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38351961970329285, "perplexity": 3809.6521718481313}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662522270.37/warc/CC-MAIN-20220518115411-20220518145411-00172.warc.gz"} |
https://www.taylorfrancis.com/chapters/mono/10.1201/9780203643426-10/section-6-1-invariants-hamiltonian-system-2-atom-bolsinov-fomenko?context=ubx&refId=321e486b-bdec-401c-8ab0-5e7999d5255b | ## ABSTRACT
Chapter 6
Classication of Hamiltonian Flows
on Two-Dimensional Surfaces
up to Topological Conjugacy
6.1. INVARIANTS OF A HAMILTONIAN
SYSTEM ON A 2-ATOM
In this section we produce a complete set of invariants giving the classication
of Hamiltonian systems with one degree of freedom in a neighborhood of a singular
level of the Hamiltonian up to topological conjugacy. When we speak of topological
conjugacy of such systems, we mean the existence of a homeomorphism that not only
conjugates ows, but also preserves orientation. | 2022-12-06 15:05:00 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9309460520744324, "perplexity": 1220.515957844716}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711108.34/warc/CC-MAIN-20221206124909-20221206154909-00557.warc.gz"} |
https://support.bioconductor.org/p/124979/ | SVA for repeated measures design
2
0
Entering edit mode
@sarablocquiaux-21717
Last seen 19 months ago
Hi all,
I have RNA-seq data (5 subjects are measured on 4 time points) and would like to do a SVA first to be able to include potential confounders into the statistical model (Deseq2 pipeline).
I am having troubles how to define my null and full model in the SVA:
Full model ~ TIME + SUBJECT.ID
Null model ~ SUBJECT.ID OR Null model ~ 1
Should the subjects ID be treated as a factor of interest or as a confounding factor?
Thanks in advance!
Best,
Sara
SVA deseq2 repeated measures • 390 views
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1
Entering edit mode
@james-w-macdonald-5106
Last seen 7 hours ago
United States
You should probably use just an intercept for your null model. In general, if you have repeated measures (which I assume you do, given the subject ID), AND given that you have complete repeated measures (where you have measurements from each subject at each time), then the subject-specific changes are orthogonal to the measure of interest, and blocking on subject is the way to go. It also makes it easier to interpret your coefficients.
Put a different way, sva is intended to generate surrogate variables for unobserved variability. The subjects are by definition observed, so if you wanted to use the sva package to do something with them, you could consider them to be batch effects and use ComBat (note that I am not advocating this, but just noting that sva is for things you don't observe and ComBat is for things you know about.)
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0
Entering edit mode
Thanks James.
Yes, I have repeated measures. Subject.ID is not my factor of interest, but of course I want to include it in the model. For one subject, I have two missing time points though.
I do not want ComBat to correct for Subject.ID, but rather want SVAseq to find confounding factors (other than Time and Subject.ID). The design model I intend to use in deseq is: ~Subject + SV1 + ... + Time.
I will use the null model ~1 in SVAseq, as suggested.
ADD REPLY
0
Entering edit mode
Robert Castelo ★ 2.7k
@rcastelo
Last seen 4 weeks ago
Barcelona/Universitat Pompeu Fabra
Hi,
I would say the answer is to include SUBJECT.ID in the null model because, as argued by Jeff Leek, author of SVA, in this thread about a similar design case, SUBJECT.ID will be used in the ultimate linear model you intend to fit to test for the effect of your variable of interest.
cheers,
robert.
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0
Entering edit mode
I agree, that was what I was thinking at first.
But subject.ID is not just a covariate, it is a random factor. So it is still not clear to me whether to include it in the null model or not. Not including it in the null model, makes it kind of a variable of interest itself.
ADD REPLY
1
Entering edit mode
If SUBJECT.ID is a random factor, then you should not put it into the design matrix and use duplicateCorrelation() and the arguments correlation and block in the call to lmFit(); see section on Multi-level experiments from the limma User's Guide. If you don't need surrogate variables, then you can just follow that documentation.
The complication comes when you want to combine it with surrogate variables estimated with SVA. You can try to have a full model with TIME only and the null with the intercept. Then, estimate surrogate variables, paste them into the design matrix and proceed with the duplicateCorrelation() blocking on SUBJECT.ID. However, it may happen that SVA has already estimated part of the SUBJECT.ID variablity and this may lead to problems with duplicateCorrelation(); see this thread about that possibility. So, I'd suggest to include SUBJECT.ID in the full and null models that you give to SVA (next to TIME), just to ensure that the SUBJECT.ID variability is not picked up by SVA. Then, place TIME and the surrogate variables in a new design matrix, i.e., without SUBJECT.ID, and proceed with duplicateCorrelation() blocking on SUBJECT.ID.
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Powered by the version 2.3.6 | 2021-08-05 05:41:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40969347953796387, "perplexity": 1837.300900047365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155322.12/warc/CC-MAIN-20210805032134-20210805062134-00269.warc.gz"} |
https://www.r-bloggers.com/introducing-imagemetrics/ | # Introducing imagemetrics
September 25, 2013
By
(This article was first published on Just Another R Blog, and kindly contributed to R-bloggers)
References
In my recent projects, I had the opportunity to work with the professor Raphaël Proulx who introduced me to several metrics commonly used in landscape ecology for quantifying image texture. In order to make my life easier, I decided to implement them as a R package. At this time, the package is still under development and so better documentation/debugging/testing are still needed. Nevertheless, here is a short tutorial on how to use the package.
Step 1: Install the package
The package is currently hosted on Bitbucket and can be installed using devtools.
library(devtools)install_bitbucket("imagemetrics", "persican")library(imagemetrics)
Step 2: Open a raster image
Indeed, the first thing to do is to open the image on which you want to calculate the metrics.
## Open the R logo and average on calculate the average on R,G,B channelsr = brick(system.file("external/rlogo.grd", package = "raster"))r = mean(r)## Plot the raster (optional)plot(r, useRaster = FALSE, col = gray((0:100)/100))
Step 3: Extract pixels from the image to calculate occurrence probabilities
It is worth mentioning that the metrics are calculated on a probability matrix that represents the chances of getting a specific pair of pixel values. For each pixel in the image, we have to choose a neighbor that is located either to the right, bottom right or bottom of a reference pixel (see the following image).
Before calculating such probabilities, we have to extract the values of both reference and neighbor pixels. To do so, the user can use getImagePixels(r, side). The function takes as parameters a raster image (r) and side, a numeric value specifying the neighbor to use. side = 1 for the right pixel, side = 2 for the lower right pixel, side = 3 the bottom pixel.
## Get bottom pixel (side = 3)v = getImagePixels(r, side = 3)str(v)
## List of 2## $reference_vector: num [1:7676] 255 255 255 255 255 255 255 255 255 255 ...##$ neighbour_vector: num [1:7676] 255 255 255 255 255 255 255 255 255 255 ...
Step 4: Calculate occurrence probabilities
To calculate the probabilities, simply use calculateHisto(reference_vector, neighbour_vector, nbins, where reference_vector and neighbour_vector are vectors returned by getImagePixels and nbins is a numerical value indicating the bin size used to compute histograms.
From Mellin et al. 2012:
“For $$M$$ classes of values, the number of possible configurations in a $$k-pixel$$ neighborhood is $$M^k$$. To ensure that each possible configuration has a reasonable probability of occurring in an image, it is generally recommended that the ratio of the total number of pixels in the image to $$M^k$$ be greater than 100.”
suggestMaximumBins® can be used to determine the number of maximum bins to use.
suggestMaximumBins(r)
## [1] 8
## Calculate probabilityr.prob = calculateHisto(reference_vector = v$reference_vector, neighbour_vector = v$neighbour_vector, nbins = 6)
Step 5: Compute the metrics
Metrics are thereafter calculated using the object returned by calculateHisto.
contagion(r.prob)
## [1] 0.3469
contrast(r.prob)
## [1] 0.02727
evenness(r.prob)
## [1] 0.8393
homogeneity(r.prob)
## [1] 0.848
jointEntropy(r.prob)
## [1] 2.341
marginalEntropy(r.prob)
## [1] 1.504
maximumMutualInformation(r.prob)
## [1] 0.3724
meanInformationGain(r.prob)
## [1] 0.4669
## References
Mellin, C., Parrott, L., Andréfouët, S., Bradshaw, C.J.A. a, MacNeil, M.A. & Caley, M.J. (2012) Multi-scale marine biodiversity patterns inferred efficiently from habitat image processing. Ecological Applications, 22, 792803.
Proulx, R. & Parrott, L. (2008) Measures of structural complexity in digital images for monitoring the ecological signature of an old-growth forest ecosystem. Ecological Indicators, 8, 270284.
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https://math.stackexchange.com/questions/1922761/order-dense-set-definition/3308478 | # Order-Dense Set Definition
The usual definitions of order-dense set I've seen are:
(1) Let $\precsim$ be an order defined on set $X$. $(X,\precsim)$ is order-dense if $\forall x,y \in X: x \precsim y, \exists z \in X: x \precsim z \precsim y$ (wikipedia, math.se).
(2) Let $\precsim$ be an order defined on set $X$. $(X,\precsim)$ is order-dense if $\forall x,y \in X: x \prec y, \exists z \in X: x \precsim z \prec y$ (some lecture notes).
(3) Let $\precsim$ be an order defined on set $X$. $(X,\precsim)$ is order-dense if $\forall x,y \in X: x \prec y, \exists z \in X: x \prec z \prec y$ (proofwiki, math.se).
where $\precsim$ is a weak partial or total order and $\prec$ denotes the strict version of it.
Although similar, these are quite different. In sum, some definitions require some or all relations between the elements to be strict ($\prec$ instead of $\precsim$) and others even specify that $z \in X \backslash \{x,y\}$.
Is there a consensual definition of an order-dense set?
• Can $z \in X$ or has to be distinct from $x,y$, i.e. $z \in X \backslash \{x,y\}$?
• Can $x\precsim z \precsim y$ or has to be $x\prec z \prec y$?
(This is not trivial, as (1) holds trivially for any partially-ordered set; (2) and (3) prevent $X$ to be a singleton and, requiring $z \in X \backslash \{x,y\}$, would prevent $(\mathbb{N},\leq)$ to be order-dense in itself)
• The wikipedia uses (3) but not (1) Jul 30, 2019 at 14:55
Actually you should differentiate between a dense order relation and a dense subset of an ordered set.
The order relation is dense when (3) holds.
A $$Y$$ is dense in an ordered set $$(X,≾)$$ iff a modified condition (1) holds: $$$$∀x,y∈X:x≺y,∃z∈Y:x≾z≾y$$$$ This reads: There are two sets: One set $$X$$ which provides order relation and works as the universe, here. And another set $$Y$$ which has a nonempty intersection with every closed order interval. This is more or less a relational description of the same idea which is behind the notion of a dense set in topology: Every element of $$X$$ has sufficiently many elements of $$Y$$ nearby. Or in other words $$Y$$ may not be the full set $$X$$, but $$Y$$ is a sufficiently close approximation of $$X$$ in some setting. And yes, $$X$$ should be dense in $$X$$ in this sense.
• A slight modification for partial order $\precsim$: $\forall x,y\in X: x\prec y, \exists z \in Y: x\precsim z \precsim y$. $x$, $y$ should be distinct elements in $X$ to avoid the case $x=y\in X/Y$. When $x=y\in X/Y$, $x\succsim y$ is true (by reflexivity) but $\nexists z\in Y$, s.t. $x\precsim z \precsim y$ (by transitivity and anti-symmetry). Feb 18, 2021 at 4:26
• @HanWang you are right. I fixed the answer. Feb 18, 2021 at 14:09
In both (1) and (2) taking $x=z$ will satisfy the conclusion, making every order a dense order. Unless we require that $z\notin\{x,y\}$, or a condition which implies that like $x\prec z\prec y$, there is no way to guarantee that some orders will not be dense.
While it is somewhat unusual for a singleton to be a "dense set", the definition does make sense if you only require that when $x\prec y$, then there is some $z$ such that $x\prec z\prec y$. Therefore often you should also require that for every $x$ there is some $y$ such that $x\prec y$ or $y\prec x$. Otherwise taking a discrete ordering will be dense vacuously.
• If I interpret you correctly, then the definition should include either $z \notin \{x,y\}$ or $x \prec z \prec y$? But which (given that the second condition is much stronger than the first if we allow for $x \precsim z \precsim y$)? Sep 11, 2016 at 16:19
• Allowing weak equality and requiring that $z\notin\{x,y\}$ is the same as requiring $x\prec z\prec y$. Sep 11, 2016 at 16:27
• I was thinking that we could have $\forall x,y \in X$, $x \precsim y$ and $y \precsim x$ and allowing for $z \in X\backslash \{x,y\}$ satisfying the first and not the second, so that the two conditions would not, in general, be the same. Am I thinking wrong? Edit: I am. I'm thinking about preorders and not orders. I apologize. Sep 11, 2016 at 16:29 | 2022-05-23 04:31:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9490664005279541, "perplexity": 161.74764810623256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662555558.23/warc/CC-MAIN-20220523041156-20220523071156-00247.warc.gz"} |
https://amirasghari.com/tag/algebra/ | # Tag: Algebra
• ## Structure, Structure, Structure!
Look at this equality: ( (a + b) + c = a + (b + c) ) , or this one: ( a . (b + c) = a.b + a.c ) . They are true structurally. In principle, You can just replace one side of the equality with the other side without any extra […] | 2022-08-09 11:47:04 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9346936345100403, "perplexity": 692.7694431480405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00664.warc.gz"} |
https://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.b200002 | - Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
Convergence properties for the partial sums of widely orthant dependent random variables under some integrable assumptions and their applications Bull. Korean Math. Soc. 2020 Vol. 57, No. 6, 1451-1473 https://doi.org/10.4134/BKMS.b200002Published online September 11, 2020Printed November 30, 2020 Yongping He, Xuejun Wang, Chi Yao Anhui University; Anhui University; Anhui University Abstract : Widely orthant dependence (WOD, in short) is a special dependence structure. In this paper, by using the probability inequalities and moment inequalities for WOD random variables, we study the $L_p$ convergence and complete convergence for the partial sums respectively under the conditions of RCI$(\alpha)$, SRCI$(\alpha)$ and $R$-$h$-integrability. We also give an application to nonparametric regression models based on WOD errors by using the $L_p$ convergence that we obtained. Finally we carry out some simulations to verify the validity of our theoretical results. Keywords : Widely orthant dependent random variables, $L_p$ convergence, complete convergence, residual Ces\{a}ro alpha-integrability, strongly residual Ces\{a}ro alpha integrability, $R$-$h$-integrability MSC numbers : 60F05, 60F15, 60F25, 62G05 Supported by : Supported by the National Natural Science Foundation of China (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1808085QA03, 1908085QA01, 1908085QA07), the Provincial Natural Science Research Project of Anhui Colleges (KJ2019A0003) and the Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (2017H123) Downloads: Full-text PDF Full-text HTML | 2021-03-07 08:31:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22057735919952393, "perplexity": 3776.6315029525754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376206.84/warc/CC-MAIN-20210307074942-20210307104942-00272.warc.gz"} |
https://math.stackexchange.com/questions/4006463/how-to-compute-trigonometric-functions-based-on-inverse-square-root/4006759 | # How to compute Trigonometric functions based on Inverse Square Root
I am writing a rational number library for performing fast math on integer-only microcontrollers. So far, I have all the basic operations, as well as a first-step estimation function for square root. It takes a positive integer input and returns a rational number approximating the square root (accurate to roughly 4 bits of precision; to be refined further by Newton-Raphson or similar method). Obviously, inverse (reciprocal) square root comes for free since the answer is a fraction that can be trivially inverted.
I read somewhere that other functions (e.g. trigonometric functions) can be calculated in terms of inverse square root (can't find the link now), but I am having a hard time finding any algorithms since search results are almost entirely about inverse trigonometric functions (mostly high school help stuff).
Does anyone have a handy resource where I can find efficient methods for calculating sin, cos, tan, atan, etc. in terms of square root or inverse square root? Is it possible or am I imagining things?
I know this is not a "code" StackExchange but I believe this is fundamentally a math question. If you're interested in my code, it's here. The sqrt_est() function is somewhat remarkable in that it is very short, uses only shifts and adds, no looping or iteration, or even multiplies or divides! (core implementation is on lines 39-45.)
• I have used (and advocated for) inverse square roots to eliminate trig functions when you already have another related trig function. But not to compute trig functions of angles from scratch. Is that what you remember? Jan 31, 2021 at 3:56
• @DanPiponi perhaps... I could probably implement at least one reasonably fast trig function. Any pointers to examples of what you're describing would be appreciated! Jan 31, 2021 at 4:57
• To satisfy my curiosity, I am going to reveal my ignorance. Consider as a starting point using Taylor Series, where you first convert $x$ to the interval $(-\pi,\pi].$ Are you saying that the corresponding computational performance here can be improved? Jan 31, 2021 at 5:52
I personally am not aware of any method utilizing the inverse square root for evaluating trig functions and without a reference, it's really hard to understand what you're referring to. With such implementations, the general idea is to go with ready-made, optimized and tested implementations, but that probably won't answer your question.
Taylor Series is certainly a good starting point and although it technically has a time complexity of $$O(n)$$ it converges really fast (you can achieve really good precision by calculating a few terms) so depending on the scope of your library it may actually be ok. Chebyshev approximation is usually prefered since it converges faster and I haven't seen many real-world applications favouring Taylor expansions over it.
Another "dirtier" option is to simply use a look-up table and either do closest-match or interpolate, losing some precision along the way. You don't need to create a particular big look-up table either, just create one for many steps along the interval $$[0, x]$$ and then partition $$[0, \pi]$$ with a step of x. Then you can calculate $$\sin(y)$$ and $$\cos(y)$$ by writing them as:
$$\sin(y_1 + \delta_1) = \sin(y_1)\cos(\delta_1) + \cos(y_1)\sin(\delta_1)$$
$$\cos(y_2 + \delta_2) = \cos(y_2)\cos(\delta_2) + \sin(y_2)\sin(\delta_2)$$
where $$\delta_1,\ \delta_2 < x$$.
For $$\arctan$$ the basic idea is quite similar. You again usually end up partitioning an interval $$[0, N]$$ (for $$x>N$$ you assume the answer to simply be $$\frac{\pi}{2}$$ or use some rough and cheap approximation method) and approximate your function in each part using something like a Chebyshev series. You again use trig identities to try to bring your argument down to $$[0, N]$$ and evaluate according to your polynomial in that range. The specifics of this can be quite tricky without an FPU so I don't know if you would want to bother with that.
An alternative solution, is to try to solve the integrals in each interval instead of evaluating the series. For example, we know that $$\arctan(x) = \int_0^x\frac{1}{1+x^2}$$. You can again create a LUT and then either use a numerical integration method (like Newton-Cotes) which would require several evaluations of $$\frac{1}{1+x^2}$$ depending on the target precision, which I'm not sure you can afford. You could also utilize the much simpler to calculate Simpson's Rule. I haven't seen those methods used though, so it's probably not worth it in terms of efficiency, ease of implementation and precision. Just throwing this in in case it might be worth considering.
Another cheap way is using the CORDIC algorithm which intuitively works by doing successive rotations around the imaginary unit circle. You can find a surprisingly in-depth explanation in Matlab's documentation
Of course this is all math. If you asked in StackOverflow they would let you know about a myriad other things that factor in your implementation like cache misses, utilizing SIMD intrinsic and other stuff this answer didn't touch on ;)
• Thanks for the variety of information. Yes, at this point I am more interested in the mathematical options than the specifics of machine implementation. If you look at my code for sqrt_est() you can see it’s already remarkably optimal, which is why I’d like to leverage it as much as possible in computing other functions. What I am really looking for is what @Dan Piponi mentioned in his first comment on my original question. Jan 31, 2021 at 15:16
Square roots are key to all sorts of computations because they let us compute arbitrary powers. If you want to compute $$2^{1/7}$$, for example, first write $$\frac17$$ in binary as $$0.001001001001001\dots$$, which tells us that $$2^{1/7} = 2^{1/8} \cdot 2^{1/64} \cdot 2^{1/512} \cdot 2^{1/4096} \cdot 2^{1/32768} \cdot \dots$$ All of these are repeated square roots: $$2^{1/2} = \sqrt 2$$, $$2^{1/4} = \sqrt{2^{1/2}}$$, $$2^{1/8} = \sqrt{2^{1/4}}$$, and so on. So we can take as many square roots as we need to, and multiply them together.
Square roots are key to trig functions because those are just powers. If you want to compute $$\cos \frac\pi7$$, for example, then it's enough to compute $$(-1)^{1/7} = \cos \frac\pi7 + i \sin \frac\pi7$$. This is done in the same way as above: $$(-1)^{\frac17} = (-1)^{1/8} \cdot (-1)^{1/64} \cdot (-1)^{1/512} \cdot (-1)^{1/4096} \cdot (-1)^{1/32768} \cdot \dots$$ Computing iterated square roots $$(-1)^{1/2} = \sqrt{-1}$$, $$(-1)^{1/4} = \sqrt{(-1)^{1/2}}$$, and so on is a bit trickier, but comes down to real square roots in the end.
But if the $$(-1)^x$$ argument seems a little bit disingenuous to you, then we can rephrase this in terms of trig identities. Start with a known $$(\cos x, \sin x)$$ pair: it's convenient to start with $$(\cos \frac\pi2, \sin \frac\pi2) = (0,1)$$, but we can start anywhere if we want to. From $$(\cos x, \sin x)$$ we can compute $$(\cos \frac x2, \sin \frac x2)$$ by the pair of identities $$\cos \frac x2 = \pm \sqrt{\frac{1 + \cos x}{2}}, \qquad \sin \frac x2 = \pm \sqrt{\frac{1 - \cos x}{2}}.$$ The angles are determined by the quadrant, which in this case is always going to be positive for both $$\cos$$ and $$\sin$$. This lets us compute $$(\cos \frac{x}{2^n}, \sin \frac{x}{2^n})$$ for many $$n \ge 0$$, which we can actually save in a table.
Then, if you want to compute $$\cos \theta$$ for an arbitrary angle $$\theta \in [0,2x]$$, where $$x$$ is our reference angle, first write $$\frac{\theta}{x}$$ in binary. This is telling us $$\theta$$ as a sum of some subset of $$\{x, \frac x2, \frac x4, \frac x8, \frac x{16}, \frac x{32}, \dots\}$$, all of whose sines and cosines we know. Angle addition formulas tell us $$\cos \theta$$ and $$\sin \theta$$.
This is very similar in flavor to CORDIC algorithm also mentioned in the other answer, but that one is more efficient (and also more complicated). | 2022-09-28 09:09:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 45, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8274616599082947, "perplexity": 230.71673690016473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335190.45/warc/CC-MAIN-20220928082743-20220928112743-00168.warc.gz"} |
http://techtagg.com/standard-error/standard-error-binary-data.html | Home > Standard Error > Standard Error Binary Data
# Standard Error Binary Data
## Contents
The asymptotic consistency of x_o/n when estimating a small p is only attained after a very large number of trials. Got a question you need answered quickly? Who can advice on this scheme compared with his own knowledge and eventually some references? Is this safe to display MySQL query error in webpage if something went wrong?
Moreover, to analyze my data, I used logistic regression indeed, while means comparisons were made by contrast analysis. I recommend it to anyone seriously interested in this rather tricky problem. Feb 11, 2013 Shashi Ajit Chiplonkar · Jehangir Hospital For binomial distribution, SD = square root of (npq), where n= sample size, p= probability of success, and q=1-p. Share Facebook Twitter LinkedIn Google+ 1 / 0 Popular Answers Todd Mackenzie · Dartmouth College If one is estimating a proportion, x/n, e.g., the number of "successes", x, in a number
## Standard Error Binary Data
This very straightforward, and apparently sound answer, can collapse when computing intervals using standard deviations (see example by R. They do not have exactly 95% coverage for all sample sizes and all observed frequencies. So, standard error for $\hat p$ (a sample statistic) is $\sqrt{pq/n}$ share|improve this answer edited Jun 29 at 2:45 Silverfish 10k104086 answered Jun 28 at 20:21 Tarashankar 1 You Feb 8, 2013 Giovanni Bubici · Italian National Research Council Ops, you're right, thanks!
The standard error of $\overline{X}$is the square root of the variance: $\sqrt{\frac{ k pq }{n}}$. Since there are $n$ tosses or Bernoulli trials in the experiment, $V(Y) = \sum V(X_i) = npq$. Quote Postby Xanthir » Wed Sep 24, 2008 1:43 pm UTC I'll rephrase the relevant parts of my post in a simpler way. ^_^All right, so you did N trials and T Test Binary Is the empty set homeomorphic to itself?
This really just gives you a way to calculate the chance that r is in any particular interval. Correlation Binary Circle Font Awesome Icons Meaning of Guns and ghee Force Microsoft Word to NEVER auto-capitalize the name of my company Do COB LEDs usually need electrically insulating from the heatsink? They are not what you think they are! What is the standard deviation of a proportion?
## Standard Deviation Binary
Standard Error Bionomial distribution.xls Feb 12, 2013 Jochen Wilhelm · Justus-Liebig-Universität Gießen I think there is some confusion what the SE refers to. i wasn't able to follow all discussions in the thread, but i think your interest is not the sum of the successes but the mean or average success (which is sum Standard Error Binary Data this is a bit special design. Standard Deviation Binary Distribution Thank you. –user3596790 Jul 26 at 9:25 @user3596790 Happy to help!
Rumsey Because the binomial distribution is so commonly used, statisticians went ahead and did all the grunt work to figure out nice, easy formulas for finding its mean, variance, and standard http://techtagg.com/standard-error/explain-the-difference-between-standard-deviation-and-standard-error-of-measurement.html The standard deviation of a single fair coin is .5 exactly.However, I gave you the stdev of a bernoulli variable, while it seems you're wanting the stdev of a binomial experiment Conroy). What are the holes on the sides of a computer case frame for? Confidence Interval Binary
Linked 0 Standard Deviation vs. Is there a good way to get from Levoča to Lviv? Feb 14, 2013 Ivan Faiella · Banca d'Italia If you're presenting averages of different replicates per each year, the standard deviation of those averages (among different replicates) is an estimate of Therefore, the total number of successes you can expect -- that is, the mean of X -- is The formula for variance has somewhat of an intuitive meaning as well.
Why write an entire bash script in functions? Standard Error Binomial Distribution This results in different standard error formulas. share|improve this answer answered Nov 17 '15 at 13:48 Stan 211 add a comment| up vote 0 down vote We can look at this in the following way: Suppose we are
## We can now easily plug in the number of trials and the probability of success to come up with our answers: Figure 2.
Is it possible to check for existence of member templates just by an identifier? Duh. Second question is not clear. Standard Deviation Of Yes No Data The SE always refers to an estimate.
Giovanni Bubici Italian National Research Council Can standard deviation and standard error be calculated for a binary variable? This means that X follows a Bernoulli(p) distribution. The variance as the average squared deviations is then (kq²+(n-k)p²)/n. The former is an intrinsic property of the distribution; the latter is a measure of the quality of your estimate of a property (the mean) of the distribution.
Then, you can model the trend on the proportion of successes and give intervals for the whole trend. Also, since the graphs will report several kinds of data I would prefer to use the same precision/variability parameter for all the graphs (=kind of variables), and by the way I | 2018-01-20 06:56:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7031528353691101, "perplexity": 1222.376176320004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889473.61/warc/CC-MAIN-20180120063253-20180120083253-00521.warc.gz"} |
https://jp.maplesoft.com/support/help/Maple/view.aspx?path=StudyGuides%2FMultivariateCalculus%2FChapter4%2FExamples%2FSection4-8%2FExample4-8-11 | Example 4-8-11 - Maple Help
Chapter 4: Partial Differentiation
Section 4.8: Unconstrained Optimization
Example 4.8.11
Obtain formulas for $a$ and $b$ so that the line minimizes , the sum of squares of the deviations from the points $\left({x}_{k},{y}_{k}\right),k=1,\dots ,n$, to the line. | 2022-12-09 23:38:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 41, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9895434975624084, "perplexity": 1918.244547533419}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711552.8/warc/CC-MAIN-20221209213503-20221210003503-00381.warc.gz"} |
https://zbmath.org/?q=an:1030.60032 | ## The first exit time of a Brownian motion from an unbounded convex domain.(English)Zbl 1030.60032
Let $$\tau_D$$ be the first exit time of a $$(d+1)$$-dimensional Brownian motion from an unbounded open domain $$D= \{(x,y)\in \mathbb{R}^{d+1}\mid y> f(x),\;x\in \mathbb{R}^d\}$$ starting at $$(x_0,f(x_0)+ 1)\in \mathbb{R}^{d+1}$$ for some $$x_0\in \mathbb{R}^d$$, where the function $$f(.)$$ on $$\mathbb{R}^d$$ is convex and $$f(x)\to\infty$$ as the Euclidean norm $$|x|\to\infty$$. The author determines a general estimation for the asymptotics of $$\log P(\tau_D> t)$$ by using Gaussian techniques.
### MSC:
60G40 Stopping times; optimal stopping problems; gambling theory 60J65 Brownian motion
### Keywords:
Brownian motion; Gaussian techniques
Full Text:
### References:
[1] ANDERSON, J. M. and PITT, L. D. (1997). Large time asy mptotics for Brownian hitting densities of transient concave curves. J. Theoret. Probab. 10 921-934. · Zbl 0890.60078 [2] BAÑUELOS, R. and SMITS, R. (1997). Brownian motion in cones. Probab. Theory Related Fields 108 299-319. · Zbl 0884.60037 [3] BAÑUELOS, R., DEBLASSIE, R. and SMITS, R. (2001). The first exit time of Brownian motion from interior of a parabola. Ann. Probab. 29 882-901. · Zbl 1013.60060 [4] BASS, R. F. and CRANSTON, M. (1983). Brownian motion with lower class moving boundaries which grow faster than t1/2. Ann. Probab. 11 34-39. · Zbl 0503.60080 [5] BERTHET, P. and SHI, Z. (2000). Small ball estimates for Brownian motion under a weighted supnorm. Studia Sci. Math. Hungar. 36 275-289. · Zbl 0973.60083 [6] BURKHOLDER, D. L. (1977). Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. in Math. 26 182-205. · Zbl 0372.60112 [7] CHEN, X. and LI, W. V. (2003). Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion. Ann. Probab. 31 1052-1077. · Zbl 1030.60026 [8] CHEN, X., KUELBS, J. and LI, W. V. (2000). A functional LIL for sy mmetric stable processes. Ann. Probab. 28 258-276. · Zbl 1044.60026 [9] CIESIELSKI, Z. and TAy LOR, S. J. (1962). First passage time and sojourn density for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 434-450. JSTOR: · Zbl 0121.13003 [10] DEBLASSIE, R. D. (1987). Exit times from cones in Rn of Brownian motion. Probab. Theory Related Fields 74 1-29. · Zbl 0586.60077 [11] DEBLASSIE, R. D. (1988). Remark on Exit times from cones in Rn of Brownian motion. Probab. Theory Related Fields 79 95-97. · Zbl 04506403 [12] DUDLEY, R. and STROOCK, D. (1987). Slepian’s inequality and commuting semigroups. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247 574-578. Springer, Berlin. · Zbl 0655.47035 [13] DURBIN, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary. Appl. Probab. 22 99-122. JSTOR: · Zbl 0576.60029 [14] DURBIN, J. (1992). The first-passage density of the Brownian motion process to a curved boundary, with an appendix by D. Williams. J. Appl. Probab. 29 291-304. JSTOR: · Zbl 0806.60063 [15] FLEMING, W. H. and SONER, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York. · Zbl 0773.60070 [16] GREENWOOD, P. E. and NOVIKOV, A. A. (1986). One-sided boundary crossing for processes with independent increments. Theory Probab. Appl. 31 221-232. · Zbl 0658.60103 [17] KESTEN, H. (1992). An absorption problem for several Brownian motions. In Sem. Stochastic Process. 59-72. Birkhäuser, Boston. · Zbl 0765.60081 [18] KUELBS, J. and LI, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133-157. · Zbl 0799.46053 [19] LAI, T. L. (1977). First exit times from moving boundaries for sums of independent random variables. Ann. Probab. 5 210-221. · Zbl 0379.60026 [20] LAI, T. L. and WIJSMAN, R. A. (1979). First exit time of a random walk from the bounds f (n)cg(n), with applications. Ann. Probab. 7 672-692. · Zbl 0413.60043 [21] LEDOUX, M. and TALAGRAND, M. (1991). Probability on Banach Spaces. Springer, Berlin. · Zbl 0748.60004 [22] LERCHE, H. R. (1986). Boundary Crossing of Brownian Motion: Its Relation to the Law of the Iterated Logarithm and to Sequential Analy sis. Lecture Notes in Statistics 40. Springer, New York. · Zbl 0604.62075 [23] LI, W. V. (1999). Small deviations for Gaussian Markov processes under the sup-norm. J. Theoret. Probab. 12 971-984. · Zbl 0961.60051 [24] LI, W. V. (2001). Small ball probabilities for Gaussian Markov processes under the Lp-norm. Stochastic Process. Appl. 92 87-102. · Zbl 1047.60031 [25] LI, W. V. and LINDE, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556-1578. · Zbl 0983.60026 [26] LI, W. V. and SHAO, Q. M. (2001). Gaussian processes: Inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods. Handbook of Statistics (C. R. Rao and D. Shanbhag, eds.) 19 533-598. North-Holland, Amsterdam. · Zbl 0987.60053 [27] LI, W. V. and SHAO, Q. M. (2002a). Lower tail probabilities for Gaussian processes. Ann. Probab. · Zbl 1052.60028 [28] LI, W. V. and SHAO, Q. M. (2002b). A normal comparison inequality and its applications. Probab. Theory Related Fields 122 494-508. · Zbl 1004.60031 [29] LIFSHITS, M. A. (1995). Gaussian Random Functions. Kluwer, Boston. · Zbl 0832.60002 [30] LIPTSER, R. S. and SHIRy AEV, A. N. (1977). Statistics of Random Processes I, General Theory. Springer, New York. · Zbl 0364.60004 [31] MOGULSKII, A. A. (1974). Small deviations in a space of trajectories. Theory Probab. Appl. 19 726-736. · Zbl 0326.60061 [32] NOVIKOV, A. A. (1979). On estimates and asy mptotic behavior of non-exit probabilities of a Wiener process to a moving boundary. Mat. Sb. 38 495-505. · Zbl 0462.60079 [33] NOVIKOV, A. A. (1981). Small deviations of Gaussian processes. Math. Notes 29 150-155. · Zbl 0479.60047 [34] NOVIKOV, A., FRISHLING, V. and KORDZAKHIA, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Probab. 36 1019-1030. · Zbl 0978.60092 [35] PORTNOY, S. (1978). Probability bounds for first exits through moving boundaries. Ann. Probab. 6 106-117. · Zbl 0379.60066 [36] ROCKAFELLAR, R. T. (1970). Convex Analy sis. Princeton Univ. Press. [37] SHI, Z. (1996). Small ball probabilities for a Wiener process under weighted sup-norms, with an application to the supremum of Bessel local times. J. Theoret. Probab. 9 915-929. · Zbl 0959.60080 [38] SLEPIAN, D. (1962). The one sided barrier problem for Gaussian noise. Bell Sy stem Tech. J. 41 463-501. [39] SPITZER, F. (1958). Some theorems concerning two-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 197. JSTOR: · Zbl 0089.13601 [40] UCHIy AMA, K. (1980). Brownian first exit from and sojourn over one-sided moving boundary and application. Z. Wahrsch. Verw. Gebiete 54 75-116. · Zbl 0431.60080 [41] NEWARK, DELAWARE 19716 E-MAIL: wli@math.udel.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2022-10-02 23:39:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.754400372505188, "perplexity": 3291.2922707461935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337360.41/warc/CC-MAIN-20221002212623-20221003002623-00798.warc.gz"} |
http://cpr-nuclex.blogspot.com/2013/05/13051523-gluex-collaboration-et-al.html | ## An initial study of mesons and baryons containing strange quarks with GlueX [PDF]
The GlueX Collaboration, A. AlekSejevs, S. Barkanova, M. Dugger, B. Ritchie, I. Senderovich, E. Anassontzis, P. Ioannou, C. Kourkoumeli, G. Voulgaris, N. Jarvis, W. Levine, P. Mattione, W. McGinley, C. A. Meyer, R. Schumacher, M. Staib, P. Collins, F. Klein, D. Sober, D. Doughty, A. Barnes, R. Jones, J. McIntyre, F. Mokaya, B. Pratt, W. Boeglin, L. Guo, P. Khetarpal, E. Pooser, J. Reinhold, H. Al Ghoul, S. Capstick, V. Crede, P. Eugenio, A. Ostrovidov, N. Sparks, A. Tsaris, D. Ireland, K. Livingston, D. Bennett, J. Bennett, J. Frye, M. Lara, J. Leckey, R. Mitchell, K. Moriya, M. R. Shepherd, A. Szczepaniak, R. Miskimen, A. Mushkarenkov, B. Guegan, J. Hardin, J. Stevens, M. Williams, A. Ponosov, S. Somov, C. Salgado, P. Ambrozewicz, A. Gasparian, R. Pedroni, T. Black, L. Gan, S. Dobbs, K. K. Seth, A. Tomaradze, J. Dudek, F. Close, E. Swanson, S. Denisov, G. Huber, D. Kolybaba, S. Krueger, G. Lolos, Z. Papandreou, A. Semenov, I. Semenova, M. Tahani, W. Brooks, H. Hakobyan, S. Kuleshov, O. Soto, A. Toro, I. Vega, R. White, F. Barbosa, E. Chudakov, H. Egiyan, M. Ito, D. Lawrence, M. McCaughan, M. Pennington, L. Pentchev, Y. Qiang, E. S. Smith, A. Somov, S. Taylor, T. Whitlatch, E. Wolin, B. Zihlmann
The primary motivation of the GlueX experiment is to search for and ultimately study the pattern of gluonic excitations in the meson spectrum produced in $\gamma p$ collisions. Recent lattice QCD calculations predict a rich spectrum of hybrid mesons that have both exotic and non-exotic $J^{PC}$, corresponding to $q\bar{q}$ states ($q=u,$ $d,$ or $s$) coupled with a gluonic field. A thorough study of the hybrid spectrum, including the identification of the isovector triplet, with charges 0 and $\pm1$, and both isoscalar members, $|s\bar{s}\ >$ and $|u\bar{u}\ > + |d\bar{d}\ >$, for each predicted hybrid combination of $J^{PC}$, may only be achieved by conducting a systematic amplitude analysis of many different hadronic final states. Detailed studies of the performance of the \gx detector have indicated that identification of particular final states with kaons is possible using the baseline detector configuration. The efficiency of kaon detection coupled with the relatively lower production cross section for particles containing hidden strangeness will require a high intensity run in order for analyses of such states to be feasible. We propose to collect a total of 200 days of physics analysis data at an average intensity of $5\times 10^7$ tagged photons on target per second. This data sample will provide an order of magnitude statistical improvement over the initial GlueX running, which will allow us to begin a program of studying mesons and baryons containing strange quarks. In addition, the increased intensity will permit us to study reactions that may have been statistically limited in the initial phases of GlueX. Overall, this will lead to a significant increase in the potential for \gx to make key experimental advances in our knowledge of hybrid mesons and excited $\Xi$ baryons.
View original: http://arxiv.org/abs/1305.1523 | 2017-12-15 23:31:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5153090953826904, "perplexity": 13067.869003930975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948580416.55/warc/CC-MAIN-20171215231248-20171216013248-00194.warc.gz"} |
https://solvedlib.com/n/circulation-and-gas-exchange-4-describe-or-draw-the-path,16097692 | # CIRCULATION AND GAS EXCHANGE 4: Describe or draw the path of blood through all partsof the mammalian heart, and major
###### Question:
CIRCULATION AND GAS EXCHANGE 4: Describe or draw the path of blood through all parts of the mammalian heart, and major arteries and veins near the heart in both systemic and pulmonary circuits (aorta, pulmonary arteries, and veins, superior, inferior vena cava); where is blood oxygenated/deoxygenated (as for lab exam)
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http://jphblog.com/2015/06/anisotropic-viscosity/ | Anisotropic Viscosity | Jonathan Perry-Houts
Jonathan Perry-Houts
### Anisotropic Viscosity
Nondimensionalized temperature is plotted over time. The model on the left has isotropic viscosity; the model on the right has anisotropic viscosity, oriented diagonally (45 degrees). The shear viscosity is 0.01 times the normal viscosity in the anisotropic case. Both models have no-slip boundaries on the top and bottom, and free slip on the sides. Note that in a confined box, the existence of uniform easy slip planes basically just reduces the effective viscosity of the entire box. The model on the right would look very similar with isotropically lower viscosity than the model on the left.
I’ve been working a bit lately on fluid flow problems involving materials with anisotropic viscosities. That is, deforming such a material in one direction is easier than in another direction. An example of such a material in the real world is a sequence of rock layers with varying strengths. In simple shear along the layered plane the stronger layers are able to slide past each other along the weak layers, but the rock is difficult to squish in “pure shear”, because it is supported by the strong layers.
Alternating strong and weak layers in sedimentary rock. Sandy Hollow, MT.
Deformation in layered rock is accommodated primarily in the weak layers. Sun River, MT.
In order to model this sort of behavior I’ve been working on extending a geodynamic modeling code, aspect, to incorporate more complex constitutive laws. Before I get too far in to ASPECT though I’d like to to step back a bit and define the equations I plan to solve. To that end we can start with the basic governing equations of fluid dynamics.
In general fluid flow obeys the Cauchy momentum equation,
$\rho \frac{D\vec{u}}{Dt} = \nabla \cdot \sigma + \rho \vec{g}$,
where $\vec{u}$ is the fluid velocity field, $\frac{D}{Dt} \equiv \frac{d}{dt} + \vec{u} \cdot \nabla$ is the material derivative, $\sigma$ is the stress tensor, $\rho$ is the fluid density, and $\vec{g}$ is the gravity vector. In geodynamics we deal with very low Reynolds number systems (i.e., momentum is negligibly small), so we set the left hand side of Cauchy’s equation to zero. This leaves us with the governing equation:
$-\nabla \cdot \sigma = \rho \vec{g}$.
We often also add the constraint that mass is conserved, $\nabla \cdot (\rho \vec{u}) = 0$, or in some cases that the flow is entirely incompressible, $\nabla \cdot \vec{u} = 0$.
In general, we don’t care so much about the stress state of the fluid, but we want to solve for the velocity. Therefore we need an equation to relate applied stress to strain rate. This type of equation is called a “constitutive law,” and is dependent on the chemical and state properties of the material. The most general constitutive law can be written in index notation, $\sigma_{ij} = - \overline P \delta_{ij} + C_{ijkl} \varepsilon_{kl}$, where $\overline P$ is the dynamic pressure; $\varepsilon$ is the strain rate, defined as the symmetric component of the gradient of the velocity, $\varepsilon \equiv \frac{1}{2} (\nabla \vec{u} + (\nabla \vec{u})^T)$; and $C_{ijkl}$ is a fourth order tensor, related to the viscosity, which maps strain rates to stresses.
Aspect assumes that the fluid in question is isotropic, and thus reduces to only two values, bulk viscosity (which dissipates energy during compaction and dilation), and shear viscosity which dissipates energy during shear deformation. Further, it is assumed that because bulk viscosity is only important in rapid compressions and dilations, such as sound waves and shock waves, it can be ignored in slow flows like geologic applications. Therefore, in Aspect $C_{ijkl}$ is reduced to a single scalar state variable, $2 \eta$, called the shear viscosity. Generalization back to a full constitutive tensor turns out to be as straightforward as changing the variable type of $\eta$ from a floating point scalar to a symmetric, fourth-order tensor. This modification is easily supported in the finite element library on which Aspect is built. With such a modification it is possible to model any arbitrary constitutive law.
Note (2016/02/16): This draft is already over half a year old, and I’ve been up to a lot since I wrote it, so I think it’s time to just post it and write more in a next installment. For now I’ll leave off with some figures to demonstrate the effects of the constitutive law I’m using:
Velocity magnitude is plotted here with velocity vectors overlain. The model setups are the same as in the animated example above except the left and right are stress-free boundaries, and the bottom has a prescribed horizontal velocity to the left. We can see the development of preferred shear directions in the velocity vectors.
Written on June 2nd , 2015 by JPH | 2018-12-16 03:28:25 | {"extraction_info": {"found_math": true, "script_math_tex": 17, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.739891529083252, "perplexity": 481.8304455543849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827252.87/warc/CC-MAIN-20181216025802-20181216051802-00617.warc.gz"} |
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### BE Agile before you Become Agile
Xebia Blog - Wed, 08/20/2014 - 20:49
People dislike change. It disrupts our routines and we need to invest to adapt. We only go along if we understand why change is needed and how we benefit from it.
The key to intrinsic motivation is to experience the benefits of the change yourself, rather than having someone else explain it to you.
Agility is almost an acronym for change. It is critical to let people experience the benefits of Agility before asking them to buy into this new way of working. This post explains how to create a great Agile experience in a fun, simple, cost efficient and highly effective way. BEing agile, before BEcoming agile!
The concept of a “Company Innovation Day”
Have you seen this clip about Dan Pinks’ Drive? According to him, the key factors for more motivation and better performance are: autonomy, mastery and purpose.
If you have some scrum experience this might sound familiar, right? That is because these 3 things really tie in nicely with agile and scrum, for example:
Autonomy = being able to self-direct;
• Let the team plan their own work
• Let the team decide how to best solve problems
Mastery = learning, applying and mastering new skills and abilities, a.k.a. "get better at stuff";
• Retrospect and improve
• Learn, apply and master new skills to get achieve goals as a team.
Purpose = understanding necessity and being as effective as possible;
• Write user stories that add value
• Define sprint goals that tie in to product- and business goals.
In the clip, the company "Atlassian" is mentioned. This is the company that makes "JIRA", one of the most popular Agile support tools. Atlassian tries to facilitate autonomy, mastery and purpose by organizing one day per quarter of “management free” innovation. They call it a “ship it day”.
Now this is cool! According to Dan, their people had fun (most important), fixed a whole array of bugs and delivered new product ideas as well. They have to ship all this in one day, again showing similarities with the time boxed scrum approach. When I first saw this, I realized that this kind of fast delivery of value is pretty much something you would like to achieve with Agile Scrum too! Doing Scrum right would feel like a continuous series of ship it days.
My own experience with innovation days
Recently I organized an innovation day with a client (for tips see on how to organize yours, click here). We invited the whole department to volunteer. If you didn’t feel like it, you could just skip it and focus on sprint work. Next we promoted the day and this resulted in a growing list of ideas coming in.
Except for the framing of the day, the formation of ideas and teams was totally self-organized and also result driven as we asked for the expected result. Ultimately we had 20 initiatives to be completed in one day.
On the day itself, almost everyone joined in and people worked hard to achieve results at the end of the day.
The day ended in presenting the results and having pizzas. Only some ideas just missed the deadline, but most were finished including usable and fresh new stuff with direct business value. When looking at the photos of that day it struck me that 9 out of ten photos showed smiling faces. Sweet!
The first innovation day was concluded with an evaluation. In my opinion evaluation is essential, because this is the perfect moment discuss deeper lessons and insights. Questions like; “how can we create the innovation day energy levels during sprints”, and “how can we utilize self-organizing abilities more” are invaluable as they could lead to new insights, inspiration and experiments for day-to-day operations.
The value of an innovation day as a starting point for Agile
All in all, I think an innovation day is the perfect way to get people experiencing the power of Agile.
Doing the innovation day on “day one” offers huge benefits when added to standard stuff like training and games. This is because the context is real. You have a real goal, a real timebox and you need to self-organize to achieve the desired result.
People doing the work get to experience their potential and the power of doing stuff within a simplified context. Managers get to experience unleashing the human potential when they focus only on the context and environment for that day.
I can only imagine the amazement and renewed joy when people experience the possibilities coming from a strong waterfall setting. All that good stuff from just a one-day investment!
Conclusion
It would be great if you would start out an Agile change initiative with an innovation day. Get people enthusiastic and inspired (e.g. motivated for change) first and then tell them why it works and how we are going to apply the same principles in day-to-day operations. This will result in less friction and resistance and give people a better sense for where they are heading.
Do you want to start doing innovation days or do you want to share your experience, feel free to leave a comment below.
### Part 2: The Cloud Does Equal High performance
This a guest post by Anshu Prateek, Tech Lead, DevOps at Aerospike and Rajkumar Iyer, Member of the Technical Staff at Aerospike.
In our first post we busted the myth that cloud != high performance and outlined the steps to 1 Million TPS (100% reads in RAM) on 1 Amazon EC2 instance for just $1.68/hr. In this post we evaluate the performance of 4 Amazon instances when running a 4 node Aerospike cluster in RAM with 5 different read/write workloads and show that the r3.2xlarge instance delivers the best price/performance. Several reports have already documented the performance of distributed NoSQL databases on virtual and bare metal cloud infrastructures: Categories: Architecture ### Creating a Company Where Everyone Gives Their Best “Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do.” —Steve Jobs What does it take to create a company where everybody gives their best where they have their best to give? It takes empathy. It also takes encouraging people to be zestful, zany, and zealous. It takes bridging the gap between the traits that make people come alive, and the traits that traditional management practices value. In the book The Future of Management, Gary Hamel walks through what it takes to create a company where everyone gives their best so that employees thrive and companies create sustainable competitive advantage. Resilience and Creativity: The Traits that Differentiate Human Beings from Other Species Resilience and creativity are what separate us from the pack. “Ask your colleagues to describe the distinguishing characteristics of your company, and few are likely to mention adaptability and inventiveness. Yet if you ask them to make a list of the traits that differentiate human beings from other species, resilience and creativity will be near the top of the list. We see evidence of these qualities every day -- in ourselves and in those around us. “ We Work for Organizations that Aren't Very Human People are adaptive and creative, but they often work for organizations that are not. “All of us know folks who've switched careers in search of new challenges or a more balanced life. We know people who've changed their consumption habits for the sake of the planet. We have friends and relatives who've undergone a spiritual transformation, or risen to the demands of parenthood, or overcome tragedy. Every day we meet people who write blogs, experiment with new recipes, mix up dance tunes, or customize their cars. As human beings, we are amazingly adaptable and creative, yet most of us work for companies that are not. In other words, we work for organizations that aren't very human.” Modern Organizations Deplete Natural Resilience and Creativity Why do so many organizations underperform? They ignore or devalue the capabilities that make us human. “There seems to be something in modern organizations that depletes the natural resilience and creativity of human beings, something that literally leaches these qualities out of employees during daylight hours. The culprit? Management principles and processes that foster discipline, punctuality, economy, rationality, and order, yet place little value on artistry, nonconformity, originality, audacity, and élan. To put it simply, most companies are only fractionally human because they make room for only a fraction of the qualities and capabilities that make us human. Billions of people show up for work every day, but way too many of them are sleepwalking. The result: organizations that systematically underperform their potential.” Adaptability and Innovation Have Become the Keys to Competitive Success There’s a great big gap between what makes people great and the management systems that get in the way. “Weirdly, many of those who labor in the corporate world--from lowly admins to high powered CEOs--seem resigned to this state of affairs. They seem unperturbed by the confounding contrast between the essential nature of human beings and the essential nature of the organization in which they work. In years past, it might have been possible to ignore this incongruity, but no longer--not in a world where adaptability and innovation have become the sine qua non of competitive success. The challenge: to reinvent our management systems so they inspire human beings to bring all of their capabilities to work every day.” The Human Capabilities that Contribute to Competitive Success Hamel offers his take on what the relative contribution of human capabilities that contribute to value creation, recognizing that we now live in a world where efficiency and discipline are table stakes. Passion 35% Creativity 25% Initiative 20% Intellect 15% Diligence 5% Obedience 0% 100% “The human capabilities that contribute to competitive success can be arrayed in a hierarchy. At the bottom is obedience--an ability to take direction and follow rules. This is the baseline. Next up the ladder is diligence. Diligent employees are accountable. They don't take shortcuts. They are conscientious and well-organized. Knowledge and intellect are on the next step. Most companies work hard to hire intellectually gifted employees. They value smart people who are eager to improve their skills and willing to borrow best practices from others. Beyond intellect lies initiative. People with initiative don't wait to be asked and don't wait to be told. They seek out new challenges and are always searching for new ways to add value. Higher still lies the gift of creativity. Creative people are inquisitive and irrepressible. They're not afraid of saying stupid things. They start a lot of conversations with, 'Wouldn't it be cool if ..." And finally, at the top lies passion.” The Power of Passion Passion makes us do dumb things. But it’s also the key to doing great things. Via Via The Future of Management: “Passion can make people do stupid things, but it's the secret sauce that turns intent into accomplishment. People with passion climb over obstacles and refuse to give up. Passion is contagious and turns one-person crusades into mass movements. As the English novelist E.M. Forster put it, 'One person with passion is better than forty people merely interested.'” Obedience is Worth Zip in Terms of Competitive Advantage Rule-following employees won’t help you change the world. “I'm not suggesting that obedience is literally worth nothing. A company where no one followed any rules would soon descend into anarchy. Instead, I'm arguing that rule-following employees are worth zip in terms of their competitive advantage they generate. In a world with 4 billion nearly distributed souls, all eager to climb the ladder of economic progress, it's not hard to find billable, hardworking employees. And what about intelligence? For years we've been told we're living in the knowledge economy; but as knowledge itself becomes commoditized, it will lose much of its power to create competitive advantage.” Obedience, Diligence, and Expertise Can Be Bought for Next to Nothing You can easily buy obedience, diligence, and expertise from around the world. But that’s not what will make you the next great company or the next great thing or a great place to work. “Today, obedience, diligence, and expertise can be bought for next to nothing. From Bangalore to Guangzhou, they have become global commodities. A simple example: turn over your iPod, and you'll find six words engraved on the back that foretell the future of competition: 'Designed in California. Made in China.' Despite the equal billing, the remarkable success of Apple's music business owes relatively little to the company's network of Asian subcontractors. It is a credit instead to the imagination of Apple's designers, marketers, and lawyers. Obviously not every iconic product is going to be designed in California, not nor manufactured in China. “ You Need Employees that are Zestful, Zany, and Zealous If you want to bring out the best in people and what they are capable of, aim for zestful, zany, and zealous. “The point, though, is this: if you want to capture the economic high ground in the creative economy, you need employees who are more than acquiescent, attentive, and astute--they must also be zestful, zany, and zealous.” If you want to bring out your best, then break our your zest and get your zane on. You Might Also Like The Principles of Modern Management How Employees Lost Empathy for their Work, for the Customer, and for the Final Product No Slack = No Innovation The Drag of Old Mental Models on Innovation and Change The New Competitive Landscape The New Realities that Call for New Organizational and Management Capabilities Who’s Managing Your Company Categories: Architecture, Programming ### Agile 2014 – speaking and attending; a summary Xebia Blog - Tue, 08/19/2014 - 17:14 So Agile 2014 is over again… and what an interesting conference it was. What did I find most rewarding? Meeting so many agile people! My first conclusion was that there were experts like us agile consultants or starting agile coaches, ScrumMasters and other people getting acquainted with our cool agile world. Another trend I noticed was the scaled agile movement. Everybody seems to be involved in that somehow. Some more successful than others; some more true to agile than others. What I missed this year was the movement of scrum or agile outside IT although my talk about scrum for marketing had a lot of positive responses. Everybody I talked to was interested in hearing more information about it. There was a talk maybe even two about hardware agile but I did not found a lot of buzz around it. Maybe next year? I do feel that there is potential here. I believe Fullstack product development should be the future. Marketing and IT teams? Hardware and software teams? Splitting these still sounds as design and developer teams to me. But what a great conference it was. I met a lot of awesome people. Some just entering the agile world; some authors of books I had read which got me further in the agile movement. I talked to the guys from Spotify. The company which is unique in its agile adoption / maturity. And they don’t even think that they are there yet. But then again will somebody ever truly BE agile ..? I met the guys from scrum.inc who developed a great new scaled framework. Awesome ideas on that subject and awesome potential to treat it as a community created open framework; keep your eyes open for that! I attended some nice talks too; also some horrible ones. Or actually 1, which should never have been presented in a 90 minute slot in a conference like this. But lets get back to the nice stories. Lyssa Adkins had a ‘talk’ about conflicts. Fun thing was that she actually facilitated the debate about scaled agile on stage. The session could have been better but the idea and potential of the subject is great. Best session? Well probably the spotify guys. Still the greatest story out there of an agile company. The key take-out of that session for me is: agile is not an end-state, but a journey. And if you take it as serious as Spotify you might be able to make the working world a lot better. Looking at Xebia we might not even be considered to be trying agile compared to them. And that is meant in a humble way while looking up to these guys! - I know we are one of the frontiers of agile in the Netherlands. The greatest question in this session: ‘Where is the PMO in your model….’ Well you clearly understand this … Another inspiring session was the keynote session from the CFO of Statoil about beyond budgeting. This was a good story which should become bigger in the near future as this is one of the main questions I get when implementing agile in a company: “how do we plan / estimate and budget projects when we go and do agile?” Beyond budgeting at least get’s us a little closer. Long story short. I had a blast in Orlando. I learnt new things and met a lot of cool people.My main take out: Our community is growing which teaches us that we are not yet there by a long run. An awesome future is ahead! See you next year! ### Sponsored Post: Apple, Tumblr, Gawker, FoundationDB, Monitis, BattleCry, Surge, Cloudant, CopperEgg, Logentries, Couchbase, MongoDB, BlueStripe, AiScaler, Aerospike, AppDynamics, ManageEngine, Site24x7 Who's Hiring? • Apple has multiple openings. Changing the world is all in a day's work at Apple. Imagine what you could do here. • Senior Software Engineer -iOS Systems. Do you love building highly scalable, distributed web applications? Does the idea of a fast-paced environment make your heart leap? Do you want your technical abilities to be challenged every day, and for your work to make a difference in the lives of millions of people? If so, the iOS Systems Carrier Services team is looking for a talented software engineer who is not afraid to share knowledge, think outside the box, and question assumptions. Please apply here. • Software Engineering Manager, IS&T WWDR Dev Systems. 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Now I went one step further by creating a set of AngularJS directives. You can read the blog on the trifork blog: http://blog.trifork.com/2014/08/19/angularjs-directives-for-c3-js-chart-library/ The post AngularJS directives for c3.js appeared first on Gridshore. Categories: Architecture, Programming ### 1 Aerospike server X 1 Amazon EC2 instance = 1 Million TPS for just$1.68/hour
This a guest post by Anshu Prateek, Tech Lead, DevOps at Aerospike and Rajkumar Iyer, Member of the Technical Staff at Aerospike.
Cloud infrastructure services like Amazon EC2 have proven their worth with wild success. The ease of scaling up resources, spinning them up as and when needed and paying by unit of time has unleashed developer creativity, but virtualized environments are not widely considered as the place to run high performance applications and databases.
Cloud providers however have come a long way in their offerings and need a second review of their performance capabilities. After showing 1 Million TPS on Aerospike on bare metal servers, we decided to investigate cloud performance and in the process, bust the myth that cloud != high performance.
We examined a variety of Amazon instances and just discovered the recipe for processing 1 Million TPS in RAM on 1 Aerospike server on a single C3.8xlarge instance - for just $1.68/hr !!! According to internetlivestats.com, there are 7.5k new tweets per second, 45k google searches per second and 2.3 Million emails sent per second. What would you build if you could process 1 Million database transactions per second for just$1.68/hr?
Categories: Architecture
### Little's Law in 3D
Xebia Blog - Sun, 08/17/2014 - 16:21
The much used relation between average cycle time, average total work and input rate (or throughput) is known as Little's Law. It is often used to argue that it is a good thing to work on less items at the same time (as a team or as an individual) and thus lowering the average cycle time. In this blog I will discuss the less known generalisation of Little's Law giving an almost unlimited number of additional relation. The only limit is your imagination.
I will show relations for the average 'Total Operational Cost in the system' and for the average 'Just-in-Timeness'.
First I will describe some rather straightforward generalisations and in the third part some more complex variations on Little's Law.
Little's Law Variations
As I showed in the previous blogs (Applying Little's Law in Agile Games and Why Little's Law Works...Always) Little's Law in fact states that measuring the total area from left-to-right equals summing it from top-to-bottom.
Once we realise this, it is easy to see some straightforward generalisations which are well-known. I'll mention them here briefly without ging into too much details.
Subsystem
Suppose a system that consists of 1 or more subsystems, e.g. in a kanban system consisting of 3 columns we can identify the subsystems corresponding to:
1. first column (e.g. 'New') in 'red',
2. second column (e.g. 'Doing') in 'yellow',
3. third column (e.g. 'Done') in 'green'
See the figure on the right.
By colouring the subsystems different from each other we see immediately that Little's Law applies to the system as a whole as well as to every subsystem ('red' and 'yellow' area).
Note: for the average input rate consider only the rows that have the corresponding color, i.e. for the input rate of the column 'Doing' consider only the rows that have a yellow color; in this case the average input rate equals 8/3 items per round (entering the 'Doing' column). Likewise for the 'New' column.
Work Item Type
Until now I assumed only 1 type of work items. In practise teams deal with more than one different work item types. Examples include class of service lanes, user stories, and production incidents. Again, by colouring the various work item type differently we see that Little's Law applies to each individual work item type.
In the example on the right, we have coloured user stories ('yellow') and production incidents ('red'). Again, Little's Law applies to both the red and yellow areas separately.
Doing the math we se that for 'user stories' (yellow area):
• Average number in the system (N) = (6+5+4)/3 = 5 user stories,
• Average input rate ($\lambda$\lambda = 6/3 = 2 user stories per round,
• Average waiting time (W) = (3+3+3+3+2+1)/6 = 15/6 = 5/2 rounds.
As expected, the average number in the system equals the average input rate times the average waiting time.
The same calculation can be made for the production incidents which I leave as an exercise to the reader.
Expedite Items
Finally, consider items that enter and spend time in an 'expedite' lane. In Kanban an expedite lane is used for items that need special priority. Usually the policy for handling such items are that (a) there can be at most 1 such item in the system at any time, (b) the team stop working on anything but on this item so that it is completed as fast as possible, (c) they have priority over anything else, and (d) they may violate any WiP limits.
Colouring any work items blue that spend time in the expedite lane we can apply Little's Law to the expedite lane as well.
An example of the colouring is shown in the figure on the right. I leave the calculation to the reader.
3D
We can even further extend Little's Law. Until now I have considered only 'flat' areas.
The extension is that we can give each cell a certain height. See the figure to the right. A variation on Little's Law follows once we realise that measuring the volume from left-to-right is the same as calculating it from top-to-bottom. Instead of measuring areas we measure volumes instead.
The only catch here is that in order to write down Little's Law we need to give a sensible interpretation to the 'horizontal' sum of the numbers and a sensible interpretation to the 'vertical' sum of the numbers. In case of a height of '1' these are just 'Waiting Time' (W) and 'Number of items in the system' (N) respectively.
A more detailed, precise, and mathematical formulation can be found in the paper by Little himself: see section 3.2 in [Lit11].
Some Applications of 3D-Little's Law
Value
As a warming-up exercise consider as the height the (business) value of an item. Call this value 'V'. Every work item will have its own specific value.
$\overline{\mathrm{Value}} = \lambda \overline{V W}$ \overline{\mathrm{Value}} = \lambda \overline{V W}
The interpretation of this relation is that the 'average (business) value of unfinished work in the system at any time' is equal to the average input rate multiplied by the 'average of the product of cycle time and value'.
Teams may ant to minimise this while at the same time maximising the value output rate.
Total Operational Cost
As the next example let's take as the height for the cells a sequence of numbers 1, 2, 3, .... An example is shown in the figures below. What are the interpretations in this case?
Suppose we have a work item that has an operational cost of 1 per day. Then the sequence 1, 2, 3, ... gives the total cost to date. At day 3, the total cost is 3 times 1 which is the third number in the sequence.
The 'vertical' sum is just the 'Total Cost of unfinished work in the system.
For the interpretation of the 'horizontal' sum we need to add the numbers. For a work item that is in the system for 'n' days, the total is $1+2+3+...+n$1+2+3+...+n which equals $1/2 n (n+1)$1/2 n (n+1). For 3 days this gives $1+2+3=1/2 * 3 * 4 = 6$1+2+3=1/2 * 3 * 4 = 6. Thus, the interpretation of the 'horizontal' sum is $1/2 W (W+1)$1/2 W (W+1) in which 'W' represents the waiting time of the item.
Putting this together gives an additional Little's Law of the form:
$\overline{\mathrm{Cost}} = \frac{1}{2} \lambda C \overline{W(W + 1)}$ \overline{\mathrm{Cost}} = \frac{1}{2} \lambda C \overline{W(W + 1)}
where 'C' is the operational cost rate of a work item and $\lambda$\lambda is the (average) input rate. If instead of rounds in a game, the 'Total Cost in the system' is measured at a time interval 'T' the formula slightly changes into
$\overline{\mathrm{Cost}} = \frac{1}{2} \lambda C \overline{W\left(W + T\right)}$ \overline{\mathrm{Cost}} = \frac{1}{2} \lambda C \overline{W\left(W + T\right)}
Teams may want to minimise this which gives an interesting optimisation problem is different work item types have different associated operational cost rates. How should the capacity of the be divided over the work items? This is a topic for another blog.
Just-in-Time
For a slightly more odd relation consider items that have a deadline associated with them. Denote the date and time of the deadline by 'D'. As the height choose the number of time units before or after the deadline the item is completed. Further, call 'T' the time that the team has taken up to work on the item. Then the team finishes work on this item at time $T + W$ T + W , where 'W' represent the cycle time of the work item.
In the picture on the left a work item is shown that is finished 2 days before the deadline. Notice that the height decreases as the deadline is approached. Since it is finished 2 time units before the deadline, the just-in-timeness is 2 at the completion time.
The picture on the left shows a work item one time unit after the deadline and has an associated just-in-timeness of 1.
$\overline{\mathrm{Just-in-Time}} = \frac{1}{2} \lambda \overline{|T+W-D|(|T+W-D| + 1)}$ \overline{\mathrm{Just-in-Time}} = \frac{1}{2} \lambda \overline{|T+W-D|(|T+W-D| + 1)}
This example sounds like a very exotic one and not very useful. A team might want to look at what the best time is to start working on an item so as to minimise the above variable.
Conclusion
From our 'playing around' with the size of areas and volumes and realising that counting it in different ways (left-to-right and top-to-bottom) should give the same result I have been able to derive a new set of relations.
In this blog I have rederived well-known variations on Little's Law regarding subsystems and work items types. In addition I have derived new relations for the 'Average Total Operational Cost', 'Average Value', and 'Average Just-in-Timeness'.
Together with the familiar Little's Law these give rise to interesting optimisation problems and may lead to practical guidelines for teams to create even more value.
I'm curious to hear about the variations that you can come up with! Let me know by posting them here.
References
[Lit11] John D.C. Little, "Little’s Law as Viewed on Its 50th Anniversary", 2011, Operations Research, Vol. 59 , No 3, pp. 536-549, https://www.informs.org/content/download/255808/2414681/file/little_paper.pdf
### Managing OpenStack security groups from the command line
Agile Testing - Grig Gheorghiu - Fri, 08/15/2014 - 20:47
I had an issue today where I couldn't connect to a particular OpenStack instance on port 443. I decided to inspect the security group it belongs (let's call it myapp) to from the command line:
# nova secgroup-list-rules myapp
+-------------+-----------+---------+------------+--------------+
| IP Protocol | From Port | To Port | IP Range | Source Group |
+-------------+-----------+---------+------------+--------------+
| tcp | 80 | 80 | 0.0.0.0/0 | |
| tcp | 443 | 443 | 0.0.0.0/24 | |
+-------------+-----------+---------+------------+--------------+
Note that the IP range for port 443 is wrong. It should be all IPs and not a /24 network.
I proceeded to delete the wrong rule:
# nova secgroup-delete-rule myapp tcp 443 443 0.0.0.0/24
+-------------+-----------+---------+------------+--------------+
| IP Protocol | From Port | To Port | IP Range | Source Group |
+-------------+-----------+---------+------------+--------------+
| tcp | 443 | 443 | 0.0.0.0/24 | |
+-------------+-----------+---------+------------+--------------+
Then I added back the correct rule:
# nova secgroup-add-rule myapp tcp 443 443 0.0.0.0/0 +-------------+-----------+---------+-----------+--------------+| IP Protocol | From Port | To Port | IP Range | Source Group |+-------------+-----------+---------+-----------+--------------+| tcp | 443 | 443 | 0.0.0.0/0 | |+-------------+-----------+---------+------------+--------------+
Finally, I verified that the rules are now correct:
# nova secgroup-list-rules myapp +-------------+-----------+---------+-----------+--------------+| IP Protocol | From Port | To Port | IP Range | Source Group |+-------------+-----------+---------+-----------+--------------+| tcp | 443 | 443 | 0.0.0.0/0 | || tcp | 80 | 80 | 0.0.0.0/0 | |+-------------+-----------+---------+-----------+--------------+
Of course, the real test was to see if I could now hit port 443 on my instance, and indeed I was able to.
### Creativity, INC. – by Ed Catmull
Gridshore - Fri, 08/15/2014 - 20:39
A colleague of mine, Ronald Vonk, recommended this book to me. It is a book by one of the founders of Pixar, you know from all those fantastic computer animated movies. At pixar they created an continuous changing environment where creativity should excel. It is a very interesting read if you are interested in management books that are not to heavy on theories. Ed explains very well and entertaining how they went from a small company with a vision to a huge company with a vision.
Without to much spoilers, you really need to reed the book yourself, I want to mention a few things that I remembered after reading the book.
The team is more important than the idea’s or the talent of the separate people. Take care of the team, make sure they function well and give them responsibility. Make them feel proud when they finished what they wanted to create. Always put people first.
This is something I ran into in my normal working life as well. I do think you have to enable the teams to adept and to stay as a good team. The challenge is to get others in to learn and later on replace team members or start their own team.
We would never make a film that way again. It is the managements job to take the long view, to intervene and protect our people from their willingness to pursue excellence at all costs. Not to do so would be irresponsible.
This was a remark after delivering a movie under great time stress. They pulled through, but at a cost.
Braintrust – Group of people giving feedback and ideas for improvements on a certain idea. Important is that the feedback is meant to improve the idea, not to bully the person(s) the idea originated from. It is very important that everybody is open to the feedback and not defensive. In the end it is not the braintrust that makes a decision, it is the person in charge for the product. Still this group of people is kind of the first user and therefore the feedback should not be taken to lightly.
This was something I had a long thought about, my conclusion was that I am not really good at this. I often do feel that my ideas are my babies that need to be defended. First persuade me I am wrong, o sorry, an idea that someone had was not the best.
I did not want to become a manager, I just wanted to be one of the boys and do research. When we became bigger I realised I became more important and new people did not see me as a peer or one of the guys. I realised things were starting to get hidden from me. It is no problem as long as you trust people will tell someone else that will tell the most important things to me again.
Couldn’t agree more.
You can have this very nice polished finely tuned locomotive. People think that being the driver of the train is giving them power. They feel that driving the train in the end is shaping the company. The truth is, it’s not. Driving the train does not set it’s course. The real job is laying the track.
This was an eye opener a well, something you know but is hard to put into words.
At pixar they do not have contracts. They feel that employment contracts both hurt the employer as well as the employee. If someone had a problem with the company, there wasn’t much point in complaining because they were under contract. If someone didn’t perform well, on the other hand, there was no point in confronting them about it; their contract simply wouldn’t be renewed, which might be the first time they heard about their need to improve. The whole system discouraged and devaluated day-to-day communication and was culturally dysfunctional. But since everybody was used to it, they were blind to the problem.
This is a long one, have thought about it for a while. I think for now I would be to scared to do this in a company, still I like the idea.
What is the point of hiring smart people if you don’t empower them to fix what’s broken? Often to much time is lost in making sure no mistakes will be made. Often however, it just takes a few days to find solutions for mistakes.
Keeps coming back to the same point, a manager is a facilitator, nothing more nothing less. It is a very important role, just like all the others. Think about it, it is the team, the complete team.
The post Creativity, INC. – by Ed Catmull appeared first on Gridshore.
Categories: Architecture, Programming
### Stuff The Internet Says On Scalability For August 15th, 2014
Hey, it's HighScalability time:
Somehow this seems quite appropriate. (via John Bredehoft)
• 75 acres: Pizza eaten in US daily; 270TB: Backblaze storage pod; 14nm: Intel extends Moore's Law
• Quotable Quotes
• discreteevent: The dream of reuse has made a mess of many systems.
• David Crawley: Don't think of Moore's Law in terms of technology; think of it in terms of economics and you get much greater understanding. The limits of Moore's Law is not driven by current technology. The limits of Moore's Law are really a matter of cost.
• Simon Brown: If you can't build a monolith, what makes you think microservices are the answer?
• smileysteve: The net result is that you should be able to transmit QPSK at 32GBd in 2 polarizations in maybe 80 waves in each direction. 2bits x 2 polarizations x 32G ~128Gb/s per wave or nearly 11Tb/s for 1 fiber. If this cable has 6 strands, then it could easily meet the target transmission capacity [60TB].
• Eric Brumer: Highly efficient code is actually memory efficient code.
• How to be a cloud optimist. Tell yourself: an instance is half full, it's not half empty; Downtime is temporary; Failures aren't your fault.
• Mother Earth, Motherboard by Neal Stephenson. Goes without saying it's gorgeously written. The topic: The hacker tourist ventures forth across the wide and wondrous meatspace of three continents, chronicling the laying of the longest wire on Earth. < Related to Google Invests In $300M Submarine Cable To Improve Connection Between Japan And The US. • IBM compares virtual machines and against Linux containers: Our results show that containers result in equal or better performance than VM in almost all cases. Both VMs and containers require tuning to support I/O-intensive applications. • Does Psychohistory begin with BigData? Of a crude kind, perhaps. Google uses BigQuery to uncover patterns of world history: What’s even more amazing is that this analysis is not the result of a massive custom-built parallel application built by a team of specialized HPC programmers and requiring a dedicated cluster to run on: in stark contrast, it is the result of a single line of SQL code (plus a second line to create the initial “view”). All of the complex parallelism, data management, and IO optimization is handled transparently by Google BigQuery. Imagine that – a single line of SQL performing 2.5 million correlations in just 2.5 minutes to uncover the underlying patterns of global society. • Fabian Giesen with an deep perspective on how communication has evolved to use a similar pattern. Networks all the way down (part2): anything we would call a computer these days is in fact, for all practical purposes, a heterogeneous cluster made up of various specialized smaller computers, all connected using various networks that go by different names and are specified in different standards, yet are all suspiciously similar at the architecture level; a fractal of switched, packet-based networks of heterogeneous nodes that make up what we call a single “computer”. It means that all the network security problems that plague inter-computer networking also exist within computers themselves. Implementations may change substantially over time, the interfaces – protocols, to stay within our networking terminology – stay mostly constant over large time scales, warts and all. Don't miss all that the Internet has to say on Scalability, click below and become eventually consistent with all scalability knowledge (which means this post has many more items to read so please keep on reading)... Categories: Architecture ### Hamsterdb: An Analytical Embedded Key-value Store In this post, I’d like to introduce you to hamsterdb, an Apache 2-licensed, embedded analytical key-value database library similar to Google's leveldb and Oracle's BerkeleyDB. hamsterdb is not a new contender in this niche. In fact, hamsterdb has been around for over 9 years. In this time, it has dramatically grown, and the focus has shifted from a pure key-value store to an analytical database offering functionality similar to a column store database. hamsterdb is single-threaded and non-distributed, and users usually link it directly into their applications. hamsterdb offers a unique (at least, as far as I know) implementation of Transactions, as well as other unique features similar to column store databases, making it a natural fit for analytical workloads. It can be used natively from C/C++ and has bindings for Erlang, Python, Java, .NET, and even Ada. It is used in embedded devices and on-premise applications with millions of deployments, as well as serving in cloud instances for caching and indexing. hamsterdb has a unique feature in the key-value niche: it understands schema information. While most databases do not know or care what kind of keys are inserted, hamsterdb supports key types for binary keys... Categories: Architecture ### Success Articles for Work and Life "Success consists of going from failure to failure without loss of enthusiasm." -- Winston Churchill I now have more than 300 articles on the topic of Success to help you get your game on in work and life: Success Articles That’s a whole lot of success strategies and insights right at your fingertips. (And it includes the genius from a wide variety of sources including Scott Adams, Tony Robbins, Bruce Lee, Zig Ziglar, and more.) Success is a hot topic. Success has always been a hot topic, but it seems to be growing in popularity. I suspect it’s because so many people are being tested in so many new ways and competition is fierce. But What is Success? (I tried to answer that using Zig Ziglar’s frame for success.) For another perspective, see Success Defined (It includes definitions of success from Stephen Covey and John Maxwell.) At the end of the day, the most important definition of success, is the one that you apply to you and your life. People can make or break themselves based on how they define success for their life. Some people define success as another day above ground, but for others they have a very high, and very strict bar that only a few mere mortals can ever achieve. That said, everybody is looking for an edge. And, I think our best edge is always our inner edge. As my one mentor put it, “the fastest thing you can change in any situation is yourself.” And as we all know, nature favors the flexible. Our ability to adapt and respond to our changing environment is the backbone of success. Otherwise, success is fleeting, and it has a funny way of eluding or evading us. I picked a few of my favorite articles on success. These ones are a little different by design. Here they are: Scott Adam’s (Dilbert) Success Formula It’s the Pebble in Your Shoe The Wolves Within Personal Leadership Helps Renew You The Power of Personal Leadership Tony Robbins on the 7 Traits of Success The Way of Success The future is definitely uncertain. I’m certain of that. But I’m also certain that life’s better with skill and that the right success strategies under your belt can make or break you in work and life. And the good news for us is that success leaves clues. So make like a student and study. Categories: Architecture, Programming ### The Easy Way of Building a Growing Startup Architecture Using HAProxy, PHP, Redis and MySQL to Handle 1 Billion Requests a Week This Case Study is a guest post written by Antoni Orfin, Co-Founder and Software Architect at Octivi In the post I'll show you the way we developed quite simple architecture based on HAProxy, PHP, Redis and MySQL that seamlessly handles approx 1 billion requests every week. There’ll be also a note of the possible ways of further scaling it out and pointed uncommon patterns, that are specific for this project. Stats: Categories: Architecture ### The AngularJS Promise DSL Xebia Blog - Mon, 08/11/2014 - 10:21 As promised in my previous post, I just pushed the first version of our "Angular Promise DSL" to Github. It extends AngularJS's$q promises with a number of helpful methods to create cleaner applications.
The project is a V1, it may be a bit rough around the edges in terms of practical applicability and documentation, but that's why it's open source now.
The repository is at https://github.com/fwielstra/ngPromiseDsl and licensed as MIT. It's the first OS project I've created, so bear with me. I am accepting pull requests and issues, of course.
Questions? Ask them on the issues page, ask me via Twitter (@frwielstra) or send me an e-mail. I'd offer you to come by my office too... if I had one.
### Introduction to big data presentation
I presented big data to Amdocs’ product group last week. One of the sessions I did was recorded so I might be able to add here later. Meanwhile you can check out the slides.
Note that trying to keep the slide visual I put some of the information is in the slide notes and not on the slides themselves.
Categories: Architecture
### Are You Doing Agile Results?
Why?
Because most people are running around, scrambling through a laundry list of too many things to do, a lack of clarity around what the end result or outcomes should be, and a lack of clarity around what the high-value things to focus on are. They are using their worst energy for their most important things. They are spending too much time on the things that don’t matter and not enough time on the things that do. They are feeling at their worst, when they need to feel at their best, and they are struggling to keep up with the pace of change.
I created Agile Results to deal with the chaos in work and life, as a way to rise above the noise, and to easily leverage the most powerful habits and practices for getting better results in work and life.
Agile Results, in a nutshell, is a simple system for mastering productivity and time management, while at the same time, achieving more impact, realizing your potential, and feeling more fulfillment.
I wrote about the system in the book Getting Results the Agile Way. It’s been a best seller in time management.
How Does Agile Results Work?
Agile Results works by combining proven practices for productivity, time management, psychology, project management, and some of the best lessons learned on high-performance. And it’s been tested for more than a decade under extreme scenarios and a variety of conditions from individuals to large teams.
Work-Life balance is baked into the system, but more importantly Agile Results helps you live your values wherever you are, play to your strengths, and rapidly learn how to improve your results in an situation. When you spend more time in your values, you naturally tap into your skills and abilities that help bring out your best.
The simplest way to think of Agile Results is that it helps you direct your attention and apply your effort on the things that count. By spending more time on high-value activities and by getting intentional about your outcomes, you dramatically improve your ability to get better results.
But none of that matters if you aren’t using Agile Results.
How Can You Start Using Agile Results?
Start simple.
Simply ask yourself, “What are the 3 wins, results, or outcomes that I want for today?.” Consider the demands you have on your plate, the time and energy you’ve got, and the opportunities you have for today, and write those 3 things down.
That’s it. You’re doing Agile Results.
Of course, there’s more, but that’s the single most important thing you can do to immediately gain clarity, regain your focus, and spend your time and energy on the most valuable things.
Now, let’s assume this is the only post you ever read on Agile Results. Let’s take a fast walkthrough of how you could use the system on a regular basis to radically and rapidly improve your results on an ongoing basis.
How I Do Agile Results? …
Here’s a summary of how I do Agile Results.
I create a new monthly list at the start of each month that lists out all the things that I think I need to do, and I bubble up 3 of my best things I could achieve or must get done to the top. I look at it at the start of the week, and any time I’m worried if I’m missing something. This entire process takes me anywhere from 10-20 minutes a month.
I create a weekly list at the start of the week, and I look at it at the start of each day, as input to my 3 target wins or outcomes for the day, and any time I’m worried if I’m missing anything. This tends to take me 5-10 minutes at the start of the week.
I barely have to ever look at my lists – it’s the act of writing things down that gives me quick focus on what’s important. I’m careful not to put a bunch of minutia in my lists, because then I’d train my brain to stop focusing on what’s important, and I would become forgetful and distracted. Instead, it’s simple scaffolding.
Each day, I write a simple list of what’s on my mind and things I think I need to achieve. Next, I step back and ask myself, “What are the 3 things I want to accomplish today?”, and I write those down. (This tends to take me 5 minutes or less. When I first started it took me about 10.)
Each Friday, I take the time to think through three things going well and three things to improve. I take what I learn as input into how I can simplify work and life, and how I can improve my results with less effort and more effectiveness. This takes me 10-20 minutes each Friday.
How Can You Adopt Agile Results?
Here is a simple recipe for adopting Agile Results and using it to get better results in work and life:
1. Add a recurring appointment on your calendar for Monday mornings. Call it Monday Vision. Add this text to the body of the reminder: “What are your 3 wins for this week?”
2. Add a recurring appointment on your calendar to pop up every day in the morning. Call it Daily Wins. Add this text to the body of the reminder: “What are your 3 wins for today?”
3. Add a recurring appointment on your calendar to pop up every Friday mid-morning. Call it Friday Reflection. Add this text to the body of your reminder: What are 3 things going well? What are 3 things to improve?”
4. On the last day of the month, make a full list of everything you care about for the next month. Alphabetize the list. Identify the 3 most important things that you want to accomplish for the month, and put those at the top of the list. Call this list Monthly Results for Month XYZ. (Note – Alphabetizing your list helps you name your list better and sort your list better. It’s hard to refer to something important you have to do if you don’t even have a name for it. If naming the things on your list and sorting them is too much to do, you don’t need to. It’s just an additional tip that helps you get even more effective and more efficient.)
5. On Monday of each week, when you wake up, make a full list of everything you care about accomplishing for the week. Alphabetize the list. Identify the 3 most important things you want to accomplish and add that to the top of the list. (Again, if you don’t want to alphabetize then don’t.)
7. On Fridays, in the morning, do your Friday Reflection. As part of the exercise, check against your weekly outcomes and your monthly outcomes that you want to accomplish. If you aren’t effective for the week, don’t ask “why not,” ask “how to.” Ask how can you bite off better things and how can you make better choices throughout the week. Just focus on little behavior changes, and this will add up over time. You’ll get better and better as you go, as long as you keep learning and changing your approach. That’s the Agile Way.
There are lots of success stories by other people who have used Agile Results. Everybody from presidents of companies to people in the trenches, to doctors and teachers, to teams and leaders, as well as single parents and social workers.
But none of that matters if it’s not your story.
Work on your success story and just start getting better results, right here, right now.
What are the three most important things you really want to accomplish or achieve today?
Categories: Architecture, Programming
### Stuff The Internet Says On Scalability For August 8th, 2014
Hey, it's HighScalability time:
Physicists reveal the scaling behaviour of exotic giant molecules.
• 5 billion: Transistors Intel manufactures each second; 396M: WeChat active users
• Quotable Quotes:
• @BenedictEvans: Every hour or so, Apple ships phones with something around 2x more transistors than were in all the PCs on earth in 1995.
• @robgomes: New client. Had one of their employees tune an ORM-generated query. Reduced CPU by 99.999%, IO by 99.996%. Server now idle.
• @pbailis: As a hardware-oriented systems builder, I'd pay attention to, say, ~100 ns RTTs via on-chip photonic interconnects
• @CompSciFact: "Fancy algorithms are buggier than simple ones, and they're much harder to implement." -- Rob Pike's rule No. 4
• @LusciousPear: I'm probably doing in Google what would have taken 5-8 engineers on AWS.
• C. Michael Holloway, NASA: To a first approximation, we can say that accidents are almost always the result of incorrect estimates of the likelihood of one or more things.
• Stephen O'Grady: More specific to containers specifically, however, is the steady erosion in the importance of the operating system.
• Wait, I thought mobile meant making single purpose apps? Mobile meant tearing down the portal cathedrals built by giants of the past. Then Why aren’t App Constellations working?: The App Constellation strategy works when you have a core resource which can be shared across multiple apps.
• Decentralization: I Want to Believe. The irony is mobile loves centralization, not p2p. Mobile IP addresses change all the time and you can't run a server on a phone. The assumption that people want decentralization has been disproven. Centralized services have won. People just want a service that works. The implementation doesn't matter that much. Good discussion on HackerNews and on Reddit.
• Myth: It takes less money to start a startup these days. Sort of. Why the Structural Changes to the VC Industry Matter: It turns out that, while it is in fact cheaper to get started and enter the market, it also requires more money for the breakout companies to win the market. Ultimately, today’s winners have a chance to be a lot bigger. But winning requires more money for geographic expansion, full-stack support of multiple new disciplines, and product expansion. And these companies have to do all of this while staying private for a much longer period of time; the median for money raised by companies prior to IPO has doubled in the past five years.
Don't miss all that the Internet has to say on Scalability, click below and become eventually consistent with all scalability knowledge (which means this post has many more items to read so please keep on reading)...
Categories: Architecture
### Extending AngularJS services with the Decorate method
Xebia Blog - Fri, 08/08/2014 - 12:00
Many large Angular applications tend to see a lot of repetition - same API endpoint, same method of dealing with and transforming data, etcetera. One technique you can use to at least alleviate that is using AngularJS's decorate method, which allows you to extend, adjust or even fully replace any existing service.
As you'll see in this post, using this allows you to modify and extend the framework you build your app in, which will lead to a cleaner, more legible codebase, written in a more functional style (the what, not the how).
Update 11/8: The follow-up is now live, along with the GitHub repository.
A feature not often used when developing AngularJS applications is the $provide service, which is the primary service used to register components with the$injector. More commonly, a developer would use methods like $provide.service() or$provide.factory to do so, but those are merely utility methods defined in the $provide service and exposed via angular.module(). The main reasons to use$provide over the service() and factory() methods is to configure the service before it's instantiated, for example. While there may be more advanced use-cases for using $provide, I haven't yet encountered them while developing regular applications and I'm sure they won't occur often. One of the methods listed at the very bottom of the$provide documentation is the decorate() method. It doesn't look like much (it's at the bottom, after all), but its documentation hints that it's very powerful:
"A service decorator intercepts the creation of a service, allowing it to override or modify the behaviour of the service. The object returned by the decorator may be the original service, or a new service object which replaces or wraps and delegates to the original service."
Nothing to add there. You can use decorate() to change, add to, or completely replace the behaviour of services without having to edit its code. This can be done on any code not your own - core AngularJS services, but also third-party libraries. It's the equivalent of overriding methods in OO languages or monkey-patching in the more dynamic languages.
“Isn’t that evil?”, I hear you ask. As with every programming-related question, the only correct answer is: it depends. I’m going to give a few practical examples of when I believe using decorate() is appropriate. In a future blog post, I'll expand on this example, showing how relatively simple code can positively influence your entire application architecture.
Here’s a practical example, a neat follow-up on my previous blog about angular promises: decorating $q to add methods to the promise object. The promise API itself defines only one method: then().$q adds a few simple methods to that like catch() and finally(), but for your own application you can add a few more.
If you’ve been working with promises for a little while in your AngularJS application, you’ve probably noticed some operations are pretty common; assigning the promise result to the scope (or any object), logging the result in the console, or calling some other method. Using decorate(), we can add methods to the promise object to simplify those. Here's a bit of code from my previous post; we'll add a method to $q to remove the need for a callback: CustomerService.getCustomer(currentCustomer) .then(CartService.getCart) .then(function(cart) {$scope.cart = cart;
})
.catch($log.error); First, we’ll need to do some boilerplate: we create a function that adds our methods to the promise object, and then we replace all the default promise methods. Note that the decorating function will also apply itself to the given promise.then method again, so that our customisations aren’t lost further down a promise chain: angular.module('ngPromiseDsl', []) .config(function ($provide) {
$provide.decorator('$q', function ($delegate,$location) {
// decorating method
function decoratePromise(promise) {
var then = promise.then;
// Overwrite promise.then. Note that $q's custom methods (.catch and .finally) are implemented by using .then themselves, so they're covered too. promise.then = function (thenFn, errFn, notifyFn) { return decoratePromise(then(thenFn, errFn, notifyFn)); }; return promise; } // wrap and overwrite$q's deferred object methods
var defer = $delegate.defer, when =$delegate.when,
reject = $delegate.reject, all =$delegate.all;
$delegate.defer = function () { var deferred = defer(); decoratePromise(deferred.promise); return deferred; };$delegate.when = function () {
return decoratePromise(when.apply(this, arguments));
};
$delegate.reject = function () { return decoratePromise(reject.apply(this, arguments)); };$delegate.all = function () {
return decoratePromise(all.apply(this, arguments));
};
return $delegate; }); }); With that boilerplate in place, we can now start adding methods. As I mentioned earlier, one of the most common uses of a then() function is to set the result onto the scope (or some other object). This is a fairly trivial operation, and it’s pretty straightforward to add it to the promise object using our decorator, too: function decoratePromise(promise) { var then = promise.then; promise.then = function (thenFn, errFn, notifyFn) { return decoratePromise(then(thenFn, errFn, notifyFn)); }; // assigns the value given to .then on promise resolution to the given object under the given varName promise.thenSet = function (obj, varName) { return promise.then(function (value) { obj[varName] = value; return value; }); }; return promise; } That’s all. Put this .config block in your application's module definition, or create a new module and add a dependency to it, and you can use it throughout your application. Here's the same piece of code, now with our new thenSet method: CustomerService.getCustomer(currentCustomer) .then(CartService.getCart) .thenSet($scope, 'cart')
.catch($log.error); This particular example can be extended in a multitude ways to add useful utilities to promises. In my current project we’ve added a number of methods to the promise object, which allows us to reduce the number of callback definitions in our controllers and thus create cleaner, more legible code. Replacing custom callbacks with named methods allows for a more functional programming style, and allows readers to read and write code as a list of “whats”, instead of “hows” - and it's also fully asynchronous. Extending$q is just the start though: Every angular service can be extended for various purposes - add performance monitoring and logging to $http, set common prefixes or fixed properties on$resource urls or template paths, you name it. Leave a remark in the comments about how you've used decorate() to create a better application.
Stay tuned for an upcoming post where I release a small open source project that extends angular’s promise objects with a number of helpful methods to perform common tasks.
### The Principles of Modern Management
Are your management practices long in the tooth?
I think I was lucky that early on, I worked in environments that shook things up and rattled the cage in pursuit of more customer impact, employee engagement, and better organizational performance.
In one of the environments, a manufacturing plant, the management team flipped the typical pyramid of the management hierarchy upside down to reflect that the management team is there to empower and support the production line.
And when I was on the Microsoft patterns & practices team, we had an interesting mix of venture capitalist type management coupled with some early grandmasters of the Agile movement. More than just Agile teams, we had an Agile management culture that encouraged a customer-connected approach to product development, complete with self-organizing, multi-disciplinary teams, empowered people, a focus on execution excellence, and a fierce focus on being a rapid learning machine.
We thrived on change.
We also had a relentless focus on innovation. Not just in our product, but in our process. If we didn’t innovate in our process, then we got pushed out of market by becoming too slow, too expensive, or by lacking the quality experience that customers have come to expect.
But not everybody knows what a great environment for helping people thrive and do great things for the world, looks like.
While a lot of people in software or in manufacturing have gotten a taste of Agile and Lean practices, there are many more businesses that don’t know what a modern learning machine of people and processes that operate at a higher-level looks like.
Many, many businesses and people are still operating and looking at the world through the lens of old world management principles.
In the book The Future of Management, Gary Hamel walks through the principles upon which modern management is based.
The Principles of Modern Management
Hamel gives us a nice way to frame looking at the modern management principles, by looking at their application, and their intended goal.
Principle Application Goal Standardization Minimize variances from standards around inputs, outputs, and work methods. Cultivate economies of scale, manufacturing efficiency, reliability, and quality. Specialization (of tasks and functions) Group like activities together in modular organizational units. Reduce complexity and accelerate learning. Goal alignment Establish clear objectives through a cascade of subsidiary goals and supporting metrics. Ensure that individual efforts are congruent with top-down goals. Hierarchy Create a pyramid of authority based on a limited span of control. Maintain control over a broad scope of operations. Planning and control Forecast demand, budget resources, and schedule tasks, then track and correct deviations from plan. Establish regularity and predictability in operations; conformance to plans. Extrinsic rewards Provide financial rewards to individuals and teams for achieving specified outcomes. Motivate effort and ensure compliance with policies and standards. What are the Principles Upon Which Your Management Beliefs are Based?
Most people aren’t aware of the principles behind the management beliefs that they practice or preach. But before coming up with new ones, it helps to know what current management thinking is rooted in.
“Have you ever asked yourself, what are the deepest principles upon which your management beliefs are based? Probably not. Few executives, in my experience, have given much thought to the foundational principles that underlie their views on how to organize and manage. In that sense, they are as unaware of their management DNA as they are of their biological DNA. So before we set off in search of new management principles, we need to take a moment to understand the principles that comprise our current management genome, and how those tenets may limit organizational performance.”
A Small Nucleus of Core Principles
It really comes down to a handful of core principles. These principles serve as the backbone for much of today’s management philosophy.
“These practices and processes of modern management have been built around a small nucleus of core principles: standardization, specialization, hierarchy, alignment, planning, and control, and the use of extrinsic rewards to shape human behavior.”
How To Maximize Operational Efficiency and Reliability in Large-Scale Organizations
It’s not by chance that the early management thinkers came to the same conclusions. They were working on the same problems in a similar context. Of course, the challenge now is that the context has changed, and the early management principles are often like fish out of water.
“These principles were elucidated early in the 20th century by a small band of pioneering management thinkers -- individuals like Henri Fayol, Lyndall Urwick, Luther Gullick, and Max Weber. While each of these theorists had a slightly different take on the philosophical foundations of modern management, they all agreed on the principles just enumerated. This concordance is hardly surprising, since they were all focusing on the same problem: how to maximize operational efficiency and reliability in large-scale organizations. Nearly 100 years on, this is still the only problem that modern management is fully competent to address.”
If your management philosophy and guiding principles are nothing more than a set of hand me downs from previous generations, it might be time for a re-think.
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The New Realities that Call for New Organizational and Management Capabilities | 2014-10-22 07:57:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2828361392021179, "perplexity": 2061.5590294087892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507446323.3/warc/CC-MAIN-20141017005726-00248-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://vpcug.net/mean-square/error-mse.html | Home > Mean Square > Error Mse
Error Mse
Contents
This is the formula to calculate the variance of a normally distributed sample: $$\frac{\sum(X - \bar{X}) ^2}{n-1}$$ This is the formula to calculate the mean squared error of observations in a Your point regarding the degree of freedoms also shows that is not quite as obvious and definitely something worth mentioning. –bluenote10 Oct 29 '15 at 11:18 add a comment| 1 Answer As we have seen before, if $X$ and $Y$ are jointly normal random variables with parameters $\mu_X$, $\sigma^2_X$, $\mu_Y$, $\sigma^2_Y$, and $\rho$, then, given $Y=y$, $X$ is normally distributed with \begin{align}%\label{} Is a larger or smaller MSE better?UpdateCancelAnswer Wiki2 Answers Peter Flom, Independent statistical consultant for researchers in behavioral, social and ...Written 134w agoSmaller error is better1.4k Views · View UpvotesRelated QuestionsMore
If so I wanna learn of it. Wird geladen... The mean squared error then reduces to the sum of the two variances. Properties of the Estimation Error: Here, we would like to study the MSE of the conditional expectation. https://en.wikipedia.org/wiki/Mean_squared_error
Mean Square Error Example
Eric NunesWritten 75w agoSmall MSE is better . Wird verarbeitet... Statisticshowto.com Apply for $2000 in Scholarship Money As part of our commitment to education, we're giving away$2000 in scholarships to StatisticsHowTo.com visitors. Step 1:Find the regression line.
Part of the variance of $X$ is explained by the variance in $\hat{X}_M$. Introduction to the Theory of Statistics (3rd ed.). By choosing an estimator that has minimum variance, you also choose an estimator that has minimum mean squared error among all unbiased estimators. What Is Meant By Mean Square Error In the formula for the sample variance, the numerator is a function of a single variable, so you lose just one degree of freedom in the denominator.
However, you are on track in noticing that these are conceptually similar quantities. Find the MMSE estimator of $X$ given $Y$, ($\hat{X}_M$). Anmelden 49 Wird geladen... Mathematical Statistics with Applications (7 ed.).
Not the answer you're looking for? Mse Values MSE is also used in several stepwise regression techniques as part of the determination as to how many predictors from a candidate set to include in a model for a given Its an error , the smaller the better.1.4k Views · View UpvotesView More AnswersRelated QuestionsIn a linear regression model, which unbiased variance does mean squared error approximate?What is the semantic difference Moreover, $X$ and $Y$ are also jointly normal, since for all $a,b \in \mathbb{R}$, we have \begin{align} aX+bY=(a+b)X+bW, \end{align} which is also a normal random variable.
The Mean Square Error (mse) Is
The mean squared error (MSE) of this estimator is defined as \begin{align} E[(X-\hat{X})^2]=E[(X-g(Y))^2]. \end{align} The MMSE estimator of $X$, \begin{align} \hat{X}_{M}=E[X|Y], \end{align} has the lowest MSE among all possible estimators. Mean Square Error Example Pearson's Correlation Coefficient Privacy policy. What Does Mean Squared Error Tell You However, as you can see from the previous expression, bias is also an "average" property; it is defined as an expectation.
Discrete vs. Generated Fri, 14 Oct 2016 05:59:12 GMT by s_ac15 (squid/3.5.20) ANOVA Test: Definition, Types, Examples → Leave a Reply Cancel reply Your email address will not be published. The usual estimator for the mean is the sample average X ¯ = 1 n ∑ i = 1 n X i {\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}} which has an expected E Mean Square Error
Misleading Graphs 10. There are, however, some scenarios where mean squared error can serve as a good approximation to a loss function occurring naturally in an application.[6] Like variance, mean squared error has the Wird verarbeitet... Veröffentlicht am 19.08.2012Overview of mean squared error for an estimator.
Belmont, CA, USA: Thomson Higher Education. Expected Value Of Mse References ^ a b Lehmann, E. It does this by taking the distances from the points to the regression line (these distances are the "errors") and squaring them.
In an analogy to standard deviation, taking the square root of MSE yields the root-mean-square error or root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being
The only difference is that everything is conditioned on $Y=y$. Wird geladen... That is why it is called the minimum mean squared error (MMSE) estimate. Mse Error Code Not the answer you're looking for?
Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the If the data are uncorrelated, then it is reasonable to assume in that instance that the new observation is also not correlated with the data. Swedes and their mean ice cream intake over a year) using chi-square analysis and the significanc...What are the real-world applications of the mean squared error (MSE)?What are the applications of the Transkript Das interaktive Transkript konnte nicht geladen werden.
Vernier Software & Technology Caliper Logo Vernier Software & Technology 13979 SW Millikan Way Beaverton, OR 97005 Phone1-888-837-6437 Fax503-277-2440 [email protected] Resources Next Generation Science Standards Standards Correlations AP Correlations IB Correlations Hot Network Questions Why are there no BGA chips with triangular tessellation of circular pads (a "hexagonal grid")? Nächstes Video MAD and MSE Calculations - Dauer: 8:30 East Tennessee State University 42.494 Aufrufe 8:30 Easy proof that MSE = variance +bias-squared - Dauer: 7:51 Phil Chan 18.987 Aufrufe 7:51 Examples Mean Suppose we have a random sample of size n from a population, X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} .
Further, while the corrected sample variance is the best unbiased estimator (minimum mean square error among unbiased estimators) of variance for Gaussian distributions, if the distribution is not Gaussian then even Sign Up Thank you for viewing the Vernier website. The minimum excess kurtosis is γ 2 = − 2 {\displaystyle \gamma _{2}=-2} ,[a] which is achieved by a Bernoulli distribution with p=1/2 (a coin flip), and the MSE is minimized | 2017-08-18 19:59:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9036126732826233, "perplexity": 1205.4233670382569}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105108.31/warc/CC-MAIN-20170818194744-20170818214744-00543.warc.gz"} |
https://electronics.stackexchange.com/questions/361024/how-to-convert-a-metal-work-table-surface-for-electronic-work/361051 | How to convert a metal work table surface for electronic work?
I have a 23" x 60" metal work table...the kind that restaurants use for prep work. I'm using it as my computer table & electronics workstation.
I want to make the surface non-conductive in the cheapest way.
I've seen cheap neoprene mats and more expensive mats with grounding wires. An expensive example out there: Static Control Work surface mat
What are some cheap alternatives to cover a large area? Can I get something from the hardware store? Rubber shelf liners?
While it is risky, if done properly, can provide an ESD safe place to work. Since you live on the beach near LA, this may be of no importance. Humidity above 60% or so makes ESD protection somewhat redundant. I'd get a sheet of 3 or 6mm plywood for the top (full sheet, not pieces which will let the table show through). An overhang of 2-3 inches on all sides is best, but will require a thicker sheet. The sides of the table top can be covered with heavy duty wide insulation tape.
Make sure no live wires are below the table. A simple stand to hold the power points can be made using wooden batons (1"x1") and fixed to the sides or rear, so all live wires are well above the table top. Do not skimp on safety.
My wooden shelf constructed with 1"x0.5" batons is behind the table and holds all my gear, and a wooden stand with power sockets and wire spools, tape rolls, multimeter probes, is attached to the table side. Makes it all very handy, and safe. I work at a metal table with 3/4" blockboard top.
edit: pics you requested
The curtains are a hazard, but it's simply too dusty in India. Definitely not recommended. [Almost all are brick houses in India, so hopefully only me and my room will burn in case of fire]
The full setup. table is 3'x2', always wish it was at least a foot wider. Rear shelf is 40" wide, 10" deep (fitted inside unused doorway). Bottom steel shelves hold multimeters, magnifying glasses soldering accessories, pliers wire cutters etc.
There is a fuse on the left, and another fuse where this strip attaches to the wall socket. Wiring is very heavy gauge (4 square mm).
Shelves lined with cardboard, but no cardboard under laptop. Keeps it very cool. The batons are 1"x0.5" shorea robusta... extremely strong, and dirt cheap.
As you can see, I do not throw away any packaging... ever :(
two 12W LED bulbs fitted on the top of the near edge of the table, so no shadows under my fingers while working. Also, 2' away from the curtains.
Can there be a fire? yes. Never leave the workplace unattended. Always switch off everything when not in the room. Do not sleep at work.
• Can you show pictures of your setup? Thanks. :) – milesmeow Mar 11 '18 at 19:19
• I actually just want to be able to do simple electronics prototyping...Arduino, Raspberry Pi, etc. on the table. Do I need all of that wood or can I just roll out some sort of material onto the surface? – milesmeow Mar 11 '18 at 19:33
• I added pics, maybe too many. Can you guarantee that stiff wires will not poke holes into any material that you roll out, even when it becomes old and starts to deteriorate? Additionally, you will eventually be spending a fair bit, so better to do it properly from the start to save being frustrated later. You can't do electronics on a shoestring budget. Expect to spend at least ~$500 to get started with basic supplies. – Indraneel Mar 11 '18 at 20:34 • This is what a man cave looks like. I rate it 5 men out of 5 men. – Harry Svensson Mar 11 '18 at 21:52 I've used tempered masonite to cover a work table. You can obtain it from the big box home improvement stores in 4' x 8' sheets. It's available unfinished as well as melamine coated. I prefer the white melamine coating because it's easy to see small surface mount parts and it cleans well. Use an ESD mat as needed when working with vulnerable components. Linoleum has nice anti-static properties and can be put on top of any table. I've done some research on self-healing cutting mats such as this one...because my wife happen to have a couple of these lying around. These are supposed to be static free, too. They're not the cheapest thing out there, but b/c I had some lying around, they cost$0 for me.
These silicone pastry mats seem good too. | 2020-11-24 21:23:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19769735634326935, "perplexity": 3770.648207243658}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141177566.10/warc/CC-MAIN-20201124195123-20201124225123-00554.warc.gz"} |
http://openstudy.com/updates/50999e67e4b085b3a90dca08 | ## xKingx Group Title How do you simplify numbers using the imaginary unit (i)? Ex: sqrt -16 one year ago one year ago
1. AccessDenied Group Title
If you have something in the form $$\sqrt{-a}$$ for some $$a > 0$$, then we can break it up using exponent / radical properties: $$\sqrt{-a} = \sqrt{-1 \times a} = \sqrt{-1} \sqrt{a} = i \sqrt{a}$$
2. xKingx Group Title
so it would be i$\sqrt{16}$?
3. AccessDenied Group Title
$$i \sqrt{16}$$ Yes, although we can go a step further since $$\sqrt{16}$$ is the square root of a perfect square. = $$4i$$
4. xKingx Group Title
Okay thanks.
5. AccessDenied Group Title
You're welcome. :) | 2014-09-15 02:08:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9405553340911865, "perplexity": 2889.5333204938497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657102753.15/warc/CC-MAIN-20140914011142-00045-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://www.veritaz.in/a-million-vfvfn/area-of-a-circle-equation-9dcb23 | Formula, Derivation, Calculate the area, Area using diameter and circumference at BYJUâS. This is illustrated below in brown. Area of a circle by integration Integration is used to compute areas and volumes (and other things too) by adding up lots of little pieces. A circle is a shape with all points at the boundary having the same distance to the centre. Investigate the cases when circle center is on the ⦠Equation of circle- $2x^2+ \lambda xy+2y^2+( \lambda -4)x+6y-5=0$ find area of the circle. This is illustrated below in brown. If a circle touches both the axes, then consider the center of the circle to be $$(r, r)$$, where $$r$$ is the The equation of a circle can be found using the centre and radius. The area of a circle is number of square units inside the circle. Geometrical shapes are fun to play with. The diameter of a circle calculator uses the following equation: Area of a circle = Ï * (d/2) 2 Where: Ï is approximately equal to 3.14. This page looks to give a general run through of how the formula for the area of a circle can be derived. Watch Area of a Circle Equation Definition Proof Examples - WORLD ENTERTAINMENT on Dailymotion Find the equation of the tangent line. Semicircle Definition A semicircle is a half circle. Attempt- For converting the equation from second degree to first degree $\lambda xy=0$. Plug the circumference into the formula. Area is defined as the region occupied by a two-dimensional shape. Circle A is given by the equation (x â 4) 2 + (y + 3) 2 = 29. Standard formula to calculate the area of a circle is: A=Ïr². By finding the area of go shopping with my old papa he is still alive!And i The surface area of circle is 4Ïr 2 .The diameter is the distance across two extreme ends of a circle passing through the center. For the equation of a line, you need a point (you have it) and the lineâs slope. Then, its radius is doubled. Look at the graph below, can you express the equation of the circle in standard form? Explanation: The general equation of a circle is (x â h) 2 + (y â k) 2 = r 2, where (h, k) represents the location of the circle's center, and r represents the length of its radius. The greeks considered the Egyptians as the inventors of geometry. Equation of a Circle: The equation of the circle can be formed using the Center-Radius Formula. You can ask me questions directly at https://www.simor.org/user/1723/saul-remi_hernandez/rooms Equation for the area of a circle is derived with integration And so: All points are the same distance from the center. We suggest to use the same formula to find the area of a circle which uses You might think that means it will have half the perimeter of a circle, but that is not true. Q. Whenever a math problem tells you the circumference C of a circle, you can use this equation to find the radius r.Replace C in the equation with the circumference of the circle in your problem: The following formulas are used for circle calculations. Ï is defined as the ratio of the circumference of a circle to its diameter.. Calculates the area and circumference of a circle given the radius or diameter. Area A = pi x radius x radius A = 3.14 x 1 x 1 A = 3.14 x 1 = 3.14 sq.cm Notice that the calculated area of the circle is same in both the methods. Show Solutions Since the radius of this this circle is 2, and its center is (3,1) , this circle's equation is. The distance between any point of the circle and the centre is called the radius. Derive Area of Circle Formula The Area of Circle formula is: AREA = Ï × radius 2 But how does that come about? Area of a Circle Segment Given the Central Angle Definition: The number of square units it takes to fill a segment of a circle Try this Drag one of the orange dots that define the endpoints of the segment. Also, know how to calculate the segment of a circle ⦠That means a semicircle will have half of the area of a circle. In National 4 Lifeskills Maths solve a problem by calculating the area and perimeter of a composite shape which will also include parts of a circle. Area of a circle = Ï * r 2 Area of a circle diameter. It doesn't matter whether you want to find the Please go through following articles of c programming, if you do not already know about these topics : The area of a trapezoid is equal to half of the product of the height and sum of the bases. Given by the equation of a circle, But that is radius '' away from central! Its diameter shapes when you get a grip over geometry all points the! Equations a circle is number of square units inside the circle and the lineâs slope shifted up five and... By the equation from second degree to first degree $\lambda xy=0$ page to! Is shifted up five units and left by six units Math Tutors available... The portion of a plane bounded by such a curve that is radius away! The ratio of the circle goes back beyond recorded history are available to area of a circle Worksheet ( have... A two-dimensional shape, pi, denoted as Ï ; Ï â 3.14159 and geometery Comment/Request i like cheetos the. \Lambda -4 ) x+6y-5=0 $find area of a circle to its diameter how that...  4 ) 2 + ( y + 3 area of a circle equation and ( 0, ). Often contain a mathematical constant, pi, denoted as Ï ; Ï 3.14159... Have half of the object are the same distance from the center having the same distance to the of. Is the distance across two extreme ends of a circle, But that is ''... By the equation of thr circle if the radius how does that about. 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Thus, the circleâs y-intercepts are ( 0, 3 ) and the centre is called the is... | 2021-10-26 18:19:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8227183222770691, "perplexity": 396.20733391363115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587915.41/warc/CC-MAIN-20211026165817-20211026195817-00346.warc.gz"} |
https://physics.stackexchange.com/questions/149599/if-entropy-of-a-closed-system-decreases-by-a-certain-amount-why-does-not-the-en | # If entropy of a closed system decreases by a certain amount, why does not the entropy of the surroundings increase by the same amount? [duplicate]
As said by the Second law of thermodynamics,
Energy spontaneously disperses from being localized to being spread out if not hindered from doing so.
Now, for systems other than isolated one, and for the surroundings , the net entropy change always is greater than $0$ . $$\Delta{S_{sys}} + \Delta{S_{surr}} > 0$$ . Net entropy of the universe always increases. As my book says
Entropy change of the universe must be positive and for this entropy change of the surroundings must be greater than that of the system when it releases thermal energy.
But, can't net entropy of the universe be $0$ ? That is, can't the entropy change in the system be equal to entropy change of the surroundings?? If not, why???
Can't the entropy change in the system be equal to entropy change of the surroundings?
Yes, it can be in a reversible process. Although most spontaneous processes in nature are irreversible, so that only happens in a minority of cases. One example is the Carnot cycle which is a reversible process with heat exchange between two sources. It is used in refrigerators for instance. You can see a more detailed description in the link, and the graphs are pretty explanatory. Of course, any reversible process is ideal, no one exactly happens in nature. Any thing in nature that looks like the step three would be an example (but see the webpage for details):
isothermal heat rejection: a reversible isothermal compression of the gas, where the surroundings do work on the gas, causing an amount of heat energy and of entropy to flow out of the gas to a low temperature reservoir.
Can't net entropy of the universe be 0 ?
For it to be zero you need all the energy concentrated at a single place, but this will not last much, because as you quoted: Energy spontaneously disperses from being localized to being spread out if not hindered from doing so.
• Thanks! So, the entropy of the universe either doesn't or will increase but not decrease, right? – user36790 Nov 30 '14 at 8:57
• Just can you give me an example of spontaneous reversible reaction where the entropy of the universe remains same? – user36790 Nov 30 '14 at 8:58
• @user36790 sure, I'll do it later (gotta go work now) , the only ones I have in mind now are the ones where there is no change in entropy in either system (such as an adiabatic expansion or contraction). But I want to find an example where the entropy decrease in one and increases on the other. – Wolphram jonny Nov 30 '14 at 12:45
• @user36790 I updated my answer with an example. But I encourage you to read the link for more details. – Wolphram jonny Nov 30 '14 at 14:42 | 2020-02-17 06:18:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6348568201065063, "perplexity": 351.93181138259916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875141749.3/warc/CC-MAIN-20200217055517-20200217085517-00373.warc.gz"} |
https://www.physicsforums.com/threads/definite-integration.180005/ | # Definite integration
1. Aug 10, 2007
### cd246
1. The problem statement, all variables and given/known data
integration 9_0 c/x^3 dx. the answer is (3/8)c.
2. Relevant equations
just distrib. then plug in the #'s
3. The attempt at a solution
=cx^-3
= (cx^-2)/-2=c/-2x^2
I know it has something to do with distribution.
Last edited: Aug 10, 2007
2. Aug 10, 2007
### G01
$$\int^9_0 \frac{c}{x^3}dx$$
If this is the case, I agree with your answer. When I evaluate the integral at the bounds I do not get the answer you stated. I get -c/162 as an answer. Are you sure that is correct?
Last edited: Aug 10, 2007
3. Aug 10, 2007
### cd246
My apologies, it is 2_1 not 9_0
4. Aug 10, 2007
### G01
Ahh if that is the case, then I do get (3/8)c as the answer. Were the bounds your problem, or is there something else?
5. Aug 10, 2007
### cd246
that was pretty much it. did u put x^3 on top? or did you left the x^3 on the bottom? Because i believe that was my problem on this one.
6. Aug 10, 2007
### G01
I think it is good practice to write any fraction of c/x^n as cx^-n. I think it makes many problems look less complicated. So, yes, if this is what you mean, I did "put the x^3 on top."
7. Aug 10, 2007
### cd246
ok i got it now. thanx
8. Aug 10, 2007 | 2017-06-27 10:45:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8550999164581299, "perplexity": 2418.724811655269}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321309.25/warc/CC-MAIN-20170627101436-20170627121436-00348.warc.gz"} |
https://dsp.stackexchange.com/questions/63821/does-a-rhp-zero-imposes-some-limitation-in-the-time-domain | # Does a RHP zero imposes some limitation in the time domain?
I am studying control systems and I have seen that a non-minimum phase zero imposes limitation in bandwidth , which is that the bandwidth has to be lower than a frequency which is half the frequency of the RHP pole.
I have also seen that it imposes in the step response an undershoot, but as far as I know there are no other influences of RHP zero in the time domain.
Does a RHP zero imposes some limitation in the time domain?
## 1 Answer
If a RHP zero imposes limitation on the bandwidth in the frequency domain, it also imposes limitation in the time domain. This is due to time-frequency duality.
For example, if you want to have a really fast step response, a slow RHP zero will prevent you from reaching that goal.
I really recommand Murray's book for more information http://www.cds.caltech.edu/~murray/amwiki/index.php/Second_Edition
• Thanks for answering. So the limitations regard only the speed of the response and the undershoot? Thanks again. – J.D. Feb 11 at 22:27
• It is a pretty important limitation, don't you think? It could prevent you from reaching your specifications. – Ben Feb 11 at 22:28
• I have found this paper that talks about a time domain integral when it is present a RHP zero: sciencedirect.com/science/article/pii/S1474667015385694. It makes a differentiation about when the zero is stable or when it is unstable. So, it is possible to compute a time until whihìch the step response has better behaviour? Sorry, maybe is not a good question, but I want to understand well. Thanks. – J.D. Feb 12 at 14:59
• For example, if I have a RHP zero at $s=1$, does it change something if I plot the step response starting at $t=0$ or at $t=5$? Thanks in advance. – J.D. Feb 12 at 15:33
• If it is time invariant, then if I shift the input , I obtain the same output shifted by the same quantity, so it means that I have the same step response but shifted. So, even if I have a RHP zero, it impses limitation on the badwidth, and so it imposes limitation on the speed and adds an undershoot in the step response. If it is time invariant it should have the same characteristics but shifted in time. I thisnk that this solves my doubts. Thanks. – J.D. Feb 12 at 16:02 | 2020-09-24 02:43:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7191669940948486, "perplexity": 588.3775610161499}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400213006.47/warc/CC-MAIN-20200924002749-20200924032749-00672.warc.gz"} |
https://homework.cpm.org/category/CON_FOUND/textbook/ac/chapter/7/lesson/7.2.1/problem/7-61 | Home > AC > Chapter 7 > Lesson 7.2.1 > Problem7-61
7-61.
On graph paper, graph the line that goes through the points $(-6,3)$ and $(-3,-1)$7-61 HW eTool (Desmos)
1. What is the slope of the line?
Create a slope triangle to find the rise over the run.
$y=-\frac{4}{3}x -5$
2. What is the $y$-intercept?
On the graph, where does the line cross the $y$-axis?
3. Find the equation of the line.
Use the table in the eTool below to graph the line.
Click the link at right for the full version of the eTool: 7-61 HW eTool (Desmos) | 2022-06-30 22:44:38 | {"extraction_info": {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4904993176460266, "perplexity": 1314.2605567694266}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103915196.47/warc/CC-MAIN-20220630213820-20220701003820-00775.warc.gz"} |
http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/IntegralClosure/html/_ic__P__Ideal.html | icPIdeal -- compute the integral closure in prime characteristic of a principal ideal
Synopsis
• Usage:
icPIdeal (a, D, N)
• Inputs:
• a, an element in R
• D, a non-zerodivisor of R that is in the conductor
• N, the number of steps in icFracP to compute the integral closure of R, by using the conductor element D
• Outputs:
• the integral closure of the ideal (a).
Description
The main input is an element a which generates a principal ideal whose integral closure we are seeking. The other two input elements, a non-zerodivisor conductor element D and the number of steps N are the pieces of information obtained from icFracP(R, Verbosity => true). (See the Singh--Swanson paper, An algorithm for computing the integral closure, Remark 1.4.)
i1 : R=ZZ/3[u,v,x,y]/ideal(u*x^2-v*y^2); i2 : icFracP(R, Verbosity => 1) Number of steps: 3, Conductor Element: x^2 u*x o2 = {1, ---} y o2 : List i3 : icPIdeal(x, x^2, 3) o3 = ideal (x, v*y) o3 : Ideal of R
Caveat
The interface to this algorithm will likely change eventually | 2023-02-01 13:25:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8655019998550415, "perplexity": 2900.161604337626}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499934.48/warc/CC-MAIN-20230201112816-20230201142816-00030.warc.gz"} |
https://lookformedical.com/en/sites/dimyristoylphosphatidylcholine | The absolute accuracy of measured order parameters is largely affected by the accuracy of the prefactor connecting the measured splittings to the order parameters (Eqs. (1) and (2)). When considering the change of an order parameter with a changing external condition (hydration, ion concentration, etc.), the prefactor can be considered to be unchanging. Therefore, the change in a given order parameter, when measured by the same people with the same technique and equipment, can be determined in a much higher accuracy than the absolute value of the order parameter. We refer to this as the relative accuracy. It is determined by the accuracy of the splitting measurement. Especially the very high spectral resolution of the ²H-measurements allows the measurement of very small order parameter changes. Let us exemplify this with a classical experiment by Akutsu et al., where the effect of different ions on the headgroup $$\alpha$$- and $$\beta$$-quadrupolar splittings was measured, see Fig. 1 ...
in Journal of Protein Chemistry (1994), 13(1), 77-88. Peptides corresponding to lipid binding domains of Apo B-100 were synthesized, purified, and incubated with dimyristoylphosphatidylcholine (DMPC) liposomes. The secondary structure of the apo B-100 ... [more ▼]. Peptides corresponding to lipid binding domains of Apo B-100 were synthesized, purified, and incubated with dimyristoylphosphatidylcholine (DMPC) liposomes. The secondary structure of the apo B-100 peptide-lipid complexes was evaluated by attenuated total reflection Fourier transform infrared spectroscopy (ATR-FTIR). Those peptides belonging to the hydrophobic core domain of apo B-100 when associated with phospholipids were rich in beta sheet structure; a predominant alpha helical conformation was shown to be associated with one peptide located in a surface region of apo B-100. IR dichroic spectra revealed, in the case of the core peptides, that the beta sheet component is the only oriented structure with respect to the ...
[140 Pages Report] Check for Discount on Global and Chinese 1,2-dimyristoyl-sn-glycero-3-phospho-ethanolamine (CAS 998-07-2) Industry, 2017 Market Research Report report by Prof Research. The Global and Chinese 1,2-dimyristoyl-sn-glycero-3-...
en] The structure, composition, and physico-chemical properties of lipid-protein complexes generated between dimyristoylphosphatidylcholine (DPMC) and the CNBr fragments of human apoA-I were studied. The fragments were separated by high performance liquid chromatography and purified on a reversed-phase column. The complexes with DMPC were isolated on a Superose column; their dimensions were obtained by gradient gel electrophoresis and by electron microscopy. The secondary structure of the protein in the complexes was studied both by circular dichroism and by attenuated total reflection infrared spectroscopy. The fragments 1 and 4 of apoA-I, containing, respectively, two and three amphipathic helices, recombined with the phospholipid to generate discoidal particles with sizes similar to that of apoA-I- and apoA-II-DMPC complexes. The infrared measurements indicated that in all complexes the apolipoprotein helical segments were oriented parallel to the phospholipid acyl chains and that the protein ...
Page contains details about PLGA nanocapsules coated with phosphatidylcholine bilayer . It has composition images, properties, Characterization methods, synthesis, applications and reference articles : nano.nature.com
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Ras proteins regulate signal transduction processes that control cell growth and proliferation. Their disregulation is a common cause of human tumors. Atomic level structural and dynamical information in a membrane environment is crucial for understanding signaling specificity among Ras isoforms and for the design of selective anti-cancer agents. Here, the structure of the full-length H-Ras protein in complex with a 1,2-dimyristoylglycero-3-phosphocholine (DMPC) bilayer obtained from modeling and all-atom explicit solvent molecular dynamics simulations, as well as experimental validation of the main results, are presented. We find that, in addition to the lipid anchor, H-Ras membrane binding involves direct interaction of residues in the catalytic domain with DMPC phosphates. Two modes of binding (possibly modulated by GTP/GDP exchange) differing in the orientation and bilayer contact of the soluble domain as well as in the participation of the flexible linker in membrane binding are proposed. ...
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Physically--Im so sore, and I become really achy super easily without doing much. Its very annoying. There are a few other symptoms as well that well keep discreet for now. Its beginning to get in the way of general daily things, so adjustments are being made. This was not at all the case with Braska. Its weird and frustrating, to be honest, but as long as Kinlees ok through it all, well deal. M carries her for me whenever possible, and I just have to plan time on my feet in small doses. I thought it might be related to some projects from a couple weeks ago, but its hung around too long to be due to that. Well see if the OB has any thoughts tomorrow. Im not really worried, but I would like to feel better for the next few months ...
How we perceive and manage stress is important to our overall health. When stress becomes too great, the damage may be irreversible.
55 matching references were found. Zhang C.; Zhang H.; Wang L.; Gao H.; Guo X.N.; Yao H.Y., Improvement of texture properties and flavor of frozen dough by carrot (Daucus carota) antifreeze protein supplementation, J. Agric. Food Chem., 2007, 55, 23, 9620-9626, https://doi.org/10.1021/jf0717034 . [all data] Johnston, H.S.; Paige, M.; Yao, F., Oxygen absorption cross sections in the Herzberg continuum and between 206 and 327 K, J. Geophys. Res., 1984, 89, 11661-11665. [all data] Yao, F.; Liu, W.; Qiu, Q., GC-MS Analysis of volatile oil from Atractylodes chinensis, Zhongyao Cai., 2001, 24, 9, 643-645. [all data] Nagano, H.; Yao, H.; Ema, K., Dynamic heat capacity at the gel to liquid-crystalline phase transition in large unilamellar vesicles of dimyristoylphosphatidylcholine in the ultralow frequency region, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interd- iscip. Top., 1995, 51, 3363-7. [all data] Hatta, I.; Yao, H.; Kato, R.; Maesono, A., Development of ac calorimetric method for ...
Slow Motion Grimentz 2013 LEquipe CAP - Vidéo tournée lors du camp dété de Grimentz 2013 organisé par lEquipe CAP : Camp Animation Partage. Cette vidéo retrace bien lactivité au cours des camps : montagne magnifique, sport, ambiance, délires etc... 80 campeurs ont trouvés cles idées pour ...
Orientational order and vesicle shape. . Biblioteca virtual para leer y descargar libros, documentos, trabajos y tesis universitarias en PDF. Material universiario, documentación y tareas realizadas por universitarios en nuestra biblioteca. Para descargar gratis y para leer online.
Two methods of measuring the misorientation of domains in oriented multilamellar stacks of lipid bilayers superficially appeared to give different values for the mosaic spread. It is first shown that the traditional rocking method and a newer ring method give the same value of the mosaic spread when the two
As a mother of a child diagnosed with this condition it is my responsibility to slowly build Nicks sensory endurance. Introducing events and new experiences slowly, building each time in intensity so that he can finally begin to function and relate better to our fast-moving world. This is a necessity if he is ever going to be able to live on his own, go to school or secure a job ...
Recently Mom took a spill in front of her house and I watched her go down -almost in slow motion. I was at least 15 feet away from her, so it wasnt possible to run to catch her somehow, but just watching her fall like that was VERY disturbing to me. Okay, now Im pregnant, so everything is a BIGGER deal than it might have actually been, but it was absolutely terrifying. I had the quick vision of mom going down and not coming back up. The fall was quite bad, and Mom was really banged up -her eyes were so bad, she couldnt tell if the huge cut on her arm was deep enough to go to the ER… It was the next day they found this tumor ...
Recently Mom took a spill in front of her house and I watched her go down -almost in slow motion. I was at least 15 feet away from her, so it wasnt possible to run to catch her somehow, but just watching her fall like that was VERY disturbing to me. Okay, now Im pregnant, so everything is a BIGGER deal than it might have actually been, but it was absolutely terrifying. I had the quick vision of mom going down and not coming back up. The fall was quite bad, and Mom was really banged up -her eyes were so bad, she couldnt tell if the huge cut on her arm was deep enough to go to the ER… It was the next day they found this tumor ...
WK 2,5 - 3 D SL. 56.503.8355.0. 50. WK 2,5 - 3 D SL-NGN. 56.503.8455.0. 50. WK 2,5 - 3 D SL-PGN. 56.503.8555.0. 50. 6 mm / 79 mm / 69 mm. 7 mm. E M D .. ...
Deuterium nuclear magnetic resonance (2H-NMR) was used to investigate the structure and dynamics of the sn-2 hydrocarbon chain of semi-synthetical choline and ethanolamine plasmalogen in bilayers containing 0, 30, and 50 mol% cholesterol. The deuterium NMR spectra of the choline plasmalogen yielded well-resolved quadrupolar splittings which could be assigned to the corresponding hydrocarbon chain deuterons. The sn-2 acyl chain was found to adopt a similar conformation as observed in the corresponding diacyl phospholipid, however, the flexibility at the level of the C-2 methylene segment of the plasmalogen was increased. Deuterium NMR spectra of bilayers composed of the ethanolamine plasmalogen yielded quadrupolar splittings of the C-2 segment much larger than those of the corresponding diacyl lipids, suggesting that the sn-2 chain is oriented perpendicular to the membrane surface at all segments. Cholesterol increased the ordering of the choline plasmalogen acyl chain to the same extent as in ...
A pulsed EPR saturation-recovery method has been used to measure spin-lattice relaxation times (T 1 ) for aminoxyl spin labels in cardiolipin bilayers, with and without cytochrome c. The relaxation time for each spin label was determined at various positions in the bilayer, which included the bilayer surface and three positions within the hydrophobia interior in the membrane, as well as a position close to the glycerol backbone of phospholipids in the bilayer. A dynamic profile for the hydrocarbon chains in bilayers was found in agreement with results from other techniques. On addition of cytochrome c to cardiolipin bilayers, changes in the spin-lattice relaxation time and order parameter were observed around the unsaturated region in the cardiolipin acyl chains. Supported by complementary lineshape analysis, these effects were interpreted as changes in the molecular dynamics around the double bonds in the acyl chains of cardiolipin bilayers, induced upon the binding of
The results of a computer program written to study rotational dynamics by following the reorientation of a carbon-deuterium (C-D) bond vector in a deuteriated molecule are described. Three-dimensional plots are constructed to show how deuterium nuclear magnetic resonance splittings vary as a function of two rotational parameters. It is suggested that these diagrams provide insight into the rotational motions of the molecule. © 1987 American Chemical Society.
We investigated the partitioning behavior of gramicidin A′ in various binary phospholipid mixtures in which gel and liquid crystalline phase coexistence had been induced. The quenching of the...
Page contains details about streptavidin-modified DNA T pores incorporated into biotinylated PC small unilamellar vesicles . It has composition images, properties, Characterization methods, synthesis, applications and reference articles : nano.nature.com
Its bath time for these cute pups....in super slow motion. Subscribe to The Pet Collective: http://full.sc/HbM62v Facebook: http://www.facebook.com/thepetco...
Download this Businesswoman Jumping In Air Doing Splits Slow Motion video now. And search more of the webs best library of royalty-free stock video footage from iStock.
Finally a program that really works capturing/recording video on my pc. BUT , whatever I do,the video are in slow motion.. why? why? why?...I cant figure this out...I've been using hours and...
Director Bob Purman used a Phantom camera at 1,000 fps to capture these expressive canines in action for a dog food commercial. The director was initially charged with shooting two spots, a Catch and a Jump execution. The director says: The Catch spot was to be a series of shots of dogs looking with anticipation as a piece of dog food is flying through the air towards them. The above video may play off the video below of dogs in slow motion: Via Creativity Online
Well, we dun dun it - fired a tracer round into ballistic gelatin and captured it in slow motion video for the world to see. We fired 18 of the 20 rounds in the box (commercially-available repacked M856) to check for trace function and all rounds lit up brightly and quickly out of the muzzle.…
Download this Hd Slow Motion Mountain Stream Crashing Against The Rocks video now. And search more of the webs best library of royalty-free stock video footage from iStock.
In Dr. Pimple Poppers new Instagram video, she extracts an arm cyst with her punch biopsy tool. Then, she squeezes it all out in super slow motion.
Human Leukocyte Membrane Tissue Lysate (Adult Membrane Normal). Tested Reactivity: Hu. Validated: WB, IP. Backed by our 100% Guarantee.
Recently, there has been a surge of interest in mesogens exhibiting the twist-bend nematic (N-TB) phase that is shown to be chiral even though formed by effectively achiral molecules. Although it now seems to be clear that the N-TB phase in the bulk is formed by degenerate domains having opposite handedness, the presence of a supramolecular heliconical structure proposed in the Dozov model has been contradicted by the Hoffmann et al. model in which the heliconical arrangement is replaced by a polar nematic phase. The evidence in support of this is that the quadrupolar splitting tensor measured in various experiments is uniaxial and not biaxial as expected for the twist-bend nematic structure. In this debate, among other evidence, the molecular translational diffusion, and its magnitude with respect to that in the nematic phase above the N-TB phase, has also been invoked to eliminate or to confirm one model or the other. We attempt to resolve this issue by reporting the first measurements of the ...
This page includes the following topics and synonyms: E/M Nature of Presenting Problem, E/M Minimal Problem, E/M Self Limited or Minor Problem, E/M Low Severity Problem, E/M Moderate Severity Problem, E/M High Severity Problem.
This figure shows that no potential difference can be established across a pure lipid bilayer that separates solutions containing different ionic concentrations.
The invention relates to liquid-crystalline compounds of the formula I ##STR1##in which R.sup.1, R.sup.2, A.sup.1, A.sup.2, A.sup.3, A.sup.4, Z.sup.1, Z.sup.2, a, b and c are as defined in claim 1, and to liquid-crystalline media comprising at least compound of the formula I and to electro-optical displays containing a liquid-crystalline medium of this type.
Many of the diseases we associate with ageing - such as heart disease, cancer, dementia and stroke - may be the result of lifelong, poorly managed stress.
Stock video footage Egg falling into flour against white background, Slow motion. 00:00:09 . From $99. Royalty free. Download now on Pond5 ||| Stock video footage Baby peeks out from behind a blue board advertising a product. Green screen. Slow motion. 00:00:13 4k. From$80. Royalty free. Download now on Pond5 ||| | 2020-12-05 17:28:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3484213054180145, "perplexity": 5646.006609042927}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141748276.94/warc/CC-MAIN-20201205165649-20201205195649-00388.warc.gz"} |
https://defelement.com/elements/direct-serendipity.html | an encyclopedia of finite element definitions
# Direct serendipity
Orders $$1\leqslant k$$ Reference elements quadrilateral Categories Scalar-valued elements
## Implementations
Symfem "direct serendipity"↓ Show Symfem examples ↓ UFL "Sdirect"↓ Show UFL examples ↓
## Examples
order 1
order 2
order 3
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = x y - x - y + 1$$
This DOF is associated with vertex 0 of the reference element.
$$\displaystyle \phi_{1} = x \left(1 - y\right)$$
This DOF is associated with vertex 1 of the reference element.
$$\displaystyle \phi_{2} = y \left(1 - x\right)$$
This DOF is associated with vertex 2 of the reference element.
$$\displaystyle \phi_{3} = x y$$
This DOF is associated with vertex 3 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = x y - x - y + 1$$
This DOF is associated with vertex 0 of the reference element.
$$\displaystyle \phi_{1} = x \left(1 - y\right)$$
This DOF is associated with vertex 1 of the reference element.
$$\displaystyle \phi_{2} = y \left(1 - x\right)$$
This DOF is associated with vertex 2 of the reference element.
$$\displaystyle \phi_{3} = x y$$
This DOF is associated with vertex 3 of the reference element.
$$\displaystyle \phi_{4} = y \left(1 - y\right)$$
This DOF is associated with edge 1 of the reference element.
$$\displaystyle \phi_{5} = \frac{4 x y \left(y - 1\right)}{x + 1}$$
This DOF is associated with edge 2 of the reference element.
$$\displaystyle \phi_{6} = x \left(1 - x\right)$$
This DOF is associated with edge 0 of the reference element.
$$\displaystyle \phi_{7} = \frac{4 x y \left(x - 1\right)}{y + 1}$$
This DOF is associated with edge 3 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = x y - x - y + 1$$
This DOF is associated with vertex 0 of the reference element.
$$\displaystyle \phi_{1} = x \left(1 - y\right)$$
This DOF is associated with vertex 1 of the reference element.
$$\displaystyle \phi_{2} = y \left(1 - x\right)$$
This DOF is associated with vertex 2 of the reference element.
$$\displaystyle \phi_{3} = x y$$
This DOF is associated with vertex 3 of the reference element.
$$\displaystyle \phi_{4} = y \left(1 - y\right)$$
This DOF is associated with edge 1 of the reference element.
$$\displaystyle \phi_{5} = y \left(- y^{2} - y + 2\right)$$
This DOF is associated with edge 1 of the reference element.
$$\displaystyle \phi_{6} = y \left(x y - x - 2 y + 2\right)$$
This DOF is associated with edge 2 of the reference element.
$$\displaystyle \phi_{7} = \frac{4 x y \left(y^{2} + y - 2\right)}{x + 1}$$
This DOF is associated with edge 2 of the reference element.
$$\displaystyle \phi_{8} = x \left(1 - x\right)$$
This DOF is associated with edge 0 of the reference element.
$$\displaystyle \phi_{9} = x \left(- x^{2} - x + 2\right)$$
This DOF is associated with edge 0 of the reference element.
$$\displaystyle \phi_{10} = x \left(x y - 2 x - y + 2\right)$$
This DOF is associated with edge 3 of the reference element.
$$\displaystyle \phi_{11} = \frac{4 x y \left(x^{2} + x - 2\right)}{y + 1}$$
This DOF is associated with edge 3 of the reference element.
## References
• Arbogast, T. and Tao, Z. Direct serendipity and mixed finite elements on convex quadrilaterals, arΧiv: 1809.02192, 2018. [BibTeX]
## DefElement stats
Element added 27 May 2021 Element last updated 13 June 2021 | 2021-10-23 02:02:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8017244338989258, "perplexity": 2725.0083572604685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585537.28/warc/CC-MAIN-20211023002852-20211023032852-00030.warc.gz"} |
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## Microstrip patch antenna design
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This chapter describes patch design cases for different applications. The use of microstrip patch antennas in wireless communication systems provides several advantages like low profile, low cost, and ease of fabrication. Moreover, microstrip patch antennas can provide a possible solution for fifth generation (5G) antenna design. Different antenna shapes can be fabricated using the rectangular patch as an initial step. This type of antenna design requires the simultaneous optimization of several different geometrical parameters. An optimization algorithm or techniques is a suitable approach for solving this problem. In the literature, there are several examples of patch antenna design and optimization using different evolutionary algorithms (BAs) [1-5]. These approaches include genetic algorithms [6], particle swarm optimization (PSO) [7-9], differential evolution (DB) [10-13], teaching-learning-based optimization (TLBO)[14], Jaya [15], and a hybrid Jaya-GWO algorithm [16].
Chapter Contents:
• 4.1 E-shaped patch antenna design
• 4.1.1 Frequency-independent design procedure
• 4.1.2 Dual-band 5G antenna design
• 4.2 Half E-shaped patch antenna design
• 4.2.1 Wireless LAN antenna design
• 4.2.2 5G antenna design
• 4.3 Arbitrary-shaped patch antenna design
• References
Preview this chapter:
Microstrip patch antenna design, Page 1 of 2
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This is a required field | 2021-09-27 11:16:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24001309275627136, "perplexity": 7293.55820511867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058415.93/warc/CC-MAIN-20210927090448-20210927120448-00709.warc.gz"} |
http://www.r-bloggers.com/a-slightly-different-introduction-to-r-part-v-plotting-and-simulating-linear-models/ | A slightly different introduction to R, part V: plotting and simulating linear models
November 11, 2013
By
(This article was first published on There is grandeur in this view of life » R, and kindly contributed to R-bloggers)
In the last episode (which was quite some time ago) we looked into comparisons of means with linear models. This time, let’s visualise some linear models with ggplot2, and practice another useful R skill, namely how to simulate data from known models. While doing this, we’ll learn some more about the layered structure of a ggplot2 plot, and some useful thing about the lm function.
11. Using points, lines and error bars to show predictions from linear models
Return to the model of comb gnome mass at time zero. We’ve already plotted the coefficient estimates, but let us just look at them with the coef() function. Here the intercept term is the mean for green comb gnomes subjected to the control treatment. The ‘grouppink’ and ‘treatmentpixies’ coefficients are the mean differences of pink comb gnomes and comb gnomes exposed to pixies from this baseline condition. This way of assigning coefficients is called dummy coding and is the default in R.
model <- lm(mass0 ~ group + treatment, data)
coef(model)[1]
(Intercept) grouppink treatmentpixies
141.56771 -49.75414 23.52428
The estimate for a pink comb gnome with pixies is:
coef(model)[1] + coef(model)[2] + coef(model)[3]
There are alternative codings (”contrasts”) that you can use. A common one in Anova is to use the intercept as the grand mean and the coefficients as deviations from the mean. (So that the coefficients for different levels of the same factor sum to zero.) We can get this setting in R by changing the contrasts option, and then rerun the model. However, whether the coefficients are easily interpretable or not, they still lead to the same means, and we can always calculate the values of the combinations of levels that interest us.
Instead of typing in the formulas ourself as above, we can get predictions from the model with the predict( ) function. We need a data frame of the new values to predict, which in this case means one row for each combination of the levels of group and treatment. Since we have too levels each there are only for of them, but in general we can use the expand.grid( ) function to generate all possible factor levels. We’ll then get the predictions and their confidence intervals, and bundle everything together to one handy data frame.
levels <- expand.grid(group=c("green", "pink"), treatment=c("control", "pixies"))
predictions <- predict(model, levels, interval="confidence")
predicted.data <- cbind(levels, predictions)
group treatment fit lwr upr
1 green control 141.56771 125.82527 157.3101
2 pink control 91.81357 76.48329 107.1439
3 green pixies 165.09199 149.34955 180.8344
4 pink pixies 115.33785 98.93425 131.7414
Now that we have these intervals in a data frame we can plot them just like we would any other values. Back in part II, we put several categorical variables into the same plot by colouring the points. Now, let’s introduce nice feature of ggplot2: making small multiples with faceting. qplot( ) takes facets argument which is a formula where the left hand side, before the tilde (‘~’), will be used to split the plot vertically, and the right hand side will split the plot horizontally. In this case, we split horizontally, each panel representing one level of the treatment variable. Also, we use a new geometry: pointrange, which draws a point with bars above and below it and is quite suitable for the intervals we’ve got.
qplot(x=treatment, facets=~group,
y=fit, ymax=upr, ymin=lwr
geom="pointrange", data=predicted.data)
That’s good, but combining the predictions from the model and the actual data in the same plot would be nice. In ggplot2, every plot is an object that can be saved away to a variable. Then we can use the addition operator to add layers to the plot. Let’s make a jittered dotplot like the above and then add a layer with the pointrange geometry displaying confidence intervals. The scatter of the data points around the confidence intervals reminds us that there is quite a bit of residual variance. The coefficient of determination, as seen in the summary earlier, was about 0.25.
qplot(x=treatment, y=mass0, facets=~group, geom="jitter", data=data) +
geom_pointrange(aes(y=fit, ymax=upr, ymin=lwr), colour="red", data=predicted.data)
In the above, we make use of ggplot2′s more advanced syntax for specifying plots. The addition operator adds layers. The first layer can be set up with qplot(), but the following layers are made with their respective functions. Mapping from variables to features of the plot, called aesthetics, have to be put inside the aes() function. This might look a bit weird in the beginning, but it has its internal logic — all this is described in Hadley Wickham’s ggplot2 book.
We should probably try a regression line as well. The abline geometry allows us to plot a line with given intercept and slope, i.e. the coefficients of a simple regression. Let us simplify a little and look at the mass at time zero and the log-transformed mass at time 50 in only the green group. We make a linear model that uses the same slope for both treatments and a treatment-specific intercept. (Exercise for the reader: look at the coefficients with coef( ) and verify that I’ve pulled out the intercepts and slope correctly.) Finally, we plot the points with qplot and add the lines one layer at the time.
green.data <- subset(data, group=="green")
model.green <- lm(log(mass50) ~ mass0 + treatment, green.data)
intercept.control <- coef(model.green)[1]
intercept.pixies <- coef(model.green)[1]+coef(model.green)[3]
qplot(x=mass0, y=log(mass50), colour=treatment, data=green.data) +
geom_abline(intercept=intercept.pixies, slope=coef(model.green)[2]) +
geom_abline(intercept=intercept.control, slope=coef(model.green)[2])
12. Using pseudorandom numbers for sanity checking
There is a short step from playing with regression functions that we’ve fitted, like we did above, to making up hypothetical regression functions and simulating data from them. This type of fake-data simulation is very useful to for testing how designs and estimation procedures behave and check things like the control of false positive rate and the power to accurately estimate a known model.
The model will be the simplest possible: a single categorical predictor with only two levels and normally distributed equal error variance, i.e. a t-test. There is a formula for the power of the t-test and an R function, power.t.test( ), that calculates it for us without the need for simulation. However, a nice thing about R is that we can pretty easily replace the t-test with more complex procedures. Any model fitting process that you can program in R can be bundled into a function and applied to pseudorandom simulated data. In the next episode we will go into how to make functions and apply them repeatedly.
Let us start out with a no effect model: 50 observations in two groups drawn from the same distribution. We use the mean and variance of the green control group. This first part just sets up the variables:
mu <- mean(subset(data, group=="green" & treatment=="control")$mass0) sigma <- sd(subset(data, group=="green" & treatment=="control")$mass0)
treatment <- c(rep(1, 50), rep(0, 50))
The rnorm( ) function generates numbers from a normal distribution with specified mean and standard deviation. Apart from drawing numbers from it, R can of course pull out various table values, and it knows other distributions as well. Look at the documentation in ?distributions. Finally we perform a t-test. Most of the time, it should not show a significant effect, but sometimes it will.
sim.null <- rnorm(100, mu, sigma)
t.test(sim.null ~ treatment)$p.value We can use the replicate( ) function to evaluate an expression multiple times. We put the simulation and t-test together into one expression, rinse and repeat. Finally, we check how many of the 1000 replicates gave a p-value below 0.05. Of course, it will be approximately 5% of them. sim.p <- replicate(1000, t.test(rnorm(100, mu, sigma) ~ treatment)$p.value)
length(which(sim.p < 0.05))/1000
[1] 0.047
Let us add an effect! Say we’re interested in an effect that we expect to be approximately half the difference between the green and pink comb gnomes:
d <- mean(subset(data, group=="green" & treatment=="control")$mass0) - mean(subset(data, group=="pink" & treatment=="control")$mass0)
sim.p.effect <- replicate(1000, t.test(treatment * d/2 +
rnorm(100, mu, sigma) ~ treatment)$p.value) length(which(sim.p.effect < 0.05))/1000 [1] 0.737 We see that with 50 individuals in each group and this effect size we will detect a significant difference about 75% of the time. This is the power of the test. If you are able to find nice and trustworthy prior information about the kind of effect sizes and variances you expect to find in a study, design analysis allows you to calculate for instance how big a sample you need to have good power. Simulation can also give you an idea of how badly a statistical procedure will break if the assumptions don’t hold. We can try to simulate a situation where the variances of the two groups differs quite a bit. sim.unequal <- replicate(1000, t.test(c(rnorm(50, mu, sigma), rnorm(50, mu, 2*sigma)) ~ treatment)$p.value)
length(which(sim.unequal < 0.05))/1000
[1] 0.043
sim.unequal.effect <- replicate(1000, t.test(c(rnorm(50, mu+d/2, sigma),
rnorm(50, mu, 2*sigma)) ~ treatment)\$p.value)
length(which(sim.unequal.effect < 0.05))/1000
[1] 0.373
In conclusion, the significance is still under control, but the power has dropped to about 40%. I hope that has given a small taste of how simulation can help with figuring out what is going on in our favourite statistical procedures. Have fun!
Postat i:computer stuff, data analysis, english Tagged: ggplot2, lm, R, simulation, tutorial | 2014-10-31 09:10:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6169472932815552, "perplexity": 1677.1065695726468}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637899290.52/warc/CC-MAIN-20141030025819-00209-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://www.moderatenerd.com/2012/03/29/the-expected-value-of-a-mega-millions-ticket/ | # The expected value of a Mega Millions ticket
As I type this post, the Mega Millions jackpot projects to be $500,000,000 (annuity value – after-tax cash value is more like$220,000,000). A single ticket costs $1. Is it worth it? By “worth it,” I mean that the expected value of a ticket is greater than zero. The expected value of a random variable $$W$$ which represents the event of winning the Mega Millions lottery with the ticket is equal to, roughly, (Probability of winning)*(Value of winning) + (Probability of losing)*(value of losing). In shorter math notation, this looks like: $$E[W]=P[W]*V+(1-P[W])*C$$ Here, $$E[W]$$ is the expected value of $$W$$, $$P[W]$$ is the probability of $$W$$, and therefore $$1-P[W]$$ is the probability of NOT $$W$$. Also, $$V$$ is the value of winning – in this case,$500,000,000 or $220,000,000, and $$C$$ is the value of losing: -$1.
What is this probability? The players are asked to pick 5 balls from a first set of 56, then 1 ball from a second set of 46, the first set and second sets being independent. The number of possibilities is $$46 \binom {56}{5} = 175,711,536$$. The probability $$P[W]$$ is then the reciprocal of this number.
Although this probability is indeed small, it would seem that a sufficiently large jackpot would make buying a ticket worth it. And, indeed, even with the after-tax cash value, the expected value is greater than zero:
$$E[W]=\displaystyle \frac{220,000,000}{175,711,536} – \displaystyle \frac{175,711,535}{175,711,536} \approx 0.25$$.
This could be taken as an encouragement to purchase Mega Millions tickets; after all, you can expect about \$0.25 of value for each ticket you buy. What a bargain! With a return on investment like that, why don’t investment houses just put their client’s money into Mega Millions tickets?
The reason is that return is wrong and based on a false assumption: no matter how many tickets got the correct number, the payout is the same. In fact, the payout is split evenly among the winners. This will of course lower the expected value of a ticket. But by how much?
Consider a population of $$N$$ tickets, where yours is among the $$k$$ winners. (Mazeltov.) That means there are $$k-1$$ winners besides yourself among the rest of the remaining $$N-1$$ tickets. What is the probability that there are $$k-1$$ winners among $$N-1$$ tickets?
We consider each event of a ticket being a winning ticket being independent from any other ticket being a winning or losing ticket. (This is clearly the case.) The probability of a particular set of $$k-1$$ winners among the $$N-1$$ tickets is then $$p^{k-1} (1-p)^{(N-1)-(k-1)}$$. That is, when there are $$k-1$$ winners, there are also $$(N-1) – (k-1) = N – k$$ losers.
That said, there are also many different ways to arrange the other $$k-1$$ winners among the remaining $$N-1$$ tickets. To be precise, there are $$\binom {N-1}{k-1}$$ different arrangements of winners and losers, on top of you as a winner. (Not literally!) I imagine that my friends literate in statistics recognize the random variable $$W$$ as having a binomial distribution.
Note that, when there are $$k$$ winners, the jackpot is split evenly among the winners; that is, each winner gets $$V/k$$, and not $$V$$. The expected value of a ticket is then a sum over all possible values of $$k$$:
$$E[W]= \displaystyle V \sum_{k=1}^{N} \frac{1}{k} \binom{N-1}{k-1} p^{k} (1-p)^{N-k} – C (1-p)$$,
where $$p = P[W]$$ as defined above. Note that there is an extra factor of $$p$$ coming from your own ticket.
The above sum is very difficult to evaluate numerically for $$N \approx 100,000,000$$, and approximations to normal or Poisson distributions do not apply. However, we can observe that we are comparing the value of the above sum to $$p$$. We can see that the value of the sum is less than $$p$$ because of the fact, well-known from binomial distributions, that
$$\displaystyle \sum_{k=1}^{N} \binom{N-1}{k-1} p^{k} (1-p)^{N-k} = p$$.
Therefore,
$$\displaystyle \sum_{k=1}^{N} \frac{1}{k} \binom{N-1}{k-1} p^{k} (1-p)^{N-k} < p[/latex], because the factor [latex,size="-2"]\frac{1}{k} < 1[/latex]. In fact, a rough calculation (it turns out you can neglect anything beyond the 6th term) gives a value of about 0.374 for the above sum. Again, compare this to 1 using the crude (and incorrect) estimate. So, in multiplying the first term in the first expected value equation by about 0.374, we get an expected value of about -0.53. That is, rather than gaining 25 cents, you should expect to lose about 47 cents for every dollar you spend on a Mega Millions ticket. Then again, we're all kind of suckers for that galactically small chance we could win, and I don't blame any of you for throwing away a few pennies chasing the dream. Update (3/31/12): I need to correct an assertion I made, and some numbers. The conclusion stands - in fact, in light of what has happened over the past 24 hours, the conclusion is even more stark. First of all, I posted some incorrect numbers that I have since corrected. For the above values of the number of tickets in circulation and probability of winning: $\displaystyle \sum_{k=1}^{N} \frac{1}{k} \binom{N-1}{k-1} p^{k-1} (1-p)^{N-k} \approx 0.763$. This is the reduction factor on the jackpot, not 0.374 as I published before. The expected value of a ticket is then about -0.05. Again, still a loser. That said, it turns out that 100,000,000 tickets was a gross underestimate; rather, 1,500,000,000 tickets were sold! For this number, the reduction factor is about 0.117, which gives an expected value of a ticket as -0.85, an even worse value than my incorrect previous numbers show. Second, although I was right in asserting that the multiple winners are not governed by a Poisson distribution (a great explanation is here), I was incorrect in ignoring the Law of Rare Events, which states that, under a certain limiting behavior, the binomial sum approaches the Poisson sum that results from assuming a Poisson distribution. Further, the limiting behavior need not be rigorously enforced: a large enough sample and a small enough probability does the trick. The mathematical statement of the Law of Rare Events in this context is [latex]\displaystyle \sum_{k=1}^{N} \frac{1}{k} \binom{N-1}{k-1} p^{k-1} (1-p)^{N-k} \approx \sum_{k=1}^{\infty} \frac{(N p)^{k-1}}{k!} e^{-N p} = \frac{1 – e^{-N p}}{N p}$$.
It turns out that this is a very good approximation out to many decimal places. So, a very simple formula for the expected value of a Mega Millions ticket is
$$\displaystyle E[W] = V \frac{1 – e^{-N p}}{N} – C (1-p)$$.
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Math nerd in his early 40’s who seems to have an opinion about everything and an inability to keep it to himself. | 2017-12-12 18:07:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8108290433883667, "perplexity": 392.59147185289436}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948517845.16/warc/CC-MAIN-20171212173259-20171212193259-00263.warc.gz"} |
http://crookedtimber.org/2006/12/19/reviewing-the-stern-review-again/ | # Reviewing the Stern Review, again
by on December 19, 2006
Following the publication of this piece in the NY Times, I’ve had a string of email exchanges with Hal Varian, cc:ing Brad DeLong in the role of interested onlooker. I was surprised by the NY Times article since it included both a correct statement of the way in which Stern treats discounting and income redistribution (roughly speaking a 1 per cent change in income has the same value whenever it is incurred and whoever receives it) with a lot of statements that were either misleading or downright wrong, implying that the near-zero rate of pure time preference in the Stern Review implied a near-zero discount rate for cash flows.
Since Varian is one of the brightest and most technically careful people in the economics profession, I was unsurprised by the correct statement, but very surprised to see errors I’d already refuted when put forward by Arnold Kling, Bjorn Lomborg, Megan McArdle and others. Email revealed that the main problems arose from editorial attempts to ‘simplify’ things for readers, but we still have a lot of disagreements about the justifiability or otherwise of inherent discounting.
In any case, all this has spurred me on to produce my long-promised review of Stern on discounting, at least in draft form. Read, enjoy and criticise.
1
Chris Edmond 12.19.06 at 10:34 am
John,
A couple of technical grumbles. The consumption Euler equation formula on page 4 is:
r = \rho + \eta \times g
and you refer to \eta as measuring the “elasticity of substitution for consumption”. But with your notation in this equation, the elasticity of substitution for consumption is the *reciprocal* \eta^{-1}. With your notation, consumers want a perfectly flat consumption path when \eta = \infty (an IES of zero) and are indifferent to the temporal pattern of consumption when \eta = 0 (an IES of \infty). Of course in the log utility case these are equal as you rightly point out. I’m being pedantic because a lot of people already seem to be confused about these things and a typo like this doesn’t help.
Also: why do you give the intuition for the elasticity of substitution in terms of “income”: we are talking about preferences for consumption and my willingness to trade consumption over time is quite different from my willingness to trade income because I can do things with my income other than consume. (Or, technically if you prefer: in a dynamic model the envelope theorem does not give the result that my marginal utility of consumption equals the derivative of my value function with respect to income; instead in a dynamic model the marginal utility of consumption each period equals the derivative of my value function with respect to my net asset position that period). Since a lot of people already seem to be confused by this discussion, I don’t think it helps to be careless with the intuition you’re providing.
Chris
2
aaron_m 12.19.06 at 10:58 am
”one per cent of income now has the same value as one per cent of income at any time in the future.”
Am I missing something? I must be because this just seems to be a huge mistake
You say
“Under the projections used in the Stern Review, average world income in 2100 is estimated at about $US 100 000. Using eta =1, a sacrifice of$70 per person (1 per cent of income) today would be justified if (and only if) it increased the income of our great-grandchildren in 2100 by at least $1 000. If this trade-off appears reasonable, then a value of eta =1 is appropriate.” You highlight that following the logic of eta=1,$1000 from the more rich to produce a $70 gain for less rich has the same moral value as visa versa. In other words if it is reasonable for the less rich to give$70 to the more rich, then the more rich out also to give a $1000 for the sake of raising the income of the less rich 70$. But the more rich should be willing to give much more than 1% of their income to raise the income of the less rich by 1% because of declining marginal utility.
When do transfers from the poor to the rich make sense?
Obviously we do not say that one percent of my income (lets say $20 000/year) has the same value to me as one percent of the income of an extremely poor person. Lets call our poor person Dave, he makes 200/year. One percent of income for both Dave and I represents a little less than four days income, but the loose of four days of income means that I can’t buy a new Ipod while the loose of fours days of income for Dave means that he will starve to death. So it is not the case that if Dave can raise my income by 1% by giving 2 dollars to the Help the Ipodless Programme (HIP) that he does or very nearly does have a moral obligation to do so. Obviously because of the starvation consequences of 2 dollars Dave will never be morally obliged to give this money to HIP. In fact the only way that it makes sense on utilitarian grounds for Dave to make a transfer to me is if, for example, his giving 1 dollar (i.e. 0.5% of his income) will prevent me from loosing a little less than 90 percent (or something like that) of my income. Obviously the value of 1% of income for Dave and I are not the same. So why these utility values become equal when the poorer person is asked to make a transfer to a richer person in the future. 3 aaron_m 12.19.06 at 11:19 am Oops forgot to include that the$70 is one percent of the current average world income, which is $7000. 4 aaron_m 12.19.06 at 12:12 pm and sorry about all the typos 5 radek 12.19.06 at 5:23 pm That’s a really nice write up John. And the Ramsey reference you’re looking for is 1928, Economic Journal, Vol. 38, “A Mathematical Theory of Saving”. 6 Tom T. 12.20.06 at 10:43 am I don’t understand why the appropriate measure of intertemporal tradeoffs is exclusively the risk-free bond rate? Any investment is an intertemporal tradeoff, and people as a whole (or society, or the market, if you prefer) invest their money in a variety of vehicles, including some percentage in higher-risk items such as junk bonds or equities. Money that is spent on (or income that is foregone due to) abatement of global warming would presumably be invested in a mixed basket of investments at varying rates of return, so why does your analysis assume that only the risk-free bond rate matters? There’s a typo on page 5. This sentence is missing a word (perhaps “decrease”): “a policy that made income (not the growth rate of income!) by one percentage point from 2000 to 2050….” 7 Tom T. 12.20.06 at 10:46 am Stern’s figure that you cite, of$100,000 average income in 2100, is that assuming that we have invested in global-warming abatement in the meantime, or that we have not?
8
Hal Varian 12.20.06 at 11:14 am
I disagree that there are “errors” in my NY Times piece. The two passages that are in dispute are the following:
“is it really ethical to transfer wealth from someone making $7,000 a year to someone making$94,000 a year?”
You claim that this “suggests a one-for-one transfer, not taking $1 from the poor person and giving around$14 to the rich person.” However, just a few sentences before I made it clear that I was thinking of percentage transfers.
(However, I agree that I could have reminded readers of that point.)
The other passage that you claim is an error is
the statement “a dollar a year over a million years is a million dollars.” This simply indicates that summing a small sum (of anything) over a long time gives you a big number.
I don’t think that either of these passages could remotely be called mistakes.
But coming back to my critique of the Stern Review, the important point is the one I make at the end: he should have done a sensitivity analysis. This would have made the report much more useful.
I’m happy to see that he subsequently did some sensitivity analysis in a follow up piece, which is definitely a step in the right direction.
9
Matt Kuzma 12.20.06 at 11:28 am
This isn’t a critique as much as an edit. In the last paragraph before the subsection Expected Utility, the first sentence is a fragment:
“The main focus of discussion of the Stern review has been the way in which future costs and benefits.”
10
conchis 12.20.06 at 8:38 pm
I’m more of a prioritarian than a utilitarian – which, if I understand things correctly, means that (given equivalent assumptions about the shape of utlity functions) I should be willing to assign an eta greater than 1, because I’m less inclined to transfer utility from people in the (relatively poor) present, to people in the (relatively richer) future.
Does it strike anyone else as interesting that the a greater preference for redistribution should lead us to be less enthusiastic about attempts to mitigate global warming? And that this seems to be precisely the opposite of the way people’s opinions correlate in practice?
(Although cutting against this somewhat are considerations of hedonic adaptation and reference effects, which suggest that, even at significantly higher incomes, people in the future aren’t likely to be that much better of in utility terms, and so transfers of utility to them are more likely to be justified at the margins.)
11
Michael Sullivan 12.20.06 at 8:58 pm
This is a good writeup john, and I finally get what eta is supposed to mean, although I haven’t yet reasoned through why the consumption equation would hold in the model.
This comment will tackle the moral question about deciding on eta as a utilitarian (or semi-utilitarian).
I see two problems with basic utility theory for these questions. The first is that real utility functions don’t have the nice properties that a function based on eta would expect.
The second is that moral questions have another component, and that is compulsion or social pressure. To me social pressure is a significant cost, and compulsion is a very large cost. If I am willing to give up my $10 for some utility maximizing exercise, not because I believe it is best, but only because of social pressure, then the cost to me is really more than the$10. In the case of compulsion (taxes, theft) the cost is *much* greater than $10. What this means is that there is a spread between the transfers a poor person could morally be asked to make for a rich person and those a rich person could morally be asked (or compelled) to make for a poor person. In the real world, I would have a lot of difficulty if asked to give up$1000 so that some poor person could have $70, when I could find a poor person on my own and give them$100, leaving me with $900 of the$1000. Or I could give the $1000 to some efficient charitable organization and expect somewhere between$800 and $900 to end up in the hands of those who are no better off than the guy who would have ended up with$70. Only if I were in a universe where there were no other ways to get my money to the person earning only $7000/year than to accept this mephistophelean exchange which increases global utility by only a tiny fraction would I consider social pressure to do so even remotely justified. Meanwhile I consider social pressure on the poor person to transfer wealth to the richer unjustified in all cases. Why? Because if some poor soul giving up$70 has the potential to earn me $1000, I ought to be able to profitably offer her$100 to do so and get nearly all the benefit while enriching her by $30 instead of taking away her money. Again, in the world this is these horribly exchanges are the only possible option to increase global utility, we might consider this exchange worthy, but I still think there’s a middle area where the social pressure would not be moral on either side, because the cost of the social pressure is greater than the utility benefit of the exchange. When you consider compulsion, that middle area becomes very large. I cannot easily conscience compelling anyone to give up money or resources to those who are richer, even if doing so increases global utility significantly. If there is a utility gain, then there is clearly a large nominal money gain as well, and the transaction should go off when the richer person is willing to pay for it and not otherwise. Similarly, I find it hard to swallow compelling anyone to give up resources even to those who are poorer, if those poor are well above subsistence level. The costs of compulsion are high enough that I regret imposing them for any but a large benefit, and the marginal utility of persons well above subsistence is not improved greatly enough by income in my estimation to justify this. Therefore I see no paradox here. The utility function is just talking about money and consumption, all else being equal. It’s not considering “everything else”, and everything else is not at all equal when there is a social obligation, or even more so, a legal obligation for transfer of wealth in one case and not in another. And the problem with using utilitarian considerations on a societal level is that some sort of obligation (usually legal) becomes necessary to enforce any change in the status quo. As I’ve long maintained, the ends do justify the means, but only if one has accounted for *all* of the ends, which often include a lot of unintended consequences of the means choice. 12 aaron_m 12.21.06 at 5:41 am M Sullivan 1. None of your alternative scenarios for wealth transfers entail a criticism. All utilitarians think providing a poor person 100 dollars at a cost of 100 dollars is better than providing 70 dollars at a cost of 1000 dollars. You change the choice situation to reject the moral demands of the example, but changing the choice situation simply is avoiding the moral dilemma at hand and not at criticism at all. What is at issue here is a situation where we have a choice to transfer wealth to the future or not to do this. And this means we need to spend$70 for $1000 in the 2100 or impose a cost of$1000 on the future so that we can keep our $70. 2. You claim that the negative utility from coercion almost always outweighs the benefits. This claim is not defended and I doubt that many would be convinced if you tried. Ultimately to defend your coercion as hugely negative for utility claim you would need to argue that utility is better maximised in conditions of real anarchy (i.e. not state coercion for property rights but not for public goods anarchy, or ordered and thus socially coercive non-state community anarchy). 3. Although there is a lot to criticise about the way a utilitarian view on justice deals with intergenerational distributive justice we should be aware that the normative claim that “a sacrifice of$70 per person (1 per cent of income) today would be justified if (and only if) it increased the income of our great-grandchildren,” by one percent (in 2100 their income will be $100 000) is MUCH MUCH MUCH more demanding than what utilitarianism would require of us for intergenerational distributive justice. Following the view that 1 percent of anybody’s income now or in the future is equal to 1 percent of anybody else’s income now or in the future regardless of how rich they are we DRAMTICALLY weaken the demands for transfers from the rich to the poor and DRAMTICALLY increases the demands from the less rich today to the more rich tomorrow. I do not know of any serious utilitarian that would accept the view that 1% of income or consumption is of equal utility value to a rich person as it is to a poor person. 13 John Quiggin 12.21.06 at 8:28 am ‘“a dollar a year over a million years is a million dollars.” This simply indicates that summing a small sum (of anything) over a long time gives you a big number.’ It seems to me that this example can only mislead readers into thinking that Stern discounts cash flows in this way, when in fact Stern’s procedure values an infinite stream of$1 payments at about $50. As a general point about adding up it’s true of all discounting procedures that a million payments, each with present value$1 will be valued at $1 million. Adding up present values is what discounting does. That’s true whether you’re adding up dollars (as in the example), marginal increments to utility (as in Stern) or time-weighted increments to utility (as in Nordhaus). So I can’t see that the example, correctly interpreted, tells us anything about Stern. Incorrectly interpreted, it suggests that Stern is using a zero discount rate, which is wrong. 14 John Quiggin 12.21.06 at 8:33 am “Does it strike anyone else as interesting that the a greater preference for redistribution should lead us to be less enthusiastic about attempts to mitigate global warming? And that this seems to be precisely the opposite of the way people’s opinions correlate in practice?” It has struck me. The real divide is, as Brad DeLong has pointed out, between those who take the ethical imperatives seriously and those whose real viewpoint is “What I have, I hold”. 15 lemuel pitkin 12.21.06 at 12:35 pm 14- Good question. Good answer. 16 Hal Varian 12.21.06 at 2:57 pm Here’s the first part of the paragraph. “Given these assumptions it is easy to see where the large numbers come from. Unchecked global warming will certainly make future generations worse off to some degree. If we add up these losses over all time using a zero social discount rate, we get a large sum.” This simply makes the point that an undiscounted sum of a small number (of utils, dollars or whatever) will be large if the time period is large. Suppose I had said: “Let us imagine that the cost of global warming was very small — equivalent in utility to$1 per year. Summed over a million years, this would be a million dollars.”
Would you have a problem with that?
17
Hal Varian 12.21.06 at 3:09 pm
I went back and look at the column again and it seems very clear that I was talking about discounting welfare, not discounting consumption.
I don’t see how one could possibly think otherwise, when reading the entire document in context.
Examples:
the “social rate of time discount,” the rate used to compare the well-being of future generations to the well-being of those alive
today….
…chooses to weigh all generations’ welfare almost equally
Unchecked global warming will certainly make future generations worse off to some degree. If we add up these losses over all time…
So, should the social discount rate be 0.1 percent, as Sir Nicholas Stern, who led the study, would have it, or 3 percent as Mr. Nordhaus
prefers? There is no definitive answer to this question because it is inherently an ethical judgment that requires comparing the well-being
of different people: those alive today and those alive in 50 or 100 years.
appropriate policy toward global warming depends heavily on how one weighs the costs and benefits it imposes on different generations.
18
John Quiggin 12.21.06 at 4:18 pm
The term “social rate of time discount” is used in general, and by the Stern Review in particular, to refer to the rate at which the value of future consumption (not the social value of utility) declines. What you are describing is the pure rate of time preference.
To quote the IPCC Third Report , whose approach is followed in the Stern Review
The ethical approach applies the so-called social rate of time discount, which is the sum of the rate of pure time-preference and the rate of increase of welfare derived from higher per capita incomes in the future.
(emphasis added). So, the claim that Stern uses a social discount rate of 0.1 is wrong. The examples tend, in my view, to imply that this incorrect statement is the one being discussed.
As you say, if the example had been expressed in utility terms, it would have been more helpful.
19
Michael E. Sullivan 12.22.06 at 3:45 pm
12:
your point 1 is fair for the consideration of global warming. In effect we are forced on a choice. Either we sacrifice $70 to save them$1000, or we are in essence asking them to sacrifice $1000 in order to save us$70. And we are effectively compelling them to do so, so the compulsion issue doesn’t really apply.
I’m thinking this stuff through, and was considering all kinds of potential redistribution, not just that of remediation/non-remediation of climate change.
2. We can have many differences about what the cost of compulsion is, but surely you will admit that there is some cost. People tend to want the things they have. Compelling them to give them up costs them utility over and above the lack of the thing. Compelling people to give things up in the absence of very good arguments and some kind of bought-into due process costs *enormous* utility.
On further thought, it turns out that the cost of compulsion is largely irrelevant in terms of climate change, unless one considers the current generation to have an inalienable right to pour CO into the air that overrides the rights of future generations. If you don’t, then it is just as accurate to say that by failing to reduce emissions, we are compelling future generations to give us a subsidy, and to do so at these very inefficient rates that we would decry if demanded of us to support the current poor. So compulsion is irrelevant to the uutility calculations of global warming question.
3. Where do I find this information on what the utilitarian academy considers the appropriate value of eta for intergenerational transfers? You are making claims about what “utilitarianism” requires, but I didn’t know there was a generally agreed definition. It’s true that log utility (eta =1) is considered a very risk-loving posture, but I’d like to know exactly what utility function you think I should be using instead and who endorses it.
Comments on this entry are closed. | 2019-08-22 18:00:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6151198148727417, "perplexity": 1270.6532517524754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317339.12/warc/CC-MAIN-20190822172901-20190822194901-00412.warc.gz"} |
https://projecteuclid.org/euclid.aos/1176345689 | ## The Annals of Statistics
### Asymptotic Lognormality of $P$-Values
#### Abstract
Sufficient conditions for asymptotic lognormality of exact and approximate, unconditional and conditional $P$-values are established. It is pointed out that the mean, which is half the Bahadur slope, and the standard deviation of the asymptotic distribution of the log transformed $P$-value together, but not the mean alone, permit approximation of both the level and power of the test. This provides a method of discriminating between tests that have Bahadur efficiency one. The asymptotic distributions of the log transformed $P$-values of the common one- and two-sample tests for location are derived and compared.
#### Article information
Source
Ann. Statist., Volume 10, Number 1 (1982), 44-64.
Dates
First available in Project Euclid: 12 April 2007
Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345689
Digital Object Identifier
doi:10.1214/aos/1176345689
Mathematical Reviews number (MathSciNet)
MR642718
Zentralblatt MATH identifier
0484.62038
JSTOR
Lambert, Diane; Hall, W. J. Asymptotic Lognormality of $P$-Values. Ann. Statist. 10 (1982), no. 1, 44--64. doi:10.1214/aos/1176345689. https://projecteuclid.org/euclid.aos/1176345689
• See Correction: D. Lambert, W. J. Hall. Corrections: Asymptotic Lognormality of $P$-Values. Ann. Statist., Volume 11, Number 1 (1983), 348--348. | 2020-02-20 13:44:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41653406620025635, "perplexity": 3109.9742384202564}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144979.91/warc/CC-MAIN-20200220131529-20200220161529-00106.warc.gz"} |
https://peeterjoot.wordpress.com/tag/stress-tensor/ | • 346,727
# Posts Tagged ‘stress tensor’
## Putting the stress tensor (and traction vector) into explicit vector form.
Posted by peeterjoot on April 8, 2012
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Motivation.
Exersize 6.1 from [1] is to show that the traction vector can be written in vector form (a rather curious thing to have to say) as
\begin{aligned}\mathbf{t} = -p \hat{\mathbf{n}} + \mu ( 2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla})\mathbf{u} + \hat{\mathbf{n}} \times (\boldsymbol{\nabla} \times \mathbf{u})).\end{aligned} \hspace{\stretch{1}}(1.1)
Note that the text uses a wedge symbol for the cross product, and I’ve switched to standard notation. I’ve done so because the use of a Geometric-Algebra wedge product also can be used to express this relationship, in which case we would write
\begin{aligned}\mathbf{t} = -p \hat{\mathbf{n}} + \mu ( 2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla}) \mathbf{u} + (\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{n}}).\end{aligned} \hspace{\stretch{1}}(1.2)
In either case we have
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{n}}=\hat{\mathbf{n}} \times (\boldsymbol{\nabla} \times \mathbf{u})=\boldsymbol{\nabla}' (\hat{\mathbf{n}} \cdot \mathbf{u}') - (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla}) \mathbf{u}\end{aligned} \hspace{\stretch{1}}(1.3)
(where the primes indicate the scope of the gradient, showing here that we are operating only on $\mathbf{u}$, and not $\hat{\mathbf{n}}$).
After computing this, lets also compute the stress tensor in cylindrical and spherical coordinates (a portion of that is also problem 6.10), something that this allows us to do fairly easily without having to deal with the second order terms that we encountered doing this by computing the difference of squared displacements.
We’ll work primarily with just the strain tensor portion of the traction vector expressions above, calculating
\begin{aligned}2 {\mathbf{e}}_{\hat{\mathbf{n}}}=2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla})\mathbf{u} + \hat{\mathbf{n}} \times (\boldsymbol{\nabla} \times \mathbf{u})=2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla})\mathbf{u} + (\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{n}}.\end{aligned} \hspace{\stretch{1}}(1.4)
We’ll see that this gives us a nice way to interpret these tensor relationships. The interpretation was less clear when we computed this from the second order difference method, but here we see that we are just looking at the components of the force in each of the respective directions, dependent on which way our normal is specified.
# Verifying the relationship.
Let’s start with the the plain old cross product version
\begin{aligned}(\hat{\mathbf{n}} \times (\boldsymbol{\nabla} \times \mathbf{u}) + 2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla}) \mathbf{u})_i&=n_a (\boldsymbol{\nabla} \times \mathbf{u})_b \epsilon_{a b i} + 2 n_a \partial_a u_i \\ &=n_a \partial_r u_s \epsilon_{r s b} \epsilon_{a b i} + 2 n_a \partial_a u_i \\ &=n_a \partial_r u_s \delta_{ia}^{[rs]} + 2 n_a \partial_a u_i \\ &=n_a ( \partial_i u_a -\partial_a u_i ) + 2 n_a \partial_a u_i \\ &=n_a \partial_i u_a + n_a \partial_a u_i \\ &=n_a (\partial_i u_a + \partial_a u_i) \\ &=\sigma_{i a } n_a\end{aligned}
We can also put the double cross product in wedge product form
\begin{aligned}\hat{\mathbf{n}} \times (\boldsymbol{\nabla} \times \mathbf{u})&=-I \hat{\mathbf{n}} \wedge (\boldsymbol{\nabla} \times \mathbf{u}) \\ &=-\frac{I}{2}\left(\hat{\mathbf{n}} (\boldsymbol{\nabla} \times \mathbf{u})- (\boldsymbol{\nabla} \times \mathbf{u}) \hat{\mathbf{n}}\right) \\ &=-\frac{I}{2}\left(-I \hat{\mathbf{n}} (\boldsymbol{\nabla} \wedge \mathbf{u})+ I (\boldsymbol{\nabla} \wedge \mathbf{u}) \hat{\mathbf{n}}\right) \\ &=-\frac{I^2}{2}\left(- \hat{\mathbf{n}} (\boldsymbol{\nabla} \wedge \mathbf{u})+ (\boldsymbol{\nabla} \wedge \mathbf{u}) \hat{\mathbf{n}}\right) \\ &=(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{n}}\end{aligned}
Equivalently (and easier) we can just expand the dot product of the wedge and the vector using the relationship
\begin{aligned}\mathbf{a} \cdot (\mathbf{c} \wedge \mathbf{d} \wedge \mathbf{e} \wedge \cdots )=(\mathbf{a} \cdot \mathbf{c}) (\mathbf{d} \wedge \mathbf{e} \wedge \cdots ) - (\mathbf{a} \cdot \mathbf{d}) (\mathbf{c} \wedge \mathbf{e} \wedge \cdots ) +\end{aligned} \hspace{\stretch{1}}(2.5)
so we find
\begin{aligned}((\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{n}} + 2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla}) \mathbf{u})_i&=(\boldsymbol{\nabla}' (\mathbf{u}' \cdot \hat{\mathbf{n}})-(\hat{\mathbf{n}} \cdot \boldsymbol{\nabla}) \mathbf{u}+ 2 (\hat{\mathbf{n}} \cdot \boldsymbol{\nabla}) \mathbf{u})_i \\ &=\partial_i u_a n_a+n_a \partial_a u_i \\ &=\sigma_{ia} n_a.\end{aligned}
# Cylindrical strain tensor.
Let’s now compute the strain tensor (and implicitly the traction vector) in cylindrical coordinates.
Our gradient in cylindrical coordinates is the familiar
\begin{aligned}\boldsymbol{\nabla} = \hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \hat{\boldsymbol{\phi}} \frac{1}{{r }}\frac{\partial {}}{\partial {\phi}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}},\end{aligned} \hspace{\stretch{1}}(3.6)
and our cylindrical velocity is
\begin{aligned}\mathbf{u} = \hat{\mathbf{r}} u_r + \hat{\boldsymbol{\phi}} u_\phi + \hat{\mathbf{z}} u_z.\end{aligned} \hspace{\stretch{1}}(3.7)
Our curl is then
\begin{aligned}\boldsymbol{\nabla} \wedge \mathbf{u}&=\left(\hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \hat{\boldsymbol{\phi}} \frac{1}{{r }}\frac{\partial {}}{\partial {\phi}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}}\right)\wedge\left(\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\phi}} u_\phi + \hat{\mathbf{z}} u_z\right) \\ &=\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\phi}}\left(\partial_r u_\phi -\frac{1}{{r}} \partial_\phi u_r\right)+\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{z}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)+\hat{\mathbf{z}} \wedge \hat{\mathbf{r}}\left(\partial_z u_r - \partial_r u_z\right)+\frac{1}{{r}} \hat{\boldsymbol{\phi}} \wedge \left((\partial_\phi \hat{\mathbf{r}}) u_r+(\partial_\phi \hat{\boldsymbol{\phi}}) u_\phi\right)\end{aligned}
Since $\partial_\phi \hat{\mathbf{r}} = \hat{\boldsymbol{\theta}}$ and $\partial_\phi \hat{\boldsymbol{\phi}} = -\hat{\mathbf{r}}$, we have only one cross term and our curl is
\begin{aligned}\boldsymbol{\nabla} \wedge \mathbf{u}=\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\phi}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)+\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{z}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)+\hat{\mathbf{z}} \wedge \hat{\mathbf{r}}\left(\partial_z u_r - \partial_r u_z\right).\end{aligned} \hspace{\stretch{1}}(3.8)
We can now move on to compute the directional derivatives and complete the strain calculation in cylindrical coordinates. Let’s consider this computation of the stress for normals in each direction in term.
## With $\hat{\mathbf{n}} = \hat{\mathbf{r}}$.
Our directional derivative component for a $\hat{\mathbf{r}}$ normal direction doesn’t have any cross terms
\begin{aligned}2 (\hat{\mathbf{r}} \cdot \boldsymbol{\nabla}) \mathbf{u}&=2 \partial_r\left(\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\phi}} u_\phi + \hat{\mathbf{z}} u_z\right) \\ &=2\left(\hat{\mathbf{r}} \partial_r u_r + \hat{\boldsymbol{\phi}} \partial_r u_\phi + \hat{\mathbf{z}} \partial_r u_z\right).\end{aligned}
Projecting our curl bivector onto the $\hat{\mathbf{r}}$ direction we have
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{r}}&=(\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\phi}}) \cdot \hat{\mathbf{r}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)+(\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{z}}) \cdot \hat{\mathbf{r}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)+(\hat{\mathbf{z}} \wedge \hat{\mathbf{r}}) \cdot \hat{\mathbf{r}}\left(\partial_z u_r - \partial_r u_z\right) \\ &=-\hat{\boldsymbol{\phi}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)+\hat{\mathbf{z}}\left(\partial_z u_r - \partial_r u_z\right).\end{aligned}
Putting things together we have
\begin{aligned}2 \mathbf{e}_{\hat{\mathbf{r}}}&=2\left(\hat{\mathbf{r}} \partial_r u_r + \hat{\boldsymbol{\phi}} \partial_r u_\phi + \hat{\mathbf{z}} \partial_r u_z\right)-\hat{\boldsymbol{\phi}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)+\hat{\mathbf{z}}\left(\partial_z u_r - \partial_r u_z\right) \\ &=\hat{\mathbf{r}}\left(2 \partial_r u_r\right)+\hat{\boldsymbol{\phi}}\left(2 \partial_r u_\phi-\partial_r u_\phi+\frac{1}{{r}} \partial_\phi u_r- \frac{u_\phi}{r}\right)+\hat{\mathbf{z}}\left(2 \partial_r u_z+\partial_z u_r - \partial_r u_z\right).\end{aligned}
For our stress tensor
\begin{aligned}\boldsymbol{\sigma}_{\hat{\mathbf{r}}} = - p \hat{\mathbf{r}} + 2 \mu e_{\hat{\mathbf{r}}},\end{aligned} \hspace{\stretch{1}}(3.9)
we can now read off our components by taking dot products to yield
\begin{subequations}
\begin{aligned}\sigma_{rr}=-p + 2 \mu \frac{\partial {u_r}}{\partial {r}}\end{aligned} \hspace{\stretch{1}}(3.10a)
\begin{aligned}\sigma_{r \phi}=\mu \left( \frac{\partial {u_\phi}}{\partial {r}}+\frac{1}{{r}} \frac{\partial {u_r}}{\partial {\phi}}- \frac{u_\phi}{r}\right)\end{aligned} \hspace{\stretch{1}}(3.10b)
\begin{aligned}\sigma_{r z}=\mu \left( \frac{\partial {u_z}}{\partial {r}}+\frac{\partial {u_r}}{\partial {z}}\right).\end{aligned} \hspace{\stretch{1}}(3.10c)
\end{subequations}
## With $\hat{\mathbf{n}} = \hat{\boldsymbol{\phi}}$.
Our directional derivative component for a $\hat{\boldsymbol{\phi}}$ normal direction will have some cross terms since both $\hat{\mathbf{r}}$ and $\hat{\boldsymbol{\phi}}$ are functions of $\phi$
\begin{aligned}2 (\hat{\boldsymbol{\phi}} \cdot \boldsymbol{\nabla}) \mathbf{u}&=\frac{2}{r}\partial_\phi\left(\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\phi}} u_\phi + \hat{\mathbf{z}} u_z\right) \\ &=\frac{2}{r}\left(\hat{\mathbf{r}} \partial_\phi u_r + \hat{\boldsymbol{\phi}} \partial_\phi u_\phi + \hat{\mathbf{z}} \partial_\phi u_z+(\partial_\phi \hat{\mathbf{r}}) u_r + (\partial_\phi \hat{\boldsymbol{\phi}}) u_\phi\right) \\ &=\frac{2}{r}\left(\hat{\mathbf{r}} (\partial_\phi u_r - u_\phi) + \hat{\boldsymbol{\phi}} (\partial_\phi u_\phi + u_r )+ \hat{\mathbf{z}} \partial_\phi u_z\right) \\ \end{aligned}
Projecting our curl bivector onto the $\hat{\boldsymbol{\phi}}$ direction we have
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\boldsymbol{\phi}}&=(\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\phi}}) \cdot \hat{\boldsymbol{\phi}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)+(\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{z}}) \cdot \hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)+(\hat{\mathbf{z}} \wedge \hat{\mathbf{r}}) \cdot \hat{\boldsymbol{\phi}}\left(\partial_z u_r - \partial_r u_z\right) \\ &=\hat{\mathbf{r}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)-\hat{\mathbf{z}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)\end{aligned}
Putting things together we have
\begin{aligned}2 \mathbf{e}_{\hat{\boldsymbol{\phi}}}&=\frac{2}{r}\left(\hat{\mathbf{r}} (\partial_\phi u_r - u_\phi) + \hat{\boldsymbol{\phi}} (\partial_\phi u_\phi + u_r )+ \hat{\mathbf{z}} \partial_\phi u_z\right)+\hat{\mathbf{r}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)-\hat{\mathbf{z}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right) \\ &=\hat{\mathbf{r}}\left(\frac{1}{r}\partial_\phi u_r-\frac{u_\phi}{r}+\partial_r u_\phi\right)+\frac{2}{r} \hat{\boldsymbol{\phi}}\left(\partial_\phi u_\phi + u_r\right)+\hat{\mathbf{z}}\left(\frac{1}{r} \partial_\phi u_z + \partial_z u_\phi\right).\end{aligned}
For our stress tensor
\begin{aligned}\boldsymbol{\sigma}_{\hat{\boldsymbol{\phi}}} = - p \hat{\boldsymbol{\phi}} + 2 \mu e_{\hat{\boldsymbol{\phi}}},\end{aligned} \hspace{\stretch{1}}(3.11)
we can now read off our components by taking dot products to yield
\begin{subequations}
\begin{aligned}\sigma_{\phi \phi}=-p + 2 \mu \left(\frac{1}{{r}}\frac{\partial {u_\phi}}{\partial {\phi}} + \frac{u_r}{r}\right)\end{aligned} \hspace{\stretch{1}}(3.12a)
\begin{aligned}\sigma_{\phi z}=\mu \left(\frac{1}{r} \frac{\partial {u_z}}{\partial {\phi}} + \frac{\partial {u_\phi}}{\partial {z}}\right)\end{aligned} \hspace{\stretch{1}}(3.12b)
\begin{aligned}\sigma_{\phi r}=\mu \left(\frac{1}{r}\frac{\partial {u_r}}{\partial {\phi}}-\frac{u_\phi}{r}+\frac{\partial {u_\phi}}{\partial {r}}\right).\end{aligned} \hspace{\stretch{1}}(3.12c)
\end{subequations}
## With $\hat{\mathbf{n}} = \hat{\mathbf{z}}$.
Like the $\hat{\mathbf{r}}$ normal direction, our directional derivative component for a $\hat{\mathbf{z}}$ normal direction will not have any cross terms
\begin{aligned}2 (\hat{\mathbf{z}} \cdot \boldsymbol{\nabla}) \mathbf{u}&=\partial_z\left(\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\phi}} u_\phi + \hat{\mathbf{z}} u_z\right) \\ &=\hat{\mathbf{r}} \partial_z u_r + \hat{\boldsymbol{\phi}} \partial_z u_\phi + \hat{\mathbf{z}} \partial_z u_z\end{aligned}
Projecting our curl bivector onto the $\hat{\mathbf{z}}$ direction we have
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\boldsymbol{\phi}}&=(\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\phi}}) \cdot \hat{\mathbf{z}}\left(\partial_r u_\phi-\frac{1}{{r}} \partial_\phi u_r+ \frac{u_\phi}{r}\right)+(\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{z}}) \cdot \hat{\mathbf{z}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)+(\hat{\mathbf{z}} \wedge \hat{\mathbf{r}}) \cdot \hat{\mathbf{z}}\left(\partial_z u_r - \partial_r u_z\right) \\ &=\hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)-\hat{\mathbf{r}}\left(\partial_z u_r - \partial_r u_z\right)\end{aligned}
Putting things together we have
\begin{aligned}2 \mathbf{e}_{\hat{\mathbf{z}}}&=2 \hat{\mathbf{r}} \partial_z u_r + 2 \hat{\boldsymbol{\phi}} \partial_z u_\phi + 2 \hat{\mathbf{z}} \partial_z u_z+\hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)-\hat{\mathbf{r}}\left(\partial_z u_r - \partial_r u_z\right) \\ &=\hat{\mathbf{r}}\left(2 \partial_z u_r -\partial_z u_r + \partial_r u_z\right)+\hat{\boldsymbol{\phi}}\left(2 \partial_z u_\phi +\frac{1}{{r}} \partial_\phi u_z- \partial_z u_\phi\right)+\hat{\mathbf{z}}\left(2 \partial_z u_z\right) \\ &=\hat{\mathbf{r}}\left(\partial_z u_r + \partial_r u_z\right)+\hat{\boldsymbol{\phi}}\left(\partial_z u_\phi +\frac{1}{{r}} \partial_\phi u_z\right)+\hat{\mathbf{z}}\left(2 \partial_z u_z\right).\end{aligned}
For our stress tensor
\begin{aligned}\boldsymbol{\sigma}_{\hat{\mathbf{z}}} = - p \hat{\mathbf{z}} + 2 \mu e_{\hat{\mathbf{z}}},\end{aligned} \hspace{\stretch{1}}(3.13)
we can now read off our components by taking dot products to yield
\begin{subequations}
\begin{aligned}\sigma_{z z}=-p + 2 \mu \frac{\partial {u_z}}{\partial {z}}\end{aligned} \hspace{\stretch{1}}(3.14a)
\begin{aligned}\sigma_{z r}=\mu \left(\frac{\partial {u_r}}{\partial {z}}+ \frac{\partial {u_z}}{\partial {r}}\right)\end{aligned} \hspace{\stretch{1}}(3.14b)
\begin{aligned}\sigma_{z \phi}=\mu \left(\frac{\partial {u_\phi}}{\partial {z}}+\frac{1}{{r}} \frac{\partial {u_z}}{\partial {\phi}}\right).\end{aligned} \hspace{\stretch{1}}(3.14c)
\end{subequations}
## Summary.
\begin{subequations}
\begin{aligned}\sigma_{rr}=-p + 2 \mu \frac{\partial {u_r}}{\partial {r}}\end{aligned} \hspace{\stretch{1}}(3.15a)
\begin{aligned}\sigma_{\phi \phi}=-p + 2 \mu \left(\frac{1}{{r}}\frac{\partial {u_\phi}}{\partial {\phi}} + \frac{u_r}{r}\right)\end{aligned} \hspace{\stretch{1}}(3.15b)
\begin{aligned}\sigma_{z z}=-p + 2 \mu \frac{\partial {u_z}}{\partial {z}}\end{aligned} \hspace{\stretch{1}}(3.15c)
\begin{aligned}\sigma_{r \phi}=\mu \left( \frac{\partial {u_\phi}}{\partial {r}}+\frac{1}{{r}} \frac{\partial {u_r}}{\partial {\phi}}- \frac{u_\phi}{r}\right)\end{aligned} \hspace{\stretch{1}}(3.15d)
\begin{aligned}\sigma_{\phi z}=\mu \left(\frac{1}{r} \frac{\partial {u_z}}{\partial {\phi}} + \frac{\partial {u_\phi}}{\partial {z}}\right)\end{aligned} \hspace{\stretch{1}}(3.15e)
\begin{aligned}\sigma_{z r}=\mu \left(\frac{\partial {u_r}}{\partial {z}}+ \frac{\partial {u_z}}{\partial {r}}\right)\end{aligned} \hspace{\stretch{1}}(3.15f)
\end{subequations}
# Spherical strain tensor.
Having done a first order cylindrical derivation of the strain tensor, let’s also do the spherical case for completeness. Would this have much utility in fluids? Perhaps for flow over a spherical barrier?
We need the gradient in spherical coordinates. Recall that our spherical coordinate velocity was
\begin{aligned}\frac{d\mathbf{r}}{dt} = \hat{\mathbf{r}} \dot{r} + \hat{\boldsymbol{\theta}} (r \dot{\theta}) + \hat{\boldsymbol{\phi}} ( r \sin\theta \dot{\phi} ),\end{aligned} \hspace{\stretch{1}}(4.16)
and our gradient mirrors this structure
\begin{aligned}\boldsymbol{\nabla} = \hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \hat{\boldsymbol{\theta}} \frac{1}{{r }}\frac{\partial {}}{\partial {\theta}} + \hat{\boldsymbol{\phi}} \frac{1}{{r \sin\theta}} \frac{\partial {}}{\partial {\phi}}.\end{aligned} \hspace{\stretch{1}}(4.17)
We also previously calculated \inbookref{phy454:continuumL2}{eqn:continuumL2:1010} the unit vector differentials
\begin{subequations}
\begin{aligned}d\hat{\mathbf{r}} = \hat{\boldsymbol{\phi}} \sin\theta d\phi + \hat{\boldsymbol{\theta}} d\theta\end{aligned} \hspace{\stretch{1}}(4.18a)
\begin{aligned}d\hat{\boldsymbol{\theta}} = \hat{\boldsymbol{\phi}} \cos\theta d\phi - \hat{\mathbf{r}} d\theta\end{aligned} \hspace{\stretch{1}}(4.18b)
\begin{aligned}d\hat{\boldsymbol{\phi}} = -(\hat{\mathbf{r}} \sin\theta + \hat{\boldsymbol{\theta}} \cos\theta) d\phi,\end{aligned} \hspace{\stretch{1}}(4.18c)
\end{subequations}
and can use those to read off the partials of all the unit vectors
\begin{aligned}\frac{\partial \hat{\mathbf{r}}}{\partial \{r,\theta, \phi\}} &= \{0, \hat{\boldsymbol{\theta}}, \hat{\boldsymbol{\phi}} \sin\theta \} \\ \frac{\partial \hat{\boldsymbol{\theta}}}{\partial \{r,\theta, \phi\}} &= \{0, -\hat{\mathbf{r}}, \hat{\boldsymbol{\phi}} \cos\theta \} \\ \frac{\partial \hat{\boldsymbol{\phi}}}{\partial \{r,\theta, \phi\}} &= \{0, 0, -\hat{\mathbf{r}} \sin\theta -\hat{\boldsymbol{\theta}} \cos\theta \}.\end{aligned} \hspace{\stretch{1}}(4.19)
Finally, our velocity in spherical coordinates is just
\begin{aligned}\mathbf{u} = \hat{\mathbf{r}} u_r + \hat{\boldsymbol{\theta}} u_\theta + \hat{\boldsymbol{\phi}} u_\phi,\end{aligned} \hspace{\stretch{1}}(4.22)
from which we can now compute the curl, and the directional derivative. Starting with the curl we have
\begin{aligned}\boldsymbol{\nabla} \wedge \mathbf{u}&=\left( \hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \hat{\boldsymbol{\theta}} \frac{1}{{r }}\frac{\partial {}}{\partial {\theta}} + \hat{\boldsymbol{\phi}} \frac{1}{{r \sin\theta}} \frac{\partial {}}{\partial {\phi}} \right) \wedge\left( \hat{\mathbf{r}} u_r + \hat{\boldsymbol{\theta}} u_\theta + \hat{\boldsymbol{\phi}} u_\phi \right) \\ &=\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\theta}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r\right)\\ & +\hat{\boldsymbol{\theta}} \wedge \hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta\right)\\ & +\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi\right)\\ & +\frac{1}{{r}} \hat{\boldsymbol{\theta}} \wedge \left(u_\theta \underbrace{\partial_\theta \hat{\boldsymbol{\theta}}}_{-\hat{\mathbf{r}}}+u_\phi \underbrace{\partial_\theta \hat{\boldsymbol{\phi}}}_{0}\right)\\ & +\frac{1}{{r \sin\theta}} \hat{\boldsymbol{\phi}} \wedge \left(u_\theta \underbrace{\partial_\phi \hat{\boldsymbol{\theta}}}_{\hat{\boldsymbol{\phi}} \cos\theta}+u_\phi \underbrace{\partial_\phi \hat{\boldsymbol{\phi}}}_{-\hat{\mathbf{r}} \sin\theta - \hat{\boldsymbol{\theta}} \cos\theta}\right).\end{aligned}
So we have
\begin{aligned}\begin{aligned}\boldsymbol{\nabla} \wedge \mathbf{u}&=\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\theta}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)\\ & +\hat{\boldsymbol{\theta}} \wedge \hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right)\\ & +\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right).\end{aligned}\end{aligned} \hspace{\stretch{1}}(4.23)
## With $\hat{\mathbf{n}} = \hat{\mathbf{r}}$.
The directional derivative portion of our strain is
\begin{aligned}2 (\hat{\mathbf{r}} \cdot \boldsymbol{\nabla}) \mathbf{u}&=2 \partial_r (\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\theta}} u_\theta + \hat{\boldsymbol{\phi}} u_\phi ) \\ &=2 (\hat{\mathbf{r}} \partial_r u_r + \hat{\boldsymbol{\theta}} \partial_r u_\theta + \hat{\boldsymbol{\phi}} \partial_r u_\phi ).\end{aligned}
The other portion of our strain tensor is
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{r}}&=(\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\theta}}) \cdot \hat{\mathbf{r}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)\\ & +(\hat{\boldsymbol{\theta}} \wedge \hat{\boldsymbol{\phi}}) \cdot \hat{\mathbf{r}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right)\\ & +(\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{r}}) \cdot \hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right) \\ &=-\hat{\boldsymbol{\theta}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)\\ & +\hat{\boldsymbol{\phi}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right).\end{aligned}
Putting these together we find
\begin{aligned}2 {\mathbf{e}}_{\hat{\mathbf{r}}}&=2 (\hat{\mathbf{r}} \cdot \boldsymbol{\nabla})\mathbf{u} + (\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\mathbf{r}} \\ &=2 (\hat{\mathbf{r}} \partial_r u_r + \hat{\boldsymbol{\theta}} \partial_r u_\theta + \hat{\boldsymbol{\phi}} \partial_r u_\phi )-\hat{\boldsymbol{\theta}}\left(\partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)+\hat{\boldsymbol{\phi}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right) \\ &=\hat{\mathbf{r}}\left(2 \partial_r u_r\right)+\hat{\boldsymbol{\theta}}\left(2 \partial_r u_\theta-\partial_r u_\theta + \frac{1}{{r}} \partial_\theta u_r - \frac{u_\theta}{r}\right)+\hat{\boldsymbol{\phi}}\left(2 \partial_r u_\phi+ \frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right).\end{aligned}
Which gives
\begin{aligned}2 {\mathbf{e}}_{\hat{\mathbf{r}}}=\hat{\mathbf{r}}\left(2 \partial_r u_r\right)+\hat{\boldsymbol{\theta}}\left(\partial_r u_\theta+ \frac{1}{{r}} \partial_\theta u_r - \frac{u_\theta}{r}\right)+\hat{\boldsymbol{\phi}}\left(\partial_r u_\phi+ \frac{1}{{r \sin\theta}} \partial_\phi u_r- \frac{u_\phi}{r}\right)\end{aligned} \hspace{\stretch{1}}(4.24)
For our stress tensor
\begin{aligned}\boldsymbol{\sigma}_{\hat{\mathbf{r}}} = - p \hat{\mathbf{r}} + 2 \mu e_{\hat{\mathbf{r}}},\end{aligned} \hspace{\stretch{1}}(4.25)
we can now read off our components by taking dot products
\begin{subequations}
\begin{aligned}\sigma_{rr}=-p + 2 \mu \frac{\partial {u_r}}{\partial {r}}\end{aligned} \hspace{\stretch{1}}(4.26a)
\begin{aligned}\sigma_{r \theta}=\mu \left(\frac{\partial {u_\theta}}{\partial {r}}+ \frac{1}{{r}} \frac{\partial {u_r}}{\partial {\theta}} - \frac{u_\theta}{r}\right)\end{aligned} \hspace{\stretch{1}}(4.26b)
\begin{aligned}\sigma_{r \phi}=\mu \left(\frac{\partial {u_\phi}}{\partial {r}}+ \frac{1}{{r \sin\theta}} \frac{\partial {u_r}}{\partial {\phi}}- \frac{u_\phi}{r}\right).\end{aligned} \hspace{\stretch{1}}(4.26c)
\end{subequations}
This is consistent with (15.20) from [3] (after adjusting for minor notational differences).
## With $\hat{\mathbf{n}} = \hat{\boldsymbol{\theta}}$.
Now let’s do the $\hat{\boldsymbol{\theta}}$ direction. The directional derivative portion of our strain will be a bit more work to compute because we have $\theta$ variation of the unit vectors
\begin{aligned}(\hat{\boldsymbol{\theta}} \cdot \boldsymbol{\nabla}) \mathbf{u} &= \frac{1}{r} \partial_\theta (\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\theta}} u_\theta + \hat{\boldsymbol{\phi}} u_\phi ) \\ &= \frac{1}{r} \left( \hat{\mathbf{r}} \partial_\theta u_r + \hat{\boldsymbol{\theta}} \partial_\theta u_\theta + \hat{\boldsymbol{\phi}} \partial_\theta u_\phi \right)+\frac{1}{r} \left( (\partial_\theta \hat{\mathbf{r}}) u_r + (\partial_\theta \hat{\boldsymbol{\theta}}) u_\theta + (\partial_\theta \hat{\boldsymbol{\phi}}) u_\phi \right) \\ &= \frac{1}{r}\left(\hat{\mathbf{r}} \partial_\theta u_r + \hat{\boldsymbol{\theta}} \partial_\theta u_\theta + \hat{\boldsymbol{\phi}} \partial_\theta u_\phi \right)+\frac{1}{r} \left( \hat{\boldsymbol{\theta}} u_r - \hat{\mathbf{r}} u_\theta \right).\end{aligned}
So we have
\begin{aligned}2 (\hat{\boldsymbol{\theta}} \cdot \boldsymbol{\nabla}) \mathbf{u}=\frac{2}{r} \hat{\mathbf{r}} (\partial_\theta u_r- u_\theta)+ \frac{2}{r} \hat{\boldsymbol{\theta}} (\partial_\theta u_\theta+ u_r) + \frac{2}{r} \hat{\boldsymbol{\phi}} \partial_\theta u_\phi,\end{aligned} \hspace{\stretch{1}}(4.27)
and can move on to projecting our curl bivector onto the $\hat{\boldsymbol{\theta}}$ direction. That portion of our strain tensor is
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\boldsymbol{\theta}}&=(\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\theta}}) \cdot \hat{\boldsymbol{\theta}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)\\ & +(\hat{\boldsymbol{\theta}} \wedge \hat{\boldsymbol{\phi}}) \cdot \hat{\boldsymbol{\theta}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right)\\ & +(\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{r}}) \cdot \hat{\boldsymbol{\theta}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right) \\ &=\hat{\mathbf{r}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)-\hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right).\end{aligned}
Putting these together we find
\begin{aligned}2 {\mathbf{e}}_{\hat{\boldsymbol{\theta}}}&=2 (\hat{\boldsymbol{\theta}} \cdot \boldsymbol{\nabla})\mathbf{u} + (\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\boldsymbol{\theta}} \\ &= \frac{2}{r} \hat{\mathbf{r}} (\partial_\theta u_r - u_\theta )+ \frac{2}{r} \hat{\boldsymbol{\theta}} (\partial_\theta u_\theta + u_r )+ \frac{2}{r} \hat{\boldsymbol{\phi}} \partial_\theta u_\phi \\ &+\hat{\mathbf{r}}\left(\partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)-\hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta + \frac{u_\phi \cot\theta}{r}\right).\end{aligned}
Which gives
\begin{aligned}2 {\mathbf{e}}_{\hat{\boldsymbol{\theta}}}=\hat{\mathbf{r}} \left( \frac{1}{r} \partial_\theta u_r + \partial_r u_\theta- \frac{u_\theta}{r}\right)+\hat{\boldsymbol{\theta}} \left( \frac{2}{r} \partial_\theta u_\theta+ \frac{2}{r} u_r\right)+\hat{\boldsymbol{\phi}} \left(\frac{1}{r} \partial_\theta u_\phi+ \frac{1}{{r \sin\theta}} \partial_\phi u_\theta- \frac{u_\phi \cot\theta}{r}\right).\end{aligned} \hspace{\stretch{1}}(4.28)
For our stress tensor
\begin{aligned}\boldsymbol{\sigma}_{\hat{\boldsymbol{\theta}}} = - p \hat{\boldsymbol{\theta}} + 2 \mu e_{\hat{\boldsymbol{\theta}}},\end{aligned} \hspace{\stretch{1}}(4.29)
we can now read off our components by taking dot products
\begin{subequations}
\begin{aligned}\sigma_{\theta \theta}=-p+\mu \left( \frac{2}{r} \frac{\partial {u_\theta}}{\partial {\theta}}+ \frac{2}{r} u_r\right)\end{aligned} \hspace{\stretch{1}}(4.30a)
\begin{aligned}\sigma_{\theta \phi}=\mu \left(\frac{1}{r} \frac{\partial {u_\phi}}{\partial {\theta}}+ \frac{1}{{r \sin\theta}} \frac{\partial {u_\theta}}{\partial {\phi}}- \frac{u_\phi \cot\theta}{r}\right)\end{aligned} \hspace{\stretch{1}}(4.30b)
\begin{aligned}\sigma_{\theta r}= \mu \left(\frac{1}{r} \frac{\partial {u_r}}{\partial {\theta}} + \frac{\partial {u_\theta}}{\partial {r}}- \frac{u_\theta}{r}\right).\end{aligned} \hspace{\stretch{1}}(4.30c)
\end{subequations}
This again is consistent with (15.20) from [3].
## With $\hat{\mathbf{n}} = \hat{\boldsymbol{\phi}}$.
Finally, let’s do the $\hat{\boldsymbol{\phi}}$ direction. This directional derivative portion of our strain will also be a bit more work to compute because we have $\hat{\boldsymbol{\phi}}$ variation of the unit vectors
\begin{aligned}(\hat{\boldsymbol{\phi}} \cdot \boldsymbol{\nabla}) \mathbf{u}&=\frac{1}{r \sin\theta} \partial_\phi (\hat{\mathbf{r}} u_r + \hat{\boldsymbol{\theta}} u_\theta + \hat{\boldsymbol{\phi}} u_\phi) \\ &=\frac{1}{r \sin\theta}(\hat{\mathbf{r}} \partial_\phi u_r+\hat{\boldsymbol{\theta}} \partial_\phi u_\theta+\hat{\boldsymbol{\phi}} \partial_\phi u_\phi+(\partial_\phi \hat{\mathbf{r}} )u_r+(\partial_\phi \hat{\boldsymbol{\theta}} )u_\theta+(\partial_\phi \hat{\boldsymbol{\phi}} )u_\phi) \\ &=\frac{1}{r \sin\theta}(\hat{\mathbf{r}} \partial_\phi u_r+\hat{\boldsymbol{\theta}} \partial_\phi u_\theta+\hat{\boldsymbol{\phi}} \partial_\phi u_\phi+\hat{\boldsymbol{\phi}} \sin\thetau_r+\hat{\boldsymbol{\phi}} \cos\thetau_\theta-(\hat{\mathbf{r}} \sin\theta+ \hat{\boldsymbol{\theta}} \cos\theta)u_\phi)\end{aligned}
So we have
\begin{aligned}2 (\hat{\boldsymbol{\phi}} \cdot \boldsymbol{\nabla}) \mathbf{u}=2 \hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \frac{u_\phi}{r}\right)+2 \hat{\boldsymbol{\theta}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_\theta-\frac{1}{{r}} \cot\theta u_\phi\right)+2 \hat{\boldsymbol{\phi}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_\phi+ \frac{1}{{r}} u_r+ \frac{1}{{r}} \cot\theta u_\theta\right),\end{aligned} \hspace{\stretch{1}}(4.31)
and can move on to projecting our curl bivector onto the $\hat{\boldsymbol{\phi}}$ direction. That portion of our strain tensor is
\begin{aligned}(\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\boldsymbol{\phi}}&=(\hat{\mathbf{r}} \wedge \hat{\boldsymbol{\theta}}) \cdot \hat{\boldsymbol{\phi}}\left( \partial_r u_\theta - \frac{1}{{r}} \partial_\theta u_r + \frac{u_\theta}{r}\right)\\ & +(\hat{\boldsymbol{\theta}} \wedge \hat{\boldsymbol{\phi}}) \cdot \hat{\boldsymbol{\phi}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right)\\ & +(\hat{\boldsymbol{\phi}} \wedge \hat{\mathbf{r}}) \cdot \hat{\boldsymbol{\phi}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right) \\ &=\hat{\boldsymbol{\theta}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right)\\ &-\hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right).\end{aligned}
Putting these together we find
\begin{aligned}2 {\mathbf{e}}_{\hat{\boldsymbol{\theta}}}&=2 (\hat{\boldsymbol{\phi}} \cdot \boldsymbol{\nabla})\mathbf{u} + (\boldsymbol{\nabla} \wedge \mathbf{u}) \cdot \hat{\boldsymbol{\phi}} \\ &=2 \hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \frac{u_\phi}{r}\right)+2 \hat{\boldsymbol{\theta}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_\theta-\frac{1}{{r}} \cot\theta u_\phi\right)+2 \hat{\boldsymbol{\phi}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_\phi+ \frac{1}{{r}} u_r+ \frac{1}{{r}} \cot\theta u_\theta\right) \\ &+\hat{\boldsymbol{\theta}}\left(\frac{1}{{r}} \partial_\theta u_\phi - \frac{1}{{r \sin\theta}} \partial_\phi u_\theta+ \frac{u_\phi \cot\theta}{r}\right)-\hat{\mathbf{r}}\left(\frac{1}{{r \sin\theta}} \partial_\phi u_r - \partial_r u_\phi- \frac{u_\phi}{r}\right).\end{aligned}
Which gives
\begin{aligned}2 {\mathbf{e}}_{\hat{\boldsymbol{\phi}}}=\hat{\mathbf{r}} \left( \frac{ \partial_\phi u_r }{r \sin\theta}- \frac{u_\phi}{r}+ \partial_r u_\phi\right)+\hat{\boldsymbol{\theta}} \left(\frac{\partial_\phi u_\theta}{r \sin\theta}- \frac{u_\phi \cot\theta}{r}+\frac{\partial_\theta u_\phi}{r}\right)+2 \hat{\boldsymbol{\phi}} \left(\frac{\partial_\phi u_\phi}{r \sin\theta}+ \frac{u_r}{r}+ \frac{\cot\theta u_\theta}{r}\right).\end{aligned} \hspace{\stretch{1}}(4.32)
For our stress tensor
\begin{aligned}\boldsymbol{\sigma}_{\hat{\boldsymbol{\phi}}} = - p \hat{\boldsymbol{\phi}} + 2 \mu e_{\hat{\boldsymbol{\phi}}},\end{aligned} \hspace{\stretch{1}}(4.33)
we can now read off our components by taking dot products
\begin{subequations}
\begin{aligned}\sigma_{\phi \phi}=-p+2 \mu \left(\frac{1}{{r \sin\theta}} \frac{\partial {u_\phi}}{\partial {\phi}}+ \frac{u_r}{r}+ \frac{\cot\theta u_\theta}{r}\right)\end{aligned} \hspace{\stretch{1}}(4.34a)
\begin{aligned}\sigma_{\phi r}=\mu \left( \frac{1}{r \sin\theta} \frac{\partial {u_r}}{\partial {\phi}}- \frac{u_\phi}{r}+ \frac{\partial {u_\phi}}{\partial {r}}\right)\end{aligned} \hspace{\stretch{1}}(4.34b)
\begin{aligned}\sigma_{\phi \theta}= \mu \left(\frac{1}{r \sin\theta} \frac{\partial {u_\theta}}{\partial {\phi}}- \frac{u_\phi \cot\theta}{r}+\frac{1}{{r}} \frac{\partial {u_\phi}}{\partial {\theta}}\right).\end{aligned} \hspace{\stretch{1}}(4.34c)
\end{subequations}
This again is consistent with (15.20) from [3].
## Summary
\begin{subequations}
\begin{aligned}\sigma_{rr}=-p + 2 \mu \frac{\partial {u_r}}{\partial {r}}\end{aligned} \hspace{\stretch{1}}(4.35a)
\begin{aligned}\sigma_{\theta \theta}=-p+2 \mu \left( \frac{1}{r} \frac{\partial {u_\theta}}{\partial {\theta}}+ \frac{ u_r }{r}\right)\end{aligned} \hspace{\stretch{1}}(4.35b)
\begin{aligned}\sigma_{\phi \phi}=-p+2 \mu \left(\frac{1}{{r \sin\theta}} \frac{\partial {u_\phi}}{\partial {\phi}}+ \frac{u_r}{r}+ \frac{\cot\theta u_\theta}{r}\right)\end{aligned} \hspace{\stretch{1}}(4.35c)
\begin{aligned}\sigma_{r \theta}=\mu \left(\frac{\partial {u_\theta}}{\partial {r}}+ \frac{1}{{r}} \frac{\partial {u_r}}{\partial {\theta}} - \frac{u_\theta}{r}\right)\end{aligned} \hspace{\stretch{1}}(4.35d)
\begin{aligned}\sigma_{\theta \phi}= \mu \left(\frac{1}{r \sin\theta} \frac{\partial {u_\theta}}{\partial {\phi}}- \frac{u_\phi \cot\theta}{r}+\frac{1}{{r}} \frac{\partial {u_\phi}}{\partial {\theta}}\right).\end{aligned} \hspace{\stretch{1}}(4.35e)
\begin{aligned}\sigma_{\phi r}=\mu \left( \frac{1}{r \sin\theta} \frac{\partial {u_r}}{\partial {\phi}}- \frac{u_\phi}{r}+ \frac{\partial {u_\phi}}{\partial {r}}\right)\end{aligned} \hspace{\stretch{1}}(4.35f)
\end{subequations}
# References
[1] D.J. Acheson. Elementary fluid dynamics. Oxford University Press, USA, 1990.
[2] Peeter Joot. Continuum mechanics., chapter {Introduction and strain tensor.} http://sites.google.com/site/peeterjoot2/math2012/phy454.pdf.
[3] L.D. Landau and E.M. Lifshitz. A Course in Theoretical Physics-Fluid Mechanics. Pergamon Press Ltd., 1987.
## Compilation of class notes for phy454h1s, continuum mechanics (so far).
Posted by peeterjoot on February 17, 2012
Have collected all my pre-midterm continuum mechanics notes into a single document. The individual pdfs below are still available, but won’t be updated further.
Feb 17, 2012 Flow in a pipe. Gravity driven flow of a film.
Feb 10, 2012 Navier-Stokes equation.
Feb 1, 2012 P-waves and S-waves.
Jan 27, 2012 Compatibility condition and elastostatics.
Jan 25, 2012 Constitutive relationship.
Jan 20, 2012 Strain tensor components.
Jan 18, 2012 Strain tensor review. Stress tensor.
Jan 13, 2012 Introduction and strain tensor.
Jan 11, 2012 Overview.
## PHY454H1S Continuum Mechanics. Lecture 12: Flow in a pipe. Gravity driven flow of a film. Taught by Prof. K. Das.
Posted by peeterjoot on February 17, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Review. Steady rectilinear flow.
\begin{aligned}\frac{\partial {}}{\partial {t}} = 0\end{aligned} \hspace{\stretch{1}}(2.1)
Rectilinear is a unidirectional flow such as
\begin{aligned}\mathbf{u} = \hat{\mathbf{x}} u( x, y, z ),\end{aligned} \hspace{\stretch{1}}(2.2)
\begin{enumerate}
\item
Utilizing an incompressibility assumption $\boldsymbol{\nabla} \cdot \mathbf{u} = 0$, so for this case we have
\begin{aligned}\frac{\partial {u}}{\partial {x}} = 0\end{aligned}
or
\begin{aligned}u = u(y, z)\end{aligned}
Note that Prof. Das called this a continuity requirement, and justified this label with the relation
\begin{aligned}\frac{d\rho}{dt} = \rho (\boldsymbol{\nabla} \cdot \mathbf{u}),\end{aligned} \hspace{\stretch{1}}(2.3)
which was a consequence of mass conservation. It’s still not clear to me why he would call this a continuity requirement.
\item Nonlinear term is zero. $(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} = 0$
\item $p = p(x)$. Since $\frac{d^2 p}{dx^2} = 0$ we also have $\frac{dp}{dx} = -G$, a constant.
\item $\mu \left( \frac{\partial^2 {{u}}}{\partial {{y}}^2} + \frac{\partial^2 {{u}}}{\partial {{z}}^2} \right) = G$
\end{enumerate}
# Solution by intuition.
Two examples that we have solved analytically are illustrated in figure (\ref{fig:continuumL12:continuumL12fig1}) and figure (\ref{fig:continuumL12:continuumL12fig2})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig1}
\caption{Simple shear flow}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig2}
\caption{Channel flow}
\end{figure}
Sometimes we can utilize solutions already found to understand the behaviour of more complex systems. Combining the two we can look at flow over a plate as in figure (\ref{fig:continuumL12:continuumL12fig3})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig3}
\caption{Flow on a plate}
\end{figure}
Example 2. Fluid in a container. If the surface tension is altered on one side, we induce a flow on the surface, leading to a circulation flow. This can be done for example, by introducing a heat source or addition of surfactant.
This is illustrated in figure (\ref{fig:continuumL12:continuumL12fig4})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig4}
\caption{Circulation flow induced by altering surface tension.}
\end{figure}
This sort of flow is hard to analyze, only first done by Steve Davis in the 1980’s. The point here is that we can use some level of intuition to guide our attempts at solution.
# Flow down a pipe.
Reading: section 2 from [1].
Recall that the Navier-Stokes equation is
\begin{aligned}\boxed{\rho \frac{\partial {\mathbf{u}}}{\partial {t}} + \rho (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} = - \boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^2 \mathbf{u} + \rho \mathbf{f}.}\end{aligned} \hspace{\stretch{1}}(4.4)
We need to express this in cylindrical coordinates $(r, \theta, z)$ as in figure (\ref{fig:continuumL12:continuumL12fig5})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig5}
\caption{Flow through a pipe.}
\end{figure}
\begin{aligned}\boldsymbol{\nabla} = \hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial {}}{\partial {\theta}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}},\end{aligned} \hspace{\stretch{1}}(4.5)
For our Laplacian we find
\begin{aligned}\boldsymbol{\nabla}^2 &= \left(\hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial {}}{\partial {\theta}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}}\right) \cdot\left(\hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial {}}{\partial {\theta}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}}\right) \\ &=\partial_{rr} + \frac{\hat{\boldsymbol{\theta}}}{r} \cdot (\partial_\theta \hat{\mathbf{r}}) \partial_r+ \frac{1}{{r}} \partial_\theta \left( \frac{1}{{r}} \partial_\theta \right)+ \partial_{zz} \\ &=\partial_{rr} + \frac{1}{{r}} \partial_r + \frac{1}{{r^2}} \partial_{\theta\theta} + \partial_{zz},\end{aligned}
which we can write as
\begin{aligned}\boldsymbol{\nabla}^2 = \frac{1}{{r}} \frac{\partial {}}{\partial {r}} \left( r \frac{\partial {}}{\partial {r}} \right) + \frac{1}{{r^2}} \frac{\partial^2 {{}}}{\partial {{\theta}}^2} + \frac{\partial^2 {{}}}{\partial {{z}}^2}.\end{aligned} \hspace{\stretch{1}}(4.6)
NS takes the form
\begin{aligned}\boxed{\begin{aligned}\rho \frac{\partial {\mathbf{u}}}{\partial {t}} &+ \rho \left(u_r \frac{\partial {}}{\partial {r}} + \frac{u_\theta}{r} \frac{\partial {}}{\partial {\theta}} + u_z \frac{\partial {}}{\partial {z}} \right) \mathbf{u} = \\ &- \left(\hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial {}}{\partial {\theta}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}}\right)p + \mu \left(\frac{1}{{r}} \frac{\partial {}}{\partial {r}} \left( r \frac{\partial {}}{\partial {r}} \right) + \frac{1}{{r^2}} \frac{\partial^2 {{}}}{\partial {{\theta}}^2} + \frac{\partial^2 {{}}}{\partial {{z}}^2} \right)\mathbf{u} + \rho \mathbf{f}.\end{aligned}}\end{aligned} \hspace{\stretch{1}}(4.7)
For steady state and incompressible fluids in the absence of body forces we have
\begin{aligned}\left(\hat{\mathbf{r}} \frac{\partial {}}{\partial {r}} + \frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial {}}{\partial {\theta}} + \hat{\mathbf{z}} \frac{\partial {}}{\partial {z}}\right)p = \mu \left(\frac{1}{{r}} \frac{\partial {}}{\partial {r}} \left( r \frac{\partial {}}{\partial {r}} \right) + \frac{1}{{r^2}} \frac{\partial^2 {{}}}{\partial {{\theta}}^2} + \frac{\partial^2 {{}}}{\partial {{z}}^2} \right)\mathbf{u},\end{aligned} \hspace{\stretch{1}}(4.8)
or, in coordinates
\begin{aligned}\frac{\partial {p}}{\partial {r}} &= \mu \left(\frac{1}{{r}} \frac{\partial {}}{\partial {r}} \left( r \frac{\partial {}}{\partial {r}} \right) + \frac{1}{{r^2}} \frac{\partial^2 {{}}}{\partial {{\theta}}^2} + \frac{\partial^2 {{}}}{\partial {{z}}^2} \right)u_r \\ \frac{1}{r} \frac{\partial {p}}{\partial {\theta}}&= \mu \left(\frac{1}{{r}} \frac{\partial {}}{\partial {r}} \left( r \frac{\partial {}}{\partial {r}} \right) + \frac{1}{{r^2}} \frac{\partial^2 {{}}}{\partial {{\theta}}^2} + \frac{\partial^2 {{}}}{\partial {{z}}^2} \right)u_\theta \\ \frac{\partial {p}}{\partial {z}}&= \mu \left(\frac{1}{{r}} \frac{\partial {}}{\partial {r}} \left( r \frac{\partial {}}{\partial {r}} \right) + \frac{1}{{r^2}} \frac{\partial^2 {{}}}{\partial {{\theta}}^2} + \frac{\partial^2 {{}}}{\partial {{z}}^2} \right)u_z\end{aligned} \hspace{\stretch{1}}(4.9)
With an assumption that we have no radial or circulatory flows ($u_r = u_\theta = 0$), and with $u_z = w$ assumed to only have a radial dependence, our velocity is
\begin{aligned}\mathbf{u} = \hat{\mathbf{z}} w(r),\end{aligned} \hspace{\stretch{1}}(4.12)
and an assumption of linear pressure dependence
\begin{aligned}\frac{dp}{dz} = -G,\end{aligned} \hspace{\stretch{1}}(4.13)
then NS takes the final simple form
\begin{aligned}\frac{1}{{r}} \frac{d}{dr} \left( r \frac{dw}{dr} \right) = - \frac{G}{\mu}.\end{aligned} \hspace{\stretch{1}}(4.14)
Solving this we have
\begin{aligned}r \frac{dw}{dr} = - \frac{G r^2}{2\mu} + A\end{aligned} \hspace{\stretch{1}}(4.15)
\begin{aligned}w = -\frac{G r^2}{4 \mu} + A \ln(r) + B\end{aligned} \hspace{\stretch{1}}(4.16)
Requiring finite solutions for $r = 0$ means that we must have $A = 0$. Also $w(a) = 0$, we have $B = G a^2/4 \mu$ so we must have
\begin{aligned}w(r) = \frac{G}{4 \mu}( a^2 - r^2 )\end{aligned} \hspace{\stretch{1}}(4.17)
# Example: Gravity driven flow of a liquid film
(This is one of our Professor’s favorite problems).
Coordinates as in figure (\ref{fig:continuumL12:continuumL12fig6})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig6}
\caption{Gravity driven flow down an inclined plane.}
\end{figure}
\begin{aligned}\mathbf{u} = \hat{\mathbf{x}} u(y)\end{aligned} \hspace{\stretch{1}}(5.18)
Boundary conditions
\begin{enumerate}
\item $u(y = 0) = 0$
\item Tangential stress at the air-liquid interface $y = h$ is equal.
\begin{aligned}\boldsymbol{\tau} \cdot (\boldsymbol{\sigma}_l \cdot \hat{\mathbf{n}}) = \boldsymbol{\tau} \cdot (\boldsymbol{\sigma}_a \cdot \hat{\mathbf{n}}),\end{aligned} \hspace{\stretch{1}}(5.19)
\end{enumerate}
We write
\begin{aligned}\boldsymbol{\tau} &= \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix} \\ \hat{\mathbf{n}} &= \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} \\ \end{aligned} \hspace{\stretch{1}}(5.20)
and seek simultaneous solutions to the pair of stress tensor equations
\begin{aligned}\sigma_{ij}^l &= - p \delta_{ij} + \mu^l \left( \frac{\partial {u_i}}{\partial {x_j}} +\frac{\partial {u_j}}{\partial {x_i}}\right) \\ \sigma_{ij}^a &= - p \delta_{ij} + \mu^a \left( \frac{\partial {u_i}}{\partial {x_j}} +\frac{\partial {u_j}}{\partial {x_i}}\right).\end{aligned} \hspace{\stretch{1}}(5.23)
In general this requires an iterated approach, solving for one with an initial approximation of the other, then switching and tuning the numerical method carefully for convergence.
We expect that the flow of liquid will induce a flow of air at the interface, but may be able to make a one-sided approximation. Let’s see how far we get before we have to introduce any approximations and compute the traction vector for the liquid
\begin{aligned}\boldsymbol{\sigma}^l \cdot \hat{\mathbf{n}} &= \begin{bmatrix}-p & \mu^l {\partial {u}}/{\partial {y}} & 0 \\ \mu^l {\partial {u}}/{\partial {y}} & -p & 0 \\ 0 & 0 & 0\end{bmatrix}\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} \\ &=\begin{bmatrix}\mu^l {\partial {u}}/{\partial {y}} \\ -p \\ 0\end{bmatrix}\end{aligned}
So
\begin{aligned}\boldsymbol{\tau} \cdot (\boldsymbol{\sigma}^l \cdot \hat{\mathbf{n}})=\begin{bmatrix}1 & 0 & 0\end{bmatrix}\begin{bmatrix}\mu^l {\partial {u}}/{\partial {y}} \\ -p \\ 0\end{bmatrix}=\mu^l \frac{\partial {u}}{\partial {y}}\end{aligned} \hspace{\stretch{1}}(5.25)
Our boundary value condition is therefore
\begin{aligned}{\left.{{\mu^l \frac{\partial {u^l}}{\partial {y}}}}\right\vert}_{{y = h}} ={\left.{{\mu^a \frac{\partial {u^a}}{\partial {y}}}}\right\vert}_{{y = h}}\end{aligned} \hspace{\stretch{1}}(5.26)
When can we decouple this, treating only the liquid? Observe that we have
\begin{aligned}{\left.{{\frac{\partial {u^l}}{\partial {y}}}}\right\vert}_{{y = h}} ={\left.{{\frac{\mu^a}{\mu^l} \frac{\partial {u^a}}{\partial {y}}}}\right\vert}_{{y = h}}\end{aligned} \hspace{\stretch{1}}(5.27)
so if
\begin{aligned}\frac{\mu_a}{\mu_l} \ll 1\end{aligned} \hspace{\stretch{1}}(5.28)
we can treat only the liquid portion of the problem, with a boundary value condition
\begin{aligned}{\left.{{\frac{\partial {u^l}}{\partial {y}}}}\right\vert}_{{y = h}} = 0.\end{aligned} \hspace{\stretch{1}}(5.29)
Let’s look at the component of the traction vector in the direction of the normal (liquid pressure acting on the air)
\begin{aligned}\hat{\mathbf{n}} \cdot (\boldsymbol{\sigma}^l \cdot \hat{\mathbf{n}}) = \hat{\mathbf{n}} \cdot (\boldsymbol{\sigma}^a \cdot \hat{\mathbf{n}}) \end{aligned} \hspace{\stretch{1}}(5.30)
or
\begin{aligned}\begin{bmatrix}0 & 1 & 0\end{bmatrix}\begin{bmatrix}\mu^l \frac{\partial {u}}{\partial {y}} \\ -p^l \\ 0\end{bmatrix}= -{\left.{{p^l}}\right\vert}_{{y = h}} = -{\left.{{p^a}}\right\vert}_{{y = h}}\end{aligned} \hspace{\stretch{1}}(5.31)
i.e. We have pressure matching at the interface.
Our body force is
\begin{aligned}\mathbf{f} = \begin{bmatrix}g \sin\alpha \\ -g \cos\alpha \\ 0\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(5.32)
Referring to the Navier-Stokes equation 4.4, we see that our only surviving parts are
\begin{subequations}
\begin{aligned}0 = -\frac{\partial {p}}{\partial {x}} + \mu \frac{\partial^2 {{u}}}{\partial {{y}}^2} + \rho g \sin\alpha \end{aligned} \hspace{\stretch{1}}(5.33a)
\begin{aligned}0 = -\frac{\partial {p}}{\partial {y}} - \rho g \cos\alpha \end{aligned} \hspace{\stretch{1}}(5.33b)
\begin{aligned}0 = -\frac{\partial {p}}{\partial {z}} \end{aligned} \hspace{\stretch{1}}(5.33c)
\end{subequations}
The last gives us $p \ne p(z)$. Integrating the second we have
\begin{aligned}p = \rho g y \cos\alpha + p_1\end{aligned} \hspace{\stretch{1}}(5.34)
Since $p = p_{\text{atm}}$ at $y = h$, we have
\begin{aligned}p_{\text{atm}} = \rho g h \cos\alpha + p_1\end{aligned} \hspace{\stretch{1}}(5.35)
Our first NS equation 5.33a becomes
\begin{aligned}0 = \mu \frac{\partial^2 {{u}}}{\partial {{y}}^2} + g \sin\alpha,\end{aligned} \hspace{\stretch{1}}(5.36)
or
\begin{aligned}\frac{\partial^2 {{u}}}{\partial {{y}}^2} = -\frac{g}{\mu} \sin\alpha\end{aligned} \hspace{\stretch{1}}(5.37)
Solving we have
\begin{aligned}u = - \rho g \frac{\sin\alpha}{2 \mu} y^2 + A y + B\end{aligned} \hspace{\stretch{1}}(5.38)
With
\begin{aligned}u(0) &= 0 \\ {\left.{{\frac{\partial {u}}{\partial {y}}}}\right\vert}_{{y = h}} &= 0\end{aligned} \hspace{\stretch{1}}(5.39)
\begin{aligned}u = \rho g \frac{\sin\alpha}{2 \mu} \left( 2 h y - y^2 \right) .\end{aligned} \hspace{\stretch{1}}(5.41)
This velocity distribution is illustrated figure (\ref{fig:continuumL12:continuumL12fig7}).
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL12fig7}
\caption{Velocity streamlines for flow down a plane.}
\end{figure}
It’s important to note that in these problems we have to derive our boundary value conditions! They are not given.
In this discussion, the height $h$ was assumed to be constant, with the tangential direction constant and parallel to the surface that the liquid is flowing on. It’s claimed in class that this is actually a consequence of surface tension only! That’s not at all intuitive, but will be covered when we learn about “stability conditions”.
# Study note.
Memorizing the NS equation is required for midterm, but more complex stuff (like cylindrical forms of the strain tensor if required) will be given.
# References
[1] D.J. Acheson. Elementary fluid dynamics. Oxford University Press, USA, 1990.
## PHY454H1S Continuum Mechanics. Lecture 10: Navier-Stokes equation. Taught by Prof. K. Das.
Posted by peeterjoot on February 11, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Review. Newtonian fluid.
Reading: section 6.* from [1].
We stated the model for a newtonian fluid
\begin{aligned}\sigma_{ij} = -p \delta_{ij} + 2 \mu e_{ij}\end{aligned} \hspace{\stretch{1}}(2.1)
and started considering conservation of mass with a volume $dV$ through an area element $d\mathbf{s}$. For the rate of change of mass flowing out of the volume $V$ is
\begin{aligned}\oint \rho \mathbf{u} \cdot d\mathbf{s} = - \frac{\partial {}}{\partial {t}} \int_V \rho dV.\end{aligned} \hspace{\stretch{1}}(2.2)
Application of Green’s theorem, for a fixed (in time) volume $V$ produces
\begin{aligned}0 = \int_V \left( \boldsymbol{\nabla} \cdot (\rho \mathbf{u}) + \frac{\partial {\rho}}{\partial {t}} \right) dV,\end{aligned} \hspace{\stretch{1}}(2.3)
or in differential form for an infinitesimal volume
\begin{aligned}0 = \frac{\partial {\rho}}{\partial {t}} + \boldsymbol{\nabla} \cdot (\rho \mathbf{u}).\end{aligned} \hspace{\stretch{1}}(2.4)
Expanding out the divergence term using
\begin{aligned}\boldsymbol{\nabla} \cdot (a \mathbf{b})&=\partial_i (a b_i) \\ &=b_i \partial_i a +a \partial_i b_i \\ &=\mathbf{b} \cdot \boldsymbol{\nabla} a+ a \boldsymbol{\nabla} \cdot \mathbf{b}\end{aligned}
\begin{aligned}0 = \frac{\partial {\rho}}{\partial {t}} + \rho \boldsymbol{\nabla} \cdot \mathbf{u}+ \mathbf{u} \cdot \boldsymbol{\nabla} \rho.\end{aligned} \hspace{\stretch{1}}(2.7)
For an incompressible fluid
\begin{aligned}\boldsymbol{\nabla} \cdot \mathbf{u} = 0\end{aligned} \hspace{\stretch{1}}(2.6)
so the conservation of mass equality relation takes the form
\begin{aligned}0 = \frac{\partial {\rho}}{\partial {t}} + \mathbf{u} \cdot \boldsymbol{\nabla} \rho.\end{aligned} \hspace{\stretch{1}}(2.7)
# Conservation of momentum.
In classical mechanics we have
\begin{aligned}\mathbf{f} = m \mathbf{a},\end{aligned} \hspace{\stretch{1}}(3.8)
our analogue here is found in terms of the stress tensor
\begin{aligned}\int_V F_i dV = \int_V \frac{\partial {\sigma_{ij}}}{\partial {x_j}} dV\end{aligned} \hspace{\stretch{1}}(3.9)
Here $F_i$ is the force per unit volume. With body forces we have
\begin{aligned}F_i = \rho \frac{du_i}{dt} = \frac{\partial {\sigma_{ij}}}{\partial {x_j}} + \rho f_i\end{aligned} \hspace{\stretch{1}}(3.10)
where $f_i$ is an external force per unit volume. Observe that $\sigma_{ij}$, through the constituative relation, includes both contributions of linear displacement and the vorticity component.
From the constitutive relation 2.1, we have
\begin{aligned}\frac{\partial {\sigma_{ij}}}{\partial {x_j}} &= - \frac{\partial {p}}{\partial {x_j}} \delta_{ij} + 2 \mu \frac{\partial {e_{ij}}}{\partial {x_j}} \\ &= - \frac{\partial {p}}{\partial {x_i}} + 2 \mu \frac{\partial {}}{\partial {x_j}} \left( \frac{1}{{2}} \left( \frac{\partial {u_i}}{\partial {x_j}}+ \frac{\partial {u_j}}{\partial {x_i}}\right)\right) \\ &= - \frac{\partial {p}}{\partial {x_i}} + \mu \left(\frac{\partial^2 u_i}{\partial x_j \partial x_j}+\frac{\partial^2 u_j}{\partial x_i \partial x_j}\right) \end{aligned}
Observe that the term
\begin{aligned}\frac{\partial^2 u_i}{\partial x_j \partial x_j}\end{aligned} \hspace{\stretch{1}}(3.11)
is the $i^{\text{th}}$ component of $\boldsymbol{\nabla}^2 \mathbf{u}$, whereas
\begin{aligned}\frac{\partial^2 u_j}{\partial x_i \partial x_j} &= \frac{\partial {}}{\partial {x_i}} \left( \frac{\partial {u_j}}{\partial {x_j}} \right) \\ &= \frac{\partial {}}{\partial {x_i}} (\boldsymbol{\nabla} \cdot \mathbf{u})\end{aligned}
is the $i^{\text{th}}$ component of $\boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{u})$.
We have therefore that
\begin{aligned}\rho \frac{du_i}{dt} = \left( -\boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^2 \mathbf{u} + \mu \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{u}) + \rho \mathbf{f}\right)_i,\end{aligned} \hspace{\stretch{1}}(3.12)
or in vector notation
\begin{aligned}\rho \frac{d\mathbf{u}}{dt} = -\boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^2 \mathbf{u} + \mu \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{u}) + \rho \mathbf{f}.\end{aligned} \hspace{\stretch{1}}(3.13)
We can expand this a bit more writing our velocity $\mathbf{u} = \mathbf{u}(x, y, z, t)$ differential
\begin{aligned}du_i = \frac{\partial {u_i}}{\partial {x_j}} \delta x_j + \frac{\partial {u_i}}{\partial {t}} \delta t.\end{aligned} \hspace{\stretch{1}}(3.14)
Considering rates
\begin{aligned}\frac{du_i}{dt} = \frac{\partial {u_i}}{\partial {x_j}} \frac{dx_j}{dt} + \frac{\partial {u_i}}{\partial {t}} .\end{aligned} \hspace{\stretch{1}}(3.15)
In vector notation we have
\begin{aligned}\frac{d\mathbf{u}}{dt} = (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} + \frac{\partial {\mathbf{u}}}{\partial {t}}.\end{aligned} \hspace{\stretch{1}}(3.16)
Newton’s second law 3.13 now becomes
\begin{aligned}\boxed{\rho (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} + \rho \frac{\partial {\mathbf{u}}}{\partial {t}} = -\boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^2 \mathbf{u} + \mu \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{u}) + \rho \mathbf{f}.}\end{aligned} \hspace{\stretch{1}}(3.20)
This is the Navier-Stokes equation. Observe that we have an explicitly non-linear term
\begin{aligned}(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} ,\end{aligned} \hspace{\stretch{1}}(3.18)
something we don’t encounter in most classical mechanics. The impacts of this non-linear term are very significant and produce some interesting effects.
## Incompressible fluids.
Incompressibility was the condition
\begin{aligned}\boldsymbol{\nabla} \cdot \mathbf{u} = 0,\end{aligned} \hspace{\stretch{1}}(3.19)
so the Navier-Stokes equation takes the form
\begin{aligned}\begin{aligned}\rho (\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} + \rho \frac{\partial {\mathbf{u}}}{\partial {t}} &= -\boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^2 \mathbf{u} + \rho \mathbf{f} \\ \boldsymbol{\nabla} \cdot \mathbf{u} &= 0\end{aligned}\end{aligned} \hspace{\stretch{1}}(3.20)
## Boundary value conditions.
In order to solve any sort of PDE we need to consider the boundary value conditions. Consider the interface between two layers of liquids as in figure (\ref{fig:continuumL9:continuumL10fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL10fig1}
\caption{Rocker tank with two viscosity fluids.}
\end{figure}
Also found an illustration of this in fig 1.13 of white’s text online.
We see the fluids sticking together at the boundary. This is due to matching of the tangential velocities at the interface.
# References
[1] D.J. Acheson. Elementary fluid dynamics. Oxford University Press, USA, 1990.
## PHY456H1S Continuum mechanics. Problem Set 1. Stress, Strain, Traction vector. Force free equilibrium.
Posted by peeterjoot on February 9, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
This problem set is as yet ungraded.
# Problem Q1.
## Statement
For the stress tensor
\begin{aligned}\sigma =\begin{bmatrix}6 & 0 & 2 \\ 0 & 1 & 1 \\ 2 & 1 & 3\end{bmatrix}\text{M Pa}\end{aligned} \hspace{\stretch{1}}(2.1)
Find the corresponding strain tensor, assuming an isotropic solid with Young’s modulus $E = 200 \times 10^9 \text{N}/\text{m}^2$ and Poisson’s ration $\nu = 0.35$.
## Solution
We need to express the relation between stress and strain in terms of Young’s modulus and Poisson’s ratio. In terms of Lam\’e parameters our model for the relations between stress and strain for an isotropic solid was given as
\begin{aligned}\sigma_{ij} = \lambda e_{kk} \delta_{ij} + 2 \mu e_{ij}.\end{aligned} \hspace{\stretch{1}}(2.2)
Computing the trace
\begin{aligned}\sigma_{kk} = (3 \lambda + 2 \mu) e_{kk},\end{aligned} \hspace{\stretch{1}}(2.3)
allows us to invert the relationship
\begin{aligned}2 \mu e_{ij} = \sigma_{ij} - \lambda \frac{\sigma_{kk}}{3 \lambda + 2 \mu} \delta_{ij}.\end{aligned} \hspace{\stretch{1}}(2.4)
In terms of Poisson’s ratio $\nu$ and Young’s modulus $E$, our Lam\’e parameters were found to be
\begin{aligned}\lambda &= \frac{ E \nu }{(1 - 2 \nu)(1 + \nu)} \\ \mu &= \frac{E}{2(1 + \nu)},\end{aligned} \hspace{\stretch{1}}(2.5)
and
\begin{aligned}3 \lambda + 2 \mu&= \frac{ 3 E \nu }{(1 - 2 \nu)(1 + \nu)} + \frac{E}{1 + \nu} \\ &= \frac{E}{1 + \nu} \left( \frac{3 \nu}{1 - 2 \nu} + 1\right) \\ &= \frac{E}{1 + \nu} \frac{1 + \nu}{1 - 2 \nu} \\ &= \frac{E}{1 - 2 \nu}.\end{aligned}
Our stress strain model for the relationship for an isotropic solid becomes
we find
\begin{aligned}\frac{E}{1 + \nu} e_{ij}&=\sigma_{ij}-\frac{ E \nu }{(1 - 2 \nu)(1 + \nu)} \frac{1 - 2 \nu}{E}\sigma_{kk} \delta_{ij} \\ &=\sigma_{ij}-\frac{ \nu }{1 + \nu}\sigma_{kk} \delta_{ij} \\ \end{aligned}
or
\begin{aligned}e_{ij}=\frac{1}{{E}}\left((1 + \nu)\sigma_{ij}-\nu\sigma_{kk} \delta_{ij}\right).\end{aligned} \hspace{\stretch{1}}(2.7)
As a sanity check note that this matches (5.12) of [1], although they use a notation of $\sigma$ instead of $\nu$ for Poisson’s ratio. We are now ready to tackle the problem. First we need the trace of the stress tensor
\begin{aligned}\sigma_{kk} = (6 + 1 + 3) \text{M Pa} = 10 \text{M Pa},\end{aligned} \hspace{\stretch{1}}(2.8)
\begin{aligned}e_{ij}&=\frac{1}{{E}}\left((1 + \nu)\begin{bmatrix}6 & 0 & 2 \\ 0 & 1 & 1 \\ 2 & 1 & 3\end{bmatrix}-10 \nu\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\right)\text{M Pa} \\ &=\frac{1}{{E}}\left(\begin{bmatrix}6 & 0 & 2 \\ 0 & 1 & 1 \\ 2 & 1 & 3\end{bmatrix}+ 0.35\begin{bmatrix}-4 & 0 & 2 \\ 0 & -9 & 1 \\ 2 & 1 & -7\end{bmatrix}\right)\text{M Pa} \\ &=\frac{1}{{2 \times 10^{5}}}\left(\begin{bmatrix}6 & 0 & 2 \\ 0 & 1 & 1 \\ 2 & 1 & 3\end{bmatrix}+ 0.35\begin{bmatrix}-4 & 0 & 2 \\ 0 & -9 & 1 \\ 2 & 1 & -7\end{bmatrix}\right)\end{aligned}
Expanding out the last bits of arithmetic the strain tensor is found to have the form
\begin{aligned}e_{ij}=\begin{bmatrix} 23 & 0 & 13.5 \\ 0 & -10.75 & 6.75 \\ 13.5 & 6.75 & 2.75\end{bmatrix} 10^{-6}.\end{aligned} \hspace{\stretch{1}}(2.9)
Note that this is dimensionless, unlike the stress.
# Problem Q2.
## Statement
Small displacement field in a material is given by
\begin{aligned}e_1 &= 2 x_1 x_2 \\ e_2 &= x_3^2 \\ e_3 &= x_1^2 - x_3\end{aligned} \hspace{\stretch{1}}(3.10)
Find
\begin{enumerate}
\item the infinitesimal strain tensor $e_{ij}$,
\item the principal strains and the corresponding principal axes at $(x_1, x_2, x_3) = (1, 2, 4)$,
\item Is the body under compression or expansion?
\end{enumerate}
## Solution. infinitesimal strain tensor $e_{ij}$
Diving right in, we have
\begin{aligned}e_{11}&= \frac{\partial {e_1}}{\partial {x_1}} \\ &= \frac{\partial {}}{\partial {x_1}}2 x_1 x_2 \\ &= 2 x_2\end{aligned}
\begin{aligned}e_{22}&= \frac{\partial {e_2}}{\partial {x_2}} \\ &= \frac{\partial {}}{\partial {x_2}} x_3^2 \\ &= 0\end{aligned}
\begin{aligned}e_{33}&= \frac{\partial {e_3}}{\partial {x_3}} \\ &= \frac{\partial {}}{\partial {x_3}} ( x_1^2 - x_3 ) \\ &= -1\end{aligned}
\begin{aligned}e_{12}&=\frac{1}{{2}} \left(\frac{\partial {e_2}}{\partial {x_1}}+\frac{\partial {e_1}}{\partial {x_2}}\right) \\ &=\frac{1}{{2}}\left(\not{{\frac{\partial {}}{\partial {x_1}} x_3^2 }}+\frac{\partial {}}{\partial {x_2}} 2 x_1 x_2\right) \\ &=x_1\end{aligned}
\begin{aligned}e_{23}&=\frac{1}{{2}} \left(\frac{\partial {e_3}}{\partial {x_2}}+\frac{\partial {e_2}}{\partial {x_3}}\right) \\ &=\frac{1}{{2}}\left(\not{{\frac{\partial {}}{\partial {x_2}} (x_1^2 - x_3 )}}+\frac{\partial {}}{\partial {x_3}} x_3^2\right) \\ &=x_3\end{aligned}
\begin{aligned}e_{31}&=\frac{1}{{2}} \left(\frac{\partial {e_1}}{\partial {x_3}}+\frac{\partial {e_3}}{\partial {x_1}}\right) \\ &=\frac{1}{{2}}\left(\not{{\frac{\partial {}}{\partial {x_3}} 2 x_1 x_2 }}+\frac{\partial {}}{\partial {x_1}} (x_1^2 - x_3 )\right) \\ &=x_1\end{aligned}
In matrix form we have
\begin{aligned}\mathbf{e} =\begin{bmatrix}2 x_2 & x_1 & x_1 \\ x_1 & 0 & x_3 \\ x_1 & x_3 & -1 \\ \end{bmatrix}\end{aligned} \hspace{\stretch{1}}(3.13)
## Solution. principle strains and axes
At the point $(1, 2, 4)$ the strain tensor has the value
\begin{aligned}\mathbf{e} =\begin{bmatrix}4 & 1 & 1 \\ 1 & 0 & 4 \\ 1 & 4 & -1\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(3.14)
We wish to diagonalize this, solving the characteristic equation for the eigenvalues $\lambda$
\begin{aligned}0 &=\begin{vmatrix}4 -\lambda & 1 & 1 \\ 1 & -\lambda & 4 \\ 1 & 4 & -1 -\lambda\end{vmatrix} \\ &=(4 -\lambda )\begin{vmatrix} -\lambda & 4 \\ 4 & -1 -\lambda\end{vmatrix}-\begin{vmatrix}1 & 1 \\ 4 & -1 -\lambda\end{vmatrix}+\begin{vmatrix}1 & 1 \\ -\lambda & 4 \\ \end{vmatrix} \\ &=(4 - \lambda)(\lambda^2 + \lambda - 16)-(-1 -\lambda - 4)+(4 + \lambda) \\ \end{aligned}
We find the characteristic equation to be
\begin{aligned}0 = -\lambda^3 + 3 \lambda^2 + 22\lambda - 55.\end{aligned} \hspace{\stretch{1}}(3.15)
This doesn’t appear to lend itself easily to manual solution (there are no obvious roots to factor out). As expected, since the matrix is symmetric, a plot (\ref{fig:continuumL8:continuumProblemSet1Q2fig1}) shows that all our roots are real
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumProblemSet1Q2fig1}
\caption{Q2. Characteristic equation.}
\end{figure}
Numerically, we determine these roots to be
\begin{aligned}\{5.19684, -4.53206, 2.33522\}\end{aligned} \hspace{\stretch{1}}(3.16)
with the corresponding basis (orthonormal eigenvectors), the principle axes are
\begin{aligned}\left\{\hat{\mathbf{p}}_1,\hat{\mathbf{p}}_2,\hat{\mathbf{p}}_3\right\}=\left\{\begin{bmatrix}0.76291 \\ 0.480082 \\ 0.433001\end{bmatrix},\begin{bmatrix}-0.010606 \\ -0.660372 \\ 0.750863\end{bmatrix},\begin{bmatrix}-0.646418 \\ 0.577433 \\ 0.498713\end{bmatrix}\right\}.\end{aligned} \hspace{\stretch{1}}(3.17)
## Solution. Is body under compression or expansion?
To consider this question, suppose that as in the previous part, we determine a basis for which our strain tensor $e_{ij} = p_i \delta_{ij}$ is diagonal with respect to that basis at a given point $\mathbf{x}_0$. We can then simplify the form of the stress tensor at that point in the object
\begin{aligned}\sigma_{ij}&=\frac{E}{1 + \nu} \left(e_{ij} + \frac{\nu}{1 - 2 \nu} e_{mm} \delta_{ij}\right) \\ &=\frac{E}{1 + \nu} \left(p_i + \frac{\nu}{1 - 2 \nu} e_{mm}\right)\delta_{ij}.\end{aligned}
We see that the stress tensor at this point is also necessarily diagonal if the strain is diagonal in that basis (with the implicit assumption here that we are talking about an isotropic material). Noting that the Poisson ratio is bounded according to
\begin{aligned}-1 \le \nu \le \frac{1}{{2}},\end{aligned} \hspace{\stretch{1}}(3.18)
so if our trace is positive (as it is in this problem for all points $x_2 > 1/2$), then any positive principle strain value will result in a positive stress along that direction). For example at the point $(1,2,4)$ of the previous part of this problem (for which $x_2 > 1/2$), we have
\begin{aligned}\sigma_{ij}=\frac{E}{1 + \nu}\begin{bmatrix}5.19684+ \frac{3 \nu}{1 - 2 \nu} & 0 & 0 \\ 0 & -4.53206+ \frac{3 \nu}{1 - 2 \nu} & 0 \\ 0 & 0 & 2.33522+ \frac{3 \nu}{1 - 2 \nu}\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(3.19)
We see that at this point the $(1,1)$ and $(3,3)$ components of stress is positive (expansion in those directions) regardless of the material, and provided that
\begin{aligned}\frac{3 \nu}{1 - 2 \nu} > 4.53206\end{aligned} \hspace{\stretch{1}}(3.20)
(i.e. $\nu > 0.375664$) the material is under expansion in all directions. For $\nu < 0.375664$ the material at that point is expanding in the $\hat{\mathbf{p}}_1$ and $\hat{\mathbf{p}}_3$ directions, but under compression in the $\hat{\mathbf{p}}_2$ directions.
(save to disk and run with either Mathematica or the free Wolfram CDF player ( http://www.wolfram.com/cdf-player/ ) )
For a Mathematica notebook that visualizes this part of this problem see https://raw.github.com/peeterjoot/physicsplay/master/notes/phy454/continuumProblemSet1Q2animated.cdf. This animates the stress tensor associated with the problem, for different points $(x,y,z)$ and values of Poisson’s ratio $\nu$, with Mathematica manipulate sliders available to alter these (as well as a zoom control to scale the graphic, keeping the orientation and scale fixed with any variation of the other parameters). This generalizes the solution of the problem (assuming I got it right for the specific $(1,2,4)$ point of the problem). The vectors are the orthonormal eigenvectors of the tensor, scaled by the magnitude of the eigenvectors of the stress tensor (also diagonal in the basis of the diagonalized strain tensor at the point in question). For those directions that are under expansive stress, I’ve colored the vectors blue, and for compressive directions, I’ve colored the vectors red.
This requires either a Mathematica client or the free Wolfram CDF player, either of which can run the notebook after it is saved to your computer’s hard drive.
# Problem Q3.
## Statement
The stress tensor at a point has components given by
\begin{aligned}\sigma =\begin{bmatrix}1 & -2 & 2 \\ -2 & 3 & 1 \\ 2 & 1 & -1\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(4.21)
Find the traction vector across an area normal to the unit vector
\begin{aligned}\hat{\mathbf{n}} = ( \sqrt{2} \mathbf{e}_1 - \mathbf{e}_2 + \mathbf{e}_3)/2\end{aligned} \hspace{\stretch{1}}(4.22)
Can you construct a tangent vector $\boldsymbol{\tau}$ on this plane by inspection? What are the components of the force per unit area along the normal $\hat{\mathbf{n}}$ and tangent $\boldsymbol{\tau}$ on that surface? (hint: projection of the traction vector.)
## Solution
The traction vector, the force per unit volume that holds a body in equilibrium, in coordinate form was
\begin{aligned}P_i = \sigma_{ik} n_k\end{aligned} \hspace{\stretch{1}}(4.23)
where $n_k$ was the coordinates of the normal to the surface with area $df_k$. In matrix form, this is just
\begin{aligned}\mathbf{P} = \sigma \hat{\mathbf{n}},\end{aligned} \hspace{\stretch{1}}(4.24)
so our traction vector for this stress tensor and surface normal is just
\begin{aligned}\mathbf{P} &=\frac{1}{{2}}\begin{bmatrix}1 & -2 & 2 \\ -2 & 3 & 1 \\ 2 & 1 & -1\end{bmatrix}\begin{bmatrix}\sqrt{2} \\ -1 \\ 1\end{bmatrix} \\ &=\frac{1}{{2}}\begin{bmatrix}\sqrt{2} + 2 + 2 \\ -2\sqrt{2} - 3 + 1 \\ 2\sqrt{2} - 1 -1\end{bmatrix} \\ &=\begin{bmatrix}\sqrt{2}/2 + 2 \\ -\sqrt{2} -1 \\ \sqrt{2} - 1\end{bmatrix}\end{aligned}
We also want a vector in the plane, and can pick
\begin{aligned}\boldsymbol{\tau} = \frac{1}{{\sqrt{2}}}\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(4.25)
or
\begin{aligned}\boldsymbol{\tau}' = \begin{bmatrix}\frac{1}{{\sqrt{2}}} \\ \frac{1}{{2}} \\ -\frac{1}{{2}}\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(4.26)
It’s clear that either of these is normal to $\hat{\mathbf{n}}$ (the first can also be computed by normalizing $\hat{\mathbf{n}} \times \mathbf{e}_1$, and the second with one round of Gram-Schmidt). However, neither of these vectors in the plane are particularly interesting since they are completely arbitrary. Let’s instead compute the projection and rejection of the traction vector with respect to the normal. We find for the projection
\begin{aligned}(\mathbf{P} \cdot \hat{\mathbf{n}}) \hat{\mathbf{n}}&=\frac{1}{{4}}\left(\begin{bmatrix}\sqrt{2}/2 + 2 \\ -\sqrt{2} -1 \\ \sqrt{2} - 1\end{bmatrix}\cdot \begin{bmatrix}\sqrt{2} \\ -1 \\ 1\end{bmatrix} \right)\begin{bmatrix}\sqrt{2} \\ -1 \\ 1\end{bmatrix} \\ &=\frac{1}{{4}}\left( 1 + 2\sqrt{2}+\sqrt{2} +1 +\sqrt{2} - 1\right)\begin{bmatrix}\sqrt{2} \\ -1 \\ 1\end{bmatrix} \\ &=\frac{1}{{2}}\left( 1 + 4\sqrt{2}\right)\hat{\mathbf{n}}\end{aligned}
Our rejection, the component of the traction vector in the plane, is
\begin{aligned}(\mathbf{P} \wedge \hat{\mathbf{n}}) \hat{\mathbf{n}} &=\mathbf{P} - (\mathbf{P} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}} \\ &=\frac{1}{{2}}\begin{bmatrix}\sqrt{2}/2 + 2 \\ -\sqrt{2} -1 \\ \sqrt{2} - 1\end{bmatrix}-\frac{1}{{4}}(1 + r \sqrt{2})\begin{bmatrix}\sqrt{2} \\ -1 \\ 1\end{bmatrix} \\ &=\frac{1}{{4}}\begin{bmatrix}\sqrt{2} \\ -3 \\ -5\end{bmatrix}\end{aligned}
This gives us a another vector perpendicular to the normal $\hat{\mathbf{n}}$
\begin{aligned}\hat{\boldsymbol{\tau}} = \frac{1}{{6}}\begin{bmatrix}\sqrt{2} \\ -3 \\ -5\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(4.27)
Wrapping up, we find the decomposition of the traction vector in the direction of the normal and its projection onto the plane to be
\begin{aligned}\mathbf{P} = \frac{1}{{2}}(1 + 4\sqrt{2}) \hat{\mathbf{n}}+\frac{3}{2} \hat{\boldsymbol{\tau}}.\end{aligned} \hspace{\stretch{1}}(4.28)
The components we can read off by inspection.
# Problem Q4.
## Statement
The stress tensor of a body is given by
\begin{aligned}\sigma =\begin{bmatrix}A \cos x & y^2 & C x \\ y^2 & B \sin y & z \\ C x & z & z^3\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(5.29)
Determine the constant $A$, $B$, and $C$ if the body is in equilibrium.
## Solution
In the absence of external forces our equilibrium condition was
\begin{aligned}\partial_k \sigma_{ik} = 0.\end{aligned} \hspace{\stretch{1}}(5.30)
In matrix form we wish to operate (to the left) with the gradient coordinate vector
\begin{aligned}0 &= \sigma \stackrel{ \leftarrow }{\boldsymbol{\nabla}} \\ &=\begin{bmatrix}A \cos x & y^2 & C x \\ y^2 & B \sin y & z \\ C x & z & z^3\end{bmatrix}\begin{bmatrix}\stackrel{ \leftarrow }{\partial}_x \\ \stackrel{ \leftarrow }{\partial}_y \\ \stackrel{ \leftarrow }{\partial}_z \\ \end{bmatrix} \\ &=\begin{bmatrix}\partial_x (A \cos x) + \partial_y(y^2) + \not{{\partial_z(C x)}} \\ \not{{\partial_x (y^2)}} + \partial_y(B \sin y) + \partial_z(z) \\ \partial_x (C x) + \not{{\partial_y(z)}} + \partial_z(z^3)\end{bmatrix} \\ &=\begin{bmatrix}-A \sin x + 2 y \\ B \cos y + 1 \\ C + 3 z^2 \end{bmatrix} \\ \end{aligned}
So, our conditions for equilibrium will be satisfied when we have
\begin{aligned}A &= \frac{2 y }{\sin x} \\ B &= -\frac{1}{\cos y} \\ C &= -3 z^2,\end{aligned} \hspace{\stretch{1}}(5.31)
provided $y \ne 0$, and $y \ne \pi/2 + n\pi$ for integer $n$. If equilibrium is to hold along the $y = 0$ plane, then we must either also have $A = 0$ or also impose the restriction $x = m \pi$ (for integer $m$).
# A couple other mathematica notebooks
Some of the hand calculations done in this problem set I’ve confirmed using Mathematica. Those notebooks are available here
These all require either a Mathematica client or the free Wolfram CDF player. Note that I haven’t figured out a way to get a browser based CDF player to play these without explicit download.
# References
[1] L.D. Landau, EM Lifshitz, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. 1960.
## PHY454H1S Continuum Mechanics. Lecture 7: P-waves and S-waves. Taught by Prof. K. Das.
Posted by peeterjoot on February 1, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Setup
Reading: section 22 from [1].
We got as far as expressing the vector displacement $\mathbf{e}$ for an isotropic material at a given point in terms of the Lam\’e parameters
\begin{aligned}\rho \frac{\partial^2 {\mathbf{e}}}{\partial {{t}}^2} = (\lambda + \mu) \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{e}) + \mu \boldsymbol{\nabla}^2 \mathbf{e}.\end{aligned} \hspace{\stretch{1}}(2.1)
## P-waves.
Operating on this with the divergence once more, and writing $\theta = \boldsymbol{\nabla} \cdot \mathbf{e}$, we have
\begin{aligned}\rho \frac{\partial^2 {{\boldsymbol{\nabla} \cdot \mathbf{e}}}}{\partial {{t}}^2} = (\lambda + \mu) \boldsymbol{\nabla} \cdot \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{e}) + \mu \boldsymbol{\nabla}^2 (\boldsymbol{\nabla} \cdot \mathbf{e})\end{aligned} \hspace{\stretch{1}}(2.2)
or
\begin{aligned}\frac{\partial^2 {{\theta}}}{\partial {{t}}^2} = \frac{\lambda + 2 \mu}{\rho} \boldsymbol{\nabla}^2 \theta.\end{aligned} \hspace{\stretch{1}}(2.3)
We see that our divergence is governed by a wave equation where the speed of the wave $C_L$ is specified by
\begin{aligned}C_L^2 = \frac{\lambda + 2 \mu}{\rho},\end{aligned} \hspace{\stretch{1}}(2.4)
so the displacement wave equation is given by
\begin{aligned}\frac{\partial^2 {{\theta}}}{\partial {{t}}^2} = C_L^2 \boldsymbol{\nabla}^2 \theta.\end{aligned} \hspace{\stretch{1}}(2.5)
Let’s look at the divergence of the displacement vector in some more detail. By definition this is just
\begin{aligned}\boldsymbol{\nabla} \cdot \mathbf{e} = \frac{\partial {e_1}}{\partial {x_1}}+\frac{\partial {e_2}}{\partial {x_2}}+\frac{\partial {e_3}}{\partial {x_3}}.\end{aligned} \hspace{\stretch{1}}(2.6)
Recall that the strain tensor $e_{ij}$ was defined as
\begin{aligned}e_{ij} = \frac{1}{{2}} \left( \frac{\partial {e_i}}{\partial {x_j}}+\frac{\partial {e_j}}{\partial {x_i}}\right),\end{aligned} \hspace{\stretch{1}}(2.7)
so we have
\begin{aligned}\frac{\partial {e_1}}{\partial {x_1}} &= e_{11} \\ \frac{\partial {e_2}}{\partial {x_2}} &= e_{22} \\ \frac{\partial {e_3}}{\partial {x_3}} &= e_{33}.\end{aligned} \hspace{\stretch{1}}(2.8)
So the divergence in question can be written in terms of the strain tensor
\begin{aligned}\boldsymbol{\nabla} \cdot \mathbf{e} = e_{11}+e_{22}+e_{33} = e_{ii}.\end{aligned} \hspace{\stretch{1}}(2.11)
We also found that the trace of the strain tensor was the relative change in volume. We call this the dilatation. A measure of change in volume as illustrated (badly) in figure (\ref{fig:continuumL7:continuumL7fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL7fig1}
\caption{Illustrating changes in a control volume.}
\end{figure}
This idea can be found nicely animated in the wikipedia page [2].
## S-waves.
Now let’s operate on our equation 2.1 with the curl operator
\begin{aligned}\rho \frac{\partial^2 {{\boldsymbol{\nabla} \times \mathbf{e}}}}{\partial {{t}}^2} = (\lambda + \mu) \boldsymbol{\nabla} \times (\boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{e})) + \mu \boldsymbol{\nabla}^2 (\boldsymbol{\nabla} \times \mathbf{e}).\end{aligned} \hspace{\stretch{1}}(2.12)
Writing
\begin{aligned}\boldsymbol{\omega} = \boldsymbol{\nabla} \times \mathbf{e},\end{aligned} \hspace{\stretch{1}}(2.13)
and observing that $\boldsymbol{\nabla} \times \boldsymbol{\nabla} f = 0$ (with $f = \boldsymbol{\nabla} \cdot \mathbf{e}$), we find
\begin{aligned}\rho \frac{\partial^2 {{\boldsymbol{\omega}}}}{\partial {{t}}^2} = \mu \boldsymbol{\nabla}^2 \boldsymbol{\omega}.\end{aligned} \hspace{\stretch{1}}(2.14)
We call this the S-wave equation, and write $C_T$ for the speed of this wave
\begin{aligned}C_T^2 = \mu,\end{aligned} \hspace{\stretch{1}}(2.15)
so that we have
\begin{aligned}\frac{\partial^2 {{\boldsymbol{\omega}}}}{\partial {{t}}^2} = C_T^2 \boldsymbol{\nabla}^2 \boldsymbol{\omega}.\end{aligned} \hspace{\stretch{1}}(2.16)
Again, we can find nice animations of this on wikipedia [3].
## Relative speeds of the p-waves and s-waves.
Taking ratios of the wave speeds we find
\begin{aligned}\frac{C_L}{C_T} = \sqrt{\frac{ \lambda + 2 \mu}{\mu}} = \sqrt{ \frac{\lambda}{\mu} + 2}.\end{aligned} \hspace{\stretch{1}}(2.17)
Since both $\lambda > 0$ and $\mu > 0$, we have
\begin{aligned}C_L > C_T.\end{aligned} \hspace{\stretch{1}}(2.18)
Divergence (p-waves) are faster than rotational (s-waves) waves.
In terms of the Poisson ratio $\nu = \lambda/(2(\lambda + \mu))$, we find
\begin{aligned}\frac{\mu}{\lambda} = \frac{1}{{2 \nu}} - 1.\end{aligned} \hspace{\stretch{1}}(2.19)
we see that Poisson’s ratio characterizes the speeds of the waves for the medium
\begin{aligned}\frac{C_L}{C_T} = \sqrt{\frac{2(1-\nu)}{1 - 2\nu}}\end{aligned} \hspace{\stretch{1}}(2.20)
## Assuming a gradient plus curl representation.
Let’s assume that our displacement can be written in terms of a gradient and curl as we do for the electric field
\begin{aligned}\mathbf{e} = \boldsymbol{\nabla} \phi + \boldsymbol{\nabla} \times \mathbf{H},\end{aligned} \hspace{\stretch{1}}(2.21)
Inserting this into 2.1 we find
\begin{aligned}\rho \frac{\partial^2 {{(\boldsymbol{\nabla} \phi + \boldsymbol{\nabla} \times \mathbf{H})}}}{\partial {{t}}^2} = (\lambda + \mu) \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot (\boldsymbol{\nabla} \phi + \boldsymbol{\nabla} \times \mathbf{H})) + \mu \boldsymbol{\nabla}^2 (\boldsymbol{\nabla} \phi + \boldsymbol{\nabla} \times \mathbf{H}).\end{aligned} \hspace{\stretch{1}}(2.22)
using
\begin{aligned}\boldsymbol{\nabla} \cdot \boldsymbol{\nabla} \phi = \boldsymbol{\nabla}^2 \phi.\end{aligned} \hspace{\stretch{1}}(2.23)
Observe that
\begin{aligned}\boldsymbol{\nabla} \cdot (\boldsymbol{\nabla} \times \mathbf{H}) &=\partial_k (\partial_a H_b \epsilon_{abk})&=0\end{aligned}
Here we make use of the fact that an antisymmetric sum of symmetric partials is zero assuming sufficient continuity. Grouping terms we have
\begin{aligned}\boldsymbol{\nabla} \left(\rho \frac{\partial^2 {{\phi}}}{\partial {{t}}^2} - (\lambda + 2\mu) \boldsymbol{\nabla}^2 \phi\right)+ \boldsymbol{\nabla} \times \left(\rho \frac{\partial^2 {\mathbf{H}}}{\partial {{t}}^2} - \mu \boldsymbol{\nabla}^2 \mathbf{H}\right)= 0.\end{aligned} \hspace{\stretch{1}}(2.24)
When the material is infinite in scope, so that boundary value coupling is not a factor, we can write this as a set of independent P-wave and S-wave equations
\begin{aligned}\rho \frac{\partial^2 {{\phi}}}{\partial {{t}}^2} - (\lambda + 2\mu) \boldsymbol{\nabla}^2 \phi = 0\end{aligned} \hspace{\stretch{1}}(2.25)
The P-wave is irrotational (curl free).
\begin{aligned}\rho \frac{\partial^2 {\mathbf{H}}}{\partial {{t}}^2} - \mu \boldsymbol{\nabla}^2 \mathbf{H} = 0\end{aligned} \hspace{\stretch{1}}(2.26)
The S-wave is solenoidal (divergence free).
## A couple summarizing statements.
\begin{itemize}
\item
P-waves: irrotational. Volume not preserved.
\item
S-waves: divergence freee. Shearing forces are present and volume is preserved.
\item
P-waves are faster than S-waves.
\end{itemize}
# References
[1] L.D. Landau, EM Lifshitz, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. 1960.
[2] Wikipedia. P-wave — wikipedia, the free encyclopedia [online]. 2012. [Online; accessed 1-February-2012]. http://en.wikipedia.org/w/index.php?title=P-wave&oldid=474119033.
[3] Wikipedia. S-wave — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 1-February-2012]. http://en.wikipedia.org/w/index.php?title=S-wave&oldid=468110825.
## PHY454H1S Continuum Mechanics. Lecture 6: Compatibility condition and elastostatics. Taught by Prof. K. Das.
Posted by peeterjoot on January 29, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Review: Elastostatics
We’ve defined the strain tensor, where assuming the second order terms are ignored, was
\begin{aligned}e_{ij} = \frac{1}{{2}} \left( \frac{\partial {e_i}}{\partial {x_j}}+ \frac{\partial {e_j}}{\partial {x_i}} \right).\end{aligned} \hspace{\stretch{1}}(2.1)
We’ve also defined a stress tensor defined implicitly as a divergence relationship using the force per unit volume $F_i$ in direction $i$
\begin{aligned}\sigma_{ij} \leftrightarrow F_i = \frac{\partial {\sigma_{ij}}}{\partial {x_j}}.\end{aligned} \hspace{\stretch{1}}(2.2)
We’ve also discussed the constitutive relation, relating stress $\sigma_{ij}$ and strain $e_{ij}$.
We’ve also discussed linear constitutive relationships (Hooke’s law).
# 2D strain.
\begin{aligned}e_{ij} = \begin{bmatrix}e_{11} & e_{12} \\ e_{21} & e_{22}\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(3.3)
From 2.1 we see that we have
\begin{aligned}e_{11} &= \frac{\partial {e_1}}{\partial {x_1}} \\ e_{22} &= \frac{\partial {e_2}}{\partial {x_2}} \\ e_{12} = e_{21} &= \frac{1}{{2}} \left( \frac{\partial {e_2}}{\partial {x_1}}+ \frac{\partial {e_1}}{\partial {x_2}} \right).\end{aligned} \hspace{\stretch{1}}(3.4)
We have a relationship between these displacements (called the compatibility relationship), which is
\begin{aligned}\boxed{\frac{\partial^2 {{e_{11}}}}{\partial {{x_2}}^2} +\frac{\partial^2 {{e_{22}}}}{\partial {{x_1}}^2} = 2\frac{\partial^2 e_{12}}{\partial x_1 \partial x_2}.}\end{aligned} \hspace{\stretch{1}}(3.7)
We find this by straight computation
\begin{aligned}\frac{\partial^2 {{e_{11}}}}{\partial {{x_2}}^2} &= \frac{\partial^2 {{}}}{\partial {{x_2}}^2}\left( \frac{\partial {e_1}}{\partial {x_1}}\right) \\ &=\frac{\partial^3 e_1}{\partial x_1 \partial x_2^2},\end{aligned}
and
\begin{aligned}\frac{\partial^2 {{e_{22}}}}{\partial {{x_1}}^2} &= \frac{\partial^2 {{}}}{\partial {{x_1}}^2}\left( \frac{\partial {e_2}}{\partial {x_2}}\right) \\ &= \frac{\partial^3 e_2}{\partial x_2 \partial x_1^2},\end{aligned}
Now, looking at the cross term we find
\begin{aligned}2 \frac{\partial^2 e_{12}}{\partial x_1 \partial x_2} &= \frac{\partial^2 e_{12}}{\partial x_1 \partial x_2} \left(\frac{\partial {e_2}}{\partial {x_1}}+ \frac{\partial {e_1}}{\partial {x_2}} \right) \\ &=\left(\frac{\partial^3 e_1}{\partial x_1 \partial x_2^2} +\frac{\partial^3 e_2}{\partial x_2 \partial x_1^2} \right) \\ &=\frac{\partial^2 {{e_{22}}}}{\partial {{x_1}}^2} +\frac{\partial^2 {{e_{11}}}}{\partial {{x_2}}^2} \end{aligned}
This is called the compatibility condition, and ensures that we don’t have a disjoint deformation of the form in figure (\ref{fig:continuumL6:continuumL6fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL6fig1}
\caption{disjoint deformation illustrated.}
\end{figure}
# 3D strain.
While we have 9 components in the tensor, not all of these are independent. The sets above and below the diagonal can be related, as illustrated in figure (\ref{fig:continuumL6:continuumL6fig2}).
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL6fig2}
\caption{continuumL6fig2}
\end{figure}
Here we have 6 relationships between the components of the strain tensor $e_{ij}$. Deriving these will be assigned in the homework.
# Elastodynamics. Elastic waves.
Reading: Chapter III (section 22 – section 26) of the text [1].
Example: sound or water waves (i.e. waves in a solid or liquid material that comes back to its original position.)
\begin{definition}
\emph{(Elastic Wave)}
An elastic wave is a type of mechanical wave that propagates through or on the surface of a medium. The elasticity of the material provides the restoring force (that returns the material to its original state). The displacement and the restoring force are assumed to be linearly related.
\end{definition}
In symbols we say
\begin{aligned}e_i(x_j, t) \quad \mbox{related to force},\end{aligned} \hspace{\stretch{1}}(5.8)
and specifically
\begin{aligned}\rho \frac{\partial^2 {{e_i}}}{\partial {{t}}^2} = F_i = \frac{\partial {\sigma_{ij}}}{\partial {x_j}}.\end{aligned} \hspace{\stretch{1}}(5.9)
This is just Newton’s second law, $F = ma$, but expressed in terms of a unit volume.
Should we have an external body force (per unit volume) $f_i$ acting on the body then we must modify this, writing
\begin{aligned}\boxed{\rho \frac{\partial^2 {{e_i}}}{\partial {{t}}^2} = \frac{\partial {\sigma_{ij}}}{\partial {x_j}} + f_i}\end{aligned} \hspace{\stretch{1}}(5.10)
Note that we are separating out the “original” forces that produced the stress and strain on the object from any constant external forces that act on the body (i.e. a gravitational field).
With
\begin{aligned}e_{ij} = \frac{1}{{2}} \left( \frac{\partial {e_i}}{\partial {x_j}}+ \frac{\partial {e_j}}{\partial {x_i}} \right),\end{aligned} \hspace{\stretch{1}}(5.11)
we can expand the stress divergence, for the case of homogeneous deformation, in terms of the Lam\’e parameters
\begin{aligned}\sigma_{ij} = \lambda e_{kk} \delta_{ij} + 2 \mu e_{ij}.\end{aligned} \hspace{\stretch{1}}(5.12)
We compute
\begin{aligned}\frac{\partial {\sigma_{ij}}}{\partial {x_j}}&=\lambda \frac{\partial {e_{kk}}}{\partial {x_j}}\delta_{ij} + 2 \mu \frac{\partial {}}{\partial {x_j}}\frac{1}{{2}} \left( \frac{\partial {e_i}}{\partial {x_j}}+ \frac{\partial {e_j}}{\partial {x_i}} \right), \\ &=\lambda \frac{\partial {e_{kk}}}{\partial {x_i}}+ \mu \left(\frac{\partial^2 {{e_{i}}}}{\partial {{x_j}}^2}+\frac{\partial^2 e_{j} }{ \partial x_j \partial x_i}\right) \\ &=\lambda \frac{\partial {}}{\partial {x_i}}\frac{\partial {e_k}}{\partial {x_k}}+ \mu \left(\frac{\partial^2 {{e_{i}}}}{\partial {{x_j}}^2}+\frac{\partial^2 e_{k} }{ \partial x_k \partial x_i}\right) \\ &=(\lambda + \mu)\frac{\partial {}}{\partial {x_i}}\frac{\partial {e_k}}{\partial {x_k}}+ \mu \frac{\partial^2 {{e_{i}}}}{\partial {{x_j}}^2}\end{aligned}
We find, for homogeneous deformations, that the force per unit volume on our element of mass, in the absence of external forces (the body forces), takes the form
\begin{aligned}\rho \frac{\partial^2 {{e_i}}}{\partial {{t}}^2} = (\lambda + \mu) \frac{\partial^2 e_i}{\partial x_i \partial x_j}+ \mu\frac{\partial^2 e_i}{\partial x_j^2}.\end{aligned} \hspace{\stretch{1}}(5.13)
This can be seen to be equivalent to the vector relationship
\begin{aligned}\boxed{\rho \frac{\partial^2 {\mathbf{e}}}{\partial {{t}}^2} = (\lambda + \mu) \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \mathbf{e}) + \mu \boldsymbol{\nabla}^2 \mathbf{e}.}\end{aligned} \hspace{\stretch{1}}(5.14)
TODO: What form do the stress and strain tensors take in vector form?
# References
[1] L.D. Landau, EM Lifshitz, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. 1960.
## PHY454H1S Continuum Mechanics. Lecture 5: Constitutive relationship. Taught by Prof. K. Das.
Posted by peeterjoot on January 28, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Review: Cauchy Tetrahedron.
Referring to figure (\ref{fig:continuumL5:continuumL5fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL5fig1}
\caption{Cauchy tetrahedron direction cosines.}
\end{figure}
recall that we can decompose our force into components that refer to our direction cosines $n_i = \cos\phi_i$
\begin{aligned}f_1 &= \sigma_{11} n_1 + \sigma_{12} n_2 + \sigma_{13} n_3 \\ f_2 &= \sigma_{21} n_1 + \sigma_{22} n_2 + \sigma_{23} n_3 \\ f_3 &= \sigma_{31} n_1 + \sigma_{32} n_2 + \sigma_{33} n_3\end{aligned} \hspace{\stretch{1}}(2.1)
Or in tensor form
\begin{aligned}f_i = \sigma_{ij} n_j.\end{aligned} \hspace{\stretch{1}}(2.4)
We call this the traction vector and denote it in vector form as
\begin{aligned}\mathbf{T} = \boldsymbol{\sigma} \cdot \hat{\mathbf{n}}=\begin{bmatrix}\sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}\end{bmatrix}\begin{bmatrix}n_1 \\ n_2 \\ n_3\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.5)
# Constitutive relation.
Reading: section 2, section 4 and section 5 from the text [1].
We can find the relationship between stress and strain, both analytically and experimentally, and call this the Constitutive relation. We prefer to deal with ranges of distortion that are small enough that we can make a linear approximation for this relation. In general such a linear relationship takes the form
\begin{aligned}\sigma_{ij} = c_{ijkl} e_{kl}.\end{aligned} \hspace{\stretch{1}}(3.6)
Consider the number of components that we are talking about for various rank tensors
\begin{aligned}\begin{array}{l l}\mbox{latex 0^\text{th}rank tensor} & \mbox{$3^0 = 1$ components} \\ \mbox{$1^\text{st}$ rank tensor} & \mbox{$3^1 = 3$ components} \\ \mbox{$2^\text{nd}$ rank tensor} & \mbox{$3^2 = 9$ components} \\ \mbox{$3^\text{rd}$ rank tensor} & \mbox{$3^3 = 81$ components}\end{array}\end{aligned} \hspace{\stretch{1}}(3.7)
We have a lot of components, even for a linear relation between stress and strain. For isotropic materials we model the constitutive relation instead as
\begin{aligned}\boxed{\sigma_{ij} = \lambda e_{kk} \delta_{ij} + 2 \mu e_{ij}.}\end{aligned} \hspace{\stretch{1}}(3.8)
For such a modeling of the material the (measured) values $\lambda$ and $\mu$ (shear modulus or modulus of rigidity) are called the Lam\’e parameters.
It will be useful to compute the trace of the stress tensor in the form of the constitutive relation for the isotropic model. We find
\begin{aligned}\sigma_{ii}&= \lambda e_{kk} \delta_{ii} + 2 \mu e_{ii} \\ &= 3 \lambda e_{kk} + 2 \mu e_{jj},\end{aligned}
or
\begin{aligned}\sigma_{ii} = (3 \lambda + 2 \mu) e_{kk}.\end{aligned} \hspace{\stretch{1}}(3.9)
We can now also invert this, to find the trace of the strain tensor in terms of the stress tensor
\begin{aligned}e_{ii} = \frac{\sigma_{kk}}{3 \lambda + 2 \mu}\end{aligned} \hspace{\stretch{1}}(3.10)
Substituting back into our original relationship 3.8, and find
\begin{aligned}\sigma_{ij} = \lambda \frac{\sigma_{kk}}{3 \lambda + 2 \mu} \delta_{ij} + 2 \mu e_{ij},\end{aligned} \hspace{\stretch{1}}(3.12)
which finally provides an inverted expression with the strain tensor expressed in terms of the stress tensor
\begin{aligned}\boxed{2 \mu e_{ij} =\sigma_{ij} - \lambda \frac{\sigma_{kk}}{3 \lambda + 2 \mu} \delta_{ij}.}\end{aligned} \hspace{\stretch{1}}(3.12)
## Special cases.
### Hydrostatic compression
Hydrostatic compression is when we have no shear stress, only normal components of the stress matrix $\sigma_{ij}$ is nonzero. Strictly speaking we define Hydrostatic compression as
\begin{aligned}\sigma_{ij} = -p \delta_{ij},\end{aligned} \hspace{\stretch{1}}(3.13)
i.e. not only diagonal, but with all the components of the stress tensor equal.
We can write the trace of the stress tensor as
\begin{aligned}\sigma_{ii} = - 3 p = (3 \lambda + 2 \mu) e_{kk}.\end{aligned} \hspace{\stretch{1}}(3.14)
Now, from our discussion of the strain tensor $e_{ij}$ recall that we found in the limit
\begin{aligned}dV' = (1 + e_{ii}) dV,\end{aligned} \hspace{\stretch{1}}(3.15)
allowing us to express the change in volume relative to the original volume in terms of the strain trace
\begin{aligned}e_{ii} = \frac{dV' - dV}{dV}.\end{aligned} \hspace{\stretch{1}}(3.16)
Writing that relative volume difference as $\Delta V/V$ we find
\begin{aligned}- 3 p = (3 \lambda + 2 \mu) \frac{\Delta V}{V},\end{aligned} \hspace{\stretch{1}}(3.17)
or
\begin{aligned}- \frac{ p V}{\Delta V} = \left( \lambda + \frac{2}{3} \mu \right) = K,\end{aligned} \hspace{\stretch{1}}(3.18)
where $K$ is called the Bulk modulus.
### Uniaxial stress
Again illustrated in the plane as in figure (\ref{fig:continuumL5:continuumL5fig2})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL5fig2}
\caption{Uniaxial stress.}
\end{figure}
Expanding out 3.12 we have for the $1,1$ element of the strain tensor
\begin{aligned}\boldsymbol{\sigma} =\begin{bmatrix}\sigma_{11} & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(3.19)
\begin{aligned}2 \mu e_{11}&= \sigma_{11} - \frac{\lambda ( \sigma_{11} + \not{{\sigma_{22}}} ) }{3 \lambda + 2 \mu} \\ &= \sigma_{11} \frac{3 \lambda + 2 \mu - \lambda }{3 \lambda + 2 \mu} \\ &= 2 \sigma_{11} \frac{\lambda + \mu }{3 \lambda + 2 \mu}\end{aligned}
or
\begin{aligned}\frac{\sigma_{11}}{e_{11}} = \frac{\mu(3 \lambda + 2 \mu)}{\lambda + \mu } = E\end{aligned} \hspace{\stretch{1}}(3.20)
where $E$ is Young’s modulus. Young’s modulus in the text (5.3) is given in terms of the bulk modulus $K$. Using $\lambda = K - 2\mu/3$ we find
\begin{aligned}E &=\frac{\mu(3 \lambda + 2 \mu)}{\lambda + \mu } \\ &=\frac{\mu(3 (K - 2\mu/3)+ 2 \mu)}{K - 2\mu/3 + \mu } \\ &=\frac{3 K \mu}{ K + \mu/3 } \end{aligned}
\begin{aligned}\boxed{E =\frac{\mu(3 \lambda + 2 \mu)}{\lambda + \mu } =\frac{9 K \mu}{ 3 K + \mu } }\end{aligned} \hspace{\stretch{1}}(3.21)
FIXME: figure (\ref{fig:continuumL5:continuumL5fig3}) reference?
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL5fig3}
\caption{stress associated with Young’s modulus}
\end{figure}
We define Poisson’s ratio $\nu$ as the quantity
\begin{aligned}\frac{e_{22}}{e_{11}} = \frac{e_{33}}{e_{11}} = - \nu.\end{aligned} \hspace{\stretch{1}}(3.22)
Note that we are still talking about uniaxial stress here. Referring back to 3.12 we have
\begin{aligned}2 \mu e_{2 2}&= \sigma_{2 2} - \lambda \frac{\sigma_{k k}}{3 \lambda + 2 \mu} \delta_{2 2} \\ &= \sigma_{2 2} - \lambda \frac{\sigma_{k k}}{3 \lambda + 2 \mu} \\ &= - \frac{\lambda \sigma_{11}}{3 \lambda + 2 \mu}\end{aligned}
Recall (3.20) that we had
\begin{aligned}\sigma_{11} = \frac{\mu (3 \lambda + 2 \mu)}{\lambda + \mu} e_{11}.\end{aligned} \hspace{\stretch{1}}(3.23)
Inserting this gives us
\begin{aligned}2 \mu e_{22} = - \frac{\lambda}{\not{{3 \lambda + 2 \mu}}} \frac{ \mu (\not{{3 \lambda + 2\mu}})}{\lambda + \mu} e_{11}\end{aligned}
so
\begin{aligned}\boxed{\nu = -\frac{e_{22}}{e_{11}} = \frac{\lambda}{2 (\lambda + \mu)}.}\end{aligned} \hspace{\stretch{1}}(3.24)
We can also relate the Poisson’s ratio $\nu$ to the shear modulus $\mu$
\begin{aligned}\mu = \frac{E}{2(1 + \nu)}\end{aligned} \hspace{\stretch{1}}(3.25)
\begin{aligned}\lambda = \frac{E \nu}{(1 - 2 \nu)(1 + \mu)}\end{aligned} \hspace{\stretch{1}}(3.26)
\begin{aligned}e_{11} &= \frac{1}{{E}}\left( \sigma_{11} - \nu(\sigma_{22} + \sigma_{33}) \right) \\ e_{22} &= \frac{1}{{E}}\left( \sigma_{22} - \nu(\sigma_{11} + \sigma_{33}) \right) \\ e_{33} &= \frac{1}{{E}}\left( \sigma_{33} - \nu(\sigma_{11} + \sigma_{22}) \right)\end{aligned} \hspace{\stretch{1}}(3.27)
These ones are (5.14) in the text, and are easy enough to verify (not done here).
### Appendix. Computing the relation between Poisson’s ratio and shear modulus.
Young’s modulus is given in 3.21 (equation (43) in the Professor’s notes) as
\begin{aligned}E = \frac{\mu(3 \lambda + 2 \mu)}{\lambda + \mu },\end{aligned} \hspace{\stretch{1}}(3.30)
and for Poisson’s ratio 3.24 (equation (46) in the Professor’s notes) we have
\begin{aligned}\nu = -\frac{e_{22}}{e_{11}} = \frac{\lambda}{2 (\lambda + \mu)}.\end{aligned} \hspace{\stretch{1}}(3.31)
Let’s derive the other stated relationships (equation (47) in the Professor’s notes). I get
\begin{aligned}2 (\lambda + \mu) \nu = \lambda \\ \implies \\ \lambda ( 2 \nu - 1 ) = - 2\mu\nu\end{aligned}
or
\begin{aligned}\lambda = \frac{ 2 \mu \nu} { 1 - 2 \nu }\end{aligned}
For substitution into the Young’s modulus equation calculate
\begin{aligned}\lambda + \mu &= \frac{ 2 \mu \nu} { 1 - 2 \nu } + \mu \\ &= \mu \left( \frac{ 2 \nu} { 1 - 2 \nu } + 1 \right) \\ &= \mu \frac{ 2 \nu + 1 - 2 \nu} { 1 - 2 \nu } \\ &= \frac{ \mu} { 1 - 2 \nu } \\ \end{aligned}
and
\begin{aligned}3 \lambda + 2 \mu &= 3 \frac{ \mu} { 1 - 2 \nu } - \mu \\ &= \mu \frac{ 3 - (1 - 2 \nu)} { 1 - 2 \nu } \\ &= \mu \frac{ 2 + 2 \nu} { 1 - 2 \nu } \\ &= 2 \mu \frac{ 1 + \nu} { 1 - 2 \nu } \\ \end{aligned}
Putting these together we find
\begin{aligned}E &= \frac{\mu(3 \lambda + 2 \mu)}{\lambda + \mu } \\ &= \mu 2 \mu \frac{ 1 + \nu} { 1 - 2 \nu } \frac{ 1 - 2 \nu}{\mu} \\ &= 2 \mu ( 1 + \nu ) \\ \end{aligned}
Rearranging we have
\begin{aligned}\mu = \frac{E}{2 (1 + \nu)}.\end{aligned} \hspace{\stretch{1}}(3.32)
This matches (5.9) in the text (where $\sigma$ is used instead of $\nu$).
We also find
\begin{aligned}\lambda &= \frac{ 2 \mu \nu} { 1 - 2 \nu } \\ &= \frac{ \nu} { 1 - 2 \nu } \frac{E }{1 + \nu}.\end{aligned}
# References
[1] L.D. Landau, EM Lifshitz, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. 1960.
## PHY454H1S Continuum Mechanics. Lecture 4: Strain tensor components. Taught by Prof. K. Das.
Posted by peeterjoot on January 21, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Stress tensor.
Reading: Portions of this lecture cover section 2 from the text [1].
For the stress tensor
\begin{aligned}\sigma_{ij},\end{aligned} \hspace{\stretch{1}}(2.1)
a second rank tensor, the first index $i$ defines the direction of the force, and the second index $j$ defines the surface.
Observe that the dimensions of $\sigma_{ij}$ is force per unit area, just like pressure. We will in fact show that this tensor is akin to the pressure, and the diagonalized components of this tensor represent the pressure.
We’ve illustrated the stress tensor in a couple of 2D examples. The first we call uniaxial stress, having just the $1,1$ element of the matrix as illustrated in figure (\ref{fig:continuumL4:continuumL4fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig1}
\caption{Uniaxial stress}
\end{figure}
\begin{aligned}\sigma = \begin{bmatrix}\sigma_{11} & 0 \\ 0 & 0\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(2.2)
A biaxial stress is illustrated in figure (\ref{fig:continuumL4:continuumL4fig2})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig2}
\caption{Biaxial stress.}
\end{figure}
where for $\sigma_{11} \ne \sigma_{22}$ our tensor takes the form
\begin{aligned}\sigma = \begin{bmatrix}\sigma_{11} & 0 \\ 0 & \sigma_{22}\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(2.3)
In the general case we have
\begin{aligned}\sigma = \begin{bmatrix}\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(2.4)
We can attempt to illustrate this, but it becomes much harder to visualize as shown in figure (\ref{fig:continuumL4:continuumL4fig3})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig3}
\caption{General strain}
\end{figure}
In equilibrium we must have
\begin{aligned}\sigma_{12} = \sigma_{21}.\end{aligned} \hspace{\stretch{1}}(2.5)
We can use similar arguments to show that the stress tensor is symmetric.
In 3D we have three components of the stress tensor acting on each surface, as illustrated in figure (\ref{fig:continuumL4:continuumL4fig5})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig5}
\caption{Strain components on a 3D volume.}
\end{figure}
We have three unique surface orientations and three components of the force for each of these, resulting in nine components, but these are not all independent. For an object in equilibrium we must have $\sigma_{ij} = \sigma_{ji}$ (FIXME: justify?). Explicitly, that is
\begin{aligned}\sigma_{12} &= \sigma_{21} \\ \sigma_{23} &= \sigma_{32} \\ \sigma_{31} &= \sigma_{13}\end{aligned} \hspace{\stretch{1}}(2.6)
## Diagonalization
We’ll look at the two dimensional case in some detail, as in figure (\ref{fig:continuumL4:continuumL4fig6})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig6}
\caption{Area element under strain with and without rotation.}
\end{figure}
Under this coordinate transformation, a rotation, the diagonal stress tensor is taken to a non-diagonal form
\begin{aligned}\begin{bmatrix}\sigma_{11} & 0 \\ 0 & \sigma_{22} \end{bmatrix}\leftrightarrow\begin{bmatrix}\sigma_{11}' & \sigma_{12}' \\ \sigma_{21}' & \sigma_{22}' \end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.9)
## How do the stress tensor and the force relate
We form a Cauchy tetrahedron as in figure (\ref{fig:continuumL4:continuumL4fig7})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig7}
\caption{Cauchy tetrahedron}
\end{figure}
\begin{aligned}\mathbf{f} = \frac{\text{external force}}{\text{unit area}} = f_j \mathbf{e}_j\end{aligned} \hspace{\stretch{1}}(2.10)
\begin{aligned}\text{internal stress} = \text{external force}\end{aligned} \hspace{\stretch{1}}(2.11)
We write $\hat{\mathbf{n}}$ in terms of the direction cosines
\begin{aligned}\hat{\mathbf{n}} = n_1 \mathbf{e}_1 + n_2 \mathbf{e}_2 + n_3 \mathbf{e}_3 \end{aligned} \hspace{\stretch{1}}(2.12)
Here
\begin{aligned}n_1 &= \hat{\mathbf{n}} \cdot \mathbf{e}_1 \\ n_2 &= \hat{\mathbf{n}} \cdot \mathbf{e}_2 \\ n_3 &= \hat{\mathbf{n}} \cdot \mathbf{e}_3,\end{aligned} \hspace{\stretch{1}}(2.13)
or
\begin{aligned}n_j = \hat{\mathbf{n}} \cdot \mathbf{e}_j = \cos\phi_j\end{aligned} \hspace{\stretch{1}}(2.16)
Force balance on $x_1$ direction, matching total external force in this direction to the total internal force ($\sigma_{ij}'s$) as follows
\begin{aligned}\begin{aligned}f_1 \times \text{area ABC} &= \sigma_{11} \times \text{area BOC} \\ &+\sigma_{12} \times \text{area AOC} \\ &+\sigma_{13} \times \text{area AOB}\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.17)
Similarily
\begin{aligned}\begin{aligned}f_2 \times \text{area ABC} &= \sigma_{21} \times \text{area BOC} \\ &+\sigma_{22} \times \text{area AOC} \\ &+\sigma_{23} \times \text{area AOB},\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.18)
and
\begin{aligned}\begin{aligned}f_3 \times \text{area ABC} &= \sigma_{31} \times \text{area BOC} \\ &+\sigma_{32} \times \text{area AOC} \\ &+\sigma_{33} \times \text{area AOB},\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.19)
We can therefore write these force components like
\begin{aligned}f_1 = \sigma_{11} \frac{BOC}{ABC} + \sigma_{12} \frac{AOC}{ABC} + \sigma_{13} \frac{AOB}{ABC} \end{aligned} \hspace{\stretch{1}}(2.20)
but these ratios are really just the projections of the areas as illustrated in figure (\ref{fig:continuumL4:continuumL4fig8})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL4fig8}
\caption{Area projection.}
\end{figure}
where an arbitrary surface with area $\Delta S$ can be decomposed into projections
\begin{aligned}\Delta S \cos\phi_j,\end{aligned} \hspace{\stretch{1}}(2.21)
utilizing the direction cosines. We can therefore write
\begin{aligned}f_1 &= \sigma_{11} n_1 + \sigma_{12} n_2 + \sigma_{13} n_3 \\ f_2 &= \sigma_{21} n_1 + \sigma_{22} n_2 + \sigma_{23} n_3 \\ f_3 &= \sigma_{31} n_1 + \sigma_{32} n_2 + \sigma_{33} n_3,\end{aligned} \hspace{\stretch{1}}(2.22)
or in matrix notation
\begin{aligned}\begin{bmatrix}f_1 \\ f_2 \\ f_3 \end{bmatrix}=\begin{bmatrix}\sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}\begin{bmatrix}n_1 \\ n_2 \\ n_3 \\ \end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(2.25)
This is just
\begin{aligned}\boxed{f_i = \sigma_{ij} n_j.}\end{aligned} \hspace{\stretch{1}}(2.26)
This force with components $f_i$ is also called the traction vector
\begin{aligned}T_i = \sigma_{ij} n_j.\end{aligned} \hspace{\stretch{1}}(2.27)
# References
[1] L.D. Landau, EM Lifshitz, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. Physics Today, 13:44, 1960.
## PHY454H1S Continuum Mechanics. Lecture 3. Strain tensor review. Stress tensor. Taught by Prof. K. Das.
Posted by peeterjoot on January 20, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Review. Strain.
Strain is the measure of stretching. This is illustrated pictorially in figure (\ref{fig:continuumL3:continuumL3fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL3fig1}
\caption{Stretched line elements.}
\end{figure}
\begin{aligned}{ds'}^2 - ds^2 = 2 e_{ik} dx_i dx_k,\end{aligned} \hspace{\stretch{1}}(1.1)
where $e_{ik}$ is the strain tensor. We found
\begin{aligned}e_{ik} = \frac{1}{{2}} \left( \frac{\partial {e_i}}{\partial {x_k}} +\frac{\partial {e_k}}{\partial {x_i}} +\frac{\partial {e_l}}{\partial {x_i}} \frac{\partial {e_l}}{\partial {x_k}} \right)\end{aligned} \hspace{\stretch{1}}(1.2)
Why do we have a factor two? Observe that if the deformation is small we can write
\begin{aligned}{ds'}^2 - ds^2 &= (ds' - ds)(ds' + ds) \\ &\approx (ds' - ds) 2 ds\end{aligned}
so that we find
\begin{aligned}\frac{{ds'}^2 - ds^2 }{ds^2}\approx\frac{ds' - ds }{ds}\end{aligned} \hspace{\stretch{1}}(1.3)
Suppose for example, that we have a diagonalized strain tensor, then we find
\begin{aligned}{ds'}^2 - ds^2 = 2 e_{ii} \left(\frac{dx_i}{ds}\right)^2\end{aligned} \hspace{\stretch{1}}(1.4)
so that
\begin{aligned}\frac{{ds'}^2 - ds^2 }{ds^2}= 2 e_{ii} dx_i^2\end{aligned} \hspace{\stretch{1}}(1.5)
Observe that here again we see this factor of two.
If we have a diagonalized strain tensor, the tensor is of the form
\begin{aligned}\begin{bmatrix}e_{11} & 0 & 0 \\ 0 & e_{22} & 0 \\ 0 & 0 & e_{33} \end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.6)
we have
\begin{aligned}{dx_i'}^2 - dx_i^2 = 2 e_{ii} dx_i^2\end{aligned} \hspace{\stretch{1}}(1.7)
\begin{aligned}{ds'}^2 = (1 + 2 e_{11}) dx_1^2+(1 + 2 e_{22}) dx_2^2+(1 + 2 e_{33}) dx_3^2\end{aligned} \hspace{\stretch{1}}(1.8)
\begin{aligned}ds^2 = dx_1^2+dx_2^2+dx_3^2\end{aligned} \hspace{\stretch{1}}(1.9)
so
\begin{aligned}dx_1' &= \sqrt{1 + 2 e_{11}} dx_1 \sim ( 1 + e_{11}) dx_1 \\ dx_2' &= \sqrt{1 + 2 e_{22}} dx_2 \sim ( 1 + e_{22}) dx_2 \\ dx_3' &= \sqrt{1 + 2 e_{33}} dx_3 \sim ( 1 + e_{33}) dx_3\end{aligned} \hspace{\stretch{1}}(1.10)
Observe that the change in the volume element becomes the trace
\begin{aligned}dV' = dx_1'dx_2'dx_3'= dV(1 + e_{ii})\end{aligned} \hspace{\stretch{1}}(1.13)
How do we use this? Suppose that you are given a strain tensor. This should allow you to compute the stretch in any given direction.
FIXME: find problem and try this.
# Stress tensor.
Reading for this section is section 2 from the text associated with the prepared notes [1].
We’d like to consider a macroscopic model that contains the net effects of all the internal forces in the object as depicted in figure (\ref{fig:continuumL3:continuumL3fig2})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL3fig2}
\caption{Internal forces.}
\end{figure}
We will consider a volume big enough that we won’t have to consider the individual atomic interactions, only the average effects of those interactions. Will will look at the force per unit volume on a differential volume element
The total force on the body is
\begin{aligned}\iiint \mathbf{F} dV,\end{aligned} \hspace{\stretch{1}}(2.14)
where $\mathbf{F}$ is the force per unit volume. We will evaluate this by utilizing the divergence theorem. Recall that this was
\begin{aligned}\iiint (\boldsymbol{\nabla} \cdot \mathbf{A}) dV= \iint \mathbf{A} \cdot d\mathbf{s}\end{aligned} \hspace{\stretch{1}}(2.15)
We have a small problem, since we have a non-divergence expression of the force here, and it is not immediately obvious that we can apply the divergence theorem. We can deal with this by assuming that we can find a vector valued tensor, so that if we take the divergence of this tensor, we end up with the force. We introduce the quantity
\begin{aligned}\mathbf{F} = \frac{\partial {\sigma_{ik}}}{\partial {x_k}},\end{aligned} \hspace{\stretch{1}}(2.16)
and require this to be a vector. We can then apply the divergence theorem
\begin{aligned}\iiint \mathbf{F} dV = \iiint \frac{\partial {\sigma_{ik}}}{\partial {x_k}} d\mathbf{x}^3 \iint \sigma_{ik} ds_k,\end{aligned} \hspace{\stretch{1}}(2.17)
where $ds_k$ is a surface element. We identify this tensor
\begin{aligned}\sigma_{ik} = \frac{\text{Force}}{\text{Unit Area}}\end{aligned} \hspace{\stretch{1}}(2.18)
and
\begin{aligned}f_i = \sigma_{ik} ds_k,\end{aligned} \hspace{\stretch{1}}(2.19)
as the force on the surface element $ds_k$. In two dimensions this is illustrated in the following figures (\ref{fig:continuumL3:continuumL3fig3})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL3fig3}
\caption{2D strain tensor.}
\end{figure}
Observe that we use the index $i$ above as the direction of the force, and index $k$ as the direction normal to the surface.
Note that the strain tensor has the matrix form
\begin{aligned}\begin{bmatrix}\sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.20)
We will show later that this tensor is in fact symmetric.
FIXME: given some 3D forces, compute the stress tensor that is associated with it.
## Examples of the stress tensor
### Example 1. stretch in two opposing directions.
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL3fig4}
\caption{Opposing stresses in one direction.}
\end{figure}
Here, as illustrated in figure (\ref{fig:continuumL3:continuumL3fig4}), the associated (2D) stress tensor takes the simple form
\begin{aligned}\begin{bmatrix}\sigma_{11} & 0 \\ 0 & 0\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.21)
### Example 2. stretch in a pair of mutually perpendicular directions
For a pair of perpendicular forces applied in two dimensions, as illustrated in figure (\ref{fig:continuumL3:continuumL3fig5})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL3fig5}
\caption{Mutually perpendicular forces}
\end{figure}
our stress tensor now just takes the form
\begin{aligned}\begin{bmatrix}\sigma_{11} & 0 \\ 0 & \sigma_{22}\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.22)
It’s easy to imagine now how to get some more general stress tensors, should we make a change of basis that rotates our frame.
### Example 3. radial stretch
Suppose we have a fire fighter’s safety net, used to catch somebody jumping from a burning building (do they ever do that outside of movies?), as in figure (\ref{fig:continuumL3:continuumL3fig6}). Each of the firefighters contributes to the stretch.
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL3fig6} | 2019-05-25 01:56:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 467, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000097751617432, "perplexity": 8235.306862523388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257845.26/warc/CC-MAIN-20190525004721-20190525030721-00024.warc.gz"} |
http://mathoverflow.net/questions/77786?sort=newest | ## Number of the Reidemeister moves needed to transform one diagram into another one
A recent question http://mathoverflow.net/questions/77570/random-reidemeister-moves-to-unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. Lagarias show that one can transform any unknot diagram with $n$ crossings into the standard unknot diagram using not more than $2^{cn}$ Reidemeister moves, with $c=10^{11}$.
[As an aside: this is quite a large bound, so the first thing that comes to mind when one looks at it is a computer falling apart with all its atoms decaying long before it manages to untie a diagram with a single crossing. As far as I understand, for those diagrams the algorithm works faster, but still it is probably impractical for untying knots that can't be untied by trial and error.]
It seems plausible that the methods of Hass and Lagarias can be adapted to give a similar explicit upper bound for the number of the Reidemeister moves needed to transform two diagrams representing isotopic links into one another. I would like to ask whether this is indeed the case, and if so, whether there is a reference for that.
A related question: given a nonnegative integer $n$, is it possible to estimate from above the minimal $m$ such that any two link diagrams with $\leq n$ crossings that represent isotopic links can be connected by a sequence of diagrams with $\leq m$ crossings such that each is obtained from the preceding one by a Reidemeister move?
-
front.math.ucdavis.edu/1104.1882 – Agol Oct 11 2011 at 4:32
Dear Ian -- thanks a lot! if you choose to post this as an answer, I'll accept it. However, the upper bound they give is absolutely huge, and I am wondering if there is a smaller one for the maximal number of crossings the sequence of diagrams must pass through. – algori Oct 11 2011 at 5:02
Not an answer, but related: arxiv.org/pdf/math/0501490 – Scott Carter Oct 11 2011 at 13:41
I haven't read this paper of Suh's but there's a stated lower bound than the Hass and Lagarias one: front.math.ucdavis.edu/1010.4101 – Ryan Budney Oct 11 2011 at 16:41
Hass and Nowik show that the best upper bound you can hope for is quadratic in the number of crossings: arxiv.org/abs/0711.2350 – bb Oct 21 2011 at 22:45
Suppose you had a much better upper bound on the number of crossings of diagrams in the sequence of moves than their bound. Then since the number of diagrams with $c$ crossings is no more than say $k^{k^c}$ for some $k$, one would get a much better bound on the number of reidemeister moves to get between two diagrams. So I think one would need a new idea to get such an estimate. | 2013-05-23 13:00:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8405142426490784, "perplexity": 229.99498338241017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703317384/warc/CC-MAIN-20130516112157-00092-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/51683/is-enthalpy-defined-under-non-isobaric-conditions | Is enthalpy defined under non-isobaric conditions?
My question is this,
Does Enthalpy have a meaning under non-isobaric conditions?
Is its existence as a property of a system independent of whether the system is under isobaric condition or not?
Edit:- I wanted to know if enthalpy as a property of a system is valid only if from its creation to current state, it undergoes only isobaric processes.
-
The question is rather vaguely formulated but if you mean "is there a systematic method for computing the enthalpy from the equations of state?", then the answer is yes. For if these are $$T=f(p,V), \quad S=g(p,V)$$ for suitable functions $f$ and $g$ of $p$ and $V$, then it follows from the defining condition $dH=TdS + V dp$ for the enthalpy $H$ and an elementary application of the inverse function theorem and the chain rule that if we regard $H$ as a function of $p$ and $V$, then $$dH = (V+f g_1)dp + fg_2 dV.$$ (We are using subscript to denote partial derivatives with respect to $p$ and $V$). This equation can then be solved by the standard methods for exact differential equations (that the equation is exact is a consequence of the Maxwell relations). For relatively simple models, this can be done by hand---otherwise there are standard numerical methods available. For example, a two-line calculation shows that $H=\dfrac \gamma{\gamma-1} T$ for the ideal gas, which we assume to have equations of state $T=pV$ and $S=\dfrac 1{\gamma-1} \ln p + \dfrac \gamma{\gamma-1} \ln V$. In the case of the van der Waals gas (in the simplified version $$T=\left( p + \frac 1 {V^2}\right ) \left (V - 1\right ), \quad S = \frac 1 {\gamma-1}\ln \left (p + \frac 1 {V^2}\right )+ \frac \gamma{\gamma-1} \ln (V-1)),$$ the calculation is a bit messier but can be carried out by hand in a few lines (this is less work than writing down the answer in TeX).
-
Enthalpy is just a measure of the total energy of a thermodynamic system, so it always exists.
I'm not sure what you mean by non-isobaric conditions. Do you mean a system that isn't the same pressure everywhere (and therefore isn't in equilibrium) or do you mean a change during which the pressure changes. In both cases the enthalpy of the whole system has a definite value, though it may not be easy to calculate.
-
Enthalpy is as follows: $$dH=TdS+VdP$$ You can get that from the famous diagrams "Good physicists have studied under very fine teachers", (it's the best mnemotechnique rule to remember all the relations of potentials/functions). Under isobaric conditions $dP=0$ and $dH=TdS$, but if the pressure is not constant during a proccess, then enthalpy is also defined, just as before: $dH=TdS+VdP$.
- | 2015-07-07 20:18:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9260633587837219, "perplexity": 172.50050908595477}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375100481.40/warc/CC-MAIN-20150627031820-00164-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://www.ck12.org/algebra/Graphs-of-Functions-based-on-Rules/lesson/Graphs-of-Functions-based-on-Rules-ALG-I/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Graphs of Functions based on Rules
## Graph functions using value tables
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Graphs of Functions based on Rules
### Graphs of Functions based on Rules
Of course, we can always make a graph from a function rule by substituting values in for the variable and getting the corresponding output value.
#### Graphing a Function based on Rules
1. Graph the following function: \begin{align*}f(x) = |x - 2|\end{align*}
Make a table of values. Pick a variety of negative and positive values for \begin{align*}x\end{align*}. Use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}. Then, graph each of the coordinate points.
\begin{align*}x\end{align*} \begin{align*}y = f(x) = \mid x - 2 \mid\end{align*}
-4 \begin{align*}\mid -4 - 2 \mid = \mid -6 \mid = 6\end{align*}
-3 \begin{align*}\mid -3 - 2 \mid = \mid -5 \mid = 5\end{align*}
-2 \begin{align*}\mid -2 - 2 \mid = \mid -4 \mid = 4\end{align*}
-1 \begin{align*}\mid -1 - 2 \mid = \mid -3 \mid = 3\end{align*}
0 \begin{align*}\mid 0 - 2 \mid = \mid -2 \mid = 2 \end{align*}
1 \begin{align*}\mid 1 - 2 \mid = \mid -1 \mid = 1\end{align*}
2 \begin{align*}\mid 2 - 2 \mid = \mid 0 \mid = 0\end{align*}
3 \begin{align*}\mid 3 - 2 \mid = \mid 1 \mid = 1\end{align*}
4 \begin{align*}\mid 4 - 2 \mid = \mid 2 \mid = 2\end{align*}
5 \begin{align*}\mid 5 - 2 \mid = \mid 3 \mid = 3\end{align*}
6 \begin{align*}\mid 6 - 2 \mid = \mid 4 \mid = 4\end{align*}
7 \begin{align*}\mid 7 - 2 \mid = \mid 5 \mid = 5\end{align*}
8 \begin{align*}\mid 8 - 2 \mid = \mid 6 \mid = 6\end{align*}
It is wise to work with a lot of values when you begin graphing. As you learn about different types of functions, you will start to only need a few points in the table of values to create an accurate graph.
2. Graph the following function: \begin{align*}f(x)=\sqrt{x}\end{align*}
Make a table of values. We know \begin{align*}x\end{align*} can’t be negative because we can't take the square root of a negative number. The domain is all positive real numbers, so we pick a variety of positive integer values for \begin{align*}x\end{align*}. Use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}.
\begin{align*}x\end{align*} \begin{align*}y = f(x) = \sqrt{x}\end{align*}
0 \begin{align*}\sqrt{0} = 0\end{align*}
1 \begin{align*}\sqrt{1} = 1\end{align*}
2 \begin{align*}\sqrt{2} \approx 1.41\end{align*}
3 \begin{align*}\sqrt{3} \approx 1.73\end{align*}
4 \begin{align*}\sqrt{4} = 2\end{align*}
5 \begin{align*}\sqrt{5} \approx 2.24\end{align*}
6 \begin{align*}\sqrt{6} \approx 2.45\end{align*}
7 \begin{align*}\sqrt{7} \approx 2.65\end{align*}
8 \begin{align*}\sqrt{8} \approx 2.83\end{align*}
9 \begin{align*}\sqrt{9} = 3\end{align*}
Note that the range is all positive real numbers.
#### Real-World Application
The post office charges 41 cents to send a letter that is one ounce or less and an extra 17 cents for each additional ounce or fraction of an ounce. This rate applies to letters up to 3.5 ounces.
Make a table of values. We can’t use negative numbers for \begin{align*}x\end{align*} because it doesn’t make sense to have negative weight. We pick a variety of positive values for \begin{align*}x\end{align*}, making sure to include some decimal values because prices can be decimals too. Then we use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}.
\begin{align*}& x \quad 0 \quad 0.2 \quad 0.5 \quad 0.8 \quad 1 \quad 1.2 \quad 1.5 \quad 1.8 \quad 2 \quad 2.2 \quad 2.5 \quad 2.8 \quad 3 \quad 3.2 \quad 3.5\\ & y \quad 0 \quad 41 \quad \ 41 \quad \ 41 \quad 41 \quad 58 \quad 58 \quad \ 58 \quad 58 \quad 75 \quad 75 \quad \ 75 \quad 75 \quad 92 \quad 92\end{align*}
### Example
#### Example 1
Graph the following function: \begin{align*}f(x)=\sqrt{x^2}\end{align*}
Make a table of values. Even though \begin{align*}x\end{align*} can’t be negative inside the square root, because we are squaring \begin{align*}x\end{align*} first, the domain is all real numbers. So we integer values for \begin{align*}x\end{align*} which are on either side of zero. Use the function rule to find the value of \begin{align*}y\end{align*} for each value of \begin{align*}x\end{align*}.
\begin{align*}x\end{align*} \begin{align*}y = f(x) = \sqrt{x^2}\end{align*}
-2 \begin{align*}\sqrt{(-2)^2} = 2\end{align*}
-1 \begin{align*}\sqrt{(-1)^2} = 1\end{align*}
0 \begin{align*}\sqrt{0^2} = 0\end{align*}
1 \begin{align*}\sqrt{1^2} = 1\end{align*}
2 \begin{align*}\sqrt{2^2}=2\end{align*}
Note that the range is all positive real numbers, and that this looks like an absolute value function.
### Review
Graph the following functions.
1. Vanson spends $20 a month on his cat. 2. Brandon is a member of a movie club. He pays a$50 annual membership and \$8 per movie.
3. \begin{align*}f(x) = (x - 2)^2\end{align*}
4. \begin{align*}f(x) = 3.2^x\end{align*}
5. \begin{align*}f(t) = 27t-t^2\end{align*}
6. \begin{align*}f(w) = \frac{w}{4}+5\end{align*}
7. \begin{align*}f(x) = t+2t^2+3t^3\end{align*}
8. \begin{align*}f(x) = (x-1)(x+3)\end{align*}
9. \begin{align*}f(x) = \frac{x}{3}+\frac{x^2}{5}\end{align*}
10. \begin{align*}f(x) = \sqrt{2x}\end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
Coordinate Plane
The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.
coordinate point
A coordinate point is the description of a location on the coordinate plane. Coordinate points are written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.
Function
A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise. | 2016-10-27 20:34:53 | {"extraction_info": {"found_math": true, "script_math_tex": 62, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 2, "texerror": 0, "math_score": 0.9916092157363892, "perplexity": 1479.4050171217864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721392.72/warc/CC-MAIN-20161020183841-00176-ip-10-171-6-4.ec2.internal.warc.gz"} |
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# During a 7-year period the profits of Midas, Inc. changed by
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During a 7-year period the profits of Midas, Inc. changed by [#permalink]
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During a 7-year period the profits of Midas, Inc. changed by what percent from the sixth year to the seventh year?
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
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28 Mar 2010, 04:23
ankithakhu wrote:
During a 7-year period the profits of Midas, Inc. changed by what percent from the sixth year to the seventh year?
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
(p7-p6)/p6??
st 1) p7 = p3 * 1.5
Not sufficient
st 2) increase is same for subsequent years.
not sufficient
combining
let increase be a
so p7=p6 + a = p5+2a = p4 + 3a = p3 + 4a = p3*1.5
a = p3/8
but still cannot figure out (p7-p6)/p6
E
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Re: Trap question [#permalink]
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28 Mar 2010, 06:06
ankithakhu wrote:
During a 7-year period the profits of Midas, Inc. changed by what percent from the sixth year to the seventh year?
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
1 & 2 alone are insufficient.
Combining, let x be increase in profit each year.
p7 = p3+4x
p7-p3 = 4x
50 = 4x
x = 12.5%
hence sufficient.
Its C.
Whats the OA.
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Re: Trap question [#permalink]
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28 Mar 2010, 10:32
This increase is the actual increase not increase in %.
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Re: Trap question [#permalink]
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28 Mar 2010, 10:36
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The OA is C.
Combining the two statements, Let P be the Profit in the third year and x be the increase in profit which is constant. Thus profit in the 6th year = P+3x
and Profit in the 7th year = P+4x
Also from statement 1 Profit in 7th year =1.5 times profit in 3rd year
thus P+4x = 1.5P
thus x=P/8
Hence the percentage increase of 7th year over 6th can now be known.
Hope this is clear.
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Re: Trap question [#permalink]
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29 Mar 2010, 01:32
Good question Ankit
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Re: Trap question [#permalink]
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05 Apr 2010, 08:16
Damn... I was caught in the trap.
Is this a relatively high level question or not?
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Re: Trap question [#permalink]
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05 Apr 2010, 09:35
No its a simple question. which reinstates the fact that one needs to be really careful, alert and consider all possibilities.
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Re: During a 7-year period the profits of Midas, Inc. changed by [#permalink]
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25 Aug 2013, 05:33
1
1
ankithakhu wrote:
During a 7-year period the profits of Midas, Inc. changed by what percent from the sixth year to the seventh year?
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
This is an interesting old question,But I think needs expert opinion.
we need to know the profits of 6th and 7th year in order to answer this question.
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
Nothing about the sixth year so Insufficient
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
Initially I thought this was sufficient as profits of 6 and 7 year seemed to be same, but that is not so.
The original profits of 6 and 7 year may be different but increase may be same, hence we do not know that actual 6 and 7 year profits.
1+2
from 1
Let 3 year profit be x and 7 year profit is 1.5x still we know nothing about 6 year
from 2 : increase in profits were the same each year , so let the increase in profits be y each year
year ---- profit
3 ------- x
4 ------- x+y
5 ------- x+y+y = x+2y
6 ------- x+3y
7 ------- x+4y
Now we know 6 year profit is x+3y and 7 year profit is x+4y
also we know x+4y=1.5x ( 7 year profit is 1.5 times 3 year profit, we have taken 3 year profit to be x)
on solving x+4y=1.5x we get x= 8y
now 6 year profit is 11y ( from chart above )and 7 year profit is 12y
so profit % increase from 6 to 7 year $$\frac{12y-11y}{11y}*100 =\frac{100}{11} = 9.09$$
We know C is is the answer, but can any one tell me if I have arrived at the correct percentage change in profit?
Thanks
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Re: During a 7-year period the profits of Midas, Inc. changed by [#permalink]
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25 Aug 2013, 09:28
1
stne wrote:
ankithakhu wrote:
During a 7-year period the profits of Midas, Inc. changed by what percent from the sixth year to the seventh year?
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
This is an interesting old question,But I think needs expert opinion.
we need to know the profits of 6th and 7th year in order to answer this question.
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
Nothing about the sixth year so Insufficient
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
Initially I thought this was sufficient as profits of 6 and 7 year seemed to be same, but that is not so.
The original profits of 6 and 7 year may be different but increase may be same, hence we do not know that actual 6 and 7 year profits.
1+2
from 1
Let 3 year profit be x and 7 year profit is 1.5x still we know nothing about 6 year
from 2 : increase in profits were the same each year , so let the increase in profits be y each year
year ---- profit
3 ------- x
4 ------- x+y
5 ------- x+y+y = x+2y
6 ------- x+3y
7 ------- x+4y
Now we know 6 year profit is x+3y and 7 year profit is x+4y
also we know x+4y=1.5x ( 7 year profit is 1.5 times 3 year profit, we have taken 3 year profit to be x)
on solving x+4y=1.5x we get x= 8y
now 6 year profit is 11y ( from chart above )and 7 year profit is 12y
so profit % increase from 6 to 7 year $$\frac{12y-11y}{11y}*100 =\frac{100}{11} = 9.09$$
We know C is is the answer, but can any one tell me if I have arrived at the correct percentage change in profit?
Thanks
Yes, the actual increase is 100/11%.
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Re: During a 7-year period the profits of Midas, Inc. changed by [#permalink]
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20 Jan 2017, 10:06
mikemcgarry
Hi Mike, I need your help here...
I think that you explain thing really well, so you might help me a lot
I can't understand what is happening here.. How can we add percents?? Aren't we supposed to multiply the starting value by the percent to get the percent change? Why are we summing percents???
Omg I am so lost with this topic and it's so important on the test. Plus I am running out of questions and I just cant seem to grab the fundamentals. I am 100% sure I understand all the basics but I keep getting these Qs wrong all the time.. Please help!
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Re: During a 7-year period the profits of Midas, Inc. changed by [#permalink]
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20 Jan 2017, 14:40
1
iliavko wrote:
mikemcgarry
Hi Mike, I need your help here...
I think that you explain thing really well, so you might help me a lot
I can't understand what is happening here.. How can we add percents?? Aren't we supposed to multiply the starting value by the percent to get the percent change? Why are we summing percents???
Omg I am so lost with this topic and it's so important on the test. Plus I am running out of questions and I just cant seem to grab the fundamentals. I am 100% sure I understand all the basics but I keep getting these Qs wrong all the time.. Please help!
Dear iliavko,
My friend, thank you for your kind words. I am happy to respond.
The first thing I'll say is that when we have a series of percent changes, one after the other---say a 40%, then a 30% increase on that, then a 20% decrease--never think addition. Think multiplication.
In fact, for any percent increase or decrease, you should be thinking about these in terms of multiplication. Read this thoroughly:
Understanding Percents on the GMAT
Thus, to accomplish the proposed series of percent changes above, we would find the multipliers for each and multiply them together (I used a calculator to get the product--that's beyond what the GMAT would expect you compute in your head!)
(1.4)*(1.3)*(0.80) = 1.456
Thus, if we have an initial amount, and we increase by 40%, then increase by 30%, then decrease by 20%, that's equivalent to a 45.6% increase over the initial amount.
Now, if you read the blog thoroughly and understand all that, we can look at this problem:
During a 7-year period the profits of Midas, Inc. changed by what percent from the sixth year to the seventh year?
(1) The profits of Midas, Inc. during the seventh year were 50 percent greater than the profits during the third year.
(2) The increase in the profits of Midas, Inc. was the same for each year during the 7-year period.
The individual statements are not sufficient. Let's deal with the combined statements. Let's say that x is the percent change from year to year, written as a decimal. Thus, x would be the answer to the prompt question, because the change from year 6 to year 7 would be the same as the change from any other year.
The multiplier for a change from one year to the next is (1 + x). Let P3 be profit in the third year, P4, profit in the 4th year, etc. We know
P4 = P3*(1 + x)
P5 = P4*(1 + x) = P3*(1 + x)^2
P6 = P5*(1 + x) = P3*(1 + x)^3
P7 = P6*(1 + x) = P3*(1 + x)^4
Now, since we also know that P7 = 1.5*P3, that allows us to set up an equation:
1.5 = (1 + x)^4
We'd need a calculator to solve for the exact value, but that doesn't matter. This is DS, it's enough to know that we could solve for the exact value of x and answer the prompt question. Both statements together are sufficient. Answer = (C).
BTW, x = .10668192..., about 10.67%
Does all this make sense?
Mike
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Re: During a 7-year period the profits of Midas, Inc. changed by [#permalink]
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Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2019-01-24 03:01:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6471967697143555, "perplexity": 3848.222964181041}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584445118.99/warc/CC-MAIN-20190124014810-20190124040810-00506.warc.gz"} |
http://mathonline.wikidot.com/n-dimensional-real-projective-spaces-as-manifolds | n-Dimensional Real Projective Spaces as Manifolds
n-Dimensional Real Projective Spaces as Manifolds
Recall from the n-Dimensional Toruses as Manifolds page that the $n$-dimensional torus in $\mathbb{R}^{n+1}$can be defined as the following product
(1)
\begin{align} \quad T^n = \prod_{i=1}^{n} S_i = S_1 \times S_2 \times ... \times S_n \end{align}
We will now look at another class of manifolds called the $n$-dimensional real projective spaces.
Definition: The $n$-Dimensional Real Projective Space denoted $\mathbb{R}P^n$ is the set of all lines passing through the origin of $\mathbb{R}^{n+1}$.
The $1$-dimensional real projective space looks like:
Each line in $\mathbb{R}P^1$ passes through two points of $S^1$ and so $\mathbb{R}P^1$ can be defined by identifying an equivalence relation on antipodal points of $S^1$.
Similarly, each line in $\mathbb{R}P^2$ passes through two points in $S^2$ and so $\mathbb{R}P^2$ can be defined by identifying an equivalence relation on antipodal points of $S^2$. | 2018-06-19 16:12:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9994964599609375, "perplexity": 185.4382530316777}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863100.8/warc/CC-MAIN-20180619154023-20180619174023-00041.warc.gz"} |
https://ccssmathanswers.com/ordered-pair/ | # Ordered Pair – Definition, Facts, Examples | Equality of Ordered Pairs | How to find Ordered Pairs?
Sets are a collection of well-defined objects. In general, we can represent a set in three forms they are statement or description form, roster form, and set builder. The other easy way to represent a set in some situations is an ordered pair. As we already learned that an ordered pair in the coordinate plane has two coordinates namely x-coordinate and y-coordinate. In the same way, ordered pair in the set theory also has two elements. Check the following sections to know more about an ordered pair definition and solved questions.
## What is an Ordered Pair in Math?
The pair of elements that occur in a particular order, separated by a comma and are enclosed in brackets is called a set of ordered pairs. If a, b are two elements of a set, then it is possible to write two different pairs are (a, b) and (b, a). In (a, b) a is called the first component and b is called the second component.
If A and B are two sets and a is an element of set A and b is an element of set B, then the ordered pair of elements is (a, b). Here a is called the 1st component and b is called the 2nd component of the ordered pair. The ordered pair is used to locate a point in the coordinate system. The first integer in the ordered pair s called either x-coordinate or abscissa and the second integer is ordinate or y-coordinate.
### Equality of Ordered Pairs
Two ordered pairs are said to be equal if and only if the corresponding 1st components and 2nd components are equal. In case their corresponding components are not equal, then the ordered pairs are not equal. It means (a, b) ≠ (b, a) as their 1st, 2nd components are not equal. For example, two ordered pairs (x, y) and (m, n) are equal if x = m and y = n. Both the elements of an ordered pair can be the same but they are not equal.
### Ordered Pairs Examples
Example 1:
Calculate the values of a and b if two ordered pairs (a, b) and (7, 8) are equal.
Solution:
Given two ordered pairs are (a, b) and (7, 8)
Two ordered pairs are equal means their corresponding elements are also equal.
So, 1st components are equal, second components are equal.
a = 7 and b = 8.
Example 2:
If (2x + 5, $$\frac { y }{ 2 }$$ – 7) = (10, 16), then find the values of x and y.
Solution:
Given that,
(2x + 5, $$\frac { y }{ 2 }$$ – 7) = (10, 16)
Two ordered pairs are equal means their corresponding first components and second components are equal.
So, 2x + 5 = 10 and $$\frac { y }{ 2 }$$ – 7 = 16
2x = 10 – 5 and $$\frac { y }{ 2 }$$ = 16 + 7
2x = 5 and $$\frac { y }{ 2 }$$ = 23
x = $$\frac { 5 }{ 2 }$$ and y = 23 x 2
x = $$\frac { 5 }{ 2 }$$ and y = 46
Example 3:
Find the values of a, b if both ordered pairs (3a, 3) and (5a – 4, b + 1) are equal.
Solution:
Given two ordered pairs are (3a, 3), (5a – 4, b + 1)
Two ordered pairs are equal means their corresponding first components and second components are equal.
So, Equate 1st components
3a = 5a – 4
5a – 3a – 4 = 0
2a = 4
a = $$\frac { 4 }{ 2 }$$
a = 2
Equate second components
3 = b + 1
b = 3 – 1
b = 2
Therefore, a = 2, b = 2.
### FAQ’s on Ordered Pairs
1. What is an ordered pair in the sets?
An ordered pair has a pair of elements that are placed in a particular order and enclosed in brackets. There are formed while doing the cross product of sets.
2. What is the structure of an ordered pair?
The structure of an ordered pair is (a, b). Where a is the 1st component, b is the 2nd component of the ordered pair and a ∈ A, b ∈ B.
3. How to say 2 ordered pairs are equal?
Two ordered pairs are equal only when their corresponding component values are equal. The ordered pair (x, y) is equal to (a, b) means x = a and y = b.
Scroll to Top | 2022-01-23 19:05:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5759223699569702, "perplexity": 323.9004935710555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304309.5/warc/CC-MAIN-20220123172206-20220123202206-00278.warc.gz"} |
http://mathhelpforum.com/calculus/227798-arithmetic-sequence.html | # Math Help - Arithmetic Sequence
1. ## Arithmetic Sequence
For a positive integer, consider the arithmetic sequence:
n, n+19, n+38, n+57,.......
The number 2013 may or may not belong to the sequence. What is the smallest positive integer n such that 2013 does belong to the sequence?
A) 4 B) 7 C) 11 D) 14 E) 18
2. ## Re: Arithmetic Sequence
Originally Posted by ConnieLam
For a positive integer, consider the arithmetic sequence:
n, n+19, n+38, n+57,.......
The number 2013 may or may not belong to the sequence. What is the smallest positive integer n such that 2013 does belong to the sequence?
A) 4 B) 7 C) 11 D) 14 E) 18
This sequence is $\{x: x=n+19k, k\in \mathbb{N}$
Divide 2013 by 19 to get
$2013 = 105 \times 19 +18$
Thus 18 will be the smallest $n$ such that 2013 is an element of the sequence.
3. ## Re: Arithmetic Sequence
use formula for AP
a(m)=a+(m-1)d
=n+(m-1)19
now here we have to find n
so here if m=106
(m-1)*19=1995
and a(m)=2013 so m=106
if m=107 n=-1 which is not possible
so n=18 | 2015-09-02 02:41:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49031862616539, "perplexity": 2014.168955135415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645241661.64/warc/CC-MAIN-20150827031401-00226-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.askiitians.com/forums/Electromagnetic-Induction/16/45804/1998-physics-paper.htm | # A small square loop of wire l is placed inside a square loop of wire of side L(L>>l).The loops are co-planer and their centres coincide.The mutual inductance of the system is proportional toA)l/LB)L2 /lC)L/lD) l2 /Lplease give me the complete and simple explanation to this answer...
Arun Kumar IIT Delhi
8 years ago
$M_{12}=M_{21}$
So we can put current on any of the conductor.
We put I on bigger loop because on small loop its effect will be almost constant (since l<<L).
Now its easier to solve this.
Thanks & Regards
Arun Kumar
IIT Delhi | 2022-09-28 03:39:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6953715682029724, "perplexity": 3392.273963699402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335059.43/warc/CC-MAIN-20220928020513-20220928050513-00198.warc.gz"} |
https://www.physicsforums.com/threads/impedance-calculation.393574/ | # Impedance calculation?
1. Apr 8, 2010
### 9.8m*s^-2
1. The problem statement, all variables and given/known data
A 25-Ω resistance is connected in series with a 30-mH inductance and a 12-µF capacitor and are connected to a 90-V (rms) ac. If the frequency of the ac source is 500 Hz, calculate
(i) the current in the circuit.
(ii) the voltage across each element
(iii) the phase angle
(iv) the power dissipated in the circuit
2. Relevant equations
3. The attempt at a solution
I believe I have to begin this problem by calculating the impedance but I do not understand how to begin doing that.
Thank you.
2. Apr 8, 2010
### xcvxcvvc
$$X_L=\omega L$$
$$X_C=\frac{1}{\omega C}$$
$$Z= R + i(X_L - X_C)$$
3. Apr 8, 2010
### Fronzbot
Well Impedence for each element is given by the equations that xcvxcvvc gave except with one little adjustment:
$$Z_L=j\omega L$$
$$Z_C=\frac{1}{j\omega C}$$
(The ones he gave were reactance)
$$\omega$$ is the angular frequency which can be calculated by setting it equal to $$2\pi f$$
From there you can solve the circuit.
4. Apr 8, 2010
### xcvxcvvc
Also, just in case you don't see it:
$$j = \sqrt{-1}$$
$$\frac{1}{j}= \frac{j}{j}\frac{1}{j} =\frac{j}{-1}=-j$$
which is why that capacitive reactance is subtracted when forming the expression for impedance. | 2018-01-16 14:14:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6581260561943054, "perplexity": 1482.377295527368}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886436.25/warc/CC-MAIN-20180116125134-20180116145134-00130.warc.gz"} |
http://cvgmt.sns.it/paper/4277/ | # $\Gamma$-convergence for functionals depending on vector fields I. Integral representation and compactness.
created by pinamonti on 11 Apr 2019
modified on 13 Apr 2019
[BibTeX]
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Inserted: 11 apr 2019
Last Updated: 13 apr 2019
Year: 2019
Abstract:
Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$. We prove an integral representation for local functionals with respect to $X$ and a result of $\Gamma$-compactness for a class of integral functionals depending on $X$. | 2019-06-16 07:44:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8195998668670654, "perplexity": 910.0949627456981}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997801.20/warc/CC-MAIN-20190616062650-20190616084650-00068.warc.gz"} |
http://www.msri.org/seminars/19085/schedules/12210 | # Mathematical Sciences Research Institute
Home » Seminar » Schedules » A problem related to the ABC conjecture | 2014-04-23 20:08:48 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8620144128799438, "perplexity": 140.09037745853325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00026-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://patriciabarber.com/when-will-waqx/452e3d-how-to-find-arc-length-with-radius-and-area | This sector has a minor arc, because the angle is less than 180⁰. However, the formula for the arc length includes the central angle. The same process can be applied to functions of ; The concepts used to calculate the arc length can be generalized to find the surface area … You always need another piece of information, just the arc length is not enough - the circle could be big or small and the arc length does not indicate this. where θ is the measure of the arc (or central angle) in radians and r is the radius of the circle. Arc length is the distance between two points along a section of a curve. Area of a circular segment and a formula to calculate it from the central angle and radius. You can also use the arc length calculator to find the central angle or the circle's radius. Figure 1. formulas for arc Length, chord and area of a sector In the above formulas t is in radians. Our calculators are very handy, but we can find the. Find the arc length and area of a sector of a circle of radius $6$ cm and the centre angle $\dfrac{2 \pi}{5}$. So arc length s for an angle θ is: s = (2π R /360) x θ = π θR /180. r 2 = 144. r =12. Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a fraction of the area of the circle. I have a math problem where I'm supposed to find the radius and central angle of a circle with an arc length of 12 cm. Differentiated objectives: Developing learners will be able to calculate the angle of a sector, given its area, arc length or perimeter. All this means is that by the power of radians and proportions, the length of an arc is nothing more than the radius times the central angle! So to find the sector area, we need to, First, let’s find the fraction of the circle’s area our sector takes up. πr 2 = 144π. A central angle which is subtended by a major arc has a measure larger than 180°. Our part is 72°. It will help to be given the sector angle. We make a fraction by placing the part over the whole and we get $$\frac{72}{360}$$. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. Calculate the area of a sector: A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm². Find angle subten If you have the sector angle #theta#, and the arc length, #l# then you can find the radius. 5:00 Problem 2 Find the length of the intercepted arc of a circle with radius 9 and arc length in radians of 11Pi/12. Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). In this lesson you will find the radian measure of an angle by dividing the arc length by the radius of a circle. = 2 ⋅ 22. #r = (180 xxl)/(pi theta)# Let’s look at both of these concepts using the following problems. To use the arc length calculator, simply enter the central angle and the radius into the top two boxes. Using the entire length of the swing arm as my radius, I get the area of the swing-arm's sector (using the conversion factor to swap radians for degrees) as being: I have to remember that this is the total area swept by the swing arm. You will learn how to find the arc length of a sector, the angle of a sector or the radius of a circle. So to find the sector area, we need to find the fraction of the circle made by the central angle we know, then find the area of the total circle made by the radius we know. Circular segment - is an area of a circle which is "cut off" from the rest of the circle by a secant (chord). So, our sector area will be one fifth of the total area of the circle. You cannot find the area of a sector if you do not know the radius of the circle. Find the radius of the circle. The video provides two example problems for finding the radius of a circle given the arc length. Find the length of arc whose radius is 10.5 cm and central angle is 36 ... Area and perimeter worksheets. An arc length is just a fraction of the circumference of the entire circle. Worksheet to calculate arc length and area of sector (radians). Note that our answer will always be an area so the units will always be squared. Please help! You can try the final calculation yourself by rearranging the formula as: L = θ * r Although Archimedes had pioneered a way of finding the area beneath a curve with his "method of exhaustion", few believed it was even possible for curves to have definite lengths, as do straight lines. Sum of the angles in a triangle is 180 degree worksheet. The circumference can be found by the formula C = πd when we know the diameter and C = 2πr when we know the radius, as we do here. In other words, it’s the distance from one point on the edge of a circle to another, or just a portion of the circumference. The central angle is a quarter of a circle: 360° / 4 = 90°. Problem one finds the radius given radians, and the second problem … into the top two boxes. The area can be found by the formula A = πr2. hayharbr. For this exercise, they've given me the radius and the arc length. Let’s try an example where our central angle is 72° and our radius is 3 meters. Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m2. Then, knowing the radius and half the chord length, proceed as in method 1 above. Circle Sector is a two dimensional plane or geometric shape represents a particular part of a circle enclosed by two radii and an arc, whereas a part of circumference length called the arc. Arc Measure Definition. Just as every arc length is a fraction of the circumference of the whole circle, the, is simply a fraction of the area of the circle. The whole circle is 360°. 7 3 2 0 5) = 44 cm. Find the length of arc whose radius is 42 cm and central angle is 60°, Here central angle (θ) = 60° and radius (r) = 42 cm, Find the length of arc whose radius is 10.5 cm and central angle is 36°, Here central angle (θ) = 36° and radius (r) = 10.5 cm, Find the length of arc whose radius is 21 cm and central angle is 120°, Here central angle (θ) = 120° and radius (r) = 21 cm, Find the length of arc whose radius is 14 cm and central angle is 5°, Here central angle (θ) = 5° and radius (r) = 14 cm. Now we multiply that by $$\frac{1}{5}$$ (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared. To calculate Sector Area from Arc length and Radius, you need Arc Length (s) and radius of circle (r). Given a circle with radius r = 8 units and a sector with subtended angle measuring 45°, find the area of the sector and the length of the arc. and sector area of 50 cm^2. The whole circle is 360°. Sometimes you might need to determine the area under an arc, or the area of a sector. Arc Length = θr. Solution : 6:32 Find central angle of a circle with radius 100 and arc length is 310. Let’s say our part is 72°. Note that our units will always be a length. Favorite Answer. Circles have an area of πr 2, where r is the radius. Be careful, though; you may be able to find the radius if you have either the diameter or the circumference. We make a fraction by placing the part over the whole and we get $$\frac{72}{360}$$. If you know the length of the arc (which is a portion of the circumference), you can find what fraction of the circle the sector represents by comparing the arc length to the total circumference. and sector area of 50 cm^2. Note that our answer will always be an area so the units will always be squared. = (60°/360) ⋅ 2 ⋅ (22/7) ⋅ 42. It should be noted that the arc length is longer than the straight line distance between its endpoints. It works for arcs that are up to a semicircle, so the height you enter must be less than half the width. Including a calculator In given figure the area of an equilateral triangle A B C is 1 7 3 2 0. The derivation is much simpler for radians: By definition, 1 radian corresponds to an arc length R. K-12 students may refer the below formulas of circle sector to know what are all the input parameters are being used to find the area and arc length of a circle sector. Learn how tosolve problems with arc lengths. Thanks! Learn how tosolve problems with arc lengths. Do I need to find the central angle to set up the proportion first? How would I find it? Note that our units will always be a length. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate. An arc is a segment of a circle around the circumference. Let’s try an example where our central angle is 72° and our radius is 3 meters. It also separates the area into two segments - the … Example 2 : Find the length of arc whose radius is 10.5 cm and central angle is 36°. where: C = central angle of the arc (degree) R = is the radius of the circle π = is Pi, which is approximately 3.142 360° = Full angle. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate. C = L / r Where C is the central angle in radians L is the arc length 1 4 and 3 = 1. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you. When the groundskeeper goes from the center of the circle to the edge, he's creating a radius, which is 12 meters. If you know any two of them you can find … You can find both arc length and sector area using formulas. Now we just need to find that area. Easy! I have a math problem where I'm supposed to find the radius and central angle of a circle with an arc length of 12 cm. A sector is a part of a circle that is shaped like a piece of pizza or pie. The arc length L of a sector of angle θ in a circle of radius ‘r’ is given by. They've given me the radius and the central angle, so I can just plug straight into the formulas, and simplify to get my answers. Worksheet to calculate arc length and area of sector (radians). Properties of parallelogram worksheet. If this circle has an area of 144π, then you can solve for the radius:. And that’s what this lesson is all about! You can’t. The whole circle is 360°. Finding the arc width and height. Use the central angle calculator to find arc length. 2 Answers. Make a proportion: arc length / full circumference = sector area / area of whole circle. In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length. The corresponding sector area is $108$ cm$^2$. Secure learners will be able to calculate the radius of a sector, given its area, arc length or perimeter. The area can be found by the formula A = πr, . Answer Save. is just a fraction of the circumference of the entire circle. The wiper blade only covers the outer 60 cm of the length of the swing arm, so the inner 72 – 60 = 12 centimeters is not covered by the blade. However, the wiper blade itself does not go from the tip of the swing arm, all the way down to the pivot point; it stops short of the pivot point (or, in this mathematical context, the center of the circle). Can calculate area, arc length,chord length, height and perimeter of circular segment by radius and angle. Find its central angle, radius, and arc length, rounding to the nearest tenth. Arc Length Formula - Example 1 Discuss the formula for arc length and use it in a couple of examples. . 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Then we just multiply them together. We make a fraction by placing the part over the whole and we get $$\frac{72}{360}$$, which reduces to $$\frac{1}{5}$$. It should be noted that the arc length is longer than the straight line distance between its endpoints. of the total circle made by the radius we know. The central angle is a quarter of a circle: 360° / 4 = 90°. Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18.8495559 m. Now we multiply that by $$\frac{1}{5}$$ (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. The arc length should be in the same proportion to the circumference of the circle as the area subtended by the arc is to the area of the complete circle. It will also calculate the area of the sector with that same central angle. Finding arc length is easy as a circle is always equal to 360° and it is consisting of consecutive points lined up in 360 degree; so, if you divide the measured arc’s degree by 360°, you discover the fraction of the circle’s circumference that the arc makes up. You can find the circumference from just this piece of information, but then you’d need some other piece of info to tell you what fraction of the circumference you need to take. . Now we just need to find that circumference. It’s good practice to make sure you know how to calculate these measurements on your own. We won’t be working any examples in this section. Hence we can say that: Arc Length = (θ/360°) × Circumference Of Circle Arc Length = θr. Now, arc length is given by (θ/360) ⋅ 2 Π r = l (θ/360) ⋅ 2 ⋅ (22/7) ⋅ 45 = 27.5. θ = 35 ° Example 3 : Find the radius of the sector of area 225 cm 2 and having an arc length of 15 cm. Remember that the circumference of the whole circle is 2πR, so the Arc Length Formula above simply reduces this by dividing the arc angle to a full angle (360). The calculator will then determine the length of the arc. And you can see this is going three fourths of the way around the circle, so this arc length is going to be three fourths of the circumference. The width, height and radius of an arc are all inter-related. Arc length. Hence, perimeter is l + 2r = 27.5 + 2(45) = 117.5cm. Explanation: . So, our arc length will be one fifth of the total circumference. In this case, they've given me the radius and the subtended angle, and they want me to find the area, so I'll be using the sector-area formula. Use the central angle calculator to find arc length. Calculate the arc length according to the formula above: L = r * θ = 15 * π/4 = 11.78 cm. A radius of a circle a straight line joining the centre of a circle to any point on the circumference. Please help! A minor arc is an arc smaller than a semicircle. In the formula, r = the length of the radius, and l = the length of the arc. How do you find the Arc Length (X degrees) of the smaller sector with the given radius: 6 and the smaller sector area: 12 Pi? Our part is 72°. The video provides two example problems for finding the radius of a circle given the arc length. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. The radius is the distance from the Earth and the Sun: 149.6 million km. Problem one finds the radius given radians, and the second problem … Whenever you want to find the length of an arc of a circle (a portion of the circumference), you will use the arc length formula: Where θ equals the measure of the central angle that intercepts the arc and r equals the length of the radius. I have not attempted this question and do not understand how to solve this. On the picture: L - arc length h- height c- chord R- radius a- angle. manually. We are learning to: Calculate the angle and radius of a sector, given its area, arc length or perimeter. Then we just multiply them together. First, let’s find the fraction of the circle’s circumference our arc length is. To find the arc length for an angle θ, multiply the result above by θ: 1 x θ corresponds to an arc length (2πR/360) x θ. 5 c m 2. Find the area of the shaded region. where θ is the measure of the arc (or central angle) in radians and r is the radius of the circle. how do you find the arc length when you are given the radius and area in terms of pi. We are given the radius of the sector so we need to double this to find the diameter. In this calculator you may enter the angle in degrees, or radians or both. Relevance. Arc Length Formula - Example 1 Discuss the formula for arc length and use it in a couple of examples. Now we just need to find that circumference. person_outlineAntonschedule 2011-05-14 19:39:53. The length of an arc of a circle is $12$ cm. Remember the circumference of a circle = \ (\pi d\) and the diameter = \ (2 \times \text {radius}\). You can try the final calculation yourself by rearranging the formula as: L = θ * r So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. We will use our new found skills of finding arc length to see how one wheel can turn another, as well as how many inches a pulley can lift a weight. With each vertex of the triangle as a center, a circle is drawn with a radius equal to half the length of the side of the triangle. In order to find the area of this piece, you need to know the length of the circle's radius. So here, instead of area, we're asked to find the arc length of the partial circle, and that's we have here in this bluish color right over here, find this arc length. Here is a three-tier birthday cake 6 6 inches tall with a diameter of 10 10 inches. Let's do another example. A chord separates the circumference of a circle into two sections - the major arc and the minor arc. Let’s say our part is 72°. An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. Types of angles worksheet. Arc Length : (θ/180°) × πr. So, our arc length will be one fifth of the total circumference. First, let’s find the fraction of the circle’s circumference our arc length is. So here, instead of area, we're asked to find the arc length of the partial circle, and that's we have here in this bluish color right over here, find this arc length. How to use the calculator Enter the radius and central angle in DEGREES, RADIANS or both as positive real numbers and press "calculate". Example 1. 100πr = … 8:20 Find sector area of a circle with a radius of 9inches and central angle of 11pi/12 10:40 Find the radius of a circle. 7:06 Finding sector area in degrees 8:00 Find sector area of a circle with radius of 12 and central angle measure of 2pi/3. A major arc is an arc larger than a semicircle. This post will review two of those: arc length and sector area. Lv 7. = (1/6) ⋅ 2 ⋅ 22 ⋅ 6. 12/ (2πr) = 50 / (π r^2) cross multiply. Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m. (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared. The question is as follows: There is a circular sector that has a 33-inch perimeter and that encloses an area of 54-inch. Circular segment. So I can plug the radius and the arc length into the arc-length formula, and solve for the measure of the subtended angle. 5:55 Find the central angle in radians 6:32 Find central angle of a circle with radius 100 and arc length is 310. Length of arc = (θ/360) x 2 π r. Here central angle (θ) = 60° and radius (r) = 42 cm. The following equation is used to calculate a central angle contained by a circular arc. Proving triangle congruence worksheet. (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. A central angle which is subtended by a minor arc has a measure less than 180°. Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). So what is the circumference? Finding the radius, given the sagitta and chord If you know the sagitta length and arc width (length of the chord) you can find the radius from the formula: where: The whole circle is 360°. In order to fully understand Arc Length and Area in Calculus, you first have to know where all of it comes from. arc length and sector area formula: finding arc length of a circle: how to calculate the perimeter of a sector: how to find the area of a sector formula: how to find the radius of an arc: angle of sector formula: how to find the arc length of a sector: how to find angle of a sector: area … We can find the length of an arc by using the formula: \ [\frac {\texttheta} {360} \times \pi~\text {d}\] \ (\texttheta\) is the angle of the sector and \ (\text {d}\) is the diameter of the circle. L = (θ/180°) × πr = (θ/360°) × 2πr = (θ/360°) × 2πr = (θ/360°) × Circumference Of Circle. So we need to, of the circle made by the central angle we know, then find the. \begin{align} \displaystyle 3. First, let’s find the fraction of the circle’s area our sector takes up. Our calculators are very handy, but we can find the arc length and the sector area manually. Now we just need to find that area. Arc Length, according to Math Open Reference, is the measure of the distance along a curved line.. You can also find the area of a sector from its radius and its arc length. Area = lr/ 2 = 618.75 cm 2 (275 ⋅ r)/2 = 618.75. r = 45 cm. the radius is 5cm . So, our sector area will be one fifth of the total area of the circle. The arc length is \ (\frac {1} {4} of the full circumference. Find angle subten Let us consider a circle with radius rArc is a portion of the circle.Let the arc subtend angle θ at the centerThen,Angle at center = Length of Arc/ Radius of circleθ = l/rNote: Here angle is in radians.Let’s take some examplesIf radius of circle is 5 cm, and length of arc is 12 cm. To find the area of the sector, I need the measure of the central angle, which they did not give me. How to Find the Arc Length An arc length is just a fraction of the circumference of the entire circle. How to Find Area of a Sector. Section 3-11 : Arc Length and Surface Area Revisited. This section is here solely for the purpose of summarizing up all the arc length and surface area … The radius is the distance from the Earth and the Sun: 149.6 million km. You will learn how to find the arc length of a sector, the angle of a sector or the radius of a circle. In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length. And you can see this is going three fourths of the way around the circle, so this arc length … Then we just multiply them together. The Sector Area from Arc length and Radius is the area of the circle enclosed between two radii of circle and the arc is calculated using Area of Sector= (Arc Length*radius of circle)/2. Let us consider a circle with radius rArc is a portion of the circle.Let the arc subtend angle θ at the centerThen,Angle at center = Length of Arc/ Radius of circleθ = l/rNote: Here angle is in radians.Let’s take some examplesIf radius of circle is 5 cm, and length of arc is 12 cm. So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. 1 decade ago. It’s good practice to make sure you know how to calculate these measurements on your own. Or you can take a more “common sense” approach using what you know about circumference and area. The formula for area, A A, of a circle with radius, r, and arc length, L L, is: A = (r × L) 2 A = ( r × L) 2. Taking a limit then gives us the definite integral formula. Remember the formula for finding the circumference (perimeter) of a circle is 2r. We make a fraction by placing the part over the whole and we get $$\frac{72}{360}$$, which reduces to $$\frac{1}{5}$$. The arc length is first approximated using line segments, which generates a Riemann sum. (Use π = 3. The distance along that curved "side" is the arc length. For example, enter the width and height, then press "Calculate" to get the radius. Also calculate the area of sector ( radians ) 8:20 find sector area is $108 cm... / full circumference won ’ t be working any examples in this calculator you may enter how to find arc length with radius and area... Arc, because the angle of a sector if you have the angle... Perimeter worksheets given the radius of an equilateral triangle a B C is 1 3! Circular segment by radius and the minor arc, because the angle of a sector if need... 22 ⋅ 6 Riemann sum sector is a quarter of a circle, our sector area / area the... And angle 've given me the radius and area sector with that same central angle is 36... and! 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Cosmic Cube Powers, Mozart Symphonies Imslp, Tata Zest Axle Price, Lepton Periodic Table, Spitting Venom Lyrics, Waterfalls In Massachusetts, Bbva Stock Price Target, Bronk Stone Unlock, Sample Recommendation Letter For Scholarship, Definition Of Hypodermis, | 2021-05-09 19:37:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8870503306388855, "perplexity": 381.45208084626285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989012.26/warc/CC-MAIN-20210509183309-20210509213309-00131.warc.gz"} |
https://echasnovski.github.io/pdqr/reference/form_trans.html | Perform a transformation of pdqr-function(s) (which assumed to be independent).
form_trans(f_list, trans, ..., method = "random", n_sample = 10000,
args_new = list())
form_trans_self(f, trans, ..., method = "random", args_new = list())
## Arguments
f_list A list consisting from pdqr-function(s) and/or single number(s). Should have at least one pdqr-function (see Details). Transformation function. Should take as many (vectorized) arguments as there are elements in f_list or a single argument for form_trans_self(). Should return numeric or logical values. Extra arguments to trans. Transformation method. One of "random" or "bruteforce". Number of elements to sample. List of extra arguments for new_*() to control density(). A pdqr-function.
## Value
A pdqr-function for transformed random variable.
## Details
form_trans_self() is a thin wrapper for form_trans() that accepts a single pdqr-function instead of a list of them.
Class of output is chosen as class of first pdqr-function in f_list. Type of output is chosen to be "discrete" in case all input pdqr-functions have "discrete" type, and "continuous" otherwise.
Method "random" performs transformation using random generation of samples:
• Generates a sample of size n_sample from every element of f_list (if element is single number, it is repeated n_sample times).
• Calls trans with all generated samples (in order aligned with f_list). Note that output should be either numeric or logical and have n_sample elements (one for each combination of input values in "vectorized" fashion). So, for example, using sum directly is not possible as it returns only single number.
• Creates output pdqr-function. If output is logical, probability of being true is estimated as share of TRUE in output, and boolean pdqr-function is created (type "discrete" with "x" values equal to 0 and 1, and probabilities of being false and true respectively). If output is numeric, one of new_*() (suitable for output class) is called with arguments from args_new list.
Method "bruteforce":
• Retypes input pdqr-function to "discrete" type (using "piecelin" method).
• Computes output for every combination of "x" values (probability of which will be a product of corresponding probabilities).
• Creates pdqr-function of type "discrete" with suitable new_*() function.
• Possibly retypes to "continuous" type if output should have it (also with "piecelin" method).
• Its main advantage is that it is not random.
• It may start to be very memory consuming very quickly.
• It is usually useful when type of output function is "discrete". In case of "continuous" type, retyping from "discrete" to "continuous" might introduce big errors.
• Used "discrete" probabilities shouldn't be very small because they will be directly multiplied, which might cause numerical accuracy issues.
Pdqr methods for S3 group generic functions for more accurate implementations of most commonly used functions. Some of them are direct (without randomness) and some of them use form_trans() with "random" method.
form_regrid() to increase/decrease granularity of pdqr-functions for method "bruteforce".
Other form functions: form_estimate(), form_mix(), form_regrid(), form_resupport(), form_retype(), form_smooth(), form_tails()
## Examples
# Default "random" transformation
d_norm <- as_d(dnorm)
# More accurate result would give use of + directly with: d_norm + d_norm
d_norm_2 <- form_trans(list(d_norm, d_norm), trans = +)
plot(d_norm_2)lines(as_d(dnorm, sd = sqrt(2)), col = "red")
# Input list can have single numbers
form_trans(list(d_norm, 100), trans = +)#> Density function of continuous type
#> Support: ~[95.80216, 104.85072] (511 intervals)
# Output of trans can be logical. Next example is random version of
# d_norm >= 0.
form_trans(list(d_norm, 0), trans = >=)#> Probability mass function of discrete type
#> Support: [0, 1] (2 elements, probability of 1: 0.4952)
# Transformation with "bruteforce" method
power <- function(x, n = 1) {x^n}
p_dis <- new_p(
data.frame(x = 1:3, prob = c(0.1, 0.2, 0.7)),
type = "discrete"
)
p_dis_sq <- form_trans_self(
p_dis, trans = power, n = 2, method = "bruteforce"
)
meta_x_tbl(p_dis_sq)#> x prob cumprob
#> 1 1 0.1 0.1
#> 2 4 0.2 0.3
#> 3 9 0.7 1.0 # Compare with "random" method
p_dis_sq_rand <- form_trans_self(p_dis, trans = power, n = 2)
meta_x_tbl(p_dis_sq_rand)#> x prob cumprob
#> 1 1 0.1010 0.1010
#> 2 4 0.1998 0.3008
#> 3 9 0.6992 1.0000
# form_trans_self() is a wrapper for form_trans()
form_trans_self(d_norm, trans = function(x) {2*x})#> Density function of continuous type
#> Support: ~[-7.70272, 7.89026] (511 intervals) | 2020-02-23 03:40:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3869159519672394, "perplexity": 8779.50928879304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145746.24/warc/CC-MAIN-20200223032129-20200223062129-00482.warc.gz"} |
https://www.proofwiki.org/wiki/Category:Axioms/Axiom_of_Specification | # Category:Axioms/Axiom of Specification
This category contains axioms related to Axiom of Specification.
For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.
Because we cannot quantify over functions, we need an axiom for every condition we can express.
Therefore, this axiom is sometimes called an axiom schema, as we introduce a lot of similar axioms.
This axiom schema can be formally stated as follows:
### Set Theory
For any well-formed formula $\map P y$, we introduce the axiom:
$\forall z: \exists x: \forall y: \paren {y \in x \iff \paren {y \in z \land \map P y} }$
where each of $x$, $y$ and $z$ range over arbitrary sets.
### Class Theory
The axiom of specification in the context of class theory has a similar form:
Let $\map \phi {A_1, A_2, \ldots, A_n, x}$ be a propositional function such that:
$A_1, A_2, \ldots, A_n$ are a finite number of free variables whose domain ranges over all classes
$x$ is a free variable whose domain ranges over all sets
Then the axiom of specification gives that:
$\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \map \phi {A_1, A_2, \ldots, A_n, x} }$
where each of $B$ ranges over arbitrary classes.
## Pages in category "Axioms/Axiom of Specification"
The following 7 pages are in this category, out of 7 total. | 2022-06-30 12:48:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9310975670814514, "perplexity": 525.735304198925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103821173.44/warc/CC-MAIN-20220630122857-20220630152857-00109.warc.gz"} |
https://stackoverflow.com/questions/14237207/source-bash-script-to-another-one | # Source bash script to another one [duplicate]
Possible Duplicate:
Reliable way for a bash script to get the full path to itself?
I have bash script test.sh which use functions from another search.sh script by following lines:
source ../scripts/search.sh
<call some functions from search.sh>
Both scripts are located in git repository. search.sh in <git_root>/scripts/ directory, test.sh is located in the same directory (but, generally speaking, could be located anywhere inside <git_root> directory - I mean I can't rely on the following source search.sh approach ).
When I call test.sh script from <git_root>/scripts/ everything works well, but as soon as I change current working directory test.sh fails:
cd <git_root>/scripts/
./test.sh //OK
cd ..
./scripts/test.sh //FAILS
./scripts/test.sh: line 1: ../scripts/search.sh: No file or directory ...
Thus what I have:
1. Relative path of search.sh script towards <git_root> directory
What I want: To have ability to run test.sh from anywhere inside <git_root> without errors.
P.S.: It is not possible to use permanent absolute path to search.sh as git repository can be cloned to any location.
• It is not duplicate. I tried methods from this link. But it's not my case as here I do not need to get path of my test.sh script. – likern Jan 9 '13 at 15:02
• Perhaps I'm stating the obvious, but if you currently source a relative path from the current directory, sourcing a relative path from the script's path seems like a good improvement. The only complication is if the user moves the script to a different location altogether, but I cannot imagine how you would fix that, other than with an environment variable. – tripleee Jan 9 '13 at 16:07
• If the search.sh script is in the same directory as test.sh, then finding the path that test.sh is in will most definitely help, as then you use that path to find your other script. – Some programmer dude Jan 9 '13 at 16:55
If both the scripts are in the same directory, then if you get the directory that the running script is in, you use that as the directory to call the other script:
# Get the directory this script is in
pushd dirname $0 > /dev/null SCRIPTPATH=pwd -P popd > /dev/null # Now use that directory to call the other script source$SCRIPTPATH/search.sh
Taken from the accepted answer of the question I marked this question a duplicatre of: https://stackoverflow.com/a/4774063/440558
• Seems a lot easier to do SCRIPTPATH=$( readlink -f$0 ) – William Pursell Jan 9 '13 at 22:41
Is there a way to identify this Git repository location? An environment variable set? You could set PATH in the script itself to include the Git repository:
PATH="$GIT_REPO_LOCATION/scripts:$PATH"
. search.sh
Once the script is complete, your PATH will revert to its old value, and $GIT_REPO_LOCATION/scripts will no longer be part of the PATH. The question is finding this location to begin with. I guess you could do something like this in your script: GIT_LOCATION=$(find $HOME -name "search.sh" | head -1) GIT_SCRIPT_DIR=$(dirname $GIT_LOCATION) PATH="$GIT_SCRIPT_DIR:$PATH" . search.sh By the way, now that $PATH is set, I can call the script via search.sh and not ./search.sh which you had to do when you were in the scripts directory, and your PATH didn't include . which is the current directory (and PATH shouldn't include . because it is a security hole).
One more note, you could search for the .git directory too which might be the Git repository you're looking for:
GIT_LOCATION=$(find$HOME -name ".git" -type d | head -1)
PATH="$GIT_LOCATION:$PATH"
. search.sh
• But how to avoid problems with similar git repositories. There is no guarantee that someone did git clone <link> first-git-repo and then git clone <link> second-git-repo. Also in general case this solution can take too much time for searching this git repository. Also in general case we should use not only $HOME dir, but / (root) dir. – likern Jan 9 '13 at 15:19 • For these reasons I simplified the task. And even for this one can't find reliable approach ;) – likern Jan 9 '13 at 15:20 • You're missing a closing parenthesis on the second line of the second code block. – Ian Stapleton Cordasco Jan 9 '13 at 15:21 • @sigmavirus24 Thanks fixed – David W. Jan 9 '13 at 15:29 • @user966467 Okay, how do you know where that Git repository is located? However you find the location of that Git repository, you'll have to let your program know. – David W. Jan 9 '13 at 15:34 You could do this: # Get path the Git repo GIT_ROOT=git rev-parse --show-toplevel # Load the search functions source$GIT_ROOT/scripts/search.sh
Or like @Joachim Pileborg says, but you have to pay attention that you must know the path of this one to another script;
# Call the other script
source $SCRIPTPATH/../scripts/search.sh # Or if it is in another path source$SCRIPTPATH/../scripts/seachers/search.sh
The Apache Tomcat scripts use this approach:
# resolve links - $0 may be a softlink PRG="$0"
while [ -h "$PRG" ] ; do ls=ls -ld "$PRG"
link=expr "$ls" : '.*-> $$.*$$$'
if expr "$link" : '/.*' > /dev/null; then PRG="$link"
else
PRG=dirname "$PRG"/"$link"
fi
done
PRGDIR=dirname "$PRG" Any way, you have to put this snippet on all scripts that use other scripts. • Does any other way, without using git features as in reliable code I must use check of existence of git command. I mean in every new script I must use some duplicated code: check git existence, getting GIT_ROOT, etc... – likern Jan 9 '13 at 15:07 • What if the user isn't in the Git repository itself? Is there a way to find a Git local repository? – David W. Jan 9 '13 at 15:09 • @DavidW. the most reliable way of finding a Git local repository is to move upwards in the directory structure until you find a .git directory or hit some defined directory, e.g., $HOME, /, etc. – Ian Stapleton Cordasco Jan 9 '13 at 15:24
• @sigmavirus24 I took it from the OP that the directory he is using may be outside of the Git repository itself. For example, he's in $HOME/bin, but the Git repo is in $HOME/work/git_repo. – David W. Jan 9 '13 at 15:37
• From OP: Both scripts are located in git repository. ... What I want: To have ability to run test.sh from anywhere inside <git_root> without errors. – Ian Stapleton Cordasco Jan 9 '13 at 15:46
For the people who would rather not use git's features for finding the parent directory. If you can be sure you'll always be running the script from within the git directory, you can use something like this:
git_root=""
while /bin/true ; do
if [[ "$(pwd)" == "$HOME" ]] || [[ "$(pwd)" == "/" ]] ; then break fi if [[ -d ".git" ]] ; then git_root="$(pwd)"
break
fi
cd ..
done
I haven't tested this but it will just loop back until it hits your home directory or / and it will see if there is a .git directory in each parent directory. If there is, it sets the git_root variable and it will break out. If it doesn't find one, git_root will just be an empty string. Then you can do:
if [[ -n "$git_root" ]] ; then .${git_root}/scripts/search.sh
fi
IHTH | 2021-04-21 12:26:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5166599750518799, "perplexity": 3724.059293310396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039536858.83/warc/CC-MAIN-20210421100029-20210421130029-00576.warc.gz"} |
https://www.physicsforums.com/threads/finding-angular-frequency-in-electric-circuit.225033/ | # Finding angular frequency in electric circuit
1. Homework Statement
At a certain angular frequency, the phase difference between $$U_{2}$$ and $$U_{1}$$ is $$180^{\circ}$$.
a) Calculate this angular frequency
b) Calculate $$U_{2}$$ at this angular frequency
See the attachment for circuit configuration.
2. Homework Equations
Ohms law: $$U = Z \cdot I$$
Voltage: $$U = \hat{U} cos(\omega t + \alpha)$$
3. The Attempt at a Solution
I can't seem to find the relation between phase difference and angular frequency.
I've tried to compute the equivalent double-pole, separating the coil with voltage $$U_2$$ from the rest och the circuit, with the following result:
Idle current:
$$U_t = \frac{\omega^2 L^2 + Rj\omega L}{R^2 + \omega^2 L^2} U_1$$
Equivalent impedance:
$$Z_0 = \frac{\omega^2 RCL - R - j\omega L}{\omega^2 LC - j\omega RC}$$
And from this I've calculated $$U_{2}$$:
$$U_{2} = \frac{j\omega L}{Z_0 + j\omega L} U_t$$
But as stated, I can't find the relation between angular frequency and phase difference. How do I use the phase difference?
#### Attachments
• 3.5 KB Views: 350
Last edited: | 2019-12-16 00:09:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9552392363548279, "perplexity": 2222.255524730717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541310970.85/warc/CC-MAIN-20191215225643-20191216013643-00295.warc.gz"} |
http://physics.stackexchange.com/questions/5073/what-are-the-alternatives-to-the-higgs-mechanism/5079 | # What are the alternatives to the Higgs mechanism?
Can someone summarize, with references if possible, all of the alternatives to the simplest model (that requires only a single scalar Higgs field with the Mexican Hat potential) of spontaneous electroweak symmetry breaking?
-
The FAQ says "ask practical, answerable questions based on actual problems that you face." Does your question fit this criterion? – pho Feb 12 '11 at 22:38
The question is practical and answerable for someone who is working in particle physics and is familiar with the Higgs mechanism and its alternatives. I'm guessing a competent phenomenologist would be. It's also an "actual problem I face" because I'm studying the Standard Model and want to know what the best alternatives to the Higgs mechanism are. – dbrane Feb 12 '11 at 22:47
They are called 'Higgsless' models and you can google for the appropriate model. I vote to close this topic absent something more specific – Columbia Feb 12 '11 at 22:48
You asked for "all" not "best". There are many alternatives to the simplest model: 2 Higgs fields, 3 Higgs fields ad infinitum; technicolor models; Susy models, top condensate models, little Higgs, Higgsless and so on and so on with hundreds of papers on each topic. This is way too broad a question to expect a short useful answer. – pho Feb 12 '11 at 23:05
@Jeff, @dbrane, @Columbia: Since there are so many approaches, maybe it would be best to make this a community wiki, so that everyone can put down their favourites. – Simon Feb 13 '11 at 0:00
I'm pretty sure that even the brief summarization of all the alternatives will take a book or two. I will try to give a review of basic things from my perspective. Let me from the beginning note that the following classification is not accurate -- different classes may and do overlap.
## More scalar doublets (multiplets)
First of all one can introduce more scalar multiplets. Two Higgs doublet model (2HDM) is the most favored, because it is also naturally arises from MSSM. NHDMs are also considered.
Doublets are usually considered, because there is a basic constraint on the quantum numbers of the fields, coming from "rho parameter":
$\rho = \frac{M_W^2}{M_Z^2\cos^2\theta_w}=1$
Which can be satisfied only if $(2T+1)^2-3Y^2=1$ with the most natural solution $Y=1,T=1/2$. Of course there are other solutions, leading to bigger values, but I've never seen anyone seriously considering those.
There is still a lot of freedom to impose some extra discrete symmetries, continuous symmetries, the way these scalars interact with fermions, e.t.c., which leads to many subclasses of such models.
## Composite Higgses
The central example is the Little Higgs model where Higgs arrives as a (pseudo-)goldstone boson from some higher global symmetries. Changing the underlying symmetry one obtains the whole class of such models.
Extra gauge symmetries are also considered -- they are usually broken dynamically. Technicolor was the most popular one -- now it not so favored, while I don't think that it was refuted completely. Top condensate is another dynamical model.
## Originated from extra dimensions
Lots of geometries, compactifications, boundary conditions -- I feel completely lost with those. Most popular are higgsless models -- attempts to get rid of Higgses completely. They are usually based on some specific boundary conditions.
Of course the list in incomplete. There are a lot of different "mixtures" between those models, usually with some new funny names.
Here is a nice recent reference that reviews some of the mentioned models going more deeply into some tehnical details.
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Thanks, that last reference is very useful. – dbrane Feb 13 '11 at 15:52
There is an important alternative to the Higgs mechanism. This will be explained by first stating that photons are only massless particles when they are freely propagating. If a photon is confined to a specific volume, for example in a hypothetical reflecting box, then the photon is forced to adopt the frame of reference of the box and the confined photon exhibits the same inertia as a mass with equal energy. If there was any difference in the inertia of energy in two different forms, this would be a violation of the conservation of momentum. The explanation is that confined light exerts uniform photon pressure on the walls of the box when the box is at rest. However, when the box is accelerated there is a Doppler shift on light propagating opposite directions and more pressure is exerted on the rear wall than on the front wall. The difference is a net force which exactly equals the force that accelerating a mass of equal energy would generate. Any confined energy propagating at the speed of light generates this inertial force without the Higgs mechanism. A particle model incorporating dipole waves in spacetime has been proposed ( http://onlyspacetime.com/ ) and it generates exactly the correct inertia.
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So basically you are saying that because confined photons have an inertial mass, then it follows that particles have mass? – Kyle Kanos Mar 8 '15 at 19:45
Uhole Model
This model is based on John A. Macken´s proposal,[2] that universe is only spacetime and can be seen as a sea of energetic waves (Dipole Waves), traveling at light speed.
From Macken´s Dipole Waves (DW model), Policarpo Y. Ulianov [5] defines a fundamental particle, named Ulianov Hole (uhole),[3][4] that can seems as an elastic tube that connect two points in spacetime, changing the DW high pressure (~10^113 J/m3).
There is two kinds of Uholes: The Uhole-S that has a mass property; The Uhole-T that has an electric charge property.
The Uhole-S Model can be an alternative to the Higgs field mass generation mechanism,[1] enabling some simple deductions of Newton´s laws, associated with matter, and also explains the fact that the inertial mass is equal to gravitational mass.
References:
1- Ulianov, P. Y. “An Alternative to the Higgs Field Mass Generation Mechanism Based on a Dipole WavePressure Model”. Asian Journal of Mathematics and Physics, Volume 2013,Article ID amp0084, ISSN 2308-3131. Available at: http://scienceasia.asia/index.php/amp/article/view/84/46
2- Macken, J. A. The Universe is Only Spacetime. Particles, Fields and ForcesDerived from the Simplest Starting Assumption. Available at: www.onlyspacetime.com
3- Ulianov, P. Y. "Ulianov Sphere Network - A Digital Model for Representation of Non-Euclidean Spaces". Available at: www.vixra.org/abs/1201.0100
4- Ulianov, P. Y. "Ulianov String Theory a New Representation for Fundamental Particles". Available at: www.vixra.org/abs/1201.0101
5- Ulianov, P. Y. "Spacetime Dipole Wave Pressure and Black Holes - A New Way to Obtain the Schwarzschild Metric, without Using General Relativity Field Equations ". Asian Journal of Mathematics and Physics, Volume 2013,Article ID amp0107, ISSN 2308-3131. Available at: http://scienceasia.asia/index.php/amp/article/view/107/56
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This might be a better post if (a) you explained more about this "Uhole" theory and (b) didn't cite vixra (it's not well regarded by many). – Kyle Kanos Mar 6 '15 at 4:31 | 2016-07-27 13:30:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49462541937828064, "perplexity": 1106.60919407894}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257826907.66/warc/CC-MAIN-20160723071026-00294-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://scikit-bio.org/docs/0.5.0/generated/skbio.alignment.global_pairwise_align_protein.html | skbio.alignment.global_pairwise_align_protein¶
skbio.alignment.global_pairwise_align_protein(seq1, seq2, gap_open_penalty=11, gap_extend_penalty=1, substitution_matrix=None, penalize_terminal_gaps=False)[source]
Globally align pair of protein seqs or alignments with Needleman-Wunsch
State: Experimental as of 0.4.0.
Parameters: seq1 : Protein or TabularMSA[Protein] The first unaligned sequence(s). seq2 : Protein or TabularMSA[Protein] The second unaligned sequence(s). gap_open_penalty : int or float, optional Penalty for opening a gap (this is substracted from previous best alignment score, so is typically positive). gap_extend_penalty : int or float, optional Penalty for extending a gap (this is substracted from previous best alignment score, so is typically positive). substitution_matrix: 2D dict (or similar), optional Lookup for substitution scores (these values are added to the previous best alignment score); default is BLOSUM 50. penalize_terminal_gaps: bool, optional If True, will continue to penalize gaps even after one sequence has been aligned through its end. This behavior is true Needleman-Wunsch alignment, but results in (biologically irrelevant) artifacts when the sequences being aligned are of different length. This is False by default, which is very likely to be the behavior you want in all or nearly all cases. tuple TabularMSA object containing the aligned sequences, alignment score (float), and start/end positions of each input sequence (iterable of two-item tuples). Note that start/end positions are indexes into the unaligned sequences.
Notes
Default gap_open_penalty and gap_extend_penalty parameters are derived from the NCBI BLAST Server [R104].
The BLOSUM (blocks substitution matrices) amino acid substitution matrices were originally defined in [R105].
This function can be use to align either a pair of sequences, a pair of alignments, or a sequence and an alignment.
References
[R104] (1, 2) http://blast.ncbi.nlm.nih.gov/Blast.cgi
[R105] (1, 2) Amino acid substitution matrices from protein blocks. S Henikoff and J G Henikoff. Proc Natl Acad Sci U S A. Nov 15, 1992; 89(22): 10915-10919. | 2018-05-24 15:54:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4261411726474762, "perplexity": 8967.899410040052}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866511.32/warc/CC-MAIN-20180524151157-20180524171157-00248.warc.gz"} |
http://blog.aylien.com/tag/text-mining/ | ## Intro
The 2016 US Presidential election was one of (if not the) most controversial in the nation’s history. With the end prize being arguably the most powerful job in the world, the two candidates were always going to find themselves coming under intense media scrutiny. With more media outlets covering this election than any that have come before it, an increase in media attention and influence was a given.
But how much of an influence does the media really have on an election? Does journalistic bias sway voter opinion, or does voter opinion (such as poll results) generate journalistic bias? Does the old adage “all publicity is good publicity” ring true at election time?
“My sense is that what we have here is a feedback loop. Does media attention increase a candidate’s standing in the polls? Yes. Does a candidate’s standing in the polls increase media attention? Also yes.” -Jonathan Stray @jonathanstray
Thanks to an ever-increasing volume of media content flooding the web, paired with advances in natural language processing and text analysis capabilities, we are in a position to delve deeper into these questions than ever before, and by analyzing the final sixty days of the 2016 US Presidential election, that’s exactly what we set out to do.
## So, where did we start?
We started by building a very simple search using our News API to scan thousands of monitored news sources for articles related to the election. These articles, 170,000 in total, were then indexed automatically using our text analysis capabilities in the News API.
This meant that key data points in those articles were identified and indexed to be used for further analysis:
• Keywords
• Entities
• Concepts
• Topics
With each of the articles or stories sourced comes granular metadata such as publication time, publication source, source location, journalist name and sentiment polarity of each article. Combined, these data points provided us with an opportunity to uncover and analyze trends in news stories relating to the two presidential candidates.
We started with a simple count of how many times each candidate was mentioned from our news sources in the sixty days leading up to election day, as well as the keywords that were mentioned most.
## Keywords
By extracting keywords from the news stories we sourced, we get a picture of the key players, topics, organizations and locations that were mentioned most. We generated the interactive chart below using the following steps;
1. We called the News API using the query below.
2. We called it again, but searched for “Trump NOT Clinton”
3. Mentions of the two candidates naturally dominated in both sets of results so we removed them in order to get a better understanding of the keywords that were being used in articles written about them. We also removed some very obvious and/or repetitive words such as USA, America, White House, candidate, day, etc.
Here’s the query;
#### Most mentioned keywords in articles about Hillary Clinton
Straight away, bang in the middle of these keywords, we can see FBI and right beside it, emails.
#### Most mentioned keywords in articles about Donald Trump
Similar to Hillary, Trump’s main controversies appear most prominently in his keywords, with terms like women, video, sexual and assault all appearing prominently.
## Most media mentions
If this election was decided by the number of times a candidate was mentioned in the media, who would win? We used the following search queries to total the number of mentions from all sources over the sixty days immediately prior to election day;
Note: We could also have performed this search with a single query, but we wanted to separate the candidates for further analysis, and in doing this, we removed overlapping stories with titles that mentioned both candidates.
Here’s what we found, visualized;
#### Who was mentioned more in the media? Total mentions volume:
It may come as no surprise that Trump was mentioned considerably more than Clinton during this period, but was he consistently more prominent in the news over these sixty days, or was there perhaps a major story that has skewed the overall results? By using the Time Series endpoint, we can graph the volume of stories over time.
We generated the following chart using results from the two previous queries;
#### How media mentions for both candidates fluctuated in the final 60 days
As you would expect, the volume of mentions for each candidate fluctuates throughout the sixty day period, and to answer our previous question – yes, Donald Trump was consistently more prominent in terms of media mentions throughout this period. In fact, he was mentioned more than Hillary Clinton in 55 of the 60 days.
Let’s now take a look at some of the peak mention periods for each candidate to see if we can uncover the reasons for the spikes in media attention;
### Donald Trump
Trump’s peak period of media attention was October 10-13, as indicated by the highest red peak in the graph above. This period represented the four highest individual days of mention volume and can be attributed to the scandal that arose from sexual assault accusations and a leaked tape showing Trump making controversial comments about groping women.
The second highest peak, October 17-20, coincides with a more positive period for Trump, as a combination of a strong final presidential debate and a growing email scandal surrounding Hillary Clinton increased his media spotlight.
### Hillary Clinton
Excluding the sharp rise in mentions just before election day, Hillary’s highest volume days in terms of media mentions occurred from October 27-30 as news of the re-emergence of an FBI investigation surfaced.
So we’ve established the dates over the sixty days when each candidate was at their peak of media attention. Now we want to try establish the sentiment polarity of the stories that were being written about each candidate throughout this period. In other words, we want to know whether stories were being written in a positive, negative or neutral way. To achieve this, we performed Sentiment Analysis.
## Sentiment analysis
Sentiment Analysis is used to detect positive or negative polarity in text. Also known as opinion mining, sentiment analysis is a feature of text analysis and natural language processing (NLP) research that is increasingly growing in popularity as a multitude of use-cases emerge. Put simply, we perform Sentiment Analysis to uncover whether a piece of text is written in a positive, negative or neutral manner.
Note: The vast majority of news articles about the election will undoubtedly contain mentions of both Trump and Clinton. We therefore decided to only count stories with titles that mentioned just one candidate. We believe this significantly increases the likelihood that the article was written about that candidate. To achieve this, we generated search queries that included one candidate while excluding the other. The News API supports boolean operators, making such search queries possible.
First of all, we wanted to compare the overall sentiment of all stories with titles that mentioned just one candidate. Here are the two queries we used;
And here are the visualized results;
What am I seeing here? Blue represents articles written in a neutral manner, red in a negative manner and green in a positive manner. Again, you can hover over the graph to view more information.
#### What was the overall media sentiment towards Donald Trump?
Those of you that followed the election, to any degree, will probably not be surprised by these results. We don’t really need data to back up the claim that Trump ran the more controversial campaign and therefore generated more negative press.
Again, similar to how we previously graphed mention volumes over time, we also wanted to see how sentiment in the media fluctuated throughout this sixty day period. First we’ll look at Clinton’s mention volume and see if there is any correlation between mention volume and sentiment levels.
## Hillary Clinton
How to read this graph: The top half (blue) represents fluctuations in the number of daily media mentions (‘000’s) for Hillary Clinton. The bottom half represents fluctuations in the average sentiment polarity of the stories in which she was mentioned. Green = positive and red = negative.
You can hover your cursor over the data points to view more in-depth information.
#### Mentions Volume (top) vs. Sentiment (bottom) for Hillary Clinton
From looking at this graph, one thing becomes immediately clear; as volume increases, polarity decreases, and vice versa. What does this tell us? It tells us that perhaps Hillary was in the news for the wrong reasons too often – there were very few occasions when both volume and polarity increased simultaneously.
Hillary’s average sentiment remained positive for the majority of this period. However, that sharp dip into the red circa October 30 came just a week before election day. We must also point out the black line that cuts through the bottom half of the graph. This is a trend line representing average sentiment polarity and as you can see, it gets consistently closer to negative as election day approaches.
#### Mentions Volume (top) vs. Sentiment (bottom) for Donald Trump
Trump’s graph paints a different picture altogether. There was not a single day when his average polarity entered into the positive (green). What’s interesting to note here, however, is how little his mention volumes affected his average polarity. While there are peaks and troughs, there were no major swings in either direction, particularly in comparison to those seen on Hillary’s graph.
These results are of course open to interpretation, but what is becoming evident is that perhaps negative stories in the media did more damage to Clinton’s campaign than they did to Trump’s. While Clinton’s average sentiment polarity remained consistently more positive, Trump’s didn’t appear to be as badly affected when controversial stories emerged. He was consistently controversial!
Trumps lowest point, in terms of negative press, came just after the second presidential debate at the end of September. What came after this point is the crucial detail, however. Trump’s average polarity recovered and mostly improved for the remainder of the campaign. Perhaps critically, we see his highest and most positive averages of this period in the final 3 weeks leading up to election day.
## Sentiment from sources
At the beginning of this post we mentioned the term media bias and questioned its effect on voter opinion. While we may not be able to prove this effect, we can certainly uncover any traces of bias from media content.
What we would like to uncover is whether certain sources (ie publications) write more or less favorably about either candidate.
To test this, we’ve analyzed the sentiment of articles written about both candidates from two publications: USA Today and Fox News.
### USA Today
Query:
Similar to the overall sentiment (from all sources) displayed previously, the sentiment polarity of articles from USA Today shows consistently higher levels of negative sentiment towards Donald Trump. The larger than average percentage of neutral results indicate that USA Today took a more objective approach in their coverage of the election.
### Fox News
Again, Trump dominates in relation to negative sentiment from Fox News. However, what’s interesting to note here is that Fox produced more than double the percentage of negative story titles about Hillary Clinton than USA Today did. We also found that, percentage-wise, they produced half as many positive stories about her. Also, 3.9% of Fox’s Trump coverage was positive, versus USA Today’s 2.5%.
### Media bias?
These figures beg the question; how are two major news publications writing about the exact same news, with such varied levels of sentiment? It certainly highlights the potential influence that the media can have on voter opinion, especially when you consider how many people see each article/headline. The figures below represent social shares for a single news article;
Bear in mind, these figures don’t represent the number of people who saw the article, they represent the number of people who shared it. The actual number of people who saw this on their social feed will be a high-multiple of these figures. In fact, we grabbed the average daily social shares, per story, and graphed them to compare;
#### Average social shares per story
Pretty even, and despite Trump being mentioned over twice as many times as Clinton during this sixty day period, he certainly didn’t outperform her when it came to social shares.
## Conclusion
Since the 2016 US election was decided there has been a sharp focus on the role played by news and media outlets in influencing public opinion. While we’re not here to join the debate, we are here to show you how you can deep-dive into news content at scale to uncover some fascinating and useful insights that can help you source highly targeted and precise content, uncover trends and assist in decision making.
## Introduction
It’s certainly an exciting time be involved in Natural Language Processing (NLP), not only for those of us who are involved in the development and cutting-edge research that is powering its growth, but also for the multitude of organizations and innovators out there who are finding more and more ways to take advantage of it to gain a competitive edge within their respective industries.
With the global NLP market expected to grow to a value of $16 billion by 2021, it’s no surprise to see the tech giants of the world investing heavily and competing for a piece of the pie. More than 30 private companies working to advance artificial intelligence technologies have been acquired in the last 5 years by corporate giants competing in the space, including Google, Yahoo, Intel, Apple and Salesforce. [1] It’s not all about the big boys, however, as NLP, text analysis and text mining technologies are becoming more and more accessible to smaller organizations, innovative startups and even hobbyist programmers. NLP is helping organizations make sense of vast amounts of unstructured data, at scale, giving them a level of insight and analysis that they could have only dreamed about even just a couple of years ago. Today we’re going to take a look at 3 industries on the cusp of disruption through the adoption of AI and NLP technologies; 1. The legal industry 2. The insurance industry 3. Customer service ## NLP & Text Analysis in the Legal industry While we’re still a long long way away from robot lawyers, the current organic crop of legal professionals are already taking advantage of NLP, text mining and text analysis techniques and technologies to help them make better-informed decisions, in quicker time, by discovering key insights that can often be buried in large volumes of data, or that may seem irrelevant until analyzed at scale, uncovering strategy-boosting and often case-changing trends. Let’s take a look at two examples of how legal pro’s are leveraging NLP and text analysis technologies to their advantage; • Information retrieval in ediscovery • Contract management • Article summarization ### Information retrieval in ediscovery Ediscovery refers to discovery in legal proceedings such as litigation, government investigations, or Freedom of Information Act requests, where the information sought is in electronic format. Electronic documents are often accompanied by metadata that is not found on paper documents, such as the date and time the document was written, shared, etc. This level of minute detail can be crucial in legal proceedings. As far as NLP is concerned, ediscovery is mainly about information retrieval, aiding legal teams in their search for relevant and useful documents. In many cases, the amount of data requiring analysis can exceed 100GB, when often only 5% – 10% of it is actually relevant. With outside service bureaus charging$1,000 per GB to filter and reduce this volume, you can start to see how costs can quickly soar.
Data can be filtered and separated by extracting mentions of specific entities (people, places, currency amounts, etc), including/excluding specific timeframes and in the case of email threads, only include mails that contain mentions of the company, person or defendant in question.
### Contract management
NLP enables contract management departments to extract key information, such as currency amounts and dates, to generate reports that summarize terms across contracts, allowing for comparisons among terms for risk assessment purposes, budgeting and planning.
In cases relating to Intellectual Property disputes, attorneys are using NLP and text mining techniques to extract key information from sources such as patents and public court records to help give them an edge with their case.
### Article summarization
Legal documents can be notoriously long and tedious to read through in their entirety. Sometimes all that is required is a concise summary of the overall text to help gain an understanding of its content. Summarization of such documents is possible with NLP, where a defined number of sentences are selected from the main body of text to create, for example, a summary of the top 5 sentences that best reflect the content of the document as a whole.
## NLP & Text Analysis in the Insurance industry
Insurance providers gather massive amounts of data each day from a variety of channels, such as their website, live chat, email, social networks, agents and customer care reps. Not only is this data coming in from multiple channels, it also relates to a wide variety of issues, such as claims, complaints, policies, health reports, incident reports, customer and potential customer interactions on social media, email, live chat, phone… the list goes on and on.
The biggest issue plaguing the insurance industry is fraud. Let’s take a look at how NLP, data mining and text analysis techniques can help insurance providers tackle these key issues;
• Streamline the flow of data to the correct departments/agents
• Improve agent decision making by putting timely and accurate data in front of them
• Improve SLA response times and overall customer experience
• Assist in the detection of fraudulent claims and activity
### Streamlining the flow of data
That barrage of data and information that insurance companies are being hit by each and every day needs to be intricately managed, stored, analyzed and acted upon in a timely manner. A missed email or note may not only result in poor service and an upset customer, it could potentially cost the company financially if, for example, relevant evidence in a dispute or claim case fails to surface or reach the right person/department on time.
Natural Language Processing is helping insurance providers ensure the right data reaches the right set of eyeballs at the right time through automated grouping and routing of queries and documents. This goes beyond simple keyword-matching with text analysis techniques used to ‘understand’ the context and category of a piece of text and classify it accordingly.
### Fraud detection
According to a recent report by Insurance Europe, detected and undetected fraudulent claims are estimated to represent 10% of all claims expenditure in Europe. Of note here, of course, is the fraud that goes undetected.
Insurance companies are using NLP and text analysis techniques to mine the data contained within unstructured sources such as applications, claims forms and adjuster notes to unearth certain red flags in submitted claims. For example, a regular indicator of organized fraudulent activity is the appearance of common phrases or descriptions of incidents from multiple claimants. The trained human eye may or may not be able to spot such instances but regardless, it would be a time consuming exercise and likely prone to subjectivity and inconsistency from the handler.
The solution for insurance providers is to develop NLP-powered analytical dashboards that support quick decision making, highlight potential fraudulent activity and therefore enable their investigators to prioritise cases based on specifically defined KPIs.
## NLP, Text Analysis & Customer Service
In a world that is increasingly focused on SLAs, KPIs and ROIs, the role of Customer Support and Customer Success, particularly in technology companies, has never been more important to the overall performance of an organization. With the ever-increasing number of startups and innovative companies disrupting pretty much every industry out there, customer experience has become a key differentiator in markets flooded with consumer choice.
Let’s take a look at three ways that NLP and text analysis is helping to improve CX in particular;
• Chat bots
• Analyzing customer/agent interactions
• Sentiment analysis
• Automated routing of customer queries
### Chat bots
It’s safe to say that chat bots are a pretty big deal right now! These conversational agents are beginning to pop up everywhere as companies look to take advantage of the cutting edge AI that power them.
Chances are that you interact with multiple artificial agents on a daily basis, perhaps even without realizing it. They are making recommendations as we online shop, answering our support queries in live chats, generating personalized fitness routines and communicating with us as virtual assistants to schedule meetings.
A recent interaction I had with a personal assistant bot, Amy
Chat bots are helping to bring a personalized experience to users. When done right, not only can this reduce spend in an organization , as they require less input from human agents, but it can also add significant value to the customer experience with intelligent, targeted and round-the-clock assistance at hand.
### Analyzing customer/agent interactions
Interactions between support agents and customers can uncover interesting and actionable insights and trends. Many interactions are in text format by default (email, live chat, feedback forms) while voice-to-text technology can be used to convert phone conversations to text so they can be analyzed.
### Listening to their customers
The voice of the customer is more important today than ever before. Social media channels offer a gold mine of publicly available consumer opinion just waiting to be tapped. NLP and text analysis enables you to analyze huge volumes of social chatter to help you understand how people feel about specific events, products, brands, companies, and so on.
Analyzing the sentiment towards your brand, for example, can help you decrease churn and improve customer support by uncovering and proactively working on improving negative trends. It can help show you what you are doing wrong before too much damage has been done, but also quickly show you what you are doing right and should therefore continue doing.
Customer feedback containing significantly high levels of negative sentiment can be relayed to Product and Development teams to help them focus their time and efforts more accordingly.
Because of the multi-channel nature of customer support, you tend to have customer queries and requests coming in from a variety of sources – email, social media, feedback forms, live chat. Speed of response is a key performance metric for many organizations and so routing customer queries to the relevant department, in as few steps as possible, can be crucial.
NLP is being used to automatically route and categorize customer queries, without any human interaction. As mentioned earlier, this goes beyond simple keyword-matching with text analysis techniques being used to ‘understand’ the context and category of a piece of text and classify it accordingly.
## Conclusion
As the sheer amount of unstructured data out there grows and grows, so too does the need to gather, analyze and make sense of it. Regardless of the industry in which they operate, organizations that focus on benefitting from NLP and text analysis will no doubt gain a competitive advantage as they battle for market share.
Most of our users will make 3 or more calls to our API for every piece of text or URL they analyze. For example if you’re a publisher who wants to extract insight from a an article or URL it’s likely you’ll want to use more than one of our features to get a proper understanding of that particular article or URL.
With this in mind, we decided to make it faster, easier and more efficient for our users to run multiple analysis operations in one single call to the API.
Our Combined Calls endpoint, allows you to run more than one type of analysis on a piece of text or URL without having to call each endpoint separately.
• Run multiple operations at once
• Speed up your analysis process
• Write cleaner, more efficient code
### Combined Calls
To showcase how useful the Combined Calls endpoint can be, we’ve ran a typical process that a lot of our news and media focused users would use when analyzing URL’s or articles on news sites.
In this case, we’re going to Classify the article in question and extract any Entities and Concepts present in the text. To run a process like this would typically involve passing the same URL to the API 3 times, once for each analysis operation and following that, retrieving 3 separate results relevant to each operation. However, with Combined Calls, we’re only making 1 call to the API and retrieving 1 set of results, making it a lot more efficient and cleaner for the end user.
Code Snippet:
var AYLIENTextAPI = require('aylien_textapi');
var textapi = new AYLIENTextAPI({
application_id: "APP_ID",
application_key: "APP_KEY"
});
textapi.combined({
"url": "http://www.bbc.com/news/technology-33764155",
"endpoint": ["entities", "concepts", "classify"]
}, function(err, result) {
if (err === null) {
console.log(JSON.stringify(result));
} else {
console.log(err)
}
});
The code snippet above was written using our Node.js SDK. SDKs are available for a variety of languages on our SDKs page.
### Results
We’ve broken down the results below into three sections, Entities, Concepts and Classification to help with readability, but using the combined calls endpoint all of these results would be returned together.
#### Entities:
{
"results": [
{
"endpoint": "entities",
"result": {
"entities": {
"keyword": [
"internet servers",
"flaw in the internet",
"internet users",
"server software",
"exploits of the flaw",
"internet",
"System (DNS) software",
"servers",
"flaw",
"expert",
"vulnerability",
"systems",
"software",
"exploits",
"users",
"websites",
"offline",
"URLs",
"services"
],
"organization": [
"DNS",
"BBC"
],
"person": [
"Daniel Cid",
"Brian Honan"
]
},
"language": "en"
}
},
#### Concepts:
{
"endpoint": "concepts",
"result": {
"concepts": {
"http://dbpedia.org/resource/Apache": {
"support": 3082,
"surfaceForms": [
{
"offset": 1261,
"score": 0.9726336488480631,
"string": "Apache"
}
],
"types": [
"http://dbpedia.org/ontology/EthnicGroup"
]
},
"http://dbpedia.org/resource/BBC": {
"support": 61289,
"surfaceForms": [
{
"offset": 1108,
"score": 0.9997923194235071,
"string": "BBC"
}
],
"types": [
"http://dbpedia.org/ontology/Agent",
"http://schema.org/Organization",
"http://dbpedia.org/ontology/Organisation",
"http://dbpedia.org/ontology/Company"
]
},
"http://dbpedia.org/resource/Denial-of-service_attack": {
"support": 503,
"surfaceForms": [
{
"offset": 264,
"score": 0.9999442627824017,
"string": "denial-of-service attacks"
}
],
"types": [
""
]
},
"http://dbpedia.org/resource/Domain_Name_System": {
"support": 1279,
"surfaceForms": [
{
"offset": 442,
"score": 1,
"string": "Domain Name System"
},
{
"offset": 462,
"score": 0.9984593397878601,
"string": "DNS"
}
],
"types": [
""
]
},
"http://dbpedia.org/resource/Hacker_(computer_security)": {
"support": 1436,
"surfaceForms": [
{
"offset": 0,
"score": 0.7808308562314218,
"string": "Hackers"
},
{
"offset": 246,
"score": 0.9326746054676964,
"string": "hackers"
}
],
"types": [
""
]
},
"http://dbpedia.org/resource/Indian_School_Certificate": {
"support": 161,
"surfaceForms": [
{
"offset": 794,
"score": 0.7811847159512098,
"string": "ISC"
}
],
"types": [
""
]
},
"http://dbpedia.org/resource/Internet_Systems_Consortium": {
"support": 35,
"surfaceForms": [
{
"offset": 765,
"score": 1,
"string": "Internet Systems Consortium"
}
],
"types": [
"http://dbpedia.org/ontology/Agent",
"http://schema.org/Organization",
"http://dbpedia.org/ontology/Organisation",
"http://dbpedia.org/ontology/Non-ProfitOrganisation"
]
},
"http://dbpedia.org/resource/OpenSSL": {
"support": 105,
"surfaceForms": [
{
"offset": 1269,
"score": 1,
"string": "OpenSSL"
}
],
"types": [
"http://schema.org/CreativeWork",
"http://dbpedia.org/ontology/Work",
"http://dbpedia.org/ontology/Software"
]
}
},
"language": "en"
}
},
#### Classification:
{
"endpoint": "classify",
"result": {
"categories": [
{
"code": "04003005",
"confidence": 1,
"label": "computing and information technology - software"
}
],
"language": "en"
}
}
You can find more information on using Combined Calls in our Text Analysis Documentation.
We should also point out that the existing rate limits will also apply when using Combined Calls. You can read more about our rate limits here.
## Introduction
This is the second edition of our NLP terms explained blog posts. The first edition deals with some simple terms and NLP tasks while this edition, gets a little bit more complicated. Again, we’ve just chosen some common terms at random and tried to break them down in simple English to make them a bit easier to understand.
#### Part of Speech tagging (POS tagging)
Sometimes referred to as grammatical tagging or word-category disambiguation, part of speech tagging refers to the process of determining the part of speech for each word in a given sentence based on the definition of that word and its context. Many words, especially common ones, can serve as multiple parts of speech. For example, “book” can be a noun (“the book on the table”) or verb (“to book a flight”).
#### Parsing
Parsing is a major task of NLP. It’s focused on determining the grammatical analysis or Parse Tree of a given sentence. There are two forms of Parse trees Constituency based and dependency based parse trees.
#### Semantic Role Labeling
This is an important step towards making sense of the meaning of a sentence. It focuses on the detecting semantic arguments associated with a verb or verbs in a sentence and the classification of those verbs into into specific roles.
#### Machine Translation
A sub-field of computational linguistics MT investigates the use of software to translate text or speech from one language to another.
#### Statistical Machine Translation
SMT is one of a few different approaches to Machine Translation. A common task in NLP it relies on statistical methods based off bilingual corpora such as the Canadian Hansard corpus. Other approaches to Machine Translation include Rule Based Translation and Example-Based Translation.
#### Bayesian Classification
Bayesian classification is a classification method based on Bayes Theorem and is commonly used in Machine Learning and Natural Language Processing to classify text and documents. You can read more about it in Naive Bayes for Dummies.
#### Hidden Markov Model (HMM)
In order to understand a HMM we need to define a Markov Model. This is used to model randomly changing systems where it is assumed that future states only depend on the present state and not on the sequence of events that happened before it.
A HMM is a Markov model where the system being modeled is assumed to have unobserved or hidden states. There are a number of common algorithms used for hidden Markov models. The Viterbi algorithm which will compute the most-likely corresponding sequence of states and the forward algorithm, for example, will compute the probability of the sequence of observations and both are often used in NLP applications.
In hidden Markov models, the state is not directly visible, but output, dependent on the state, is visible. Each state has a probability distribution over the possible output tokens. Therefore the sequence of tokens generated by an HMM gives some information about the sequence of states.
#### Conditional Random Fields (CRFs)
A class of statistical modeling methods that are often applied in pattern recognition and machine learning, where they are used for structured prediction. Ordinary classifiers will predict labels for a sample without taking neighboring samples into account, a CRF model however, will take context into account. CRF is commonly used in NLP (e.g. in Named Entity Extraction) and more recently in image recognition.
#### Affinity Propagation (AP)
AP is a clustering algorithm commonly used in Data Mining, unlike other clustering algorithms such as, k-means, AP does not require the number of clusters to be estimated before running the algorithm. A semi-supervised version of AP is commonly used in NLP.
#### Relationship extraction
Given a chunk of words or a piece of text determining the relationship between named entities.
### What’s Blockspring?
Blockspring is a really exciting, YC backed startup, who pitch themselves as “The world’s library of functions, accessible from everywhere you do work.” Their platform allows you to interact with a library of various APIs through a spreadsheet, simple code snippets and soon a chat interface.
The platform lets you run 1000+ functions directly from your spreadsheet or through simple code snippets for the more technically inclined. Accessing APIs with Blockspring is done through the concept of functions and they certainly have some cool APIs available to interact with in their library.
Where Blockspring gets really interesting though, is when you start to combine multiple functions. Your spreadsheet pretty much becomes a playpen where you can interact with one or multiple APIs and create powerful applications and “mashups”. Some of the examples of what can be done with Blockspring include, automating social activity and monitoring, gathering marketing data about user segments and usage, accessing public datasets, scraping websites and now even analyzing text and unstructured data all of which are really nicely showcased on their getting started page.
### AYLIEN and Blockspring
Like Blockspring, we want to get the power of our API into the hands of anyone that can get value from it. We launched our own Text Analysis Add-on for google sheets last year. The add-on works in the same way as Blockspring, through simple functions and acts as an interface for our Text Analysis API. Integrating with Blockspring, however, means our users can now open up their use cases by combining our functions with other complementary APIs to create powerful tools and integrations.
All of the AYLIEN end-points are available through Blockspring as simple snippets or spreadsheet functions and getting started with AYLIEN and Blockspring is really easy.
#### It’s simple to get up and running:
Step 1.
Step 2.
Grab your AYLIEN APP ID and API key and keep it handy. If you don’t have an AYLIEN account just sign up here.
Step 3.
Explore the getting started section to see examples of the functions and APIs available.
Step 4.
Try some of the different functions through their interactive docs to get a feel for how they work.
Step 5.
Go wild and start building and creating mashups of functions with code snippets or in Google Sheets.
PS: Don’t forget to add your AYLIEN keys to your Blockspring account in the Secrets section of your account settings. Once they’ve been added, you won’t have to do it again.
We’re really excited to see what the Blockspring community start to build with our various functions. Over the next couple of weeks, we’ll also be showcasing some cool mashups that we’ve put together in Blockspring so keep your eyes peeled on the blog.
### Introduction
As you may already know, we like to feature interesting and useful examples of what the AYLIEN community are building with our API. Previously, we’ve showcased a PowerShell wrapper and a Laravel wrapper. For this edition of our blog, we’re going feature a data scientist who spent some time building an R binding for our Text Analysis API.
Arnab is a solutions architect and data analytics expert based in India. He recently developed an R binding for our Text Analysis API that we think is going to be very popular amongst our R users. This R Binding makes it really quick and easy for the R community, to get up and running with our API and we’re glad he spent the time to putting it together.
### Setup
If you’re new to AYLIEN and Text Analysis, the first thing you’ll need to do is sign up for free access to the API. Take a look at our getting started page, which will take you through the signup process. We have a free plan available which allows you to make up to 1,000 calls to the API per day, for free.
The second thing you need to do is, install the following packages from your R console using the following commands:
install.packages("XML")
install.packages("plyr")
Point to file:
source("/Users/parsa/Desktop/aylienAPI-R.R")
Note: You must have XQuartz installed to view the results which you can download here.
### Utilisation
To show how easy it is to use the API with R, we’re going to run a simple analysis using the binding, by analyzing the following article: (http://www.bbc.com/sport/0/football/25912393), extracting the main body of text from the page and classifying the article based on IPTC news-codes.
Code Snippet:
aylienAPI<-function(APPLICATION_ID, APPLICATION_KEY, endpoint, parameters, type)
{
url = paste0('https://api.aylien.com/api/v1/',endpoint)
httpHeader = c(Accept="text/xml", 'X-AYLIEN-TextAPI-Application-ID' = APPLICATION_ID,
'X-AYLIEN-TextAPI-Application-Key'= APPLICATION_KEY,
'Content-Type'="application/x-www-form-urlencoded; charset=UTF-8")
paramPost<-paste0(type,"=",parameters)
paramEncode<-URLencode(paramPost)
postfields=paramEncode, verbose = FALSE)
resp
}
APPLICATION_ID = 'YOUR_APPLICATION_ID'
APPLICATION_KEY = 'YOUR_APPLICATION_KEY'
Arnab has made it really easy to call each end point, all you need to do is specify the endpoint in the code. To call the classification endpoint for example, we simply use “classify”.
endpoint = "classify"
parameters = "http://www.bbc.com/sport/0/football/25912393"
type = "url"
### Results
It’s up to you, how you want to display your results, but using the following command, displays them nice and clearly in a table format converting the output to a data frame, as shown in the image below.
resultsdf<-ldply(xmlToList(results), data.frame)
View(resultsdf)
As you can see from the Results, the API returned an accurate two-tier classification of “Sport – Soccer”.
You can also choose to retrieve data using Xpath from the XML result with the following request.
PARSED<-xmlInternalTreeParse(results)
View(xpathSApply(PARSED, "//category",xmlValue))
If you have an app or a wrapper you’ve built with our API’s, we’d love to hear about it and feature it on our blog. Get in touch at hello@aylien.com and tell us what you’re building.
### Introduction
There is a wealth of information in a LinkedIn profile. You can tell a lot about someone and how well they are suited to a role by analyzing their profile on LinkedIn, and let’s face it, LinkedIn in the number one platform to showcase yourself to potential employers and recruiters.
However, there are a number of issues that arise in relying on Linkedin profiles to understand a candidate’s suitability for a role and their current job function.
### The Experiment
We set out to find out what section of a LinkedIn profile contains the most insight into an individual’s job function by using Semantic Labeling to try and predict an individual’s job title based on the info they have on their profile.
#### How did we do it?
We scraped and parsed a number of well known LinkedIn profiles. Using the information we extracted from the profile such as keywords, summaries, job titles and skills we attempted to predict an individual’s job function from each information section to understand which best represents an individual’s ability or function.
We started out by choosing 4 general tags or labels for an individual’s profile that would point towards their high-level job function:
• Information Technology
• Finance
• Marketing
• Operations
Using the Semantic Labeling feature to check how related a tag or label, like Marketing, was to an individual’s actual job function, we could essentially predict what an individual’s actual function is.
Our findings are displayed in the sheet embedded below. The first section of the sheet contains the profiles and information extracted. The Yellow section is the prediction results based on the skills section, red is the Summary section and Green is the Job Title results.
When a label/job function is assigned following our analysis it is also accompanied by a confidence score, which indicates how confident we are in the results. This is important to note as we dive into some of the results. The “winning” results with the highest scores are marked in green.
Note:
For this blog, we kept the functions quite general but you can get quite specific as we have with Gregory’s account below.
### But what section of a profile provides the most insight?
#### Content
When analyzing a Linkedin profile or even using the search feature we’re primarily focusing on keywords mentioned in the content of that profile. Educational institutes, companies, and technologies mentioned for example.
Relying on keywords can often cause problems, there is huge a amount of latent information in a profile that is often overlooked when scanning profiles for keywords. A major problem with keyword search is that it misses related skills, e.g. someone might have “Zend Framework” on their profile, but not PHP – which is inherent, ‘cause Zend is a PHP framework. Good recruiters or somenone with programming knowledge would know this, average recruiters, however, may not.
The same could be said for someone who mentions Image Processing in their profile there is no obvious connection to other inherent knowledge such as Facial Recognition. A knowledge base such as Wikipedia, DBpedia or Freebase can be used to discover these latent connections:
#### Job Titles
Relying on job titles can also cause problems. They can be inaccurate, misleading or even made up. Especially today, as people often create their own titles. Take Rand Fishkin’s profile on LinkedIn as an example. Unless you know of MOZ and Rand’s wizardry you would have no idea he is at the forefront of Inbound, Social and SEO.
Another good example is Dharmesh Shah, founder of HubSpot’s profile. Dharmesh’s title is Founder and CTO of HubSpot. Running our analysis on the extracted title, Information Technology with a score of .016 is the job function returned for Dharmesh, which is somewhat accurate. However, running the same analysis on his skills section gives a far more accurate result suggesting Dharmesh is actually a Marketer with a “winning” score of .23.
#### Summaries
A profile Summary can be quite insightful and can provide a strong understanding of someone’s ability and function, but the problem is they aren’t always present or they often contain very little information causing them to be overlooked or disregarded. As was the case in many of the example profiles we used.
The ones that do have a detailed summary provided some strong results. With Rand Fishkin’s profile summary returning some accurate results of Marketing and a score of .188.
There was one section that outperformed the others when providing relevant tags and confidence scores.
#### Skills
The Skills section on a LinkedIn profile is a gold mine of insight. Based on the information extracted from the skills section, we could more accurately predict an individual’s job function.
Comparing the results and labels assigned across all the information sections and on every profile we used, the Skills section produced the most accurate relationships and the highest confidence scores, which can be seen marked green in the sheets above.
### Conclusion
We don’t have an exact science or formula for deciding whether a label is accurate or not, however, our experiment still does a good job of highlighting the fact that, a lot more information and insight can be gleaned from the skills section of a linkedIn profile in deciding at first glance, or automatically how well a candidate is suited to a particular job function. We will explore these ideas in future posts.
This is the sixth in our series of blogs on getting started with AYLIEN’s various SDKs.
If you’re new to AYLIEN and you don’t have an account yet, you can go directly to the Getting Started page on the website which will take you through the signup process. We have a free plan to get started with that allows you to make up to 1,000 calls per day for free.
All of our SDK repositories are hosted on Github. For the Java SDK, the Text Analysis API is published to Maven Central, so simply add the dependency to the POM:
<dependency>
<groupId>com.aylien.textapi>
<artifactId>client>
<version>0.1.0>
</dependency>
Once you’ve installed the SDK you’re ready to start coding. For the remainder of this blog we’ll walk you through making calls and show the output you should receive in each case. Taking simple examples we’ll showcase some of the API’s endpoints like Language detection, Sentiment Analysis and hashtag suggestion.
### Configuring the SDK with your AYLIEN credentials
Once you’ve received your AYLIEN APP_ID and APP_KEY from the signup process and have downloaded the SDK you can begin making calls with the following imports and configuration code.
import com.aylien.textapi.TextAPIClient;
import com.aylien.textapi.parameters.*;
import com.aylien.textapi.responses.*;
TextAPIClient client = new TextAPIClient(
"YourApplicationId", "YourApplicationKey");
When calling the various API endpoints you can specify a piece of text directly for analysis or you can pass a url linking to the text or article you wish to analyze.
### Language Detection
First off, let’s take a look at the language detection endpoint. As a simple example we’re going to detect the language of the following sentence: ‘What language is this sentence written in?’
To do this, you can call the endpoint using the following piece of code.
String text = "What language is this sentence written in?";
LanguageParams languageParams = new LanguageParams(text,null);
Language language = client.language(languageParams); System.out.printf("nText : %s",language.getText());
System.out.printf("nLanglanguage : %s",language.getLanguage());
System.out.printf("nConfidence %f",language.getConfidence());
You should receive an output very similar to the one shown below. This shows that the language detected was English and the confidence that it was detected correctly (a number between 0 and 1) is very close to 1, which means you can be pretty sure it is correct.
#### Language Detection Results
Text : What language is this sentence written in?
Langlanguage : en
Confidence 0.999997
### Sentiment Analysis
Next, we’ll look at analyzing the sentence “John is a very good football player” to determine it’s sentiment i.e. whether it’s positive, neutral or negative. The Sentiment Analysis endpoint will also determine if the text is subjective or objective. You can call the endpoint with the following piece of code
text = "John is a very good football player!";
SentimentParams sentimentParams = new SentimentParams(text,null,null);
Sentiment sentiment = client.sentiment(sentimentParams);
System.out.printf("nText : %s",sentiment.getText());
System.out.printf("nSentiment Polarity : %s",sentiment.getPolarity());
System.out.printf("nPolarity Confidence : %f",sentiment.getPolarityConfidence());
System.out.printf("nSubjectivity : %s",sentiment.getSubjectivity());
System.out.printf("nSubjectivity Confidence: %f",sentiment.getSubjectivityConfidence());
You should receive an output similar to the one shown below which indicates that the sentence is objective and is positive, both with a high degree of confidence.
#### Sentiment Analysis Results
Text : John is a very good football player!
Sentiment Polarity : positive
Polarity Confidence : 0.999999
Subjectivity : objective
Subjectivity Confidence: 0.989682
### Hashtag Suggestion
Finally, we’ll look at analyzing a BBC article to extract hashtag suggestions for it with the following code.
HashTagsParams hashtagsParams = new HashTagsParams(null,url,null);
HashTags hashtags = client.hashtags(hashtagsParams);
System.out.print("Hashtags : n");
System.out.print(hashtags + "n");
You should receive the output shown below.
#### Hashtag Suggestion Results
Hashtags :
#Planet #JohannesKepler #Kepler #Birmingham #Earth #Astronomy #TheAstrophysicalJournal #Warwick #Venus #Orbit #Mercury #SolarSystem #Resonance #TerrestrialPlanet #Lightyear #Imagine
If Java’s not your preferred language then check out our otherSDKs for node.js, Go, PHP, Python, Ruby and .Net (C#). For more information regarding the APIs go to the documentation section of our website.
We’ve just added support for microformat parsing to our Text Analysis API through our Microformat Extraction endpoint.
Microformats are simple conventions or entities that are used on web pages, to describe a specific type of information, for example, Contact info, Reviews, Products, People, Events, etc.
Microformats are often included in the HTML of pages on the web to add semantic information about that page. They make it easier for machines and software to scan, process and understand webpages. AYLIEN Microformat Extraction allows users to detect, parse and extract embedded Microformats when they are present on a page.
Currently, the API supports the hCard format. We will be providing support for the other formats over the coming months. The quickest way to get up and running with this endpoint is to download an SDK and checkout the documentation. We have gone through a simple example below to showcase the endpoints capabilities.
### Microformat Extraction in Action
The following piece of code sets up the credentials for accessing our API. If you don’t have an AYLIEN account, you can sign up here.
var AYLIENTextAPI = require('aylien_textapi');
var textapi = new AYLIENTextAPI({
application_id: YOUR_APP_ID,
application_key: ‘YOUR_APP_KEY'
});
The next piece of code accesses an HTML test page containing microformats, that we have setup in codepen to illustrate how the endpoint works (check out http://codepen.io/michaelo/pen/VYxxRR.html to see the raw HTML). The code consists of a call to the microformats endpoint and a forEach statement to display any hCards detected on the page.
textapi.microformats('http://codepen.io/michaelo/pen/VYxxRR.html',
function(err, res) {
if (err !== null) {
console.log("Error: " + err);
} else {
res.hCards.forEach(function(hCard) {
console.log(hCard);
console.log("n****************************************");
console.log("End Of vCard");
console.log("******************************************");
});
}
});
As you can see from the results below, there are two hcards on the page, one for Sally Ride and the other for John Glenn. The documentation for the endpoint shows the structure of the data returned by the endpoint and lists the optional hCard fields that are currently supported. You can copy the code above and paste it into our sandbox environment to view the results for yourself and play around with the various fields.
#### Results
{ birthday: '1951-05-26',
organization: 'Sally Ride Science',
telephoneNumber: '+1.818.555.1212',
location:
{ id: '9f15e27ff48eb28c57f49fb177a1ed0af78f93ab',
latitude: '37.386013',
longitude: '-122.082932' },
photo: 'http://example.com/sk.jpg',
email: 'sally@example.com',
url: 'http://sally.example.com',
fullName: 'Sally Ride',
structuredName:
{ familyName: 'van der Harten',
givenName: 'Sally',
honorificSuffix: 'Ph.D.',
honorificPrefix: 'Dr.' },
logo: 'http://www.abc.com/pub/logos/abccorp.jpg',
id: '7d021199b0d826eef60cd31279037270e38715cd',
note: '1st American woman in space.',
countryName: 'U.S.A',
postalCode: 'LWT12Z',
id: '00cc73c1f9773a66613b04f11ce57317eecf636b',
region: 'California',
locality: 'Los Angeles' },
category: 'physicist' }
****************************************
End Of vCard
****************************************
{ birthday: '1921-07-18',
telephoneNumber: '+1.818.555.1313',
location:
latitude: '30.386013',
longitude: '-123.082932' },
photo: 'http://example.com/jg.jpg',
email: 'johnglenn@example.com',
url: 'http://john.example.com',
fullName: 'John Glenn',
structuredName:
{ familyName: 'Glenn',
givenName: 'John',
id: 'a1146a5a67d236f340c5e906553f16d59113a417',
honorificPrefix: 'Senator' },
logo: 'http://www.example.com/pub/logos/abccorp.jpg',
id: '18538282ee1ac00b28f8645dff758f2ce696f8e5',
note: '1st American to orbit the Earth',
countryName: 'U.S.A',
postalCode: 'PC123',
id: '8cc940d376d3ddf77c6a5938cf731ee4ac01e128',
region: 'Ohio',
locality: 'Columbus' } }
****************************************
End Of vCard
****************************************
Microformats Extraction allows you to automatically scan and understand webpages by pulling relevant information from HTML. This microformat information is easier for both humans and now machines to understand than other complex forms such as XML.
Our development team have been working hard adding additional features to the API which allow our users to analyze, classify and tag text in more flexible ways. Unsupervised Classification is a feature we are really excited about and we’re happy to announce that it is available as a fully functional and documented feature, as of today.
#### So what exactly is Unsupervised Classification?
It’s a training-less approach to classification, which means, unlike our standard classification, that is based on IPTC News Codes, it doesn’t rely on a predefined taxonomy to categorize text. This method of classification allows automatic tagging of text that can be tailored to a users needs, without the need for a pre-trained classifier.
#### Why are we so excited about it?
Our Unsupervised Classification endpoint will allow users to specify a set of labels, analyze a piece of text and then assign the most appropriate label to that text. This allows greater flexibility for our users to decide, how they want to tag and classify text.
There are a number of ways this endpoint can be used and we’ll walk you through a couple of simple examples; Text Classification from a URL and Customer Service Routing of social interactions.
### Classification of Text
We’ll start with a simple example to show how the feature works. The user passes a piece of text or a URL to the API, along with a number of labels. In the case below we want to find out which label, Football, Baseball, Hockey or Basketball, best represents the following article: ‘http://insider.espn.go.com/nfl/story/_/id/12300361/bold-move-new-england-patriots-miami-dolphins-new-york-jets-buffalo-bills-nfl’
#### Code Snippet:
var AYLIENTextAPI = require('aylien_textapi');
var textapi = new AYLIENTextAPI({
application_id: 'YourAppId',
application_key: 'YourAppKey'
});
var params = {
url: 'http://insider.espn.go.com/nfl/story/_/id/12300361/bold-move-new-england-patriots-miami-dolphins-new-york-jets-buffalo-bills-nfl',
};
textapi.unsupervisedClassify(params, function(error, response) {
if (error !== null) {
console.log(error, response);
} else {
console.log("nThe text to classify is : nn",
response.text, "n");
for (var i = 0; i < response.classes.length; i++) {
console.log("label - ", response.classes[i].label,
", score -", response.classes[i].score, "n");
}
}
});
#### Results:
The text to classify is:
"Each NFL team's offseason is filled with small moves and marginal personnel decisions... "
label - football , score - 0.13
label - baseball , score - 0.042
label - hockey , score - 0.008
label - basketball , score - 0.008
Based on the scores provided, we can confidently say, that the article is about football and should be assigned a “Football” label.
### Customer Service Routing
As another example, let’s say we want to automatically determine whether a post on social media should be routed to our Sales, Marketing or Support Departments. In this example, we’ll take the following comment: “I’d like to place an order for 1000 units.” and automatically determine whether it should be dealt with by Sales, Marketing or Support. To do this, we pass the text to the API as well as our pre-chosen labels, in this case: ‘Sales’, ‘Customer Support’, ‘Marketing’.
#### Code Snippet:
var AYLIENTextAPI = require('aylien_textapi');
var textapi = new AYLIENTextAPI({
application_id: 'YourAppId',
application_key: 'YourAppKey'
});
var params = {
text: "I'd like to place an order for 1000 units.",
'class': ['Sales', 'Customer Support', 'Marketing']
};
textapi.unsupervisedClassify(params, function(error, response) {
if (error !== null) {
console.log(error, response);
} else {
console.log("nThe text to classify is : nn",
response.text, "n");
for (var i = 0; i < response.classes.length; i++) {
console.log("label - ",
response.classes[i].label,
", score -", response.classes[i].score, "n");
}
}
});
#### Results:
The text to classify is:
I'd like to place an order for 1000 units.
label - Sales , score - 0.032
label - Customer Support , score - 0.008
label - Marketing , score - 0.002
Similarily, based on the scores given on how closely the text is semantically matched to a label, we can decide that this inquiry should be handled by a sales agent rather than, marketing or support.
### Divide and Conquer
Our next example deals with the idea of using the unsupervised classification feature, with a hierarchical taxonomy. When classifying text, it’s sometimes necessary to add a sub-label for finer grained classification, for example “Sports – Basketball” instead of just “sports”.
So, in this example we’re going to analyze a simple piece of text: “The oboe is a woodwind musical instrument” and we’ll attempt to provide a more descriptive classification result, based on the following taxonomy;
• ‘music’: [‘Instrument’, ‘composer’],
• ‘technology’: [‘computers’, ‘space’, ‘physics’],
• ‘health’: [‘disease’, ‘medicine’, ‘fitness’],
The taxonomy has a primary label and a secondary label, for example ‘music’ (primary) and ‘instrument, Composer’ (secondary)
#### Code Snippet:
var AYLIENTextAPI = require('aylien_textapi');
var textapi = new AYLIENTextAPI({
application_id: 'YourAppId',
application_key: 'YourAppKey'
});
var _ = require('underscore');
var taxonomy = {
'music': ['Instrument', 'composer'],
'technology': ['computers', 'space', 'physics'],
'health': ['disease', 'medicine', 'fitness'],
};
var topClasses = ['technology', 'music', 'health', 'sport'];
var queryText = "The oboe is a woodwind musical instrument.";
var params = {
text: queryText,
'class': topClasses
};
textapi.unsupervisedClassify(params, function(error, response) {
if (error !== null) {
console.log(error, response);
} else {
var classificationResult = '';
console.log("nThe text to classify is : nn",
response.text, "n");
classificationResult = response.classes[0].label +
" (" + response.classes[0].score + ") ";
params = {
text: queryText,
'class': _.values(
_.pick(taxonomy, response.classes[0].label)
)[0]
};
textapi.unsupervisedClassify(params,
function(error, response) {
if (error !== null) {
console.log(error, response);
} else {
classificationResult += " - " +
response.classes[0].label +
" (" + response.classes[0].score +
") ";
console.log("Label: ", classificationResult);
}
}
);
}
});
#### Results:
The text to classify is :
The Obo is a large musical instrument
Label : music (0.076) - Instrument (0.342)
As you can see from the results, the piece of text has been assigned ‘music’ as its primary label and ‘instrument’ as its secondary label.
All the code snippets in our examples are fully functional and can be copied and pasted or tested in our sandbox. We’ll also be adding some of these and more interesting apps to our sandbox over the next week or so that will showcase some interesting use cases for Unsupervised Classification. We’d also love to hear more about how you would use this feature, so don’t hesitate to get in touch with comments or feedback.
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https://golem.ph.utexas.edu/category/2007/10/on_weak_cokernels_for_2groups.html | ## October 17, 2007
### On Weak Cokernels for 2-Groups
#### Posted by Urs Schreiber
In Detecting higher order necklaces I mentioned how
P. Carrasco A. R. Garzó́n and E. M. Vitale
On categorical crossed modules
Theory and Applications of Categories, Vol. 16, 2006, No. 22, pp 585-618
is related to
David Roberts, U.S.
The inner automorphism 3-group of a strict 2-group
arXiv:0708.1741
(html).
with the relation becoming obvious after drawing some diagram. This discussion, quite brief, but the picture provided there is useful to keep in mind, I have now prepared here:
This blog entry is hence mainly a private message to David Roberts (since it builds on our discussion of inner automorphism $(n+1)$-groups). And maybe to Todd Trimble (since it builds on our discussion of tangent categories (pdf, html). And to Jim Stasheff (since it is going to be applied to obstructions to lifts of $n$-Cartan connections (slides, BIG diagram)). And to John Baez of course, from whome I am hoping to receive more hints on how to think of the big picture . And to Bruce Bartlett, with whom I was talking about this here and by email.
Hence a blog post.
You’ll see that this issue of weak cokernels of 2-groups is the integral version of what I had started to discuss at the level of Lie $n$-algebras in
Obstructions and cokernels of Lie n-algebra morphisms
(pdf)
(html).
To whetten your appetite, a degenerate (low $n$) version of what I am trying to get at here is related to Schreier theory:
Given a sequence
$K \to G \to B$
of groups, we cannot, in general, find a splitting $G \leftarrow B$ which is a group homomorphism. But the failure is “coherent”.
One way to say this is:
denote by
$(K \to G)$
the weak cokernel of the map $K \to G$. This is nothing but the 2-group defined by the crossed module of the same name $K \to G$!
And this 2-group is equivalent to $B$
$(K \to G) \simeq B$
This means we always have morphisms
$B \hookrightarrow (K \to G)$
which are equivalences.
These are best thought of as pseudofunctors on the corresponding one-object 2-groupoids:
on morphisms these functors are nothing but lifts
$G \leftarrow B$
that may fail to respect composition, strictly. But the failure – the compositor – is something in $K$, indeed a 2-morphism in $\Sigma(K \to G)$.
Now, the 2-group $(K \to G)$ sits inside the 2-group $\mathrm{AUT}(K)$,
$(K\to G ) \to \mathrm{AUT}(K)$
of automorphic functors on $\Sigma K$
Notice how this is a nice way to understand lifting gerbes:
A $B$-bundle is given by a cocycle which can be thought of as lots of triangles labeled in $B$.
We may ask if we can lift these labels to labels in $G$. In general we cannot. But what we can always do is lift them to labels in $(K \to G)$: because that simply means that we choose any lifts, and then label the interior of the triangle with the mistake in K that we made thereby.
So every $B$-cocycle is a $(K \toG)$- cocycle.
We may want to characterize the “mistakes in $K$” that we made by this attempted lift. To find these, we check which part of our failed lift doesn’t sit entirely in B, so we look at the cokernel of the canonical inclusion
$G \hookrightarrow (K \to G)$
(notice that we are using here that the “puffed up” version $(K \to G)$ of $B$ has the property that what used to be a surjection $G \to B$ now becomes an inclusion!)
which is simply $\Sigma K$
$G \to (K\to G ) \to \Sigma K$
Postcomposing our labelling of triangles with this morphism
$(K \to G) \to \Sigma K$
simply forgets the labels of the boundaries of the triangle and only remembers the $K$-labels in their interior. This is now the 2-cocycle describing a $\Sigma K$- 2-bundle. And this is the lifting 2-bundle (lifting gerbe).
If this 2-cocycle is nontrivial it means that the failures to lift that we made are real and cannot be undone by a more clever lift. If that 2-cocycle however is a 2-coboundary it means that there is a more clever choice of lift which produces no failure: the lifting gerbe trivializes.
You have maybe seen the pictures illustrating this on slides 322 onwards in my slides.
Posted at October 17, 2007 10:34 PM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1467
### Re: On Weak Cokernels for 2-Groups
I’m on it.
Processing…
Posted by: David Roberts on October 18, 2007 6:05 AM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
I’m on it.
Processing…
Okay, great. Let me point out more clearly which points need clarification:
while the discussion of the mapping cone $(K \to G)$ itself can be done entirely in the world of strict 2-groups, the morphism
$B \to (K \to G)$
is not only pseudo (a weak 3-functor) but needs to be something like “ana-pseudo” for Lie 2-groups: even the failed lifts don’t exist globally.
I am confident that this has to all work out, since it did in the Lie $n$-algebra version of the same discussion (beware, though, this is the old version of my file which needs some rectification somewhere, will try to update it) but it certainly requires further discussion.
I need to further study CarrascoGarzónVitale to understand to which degree this is addressed by them.
Posted by: Urs Schreiber on October 18, 2007 11:55 AM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
What you explain in your note seems to be an application of the following fact : the weak cokernel of a morphism of 2-groups seen as a morphism of discrete 3-groups is the same as the weak kernel of the suspension of this morphism.
Let me explain this one dimension lower. If you start with a morphism of groups $t:G\to H$, you can on the one hand take the weak cokernel of $t$ seen as a morphism of discrete 2-groups, you get a pointed groupoid $\Coker t$, which is a 2-group if you start with a crossed module (this is the “realization” of the crossed module as a 2-group).
On the other hand, you can take the weak kernel of the suspension of $t$. In general, the (weak) kernel of a morphism of pointed groupoids $T:\mathbf{G}\to\mathbf{H}$ is the pointed groupoid $\mathrm{Ker T}$ such that
• the objects are given by an object $A$ of $\mathbf{G}$ and an arrow $\phi:FA\to *$ in $\mathbf{H}$ (where $*$ is the point of $\mathbf{H}$);
• the arrows $(A,\phi)\to(A',\phi')$ are arrows $a:A\to A'$ in $\mathbf{H}$ such that $\phi' Fa = \phi$.
In the case of $\Sigma t:\Sigma G\to\Sigma H$, you get what you call $T^t \Sigma G$, and so you get back the cokernel of $t$ seen as a morphism of discrete 2-groups.
There is then a sequence of pointed groupoids $G\overset{t}\to H\to \mathrm{Coker} t\simeq\mathrm{Ker} \Sigma t\to\Sigma G\overset{\Sigma t}\to\Sigma H.$
If you start with a morphism of 2-groups, you’ll get the same situation, with pointed 2-groupoids.
Posted by: Mathieu Dupont on October 18, 2007 12:47 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Thanks for this comment. You seem to be well familiar with this. Can you suggest more literature? Is this standard?
There is then a sequence of pointed groupoids
$G \stackrel{t}{\to} H \to \mathrm{Coker}(t) \simeq \mathrm{Ker}( \Sigma t) \to \Sigma G \stackrel{\Sigma t}{\to} \Sigma H$
Ah, that’s great. I hadn’t thought about it quite this way yet.
But it seems to point in a direction that I was wondering about a bit:
shouldn’t that situation be closely related to the way every short exact sequence of topological groups
$0 \to A \to B \to C \to 0$
gives rise to a long exact sequence in homotopy
$\cdots \to \pi_n(A) \to \pi_n(B) \to \pi_n(C) \stackrel{\delta}{\to} \pi_{n-1}(A) \to \cdots$ ?
It’s not entirely clear to me yet, but my gut feeling is that if we replace homotopy groups here with the corresponding fundamental $n$-groups of a space, then the long sequence in homotopy should be a special case of what we’d obtain from the picture you just sketched in the case that further suspensions are possible
$\cdots \to G \stackrel{t}{\to} H \to \mathrm{Coker}(t) \simeq \mathrm{Ker}( \Sigma t) \to \Sigma G \stackrel{\Sigma t}{\to} \Sigma H \to \mathrm{Coker}(\Sigma t) \simeq \mathrm{Ker} (\Sigma^2 t) \to \Sigma^2 G \stackrel{\Sigma^2 t}{\to} \Sigma^2 H \to \cdots \,.$
Could that be right?
Posted by: Urs Schreiber on October 18, 2007 1:39 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Thanks for this comment. You seem to be well familiar with this. Can you suggest more literature? Is this standard?
I’m finishing my PhD thesis with Enrico Vitale.
There is a connection with the long exact sequence in homotopy : they are both a special case of the following fibration sequence of pointed groupoids (see Gabriel and Zisman, “Calculus of fractions and homotopy theory”, chapter V). For a pointed groupoid $X$, let $\Omega X$ be the weak kernel of the unique pointed functor $0\to X$ (where $0$ is the one object, one arrow pointed groupoid), i.e. $\Omega X$ is the group $X(*,*)$ (where $*$ is the point of $X$) seen as a discrete pointed groupoid. Then for every pointed functor $F:X\to Y$, there is a sequence $0\simeq \Omega(\Omega Y)\to\Omega\Ker F\to\Omega X\overset{\Omega F}\to\Omega Y\to\Ker F\to X\overset{F}\to Y,$ where each arrow is equivalent to the weak kernel of the following one.
If you apply $\pi_0$ to this sequence, you get a “long homotopy sequence” of pointed sets and groups, since $\pi_0\Omega=\pi_1$ : $0\to\pi_1(\Ker F)\to\pi_1(X)\overset{\pi_1(F)}\to\pi_1(Y)\to\pi_0(\Ker F)\to\pi_0(X)\overset{\pi_0(F)}\to\pi_0(Y)$
Now, if you have a group homomorphism $f:X\to Y$, you can apply the first sequence above to $\Sigma f$, and since $\Omega\Sigma X$ is $X$ seen as a discrete groupoid, you get the following sequence of pointed groupoids. $0\to\Ker f\to X\overset{f}\to Y\to\Ker\Sigma f\to\Sigma X\overset{\Sigma f}\to\Sigma Y$
So these are not the same sequences, one is a sequence of pointed sets constructed from a morphism of pointed groupoids, and the other a sequence of pointed groupoids constructed from a morphism of groups.
You can extend these sequences with the weak cokernel. Corollary 2.7 in Bourn and Vitale, Extensions of symmetric cat-groups* gives for a morphism of symmetric 2-groups $F:X\to Y$ the following sequence of abelian groups : $0\to\pi_1(\mathrm{Ker} F)\to\pi_1(X)\overset{\pi_1 F}\to\pi_1(Y)\to\pi_1(\mathrm{Coker} F)\simeq\pi_0(\mathrm{Ker} F)\to\pi_0(\X)\overset{\pi_0(F)}\to\pi_0(Y)\to\pi_0(\mathrm{Coker} F)\to 0$
For higher dimensions, Beppe Metere has studied the “ziggurat of exact sequences of n-groupoids”, but there is still no available reference as far as I know.
*On the same page, there is also “A higher dimensional homotopy sequence”, by Grandis and Vitale.
Posted by: Mathieu Dupont on October 22, 2007 12:33 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Mathieu,
this is most interesting. I am grateful that you took the time to share this information. This looks like it is the answer to something I have been struggling with for a while now.
I’m printing the references you provided, will have a look at them on the plane tomorrow (unless I fall asleep, as usual ;-).
For higher dimensions, Beppe Metere has studied the “ziggurat of exact sequences of $n$-groupoids”, but there is still no available reference as far as I know.
Googling, I found at least
Beppe Metere, (The Ziggurath of) exact sequences of $n$-groupoids (handwritten notes)
Thanks again! I might get back to you with further questions/comments here once I have had a chance to look into this.
Posted by: Urs Schreiber on October 23, 2007 10:26 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
The following may or may not be implicit in what Mathieu Dupoint said above, or in the literature which he provided, I haven’t fully absorbed that yet.
But today I realized that the construction of weak cokernels that I am talking about, is apparently nicely related to relative homotopy groups, in a very vivid way.
So recall the construction I am talking about:
given two 2-groupoids $C$ and $D$, and a 2-functor $t : C \to D$ which is “sufficiently injective” (let’s assume for simplicity that its component map is an injective maps on objects, morphisms and $2$-morphisms), then the weak cokernel of $t$ is the 2-groupoid
a) whose objects are 2-functors from the “fat point” $(\bullet \stackrel{\simeq}{\to} \circ)$ into $D$
b) whose morphisms are transformations of such 2-morphisms restricted to be trivial on $\bullet$ and to lie in the image of $t$ on $\circ$
c) 2-morphisms are modifications of such transformations, whose components lie in the image of $t$.
So a 2-morphism in $\mathrm{wcoker}(t)$ looks like a diagram of the form
in $D$, with $L$ in $C$.
Notice that this looks a bit like a 2-disk in $D$ whose boundary (2-1)-sphere is in the image of $t$.
But this is the way relative homotopy groups are defined.
Let $X$ be a topological space and $t : A \hookrightarrow X$ a subspace, then the elements of the $n$-homotopy group of $X$ relative to $A$ are those maps into $X$ which look like maps of the $n$-sphere into $X$ once we think of $A$ as being a single point.
More precisely, $\pi_n(X,A) = \left\lbrace f : D^n \to X | \partial D^n \subset t(A) \right\rbrace$ (I am suppressing the basepoints and the respect for them).
Clearly, this formula expresses the same kind of condition that enters the construction of the diagram displaeyed above.
In fact, I think that if I assume my 2-groupoids $C$ and $D$ to be the fundamental 2-groupoids $\Pi_2$ of $A$ and $X$, respectively, then then the weak cokernel of $\Pi_2(A) \stackrel{\Pi_2(t)}{\to} \Pi_2(X)$ is pretty much the 2-groupoid version of the $t$-relative homotopy 2-group.
If you see what I mean.
And indeed, the relative homotopy groups fit into a long exact sequence $\pi_n(A) \stackrel{t}{\to} \pi_n(X) \to \pi_n(X,A) \to \pi_{n-1}(A) \to \cdots$
which seem to correspond to precisely the corresponding long sequences of $n$-groupoids that we were talking about above.
(Sorry for the vagueness of these statements, I am just trying to quickly communicate a rough observation here.)
Posted by: Urs Schreiber on November 1, 2007 6:26 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
I am suppressing the basepoints and the respect for them.
Gosh Urs - are you disrespecting basepoints? The homotopy police are not going to take kindly to this…!
Posted by: Bruce Bartlett on November 2, 2007 10:39 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Gosh Urs - are you disrespecting basepoints?
No, suppressing them, notationally that is.
Posted by: Urs Schreiber on November 4, 2007 4:18 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Thanks for constructing at least half the bridge between cats and homtopy theory.
0≃Ω(ΩY)→ΩKerF→ΩX→ΩFΩY→KerF→X→FY,
where each arrow is equivalent to the weak kernel of the following one.
In homotopy theory, as I’m sure you know but let’s let the rest of the blog in on the story:
for any map of spaces f: X –> Y
we have the homtopy fibre H_f (apparently to be called weak kernel) and a long sequence
…–> Omega H_f –> Omega X —> Omega Y –> H_f –> X –> Y
where each space is homotopy equvalent to the corresponding homotopy fibre
on the other hand, we have the mapping cone C_f and the Barratt-Puppe sequence
X –> Y –> C_f –> Sigma X –> Sigma Y –>…
N.B. Here Sigma means topological suspension NOT BX
also any map can be decomposed via a
fibration…
Posted by: jim stasheff on November 3, 2007 12:26 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Having seen no response to this, I’ll auto-respond!
In so far as hihger cats mimic (part of?) homotopy theory, I think it’s really important that we agree or establish an alternative:
weak kernel = homotopy fibre?
weak cokernel = homotopy quotient?
mapping cone = cofibre = ??
e.g. for ordinary top groups
Sigma G is the suspension = projective line whihc is a subspace of BG = `infinite proj space’
speak now or foreever…
Posted by: jim stasheff on November 6, 2007 5:13 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Perhaps one can write something like $A^{[n]}$ for something where the $n$-cells are given by the Abelian group $A$? You can then also write things like $G^{[1,2]}$ to mean a group $G$, thought of as a one-object 2-category with the elements of $G$ being the one-cells, and only identity 2-cells.
Posted by: Bruce Bartlett on November 7, 2007 12:03 AM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Sorry for not replying sooner. It’s not due to lack of interest, but rather the opposite: I feel I need to think about the answer but am distracted due to travelling.
My personal feeling is this:
all the identifications you indicated are certainly morally right, and probably can be made into precise statements about certain things being equivalent.
But at the same time, my personal impression is that we are not quite at that point yet where we don’t run into trouble by using all these terms, topological ones on one side and category theoretic ones on the other, as if we had already fully established that and how they are precisels “the same”.
For instance: while maybe the choice of “$\Sigma$” for the operation of
$\Sigma$ : turning a $k$-tuply monoidal $n$-catgegory into a $(k-1)$-tuply monoidal $(n+1)$-category
is not optimal (I seem to remember that I adopted this notation from Aaron Lauda), I do think that it is not unimportant not to write $B$ for $\Sigma$.
It is true that under the classifying space functor $|\cdot|$, $\Sigma G$ turns into $|\Sigma G| = B G \,,$ but my feeling is that until we have reached the point – maybe in one or trwo decades (?!) – that we all feel safe with completely identifying the world of $n$-categories with the world of spaces, we will run into the risk of confusing ourselves and our audiences by treating these terms all as established synonyms.
While from some big bird’s eye point of view a category (or a groupoid, anyway) is a space, for my practical purposes I certainly want to distinguish between, for instance, the finite category $\Sigma G$, which is really just a finite set with a composition operation on it, and the “huge” space $B G$ obtainable from it.
I am well aware that at various places in the literature the symbol $B G$ is used for things less “spacelike” than the classifying space of $G$, but I am feel hesitant to add to that by writing $B G$ for $\Sigma G$.
While it is certainly true that weak cokernels play the role of homotopy fibers, etc., I wouldn’t want – not until some point in the future when we all agree by default on that and how spaces are the same as $n$-categories generally – to forget that while morally the same, they differ at least in terms of which they are defined.
Maybe it’s just me being slow and hesitant and overcautious. But in any case, that’s my comment to your remark.
Posted by: Urs Schreiber on November 7, 2007 7:34 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
I see your point but still would like something onther that Sigma for you cat construction. Even S or s for shift.
as for something versus it’s realization
some people get by with a change in font
BG for the realization of say {\mathcal B} G
If weak kernel is not always satisfying the analogous universal property in cats to homotopy fibre in spaces,
then let’s establish when and where they do agree
Posted by: jim stasheff on November 8, 2007 3:04 AM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
As a homotopy theorist, I see
G→tH→Coker(t)≃Ker(Σt)→ΣG→ΣtΣH
as corresponding but roughly to what I’m familiar with:
G –> H –> H_G meaning the homotopy quotient which is of the homtopy type of H/G
when G acts nicely
up to homotopy, the sequence continues
G –> H –> H_G –> BG –> BH
in which any three terms in a row can be regarded as fibre –> total space –> base
IF G is a normal subgroup of H
we can go one step further
G –> H –> H_G –> BG –> BH –> B(H_G)
If G is abelian so BBG exists, yet one more step
If G and H are abelian, we can go on forever
on the other side, we can take based oop spaces and back up forever
–> \Omega G –>\Omega h –> \Omega(H_G)–> G –> H –> H_G –> BG –> BH
Posted by: jim stasheff on December 15, 2007 10:07 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Given a short exact sequence of groups
$0 \to K \stackrel{t}{\to} G \to B \to 0$
what do you know about the (weakly) universal morphism $f$
$\array{ K &\stackrel{t}{\to}& G &\to& \mathrm{coker}(t) \\ && \downarrow & \swarrow_{f} \\ && B }$
in the case of Lie groups, with
$\mathrm{coker}(t) = (K \stackrel{t}{\to} G)$
the weak cokernel?
The morphism $f : (K \stackrel{t}{\to} G) \to B$
should be an equivalence in a suitable sense, but its weak inverse may not be smooth.
I’d think we’d need ana-2-functors to handle this, or something similar. But maybe you know more about this.
Posted by: Urs Schreiber on October 18, 2007 2:17 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Thanks for taking the time to explain this, Urs – I’ll try to read this more carefully today.
Allow me to make one obnoxious comment about an idiomatic English expression: it should be “whet your appetite”, meaning “stimulate your appetite”. “Whet” literally means “sharpen”: a whetstone is something used to sharpen knives or tools.
Feel free to reciprocate if you see me butchering your native language, e.g., by omitting the umlaut from “doppelgänger”, as I did recently. (My wife is of German ancestry and speaks fluently, but I never picked it up.)
Posted by: Todd Trimble on October 18, 2007 1:30 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
one obnoxious comment about an idiomatic English expression
Thanks a lot! I appreciate it. It’s fixed now.
I am actually afraid that you all are being way too polite with me in not correcting me more often.
This entire blog here serves one big purpose for me: find my mistakes. :-) Mostly mathwise, but if I eventually improve my English this way, it wouldn’t hurt either.
Posted by: Urs Schreiber on October 18, 2007 1:44 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
Improved version of
On weak cokernels
now avaialble, incorporating a couple of improvements made by David Roberts:
The pullback property of $T^t \Sigma G_{(2)}$ has been put in a nicer way, the condition on the morphism $H_{(2)} \to G_{(2)}$ stated better (needs to be faithful on the underlying 1-groupoids). And my funny comment in the introduction has been clarified a little more.
Posted by: Urs Schreiber on October 18, 2007 4:40 PM | Permalink | Reply to this
### Re: On Weak Cokernels for 2-Groups
I have now polished my former discussion of weak cokernels of Lie $n$-algebras and merged it with the discussion of weak cokernels of Lie 2-groups by means of mapping cones/tangent categories. The result is here:
On weak cokernels
Then I have incorporated this discussion more fully in my Wiki-substitute called
String- and Chern-Simons $n$-Transport
See
- section $n$-Categorical background, subsection Mapping cones
as well as
- section String- and Chern-Simons $n$-transport, subsection Obstruction theory
Posted by: Urs Schreiber on October 19, 2007 4:33 PM | Permalink | Reply to this
Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 4:33 PM
Read the post BV for Dummies (Part V)
Weblog: The n-Category Café
Excerpt: Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
Tracked: October 30, 2007 10:09 PM
Read the post n-Bundle Obstructions for Bruce
Weblog: The n-Category Café
Excerpt: On the global description of n-bundles obstructing lifts through shifted central extensions.
Tracked: November 4, 2007 5:55 PM
Read the post Lie oo-Connections and their Application to String- and Chern-Simons n-Transport
Weblog: The n-Category Café
Excerpt: A discussion of connections for general L-infinity algebras and their application to String- and Chern-Simons n-transport.
Tracked: January 9, 2008 10:20 AM
Read the post Construction of Cocycles for Chern-Simons 3-Bundles
Weblog: The n-Category Café
Excerpt: On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
Tracked: February 12, 2008 1:42 PM
Post a New Comment | 2019-07-17 19:14:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 157, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.821232259273529, "perplexity": 1103.2417083536595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525374.43/warc/CC-MAIN-20190717181736-20190717203736-00541.warc.gz"} |
https://laravelquestions.com/category/mongodb/ | #### Category : mongodb
$model =$user->find(1); $model->where(‘test’,"like",’%23%’); Sourc.. Read more i am NOT getting result thought the data is there in DB$conditions = collect($this->getColumns())->map(function ($column, $key) use ($keyword, $or_conditions) { return$or_conditions[] = [ $column => [ ‘$regex’ => "$keyword", ‘$options’ => ‘i’ ], ]; }); $final_or_condition = [ ‘$or’ => $conditions->toArray() ];$this->collection = $this->collection->whereRaw($final_or_condition); when i search in Datatable input field on ..
In my project, I use Laravel 8 with Mongodb as my database, and jenssegers/laravel-mongodb as its driver. Let say I have a collection named products which has relationship embedsOne to vendors. when I create a new product, how can I insert existing vendor to this new product? I tried the following $vendor = Vendor::find($request->vendor); $product .. Read more I try to query my collection with only one query and 3 potentials search method : fulltext search, classic search and search regex. This 3 matches can be executed at the same time or just one of them. The fulltext search is the first stage pipeline as we know. Does this fulltext search can be .. Read more i am having collection of employee assosiated with company use IlluminateDatabaseEloquentModel; use JenssegersMongodbEloquentModel as Eloquent; class Employee extends Eloquent { protected$casts = [ ‘has_pp’ => ‘Boolean’, ‘has_eal’ => ‘Boolean’, ‘support’ => ‘String’, ‘has_cin’ => ‘Boolean’, ‘has_lac’ => ‘Boolean’ ]; protected $fillable = [ ‘first_name’, ‘last_name’, ‘has_pp’, ‘has_eal’, ‘support’, ‘has_cin’, ‘has_lac’ ]; public function company(){ .. Read more I was using "jenssegers/mongodb": "2.1" and switched to "jenssegers/mongodb": "3.0.x" while upgrading from Laravel5.0 to Laravel 5.1 with php 5.6.x I modified my codes for the upgraded version, but I’m no longer able to get the results count as used to before with this code:$results = DB::connection($connection) ->collection("listing") ->timeout(-1) ->raw(function ($collection) use ($filters,$cols) ..
I am having multiple forms as collection and needs to validate their ids in one request for which i am taking from*name as name of forms and form id wrt to it. but i am not able to validate form_id wrt to formname provided. any idea to achive this would be very grateful. $v = .. Read more I am new to mongodb and using the aggregate function to fetch some records Query I used is:$records = $this->raw()->aggregate([ [‘$match’ => ["ts" => [‘$gte’=>$startTime,’$lt’=>$endTime]]], [‘$group’ => ["_id" =>["config" => ‘$conf’, "status" => ‘$s’ ], "count" =>[‘$sum’ =>1]], "ts" => [ ‘$dateToString’ => [ ‘format’ => ‘%Y-%m-%d’, ‘date’ =>$startTime, ‘timezone’ => ..
It must be a very stupid question but I am unable to fetch a date key in the mongo aggregate function in laravel framework I used this query to fetch the records but I am not getting how to store the "ts" key in the records too. $records =$this->raw()->aggregate([ [‘$match’ => ["ts" => [‘$gte’=> ..
I’m trying to crear a seeder on laravel 6 using mongodb this is database.php use IlluminateSupportStr; return [ ‘default’ => env(‘DB_CONNECTION’), ‘connections’ => [ ‘mongodb’ => [ ‘driver’ => ‘mongodb’, ‘host’ => env(‘DB_HOST’), ‘port’ => env(‘DB_PORT’), ‘database’ => env(‘DB_DATABASE’), ‘username’ => env(‘DB_USERNAME’), ‘password’ => env(‘DB_PASSWORD’), ‘options’ => [ ‘database’ => ‘mydb’ ] ], ], and ..
I am working on a project using Laravel and MongoDB from jenssegers/laravel-mongodb package and I am struggling with crud operations on cart. Before changing the database to a NoSQL one, my functions were working, it was searching the product in my products table by id and adding, removing or updating as the case may be. ..
hello so this is my first laravel project and i’m lost.. i have a mongodb database that has a users collection and a posts collection this is my Post model: <?php namespace AppModels; use IlluminateDatabaseEloquentFactoriesHasFactory; use IlluminateDatabaseEloquentModel; use JenssegersMongodbEloquentSoftDeletes; /** * @method static create(array $array) * @method static firstWhere(string$string, int $id) */ class Post .. Read more I tried to create a mongodb logging channel ‘mongo’ => [ ‘driver’ => ‘monolog’, ‘handler’ => MonologHandlerMongoDBHandler::class, ‘formatter’ => MonologFormatterMongoDBFormatter::class, ‘handler_with’ => [ ‘mongo’ => new MongoDBClient("mongodb://localhost:27017"), ‘database’ => ‘testdb’, ‘collection’ => ‘testcoll’ ] ], However, im getting error: IlluminateContractsContainerBindingResolutionException(code: 0): Unresolvable dependency resolving [Parameter #0 [ <required>$mongodb ]] in class MonologHandlerMongoDBHandler The error ..
this code working fine when i not change j.acc_no value, If we change j.acc_no value it does not update the document. any one suggest any idea to fix this $updateData = [ ‘bank.$[i].details.$[j].acc_name’ => ‘raja K’, ‘bank.$[i].details.$[j].acc_no’ => ‘35252555’, ‘bank.$[i].details.$[j].min_deposit’ => ‘10000’, ‘bank.$[i].details.$[j].max_deposit’ => ‘1000000’, ‘bank.$[i].details.$[j].status’ => 1, ];$filters = [ [‘i.bankname_slug’ => [‘$eq’ .. Read more I have an array(or collection) like this for my notifications. Each notification has an array field for all users who has seen the notification. array:2 [ 0 => array:8 [ "_id" => "604485ddf92e0000050079c6" "title" => "Notification title" "message" => "Notification message" "type" => "success" "expire_date" => "2021-05-09T19:30:00.000000Z" "read_by_users" => array:2 [ 0 => array:2 [ .. Read more Let’s say I declared a variable$a = "users" and I know there is a collection named users in my mongodb. How can I fetch the collection’s instance in folders like commands? I used Schema facade but it only returns the collection info or it can return if collection exists. Sourc..
When using jenssegers/laravel-mongodb, the belongsTo relationship properly always returns null, despite, when dumping, the value appearing. How can I get the belongsTo relationship? I have two models, a post and a user. Where the post has a belongs to relationship with user. When I use $post->user, I always get null. Despite the fact that when .. Read more I’m trying to setup MongoDB in a Laravel Jetstream + Sail. What I’m doing is trying to install the following packages: mongodb/mongodb jenssegers/mongodb but I get the following errors : Problem 1 – jenssegers/mongodb[v3.8.0, …, v3.8.2] require mongodb/mongodb ^1.6 -> satisfiable by mongodb/mongodb[1.6.0, …, 1.9.x-dev (alias of dev-master)]. – mongodb/mongodb 1.9.x-dev is an alias of .. Read more i have a lumen project that is using mongo db now i want to access the data and do some logic on them my object in mongo is like below : _id: ObjectId(‘602cfb30bc865100073f0e56’), serviceType: ‘normal’, segment: ‘Basic’, steps: { ‘step1’: 1, ‘step2’: 2 } nnow in my laravel application i do it like below : .. Read more I would like to save the following object to a field on a mongodb collection via eloquent "project_settings" : { "resource" : true, "zone" : true, "contractor" : true, "responsible_person" : false } this is my php code: return ProjectModel::where($project_id)->first()->update($project_data); dont worry about the model it works fine i need to make a query that .. Read more I am running following query in mongodb collection and I am using this package https://github.com/jenssegers/laravel-mongodb to communicate with mongodb in laravel.$ibQuery = $this->model->project([‘message_count’ => [ ‘$size’ => [ ‘$ifNull’ => [‘$messages’, []] ] ] ]) ->get(); return $ibQuery; and I am getting this error. exception: "MongoDBDriverExceptionCommandException" file: "/opt/lampp/htdocs/my-project/vendor/mongodb/mongodb/src/Operation/Find.php" line: 337 message: "Unsupported projection option: .. Read more So I’ve managed to do a clean install of Laravel in Windows using WSL2. One of the requirements of my project is to have two databases running at the same time mysql and mongodb. I added an image of MongoDB in my docker-compose.yml as the dock-hub tells me, but when I try to add the .. Read more i have data in mongodb it is a road object that has a property and an array of points that it consists of: my model in laravel <?php namespace AppModels; use AppTraitsUuids; use GraphQLTypeDefinitionType; use RebingGraphQLSupportFacadesGraphQL; use JenssegersMongodbEloquentModel as Eloquent; class Road extends Eloquent { //use HasFactory; use Uuids; protected$connection = ‘mongodb’; protected $collection .. Read more I get wrong date from MongoDB using Eloquent in Laravel. My record in database looks like this "created_at" : ISODate("2020-11-17T15:30:42.131+01:00") Code to get records from MongoDB$taskObj = TaskComments::where(‘task_id’, \$task_id)->get()->toArray(); Result date for created_at is 1970-01-25 20:31:23 which is wrong, I would like to get in this format 2020-11-17 15:30:42 I would like to create .. | 2021-04-14 15:58:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1836722195148468, "perplexity": 11197.588104155528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077843.17/warc/CC-MAIN-20210414155517-20210414185517-00374.warc.gz"} |
https://www.physicsforums.com/threads/magnitude-of-current-through-a-rectangular-loop-given-the-magnetic-field.588102/ | # Magnitude of Current through a Rectangular Loop, given the Magnetic field
1. Mar 18, 2012
### blastoise
1. The problem statement, all variables and given/known data
A rectangular loop with dimensions 4.2 cm by 9.5 carries current I. The current in the loop produces a magnetic field at the center of the loop that has magnitude 5.60×10−5 T and direction away from you as you view the plane of the loop.
2. Relevant equations
$$\vec{B}=\frac{μ_0}{4\pi} \frac{Id\vec{l}\times \vec{r}}{r^2} (1.0) \ \ :|\vec{B}| =5.60 \times 10^{-5}; μ_0:= 4\pi \times 10^{-7} .$$
Where, dl is length, r hat is the unit vector, r^2 is source to point (would like to get this checked though not sure).
3. The attempt at a solution
Coordinate system: Holding up our left hand making an L with the index finger and thumb, let the thumb point in the + X direction index finger point in the +Y direction. Then the -Z direction will be defined in the direction we look.
Recall the unit vector is written in the form <i,j,k>. It becomes apparent using the "Right hand rule" the B field is pointing in the -k direction at all points; Thus, using the Principle of Superposition of Magnetic fields - The total magnetic field caused by several moving charges is the vector sum of the fields caused by the individual charges one can sum the magnetic field B created by the 4 wire segments(the dl vector is a vector with length dl, in the same direction as the current in the conductor).
Let B_1 be when L = 9.5 cm = .095m and r = 4.2cm /2 = .021 m is my r correct?
Then,
$$B_1=\frac{μ_0}{4\pi} \frac{I*L}{r^2} = \frac{μ_0}{4\pi} \frac{I*.095m}{.021^2m^2} .$$
Since there are two sides with 9.5 cm that create a B field in the -K direction the B field produced them is 2*B_1.
Let B_2 be when L = 4.5 cm = .045m and r = 9.5 cm / 2 = .0475m; Then,
$$B_2=\frac{μ_0}{4\pi} \frac{I*L}{r^2} = \frac{μ_0}{4\pi} \frac{I*.045m}{.0475^2m^2} .$$
Since there are two sides with 4.5cm that create a B field in the -K direction the B field produced by them is 2*B_2.
The Total B field is then 2B_1 + 2B_2
$$5.60×10^{-5}T=\frac{μ_0}{4\pi} I2[ \frac{.095 m}{.021^2m^2}+ \frac{.042m}{.0475^2m^2}] .$$
^ was wondering if that is correct.
(1.0) is Law of Biot and Savart.
Last edited: Mar 18, 2012 | 2018-02-26 00:28:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5357728600502014, "perplexity": 772.8897745677194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891817523.0/warc/CC-MAIN-20180225225657-20180226005657-00681.warc.gz"} |
https://psspy.org/psse-help-forum/answers/1724/revisions/ | Perhaps you could move the load to a new bus, which is connected to the original bus via a zero impedance line. Use DISTBRANCHTRIP DIST_BRANCH_TRIP to trip the branch (and island the load), then use busdata2 busdata2 to change the bus code to 1 before using DISTBRANCHCLOSE DIST_BRANCH_CLOSE to reconnect it. | 2022-12-08 09:43:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4323248267173767, "perplexity": 3883.475174098273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711286.17/warc/CC-MAIN-20221208082315-20221208112315-00018.warc.gz"} |
https://nodus.ligo.caltech.edu:8081/40m/?id=16192&sort=Subject | 40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
40m Log, Page 217 of 329 Not logged in
ID Date Author Type Category Subject
3734 Mon Oct 18 11:22:13 2010 JenneUpdateComputersShame on people not elogging! FrameFiles backups not working.
On the one hand, SHAME ON ALL PEOPLE WHO DON'T ELOG THINGS, such as the moving of scripts directories (it was a pain to figure out that that's part of why the backup scripts are broken). On the other hand, the moving of the scripts directories brought to light a critical problem in the backup scheme. None of the frame files have been backed up since Joe replaced fb40m with fb, on ~23 Sept (I think).
What went down:
The frame builder was replaced, and no backup script was started up on the new machine. Sadface. Crontab doesn't work yet on the new machine, and also the 'ssh-add' commands give an error:
controls@fb /cvs/cds/rtcds/caltech/c1/scripts/backup $ssh-add ~/.ssh/id_rsa No such file or directory controls@fb /cvs/cds/rtcds/caltech/c1/scripts/backup$ ssh-add ~/.ssh/backup2PB
No such file or directory
Thus, I know that the backup was never running on the new fb. However, the check-er script runs on nodus, and looks at the logfile, and since there was no script running, it wasn't adding to the log file, so the last log was an "Okay, everything worked" entry. So, the check-er script kept sending me daily emails saying that everything was okie-dokey.
Since all of the scripts were moved (Joe said this happened on Friday, although there's no elog about it), the check-er script, and all of the rest of the backup scripts point to the wrong places (the old scripts/backup directory), so I didn't receive any emails about the backup either way (usually it at least sends a "Hey, I'm broken" email). This clued me in that we need to check things out, and I discovered that it's all gone to hell.
Since I can't add the ssh clients to the new fb, I can't actually log in to the backup computers over in Powell-Booth to check when the last legitimately successful backup was. But I suspect it was just before the fb was replaced.
So, we need to get Crontab up and running on the new Frame Builder machine so that we can run cron jobs, and we also need to figure out this backup hullabaloo. I think I'll email / call Dan Kozak over in downs, who was talking about upgrading our backup scheme anyway.
13906 Thu May 31 22:59:27 2018 KojiConfigurationComputersShorewall on nodus
[Jonathan Koji]
Shorewall (http://shorewall.net/), a tool to configure iptables, was installed on nodus.
The description about shorewall setting on nodus can be found here: https://wiki-40m.ligo.caltech.edu/NodusShorewallSetting
NDS (31200) on megatoron is not enabled outside of the firewall yet.
3084 Thu Jun 17 17:09:44 2010 AlbertoUpdateLSCShort Cavity Length Adjustments
I calculated the phase shifts that the sidebands would pick up in the arms in the case we changed the arm length to 38.4m as proposed. I obtained the following values (in degrees):
phi(-f2) = 0.66; phi(-f1) = -0.71; phi(f1) = 0.71; phi(+f2) = -0.66
These are the plots with the results as I obtained from an Optickle simulation (the second zooms in around 38.4m).
These values agree with what Koji had already estimated (see elog entry 3023).
Since we can't make the arm longer than that, to increase the distance from the resonance, we would like to adjust the length of the short cavities to compensate for that. For f2 (=55MHz), 0.7 degrees correspond to about 5cm. That is about the length change that we expect to make to the design.
I simulated with Optickle the effect of changing the length of either the SRC or the PRC. The best way I found to do that, was to measure the cavity circulating power when the macroscopic lengths change.
The following plots show the effect of changing either the PRC or SRC length (left or right figure), on the circulating power of both cavities at the same time (top and bottom plots).
You can compare these with the case of perfect antiresonance as in the following plots:
It seems that the design length for the short cavities are not too bad. f1 is not optimized in the PRC, but changing the length of the cavity wold just make f2 worse in SRC.
These simulations seem to support the choice of not changing the design cavity lengths for PRC and SRC.
Of course these are only an "open loop" simulations. At the moment we don't know what would be the effect of closing the control loops. That is something I'm going to do later. It'll be part of my studies on the effects of cavity absolute length on the whole IFO.
3086 Fri Jun 18 13:47:20 2010 KojiUpdateLSCShort Cavity Length Adjustments
You should have been in my lecture yesterday!
Power in the cavity is not a good index (=error signal) to judge the optimal length.
You should look at the phases of the length signals. (i.e. demodulation phase which gives you the maximum amplitude for CARM, PRC, SRC, etc)
You must move the SRC and PRC lengths at the same time.
The resonance of f1 (mostly) depends on the PRC length, but that of f2 depends on both the PRC and SRC lengths.
3087 Fri Jun 18 15:07:26 2010 AlbertoUpdateLSCShort Cavity Length Adjustments
Quote: You should have been in my lecture yesterday! Power in the cavity is not a good index (=error signal) to judge the optimal length. You should look at the phases of the length signals. (i.e. demodulation phase which gives you the maximum amplitude for CARM, PRC, SRC, etc) You must move the SRC and PRC lengths at the same time. The resonance of f1 (mostly) depends on the PRC length, but that of f2 depends on both the PRC and SRC lengths.
Right. Ultimately the phase gain inside the cavity is what we look at. Calculating that for the SBs inside PRC and SRC is actually the first thing I did.
But I kept getting very small angles. Too small, I thought. Maybe there was some problem in the way I calculated it.
Then I made a power analysis to check if the SBs were getting affected at all by that 0.7degree phase shift they're picking up in the arms.
I wanted to show the point where I am, before leaving. But, I keep working on it.
1089 Fri Oct 24 21:49:15 2008 JenneConfigurationPEMShort Seismometer Cable
Bad news regarding the cable that goes between the Guralp seismometer and the box that I've been working on: it's too short by about a factor of 2. Dang it. I've placed the seismometer underneath the Beam Splitter Chamber (where it needs to go), and started running the cable toward the ADC rack where box was planned to go, and as Rana guessed earlier tonight, the cable isn't nearly long enough. We have some options: the seismometer can go back into the half-height rack near the BS, SRM, PRM oplev's optical table where I think it used to be, or it can go into the rack with the Kepco high voltage power supplies and the laser's supply. The cable won't reach any farther than that.
I think that we can just add BNC extensions onto the octopus cable that Bob made for the Guralp box, so all we need to figure out after we decide on a rack is the power for the box.
2219 Mon Nov 9 16:32:36 2009 AlbertoFrogsEnvironmentShot of the white board yesterday before erasing
Yesterday Rana and I needed some room on the white board in the Control Room. We had to erase some of the stuff present on the board despite the bif warning "Do Not Erase".
This is how it looked like before erasing.
Attachment 1: DSC_0980.JPG
11386 Wed Jul 1 09:33:31 2015 KojiUpdateGeneralShutters closed, watch dogs disabled for the RCG upgrade
I closed the PSL/GREEN shutters and shut off the LSC feedback/SUS watch dogs at 9AM PDT, to allow Jamie to start his disruptive work.
4837 Mon Jun 20 09:28:19 2011 JamieUpdateCDSShutting down low-voltage DC power in 1X1/1X2 racks
In order to install the BO module in 1X2, I need to shut down all DC power to the 1X1 and 1X2 racks.
11073 Thu Feb 26 01:51:39 2015 ericqUpdateLSCSideband HOMs
So, my previous post suggested that 44*11 products might be the dominating signals in our "nominal" setup. I suppose that this could be not so bad, since it's not too unlike -11*22; the 11MHz field couples into the PRC and reflects with a rapidly changing phase around PRC resonance, and 44MHz is antiresonant, so it is a good local oscillator for REFL33.
However, it then occured to me that my previous HOM analysis only looked at the 11MHz and 55MHz sidebands.
When extending this to all of the sidebands within 55MHz, I discovered a troubling fact:
With the IFO parameters I have, the second spatial order +22MHz and fourth spatial order +44MHz fields almost exactly coresonate with the carrier in the PRFPMI! (DR, too)
If this is true, then any REFL33 signal seems to be in danger if we have an appreciable amount of these modes from, say, imperfect modematching.
On the other hand, we've been able to hold the PRMI with REFL33 when ALS is "on resonance," so I'm not sure what to think. (As a reminder, this analysis is done with analytic formulae for the complex reflectivities of the arm cavities and coupled recycling cavities as a function of CARM, spatial order and field frequency. Source is attached.)
It seems the Y arm geometry is to blame for this.
Maybe we should try to measure/confirm the Y arm g-factor...
Attachment 1: C1_HOMcurves_PR.png
Attachment 2: C1_HOMcurves_Y.png
Attachment 3: C1_HOMcurves_X.png
Attachment 4: C1_HOMlist.zip
12234 Thu Jun 30 16:21:32 2016 gautamUpdateCOCSideband HOMs resonating in arms
[EricQ, gautam]
Last night, we set about trying to see if we could measure and verify the predictions of the simulations, and if there are indeed HOM sidebands co-resonating with the carrier. Koji pointed out that if we clip the transmitted beam from the arm incident on a PD, then the power of the higher order HG modes no longer integrate to 0 (i.e. the orthogonality is broken), and so if there are indeed some co-resonating modes, we should be able to see the beat between them on a spectrum analyzer. The procedure we followed was:
1. Choose a suitable PD to measure the beat. We chose to use the Thorlabs PDA10CF because it has ~150MHz bandwidth, and also the responsivity is reasonable at 1064nm.
2. We started our measurements at the Y-end. There was a sufficiently fast lens in the beam path between the transmon QPD and the high gain PD at the Y end, so we went ahead and simply switched out the high gain thorlabs PDA520 for the PDA10CF. To power the PDA10CF, we borrowed the power cable from the green REFL PD temporarily.
3. We maximized the DC power of the photodiode signal using an oscilloscope. Then to introduce the above-mentioned clipping and orthogonality-breaking, we misaligned the beam on the PD until the DC power was ~2/3 the maximum value.
4. We then hooked up the PD output to the Agilent network analizyer (with a DC block).
5. We measured the spectrum of the PD signal around 11.066MHz (with 100kHz span) and higher harmonics up to 55MHz and used a narrow bandwidth (100Hz) and long integration time (64 averages) to see if we could find any peaks. More details in the results section.
6. Having satisfied ourselves with the Y-end measurements, we
• restored the power cable to the green beat PD
• re-installed the thorlabs PDA520
• verified that both IR and green could be locked to the arm
We then repeated the above steps at the X-end (but here, an additional lens had to be installed to focus the IR beam onto the PDA10CF - there was, however, sufficient space on the table so we didn't need to remove the PDA520 for this measurement).
Results:
Y-end: DC power on the photodiode at optimal alignment ~ 200mV => spectra taken by deliberately misaligning the beam incident on the PD till the DC power was ~120mV (see remarks about these values).
RF sideband (Y-arm) Peak height (uV) Beat power (nW) RF sideband (X-arm) Peak height (uV) Beat Power (nW)
11 1.55 0.52 11 1.2 0.4
22 10.6 3.53 22 none seen N.A.
33 none seen N.A. 33 none seen N.A.
44 22.0 7.33 44 7 2.33
55 8.6 2.97 55 5 1.67
I converted the peak heights seen on the spectrum analyzer in volts to power by dividing by transimpedance (=5*10^3 V/A into a 50ohm load) * responsivity at 1064nm (~0.6A/W for PDA10CF).
Remarks:
1. This effect flagged by the simulations seems to be real. Unfortunately I can't get a more quantitative picture because we can't quantify the mode-overlap between the carrier 00 mode and any higher order mode on the beat PD (as we know nothing about the profile of these modes), but the simulations did suggets that the 2nd order 22MHz and 4th order 44MHz HOMs are the ones closest to the carrier 00 resonance (see Attachments #2 and #3), which is kind of borne out by these results.
2. I disbelieve the conversions into power that I have done above, but have just put them in for now, because a DC power of 200mW at the Y-end suggests that there is >160uW of light transmitted from the arm, which is at least twice what we expect from a simple FP cavity calculation with the best-known parameters. If I've missed out something obvious in doing this conversion, please let me know!
3. For the Y-arm, the region around 55MHz had a peak (presumably from the sideband HOM beating with the carrier) but also a bunch of other weird sub-structures. I'm attaching a photo of the analyzer screen. Not sure what to make of this...
Attachment 1: image.jpeg
Attachment 2: C1_HOMcurves_Y.pdf
Attachment 3: C1_HOMcurves_X.pdf
3360 Wed Aug 4 16:52:59 2010 Razib, AidanUpdatePhase CameraSideband power measurement (updated)
Aidan and I made some attempt to measure the power of the sidebands so that we can calculate our expected signal strength.
Our setup looks like the following:
As light from the laser is split into two at BS1, the transmitted beam has higher power as our BS1 is only coated for 1064nm. We get two reflected beams from BS1, one reflected of the front surface and the other from the back surface. We took the stronger back reflected beam to the EOM driven at 40 MHz (also at 25 MHz at a later time). The AOM produced a reference beam with 40 .000 005 MHz offset which we recombined with the sidebands obtained from the EOM. The beat produced is sent off to PDA 10CF connected to 4395A spectrum analyzer.
The plots for 40MHz sidebands and 25 MHz sidebands looks like this:
From the above spectra, at 40 MHz sideband regime:
Power of the carrier @ 40 MHz = -39.72 dBm
Power of the sideband @ 80 MHz = -60.39 dBm
At 25 MHz sideband regime,
Power of the carrier @ 40 MHz = -40.22 dBm
Power of the upper sideband @ 65 MHz = -61.72 dBm
Power of the lower sideband @ 15 MHz = -60.99 dBm
Power Measurement:
We made some necessary power measurement using a PD connected to a voltmeter after the EOM and the AOM when the EOM is driven at 40 MHz:
___________________________________________________________
Dark : 0.025 V
AOM on: 4.10 V (EOM blocked)
EOM : 2.425 V (AOM blocked)
___________________________________________________________
From the earlier calculation (ref: Elog entry July 28) the power that we expect to see at the PD is,
P= A_c ^2 + A_r^2 + A_(-sb)^2+ A_sb ^2 +2* A_r* A_sb * cos ( w_(r,sb) t ) , where A_c= carrier; A_r= reference beam; A_sb=Upper sideband; A_(-sb)= Lower sideband, w_(r,sb) = w_r - w_sb
P = A_c ^2 + A_r^2 + A_(-sb)^2+ A_sb ^2 +2* A_r* A_sb , letting cos (w_(r,sb) go to 1) is order to approximate the maximum signal
So the signal that we expect to see relative to the DC ( i.e A_c ^2 + A_r^2 + A_(-sb)^2+ A_sb ^2, the first four terms of the power equation) is,
Sig = 2* A_r* A_sb / { A_c ^2 + A_r^2 + A_(-sb)^2+ A_sb ^2 },
Since the modulation index is small, the power in the sideband is very small compared to carrier and the reference beam. So we can ignore the sideband power for the signal expression.
So,
Sig = 2* A_r* A_sb / ( A_c ^2 + A_r^2 )
So if we want to maximize this signal w.r.t the reference then,
d (sig)/ d(A_r) = 2 { ( A_c ^2 - A_r^2) *A_sb } / {( A_c^2 + A_r^2)} ^2
Thus, the signal is maximized when,
A_r^2 = A_c^2
We adjusted the AOM to be driven at + 7.7 dBM so that the new power at the AOM matched the EOM power, which is 2.397 in the voltmeter.
So the power at both the AOM and the EOM are:
P_AOM = ( V_AOM - V_dark) / (PD responsitivity * Transimpedance gain)
= ( 2.397 - 0.025 ) / ( 0.45 * 1.5 x 10 ^5 )
= 3.51 x 10 ^ - 5 W
P_EOM = (V_EOM - V _dark) / (PD responsitivity * Transimpedance gain)
= ( 2. 425 - .0.025) / ( 0.45 * 1.5 x 10 ^5 )
= 3.55 x 10^ - 5 W
From the spectra of the 40 MHz sideband above, the ratio of the carrier and the sideband amplitude is: A_c / A_sb = 10.8 .
P_EOM = A_c ^2 + 2 A_sb ^2
Therefore, A_sb = sqrt ( P_EOM / 118.64) = 5.47 x 10^ - 4 V/m
Thus, A_c = 5.908 x 10^ -3 V/m
and A_r = sqrt ( P_AOM) = 5.92 x 10 -3 V/m.
This measurement can be used to calculate the signal to contrast ratio (SCR) that we expect to see:
SCR = 2 A_r * A_sb / ( A_c^2 + A_r^2 ) = 0.09
Our next step is to measure the actual signal to constrast ratio as seen by the camera. Details of that will be posted soon.
3411 Thu Aug 12 16:52:02 2010 RazibUpdatePhase CameraSideband power measurement (updated)
I made some measurement of the SCR (signal to contrast ratio) from the signal from the EOM and the AOM.
The recipe for that was:
1. Trigger the camera at 20 Hz (from function generator).
2. Take a series of 20 images.
3. Do FFT to take out the DC component.
4. Extract the beat signal out of the FFT'd data.
5. Block the EOM.
6. Take another set of images of the AOM beam.
7. Take more(!) images, but this time of the background (blocking both EOM and AOM).
So the SCR is calculated by the ratio of the FFT'd DC and the 5 Hz signal. Using the CCD, I obtained the SCR to be 0.075 ± 0.01. Previously, we expected our SCR to be 0.09 as in the previous e-log entry.
The plot for that is:
After measuring the SCR, I also measured the amplitude of the sideband and made an amplitude profile of the 40 MHz sideband.
The amplitude measurement is done as follows:
We know that the our 5 Hz signal consists of,
Sig = A_r * A_sb where A_r = amplitude of the reference beam, A_sb= amplitude of the sideband
So, A_sb = Sig / A_r .
But, A_r = sqrt ( P_AOM - Background),
Thus A_sb = Sig / sqrt( P_AOM - Background) .
So the amplitude profile is done by taking the 5 Hz beat signal and dividing by the square root of the AOM beam minus the background light.
The plots looks like this:
The solo sideband profile looks like this:
Next we plan to trigger the camera with a 1 KHz signal and take snaps at n* T/4 (where n=0,1,2,3) of the period of the beat signal. So the plan is to trigger the camera at the point where the red dots appear in following cartoon.
Some more details of this setup will be posted later.
Quote:
Attachment 4: sine_trig.jpg
3412 Thu Aug 12 17:10:07 2010 KojiUpdatePhase CameraSideband power measurement (updated)
This sounds very relieving although this could be a lower bound of the number.
Why didn't you use the output on the PD which just give us the direct observation of your so-called SCR.
Ed: I meant time series of the PD output
Quote: So the SCR is calculated by the ratio of the FFT'd DC and the 5 Hz signal. Using the CCD, I obtained the SCR to be 0.075 ± 0.01. Previously, we expected our SCR to be 0.09 as in the previous e-log entry.
3413 Thu Aug 12 17:28:28 2010 RazibUpdatePhase CameraSideband power measurement (updated)
Quote:
This sounds very relieving although this could be a lower bound of the number.
Why didn't you use the output on the PD which just give us the direct observation of your so-called SCR.
Quote: So the SCR is calculated by the ratio of the FFT'd DC and the 5 Hz signal. Using the CCD, I obtained the SCR to be 0.075 ± 0.01. Previously, we expected our SCR to be 0.09 as in the previous e-log entry.
The SCR was at first measured using the output of the PD. That was exactly from where we got our 0.09 (previous elog entry). The second measurement was from the CCD.
6167 Wed Jan 4 05:02:58 2012 kiwamuUpdateLSCSidebands measurement at POP
Just a quick report:
I did the first attempt to measure the recycling gains of the sidebands in the DRMI configuration (sidebands resonant condition)
by looking at the output of the POP22/110 RFPD.
Because this time what I measured is some absolute values of the sidebands power,
it doesn't tell us anything quantitatively until we calibrate it or compare it with similar data.
So I need to measure the same things in some different configurations (e.g. PRMI, SRMI, etc.)
in order to extract some useful information from the measurement.
The attached picture is the display of a power spectrum analyzer looking at the output of the POP22/110 broadband RFPD
while the DRMI (in the sideband resonant condition) was kept locked.
You can see that 111 MHz (twice of 55 MHz) is prominent. Also there are several peaks at 11, 22, 44 and 66 MHz.
1297 Thu Feb 12 14:39:07 2009 ranaSummaryGeneralSilicon Beam Dump test
Yesterday evening, Ken Mailand and I tested the reflectivity of a piece of polished Silicon. Since Silicon has such a high thermalconductivity (compared to stainless and fused silica) and can take much more heat than black glass and should have a very good BRDF and should absorb most of the 1064 nm light if we hit it at Brewster's angle, we want to try it out in the first version high power, low scatterbeam dump. This dump will be a 'V' style dump like what Steve has tested nearly a year ago, but the incoming beam will first hit this piece of Silicon.
The pictures of the setup and the Silicon with the beam hitting it are here.
Brewster's angle for p-pol at 1064 nm is 74.2 deg (n = 3.53 @ 1064 nm). We set up a cube polarizer on the output of the ~1064 nm CrystaLaser. 144 mW got to the Si; the reflected beam was ~1.9-2.0 mW after tuning the angle for minimum power. Via eyeball and protractor it seems that we're at ~74 deg. So the reflectivity is < 1.5-2%. This is good enough; the reflected power will be < 1 W in all cases for eLIGO and that can be absorbed by the rest of the stainless V dump. The 35 W of heat in the silicon will be mostly gotten rid of through conduction into the attached heat sink fins.
This kind of dump would go into places like the PMC-REFL, MC-REFL, & IFO-REFL, where we occasionally need to take high power, but also are sensitive to backscatter.
2752 Thu Apr 1 16:34:29 2010 HartmutUpdateGreen LockingSilicon PDs
just a few infos on Silicon PDs I looked up.
If you want to go beyond the 100MHz achievable with the device I worked on,
the one thing to improve is the opamp, where Steve is trying to find OPA657.
This is a FET with 1.6GHz BWP, minimum stable gain of 7, and 4.8nV/rt(Hz) noise.
Should be ok with 750-1000 Ohm transimpedance.
The other thing you might want to change is the PD
(although it might be the 1cm PD with high bias is as fast as smaller ones with lower bias).
There are two types of other Si diodes at the 40m right now (~3mm):
-Rana and I found a Centronic OSD 15-5T in the old equipment
-Frank gave me a Hamamatsu S1223-01 on a Thorlabs pre-amp device (could be taken out).
The Centronic OSD 15-5T has up to 80pF with 12 V bias according to the datasheet.
The Hamamatsu S1223-01 is stated with 20pF only, but stated to have a max. frequency resp. of 20MHz ('-3db point').
I dont know what this means, as the corner freq. of 10pF into 50Ohm is still 160MHz.
In any case there are faster 3mm types to start with, as for example Hamamatsu S3399 (~ 90\$),
which is stated to have the corner at 100MHz with 50 Ohm load.
For this type the stated capacity (20pF) looks consistent with ~100MHz corner into 50 Ohm.
So probably you can get higher BW with this one using much smaller load, as in transimpedance stage.
7181 Tue Aug 14 16:33:51 2012 SashaUpdateComputer Scripts / ProgramsSimPlant indicator added
I added an indicator to the watch dog screen so that a little "SP" icon appears whenever the SimPlant is on. Since we only have one simplant (ETMX), only ETMX has the simPlant indicator. However, since assymetry is ugly, I moved all of the OL icons over so that they're in a line and so that there is room for future SP icons.
I also fixed the link to the Watchdogs on the main SUS screens (it was dead, but now it is ALIVE).
4394 Thu Mar 10 01:28:47 2011 joe, jamie, rana, chrisSummaryCDSSimSuspension !
Today was a banner day for Simulated Plants.
Joe and Jamie have been working to get it all happening and this afternoon we started stuffing filters into the plant to make it act like the:
We put in the following features so far:
1. Anti-Imaging filters (these are hacked to be approximate since the real ones are 7570 Hz LP filters and the SimAI only can have filters up to 8192 Hz).
2. Dewhitening filters (copied from the SimDW in the SUS-ETMY screens)
3. Coil Driver transimpedance (1 / 200 Ohms)
4. Magnet-coil force constant (0.016 N/A)
5. Conversion from Coil to DOF Basis
6. All DOFs of the mechanical model are represented as simple harmonic oscillators with Q~100 and f ~ measured free swinging peaks.
7. Signals/Noise can be injected either as force noise on the test mass or as displacement noise at the suspension point.
8. Conversion from DOF to Shadow Sensor basis.
9. Optical Levers (with whitening)
We have also changed the switching logic for the SUS and SimETMs for the shadow sensor whitening. It used to be that either the hardware OR the software whitening was on. Now we have made it like all of the other whitening/antiwhitening in LIGO and the whitening/antiwhitening come on together. Joe and Jamie are going to propagate this to the other SUS. The hardware filter is a 30,100:3 (poles:zeros) whitening filter. The digital filter used to also be 30,100:3 with a DC gain = 1. I've changed the FM1 filter in the XXSEN filter banks into a 3:30 for the ETMY so that it now comes on and just compensates the hardware filter. This change should be propagated to all other SUS and the MEDM screens updated to show the new situation.
After this change, we decided to calibrate the {UL,UR,LL,LR,SD}SEN channels into units of microns. To do this we have made an FM6 filter called 'cts2um' that accounts for the OSEM gain and the ADC conversion factors. These channels are now in units of microns without applying any calibration in the DTT or Dataviewer. The plot below shows the OSEM shadow sensor time series with all damping loops ON and a very rough version of seismic noise being injected in all 6 DOFs (note that the y-axis is microns and the x-axis is seconds).
Next, Jamie is adding the angular calibrations (so that SUSPIT and SUSYAW are in rads) and Chris is making vectift quality seismic noise injectors.
We also need to add coating thermal noise, suspension thermal noise, substrate thermal noise, ADC/DAC noise and a lot of MEDM screen indicators of what state we're in. I myself can't tell from the OSEM time series if its real or Sim.
7151 Sat Aug 11 01:10:26 2012 SashaUpdateSimulationsSim_Plant Working!
Sim_Plant going okay. Adding seismic noise tomorrow - we'll see what happens. The gain is still semi-off, but I know how to fix it - its just nice to have it gained up while I play with noise.
P.S. JAMIE DO YOU NOTICE HOW PRETTY MY GRAPH IS?
Attachment 1: Plant_sen.jpg
7152 Sat Aug 11 18:05:49 2012 SashaUpdateSimulationsSim_Plant Working!
Quote: Sim_Plant going okay. Adding seismic noise tomorrow - we'll see what happens. The gain is still semi-off, but I know how to fix it - its just nice to have it gained up while I play with noise. P.S. JAMIE DO YOU NOTICE HOW PRETTY MY GRAPH IS?
Developed some seismic noise. I adapted the seismic noise filters we used for the MC model way back when. They looked questionable to begin with, but I added some poles/zeroes to make it more accurate (see Attached).
Attachment 1: seismic_noise1.jpg
7167 Mon Aug 13 23:06:08 2012 JenneUpdateSUSSimplant left on
Simplant for ETMX was left on, so I didn't have control of ETMX. Not cool. The IFO should be left in it's 'regular' state (all optics restored to saved alignments, no simplant, LSC/ALS/ASS loops off) if you're not actively working on it.
What this did point out, however, is that we need a big ol' indicator on the IFO_ALIGN / LSC / Watchdog / Overview screens to indicate that simplant is on for a particular optic, or whatever simplant might be controlling that takes away 'regular' control. I probably would have continued being frustrated and confused for a lot longer if Eric didn't mention that simplant could have been left on. Thanks Eric!
Symptoms, which perhaps would have eventually pointed me to simplant, were that there was some weird moving beam on the AS camera that was flashing fabry-perot fringes, and the POX signal looked like junk. After some looking around, I noticed that ETMX, while it claimed to have all the damping loops on, and the oplev on, was swinging a lot (rms levels of 4 - 7, rather than the usual < 2 ). I said something out loud, and Eric suggested looking at Simplant. After putting Simplant back to Reality, things are back to normal.
3276 Fri Jul 23 14:26:01 2010 GopalUpdateOptic StacksSimple Frequency Response Measurements in COMSOL
Over the past couple days, I discovered a simple, direct method for calculating frequency responses with a combination of COMSOL and any plotter such as Excel or MatLab. The simple case of rectangular prism of steel was analyzed using this method; details will be posted shortly on the COMSOL Wiki page. The frequency response matched theoretical reasoning: the bar acts as a simple mechanical low-pass filter, rapidly attenuating driving frequencies at the base beyond the first eigenmode.
It therefore shouldn't be too difficult to extend this analysis to the MC1/MC3 stack. The many eigenfrequencies will produce a more complicated transfer function, and so more data points will be taken.
The major shortcoming of this method involves dealing with the imaginary components of the eigenfrequencies. As of now, I haven't found a way of measuring the phase lag between the drive and the response. I also haven't found a way of changing the damping constants and therefore playing with phase components.
4388 Tue Mar 8 16:59:47 2011 josephbUpdateCDSSimulated Plant Work
The screens for the simplified c1spx model have been updated. I re-introduced the suspension point information into the sensor output matrix so we can take into account the fact that as the entire supporting structure moves, the osems moves relative to the optic.
Master screens for the noise filters (i.e. 60 Hz, suspension point motion, and optic noise) have been created.
I have currently set the matrix values of the c1spx model to handle just longitudinal motion. I.e. Coils drive only in the POS degree of freedom and sensor read outs are also only in the POS degree of freedom. I've turned off all the noise inputs.
I added a simple double pole at 1 Hz in the C1:SUP_ETMX_PL_F2P_0_0 filter bank.
14667 Wed Jun 12 22:02:04 2019 MilindUpdateCamerasSimulation enhancements
Today, Rana asked me to work on improving simulations based on the ideas we discussed last week. As of the previous elog the simulation accomodated only
1. Simulation of Gaussian beam spot.
2. Arbitrary motion.
Today, I added the simulation of point scatterers.
What?
The image on the sensor (camera) is produced in roughly the following steps.
1. Motion of the Gaussian beam on the optic (X,Y coordinates) which is what has been simulated so far.
2. Reflection from the surface of the optic which can be modeled using knowledge of the BRDF has not been included as of this elog as I wish to do a little more reading before doing so.
3. Reflection from point scatterers (dust particles burnt into the optic surface by the laser and so forth) which are characterised as peaks (impulses) in the TIS vs position plot. The laser beam is incident nearly normally on the optic and this behaviour is independent of the angle of observation. This is what has been added to the simulation.
How?
1. Increased the frame resolution to 720 x 480.
2. Defined an array of the same size and set values of at most "num_scatter" number of points at random positions to values determined randomly between 1 and "scatter_amp" + 1 where scatter_amp is non-negative.
3. Multiplied the resulting array by the resulting Gaussian beam. The motivation was to imitate the bright specks obtained on various camera feeds in the lab. Physically, this also implies normal incidence and normal observation which is not the real case at all. I shall add these features in a day or two.
Herewith, in attachments #1, #2, #3 I am attaching videos obtained by varying scattering amplitude and number of scattering points in a vain attempt to reproduce this data. I shall work more on this simulation on Friday.
Scripting stuff:
1. Previous elogs detail how to take gige images at various exposure times. I am still waiting on Kruthi to use the script.
2. Tomorrow I shall work on the scripting software to interact with the GigE and take video for a fixed duration etc. I shall also begin working on a script to autolock the PMC based on what Rana showed me on Monday. I will also take a look at the the contents of this elog and try to pick up from there. I hope to make significant progress by the next lab meeting.
Neural network stuff:
GANs for simulation:
1. Other than putting the physics into simulation i.e the first portion of this elog, GANs can be trained to generate images similar to the original data. I am unfamiliar with training GANs and the various tricks that are used specifically for them. I will do a bit of reading and make an update by Friday. As of now, the data I plan to use is this and I will train it using the GTX 1060 on my machine.
Networks for beam tracking:
1. I will use the architectures suggested in this work with a few modifications. I will use MSE loss function, Adam optimizer and my local GPU for training.
Attachment 1: simulated_motion0.mp4
Attachment 2: simulated_motion0.mp4
Attachment 3: simulated_motion0.mp4
14698 Tue Jun 25 23:52:37 2019 MilindUpdateCamerasSimulation enhancements
Yesterday, Rana asked me to look at Hiro Yamamoto's docs on the DCC to improve the simulation. I'm performing a first pass (=> Just skimming through to see if they're relevant, I will go through them more carefully soon!) and putting up stuff here for future reference. @Kruthi's help much appreciated!
14714 Mon Jul 1 20:11:34 2019 MilindUpdateCamerasSimulation enhancements
Today, I read a lot more about BRDF and modelling but could not make much headway regarding the implementation in the simulation. I've stopped for now and I'll take a crack at it tomorrow again.
Quote: Yesterday, Rana asked me to look at Hiro Yamamoto's docs on the DCC to improve the simulation. I'm performing a first pass (=> Just skimming through to see if they're relevant, I will go through them more carefully soon!) and putting up stuff here for future reference. @Kruthi's help much appreciated!
14635 Thu May 23 15:37:30 2019 MilindUpdateCamerasSimulation enhancements and performance of contour detection
1. Implemented image level noise for simulation. Added only uniform random noise.
2. Implemented addition of uniform random noise to any sinusoidal motion of beam spot.
3. Implemented motion along y axis according to data in "power_spectrum" file.
4. Impelemented simulation of random motion of beam spot in both x and y directions (done previously by Pooja, but a cleaner version).
5. Created a video file for 10s with motion of beam spot along the y direction as given by Attachment #1. This was created by mixing four sinusoids at different amplitudes (frequencies (0.1, 0.2, 0.4, 0.8) Hz Amplitudes as fractions of N = 64 (0.1 0.09 0.08 0.09). FPS = 10. Total number of frames = 100 for the sake of convenience. See Attachment #5.
6. Following this, I used the thresholding (threshold = 127, chosen arbitrarily), contour detection and centroid computation sequence (see Attachment #6 for results) to obtain the plot in Attachment 2 for the predicted motion of the y coordinate. As is evident, the centering and scale of values obtained are off and I still haven't figured out how to precisely convert from one to another.
7. Consequently, as a workaround, I simply normalised the values corresponding to each plot by subtracting the mean in each case and dividing the resulting series of values by their maximum. This resulted in the plots in Attachments 3 and 4 which show the normalised values of y coordinate variation and the error between the actual and predicted values between 0 and 1 respectively.
Things yet to be done:
Simulation:
1. I will implement the mean square error function to compute the relativer performance as conditions change.
2. I will add noise both to the image and to the motion (meaning introduce some randomness in the motion) to see how the performance, determined by both the curves such as the ones below and the mean square error, changes.
3. Following this, I will vary the standard deviation of the beam spot along X and Y directions and try to obtain beam spot motion similar to the video in Attachment #2 of elog post 14632.
4. Currently, I have made no effort to carefully tune the parameters associated with contour detection and threshold and have simply used the popular defaults. While this has worked admirably in the case of the simple simulated videos, I suspect much more tweaking will be needed before I can use this on real data.
5. It is an easy step to determine the performance of the algorithm for random, circular and other motions of the beam spot. However, I will defer this till later as I do not see any immediate value in this.
6. Determine noise threshold. In simulation or with real data: obtain a video where the beam spot is ideally motionless (easy to do with simulated data) and then apply the above approach to the video and study the resulting predicted motion. In simulation, I expect the predictions for a motionless beam spot video (without noise) to be constant. Therefore, I shall add some noise to the video and study the prediction of the algorithm.
7. NOTE: the above approach relies on some previous knowledge of what the video data will look like. This is useful in determining which contours to ignore, if any like the four bright regions at the corners in this video.
Real data:
1. Obtaining real data and evaluate if the algorithm is succesful in determining contours which can be used to track the beam spot.
2. Once the kind of video feed this will be used on is decided, use the data generated from such a feed to determine what the best settings of hyperparameters are and detect the beam spot motion.
3. Synchronization of data stream regarding beam spot motion and video.
4. Determine the calibration: anglular motion of the optic to beam spot motion on the camera sensor to video to pixel mapping in the frames being processed.
Other approaches:
1. Review work done by Gabriele with CNNs, implement it and then compare performance with the above method.
Attachment 1: actual_motion.pdf
Attachment 2: predicted_motion.pdf
Attachment 3: normalised_comparison.pdf
Attachment 4: residue_normalised.pdf
Attachment 5: simulated_motion1.mp4
Attachment 6: elog_22may_contours.mp4
14638 Sat May 25 20:29:08 2019 MilindUpdateCamerasSimulation enhancements and performance of contour detection
1. I used the same motion as defined in the previous elog. I gradually added noise to the images. Noise added was uniform random noise - a 2 dimensinoal array of random numbers between 0 and a predetermined maximum (noise_amp). The previous elog provides the variation of the y coordinate. In this, I am also uploading the effect of noise on the error in the prediction of the x coordinate. As a reminder, the motion of the beam spot center was purely vertical. Attachement #1 is the error for noise_amp = 0, #2 for noise_amp = 20 and #3 for noise_amp = 40. While Attachment #3 does provide the impression of there being a large error, this is not really the case as without normalization, each peak corresponds to a deviation of one pixel about the central value, see Attachement #4 for reference.
2. While the error does increase marginally, adding noise has no significant effect on the prediction of the y coordinate of the centroid as Attachment #5 shows at noise_amp = 40.
3. I am currently running an experiment to obtain the variation of mean square error with different noise amplitudes and will put up the plots soon. Further, I shall vary the resolution of the image frames and the the standard deviation of the Gaussain beam with time and try to obtain simulations very close to the real data available and then determine the performance of the algorithm.
4. The following videos will serve as a quick reference for what the videos and detection look like at
1. noise_amp = 20
2. noise_amp = 40
5. I also performed a quick experiment to see how low the amplitude of motion could be before the algorithm falied to detect the motion and found it to occur at 2 orders of magnitude below the values used in the previous post. This is a line of thought I intend to pursue more carefully and I am looking into how opencv and python handle images with floats as coordinates and will provide more details about the previous trial soon. This should give us an idea of what the smallest motion of the beam spot that can be resolved is.
Quote: Implemented image level noise for simulation. Added only uniform random noise. Implemented addition of uniform random noise to any sinusoidal motion of beam spot. Implemented motion along y axis according to data in "power_spectrum" file. Impelemented simulation of random motion of beam spot in both x and y directions (done previously by Pooja, but a cleaner version). Created a video file for 10s with motion of beam spot along the y direction as given by Attachment #1. This was created by mixing four sinusoids at different amplitudes (frequencies (0.1, 0.2, 0.4, 0.8) Hz Amplitudes as fractions of N = 64 (0.1 0.09 0.08 0.09). FPS = 10. Total number of frames = 100 for the sake of convenience. See Attachment #5. Following this, I used the thresholding (threshold = 127, chosen arbitrarily), contour detection and centroid computation sequence (see Attachment #6 for results) to obtain the plot in Attachment 2 for the predicted motion of the y coordinate. As is evident, the centering and scale of values obtained are off and I still haven't figured out how to precisely convert from one to another. Consequently, as a workaround, I simply normalised the values corresponding to each plot by subtracting the mean in each case and dividing the resulting series of values by their maximum. This resulted in the plots in Attachments 3 and 4 which show the normalised values of y coordinate variation and the error between the actual and predicted values between 0 and 1 respectively. Things yet to be done: Simulation: I will implement the mean square error function to compute the relativer performance as conditions change. I will add noise both to the image and to the motion (meaning introduce some randomness in the motion) to see how the performance, determined by both the curves such as the ones below and the mean square error, changes. Following this, I will vary the standard deviation of the beam spot along X and Y directions and try to obtain beam spot motion similar to the video in Attachment #2 of elog post 14632. Currently, I have made no effort to carefully tune the parameters associated with contour detection and threshold and have simply used the popular defaults. While this has worked admirably in the case of the simple simulated videos, I suspect much more tweaking will be needed before I can use this on real data. It is an easy step to determine the performance of the algorithm for random, circular and other motions of the beam spot. However, I will defer this till later as I do not see any immediate value in this. Determine noise threshold. In simulation or with real data: obtain a video where the beam spot is ideally motionless (easy to do with simulated data) and then apply the above approach to the video and study the resulting predicted motion. In simulation, I expect the predictions for a motionless beam spot video (without noise) to be constant. Therefore, I shall add some noise to the video and study the prediction of the algorithm. NOTE: the above approach relies on some previous knowledge of what the video data will look like. This is useful in determining which contours to ignore, if any like the four bright regions at the corners in this video. Real data: Obtaining real data and evaluate if the algorithm is succesful in determining contours which can be used to track the beam spot. Once the kind of video feed this will be used on is decided, use the data generated from such a feed to determine what the best settings of hyperparameters are and detect the beam spot motion. Synchronization of data stream regarding beam spot motion and video. Determine the calibration: anglular motion of the optic to beam spot motion on the camera sensor to video to pixel mapping in the frames being processed. Other approaches: Review work done by Gabriele with CNNs, implement it and then compare performance with the above method.
Attachment 1: residue_normalised_x.pdf
Attachment 2: residue_normalised_x.pdf
Attachment 3: residue_normalised_x.pdf
Attachment 4: predicted_motion_x.pdf
Attachment 5: normalised_comparison_y.pdf
9326 Fri Nov 1 17:01:46 2013 GabrieleSummaryLSCSimulation of REFL_3f signal when the arms come in
I simulated how the 3f signal is affected by the resonance condition of the arms.
To keep it simple, I only simulated a double cavity. The attached plot shows the result. In x there is the arm cavity detuning from resonance (in log scale to show what happens close to the 0 value). In the y axis there is the PRC detuning. So every vertical slice of the upper plot gives a PDH signal for a given arm detuning. The bottom plot shows the power build up inside the arm, which is dominated by the carrier.
The 3f signal is not perturbed in any significant way by the arm resonance condition. This is good and what we expected.
However, in this simulation I had to ensure that the 1f sidebands are not perfectly anti-resonant inside the arms. They are indeed quite far away from resonance. If the modulation frequency is chosen in order to make the 1f sidebands exactly ant-resonant, the 2f will be resonant. This screws up the signal: REFL_3f is made of two contributions of equal amplitude, one on the PRC sidebands resonance and the other on the PRC carrier resonance. When the arm tuning goes to zero, these two cancels out and there is no more PDH...
However, this is a limit case, since the frequency show match perfectly. If the modulation frequency is few arm line widths away from perfect anti-resonance, we have no problem.
9327 Fri Nov 1 17:44:06 2013 KojiSummaryLSCSimulation of REFL_3f signal when the arms come in
Yes, the resonance of the 2nd-order sidebands to the IFO screws up the 3f scheme.
2f (~22MHz) and 10f (~110MHz) are at x 5.6 and x 27.9 FSR from the carrier, so that's not the case.
Could we also see how much gain fluctuation of the 3f signals we would experience when the arm comes into the resonance?
9337 Mon Nov 4 14:11:23 2013 GabrieleSummaryLSCSimulation of REFL_3f signal when the arms come in
Quote: Yes, the resonance of the 2nd-order sidebands to the IFO screws up the 3f scheme. 2f (~22MHz) and 10f (~110MHz) are at x 5.6 and x 27.9 FSR from the carrier, so that's not the case. Could we also see how much gain fluctuation of the 3f signals we would experience when the arm comes into the resonance?
From the simulation there is no visible change in the gain.
11564 Thu Sep 3 02:12:08 2015 ranaUpdateCDSSimulink Webview updated
Back in 2011, JoeB wrote some entries on how to automatically update the Simulink webview stuff.
Somehow, the cron broke down over the years. I reran the matlab file by hand today and it worked fine, so now you can see the up to date models using the internet.
https://nodus.ligo.caltech.edu:30889/FE/
11567 Thu Sep 3 13:25:40 2015 ranaUpdateCDSSimulink Webview updated
added the cron script for this to megatron to run at 8:44 AM each morning. Here's the new MegaCron attached :-()-
** it takes ~13 minutes to complete on megatron
Attachment 1: crontab_150903.rtf
MAILTO=ericq@caltech.edu
# m h dom mon dow command
#0 */1 * * * bash /home/controls/public_html/summary/bin/c1_summary_page.sh > /dev/null 2>&1
#15 5 * * * /ligo/apps/nds2/nds2-megatron/test-restart
# MEDM Screen caps for the webpage
2,13,25,37,49 * * * * /cvs/cds/project/statScreen/bin/cronjob.sh
# op340m transplants -ericq
... 18 more lines ...
12727 Tue Jan 17 20:47:23 2017 ranaUpdateCDSSimulink Webview updated
Seems like this stops working every ~2 years. Its been busted since early 2016 according to cron, so I fixed up the paths and restored some missing files and committed things to the SVN (with comments!) and now its working and grabbing the Web viewable versions of the front end models. Just need to restore its viewability and then the world can watch our models any time.
Quote: Back in 2011, JoeB wrote some entries on how to automatically update the Simulink webview stuff. Somehow, the cron broke down over the years. I reran the matlab file by hand today and it worked fine, so now you can see the up to date models using the internet. https://nodus.ligo.caltech.edu:30889/FE/
8299 Fri Mar 15 02:14:27 2013 JenneUpdateCDSSimulink linking to wrong library part
Jamie and I discovered a problem with Matlab/Simulink earlier today.
In the end suspension models, there is a subblock (with top_names) for ALS stuff. Inside there, we use a library part called "ALS_END". When the model was created, it included the part ...../userapps/release/isc/c1/models/ALS_END.mdl . However, if you open up the c1scy diagram and look in the ALS block for this part, you see the part that is in ..../userapps/release/isc/common/models/ALS_END.mdl . Note the difference - the one we want is in the c1 directory, while the one that was created (by Jamie) for the LHO One Arm Test is in the common directory.
If you compile the c1scy model, the RCG is using the correct library part, so the information regarding which part we want is still in there.
However, if you delete the ALS_END part from the model, put the correct one in, save, close, then reopen the model, it once again displays the wrong model. The right click "go to library part" option brings you to the library part that is displayed, which is currently the wrong one. THIS IS BAD, since we could start modifying the wrong things. You do get a warning by Matlab about the file being "shadowed", so we should take heed when we see that warning, and make sure we are getting the file we want.
We are currently running Matlab version 7.11.0.584, which is r2010b. Step 1 will be to update Matlab to the latest version, in hopes that this fixes things. We also should change the name of our c1 part, so that it does not have the same name as the one for the sites. This is not a great solution since we can't guarantee that we will never choose the same names as other sites, but it will at least fix this one case. Again, if you see the warning about "shadowed" filenames, pay attention.
16168 Fri May 28 17:32:48 2021 AnchalSummaryALSSingle Arm Actuation Calibration with IR ALS Beat
I attempted a single arm actuation calibration using IR beatnote (in the directions of soCal idea for DARM calibration)
## Measurement and Inferences:
• I sent 4 excitation signals at C1:SUS-ITM_LSC_EXC wit 30cts at 31Hz, 200cts at 197Hz, 600cts at 619Hz and 1000cts at 1069 Hz.
• These were sent simultaneously using compose function in python awg.
• The XARM was locked to mai laser and alignment was optimized with ASS.
• The Xend Green laser was locked to XARM and alignment was optimized.
• Sidenote: GTRX is now normalized to give 1 at near maximum power.
• Green lasers can be locked with script instead of toggling.
• Script can be called from sitemap->ALS->! Toggle Shutters->Lock X Green
• Script is present at scripts/ALS/lockGreen.py.
• C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ was measured for 60s.
• Also, measured C1:LSC-XARM_OUT_DQ and C1:SUS-ITMX_LSC_OUT_DQ.
• Attachment 1 shows the measured beatnote spectrum with excitations on in units of m/rtHz.
• It also shows resdiual displacement contribution PSD of (output referred) XARM_OUT and ITMX_LSC_OUT to the same point in the state space model.
• Note: that XARM_OUT and ITMX_LSC_OUT (excitation signal) get coherently added in reality and hence the beatnote spectrum at each excitation frequency is lower than both of them.
• The remaining task is to figure out how to calculate the calibration constant for ITMX actuation from this information.
• I need more time to understand the mixture of XARM_OUT and ITMX_LSC_OUT in the XARM length node in control loop.
• Beatnote signal tells us the actual motion of the arm length, not how much ITMX would have actuated if the arm was not locked.
• Attachment 2 has the A,B,C,D matrices for the full state space model used. These were fed to python controls package to get transfer functions from one point to another in this MIMO.
• Note, that here I used the calibration of XARM_OUT we measured earlies in 16127.
• On second thought, maybe I should first send excitation in ETMX_LSC_EXC. Then, I can just measure ETMX_LSC_OUT which includes XARM_OUT due to the lock and use that to get calibration of ETMX actuation directly.
Attachment 1: SingleArmActCalwithIRALSBeat.pdf
Attachment 2: stateSpaceModel.zip
16171 Tue Jun 1 16:55:32 2021 Anchal, PacoSummaryALSSingle Arm Actuation Calibration with IR ALS Beat
Rana suggested in today's meeting to put in a notch filter in the XARM IR PDH loop to avoid suppressing the excitation line. We tried this today first with just one notch at 1069 Hz and then with an additional notch at 619 Hz and sent two simultaneous excitations.
## Measurement and Analysis:
• We added notch filters with Q=10, depth=50dB, freq=619 Hz and 1069 Hz using foton in SUS-ETMX_LSC filter bank at FM10.
• We sent excitation signals with amplitudes 600cts and 1000 cts for 619 Hz and 1069 Hz signals respectively.
• We measured time series data of C1:SUS-ITMX_LSC_OUT_DQ and C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ for 60s.
• Then, spectrum of both signals is measured with Hanning window using scipy.welch function with scaling set to 'spectrum', binwidth=1Hz.
• The beatnote signal was converted into length units by multiplying it by 1064nm * 37.79m / c.
• The ratio of the two spectrums at teh excitation frequency multiplies by excitation frequency squared gives us teh calibration constant in units of nm Hz^2/cts.
• At 619 Hz, we got $\frac{5.01}{f^2}$nm/cts
• At 1069 Hz, we got $\frac{5.64}{f^2}$nm/cts.
• The calibration factor in use is from $\frac{7.32}{f^2}$ nm/cts from 13984.
• So, the calibration factor from this methos is about 23% smaller than measured using freeswinging MICH in 13984.
• One possiblity is that our notch filter is not as effective in avoiding suppresion of excitation.
• We tried increasing the notch filter depths to 100 dB but got the same result within 2%.
• We tried changing the position of notch filters. We put them in POX filter banks. Again the result did not change more than 2%.
• The open loop gain of green PDH at 619 Hz and 1069 Hz must be large enough for our assumption of green laser perfectly following length motion to be true. The UGF of green laser is near 11 kHz.
• The discrepancy could be due to outdated freeswinging MICH measurement that was done 3 years ago. Maybe we should learn how to do the ITMX calibration using this method and compare our own two measurements.
Attachment 1: SingleArmActCalwithIRALSBeat-1306624785.pdf
16192 Tue Jun 8 11:40:53 2021 Anchal, PacoSummaryALSSingle Arm Actuation Calibration with IR ALS Beat
We attempted to simulate "oscillator based realtime calibration noise monitoring" in offline analysis with python. This helped us in finding about a factor of sqrt(2) that we were missing earlier in 16171. we measured C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ when X-ARM was locked to main laser and Xend green laser was locked to XARM. An excitation signal of amplitude 600 was setn at 619 hz at C1:ITMX_LSC_EXC.
## Signal analysis flow:
• The C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ is calibrated to give value of beatntoe frequency in Hz. But we are interested in the fluctuations of this value at the excitation frequency. So the beatnote signal is first high passed with 50 hz cut-off. This value can be reduced a lot more in realtime system. We only took 60s of data and had to remove first 2 seconds for removing transients so we didn't reduce this cut-off further.
• The I and Q demodulated beatntoe signal is combined to get a complex beatnote signal amplitude at excitation frequency.
• This signal is divided by cts amplitude of excitation and multiplied by square of excitation frequency to get calibration factor for ITMX in units of nm/cts/Hz^2.
• The noise spectrum of absolute value of the calibration factor is plotted in attachment 1, along with its RMS. The calibration factor was detrended linearly so the the DC value was removed before taking the spectrum.
• So Attachment 1 is the spectrum of noise in calibration factor when measured with this method. The shaded region is 15.865% - 84.135% percentile region around the solid median curves.
We got a value of $\frac{7.3 \pm 3.9}{f^2}\, \frac{nm}{cts}$. The calibration factor in use is from $\frac{7.32}{f^2}$ nm/cts from 13984.
Next steps could be to budget this noise while we setup some way of having this calibration factor generated in realitime using oscillators on a FE model. Calibrating actuation of a single optic in a single arm is easy, so this is a good test setup for getting a noise budget of this calibration method.
Attachment 1: ITMX_Cal_Noise_Spectrum_1307143423.pdf
16242 Fri Jul 9 15:39:08 2021 AnchalSummaryALSSingle Arm Actuation Calibration with IR ALS Beat [Correction]
I did this analysis again by just doing demodulation go 5s time segments of the 60s excitation signal. The major difference is that I was not summing up the sine-cosine multiplied signals, so the error associated was a lot more. If I simply multpy the whole beatnote signal with digital LO created at excitation frequency, divide it up in 12 segments of 5 s each, sum them up individually, then take the mean and standard deviation, I get the answer as:
$\frac{6.88 \pm 0.05}{f^2} nm/cts$as opposed to $\frac{7.32 \pm 0.03}{f^2} nm/cts$that was calculated using MICH signal earlier by gautum in 13984.
Attachment 1 shows the scatter plot for the complex calibration factors found for the 12 segments.
My aim in the previous post was however to get a time series of the complex calibration factor from which I can take a noise spectral density measurement of the calibration. I'll still look into how I can do that. I'll have to add a low pass filter to integrate the signal. Then the noise spectrum up to the low pass pole frequency would be available. But what would this noise spectrum really mean? I still have to think a bit about it. I'll put another post soon.
Quote:
We attempted to simulate "oscillator based realtime calibration noise monitoring" in offline analysis with python. This helped us in finding about a factor of sqrt(2) that we were missing earlier in 16171. we measured C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ when X-ARM was locked to main laser and Xend green laser was locked to XARM. An excitation signal of amplitude 600 was setn at 619 hz at C1:ITMX_LSC_EXC.
## Signal analysis flow:
• The C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ is calibrated to give value of beatntoe frequency in Hz. But we are interested in the fluctuations of this value at the excitation frequency. So the beatnote signal is first high passed with 50 hz cut-off. This value can be reduced a lot more in realtime system. We only took 60s of data and had to remove first 2 seconds for removing transients so we didn't reduce this cut-off further.
• The I and Q demodulated beatntoe signal is combined to get a complex beatnote signal amplitude at excitation frequency.
• This signal is divided by cts amplitude of excitation and multiplied by square of excitation frequency to get calibration factor for ITMX in units of nm/cts/Hz^2.
• The noise spectrum of absolute value of the calibration factor is plotted in attachment 1, along with its RMS. The calibration factor was detrended linearly so the the DC value was removed before taking the spectrum.
• So Attachment 1 is the spectrum of noise in calibration factor when measured with this method. The shaded region is 15.865% - 84.135% percentile region around the solid median curves.
We got a value of $\frac{7.3 \pm 3.9}{f^2}\, \frac{nm}{cts}$. The calibration factor in use is from $\frac{7.32}{f^2}$ nm/cts from 13984.
Next steps could be to budget this noise while we setup some way of having this calibration factor generated in realitime using oscillators on a FE model. Calibrating actuation of a single optic in a single arm is easy, so this is a good test setup for getting a noise budget of this calibration method.
Attachment 1: ITMX_calibration_With_ALS_Beat.pdf
1757 Thu Jul 16 10:52:58 2009 ClaraUpdatePEMSingle Channel TRS-RNC Cable
I made and tested a female-to-female TRS(audio)-RNC cable. It only has a single channel, so it won't work for stereo speakers or anything, but I should only need one speaker for testing the microphones. The tip of the plug is the signal, the sleeve is ground, and the ring is null.
1432 Thu Mar 26 04:09:38 2009 YoichiUpdateIOOSingle X arm lock spectra with different MC lock schemes
The attached plots show MC_F, FSS_FAST_F and XARM IN/OUT spectra with different MC locking modes.
The conventional locking means the FSS is used. The direct frequency lock is the new way.
You can see that at low frequencies, the frequency actuator is working hard to suppress the MC pendulum motions.
The X-arm also sees a lot of frequency noise at low frequencies because of this.
The transmitted power of the X-arm fluctuates a lot making it difficult to align the mirrors.
The zoomed plots show that the structures in the kHz band are also present in the case of the direct frequency lock, although the frequencies are somewhat different.
Attachment 1: XarmSpectra.pdf
Attachment 2: XarmSpectraZoom.pdf
11061 Tue Feb 24 18:54:26 2015 ericqUpdateASCSingle arm QPD ASC stability
I've lowered the UGFs for the transmission QPD servos to ~1-2Hz, and made it just an integrator. I left the arms locked with the QPD servos on for a few hours during the daytime today, and they succesfully prevented the Y arm from losing power from alignment drift for ~4 hours. Turning the servo off caused TRY to drop to ~0.6 or so.
The X arm was only held for 2 hours or so, because after some unlock/drift event the power was below the servo trigger threshold. However, after gently nudging ETMX to get the transmission above the threshold, the servo kicked in, and brought it right back to TRX=1.0
Unfortunately, daqd was dead for much of the day, so I don't have much data to show; the trend was inferred from the wall striptool.
It is not proven that there aren't further issues that prevent this from working with higher / more dynamic arm powers, but this is at least a point in favor of it working.
EDIT: Here's a screenshot of the wall StripTool. Brown is TRY, blue is TRX. The downturn at the very end is me deactivating the servos.
There is no scientific justifcation for the 0.9 threshold. Really, I should look at the noise/SNR again, now that there is some ND filtering on the QPDs.
Attachment 1: trend.png
15611 Mon Oct 5 00:37:19 2020 gautamUpdateBHDSingle bounce interferometer locked
Summary:
The simple interferometer, composed of a single bounce reflection from ITMY and the LO beam deilvered via fiber to the AS table, can be locked - i.e. the phase of the LO beam can be controlled such that the DC light level on the DCPDs after the two beams are interfered can be stabilized. This test allows us to confirm that various parts of the sensing and actuation chain (e.g. PI PZT for homodyne phase control, Trek amplifier etc etc) are working.
I will post more quantitative analysis tomorrow.
Optical configuration:
• LO beam is a pickoff of the main PSL beam from just before it goes into the vacuum. The optical power arriving on each DCPD after the various beamsplitters, coupling loss etc is ~200 uW.
• IFO beam is the single bounce reflection from ITMY. For this test, ETMY, ITMX and ETMY are misaligned. Optical power arriving on each DCPD is ~80uW.
• The two beams are interfered on a 50-50 beamsplitter. The mode-matching efficiency was estimated to be ~50% which isn't stellar, but should be fine for this test.
• So, at half-fringe, we expect the signal on each DCPD to be linearly proportional to the phase difference between the two fields, and so we can use that as an error signal.
Servo topology:
Attachment #2 shows the servo topology.
• For a first attempt to close the feedback loop, we can consider the two blocks labelled "Sensing Chain" and "Actuation chain" to have a flat frequency response. While this isn't true, for a taget loop with ~100 Hz UGF, I think the approximation is reasonable.
• From the peak-to-peak value (160 cts) of the DCPD signals when the homodyne phase is uncontrolled, I estimate a sensing response (at half-fringe) of approximately 0.3 ct/nm, since this corresponds to 532nm of relative phase between the two beams.
• An inverting summing amplifier is used to map the +/- 2^15 ct DAC range to 0-125V on the PI PZT. Assuming the full stroke of the PZT is 10um per the datasheet, and that this voltage range drives half of the full stroke (this is just a guess since all the old PI PZT circuits were designed to work at 0-250 V), we get an actuation coefficient of 0.075 nm/ct.
• Using these two numbers, we can then design a digital feedback loop that gives an open loop transfer function with ~100 Hz UGF, and sufficient stability margin.
• From the earlier measurements, we have an estimate for the amount of phase fluctuations caused by (i) seismic disturbances and (ii) fiber phase noise. This is the quantity we wish to suppress, and the suppression factor will be 1/(1+L), where L is the open loop gain.
• I didn't do this in any systematic way, but the loop in Attachment #3 seemed like a reasonable shape that would suppress the error signal RMS by ~10x, as shown in Attachment #4. So I decided to try this out.
Other notes:
1. The idea of offloading the DC control voltage to the ITMY suspension seemed to work fine.
2. It also seems like the relative phase between the two beams doesn't drift by so large an amount in short time scales, at least at night/quiet seismic conditions. So it is possible to maintain the lock for several seconds without having to offload the DC signal to the suspensions.
3. I didn't bother adapting the FSS Slow PID script to do this offloading in an automated way, seemed like more trouble than was just doing it by hand. But we may want to automate this in the future.
4. I couldn't make a clean measurement of the loop transfer function using the usual IN1/IN2 method. Introducing a step offset at the error point, the servo is able to track it (I didn't fit the step response time, but it's not as if the loop bandwidth is <1 Hz or something). I have to compare the measured in-loop error signal ASD to the free-running one to get a feel for what the UGF is, I guess, to rule out a weird loop.
5. Update 1100 Oct 6 2020: I have now added measured, in-loop, error point spectra to Attachment #4. Looks like there might be significant sensing noise re-injection.
• Initially, I forgot to turn the HEPA on the PSL down for the measurement. So I have the two traces to compare. Looks like with the HEPA turned up to full, there is more noise in the 50-200 Hz range.
• The trace marked "highGain" was taken with an overall loop gain that was 3dB higher than the nominal value - I could see some oscillations start to appear, and in the spectrum, maybe the feature at ~150 Hz is evidence of some gain peaking?
Conclusions:
1. The PI PZT seems to work just fine.
2. Need to look into the loop shape. I guess it's not reasonable to expect a UGF much higher than 100-200 Hz, because of the various delays in the system, but maybe the low frequency suppression can be made better.
3. What are the next steps?? What does this mean for the RF44 sensing scheme?
Attachment 1: simpleHomodyne.png
Attachment 2: singleBounceIFO.pdf
Attachment 3: proposedController.pdf
Attachment 4: freeRunningSuppressed.pdf
15612 Mon Oct 5 00:53:16 2020 KojiUpdateBHDSingle bounce interferometer locked
🤘🤘🤘
15290 Wed Apr 1 00:51:41 2020 gautamUpdateWienerSlightly improved MCL FF
Summary:
Retraining the MCL filters resulted in a slight improvement in the performance. Compared to no FF, the RMS in the 0.5-5 Hz range is reduced by approximately a factor of 3
Details:
Attachment #1 shows my re-measurement of the MC2 position drive to MCL transfer function.
• The measurement was made using DTT swept sine, with the amplitude enveloped appropriately to avoid knocking the IMC out of lock.
• Coherence was >0.97 for all datapoints.
• Fitting was done using Lee's IIRrational, with the weighting being the coherence. I think there are some features of the fitting I don't fully understand, but I wanted to try and do everything in python and for this simple fit, it came out nicely I think.
Attachment #2 shows the IIR fits to the FIR filters calculated here
• Again, IIRrational was used.
• In the frequency band where subtraction is possible, the fit is good.
• But there is definitely room for improvement in the way this is done, for now, I did quite a bit "by eye" and tweaked the order of the filter and the minimum number of excess poles relative to zeros to get the AC coupling, but it'd be nice to make all of this iterative and quantitative (e.g. by minimizing a cost function).
• One nice feature of IIRrational is that it directly gives me a formatted string I can paste into foton. The order of these fits were 22, so I split them into two 19+3 order filters to be compatible with the realtime system before loading the coefficients (the overall gain was allocated to a single filter arbitrarily, with the other filter in the pair set to have unity gain in the zpk representation).
Attachment #3 shows several MCL spectra.
• Blue trace is the unsubtracted test dataset.
• Red is the performance of the calculated FIR filter, but the filtering is done offline.
• Gold is the performance of the IIR fit to the FIR filter, as shown in Attachment #2, applied offline to the test dataset.
• Green is the calculated ASD of MCL from a ~1 hour stretch from earlier tonight, when I left the feedforward loop on. So this is an actual measurement of the online performacne of the filter.
• Grey is the performance of the old filter loaded in the CDS system - the filtering is done using scipy, and the sos coefficients from the C1OAF.txt file.
Conclusions + next steps
1. Retraining the filters has resulted in a slight improvement, especially at ~3 Hz.
2. More tests need to be done to confirm that noise isn't being reinjected in the frequency bands where subtraction isn't possible (e.g. using arm cavities as OOL sensors).
3. The online filter isn't quite as good as what we would expect from calculations (green trace is noisier than gold). Need to think about why this is.
4. Why can't we get more subtraction at 1 Hz?
5. Now that I have the infrastructure ready, I will attempt to revive the PRC angular FF loops, which was the whole point of this exercise.
Attachment 1: MC2_act_calib.pdf
Attachment 2: IIR_fit_to_FIR.pdf
Attachment 3: FIRvIIR.pdf
12290 Tue Jul 12 09:18:12 2016 ericqUpdateGeneralSlippery substance mystery
I found a note on Steve's desk that R. Abbott left yesterday afternoon about an unidentified slippery substance being present on the floor by cabinet S12, along the X arm. (Steve is away this week)
Just now, I found no trace of the substance in the vicinity of that cabinent (which is one of the cabinets for clean objects). Maybe the janitor cleaned it already?
12291 Tue Jul 12 09:35:51 2016 JohannesUpdateGeneralSlippery substance mystery
I've noticed the spot that Rich means before, too. I think you only notice this when you're wearing the shoe covers, not sneakers or crocs. I didn't see any 'substance', it seems more like the floor finish (wax?) seems to be more slippery in that area than others.
Quote: I found a note on Steve's desk that R. Abbott left yesterday afternoon about an unidentified slippery substance being present on the floor by cabinet S12, along the X arm. (Steve is away this week) Just now, I found no trace of the substance in the vicinity of that cabinent (which is one of the cabinets for clean objects). Maybe the janitor cleaned it already?
13443 Wed Nov 22 00:54:18 2017 johannesOmnistructureComputersSlow DAQ replacement computer progress
I got the the SuperMicro 1U server box from Larry W on Monday and set it up in the CryoLab for initial testing.
The processor is an Intel D525 dual core atom processor with 1.8 GHz (i386 architecture, no 64-bit support). The unit has a 250GB SSD and 4GB RAM.
I installed Debian Jessie on it without any problems and compiled the most recent stable versions of EPICS base (3.15.5), asyn drivers (4-32), and modbus module (2-10-1). EPICS and asyn each took about 10 minutes, and modbus about 1 minute.
I copied the database files and port driver definitions for the cryolab from cryoaux, whose modbus services I suspended, and initialized the EPICS modbus IOC on the SuperMicro machine instead. It's working flawlessly so far, but admittedly the box is not under heavy load in the cryolab, as the framebuilder there is logging only the 16 analog channels.
I have recently worked out some kinks in the port driver and channel definitions, most importantly:
• mosbus IOC initialization is performed automatically by systemd on reboot
• If the IOC crashes or a system reboot is required the Acromag units freeze in their last current state. When the IOC is started a single read operation of all A/D registers is performed and the result taken as the initial value of the corresponding channel, causing no discontinuity in generated voltage EVER (except of course for the rare case when the Acromags themselves have to be restarted)
Aaron and I set 12/4 as a tentative date when we will be ready to attempt a swap. Until then the cabling needs to be finished and a channel database file needs to be prepared.
ELOG V3.1.3- | 2022-01-17 21:49:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6289424896240234, "perplexity": 2375.4007659030744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300624.10/warc/CC-MAIN-20220117212242-20220118002242-00611.warc.gz"} |
https://math.stackexchange.com/questions/2919313/conditional-probability-for-dependent-events | # Conditional probability for dependent events
There are two bags, the first bag contains 2 white balls and 5 black balls and the second contains 3 white balls and 4 black balls. What is the probability that the first bag was chosen given a white ball was chosen? Now suppose the ball was put back into its original bag, and another ball is picked from the same bag randomly. What is the probability that the second ball is also white?
For the first question I got 2/7 by doing probability of the first bag and a red ball being chosen divided by the probability of the first bag being chosen. ((1/2)*(2/7))/(1/2). (Don't know if this is right)
I'm not sure how to go about doing the second part.
• Your definition of the two bags do not contain any red balls. It's not possible for the second ball to be "also" red, as the first chosen ball was white. – Steve Schroeder Sep 16 '18 at 19:24
Before answering your question, I am assuming that where you meant white where you've written red.
When dealing with probabilities it helps to think of actions as events, and assigning an event a probability. In your example,
Let $P(W)=5/14$ be the probability of drawing a white ball, as 5 of 14 balls are while.
$P(W')=9/14$, i.e. the probability of drawing a black ball.
Let $B_1$, and $B_2$ be the event that the ball was taken from bag 1, or bag 2, respectively.
$P(W\cap B_1)= 2/14$, and $P(W\cap B_2)=3/14$.
$P(W'\cap B_1)=5/14$, and $P(W'\cap B_2)=4/14$.
All of this information is given to you in the context of the exercise. The conditional probability, "what is the probability event $B$ happens, given that the event $A$ has happened" is written as $$P(B|A)=\dfrac{P(B\cap A)}{P(A)}$$ Use this, along with the information above to answer question 1.
Question two resets by putting the white ball back into it's bag (i.e. bag 1), and asks what the probability of getting a white ball from bag one. In this case, just ignore bag two.
• yeah I meant white, sorry I'm working on a couple different problems at once – Ethan Smith Sep 16 '18 at 20:14 | 2019-11-22 14:56:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8938292264938354, "perplexity": 170.78044900516588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671363.79/warc/CC-MAIN-20191122143547-20191122172547-00396.warc.gz"} |
http://googology.wikia.com/wiki/User_blog:Nayuta_Ito/Does_this_hydra_halt%3F | FANDOM
10,821 Pages
This is Buchholz's hydra labeled with ordinals that can be larger than omega.
Modified rule:
1. If $$a$$ has label 0, we proceed as in Kirby-Paris' game. Call the node's parent $$b$$, and its grandparent $$c$$ (if it exists). First we delete $$a$$. If $$c$$ exists (i.e. $$b$$ is not the root), we make $$n$$ copies of $$b$$ and all its children and attach them to $$c$$.
2. If $$a$$ has successor label $$\alpha + 1$$, we go down the tree looking for a node $$b$$ with label $$v \leq \alpha$$ (which is guaranteed to exist, a every child of the root node has label 0). Consider the subtree rooted at $$b$$ — call it $$S$$. Create a copy of $$S$$, call it $$S'$$. Within $$S'$$, we relabel $$b$$ with $$\alpha$$ and relabel $$a$$ with $$0$$. Back in the original tree, replace $$a$$ with $$S'$$.
3. If $$a$$ has transfinite label $$\beta$$, we simply relabel it with $$\beta[n + 1]$$. | 2017-08-17 17:38:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8725629448890686, "perplexity": 307.60934020666065}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103891.56/warc/CC-MAIN-20170817170613-20170817190613-00078.warc.gz"} |
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However, the basic conclusion of these two theories is same price level varies directly with and proportionally to money supply. {\displaystyle P} This additional expenditure, given full employment, raises the price level. Reasons were that interest targeting turned out to be a less effective tool in low-interest phases and it did not cope with the public uncertainty about future inflation rates to expect. An important implication of this is that inflation is more often than not a monetary phenomenon. This Theory is fundamental. Changes in the money supply o affect nominal variables o do not affect real economic variables ° The velocity of money the speed at which the typical dollar bill travels around the economy from pocket to pocket. Before the war (and indeed since) there was a considerable element of what was conventional and arbitrary in the reserve policy of the banks, but especially in the policy of the State Banks towards their gold reserves. An alternative version, known as cash balance version, was developed by a group of Cambridge economists like Pigou, Marshall, Robertson and Keynes in the early 1900s. This also means that the average number of times a unit of money exchanges hands during a specific period of time. It is supported and calculated by using the Fisher Equation on Quantity Theory of Money. Introduction to Quantity Theory . [26] The Classical Theory of Inflation is also known as. Supply of bank money or credit money is influenced largely by the interest rate. Critics say that the relationship is not a direct one. Classical explanation for charges in the value of money is embodied in the quantity theory of money. The Quantity Theory is often stated in this, or a similar, form. {\displaystyle M^{\textit {d}}=M} to the quantity of money In its earliest form the theory was presented by Davanzal in 1588. {\displaystyle P\cdot Q} Consequently, when gold became relatively abundant they tended to hoard what came their way and to raise the proportion of the reserves, with the result that the increased output of South African gold was absorbed with less effect on the price level than would have been the case if an increase of n had been totally without reaction on the value of r. {\displaystyle Y} Economic SYNOPSES short essays and reports on the economic issues of the day 2006 Number 25 T he quantity theory of money (QTM) asserts that aggre-gate prices (P) and total money supply (M) are relatedaccording to the equation P = VM/Y, where Y is real output and V is velocity of money… {\displaystyle M} This is the level of real GDP which is obtained when the economy’s resources are fully employed. Sixthly, the classical theory establishes a direct and proportional relationship between money supply and price level. Keynes had originally been a proponent of the theory, but he presented an alternative in the General Theory. Quantity Theory of Money: Cambridge Version. {\displaystyle P\cdot Q} price level. 2 The Quantity Theory of Money. Back . Q 5 crore and the quantity of money is Rs. His introduction of the central bank's ability to influence the price level was a major contribution to the development of the quantity theory of money. There was a decided tendency on the part of these banks between 1900 and 1914 to bottle up gold when it flowed towards them and to part with it reluctantly when the tide was flowing the other way. But this long run is a misleading guide to current affairs. Simply put, this theory states that the supply (or quantity) of money determines the level of prices (or, general price level) in the economy. The Quantity Theory of Money is an economic theory that states that the level of money supply in an economy is directly proportional to the general price level. An increase in money supply, from M1 to M2 leads to a shift in the aggregate demand curve, from AD to AD’. the quantity theory of money (classical theory) prices rise when the government prints to much money ... ***** this explanation of how the price level is determined and why it might change over time is called the Quantity Theory of Money. Keynes argued that the price level was not strictly determined by the money supply. According to Fisher, MV = PT. P , This bank money has also a stable velocity of circulation, V’. Henry Thornton: Seminal Monetary Theorist and Father of the Modern Central Bank (n.d.): 1. According to Cambridge economists, people wish to hold cash to finance transactions and for security against unforeseen needs. In such an economy, a change in any of the variables may cause a change in price level, even if money supply does not change. He argued... .mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}. In the stylized classical model we have developed, the quantity of money is irrelevant for the determination of the real variables. One of the primary research areas for the branch of economics referred to as monetary economics is called the quantity theory of money. When the quantity of money rises rapidly in almost any country, velocity also rises rapidly. This theory dates back at least to the mid-16th cen- The theory above is based on the following hypotheses: An application of the quantity-theory approach aimed at removing monetary policy as a source of macroeconomic instability was to target a constant, low growth rate of the money supply. ), Milton Friedman (1956), "The Quantity Theory of Money: A Restatement" in, Roy Green (1987), "real bills doctrine", in. ⋅ Unreliable citations may be challenged or deleted. Thus the number of notes which the public ordinarily have on hand is determined by the purchasing power which it suits them to hold or to carry about, and by nothing else. First published by the Institute of Economic Affairs, London, 1970. They highlighted the transactions demand for money of exchange and smooth the progress of the exchange of goods and services. Let us assume that the supply of money, MS’ is determined by the monetary authority, i.e., Equilibrium requires that the supply of money must equal the demand for money, or, k and Y are determined independently of the money supply. This means After attaining the stage of full employment, an increase in effective demand which is the sum of consumption expenditure, investment expenditure and government expenditure (i.e., C + I + G) will raise the price level, but not proportionately. In monetary economics, the quantity theory of money (QTM) states that the general price level of goods and services is directly proportional to the amount of money in circulation, or money supply. Fisherian Approach: To the classical economists, the demand for money is transactions demand for money. In this sense, these are not independent variables, although the authors of this theory assumed quantity of money as independent of other elements of the equation. If M is reduced to half, P will decline by the same amount. The classical theory of demand for money is presented in the classical quantity theory of money and has two approaches: the Fisherman approach and the Cambridge approach. 2.2 THE CLASSICAL QUANTITY THEORY OF MONEY One of the basic tenets of classical macroeconomics is the quantity theory of money. The Classical Approach The classical economists did not unambiguously devise demand for money thesis but their outlooks are intrinsic in the volume of thesis of money. The Classical Quantity Theory of Money. The important determinant of money supply is the income level and the total expenditure of the country. Q They highlighted the transactions demand for money of exchange and smooth the progress of the exchange of goods and services. P The law, that the quantity of the circulating medium is determined by the sum of the prices of the commodities circulating, and the average velocity of currency may also be stated as follows: given the sum of the values of commodities, and the average rapidity of their metamorphoses, the quantity of precious metal current as money depends on the value of that precious metal. The communication of inflation targets helps to anchor the public inflation expectations, it makes central banks more accountable for their actions, and it reduces economic uncertainty among the participants in the economy. It is determined by the central bank (as discussed in the monetary base and the supply of money). 42–43, 55–56). could be accompanied by a change of 1/(1 + 10%) in {\displaystyle P\cdot Y} MV and PT are always equal. The classical theory is rather ambiguous and indefinite. This Friedmanian words are enough to establish the essence of quantity theory of money inflation is largely caused by the excessive growth of money supply and by nothing else. C) only partially from changes in the quantity of money. and Q M The classical dichotomy is, essentially, a derivation of the quantity theory of money, which is captured by the formula MV = PY, where M stands for the money stock, V is the velocity of money circulation, P is the price level, and Y is the level of income. The classical economists did not unambiguously devise demand for money thesis but their outlooks are intrinsic in the volume of thesis of money. In examining the trade data between more and less developed countries (for example, in the trade between Greece and her main trading partner, Germany), we find significant transfers of value which, The project summarizes issues related to the theory of capital and brings to the fore the more recent developments. real output (which equals real expenditure in macroeconomic equilibrium) with The quantity theory of money states that the supply of money times the velocity of money equals nominal GDP. "Keynes' Theory of Money and His Attack on the Classical Model", L. E. Johnson, R. Ley, & T. Cate (International Advances in Economic Research, November 2001), "The Counter-Revolution in Monetary Theory", Milton Friedman (IEA Occasional Paper, no. There are debates about the extent to which each of these variables is dependent upon the others. Y [19], amounts to a statement of the theory,[20] while other economic historians date the discovery later, to figures such as Jean Bodin, David Hume, and John Stuart Mill. [29] But monetary aggregates remain a leading economic indicator. M What matters is the part of total spending which is independent of current income, what has come to be called autonomous spending and to be identified in practice largely with investment by business and expenditures by government. 1 Monetary theory was an integral part of classical value theory. In other words, the quantity theory of money states that a given percentage change in the money supply results in an equivalent level of inflation or deflation. With V and T constant, the above identity is modified as: where the bar sign over the heads of ‘V’ and ‘T’ indicates that these two are fixed. It is to determine the general level of prices at which goods and services will be exchanged. The quantity theory was developed by Simon Newcomb, Alf… Fifthly, change in price level is caused by various factors, besides money supply. This page was last edited on 4 January 2021, at 15:24. ", Federal Reserve Bank of San Francisco. The QTM states that the general price level should, over the long-run, co-move with the quantity of money available in the economy. ⋅ He said the theory "fails to explain the mechanism of variations in the value of money". This equation is an identity that always holds true: It tells us that the total stock of money used for transactions must equal to the value of goods sold in the economy. In the 1950s, Milton Friedman came out with a thesis that ‘inflation is always and everywhere a monetary phenomenon’. {\displaystyle M} Q As the price level decreases, the value of money. than the price level Nicolaus Copernicus (1517), memorandum on monetary policy. 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Considerations involved serious changes as to the scope of countercyclical economic policy with fact is not influenced by the policy... | 2021-03-03 21:49:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5991514325141907, "perplexity": 1925.0371209127727}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178367790.67/warc/CC-MAIN-20210303200206-20210303230206-00182.warc.gz"} |
https://www.ques10.com/p/23562/compare-full-scaling-constant-voltage-scaling-mode/ | 0
Compare Full Scaling & Constant Voltage Scaling models of MOSFET .Demonstrate clearly the effects of scaling on the device density, speed of the circuit, power consumption and current density of the
gates. OR Demonstrate the effect of Scaling on Area ,delay ,power Consumption ,Current Density
Subject: Basic VLSI Design
Topic: Technology Trend
Difficulty: Medium
bvlsi(46) • 1.8k views
0
Effects of scaling:
• The Full scaling technique attempts to preserve the magnitude of internal electric fields in the MOSFET, while the dimensions are scaled down by factor of ‘s’.
• In Constant voltage scaling, all dimensions of the MOSFET are reduced by a factor of ‘s’, but power supply & terminal voltage remain unchanged.
• Device density is the area i.e. Area = L.W, therefore after scaling it becomes, Area’ = Area/ $s^2$
• Also current density is given by J = $I_D$/Area. Therefore for Full scaling, J’ = s.J and for Constant voltage scaling, J’ = $s^3$ .J
• In Constant voltage scaling, drain current density and power density are increased which may eventually cause serious reliability problem for the scaled transistor such as electro-migration, hot carrier degradation, oxide breakdown & electrical overstress. | 2020-01-20 14:41:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5946875810623169, "perplexity": 5649.841883007315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250598800.30/warc/CC-MAIN-20200120135447-20200120164447-00008.warc.gz"} |
http://djalil.chafai.net/blog/2011/10/ | # Month: October 2011
Let ${\mathcal{A}}$ be a ${\star}$-algebra over ${\mathbb{C}}$ with zero ${0}$, unit ${1}$, and involution ${a\mapsto a^*}$ such that ${(ab)^*=b^*a^*}$ for every ${a,b\in\mathcal{A}}$. Let ${\tau:\mathcal{A}\rightarrow\mathbb{C}}$ be a linear form such that ${\tau(aa^*)\geq0}$ for every ${a\in\mathcal{A}}$, and ${\tau(1)=1}$. We say then that ${(\mathcal{A},\tau)}$ is an algebraic probability space. We do not assume that ${\tau(ab)=\tau(ba)}$ for every ${a,b\in\mathcal{A}}$ or that ${\tau(aa^*)=0}$ iif ${a=0}$, even if it is the case in the following couple of examples. The simplest example, commutative, is given by
$\mathcal{A}=\bigcap_{1\leq p<\infty}\mathrm{L}^p(\Omega,\mathcal{F},\mathbb{P},\mathbb{C}) \quad\text{and}\quad \tau=\mathbb{E} \quad\text{and}\quad a^*=\bar{a}$
where ${(\Omega,\mathcal{F},\mathbb{P})}$ is a classical probability space. A non commutative example is given by
$\mathcal{A}=\mathcal{M}_n(\mathbb{C}) \quad\text{and}\quad \tau=\frac{1}{n}\mathrm{Trace} \quad\text{and}\quad a^*=\bar{a}^\top.$
One can mix the two by considering integrable random matrices equipped with ${\tau=\frac{1}{n}\mathbb{E}\mathrm{Trace}(\cdot)}$. Here we focus on the purely algebraic notion of ${\star}$-algebras, and we should not confuse this notion with the algebraic-analytic notions of ${C^*}$-algebras or von Neumann ${W^*}$-algebras.
Algebraic random variables. An element ${a\in\mathcal{A}}$ is called an algebraic random variable, and its ${\star}$-distribution is the collection of ${\star}$-moments
$\tau(a^{\varepsilon_1}\cdots a^{\varepsilon_n})$
for every ${n\geq1}$ and every ${\varepsilon_1,\ldots,\varepsilon_n}$ in ${\{1,\star\}}$. When ${a=a^*}$ (we say that ${a}$ is real) then the ${\star}$-distribution of ${a}$ is characterized by the sequence of moments ${\tau(a^n)}$, ${n\in\mathbb{N}}$. In this case, and thanks to the Hamburger moment theorem, this sequence of moments of ${a}$ is the sequence of moments of some probability measure ${\mu}$ on ${\mathbb{R}}$. This probability distribution ${\mu}$ is not unique in general, the Carleman condition says that uniqueness holds if
$\sum_n(\tau(a^{2n}))^{-1/(2n)}=\infty.$
Now we define four algebraic notions of independence, which corresponds actually to simplification rules for the computation of mixed moments. The first notion matches the classical notion of commutative probability theory, while the second notion is the one of free probability theory.
Commutative independence. A family ${(\mathcal{A}_i)_{i\in I}}$ of sub-${\star}$-algebras of ${\mathcal{A}}$ is commutative independent when for every ${i_1,\ldots,i_n\in I}$, and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,
$\tau(a_1\cdots a_n) = \left\{ \begin{array}{ll} \tau(a_1)\tau(a_2\cdots a_n) & \mbox{if } i_1\not\in\{i_2,\ldots,i_n\} \\ \tau(a_2\cdots a_{r-1}(a_1a_r)a_{r+1}\cdots a_n) & \mbox{if } r=\min\{j>1:i_1=i_j\} \end{array} \right.$
This allows first to group the ${a_i}$ belonging to the same sub-${\star}$-algebra and then to break ${\tau}$. The ${\star}$-distribution of ${a}$ is in this case uniquely determined by the law of ${a}$ as a classical random variable (the converse is not true in general, since the moment problem may not have a unique solution).
Free independence. A family ${(\mathcal{A}_i)_{i\in I}}$ of sub-${\star}$-algebras of ${\mathcal{A}}$ is free independent when for every ${i_1,\ldots,i_n\in I}$ with ${i_1\neq\cdots\neq i_n}$ (any two consecutive indices are different), and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,
$\tau((a_1-\tau(a_1))\cdots(a_n-\tau(a_n))) =0.$
Note that if ${a,b}$ are free independent (i.e. their generated algebra are free independent), and if ${\tau(a)=\tau(b)=0}$, then ${\tau(abab)=0}$, while for the commutative independence, ${\tau(abab)=\tau(a^2b^2)=\tau(a^2)\tau(b^2)}$ which is not zero in general.
Boolean independence. A family ${(\mathcal{A}_i)_{i\in I}}$ of sub-sets of ${\mathcal{A}}$ closed for the algebraic operations and the involution ${\star}$, but which may not contain the unit ${1}$, is Boolean independent when for every ${i_1,\ldots,i_n\in I}$ with ${i_1\neq\cdots\neq i_n}$ (any two consecutive indices are different), and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,
$\tau(a_1\cdots a_n)=\tau(a_1)\tau(a_2\cdots a_n).$
For instance, if ${a}$ and ${b}$ are Boolean free (i.e. their generated sub-sets are Boolean free) then ${\tau(abab)=\tau(a)^2\tau(b)^2}$ and ${\tau(aba^2b)=\tau(a)\tau(b)^2\tau(a^2)}$.
Monotone independence. Recall that if ${i_1\neq \cdots \neq i_n}$ is a sequence of integers where any two consecutive are different, then ${i_k}$ is a peak when either ${i_1>i_2}$ (if ${k=1}$), ${i_{n-1}<i_n}$ (if ${k=n}$), ${i_{k-1}<i_k>i_{k+1}}$ (if ${1<k<n}$). A family ${(\mathcal{A}_i)_{i\in\mathbb{N}}}$ of sub-sets of ${\mathcal{A}}$ closed for the algebraic operations and the involution ${\star}$, but which may not contain the unit ${1}$, is monotone independent when for every ${i_1,\ldots,i_n\in\mathbb{N}}$ with ${i_1\neq\cdots\neq i_n}$ (any two consecutive indices are different), and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,
$\tau(a_1\cdots a_n)=\tau(a_k)\tau(a_1\cdots \check{a}_k \cdots a_n)$
when ${k}$ is a peak of the sequence ${i_1,\ldots,i_n}$, and where ${\check{a}_k}$ is the removal of ${a_k}$.
Note. The Boolean and monotone independence are trivial when ${1\in\mathcal{A}_i}$ i.e. when ${\mathcal{A}_i}$ is a sub-${\star}$-algebras. The notion of free independence was introduced by Voiculescu and is at the heart of free probability theory. The notion of Boolean independence was developed by Bozjeko and his followers. The notion of monotone independence is due to Lu and Muraki. Various other notions of independence, not considered here, are studied in the literature.
Convolutions. If ${a}$ and ${b}$ are independent for one of the four notions of independence, then the ${\star}$-distribution of ${a+b}$ depends only on the ${\star}$-distribution of ${a}$ and of ${b}$, and is called the convolution of these distributions. We recover the classical notion of convolution for the commutative independence, and the Voiculescu notion of free convolution for the free independence.
Singleton property. The four notion of independence satisfy to the singleton property: if ${a_1\in\mathcal{A}_{i_1}\ldots,a_n\in\mathcal{A}_{i_n}}$ where ${\mathcal{A}_1,\ldots,\mathcal{A}_n}$ are independent (for any of these four notions), and if ${a_i=a_i^*}$ for any ${1\leq i\leq n}$, and if ${\tau(a_1)=\cdots=\tau(a_n)=0}$, and if there exists ${1\leq k\leq n}$ such that ${\{1\leq j\leq n:i_j=i_k\}=\{k\}}$ (i.e. ${a_k}$ is the unique element of ${\mathcal{A}_{i_k}}$ in the sequence ${a_1,\ldots,a_n}$), then ${\tau(a_1\cdots a_n)=0}$.
Central limit theorems. Let ${a_1,a_2,\ldots\in\mathcal{A}}$. We have, for every ${n,m\geq1}$,
$\tau((a_1+\cdots+a_n)^m)=\sum_{i_1,\ldots,i_m=1}^n\tau(a_{i_1}\cdots a_{i_m}).$
The mixed moment ${\tau(a_{i_1}\cdots a_{i_m})}$ can be computed using the notions of independence. Let us make the following assumptions on the variables:
• the variables are real: ${a_i=a_i^*}$ for any ${i\geq1}$
• the variables are centered and normalized: ${\tau(a_i)=0}$ and ${\tau(a_i^2)=1}$ for all ${i\geq1}$
• the variables have bounded mixed moments: for all ${n\geq1}$,
$\sup_{i_1\geq1,\ldots,i_n\geq1}|\tau(a_{i_1}\cdots a_{i_n})|<\infty$
• the variables are independent (for one of the four notions of independence).
Then it can be shown that for every ${m\geq1}$,
$\lim_{n\rightarrow\infty}\tau\left(\left(\frac{a_1+\cdots+a_n}{\sqrt{n}}\right)^m\right) = \int\!x^m\,d\mu$
where the limiting distribution ${\mu}$ is${\ldots}$
• for commutative independence, the standard Gaussian distribution
$\frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\,dx$
Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=\frac{(2k)!}{2(k!)}=(2k-1)!!}$
• for free independence, the Wigner semi-circle distribution on ${[-2,2]}$
$\sqrt{4-x^2}\frac{\mathbf{1}_{[-2,2]}(x)}{2\pi}\,dx$
Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=\frac{1}{k+1}\binom{2k}{k}=\frac{(2k)!}{k!(k+1)!}}$ (Catalan numbers)
• for Boolean independence, the symmetric Bernoulli distribution on ${\{-1,1\}}$
$\frac{1}{2}(\delta_{-1}+\delta_1)$
Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=1}$
• for monotone independence, the arc-sine distribution on ${[-\sqrt{2},\sqrt{2}]}$
$\frac{\mathbf{1}_{[-\sqrt{2},\sqrt{2}]}(x)}{\pi\sqrt{2-x^2}}\,dx$
Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=2^{-k}\binom{2k}{k}=\frac{(2k)!}{k!^22^k}}$
Stability. The Gaussian distribution is stable by the commutative convolution, the Wigner semi-circle distribution is stable by the free convolution, the Bernoulli distribution is stable by the Bernoulli convolution, while the arc-sine distribution is stable by the monotone convolution.
Open problem. Note that in the four cases, the second moment is constantly equal to ${1}$ along the central limit theorem (conservation law). In the case of commutative independence, it has been conjectured by Shannon and proved few years ago that the Boltzmann-Shannon entropy is monotonic along the central limit theorem (additionally, its maximum under a second moment constraint is achieved by the standard Gaussian law). Similarly, in the case of free independence, it has been proved by Shlyakhtenko few years ago that the Voiculescu entropy is monotonic along the central limit theorem (additionally, its maximum under a second moment constraint is achieved by the Wigner semi-circle distribution). Both entropies are additive for tensor products of random variables. The existence of such entropies for the Boolean and monotonic independence constitutes a natural problem (still open at the time of writing – any ideas?).
Recently, a French friend of mine, Mr C, was visiting Italy. He wanted to watch a streaming video on a French media website, say www.media.fr. The problem was that this website blocks accesses from outside France due to capitalistic reasons. So Mr C asked if he can use a sort of proxy based in France and accessible from Italy in order to watch his video. We have thus two constraints: throughput and location. This post is devoted to the quick description of two possible solutions using some knowledge in IPv4 TCP/IP networking.
IP level solution : Virtual Private Network. This is the best solution in principle, since it solves the problem at the IP level, for all services, not only for the web. For this solution, Mr C needs a machine in France, say machine.domain.fr, connected to the Internet, with a high capacity in upstream and downstream (this excludes machines connected with commercial ADSL due to the limited upstream). On machine.domain.fr, Mr C may install for instance as root the free software OpenVPN (this is quite easy on Debian GNU/Linux for instance). Mr C can then connect his laptop in Italy to this Virtual Private Network (VPN). The main problem for this solution is to find such a machine. Most machines in academic networks are protected by a firewall, blocking arbitrary connections from outside the academic network. Of course, one can break the firewall using SSH, but this complicates things and produces an ugly solution. Mr C is lucky if his university provides a VPN service. Some universities do. Mine does not. The Mathrice VPN (CNRS) allows connection on MathSciNet for instance but it seems that it does not allow connections on video streaming sites! (test by yourself).
Application level solution : SOCKS server over SSH. This is the simplest solution. Suppose that Mr C has access to an OpenSSH server located in his French University, say ssh.uni.fr. From his laptop in Italy, Mr C can connect to this server, say using the command ssh -D 6666 ssh.uni.fr. It remains for him to configure his favorite web browser(1) to use a SOCKS proxy with IP 127.0.0.1 and port 6666. One may replace 6666 by any number in [1024,65535]. This solution at the application level works very well for all applications able to use the SOCKS v5 protocol. For other applications, one can use a sockifing wrapper. Of course, this solution will not work if the server ssh.uni.fr blocks the SOCKS feature of the SSH server (test by yourself).
(1) for Firefox: Edit/Preferences/Advanced/Network/Parameters/SOCKS_Host (not HTTP_Proxy).
Note. It is also possible to use other proxies available on the Internet, either generic purpose proxies or specific proxies dedicated to video streams. Some of them are free. Personally, I prefer the solutions above since they do not involve untrusted third parties.
IPv6. Both solutions can be adapted to IPv6 (exercise!).
Thanks to my former teacher and now colleague Philippe Carmona – nephew of René Carmona – I had recently the opportunity to learn the basics of the job of Managing Editor for an electronic mathematical journal. The articles published in this journal are written in $\LaTeX$. Even if I do not pretend that my own $\LaTeX$ files are perfect – I am constantly learning! – I was disappointed by the poor quality of the $\LaTeX$ code produced by many young and old mathematicians. Take a tour on arXiv to be convinced if needed : the source code is always available on this site, a museum of horrors. Here are some basic guidelines for sane habits:
• never use \def for defining macros, use instead \newcommand
• never use for displayed equations, use instead the brackets $\backslash[\backslash]$
• use \textbf{}, \textit{}, and \emph{} instead of {\bf }, {\it }, and {\em }
• never use one letter names for macros or for environments
• never use strange names for environments and macros
• use the environment proof provided by amsmath
• use \newenvironment to define new environments
• use \binom{n}{k} instead of n \choose k
• use \frac{a}{b} instead of a \over b
• never use an exotic package if you do not *really* need it
• indent your code and avoid too long lines
• use prefixed labels such as eq: for equations and th: for theorems
• to produce graphics, avoid using psfrag or xfig and use instead ipe
• learn how to use the error messages produced during compilation
• always use your bright sense of esthetics, not a dark laziness
• read the wiki-books on LaTeX and LaTeX Maths without moderation
All mathematicians believe that the mathematical result is the most important. Many of them believe that the esthetic of the proof is also important. Some of them believe that even the writing style is important. Few of them believe that the $\LaTeX$ code should also be nice. It is a matter of taste after all. To me, a mathematical proof and a $\LaTeX$ file are both programs, and I like nice programs. A good $\LaTeX$ code is easier to maintain, easier to convert, easier to read. Last but not least, a good $\LaTeX$ code helps your co-authors and helps to speed up the publication process. $\TeX$ is a good program, but an author using $\LaTeX$ should write a genuine $\LaTeX$ code, not an ugly mixture of both $\TeX$ and $\LaTeX$. Nothing is perfect, but this is not a reason to leave things ugly 😉 If you think that you have learned $\LaTeX$ once for all in your youth, you are wrong. Every non trivial language needs constant learning and practice.
The Heisenberg group is a remarkable simple mathematical object, with interesting algebraic, geometric, and probabilistic aspects. It is available in tow flavors: discrete and continuous. The (continuous) Heisenberg group ${\mathbb{H}}$ is formed by the real ${3\times 3}$ matrices of the form
$\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \\ \end{pmatrix}, \quad x,y,z\in\mathbb{R}.$
The Heisenberg group is a non-commutative sub-group of ${\mathrm{GL}_3(\mathbb{R})}$:
$\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} 1 & x’ & z’ \\ 0 & 1 & y’ \\ 0 & 0 & 1 \\ \end{pmatrix} = \begin{pmatrix} 1 & x+x’ & z+z’+xy’ \\ 0 & 1 & y+y’ \\ 0 & 0 & 1 \\ \end{pmatrix}$
The inverse is given by
$\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \\ \end{pmatrix}^{-1} = \begin{pmatrix} 1 & -x & -z+xy \\ 0 & 1 & -y \\ 0 & 0 & 1 \\ \end{pmatrix}.$
(the discrete Heisenberg group is the discrete sub-group of ${\mathbb{H}}$ formed by the elements of ${\mathbb{H}}$ with integer coordinates). The Heisenberg group ${\mathbb{H}}$ is a Lie group. Its Lie algebra ${\mathfrak{H}}$ is the sub-algebra of ${\mathcal{M}_3(\mathbb{R})}$ given by the ${3\times 3}$ real matrices of the form
$\begin{pmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \\ \end{pmatrix}, \quad a,b,c\in\mathbb{R}$
The exponential map ${\exp:A\in\mathcal{L}\mapsto\exp(A)\in\mathbb{H}}$ is a diffeomorphism. This allows to identify the group ${\mathbb{H}}$ with the algebra ${\mathfrak{H}}$. Let us define
$X= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}, \quad Y= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix}, \quad\text{and}\quad Z= \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}.$
We have then
$[X,Y]=XY-YX=Z\quad\text{and}\quad [X,Z]=[Y,Z]=0.$
The Lie algebra ${\mathfrak{H}}$ is nilpotent of order ${2}$:
$\mathfrak{H}=\mathrm{span}(X,Y)\oplus\mathrm{span}(Z).$
This makes the Baker-Campbell-Hausdorff formula on ${\mathfrak{H}}$ particularly simple:
$\exp(A)\exp(B)=\exp\left(A+B+\frac{1}{2}[A,B]\right).$
The names Heisenberg group and Heisenberg algebra come from the fact that in quantum physics, and following Werner Heisenberg and Hermann Weyl, the algebra generated by the position operator and the momentum operator is exactly ${\mathfrak{H}}$. The identification of ${\mathbb{H}}$ with ${\mathfrak{H}}$
$\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \\ \end{pmatrix} \equiv \exp \begin{pmatrix} 0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \\ \end{pmatrix} =\exp(xX+yY+zZ)$
allows to identify ${\mathbb{H}}$ with ${\mathbb{R}^3}$ equipped with the group structure
$(x,y,z)(x’,y’,z’)=(x+x’,y+y’,z+z’+\frac{1}{2}(xy’-yx’))$
and
$(x,y,z)^{-1}=(-x,-y,-z).$
This is the exponential coordinates of ${\mathbb{H}}$. Geometrically, the quantity ${\frac{1}{2}(xy’-yx’)}$ is the algebraic area in ${\mathbb{R}^2}$ between the piecewise linear path
$[(0,0),(x,y)]\cup[(x,y),(x+x’,y+y’)]$
and its chord
$[(0,0),(x+x’,y+y’)].$
This area is zero if ${(x,y)}$ and ${(x’,y’)}$ are colinear. The group product encodes the sum of increments in ${\mathbb{R}^2}$ and computes automatically the generated area.
The Heisenberg group ${\mathbb{H}}$ is topologically homeomorphic to ${\mathbb{R}^3}$ and the Lebesgue measure on ${\mathbb{R}^3}$ is a Haar measure on ${\mathbb{H}}$. However, as a manifold, its geometry is sub-Riemannian: the tangent space (at the origin and thus everywhere) is of dimension ${2}$ instead of ${3}$, putting a constraint on the geodesics (due to the lack of vertical speed vector, some of them are helices instead of straight lines). The Heisenberg group ${\mathbb{H}}$ is also a metric space for the so called Carnot-Carathéodory sub-Riemannian distance. The Heisenberg group is a Carnot group. Its Hausdorff dimension with respect to the Carnot-Carathéodory metric is ${4}$, in contrast with its dimension as a topological manifold which is ${3}$.
The dilation semigroup of automorphisms ${(\mathrm{dil}_t)_{t\geq0}}$ on ${\mathbb{H}}$ is defined by
$\mathrm{dil}_t \exp \begin{pmatrix} 0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \\ \end{pmatrix} = \exp \begin{pmatrix} 0 & tx & t^2z \\ 0 & 0 & ty \\ 0 & 0 & 0 \\ \end{pmatrix} .$
Let ${(x_n,y_n)_{n\geq0}}$ be the simple random walk on ${\mathbb{Z}^2}$ starting from the origin. If one embed ${\mathbb{Z}^2}$ into ${\mathbb{H}}$ by ${(x,y)\mapsto xX+yY}$ then one can consider the position at time ${n}$ in the group by taking the product of increments in the group:
$\begin{array}{rcl} S_n=(x_1,y_1)\cdots(x_n,y_n) &=&(s_{n,1},s_{n,2},s_{n,3}) \\ &=&(x_1+\cdots+x_n,y_1+\cdots+y_n,s_{n,3}). \end{array}$
These increments are commutative for the first two coordinates (called the horizontal coordinates) and non commutative for the third coordinate. The first two coordinates of ${S_n}$ form the position in ${\mathbb{Z}^2}$ of the random walk while the third coordinate is exactly the algebraic area between the random walk path and its chord on the time interval ${[0,n]}$. We are now able to state the Central Limit Theorem on the Heisenberg group:
$\left(\mathrm{dil}_{1/\sqrt{n}}(S_{\lfloor nt\rfloor})\right)_{t\geq0} \quad \underset{n\rightarrow\infty}{\overset{\text{law}}{\longrightarrow}} \quad \left(B_t,A_t\right)_{t\geq0}$
where ${(B_t)_{t\geq0}}$ is a simple Brownian motion on ${\mathbb{R}^2}$ and where ${(A_t)_{t\geq0}}$ is its Lévy area (algebraic area between the Brownian path and its chord, seen as a stochastic integral). The stochastic process ${(\mathbb{B}_t)_{t\geq0}=((B_t,A_t))_{t\geq0}}$ is the sub-Riemannian Brownian motion on ${\mathbb{H}}$.
$\begin{array}{rcl} \mathbb{B}_t &=& (B_t,A_t) \\ &=&(B_{t,1},B_{t,2},A_t) \\ &=& \exp \begin{pmatrix} 0 & B_{t,1} & \frac{1}{2}\left(\int_0^t\!B_{s,1}dB_{s,2}-\int_0^t\!B_{t,2}dB_{t,1}\right) \\ 0 & 0 & B_{t,2} \\ 0 & 0 & 0 \end{pmatrix} \\ &=& \begin{pmatrix} 1 & B_{t,1} & \int_0^t\!B_{s,1}dB_{s,2} \\ 0 &1 & B_{t,2} \\ 0 &0 &1 \end{pmatrix}. \end{array}$
The process ${(\mathbb{B}_t)_{t\geq0}}$ has independent and stationary (non-commutative) increments and belong the class of Lévy processes, associated to (non-commutative) convolution semigroups on ${\mathbb{H}}$. The law of ${\mathbb{B}_t}$ is infinitely divisible and maybe seen as a sort of Gaussian measure on ${\mathbb{H}}$. The process ${(\mathbb{B}_t)_{t\geq0}}$ is also a Markov diffusion process on ${\mathbb{R}^3}$ admitting the Lebesgue measure as an invariant reversible measure, and with infinitesimal generator
$L=\frac{1}{2}(V_1^2+V_2^2)=\frac{1}{2}\left((\partial_x-\frac{1}{2}y\partial_z)^2+(\partial_y+\frac{1}{2}x\partial_z)^2\right).$
We have ${V_3:=[V_1,V_2]=\partial_z}$ and ${[V_1,V_3]=[V_2,V_3]=0}$. The linear differential operator ${L}$ on ${\mathbb{R}^3}$ is hypoelliptic but is not elliptic. It is called the sub-Laplacian on ${\mathbb{H}}$. A formula for its heat kernel was computed by Lévy using Fourier analysis (it is an oscillatory integral).
Note that ${L}$ acts like the two dimensional Laplacian on functions depending only on ${x,y}$. Note also that the Riemannian Laplacian on ${\mathbb{H}}$ is given by
$L+\frac{1}{2}V_3^2=\frac{1}{2}\left(V_1^2+V_2^2+V_3^3\right) =\frac{1}{2}\left((\partial_x-\frac{1}{2}y\partial_z)^2+(\partial_y+\frac{1}{2}x\partial_z)^2+(\partial_z)^2\right).$
Open question. Use the CLT to obtain a sub-Riemannian Poincaré inequality or even a logarithmic Sobolev inequality on ${\mathbb{H}}$ for the heat kernel. It is tempting to try to adapt to the sub-Riemannian case the strategy used by L. Gross (for Riemannian Lie groups). This question is naturally connected to my previous work on gradient bounds for the heat kernel on the Heisenberg group, in collaboration with D. Bakry, F. Baudoin, and M. Bonnefont.
Related reading. (among many other references)
Syntax · Style · Tracking & Privacy. | 2019-09-17 14:35:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8563185334205627, "perplexity": 433.10270598547504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573080.8/warc/CC-MAIN-20190917141045-20190917163045-00263.warc.gz"} |
https://socratic.org/questions/how-do-you-factor-and-solve-x-3-216-0 | # How do you factor and solve x^3-216=0?
Apr 2, 2018
$\implies x = \left\{6 , - 3 \pm 3 \sqrt{3} i\right\}$
#### Explanation:
${x}^{3} = 216 \implies x = 6$
$\implies \left(x - 6\right)$ is a factor
$\left(x - 6\right) \left({x}^{2} + b x + 36\right) = {x}^{3} - 216$
$\implies - 6 {x}^{2} + b {x}^{2} = 0 {x}^{2}$
$\implies b - 6 = 0 \implies b = 6$
$\implies \left(x - 6\right) \left({x}^{2} + 6 x + 36\right) = 0$
$\implies x = \left\{6 , - 3 \pm 3 \sqrt{3} i\right\}$
Apr 2, 2018
See below:
#### Explanation:
According to the fundamental theorem of Algebra, a polynomial to the degree $n$ will have $n$ roots. In our case, a cubic will have three roots.
$x = 6$ is the first solution, which is obtained by solving the equation (taking the cube root of $216$).
If$x = 6$ is a solution, then $\left(x - 6\right)$ must be a factor of this polynomial. So, if you divide this polynomial by $\left(x - 6\right)$ using algebraic division, you will get the quadratic:
${x}^{2} + 6 x + 36$
Solving this quadratic will give you the remaining two roots:
${x}^{2} + 6 x + 36 = 0$
${\left(x + 3\right)}^{2} - 9 + 36 = 0$
${\left(x + 3\right)}^{2} + 27 = 0$
$\left(x + 3\right) = \sqrt{- 27}$
(x+3)= ±3sqrt3i
x= -3±3sqrt3i
The catch here is to remember that if you take the square root of any number you get two answers(± the answer). I solved this quadratic by 'completing the square', there is also an alternative quadratic formula which can be used to solve quadratics. The non-real solutions exist in conjugate pairs (a±bi).
Therefore, the solutions are:
$x = 6 , \left(- 3 + \sqrt{3} i\right) , \left(- 3 - \sqrt{3} i\right)$ | 2020-01-17 21:25:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 25, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8964626789093018, "perplexity": 377.59736952032944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250591234.15/warc/CC-MAIN-20200117205732-20200117233732-00521.warc.gz"} |
https://stats.stackexchange.com/questions/10159/find-expected-value-using-cdf/10168 | # Find expected value using CDF
I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing.
Let $$X$$ have the CDF $$F(x) = 1 - x^{-\alpha}, x\ge1$$.
Find $$E(X)$$ for those values of $$\alpha$$ for which $$E(X)$$ exists.
I have no idea how to even start this. How can I determine which values of $$\alpha$$ exist? I also don't know what to do with the CDF (I'm assuming this means Cumulative Distribution Function). There are formulas for finding the expected value when you have a frequency function or density function. Wikipedia says the CDF of $$X$$ can be defined in terms of the probability density function $$f$$ as follows:
$$F(x) = \int_{-\infty}^x f(t)\,dt$$
This is as far as I got. Where do I go from here?
EDIT: I meant to put $$x\ge1$$.
Edited for the comment from probabilityislogic
Note that $F(1)=0$ in this case so the distribution has probability $0$ of being less than $1$, so $x \ge 1$, and you will also need $\alpha > 0$ for an increasing cdf.
If you have the cdf then you want the anti-integral or derivative which with a continuous distribution like this
$$f(x) = \frac{dF(x)}{dx}$$
and in reverse $F(x) = \int_{1}^x f(t)\,dt$ for $x \ge 1$.
Then to find the expectation you need to find
$$E[X] = \int_{1}^{\infty} x f(x)\,dx$$
providing that this exists. I will leave the calculus to you.
• @henry - $F(1)=1-1^{-\alpha}=1-1=0$, so support can't be below 1 (as CDF is a non-decreasing function) – probabilityislogic Apr 30 '11 at 6:59
• @probabilityislogic: You may be correct in terms of the book. I will change my response. – Henry Apr 30 '11 at 9:49
• Thanks for the response. What does f(x) represent? The probability density function? Is the derivative of the cdf always f(x)? – styfle Apr 30 '11 at 20:23
• $f(x)$ is indeed supposed to be the probability density function. If the cdf has a derivative then it is the density, though there are distributions (for example discrete) where the cdf does not have a derivative everywhere – Henry Apr 30 '11 at 20:24
• @styfle: If it exists then $E[X^2] = \int_{1}^{\infty} x^2 f(x)\,dx$, and similarly for the expectations of other functions of $x$. – Henry Apr 30 '11 at 22:08
# Usage of the density function is not necessary
## Integrate 1 minus the CDF
When you have a random variable $X$ that has a support that is non-negative (that is, the variable has nonzero density/probability for only positive values), you can use the following property:
$$E(X) = \int_0^\infty \left( 1 - F_X(x) \right) \,\mathrm{d}x$$
A similar property applies in the case of a discrete random variable.
## Proof
Since $1 - F_X(x) = P(X\geq x) = \int_x^\infty f_X(t) \,\mathrm{d}t$,
$$\int_0^\infty \left( 1 - F_X(x) \right) \,\mathrm{d}x = \int_0^\infty P(X\geq x) \,\mathrm{d}x = \int_0^\infty \int_x^\infty f_X(t) \,\mathrm{d}t \mathrm{d}x$$
Then change the order of integration:
$$= \int_0^\infty \int_0^t f_X(t) \,\mathrm{d}x \mathrm{d}t = \int_0^\infty \left[xf_X(t)\right]_0^t \,\mathrm{d}t = \int_0^\infty t f_X(t) \,\mathrm{d}t$$
Recognizing that $t$ is a dummy variable, or taking the simple substitution $t=x$ and $\mathrm{d}t = \mathrm{d}x$,
$$= \int_0^\infty x f_X(x) \,\mathrm{d}x = \mathrm{E}(X)$$
I used the Formulas for special cases section of the Expected value article on Wikipedia to refresh my memory on the proof. That section also contains proofs for the discrete random variable case and also for the case that no density function exists.
• +1 great result: the integral of the cdf is really simple, moreover, it is wise to avoid derivatives, whenever we can (they are not as well behaved as integrals ;)). Additional: using the cdf to calculate the variance see here math.stackexchange.com/questions/1415366/… – loved.by.Jesus May 30 '17 at 21:46
• When you change the order of integration, how do you get the integration limits? – Zaz Oct 1 '17 at 16:40
• The standard proof does not assume that $X$ has a density. – ae0709 Sep 27 '18 at 23:19
• @Zaz we set the integration limits so that the same part of (t, x) space is covered. The original constraints are x >0 and t > x. We can't have the outer limits depend on the inner variable, but we can define the same region as t > 0 and 0 < x < t. Good examples of this process here: mathinsight.org/… – fredcallaway Oct 7 at 21:16
The result extends to the $k$th moment of $X$ as well. Here is a graphical representation:
I think you actually mean $x\geq 1$, otherwise the CDF is vacuous, as $F(1)=1-1^{-\alpha}=1-1=0$.
What you "know" about CDFs is that they eventually approach zero as the argument $x$ decreases without bound and eventually approach one as $x \to \infty$. They are also non-decreasing, so this means $0\leq F(y)\leq F(x)\leq 1$ for all $y\leq x$.
So if we plug in the CDF we get:
$$0\leq 1-x^{-\alpha}\leq 1\implies 1\geq \frac{1}{x^{\alpha}}\geq 0\implies x^{\alpha}\geq 1 > 0\implies x\geq 1 \>.$$
From this we conclude that the support for $x$ is $x\geq 1$. Now we also require $\lim_{x\to\infty} F(x)=1$ which implies that $\alpha>0$
To work out what values the expectation exists, we require:
$$\newcommand{\rd}{\mathrm{d}}E(X)=\int_{1}^{\infty}x\frac{\rd F(x)}{\rd x}\rd x=\alpha\int_{1}^{\infty}x^{-\alpha} \rd x$$
And this last expression shows that for $E(X)$ to exist, we must have $-\alpha<-1$, which in turn implies $\alpha>1$. This can easily be extended to determine the values of $\alpha$ for which the $r$'th raw moment $E(X^{r})$ exists.
• (+1) Particularly for the sharp-eyed recognition that the given support was incorrect. – cardinal Apr 30 '11 at 14:00
• Thanks for the response. I fixed the question. I meant to put x>=1. How did you know to first differentiate the cdf to get the density function? – styfle Apr 30 '11 at 20:37
• @styfle - because that's what a PDF is, whenever the CDF is continuous and differentiable. You can see this by looking at how you have defined your CDF. Differentiating an integral just gives you the integrand when the upper limit is the subject of the differentiation. – probabilityislogic May 1 '11 at 1:00
• @styfle - the PDF can also be seen as the probability that a RV lies in an infinitesimal interval. $Pr(x<X<x+dx)=F(x+dx)-F(x)\to \frac{dF(x)}{dx}dx=f(x)dx$ as $dx\to 0$. This way holds more generally, even for discrete RV and RV without a density (the limit is just something other than a derivative) – probabilityislogic May 1 '11 at 1:04
The Answer requiring change of order is unnecessarily ugly. Here's a more elegant 2 line proof.
$$\int udv = uv - \int vdu$$
Now take $$du = dx$$ and $$v = 1- F(x)$$
$$\int_{0}^{\infty} [ 1- F(x)] dx = [x(1-F(x)) ]_{0}^{\infty} + \int_{0}^{\infty} x f(x)dx$$
$$= 0 + \int_{0}^{\infty} x f(x)dx$$
$$= \mathbb{E}[X] \qquad \blacksquare$$
• I think you mean to let du-dx so that u=x. – Michael Chernick Aug 30 at 19:20 | 2019-11-15 17:33:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9099330902099609, "perplexity": 284.55953532867295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668699.77/warc/CC-MAIN-20191115171915-20191115195915-00409.warc.gz"} |
https://mobile.surenapps.com/2020/09/ratio-and-proportion-solutions-ex-122.html | ### Ratio and Proportion-Solutions Ex-12.2
CBSE Class –VI Mathematics
NCERT Solutions
Chaper 12 Ratio And Proportion (Ex. 12.2)
Question 1. Determine the following are in proportion:
(a) 15, 45, 40, 120
(b) 33, 121, 9, 96
(c) 24, 28, 36, 48
(d) 32, 48, 70, 210
(e) 4, 6, 8, 12
(f) 33, 44, 75, 100
Answer: (a) 15 : 45 = 1 : 3
40 : 120 = 1 : 3
Since 15 : 45 = 40 : 120
Therefore 15, 45, 40, 120 are in proportion.
(b) 33 : 121 = 3 : 11
9 : 96 = 3 : 32
Since 33 : 121 $\ne$ 9 : 96
Therefore, 33, 121, 9, 96 are not in proportion.
(c) 24 : 28 = 6 : 7
36 : 48 = 3 : 4
Since 24 : 28 $\ne$ 36 : 48
Therefore 24, 28, 36, 48 are not in proportion.
(d) 32 : 48 = 2 : 3
70 : 210 = 1 : 3
Since 32 : 48 $\ne$ 70 : 210
Therefore 32, 48, 70, 210 are not in proportion.
(e) 4 : 6 = 2 : 3
8 : 12 = 2 : 3
Since 4 : 6 = 8 : 12
Therefore 4, 6, 8, 12 are in proportion.
(f) 33 : 44 = 3 : 4
75 : 100 = 3 : 4
Since 33 : 44 = 75 : 100
Therefore 33, 44, 75, 100 are in ratio.
Question 2. Write True (T) or False (F) against each of the following statements:
(a) 16 : 24 : : 20 : 30
(b) 21 : 6 : : 35 : 10
(c) 12 : 18 : : 28 : 12
(d) 8 : 9 : : 24 : 27
(e) 5.2 : 3.9 : : 3 : 4
(f) 0.9 : 0.36 : : 10 : 4
Answer: (a) 16 : 25 : : 20 : 30 $⇒$ $⇒$$\frac{2}{3}=\frac{2}{3}$
Hence, it is True.
(b) 21 : 6 : : 35 : 10 $⇒$ $⇒$$\frac{7}{2}=\frac{7}{2}$
Hence, it is True.
(c) 12 : 18 : : 28 : 12 $⇒$ $⇒$$\frac{2}{3}\ne \frac{7}{3}$
Hence, it is False.
(d) 8 : 9 : : 24 : 27 $⇒$ $⇒$$\frac{8}{9}=\frac{8}{9}$
Hence, it is True.
(e) 5.2 : 3.9 : : 3 : 4 $⇒$ $⇒$$\frac{4}{3}\ne \frac{3}{4}$
Hence, it is False.
(f) 0.9 : 0.36 : : 10 : 4 $⇒$ $⇒$$\frac{5}{2}=\frac{5}{2}$
Hence, it is True.
Question 3. Are the following statements true:
(a) 40 persons : 200 persons = Rs. 15 : Rs. 75
(b) 7.5 liters : 15 liters = 5 kg = 10 kg
(c) 99 kg : 45 kg = Rs. 44 : Rs. 20
(d) 32 m : 64 m = 6 sec. : 12 sec.
(e) 45 km : 60 km = 12 hours : 15 hours
Answer: (a) 40 persons : 200 persons :: 1 : 5
Rs. 15 : Rs. 75 = = 1 : 5
Since, 40 persons : 200 persons = Rs. 15 : Rs. 75
Hence, the statement is true.
(b) 7.5 liters : 15 liters = = 1 : 2
5 kg : 10 kg = = 1 : 2
Since, 7.5 liters : 15 liters = 5 kg : 10 kg
Hence, the statement is true.
(c) 99 kg : 45 kg = = 11 : 5
Rs. 44 : Rs. 20 = = 11 : 5
Since, 99 kg : 45 kg = Rs. 44 : Rs. 20
Hence, the statement is true.
(d) 32 m : 64 m = = 1 : 2
6 sec : 12 sec = = 1 : 2
Since, 32 m : 64 m = 6 sec : 12 sec
Hence, the statement is true.
(e) 45 km : 60 km = = 3 : 4
12 hours : 15 hours = = 4 : 5
Since, 45 km : 60 km $\ne$ 12 hours : 15 hours
Hence, the statement is false.
Question 4. Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion:
(a) 25 cm : 1 m and Rs. 40 : Rs. 160
(b) 39 liters : 65 liters and 6 bottles : 10 bottles
(c) 2 kg : 80 kg and 25 g : 625 g
(d) 200 ml : 2.5 ml and Rs. 4 : Rs. 50
Answer: (a) 25 cm : 1 m = 25 cm : (1 x 100) cm = 25 cm : 100 cm = = 1 : 4
Rs. 40 : Rs. 160 = = 1 : 4
Since the ratios are equal, therefore these are in proportion.
Middle terms = 1 m, Rs. 40 and Extreme terms = 25 cm, Rs. 160
(b) 39 liters : 65 liters = 3 :5
6 bottles : 10 bottles = = 3 : 5
Since the ratios are equal, therefore these are in proportion.
Middle terms = 65 liters, 6 bottles and Extreme terms = 39 liters, 10 bottles
(c) 2 kg : 80 kg = = 1 : 40
25 g : 625 g = = 1 : 25
Since the ratios are not equal, therefore these are not in proportion.
(d) 200 ml : 2.5 liters = 200 ml : (25 x 1000) liters = 200 ml : 2500 ml = = 2 : 25
Rs. 4 : Rs. 50 = = 2 : 25
Since the ratios are equal, therefore these are in proportion.
Middle terms = 2.5 liters, Rs. 4 and Extreme terms = 200 ml, Rs. 50 | 2021-01-21 20:09:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 22, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5506255626678467, "perplexity": 2052.7884319373106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703527850.55/warc/CC-MAIN-20210121194330-20210121224330-00244.warc.gz"} |
http://www.rddl.com.cn/EN/10.13284/j.cnki.rddl.003237 | Tropical Geography ›› 2020, Vol. 40 ›› Issue (2): 314-322.
### Estimating Inland Water Depth Based on Remote Sensing and Machine Learning Technique
Wen Kaixiang1,2,5, Li Yong1,2,5(), Wang Hua1, Yang Ji2,5, Jing Wenlong2,5, Yang Chuanxun2,3,4,5
1. 1.School of Civil and Transportation Engineering, Guangdong University of Technology, Guangzhou 510070, China
2.Key Lab of Guangdong for Utilization of Remote Sensing and Geographical Information System//Guangdong Open Laboratory of Geospatial Information Technology and Applicaton//Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou)//Guangzhou Institute of Geography, Guangzhou 510070, China
3.Guangzhou Institute of Geochemistry, Chinese Academy of Sciences, Guangzhou 510640, China
4.University of Chinese Academy of Sciences, Beijing 100049, China
5.Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou 511458, China
• Received:2019-12-02 Revised:2020-03-31 Online:2020-03-10 Published:2020-05-15
• Contact: Li Yong E-mail:59525546@qq.com
Abstract:
This study investigates the application of three machine learning algorithms, e.g. the Random Forest (RF), Extreme Gradient Boosting (XGBoost), and Support Vector Machine (SVM) algorithms, in water depth estimation using Sentinel-2 multispectral satellite images. A case study is conducted in the Meizhou Reservoir in Guangdong, China, and a comparative analysis of the inversion results is performed. The results are validated against in-situ measured data using an unmanned ship, in which a global positioning system and a single wave velocity sounding system are integrated. The experimental results based on our water depth inversion models demonstrate good inversion accuracy and efficiency of the machine learning water depth inversion model constructed using 7 925 water depth data samples and satellite multispectral images. By adjusting the key parameters of each model such that the model reaches the optimal state, the determination coefficient (R 2), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Bias were used to evaluate the accuracy of the models. For water depth inversion, the RF model indicated that R 2 = 0.80, RMSE = 2.21, MAE = 1.51, and Bias = 0.00; the XGBoost model indicated that R 2 = 0.79, RMSE = 2.22, MAE = 1.50, and Bias = -0.01; and the SVM model indicated that R 2 = 0.78, RMSE = 2.23, MAE = 1.60, and Bias = 0.01. To determine the efficiency of each model, the models’ running times were obtained: 3.92 s for RF, 4.26 s for XGBoost, and 6.66 s for SVM. Notably, the RF model is superior to the XGBoost and SVM models in terms of inversion accuracy and efficiency. Moreover, the advantages and disadvantages of each model can be inferred from the prediction result graph. The prediction results of the RF model are more detailed, and the terrain is clearer compared with those of the XGBoost model. The SVM model does not achieve ideal prediction using the same dataset, and the error at a shallow water depth is much worse than those of the other two models. The scatter plots indicate that the inversion deviation is not obvious between the RF and XGBoost models, and the prediction results are more consistent with the actual results. However, the scatter plot of the SVM has a horizontal bar near 11 m; therefore, it predicts the water depth value in this interval incorrectly. Furthermore, the errors of the RF and XGBoost models evaluated by the standard evaluation method are lower than that of the SVM model, indicating that those models have a higher prediction accuracy for water depth inversion. In summary, the machine learning models for water depth estimation yielded good performances in this study, and they are suitable for estimating the water depth using remote sensing images with lower economy and time cost, especially in unreachable waters.
CLC Number:
• P715.7 | 2022-06-27 09:16:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23065462708473206, "perplexity": 1653.6398284642512}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103329963.19/warc/CC-MAIN-20220627073417-20220627103417-00749.warc.gz"} |
http://etheses.bham.ac.uk/1537/ | eTheses Repository
# MAPS calorimetry for a future linear collider
Miller, Owen Daniel (2011)
Ph.D. thesis, University of Birmingham.
PDF (11Mb)
## Abstract
The International Linear Collider (ILC) is intended to be a new high precision electron-positron collider operating at the TeV scale. The studies detailed within this thesis deal primarily with the development of Monolithic Active Pixel Sensors (MAPS) which could be used as the sensitive component of an ILC Electromagnetic Calorimeter (ECAL). These include simulations of a MAPS ECAL to investigate pixel saturation as a function of reset time, and energy resolution for single electrons. The energy resolutionσ (E)/E was found to be (0.1147±0.0004)/E$$^{1/2}$$(GeV$$^{1/2}$$). The remaining studies describe part of the calibration process for prototype MAPS sensors and how those sensors were used to characterise the response of MAPS pixels to a 120 GeV pion beam. This resulted in a measurement of the MIP (Minimum Ionising Particle) efficiency of MAPS pixels of 75% to 95% with some variation due to the properties of the epitaxial layer.
Type of Work: Ph.D. thesis. Watson, Nigel Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Physics and Astronomy Q Science (General)QC Physics University of Birmingham 1537
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
Repository Staff Only: item control page | 2016-05-31 10:06:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3759506046772003, "perplexity": 3058.760440901663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464051268601.70/warc/CC-MAIN-20160524005428-00207-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://brilliant.org/problems/post-this-for-two-years-later/?group=j4ihOzZ0WbPn | Post this for two years later
Geometry Level 4
$\tan \left ( \frac {\pi}{2017} \right ) \times \tan \left ( \frac {2\pi}{2017} \right ) \times \tan \left ( \frac {3\pi}{2017} \right ) \times \cdots \times \tan \left ( \frac {2016\pi}{2017} \right )$
What is the value of the product above?
× | 2019-09-21 12:00:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5133336186408997, "perplexity": 2640.847152741242}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574409.16/warc/CC-MAIN-20190921104758-20190921130758-00038.warc.gz"} |
http://mathhelpforum.com/math-puzzles/199367-another-interesting-fraction-print.html | # Another interesting fraction
• May 28th 2012, 07:58 AM
Soroban
Another interesting fraction
$\text{We have: }\:\dfrac{1}{89} \;=\;0.01123595\,\,.\,.\,.$
$\text{The decimal is formed like this:}$
. . $0.0{\bf1}$
. . $0.00{\bf1}$
. . $0.000{\bf2}$
. . $0.0000{\bf3}$
. . $0.00000{\bf5}$
. . $0.000000{\bf8}$
. . $0.000000{\bf{13}}$
. . $0.0000000{\bf{21}}$
. . $0.00000000{\bf{34}}$
. . . . . . $\vdots$
$\displaystyle\text{It seems that: }\:\frac{1}{10}\sum^{\infty}_{n=1} \frac{F_n}{10^n} \;=\;\frac{1}{89}$
. . $\text{where }F_n\text{ is the }n^{th}\text{ Fibonacci number.}$
$\text{Care to prove it?}$
• May 29th 2012, 01:56 AM
BobP
Re: Another interesting fraction
Let $S=\frac{1}{10}\sum^{\infty}_{n=1}\frac{F_{n}}{10^{ n}},$ then
$10S=\frac{F_{1}}{10}+\sum^{\infty}_{n=2}\frac{F_{n +1}-F_{n-1}}{10^{n}}$
$=\frac{1}{10}+10\sum^{\infty}_{n=2}\frac{F_{n+1}}{ 10^{n+1}}-\frac{1}{10}\sum^{\infty}_{n=2}\frac{F_{n-1}}{10^{n-1}}$
$=\frac{1}{10}+10\sum^{\infty}_{n=1}\frac{F_{n}}{10 ^{n}}-10\{\frac{F_{2}}{10^{2}}+\frac{F_{1}}{10}\}}-\frac{1}{10}\sum^{\infty}_{n=1}\frac{F_{n}}{10^{n} }$
$=\frac{1}{10}+100S-\frac{1}{10}-1-S$
$S=\frac{1}{89}.$
• May 30th 2012, 07:09 AM
Soroban
Re: Another interesting fraction
Elegant proof, BobP !
• Jun 13th 2012, 11:56 PM
richard1234
Re: Another interesting fraction
I like this one:
$\frac{1000}{998999} = 0.001001002003005008013...$
It's very easy to prove. Let
$S = .001001002003005008... = \sum_{i=1}^{\infty} (\frac{1}{1000})^{i} F_i$
$\frac{S}{1000} = .000001001002003005...$
Adding them up, we get $\frac{1001S}{1000} = .001002003005008... = 1000S - 1$
Solving for S, we obtain $S = \frac{1000}{998999}$. According to WolframAlpha, this decimal repeats with period 496,620. | 2016-09-28 23:07:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 26, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9948647022247314, "perplexity": 4995.042176653921}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661768.10/warc/CC-MAIN-20160924173741-00048-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.splashlearn.com/math-vocabulary/estimation-in-maths | # What is Estimation in Maths? Definition With Examples
## Introduction
There’s no single way to define estimation. The closest meaning of an estimation is a guess or an assumption about a value that isn’t too far off. What is an estimate in Math? Definition for estimate in math is an approximate value close enough to the correct value. A lot of guesses are made to make math easier and clearer.
## What is Estimation in Maths?
Most of the time, estimation in math is done by rounding off the numbers to their closest whole value to get a quick and simple rough answer. It saves time and effort.
For instance, kids use estimation many times a day without realizing it. They tend to figure out how long they play and how long they have for recess.
## Examples of Estimation
• Lily counted the number of flowers in the garden by counting the first row and then multiplying it by the number of rows. She estimated that there were 300 flowers in the garden.
## The General Rule of Estimation
• As soon as you pick the number you want to estimate/round, first find the digit at the place you want to round the number to.
• Next look at the digit right to that place to see which way you will round.
• If the digit to the right is four or less (as in 4, 3, 2, 1, 0), your number remains the same.
• If the digit to the right is five or higher (as in 5, 6, 7, 8, or nine), your number increases by one.
• In the end, all the extra digits that weren’t rounded off, should be removed.
A quick tip to remember this rule is:
## Conclusion
Our math specialists are committed to making studying enjoyable for our most ardent readers, the students!
Teachers investigate all sides of a topic using a compelling and fun learning approach. To know more about estimation concepts and other similarly interesting concepts.
## Solved Examples
Example 1. Aaron scored 87.91% in his seventh grade Science finals. Round off his percentage to the nearest whole number.
Solution: Aaron scored approximately 88% in seventh grade Science finals. This estimation was made after rounding off the number 87.91% to the nearest whole number. Since the number to the right after the decimal is 9 which is greater than 5, the number to the left(7) increases by one digit.
Example 2. Gina collected 56 coins and Carol collected 86 coins. Choose a better estimate to find the number of coins they own together.
Solution: Gina collected 56 coins. We can round off 56 to 60 coins and we can round off 86 to 90 coins. After adding 60 and 90, we get 150. The estimated answer is 150.
Example 3. Sam wants to buy 5 books, each costs $\$$2.95. The bundle of the 5 books together costs \$$20. Is that correct? Solution:No, that doesn’t seem correct. Each book costs approximately$\$$3. Five such books should cost 5 X \$$3 = \$\$$15. \$$20 seems too much.
## Practice Problems
1
### The seventh grade in a school has 54 students and the eighth grade has 77 students. Which of the following is a better estimation for the total number of students in both grades?
130
140
135
125
CorrectIncorrect
We can round off 54 to 50 students and 77 to 80 students. Adding 50 and 80 gives us 130. The correct estimation is 130.
2
### Which number is obtained when we round off 57.26 to the nearest whole number?
56
57
55
54
CorrectIncorrect
Rounding off 57.26 to the nearest whole number gives us 57. This is because the number to the right of the decimal is less than 5. When the number to the right of the decimal is less than 5, the number to the left stays the same after rounding off.
3
### The moon takes 27.322 days to complete one earth rotation. What is the better estimate?
27 days
28 days
29 days
30 days
CorrectIncorrect
The number to the right of the decimal is less than 5. When the number to the right of the decimal is less than 5, the number to the left stays the same after rounding off.
4
368
369
400
399
CorrectIncorrect | 2022-11-28 08:13:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4257332682609558, "perplexity": 650.1785221308487}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710488.2/warc/CC-MAIN-20221128070816-20221128100816-00283.warc.gz"} |
https://chaosatthesky.wordpress.com/2015/01/08/intro-to-cgt-3/ | # Introduction to Combinatorial Game Theory, Part 3
Follow-up of the first part and the second part.
Remember our discussion about impartial games and nimbers? Why do they matter? Well, a theorem called Sprague—Grundy theorem states that every impartial game is equivalent to a nimber; that is, its value is a nimber. Got any terribly complicated impartial game? It’s actually just Nim in disguise. This theorem is very powerful to analyze most impartial games, allowing all sorts of results from Nim to be borrowed in, which is exactly what we’re going to do here with Monorail (also known as Venice Connection). The rules of Monorail have been posted when I made my post a month or two ago; basically, you place tiles and attempt to produce a cycle using all tiles.
Before we start, let’s create a new game which we will call “Straight” for the lack of a better name. We have a board of an irregular shape, but the board is composed of squares and is gridded. (Basically you can cut off the board from a graph paper.) Some of the squares are initially marked. A move is to mark 1-3 squares that are contiguous and are in a straight line (horizontal/vertical), such that at least one of the new marked squares is orthogonally adjacent to a previously marked square. Here’s an example board with the marked squares in dark blue and other squares in dark blue:
Here are several legal moves, marked in green:
And several illegal moves, marked in red:
Note that this is an impartial game. Also note that as long as the board is connected and there is at least one marked square in the beginning, the whole board will eventually be filled. The player that cannot move (because the board is full) loses.
Let’s begin analyzing this game…
First, if the board is completely full, clearly the first player loses because it’s already full. The next simplest is when there is one empty square. Since we need a marked square and the board is connected, this empty square will be adjacent to some marked square, allowing the first player to mark it to win.
The next few simple games are two and three empty squares in a straight line. They are also trivially winning, as long as these squares are adjacent to some previously marked square. Note that we only need one marked square. In all cases, these positions are winning.
But what are their values? The full board is clearly $*0$. The single empty square board is $*1$, as the only move is to leave it to a full board with is $*0$. (Remember that $*1 = \{*0\}$.) The double empty squares board is $*2$; we can mark one square, leaving a $*1$, or two squares, leaving a $*0$. Similar with the triple empty squares giving $*3$.
What about four empty squares? Welcome to the first weirdness of the game!
The left position is losing. The only available moves are $*3, *2, *1$, and thus since it’s missing $*0$, its value is $*0$. Remember how $\{*3, *2, *1\} = *0$. The right position is winning; indeed, its value is as we expected, $*4$, since it has the additional available move of marking the two middle squares to leave $*1 + *1 = *0$.
…got confused? Try digesting it more slowly. The left position is losing, as it is $\{*3, *2, *1\} = *0$.
But the right position is winning, as it is $\{*1+*2, *1+*1, *1, *2, *1\} = \{*3, *0, *1, *2, *1\} = *4$.
That’s more or less everything required! Simply carefully break down every move, compute its value, and find the minimum excluded value to find the original board’s value.
Note the second position above (the $*4$ one), and among its choices, see the one that separates into $*1+*1$. With nimbers, we can clearly see that they add up to $*0$. But sometimes the original position makes it more obvious: they are two identical shapes! So certainly they equate to the same value, summing up to $*0$. In winning strategy terms, this is “play symmetrically; whatever your opponent plays in one part, copy it in the other part”. Thus you can see how the following position is winning, by simply finding a single move that breaks the rest into pairs of identical pieces:
Evaluating the value is harder, but sticking with the same principle, we can try searching for moves that break into pairs of identical pieces, except for some piece whose value we already know. Of course, we cannot always get such move, and sometimes we need to look for pretty much all moves to determine that some value can or cannot be obtained.
Also note that the value is never more than the number of squares. This is not a coincidence and can be proven by strong induction.
With that, we’ll do one more board:
You might be familiar with the shape; indeed, this is the exact Monorail board that I suggested in the original discussion linked above. As it turns out, the left L-part has value $*6$ as one can move in it to reach all of $*0, *1, \ldots, *5$ positions.
However, the d-part has value $*2$! Yes, I made a mistake in the original analysis. (I think. Someone please verify? I got the d-part is $\{*0, *0, *0, *1, *4, *4, *5, *5, *5, *5, *6, *6\} = *2$.) Thus the value of the original board is $*6+*2 = *4$, and by using Nim strategy, the player to move wins by playing on $*6$ to reduce it to $*2$.
Of course, in Monorail, the first player was the one that forced the above configuration and now it’s second player’s turn, so the first player made a fatal mistake of playing that move. But again, the reason I suggested this move was more psychological than logical. The fact that you know exactly how to respond on any opponent’s mistake also helps.
Then, another game for Jinho versus Hyunmin. The track is completely determined at this point, and so we convert to a Straight position:
I haven’t completely evaluated the above. Anyone interested on figuring it out? I know when Hyunmin claimed the win (1:25), it was actually a loss (a win for player to move, Jinho).
That’s pretty much all I have about Monorail and Straight at the moment. I might write more, but it would be about some other impartial game and interesting observations from it.
## 8 thoughts on “Introduction to Combinatorial Game Theory, Part 3”
1. woll says:
How would you go about evaluating this board state:
Where blue is the original station, red is P1s move that forces a particular path, and black is empty space (ie, putting a tile over black results in opponent calling impossible)
My current hunch is that Monorail as a game is a win for P2 given perfect play, but I’m not sure how to go about proving that.
• I’m not sure it’s possible to force that particular path:
However, in either case, the domino at top-right is a *2. The giant 11-square area below needs massive case checking, much more than the L-shape above; this is the problem with nimbers (you need to test everything to get a perfect count). Of course, you actually only need to find if it’s possible to make the lower one a *2 (so that the sum is *2 + *2 = *0, meaning the next player to move loses, which means by playing that move you’ll win), but still checking everything.
• woll says:
Ah, I missed the different paths you could take in the bottom left! Thanks for that, it’s clear now that there’s a forced win for player 2:
If player 2 plays in green:
He leaves player one with two 1×2 units and a 2×4 unit. Any movement by player one in the 1×2 units can be mirrored by player 2, and the 2×4 unit is a forced win for the player to move second (invariably player 2), since after any move by the first player, the second can simply divide it to form *0.
• I’m interested on what reply there is for the move R4C3-4 (a domino along row 4, column 3-4 in the above picture). (Too lazy to make another picture.)
• woll says:
That moves loses to R3C4-3+R4C4-3. This produces one 1×1 and one 2×2 (where one square in the 2×2 is inaccessible). No move P1 makes in the 2×2 can stop P2 from reducing the 2×2 into a 1×1, leaving a symmetric, *0 position. If P1 plays in the 1×1, P2 responds with R3C5-2!, which wins because P1 cannot respond in the bottom right corner.
The other “feisty” move P1 has is R3C6-1, but this ends up losing to R3C4-2.
• Remember that the 2×4 unit is not a rectangle (it’s 7 squares, missing one corner).
On the other hand, that’s true, the reply would be R3C4-5 to make an L of 3 squares with the elbow inaccessible.
What is R3C6-1? R3C6, playing one tile? R4C3-4 means Row 4, Column 3-4, not Row 4, Column 3, playing 4 tiles.
• woll says:
Ah, my apologies, I misunderstood your syntax. I intended “R3C6-1” to mean Row 3, Column 6 (from the left), Column 1 (from the right), playing one tile. Looking back, an alternate response to your move could be Row 3, far right + Row 3, middle right, (columns 5 and 6) which produces 2 2×1 units.
• That response doesn’t quite work for the same reason: the 2×4 is not a rectangle.
Playing there makes a domino and a single square. | 2018-03-17 19:57:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 27, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6041818261146545, "perplexity": 846.4776916640792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645310.34/warc/CC-MAIN-20180317194447-20180317214447-00354.warc.gz"} |
https://mathoverflow.net/questions/312411/products-coproducts-and-equalizers-in-category-of-lattices | # Products, coproducts and equalizers in category of lattices
Background: Let $$\mathbf{Lat}$$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $$\mathbf{Pos}$$, that is, objects in $$\mathbf{Pos}$$ that have all finite products and coproducts (a.k.a meets and joins in lattice-speak); we may or may not require 2-morphisms in $$\mathbf{Lat}$$ to preserve meets and joins (i.e continuous and cocontinuous).
I am considering (what I call) cellular sheaves valued in lattices which are just functors $$F: X \rightarrow \mathbf{Lat}$$ where $$X$$ is the face relation poset of a cell complex. In order to do "sheaf theory" with sheaves valued in $$\mathbf{Lat}$$, it would be nice to have a notion of a coproduct and product in this category. I think product is fairly clear: just use the product partial order; meets and joins are what you think they would be. As far as a coproduct, I am not sure. If anyone has any suggestions? I have heard of a "free product of lattices" but it is not defined in a language I can understand. Not even clear to me that the "free product" that Gratzer defines is unique.
By "sheaf theory", I mainly mean taking limits and colimits over all the stalks. Another property I would like (or like to know doesn't hold) is the existence of all equalizers and (maybe if coproducts exist) coequalizers which would guarantee the existence of all small (co)limits. That would be lovely.
In general, looking for references on lattice theory, "non-abelian" sheaf theory, or anything about categories and lattice theory.
• Do lattices have top and bottom elements for you? Oct 9 '18 at 22:55
• @TimCampion if empty joins and meets count as finite joins and meets, then yes, they should. Oct 10 '18 at 0:14
• I suspect that if you require that maps of lattices are compatible with meets/joins, then the coproduct of lattices might again just be the cartesian product of the underlying lattice, with the left and right injections being defined by sending $x \mapsto (x,\top)$ and $y \mapsto (\top,y)$ respectively. But this might require all joins, not just finite joins. Oct 10 '18 at 0:19
• @TimCampion and I see you have a similar answer here: math.stackexchange.com/a/1195002/3835 Oct 10 '18 at 0:21
• @TimCampion Say lattices have bottom elements (not sure I want to restrict to lattices with top elements yet) which I assume is $\perp$ in DavidRoberts's notation.
– Hans
Oct 10 '18 at 21:26
I'm not sure about sheaf theory, but limits and colimits in categories of lattices are routine to construct. You just have to be clear about your categorical setup. Consider the following categories:
• $$sLat^{\vee}$$, the category of posets with $$(\vee,\bot)$$ and morphisms which preserve these.
• $$Lat$$, the category of posets with $$(\vee,\wedge,\bot,\top)$$ and morphisms which preserve these.
• $$DistLat$$ the full subcategory of $$Lat$$ on the lattices where $$\wedge$$ distributes over $$\vee$$.
• ...
and variants like
• $$Lat^{unbbd}$$, the category of posets with $$(\vee,\wedge)$$ and morphisms which preserve these.
• $$Lat^{\uparrow bdded}$$, the category of posets with $$(\vee,\wedge,\top)$$ and morphisms which preserve these.
• ...
In each case, we have a variety in the sense of universal algebra, i.e. a category where an object is a set with some $$n$$-ary functions on it satisfying some universal equations (note that the poset structure can be recovered from $$\vee$$ or $$\wedge$$) and morphisms being functions which commute with all the functions. Other examples are categories like groups, rings, etc.
In any such category $$\mathcal C$$, one has all limits and colimits. They can be constructed in the following way. There is a forgetful functor $$U: \mathcal C \to Set$$. This functor has a left adjoint $$F: Set \to \mathcal C$$, which just sends a set to the set of all words in the function symbols, quotiented by the universal relations that hold in objects of $$\mathcal C$$. For example in the case of groups, $$F(X)$$ is the free group on the set $$X$$.
This adjunction $$F \dashv U$$ yields a monad $$UF: Set \to Set$$ from which the category $$\mathcal C$$ can be recovered as the category of algebras. Limits and colimits can be constructed in a standard way. For limits, take the limit of the underlying sets and extend the functions in the obvious way. Colimits are a bit more complicated, but you can read about them here. Filtered colimits, though, are easy -- just take the colimit of the underlying sets and extend the operations in the obvious way (these are what you need to compute stalks). Reflexive coequalizers are likewise computed at the level of the underlying sets. The trickiest class of colimits are coproducts. The coproduct $$\amalg_i C_i$$ is constructed by taking $$F(\amalg_i U(C_i))$$ and quotienting by an equivalence relation (for instance, the coproduct of groups is the free amalgam). Alternative descriptions may be available depending on $$C$$ -- for example see here.
• Even coproducts are not too bad, once the free algebras are at hand. The general construction is a reflexive coequalizer as described in Theorem 2.2 here: ncatlab.org/nlab/show/… On the other hand, the explicit concrete description of free lattices takes some effort (I don't mean as abstract development, but in terms of whether two syntactic descriptions are equivalent as lattice elements), and was originally given by Whitman. See Theorem 6.2 here, for instance: math.hawaii.edu/~jb/math618/os6uh.pdf. Oct 13 '18 at 17:53
• Also, for the OP: I think "free products" mean the same thing as coproducts, if group-theoretic terminology is any guide at all. They are certainly well-defined, but the explicit concrete description is about as easy or as hard as understanding free lattices, as hinted by my previous comment -- similar to the situation in group theory. Oct 13 '18 at 17:56
• @TimCampion Warm thanks for your extremely detailed answer. I will take some time to process what you have said and will return here for a follow-up questions if you would be so kind.
– Hans
Oct 14 '18 at 18:44
• Also, if a "sheaf" is just a functor to ${\bf Lat}$ as the OP says (most people would call that a "presheaf"), then their limits and colimits are just pointwise. Jan 3 '19 at 18:47 | 2021-11-28 11:16:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 34, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.86484295129776, "perplexity": 287.6700340347143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358520.50/warc/CC-MAIN-20211128103924-20211128133924-00305.warc.gz"} |
https://www.physicsforums.com/threads/pressure-in-u-tube.332464/ | # Homework Help: Pressure in U-tube
1. Aug 25, 2009
### songoku
1. The problem statement, all variables and given/known data
Two liquids A and B of densities 1.0 g cm-3 and 0.8 g cm-3 are poured into an U-tube as shown in the figure below. Find x !
2. Relevant equations
$$P = \rho g h$$
3. The attempt at a solution
Is it possible liquid A is on top of liquid B because A has higher density than B?
Or do I misinterpret the question? Is it possible the question means density of A is 0.8 g cm-3 and B is 1.0 g cm-3 because there is no word "respectively" ?
Thanks
2. Aug 25, 2009
### Staff: Mentor
Sure, why not? Look at the diagram. Note that the height of liquid A is less than that of liquid B. There's a reason for that!
3. Aug 25, 2009
### kuruman
The statement looks correct, A is more dense than B. At the line PQ the pressure is the same. So more height of liquid above point P is needed to maintain the pressure the same as at point A.
4. Aug 25, 2009
### songoku
Hi Doc Al and kuruman
Oh so it's possible that liquid with higher density is on top of liquid with lower density. I always think it's impossible.
I get it now. Thanks a lot Doc Al and kuruman ^^
5. Aug 25, 2009
### ideasrule
Just imagine an iron piston pressing down on water in the U-tube and you'll see why it's possible. | 2018-05-22 03:12:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2625982165336609, "perplexity": 1448.2135742015103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864622.33/warc/CC-MAIN-20180522014949-20180522034949-00429.warc.gz"} |
http://www.ms.lt/sodas/Book/Geometry | Notes
Math 数学
• ms@ms.lt
• +370 607 27 665
• My work is in the Public Domain for all to share freely.
Introduction E9F5FC
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Notes EEEEEE
Software
Overall goals:
• To understand what geometry contributes to the overall map of mathematics.http://www.ms.lt/sodas/Book/Geometry?action=diff
• To distinguish four geometries: affine, projective, conformal and symplectic.
• To understand the relationship between geometries and logic, the classical Lie groups/algebras, category theory, etc.
• To have a better understanding of mathematical concepts, tools, theorems and examples that would serve me in understanding all branches of mathematics.
I should
• Make a list of geometry theorems and sort them by geometry.
• Make a list of geometries and show how they are related.
• Why are rings important for geometry rather than just groups? Because want to work with ideals and not subrings, because we are dealing with what is not as well as what is, because we are constructing both top-down and bottom-up.
Four kinds of geometry
General notions
• Geometry is the regularity of choice.
• Geometry is about defining equivalence (of shapes), thus the transformations that maintain equivalence, and the symmetries of those transformations.
Path geometry
• Affine geometry: Point + Vector = Point. Vector + Vector = Vector. Point - Point = Vector. But we can't add two points because we don't have any origin for them to reference.
• Unions of spaces.
Line geometry
• Map lines to lines. Projective geometry additionally maps zero to zero. And infinity to infinity? And do the lines have an orientation? And is that orientation preserved?
• Linear equations are intersections of hyperplanes.
• Projective geometry is linear algebra.
• Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
• In projective geometry, vectors are points and bivectors are lines.
• Projective geometry transforms conics into conics.
• Fundamental theorems of affine and projective geometry
• Fundamental Theorem of Affine geometry. let {$X,X'$} be two finite dimensional affine spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:X\to X'$} be a bijection that sends collinear points to collinear points, i.e. such that for all {$a,b,c\in X$} that are collinear, {$f(a),f(b),f(c)$} are collinear too. Then {$f$} is a semi-affine isomorphism.
• This means that there is a field isomorphism {$\sigma:K\to K'$} such that for any point {$a\in X$} the map induced by {$f_a: X_a\to X'_{f(a)}$} is a {$\sigma$}-semi-linear isomorphism.
• Fundamental Theorem of Projective geometry. let {$P(X),P(X')$} be two finite dimensional projective spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:P(X)\to P(X')$} be a bijection that sends collinear points to collinear points. Then {$f$} is a semi-linear isomorphism.
• Projective geometry: Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).
Angle geometry
• A metric yields distance, an inner product and angles.
Oriented area geometry
Geometry to study
Linear Algebra
• What does it mean that the eigenvalues of a matrix are the zeros of its characteristic polynomial? And that the matrix itself is a zero of its characteristic polynomial? And then what doe the symmetric functions of the eigenvalues of a matrix mean? The coefficients of the polynomial can be expressed in terms of the same eigenvalues that are its solutions. So in what sense are they dual? Ask at Math Overflow.
Plane Geometry
• Look at Wildberger's three binormal forms.
Videos
Affine and Projective Geometry
• Introduction to Geometry by Coxeter.
• Norman Wildberger
• Geometry at Cut-the-Knot
• Eccentricity defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
• Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
Affine geometry
• Allowing only positive "coefficients" is related to positive definiteness, convexity.
• Does not assume Euclid's third and fourth axioms.
• Different coordinate systems don't agree on any origin.
• Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
• In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
• Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
• In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
• https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume
• https://en.m.wikipedia.org/wiki/Motive_(algebraic_geometry) related to the connection between affine and projective space
• Tiesė perkelta į kitą tiesę išsaugoja trijų taškų paprastą santykį (ratio).
• Affine varieties correspond to prime ideals and as such are irreducible. So they are the building blocks of the closed subsets of the Zariski topology.
• Affine geometry - free monoid - without negative sign (subtraction) - lattice of steps - such as Young tableaux as paths on Pascal's triangle.
• What does it mean that the point at infinity is a zero of a polynomial? Is that establishing and modeling the limiting process?
• The link between projective geometry and fractions (as equivalence classes).
Projective geometry
• Projective geometry relates one plane (upon which the projection is made) with another plane (where the "eye" is, the zero where all the lines come from). And thus the line through the eye which is parallel to the plane needs to be added. Thus we can have homogeneous coordinates. And we have the decomposition of projective space into a sum of affine spaces of each dimension. Projective geometry is the space of one-dimensional subspaces, and they all include zero, thus they are the lines which go through zero. Or the hyperplanes which go through zero.
• projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
• "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
• [1:2:0] is a point that is a "direction" (two directions)
• A vector subspace needs to contain zero. How is this related to projective geometry? Vector spaces: Two different coordinate systems agree on the origin 0.
• Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
• https://en.m.wikipedia.org/wiki/Homography Homography two approaches to projective geometry with fields or without
• A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
• Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
• Sylvain Poirier: Some key ideas, probably you know, but just in case:
The (n+p-1)-dimensional projective space associated with a quadratic space with signature (n,p), is divided by its (n+p-2)-dimensional surface (images of null vectors), which is a conformal space with signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p) and positive curvature, the other with dimension (n,p-1) and negative curvature. Just by changing convention, the one with signature (n-1,p) and positive curvature can also seen as a space with signature (p,n-1) and negative curvature; and similarly for the other.
• Sylvain Poirier: Affine representations of that quadric are classified by the choice of
the horizon, or equivalently the polar point of that horizon (the point representing in the projective space the direction orthogonal to that hyperplane). So there are 3 possibilities. The null one sees it as a paraboloid and gives it an affine geometry. The 2 others, with the different signs, see it as a quadric whose center is the polar point, and give it 2 different curved geometries
• Sylvain Poirier: We can understand the stereographic projection as the effect of the
projective transformation of the space, which changes the sphere into a paraboloid, itself projected into an affine space.
• (1 + ti)(1 + ti) = (1 - t2) + (2t) i is the rational parametrization of the circle.
• What about the sphere? The stereographic projection of the circle onto the plane in Cartesian coordinates is given by (1 + xi + yj)(1 + xi + yj) where ij + ji = 1, that is, i and j anticommute.
• Note also that infinity is the flip side of zero - they make a pair. They are alternate ways of linking together the positive and negative values.
• square-root-of-pi is gamma-of-negative-one-half (relate this to the volume of an odd-dimensional ball: pi-to-the-n/2 over (n/2)!
Classical Algebraic Geometry
• How is the Zariski topology related to the Binomial theorem?
Videos
Books
Modern Algebraic Geometry
Intuition
Ravi Vakil: The intuition for schemes can be built on the intuition for affine complex varieties. Allen Knutson and Terry Tao have pointed out that this involves three different simultaneous generalizations, which can be interpreted as three large themes in mathematics.
• (i) We allow nilpotents in the ring of functions, which is basically analysis (looking at near-solutions of equations instead of exact solutions).
• (ii) We glue these affine schemes together, which is what we do in differential geometry (looking at manifolds instead of coordinate patches).
• (iii) Instead of working over C (or another algebraically closed field), we work more generally over a ring that isn’t an algebraically closed field, or even a field at all, which is basically number theory (solving equations over number fields, rings of integers, etc.).
Ideas
• Fiber is a Zero.
Videos
Books
Sheaves
Schemes
Algebraic Topology
• In topology product rule d(MxN) = dM x N union MxN addition is union (whereas in the Zariski topology multiplication is union). Why? The product rule is related to the deRham cohomology.
• What happens to the corners of the shapes?
• What is the topological quotient for an equilateral triangle or a simplex?
• Topological product (for a torus) is a list, has an order. In general, a Cartesian product is a list. What if such a product is unordered? How do we get there in the limit to F1?
• How can you cut in half a topological object if you have no metric? How can you be sure whether you will get two or three pieces?
• Try to imagine what a 3-sphere looks like as we pass through it from time t = -1 to 1.
Videos
Books
Ideas
• Tadashi Tokieda: Basic strategy of topology. When a problem has degeneracies, then deform (or perturb) to a problem without degeneracies, then deform back. We can use the same approach to show some problems are unsolvable.
• Quotient is gluing is equivalence on a boundary. Topology is the creation of a smaller space from a larger space.
• Cross cap introduces contradiction, which breaks the segregation between orientations, whether inside and outside, self and world, or true and false.
• Prieštaravimu panaikinimas išskyrimas išorės ir vidaus, (kaip kad ramybe - lūkesčių nebuvimu), tai sutapatinama, kaip kad "cross-cap".
• If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
• Constructiveness - closed sets any intersections and finite unions are open sets constructive
• A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other.
Homology and Cohomology
• Weibel, Homological Algebra
• Coherent sheaf cohomology
• Motives and Universal cohomology. Weil cohomology theory and the four classical Weil cohomology theories (singular/Betti, de Rham, l-adic, crystalline)
• spectrum - topology, cohomology
• Our Father relates a left exact sequence and a right exact sequence.
• Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
• Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.
• Long exact sequence from short exact sequence: derived functors.
• Dievas žmogui yra skylė gyvenime, prasmė - neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje - homologijoje.
• Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?
Geometry and Logic
• Sheaves in Geometry and Logic, Medak and Macleigh
Differential Geometry
• Relate sheaves and vector bundles.
Other Geometry
Videos
Noncommutative geometry
Books
Relating Geometries
History of Geometry
Books
Organizing Geometry
Intuition
Symmetry
Different geometries
Conformal Geometry
Conformal geometry
• In conformal geometry (Euclidean geometry), we have inversions. The (infinite) horizon line is a circle that we are within. Reflection takes us in and out of this circle.
• An example of conformal geometry is (universal conformal) stereographic projection. The infinite line (of the horizon) is reduced to a point (the top of the sphere).
• Algebraic geometry presumes orthogonal basis elements, thus, perpendicularity and angles. Thus affine geometry and projective geometry should be restricted to not using algebraic geometry.
• Universal hyperbolic geometry (projective geometry with a distinguished circle) is perhaps conformal geometry. It relates two different spaces, the inside and the outside of the circle.
• Moebius transformations revealed.
Hyperbolic geometry: projective plane (empty space) + distinguished circle + tools: straightedge = projective relativistic geometry
• perpendicularity via Appolonius pole-polar duality: dual of point is line and vice versa
• orthocenter - exists in Universal Hyperbolic Geometry but not in Classical Hyperbolic Geometry - need to think outside of the disk.
• most important theorem: Pythagoras q=q1+q2 - q1q2
• second most important theorem: triple quad formula (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3) + 4q1q2q3
Compare to: Beltrami-Klein model of hyperbolic geometry
Books
Symplectic Geometry
• How does the geometric product in a Clifford Algebra model angular momentum, the basis for symplectic geometry, which is otherwise typically described by the cross product?
Videos
Symplectic geometry
• Symplectic geometry is an even dimensional geometry. It lives on even dimensional
spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry. It is naturally associated with the field of complex rather than real numbers. However, it is not as rigid as complex geometry: one of its most intriguing aspects is its curious mixture of rigidity (structure) and flabbiness (lack of structure). What is Symplectic Geometry? by Dusa McDuff
• McDuff: First of all, what is a symplectic structure? The concept arose in the study of classical
mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a falling apple. The trajectory of such a system is determined if one knows its position and velocity (speed and direction of motion) at any one time. Thus for an object of unit mass moving in a given straight line one needs two pieces of information, the position q and velocity (or more correctly momentum) p:= ̇q. This pair of real numbers (x1,x2) := (p,q) gives a point in the plane R2. In this case the symplectic structure ω is an area form (written dp∧dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action pdq round the boundary ∂S.
• momentum x position is angular momentum
• McDuff: This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a
single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld quantization principle says that the area of this region is quantized, i.e. it has to be an integral multiple of a number called Planck’s constant. Thus one can think of the symplectic area as a measure of the entanglement of position and velocity.
• Symplectic area is orientable.
• Area (volume) is a pseudoscalar such as the scalar triple product.
• Symplectic geometry is naturally related to time because it is swept out (in one dimension) in time. And so the time (one-)dimension thereby "defines" the geometry of the area.
• Symplectic "sweep" is related to equivalence (for example, natural transformation) relevant for arguments of equality by continuity (for example, the Fundamental Theorem of Calculus, integration).
• Symplectic geometry relates a point and its line, that is, it treats the moving point as a line with an origin, and relates the relative distance between the origins and the relative momentum between the origins. Thus it is a relation between two dimensions. And the boundary of the curve can be fuzzy, as in quantum mechanics and the Heisenberg principle.
Books
Defining Geometry
Geometry is:
• the ways that our expectations can be related, thus how we are related to each other
• the relationship between our old and new search. And search is triggered by constancy, which is the representation of the nullsome which is related to anything and thus to calm and expectations, space and time, etc.
• how to expand our vision (from a smaller space to a larger space) (Tadashi Tokieda)
• how to embed a lower dimensional space into a higher dimensional space
• the ways that a vector space is grounded
• the relationship between two spaces, for example, points, lines, planes
• the construction of sets of roots of polynomials
• Grothendieck categories
Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.
• Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
• Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
• A geometric embedding is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.
Definitions of geometry
• Geometry is the study of curvature (Atiyah's video talk on Geometry in 2, 3 and 4 dimensions. Intrinsic and extrinsic curvature. Sphere has constant curvature. Sphere - positive - genus 0. Torus (cylinder) - zero curvature - genus 1. Higher genus - negative curvature.
• 2 dimensions - Scalar curvature R
• 3 dimensions - Ricci curvature Rij
• 4 dimensions - Riemann curvature Rijk
• Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n-1. Icosahedron is the fake sphere in 3-dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture. Icosahedron would be in A5 but reality is given by A4 and so A5 is insolvable!
• Wikipedia defines geometry as "concerned with questions of shape, size, relative position of figures, and the properties of space".
• MathWorld defines geometry as "the study of figures in a space of a given number of dimensions and of a given type", and formally, as "a complete locally homogeneous Riemannian manifold".
• nLab seems to define it as part of an Isbell duality between geometry (presheaves) and algebra (copresheaves) where presheaves (contravariant functors C->Set) and copresheaves (functors on C) are identified with each other and thus glued together (for some category C).
• At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
• Algebraic geometry is the study of spaces of solutions to algebraic equations.
Geometry challenges
• Dimension 3: relate Jones quantum invariants (knots, any manifold) with Perlman-Thurston.
• Dimension 4: understand the structure of simply-connected 4-manifolds and the relation to physics.
• Atiyah speculation: Space + Circle = 4 dimensions (Riemannian). Donaldson theory -> geometric models of matter? Signature of 4-manifold = electric charge. Second Betti number = number of protons + neutrons.
Construction of the continuum
• Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
A System of Geometries
Ideas
• Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity. And a different way of relating two dimensions.
• Each geometry is the action of a monoid, thus a language. But that monoid may contain an inverse, which distinguishes the projective geometry from the affine geometry.
• In a free monoid the theorems are equations and they are determined by what can be done with associativity. This is first order logic. A second order logic or higher order logic would be given by what can be expressed, for example, by counting various possibilities.
• Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.
Four Basic Geometries
Center and Totality
• Relate the first Betti number with my version of the Euler characteristic, C - V + E - F + T.
• Study Bezier curves and Bernstein polynomials.
• Bernstein polynomials x = 1/2 get simplex, x = 1/3 or 2/3 get cube and cross-polytope.
• Try to use the tetrahedron as a way to model the 4th dimension so as to imagine how a trefoil knot could be untangled.
Generalize this result to n-dimensions (starting with 4-dimensions): Full finite symmetry groups in 3 dimensions
Triangles
• What is the significance of a triangle or a trilateral? They are the fourth row of Pascal's triangle.
• A triangle on a sphere together with its antipodes (defined in terms of the center) defines eight triangles, an octahedron. A triangle in three dimensional space defines a demicube (simplex) in terms of the origin. A triangle with its center defines a simplex. How is a triangle related to a cube?
• Trikampis - išauga požiūrių skaičius apibudinant: affine-taškai-0, projective-tiesės-1, conformal-kampai-2, symplectic-plotai-3.
Consider a triangle with 3 directed sides A, B, C:
• Path geometry is given by A + B + C = [0] gets you back where you started from. It is geometry without space, as when God thinks why, so that everything is connected by relationships, and God of himself only thinks forwards, unfolding.
• Line geometry embeds this in a plane, which gives it an orientation, plus or minus. +0 or -0 We have A and -A, etc. Barycentric coordinates for vectors v1, v2, v3 (with scalars lambda l1, l2, l3) where the scalars are between 0 and 1 and the sum l1v1 + l2v2 + l3v3 = 1 on the triangle and <1 within it and all are 1/3 to get the center, the average. For example, a line in a plane splits that plane into two sides, just as a plane splits a three-dimensional space. Thus this is where "holes" come from, disconnections, emptiness, homology.
• Angle geometry gives this a total value of 1, the total angle. And so we can accord to A, B, C a ratio that measures the opposite angle. This creates the inside and the outside of the triangle. Indeed, the three lines carves the plane into spaces. It's not clear how they meet at infinity.
• Area geometry assigns an oriented area AREA to the total value. Time arises as we have one side and the other swept by it.
• Victor Kac's paper: “Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”
• John Baez: Whenever we pick a Dynkin diagram and a field we get a geometry: An projective, Bn Cn conformal, Dn symplectic.
Four infinite families of polytopes can be distinguished by how they are extended in each new dimension. They seem to relate to four different geometries and four different classical Lie algebras:
• An - Simplexes are extended when the Center (the -1 simplex) creates a new vertex and thereby defines direction, which is preserved by affine geometry. Simplexes have both a Center and a Totality. This is geometry without any field, and without any zeros - what does this mean for the correspondence with the polynomial ring?
• Cn - Cross-polytopes (such as the octahedron) are extended when the Center creates two new vertices ("opposites") and thereby defines a line in two directions, which is preserved by projective geometry. Cross-polytopes have a Center but no Totality.
• Bn - Cubes are extended when the Totality introduces a new mirror and thereby defines right angles with previous mirrors, and the angles are preserved by conformal geometry. Cubes have a Totality but no Center. They ground infinite limits, thus the reals.
• Dn - Demicubes have neither a Center nor a Totality. Instead of a Center they have a collection of Origins and coordinate systems which define simplexes that fit together to bound a space. We can think of the demicubes as arising by introducing with each dimension a duality mirror, that is, a mirror in which Origins become vertices and vertices become Origins, and the new and old diagrams are joined. I don't yet know but I suppose that the ambiguity of these demicubes could somehow define areas, perhaps as oriented bounded spaces, in which case they would be preserved by symplectic geometry. The duality mirror grounds the duality between points (vertices) and lines (origins).
• Simplex (1+1)^N
• Cross-polytopes (1+2)^N
• Cubes (2+1)^N
• Half-cubes (2+2)^N
• Affine geometry supposes the natural numbers
• Projective geometry supposes the rationals
• Conformal (Euclidean) geometry supposes the reals
• Symplectic geometry supposes the complexes
Lines, Angles, Areas require a Field whereas directions do not. Lines are translations and Angles are rotations. Together they define the Complexes. Are they key to Dn? Study visual complex analysis. A Field allows, for example, proportionality and other transformations - multiplications - consider!
Projective geometry adds points at infinity to affine geometry. Conformal geometry or inversive geometry adds a distinguished circle. Symplectic geometry adds an area product. Moebius strip plays with the distinguished circle changing orientation if you go around.
• Unmarked opposites: cross-polytope. Each dimension independently + or - (all or nothing).
• Cube: all vertices have a genealogy, a combination of +s and -s.
• Half-cube defines + for all, thus defines marked opposites.
Cross-polytope
• A 0-sphere is 2 points, much as generated by the center of a cross-polytope. We get a product of circles. And circles have no boundary. So there is no totality for the cross-polytope.
• Symmetric group action on an octahedron is marked, 1 and -1, the octahedron itself is unmarked.
• Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the cross-polytope. (Or consider Bernstein's polynomials.)
Understanding the demicubes
• Is the fusion of vertices in the demicube related to the fusion of edges of a square to create a torus, or of vertices to create a circle, etc.?
Defining my own demicubes
• Each vertex is plus or minus. Can we think of that as the center being inside or outside of it? As the vertex being either an inner point or an outer point? With the center being inside or outside? Or does the vertex exist or not? (Defining a subsimplex.) Is it filled or not? (As with the filling of a cycle in homology so that it is a "boundary".)
• For the distinguished point, is it necessarily an outer point, so that the center is on the outside?
• In homology, we have edges defining the vertices on either end as positive and negative. How does that work for vertices? What does it mean for a vertex to be positive or negative? And how does that relate to defining the inside or outside of a cycle?
• The ambiguity 2 may arise upon thinking of the axes of the cube, defined by pairs of opposite vertices.
• Or the ambiguity may come from the orientation of any cycle being ambiguous, and defining the inside or outside.
Dual:
• Cubes: Physical world: No God (no Center), just Totality. Descending chains of membership (set theory).
• Cross-polytopes: Spiritual world: God (Center), no Totality. Increasing chains of membership (set theory).
• Simplexes consists of cycles with fillings.
• Cross polytopes are cycles without fillings.
• Cubes are fillings without boundaries.
• Demicubes should be without fillings and without boundaries.
• Think of demihypercubes (coordinates sytems) Dn given by simplexes (like An) but in coordinate system presentation (standard simplexes rather than barycentric). So this requires an extra dimension. But then Dn and An are "dual" to each other in some sense.
• Understand symmetry groups, especially for the polytopes, such as octahedral symmetry. Try to define an infinite family of "coordinate systems", simplexes with distinguished element, for which Dn is the symmetry group. Figure out how to count the subsimplexes and see what is the analogue for Pascal's triangle. Understand octahedron as composed of pairs of vertices.
• Note that the orientation of the simplexes, positive and negative, distinguishes inside and outside. On common edges they go in opposite directions. Also, this seems to relate the coordinates x1, x2, x3 etc. in terms of their canonical order. What does all this mean for cross polytopes?
• Boundaries distinguish inside and outside. So then how does it follow that boundaries don't have boundaries?
Distinct Geometries
Special geometries
• Euclidean geometry: empty space + tools: straightedge, compass, area measurer
• most important theorem: Pythagoras q=q1+q2
• (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
• Ordered geometry features the concept of intermediacy. It is a common foundation for affine, Euclidean, absolute geometry and hyperbolic geometry, but not projective geometry. Like projective geometry, it omits the notion of measurement.
• Absolute geometry, also known as neutral geometry, is based on the axioms of Euclidean geometry (including the first four of Euclid's axioms) but with the parallel postulate removed.
These geometries show how to relate (ever more tightly) two distinct dimensions.
Pairs of Geometries
Each of the four geometries would serve to define what we mean by perspective, but especially, how a view from outside of a system (from a higher dimension) and a view inside of a system (a lower dimension) can be considered one and the same. In general, I am thinking that geometry can be thought of as the ways of embedding one space into another space, that is, a lower dimensional space into a higher dimensional space. I imagine that tensors are important as the trivial, "plain vanilla" version of this.
Think of pairs of geometries as defining equivalence classes variously. Equivalence classes are related to actions of symmetry groups.
6 Specifications
• Think of how transformations act on 0, 1, infinity, for example, translations can take 0 to 1, but infinity to infinity.
• Do the six natural bases of the symmetric functions correspond to the six transformations?
• Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?
• In category theory, where do symmetric functions come up? What are eigenvalues understood as? What would be symmetric functions of eigenvalues?
The 6 specifications between 4 geometries are transformations which make one geometry more specific than another geometry by introducing orientation, angles and areas. This also makes distance more sophisticated, allowing for negative (oriented) numbers, rational (angular) numbers, and real (continuous) numbers.
• Sylvain Poirer's list of permutations which I used.
• Grothendieck's six operations:
• pushforward along a morphism and its left adjoint
• compactly supported pushforward and its right adjoint
• tensor product and its adjoint internal hom
• The various ways that we interpret multiplication in arithmetic.
• Möbius transformation combines translation, inversion, reflection, rotation, homothety. See the classification of Moebius transformations. Note also that the classification of elements of SL2(R) includes elliptic (conjugate to a rotation), parabolic (shear) and hyperbolic (squeeze).
• The six transformations in the anharmonic group of the cross-ratio. If ratio is affine invariant, and cross-ratio is projective invariant, what kinds of ratio are conformal invariant or symplectic invariant?
• The 6 specifications can be compared with cinematographic movements of a camera. But I don't know how to think of shear or squeeze mappings in terms of a camera. However, consider what a camera would do to a tiled floor. Shear? Squeeze: the camera looks out onto the horizon?
• Reflection: a camera in a mirror, a frame within a frame...
• Rotation: a camera swivels from left to right, makes a choice, like turning one's head
• Dilation: a camera zooms for the desired composition.
• Translation: a camera moves around.
• Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.
• Reflection introduces the action of Z2. It is the reflection across the boundary of self and world. (We can later also think of reflection across the horizon around us, as inversion.) This is the parity of multisets (element or not an element). And that circle S02 is then referenced by rotations and shear mapping and all work with angles. And then the relationship between two dimensions is given perhaps by Z2 x S02, the relationship between two axes: x vs. x (dilation), x vs. 1/x (squeeze) and x vs. y (translation).
• Squeeze specification draws a hyperbola (x vs. 1/x). Dilation draws a line (x vs. x). Are there specifications that draw circles (rotation?), ellipses? parabolas?
• Transformacijos sieja nepriklausomus matus.
Reflection
• Flip around our search, turn vector around: (reflection)
Shear
• Shear map takes parallelogram to square, preserves area
• Turn a corner into another dimension
Rotation
• Harmonic analysis, periodic functions, circle are rotation.
• Rotations are multiplicative but not additive. This brings to mind the field with one element.
Dilation
• Dilation (scaling) including negative (flipping). Dilations add absolutely and multiply relatively.
Complex number dilation (rotating).
• Homothety is related to dilation. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic in
• https://en.wikipedia.org/wiki/Homothetic_transformation a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends {\displaystyle M\mapsto S+\lambda {\overrightarrow {SM}},} M\mapsto S+\lambda {\overrightarrow {SM}}, in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ.[1] In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if λ > 0) or reverse (if λ < 0) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line L is a line parallel to L.
• Dilation brings to mind the Cartesian product A x B. There is also the inner (direct) product A + B. How is it related to the disjoint union? And there is the tensor product which I think is like an expansion in terms of A.B and so is like multiplication.
Squeeze
• Squeeze mapping
• Squeeze transformacija trijuose matuose: a b c = 1. Tai simetrinė funkcija.
Translation
• Homotopy is translation.
• Sweep a new dimension in terms of an old dimension (translation)
• Translation - does not affect vectors
Other transformations
• Special conformal is reflection and inversion
• Isometry
• Homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation.
• Affine transformation
• Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).[1] The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection.
How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is speciWed in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a Riemann surface; see §8.2), etc. In all the examples just referred to, vector-space notions are what are needed to tell us what this local geometry is, the vector space in question being the n-dimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6). Penrose, Road to Reality, page 293, §14.1.
1999. I asked God which questions I should think over so as to understand why good will makes way for good heart. He responded:
• What captures attention and guides it? mažėjantis laisvumas
• What drops down upon reality and bounces away in random paths? didėjantis laisvumas
• What is wound in one direction, and lives through spinning in the opposite direction? prasmingas - kodėl
• What falls as rain day and night until there sprout and grow plants that will bear fruit? pastovus - kaip
• What like a ray reflects off of society and does not return? betarpiškas - koks
• What by its turning (in the direction of winding) commands our attention and then slips away to the side? tiesus - ar
Geometry
Naujausi pakeitimai
Puslapis paskutinį kartą pakeistas 2019 sausio 19 d., 11:14 | 2019-03-24 20:04:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7829084396362305, "perplexity": 1125.5828570683298}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203491.1/warc/CC-MAIN-20190324190033-20190324212033-00268.warc.gz"} |
https://thatsmaths.com/tag/number-theory/ | ## Posts Tagged 'Number Theory'
### Fractions of Fractions of Fractions
Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number ${x}$ can be expanded as a continued fraction:
$\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]$
where all ${a_n}$ are integers, all positive except perhaps ${a_0}$. If ${a_n=1}$ we add it to ${a_{n-1}}$; then the expansion is unique.
### It’s as Easy as Pi
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see TM120 or search for “thatsmaths” at irishtimes.com].
The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).
### A Remarkable Pair of Sequences
The terms of the two integer sequences below are equal for all ${n}$ such that ${1, but equality is violated for this enormous value and, intermittently, for larger values of ${n}$.
### A Geometric Sieve for the Prime Numbers
In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying.
### Numerical Coincidences
A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.
Cosine of 355 radians is almost exactly equal to -1. Is this a coincidence? Read on!
### Brun’s Constant and the Pentium Bug
Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:
$\displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty$
Obviously, this could not happen if there were only finitely many primes.
### The Shaky Foundations of Mathematics
The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].
Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]
The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since. | 2017-09-19 19:01:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5875394940376282, "perplexity": 756.6774016507378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818685993.12/warc/CC-MAIN-20170919183419-20170919203419-00200.warc.gz"} |
https://quant.stackexchange.com/tags/copula/hot | # Tag Info
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This may not be a consequence of biased estimators or sampling error. I don't think it is a coincidence that $$\frac{6}{\pi} \arcsin\left(\frac{0.9}{2} \right) = 0.891457\ldots \approx 0.891$$ Copula construction involves applying nonlinear transformations to random variables which need not preserve correlation. If random variables $X$ and $Y$ are ...
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No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues. This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention: an auto-callable ...
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The best introduction to copulas I know, i.e. with rigour and intuition, is the following. THE QUANT CLASSROOM BY ATTILIO MEUCCI A Short, Comprehensive, Practical Guide to Copulas Visually introducing a powerful risk management tool to generalize and stress-test correlations
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In general you don't need copulas to calculate VaR on portfolio. You can use historical method if you have time series of returns for the assets in your portfolio. If you have sufficiently enough data this will allow you to take into account correlation risk, non-normality of returns. Example of code in R for equally weighted portfolio without assuming any ...
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The algorithm is certainly useful in that it is non-parametric, fast, and versatile. Meucci summarizes the advantages nicely: Unlike traditional copula techniques, CMA a) is not restricted to few parametric copulas such as elliptical or Archimedean; b) never requires the explicit computation of marginal cdf’s or quantile functions; c) does not ...
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I found Coping With Copulas by Thorsten Schmidt really helped me to get a more basic understanding of copulas. As well as looking at some simple examples in R and thinking about different directions the transformations can happen. To answer your actual question I'll attempt to describe the steps involved as simply as I can. Let's say you use the copula ...
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In the theory of copulas you want to model a multivariate (often bivariate) distribution and keep the marginals fixed. Thus you have random variables $X$ and $Y$ with cdf $F_X(x) = P[X \le x]$ and $F_Y(y) = P[Y\le y]$ and you want to find some $F_{X,Y}(x,y) = P[X \le x, Y\le y]$ such that when you look at marginals you get $F_{X,Y}(x,\infty) = F_X(x)$ and ...
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Here is a working paper that you may be interested in.
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In general setting this is quite a tough problem and it looks like just switching from regular multivariate probability to copulas doesn't make it easier. In general case you need to rely on numerical methods for integration. There is a nice overview of the problem in Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 ...
5
If the density of $(X,Y)$ is known, then you may obtain the density of the sum $X+Y$ simply by applying the Jacobi's transformation formula, which describes the density of the transformed random variable $g(X,Y)$ for $g(x,y) = (x+y, x)$. Integrating out the $x$-component yields the density of $X+Y$. See Jacod/Protter Probability Essentials ch. 12 for details....
4
You don't really have a multivariate case: we can only define VaR (in its usual sense) for a one-dimensional output. Recall that $$\operatorname{VaR}_\alpha(X) = \inf\{v:F_X(v)\geq \alpha\}$$ and since in your case $X = X_1+X_2$ you just need to compute $F_X$ in terms of $X_1$ and $X_2$. For the notation of partial derivatives, I denote the generic ...
4
Look here for multivariate distribution on the positive quadrant ... quite difficult. http://xianblog.wordpress.com/tag/multivariate-analysis/ I have been thinking about this for weeks and months in the context of credit risk (modelling default intensities jointly) and I think this does not work.
4
I would guess you are calculating the maximum likelihood estimator: $\hat{\theta} = \frac{1}{N} \sum (x_i - \bar{x}) (y_i - \bar{y})$ instead of the unbiased estimator: $\hat{\theta} = \frac{1}{N-1} \sum (x_i - \bar{x}) (y_i - \bar{y})$ The unbiased estimator has a bias of zero, i.e. : $E_{x|\theta}[\hat{\theta}] - \theta = 0$ The unbiased ...
3
$$C(u,v) = \mathbb{P}\left(X\leq N^{(-1)}(u),\quad \rho X + \sqrt{1-\rho^2}X^\perp \leq N^{(-1)}(v)\right)$$
3
It depends on the assets which copula is best and other methods may still be better and comparable in complexity. If you want to use copula's for equities you can take a look at Clayton copula. While the Gaussian copula is symmetric the Clayton copula has asymmetric tail dependency. This makes modeling the increase in correlation during a crisis possible.
3
Do you refer with 'negative tail dependence' to the case that one variable has a extremely low value and the other random variable has an extremely large value, i.e., $$\tau=\lim_{p \rightarrow 0} \frac{Pr[x>Q_x(1-p),y<Q_y(p)]}{p},$$ where $Q_x(1-p)$ and $Q_y(p)$ refer to the $(1-p)$-th quantile of the random variable $x$ and the $p$-th quantile of ...
3
As you know, simulating AR(1) is to simulate the distributed error path. Assume the bivariate errors distributed $\sim F(x),\sim F(y)$ with copula $C(u,v)$ to model their dependence. Then the bivariate joint error distribution is given by Sklar's theorem: $$F(x,y)=C(F(x),F(y))$$ You can simulate from this distribution using Conditional Sampling: To ...
3
Is'nt it true that your first line can be written as $$F_{X,Y}(x,y_2) - F_{X,Y}(x,y_1),$$ where $F_{X,Y}$ is the joint cdf of $(X,Y)$. If we assume that the distributions of $X$ and $Y$ are continuous without atoms (I have to check the exact formulation), then it is clear from Sklar's theorem that there is exactly one copula $C$ such that F_{X,Y}(x,y) = ... 3 Note that, you only need to show that \begin{align*} A\left(\frac{\log(u_2)}{\log(u_1u_2)}\right)-\frac{\log(u_2)}{\log(u_1u_2)}A'\left(\frac{\log(u_2)}{\log(u_1u_2)}\right) \ge 0, \end{align*} or, for any t \in (0, 1), \begin{align*} A(t) - t A'(t) \ge 0. \end{align*} Recall that A is a convex function from [0,\, 1] to [1/2,\, 1], A(0)=A(1)=1, and ... 3 This is an interesting observation that you have. The interesting part is "consistently smaller". The normal copula is based on a multivariate normal distribution. The correlation you get out is the correlation parameter you put in. Everything else is most probably due to an issue in your approach. If you did not say "consistently smaller", I would say it ... 2 if you agree that the marginal probability P(u\le Y\le v)=F_Y(v)-F_Y(u), then your formula follows immediately, because next you simply plug the marginals into the copula. your 3rd equation for the joint probabilities is incorrect for P(Z\le z,u\le Y\le v), I'm not sure where you got it from 2 Implementations of the BBx families are available from the VineCopula R-package from CRAN. Spatially and spatio-temporally varying bivariate copulas are provided through the R-package spcopula from r-forge. Temporal support will need some additional work as it was not part of the initial design. The tuning of the copulas' parameter can be done via a ... 2 You can express the Normal distribution by Sklar's Theorem in terms of Gaussian Marginals and Gaussian Copula as follows:F(x_1,...,x_n)=C(F(x_1),...,F(x_n))=C^{Gau}(N(x_1),...,N(x_n))$$So the distribution equals the copula function with the respective inverse marginals as arguments. You can aswell combine any types of Copula and (continuous) different ... 2 There is a brief and not overly technical introduction here: http://prescientmuse.blogspot.co.uk/2015/01/a-brief-introduction-to-copula.html And an application of use in a trading system with full R code here: http://prescientmuse.blogspot.co.uk/2015/02/vanilla-trading-algorithm.html Hope that helps! 2 For non-normal asset price models you could look at the theory of Lévy-processes. If we assume that you work in the physical probability measure P and that the random numbers that you have generated are daily log-returns, then you can do the following: Asset i has starting price S_0^i and for the future prices you can put$$ S_t^i = S_0^i \exp(\sum_{k=...
2
$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$ $$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$$
2
"convoluted expression" in American usage just means a complicated, big mathematical expression, sometimes also called "hairy" or "messy". It is ugly to work with and to look at, so you prefer not to deal with it if possible. Nothing more than that. There is also a mathematical operation called "convolution ("Faltung" in German) but it has nothing to do ...
2
Suppose you have the copula $C(u_1,u_2)$, then you could compute the conditional copula $$c_{u_1}(u_2)=\frac{\partial C(u_1,u_2)}{\partial u_1} \; .$$ Now, you can generate a pair of independent uniformly distributed random values $(U,V)$. Let's say a particular realistation is $(u,v)$. Then the pair $$(u,c_u^{-1}(v))$$ will be distributed according to ...
2
It's very difficult to find accessible material on copulas. I'm still struggling to understand them myself. While I haven't come across any videos that explain copulas well, I have found the following resources very helpful. A blog post: An intuitive, visual guide to copulas, by Thomas Wiecki, is truly introductory with absolute minimal math but it doesn't ...
2
Your reasoning for the first property does not look correct or at least I do not understand it. Your arguments for the second property seem sound. But your wording of the second property is a bit fuzzy. You should state this more clearly, for example: $C(1,\ldots,1,u_j,1,\ldots,1) = u_j$ for all $u_j\in [0,1]$ and $j\in 1,\ldots, d.$ You don't mention it ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2019-06-18 19:33:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.840969443321228, "perplexity": 510.8245652598416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998813.71/warc/CC-MAIN-20190618183446-20190618205446-00110.warc.gz"} |
https://www.gamedev.net/forums/topic/687991-alpha-value-in-mtl-file-format/ | # Alpha value in MTL file format
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I want to use MTL files for my materials. Here, ambient, diffuse, specular colors are defined as RGB or XYZ (or custom spectrum values which I just ignore). In the MTL file format, the dissolve factor d is defined as:
d factor
Specifies the dissolve for the current material.
"factor" is the amount this material dissolves into the background. A
factor of 1.0 is fully opaque. This is the default when a new material
is created. A factor of 0.0 is fully dissolved (completely
transparent).
I assume that this is the "alpha" channel of all the RGB/XYZ values? Does one normally pass ambient, diffuse, specular colors as float3 with one separate float for the alpha channel to the shaders or does one use float4s (some redundancy)?
Edited by matt77hias
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One doesn't normally pass ambient, diffuse, and specular anymore. That is an antiquated setup for replicating the fixed-function pipeline.
The good news is that you can do it however you see fit, since the only common way to do it is not to do it.
I would assume an alpha channel on specular and ambient would be multipliers of the alpha on diffuse, so if you have Diffuse.A = 0.5 and Specular.A = 0.5, final alpha = 0.25. Of course you can do this math offline rather than in the shader and then you are still stuck with 2 floats you are not using.
vec3's are still aligned on 16-byte boundaries, so if you pass a vec3 for each of them you gain nothing. I would pass vec4's and either use the alpha channels or not. Tweak them in your 3D modeling software and see what results from it, then decide how to use them based off that.
L. Spiro
Edited by L. Spiro
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I would assume an alpha channel on specular and ambient would be multipliers of the alpha on diffuse, so if you have Diffuse.A = 0.5 and Specular.A = 0.5, final alpha = 0.25. Of course you can do this math offline rather than in the shader and then you are still stuck with 2 floats you are not using.
So only the diffuse color should be extended with the dissolve as alpha channel, since these contributions will be added in the end?
float4 Ka = Ka_mtl 0
float4 Kd = Kd_mtl d
float4 Ks = Ks_mtl 0
My Ka, Kd and Ks are the multiplies for their associated textures map_Ka, map_Kd, map_Ks. (I have a ray tracing background so maybe I use a far too many (mip mapped) textures for each material, but FPS/MEM is still ok.) One doesn't normally pass ambient, diffuse, and specular anymore.
Edited by matt77hias
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Register a new account | 2018-01-18 14:30:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37650492787361145, "perplexity": 3543.3213840681397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887414.4/warc/CC-MAIN-20180118131245-20180118151245-00084.warc.gz"} |
https://stats.stackexchange.com/questions/501449/nagelkerke-pseudo-r-squared-for-proportional-odds-model-and-the-effect-size | # Nagelkerke Pseudo-R-squared for proportional odds model and the effect size
I have computed the Nagelkerke Pseudo-R-Squared for the proportional odds model with two variables. I report ORs with p-values and confidence intervals and would like to quantify the strength of the association too. The idea is to calculate the effect size according Cohen (1992) https://journals.sagepub.com/doi/10.1111/1467-8721.ep10768783 He suggests to calculate the effect size for multiple and multiple partial correlation (for linear models) with the formula $$f^2=\frac{R^2}{1-R^2}$$ My question is if this method of determination of the effect size would be also make sense for the Nagelkerke pseudo-R-squared? (It is also scaled as the R-squared for linear regression models from 0 to 1). | 2021-08-02 12:48:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8053264021873474, "perplexity": 738.4134600153809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154320.56/warc/CC-MAIN-20210802110046-20210802140046-00062.warc.gz"} |
http://www.hashprompt.com/mexicali-blues-dxv/zkfye2.php?tag=6cea51-why-was-calculus-introduced-in-economics | For Newton, variable magnitudes are not aggregates of infinitesimal elements, but are generated by the indisputable fact of motion. Γ Where Newton over the course of his career used several approaches in addition to an approach using infinitesimals, Leibniz made this the cornerstone of his notation and calculus. Whether it is Micro economics, Production Systems, Economics growth, Macro economics, it is hard to explain as well as understand the theory without the use of mathematics. {\displaystyle {\frac {dF}{dx}}\ =\ {\frac {1}{x}}.}. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Although Adam Smith's (considered the Father of Economics) famous work - ‘The Wealth of Nations’ published in 1776 has almost no mathematics in it. Usually, you would want to choose the quantity that helps you maximize profits. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. To Lagrange (1773) we owe the introduction of the theory of the potential into dynamics, although the name "potential function" and the fundamental memoir of the subject are due to Green (1827, printed in 1828). The subject was properly the invention of two…, Monge’s educational ideas were opposed by Joseph-Louis Lagrange, who favoured a more traditional and theoretical diet of advanced calculus and rational mechanics (the application of the calculus to the study of the motion of solids and liquids). and To this day the Calculus is widely read and cited, and there is still much to be gained from reading and rereading this book. log F When examining a function used in a mathematical model, such geometric notions have physical interpretations that allow a scientist or engineer to quickly gain a feeling for the behaviour of a physical system. ∫ Depending on the context, derivatives may be interpreted as slopes of tangent lines, velocities of moving particles, or other quantities, and therein lies the great power of the differential calculus. ( This means that undergraduates thinking about graduate school in economics should take 1-2 mathematics courses each semester. A video from njc314 about using derivatives to solve Economic … Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential equations can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. It was during his plague-induced isolation that the first written conception of fluxionary calculus was recorded in the unpublished De Analysi per Aequationes Numero Terminorum Infinitas. Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. Γ As h approaches 0, this formula approaches gt, which is interpreted as the instantaneous velocity of a falling body at time t. This expression for motion is identical to that obtained for the slope of the tangent to the parabola f(t) = y = gt2/2 at the point t. In this geometric context, the expression gt + gh/2 (or its equivalent [f(t + h) − f(t)]/h) denotes the slope of a secant line connecting the point (t, f(t)) to the nearby point (t + h, f(t + h)) (see figure). As with many of his works, Newton delayed publication. Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows: although these were not the exact forms of Euler's study. It helps us to understand the changes between the values which are related by a function. Algebra is used to make computations such as total cost and total revenue. f This discipline has a unique legacy over the history of mathematics. d I was first introduced to Austrian economics during my senioryear in high school, when I first read and enjoyed the writingsof Mises and Rothbard. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. Methodus Fluxionum was not published until 1736.[27]. ∫ 1.1. Having a good understanding of mathematics is crucial to success in economics. [5] It should not be thought that infinitesimals were put on a rigorous footing during this time, however. In this book, Newton's strict empiricism shaped and defined his fluxional calculus. To this discrimination Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Denis Poisson (1831), Mikhail Vasilievich Ostrogradsky (1834), and Carl Gustav Jakob Jacobi (1837) have been among the contributors. [17][18] The first full proof of the fundamental theorem of calculus was given by Isaac Barrow. Infinitesimals to Leibniz were ideal quantities of a different type from appreciable numbers. [9], The mathematical study of continuity was revived in the 14th century by the Oxford Calculators and French collaborators such as Nicole Oresme. Independently, Newton and Leibniz established simple rules for finding the formula for the slope of the tangent to a curve at any point on it, given only a formula for the curve. So F was first known as the hyperbolic logarithm. By the end of the 17th century, each scholar claimed that the other had stolen his work, and the Leibniz–Newton calculus controversy continued until the death of Leibniz in 1716. Ancient Greek geometers investigated finding tangents to curves, the centre of gravity of plane and solid figures, and the volumes of objects formed by revolving various curves about a fixed axis. Three hundred years after Leibniz's work, Abraham Robinson showed that using infinitesimal quantities in calculus could be given a solid foundation.[35]. In effect, the fundamental theorem of calculus was built into his calculations. x With the technical preliminaries out of the way, the two fundamental aspects of calculus may be examined: Get exclusive access to content from our 1768 First Edition with your subscription. ˙ Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Kerala school of astronomy and mathematics, De Analysi per Aequationes Numero Terminorum Infinitas, Methodus Fluxionum et Serierum Infinitarum, "Signs of Modern Astronomy Seen in Ancient Babylon", "Fermat's Treatise On Quadrature: A New Reading", Review of J.M. n ) Finding the formula of the derivative function is called differentiation, and the rules for doing so form the basis of differential calculus. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Charles James Hargreave (1848) applied these methods in his memoir on differential equations, and George Boole freely employed them. He was a polymath, and his intellectual interests and achievements involved metaphysics, law, economics, politics, logic, and mathematics. Before Newton and Leibniz, the word “calculus” referred to any body of mathematics, but in the following years, "calculus" became a popular term for a field of mathematics based upon their insights. Motion under constant gravity I think is a counterexample to the necessity of calculus to solve concrete problems, and only reinforces the OP's question rather than answering it. For example, the Greek geometer Archimedes (287–212/211 bce) discovered as an isolated result that the area of a segment of a parabola is equal to a certain triangle. He then reasoned that the infinitesimal increase in the abscissa will create a new formula where x = x + o (importantly, o is the letter, not the digit 0). Calculus is used to find the derivatives of utility curves, profit maximization curves and growth models. For Leibniz the principle of continuity and thus the validity of his calculus was assured. , both of which are still in use. They proved the "Merton mean speed theorem": that a uniformly accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body. Calculus is now the basic entry point for anyone wishing to study physics, chemistry, biology, economics, finance, or actuarial science. , d 1. so that a geometric sequence became, under F, an arithmetic sequence. ( Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis of infinite series. x All through the 18th century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. y For example, if 1 One of the initial applications areas is the study of a firm, a He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. In the Methodus Fluxionum he defined the rate of generated change as a fluxion, which he represented by a dotted letter, and the quantity generated he defined as a fluent. The initial accusations were made by students and supporters of the two great scientists at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism. Understand, apply, and analyze calculus-based economic models Translate economic principles to the investigation of a wide range of real world problems Elaborate on an in-depth understanding of basic economics and its applications Expand what you'll learn Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Γ Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. It all depends on your situation. Calculus makes it possible to solve problems as diverse as tracking the position of a space shuttle or predicting the pressure building up behind a dam as the water rises. ( Honors Contract Project The Role of Calculus in Accounting and Finance Created by: Maria Paneque MAC2311 Prof. Gonzalez Applications in the Finance Field Applications in the Finance Field Calculation of Income Stream Calculation and prediction of future total sales Applications "Ideas of Calculus in Islam and India.". He viewed calculus as the scientific description of the generation of motion and magnitudes. This is called the (indefinite) integral of the function y = x2, and it is written as ∫x2dx. [1] Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter.[2][3]. The roots of calculus lie in some of the oldest geometry problems on record. [7] In the 5th century, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Calculus in Economics Economists use calculus to predict supply, demand, and maximum potential profits. Calculus use to determine the right time for buying and selling of products, how many people buy it, the margin value of a particular product and other requires instance changes. An illustration of the difference between average and instantaneous rates of changeThe graph of, With the technical preliminaries out of the way, the two fundamental aspects of calculus may be examined:…, The historian Carl Boyer called the calculus “the most effective instrument for scientific investigation that mathematics has ever produced.” As the mathematics of variability and change, the calculus was the characteristic product of the scientific revolution. {\displaystyle \Gamma (x)} Γ [19]:p.61 when arc ME ~ arc NH at point of tangency F fig.26[20], One prerequisite to the establishment of a calculus of functions of a real variable involved finding an antiderivative for the rational function x Specific importance will be put on the justification and descriptive terms which they used in an attempt to understand calculus as they themselves conceived it. Calculus-based economics is a quantitive version of economics that uses the more advanced mathematical topics included in calculus. 1 ", In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 the Methodus Fluxionum et Serierum Infinitarum. Newton developed his fluxional calculus in an attempt to evade the informal use of infinitesimals in his calculations. ( It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curve t Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinates dx and dy, and the summation of infinitely many infinitesimally thin rectangles as a long s (∫ ), which became the present integral symbol Importantly, the core of their insight was the formalization of the inverse properties between the integral and the differential of a function. The earliest economists were philosophers who used deduction and logic to explain the market. {\displaystyle \int } Actually when I was in highschool this problem was solved without resorting to calculus because we hadn't learned it yet. The rate of change of a function f (denoted by f′) is known as its derivative. You will get lots of exposure to simple calculus problems. In order to understand Leibniz’s reasoning in calculus his background should be kept in mind. In economics and business there are some uses for calculus. Calculus, a branch of Mathematics, developed by Newton and Leibniz, deals with the study of the rate of change. . Frullani integrals, David Bierens de Haan's work on the theory and his elaborate tables, Lejeune Dirichlet's lectures embodied in Meyer's treatise, and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf and Kronecker are among the noteworthy contributions. Author of. A. de Sarasa associated this feature with contemporary algorithms called logarithms that economized arithmetic by rendering multiplications into additions. With its development are connected the names of Lejeune Dirichlet, Riemann, von Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami, and many of the leading physicists of the century. From these definitions the inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. . Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. {\displaystyle \Gamma } His course on the theory may be asserted to be the first to place calculus on a firm and rigorous foundation. "[29], In 1672, Leibniz met the mathematician Huygens who convinced Leibniz to dedicate significant time to the study of mathematics. [32], While Leibniz's notation is used by modern mathematics, his logical base was different from our current one. By 1664 Newton had made his first important contribution by advancing the binomial theorem, which he had extended to include fractional and negative exponents. t x This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created. Most economics Ph.D. programs expect applicants to have had advanced calculus, differential equations, linear algebra, and basic probability theory. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. but the integral converges for all positive real In the limit, with smaller and smaller intervals h, the secant line approaches the tangent line and its slope at the point t. Thus, the difference quotient can be interpreted as instantaneous velocity or as the slope of a tangent to a curve. Marginal analysis in Economics and Commerce is the most direct application of differential calculus. The first proof of Rolle's theorem was given by Michel Rolle in 1691 using methods developed by the Dutch mathematician Johann van Waveren Hudde. They could see patterns of results, and so conjecture new results, that the older geometric language had obscured. His aptitude was recognized early and he quickly learned the current theories. Newton completed no definitive publication formalizing his fluxional calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the Principia and Opticks. In this paper, Newton determined the area under a curve by first calculating a momentary rate of change and then extrapolating the total area. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. [11], Some ideas on calculus later appeared in Indian mathematics, at the Kerala school of astronomy and mathematics. for the integral and wrote the derivative of a function y of the variable x as for the derivative of a function f.[36] Leibniz introduced the symbol . This subject constitutes a major part of modern mathematics education. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. Our editors will review what you’ve submitted and determine whether to revise the article. Calculus Math is generally used in Mathematical models to obtain optimal solutions. Like Newton, Leibniz saw the tangent as a ratio but declared it as simply the ratio between ordinates and abscissas. Introduction: Brief History Of Calculus. In physical terms, solving this equation can be interpreted as finding the distance F(t) traveled by an object whose velocity has a given expression f(t). The branch of the calculus concerned with calculating integrals is the integral calculus, and among its many applications are finding work done by physical systems and calculating pressure behind a dam at a given depth. The name "potential" is due to Gauss (1840), and the distinction between potential and potential function to Clausius. Newton would begin his mathematical training as the chosen heir of Isaac Barrow in Cambridge. , and it is now called the gamma function. {\displaystyle f(x)\ =\ {\frac {1}{x}}.} [12] However, they were not able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today. He began by reasoning about an indefinitely small triangle whose area is a function of x and y. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. Calculus is at the backbone of economics because it provides an analytically efficient way to understand the intricacies of decision-making and optimal choices. ) This revised calculus of ratios continued to be developed and was maturely stated in the 1676 text De Quadratura Curvarum where Newton came to define the present day derivative as the ultimate ratio of change, which he defined as the ratio between evanescent increments (the ratio of fluxions) purely at the moment in question. One of the foremost branches of mathematics is calculus. [9] Madhava of Sangamagrama in the 14th century, and later mathematicians of the Kerala school, stated components of calculus such as the Taylor series and infinite series approximations. p.61 when arc ME ~ arc NH at point of tangency F fig.26, Katz, V. J. x Many applicants have completed a course in real analysis. The types of math used in economics are primarily algebra, calculus and statistics. CALCULUS: THE CALCULUS OF OPTIMIZATION 15 Economists in the late 1900s thought that utility might actually be real, some-thing that could be measured using “hedonometers” or “psychogalvanometers”. If you're a seamstress, calculus probably won't be that important to you. log To it Legendre assigned the symbol Two mathematicians, Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany, share credit for having independently developed the calculus in the 17th century. Newton and Leibniz discovered that integrating f(x) is equivalent to solving a differential equation—i.e., finding a function F(t) so that F′(t) = f(t). Although the use of calculus can be found in the work of ancient Egyptians (1800 BC) and Greeks (400 BC), but the modern calculus was introduced … The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter. Torricelli extended this work to other curves such as the cycloid, and then the formula was generalized to fractional and negative powers by Wallis in 1656. [21] The mean value theorem in its modern form was stated by Bernard Bolzano and Augustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Importantly, Newton and Leibniz did not create the same calculus and they did not conceive of modern calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory of infinitesimal calculus in the later 17th century. Descartes’s method, in combination with an ancient idea of curves being generated by a moving point, allowed mathematicians such as Newton to describe motion algebraically. By the mi… He used math as a methodological tool to explain the physical world. Articles from Britannica Encyclopedias for elementary and high school students. ˙ He admits that "errors are not to be disregarded in mathematics, no matter how small" and that what he had achieved was “shortly explained rather than accurately demonstrated. Particularly, his metaphysics which described the universe as a Monadology, and his plans of creating a precise formal logic whereby, "a general method in which all truths of the reason would be reduced to a kind of calculation. While they were both involved in the process of creating a mathematical system to deal with variable quantities their elementary base was different. Let us know if you have suggestions to improve this article (requires login). While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve. [16] Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. are fluents, then Leibniz introduced this because he thought of integration as finding the area under a curve by a summation of the areas of infinitely many infinitesimally thin rectangles between the x-axis and the curve. Furthermore, infinitesimal calculus was introduced into the social sciences, starting with Neoclassical economics. From here it was not difficult for them to guess that the general formula for the area under a curve y = xn is an + 1/(n + 1). So in a calculus context, or you can say in an economics context, if you can model your cost as a function of quantity, the derivative of that is the marginal cost. 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