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How to prove this limit exists almost everywhere, in a space of sequences?

Let $\mathscr{A}$ be a finite alphabet of symbols, equipped with the discrete topology and a probability measure $\mu = (\mu_\alpha)_{\alpha\in \mathscr{A}}$. Then $\mathscr{A}^\infty = \{(p_i)_{i=0}^\...
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2 views

Prove $\forall xA(x) \to B \therefore \exists x(A(x) \to B)$.

Working on P.D. Magnus. "forallX: an Introduction to Formal Logic" (p. 297, exercise C. 1): $ \def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\}...
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0answers
8 views

Asymptotics (big-O) for power sum

I am trying to prove the following equality, for $\alpha <-1$ and $x \geq 1$ $$\sum_{n \leq x}n^\alpha=\sum_{n=1}^\infty n^\alpha+\mathcal{O}(x^{\alpha+1})$$ and have tried rearranging $$\sum_{n \...
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10 views

Is there a chain rule variant for conditional probability?

For joint probability, we have the following: $$ P(x_1,x_2, \cdots, x_n) = P(x_1|x_2,...,x_n)P(x_2|x_3,...,x_n) \cdots P(x_n-1|x_n)P(x_n) $$ If we have a conditional probability like $$ P(x_1, x_2 | ...
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8 views

About a theorem of dual space

Thm: Suppose V is a finite-dimensional vector space with the ordered basis $\beta={x_1,...x_n}$. Let $f_i(1 \leq i \leq n)$ be the $i^{th}$ coordinate function with respect to $\beta$ as defined. Let $...
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6 views

Combination or permutation arithmatics problem.

There are 16 person, who are asked to sit down on a 4 different chair. Every person have to sit with 3 different person from every other 16 person. They are unable to sit again meeting the same ...
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0answers
5 views

Probability Problem from AP Statistics

Assume a test for cancer correctly identifies 98% of the people tested who do have cancer. Unfortunately, the test gives a false positive reading 1.5% of the time. (A false positive is when the test ...
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2answers
18 views

How can I find a $2x2$ matrix with no real eigenvalues?

I am trying to see if there is a process to finding a matrix with no real eigenvalues. I know when we get to the point of $\lambda^{2} + 1 = 0$ then this will have no real solution. Is there a way ...
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5 views

Bayes Theorem in HMM

Through reading https://en.wikipedia.org/wiki/Baum%E2%80%93Welch_algorithm I have come to the expectation calculation where they argue the following is of a Bayes form. I understand the final part of ...
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5 views

Inner regularity on the pre-image measure

Let $X$ and $Y$ be Hausdorff spaces, $f:X\rightarrow Y$ a continuous function and $\mu$ an inner regular measure on $\mathscr{B}(X)$. Show that the image measure $\mu f^{-1}$ is inner regular. I have ...
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1answer
14 views

Proof by Induction - for T(n)

Got stuck on this while doing homework. Prove by induction that T(n) = T(n – 1) + lgn = O(nlgn). Any help would be great to solve this.
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2 views

Solving the Cauchy Problem for the Wave Equation using Neumann Boundary Conditions

I have been trying to crack this problem for a while but cannot seem to figure out what to do with the Neumann Boundary Condition. The problem is to solve the following: \begin{equation*} u_{tt}-u_{...
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3answers
27 views

For all real numbers $a$ and $b$. Prove that if $a>0$ and $b>0$, then $ \frac{2}{a}+\frac{2}{b} \neq \frac{4}{a+b}$

For all real numbers $a$ and $b$. Prove that if $a>0$ and $b>0$, then $$ \frac{2}{a}+\frac{2}{b} \neq \frac{4}{a+b} $$ I am very confused please help , thanks.
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7 views

$C^*$-algebra books recommendations

The maximal ideal space of a commutative $C^*$-algebra is a compact Hausdorff space and it is one way to construct a compactification of a topological space. I am looking for an introductory book in ...
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8 views

About dual space and linear functional

Definition: For a vector space V over F, we define the dual space of V to be the vector space L(V,F), denoted by $V^*$. Question: does that mean dual space is a set of linear functionals?
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0answers
10 views

How can I show this vector identity?

Is there an easy way to show that $$\vec u\times(\nabla \times \vec v)+\vec v\times (\nabla\times \vec u)+ (\vec v\cdot \nabla)\vec u+(\vec u \cdot \nabla )\vec v$$ is equal to $$(\vec u \times \...
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1answer
10 views

Show that if $K$ is a compact set of real numbers, then its inverse image $f^{-1}(K)$ also is compact under conditions.

Suppose $f$ is a continuous real-valued function on Euclidean space $\bf{R}^n$ with the property that there is a positive number $c$ such that $|f(x)| \ge c \cdot ||x||$ for all $x \in \bf{R}^n$. Show ...
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1answer
17 views

Want to prove $\det(P\sigma) = \text{sgn}(\sigma)$.

Let $P$ be a permutation matrix. I'm having trouble trying to prove that $\det(P\sigma) = \text{sgn}(\sigma)$. I think the best way to do this is using the definition of the determinant that involves ...
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1answer
5 views

Prove Lindeberg's condition for a 'modified' Poisson-Binomial random variable.

Prove Lindeberg's condition for $X_{n,k} = \frac{(B_{k} - \frac{1}{k})}{\sqrt{log(n)}}$, where $B_{k}$ ($k \geq 1$) are independent Bernoulli random variables such that $P(B_{k} = 1) = \frac{1}{k} = 1 ...
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0answers
4 views

Trying to prove that a subset of a projective plane is an affine plane.

[ Let S be a projective plane and let m be any line in S. Define a set S, as follows. The points of S, are the points of S that are not on m. The lines in S, are the lines in S with points in common ...
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0answers
5 views

Show that $\mathbb{E}(\Delta_i^2\mid \epsilon_1,…,\epsilon_{i-1})\leq(1-p)pc^2$

Let $\epsilon_1,...,\epsilon_n$ be independent bernoulli random variables with $Pr(\epsilon_i = 1) = p$. Let $f:\{0,1\}^n\rightarrow \mathbb{R}$ be a function which satisfies $\mid f(z) - f(z')\mid\...
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0answers
10 views

Can anyone help me find the mean given the sample size and a percentage?

This is the problem I'm having trouble with: "I polled $40$ of my students, and a whopping $80\%$ of you claim that I am you favourite teacher!" For the given problem I'm also being asked to find the ...
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2answers
15 views

Closed form for $\sum_{k = 1}^n k \cos\left(\alpha n + \beta k\right)$

I'm trying to find a closed form solution for $$\sum_{k = 1}^n k \cos\left(\alpha n + \beta k\right).$$ Is this possible? Thank you very much!
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0answers
11 views

How to show that addition and multiplication of two complex numbers are smooth?

I know that in $\mathbb{R}^n$, the maps such as $(x,y) \mapsto x+y$, $(x,y) \mapsto xy$ are smooth functions. Each functions is smooth, because each coordinate map (or component map) is smooth. ...
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0answers
4 views

Brownian Motion realized on the “Canonical Space”

I see authors make statements such as: Suppose $W$ is Brownian motion realized on the canonical space $(C(\mathbb{R}_+; \mathbb{R}), \mathscr{B}(C(\mathbb{R}_+; \mathbb{R})))$. I'm quite confused ...
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2answers
11 views

ONTO and 1-1 for F(x,y)

For the function $f: {\rm I\!R}^2 \rightarrow {\rm I\!R} $ where $F(x,y)$ = $ax^2 + bxy + cy^2 +dx +ey$ it's unclear to me how to determine which parameters ($a,b,c,d,e$) for $f(.)$ are onto and/or 1-...
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1answer
4 views

Subgame Perfect Nash equilibrium (Mixed strategy)

The following extensive form game is given: Find a Subgame Perfect Nash equilibrium of the game featuring one player using a mixed strategy. I know that in order to find a SPNE (Subgame Perfect Nash ...
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0answers
16 views

Find the number of solutions to the equation $x^2 -2 y^2=1\ (mod p)$ for $p=7297$

The number of elements $(x,y),\ x,y=0,1,...,p-1$, which satisfy $x^2 - 2 y^2=1\ (mod p).$
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2answers
17 views

weaker version of fundamental theorem of calculus for non-compact intervals

Fundamental thm of calculus states: Let $f \in C^0( [a, b], \mathbb{R})$. Let $F: [a, b] \mapsto \mathbb{R}$ by $$ F(x) = \int_a^x f(s) ds. $$ Then $F$ is uniformly continuous on $[a, b]$. Moreover, $...
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0answers
13 views

Large powers of matrices using eigenvalues

So I am asked to compute the power of a matrix using eigenvalues. Here is the question. You are given that the vectors $p_1=(1,1,1), p_2=(2,1,1)$ and $p_3=(1,1,2)$ are eigenvectors to the matrix $$...
1
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2answers
12 views

Graph Theory: Find a $K_{3,3}$ subdivision to this graph

This is my first post here, so I apologise in advance if some formatting is wrong. In my graph theory class today we were challenged to find a $K_{3,3}$ (complete bipartite graph) homeomorphic to ...
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0answers
8 views

Is it possible to use MATLAB's linprog or GNU Octave's glpk to solve quadratic problems?

In MATLAB, there is a function named linprog. Same in GNU Octave. There is a function named glpk Those are for solving linear ...
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1answer
16 views

Is this identity involving the curl correct?

I was wondering if the following step is correct. Suppose I have two vectors $\vec u$ and $\vec v$ $$\vec u \times (\nabla \times \vec v)=-(\nabla \times \vec v)\times\vec u \stackrel{?}{=} -(-(\vec ...
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0answers
20 views

AIME series math question

For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6)=2$ and $b(23)=5$. Find $S=\sum_{p=1}^{2007} b(p)$. I ...
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2answers
33 views

Self study recommendations for a graduate linear algebra course?

I´m looking for a book for self study about linear algebra for a graduate student. Basically in the course we want to cover the next: Vector spaces Linear transformations Inner product spaces Linear ...
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0answers
7 views

How to verify whether the conjugacy is $C^1$ or $C^2$

Can I please receive feedback on the solutions and help to solve part c? Thank you. The Hartman-Grobman Theorem guarantees the existence of a homeomorphism that conjugates the nonlinear equation to ...
3
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1answer
39 views

Homomorphisms from $\prod_{i\in\mathbb Z}\mathbb Z $ to $\oplus_{i\in\mathbb Z}\mathbb Z$ that fixes $\oplus_{i\in\mathbb Z}\mathbb Z$

I'm trying to verify that $\prod_{i\in\mathbb Z}\mathbb Z $(the direct product of countably many $\mathbb Z$) is not a coproduct in the category of abelian groups. We know that the coproduct object is ...
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0answers
22 views

Is this a way to make $\Bbb R^n$ compact?

Consider the operator $$f:\Bbb R^n\to \Bbb R^n$$ with variations on $f(x_1,x_2,\cdot\cdot\cdot, x_n)=\big(\exp(x_1),\exp(x_2),\cdot\cdot\cdot, \exp(x_n) \big)$ such that each component of $\Bbb R^n$ ...
-1
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1answer
16 views

Prove using principle of mathematical induction

The game Nim is played with two players and two piles of matches. In each turn, each player removes some non-zero number of matches from one of the two piles. The winning player is the player who ...
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0answers
7 views

Analytically solving (finding a maxima) a system of equations involving PolyLog functions (Fermi-Dirac Integrals)

I have the following system of equations involving PolyLog functions (Fermi Dirac Integrals) where $d,t\in \mathbb{Z}$ and $d,t >0$ such that $$ J = J_0 \cdot \left[F_{\frac{d-1}{t}}\left(\eta\...
0
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2answers
10 views

Least squares and null space

I want to solve a least squares problem, $$ \min_x ||y - A x ||^2 $$ with $A \in \mathbb{R}^{m\times n}$. Suppose I were to find two distinct solutions $x_1,x_2$, which solve the problem, so that $$ ||...
0
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1answer
19 views

Find the jordan canonical form of the following matrix

A = $$ \begin{matrix} 2 & 1 & 1 & 1 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \\ \end{matrix} $$ I calculated the ...
0
votes
1answer
14 views

Fixed-point iteration when image and domain are not the same

I have a function $f(x)$ defined on a domain $D$, but such that the image $f(D)$ may contain extra regions not included in its domain. I am interested in solving the fixed-point equation $x=f(x)$. If ...
2
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1answer
13 views

Existence of a factorization with some conditions [duplicate]

Question: Let $n \in \mathbb N$ not a prime power and not twice a prime power. Let $s \in \mathbb Z_n$ such that $s^2 \equiv 1 \pmod n$ but $s \not\equiv \pm1 \pmod n$. Is it true that always there ...
0
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1answer
15 views

proof verification - limit and infinite series

I need to prove the following Let $b_n \geq 0$ for $n \in \mathbb{N}$. If for every $r \in (0,1)$ is $\sum_{n \geq1}b_n r^{n}\leq M < \infty$ then $\sum_{n\geq 1} b_n \leq M$ The book I read ...
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1answer
16 views

A compact subspace of a CW complex is contained in a finite subcomplex

In Hatcher's Algebraic topology p.520 he gives the following proposition and proof about CW-complexes which I'll copy partially for clarity sake. Propostion A.1: A compact subspace of a CW complex is ...
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0answers
10 views

Two uncorrelated random variables with variances dependent on each other?

Can anyone give an example of two uncorrelated random variables that have variances dependent on each other?
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0answers
12 views

Cancellation for Field Extensions

Suppose $K,L$ are number fields with coprime discriminants. Let $N/\mathbb{Q}$ be the galois closure of $K/\mathbb{Q}$. Is it true that $$[NL:KL] = [N:K] $$ I feel that this would be true, with some ...
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0answers
10 views

Explanation of Rudin's proof on change of variable

Theorem: (change of variable) suppose $\phi$ is a strictly increasing continuous function that maps in interval [A,B] onto [a,b]. Suppose $\alpha$ is monotonically increasing on [a,b] and f is ...
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3answers
18 views

Rudin exercise 2.21 Clarification about convex sets being connected

Let $A$ and $B$ be separated subsets of $\mathbb{R}^k$, suppose $a \in A$, $b \in B$, and define: $$p(t) = (1-t)a + tb$$ for $t \in \mathbb{R}^1$. Put $A_0 = p^{-1} (A)$, $B_0 = p^{-1} (B)$ (Thus $...

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