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Proving this sequence converges to 0

Let $a_0 := 1$ and $b_0 := \sqrt{2}$, and define \begin{align*} a_{i+1} &:= |b_i - a_i| \\ b_{i+1} &:= |a_i - a_{i+1}| \end{align*} Prove that $\lim_{i \to \infty}{a_i} = 0$ and $\lim_{i \to \...
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  • 490
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Mistake in proof of "double derivative test" in calculus textbook

I'm currently studying for a semester test in advanced calculus, and one of the topics covered is finding the local minima and maxima of a 3 dimensional surface. The first theorem that was proved was ...
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Good day,everyone.Could someone teach me how to find the value of x in this type of question?😀Thank you.

9x=3(mod 5) 5x=11(mod 12) x=???
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In the proof of corollary associated to the closed subgroup theorem

I'am reading the John M. Lee, Introduction to Smooth manifolds and I have a question. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. And let $\operatorname{exp} : \mathfrak{g} \to G$ be the ...
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Moving from Finite Case to Infinite Case in proof that if $\cup_k I_k \subset I$ and $I_k$ are disjoint then $\Sigma_k |I_k| \leq |I|$

I'm self-studying Billingsley's Probability and Measure and I find myself wondering why we need to use Heine Borel in the second case but not the first. (I attached a picture of the relevant theorem ...
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0 answers
9 views

Coinciding Limsup and limit

Let $(x_n)_{n \geq 1}$ be a sequence of real numbers. Suppose that we are able to show that for a fixed number $m$, $(y_n)_{n \geq 1}:= (x_{n+m})$ and we know that $\lim_{n\to\infty}(y_n)=x$ for some ...
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  • 101
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0 answers
9 views

For every commutative local ring R, the maximal ideal is nilpotent

A commutative ring $R$ is said to be a local ring if it has a unique maximal ideal. My question is: for every commutative local ring $R$, always the maximal ideal has nilpotency that is the maximal ...
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1 answer
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$x=y\Rightarrow \arg x=\arg y$

I was wondering if the implication in the title holds. Obviously it would depend on the definition of $\arg$, so my question is the following: is there any known definition that makes the implication ...
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  • 1,047
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9 views

Sources that representations on $S_n$ are real-valued

I've been told that representations of $S_n$ are real-valued and that there are some sources for that. However, I can't find anything, does anybody have a book / paper that proves this? Thanks in ...
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Sum of adjoints of two operators is a subset of adjoint of sums.

Let $T : \mathcal{D} (T) \subset H \to H$ and $A: \mathcal{D} (A) \subset H \to H$ two densely defined operators and $H$ a complex Hilbert space. Prove $T^*+A^* \subseteq (T+A)^*$. My solution is as ...
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General Reaction Diffusion Model and Turing Instability

I was reading some paper where the following system was solved numerically but I was wondering about the analytic solution. $\partial_t u - d_1 \Delta u = a - (b+1)u +u^p v$ $\partial_t v - d_2 \Delta ...
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  • 111
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Solutions of Two Similar Quadratic Programs

I am solving a very large number of quadratic programs with the same objective function, and very similar constraints. Given a positive definite matrix $\Sigma$ and constraints $A,B,a,b$, let $$ x = \...
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Fiedler Vector for directed and not strongly connected graph

Anyone know if there is any work that deal with Fiedler vector graph partition, for the case of connected, directed, but not strongly connected graph? Thanks, Evyatar
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Recovering matrix from rotated version

I'm dealing with matrices that came from a software, 3ds Max. It uses 4x3 matrices to represent transformations https://documentation.help/3DS-Max/idx_AT_matrix_representations_of_3d.htm $$\begin{...
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0 answers
17 views

The proof of Lemma 10.1 in Nik’s book about forcing

I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set. In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ...
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How to study limits of the extrema of domain for $y'(x) = y(x)(y(x)-1)^{1/3}$?

Given a cauchy problem, $$y'(x) = y(x)(y(x)-1)^{1/3}$$ with initial conditions $y(0) = k$ Since this equation is a separable-variable equation such that $$ G_k(y)= \int_k^y \frac{ds}{s(s-1)^{1/3}} = t$...
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0 answers
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integration of conditional probability

I've tried to understand this paper https://arxiv.org/pdf/2110.14953.pdf $$P(Y|X,C) = \int P(Y|X, z)P(z|C)dz$$ How to solve the right hand side integral?
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1 answer
13 views

How to show the coprimality?

Let $a,b,r\in \mathbb{N}$ such that $ab+1=r^2$ and $$m_1 = 2r(a+r)-1\\ m_2=2r(b+r)-1\\ m_3=2r(a+b+2r)-1.$$ I want to know the possibilities of $\gcd(m_i,m_j)$ and $\gcd(a_i,m_i)$ for $i,j \in\{1,2,3\}$...
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M/M/1/10 queueing process with two different classes

I'm looking at a problem where we have calls queueing under two different classes, new calls and handovers. The number of calls arriving follow a Poisson process with $\lambda_{1} = 125$ per hour ...
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The figure eight knot complement in $S^3$.

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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1 vote
1 answer
29 views

Proof verification that $\lim_{n \to \infty}\frac{n!}{n^n}=0$ (sequence)

Using an epsilon-N approach (since this is supposed to be a sequence), we require $$\forall \varepsilon>0, \exists N \hspace{1mm}\text{s.t} \hspace{2mm}n>N \implies |a_n-L|<\varepsilon$$ Now, ...
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  • 89
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Showing the existence of starting two vectors satisfying the below conditions for cardinality 4,5

Given two distinct nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in 3 dimensions, define a sequence of vectors by $$ \mathbf{v}_{n+2}=\mathbf{v}_{n} \times \mathbf{v}_{n+1}\left(\text { so } \...
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3 votes
1 answer
22 views

Inequality involving sums with binomial coefficient

I am trying to show upper- and lower-bounds on $$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$ (where $n\geq 1$) in order to show that it basically grows as $O(n)$. The upper-bound is easy to ...
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0 answers
8 views

Explaining Conditional Probability and the Urn problem with white and black balls

I am trying to interpret the meaning behind the conditional probabilities when setting up the problem. Can someone confirm with me if I am understanding the problem correctly? Problem Statement: There ...
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  • 3
4 votes
2 answers
37 views

Show that $ \binom{n}{i\;j\;k} \le \binom{n}{m\;m\;m} $

Is it true that for integers $i+j+k= 3m = n$ where $i , j, k , m , n\ge 0$ the inequality holds ? $$ \binom{n}{i\;j\;k} \le \binom{n}{m\;m\;m} $$ I tried to show $$ \frac{n!}{m!m!m!} \Big/ \frac{n!}{...
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  • 323
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0 answers
6 views

When should natural spline be used over not-a-knot

Given the functions $f_1(x)=\sin(2\pi x), f_2(x)=\cos(2\pi x)$ i have to interpolate them with the bounds $a=0,b=1$ over a bunch (not important) points and calculate the max error. The exercise shows ...
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0 answers
14 views

To study the local and global existence of solutions for $y'(x) = y(x)(y(x)-1)^{1/3}$?

Given a Cauchy problem with initial conditions where $k ∈ R$ $$y'(x) = y(x)(y(x)-1)^{1/3}$$ $$ y(0) = k$$ $Q1.$ Discuss the local and global existence and uniqueness of solutions, depending on k? I ...
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0 answers
26 views

What real analysis wants to convey and how it developed?

Most of us have studied real analysis as undergrad. But i cannot understand the main convincing point to study real analysis and how to summerise it. I know this question is little tricky and everyone ...
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2 votes
0 answers
11 views

What does such notations $\mathbb{E}_{X\sim \mu}$ mean?

I read a notation on a paper about statistics and machine learning ("High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models" by Tengyuan Liang, Subhabrata Sen, and Pragya ...
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0 answers
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Find the cheapest arrangement of non-overlapping colored rectangles necessary to achieve a sequence of colors behind holes

Input Let the input be a sequence of colors with any length at least 2. For example, (red, blue, red, green, blue). Each color is represented in the input as a string, not as any abstract notion of a ...
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-2 votes
2 answers
23 views

How many 4 digit numbers are there between 3200 and 7300 in which 6,8,9 does not appear?

How many 4 digit numbers are there between 3200 and 7300 in which 6,8,9 does not appear? The actual answer is 1077 but I'm not sure how to approach it
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0 votes
0 answers
21 views

On $\bigcap_{n=1}^\infty I^n$ for a proper ideal $I$ of a commutative Noetherian ring

Let $I$ be a proper ideal of a commutative Noetherian ring $R$. Let $J:=\bigcap_{n=1}^\infty I^n$. Then, is it true that $J \subseteq P$ for some minimal prime $P$ of $R$? By prime avoidance, I am ...
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  • 123
3 votes
4 answers
65 views

How do I integrate $\dfrac{\sin(x)}{e^x+1}$?

I need to find $$I=\int^{\infty}_0\frac{\sin(x)}{e^x+1}\text{ d}x$$ I have tried using Feynman's method by introducing $$I(t)=\int^{\infty}_0\frac{\sin(tx)}{e^x+1}\text{ d}x$$ but this has led me ...
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  • 1,638
1 vote
0 answers
15 views

Integral representation of $\frac{1}{\Gamma(z)}$

I am trying to find the integral representation of $\frac{1}{\Gamma(z)}$ in the real axis and cant seem to find it. I know that this must have a standard representation but still cant find it.
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  • 391
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0 answers
27 views

Solution check request for the question: If $G=AB$ and $A\cap B=N,$ prove that $G/N\cong A/N\times B/N\\$

Can someone please check my solution to the following question please. Let $A$ and $B$ $N$ be normal subgroups of a group $G$ such that $N\subset A$, $N \subset B.$ If $G=AB$ and $A\cap B=N,$ prove ...
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  • 1,519
1 vote
0 answers
13 views

Is the absolute value of the derivative of a BV function equal to the derivative of its total variation?

For a given function $f\in BV([0,1])$, its function of total variation $V_f(x)$ is defined by $V_f(x)=sup_{0=t_0<t_1<...<t_n=x,\ n\in\mathbb N}\{\Sigma_{i=1}^n|f(t_i)-f(t_{i-1})|\}$, then $f$...
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  • 75
2 votes
0 answers
17 views

What is an instanton? (On a complex surface or a differentiable 4-manifold )

The question is as in the title. I have browsed online (Wikipedia, etc) and while they do give me the definition, it gets a bit too much physics-y for me. Therefore I would appreciate it if someone ...
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0 answers
9 views

Why does not the second recursion theorem guarantee a lest fixed point?

What does it mean that the first recursion theorem guaranties the existence of the lest fixed point, but the second one does not? If there is at least one (pseudo) fixed point then there must be a ...
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0 answers
10 views

Harmonic Distribution of prime numbers

I developed a sieve that depicts the distribution of prime numbers as contained in harmonic (repetitive) patterns. Published it here What would be the process to know if I’m rightfully thinking this ...
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  • 101
0 votes
1 answer
21 views

Aren't all eigenvalues of $T^*T$ non-negative? (Clarification regarding the definition of singular values)

I'm having some confusion with the definition of the singular value of a matrix. As per Wikipedia: In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact ...
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  • 317
1 vote
1 answer
17 views

Proof of bijective bounded linear operator map closed set to closed set

I am trying to prove a bijective, bounded linear operator $T$ of Hilbert space $H$ maps closed set to closed set. Here is my attempt, let $A$ be a closed subset of $H$, then $H-A$ is open, by open ...
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  • 927
-2 votes
0 answers
27 views

examine the integral x^2(cos(e^x)) for convergence

Examine the integral $$\int\limits_0^{+∞} x^2cos(e^x)\,dx$$ for convergence. My ideas - is to use Abel's test, but it doesn't work directly
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1 vote
4 answers
45 views

Let A be a 2x2 matrix. Prove/disprove if $A^2$=A then either A=0 or A=I

I'm really struggling with this problem. I feel that the statement is true because I can't seem to come up with a specific counterexample. Nevertheless, I don't know how to come up with a proof. Any ...
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  • 31
2 votes
0 answers
12 views

Combinatorics and symmetries of Kronecker Powers

Consider the $k$-th Kronecker power $A^{\otimes k}\in\mathbb{R}^{n^k\times n^k}$ of some square matrix $A\in \mathbb R^{n\times n}$. Here we define the Kronecker power inductively as $$ A^{\otimes k} ...
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  • 144
-1 votes
2 answers
19 views

Finding a spanning set of a null space

$$ A= \begin{bmatrix} -3& 6 &-1& 1 &-7\\ 1 &-2& 2& 3&-1\\ 2&-4& 5& 8& -4 \end{bmatrix} $$ Please I have a problem finding the spanning set of a null ...
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-4 votes
0 answers
22 views

Can imaginary i and -i be the elements of the group Z_2?

We know that the elements of the group Z_2 are -1 and 1. Can imaginary i and -i be considered the elements of the group $Z_2$?
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1 vote
0 answers
28 views

The integral of $\frac{\sin(x^2)}{x}\, \text{d}x$ [duplicate]

I need to calculate the following integral; $$\int\limits_0^{∞} \frac{\sin(x^2)}{x}\,\text{d}x$$ But, I don't really have any ideas. Maybe using a new variable or integrating by parts would work. Can ...
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0 votes
1 answer
18 views

Question about differential

I am trying to do the following question: Determine whether each of the following line integrals is independent of path. If it is, find a function $h$ such that $d h=P d x+Q d y$. If it is not, find a ...
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  • 121
-1 votes
0 answers
26 views

Complex number Method verification for finding the sum of $\arctan 1+\arctan2 + \arctan3$

In the complex number method posted by a fellow MSE member here : https://math.stackexchange.com/a/272244/ , I think its not true always as what he/she did , assuming $\operatorname{Arg}$ represents ...
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  • 359
0 votes
1 answer
14 views

What is the distribution of a quadratic function of normal distributions?

Suppose $Z_i$ is independent standard normal distributions, i.e. $Z_i\sim N(0,1)$, $i=1,2,\cdots, d$. What is the distribution of $$ \sum_{i=1}^d (a_iZ_i+b_iZ_i^2). $$ I know when $a_i=0$, it is the ...
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