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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12 views

Theorem 3.6-2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications:” Does the converse hold if the space is not complete?

First, a definition: Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$. Now theorem 3.6-2 in Kreyszig states the ...
2
votes
1answer
30 views

IVT and fixed point theorem

Suppose that $f:[0,1]ā†’[0,2]$ is continuous. Use the Intermediate Value Theorem to prove that there exists $cāˆˆ[0,1]$ such that $f(c)=2c^2$ The answer to this goes from the Fixed point theorem. But in ...
1
vote
1answer
10 views

Biorthogonal complement of subspace of subspace.

I'm taking a course on Banach and Hilbert spaces. The teacher who guides the exercise sessions is often a bit fast, so only when revising my notes at home I realize I do not fully understand them. We ...
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0answers
15 views

On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
0
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1answer
30 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
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0answers
7 views

Harmonic functions locally null on connected open set

Let $u$ be a harmonic function on $U$ connected open set of $\mathbb{R}^n$ and suppose there is a open set $V\subset U$, such that $u(x)=0$ for every $x\in V.$ Show that $u=0$ in $U$. So, I tried to ...
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0answers
26 views

Positive linear functionals on the space of positive semidefinite matrices

Suppose $f: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ is a linear functional with the property that $f(A) \geq 0$ whenever $A$ is positive semidefinite. Is it true that there exist vectors $v_1, ...
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0answers
33 views

Let $f$ be a non-constant entire function. Prove that $f(z)=cz^n$ for some constant $c$ and positive integer $n$

Let $f$ be a non-constant entire function satisfying the following conditions: $$f(0)=0$$ for each $M \gt 0$ the set $$\{z| |f(z)| \lt M\}$$ is connected. Prove that $f(z)=cz^n$ for some constant ...
2
votes
2answers
48 views

How to prove the elementary inequality?

The inequality is the following: $$\frac{(1+x)^q-1}{x+x^q} \leq C(q),$$ where $q\in [1,+\infty)$ and $x > 0$, and the constant $C$ depends only on $q$. It's very nice if someone can provide the ...
0
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0answers
8 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
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0answers
21 views

How can I show the equivalent condition for $\mu^*$-measurability

This is Exercise 4.15 from "Real Analysis for Graduate Students": Let $X$ be a set and $A$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a pre-measure on $A$ such that ...
0
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0answers
10 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
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0answers
16 views

Let $f:[-1,1]\to \mathbb{R}$ by $f(x)=x^4$. Determine the polynomial $p_2$ of degree less than or equal to 2 such that $||f-p_2||_2$ is minimal

also compute $||f-p_2||_2$. Write $p_2$ with respect to $\{P_0,P_1,P_2\}$ and $\{1,x,x^2\}$ I know its helpful to show what I have so far but I really don't know where to start. I'm looking at ...
2
votes
2answers
42 views

Confused about proof that diameter of a closure of a set is the same as the diameter of the set.

Definition Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$, with $p \in E$ and $q \in E$. The supremum of $S$ is called the diameter ...
0
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4answers
32 views

$\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$

Why is $\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$ (here is $\mathbb{R}$ endowed with the standard topology? I could use the criterion: Is $(x_n)\subseteq M$ such that $x_n\to ...
1
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0answers
14 views

Notation for vector valued integration

Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts ...
2
votes
0answers
11 views

Series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$

As stated in the title, I'd like to compute the series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$ where $z$ is a (small enough) complex number. More generally, for any real value $\alpha$, is there a ...
0
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2answers
36 views

Could someone explain me the task?

this is the question: Show that for each linear map $f:\mathbb R^d ā†’ \mathbb R^e$ there exists $a < \infty$ so that $\|fw\|< a\|w\|$ for each $w$ in $\mathbb R^d.$ And my problem is that $f$ ...
0
votes
0answers
12 views

Additive and multiplicative model

I'm new in study of survival analysis. There are some basic-questions that I wanna ask and it will be great if somebody could help me to aswer these following questions: What the differences between ...
0
votes
1answer
36 views

Compute the volume element in a differentiable manifold.

Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable manifold $ M = g^{-1}(0)$. The thing is that ...
2
votes
1answer
17 views

Question about disconnected metric spaces

The definition of disconnectedness that I've been taught is that a metric space $(X,d)$ is disconnected if there exists two non-empty disjoint open sets $A$ and $B$ such that $X=A\cup B$. My ...
1
vote
1answer
22 views

Incomplete beta integral

Let n be greater than one, and B be the beta integral, $$\sum _{j=0}^{\infty } C_j B_{\frac{1}{n}}(j+1,j+2)=\frac{1}{n}$$ Is it correct to call this an inversion formula? What possible ideas are ...
1
vote
1answer
30 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
1
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0answers
21 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
1
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2answers
53 views

How to find the limit points of the set $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$

How to find the limit points of the set $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$ limit point: A point $x$ is said to be a limit point of a non empty ...
3
votes
3answers
69 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
0
votes
0answers
43 views

Proving the existence of a sequence such that

I am trying to prove that there exists a sequence, for example: $$ f(n) = n! $$ (or we can select any sequence we need to prove the existence of just one), with the following property: edit: for ...
2
votes
1answer
27 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
2
votes
0answers
62 views

Finding the maximum of two functions with complicated formulas

Let $$ f(\omega)=1+\frac{m(a+\omega^2)}{a^2+\omega^2}+\alpha\left(\frac{a^2+\omega^2-ma}{a^2+\omega^2}\right)\cos(\omega\tau)+\frac{\alpha m\omega}{a^2+\omega^2}\sin(\omega\tau)\;, $$ and $$ ...
-3
votes
0answers
30 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
1
vote
1answer
28 views

A domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an “outer spherical condition”

Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e. $\Omega=\overline{\Omega}^\circ$ For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ ...
2
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2answers
32 views

A Chain of Subsets of $\mathbb{R}$ Without any Good Countable Subchain

Consider which $\bigl{(} A_i \bigr{)}_{i\in I}$ is a chain of subsets of $\mathbb{R}$. We say that a countable chain like $\bigl{(} B_n \bigr{)}_{n\in \mathbb{N}}$ is good if : for every $n\in ...
3
votes
1answer
27 views

Connections between Cesaro summation and Borel summation of series

Let $\sum_{n=0}^\infty x_n$ be a given series of numbers, let $S_n=\sum_{k=0}^n x_k$, $n=0,1,2,...$, let $g\in \mathbb R$. We say that this series is convergent to $g$ in the sense of Cesaro if $$ ...
0
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0answers
15 views

Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
4
votes
1answer
28 views

Revolving a $k$-manifold around an axis gives a $(k+1)$-manifold

I want to solve the following problem from M. Spivak's Calculus on Manifolds: Let $\mathbb{K}^n=\{x \in \mathbb{R}^n:x^1=0 \text{ and }x^2>0,\dots,x^{n-1}>0\}$. If $M \subseteq \mathbb{K}^n$ ...
0
votes
1answer
27 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
2
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0answers
35 views

Continuous function on compact subset of $\mathbb R$ to itself has a fixed point.

Let $f:[a,b] \to [a,b]$ be continuous. Then $f$ has at least a fixed point. I read the following proof from Limaye book. Define $F(x)=f(x)-x.$ Since $a \leq f(x) \leq b,\ \quad F(a)\leq 0 \ \quad ...
1
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1answer
18 views

modulation-translation operator continuous in $L^{p}$ norm?

We put, $T_{y}f(x):=f(x-y), \ (x, y\in \mathbb R^{n}).$ It is well-known that $\|T_yf-f\|_{L^{p}} \to 0$ as $y\to 0$ for $1\leq p <\infty.$ Next we put, $M_tT_yf(x):= f(x-ty) e^{i t (x\cdot y)}, ...
1
vote
1answer
36 views

Prove or disprove regarding continuity of $f$ and $g$

Prove or disprove: Let, $f,g:[a,b]\to \mathbb R$ be continuous in $[a,b]$ and are non-zero at any point. There exists $c\in [a,b]$ such that $$g(c)\int_a^bf(x)\,dx=f(c)\int_a^b g(x)\,dx.$$ ...
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0answers
41 views

The Coin-Exchange Problem (Application of the Residue Theorem) [on hold]

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...
1
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1answer
22 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
1
vote
2answers
58 views

Find functions $f$ and $\alpha$ such that $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?

Find functions $f$ and $\alpha$ such that the improper Riemann-Stieltjes integral $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist? I'm really not sure how to start ...
1
vote
1answer
32 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
5
votes
0answers
80 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
-1
votes
1answer
63 views

Example of projection sequence on Hilbert space with strong limit P

Let $P_n$ be strongly convergent with limit $P$, where $P_n$'s are projections on a Hilbert space $H$.Suppose that $P_n(H)$ is infinite dimensional. Show by example that P(H)$ may be finite ...
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votes
4answers
32 views

Proof $log_{r} a = log_r s \cdot log_s a $ [on hold]

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a $
2
votes
0answers
36 views

How to prove that $\|x+y\|_p<\|x\|_p+\|y\|_p$? [duplicate]

Let $l_p=\{(x_n)\in\mathbb{R}^\mathbb{N}: \sum_{n=0}^\infty |x_n|^p<\infty\}$ and consider the following norm in $l_p$: $$\|x\|_p=\left(\sum_{n=0}^\infty|x_n|^p\right)^{1/p}$$ for ...
3
votes
1answer
34 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...
0
votes
0answers
17 views

Second derivative and Hessian-Matrix

Suppose $f$ is twice differentiable. Why is $Df(a)[v] + \frac12 D^2f(a)[v,a] = \frac 12 \langle H_f(a) v, v \rangle $ ? $Df(a)[v]$ is the multidimensional derivative of $f$ at point $a$ in ...
4
votes
2answers
35 views

On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...