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
votes
1answer
49 views

Is it possible to develop Analysis solely from Peano's axioms

...and a few definitions on the way? When I studied Calculus using Spivak's book It was clearly shown that, in order to prove some fundamental theorems (intermediate value theorem being one of them), ...
0
votes
3answers
19 views

Fixpoints and continuity

I don't understand why this is true: If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$ I don't understand why such a point exist. Why is there not ...
5
votes
2answers
36 views

On any continuous map $f:S^1 \to \mathbb R$

Let $f:S^1 \to \mathbb R$ be any continuous map , where $S^1$ is the unit circle in the plane . Let $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ ; then how to prove $A$ is uncountable , or ...
0
votes
1answer
27 views

How to know which notion of convergence to use when proving density of a subspace

My question might be a little vague, but is there a way to know which type of convergence (i.e pointwise, uniform) to use when proving that a subspace is dense in a certain space. For example if we ...
1
vote
1answer
12 views

Formula to calculate adjusted score within range

I'm trying to come up with a formula to combine two different score types, where the second score will weight the first score without exceeding the upper or lower ranges of that score. In the interest ...
3
votes
1answer
64 views

$|f(x)-f(y)| \geq \frac{|x-y|}{2}$

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $|f(x)-f(y)| \geq \frac{|x-y|}{2}$ then prove $f$ is onto. I can prove it just using IVT, but looking for some short solution which ...
1
vote
0answers
43 views

Under what conditions is this true: $\lim_{r \to 0} \frac{1}{r} \int_{0}^{2\pi} f(r,x) dx = 2\pi f(0,0)$

I will like to know under what hypothesis the following is true, and maybe a sketch of the proof. I saw it in a solution of an exercise. In this exercise, $f$ was harmonic, but I don't know if that is ...
-1
votes
1answer
23 views

Another question on finiding special kind of power series [on hold]

Let $\sum a_nx^n$ be a real power series with finite positive radius of convergence $R$ ; then does there exist a non-constant real sequence $\{b_n\}$ such that $\sum b_nx^n$ is convergent for at ...
4
votes
1answer
30 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
2
votes
0answers
21 views

Need help with this calculus inequality

I'm currently studyding Classical and Multilinear Harmonic Analysis. Vol. 1 by Camil Muscalu, Wilhelm Schlag. I need to verify following calculus inequality(Eq. 9.27, at page 255) ...
1
vote
0answers
13 views

Continuity of a parametrized surface integral of a sobolev function

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain and let $v\in H^1(\Omega)$. Furthermore, let $S=(0,T)$ denote a time interval and let $s\in ...
7
votes
2answers
91 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
27 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
11 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
20 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
16 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
16 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 ...
3
votes
0answers
53 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
71 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
47 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
76 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
31 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
37 views

Tough problem on sum of infinite series [on hold]

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
30 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
16 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
54 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
106 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
34 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
25 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
78 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
78 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 ...
7
votes
1answer
51 views
+50

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
833 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
104 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 $I_1$ $\supseteq$ $I_2$ ...