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Fractions and binomial coefficients are common mathematical elements with similar characteristics - one number goes on top of another. This article explains how to typeset them in LaTeX.
Contents
Using fractions and binomial coefficients in an expression is straightforward.
The binomial coefficient is defined by the ne... |
I am reading V.I. Arnold's book "Mathematical methods of Classical mechanics". At the beginning of chapter 8 section B I can find the theorem (I limit the question to the $1$-dimensional case):
The cotangent bundle $T^*V$ has a natural sympletic structure. In local coordinates described above, this sympletic structure ... |
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Up a level 51. Article
Aad, G, Abbott, B, Abdallah, J et al. (2883 more authors) (2016)
Addendum to ‘Measurement of the tˉt production cross-section using eμ events with b-tagged jets in pp collisions at √s = 7 and 8 TeV with the ATLAS detector’. European Physical Journal C: Particles and Fields, 76. 6... |
Following the article's notation, I write $\mathcal{F}$ for thecategory of presheaves on a (suitable) category $\mathbb{F}$, $TV$ for thepresheaf of terms, $\delta$ for the context extension, and$\bullet$ for the product of the $\mathcal{F}$-monoid.
In Abstract Syntax and Variable Binding [Fiore, Plotkin, Turi], the au... |
In the first post in this series, I gave some background to the “Big Question” on limit operators which it appears that Lindner and Seidel have solved for the case of free abelian groups. In the next couple of posts I want to sketch some of the key ideas of their proof and to explore to what extent it can also be gener... |
Category: Ring theory Problem 624
Let $R$ and $R’$ be commutative rings and let $f:R\to R’$ be a ring homomorphism.
Let $I$ and $I’$ be ideals of $R$ and $R’$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I’)}=f^{-1}(\sqrt{I’})$
Add to solve later
(c) Suppose that $f$ i... |
Convolutions from a DSP perspective
I'm a bit late to this but still would like to share my perspective and insights. My background is theoretical physics and digital signal processing. In particular I studied wavelets and convolutions are almost in my backbone ;)
The way people in the deep learning community talk abou... |
Let $A$ be a nonempty subset, $k$ a positive integer such that $k \le |A|$, and $B = \{S \subseteq A ~|~ |S| = k \}$. Consider the action of $S_A$ on $B$ defined by $\sigma \cdot \{a_1,...,a_k\} = \{\sigma(a_1),...,\sigma(a_k)\}$. For what $k$ is this action faithful?
My conjecture is that the action is faithful if and... |
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a homomorphism of $A$ - modules defined as follows:
Every $b\in B$ acts on $B$ (when viewed as an $A$ - module) by multiplication.... |
Images are essential elements in most of the scientific documents. LaTeX provides several options to handle images and make them look exactly what you need. In this article is explained how to include images in the most common formats, how to shrink, enlarge and rotate them, and how to reference them within your docume... |
What is the history behind the choice of the name "Adam" as used in Adam: A Method for Stochastic Optimization?
On p.1 of the document you cite: "the name Adam is derived from adaptive moment estimation".
Although Nick already answered the question, I would like to elaborate a bit.
From the introduction of Adam: A Meth... |
I'm trying to solve the following exercise, but I'm totally new to Lagrangian mechanics and I don't have any solution
Let $P$ be a particle of mass $m$ in the space which is under the action of the gravity (w.r.t the $z$-axis) and of an elastic force w.r.t a fixed point $O$, and $K>0$ elastic constant
(i) Choose Lagran... |
We define a new class of total search problems as a subclass of Megiddo and Papadimitriou’s class of total $\NP$ search problems, in which solutions are verifiable in $\AC^0$. We denote this class $\forall\exists\AC^0$. We show that all total $\NP$ search problems are equivalent wrt. $\AC^0$-many-one reductions to sear... |
I'm working on question 7.4 of Chapter III.7 in Hartshorne's Algebraic Geometry. The question is about the cohomology class of a subvariety.
The setup is as follows: $X$ is an $n$-dimensional non-singular projective variety over an algebraically closed field $k$. $Y\subset X$ is a non-singular subvariety of codimension... |
By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is called a
difference set. The cardinality restrictions imply that each non-unit element $x\in C_n$ can be uniquely written as ... |
The physical 'meaning' of the imaginary part of the impedance is that it represents the energy storage part of the circuit element.
To see this, let the sinusoidal current $i = I\cos(\omega t)$ be the current through a series RL circuit.
The voltage across the combination is
$$v = Ri + L\frac{di}{dt} = RI\cos(\omega t)... |
How do you lift a body up from the ground?
The body remains at equilibrium under the acting of opposite forces on it: the $\uparrow$
normal force from the ground & the $\downarrow$ gravitational force . In order to move it up, you have to give some infinitesimal $\uparrow$ force greater than gravitational force which w... |
UPDATE 2Okay, I think I understand now why the defined CRLB is not applying to my use case.In my use case, we have very high SNR, and the the first signal $x_1$ is always the same. So the classical implementation in the literature of$$ x_1 = s(t)+n_1 (t)$$$$ x_2 = s(t-D)+n_2 (t)$$is not entirely correct, since this cas... |
Most often, the $S$-matrix is defined as an operator between asymptotic initial and final Hilbert spaces for a
time-dependent scattering process, i.e. between $t\to-\infty$ and $t\to\infty$. There unitarity encodes conservation of probabilities over time. On the other hand, the book that OP mentions, Ref. 1, talks abou... |
This equation is incredibly generic and describes many phenomena outside of nuclear phenomena.
For example: place the the following setup in a box (a spring and some bars) and weigh them:
.
Now, loosen the spring and repeat. You should measure a smaller mass because you've removed some of the energy.
In reality you cou... |
I will not give you the numerical solution, but I will below explain some analytical simplifications that I believe are required to solve the numerical problem. The strategy is simple: try to express all the parameters of the integral in term of dimensionless variables. To achieve a discussion in term of $\delta = \Del... |
Tagged: invertible matrix Problem 583
Consider the $2\times 2$ complex matrix
\[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$.
Add to solve later
(d) Using the result of the diagonalization, c... |
In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras to the category of $L(m)$-algebras. Then there is a left adjoint $\mathcal{U}_L$for that functor. They define the
univers... |
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Now showing items 1-10 of 19
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... |
I have a fully-connected graph $G=(V,E)$ with edge weights $w(v)\in\mathbb{R};v\in V$ and I need to find a spanning tree $T=(V_t\subseteq V,E_t\subseteq E)$ where the set of edge weights in the tree have as close as possible to equal separation. We quantify the notion of
equal separation by minimizing the variance of d... |
I am aware of the derivation of the Fresnel reflection and transmission coefficients for a single plane wave incident on a planar interface between two dielectrics. So I wondered what happens if there are two plane waves of the same frequency incident on a planar interface from opposite sides, with different angles of ... |
From Bernoulli's principle we know for an incompressible fluid (constant density $\rho$) in a gravitational potential $\psi=gz$, that we can state the equation along a streamline from point 1 to point 2:
$$\int_1^2\dfrac{\partial u}{\partial t}ds+\frac{u_2^2}{2}+\frac{p_2}{\rho}+gz_2=\frac{u_1^2}{2}+\frac{p_1}{\rho}+gz... |
Help:Wiki Mark-up Wiki Markup
In the left column of the table below, you can see what effects are possible. In the right column, you can see how those effects were achieved. In other words, to make text look like it looks in the left column, type it in the format you see in the right column.
You may want to keep this p... |
If not, then what does it mean when for some state $q$ and some symbol $a$, $\delta(q, a)$ does not exist?
You seem to have stumbled on a contentious issue. Apparently computer scientists like to argue. I certainly like to argue, so here goes!
My answer is an unequivocal: No. A deterministic finite automata does not ne... |
In this problem the random variable is theta and according to the formula there should be two integrations but in the solution there is only one . Nor am i able to understand the meaning of x1 and x2 ...
I am studying the role of an auto-correlation matrix for random signals and the difference of energy between a lag 0... |
Oscillations and Waves Forced, Damped oscillations and Resonance The oscillations of a body whose amplitude goes on decreasing with time are defined as damped oscillations where F = −bV where 'b' is damping constant. In forced oscillations the amplitude of oscillations decreases exponentially due to damping force like ... |
It is common to employ several clock signals in a digital system. Since the clock signals of different clock domains are independent in general, transferring data between the different clock domains can be a challenging task. This article will discuss a well-known technique called “double flopping” to transfer a single... |
I am solving the next exercise, but I have spent a lot of time and I can´t. For random vectors, $X_1,X_2,...\in\mathbb{R}^p$ The sample covariance with $\Sigma_1=0$ is given by: $$\hat\Sigma_n=\frac{1}{n-1}\sum_{i=1}^n(X_i-\hat{\mu_n})(X_i-\hat{\mu_n})^T$$ Prove that $\hat\Sigma_n=\frac{n-1}{n-2}\hat\Sigma_{n-1}+\frac{... |
Supporting Information Discrete Semiconductor Circuits: Differential Amplifier Discrete Semiconductor Circuits: Simple Op-Amp Insulated-Gate Field-Effect Transistors (MOSFET)
Differential or Single-Ended?
Introductory studies of active circuits generally devote a significant amount of time to standard single-ended ampl... |
In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon the field and its derivatives. When we derive Euler-Lagrange equation of motion we vary the action,
In particle mechanic... |
Solutions Van't Hoff Factor and Modified Colligative Properties Abnormal colligative properties:- When solute particle associated or dissociated in solvent. (a) i = Vant Hoff factor = \tt \frac{Actual \ moles \ of \ solute}{Moles \ of \ solute \ without \ dissociation \ or \ association} (b) \tt i = \frac{observed \ (o... |
We can treat this sum using the method of
annihilated coefficientextractors.
Introduce the generating function$$Q(z) = \sum_{m\ge 0} \frac{z^m}{m!} \sum_{j=0}^m j! \times {m \brace j} \times {x\choose j}$$
We require the bivariate generating function of the Stirling numbersof the second kind.
Recall the species for set... |
Is antigravity the source of accelerating expansion(dark energy)?
From the observation of 1998, we found that our universe has been continuing accelerating expansion, and the unknown cause for this accelerating expansion was named dark energy.
However, nobody is sure whether dark energy truly originates from antigravit... |
In this chapter, we will understand in detail regarding the Robertson-Walker Metric.
Suppose a photon is emitted from a distant galaxy. The space is forward for photon in all directions. The expansion of the universe is in all the directions. Let us see how the scale factor changes with time in the following steps.
Ste... |
I'm implementing Fourier transformation in my analysis and I wanted dig a bit deeper on the reasons why the absolute value of Fourier transformation is usually multiplied by the constant $2/N$ to get the peak amplitude value of a sinewave with certain frequency.
In the book Understanding Digital Signal Processing by Ly... |
Qualitative properties of solutions for an integral equation
1.
Department of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033, United States
2.
Department of Applied Mathematics, University of Colorado at Boulder
3.
Department of Mathematics, University of Toledo, Toledo OH 43606
$ u(x) = \int_{... |
I was trying to learn some more window functions and was referencing the Wikipedia article on the subject. I've noticed on a lot of the plots, there appears to be a capitalized and italicized letter B. What is this parameter? I can't find any reference to it on the page. link to wikipedia article And here's a photo tha... |
Structure of the General Solution
The nonhomogeneous differential equation of this type has the form
\[{y^{\prime\prime} + py’ + qy }={ f\left( x \right),}\]
where \(p, q\) are constant numbers (that can be both as real as complex numbers). For each equation we can write the related homogeneous or complementary equatio... |
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Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions... |
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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 ... |
Suppose I have a uniform magnetic field through all of space $$\textbf{B}(x,y,z)=\hat{\textbf{z}}$$ and a charge $q$ moving at a velocity of $v\hat{\textbf{x}}$. In this frame of reference, a magnetic force will act on $q$, pushing it up (or down, but let's assume up). Suppose instead I change my frame of reference so ... |
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hep-ph
Change to browse by: Bookmark(what is this?) High Energy Physics - Phenomenology Title: Inflection-point Higgs Inflation
(Submitted on 28 Oct 2016 (v1), last revised 24 Feb 2017 (this version, v3))
Abstract: Inflection-point inflation is an interesting possibility to realize a successful ... |
Following David Silver's course, I came across the actor-critic policy improvement algorithm family.
It holds For one-step Markov decision processes that
$$\nabla_{\theta}J(\theta) = \mathbb{E}_{\pi_{\theta}}[log \pi_{\theta}(s,a)*r]$$
where $J$ is the MDP's value function, and $\pi_\theta$ is the policy parameterized ... |
During malaria, why don't Kupffer cells (hepatic macrophages) attack the
Plasmodium sporozoites and stop schizogony, thus saving us from the disease?
Indeed the process is even more interesting than thought:
sporozoites actively attack Kupffer cells and let themselves get eaten! In a 2001 research, Pradel et al found t... |
FINAL EDIT : This edit cleans up the first proof (and simplifies it -- there are no longer any references to the free nilpotent group) and adds some remarks to the second proof following the discussion in the comments.
PROOF 1.
Here's a low-tech way to see that a surface group is not free (though cohomology is secretly... |
Sanaris's answer is a great, succinct list of what each term in the free energy expression stands for: I'm going to concentrate on the $T\,S$ term (which you likely find the most mysterious) and hopefully give a little more physical intuition. Let's also think of a chemical or other reaction, so that we can concretely ... |
Source:
Chapter 2 of Classical Dynamics by David Tong (p14)
In the text the author attempts to derive the equations of motion for the system of a free particle in a frame rotating with angular velocity $\,\boldsymbol{\omega} = (0,0,\omega)$ about the $z$ axis. In the new frame our new/primed coordinates are related to ... |
LaTeX & TeX Information LaTeX / TeX
TeX Users Group Home Page (TUG)
The Comprehensive TeX Archive Network (CTAN) Latex/Tex resources page from Goddard Institute for Space Studies A list of sites related to TeX, LaTeX, mathematics, and teaching from the software publisher MacKichan Software, Inc. This is a very extensiv... |
Hi everyone!
I have a problem at hand that needs to calculate a series of covariant derivatives like
\nabla_{\alpha} \nabla_{\beta} \nabla_{\gamma} \nabla_{\lambda}\partial_{\nu}\phi
Where \nabla is the usual covariant derivative in GR and \partial is the usual partial derivative.
In fact, this expression is a term of ... |
So, I skimmed through the papers, and this is what I gleamed. If there is anyone more knowledgable in the subject-matter, please correct me if I'm wrong (or add your own answer, and I will accept it instead!).Links to each paper can be found in the question-post, above.Simple recalculationsD* (aka Dynamic A*) (1994): O... |
I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions". Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. Let $X$ be a complex compact 3-fold with trivial canonical bundle such that the $\partial\bar{\partial}$-lemma holds, (fo... |
The electrochemical potential of an ion i in an electrolytic solvent is given by:
\begin{align}\mu_i(\vec r) &= \mu_i^0 + RT \ln(a_i(\vec r))\\& = \mu_i^0 + RT \ln(\gamma_i(\vec r) c_i(\vec r)/c_0) \\&= \mu_i^0 + RT \ln(c_i(\vec r)/c_0) + RT \ln(\gamma_i(\vec r)) \\&= \mu_i^0 + RT \ln(c_i(\vec r)/c_0)+z_i e \phi(\vec r... |
First a demonstration that the squares of both$$\begin{align}&[\dots, 0, 0, 1,\hphantom{-}1, 0, 0, \dots] \text{ and}\\&[\dots, 0, 0, 1, -1, 0, 0, \dots]\end{align}$$
equal
$$[\dots, 0, 0, 1, \hphantom{-}1, 0, 0, \dots]\hphantom{\text{ and}}$$
but the squares of their sinc interpolations differ (Fig. 1):
Figure 1. Squa... |
Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is
$$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \theta \dot{\phi}^2 $$
The canonical momenta are $$P_\theta = \frac{\partial L }{\partial \dot{\theta }} = m R^2 \dot{\theta ... |
M Jutila
Articles written in Proceedings – Mathematical Sciences
Volume 97 Issue 1-3 December 1987 pp 157-166
Let τ(n) be the arithmetical function of Ramanujan, α any real number, and x≥2. The uniform estimate$$\mathop \Sigma \limits_{n \leqslant x} \tau (n)e(n\alpha ) \ll x^6 \log x$$ is a classical result of J R Wil... |
I think many of the other answers boil down to the same underlying idea: Sections of vector bundles are "generalized functions" or "twisted functions" on your manifold/variety/whatever.
For example, Charles mentions subvarieties, which are roughly "zero loci of functions". However, there are no non-constant holomorphic... |
Definition of Linear Equation of First Order
A differential equation of type
\[y’ + a\left( x \right)y = f\left( x \right),\]
where \(a\left( x \right)\) and \(f\left( x \right)\) are continuous functions of \(x,\) is called a linear nonhomogeneous differential equation of first order. We consider two methods of solvin... |
Analysis of Rugates
\(n(x)=n_a+0.5 n_p A(x) \sin\left(\displaystyle\frac{4\pi x}{\lambda_0}\right)\)
\(A(x)=10 t^3-15 t^4+6 t^5 \)
\(t=\displaystyle\frac{2x}{TOT},\;\;\mbox{ if}\;\; x<\frac{TOT}{2} \)
\(t=\displaystyle\frac{2(TOT-x)}{TOT},\;\;\mbox{ if}\;\; x>\frac{TOT}{2}\)
Here
is total optical thickness. TOT
Design ... |
First formula :
Here are two different proofs :
Method 1 :
Let $u \cdot v$ denote the dot product of $u$ and $v$.
Using expansion :
$$(n \cdot (a-b))^2 = (n \cdot a)^2 + (n \cdot b)^2 - 2 (n \cdot a)(n \cdot b),$$
we can express :
$$(n \cdot a)(n \cdot b) = \frac12 \left((n \cdot a)^2 + (n \cdot b)^2 - (n \cdot (a-b))^... |
As a beginner in signal processing, i'll try to explain my problem most thoroughly. I'll firstly explain the idea of the transformation the real valued data has undergone. This includes the ideas and reasons why it does work and the way to revert the procedure. Then i'll explain my difficulties in reverting it.
First o... |
I'm working on a numerical optimization problem that naturally lives on the Grassmann Manifold Gr$_N(\mathbb{C^M})$, however the objective function is defined on the alternating algebra given by the Plucker embedding $\Omega: $Gr$_N(\mathbb{C^M}) \rightarrow \mathbb{P}(\wedge^N \mathbb{C}^N)$.
A lot work has been done ... |
I have seen similar post asking for interpretation of the Maurer-Cartan form, but I am still struggling to understand it, so let me try to work a specific example and pose a specific question.
Let $G$ be the group containing all matrices of the form
\begin{bmatrix} x &y \\ 0 &1/x \end{bmatrix}
where $x \neq 0$.
I read ... |
Using the Euler product,
$$\zeta(s)=\prod_p \big(1-p^{-s}\big)^{-1},$$
this identity amounts to rearranging polynomials. Indeed, we have
$$\frac{\zeta(s)\zeta(2)}{\zeta(2s)}=\prod_p \frac{(1-p^{-2s})}{(1-p^{-s})(1-p^{-2})}=\prod_p\frac{1+p^{-s}}{1-p^{-2}}$$
and
$$\prod_p \frac{1-p\cdots\pm p^{s-1}}{p^{s-1}-p^{s-2}}=\pr... |
First of all, I would suggest you to read the comments I have made in the Danu's answer to check whether I have understood your question or not.
See,$\oint B\cdot d\ell~=~ \mu_0I$ has been derived
only on the basis of $\vec{\nabla}\times \vec{B}=\mu_0 \vec{J}$. But actually the Maxwell equation is $\vec{\nabla}\times \... |
Let’s start by recalling some background about
modules.
Suppose that \(R\) is a ring and \(1_R\) is its multiplicative identity. A left
\(R\)-module \(M\) consists of an abelian group \((M, +)\) and an operation \(R \times M \rightarrow M\) such that for all \(r, s \in R\) and \(x, y \in M\), we have: \(r \cdot (x+y)= ... |
Null hypothesis testing is a procedure to evaluate the strength of evidence against a null hypothesis. Given/assuming the null hypothesis is true, we evaluate the likelihood of obtaining the observed evidence or more extreme, when the study is on a randomly-selected representative sample. The null hypothesis assumes no... |
A general class of such groups are maximal subgroups $H<\Gamma$ of infinite index. Such a subgroup $H$ must surject any finite quotient of $\Gamma$. For finitely generated linear groups like $SL_n(\mathbb{Z})$ which are not virtually solvable, maximal subgroups of infinite index were constructed by Margulis-Soifer. I d... |
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... |
30 4 Homework Statement In the figure below an electron is shot directly toward the center of a large metal plate that has surface charge density -1.20 × 10^-6 C/m2. If the initial kinetic energy of the electron is 1.60 × 10^-17 J and if the electron is to stop (due to electrostatic repulsion from the plate) just as it... |
Legacy code is a concern for any company with a reasonably size team pretty quickly. In this article Wayne Lobb from Foliage provides us with industry metrics with respect to code growth (Unfortunately need to register to download). The quick summary is that on average a developer can either create…
Blog
The first assu... |
I am in the following setting:
Let $X=[0,1]$, $Y=[0,1]^2$ and $Z=Y^n/\sim$, where $\sim$ is the equivalence relation given by $(x_1,\ldots,x_n)\sim (y_1,\ldots,y_n)$ if and only if there is a permutation $\sigma$ on $\{1,\ldots,n\}$ with $x_i=y_{\sigma(i)}$. Let $\pi\colon Y\to Z$ be the quotient map.
Does every path $... |
We consider real-valued functions.
A real-valued function \(f : I \to \mathbb R\) (where \(I \subseteq\) is an interval) is continuous at \(x_0 \in I\) when: \[(\forall \epsilon > 0) (\exists \delta > 0)(\forall x \in I)(\vert x- x_0 \vert \le \delta \Rightarrow \vert f(x)- f(x_0) \vert \le \epsilon).\] When \(f\) is c... |
Assumption 2.1 - If the payoff $P$ of a financial instrument is non negative, then the price $p$ of the financial instrument is non negative.
Assume $C$ is just the price of the call option, and $C^{*}$ is also the price of a different call option with the same parameters as $C$ except that $\tau = T - t$ where $T$ is ... |
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Physical Review Letters, ISSN 0031-9007, 03/2018, Volume 120, Issue 12, p. 121801
Journal Article
2. Observation of J /ψφ Structures Consistent with Exotic States from Am... |
Hopfield neural network was invented by Dr. John J. Hopfield in 1982. It consists of a single layer which contains one or more fully connected recurrent neurons. The Hopfield network is commonly used for auto-association and optimization tasks.
A Hopfield network which operates in a discrete line fashion or in other wo... |
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
We know that $$\mathsf{NP\subseteq P/Poly \iff coNP\subseteq P/Poly\implies PH=\Sigma_2^P}=\mathsf{\Pi_2^P}$$ $$\mathsf{NP\subseteq P/Log\iff coNP\subseteq P/Log\implies PH=\Sigma_0^P=\Pi_0^P}$$
Is there any circuit result that could imply $$\mathsf{PH=\Sigma_1^P=\Pi_1^P}$$ but still $$\mathsf{PH\neq\Sigma_0^P=\Pi_0^P}... |
We assume a Black-Scholes world
except the dynamics of the stock price, namely: No arbitrage opportunities. No dividend payments from the stock. Existence of a riskless asset yielding the risk free rate $-$ which here we assume non-constant, $(r_t)_{t \geq 0}$. Possibility to borrow and lend infinitely at the risk-free... |
Consider standard FLRW cosmology. Usually, the relation between energy density $\rho$ and pressure $p$ of a cosmological fluid component is
linear:\begin{equation}\tag{1}p = w \, \rho,\end{equation}where $w$ is just a constant ($w = 0$ for the dust model, $w = \frac{1}{3}$ for ultra-relativistic radiation, $w = - 1$ fo... |
My question is basically given three languages A, B and L, where L is A and B concatenated together and B is proven to be non regular, is it possible to find an A that makes L regular?
If we allow $A$ to be the empty language, which is regular, then we have that $L = \{w_1w_2 | w_1 \in A, w_2 \in B\} = \emptyset = A$.
... |
Complex Numbers and Quadratic Equations The Modulus and the Conjugate of a Complex Number Tips: Conjugate of a complex number a + ib is defined as a – ib Geometrically the conjugate of ‘z’ is the reflection or point image of ‘z’ in the real axis. \tt z = \overline{z} if and only if ‘z’ is purely real if and only if ‘z’... |
\(\text{FIGURE V.24A}\)
The potential
outside a solid sphere is just the same as if all the mass were concentrated at a point in the centre. This is so, even if the density is not uniform, and long as it is spherically distributed. We are going to find the potential at a point \(\text{P}\) inside a uniform sphere of ra... |
I've been researching the question for quite some time, as I understand it the phase function is actually an approximation due to the particle-wave duality inherent in participating media such as the atmosphere or anything else.
As they are reemitted from particles, some EM waves mingle with their neighbours and amplif... |
Facial Recognition & Verification using Convolutional Neural Network
Understanding the algorithm behind the Facial Recognition & Facial Verification technologies and the associated loss functions and technical details. I will also be building a code from scratch (will be posted separately - this post is mostly algorith... |
What you can do, is to employ a method called rejection sampling:Flip the coin 3 times and interpret each flip as a bit (0 or 1).Concatenate the 3 bits, giving a binary number in $[0,7]$.If the number is in $[1,6]$, take it as a die roll.Otherwise, i.e. if the result is $0$ or $7$, repeat the flips.Since $\frac 68$ of ... |
Notes takes for complexity module of CSCB63 course at UTSC.
Ordered Dictionary
Finite map from keys to values, assume keys are comparable.
Some operations supported:
insert(k, v) lookup(k) delete(k)
Ordered set is the same, just with no values, only keys.
If we implement this using a binary tree, lookup can be very slo... |
The so-called tyranny of the rocket equation, which becomes clear by examining Tsiolkovsky rocket equation, defining maximum change in velocity $\Delta v$ as:
$$\Delta v = v_\text{e} \ln {\left(\frac{m_0}{m_1}\right)}$$
where $m_0$ is the wet mass, $m_1$ dry mass, and $v_\text{e}$ effective exhaust velocity of reaction... |
If I understood correctly, you want an
itemize environment each of whose items consists of a labelled display equation followed by a short description. And you want the bullet/dash to be aligned with the first line of the description.
Here is one way to do precisely that by encapsulating the equation and the descriptio... |
Side of the base: \(a\)
Lateral edge: \(b\) Height of a pyramid: \(h\) Slant height: \(m\) Number of sides in the base: \(n\) Volume: \(V\)
Lateral edge: \(b\)
Height of a pyramid: \(h\)
Slant height: \(m\)
Number of sides in the base: \(n\)
Volume: \(V\)
Radius of the inscribed circle in the base: \(r\)
Semiperimeter ... |
Consider the PDE,
$$\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}\bigg(\sin{\theta}\frac{\partial u}{\partial\theta}\bigg)+\frac{1}{\sin^2{\theta}}\frac{\partial^2 u}{\partial\phi^2}=-\lambda u.$$ We can do separation of variables to obtain the following ODEs: $$\frac{1}{f}\bigg(\sin{\theta}\frac{d}{d\theta}\bi... |
I believe that the authors, of the reference you provided, explain the reason behind introducing these so-called quasi-Fermi levels at the end of section 4.3.3. For simplicity, let me just repeat it here; perhaps explaining it in different words, with a little more elaboration, would help. I’m sure you're aware of the ... |
Side of a regular hexagon: \(a\)
Interior angle: \(\alpha\) Apothem of a regular hexagon: \(m\) Area: \(S\)
Interior angle: \(\alpha\)
Apothem of a regular hexagon: \(m\)
Area: \(S\)
Radius of the inscribed circle: \(r\)
Radius of the circumscribed circle: \(R\) Perimeter: \(P\) Semiperimeter: \(p\)
Radius of the circu... |
Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form: \begin{equation} dX_t = f(X_t)dt + a dW_t \end{equation} where $a$ is a constant, and $f$ is some function.
Yes. Unexpected weak solutions to the SDE... |
For the case of two particles colliding totally inelastically, conservation of momentum gives:
\[m_{1} v_{1}+m_{2} v_{2}=\left(m_{1}+m_{2}\right) v_{\mathrm{f}}\]
If the masses and initial velocities of the particles are known, calculating the final velocity of the composite particle is thus straightforward.
4.6.1. Wor... |
The language Critical SAT is defined as the set of $CNF$ boolean formulas $f$ such that $f \in UNSAT$ but removing any clause from $f$ makes it satisfiable. It is known that Critical SAT is $DP$-complete. I wonder about the following variant: given a $CNF$ formula $f$, is it the case that $f$ is in $UNSAT$ and that the... |
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