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
2answers
220 views

Is this series convergent?

$a_n=(-1)^{k_n}\frac{1}{n}$ where $(k_n-1)^2<n\leq k_n^2$. Is the series $\sum_n a_n$ convergent? I tried with all the classical methods, but they seem to fail, any hint? EDIT: I had an idea: ...
1
vote
0answers
66 views

Fixed Point Theorem for Set-to-Set Mappings

Is there a fixed-point theorem regarding mappings of the form $T:2^S\to 2^S$, i.e. mappings that map subsets of $S$ to other subsets in $S$: $$ T:S\supseteq A \mapsto T(A)\subseteq S $$ Where $S$ ...
8
votes
7answers
896 views

Proving the positivity of a twice-differentiable real-valued function

This is a problem from Berkeley prelim exams, Spring '99 Suppose that $ f $ is a twice differentiable real-valued function on $\mathbb{R}$ such that $ f(0) = 0 $, $ f'(0) > 0 $, and $ ...
2
votes
1answer
158 views

What locally integrable function $f$ satisfies $\int_a^ b f(x) \phi'(x)dx=0 $ for each $\phi \in C_0^\infty(a,b)$

Let $f:(a,b) \rightarrow \mathbb{R}$ be locally integrable and such that $$\int_a^ b f(x) \phi'(x)dx=0 \textrm{ for each } \phi \in C_0^\infty(a,b).$$ How to show, without help of distribution ...
3
votes
1answer
439 views

Minimize distance between 2 functions

Just so you know, this is a homework question, and I basically need help with the steps to solve this problem. I understand what it's asking; however, my attempts haven't worked out, and I'm probably ...
14
votes
3answers
825 views

Intersection between orthogonal complement of a subspace and a set

Consider the normed vector space $E=\mathbb{R}^n$. Define $ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$. Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap ...
3
votes
1answer
589 views

Implicit Function Theorem computation problem

Problem 1, page 78 of Munkres (Analysis on Manifolds): Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x,y_1,y_2)$. Assume that $f(3,-1,2) = ...
1
vote
1answer
692 views

Determinant of the “bordered” Hessian of a composition

Write $H_{f}$ for the Hessian of a real function $f:\mathbb{R}^n\mapsto \mathbb{R}$, and define the bordered Hessian as $$ H_{f} = \left(\begin{matrix}0 & \nabla f' \\ \nabla f & H ...
1
vote
0answers
74 views

Relationship between Number of circles required to surround a circles and the distance function?

In Why is a circle in a plane surrounded by 6 other circles, the implicit assupmtion is the distance is Euclidean, my question is: Are there any relation between the distance function being used and ...
8
votes
5answers
1k views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
0
votes
1answer
57 views

The meaning of the differential map between two functions space

... a continuous differential map $\dfrac{d}{dx} : C^k(\mathbb R)\rightarrow C^{k-1}(\mathbb R)$ ... I was wondering why a differential map could from $C^k(\mathbb R)$ to $C^{k-1}(\mathbb R)$, ...
1
vote
1answer
367 views

Problem involving a hyperplane and affine subspace II

I am trying to solve this little problem. Suppose you have a normed vector space $E$. Let $H$ be a hyperplane ( $H=\{x\in E: f(x)= \alpha\}$ for some linear functional $f$ and some real number ...
1
vote
2answers
155 views

help with the Riemann - Stieltjes

why can I say that $$ \int_0^a t^2 dF(t) = \int_0^a t^2 d(F(t) -1) $$ unfortunately my experience with the Riemann -Stieltjes is practically non existent, so for instance I do not understand, why ...
0
votes
1answer
198 views

Analytically solving limits

I read the theory of limits and i have some misunderstanding. For example we have simple limit expression: $$\lim _{x\rightarrow \infty}{\frac{1}{x}}$$ I see that this limit is 0 and if build graph ...
2
votes
1answer
101 views

Equality of integrals of differential forms

I have two $(n-1)$-forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^n$ and a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ ($dg$ doesn't vanish anywhere) such that $dg \wedge \omega_1 = ...
2
votes
2answers
331 views

About example of continuous function on $\mathbb{R}$ which cannot be uniformly approximated by polynomials? [duplicate]

Possible Duplicate: Weierstrass approximation does not hold on the entire Real Line If a function $f: \mathbb{R}\rightarrow \mathbb{R}$ is continuous then $f$ can be uniformly approximated ...
3
votes
2answers
168 views

Differentiablility of a function of two variables

Here is a problem from an old comprehensive exam that I am trying to solve Problem: let $f:\mathbb{R}^{2} \to \mathbb{R}$ be a function defined as follows: $f\left ( x,y \right )=\frac{\left ( ...
4
votes
1answer
193 views

The set of diffeomorphisms preserving some metric.

Let $M$ be a finite-dimensional, smooth manifold. Call a diffeomorphism $f : M \rightarrow M$ diagonalizable if there exists a Riemannian metric $g$ on $M$ such that $f : (M, g) \rightarrow (M, g)$ is ...
2
votes
1answer
732 views

Supremums of measurable functions

According to my textbook, supremums of measurable functions exist and are measurable. But what about the sequence of functions $f_n: [0, 1] \to \mathbb{R}$ given by $f_n = n$? I don't think this ...
6
votes
1answer
786 views

How smooth is a smooth function?

Let's say a smooth function is a $\mathcal{C}^\infty$ function on $\mathbb{R}$. Some smooth functions are not analytic, the most notorious example being the bump functions. A non-analytic ...
2
votes
1answer
527 views

Definition of Borel sets

MathWorld says: Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable unions and intersections. Formally, the class of Borel ...
1
vote
1answer
92 views

How to solve quadratic over square root of quartic equals constant?

We need to solve the following equation for l. $$\frac{n_1 - \ell n_2 + \ell ^2 n_3}{\sqrt{d_1 - \ell d_2 + \ell ^2d_3 - \ell ^3d_4 + \ell ^4d_5}} - \cos{a_0} = 0$$ We have already tried ...
4
votes
1answer
179 views

Is this set closed?

$X=(C[0,1],\rho_\infty)$ where $\rho_\infty$ is the uniform norm. $M\in(0,\infty)$, define $A=\{f\in X:f(0)=0, f\;\mathrm{differentiable\;on}\;(0,1)\;,|f^\prime(x)|\leq M\;\;\forall x\in(0,1)\}$. I ...
6
votes
3answers
2k views

How to prove convex+concave=affine?

Suppose $f:R^n\to R$ is both convex and concave, how to prove that $f$ is linear? or exactly speaking, $f$ is affine? I thought for the whole day, but I cannot figure it out. When I was working on ...
2
votes
0answers
298 views

Characterization of Asymptotic Stability via KL-class functions

Let us adopt the following definition of stability and asymptotic stability of a dynamical system of the form: $$ \dot{x}=f(x) $$ The trajectory of this system starting from the initial point ...
3
votes
1answer
322 views

Solving integral equation with Laplace's Transform.

I'm trying to prove the following $$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$ The original problem ...
2
votes
0answers
143 views

Help with understanding a proof in Fourier Analysis

I have a stack of lecture notes that I am currently going through to teach myself a little bit about Fourier Analysis. Now I struggle with the following Lemma, which is needed to talk about the ...
2
votes
0answers
75 views

(RESOLVED) Interpreting a holomorphic function

The equation for and electric field is given by $E=−∇ψ$ where $\psi$ is the potential, and in this case $ψ=−Q\ln r$ where $Q$ is just some constant. I have found its harmonic conjugate to be $−Qθ+c$ ...
4
votes
0answers
166 views

Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion ...
1
vote
0answers
89 views

Is the following function convex-\cap?

Let $p=(p_1,\ldots,p_n)$ be a given nondegenerate (i.e., all $p_i> 0$) probability distribution on $n$ points. Define the following function $$\Phi(b_1,\ldots,b_n)=\frac{\left(\sum_{k=1}^n b_k ...
0
votes
1answer
216 views

Harmonic conjugate

I have been asked the following question and would appreciate an explanation. Suppose we have to find an analytic function $F(z)$ where $z=x+iy\in \mathbb C$ and its real part is $g(x,y)$. Question: ...
2
votes
1answer
83 views

P-adic “Norm” and scalability criterion

I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ? What I mean ...
4
votes
0answers
117 views

Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
7
votes
2answers
172 views

Help to finish my proof: inequality with norm and Schwarz ineq

Let $A=\left [ a_{ij} \right ]$ be the matrix of a linear mapping $A\in L\left ( \mathbb{R}^{n},\mathbb{R}^{m} \right )$. -Prove that: $\left \| A \right \|\leq \left ( ...
1
vote
1answer
408 views

Proof involving norm of an integral

I am totally stuck and have no idea whatsoever on how to prove the following inequality (by the way this is a problem from an undergraduate book in multivariable advanced calculus at Junior/Senior ...
3
votes
0answers
94 views

Iterating Arithmetic, Harmonic and Geometric Means

Starting with a data set $X_{0}$, compute its arithmetic, geometric and harmonic means, $A(X_{0}), G(X_{0})$ and $H(X_{0})$ respectively. Let $X_{1} = \{A(X_{0}),G(X_{0}),H(X_{0})\}$, and compute ...
1
vote
1answer
207 views

differentiability-continuity of derivatives

I am trying to come up with a function $g:\mathbb{R}^{2} \to\mathbb{R}$ which is differentiable at each point $(x,y)$ in $\mathbb{R}^{2}$ but whose partial derivatives are not continuous at $(0,0)$. ...
3
votes
1answer
118 views

Differentiability in $\mathbb{R}^{2}$

Here is my question: Find all the points $\left ( x,y \right )$ in $\mathbb{R}^{2}$ where the following function is differentiable: $f\left ( x,y \right )=\left | e^{x}-e^{y} \right |.\left ( x+y-2 ...
2
votes
1answer
111 views

Differentiability in $\mathbb{R}^n$ and chain rule

I have a question: Consider a function $g:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is differentiable. Find the derivative of the function: $G(x)=[ g\left ( x,x^{2},...,x^{n} \right )]^{2}$ where ...
1
vote
1answer
415 views

A problem on Lagrange interpolation polynomials

Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample. Consider the polynomial $$ ...
2
votes
1answer
360 views

Derivative of multivariable function

If $f$ is a function from $\mathbb{R}^n$ to $\mathbb{R}$, then its derivative at a point $\mathbf{u}$, $f'(\mathbf{u})$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}$. But we know ...
2
votes
1answer
434 views

Rudin Theorem 1.11

After spending a few hours trying to understand Theorem 1.11 in Rudin's Principles of Mathematical Analysis, I still don't follow the proof. 1.11 Theorem Suppose $S$ is an ordered set with the ...
4
votes
1answer
149 views

If $f$ is a $C^k$ diffeomorphism, then $f^{-1}$ is $C^k$ too.

I don't understand the reason for the conclusion, written in boldface at the end, in the following argument (taken from Elon LIMA, Curso De Análise, Vol .2). If $f:U\longrightarrow V\subset ...
1
vote
1answer
221 views

passing the Derivative inside an integral

Question: Suppose we have: $F(x)=\int_{a(x)}^{b(x)}e^{h(x,t)}dt$. Is it true that $F^{'}(x)=\int_{a(x)}^{b(x)}\frac{\partial h(x,t)}{\partial x}.e^{h(x,t)}dt$ ? Please tell me under what ...
4
votes
0answers
135 views

$\max_{y} \min_{x} f(x,y)$ as motif for exploring mathematics

It's been several years since my undergraduate math days, and I would like to spend a bit of time refreshing and then tackling a few things I never completely mastered. Rather than proceeding topic ...
2
votes
0answers
120 views

Analytical solution for an almost geometric series

Is there any way of solving explicitly the limit of the series $\sum_{n=0}^\infty q^n a^{p ^ n}$ where $0<p,q<1$ and $a>0$? The series is obviously convergent as $a^{p ^ n} < ...
3
votes
1answer
60 views

Functional disequality

Let $f \in C^{2}([a,b]) \ $, $f(a)= f(b) = 0 \ $, $f(x) > 0 \ \forall x \in (a,b) \ $, $f(x) + f(x)''>0 \ $. Then $b-a \ge \pi $. Any hint?
2
votes
1answer
133 views

Growth rate of analytical functions

Given any computable function $f(x)$, is there an algorithm to find a set of coefficients $a_n$, such that i) $g(x)=\sum_{n=1}^{n=\infty} a_nx^n$ converges for all $x>1$ ii) $g(x)$ eventually ...
1
vote
1answer
160 views

convergence / fixed point method

Any help with the following: Problem: Consider the fixed point problem: $x=f(x)$ and given: $x_{n+1}=\frac{n}{n+1}f\left ( x_{n} \right )$. If $x_{0}$ is a fixed point where $\left | f^{'}\left ( ...
2
votes
1answer
95 views

At which points is this function continuous?

At which points is the following function continuous? $$\begin{eqnarray*} f(x) = \begin{cases} 5x, &\text{if }x \in\mathbb Q, \\ x^2-6, &\text{if }x \notin\mathbb Q. \end{cases} ...