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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100 views

Is a continuously differentiable function convex if all its partial second derivatives are non-negative?

I'm having trouble understanding the relevant Wikipedia article which begins with a convex set $X$ and then uses functions of single variables for succeeding examples; the MathWorld article seems to ...
0
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2answers
87 views

$a_j \geq 0, \sum_{j=1}^{\infty} a_j$ divergent $\implies \sum_{j=1}^{\infty}\frac{a_j}{1 + a_j}$

Suppose that $a_j \geq 0$ and that the $\sum_{j=1}^{\infty} a_j$ diverges. Prove that the following series diverges: $$\sum_{j=1}^{\infty}\frac{a_j}{1 + a_j}$$ Hint: first show that if it converges, ...
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3answers
87 views

Show and prove if the following series converges or diverges

Show and prove if the following series converges or diverges $$\sum_{j=1}^\infty{ \frac{(1+(1/j))^{2j}}{e^j}}$$ "I tried the comparison test, the root test, and the ratio test, but got messed ...
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4answers
532 views

Prove the set is not open.

Prove that the set $\mathbb{R}-\{1/n|n \in\mathbb{N}\}$ is not open. OK, so I am having a little trouble. I know that the definition of open set is : iff every point of $A$ is an interior point of ...
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1answer
72 views

Convergence of $|a_{n+1} - a_n| \le Cq^n$

Be $C\gt 0$, $0\le q\lt 1$ and $(a_n)_{n\ge 1}$ a sequence in $\mathbb R$ with $$|a_{n+1} - a_n| \le Cq^n$$ Show that $(a_n)_{n\ge 1}$ converges.
0
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2answers
61 views

simple undergraduate series quesiton

consider $ \displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n $ where $ (a_n) $ is a monotone decreasing sequence of nonnegative numbers with $ a_n \rightarrow 0 $ by the alternating series test, series ...
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0answers
21 views

Compute tetrahedral region

Show that the volume of region $A$ is $1/6$. Region $A$ is a tetrahedral region in $\mathbb R^3$. $$A=\{(x,y,z)∈R^3 \mid x\ge 0, y\ge 0, z\ge 0, \text{ and } x+y+z\le 1\}$$
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2answers
370 views

Why is every continuously differentiable function with a uniform bounded derivative lipschitz continuous

I only know how to prove this for functions on a convex set by using the mean value theorem, but is this also true for this general case when nothing is said about the domain of the function besides ...
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1answer
49 views

$|a_n - a_m| \le \sum_{k=m}^{n-1} |a_{k+1} - a_k|$

How to show, that every for sequence $(a_n)_{n\ge_1}$ with $n \gt m$ following holds ? $$|a_n - a_m| \le \sum_{k=m}^{n-1} |a_{k+1} - a_k|$$
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1answer
84 views

Continuity of a constant function

If f assumes only finite many values, then f is continuous at a point $x_0$ if and only if f is constant on some interval $(x_0 - \delta, x_0 + \delta)$ I know how to prove continuity for a given ...
2
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1answer
113 views

Need a counter example for series convergence

I need some advice for constructing a counter example for $\sum\limits_{i=1}^\infty a_i$ converge but $\sum\limits_{i=1}^\infty \frac{a_i}{i}$ diverges.
0
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1answer
53 views

Counterexample to show that the change of variables for integral does not hold

Give a Counterexample to show that the change of variables formula does not hold if $g$ is not one-to-one even though $Jg(x) \neq 0$ where $Jg$ is Jacobian Matrix. [Hint: Take $f = 1$ and $g(x,y) = ...
0
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1answer
193 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
2
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1answer
50 views

Functions, Continuity and IVT

Suppose that $g$ is a function defined and continuous on $\mathbb{R}$ and $n$ is a positive integer such that $$\lim_{x\to \infty} \dfrac{g(x)}{x^n} = 0 = \lim_{x\to -\infty} \dfrac{g(x)}{x^n}$$ (i) ...
0
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2answers
51 views

Functions and the IVT

Let $g, h$ be continuous functions defined on some interval $J$ and suppose that $g(x) \neq 0$ for any $x \in J$. If $g(x)^2 = h(x)^2$ for all $x \in J$, show that either $g(x) = h(x)$ for all $x \in ...
0
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0answers
29 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
2
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2answers
44 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
2
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1answer
154 views

Does the Weierstrass $\wp$ function have any double values besides $\infty$?

Given nonzero complex constants $\omega_1,\omega_2$, with nonreal ratio, we define $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_\omega \frac{1}{(z-\omega)^2}-\frac{1}{\omega^2} $$ where the sum is ...
2
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3answers
330 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
2
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1answer
552 views

Any even elliptic function can be written in terms of the Weierstrass $\wp$ function

Given two nonzero complex numbers $\omega_1, \omega_2$, with nonreal ratio, we define the period module $$M= \omega_1 \mathbb Z+ \omega_2 \mathbb Z= \{n_1 \omega_1+ n_2 \omega_2:n_1,n_2 \in \mathbb Z ...
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1answer
137 views

Prove that an eigenvector is the maximum of a symmetric matrix

Let $f : S^{n-1} \rightarrow \mathbb{R}, x \mapsto x^TAx$ ( A is a symmetric matrix), then an eigenvector $\xi$ of A is a local maximum of this function. We are supposed to prove this in 6 steps and ...
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1answer
198 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
2
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1answer
79 views

Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots+ b_n)$ convergent?

Let $\sum_{n=1}^{\infty}a_n$ be a convergent series of positive terms (so $a_i > 0$ for all $i$) and set $b_n = 1/(na_n^2)$ for $n\ge1$. Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots + b_n)$ ...
5
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1answer
127 views

Proving $\int^{\infty}_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0$

I've been asked to prove that $$ \int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$ My approach so far has been to use a theorem proved in class ...
7
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1answer
148 views

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by ...
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1answer
50 views

A question in the proof of connectedness in $\mathbb{R}$

In my textbook there is a theorem saying that "If $S\subset\mathbb{R}$ is an interval, then $S$ is connected." I can follow most of the arguments provided there except the one indicated below. Can ...
10
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2answers
201 views

Regular $n$-gon in the plane with vertices on integers?

For which $n \geq 3$ is it possible to draw a regular $n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have integer coordinates? I figured out that $n=3$ is not possible. Is $n=4$ the only ...
1
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2answers
207 views

Power series representation/calculation

I am struggling a bit with power series at the moment, and I don't quite understand what this question is asking me to do? Am I meant to form a power series from these, or simply evaluate that series? ...
2
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1answer
81 views

basic exercise distribution theory

Consider $f \in L^{2}(R^n)$ with $\Delta f \in L^{2}(R^n) $. Show that ${\partial}^{|\alpha| } , (|\alpha| \leq 2 )f \in L^{2}(R^n)$. (the derivatives is in the distribution sense). My book gives ...
3
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1answer
111 views

Functions that satisfy $f(x,z) = f(x,y) f(y,z)$

I am specifically looking for solutions that are NOT of the form: $f(x,y) = g(x)/g(y)$ since that is an obvious solution, as is $f(x,y) = 0$. I have a suspicion that there may be answers to this ...
0
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2answers
62 views

analysis proof about convergence. What is limit definition?

I'm stuck on Question 1. Obviously since A_n goes to zero, b_n/a_n will go to zero but how do I prove it? I don't know what they mean by LIMIT DEFINITION. Could you give me hints? Thanks.
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1answer
61 views

Hölder's Inequality and step functions

Define functions $f(x) = \sum_{k=0}^\infty a_k \chi_{[k,k+1)}(x)$ and $g(x) = \sum_{k=0}^\infty b_k \chi_{[k,k+1)}(x)$ where $\chi_ {[k,k+1)}$ is the indicator function for the given interval. Let ...
1
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1answer
29 views

Prove that $Lip [a,b]^{\circ}=\emptyset$

Let $Lip[a,b]=\{f \in C[a,b] : \exists k>0, |f(x)-f(y)|\leq k|x-y|\}$, Prove that $Lip[a,b]^{\circ}=\emptyset$ in $C[a,b]$. Suppose there exists $f \in Lip[a,b]^{\circ}$, then $\exists ...
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1answer
86 views

Rewriting $\delta(x,y)$ in terms of $\delta(r)$.

On my textbook is written: The function $\tau^{-1} u(x/\tau)$ is a rectangle function of height $\tau^{-1}$ and base $\tau$ and has unit area; as $\tau$ tends to zero a sequence of unit-area pulses ...
1
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1answer
63 views

Partial derivative in higher dimension .

Let us consider $g: \mathbb R^n \to \mathbb R^m$ , $f:\mathbb R^m \to \mathbb R^k$ be $C^1$ functions . Define $F= f\circ g$. I am having the problem in finding the partial derivative of $F$, ...
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1answer
59 views

Is this equivalent to continuity?

I played around a little bit with the definition of continuity and I think I got the following relations that may be equivalent to continuity. Maybe there is somebody who can check this: $$ ...
9
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1answer
218 views

Rigorous separation of variables.

Let $I \subseteq \mathbb{R}$ denote an open, non-empty subinterval of the real line. We're given functions: $$f : I \rightarrow \mathbb{R}, \;\;g : \mathbb{R} \rightarrow \mathbb{R}.$$ Now suppose ...
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1answer
31 views

Estimating the number of nodes of a discretization scheme

Working on a problem in partial differential equations, I have come across a function $$f\colon [0, T]\to \mathbb{R}_{\ge 0}$$ which is continuous and non-decreasing, and whose maximum is $M=f(T)$. I ...
0
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1answer
80 views

To show continuity at a point using boundedness of partial derivatives

Suppose $$U = (-1,1) \times (-1,1) \subset \mathbb R^2,\ f: U\to \mathbb R\ .$$ Assume that $\partial f/\partial x$ and $\partial f/\partial y$ exist at each point of U and are bounded on U. Show ...
7
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3answers
220 views

2013th derivative of rational function

I am struggling to find $f^{(2013)}(0)$ for $$f(x) = \frac{1}{1 + x + x^3 + x^4}$$ I know that I should use power series, and following a hint I rewrote the equation as the following: $$1 = (1 + x + ...
1
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0answers
45 views

If $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are bounded on $U$, then $f$ is continuous at $(0,0)$.

Suppose $U = (-1,1)\times(-1,1) \subset \mathbb{R}^2$, $f: U \to \mathbb{R}$. Assume that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist at each point of $U$ and are bounded ...
3
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1answer
140 views

Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
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0answers
33 views

Is there a 'simple' test one can perform to see if a function maps measurable sets to measurable sets?

I know I'm probably missing a huge key point somewhere in my analysis background, but is there such a thing as a 'simple' test to check if a function does this? (Note: I am specifically considering ...
0
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1answer
51 views

What is the gradient of this function

Imagine you have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ $f(z):=z^TAz$, where $A$ is a symmetric matrix. Now I was wondering what $\nabla f(x_1,...,x_{n-1},\gamma(x_1,...,x_{n-1}))$, where ...
3
votes
1answer
69 views

Show that $\operatorname{div} X = - \delta X^\flat$

I want to show the equality $\operatorname{div} X = -\delta X^\flat$, where $X \in \Gamma(TM)$ and $M$ is some Riemannian manifold with metric tensor $g_{ij}$. If I'm not mistaken it holds for the ...
0
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2answers
42 views

A math analysis exercise question.

my teacher gave me this exercise: let $f:I\to\Bbb R$ a non decreasing function, where I is an open set in $\Bbb R$, and let $S=\{x\in \Bbb R : f $ is not continuous in $x\}$; knowing that every $x ...
0
votes
1answer
93 views

How is min max f(x,y) defined when solving a dual problem?

I am trying to solve a dual problem. And it is said that min max f() is always smaller or equal to max min f(). For example, $\max_{y \in Y} \min_{x \in X} f(x,y)$ is always smaller or equal to $ ...
0
votes
1answer
31 views

$W = \{x\in l_0 : <x,a>=0\}$ where $a=(1,\dfrac12,\dfrac13,…)$ and $l_0$ is sequences with finitely many non-zero terms. Show $W$ is separable

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
0
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1answer
36 views

Show that $\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(x+\sqrt{n})}{n}$ does not converge absolutely for $x\in [-1,1]$

Show that $\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(x+\sqrt{n})}{n}$ does not converge absolutely for $x\in [-1,1]$ Consider ...
0
votes
1answer
28 views

Inequality with partial integration in one dimension

Is it possible to prove $ \| u \|_{L^2(0,1)} \leq \| u' \|_{L^2(0,1)} $ for $u \in C^1([0,1])$ with $u(0)=0$ by using partial integration?