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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7
votes
2answers
82 views

Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?

I checked these What is the difference between square of sum and sum of square? Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. It is easy to see $p$-th power ...
1
vote
2answers
23 views

Find supremum and infimumm of a set with two variables

$$A= \left\{\frac{m}{n}+\frac{4n}{m}:m,n\in\mathbb{N}\right\}$$ Since $m,n\in \mathbb{N}$, infimum is zero because $m,n$ both are increasing to infinity. Then the supremum is $5$ when $m,n$ are ...
0
votes
0answers
8 views

PDE reduced to ODE Uniqueness??

Could you please help me with the following problem. As a first help, I know the solution of the following ODE: \begin{align} j_1(t)[r \log(j_1(t)) + \beta] &= j_1'(t) \\ \nonumber j_1(T) ...
0
votes
1answer
19 views

Prove that regular curves are locally invertible

Consider the function $F = (F_1, F_2)$ from $I = (a, b) \subset \mathbb{R}$ to $\mathbb{R}^n$ (without loss of generality, assume $n = 2$). Suppose $F$ is differentiable (i.e $F_1' = f_1$ and $F_2' = ...
1
vote
0answers
14 views

Existence and uniqueness solution of a differential equation

If I have the following equation: $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'e^{\int_0^t\int_0^1G(|r-r'|)y(s,r')dr'ds}-y(t,r)$ $ y(0,r)=a(r)$ where $G:\mathbb{R}^+\to\mathbb{R}$ is ...
1
vote
0answers
14 views

Let S be a set. Let X be the set of bounded functions S $\times$ S to R with the supremum metric. Is the subset T of bounded metrics closed in X?

Let S be a set. Let us say that a metric d on S is bounded if there exists a real number R such that d(x, y) ≤ R for all x, y ∈ S. Let X be the set of all bounded functions S × S → R regarded as a ...
1
vote
0answers
43 views

Integral $\int z^2\Re(J_1(z))dz$

$$ \int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} \left(\frac{z}{2}\right)^{2k}\right]dz $$ where $a\in \mathbb{C}$ and ...
0
votes
1answer
13 views

Inverse function for a sort of negative binomial distribution

I am trying to find the inverse function of $f(p) = \sum_{k=0}^{6}{\binom{6-H+k}{k} p^{7-H} (1-p)^k}$, where $0 \leq H \leq 6$ is a constant integer. Any ideas on how to do this? Or perhaps equally ...
1
vote
2answers
67 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
1
vote
1answer
46 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
1
vote
3answers
72 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
1
vote
1answer
21 views

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
0
votes
0answers
27 views

Prove that a bijective entire function is uniformly continuous

Let $f$ be a bijective entire function. Prove that $f$ is uniformly continuous. I want a direct proof of this without using the fact that $Aut(\Bbb C)$ is the collection of linear polynomials ...
1
vote
1answer
35 views

Tough problem on sum of infinite series

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
0
votes
1answer
28 views

X and Y are compact metric spaces. Show that X $\times$ Y is compact

$X$ and $Y$ are compact metric spaces with metrics $d_X$ and $d_Y$. $X \times Y$ is a metric space with the metric $d((x, y),(x , y' )) := \max\{d_x(x, x' ), d_y (y, y' )\}$. I want to show that ...
0
votes
0answers
15 views

Is the closure of every bounded convex set in $\mathbb R^n (n>1)$ homeomorphic to a closed ball ? [on hold]

Is the closure of every bounded convex set in $\mathbb R^n (n>1)$ homeomorphic to a closed ball ?
2
votes
1answer
51 views

Is a bijective entire function uniformly continuous?

Let $f$ be an entire function such that $f$ is bijective. Is then $f$ uniformly continuous? I am thinking on this when trying to compute the analytic automorphisms $Aut(\Bbb C)$. I know that ...
1
vote
0answers
31 views

Prove that $F_1$ and $F_2$ are continuous and that $\int_{\gamma_1}F_1(z) dz = \int_{\gamma_2}F_2(z) dw$

Let $\Omega_1, \Omega_2 \subseteq \mathbb{C}$ and let $\gamma_1: [a,b] \to \Omega_1$, $\gamma_2: [c,d] \to \Omega_2$ be paths. Let $f$ be a continuous function defined on $\gamma_1 \times \gamma_2$ ...
6
votes
0answers
100 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
1
vote
0answers
13 views

Basic examples of functions in Hörmander class

The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with $$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq ...
7
votes
1answer
127 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
5
votes
1answer
180 views

Proving existence of at least one root

The function $f:\mathbb{R}\to\mathbb{R}$, is continuous and $a>0$. How can I prove that there is at least one root of this equation: $f(x)=f(\sqrt{|x^2-a|})$
1
vote
2answers
48 views

Finding all complex roots of this equation

So i have this equation: $z^5-4z^4+11z^3+12z^2-42z+52=0 \text{ for }z\in\Bbb{C}$ One root is: $z=1+i$ That gives us also the 2nd root. $z=1-i$ But i am stuck with how to get other 3. I thought i ...
-1
votes
1answer
31 views

prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$

$\displaystyle\lim_{x \to a}f(x) = \lim_{x \to a}g(x)$ exist and $\displaystyle\lim_{x \to a}(h_1(x)g(x)+h_2(x)f(x))$ exist prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$ exist I would ...
3
votes
1answer
23 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
1
vote
1answer
74 views

How to rigorously establish this limit of sums

Assuming that $$\lim_{n}\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)=\int_{\mathbb{R}} f(u)g(u)\mathsf du,$$ (where $f$ is $C^2$ and $g$ and $g_n$ are probability distribution functions) I ...
1
vote
0answers
16 views

Representing numbers by quasilexicographic ordered strings, formula for size of conversion between different alphabets

Let $X_r = \{ 0, 1, \ldots, r-1 \}$ and $X_b = \{ 0, 1, \ldots, b-1 \}$ be two finite alphabets with order's given by their numerical value. Consider the quasilexicographic (or shortlex) order on ...
0
votes
1answer
21 views

Egorov's theorem and Baire class $1$ function

Suppose $f$ is Baire class 1. Then there exists $f_n$ each one is continuous and that $f_n \to f$. By Egorov's theorem, a measurable $\mu(B) < \varepsilon$, and $(f_n)$ converges to $f$ uniformly ...
-1
votes
0answers
24 views

Constructing uniform convergence.

Suppose $f_n\rightarrow f$ point wise and each $f_n$ is continuous. Can we construct a sequence of continuous functions $(g_n)$ that converges to $f$ uniformly and each $g_n$ is continuous?
1
vote
4answers
77 views

Prove that there is no strictly increasing and surjective function from $\mathbb Q$ into $\mathbb N$

Let $f: \mathbb{Q} \rightarrow \mathbb{N}$ be such that for $x<y$ in $\mathbb{Q}$ one has $f(x) < f(y)$ in $\mathbb{N}$. Prove that $f$ is not surjective. I tried a proof by ...
4
votes
0answers
30 views

When does interchangibility of limit and Riemann integral imply uniform convergence?

Let $\{f_n\}$ be a sequence of real-valued functions defined on an interval $[a,b]$ such that each $f_n$ is Riemann integrable, $\{f_n\}$ converges point-wise to $f$, $f$ is Riemann integrable and ...
4
votes
2answers
64 views

$n$-th derivative of $(x^2-1)^n$ has distinct real roots in $[-1,1]$.

For $n=1,2,3,\ldots$, let $$f(x) = (x^2-1)^n .$$ Show that the $n$-th derivative $f^{(n)}$ has distinct real roots in $[-1,1]$. I have no idea about the problem. Could I have a hint?
6
votes
2answers
823 views

How can I obtain this division's limit without using derivatives?

$$\lim_{y\to 0} \frac{y}{\cos(\frac{\pi}{2}(1+y))}$$ Can anybody help me? I can use basic properties of limits, and some of those basic known limits. I know it would be easier with derivatives, but i ...
6
votes
4answers
103 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
-2
votes
1answer
24 views

product of two sequences [on hold]

Let $X$ be a Banach space. Let $C$ be nonempty,closed and convex subset of $X$. Let $x_n$ be a convergent sequence in C and $t_n$ a convergent sequence in $\mathbb{R}^+$. Is it true that $t_nx_n$ ...
0
votes
1answer
33 views

Explain about proof

Let $0 \leq R_1 \leq R_2 \leq \infty$ and let $f$ be holomorphic in the annulus $R_1 < |z - z_0| < R_2 $. Then, for any $r_1, r_2, z $ such that $R_1 < r_1 <|z-z_0| < r_2 < R_2$, we ...
4
votes
1answer
28 views

Estimates for parabolic vs elliptic PDE

Elliptic and parabolic PDE share many properties. They each, for example, have an associated maximum principle and their value at any point depends on the entirety of the boundary data. I have been ...
2
votes
0answers
43 views

Variable coefficient wave equation

Consider the equation $$u_{tt} - f(x)^{2}u_{xx} + u_{t} = 0$$ for $(x,t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0) = 0$ and $u_{t}(x,0) = 0$ for all $x \in \mathbb{R}$. Furthermore, suppose ...
0
votes
2answers
32 views

Proving the last part of Nested interval property implying Axiom of completeness

I took a non-empty set A that is bounded above. And I went on with the regular algorithm, which either gave us a LUB or gave us an infinite chain of nested intervals $I_1$ $\supseteq$ $I_2$ ...
4
votes
2answers
177 views

Compact support vs. vanishing at infinity?

Consider the two sets $$ C_0 = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and } \lim_{|x|\to \infty} f(x) = 0\}$$ $$ C_c = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous ...
1
vote
0answers
20 views

Check / Improve a Proof - Convex functional on convex domain is continuous throughout its interior

I am working through Luenberger's Vector Space Methods for Optimization book. I am working on a corollary left to the reader, and I'm not quite sure that my proof is correct/the sharpest proof out ...
2
votes
3answers
82 views

$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$

Prove that for all $\xi \in \mathbb{C}$, $$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$$ I don't really know how to compute this integral. Can you please help me?
0
votes
1answer
29 views

Characterization of contraction mapping

Let $T$ be a mapping from $\mathbb{R}^n \to \mathbb{R}^n$. Fix $x^\star \in \mathbb{R}^n$, and suppose that the Jacobian matrix of $T(x) $ at $x = x^\star$is symmetric. Then, I know that if all the ...
1
vote
1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
0
votes
2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
1
vote
2answers
24 views

denseness of polynomials in bounded borel measurable functions

Let $K\subseteq \mathbb{R}$ be compact, consider $B(K)$ the set of all bounded borel measurable functions $f:K\to \mathbb{C}$ and endow $B(K)$ with the uniform norm, so you obtain a Banach space. My ...
1
vote
1answer
23 views

On the projection onto the image set of an $m\times n$ matrix

I came accross as statement that: "If $K$ is the image set of an $m\times n$ matrix $A$ with full column rank, then $$P_Kx=A(A^TA)^{-1}A^Tx."$$ How do I verify this? I know that the inequality ...
-1
votes
2answers
78 views

Improper rational/trig integral comes out to $\pi/e$ [on hold]

During my studying to integration I find this integration. So I tried to prove but I got stuk. So I need help in this integration. $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2} ...
3
votes
1answer
39 views

the sequence of derivative cannot satisfy $|f^{(n)}(z_0)| > n!n^n$

Let $f: \Omega \to \mathbb{C}$. Prove that for any $z_0 \in \Omega$, the sequence of derivatives cannot satisfy $|f^{(n)}(z_0)| > n!n^n$ In this problem, I intend to prove by contradiction, and I ...
5
votes
2answers
58 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...