1
vote
2answers
117 views

Can Watson's Lemma be applied on multiple integrals simultaneously?

I need to calculate the asymptotics of the integral $$\left(\int_0^1 \mathrm e^{-tx} f(t)\right)^j$$ for $x\to\infty$. I suspect (and would like to prove), that this behaves like ...
1
vote
2answers
89 views

Integrability on R

1) What conditions on the integrand make it integrable over $\mathbb{R}$? I know if a function is continuous and bounded on a closed interval $[-a,a]$ then this is enough for the function to be ...
0
votes
1answer
380 views

Horizontal translations of piecewise-defined functions

I understand that the graph of a real-valued function $g$ where $g(x)=f(x-h)$ is a horizontal translation of the graph of $f$. But is this true for certain piecewise-defined functions? In particular, ...
1
vote
0answers
99 views

Norms on Dual Spaces

Suppose that $\varphi$ is a norm on $\mathbb{R}^n$ such that the set $$\varphi_1 = \{x \in \mathbb{R}^n : \varphi(x) = 1 \}$$ is a polyhedron. Let the dual norm $\varphi^*$ be defined as usual: ...
43
votes
3answers
1k views

Alice and Bob matrix problem.

Alice and Bob play the following game with an $n*n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number. then Bob fills one. Then Alice and so on so forth ...
0
votes
3answers
88 views

Monotone Subseqences

If limsup$(p_k) \neq \infty$ and liminf$(p_k) \neq -\infty$, prove or disprove that $(p_k)$ has monotone increasing and monotone decreasing sub sequences.
0
votes
0answers
89 views

finite p-groups admit a central series

If a central series is considered as $$G = G_0 \supset G_1 \supset \cdots \supset G_m = \{1\}$$ such that $$G_{i+1} \triangleleft G_i$$ and $$G_i/G_{i+1} \subset Z(G/G_{i+1})$$ then, Show that finite ...
2
votes
2answers
80 views

characterisation of compactness in the space of all convergent sequences

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) ...
1
vote
0answers
60 views

Proof of a Continued Fraction Identity using basic CF definition.

Two definitions (the first is informal) of continued fraction: This is the basic Continued Fraction algorithm for real numbers. Let $\alpha \in \mathbb{R}$. If $[\alpha]=\alpha$, then we are done. ...
2
votes
2answers
196 views

Finding the limit of a function $n^{n}/e^{n^{3/2}}$

What is the limit $$\lim_{n\to\infty}\frac{n^{n}}{e^{n^{3/2}}}?$$
0
votes
2answers
64 views

show sequence converges

If $(x_n)$ and $(y_n)$ are Cauchy sequences in a metric space X with metric d, how would I show the sequence $(d(x_n,y_n))$ converges? I'm supposed to use $d(x_n,y_n)\leq ...
1
vote
3answers
149 views

Number Theory Problem $ax+by=n$ for $n>ab$

Let $a,b \in \Bbb N$ with $\gcd(a,b)=1$. Show that for every integer $n>ab$ the equation $ax+by=n$ has a solution in positive integers $x,y$. (Take $(x,y)$ with $y \leq 0$ and $x$ minimal).
9
votes
5answers
586 views

Am I fit for higher studies/teaching in mathematics? [closed]

Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I ...
4
votes
1answer
526 views

Convergence in measure implies convergence in $L^p$ under the hypothesis of domination

Given a sequence $f_n \in L^p$ and $g \in L^p$, with $|f_n| \leq g$, I am trying to show that $f_n \to f$ in measure implies $f_n \to f$ in $L^p$. Firstly, I know that if $f_n \to f$ in measure, then ...
1
vote
1answer
255 views

Show that there exists a bijection from $(A^B)^C$ into $A^{B \times C} $

Notation: Let A and B be sets. The set of all functions $f:A \rightarrow B$ is denoted by $B^A$. Problem: Let A, B, and C be sets. Show that there exists a bijection from $(A^B)^C$ into $A^{B \times ...
-1
votes
2answers
207 views

What is a formal polynomial?

I'm starting to study Field Theory by myself, the books don't say explicitly what a polynomial is, I mean, what the $x$ of $f(x)$ in $F[x]$ is? $x\in F$? When I take $f(\alpha)$ am I taking the ...
2
votes
2answers
472 views

Big - O estimation

I want to establish a Big-O estimate for the following: $$(n! + 2^{n+3})(111n^3 + 15\log(n^{201} +1))$$ Would the following be correct? $n! = O(n^{n})$ $2^{n+3}=O(2^{n+3})$ $111n^{3}=O(n^{3})$ ...
4
votes
2answers
179 views

Compact and self-adjoint operator

It is true that if $T:H \to H$ is a compact operator ($H$ Hilbert space) then $T^\ast T$ is algo compact and indeed self-adjoint. Conversely, is it true that every compact and self-adjoint operator ...
2
votes
0answers
114 views

Prove metric space…

Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$ If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by $$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$ Prove that $d$ is ...
1
vote
1answer
82 views

A question about the definition of fibre bundle

The canonical definition of fibre bundle is the following: Let $B,X,F$ be three topological spaces and $\pi:X\rightarrow B$ a continuous surjective map; then $(X,F,B,\pi)$ is a fibre bundle on $B$ ...
2
votes
2answers
92 views

General strategy for solving joint distribution problems

A joint distribution in $X$ and $Y$ is given as; $$f(x,y) = 2e^{-x}e^{-2y} \ \ \ 0< x< \infty, \ \ 0< y< \infty $$ a) Compute $P\{X>1, Y<1\}$ b) Compute $P\{X < Y\}$ ...
1
vote
0answers
55 views

Maximum value for parameter

I am facing the following problem: A number of a adults, b children older than 12, and c children younger than twelve attend an event. The sum of all people a+b+c=100. The prices are \$6 per adult, ...
3
votes
0answers
205 views

An Upper Bound for an $[n,k,d]$ Linear Binary Code.

I've been reading about the various upper bounds for different types of codes. Recently, I came across a statement that is similar to the Singleton Upper Bound that I am having trouble proving. The ...
3
votes
3answers
473 views

Proving an isomorphism

Let $N$ be a normal subgroup of $G$ and $G$ a group. Let $H$ be a subgroup of $G$ with $|(N)||(H)|=|(G)|$ and $N\cap H=\{1\}$. Why is $G=HN$ and why are two subgroups of $G$ isomorphic? Thanks a ...
0
votes
1answer
586 views

Finding the equation that is described by this block diagram

I have the following block diagram The exercise asks to find the equation that describes the system. What I did: I called what going into the $-1$ multiplier as $x_{1}$and I got $2$ equations ...
0
votes
0answers
79 views

Proof for Efron Stein-Inequality

From a note, in the proof for Theorem 1 Efron Stein-Inequality: Suppose that $X_1 , \dots, X_n , X_1' , \dots, X_n'$ are independent with $X_i$ and $X_i'$ have the same distribution for all $i$. ...
1
vote
0answers
58 views

Finding “c” of this joint distribution

A textbook example asks me to solve for $c$ in the joint distribution function; $$ f(x,y) = c(y^2-x^2)e^{-y} \ \ \ -y \le x \le y, \ \ 0 < y <\infty $$ The answer given involves integrating ...
0
votes
1answer
52 views

Question on proof of Schoenberg correspondence from Lévy Process and Stochastic Calculus by Applebaum

I quote the proof here from Applebaum's Lévy Processes and stochastic calculus (and the things before it to present the full picture) We say $\phi:\mathbb{R}^d\rightarrow\mathbb{C}$ is conditionally ...
2
votes
2answers
89 views

Complete DVR with finite residue field is compact?

How do I go about proving this? Do I have to show total boundedness (I don't know how to use the finiteness of the residue field, and this seems like something that it might pertain to).
2
votes
0answers
214 views

Helmholtz and Biharmonic equation examples with exact solution

I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare results of my numerical solutions with it. I was able to find quite ...
2
votes
3answers
921 views

Prove the Distributive Law of indexed sets

$(i) X\cap(\bigcup\limits_{n=1}^\infty A_n) = \bigcup\limits_{n=1}^\infty (X\cap A_n)$ $(ii) X\cup(\bigcap\limits_{n=1}^\infty A_n) = \bigcap\limits_{n=1}^\infty (X\cup A_n)$ Any suggestions on how ...
4
votes
1answer
140 views

Prove that f is continuous

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex function on $\mathbb{R}^n$. How to prove that $f$ is continuous?
1
vote
1answer
220 views

Plotting a Joint Probability Density function

I have a problem where I have two independent variables each having a probability density function given by: $p(s_1) = \frac{1}{2}\sqrt{3}$, when $s_1\leq\sqrt{3}$ and $0$, otherwise And the ...
2
votes
2answers
171 views

How to show that $\frac{\sin(n)}{n}$ is $1$ as $n \rightarrow 0$? [duplicate]

Possible Duplicate: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How to show that $\frac{\sin(n)}{n}$ is $1$ as $n \rightarrow 0$? just hint.
1
vote
3answers
83 views

probability that Janani will win the game?

Two people are playing a coin toss game with a fair penny. Manu gets a point if the penny lands on heads. Janani gets a point if the penny lands on tails. The score is Janani 9, Manu 7, in a game to ...
1
vote
2answers
127 views

Linear algebra problem involving the characteristic of a field

I'm having trouble with the following problem: Let $\tau_A: F^2\times F^2 \rightarrow F$ be a symmetric bilinear form given by $\tau_A (v,w)=v^tAw$, $\forall v,w\in F^2$ and ...
0
votes
0answers
116 views

necessary and sufficient condition for continuous function to be monotonic at certain domain

Lately I found there exists a function which is "continuous but nowhere monotonic". So, now I want to know that (the title). I'm really thank you if you give me a proof of it.
1
vote
1answer
45 views

A method to evaluate rounding errors

I'd like to learn to evaluate how much there is error if I compute algebraic expressions and round my intermediate steps. For example, I had data of length of couples height as (167,183), (165,165), ...
3
votes
2answers
659 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
1
vote
1answer
238 views

Universal cover via paths vs. ad hoc constructions

I'm looking for some intuition regarding universal covers of topological spaces. $\textbf{Setup:}$ For a topological space $X$ with sufficient adjectives we can construct a/the simply connected ...
0
votes
1answer
63 views

This intersection has at most 2 components

Let $U$ and $V$ two sets homeomorphic either to the open interval (0,1) or the half-open interval [0, 1), then their intersection has at most two components. My attempt was show that the open connect ...
10
votes
4answers
673 views

$AB \neq 0$ but $BA=0$

Do there exists to matrices or objects such that $AB \neq 0$ but $BA=0$? Another way to ask this question is if there exists objects or matrices $A$ and $B$ such that... $[A,B]=AB$ where $[ \, , \, ]$ ...
1
vote
2answers
233 views

Centroids of triangle

On the outside of triangle ABC construct equilateral triangles $ABC_1,BCA_1, CAB_1$, and inside of ABC, construct equilateral triangles $ABC_2,BCA_2, CAB_2$. Let $G_1,G_2,G_3$ $G_3,G_4,G_6$be ...
2
votes
1answer
86 views

To solve $U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0 $

$$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$$ $$\frac{\partial}{\partial t }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$$ ...
5
votes
4answers
727 views

Counting the number of surjections.

How many functions from set $\{1,2,3,\ldots,n\}$ to $\{A,B,C\}$ are surjections? $n \geq 3$ Attempt I was hoping to count the number of surjections by treating $A,B,C$ like bins, and counting the ...
1
vote
2answers
874 views

Related Rates - Calculus - Filling a cylinder

If there is a cylinder with a base radius of 70cm, and water is being poured into it at 10 liters per minute, how fast is the water level rising? I've gotten myself horribly confused as to how to do ...
0
votes
1answer
197 views

Euler's Criterion and Wilsons Theorem

I am trying to prove: if $m = p_1p_2\cdots p_r$ with $2 < p_1 < \cdots < p_r$ prime, then $$x^2 \equiv 1\mod m$$ has $2^r$ solutions modulo $m$. I know Euler's Criterion: $p$ is an odd ...
1
vote
1answer
72 views

Do I understand this question correctly?

My book has an exercise: "Suppose that $W$ is a subspace of a finite-dimensional vector space $V$. a) Prove that there exists a subspace $W'$ and a function $T:V\longrightarrow V$ such that $T$ is ...
12
votes
1answer
392 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
0
votes
1answer
27 views

What type of question should we use $\binom{n + k - 1}{k - 1}$ and others

I know there are some questions with solutions of the form $\binom{n + k - 1}{k - 1}$ There are also questions with solution of the form $\binom{n + k - 1}{k}$ and there are questions with ...

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