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Prove or disprove the following with reasons:
If $X,Y,Z$ are iid standard exponential random variables then $X + Y$ and $Y + Z$ are both gamma random variables. If $X,Y$ are continuous random variables with joint density $f(x,y)=2, x, y \ge 0, x+y \le 1$ then X,Y are marignally both $\operatorname{U}[0,1]$ Suppose $X,Y... |
I'd like to draw a diagram with tikz-cd for a composition of natural transformations between functors that looked like the following
The best I could do so far is given by the following code:
\documentclass{article}\usepackage{amsmath,amsfonts}\usepackage{tikz-cd}\begin{document} \[ \begin{tikzcd}[row sep=huge] \mathca... |
There is the statement that $\beta$-function vanishes for super-renormalizable theories. In $D=2$, scalar field has mass dimension zero. So any polynomial interaction is super-renormalizable. Then shouldn't all of them have vanishing $\beta$-functions? But there are many theories (e.g, sine-Gordon) in $2D$ which have n... |
Here is a well-known interview/code golf question: a knight is placed on a chess board. The knight chooses from its 8 possible moves uniformly at random. When it steps off the board it doesn’t move anymore. What is the probability that the knight is still on the board after \( n \) steps?
We could calculate this direct... |
Second version, hopefully correct.
I claim that solving the feasibility problem $\exists? x: Ax \le b$ reduces in strongly polynomial time to finding a linear separator. Then it's easy to reduce linear programming to the feasibility problem.
Let us first reduce the strict feasibility problem $\exists? x: Ax < b$ to fin... |
Here is a complete proof. (It is inspired by a similar proof that an ideal in a ring that is maximal with the property of not being finitely generated, must be prime.)
Assume that there exists subgroups that are not finitely generated. They are ordered by inclusion and by Zorn's lemma there exists a maximal subgroup th... |
I was reading the following notes on tensor products: http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf
At some point (p. 39) there is the following example
In the last paragraph, he says that using exterior powers it can be proved that if $I\oplus I\simeq S^2$ as $S$-module, then $I\otimes_S I\simeq... |
Overview of the Model
As you begin this chapter, listening in lecture and working in DL, it must seem, at least at first, that you are being introduced to a lot of new concepts. The representation of the motion of an object and the forces acting on an object are necessary ideas to understand before we can fully underst... |
Let's review the design of ChaCha to see how the nonce, the counter, and te number of rounds all fit into it.
How do we encrypt a sequence of
messages $m_1, m_2, \dots, m_\ell$? One way is to pick a sequence of message-length pads $p_1, p_2, \dots, p_\ell$ independently and uniformly at random, and encrypt the $n^{\mat... |
I am asked to verify that the sequence $\left(\frac{1}{6n^2+1}\right)$ converges to $0$:
$$\lim \frac{1}{6n^2+1}=0.$$
Here is my work:
$$\left|\frac{1}{6n^2+1}-0\right|<\epsilon$$
$\frac{1}{6n^2+1}<\epsilon$, since $\frac{1}{6n^2+1}$ is positive
$$\frac{1}{\epsilon}<6n^2+1$$
$$\frac{1}{\epsilon}-1<6n^2$$
$$\frac{1}{6\e... |
I have to agree the exposition in this paper was quite sparse. I think answering these two questions requires a bit of review of previous work, but you can skip to the bottom if you just want the answers.
Recall the energy function of binary RBMs.
$$E(v,h) = v^TWh + v^Ta + h^Tb $$
In the binomial RBM work, they duplica... |
I haven't been taught tensor product in class but they have taught us addition of spin. I looked up online in this link->http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf#page=10 (pg 148, pg 10 in the pdf) and found an explanation. I think I understand most of it except this step:
$$ \vec{S}^{(A)}... |
Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps
1.
Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China
2.
School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, China
$\l... |
Use partial fraction decompostion. This requires factoring the quadratic first.
$$I(s)=\frac{6}{L}\frac{1}{s^2 + \frac{R}{L}s + \frac{1}{LC}}$$
The roots of that quadratic are$$s_{1,2} = -\frac{R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}}$$
If $s_{1,2}$ are distinct, your function can be represented as$$I(s... |
I will attempt to atone for my previous error by showing something opposite -- that $\tilde{\Theta}\left(\frac{1}{\epsilon^2}\right)$ samples are sufficient (the lower bound of $1/\epsilon^2$ is almost tight)! See what you think....
The key intuition starts from two observations. First, in order for distributions to ha... |
You shouldn't dismiss Karo's graph. Drawing a graph to get a feel for the equation (when you don't know how to proceed) is most helpful. And this isn't nonsense, in fact the graph is the key for the solution, as it makes it clear that we can consider only $x$ for which $-2 \leq x \leq 2$.
This can be conveniently rewri... |
Darcy Weisbach Equation statement
It is an empirical equation in fluid mechanics named after Henry Darcy and Julius Weisbach. The Darcy Weisbach Equation relates the loss of pressure or head loss due to friction along the given length of pipe to the average velocity of the fluid flow for an incompressible fluid.
Darcy ... |
In model theory, the satisfiability relation $ \vDash$ between a model $M= (D,f)$ and a set of formulas tells us when a formula $\varphi$ is true or not in the model ("interpretation") $M$.
This relation is usually defined recursively in the following way. I won't be 100% precise giving the definition, but it's somethi... |
Background: A real
period is defined to be the value of an integral of the form
$$\int_D R(x_1,\cdots,x_n)dV$$
where $R$ is a rational function with rational ceofficients, and $D\subseteq\Bbb R^n$ is a set of points satisfying a system of inequalities involving rational functions also with rational coefficients (combin... |
Spurious points in WMAP3 likelihood code
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2 posts • Page
1of 1
Apologies if this has already been discussed -but I've seen some spurious model likelihoods from the WMAP code, specifically beyond about 2sigma there are some parameter combinations that give a likelihood in excess of 200 ln units better than the... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
What's in a language? Languages in abstraction
This post is about Languages from a mathematical and abstract linguistics point of view. Not much more to say about that, so let’s get right to it!
The Rigorous definition:
Let \( \Sigma \) be an alphabet and let \( \Sigma^k \) be the set of all strings of length k over th... |
Don't be intimidated, semiclassical quantization is very simple, and it can be straightforwardly understood from a few examples which lead to the general case.
Consider a particle in a box. The classical motions are reflections off the wall. These make a box in phase space, as the particle goes left, hits the wall, goe... |
Don't be intimidated, semiclassical quantization is very simple, and it can be straightforwardly understood from a few examples which lead to the general case.
Consider a particle in a box. The classical motions are reflections off the wall. These make a box in phase space, as the particle goes left, hits the wall, goe... |
A transition element is defined as the one which has incompletely filled d orbital in its ground state or in any one of its oxidation state. Zinc , cadmium and mercury are not typical transition elements because they have fill filled d orbital in their ground state as well as in their common oxidation state. These elem... |
I was puzzling: "What is the shortest proof of $\exists x \forall y (P(x) \to P(y)) $?" (a variation of the drinkers paradox see Proof of Drinker paradox) given a certain set of inference rules and using natural deduction.
I managed to proof it in 23 lines, but I am not sure if this is the shortest possible proof.
That... |
I've started learning sequences and I'm having a hard time calculating the following, for $a > 0$:
$$\lim_{n\to ∞}{\frac{\lfloor na\rfloor}{n}} $$
Using Heine’s Lemma I'm trying to solve it analogous to the corresponding limit definitions for functions, but I get stuck. I've tried mostly with the
Squeeze theorem.
Any h... |
We highly recommend you (I can not stress enough) to download the LaTeX fonts (just 151 KB) to your PC, and install them. This will not only improve the look of the website, but also save bandwidth as you do not need to download the images every time you see an equation.
Instructions to install and enable fonts:
A. How... |
I am given a zip-inflated Poisson (ZIP) model, where random data $ X_1, .., X_n$ are of the form $ X_i=R_iY_i$ , where the $ Y_i$ ‘s have Poisson distribution ($ \lambda$ ) and the $ R_i$ ‘s have Bernoulli distribution ($ p$ ), and all independent from each other. If given an outcome $ x = (x_1, .., x_n)$ , the objecti... |
I'm trying to price a "power contract" and would appreciate guidance on the next step. The payoff at time $T$ is $(S(T)/K)^\alpha$, where $K > 0$, $\alpha \in \mathbb{N}$, $T > 0$. $S$ is adapted to $\mathscr{F}$, and we are currently at time $t \in [0,T)$. Let $Q$ denote the risk-neutral measure and $\beta(t) = e^{\in... |
Let's Cover The Fundamental Group (pt 1)Mathematics · Jumping right in Prerequisites:
Firstly, this post builds off of my previous post, so if you’re learning these things for the first time and find yourself kinda lost, start there.
In addition to the previous post, the following are mathematical definitions you shoul... |
Let
$\beta \in K \setminus F; \tag 1$
then, since $[K:F] = 2$, there must exist a linear dependence over $F$ 'twixt$1$, $\beta$, and $\beta^2$; that is, there are $a, b, c \in F$, not all zero, such that
$a \beta^2 + b\beta + c = 0; \tag 2$
if now $a = 0$,
$b\beta + c = 0; \tag 3$
then $b = 0$ forces $c = 0$, so $a = b... |
Please excuse the non-specific title, this is a rather long problem.
So on our last exam in multivariable calculus, our professor gave us a very lengthy vector manipulation problem as a bonus. Seeing as it's no longer worth any points to me, I was wondering if someone could help me understand how to solve it.
So let $ ... |
A parallel plate capacitor is located horizontally so that one of its plates is submerged into the liquid while the other is over its surface. The permittivity of the liquid is equal to $\epsilon$, its density is equal to $\rho$. To what height will the level of the liquid in the capacitor rise after its plates get a c... |
Definitions
Let $\mathbb{F}$ be the set of floating point numbers in a given format, which could be either IEEE-754 binary32 (single precision) or binary64 (double precision).
Let $m(Z)$ the floating point number obtained rounding the real number $Z$.
Let $\phi(Z)$ the next floating point number greater than $Z$.
Let $... |
Homology, Homotopy and Applications Homology Homotopy Appl. Volume 5, Number 1 (2003), 407-421. Extensions of homogeneous coordinate rings to $A_ \infty$-algebras Abstract
We study $A_\infty$-structures extending the natural algebra structure on the cohomology of $\oplus_{n\in\mathbb{Z}} L^n$, where $L$ is a very ample... |
This question assumes some familiarity with Jensen's fine structure analysis of the constructible universe L (https://en.wikipedia.org/wiki/Jensen_hierarchy, http://www.math.cmu.edu/~laiken/papers/FineStructure.pdf).
Everything to follow is in L.
Given some $J_\alpha$, the n-th projectum of $J_\alpha, \rho_n(J_\alpha)$... |
The following answer doesn't really answer the question given, because it's not about counting spanning subgraphs but about counting all subgraphs where some edges are
fixed (i.e., cannot be removed). However, I'm posting it because it gives me the strong impression that having different path lengths is not crucial to ... |
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I don't know why,but when I wanted to upgrade with Brilliant2Brilliant^{2}Brilliant2 ,there's this problem I faced.I have a photo attached about what's the problem?Can you shed some light for me in this problem?
Thank you :)
Note by Vaibhav Reddy 5 years, 4 month... |
Difference between revisions of "Sine-cubed function"
(→Repeated antidifferentiation)
(→Computation of power series)
Line 117: Line 117:
We have the power series:
We have the power series:
−
<math>\! \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}(2k+1)!}</math>
+
<math>\!... |
Why do negative even numbers plugged into the Zeta function produce a zero? The Riemann Hypothesis implies that the non-trivial zeros are connected to the primes, so how does that fit with negative even numbers?
The functional equation of the zeta function, needed to extend anatically that function to $\;\Bbb C\setminu... |
My current lecture states the theorem of monotone convergence:
Let $(X, \mathcal{E}, \mu)$ a measure space and $\varphi_n: X \rightarrow \mathbb{R}$ an increasing sequence of $\mu$-integrable functions with:
$$\exists M \in \mathbb{R}: \forall n \in \mathbb{N}: \int_X\varphi_n\,d\mu \leq M$$
Then $\varphi := \lim \varp... |
I don't have an answer to the question "why would one want to consider such crazy stuff in
physics?" since I don't know much physics, but as a mathematics student I do have an answer to the question "why would one want to consider such crazy stuff in mathematics?"
What physicists call Grassmann numbers are what mathema... |
But if you don't want to have a Google account: Chrome is really good. Much faster than FF (I can't run FF on either of the laptops here) and more reliable (it restores your previous session if it crashes with 100% certainty).
And Chrome has a Personal Blocklist extension which does what you want.
: )
Of course you alr... |
I have to prove the following problem in propositional logic:
Let $F$ be a set of clauses and let $F' = F \cup \{res(C_1,C_2,A_i)\}$ be the extension of $F$ by a resolvent of some clauses $C_1,C_2 \in F$ where $A_i$ is a literal occuring positively in $C_1$ and negatively in $C_2$.
Prove that: If $F$ is valid, then $F'... |
Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b \times d},$$ And why $$\frac{a}{b}\times \frac{c}{c}=\frac{a}{b},$$ Also why $$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}.$$ I understand it, but I want a mathematical approach as a math student ... |
Elliptic boundary value problems in spaces of continuous functions
1.
Dipartimento di Matematica Applicata, Università di Pisa, Via Buonarroti 1/C, 56127 Pisa
full regularityoccurs. namely, for each $\,\lambda>\,0\,$ and arbitrary real $\,\alpha\,,$ $\,\nabla^2\,u $ and $\,f\,$ enjoy the same $\, C^{0,\,\lambda}_\alpha... |
Show that $f$ is continuous and in neigbourhood of $(0,0)$ has bounded partial derivatives but is
not differentiable.$$f(x,y) = \begin{cases}\frac{xy}{\sqrt{x^2+y^2}} & x^2+y^2 \neq 0 \\0 & x = y = 0\end{cases}$$This seems like basic introductory material and is also a copy of the same question on StackExchange, but un... |
Just as we did for linear momentum conservation, we will summarize the main ideas of the angular momentum conservation model/approach by listing the
constructs, i.e., the “things” or ideas that are get “used” in the model, the relationships–in mathematical or sentence form– that connect the constructs in meaningful way... |
In mathematics, the radian is defined as the standard unit for measurement of angles. The measurement in angles is usually equal to the total length of a corresponding arc in a unit circle. The relationship between arch length and radius of a circle is generally defined with the help of radian of a circle. Degree and r... |
Learning Objectives
Formulate the principle of conservation of mechanical energy, with or without the presence of non-conservative forces Use the conservation of mechanical energy to calculate various properties of simple systems
In this section, we elaborate and extend the result we derived in Potential Energy of a Sy... |
I needed to solve the following equation: $$\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$$
Now, the steps that I followed were as follows.
Transform the LHS first: $$\begin{split} \tan\theta + \tan 2\theta+\tan 3\theta &= (\tan\theta + \tan 2\theta) + \dfrac{\tan\theta + \tan 2\theta} {1-\tan... |
Drábek, Pavel , Takáč, Peter Convergence to travelling waves in Fisher’s population genetics model with a non-Lipschitzian reaction term
We consider the Fisher equation for advance of advatageous gene with non-Lipschitzian reaction term. We prove new travelling wave profiles and the convergence of the Cauchy problem to... |
Bulk Stress, Strain, and Modulus
When you dive into water, you feel a force pressing on every part of your body from all directions. What you are experiencing then is bulk stress, or in other words,
pressure. Bulk stress always tends to decrease the volume enclosed by the surface of a submerged object. The forces of th... |
I note $L_{t}^{[T_s, T_e]}$ the forward rate at time $t$ for the period $[T_s, T_e]$. Recall it is the strike making equal to $0$ the value at time $t$ of a forward contract for the period $[T_s, T_e]$.
The strategy I am looking at is the following : I enter (for $0$) at time $t$ in a payer (I pay the strike) forward c... |
This site contains various discussions of one-way functions and their relation to P versus NP.
Some of these discussions use a language $L=\{(x',y) ~\mid~ x'\le x \text{ and } f(x)=y \}$, where $f:\Sigma^*\to\Sigma^*$ is the one-way function and $x'\le x$ is the prefix relation. Now one central claim is that this langu... |
Exercise
If $H$ is the Heaviside function, prove, using the definition below, that $\lim \limits_{t \to 0}{H(t)}$ does not exist.
Definition
Let $f$ be a function defined on some open interval that contains the number $a$, except possible $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, an... |
This paper presents the two-dimensional (2D) steady incompressible flow in a lid-driven sectorial cavity. In order to analyze the flow structures, the 2D Navier–Stokes equations are solved by using the finite element method. Different cases of the cavity aspect ratio A and three cases of the speed ratios \((S=-1,0,1)\)... |
Here's what I say when I'm teaching this. These comments are usually spread over several lectures, but I'll say them all at once here.
The first use of partitions of unity is usually to construct integrals over manifolds. For example, let $S$ be a surface in $\mathbb{R}^3$, and say we want to integrate a $2$-form $\ome... |
$$\frac{1+\cos 5x+i\sin 5x}{1+\cos 5x-i\sin 5x}=\cos 5x+i\sin 5x$$
When I attempted this I first tried multiplying top and bottom of the LHS by the complex conjugate of what's on the bottom, $1+\cos 5x+i\sin 5x$. After simplification I got:
$$LHS=\frac{1+2\cos 5x+2i\sin 5x+\cos^25x+2i\sin 5x \cos5x-\sin^2x}{2\cos 5x+si... |
Let $\Sigma\subset\mathbb{R}^3$ be a regular surface. Prove that $\Sigma$ is minimal if and only if its coordinate functions are harmonic.
I know that every regular surface admits an isothermal parametrization. Assuming that a parametrization $X:U\subset\mathbb{R}^2\to\Sigma$ is isothermal (i.e., $<X_u,X_u>=<X_v,X_v>$ ... |
How can I solve this system of trigonometric equations analytically? It is from physics class. $$ \begin{cases} 30t\cos{\alpha}=50\\ -30t\sin{\alpha}-4.9t^2=0 \end{cases} $$
Hint: Squaring both the equations, you will get $900t^2\cos^2{\alpha}=2500\\ 900t^2\sin^2{\alpha}={4.9}^2t^4$.
Note that $\sin^2{\alpha}+\cos^2{\a... |
A very general (but extremely useful) approach is by noting the following
$$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$$$$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$
and since $\tan(x) = \frac{\sin(x)}{\cos(x)}$
$$\tan(x) = \frac{1}{i}\frac{e^{ix} - e^{-ix}}{e^{ix} + e^{-ix}} = -i \frac{e^{ix} - e^{-ix}}{e^{ix} + e^{-ix}} $$
We ... |
It is widely known that $0^0$ is usually defined to be 1. I wonder why we cannot employ a similar technique to ascribe values to functions having poles in a point.
Now take the Gamma function. The real part $\Re(\Gamma(i x))$ is equal to $\gamma$ (Euler-Masceroni constant). We thus can use the formulas:
$$\Gamma(-n)=\l... |
The displacement of a particle performing simple harmonic motion is given by $x = A \sin(\omega t + \phi)$ , where $A$ is the amplitude, $\omega$ is the frequency, $t$ is the time, and $\phi$ is the phase constant. What is the significance of $\phi$. How is it used? Please explain the meaning of the phase constant
The ... |
Let me remind you about the following classical examples in quantum mechanics.
Example 1. Bound states in 1-dim potential V(x). Let $V(x)$ be a symmetric potential i.e. $$V(x) = V(-x)$$ Let us introduce the parity operator $\hat\Pi$ in the following way: $$\hat\Pi f(x) = f(-x).$$ It is obvious that $$[\hat H,\hat\Pi] =... |
A dirty little ditty on Finite AutomataMathematics ·
This post builds on the previous post about Formal Languages.
Some Formal Definitions A Deterministic Finite Automata (DFA) is A set \( \mathcal{Q} \) called “states” A set \( \Sigma \) called “symbols” or “alphabet” A function \( \delta_F:\mathcal{Q}\times\Sigma \to... |
Consider the function $f: [0,1] \to \mathbb{R}$ where f(x)= \begin{cases} \frac 1q & \text{if } x\in \mathbb{Q} \text{ and } x=\frac pq \text{ in lowest terms}\\ 0 & \text{otherwise} \end{cases}
Determine whether or not $g$ is in $\mathscr{R}$ on $[0,1]$ and prove your assertion. For this problem you may consider $0= 0... |
Based on Leo's answer but I removed one column and replaced the
\big) with a vertical rule. Now I want to make all columns have the same width. How to do this?
\documentclass{article}\usepackage{amssymb}\begin{document}\[x^3 - x + 1 = (x-1)(x^2+x) + 1 \in \mathbb{F}_3[x]\]\[\renewcommand\arraystretch{1.2}\begin{array}{... |
I am currently working on my bachelor's thesis on the anapole / toroidal moment and it seems that I am stuck with a tensor decomposition problem.
I have actually never had a course about tensors, so I am a complete newbie.
I need to expand a localized current density, which is done by expressing the current via delta d... |
Given a translationally invariant Matrix Product State (assuming periodic boundary condition) on $N$ sites of dimension $d$ each, which takes the form
$\sum_{i_1,i_2\ldots i_N=1}^dTr(A_{i_1}A_{i_2}\ldots A_{i_N})|i_1,i_2\ldots i_N\rangle$,
with $A_i$ being $D\times D$ matrices. If $d<D^2$, then matrices $A_i$ do not sp... |
As some people on this site might be aware I don't always take downvotes well. So here's my attempt to provide more context to my answer for whoever decided to downvote.
Note that I will confine my discussion to functions $f: D\subseteq \Bbb R \to \Bbb R$ and to ideas that should be simple enough for anyone who's taken... |
Let $u_0\ge 2$ and $\alpha\ge 1$ integers.
I'm trying to study the sequence $(u_n)_{n\ge 0}$ defined by : $$\forall n\ge 0,\quad u_{n+1}=d(u_n)^{\alpha}, $$ where $d$ is the number-of-divisors function.
If $\alpha=1$, it's not hard to prove that $$\lim\limits_{n\rightarrow +\infty}u_n=2,$$ And if $\alpha=2$ we also hav... |
How to get the optimal value for $k$?
You have to define a measure for optimiality. The problem with that is that with bigger $k$ most measures become smaller (better). One measure which is independent from $k$ is the silhouette coefficient:
Let $C = (C_1, \dots, C_k)$ be the clusters. Then:
Average distance between an... |
A tantalizing open question in computational complexity is to understand the 'behavioral differences' between the determinant and the permanent. While the former is computable in polynomial time with Gauss pivot, the second is $\# P$-hard by Valiant's result. There have been attempts to generalize the notions of determ... |
Let $T:X\to Y$ a linear bounded operator between Banach spaces. Let $U$ a neighbourhood of $0\in Y$, $t\in(0,1)$ and suppose that $\forall u\in U$ $\exists \bar x\in X$ with $\|\bar x\|\le1$ and $\bar u\in U$ such that $$u=T\bar x +t\bar u.$$Then if I take $u\in U$, I can write
$$u=T\left(\sum_{i=0}^{\infty} t^ix_i\rig... |
Here is a complete proof. If comes from modifying this answer. We also make use of the lemma $\theta(x)\leq 3x$ shown in this answer. Throughout, $\theta(x)$ refers to the weighted prime counting function $\sum_{p\leq x}\log p$.
Consider $\binom{2N}{N}.$ For each prime $p$, let $v_{p}$ denote the number of times it div... |
Let $n \in \mathbb{N}$ be a composite number, and $n = pq$ where $p,q$ are
distinct primes. Let $F : \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ (*) be an algorithm which takes as an input $x \in \mathbb{N}$ and returns two primes $u, v$ such that $x = uv,$ or returns FAIL if there is no such factorization ($F... |
This question already has an answer here:
As the title says, I want to prove that there is a natural number $k$ such that $2^k$ is starting with $999$. Can you help me please ?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only tak... |
How can I show that the following holds?
$$\langle nlm\mid \partial_z^2\mid nlm\rangle=-\int_0^{4\pi}d\Omega\int_0^{\infty}drr^2\left|\partial_z\psi_{nlm}\right|^2$$
The wave functions of a free particle are: $\mid nlm\rangle=\psi_{nlm}$.
This conversion is stated in the Quantum Mechanics book by Landau, Lifshitz (Vol.... |
The aim is to determine the group velocity of a wave packet with the general form
$$\Psi\left(x,t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(x\right)e^{i\left(kx-\omega t\right)}dk.$$
Quoting from
Introducting to Quantum Mechanics by David Griffiths:
Since the integrand is negligible except in the vi... |
I was reading up on De Sitter spaces, which states that the gravitational effects from a black hole is indistinguishable from any other spherically symmetric mass distribution. This makes a lot of sense to me.
I'm now super curious, can we just formulate all the properties of a black hole beyond the limit at which GR i... |
2017-04-19 08:09
Flavour anomalies after the $R_{K^*}$ measurement / D'Amico, Guido (CERN) ; Nardecchia, Marco (CERN) ; Panci, Paolo (CERN) ; Sannino, Francesco (CERN ; Southern Denmark U., CP3-Origins ; U. Southern Denmark, Odense, DIAS) ; Strumia, Alessandro (CERN ; Pisa U. ; INFN, Pisa) ; Torre, Riccardo (EPFL, Laus... |
Suppose I have a finitely generated group $G$ of known rank $n$, and a set $\{s_i\}$ of $n$ group elements. Are there some simple necessary and sufficient conditions to determine whether $s_i$ generates $G$? (Suppose that I don't have any known generating set which I can try to generate with the $\{s_i\}$.)
For example... |
Simulation Tools for Solving Wave Electromagnetics Problems
When solving wave electromagnetics problems with either the RF or Wave Optics modules, we use the finite element method to solve the governing Maxwell’s equations. In this blog post, we will look at the various modeling, meshing, solving, and postprocessing op... |
Or : Multiplying by \(i\).
The beginning of complex numbers is often attributed to Cardano (1501 – 1576). Complex numbers provide a solution to the equation \(x^2 = -1\).
They are an extension of the number system from \(\Bbb R\) (the real numbers) to \(\Bbb C\) (the complex numbers).
Complex numbers take the form :
\(... |
Definition:Connected Relation
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Definition Then $\mathcal R$ is connected if and only if: $\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$ Also known as
Some sources use the term
weakly ... |
Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
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don't post problems (by starting a topic)in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisi... |
Cauchy test
The Cauchy criterion for the convergence of a series: Given a series $\sum_{n=1}^{\infty}u_n$ with non-negative real terms, if there exists a number $q$, $0\leq q<1$, such that, for all sufficiently large $n$, one has the inequality $(u_n)^{1/n}\leq q$, which is equivalent to the condition , then the series... |
As Maarten Bodewes points out, you should use a known key derivation function instead of this method (i.e. don't roll your own crypto). Having said that we can still try to understand what happens if you do use this method.
I assume that the method you're describing is something as follows. You take a block cipher $E:\... |
One way of summarizing the main ideas in a model/approach is to list the \((1)\) constructs, i.e., the “things” or ideas that are “used” in the model, \((2)\) the relationships–in mathematical or sentence form–that connect the constructs in meaningful ways, and \((3)\) the ways of representing the relationships. During... |
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1. Search for a Heavy Neutral Particle Decaying to eμ, eτ, or μτ in pp Collisions at √s = 8 TeV with the ATLAS Detector
Physical Review Letters, ISSN 0031-9007, 2015, Vol... |
For Confidential Transactions a Pedersen commitment is being used. The commitment preserves addition and the commutative property applies: $$C(\text{BF}_1, \text{data}_1) \oplus C(\text{BF}_2, \text{data}_1) = C(\text{BF}_1 + \text{BF}_2, \text{data}_1 + \text{data}_1)$$
$\oplus$ doesn't denote XOR here but more genera... |
Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
Forum rules
Please
don't post problems (by starting a topic)in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisi... |
An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
1.
Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210
2.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
3.
School of Mathematics, University of Minnesota, Minneapo... |
I've got a fun question, which is somewhat testing my topology skills.
The space we're working with is $\mathbb{R} \rightarrow \mathbb{R}/\sim$, which sends $x$ to $[x] = \{y \in \mathbb{R}: x-y \in \mathbb{Q} \}$, and what I'm trying to show is that $\mathbb{R}/\sim$ isn't Hausdorff.
What I'm struggling with is provin... |
Implicit in that explanation is 1) some state space $(\cal{X},\mathscr{X})$ in which $X$ takes values, and 2) that $P(\cdot\mid H)=P(\cdot,H)$ is a probability measure on $(\cal{X},\mathscr{X})$. The existence of a $P(\cdot,H)$ satisfying 2) and also satisfying the relation mentioned toward the end of the screenshot $P... |
I have the following theorem:
Theorem: Let$(S,\mathcal{A},\mu)$ be a measurable space and let $(A_n)_{n\geq 1}$ be a sequence in $\mathcal{A}$.
i) If $A_n \uparrow A$, then $\mu(A_n) \uparrow \mu(A)$.
ii) If $A_n \downarrow A$ and $\mu(A_1)<\infty$, then $\mu(A_n)\downarrow \mu(A)$.
(Note that in this definition $A = \... |
I have to solve the following SDE. $$ \mathrm dY_t= f(X_t) \mathrm dt, \tag{1} $$ where $X_t$ is a two-state Markov Process possesses states $a$ and $b$.
Moreover, I would like to solve $$ \mathrm dY_t= Z_t \mathrm dt, \tag{2} $$ where $Z_t$ is an rv conditioned on $X_t$, i.e., $Z_t \mid X_t \sim Bin(n,g(X_t))$.
After ... |
Inelastic Electron Spectroscopy of an H 2 molecule placed between 1D Au chains¶ Version: 2017
This tutorial is focused on simulating inelastic electron tunneling spectroscopy (IETS) [Ree08] for a device consisting of two one-dimensional (1D) gold electrodes and an H
2 molecule placed in between. After DFT calculations ... |
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