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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
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The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
I think
I can safely say that nobody understands quantum mechanics. ~ Richard Feynman. Hilbert space
Hilbert space is generalization of the Euclidean space. In a Hilbert space we can have infinite number of dimensions.
A vector in an infinite-dimensional Hilbert space is represented as an infinite vector: \((x_1, x_2, ... |
By generalizing the approach in Integral involving a dilogarithm versus an Euler sum. meaning by using the integral representation of the harmonic numbers and by computing a three dimensional integral over a unit cube analytically we have found the generating function of cubes of harmonic numbers. We have: \begin{eqnar... |
$(p,q)$th order oriented growth measurement of composite $p$-adic entire functions Abstract
Let us consider $\mathbb{K}$ be a complete ultrametric algebraically closed field and suppose $\mathcal{A}\left( \mathbb{K}\right) $ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire functi... |
Suppose that an object is moving along the x-axis with an initial velocity \(v_i\) (which could be \(0m/s\) but, in general, can be any arbitrary value) and is displaced by an amount \(Δx\). Let's suppose that this object is acted upon by any arbitrary force \(\vec{F}\) as it is being displaced as illustrated in Figure... |
In classical mechanics, the condition to fix the variation of the trajectory at the endpoints has a clear-cut meaning. We want the system to propagate from $x\in\mathcal{C}$ to $y\in\mathcal{C}$, therefore, we only consider curves that have these two points as endpoints, so the variations also have to respect this. It ... |
Your question is,
If the average of the first $n$ terms of a sequence tends to a limit, does the sequence itself tend to a limit?
The answer is no in general, as is discussed in the comments. The simplest counterexamples are the sequences which oscillate between two different values $\alpha$ and $\beta$; we would expec... |
Maximum Principle:Let $\Omega$ be a boundedconnected region of $\mathbb{R}^m$ with $u$ defined and continuous in $\bar{\Omega}, \, \Delta u = 0$. Then $u$ achieves its maximum(and minimum) on $\partial \Omega$. Proof:
$$ \Delta u = 0, \, \Omega $$ $$ \exists \, x^0: \max_{x \in \bar{\Omega}}u(x)=u(x^0)=M < +\infty $$
I... |
I think the Op's proof is correct assuming both functions are entire. However, even if $F(x)$ is entire, the fractional iterate is in general not entire. The Op's result does not hold if the fractional iterate is not entire
$$F^{o\frac{1}{n}}(x)\;\;\;\;h(x)=F^{o \frac{1}{2}}(x)\;\;\;\;h(h(x))=F(x)$$
If the half iterate... |
On page 75 in Sutskever's thesis http://www.cs.utoronto.ca/~ilya/pubs/ilya_sutskever_phd_thesis.pdf
In equation (7.5) setting $a_0=1$,
$a_{t+1} = (1+\sqrt{4 a_t^2 + 1})/2 $
The author said, "to understand the sequence $a_t$ we note that the function $x \rightarrow (1+ \sqrt{4x^2+1})/2$ quickly approaches $x \rightarrow... |
I am new to integral equations. In this field, people study the Fredholm equation
$$\phi(x) + \int_0^1 K(x, y) \phi(y) dy = f(x). $$
I am a bit surprised to see the first term on the left hand side. In linear algebra, we have the equation
$$ \sum_j A_{ij} x_j = b_i . $$
Here $A$ is the counterpart of $K$, and $b$ the c... |
IIs it valid to apply Einstein's Relativity to scenarios involving expansion of space? For a practical example of this: Is it legitimate to speak of distant red-shift galaxies as experiencing time more slowly in relation to our experience of time? I appreciate that isn't sensible in other ways, but by explaining if it ... |
it is a question Convergence/Divergence of calculus II! Please give me a hand!
Determine convergence or divergence using any method covered so far.
$$\sum_{n = 1}^{\infty} \sin (1/n)$$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It ... |
The Ideal Gas in a Field: Transmembrane Ionic Gradients A Half-Step Beyond Ideal: Ion Gradients and Transmembrane Potentials
One key way the cell stores free energy is by having different concentrations of molecules in different "compartments" - e.g., extra-cellular vs. intracellular or in an organelle compared to cyto... |
I have a real $ 4\times4$ matrix of the form $$ C = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ c_{31} & c_{32} & 0 & c_{34} \\ c_{41} & c_{42} & c_{43} & c_{44} \end{pmatrix} $$ The coefficients satisfy \begin{align*} c_{31}, c_{32}, c_{41} &\ge 0,\\ c_{42} &> 0,\\ c_{34}, c_{43}, c_{44} &\le 0, \end{align*} and... |
A matrix $Q$ is orthogonal if and only if its columns forms a orthonormal basis, if and only if $Q^{-1}=Q^T$.
Therefore, if there exists an orthornomal basis of eigenvectors of $A$, we havethat the matrix of change of basis if ortogonal. That is to say, there is $Q$ orthogonal so that
$Q^{-1}AQ=\Lambda$
But then $A=Q^{... |
Suppose the theory of chiral Weyl fermion (say, left) $\psi_{L}$, which interacts with abelian gauge field. This theory contains the
gauge anomaly, which I write in the form $$ \tag 1 \frac{dQ_{L}}{dt} = \text{A}, $$ where $Q_{L}$ is the left charge and $A$ is anomaly function.
The same thing is true about right fermio... |
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Last time we studied meets and joins of partitions. We observed an interesting difference between the two.
Suppose we have partitions \(P\) and \(Q\) of a set \(X\). To figure out if two elements \(x , x' \in X\) are in the same par... |
Increasing the amount of installed renewable energy sources such as solar and wind is an essential step towards the decarbonization of the energy sector.
From a technical point of view, however, the stochastic nature of distributed energy resources (DER) causes operational challenges. Among them, unbalance between prod... |
Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $\text{Bool}$ of boolean algebras and the category $\text{Stone}$ of stone spaces. For those who are not familiar with this, here is a brief st... |
I learnt Ardens theorem and its usage as follows:
Ardens Theorem Let $P$ and $Q$ be two regular expressions over alphabet $Σ$. If $P$ does not contain null string, then $R = Q + RP$ has a unique solution that is $R = QP^*$
Using Ardens theorem (source):
Using Arden's theorem to find regular expression 1. For getting th... |
Counterexample: $[0,1]\times[0,1]$ with induced subspace topology from $\mathbb{R}^2$ is compact. The open cover $\mathscr{U}$ is just the two circular sectors. When we look at the up-left corner and down-right corner, it fails - there is no such $\delta$, such that let the open ball be only in one of circular sectors.... |
This is the sine-Gordon action:$$\frac{1}{4\pi} \int_{ \mathcal{M}^2} dt \; dx \; k \partial_t \Phi \partial_x \Phi - v \partial_x \Phi \partial_x \Phi + g \cos(\ell_{}^{} \cdot\Phi_{}) $$Here $\mathcal{M}^2$ is a 1+1 dimensional spacetime, where 1D space is a $S^1$ circle of length $L$.
At $g=0$ : it is a chiral boson... |
RD Sharma Solutions Class 9 Chapter 10 Ex 10.5
(1) ABC is a triangle and D is the mid-point of BC. The perpendiculars from B to AB and AC are equal. Prove that the triangle is isosceles.
Sol:
Given that, in two right triangles one side and acute angle of one are equal to the corresponding side and angle of the other
We... |
Let $k$ be a field and $M$ a finitely generated, graded module over the graded ring $S=k[x_1,\dots,x_n]$. Let $\cdots \rightarrow F_j \rightarrow F_{j-1} \rightarrow \cdots F_1 \rightarrow F_0 \rightarrow M \rightarrow 0$ be the minimal, free, graded resolution of $M$. Define $b_j$ to be the maximum among the degrees t... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
Peter Saveliev
Hello! My name is Peter Saveliev. I am a professor of mathematics at Marshall University, Huntington WV, USA.
My current projects are these two books:
In part, the latter book is about
Discrete Calculus, which is based on a simple idea:$$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ discrete }\\ \te... |
Now let's think about emergent conservation laws!
When a heavy rock connected to a lighter one by a pulley falls down and pulls up the lighter one, you're seeing an emergent conservation law:
Here the height of the heavy rock plus the height of light one is a constant. That's a conservation law! It forces some of the p... |
To begin our analysis of rotational mechanics one should first remark that the basic equations of translational kinematics (forming the “starting point” for a description of translational mechanics) cannot be used to describe the motion of a rotating body. This is because in our analysis of translational kinematics we ... |
You can see the correct estimate
$a_n\leq n^2/(2n^2+1)$ since you have a sum of $n$ things less than or equal to $n/(2n^2+1)$.
$n^2/(2n^2+n) \leq a_n$ since you have a sum of $n$ things greater than or equal to $n/(2n^2+n)$.
In general, you try to reduce the complexity of the expression for the $n$th term of a sequence... |
I'm learning about functions of random variables and am trying to work out an example I made up. If $y = \sin(x)$ and $x$ has domain $[0, 4\pi]$, is the following the correct expression for the pdf of $y$: $$\begin{align*} f_Y(y) &= \frac{d}{dy}F_Y(y)\\ &= \frac{d}{dy}[F_X(2\pi) - F_X(\pi) + F_X(4\pi) - F_X(3\pi)]\\ &=... |
I had acquired a ballscrew assembly from one of the loading docks, and was really excited about using it as the main actuator for this desk. (This is the same ballscrew from Kris's first seek&geek) Even though there was no obvious part number or datasheet, I could estimate the stiffness by looking at similar ballscrews... |
I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,T_e)$ where $T_p$ is the time of payment and $T_e$ is the time where the interest of the forward rate ends.
So I have t... |
Would it be possible to construct a reflector such that, for a given wavelength (perhaps part of the microwave spectrum?), the reflected wave interferes constructively with itself?
Ideally, this would work for any two arbitrary reflection points that are "close" to one another.
My question has two very interrelated ide... |
№ 8
All Issues Volume 60, № 6, 2008 On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations
Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 723–736
In this work, the first boundary-value problem is considered for second-order degenerate elliptic-parabolic equa... |
Niels Bohr introduced the atomic Hydrogen model in 1913. He described it as a positively charged nucleus, comprised of protons and neutrons, surrounded by a negatively charged electron cloud. In the model, electrons orbit the nucleus in atomic shells. The atom is held together by electrostatic forces between the positi... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Search
Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Warning
Only the calculation of the density is tested for open shell configurations (and relies on a correct .OCCUPATION). All other properties are only tested for closed shell systems and should not be trusted for open shell systems without a thorough testing.
**VISUAL¶ Sampling¶ .LIST¶
Calculate various densities in ... |
Codeforces Round #548 (Div. 2) Finished
Vivek initially has an empty array $$$a$$$ and some integer constant $$$m$$$.
He performs the following algorithm:
Find the expected length of $$$a$$$. It can be shown that it can be represented as $$$\frac{P}{Q}$$$ where $$$P$$$ and $$$Q$$$ are coprime integers and $$$Q\neq 0 \p... |
Consider the following decision problem:
Given: Two (3CNF-)formulas $\varphi_1$, $\varphi_2$ on a shared set $X\cup Y$ of variables ($X$ and $Y$ disjoint).
Question: $\exists$ assignment $\tau_X$ on $X$ such that $\varphi_1$ is satisfiable and $\varphi_2$ is unsatisfiable?
(The "satisfiable" and "unsatisfiable" conditi... |
How many $(a,b)$ for $a,b \in \Bbb{N}$ pairs can satisfy the following equation: $$\log_{2^a}\left(\log_{2^b}\left(2^{1000}\right)\right)=1$$ The answer is $3$, but I can't figure out how to get that answer.
This is my attempt.
$$\log_{2^a}\left(\log_{2^b}\left(2^{1000}\right)\right)=1$$$$\frac{1}{a}\log_2\left(\log_{2... |
I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is.
No apology is necessary since your question is by no means silly. It is not at all surprising that you are puzzled by the cited exposition since it is incredibly sloppy. Kudos to you for reading it very carefully and noticing these... |
Van der Waals forces are driven by induced electrical interactions between two or more atoms or molecules that are very close to each other. Van der Waals interaction is the weakest of all intermolecular attractions between molecules. However, with a lot of Van der Waals forces interacting between two objects, the inte... |
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Okay, now I've rather carefully discussed one example of \(\mathcal{V}\)-enriched profunctors, and rather sloppily discussed another. Now it's time to build the general framework that can handle both these examples.
We can define \(... |
Double Integral: Volume Under Surface
In this lesson, we'll take a look at double integrals and see that they aren't that much
more complicated than regular integrals. Regular integrals (that is, integrals of the form \(\int{f(x)dx}\)) give the area underneath a curve. In multi-variable calculus, double integrals are w... |
If you go through the process of non-dimensionalizing the equations, the math becomes more clear. If you start with the momentum equation (ignoring viscous forces because they aren't important for the analysis):$$\frac{\partial u_i}{\partial t} + \frac{\partial u_i u_j}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}... |
Many times during the course of the Chemistry 105 laboratory you will be asked to report an average, relative deviation, and a standard deviation. You may also have to analyze multiple trials to decide whether or not a certain piece of data should be discarded. This section describes these procedures.
Average and Stand... |
The
of a matrix is$$A_{m\times n} = U_{m\times m}\Lambda_{m\times n} V_{n\times n}'$$where $U$ and $V$ are orthogonal matrices and $\Lambda$ has ( Singular Value Decomposition (SVD) i, i) entry $\lambda_i \geq 0$ for $i = 1, 2, \cdots , min(m, n)$ and the other entries are zero. Then the left singular vectors$U$ for ro... |
The capacitance of
any capacitor is defined as \(C≡\frac{ΔV}{Q}\). In this lesson, we'll be interested in finding the capacitance for what is known as a parallel-plate capacitor. A parallel-plate capacitor is a capacitor whose conductors are two thin plates which are parallel to one another and seperated by an insulato... |
Let $\Omega \subseteq \mathbb{R}^n$ be open. For any compactly supported distribution $u \in \mathcal{E}'(\Omega)$, the distributional Fourier transform $\hat{u}$ is in fact a $C^\infty$ function on $\mathbb{R}^n$ with formula given by $$\hat{u}(\xi) = \langle u(x), \chi(x) e^{i x \cdot \xi} \rangle, $$
where $\chi$ is... |
OpenCV 4.0.1
Open Source Computer Vision
class cv::DenseOpticalFlow class cv::DISOpticalFlow DIS optical flow algorithm. More... class cv::FarnebackOpticalFlow Class computing a dense optical flow using the Gunnar Farneback's algorithm. More... class cv::KalmanFilter Kalman filter class. More... class cv::SparseOptical... |
Overview
The definite integral \(∫_a^bf(x)dx\) that we're all familiar with gives us the area between the following two line segments: the curvilinear line segment \(f(x)\) along the interval \([f(a),f(b)]\) and the straight line segment along the interval \([a,b]\) on the x-axis. In other words, if you imagined buildi... |
This question already has an answer here:
I asked a similar question yesterday, but I didn't really get the info I wanted so maybe if I post a question and get an answer I will understand this concept better. Just some background info the topic is using Riemann sums approximation to find upper/lower sums.
The question ... |
Z F XU
Articles written in Bulletin of Materials Science
Volume 39 Issue 2 April 2016 pp 519-523
The effect of plating temperatures between 60 and 90$^{\circ}$C on structure and corrosion resistance for electroless NiWP coatings on AZ91D magnesium alloy substrate was investigated. Results show that temperature has a si... |
It is a well-known fact that language generated by a context-free grammar is the minimal solution of a particular system of equations, for example:
$$\begin{align*} X &=\{{\epsilon}\} \cup Y\\ X &=\{{a}\}Y \{{v}\} \end{align*}$$
generates the language $\{a^nb^n : n \in \mathbb{N}\}$.
How can I prove that fact? How do I... |
Working on the two explanations given above, the natural transformation \\(\beta\_\text{Italians}\\) maps *every* Italian in \\(F\\) to *the* Italian in \\(\mathrm{Ran}\_G (H)\\), of which there can be only one unique possible function.
We can therefore reason that \\(\beta\_\text{Germans}\\) and \\(\alpha\_\text{Germa... |
Conformational Statistical Mechanics
Conformational transitions play a key role in biomolecular processes: enzymes tend to close up over their substrates, transporters such as antiporters often undergo an outward to inward-facing transition (facing one side of the bilayer or the other), and the steps taken by molecular... |
This post describes some geometric machine learning algorithms.
K-Nearest Neighbor Regression
k-nearest neighbors regression algorithm (k-NN regression) is a non-parametric method used for regression. For a new example x, predict y as the average of values \(y_1, y_2, …, y_k\) of k nearest neighbors of x.
K-Nearest Nei... |
In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in configuration space?
The number of
physical degrees of freedom (DOF) or dynamical variables is simply the number of generalized po... |
Suppose you have an action $S(\epsilon) = S_1 + S_2 + \epsilon\, S_\mathrm{int}$. Assume that $S_1$ is gauge invariant under the action of the group $G$ and $S_2$ is gauge invariant under the action of the group $H$, such that the action $S_1$ + $S_2$ is gauge invariant under the action of $G\times H$. Suppose that $S_... |
Speaking of LaTeX, if you, like us, want to write LaTeX math code in your blog, you should have a look at the LaTeX WP plugin.
The output will become something like this:
Or maybe like this:
These are not as pretty as real output, but they sure are prettier than writing math the hard way:
f(x) = \sum_{n=0}^\infty \frac... |
The Torque in rotational motion is equivalent to force in linear motion. It is the prime parameter which keeps an object under rotatory motion. The torque applied to an object begins to rotate it with an acceleration inversely proportional to its moment of inertia. Mathematically given by-
Torque
\(\tau =I\alpha\)
Wher... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
The solubility of gases depends on the pressure: an increase in pressure increases solubility, whereas a decrease in pressure decreases solubility. This statement is formalized in Henry's Law, which states that
the solubility of a gas in a liquid is directly proportional to the pressure of that gas above the surface of... |
Document Type: Original Article
Authors
1
Ivan Franko National University of Lviv, Ukraine
2
Jan Kochanowski University, Poland
Abstract
For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\in\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n... |
Find the eigenvalues and eigenvectors of the following matrix and express the
matrix in the form of $P=Ee^{\lambda t}E^{-1}$ where $E$ are the eigenvectors and $\lambda$ are the eigenvalues
\begin{bmatrix}0 & 1 & 0 & 0\\ -a^2-b & 0 & b & 0\\ 0 & 0 & 0 & 1\\b & 0 & -a^2-b & 0\end{bmatrix} Find the matrix $P$
What i trie... |
I should calculate the Limit $\lim \limits_ {x \to 2} \left(\frac{x^2+2x-8}{x^2-2x}\right)$, although I noticed, that $x\neq 2$ must apply. Is the limit undefined? Otherwise, with which steps should I go on to calculate the limit?
We have that
$$\frac{x^2+2x-8}{x^2-2x}=\frac{\color{red}{(x-2)}(x+4)}{x\color{red}{(x-2)}... |
https://doi.org/10.1351/goldbook.C01036
The fractional variation of the @R05326-1@ frequency of a @N04256@ in nuclear magnetic @R05326-1@ (NMR) @S05848@ in consequence of its magnetic environment. The chemical shift of a @N04256@, \(\delta \), is expressed as a ratio involving its frequency, \(\nu _{\mathrm{cpd}}\), re... |
Skills to Develop
Set up a linear equation to solve a real-world application. Use a formula to solve a real-world application.
Josh is hoping to get an \(A\) in his college algebra class. He has scores of \(75\), \(82\), \(95\), \(91\), and \(94\) on his first five tests. Only the final exam remains, and the maximum of... |
In two dimensions, Poisson's equation has the fundamental solution,
$$G(\mathbf{r},\mathbf{r'}) = \frac{\log|\mathbf{r}-\mathbf{r'}|}{2\pi}. $$
I was trying to derive this using the Fourier transformed equation, and the process encountered an integral that was divergent. I was able to extract the correct function event... |
Since you know the center and the point of tangency, you can compute the slope of the radius to the point of tangency. Since the center is $(3, 0)$ and the point of tangency is $(2, 2\sqrt{2})$, the slope of the radius to the point of tangency is
$$m_r = \frac{2\sqrt{2} - 0}{3 - 2} = 2\sqrt{2}$$
The slope of the tangen... |
Category:Uniformly Continuous Functions
This category contains results about Uniformly Continuous Functions.
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: d_1 \left({x, y}\right) < \delta \implie... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Every $LL(k)$ grammar is $LR(k)$, but there are $LL(k)$ grammars which are not $LALR(k)$.
There's a simple example in Parsing Theory by Sippu&Soisalon-Soininen
$$\begin{align}S &\to a A a \mid b A b \mid a B b \mid b B a\\A &\to C \\B &\to C\end{align}$$$$\begin{align}S &\to a A a \mid b A b \mid a B b \mid b B a\\A &\... |
Slightly delayed, Feb’s open problem is by one of the three possible “yours truly”s, Sesh. Looking for more reader participation, hint, hint. Basic setting: Consider \(f:\{0,1\}^n \rightarrow R\), where \(R\) is some ordered range. There is a natural coordinate-wise partial order denoted by \(\prec\). The function is m... |
I think the most obvious way to do this is to treat entropy as the analogue to variance, if mutual information is the analogue to covariance. Notice that one similarity is that $\mathbb{I}[X;X] = \mathbb{H}[X]$ (as for variance vs covariance) one difference is that (unlike covariance) we have $\mathbb{I}[X;Y] \geq 0 $.... |
Introduction to kinematics
Kinematics is the study of how to describe the motion of objects using mathematics. Galileo long ago thought about the parabolic motion of cannonballs. This is one of the earliest applications of kinematics that I'm aware of and this will be our starting point. The study of the motion of obje... |
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Now showing items 1-2 of 2
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
For the first part take any $\alpha \in L$. Then let $\{\alpha_1,\dots \alpha_n\}$ be the set of distinct elements obtained by $\text{Aut}(L/K)$ acting on $\alpha$. Note that this set is finite, as the extension is algebraic. Now consider:
$$h(x) = \Pi_{i=1}^{n} (x-\alpha_i)$$
Now it's not hard to see that $h(x)$ is fi... |
If $(W_t^1)_{t \geq 0}$ and $(W_t^2)_{t \geq 0}$ are independent Brownian motions, then $W_t^1$ and $W_t^2$ are independent for any $t \geq 0$. Hence,
$$\mathbb{E}(F) = A \cdot \mathbb{E} \exp \bigg[ \sigma (\varrho W_t^1+\sqrt{1-\varrho^2} W_t^2) \bigg] = A \cdot \mathbb{E}\exp(\sigma \varrho W_t^1) \cdot \mathbb{E}\e... |
Friends, a colleague of mine showed me a book with a intriguing chapter structure:
I was wondering how we could achieve something similar. I understand that this unusual numbering will break the counters, and I don't expect the chapter numbers to be set automatically, but I'd like to see how it could work even with a m... |
1. What is a base
b logarithm? Discuss the meaning by interpreting each part of the equivalent equations [latex]{b}^{y}=x[/latex] and [latex]{\mathrm{log}}_{b}x=y[/latex] for [latex]b>0,b\ne 1[/latex].
2. How is the logarithmic function [latex]f\left(x\right)={\mathrm{log}}_{b}x[/latex] related to the exponential funct... |
The fourth method of solving a
quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substit... |
Are there any analytical proofs for the 2nd law of thermodynamics?
Or is it based entirely on empirical evidence?
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It's simple to "roughly prove... |
Defining parameters
Level: \( N \) = \( 8 = 2^{3} \) Weight: \( k \) = \( 21 \) Nonzero newspaces: \( 1 \) Newforms: \( 2 \) Sturm bound: \(84\) Trace bound: \(0\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{21}(\Gamma_1(8))\).
Total New Old Modular forms 43 21 22 Cusp forms 37 19 1... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
Existence of entire solutions for semilinear elliptic problems on ${\mathbb R}^{N}$
DOI: http://dx.doi.org/10.12775/TMNA.1999.001
Abstract
In this paper, we consider the existence of positive and negative
entire solutions of semilinear elliptic problem $$ -\Delta u + u = g(x,u), \quad u \in H^{1}({\mathbb R}^{N})\tag{P... |
Make sure that the dev and test sets come from the same distribution。
Not having a test set might be okay.(Only dev set.)
So having set up a train dev and test set will allow you to integrate more quickly. It will also allow you to more efficiently measure the bias and variance of your algorithm, so you can more effici... |
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Okay, now I've rather carefully discussed one example of \(\mathcal{V}\)-enriched profunctors, and rather sloppily discussed another. Now it's time to build the general framework that can handle both these examples.
We can define \(... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
How to Perform a Nonlinear Distortion Analysis of a Loudspeaker Driver
A thorough analysis of a loudspeaker driver is not limited to a frequency-domain study. Some desirable and undesirable (but nonetheless exciting) effects can only be caught by a nonlinear time-domain study. Here, we will discuss how system nonlinear... |
Last time we tackled von Neumann's minimax theorem:
Theorem. For every zero-sum 2-player normal form game,
where \( p'\) ranges over player A's mixed strategies and \( q'\) ranges over player B's mixed strategies.
We reduced the proof to two geometrical lemmas. Now let's prove those... and finish up the course!
But fir... |
For a transverse wave(or for pressure waves required to produce longitudinal waves), the motion perpendicular to the direction of propagation of the wave is governed by an equation like $y = Asin(\omega t)$ in case of harmonic waves(here $\omega$ is angular frequency of simple harmonic motion). The time period ($T$) of... |
For this process, the interaction Hamiltonian is given by:
$$\mathcal{H}_{\rm int}=-\frac{g}{\sqrt 2}\left(V_{cb}\bar{b}_L\gamma^\mu c_L W^-_\mu+\bar{\nu}_L\gamma^\mu\ell_L W^+_\mu\right).$$
After integrating-out the heavy bosons, we obtain the following Hamiltonian
$$\mathcal{H}_{\rm eff}=-\dfrac{G_F}{\sqrt{2}}V_{cb}[... |
The Ideal Gas: The basis for "mass action" and a window into free-energy/work relations The Ideal Gas: The basis for "mass action" and a window into free-energy/work relations
The simplest possible multi-particle system, the ideal gas, is a surprisingly valuable tool for gaining insight into biological systems - from m... |
These are homework exercises to accompany Miersemann's "Partial Differential Equations" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Partial differential equations are differential equations that contains unknown multivariable functions a... |
Cosecant Function is Odd
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Theorem
Let $x \in \R$ be a real number.
Let $\csc x$ be the cosecant of $x$.
Then, whenever $\csc x$ is defined: $\csc \left({-x}\right) = -\csc x$ Proof
\(\displaystyle \csc \left({-x}\right)\) \(=\) \(\displaystyle \frac 1 {\sin \left({-x}\right)}\) Cosecant... |
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