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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2
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3answers
255 views

A robust convex optimization problem

Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
1
vote
2answers
99 views

$|f(x)-f(y)|\ge c|x-y|$, show that $Jf(x)\ne 0 \forall x\in \mathbb{R}^n$ and $f(\mathbb{R}^n)=\mathbb{R}^n$

$f$ is of class $C^{(1)}$ and there exist a number $c>0$ s.t. $|f(x)-f(y)|\ge c|x-y|$ $\forall x,y\in \mathbb{R}^n$. Show that $Jf(x)\ne 0 \forall x\in \mathbb{R}^n$ and ...
0
votes
1answer
72 views

Simple question during a proof. Reducing a factorial…

So I'm reading over a proof-review and stuck on how they managed to convert: $\limsup \displaystyle \frac{|n!|^\frac{1}{2}}{|(n+1)!^\frac{1}{2}|} = \limsup \displaystyle\frac{1}{(n+1)^\frac{1}{2}}$ ...
5
votes
1answer
273 views

Highly Oscillating Integrals

I'd like to know the behavior of integrals of the form: $$ \int_0^1 f(x) \cos(k x) dx $$ as $ k \rightarrow \infty $ where f is a smooth function. It is easy to see, by expanding f in power series, ...
5
votes
1answer
240 views

Taylor series and tempered distributions

Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid? When we interpret ...
5
votes
1answer
126 views

Sum of Cauchy sequences in an abelian topological group (first countability hypothesis)

We know that, given a first countable abelian topological group $G$, the sum of two Cauchy sequences gives yet another Cauchy sequence (see, e.g., this answer). For those wondering, we say that a ...
5
votes
0answers
151 views

Operator completly continuous

For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$ and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
0
votes
1answer
84 views

Integral over a Hypercube

Let $C_R$ be a hypercube in $\mathbb{R}^n$ of side length $2R$ and $B_R$ a ball in $\mathbb{R}^n$ of radius $R$. I know I can say that $$\int_{B_R}f(x)\,dx \sim \int_{C_R} f(x)\,dx $$ Could anyone ...
0
votes
2answers
77 views

limit, low or high bound, convergence for recursive sequence [duplicate]

given is the following sequence: $a_1 > 0$ $a_n = \frac{1}{1+a_n}$ I succeeded in finding a (possible?) limit by guessing that the sequence is limited by a; then the sequence $a_n$ converges to ...
1
vote
1answer
44 views

Show that the sum can be written as:

How can the left side be expressed as the right one? $$\sum_{n\in \mathbb{N}} \frac{1}{n^2}-\sum_{n\in \mathbb{N}} \frac{1}{(2n)^2}=\sum_{n\in \mathbb{N\cup \{0\}}} \frac{1}{(2n+1)^2}$$ Thanks in ...
2
votes
3answers
68 views

Uniform Convergence and differentiable functions

I have been working on this textbook question and am not sure what to do. Is there a sequence of differentiable functions on some interval, say [0,2], converging to 0 uniformly, but where $f'_n(1)$ ...
0
votes
0answers
101 views

Complete normed vector space

I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
0
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1answer
220 views

How do I prove that an uncountable subset $S$ of $\mathbb{R}$ has the in-between property?

The in-between property is that between any two distinct reals in the set, there is another real number. Also, $S$ has no discontinuities. It's not an interval such as $[0, 1] \cup [2, 3]$, for ...
3
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0answers
98 views

Geometrical Inequality

Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals $AC$ and $BD$ intersects at $E$. If the shortest height of the triangle $ACD$ equals the radius of the incircle of the triangle ...
0
votes
2answers
30 views

Affine maps problems

How to find out a particular affine map when some points are given, say if it takes (0,0) to (1,1), (1,0) to (3,2) and (0,1) to (2,4)?
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votes
4answers
148 views

How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?

How do you find $$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$ I know it's $-1$, but I had to plot it.
4
votes
1answer
102 views

Suppose that the function f(x)

Suppose that a function $f(x)$ defined on $[0,1]$ satisfies $f(1/n)\to 0$ as $n\to\infty$. Can we say that $f(x)\to 0$ as $x\to 0^+$ if $f$ is continuous on $[0,1]$ ? and again is it true $f(x)\to ...
6
votes
1answer
6k views

Proof to show function f satisfies Lipschitz condition when derivatives f' exist and are continuous

The question is as follows: Given a function $f$, 2 known information: (1) $f'(x)$ exist (2) $f'(x)$ are continuous Goal: function $f$ satisfies Lipschitz condition on any ...
4
votes
1answer
41 views

Can anyone show or clarify

Can any anyone clarify or prove that if the derivative of a function $f$ is strictly positive then the function $f$ is strictly monotone increasing. I am really sure that the converse is not true as ...
5
votes
2answers
76 views

Proof using Rolle's Theorem to show there is c such that f$^4$(c) = 0, for a < c < b

The question is as follows: Give 3 information: (1) f is a polynomial (thus I claim f is continuous at every point) (2) $f(a) = f'(a) = f''(a) = f'''(a) = 0$ (3) $f(b) = 0$ ...
3
votes
1answer
157 views

Is a continuous function in two variables necessarily equicontinuous in one variable?

Suppose $K \in \mathcal{C}\left(\left[0, 1\right]\times\left[0, 1\right]\right)$. Then, is it necessarily the case that the set of functions $\left\{g_y(x):g_y(x) = K(x,y), \forall y \in ...
2
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1answer
81 views

assume $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$?

let $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $$\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$$ Thanks in advance
3
votes
3answers
585 views

Prove that there is at least one real solution to the equation…

$x^{17}+\frac{243}{1+x^4}=120$ Can anyone show me how to approach this problem..? Any help would be great, thanks.
1
vote
1answer
88 views

Convexity of $x^2f(x)$

Given a function $f$ which is decreasing and convex on $(0,\infty)$, is it possible to find a simple condition on $f$ such that \begin{equation} 2f(x) + 4xf^\prime(x) + x^2f^{\prime\prime}(x) \geq 0. ...
0
votes
1answer
98 views

Analysis - finding local extrema?

I must find and identify (max or min) the local extrema of $f(x) = x^2 e^{-x}$ This is a simple problem if it was in a calculus exam - but it's not. I'm not sure how to structure the solution for an ...
0
votes
2answers
43 views

Question regarding infinite sets of a metric space.

Let $A_1, A_2, A_3, ...$ be subsets of a metric space $X$. (a)If $B_n = \cup_{i=1}^n A_i$, prove that closure $\overline {B_n} = \cup_{i=1}^n \overline {A_i}$. (b)If $B = \cup_{i=1}^{\infty} A_i$, ...
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6answers
1k views

To construct a set with a limit point.

I learned how to construct a Cantor Set, and I am asked to do the following. "Construct a bounded set with exactly 3 limit points." Since the Cantor set contains infinitely many points, I don't ...
0
votes
2answers
203 views

Problems regarding countable sets.

I am required to prove that the set of algebraic numbers is countable. My understanding of an algebraic number is the following. (1) A solution $z$ to the equation ...
2
votes
1answer
351 views

Proof on showing a uniformly continuous function has limit at every cluster point of the domain

The question is as follows: Given: (a) f is uniformly continuous on a subset D of $\mathbb R^n$ and (b) $x_0$ is a cluster point of D Show: The limit of f(x), as x approaches ...
0
votes
1answer
72 views

Roots of a polynomial satisfying $f(x^{2}+1) = f(x) \cdot g(x)$

Let $f(x), g(x)$ be $2$ real polynomials of degrees ($m\ge 2$) and $(n\ge 1 )$ respectively satisfying $$f(x^{2}+1) = f(x) \cdot g(x)$$ for every $x \in \mathbb{R}$. Then which of the below options ...
1
vote
1answer
330 views

Show there exists a sequence of positive real numbers s.t. …

Let $f_n$ be a sequence of measurable functions on $[0,1]$ with $|f_n(x)|\lt\infty$ a.e. Show there exists a sequence $c_n$ of positive real numbers s.t. $f_n(x)/c_n\to0$ for almost every $x$ in ...
0
votes
1answer
51 views

About differentiability and partial differentials of function.

Problem Statement: Given:$$f: \mathbb {R^2} \rightarrow \mathbb {R},(x,y)\rightarrow \begin{cases} 0 & (x,y)=(0,0) \\ \frac{x^3}{x^2+y^2} & (x,y)\neq (0,0)\end{cases} $$ Need to show that it ...
1
vote
1answer
575 views

Show that this is a diffeomorphism

I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$ with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
0
votes
1answer
129 views

Nondegenerate critical point

I don't understand this part from the book of Zeidler , can someone help me to understand it ? Please Thank you
1
vote
1answer
212 views

proving differentiability for a function in a point

Given a differentiable function $f:D\backslash\{a\}\rightarrow\mathbb R$ and $\lim_{x\rightarrow a}f'(x)=c$ and $f$ is continous in $a$, I want to prove that $f$ is differentiable in $a$ and ...
5
votes
1answer
257 views

Class $C^{- \infty }$ functions?

If my understanding is correct, a class $C^{-1}$ function (in terms of smoothness, of course) can be thought of as a function which integrates to a class $C^{0}$ function. And when we differentiate ...
4
votes
0answers
66 views

Some questions regarding Ramanujan summation — Part I

The Ramanujan Summation method, is a method through which divergent series can be summed to convergent values. I have several questions regarding this summation method. For more info about the words ...
1
vote
1answer
155 views

Are Trigonometric Functions Dense in $C^k(S^1)?$

Consider the functions $\{e^{2\pi i nx}\}_{n \in \mathbb{Z}}$ defined on the interval $[0,1].$ These are all smooth periodic functions (so functions on $S^1)$ and by the Stone-Weierstrass theorem ...
6
votes
1answer
121 views

Minimal definition of the derivative

The definition of the Fréchet derivative according to Wikipedia is: Let $V$ and $W$ be Banach spaces, and $U\subset V$ be an open subset of $V$. A function $f : U \to W$ is called Fréchet ...
0
votes
3answers
344 views

Function Continuity on an Interval.

I must show that $f(x)=p{\sqrt{x}}$ , $p>0$ is continuous on the interval [0,1). I'm not sure how I show that a function is continuous on an interval, as opposed to at a particular point.
3
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3answers
157 views

Basic topology question regarding the complex plane.

Prove that the Complex plane is closed, open and perfect. My intuition is destroyed by the fact that a set can be open and closed at the same time. The following is my understanding. open: If all ...
4
votes
3answers
325 views

A minor question about the Cantor Set

I'm self teaching analysis and the second chapter is about some basic topology. According to the book "Principles of Mathematical Analysis (3rd)" from Walter Rudin, the Cantor Set is constructed as ...
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vote
0answers
121 views

Deformation retract

How to prove that $r_t$ is a deformation retract $M^a=\lbrace q\in M ; f(q)\leq a\rbrace$ We have the definition : $r_t$ is a difformation retract if: $r_t$ is a continius ,onto application ...
2
votes
1answer
147 views

Computing an explicit solution to an integral equation via the Neumann Series.

I am hoping that someone can provide guidance for solving the integral equation $$u=f+\lambda Au$$ where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
1
vote
2answers
34 views

Minima for a Sum

Let $A=\{|a_i|:a_i\in\mathbb{Z}\land1\leq i\leq n\}$ and $n\geq 1$ Let $b_i=\frac{\max A}{|a_i|}.$ How can one prove that the minimum possible value for$\sum\limits_{i=1}^n b_i$ is $n$?
3
votes
1answer
164 views

A combinatorial identity with Pochhammer's symbol

Let $m,k$ be an positive integers with $k\le m$. I am trying to prove $$\sum_{j=0}^k{\frac{1}{2}\choose k-j}\frac{2^{2j}(m+j)!}{(m-j)!(2j)!}=\frac{P(n,k)}{(2k)!}$$ where $n=2m+1$ and ...
2
votes
3answers
110 views

Prove uniform convergence of $x^{\frac{1}{n}}+(1-x)^{n}$

Is it true or not that the this succession converges uniformly on $(C[0, 1],\Vert . \Vert_{\infty})$: $$f_{n}=x^{\frac{1}{n}}+ (1-x)^{n}$$I have found an elementary solution, but I would like to ...
3
votes
4answers
280 views

Finding a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R})$ is neither open nor closed

Find a bounded, continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R})$ is neither open nor closed?
2
votes
1answer
82 views

A combinatorial identity related to Chebyshev differential equation

Let $m,k$ be an positive integers with $k\le m$. Does anyone have a proof that $$\sum_{j=k}^m {2m+1\choose 2j+1}{j\choose k}=\frac{2^{2(m-k)}(2m-k)!}{(2m-2k)!k!}?$$ This is related to Chebyshev ...
1
vote
1answer
69 views

Prove that $X_{v+w} \subset X_v+X_w$

Let $B \subset \mathbb{R}^d$ a set convex and simetric, ($B=-B$). Prove that $X_{v+w} \subset X_v+X_w$, where $$X_v= \{ \alpha >0 \ ; \ \frac{1}{\alpha}v \in B \}$$ $$X_w=\{\varepsilon >0 \ ; \ ...