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Expectations and conditional expectations are either random or fixed points. It depends upon your choice of axioms. Unfortunately, I have never found a good book that fairly describes both well.
On the Bayesian side, I would suggest the very polemical "Probability Theory: The Language of Science" by E.T. Jaynes. On the... |
In Chapter 3 we illustrated how
intraindividual covariation is examined within the multilevel modeling framework. We now build on that foundation in various ways. In particular, this tutorial demonstrates how the generalized multilevel model is used when the outcome variable is binary (or Poisson). Using link functions... |
Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations:
$$ k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right| $$
$$ k_2 = \left| \xi^T \mathcal{G}^T \mathcal{G} \xi \right| $$
where:
$$ \mathcal{F}= \left( \begin{matrix} c_{11} & c_{12} \... |
Given an SDE for an underlying:
$$dS(t) = \mu(S,t)dt+\sigma(S,t)dW(t)$$
the SDE for the value of the option $V=V(S,t)$ is given via Ito's lemma as:
$$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\sigma^2(S,t)dt+V_S\sigma(S,t)dW(t)$$
It seems that this would results in an SDE containing $S(t)$.
How does one then obtain an ... |
How to type this expression, where has
\min on top and
\forall at the bottom?
TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. It only takes a minute to sign up.Sign up to join this community
The bottom part is a TeX subscript (like
\sum):
\min_... |
Search
Now showing items 1-10 of 26
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has ... |
Let $\Delta$ be a simplicial complex (or more generally, a regular CW complex). Let $\mathcal{M}$ be a Morse matching (or equivalently, a discrete Morse function) on $\Delta$.
By Forman's theorems, $\Delta$ is homotopy equivalent to a CW-complex whose cells are $\mathcal{M}$-critical (=unmatched) simplices. Its homolog... |
Why is the work done on a charge calculated from infinity to a point? Why not from one particular point to other?
Consider the form of the potential energy between two point charges in the case that I use a reference distance $r_0$ as the zero (written here in SI units). $$ U_{r_0} = \frac{q_1 \,q_2}{4 \pi \epsilon_0} ... |
Closure (闭包) $\text{cl}(C)$ of a set is the collection of its limit points. Projection of a vector on a closed set isthe set of vectors in the set closest to the vector: $P_C(x) = \arg\min_{y \in C} \| y - x \|$. Distance of a vector to a set is the smallest distance from it to points in the set:$d_C(x) = \min_{y \in C... |
The fact that the period of the gravity train that passes through the center of the Earth is equal to the period of an orbit that skims the Earth's surface is not a coincidence. Consider a polar orbit around Earth (i.e., one that passes directly over the coordinate north and south poles). Now, consider only the north-s... |
1. Example
Here is an example.
Automatic machine learning greatly facilitate applying machine learning: there is no need to select among algorithms, no need to tune hyperparameters, etc.
2. Hyperparameter Optimization
Let $\lambda$ be the hyperparameters of an ML algorithm $A$ with domain $\Lambda.$ Let $\mathcal{L}(A_... |
8.5. Implementation of Recurrent Neural Networks from Scratch¶
In this section we implement a language model introduce inSection 8 from scratch. It is based on a character-levelrecurrent neural network trained on H. G. Wells’
The Time Machine. Asbefore, we start by reading the data set first, which is introduced inSect... |
I have recently studied Fourier and Laplace transformation in maths. I wanted to understand the utility in physics with some examples that requires this change in dimension and the reason why.
closed as too broad by Brandon Enright, jinawee, Qmechanic♦ Apr 26 '14 at 9:47
Please edit the question to limit it to a specif... |
I have to compute the surface brightness as a function of the radius from the following set of data: {right ascension ($\alpha$), declination ($\delta$), magnitude (m)}. I also know the center of the set $\left(\alpha_c, \delta_c \right)$.
The surface brightness in logarithmic units (mag/arcsec$^2$) can be computed by ... |
I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth is as high as possible. Formally my problem is the following: having chosen a degree bound $d \in \mathbb{N}$, what is ... |
How would you describe the set $\{1, 5, 9, 13, 17, 21,\dots\}$ in the style of $x:P(x)=$? I know that the sequence is "the last number + 4" or $4n-3$.
$\{n \in \mathbb N\; :\; n \equiv 1 \mod 4\}$
$\{4k-3 \mid k \in \mathbb{N} \}$ or $\{4k+1 \mid k \in \mathbb{N} \}$, depending on whether you consider $0$ a natural num... |
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the
quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.
One version of the $abc$-Conjecture is t... |
In Effective Field Theory video lectures found here, the professor explained power counting in effective field theories and the difficulties of power counting associated with loop diagrams. He then mentions that introducing a cutoff ($\Lambda_{UV}$) to regulate our divergences does not preserve power counting due to th... |
Answer
Please see the work below.
Work Step by Step
We know that $H=kA\frac{\Delta T}{\Delta x}$ We plug in the known values to obtain: $H=(46)(0.90\times 0.40)(\frac{310-295}{0.0045})$ $H=55000W$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll hav... |
Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$.
Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$.
(problem composed by
Laurentiu Panaitopol)
So far no idea.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and ... |
Under this menu option you will find other information related to me, my collaborators, life in general and so on. In the long run I hope to include pictures, tutorials and any other piece of information ...
... s could be added, or changed, depending on the processing needs. The image processing pipeline is responsibl... |
There exists the following metric counterpart of the Harris result:
Theorem (Banakh, Vovk, Wojcik): Each metric space $X$ can be (canonically) identified with the space $\pi_0(\circledast(X))$ of path-components of some complete metric space $\circledast(X)$.
The space $\circledast(X)$ is a closed subspace of the compl... |
Consider the situation in which one has to solve a specific instance of a parametrized family of polynomial systems
$$ P = \{F(x,p) = (f_1(x,p), \ldots, f_n(x,p)) \mid p \in \mathbb{C}^m\}. $$
To not destroy the solution structure it is desirable to not leave $P$ during the homotopy. This can be accomplished by using t... |
Who should read this document
I recently learned about org mode for Emacs. Among other things, it allows to mix text and R source code similar to SWeave, but the markup for text is very simple.
To get things working can be complicated, especially if one is not an experienced user. Therefore this documents is intended t... |
3.6. Implementation of Softmax Regression from Scratch¶
Just as we implemented linear regression from scratch, we believe that multiclass logistic (softmax) regression is similarly fundamental and you ought to know the gory details of how to implement it from scratch. As with linear regression, after doing things by ha... |
I know that a band-limited signal is a signal which have a limited set of frequencies.
What is then the relation between band limiting and $[-\pi \; \pi]$ since $-\pi$ and $\pi$ are related to angles and frequencies are in Hertz ?
Signal Processing Stack Exchange is a question and answer site for practitioners of the a... |
Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, B=M_2(\mathbb{Q}), \Gamma_1(p^n)$ one can associate to a finite place $\mathfrak{p}$ of $F$, where $B$ is split a geome... |
(
This article is based on a mathematics workshop for 12 year olds given by Warwick Evans and Alison Clark Jeavons in Cambridge in February 2000. The photos were taken at the ATE / NRICH Mathematics Superweek holiday for 10 to 13 year olds at Southam in August 2000. )
In this article we shall find out why there are onl... |
the spectrum of the Gamma and Alpha decays are both discrete, i.e. the $\alpha$-particles and the $\gamma$-rays take on only discrete values when emitted from a decaying nucleus.
Why is it then, that the $\beta^{\pm}$ can take on continuous values?
The main thing that distinguishes the beta decay from the other two is,... |
I am trying to graphically simulate a series of springs in 2D. Now one of the forces I am stuck with calculating is the damping force. The given formula is $F = -k_d v$. I know that $v$ is the velocity of the vectors, but I can't seem to find how to calculate $k_d$.
For a viscous damper, the decay in the free oscillati... |
The JKY ABMC Model (taken from Jabbour, et al. 2001) parameterizes the binomial model (in a risk-neutral world) such that,
$u = e^{r\Delta t} + e^{r\Delta t}\sqrt{e^{\sigma^2\Delta t} - 1}$
$d = e^{r\Delta t} - e^{r\Delta t}\sqrt{e^{\sigma^2\Delta t} - 1}$
JKY continue and say that this is equivalent to,
$u = 1 + \sigm... |
I have a few aligned lists of coefficients. One set of coefficients has three columns and the other has four. At the moment I display them with two
align environments:
\begin{align} \gamma_1 &= 8/15 & \gamma_2 &= 5/12 & \gamma_3 &= 3/4 \\ \zeta_1 &= 0 & \zeta_2 &= -17/60 & \zeta_3 &= -5/12 \\ \beta_1 &= 4/15 & \beta_2 ... |
Least Median of Squares regressionproblem:
\[\begin{align}\min\>&z\\&\delta_i=1 \Rightarrow \> –z \le r_i \le z\\&\sum_i \delta_i = h\\&r_i = y_i - \sum_j X_{i,j} \beta_j\\&\delta_i \in \{0,1\} \end{align}\]
Here \(X\) and \(y\) are data. Modern solvers like Cplex and Gurobi support
indicator constraints. This allows u... |
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. Show that $E[B_t|\mathscr{F}_s] = B_s$ whenever $s < t$.
I think this all comes down to manipulation since there are marting... |
Let $\{N(t), t\geq 0\}$ be a Poisson process with rate $\lambda$, $S_n$ the instant of the $n$-th arrival and $T_n$ the $n$-th interarrival time, that is, $T_n = S_n - S_{n-1}$, $n \geq 1$.
Now consider the following result:
Theorem. Given that $N(t) = n$, the $n$ arrival times $S_1, S_2, \dots, S_n$ have the same dist... |
An accurate plant model is the linchpin of control system development using Model-Based Design. With a well-constructed plant model, engineers can verify the functionality of their control system, conduct closed-loop model-in-the-loop tests, tune gains via simulation, optimize the design, and run what-if analyses that ... |
We usually call equations like
$$\frac{d}{dt} \frac{\partial L}{\partial \dot{q_i}} - \frac{\partial L}{\partial q_i} = 0$$
"equations of motion," because they are equations that tell us how the variables of our system (here $q_i$) evolve in time. Indeed, in general, the solution to $n$ second order differential equati... |
1. Perturbations
As already mentioned, the Jahn–Teller effect has its roots in group theory. The essence of the argument is that the energy of the compound is stabilised upon distortion to a lower-symmetry point group. This distortion may be considered to be a normal mode of vibration, with the corresponding vibrationa... |
Is the universal covering of a connected open subset $U$ of ℝ
n diffeomorphic to an open subset of ℝ n (standard differentiable structure)?
If not true in general, is there any condition on $U$ which guarantees a positive answer?
MathOverflow is a question and answer site for professional mathematicians. It only takes ... |
In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle v^2, w^2 \rangle \le \|v\|^2 \|w\|^2 - \langle v,w \rangle^2; \quad (*) \end{align*} here, $\langle\cdot,\cdot\rangle$ de... |
The passage from any small category
C to its set-valued functor categoy$\hat{\mathbf C}:=\mathrm{Fun}(C^{\ast},\mathrm{\bf Set})$ i.e.the full Yoneda-embedding $Y\colon \mathrm{\bf C} \to \hat{\mathrm{\bf C}}$ into the presheaf category can be considered as an universal completion-process.
A functor category such as $\... |
I) A Lagrangian variational principle for Euler's equations for a rigid body
$$ \tag{1} (DL)_i ~=~M_i, \qquad i\in\{1,2,3\}, $$
is e.g. explained in Ref. 1. Here the angular momentum $L_i$, $i\in\{1,2,3\}$, along the three principal axes of inertia is tied to the angular velocity $\omega_i$, $i\in\{1,2,3\}$, by the for... |
I chose to implement Scalable Ambient Obscurance [MML12] for my project, which is a fast improved screen space ambient occlusion method. This method improves performance and quality of a method introduced previously by McGuire et al., the Alchemy Ambient Obscurance algorithm [MOBH11]. Ambient occlusion methods attempti... |
6.1. From Dense Layers to Convolutions¶
The models that we’ve discussed so far are fine options if you’redealing with
tabular data. By tabular we mean that the data consistsof rows corresponding to examples and columns corresponding to features.With tabular data, we might anticipate that pattern we seek couldrequire mo... |
What follows is not rigorous, but hopefully has the main idea. First, you probably want to justify$$X_t-X_s \sim N\left(0, \int_s^t f(u)^2 du \right), $$which can be done by approximating $f$ by a simple function $g = a_1 1_{(s,t_1)} + a_21_{(t_1,t_2)} + \ldots + a_n1_{(t_{n-1},t)}$ and then using$$\int_s^t g(u)dW_u = ... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
The Fundamental Theorem of Finitely Generated Abelian groups is like the Fundamental Theorem of Arithmetic: it describes a "canonical way" of expressing a finitely generated abelian group as a direct sum (in fact, two different ways), in a way that is "essentially unique", and where two groups are isomorphic if and onl... |
I have a line $L$ in the plane expressed as the points in $L = \{(x,y) \in {\mathbb{R}}^2 : x \cos \theta + y \sin \theta = r \; \wedge \; 0 > \theta > \pi/2 \}$ (note that the line cannot be fully horizontal or fully vertical). This line can possibly intersect the $y$-axis in $w-m \leq y \leq w+m$ for a fixed frame wi... |
Take $x>0$ large, $t\in \mathbb R$, $q\in \mathbb N$ and a non-principal character $\chi $ mod $q$. If you want, take $t\leq x$. How do I bound
\[ \sum _{n\leq x}\frac {\chi (n)}{n^{it}}?\]
My guess was that this is $\ll \sqrt {qt}$, based on thinking about the $q=1$ case, which is the relatively well known statement
\... |
I'm trying to play with bond-future options. Bond future is a future contract on a
basket of bonds. The short-side will deliver the so-called bond cheapest-to-deliver (CTD).
A bond-future option is therefore an option on this basket. Let's simplify things such that:
the option is directly struck on the CTD; CTD is a ze... |
TL;DR Your Maxwell–Boltzmann diagram up there is not sufficient to describe the variation of rate with $E_\mathrm{a}$. Simply evaluating the shaded area alone does not reproduce the exponential part of the rate constant correctly, and therefore the shaded area should not be taken as a quantitative measure of the rate (... |
I am reading "Introduction to Quantum Mechanics" by David Griffiths and I am having trouble understanding part of a derivation of $\frac{d\langle x\rangle }{dt}$ in section 1.5 - Momentum - of the text.
The Author gives EQN 1.29 as
$$ \frac{d\langle x\rangle }{dt} = \frac{i \hbar}{2m} \int _{-\infty} ^{\infty} x \frac{... |
There is some super-heated $NH_3$ in an isolated tank with initial pressure and temperature of $1600 \ kPa$ and $50^{\circ} C$ respectively. Gas leaks out from a valve on tank very slowly such that some of gas exits reversibly. In the final state, pressure is $1400 \ kPa$ and $NH_3$ is still super-heated.
The problem s... |
What is the instantaneous P&L of a variance swap.
Is it $(\sigma^{2}_{t}-\sigma^{2}_{implied})dt$?
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up.Sign up to join this community
A variance swap has a set of fixing times, and th... |
80 1
Hello Everybody,
Instead of solving the geodesic equations for the Schwarzschild metric, in many books (nearly in all books that I consulted), conserved quantities are looked at instead.
So take for eg. Carroll, he looks at the killing equation and extracts the equation
[tex] K_\mu \frac{dx^\mu}{d \lambda}= consta... |
Genus¶
This file contains a moderately-optimized implementation to compute the genus of simple connected graph. It runs about a thousand times faster than the previous version in Sage, not including asymptotic improvements.
The algorithm works by enumerating combinatorial embeddings of a graph, and computing the genus ... |
I am reading Steven Shreve's book "Stochastic Calculus for Finance 2 Continuous-Time Models", page 304. My intuition is that when the stock price gets closer to the barrier, it will be more and more likely that the price will exceed the barrier in a near future, hence it has a large probability to become worthless. Thi... |
Current browse context:
astro-ph.GA
Change to browse by: References & Citations Bookmark(what is this?) Astrophysics > Astrophysics of Galaxies Title: CO Multi-line Imaging of Nearby Galaxies (COMING). VII. Fourier decomposition of molecular gas velocity fields and bar pattern speed
(Submitted on 3 Jan 2019)
Abstract: ... |
14.3. Matrix Factorization¶
The first version of matrix factorization model is proposed by Simon Funk in a famous blog post in which he described the idea of factorizing the interaction matrix. It then became widely known due to the Netflix contest. In 2006, Netflix, a media-streaming and video-rental company, announce... |
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Now showing items 1-10 of 18
J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in... |
Search
Now showing items 1-5 of 5
Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c wit... |
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Now showing items 1-10 of 32
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... |
Search
Now showing items 1-10 of 27
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has ... |
I'm trying to teach some secondary school students on how to complete the square. The goal is to rewrite: $$y = ax^2 + bx + c \ \ \Rightarrow \ \ y = a(x-h)^2 + k$$ The first thing I did was to ask them to confirm, via FOIL: $$\left( x + \tfrac{b}{2} \right) \left( x + \tfrac{b}{2} \right) \ \ = \ \ x^2 + bx + \tfrac{b... |
Illinois Journal of Mathematics Illinois J. Math. Volume 59, Number 1 (2015), 235-276. Limit theorems for some critical superprocesses Abstract
Let $X=\{X_{t},t\ge0;\mathbb{P}_{\mu}\}$ be a critical superprocess starting from a finite measure $\mu$. Under some conditions, we first prove that $\lim_{t\to\infty}t{ \mathb... |
Below is an approach I've been exploring for connecting the prime counting function with the logarithmic integral and expressing the error term between the two. I find it beguiling, but I've largely run out of ideas for further developing this approach.
Hence this question. What I'd really, really appreciate is either ... |
\documentclass[12pt,a4paper,onecolumn]{article}\usepackage{chemfig}\usepackage{tikz} \begin{document}\begin{tikzpicture} \path node(A){\chemfig{@{H1}H-[@{HB1}:30]@{C1}C(=[@{CDB1}:90]@{O1}O)-[@{CB2}:-30]@{O2}O-[@{OBH1}:30]@{H2}H}};\chemmove{\draw (HB1) node[above] {$\alpha$};\draw (CDB1) node[right] {$\beta$};\draw (CB2... |
Is there always a frame in which
spatially separated events are
simultaneous?
The answer is
no.
Two events that are spatially separated in one frame of reference
(1) will be co-located in another frame of reference and not simultaneous in any frame if the interval is time-like
(2) will be simultaneous in another frame ... |
(30 intermediate revisions by 3 users not shown) Line 1: Line 1:
__NOTOC__
__NOTOC__
−
=
Spring 2019 = +
= 2019 =
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.
Line 9: Line 9:
[mailto:join-probsem@lists.wisc.edu join... |
I first learned big-Oh (little-Oh, big-Theta.....) complexity for growth of functions using CLRS in a computer science class.
Now I am doing a project on optimization. In our optimization class, we were introduced to the notion of rate of convergence which are characterized by the ratio:
$\lim\limits_{k \to \infty} \df... |
I'd like to reproduce results from this paper in which the authors used Group Delay-grams as an input to the Machine Learning model. The Machine Learning side is not the one that I have problems with. I have a problem with computing a Group Delay function of an audio file.
To begin with, I know that a group delay per s... |
EDIT :
I read more about it and I get some help with someone else, here is the correct answer :
The density forecast is the predictive likelihood value of the process
estimated at the realized value computed in a one step ahead way.
Thus for instance for a standard arma garch process with normal errors:
You forecast th... |
10.10. Adam¶
Created on the basis of RMSProp, Adam also uses EWMA on the mini-batch stochastic gradient[1]. Here, we are going to introduce this algorithm.
10.10.1. The Algorithm¶
Adam [Kingma.Ba.2014] uses the momentum variable \(\boldsymbol{v}_t\) and variable \(\boldsymbol{s}_t\), which is an EWMA on the squares of ... |
Cython helper functions for congruence subgroups¶
This file contains optimized Cython implementations of a few functions related to the standard congruence subgroups \(\Gamma_0, \Gamma_1, \Gamma_H\). These functions are for internal use by routines elsewhere in the Sage library.
sage.modular.arithgroup.congroup.
degene... |
Subschemes of affine space¶
AUTHORS:
David Kohel (2005): initial version. William Stein (2005): initial version. Ben Hutz (2013): affine subschemes class
sage.schemes.affine.affine_subscheme.
AlgebraicScheme_subscheme_affine(
A, polynomials, embedding_center=None, embedding_codomain=None, embedding_images=None)¶
Constr... |
Suppose you are given two oriented manifolds with boundary $M$ say $B, B'$ and $\partial B = M = \partial B'$. Identify the boundaries and form $C = B \sqcup_{Id: M \to M} B'$. I want to see why, with the orientation induced by being submanifolds of $C$, $B$ and $B'$ induce opposite orientations to their boundary $M$. ... |
Let $\{ e_{\alpha} \}_{\alpha\in\Lambda}$ be an orthonormal subset of a Hilbert space $H$, and let $x\in H$ be given. Then the following are equivalent:
$\sum_{\alpha} \langle x,e_\alpha \rangle e_{\alpha} =x$. $\sum_{\alpha} |\langle x,e_{\alpha}\rangle|^2 = \|x\|^2$. $x$ is in the closure of the linear span of $\{ e_... |
In the paper Nerovny et al. (2017), the commentaries about a convergence of series which represent the absolute value function and corresponding equations contain several mistakes (Sect. 2, from Eqs. (4) to (6)).
The series Eq. (3)
$$\begin{aligned} |\hat{\mathbf {n}}\cdot \hat{\mathbf {s}}|=|x| = \frac{2}{\pi } - \fra... |
There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle X\mid R\rangle$ to be $FG(X)/N$ where $N$ is the intersection of all
finite index normal subgroups of $FG(X)$ containing $R... |
Articles
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Full-Text Articles in Physics
Unitarized Pseudoscalar Meson Scattering Amplitudes In Three Flavor Linear Sigma Models, Joseph Schechter, Deirdre Black, Amir H. Fariborz, Sherif Moussa, Salah Nasri
Unitarized Pseudoscalar Meson Scattering Amplitudes In Three Flavor Linear Sigma Models, Joseph Schec... |
I got a problem of calculating $E[e^X]$, where X follows a normal distribution $N(\mu, \sigma^2)$ of mean $\mu$ and standard deviation $\sigma$.
I still got no clue how to solve it. Assume $Y=e^X$. Trying to calculate this value directly by substitution $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\, e^{\frac{-(x-\mu)^2}{2\sig... |
I wish to add a remark to Noam D. Elkies' beautiful answer. From the integral representation for $f$, putting $e^{-t}=s$ in the integral,$$f(-x)=1-x\int_0^\infty e^{-xte^{-t}} e^{-t}dt = 1-x\int_0^1 s^{sx}ds\, ,$$so that, for $x\to \infty$, $ f(-x)=o(1)$ is equivalent to $$\int_0^1 xu(s)^xds=1+o(1)\, ,$$ where $u\in C(... |
This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the strong Bruhat order) on the symmetric group $S_n$ a lattice?
Details:
Let $n$ be a nonnegative integer. Consider the symmetr... |
Question
Let $$ \pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1, $$ where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to those values such that $(n+a,P(\sqrt{n}))\leq n$ for all $n\leq x$.
What is the asymptotic behaviour of the expected valu... |
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Purchase individual online access for 1 year to this journal.
Impact Factor 2019: 1.204 Fundamenta Informaticae is an international journal publishing original research results in a... |
I'm trying to create a diagram like this with Latex. (I'm trying to learn to write in Latex by myself.)
The best code I can do is this:
\documentclass{article}\begin{document}\usepackage[all]{xy}$$\xymatrix{\overset{\supset ker(f)}{ker(w)\subset E} \ar[d]_f \ar[dr]_{\pi_f} \ar[r]^w &\mathbb{K} \\F & E/ker(f) \simeq Im(... |
Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed?
$20°$ is a third of $60°$, for which the value of the tangent is well known to be $\sqrt3$. Let... |
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as: $$ \begin{eqnarray} P & = & \dfrac{i\hbar}{2mc^2}\left(\Phi^*\dfrac{\partial\Phi}{\partial t}-\Phi\dfrac{\partial\Phi^*}{\partial t}\right) \\ \vec{j} &=& \dfrac{\... |
GolfScript (23 chars)
{:^((1${\.**2^?%}+*}:f;
The sentinel result for a non-existent inverse is
0.
This is a simple application of Euler's theorem. \$x^{\varphi(2^n)} \equiv 1 \pmod {2^n}\$, so \$x^{-1} \equiv x^{2^{n-1}-1} \pmod {2^n}\$
Unfortunately that's rather too big an exponential to compute directly, so we have... |
Sufficient conditions for a discrete energy spectrum:
In one-dimension Sturm-Louville theory implies that the spectrum is purely discrete provided that the system is restricted to a finite interval [a,b] (with appropriate boundary conditions). I assume (but don't know) this can be extended to the case of a system restr... |
I think, I understood the main idea behind this algorithm:
We have the spatial coordinates of points that we want to bisect. We find a line (or, hyperplane, in general) $L$ through these points, such that it is a total least squares fit of the nodes. We project the points onto this line. We find the "median projection"... |
Given a polynomial with real coefficients is there a method (e.g. from algebra or complex analysis) to calculate the number of complex zeros with a specified real part?
Background. This question is motivated by my tests related to this problem.
Let $p>3$ be a prime number. Let $G_p(x)=(x+1)^p-x^p-1$, and let $$F_p(x)=\... |
First, let me say I'm not sure what is meant by a BEC with $T\gt 0$. Condensation is a finite temperature phenomenon, which occurs due to the presence of pair-wise interactions (generally attractive, but pairing can happen even for repulsive potential) in a many-body system. For instance, in a superconductor below some... |
For a function $f:\{0,1\}^n \rightarrow \{0,1\}^m$, let $C(f)$ be the circuit complexity (for concreteness, constants and NOT gates are free, while 2-input AND gates cost 1).
Let $k{\times}f : \{0,1\}^{kn} \rightarrow \{0,1\}^{km}$ be the function which computes $k$ copies of $f$ on independent inputs, so $k{\times}f(x... |
I am trying to understand the calculation upper bound that is given in book.
Edit 1: I added 3,44 Can someone explain to me how to come from (5.5) to (5.7) ?
Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a min... |
There are reasons that any modern example is likely to resemble the status of Legendre's constant. Most (but not all) interesting numbers admit a polynomial-time algorithm to compute their digits. In fact, there is an interesting semi-review by Borwein and Borwein that shows that most of the usual numbers in calculus (... |
Little explorations with HP calculators (no Prime)
03-27-2017, 07:39 PM
Post: #41
RE: Little explorations with the HP calculators
Let's divide one of the right-triangles into four right-triangles (two with sides lengths equal to r and x and two with side lengths r and 1-x) and a square with side r. Then it's area is
S ... |
By results of Aoki and Ibukiyama [MR:2130626, 10.1142/S0129167X05002837], and of Hayashida and Ibukiyama [MR:1840071], the ring $M_{*}(\Gamma_0(4))$ of Siegel modular forms of degree 2 with respect to the group $\Gamma_0(4)$ is generated by the following functions, which are specified in terms of theta constants:
$X$, ... |
Let $a > 0$. We consider the function: $f: (0, \infty) \to (0, \infty)$, defined by $f(x) = \frac{1}{2}(x + \frac{a}{x})$.
Let $(x_n)_{n \in \mathbb{N}_0}$ be defined by:
$x_0 \in (0, \infty)$, $x_{n+1} := f(x_n)$
What is the smallest $b > 0$ so that f is contracting on $[b, \infty)$, and what is $f's$ Lipschitz-consta... |
$(a+\sqrt{a^2+1})\,(b+\sqrt{b^2+1})=1$ is supposed to equal: $b = -a$ but how do i get that? I've been trying to solve for like 2 days now.
If $$ (a+\sqrt{a^2+1})(b+\sqrt{b^2+1})=1$$ we have $$ a+\sqrt{a^2+1}=\sqrt{b^2+1}-b \ \ \ \ \ (1) $$ $$ b+\sqrt{b^2+1}=\sqrt{a^2+1}-a \ \ \ \ \ (2) $$ $(1)+(2)$ then $a=-b$.
First ... |
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