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Integral $\int_0^{\frac{-\pi}{2}} \frac{(2\pi i + \epsilon e^{i\theta})^4i\epsilon e^{i\theta} d \theta} {e^{2 \pi i + \epsilon e^{i \theta}} - 1}$

I'm not sure to see how the integral $$\int_0^{\frac{-\pi}{2}} \frac{(2\pi i + \epsilon e^{i\theta})^4i\epsilon e^{i\theta} d \theta} {e^{2 \pi i + \epsilon e^{i \theta}} - 1}$$ yields a finite value ...
4 votes
1 answer
89 views
+100

How can I solve to a closed form of this sequence?

I was studying some things and I’m stumped with the sequence $$c_n=-\frac{1}{n}\sum_{k=0}^{n-1}c_{n-k-1}a_{k+1}, c_0=1$$ Does it have a closed form? I know it needs to have, because we can see by ...
0 votes
0 answers
3 views

Solve $\dfrac{x-1}{2}\geq \dfrac{2}{3} \geq 1-\dfrac{x}{6}$

Solve $\dfrac{x-1}{2}\geq \dfrac{2}{3} \geq 1-\dfrac{x}{6}$ Multiply everything by $6$: $3x-3 \geq 4 \geq 6-x$ Evaluate separately: $3x-3 \geq 4 \Rightarrow 3x \geq 7 \Rightarrow x \geq \dfrac{7}{3}$ ...
1 vote
2 answers
827 views

Circumcentre of a triangle given the radius vectors of the vertices

If the radius vectors of the three vertices A, B and C of a triangle ABC are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively, the radius vector of the circumcenter is given by $\frac{\vec{a}\sin(2A) +\...
1 vote
1 answer
42 views

Little-o allows two functions be little-o relatively to each other? Is not it a nonsense?

I will use Wikipedia article about Big-O, little-o as a reference. Please note, I am asking about "little-o" exactly, not "big-o". Function $f(x)$ is $o(g(x))$ at infinity, if ...
1 vote
1 answer
32 views

(p,n)=1 and if $v \in F$ and $n v \in K$ then prove that $v \in K$

This question is part of an assignment I am solving and I am unable to work this out. Let char K = p $\neq 0 $ and let n > 1 be an integer such that (p,n)=1 . If $v \in F$ and $ n v \in K$ , ...
1 vote
2 answers
14 views

Does the monoid of non-zero representations with the tensor product admit unique factorization?

Let $(M, \cdot, 1)$ be a monoid. We will now define the notion of unique factorization monoid. A non-invertible element in $M$ is called irreducible if it cannot be written as the product of two other ...
0 votes
0 answers
3 views

Prove: If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.

I need to prove the following result: Let $(X,\mathscr{A},\mu)$ be a measure space. If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$. Here is my ...
0 votes
0 answers
2 views

Spectrum of operator in $l^2$ equals closure of $\{b_k : k \in \mathbb{Z}^+\}$

I'm trying to understand why, in the following example, the spectrum of T includes limit points. Let $b_k \rightarrow l$, yet $(T-l \cdot I )(a) = (a_1(b_1-l)\cdots, a_k (b_k-l), \cdots )$. Since $\...
-1 votes
1 answer
50 views

Inverse Fourier Transform - convolution of exponential and rectangular window

I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
0 votes
2 answers
27 views

Weird error behavior when implementing Euler's method for solving ODEs in Python

I am solving this particular ODE: $$ \begin{cases} u'(t)=-2\,tu^2/20, \quad t \in [0,\sqrt{20}]\\ u(0) = 1 \end{cases} $$ which the analytical solution is given: $$ u(t) = \frac{1}{1 + t^2/20}. $$ ...
1 vote
1 answer
3k views

Bootstrap estimation of the 95% confidence intervals for the 95% quantile for gamma distribution

I cant find any where information or algorithm how to apply in steps the bootstrap procedure to estimate the 95% confidence intervals for the 95% quantile from a random sample. Does anyone knows how ...
1 vote
2 answers
96 views

Probability of n Periodic Events per Attempt, Skipping m = 20 Attempts in a row

The problem: There is a system with a clock ticking at a constant rate measured every second. Within this system there are n loops that are ongoing. The period of the loop is between 80 and 120 ...
0 votes
0 answers
12 views

polynomial $f$ having $n$ distinct roots implies either $f(x)\neq 0$ or $\deg f\geq n$

This is about two different results I have read in the note, where it says Theorem 5. If $f(x)\in F[x]$ has distinct roots $a_1, a_2,\dots, a_n$, then $f(x)$ is divisible by $(x − a_1)(x − a_2) · · · ...
0 votes
2 answers
55 views

Does $\prod_{n=1}^\infty (1-\frac{z^2}{(2n\pi i)^2})=\frac{e^z-1}{z}$?

I was thinking if the next statement is true: Let $f: \mathbb{C} \rightarrow \mathbb{C}$, $f(0)=1$, $f(\pm a_n)=0$, for all $n \in [1, \infty]$. If this is true, so $f(z)=\prod_{n=1}^\infty (1-\frac{z^...
1 vote
1 answer
152 views

A question based on intermediate fields and separability

This problem was asked in my Abstract algebra assignment and I was unable to solve it so I am asking for help here. Let E be an intermediate field . (a) If u $\in$ F is separable over K, then u is ...
9 votes
0 answers
154 views

Closed form for $\prod_{n=1}^{\infty} \frac{e^{\frac{x}{2^n}}-1}{\frac{x}{2^n}}$

I would like to know if there is a closed-form for \begin{align*} \prod_{n=1}^{\infty} \frac{e^{\frac{x}{2^n}}-1}{\frac{x}{2^n}}, \end{align*} where $x\in\mathbb{R}$. The closest results I could ...
0 votes
0 answers
27 views

Fourier analysis of sine: magnitude result clarification

I want to calculate the Fourier transform of $sine(2 \pi 60 t)$ and I have difficulty understanding/calculating the magnitude result. It is already known that the only frequency on this sine function ...
0 votes
0 answers
4 views

One dimensional regularity obstacle problem ($H^2$ regularity)

I am reading the proof of the $H^2$ regularity of the obstacle problem in the book Numerical Methods for Nonlinear Partial Differential Equation by Sören Bartels. I understand everything he does in ...
3 votes
5 answers
267 views

Find $\lim_{x\to \infty}\frac{\ln x}{x}=0$ without l'Hopital's rule

Can we find the limit $$\lim_{x\to \infty}\dfrac{\ln x}{x}=0$$ without using the l'Hopital's rule ? I did the change of variables $X=\ln x$ but it seems to be the same problem of finding the limit $\...
1 vote
0 answers
54 views

Kalman Filter to minimize weighted errors: what's wrong with my derivation

I am thinking about how to implement a "weighted Kalman Filter". Borrowing wikipedia convention update function is \begin{aligned} \mathbf {P} _{k\mid k-1}&=\mathbf {F} _{k}\mathbf {P} _{...
3 votes
2 answers
38 views

What is the proper way of interpreting "linearizing a differential equation"?

Suppose we have the following differential equation: $$\ddot{\theta} + \cos{\theta} = 0 \quad,$$ where $\theta$ is an unknown function of a real variable. It isn't uncommon to say we will "...
3 votes
1 answer
344 views

Is every $f(k)$-vertex-connected graph the edge-disjoint union of two $k$-vertex-connected graphs?

Does there exist a function $f: \mathbf{N} \to \mathbf{N}$ such that every $f(k)$-vertex-connected graph $G$ can have its edges partitioned into two spanning subgraphs $G_1$, $G_2$ such that both of ...
0 votes
0 answers
11 views

Example of a proof system that is complete but not sound

i'm trying to give an example of a proof system that is complete but not sound . i tried this : Axioms : all WFF rule of infernce : MP . then i claimed : given set $A$ and $a$ in WFF , if $ A |= a $ ...
0 votes
0 answers
10 views

Inequality in open set

See picture If the green (bigger) circle is a unit sphere, and tangents on the smaller circle are symmetric with line containing the origin, is it possible to prove that there is a constant $A$ such ...
4 votes
0 answers
48 views

When does the normality of $T^3$ imply that of $T^2$, where $T\in B(H)$?

Let $T\in B(H)$. Assume that $T^3$ is normal, i.e., $T^{*3}T^3=T^3T^{*3}$. When is $T^2$ normal? Here is a known related result, which may be more or less trivial: If $T^2$ is normal and $T$ has a ...
0 votes
0 answers
7 views

Axler Theorem 5.17, part (b)

I am trying to understand the proof of part (b) of Theorem 5.17 in Axler's Linear Algebra Done Right. He cites part (a) in his proof of (b), so I've written out the full theorem statement below. $\...
29 votes
4 answers
19k views

Example of filtration in probability theory

I'm studying Martingales and before them filtrations. Given a probability space $(\Omega, F, P)$ I define a filter $(F_n)$ as a increasing sequence of $\sigma$-algebras of $F$, such that $F_t \subset ...
0 votes
0 answers
49 views

Calculate $\lim_{n\to \infty}\int_0^1 \cos^nx \, dx$ without using newton leibniz [duplicate]

I'm asked to calculate: $$ \lim_{n\to \infty} \int_0^1 \cos^n x \, dx, $$ So far I have reached this: $$ \lim_{n\to \infty} \int_0^1 \cos^n x \, dx = \lim_{n\to \infty}\frac{n-1}{n} \int_0^1 \cos^{...
0 votes
0 answers
10 views

A subgroup with finitely many conjugates which has finite index in its normal closure is finite

I was wondering if the following statement is true: let $G$ be a group and $H$ a subgroup of $G$ such that $H$ is almost normal in $G$ (i.e. $H$ has finitely many conjugates in $G$ or equivalently the ...
0 votes
1 answer
19 views

FInd closest matrix RD to A where R and D are orthogonal and diagonal matrices respectively

Given a $3 \times 3$ matrix $A$ I wish to find the matrices $R$ and $D$ (which are orthonormal and diagonal matrices respectively) where the product $RD$ is as close to $A$ as possible (under an ...
0 votes
1 answer
26 views

How to state that a function has a certain trend in a limit

Assuming we have a function $f(r)$ that has the following limit $$ \lim_{r\to0} f(r) = \frac{5}{3 r^2} \,.$$ What is the correct symbol to express that the denominator goes like $r^2$? Is the ...
0 votes
0 answers
33 views

Trajectory forumla taking earth's curvature into account

I am aware that there are bunch of equations to determine the flight path of a moving object when it's influenced by gravity. It's clear that these equations don't take the curvature of the earth ...
1 vote
0 answers
7 views

What action on $\mathbb{R}^2$ yields a closed genus 4 surface

The closed, connected, orientable surface of genus 3 has universal cover $\mathbb{R}^2 \rightarrow \Sigma_3$. As such, $\Sigma_3$ can be described as a quotient of $\mathbb{R}^2$ by some properly ...
1 vote
1 answer
69 views

Why doesn't this diagonal argument work?

I have a question about the standard rules for computing p.r. terms (see below). It seems pretty clear that these rules could be used to define a p.r. operation that evaluates any p.r. term of the ...
0 votes
1 answer
51 views

Non visual Cauchy product demonstration.

I try to find a demonstration of this Cauchy product formula: $$(\sum_{n=0}^{\infty}A_n)*(\sum_{k=0}^{\infty}B_k)=\sum_{n=0}^{\infty}\sum_{k=0}^{n}(A_k)(B_{n-k})$$ Unfortunately all the proof that I ...
0 votes
0 answers
2 views

How to prove a differential submanifold is a Lorentzian manifold?

Consider the space $\mathbb R^{p+2}$ endowed with the bilinear form $$\langle x, y \rangle_{p,2} = \sum_{i = 1}^{p+2} x_iy_i -x_{p+1}y_{p+1}-x_{p+2}y_{p+2}$$ We define the quadric model of anti-de ...
1 vote
2 answers
75 views

Is this correct solution to arranging consecutive flowers?

Suppose I have 8 indistinguishable white flowers and 2 indistinguishable red flowers. Out of all the distinguishable arrangements, what's probability of selecting an arrangement with at least 6 ...
0 votes
1 answer
13 views

Follow-up question regarding a constant polynomial and eigenvalues.

I have a question regarding the answer given to this question: Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T In showing the second direction (the ...
0 votes
0 answers
10 views

Associativity of Convolutions

In Folland's real analysis textbook, there are the following propositions: Assuming that all integrals in question exist, we have $$ (f*g)*h=f*(g*h) $$ The proof is based on the Fubini's theorem.But ...
1 vote
1 answer
34 views

A criterion for proving that an algebraic extension $E \subset F$ of fields is normal.

Let $K \subset F$ be fields such that $F$ is an algebraic extension of $K$, if for all elements $\alpha \in F$ there is a field $K \subset E \subset F$ such that $\alpha \in E$ and $E$ is normal over $...
0 votes
0 answers
9 views

Reference for a normal CDF/PDF inequality

I am looking for references or any paper/book/report where the following inequality involving the standard Gaussian CDF and PDF appears or was used: $$ \Phi(x)\big(1-\Phi(x)\big) > \phi^2(x), \...
0 votes
0 answers
8 views

Use Taylor expansion to find out the convergence of a indefinite integral

I'm trying to decide the following integral $$\begin{align*} \int _{1}^{\infty} \frac{(e^{1/x^{2}}-1)^{\alpha}}{\log^{\beta}\left( 1+ \frac{1}{x} \right)} \ dx \end{align*}$$ is convergent or not. For ...
17 votes
2 answers
494 views

What is the minimum number of avoids for Dota 2 never ever have a match?

In Dota2, players can avoid another player. So they will never be in the same team again. Every match is a 5x5. Five players against five players. And must have 10 players in total. Let's considerate ...
4 votes
1 answer
52 views

Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
3 votes
0 answers
32 views

Why is $\{A \land \neg A\} \vdash \neg\neg B$ in this deduction tree?

I'm reading through an introduction to logic textbook and cannot follow an example on natural deduction rules. I couldn't find a way to nicely typeset the deduction tree in MathJax, so I'll instead ...
1 vote
2 answers
82 views

Imagining Rational Cauchy Sequences as Dancing Around a Real Number Instead of Converging to One

I'm trying to build my intuition regarding the Cauchy-sequence construction of the reals. Essentially, do you think that it is more accurate to visualize a real number as being defined by sequences of ...
1 vote
1 answer
23 views

What is extension of an isometry?

I am reading Lorentzian geometry. I found the following definition. Definition: A Lorentzian manifold $M$ has maximal isometry group if the action of $\text{Isom(M)}$ is transitive and, for every ...
2 votes
1 answer
4k views

Confusion in solving $(D^2+a^2)y= \tan ax$

Solve $(D^2+a^2)y= \tan ax$ I know the complete integral. I have a confusion regarding particular integral(PI). $$\text{PI} = \frac 1{D^2+a^2}\tan ax = \frac 1 {2ai} \left(\frac 1 {D-ai}-\frac 1 {D+...
0 votes
0 answers
29 views

Infinite Series : $1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\cdot\cdot\cdot\infty$

I need Help to evaluate infinite series : $$S=1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\cdot\cdot\cdot\infty$$ My try: Let $$c_n:= \left({\frac{1-i}{2}}\right)(i)^n+\left({\frac{1+i}{2}}\right)(-i)^...

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