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The responses of division by 0 that we have been taught is undefined I actually find to be unintelligible and completely unacceptable. Take this into consideration, when we look at a fraction $\frac{n}{d}$ where $n$ is the numerator and $d$ is the denominator and $d$ happens to be $0$ let's apply this concept to a comb... |
I am stuck in problem 53.4 from Munkres:
Let $ q : X \to Y $ and $ r : Y \to Z $ be covering maps; let $ p = r \circ q $. Show that if $ r^{-1}(z) $ is finite for each $ z \in Z $, then $ p $ is a covering map.
This is my work so far:
First, $ p $ is both continuous and surjective, because the composition of continuous... |
Definition:Projection (Mapping Theory)/Second Projection Definition
Let $S$ and $T$ be sets.
Let $S \times T$ be the Cartesian product of $S$ and $T$.
The second projection on $S \times T$ is the mapping $\pr_2: S \times T \to T$ defined by: $\forall \tuple {x, y} \in S \times T: \map {\pr_2} {x, y} = y$ Also known as
... |
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... |
3
Orlp gives a solution using $O(n)$ words of space, which are $O(n\log n)$ bits of space (assuming for simplicity that $n=m$). Conversely, it is easy to show that $\Omega(n)$ bits of space are needed by reducing set disjointness to your problem.
Suppose that Alice holds a binary vector $x_1,\ldots,x_n$ and Bob holds a... |
$\newcommand{\dd}{\mathrm{d}}$You basically have two ODEs to solve:\begin{align}\frac{\dd v^\mu}{\dd t}&=\frac{1}{m}F(x^\mu,v^\mu) \tag{1} \\\frac{\dd x^\mu}{\dd t}&=v^\mu\tag{2}\end{align}which is pretty much the case for most forces in Newtonian mechanics. In order to solve this numerically, you want to discretize sp... |
Disclaimer: I do particle physics / cosmology, so this is definitely outside my field, apply grains of salt to this answer appropriately.
I think Reference [29] (Lin et al, arxiv reference: 1008.4864) honestly does a better job of explaining what is going on (which makes sense, the impression I get is that 1008.4864 is... |
>**Puzzle 186.** I said that \\(\infty\\) plays the same role in \\(\textbf{Cost}\\) that \\(\text{false}\\) does in \\(\textbf{Bool}\\). What exactly is this role?
Both play the role of showing that two objects are disconnected.
In the case of \\(\textbf{Bool}\\), \\(\Phi(x,y) = \text{false}\\) means \\(x \not\leq y\\... |
On a generalized critical point theory on gauge spaces and applications to elliptic problems on ${\mathbb R}^N$
DOI: http://dx.doi.org/10.12775/TMNA.2001.005
Abstract
In this paper, we introduce some aspects of a critical
point theory for multivalued functions $\Phi : E \to {\mathbb R}^{\mathbb N} \cup \{\infty\}$ defi... |
Can we exchange the permutation of a sponge construction?
Part of a sponge construction (like SHA3 uses) is a fixed permutation $p$; which is clearly not one-way.
Could we, theoretically, exchange the permutation $p$ with any other permutation? What basic characteristics should such a permutation have – or would, for e... |
Radio astronomy
When we view the Earth or the night sky, we
can see plants, people, cars and cities, your coffee mug, the Moon, the stars. This is all we have ever seen with our two eyes sine we've lived our whole lives seeing things with visible light. But a bee can see something that we cannot see. They see light—inv... |
From wikipedia:
The discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT)
also from wikipedia (DTFT):
It produces a function of frequency that is a periodic summation of the ... |
Weird conjecture - Irrational Triangle About: title not given
I wonder if it is true for some base changes of the numbers.
Comments Mark de LA says
https://www.wolframalpha.com/input/?i=%28natural+log+base%29%5E2+%2B+pi%5E2+%3D+phi+%5E1%2F2
??
https://www.open.wolframcloud.com/env/36e6f6cc-f2d4-4648-9457-e50d878c923f#s... |
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We've been looking at feasibility relations, as our first example of enriched profunctors. Now let's look at another example. This combines many ideas we've discussed - but don't worry, I'll review them, and if you forget some defin... |
Have you ever wondered what is expressed just in x? Probably not, but it might actually sometimes be useful to know. The expression might for example turn up when you’ve done a trigonometric substitution in an integral. So I decided how to go about to show the trigonometric identities
,
,
.
Continue reading
The first “... |
I try to express the following statements in first order logic:
X is a subset of Y. A set can be uniquely characterised by its elements. The power set p(X) contains all subsets of X. A set X is the union of all its subsets containing just one element.
Thus far I managed:
$\forall x \in X \Rightarrow x \in Y$ $X=Y \Left... |
The following is an assignment, I just need some help understanding what I've done so far.
Suppose $X_1 \sim N(0, \sigma^2 / (1-\rho^2))$ and $U_2, \dots, U_n \sim N(0, \sigma^2)$, independent between them and with $X_1$. We define:
$$ X_k = U_k + \rho X_{k-1},$$
for $k=2,\dots,n$.
Show using induction that $(X_{k-1}, ... |
All matter has intrinsic wave properties. These are described mathematically by the Schrodinger Equations and it's solutions. The wavenature of electrons and other fundamental principles (eg charge and momentum) together produce the wave mechanics of electron. The effects of electron wave mechanics are far reaching, re... |
Fermat left only one proof.
The area of a Pythagorean triangle is never a square number.
Fermat wrote , “If the area of a right-angled triangle were a square, there would exist two biquadrates (fourth powers) the difference of which would be a square number.”
That is,
[latex] a^4\, -\, b^4 = c^2[/latex]
He used the met... |
Euler Buckling Formula
Jump to navigation Jump to search
The
Theorem $F = \dfrac {\pi^2 E I} {\left({K L}\right)^2}$
where:
$F$ = maximum or critical force (vertical load on column) $E$ = modulus of elasticity $I$ = area moment of inertia of the cross section of the rod $L$ = unsupported length of column $K$ = column e... |
We know that there is only one non-trivial ring homomorphism from $\mathbb{Z}$ or $\mathbb{Q}$ to another unital ring $S$.
What’s more,when we consider the automorphism of $\mathbb{R}$,it is unique determined,too.More explicitly speaking: let $\varphi \in Aut(\mathbb{R})$ ,then$\varphi|_{\mathbb{Q}}=Id_{\mathbb{Q}}$, a... |
I am looking for interesting functional equations of a specific type, and I thought that perhaps the math SE community would be able to deliver a good amount of them.
When I look up "functional equation problems", I usually get problems like $$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$$ and they usually have rather boring answer... |
I need advise or correction if something is incorrect with my proof.
Your proof is good!
Would appreciate any correction in proof writing also!
To this, I would respond: its good to read different people's writing just for style. So here's my version of the proof, which is logically similar to yours but just differs on... |
We have to do it without calculus or any complex inequality. Level of complexity is that we cannot even use the AM-GM inequality. So I tried, $$(\sin\theta-\cos\theta)^2\geq0$$ $$1-2\sin\theta\cos\theta\geq0$$ $$\frac12\geq\sin\theta\cos\theta$$ Reverting back to the previous step, $$(\sin\theta+\cos\theta)^2\geq4\sin\... |
→ → → → Browse Dissertations and Theses - Mathematics by Title
Now showing items 839-858 of 1147
(1999)What is the maximum number of edges in a multigraph on n vertices if every k-set spans at most r edges? We asymptotically determine this maximum for almost all k and r as n tends to infinity, thus giving a generalizat... |
In section 13.6 of Nakahara, the parity anomaly is in odd dimensional spacetime.
From the paper "Fermionic Path Integral And Topological Phases"
https://arxiv.org/abs/1508.04715
by Witten, the problem appears as one cannot define the sign of the path integral,
$$S[\bar{\psi},\psi;A]=\int d^{2n+1}x\bar{\psi}iD \!\!\!\!/... |
OpenCV #005 Averaging and Gaussian filter Digital Image Processing using OpenCV (Python & C++) Highlights: In this post, we will learn how to apply and use an Averaging and a Gaussian filter. We will also explain the main differences between these filters and how they affect the output image. What does make a good filt... |
Hyperbolic spiral
A plane transcendental curve whose equation in polar coordinates is
$$\rho=\frac a\phi.$$
It consists of two branches, which are symmetric with respect to a straight line $d$ (see Fig.). The pole is an asymptotic point. The asymptote is the straight line parallel to the polar axis at a distance $a$ fr... |
Multivariable Limits
1 Attachment(s)
Has anybody idea on what techniques I can apply on these limits?
at c and d I did direct substitution since there is no zero at the denominator. Is that correct?
a) I converted polar form and denominator became cos^2Theta+Sin^2theta = 1 so that limit exists
how about b?
no technique... |
Search
Now showing items 1-10 of 190
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 i... |
Equilibrium Means Detailed Balance Equilibrium and Detailed Balance Equlibrium has a very precise meaning in statistical physics, which also applies to biology. Equilibrium describes the average behavior (averaged over many systems under identical conditions) in which there is no net flow of material, probability or re... |
Let’s say that we have some classification dataset where items can be categorised as being in one of $N$ classes: $C_1, C_2, \ldots, C_N$. Each example in the dataset has a vector of features, $\boldsymbol{x}$, and a class, $C_k$. For many public datasets, the examples are balanced amongst classes such that $\Pr(C_1) =... |
Hello guys! I was wondering if you knew some books/articles that have a good introduction to convexity in the context of variational calculus (functional analysis). I was reading Young's "calculus of variations and optimal control theory" but I'm not that far into the book and I don't know if skipping chapters is a goo... |
Spectroscopy is the study of the interaction of radiation with matter.
We know that radiation of frequency υ consists of photons whose energy is given by Planck’s law: \(E = hv\). Where, \(h\) is Planck’s constant = \(6.626068 \times 10^{-34} \;m^2 kg/s\). The speed of the electromagnetic radiation in vacuum is \(c\), ... |
Given a directed graph $G = (V,E)$ with positive edge weights, find the minimum cost path between $s$ and $t$ that traverses exactly $k$ edges.
Here is my attempt using a flow network: \begin{align} \min \sum_{(i,j) \in E} c_{ij}x_{ij} \end{align} \begin{align} \sum_{j \in V} x_{ji} - \sum_{j \in V} x_{ij} &= \begin{ca... |
Architecture
RNNs are generally used for sequence modeling (e.g. language modeling, time series modeling,…).
Unfolding a RNN network (many to many scenario) could be presented as follow:
When training the model, we need to find W that minimizes the sum of losses.
Multilayer RNN
Multilayer RNN is a neural network with m... |
SciPost Submission Page Quantum robustness and phase transitions of the 3D Toric Code in a field by D. A. Reiss, K. P. Schmidt This is not the current version. Submission summary
As Contributors: Kai Phillip Schmidt Arxiv Link: https://arxiv.org/abs/1902.03908v1 Date submitted: 2019-02-14 Submitted by: Schmidt, Kai Phi... |
I am interested in the relation between the Atiyah-Patodi-Singer-$\eta$ invariant and spin topological quantum field theory. In the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling"
https://arxiv.org/abs/1602.04251
by Seiberg and Witten, they presented such a relation between the two.
Let the $... |
Mini-course on multiplicative functions Speaker(s):Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University) Location:MSRI: Simons Auditorium Tags/Keywords
Multiplicative functions
smooth numbers
prime numbers
Chowla's Conjecture
Primary Mathematics Subject Classification Secondary Mathematics Subject ... |
The derivation begins by expressing the problem (which is to find the minimum value of a functional \(S(q_j(x),q_j’(x),x)\)) in the language of single-variable calculus—meaning, we’ll want to express the functional \(S(q_j(x),q_j’(x),x)\) as a function of the single variable \(ε\) (which I’ll describe later) so that we... |
Learning Objectives
Make sure you thoroughly understand the following essential ideas:
Define Avogadro's number and explain why it is important to know. Define the mole. Be able to calculate the number of moles in a given mass of a substance, or the mass corresponding to a given number of moles. Define molecular weight... |
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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 \(... |
Question:
Evaluate {eq}\displaystyle \int xdx+ydy-zdz {/eq}
Integration:
Integration is the sum under the curve under defined limits which is given by:
{eq}\iiint f(x,y,z)dxdydz {/eq}
Answer and Explanation:
{eq}\int_c xdx+zdy-ydz\\ \text{Using Linearlity for integration}\\ \int(a f(x)+b g(x)) d x=a \int f(x) d x+b \in... |
A three-round Feistel network is a good example of a realistic construction that is a secure "weak" PRP, but not a "strong" PRP.
A Feistel network uses the permutation $P_f(L, R) = R, (L\oplus f(R))$, where $f$ is an element of a pseudorandom function family. This PRP will be keyed with three keys $k_1, k_2, k_3$, whic... |
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... |
Let:
$\left\{m_1, ~...~, m_k\right\}$ be a set of coprime natural numbers, $M=\prod_{i=1}^{k} m_i$ $X$ be a natural integer, such that $X < M$
Then $X$ can be expressed in the Residue Number System as:
$$X={\left(x_1, ~...~, x_k\right)}_{RNS\left(m_1, ~...~, m_k\right)}~.$$
Where $\forall_{m_i} \left[\left(x_i \equiv X... |
This question already has an answer here:
Riccati Equation in spot rate model 1 answer
I am trying to understand the derivation of the Cox-Ingersoll-Ross interest rate model. This has a stochastic differential equation of the form
$$dr=(\eta-\gamma r)dt + \sqrt{\alpha r} \space dX$$
With an affine solution of the form
... |
Evolvent of a plane curve involute
A curve $\bar\gamma$ assigned to the plane curve $\gamma$ such that $\gamma$ is the evolute of $\bar\gamma$. If $\mathbf{r} = \mathbf{r}(s)$ (where $s$ is the arc length parameter of $\gamma$) is the equation of $\gamma$, then the equation of its evolvent has the form $$ \bar{\mathbf{... |
You should be able to recognize from the form of the electronic Hamiltonian, Equation that the electronic Schrödinger equation, Equation, cannot be solved. The problem, as for the case of atoms, is the electron-electron repulsion terms. Approximations must be made, and these approximations are based on the idea of usin... |
https://doi.org/10.1351/goldbook.P04711
The ease of distortion of the electron cloud of a @M03986@ by an electric field (such as that due to the proximity of a charged @R05190@). It is experimentally measured as the ratio of induced @D01761@ (\(\mu _{\mathrm{ind}}\)) to the field \(E\) which induces it: \[\alpha =\frac... |
Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for
only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes?
My foundings:
Let $0 < T_0 < T_1 < \ldots < T_N$ be a tenor structure. Consider a payer swaption that gives the right to enter into a payer int... |
You can't make any concrete statements about the monotonicity, convexity or even sign of the yield curve.
Yields are almost always positive, and in the past (2007 and earlier) you could find people who would argue that yields
must be positive, typically using a no-arbitrage argument. But recent history has shown us tha... |
The average equilibrium temperature can be obtained from the Stefan-Boltzmann law, for your data 293.5 K (20 C). Compensating for the Earth-like atmosphere, (+15 K for Earth, closer to +12.5 K for this planet), we have an average temperature of approximately 306 K (33 C). Quite hot, as expected from a higher solar flux... |
Skills to Develop
Add and subtract complex numbers. Multiply and divide complex numbers. Solve quadratic equations with complex numbers
Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of... |
Waring problem
A problem in number theory formulated in 1770 by E. Waring in the following form: Any natural number is a sum of 4 squares, of 9 cubes and of 19 fourth-powers. In other words, for all $k\geq2$ there exists a $s=s(k)$, depending only on $k$, such that every natural number is the sum of $s$ $k$-th powers o... |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane... |
This is a welcome opportunity to discuss and clarify what statistical models mean and how we ought to think about them. Let's begin with definitions, so that the scope of this answer is in no doubt, and move on from there. To keep this post short, I will limit the examples and forgo all illustrations, trusting the read... |
Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian
1.
Université de Pau et des Pays de l'Adour, CNRS, E2S, LMAP UMR 5142, avenue de l'université, 64013 Pau cedex, France
2.
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016,... |
The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a fixed point - someone look into this
How should one go about dealing with this question?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in... |
@Secret et al hows this for a video game? OE Cake! fluid dynamics simulator! have been looking for something like this for yrs! just discovered it wanna try it out! anyone heard of it? anyone else wanna do some serious research on it? think it could be used to experiment with solitons=D
OE-Cake, OE-CAKE! or OE Cake is ... |
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... |
Following the book of Friedrich "Dirac operators and riemannian geometry" (AMS, vol 25), I define the generalized Seiberg-Witten equations for $(A,A',\psi , \phi)$, with $A,A'$ two connections and $\psi, \phi$, two spinors:
1)
$D_A ( \psi)=0$
2)
$D_{A'} ( \phi)=0$
3)
$F_+ (A)=-(1/4) \omega (\psi)$
4)
$F_+ (A')=-(1/4) \... |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane... |
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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... |
№ 8
All Issues Asymptotic Discontinuity of Smooth Solutions of Nonlinear $q$-Difference Equations Abstract
We investigate the asymptotic behavior of solutions of the simplest nonlinear
q-difference equations having the form x( qt+ 1) = f( x( t)), q> 1, t∈ R +. The study is based on a comparison of these equations with ... |
Table of Contents
Radial boundary values of lacunary power series Article References I. V. Andrusyak, P. V. Filevych 4-7
Symmetric continuous linear functionals on complex space $L_\infty[0,1]$ Article References T. V. Vasylyshyn 8-10
On the infinite remains of the Nörlund branched continued fraction for Appell hyperge... |
Emergent compact gauge fields are ubiquitous in condensed matter theory (like $U(1)$). Are there any examples of an emergent non-compact gauge field, in which case there won't be any quantization conditions and there would be conserved currents and charges which might or might not be physical.
Emergent gauge fields com... |
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... |
In this chapter we developed the quantum mechanical description of the harmonic oscillator for a diatomic molecule and applied it to the normal modes of molecular vibrations. We examined the functional form of the wavefunctions and the associated energy level structure. We can calculate expectation values (average valu... |
Let $f, g : U\rightarrow V$ be linear maps and $\lambda\in F$. Then the maps $f + g : U\rightarrow V$ and $\lambda f : U \rightarrow V$ are linear.
My attempt at the proof for the first statement is as follows:
Let $u,z\in U$ and $a\in F$, using a linearity check
by definition of $f + g$ $$(f + g)(au + z) = f(au + z) + ... |
I have $$ A = \left( \begin{array}{ccc} -4 & 9 & -4 \\ 0 & 0 & 0 \\ 6 & -13 & 6 \end{array} \right) $$ whose eigenvalues are $\{0,0,2\}$.
For $\lambda=2$, I have $(A-\lambda\,I)\,\vec{v^{(3)}} = \bf{0}$ leading to $v^{(3)}=\left( \begin{array}{c} 1 \\ 0 \\ -3/2 \end{array} \right)$.
For $\lambda_{1,2} = \{0,0\}$, I get... |
How to Compute the Acoustic Radiation Force
Acoustic radiation force is an important nonlinear acoustic phenomenon that manifests itself as a nonzero force exerted by acoustic fields on particles. Acoustic radiation is an acoustophoretic phenomenon, that is, the movement of objects by sound. One interesting example of ... |
So no one answered this question, so I finally went ahead and solved it in case anyone is every curious about this. If there are any errors, please let me know. Also, I copied/pasted this from my own Latex document, so I may have missed some of the translation to MathJax.
In the most general case, the terminating imped... |
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Now showing items 1-6 of 6
Pion, Kaon, and Proton Production in Central Pb--Pb Collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(American Physical Society, 2012-12)
In this Letter we report the first results on $\pi^\pm$, K$^\pm$, p and pbar production at mid-rapidity (|y|<0.5) in central Pb-Pb collisions at $\sqrt{s_{N... |
I'm study 't Hooft many instanton solutions of self-duality equation. In this method $A^a_\mu=-\bar{\eta}^{a}_{\mu\nu}\partial^\nu \ln{\Phi}$. After substitution in self-duality equation I've proven that
$$\Phi^{-1}\square{\Phi}=0$$
We can write the solution of this equation as $\Phi=1+\sum^q_{i=1}\frac{\lambda_i}{(x-x... |
GCD from Prime Decomposition/General Result Contents Theorem
Let $n \in \N$ be a natural number such that $n \ge 2$.
Let $\N_n$ be defined as:
$\N_n := \set {1, 2, \dotsc, n}$ $\displaystyle \forall i \in \N_n: a_i = \prod_{p_j \mathop \in T} {p_j}^{e_{i j} }$
where:
$T = \set {p_j: j \in \N_r}$
such that:
$\forall j \... |
This is a summary of the paper
On the Number of Distinct Languages Accepted by Finite Automata with n States. The paper provides relatively easy, yet far from tight, lower and upper bounds on the number of distinct languages accepted by NFA's. Their discussion on the number of distinct DFA's is very insightful, so I wi... |
Although microlensing incorporates a fairly large number of parameters, most events can be understood quite intuitively. This glossary is intended as a quick reference, particularly to disambiguate the different symbol sets used by different authors over time. Interested readers are referred to the references at the bo... |
Happy near year, and best wishes to those close and \(\varepsilon\)-far! December concluded the year with 4 new preprints, spanning quite a lot of the property testing landscape:
Testing Stability Properties in Graphical Hedonic Games, by Hendrik Fichtenberger and Anja Rey (arXiv). The authors of this paper consider th... |
Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China
$\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),&x ∈ Ω,t>0, \\... |
Suppose there is the (pseudo)scalar field $\hat{\theta}$ with non-zero VEV $\theta$, which effectively emerges at energy scale $\Lambda$ (for example, the mass of some fermion, the scale of SSB and so on). An example is axion-like field $\theta$, which is present as
$$ \int \frac{\theta}{f_{\gamma}}F\wedge F, \quad f_{... |
Inequality comes with a signs < (less than), > (greater than),\(( \leq ) \; (less \; than \; equal \; to), \; (\geq ) \; (greater \; than \; equal \; to)\)
In GMAT, the questions will be completed and will be asked with modulus function.
Steps to solve inequality question
1. Make an equation that corresponds to the ine... |
In Quantum Electrodynamics by Landau and Lifshiz there is the following:
The correspondence between the spinor $\zeta^{\alpha \dot{\beta}}$ and the 4-vector is a particular case of a general rule: any symmetrical spinor of rank $(k,k)$ is equivalent to a symmetrical 4-tensor of rank $k$ which is irreducible (i.e. which... |
Principle 1: Whenever you measure any physical quantity \(L\), there is a Hermitian linear operator \(\hat{L}\) (called an
observable) associated with that measurement.
Principle 2: Any arbitrary state of a quantum system is represented by a ket vector \(|\psi⟩\).
Principle 3: The possible measurable values of any quan... |
It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it?
Let $p$ be a prime congruent to 1 mod 4. Then to write $p = x... |
We’re seeing lots of papers in the summer. I guess the heat and sun (and more time?) is bringing out the good stuff. Distribution testing, from testing to streaming, hypergraph testing, and bunch of papers on graph testing. And just for the record: in what follows, \(n\) is the support size in distribution testing, it’... |
Synthesis of ATP by ATP synthase Making the Miracle Molecule: ATP Synthesis by the ATP Synthase
ATP is the most important energized molecule in the cell. ATP is an activated carrier that stores free energy because it is maintained out of equilibrium with its hydrolysis products, ADP and Pi. There is a strong tendency f... |
Document Type: Original Article
Author
Abstract
We show that if $T$ is a bounded linear operator on a complex Hilbert space, then
\begin{equation*} \frac{1}{2}\Vert T\Vert\leq \sqrt{\frac{w^2(T)}{2} + \frac{w(T)}{2}\sqrt{w^2(T) - c^2(T)}} \leq w(T), \end{equation*} where $w(\cdot)$ and $c(\cdot)$ are the numerical radi... |
OpenCV 4.1.1
Open Source Computer Vision
GMat cv::gapi::concatHor (const GMat &src1, const GMat &src2) Applies horizontal concatenation to given matrices. More... GMat cv::gapi::concatHor (const std::vector< GMat > &v) GMat cv::gapi::concatVert (const GMat &src1, const GMat &src2) Applies vertical concatenation to give... |
If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary?
Is it used or considered often in mathematics? How is it notated?
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Let $z=(... |
\section{Focused Sequent Calculus} Focused sequent calculus has several class of proposition\footnote{We are writing $\Neg{a},\Pos{a}$ for negative and positive atoms respectively. You may also see these written as $\Neg{P},\Pos{P}$.} and context. We summarize them below, noting that $\Pos{\Omega}$ ranges over ordered ... |
This is a basic question I haven't see answered anywhere and I can't seem to figure out.
The usual statement of the 1+1D chiral anomaly Ward identity is that the divergence of the chiral current is the background field strength:
$\partial_\mu \langle j^\mu\rangle = \epsilon^{\mu \nu} F_{\mu \nu}/2\pi.$
I want to rewrit... |
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... |
I am aware that the minimum variance portfolio of a market with $n$ securities can be shown to be:
\begin{equation} w^* = (1^T_n\Sigma^{-1}1_n)^{-1}\Sigma^{-1}1_n, \\ s.t. \ \ 1^T_nw = 1 \end{equation}
by using the method of Langrange multipliers or other. I am interested in demonstration of the extension: \begin{equat... |
Given a Turing machine $M$, we associate a partial function $f_M : \Sigma^{\ast} \to \Sigma^{\ast}$ to it (this is called the function computed by the machine), where $\Sigma$ denotes the finite input and output alphabet, defined as$$ f(u) = v :\Leftrightarrow \mbox{The machine halts on input $u$ with output $v$}.$$The... |
In file pohex2.dat the nodes on the left-hand vertical edge have been given Dirichlet boundary fixities; the mesh has then been refined, and doubled by reflection to the right and upwards, and finally triangulated, to create an H-shaped region of width 4.0 and heaight 3.0, containing 384 linear triangles, with Dirichle... |
Question:
Find the centroid of the thin plate bounded by the graphs of {eq}f(x) = x^{2} {/eq} and {eq}g(x) = x + 6 {/eq}.
Centroid of a Region Bounded by Two Functions
In getting the centroid of a region, integration method can be applied. If a region {eq}R {/eq} lies between two functions or curves defined by {eq}y = ... |
In a comment elsewhere you write that you're interested in understanding how quantum-mechanical theory describes the radiation that a hydrogen atom does and does not emit.In your question you ask about another answer that suggests some significance to the electron having zero total momentum; I think that's a feature of... |
Codeforces Round #553 (Div. 2) Finished
A girl named Sonya is studying in the scientific lyceum of the Kingdom of Kremland. The teacher of computer science (Sonya's favorite subject!) invented a task for her.
Given an array $$$a$$$ of length $$$n$$$, consisting only of the numbers $$$0$$$ and $$$1$$$, and the number $$... |
Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-sized and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,
1... |
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