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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4
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1answer
346 views

Difference between soft analysis and hard analysis

I have sometimes overheard people using the terms hard analysis and soft analysis.I am not a particularly well-read person in mathematics but I have wondered what that is all about.I hope there ...
2
votes
0answers
50 views

About function which Fourier coefficients satisfy $a_n=o(n^{-2}), b_n=o(n^{-2})$

Assume that a function $f: R\rightarrow R$ is $2 \pi$ -periodic and integrable on $[ -\pi,\pi] $. Let $(a_n)$, $(b_n)$ are its Fourier coefficients and $n^2 a_n, n^2 b_n \rightarrow 0$. Then by ...
0
votes
1answer
946 views

Showing that $\cos(x)$ is a contraction mapping on $[0,\pi]$

How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here. So I think ...
4
votes
1answer
100 views

$K$ compact and $\Omega$ is open, then $\inf\{\rho(x,x') \mid x \in K \textrm{ and } x' \in \Omega^c\} > 0$

I have to show the following: $(V,\rho)$ be a metric space, $K\subset V$ compact and $\Omega \subset V$ is open, then $d(K,\Omega^c) = \inf\{\rho(x,x') \mid x \in K \textrm{ and } x' \in \Omega^c\} ...
1
vote
0answers
179 views

A metric space is path connected and countable then it is complete

I have to show that if a metric space is path connected and countable then it is complete. I'm pretty lost where to start this at all. I have the basic definitions of complete, path-connected, compact ...
1
vote
0answers
210 views

Strongly exposed points/Exposed points

I was studying and I got the next doubt: We suppose that $(X,\|\cdot\|)$ is a Banach space and $C$ it is a convex closed subset of X. We say that $x\in C$ it is an exposed point of $C$ if $\exists ...
5
votes
1answer
323 views

An Exercise on Inverse Function Theorem

Consider the mapping $f:R^2\rightarrow R^2$ given componentwise by: $f_1(x,y)=x+a_1x^2+2b_1xy+c_1y^2\\ f_2(x,y)=y+a_2x^2+2b_2xy+c_2y^2$ Determine a neighbourhood of $(0,0)$ as large as possible on ...
4
votes
2answers
280 views

Contraction Mapping question

Let X be the set of continuous real valued functions defined on $[0,\frac{1}{2}]$ with the metric $d(f,g):=\sup_{x\in[0,\frac{1}{2}]} |f(x)-g(x)|$. Define the map $\theta:X\rightarrow X$ such that ...
3
votes
3answers
110 views

When I can't convert a piece of a limit in a notable limit?

When I solve limits I often convert a block in a notable limit multiplying and dividing it by the same quantity. For example: $$ \lim_{x \to 0} \frac{e^{\sin(x)}-1}{x} = \lim_{x \to 0} ...
11
votes
8answers
6k views

Continuous versus differentiable

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same ...
3
votes
1answer
51 views

Periodic solutions of $f^{(m)}=g$

Let $g:R \rightarrow R$ be continuous and $ 2\pi$-periodic, let $m \in N$. How many solution in class of $m$-times continuously differentiable $2\pi$-periodic functions has equation $$f^{(m)}=g ?$$ ...
1
vote
1answer
75 views

proving a solution of a differential equation (dirichlet)

I just have found out these information in my presentation work: a) $V\subset\mathbb R^n $ open, $D:=\{(x,x)|x\in\mathbb R^n\}$ and a function $\alpha:(V\times V)\backslash D\rightarrow \mathbb R$ ...
2
votes
3answers
312 views

The concept of gradient, related to lagrange multipliers, surface areas, tangent hyper planes

As we all know, gradient is always perpendicular to the level curve. On the other hand, $\nabla f(a,b) \dot\ h$ where $h=(x-a\ \ \ \ \ \ y-b)^T$, give a tangent hyper plane which is tangent to ...
1
vote
1answer
407 views

Proving that if $u \in A$ is an upper bound of $A$, then $u = \sup A$

Let $A\subset\mathbb{R}$ a nonempty set of real numbers bounded above and $u$ be an upper bound of $A$. Prove that if $u\in A$, then $u=\sup A$.
3
votes
3answers
197 views

Differentiable and $f(0)=0$

I would like to show that if $f\colon\mathbb{R}^n\to \mathbb{R}$ is differentiable and $f(0)=0$ that there exists $g_i\colon\mathbb{R}^n\to\mathbb{R}$ such that $f(x) = \sum_{i=1}^n x^i g_i(x)$. The ...
2
votes
1answer
179 views

Intermediate value theorem is the solution unique,

Have the following $f(x)=x^2 \exp(\sin(x))-\cos(x)$ on the interval $[0,\pi/2]$, I have shown the function is continuous and that there is at least one solution on the interval via using IVT, I know ...
0
votes
0answers
543 views

$L^2$ norm and its discrete analogy

$L^2$ is space of functions with $||e||^2$ as usual integral, so it is infinite dimensional, so there is no norm equivalence rule holds there. the discrete analog of its norm is $||e||^2_2=h\sum{_1^m ...
2
votes
2answers
229 views

A question about solving multiple integral

What i wanna ask is that how do you tackle the problem oif multiple integral when you are not able to draw the diagram?Also how you would determine the order of integrand. For instance, what is the ...
1
vote
1answer
58 views

Minima of Binary Forms of Degree n

Does anyone know any upper bounds or known results on LOWER BOUNDS for binary forms i.e. if you have F(X,Y)=$X^n+YX^{n-1}+Y^2X^{n-2}+...+XY^{n-1}+Y^{n}$, I need to find a lower bound for F interms ...
2
votes
2answers
169 views

Newtonian potential of a rotationally-invariant function

Lately I read up in the wikipedia article about the Newtonian potential, that for any compactly supported continuous function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ that is rotationally invariant ...
0
votes
3answers
134 views

Find an object $x$ such that $1=\sum_{n>0} n^n\cdot x^n$

$1=\sum_{n>0} (nx)^n$ Dont have any solution in $\mathbb C$? Are there other types of equations with no solutions in $\mathbb C$? Can one define an object that satisfyes these equations? Does it ...
1
vote
1answer
67 views

to show $g$ attains maxima and minima

Let $A$ be a symmetric $n\times n$ real matrix and define $G:\mathbb{R}^n\rightarrow \mathbb{R}$ by $G(t)=\langle At,t\rangle$; let $g:S^{n-1}\rightarrow \mathbb{R}$ be the restriction of $G$ to the ...
0
votes
0answers
102 views

Probably easy recursion formula solving $x^x=a$

Let $a,x_0\in\mathbb{C}$ and set $$ x_{n+1}=\sqrt{x_n\cdot \sqrt[x_n]{a}}$$ For which $a$ and $x_0$ does $x=\lim\limits_{n\rightarrow\infty} x_n$ exist with $$x^x=a$$ Approximating it with a computer ...
6
votes
4answers
2k views

Books, Video lectures, other resources to Teach Yourself Analysis

So my limited mathematics education has been especially ignorant of analysis. In this vein, I'd like to teach myself some of the introductory basics. I'm intrigued by sources that might contain ...
1
vote
2answers
469 views

Finding the values of x for which the series converges

Have the following: $$\sum 2^{n}\log(1+\frac{1}{3^{n}})$$ Now I was thinking the best way to approach would be via the ratio test, doing so I got to the following, $\rvert\frac{a_{n+1}}{a_n}\rvert= ...
1
vote
1answer
117 views

Inequality holds?

Can anyone prove that $$ \frac{\sum\limits_{i=1}^{k*} a_i i (x+\epsilon)^{(i-1)}}{\sum\limits_{i=1}^{k*} a_i (x+\epsilon)^{i}+k^*-1}>\frac{\sum\limits_{i=1}^{k*} a_i i ...
1
vote
1answer
102 views

Convergent rate for a class of functions

I want to find a class of function $h(\tau)$ which makes the following limitation converges to zero. $\lim_{t\rightarrow \infty }\int_{0}^{t}e^{-M\left( t-\tau \right) }h\left( \tau ...
2
votes
1answer
122 views

Limit inferior taken on the norm of a sequence

Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| ...
1
vote
2answers
135 views

Rolle's theorem for showing that $(x-a)^k$ divides $p(x)$ . . .

Have the following I'm stuck on: Suppose $p(x)=p_0+p_1 x+p_2 x^2+\cdots+p_n x^n$ is a polynomial of degree $n \geq 1$. Show that if $(x-a)^k$ divides $p(x)$ for some $a\in\mathbb R$ and some ...
2
votes
1answer
123 views

When expressing area of $f(D)$ using Jacobian, why exactly must $f$ be one-to-one?

I was working on a question very similar to this one: Expressing the area of the image of a holomorphic function by the coefficients of its expansion Clearly the key lies in the formula ...
1
vote
1answer
304 views

integral over unit ball?

Problem: Let's consider the collection of $C^1$ functions, where $k=1,2,\ldots,(n-1)$: $$g_k:\mathbb{R}^k\rightarrow \mathbb{R},$$ where: $$ g_k=g_k(x_1,x_2,\ldots,x_k)$$ Then a new map $f$ is ...
0
votes
2answers
1k views

big o notation / asymptotic for factorial

I want to write $g(x)=x!\cdot(x^4-1)$ in the big O notation $g\in \mathrm O(???)$ for $x\rightarrow\infty$. But I have no idea how to do this. Thanks for helping!
-2
votes
1answer
353 views

Proof involving a double integral?

Problem: I am stuck on the following problem and I appreciate if someone is willing to help solving it. The problem is as follows: I am given a uniformly continuous function : $f:\mathbb{R}^2 ...
1
vote
0answers
193 views

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null. I think it has something to do with the fact that $f'$ is bounded in any ...
2
votes
2answers
327 views

What is $\limsup\limits_{n\to\infty} \cos (n)$, when $n$ is a natural number?

I think the answer should be $1$, but am having some difficulties proving it. I can't seem to show that, for any $\delta$ and $n > m$, $|n - k(2\pi)| < \delta$. Is there another approach to this ...
3
votes
3answers
272 views

The ratio of two strictly increasing functions

although it seems very simple and obvious, I have no idea how to give an analytical proof for this problem. I will be very happy if there are some smart ideas... Given, $f_1(a), f_2(a),..., f_n(a)$ ...
1
vote
1answer
99 views

How to show this one $C^\infty$

Could you give me a hint: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a $C^\infty$ function with $f(0,0)=0.$ Define $g(t,u)= f(t,tu)/t$ for $t\neq 0$ and $0$ when $t=0.$ How I will show that $g$ is ...
3
votes
2answers
134 views

Are there any counterexamples to the Dirichlet criterion if g(x) is not monotone?

We know that the Dirichlet criterion testing the convergence of a series needs one sequence monotonically tends to zero, and the sequence of partial sums of the the other series is bounded. I wonder ...
2
votes
2answers
453 views

Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
4
votes
1answer
164 views

A problem in elementary analysis

A friend asked me this problem but I couldn't solve. Not homework. Given reals $x>1$, $\epsilon>0$, does there always exist $i,n \in \mathbb N$ such that $|x^i -n |<\epsilon$. If the ...
4
votes
1answer
1k views

Hölder continuous but not differentiable function

How can it be proved that the function $$f(x)=\sum_{n=1}^{\infty}2^{-n\alpha}\cos(2^nx)$$ for $\alpha \in ]0,1[$ is $\alpha$-Hölder continuous but not differentiable at any point of $[0,1]$? I ...
6
votes
1answer
370 views

Weak convergence in $L^2$ and uniform covergence

I have this problem: let $f_n$ converge weakly to $f$ in $L^2[0,1]$ and let $$F_n(x)=\int_0^xf_n(t) \, \textrm{d}t,$$ $$F(x)=\int_0^xf(t) \, \textrm{d}t.$$ Then $F_n,F$ are continuous and $F_n$ ...
3
votes
0answers
99 views

Properties and extensions of the $n!$ formula for $e^{-1}$? [closed]

So I recently re-encountered the following limit: $$\lim_{n\rightarrow \infty} \dfrac{(n!)^{1/n}}{n}=\dfrac{1}{e}$$ I began to wonder about a few things from this relation: (i) I notice that ...
5
votes
3answers
790 views

Example of discontinuous function having all partial derivatives

Is it possible to a real-valued function of two variables defined on an open set to have partial derivatives of all order and to be discontinuous at some point or maybe at each point?
1
vote
1answer
127 views

Equilibrium distance formula proof

Let $$d: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}$$ be defined by $$d(x_i,x_j)=\frac{|x_i-x_j|}{\sqrt{M(i)M(j)}},$$ where $M(i)$ represents the average distance between $x_i$ and ...
0
votes
1answer
81 views

deRham groups/MV Theorem

I had a question on de Rham groups and would appreciate your help. I'm trying to compute the deRham groups of $M$: \[ M = \{(x, y, z) \in \mathbb R^3 : x^2 + y^2 = 1, 0 < z < 1\} \] I'm not ...
3
votes
1answer
212 views

Interchange of Limsup and sup

Let $\phi_n: U(\subset\mathbb R^2\sim\mathbb C)\to \mathbb R$, be sequence of continuous functions. Then for any compact set $K\subset U$, what are the necessary and sufficient condition on $\phi_ns$ ...
0
votes
1answer
104 views

Singular “Escape” in Monomial Integration [duplicate]

Possible Duplicate: Why can't you integrate all power functions without a log function? We know that $f(r,x)=\int_{1}^xt^r dt=\frac x{1+r}+C$ if $r\neq -1$ and $=\log x+C$ if $r=-1$. I ...
1
vote
4answers
1k views

Determine whether or not the cube root of x is Lipschitz.

Determine whether or not the cube root of x is Lipschitz. x^(1/3) I understand that this is NOT Lipschitz. I am having trouble properly proving this. I know for other problems I have shown ...
6
votes
2answers
406 views

Elementary bound of binomial coefficient

I'm working my way through an Erdős paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...