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How should I make someone understand the term log or logarithm and how its values are determined who is standard VI? like:
$$\log_2 8 = 3$$
or
$$\log_{2x+5}(10x^2+29x+10)=5−\log_{5x+2}(4x^2+20x+25)$$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in rela... |
My friends working on Thermalization of Black Holes explained solutions to their matrix-valued differential equations (from numerical implementation of the Berenstein-Maldacena-Nastase matrix model) result in chaotic solutions. They are literally getting random matrices. For the eigenvalue spectrum, would expect a semi... |
Let $K$ be a field and $f,g$ be (1 variable) polynomials over $K$, and suppose that $g=p_{1}^{e_1} p_{2}^{e_2} \cdots p_{k}^{e_k}$ where each $p_i$ is irreducible over $K$ and $e_i \geq 1$. Does there exist polynomials $b$ and $a_{ij}$ with the following properties?
$\deg{a_{ij}}<\deg{p_i}$ for all $i=1,\ldots,k$ $\dis... |
Is there a math equivalent of the ternary conditional operator as used in programming?
a = b + (c > 0 ? 1 : 2)
The above means that if $c$ is greater than $0$ then $a = b + 1$, otherwise $a = b + 2$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in rela... |
Resistance
\[R\], measured in Ohms
\[\Omega\]is expressed as the ratio of the driving force - the voltage
\[V\], measured in Volts, V - to the current
\[I\], measured in Amps, A:
\[R=\frac{V}{I}\].
The resistance
\[R\]of an object is itself proportional to the length
\[l\]of the object and inversely proportional to the... |
On page 48 of Calculus on Manifolds Spivak defines (Riemann) integration over
rectangles $[a_{1},b_{1}]\times\cdots\times[a_{n},b_{n}]\subset\mathbb{R}^{n}$. Then on page 55 he extends this integral to bounded subsets $C\subset\mathbb{R}^{n}$ via characteristic functions. This integral is the usual one.
Then, on page 6... |
The idea is using a "covariant" formalism in classical mechanics. I do not know how accommodate your equations in particular, but a good notion of
covariant derivative that physically encompasses the "inertial forces" (as in GR) is at disposal also in classical physics.
Consider the classical spacetime $M$. There there... |
I am currently trying to perform MCMC sampling using a (stochastic) model, for which I cannot derive a likelihood function, but which allows me to draw samples $y_\theta \sim p_{y|\theta}$, where $p_{y|\theta}$ is the distribution defined by the model with parameters $\theta$.
I already looked at several papers for lik... |
I was looking through an old schematic and found two symbols that I didn't recognize:
Is this a PNP transistor? looking up the model number doesn't give much information.
Is this some kind of variable resistor or variable inductor?
Electrical Engineering Stack Exchange is a question and answer site for electronics and ... |
I'm having difficulty proving Leibniz' formula for weak derivatives as presented in Evans PDE. I was hoping that someone could maybe shed some light on it for me? To be honest I'm not entirely comfortable with weak derivative concept. I'm actually an economist by trade hoping to improve my maths. My question is very cl... |
I am working on an model of a permanent magnet synchronous machine. Right now I am stuck with calculating the eddy current losses caused by the harmonics of the stator magnetic field in the electrical steel of the rotor. Or to put it differently. How do I calculate the eddy current in electric steel at high frequencies... |
Any family of language that is a trio isclosed under interleaving with a regular set.
This includes of course interleaving of 2 regular sets, since regular sets form a trio.
Proving the result (and more) only with closure properties
Note: I created the definitions below for the purpose of thisquestion. I do not know wh... |
Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin.
How would you prove that the Earth is, in fact, round?
Physics Stack Exchange is a question and answer site for active researchers,... |
Are mobile phone masts bad for health?
Character Questionnaire An Introduction to E-Cigarettes Why I Hate DRM Debugging The Manuscript On Being Stuck Writing and Roleplay Murphy's Law Apocryphal Tales About Rebuilding the Garden Pond About Me Pictures Of Cats Cosmology for the Layman Magnatune Demo Recommendations Hard... |
I came across John Duffield Quantum Computing SE via this hot question. I was curious to see an account with 1 reputation and a question with hundreds of upvotes.It turned out that the reason why he has so little reputation despite a massively popular question is that he was suspended.May I ...
@Nelimee Do we need to m... |
Analytic function Set
context $\mathcal O\subset \mathbb C$ definiendum $f\in \mathrm{it}$ inclusion $f:\mathcal O\to\mathbb C$ for all $c$ … series in $\mathbb C$ todo, roughly $\exists c.\ \forall z.\ f(z)=\sum_{n=-\infty}^\infty c_n\,z^n$ Discussion
Picture a continuous function $f:\mathbb R^2\to\mathbb R$ as a surf... |
Difference between revisions of "Lebesgue space"
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A [[Measure space|measure space]]
+
A [[Measure space|measure space]] (where is a set, is a -algebra of subsets of , called measurable sets, and is a measure defined on the measurable sets), isomorphic to the... |
This post is based on problems 2.10 and 2.11 in, “Heard on the Street” by Timothy Falcon Crack. I was asked how to price a digital option in a job interview - and had no idea what to do!
European Call Options
A European call option is the right to buy an asset at the strike price, , on the option’s expiration date, . A... |
Given an ordinal $\alpha$ let $\operatorname{Fn}(\omega, \alpha)$ be the set finite partial functions from $\omega$ to $\alpha$. Given a cardinal $\kappa$ let $${\prod_{\alpha<\kappa}}^{\text{fin}}\operatorname{Fn}(\omega,\alpha) = \{\vec{p}\in {\textstyle \prod_{\alpha<\kappa}}Fn(\omega,\alpha): |\{\alpha<\kappa: p_\a... |
So I want to prove that $L$ is continuous in the strong topologies, ie, $L$ takes bounded sets to bounded sets.
Using bounded sets is a good plan. We have the fact that a continuous linear map between topological vector spaces maps bounded sets to bounded sets: Let $E,F$ be topological vector spaces (real or complex), ... |
The Hamiltonian operator of Loop quantum gravity is a totally constraint system
$$H = \int_\Sigma d^3x\ (N\mathcal{H}+N^a V_a+G)$$
Here, $\Sigma$ is a 3-dimensional hypersurface; a slice of spacetime. Moreover, $\mathcal{H}$ is the Hamiltonian constraint, $V_a$ the diffeomorphism constraint, $G$ the Gauss law term and ... |
Quality factor, or \(Q\), is one of the more mysterious quantities of seismology. It's right up there with Lamé's \(\lambda\) and Thomsen's \(\gamma\). For one thing, it's wrapped up with the idea of attenuation, and sometimes the terms \(Q\) and 'attenuation' are bandied about seemingly interchangeably. For another th... |
Frequency Translation by Way of Lowpass FIR Filtering
Some weeks ago a question appeared on the dsp.related Forum regarding the notion of translating a signal down in frequency and lowpass filtering in a
single operation [1]. It is possible to implement such a process by embedding a discrete cosine sequence's values wi... |
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The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
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Now showing items 1-10 of 26
Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
This question is largely about definitions of PCA/FA, so opinions might differ. My opinion is that PCA+varimax should not be called either PCA or FA, bur rather explicitly referred to e.g. as "varimax-rotated PCA".
I should add that this is quite a confusing topic. In this answer I want to explain what a rotation actua... |
I'm not entirely sure of the extent of their power, whether it is limited to simple compound manipulation, but theoretically, the amount of energy in a 10m sphere is more than he will ever need.
Matter can be converted directly to energy, for example, during matter-antimatter collisions, so if we know the amount of mat... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
This post is about Neural Combinatorial Optimization that tackles NP-hard combinatorial optimization problems using recurrent neural networks and reinforcement learning.
Most NP-hard combinatorial problems are tackled by algorithms that rely on handcrafted heuristics. The performances of such heuristics vary widly depe... |
NTS ABSTRACTSpring2019
Return to [1]
Jan 23
Yunqing Tang Reductions of abelian surfaces over global function fields For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the pr... |
I was told today by someone smarter than myself that the time-dependent Schroedinger equation in one dimension was invariant under a Galilean transformation of $(x,t)$, namely under
$$\begin{cases}x'=x+ut\\t'=t\end{cases}.\tag{1}$$
Going to check this, I looked at the time dependent Schroedinger equation of a free part... |
Disclaimer: This answer derives the prices of two different binary options within the Black/Scholes framework. Note that this is not an appropriate valuation model to use for non-European contracts in most real-world markets.
Up-and-In Binary Call
After reading your question for a second time, I agree with Quantuple's ... |
ä is in the extended latin block and n is in the basic latin block so there is a transition there, but you would have hoped \setTransitionsForLatin would have not inserted any code at that point as both those blocks are listed as part of the latin block, but apparently not.... — David Carlisle12 secs ago
@egreg you are... |
The simplest malaria model is given by $$\frac{dI}{dt} = \frac{\alpha \beta I}{\alpha I + r} (1-I) - \mu I$$ where $\mu$ is the death rate of humans, $\alpha$ is the transmission rate from humans to mosquitoes, $\beta$ is the transmission rate from infected mosquitoes to susceptible humans, and $r$ is the natural death... |
Define the vector operator: $$ H(\textbf{x}) \equiv \alpha \textbf{n} \times\textbf{x}$$ For unit vector $\textbf{n}$ and some constant $\alpha$. We define further the operator: $$G \equiv I + H + \frac{H^2}{2!} + \frac{H^3}{3!} + ... $$ Where powers of $H$ represent iterations and $I$ is defined as the identity such t... |
Why is $\arctan\frac{x+y}{1-xy} = \arctan x +\arctan y$?
It is said that this is derived from trigonometry, but I couldn't find why this is the case.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign... |
I kind of have an elementary solution, it seems to be fine but I am not sure if everything is correct; please point out the mistake(s) I'm making, if any.
Define $$H_n:=\sum_{i=1}^n \frac{1}{i}$$Since $0<H_n<n$, if $\exists$ some $n$ for which $H_n$ is integral then $H_n=k$ where $0<k<n$. Then $$H_n=k=1+\frac{1}{2}+\fr... |
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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... |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphe... |
Added: a Stanford course on neural networks,cs231n,gives yet another form of the steps:
v = mu * v_prev - learning_rate * gradient(x) # GD + momentum
v_nesterov = v + mu * (v - v_prev) # keep going, extrapolate
x += v_nesterov
Here
v is velocity aka step aka state,and
mu is a momentum factor, typically 0.9 or so.(
v,
x... |
Bravais Lattice¶
Non-Scalar Structural
In crystallography, a
Bravais lattice 1 represents an infinite array of discrete points, constituting the underlying framework for a crystal structure. These points are generated by a set of discrete translation operations, which can be expressed in three dimensional space as foll... |
Here is a report of the Ray Tracer written by myself Christopher Chedeau. I've taken the file format and most of the examples from the Ray Tracer of our friends Maxime Mouial and Clément Bœsch. The source is available on Github.
Check out the
demo, or click on any of the images. Objects
Our Ray Tracer supports 4 object... |
Affine transformation is a linear mapping method that preserves points, straight lines, and planes. Sets of parallel lines remain parallel after an affine transformation.
The affine transformation technique is typically used to correct for geometric distortions or deformations that occur with non-ideal camera angles. F... |
Your understanding of what makes chess NP-Hard is slightly flawed. Yes, a nondeterministic machine is able to "play perfectly". But the language of chess is,
$$Chess = \{Pos \quad | \quad \text{White wins with perfect play on an }n\times n \\ \text{ chess board, starting from position } Pos \quad \}$$
Does a certificat... |
I need to prove the following that
$e^{\bar{z}} = \bar{e^{z}}$
with $e^z := \Sigma_{k = 0}^{k = \infty} \frac{z^k}{k!}$
The problem that I am having is first we know $e^z$ defined that way always converge, but what I don't understand is we have to see that $\bar{z^k}$ put into the formula above converges to same number... |
First, I am not exactly sure what you are asking. I am assuming that you know that the
minimal norm solution to the least squares problem $\min_x {1 \over 2} \|Ax-y\|^2$ is given by $x=A^{\dagger} y$ and that the issue to show that the solution to problem$\min_X {1 \over 2} \| C-AXB\|_F^2$ is given by $A^{\dagger} C B^... |
Let $M,N,P$ be smooth manifolds, $F\in\mathcal{C}^\infty(M,N)$ a smooth map and $i\in\mathcal{C}^\infty(P,N)$ an injective immersion, such that $F(M)\subset i(P)$. We define a map $G:M\rightarrow P$ by $F(p)=i(G(p))$.
I am to show the following:
If $i$ is an embedding, then $G$ is continuous.
My ideas so far:
Let $i$ b... |
We need some notation before we state the problem. Let $K$ be a quadratic number field, $d$ its discriminant. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$. Let $D$ its discriminant. Let $x_1,\cdots, x_n$ be a sequence of elements of $R$. We denote by $[x_1,\cdots,x_n... |
I'm trying not to be offended, I know how spam filters work and sometimes someone gets wrongly accused and it just happens to be me. I'm just a little frustrated because I carefully craft this question for over an hour, then I get "this looks like spam" with absolutely no suggestion to make it look less like it.
My Not... |
Definition:Two (Boolean Algebra) Definition
Denote with $\top$ the canonical tautology.
Denote with $\bot$ the canonical contradiction.
Define $\mathbf 2 := \left\{{\bot, \top}\right\}$, read two.
When endowed with the logical operations $\lor$, $\land$ and $\neg$, $\mathbf 2$ becomes a Boolean algebra.
These operation... |
H S Bhatti
Articles written in Pramana – Journal of Physics
Volume 65 Issue 3 September 2005 pp 541-546
Singly and doubly doped ZnS phosphors have been synthesized using flux method. Laser-induced photoluminescence has been observed in ZnS-doped phosphors when these were excited by the pulsed UV N
2 laser radiation. Du... |
This could be a math question. But if you think about the physical aspect of the question, it is interesting to look at the Schrodinger equation:
For free fields (without potential), you have (in units $\bar h = m = \omega = 1$):
$$i\frac{\partial \Psi( k, t)}{\partial t} = \frac{ k^2}{2}\Psi( k, t)$$ or
$$ E \tilde \P... |
Let $f\ge 0$ be a measurable real-valued function defined on $\mathbb R$. For each $c>0$, Let $E_c:=\{x \in\mathbb R :f(x)\ge c \}$. Prove that \
$$\int_{E_c}f\,dm\ge c \cdot m(E_c).$$
My idea is to construct a sequence of simple functions approximation of $f$\
$\Phi_n=\sum_{i=1}^N a_i\chi_{E_i}$. since we are integrat... |
String theory may be considered as a framework to calculate scattering amplitudes (or other physically meaningful, gauge-invariant quantities) around a flat background; or any curved background (possibly equipped with nonzero values of other fields) that solves the equations of motion. The curvature of spacetime is
phy... |
The suspension isomorphism for homology index braids
DOI: http://dx.doi.org/10.12775/TMNA.2006.028
Abstract
Let $X$ be a metric space, $\pi$ be a local
semiflow on $X$, $k\in\mathbb N$, $E$ be a $k$-dimensional normed space and $\widetilde\pi$ be the semiflow generated by the equation $\dot y=Ly$, where $L\co E\to E$ i... |
I thought of this question somewhat randomly on a walk, and have discussed it with another friend of mine (we both have pure mathematics degrees). We have made some headway, and we think we have generated a proof, but we would appreciate any additional insight and proof verification.
Let $f:\mathbb{R}\to\mathbb{R}$ be ... |
I have data tuples $(x_i,\varepsilon_i,t_i)$ generated from some observations and I suspect that $\varepsilon \sim \mathcal{N}\left(0,\sigma(t)^2\right)$, where $\sigma(t)$ is an increasing function of time, i.e. the data is zero-mean but clearly manifests increasing variance as time progresses.
The first approach is t... |
Since buying computation power is much affordable than in the past, are the knowledge of algorithms and being efficient getting less important? It's clear that you would want to avoid an infinite loop, so, not everything goes. But if you have better hardware, could you have somehow worse software?
I really like the exa... |
I see entropy as a number that gives you an idea of
how random an outcome will be based on the probability values of each of the possible outcomes in a situation.
Let's start with a simple case. Suppose only a single outcome is possible, then there is only one value of $i$ ($=1$) and $p_{1}=1$. From the formula, the en... |
Let $R$ be a commutative unital ring and $M$ an $R$-module. Then $M$ is projective iff $\operatorname{Hom}(M,-)$ is exact, injective iff $\operatorname{Hom}(-,M)$ is exact, and flat iff $M\otimes-$ is exact. Furthermore, $M$ is faithfully flat when every chain complex is exact iff the induced $M\otimes-$ chain complex ... |
Find the radius of convergence of this series and study what happens in the border. $\sum_{n=1}^{\infty}\frac{z^n}{n}$ ($z\in \Bbb{C}$)
I easily found that the radius of convergence is $\rho =1$, therefore the series doesn't converge absolutely for $|z|=\rho=1$ , since $\sum|\frac{z^n}{n}|$ diverges in this case.
There... |
Find all $\theta \in \Bigl (-\dfrac{\pi}{2},\dfrac{\pi}{2}\Bigr)$ satisfying:
$$\sec ^{2} \theta(1-\tan \theta)(1+\tan \theta)+2^{\tan^{2}\theta}=0$$
I have tried a lot but couldn't crack this one. I could only bring it down to the following problem (Solving the following problem is equivalent to solving the above equa... |
Product of Coprime Factors
Jump to navigation Jump to search
Theorem
Let $a, b, c \in \Z$ such that $a$ and $b$ are coprime.
Let both $a$ and $b$ be divisors of $c$.
Then $a b$ is also a divisor of $c$.
That is:
$a \perp b \land a \divides c \land b \divides c \implies a b \divides c$ Proof
We have:
$a \divides c \impl... |
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abstract
Session 36 - Active Galactic Nuclei.
Display session, Tuesday, June 09
Atlas Ballroom,
We present initial results from an Owens Valley millimeter array survey of the molecular gas in ten nearby active galaxies. Our sample is drawn from those systems in whose nuclei Ho et al. (1997, ApJS,... |
I'm trying to solve:
$ y'' -4xy' + (4x^2 -1)y = -3e^{x^2} \sin(2x)$
Which has a general form of
$y'' + P(x) y' + Q(x) = R(x)$
I reduced it to the normal form by using the substitution $y = u e^{ - \frac{1}{2} \int P(x)} dx$
$u'' + I(x) u = S(x)$
Where $I(x) = Q - \frac {p'}{2} - \frac {p^2}{4}$
and $S(x) = R(x) e^{ \fr... |
I am attempting to fit a model to a dataset with frequency (Hz) is the dependent variable. Using a generalized linear model based on a gamma distribution seems appropriate since the values of the dependent variable are $0 \rightarrow \infty$ and I have confirmed that the observed values align with a gamma distribution ... |
To every rational complex curve C \subset (\mathbf{C}^\times)^n we associate a rational tropical curve \Gamma \subset \mathbf{R}^n so that the amoeba \mathcal{A}(C) \subset \mathbf{R}^n of C is...
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Given n \geq 2 and 1<p<n, we consider the critical p-Laplacian equation \Delta_p u + u^{p^*-1}=0, which corresponds t... |
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Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
CDS 140b Spring 2014 Homework 2
R. Murray, D. MacMartin Issued: 9 Feb 2014 (Wed) CDS 140b, Spring 2014 Due: 16 Feb 2014 (Wed) Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent onthis homework set (including reading).
WARNING: UNDER CONSTRUCTION, D... |
Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots.
The first solution (on AoPS) involves the use of Vieta's formula's and is quite clear.
The third solution states the following :
Note that if $r$ is a root, then $\frac{1}{2... |
A fairly random example of a great pedagogical technique for mathematics -
examples first.
For example, suppose I was trying to explain to you what a
ring is in algebra. I could start by telling you that a ring is a set $R$ together with two binary operations $+$ and $.$ such that: $r+s=s+r$ and $(r+s)+t$=$r+(s+t)$ for... |
Open set in a topological space
An element of the topology (cf. Topological structure (topology)) of this space. More specifically, let the topology $\tau$ of a topological space $(X, \tau)$ be defined as a system $\tau$ of subsets of the set $X$ such that:
$X\in\tau$, $\emptyset\in\tau$; if $O_i\in\tau$, where $i=1,2$... |
BERTRAND-B CURVES IN THE THREE DIMENSIONAL SPHERE DOI Number First page Last page Abstract
We dene a Bertrand-B curve in Riemannian manifold M such that there
exists an isometry \phi of M, that is, \left( \phi \circ \beta \right) (s)=X\left( s,t(s)\right) and the binormal vector of another curve \beta is the paralel ve... |
Revisiting the Work-Energy Theorem
The work-energy theorem discussed in the previous section was derived from Newton's second law, which carries with it a reference to the center of mass of the object on which the force is acting. So technically, the velocity and displacement that appear in the work-energy theorem are ... |
\[
\int_{-\infty}^{\infty} \mathrm{d}x \exp\left(-\frac{(x-\mu)^4}{2}\right) \]
was given. Using RcppNumerical is straight forward. One defines a class that extends
Numer::Func for the function and an interface function that calls
Numer::integrate on it:
// [[Rcpp::depends(RcppEigen)]]// [[Rcpp::depends(RcppNumerical)]... |
In the derivation of the Miller-Rabin Primality (or actually, "probably composite") test, Wikipedia http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test as well as other sites (including several Stack Exchange answers, as a lot of people have asked about this test) have a couple of small details that I am a... |
This is a cross-post of someone else's question.
I am cross-posting this question from MSE since it hasn't received any answers.
In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{... |
I am trying to understand how one obtains the Galilean algebra from the Poincare algebra, specifically through the method of central extension. I'm doing this by imposing that the generators of the Poincare group scale with the velocity in certain ways, and then taking the small velocity limit. Also I was told I should... |
We use
MOEADr to optimize a simple 2-objective problem composed of the Sphere function and the Rastrigin function in \(X = \{x_1,x_2,\dots,x_D\} \in\mathbb{R}^D\).
For this example, we define the sphere and rastrigin function in \(\mathbb{R}\) as: \[ \text{sphere}(X) = \sum_{i=1}^D x_i^2\\ \text{rastrigin}(X) = \sum_{i... |
We study randomly growing trees governed by the affine preferential attachment rule. Starting with a seed tree S, vertices are attached one by one, each linked by an edge to a random vertex of...
Pages
We consider the S-matrix bootstrap of four dimensional scattering amplitudes with O(3) symmetry and no bound-states. W... |
Recall: Fourier series representation of a periodic signal $\tilde{x(t)}$ with time period $'T'$ is given by:-
Suppose, $x(t)$ is not periodic.Is there a representation for $x(t)$ as a linear combination of complex exponentials?
The main idea is to think of $x(t)$ as the limit of $\tilde{x}(t)$ when $T \rightarrow \inf... |
The 4D SUSY algebra can be written as
$$\{ Q_{\alpha}^{A} , Q_{\beta}^{B \dagger} \} = 2 m \delta^{AB} \delta_{\alpha \beta} + 2 i Z^{AB} \Gamma^0_{\alpha \beta}, \tag{B.2.37} $$
in a particular reference frame. One can find this formula in the Appendix B, page 448 of Polchinski's String Theory vol.II.
I am confused wi... |
ok, suppose we have the set $U_1=[a,\frac{a+b}{2}) \cup (\frac{a+2}{2},b]$ where $a,b$ are rational. It is easy to see that there exists a countable cover which consists of intervals that converges towards, a,b and $\frac{a+b}{2}$. Therefore $U_1$ is not compact. Now we can construct $U_2$ by taking the midpoint of eac... |
Why? How do we know how these definitions are related?
Fix a set $S$
Consider an equivalence relation $\sim$ on $S$. Consider the equivalence classes $[a]=\{ x \in S : x \sim a \}$. By definition these are subsets of $S$. Their union is $S$ because by reflexivity $a\in[a]$ for every $a\in S$. Finally, they are disjoint... |
Unit 02: Differentiation
Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Sof... |
So I am studying spring mass systems, and I'm confused by why the solution, which is complex, becomes real. Such that
$$x(t) = C_1\cos(\omega t)+iC_2\sin(\omega t) = C_1\cos(\omega t)+C_2\sin(\omega t)$$
Why does the imaginary term just become an arbitrary constant?
Also the general solution is also simplified to:
$$x(... |
So, given some data,
Mathematica 10.2 can now attempt to figure out what probability distribution might have produced it. Cool! But suppose that, instead of having data, we have something that is in some ways better -- a formula. Let's call it $f$. We suspect -- perhaps because $f$ is non-negative over some domain and ... |
Numerical method for image registration model based on optimal mass transport
1.
Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
2.
David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1... |
Here's a way of thinking about it. To be clear however, this does
not prove the Pythagorean theorem.
Given two 2-dimensional vectors $\mathbb{x},\mathbb{y} \in \Bbb R^2,$ with entries $\mathbb{x} = (x_1,x_2)$ and $\mathbb{y} = (y_1,y_2),$ we define a
norm as,$$ |\mathbb{x}| = \sqrt{x_1^2 + x_2^2} $$We say two vectors a... |
From WolframAlpha it seems that $$ \frac{1}{2}=\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1) $$
Could someone provide a proof for this?
Thanks.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to joi... |
I have the following sum $$\sum_{a=0}^{[p/2]}\frac{p!}{(a!)^2(p-2a)!}2^{p-2a},$$ where $[p/2]$ denotes the closest integer and $p \in Z^+$.
Is there any way to make the sum look simpler? Especially in regards to the fact that the upper bound bust be an integer.
Any suggestion will be appreciated.
(This is what I found ... |
That formula doesn't break down. It can be proven that the right-hand side is equal to $\pi$. One reason it comes up in connection with
approximation is that the arctangent terms can be approximated well by adding terms from the Taylor expansion of the arctangent function,
$$\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\fr... |
For a prediction interval in linear regression you still use $\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ to generate the interval. You also use this to generate a confidence interval of $E[Y|x_0]$. What's the difference between the two?
Your question isn't quite correct. A confidence interval gives a range for $\te... |
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Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 ... |
Many interesting infinite graphs are given by taking a ground set $X$ and placing edges between pairs of vertices in $X$ at certain prescribed distances.The Hadwiger-Nelson problem is to determine the chromatic number of the unit-distance graph over the plane.As a modified version of this problem, Eggleton, Erdős, and ... |
I will implement some calculators to estimate the proper settings for the boundary prism layer meshing.
Simple mathematical calculations, such as the four arithmetic operations and power, can be done using HTML and JavaScript.
Boundary Layer Mesh
\begin{align}
l_{n} = l_{1} r^{n-1} \end{align} \begin{align} l_{tot} = \... |
I have encountered a problem with Mathematica that I really don't know how to solve. I'm sorry if I am going to be
verbose but I need it to explain the problem properly (and also make the code I'm going to show more comprehensible). very
I'm using Mathematica to work for my master's thesis and in these days I'm dealing... |
I'm trying to perform a linear regression in a Bayesian way.
The response is normal,the prior I would like to put over $\mathbf{\beta}$ (vector of regression coefficients) and $\Sigma^2$ (variance of the error term) is a Normal inverse-gamma one: Pi(Beta,Sig^2)=P(Beta|Sig^2)*P(Sig^2)
$P(\mathbf{\beta}|\Sigma^2) \sim N_... |
The maximal tensor product satisfies the following universal property:
Let $A, B,$ and $C$ be C$^*$-algebras. If $\phi: A \rightarrow C$ and $\psi: B \rightarrow C$ are $*$-homomorphisms whose images commute, then there's a unique $*$-homomorphism $\phi \otimes \psi : A\otimes_{max} B \rightarrow C$ such that $(\phi\ot... |
This will be a short tutorial as the ideas are very simple. I have previously discussed standardized survival functions. In survival analysis we know that there is a simple mathematical transformation from hazard to survival function and vice versa. The idea here is to transform to a hazard function from the
standardiz... |
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