Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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5
votes
1answer
141 views

Show that $\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$

Let $g:[0,1]\mapsto\mathbb{R}$ be a continuous function, and $\lim_{x\to0^+}g(x)/x$ exists and is finite. Prove that $\forall f:[0,1]\mapsto\mathbb{R}$, ...
4
votes
0answers
131 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
1
vote
1answer
104 views

Uniform contraction proof

Prove that for every uniform contraction function $f$ there exists a unique real $z$ such that $f(z)=z$. A function $f:\mathbb R\to\mathbb R$ is called a uniform contraction if there exists an $a$ in ...
1
vote
1answer
214 views

Find limit of quite complex function

Tomorrow I have an exam on mathematical analysis and I solving exams task from earlier years, and here is: Find limit of: $$\lim_{n \to \infty} \frac{(4n^3 + 1)(4n - 2)!n\sin{\frac{2}{n}}}{(4n + ...
2
votes
1answer
129 views

Complex analysis simultaneous mapping

I need to find a map that takes the region between two circles $|z|=1$ and $|z-1/4| = 1/4$ to an annulus $a<|z|<1$. Now I know that the bilinear transform $f(z) = ...
1
vote
1answer
55 views

Is there a constant for this?

Suppose that $\sum_{i=1}^{n}\lambda_{i}=1$, where $\lambda_{i}>0$, and $\sum_{i=1}^{n}x_{i}^{2}=1$, where $x_{i}>0$. Does one have $n^{3/2}\min_{1\le i\le n}\lambda_{i}x_{i}\le B$ for some ...
1
vote
1answer
80 views

Can one use Holder's inequality or some other method for this?

Suppose that $\sum_{i=1}^{n}\lambda_{i}=1$, where $\lambda_{i}>0$, and $\sum_{i=1}^{n}b_{i}^{2}=1$, where $b_{i}>0$. Does one have $\sqrt{n}\sum_{i=1}^{n}\lambda_{i}b_{i}\le B$ for some constant ...
1
vote
1answer
37 views

Showing that a certain function is $C^1$

For an exercise in my analysis course, I have to show that the function $$\newcommand{\sgn}{\operatorname{sgn}}f: (x,y) \mapsto \begin{cases} \frac{(x \sin y)^2}{|x|+|y|},&(x,y) \neq (0,0) \\ ...
0
votes
1answer
96 views

Absolute continuity and integration formula (explain a statement please)

I read this: For $v$, $w$ in $L^2(0,T;H^1(S))$ (with weak derivatives in $H^{-1}(S)$ for each time), the product $(v(t), w(t))_{L^2(S)}$ is absolutely continuous wrt. $t \in [0,T]$ and ...
0
votes
1answer
103 views

How to integrate over curved extrusion paths?

Given a two-dimensional area $A\subset\mathbb R^2$ lying in the $xy$-plane, i.e. $A:\mathbb R^2\supset U\to A\subset \mathbb R^2, (u,v)\mapsto (x(u,v),y(u,v))$, a straight extrusion along the $z$-axis ...
0
votes
0answers
67 views

Consultation on point extreme

Let $C$ be a convex subset of a vector space $X$. A point $x\in C$ is called an extreme point if and only if whenever $x=ty+(1−t)z$, $t\in (0,1)$, implies $x=y=z$. It is known that the boundary ...
3
votes
1answer
119 views

Someone know a specific maximum principle…? [Solved]

I need a maximum principle in unbounded domains: if $u$ is a solution, bounded in $\Omega$, satisfying $$\Delta u+c(x)u=0, \ \ in \ \Omega,$$ $c\in L^\infty$, $$u\leq0 \ \ in \ \Omega$$ $$u(x_0)=0, \ ...
1
vote
3answers
379 views

Does the series $\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$ converge?

Does the series $$\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$$ converge?
2
votes
1answer
176 views

Proving $\lim\limits_{n\to\infty}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$ implies $\lim\limits_{n\to \infty}\frac{a_n}{b_n}=L$

Suppose $(a_n)$ and $(b_n)$ are sequences where $b_n$ is increasing and approaching positive infinity. Assume that $\lim_{n\to \infty}$ $\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$, where $L$ is a real number. ...
1
vote
1answer
73 views

For any function f from a countable subset M of $\mathbb{R}$ there exists a sequence of continuous functions that converges pointwise to f on M.

Problem: Prove that for any function f from a countable subset M of $\mathbb{R}$ there exists a sequence of continuous functions {f_n} that converges pointwise to f on M. Context: This was put ...
0
votes
1answer
89 views

How to prove or disprove this proposition: convex combination of two convex function is non-negative (under some assumption)

Let $f,g$ be two convex function on $D\in\mathbb{R}$ (what about $\mathbb{R}^n$?) satisfying that there is no point $x$ in $D$ such that $f(x)<0$ and $g(x)<0$ at the same time. I want to prove ...
1
vote
1answer
21 views

Question concerning solution set of an inequation

Following inequation is given: $ \frac{2-x}{3+x} < 4 $ If $ 3+x > 0$ then $ x > -2$ and if $3+x < 0 $ then $ x < -2$. Till here I understand everything. The solution set is: ...
0
votes
1answer
45 views

component of a differentiable function differentiable?

in differential geometry we define a function $f:M \rightarrow N$ between differential manifolds to be differentiable, if the function $y \circ f \circ x^{-1}$ (where $y$ and $x$ are appropriate ...
2
votes
3answers
317 views

Epigraph of a function.

I hope you can give me some suggestions on convex functions. the function $f:(0,\infty)\rightarrow \mathbb{R}$ given by $f(x)=\dfrac{1}{x}$ is convex and continuous, but its epigraph is closed in ...
2
votes
1answer
131 views

Diffeomorphisms, Vector fields and Push Forwards.

Studying for an exam, and trying to get my head around the concepts of push forwards. The question I'm attempting to answer is: "Give an example of a continuously differentiable diffeomorphism F and ...
1
vote
1answer
35 views

is there a trivial equation for the sum of $m^0,m^1 m^2…m^h$

Is there a similar trivial equation for solving $\sum^h_{k=0}(m^k)$ like for solving $\sum^h_{k=0}(k) = \frac{n(n+1)}{2}$ as I want to evaluate the smallest h which when used will fulfill the equation ...
3
votes
1answer
51 views

Continuous Connection?

Consider two compact convex sets $C_1, C_2 \subset \mathbb{R}^n$ such that $C_2 \subset C_1$. Let us denote by $\partial C_1$ and $\partial C_2$ their boundaries, that satisfy and $\partial C_1 \cap ...
0
votes
1answer
26 views

Dimension of coefficents in a density equation

The density throughout a composite material is given by $T(x, y, z) = Axy^2 + Bxz^3 + Cy^2z^3,$ where $x$, $y$ and $z$ are the cartesian coordinates of the position inside the material. (a) Find the ...
4
votes
2answers
82 views

Showing a function has only one point of continuity.

Let $$f(x) = \begin{cases}\;\;\, x\;\;,\;\text{ if } x \in \mathbb{Q}\\ -x\;\;,\; \text{ if } x \in \mathbb{R}\setminus \mathbb{Q} \end{cases}$$ (i) Determine the point or points of continuity of ...
1
vote
1answer
206 views

Scalar product rule for limits

Let $X$ be a metric space, $E$ be a subset of a metric space $X$, $p$ be a limit point of $E$, $f:E \to \mathbb{C}$, and $\lim\limits_{x \to p} f(x)=L$. Finally let $c \in \mathbb{C}$. Show that ...
1
vote
0answers
69 views

Trying to prove logarithms preserve limits without any notion of continuity

I am trying to prove that if ${s_n}$ is a convergent sequence, $\lim_{ n \to \infty} s_n = s$ iff $\lim_{ n \to \infty} \log(s_n) = \log(s)$. I don't have any notion of continuity yet (although I ...
2
votes
1answer
58 views

How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?

I would like to prove the existence of $b \in \mathbb Q$ such that $a<b^2<c$ for any given $a,c \in \mathbb Q$ with $a,c>0$ I want to use the statement above to prove a statement in a link I ...
2
votes
2answers
130 views

Compactness of unit ball in weak-operator topology.

I'm reading Richard Kadison's book about operator algebras, and in the demonstration that the unit ball is compact in weak-operator topology, the author defines a function from the set of bounded ...
0
votes
0answers
56 views

Compute the transition map of two charts.

Need to know how to compute the transition map of $h = g^{-1}\circ f$ The unit circle has charts $f(s) = (\cos(s),\sin(s))$ is element of $\Bbb R^2$, for $-\pi < s < \pi$, and $g(t) = ...
1
vote
1answer
75 views

question related to outer measure and pseudometric.

I want to show that if O is collection of open subsets of (0,1) what is the closure of O in the associated metric space of equivalence classes? The metric associated with this collection is ...
1
vote
2answers
147 views

Are these charts on the circle compatibly oriented?

I've tried a few methods but I can't seem to work this one out. Consider the charts $$f(s) = (\cos s, \sin s) \in \mathbb{R}^2$$ for $-\pi < s < \pi$ and $$g(t)=(\frac{2t}{t^2 + 1}, \frac{t^2 ...
2
votes
0answers
45 views

Graph of a set homeomorphic

I would like to please guide me on this question: Let $S_+$ denote the set of semi positive definite matrices in $\mathbb{R}^{2\times 2}$ is known that $S_+\subseteq Sym \simeq\mathbb{R}^{3}$,wherein ...
0
votes
1answer
83 views

Suppose that $f: \mathbb R^q \to \mathbb R^p$ is a linear map. Prove that $f$ is differentiable and that $f'(x) = f$ for every $x \in \mathbb R^q$

Suppose that $f: \mathbb R^q \to \mathbb R^p$ is a linear map. Prove that $f$ is differentiable and that $f'(x) = f$ for every $x \in \mathbb R^q$. I don't know of any way to prove this?
1
vote
1answer
132 views

Prove that a sequence $\{x_k\}_{k=1}^\infty\subset \mathbb{R}^n$ converges to $x$ if and only if the map $ f(j) = x_j$ is continuous.

Need to know how to prove that a sequence $\{x_k\}_{k=1}^\infty\subset \mathbb{R}^n$ converges to $x$ if and only if the map $ f:\{1,2,3...\} \to\mathbb{R}^n$, $ f(j) = x_j$, is continuous. It's been ...
2
votes
3answers
362 views

How to prove $r^2=2$ ? (Dedekind's cut)

Let a (Dedekind) cut $r=\{p \in \mathbb{Q} :p^2<2 \text{ or } p<0\}$ and a cut $2^*=\{t\in \mathbb{Q} : t<2\}$. I want to prove $r^2=2^*$. I could show that $r^2 \subset 2^*$ easily, but I ...
0
votes
1answer
898 views

Prove that every linear map is continuous.

Need to know how to prove that every linear map is continuous. It's one of the problems I'm currently working on as revision for an upcoming test. I know that that a map is continuous if the preimage ...
0
votes
1answer
119 views

Prove that every ball of radius r lies inside a box all of whose sides has length at most 2r. Prove that every bounded set lies inside a box.

Need to prove that every ball of radius $r$ lies inside a box all of whose sides has length at most $2r$. Also need to prove that every bounded set lies inside a box. I know that a ball of radius $r$ ...
0
votes
2answers
1k views

Prove that any continuous integer-valued function of a real variable is constant.

I'm stuck on a question which asks to prove that any continuous integer-valued function of a real variable is constant. In my lecture notes we are told that a map is continuous if the preimage of any ...
3
votes
1answer
258 views

Prove that there is no $1-1$ continuously differentiable map $f : \mathbb{R}^2 \to \mathbb{R}.$ [duplicate]

Possible Duplicate: existence of a map between $\mathbb R^2$ and $\mathbb R$ Prove that there is no $1-1$ continuously differentiable map $f : \mathbb{R}^2 \to \mathbb{R}$. I know it's ...
0
votes
1answer
104 views

Suppose that $f''(0)$ exists and is finite, show that $f''(0)=\lim\limits_{h \to 0}\dfrac{f(h)-2f(0)+f(-h)}{h^{2}}$

I read the following from Chung's Probability: ($f$ is the characteristic function of some distribution function $F$) Suppose that $f''(0)$ exists and is finite, then we have $f''(0)=\lim\limits_{h ...
1
vote
0answers
37 views

How to prove this estimate?

Suppose $f(X) = (X − α)^r\cdot g(X)$, where $α ∈ \Bbb C$ is nonzero, $r ∈ \Bbb Z^+$, and $g ∈ \Bbb C[X]$ is nonzero. Prove that $$||g|| < (1 + \deg g)\cdot (2 \max(1, |α|^{−1}))\cdot \deg f\cdot ...
1
vote
1answer
439 views

smooth approximations of indicator function

How would I construct Schwartz functions $f_1^\epsilon$, $f_2^\epsilon$ on $\mathbb{R}$ such that $f_1^\epsilon(x)\leq\mathbb{1}_{[a,b]}(x)\leq f_2^\epsilon(x)$, and ...
0
votes
1answer
77 views

Mathematical Analysis-Cluster points-Bolzano Weistrass THM

I am stuck in my one of the homework problems, the question is like the following: Let $(x_n)$ be a bounded sequence, and let $c$ be the greatest cluster point of $(x_n)$: (a) Prove that for every ...
6
votes
1answer
187 views

Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?

This is a follow-up to Continuous partials at a point but not differentiable there?, but I'll make this question self-contained. Throughout, $f$ will denote a function $\mathbb{R}^2\to\mathbb{R}$. An ...
4
votes
4answers
215 views

Proof of the irrationality of $\sqrt{3}$ - logic question

Prove $\sqrt{3}$ is irrational. (Proof by contradiction). Let $\sqrt{3}$ be a rational number in simplest form $\frac pq$. So squaring both sides of $\sqrt{3}=\frac pq$ we get $3=(\frac {p}{q})^2$ ...
1
vote
2answers
134 views

Prove the following inequality

Prove the following: The summation ranges from 1 to n. ${\sum{a_i^2}\ge \frac1n }$ provided $\sum a_i$=1. without using the method used to prove chebyshev's inequality. I can do this very easily ...
2
votes
1answer
153 views

A functional equation related to the exponential function

Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation \begin{equation} f(xy)=f(y)^{g(x)} \end{equation} ...
1
vote
0answers
32 views

$L^2$ of a fiber bundle

For spaces $X$ and $Y$ with measures there is an isomorphism of Hilbert spaces $$L^2(X) \otimes L^2(Y) \to L^2(X\times Y), ~~ f\otimes g\mapsto \left((x,y)\mapsto f(x)g(y)\right).$$ Now suppose $E ...
0
votes
1answer
26 views

Are we allowed to arrange the terms of a converging sequence?

I have a sequence $f_n$ of measurable, positive functions over some common domain $A$. I don't know whether they are increasing, yet I know that they converge to an integrable function $f$ on $A$ for ...
1
vote
1answer
195 views

Exponential operator on a Hilbert space

Let $T$ be a linear operator from $H$ to itself. If we define $\exp(T)=\sum_{n=0}^\infty \frac{T^n}{n!}$ then how do we prove the function $f(\lambda)=exp(\lambda T)$ for $\lambda\in\mathbb{C}$ is ...