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
4answers
13 views

what is the subset of $\Bbb{R}^2$ that corresponds to set of complex numbers $z$ such that $|z|\leq 1$

I am trying to understand what is the subset of $\Bbb{R}^2$ that corresponds to the set of complex numbers $z$ such that $|z|\leq 1$ Can you help?
1
vote
2answers
23 views

Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
0
votes
3answers
36 views

Find bounded function satisfying f(0)=0, f'(0)=0, and bounded first and second derivatives

I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or ...
-3
votes
2answers
19 views

$A$ is a convex subset with non-empty interior and $D$ is dense in $\mathbb R^n$ ; then $\mathbb R^n$ , $U\cap D \cap A \ne \phi$? [on hold]

Let $A$ be a convex subset , with non-empty interior , of $\mathbb R^n$ and $D$ be a dense subset of $\mathbb R^n$ ; then is it true that for every open subset $U$ of $\mathbb R^n$ , $U\cap D \cap A$ ...
0
votes
3answers
10 views

$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
1
vote
2answers
37 views

Sum over values of auxiliary function gets arbitrary big, justification

Let $f : \mathbb N_{>0} \to \mathbb R_{\ge 0}$ be a function satisfying $\sum_{n=1}^{\infty} 2^{-f(n)} = \infty$ (like $f(n) = \log n$). Define $$ F(n) = \left\lfloor \log_2\left( \sum_{i=1}^n ...
3
votes
1answer
26 views

Complement of the union of countably many , mutually disjoint , non-empty open balls in $\mathbb R^n , (n >1) $ is path connected?

Let $n \ge 2$ and $\{B_m\}_{m=1}^\infty$ be countably infinitely many , mutually disjoint , non-empty open balls in $\mathbb R^n$ , then is $\mathbb R^n \setminus \cup_{m=1}^\infty B_m$ ...
-1
votes
1answer
31 views

prove $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$

Can I prove the following (with or without assumptions, e.g. all the elements in $\mathbf{a}$ or $\mathbf{b}$ are positive? $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$ where ...
0
votes
0answers
15 views

Is it possible to prove $\arg\min_a f(\max(\mathbf{a}^T\mathbf{b}_i)) = \arg\min_a f(\mathbf{a}^T\max(\mathbf{b}_i))$

I have the following optimization problem, $ \arg\min_\mathbf{a} f(\max(\mathbf{a}^T\mathbf{b}_i))\;\; i=1, \dots ,N$ where $\mathbf{a}$ and $\mathbf{b}_i$ are vectors of dimension $d$. Let $B = ...
0
votes
0answers
21 views

Prob. 1, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: An infinite subset of $[0,1]^\omega$ without limit points in the uniform topology?

Let $[0,1]^\omega$ denote the set of all sequences of real numbers in the closed unit interval $[0,1]$, and let the uniform metric $d$ on $[0,1]^\omega$ be given by $$d\left( (x_n)_{n\in\mathbb{N}} , ...
3
votes
1answer
45 views

A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
0
votes
1answer
27 views

Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. [duplicate]

Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. Hint. Consider the $N+1$ numbers $tx-[tx]$ for $t=0,1,2,\dots, N$ ...
1
vote
0answers
33 views

Integral of a function which is everywhere discontinuous?

Yesterday, I tried to carry out a little thought experiment when it came to taking limits and have found that it has pushed my understanding of them to the breaking point. I tried considering the ...
0
votes
0answers
18 views

Asymptotic series of a matrix-valued function.

Consider the following matrix $$f(\lambda)=\left( \frac{\lambda-1}{\lambda + 1} \right)^{\nu \sigma_3} \ \ \ \lambda \in \mathbb{C} \setminus [-1,1]$$ where $\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 ...
-5
votes
0answers
42 views

Would you please give me your opinion about solving this equation? [on hold]

Would you please give me your opinion about solving this equation? [![enter image description here][1]][1] $\sum_{i=0}^{1}\left (-X \right )^{i}*\sum_{J=0}^{7}\sum_{i=0}^{j}\, \gamma _{j}*X^{i}=-T$ ...
0
votes
3answers
43 views

Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.

I need help justifying that $|x-a|$ is continuous and non-differentiable at $x=a$. I would also like to prove that it achieves a minimum at $x=a$, but I do not know if that is already clear enough.
2
votes
3answers
63 views

A question about complex integration of $\frac{1}{p(z)}$

Let $p(z)$ be a polynomial of degree $n\ge 2$. Is it true that, there is a $R>0$ such that $$\int\limits_{|z|=R}{\frac{1}{p(z)}dz}=0?$$ My attempt is: there is a $R>0$ such that $|p(z)|\ge ...
0
votes
1answer
20 views

Show the graph is Jordan region with volume 0

Let $f \colon [a, b] →\mathbb R$ be a continuous function. Then prove that the graph of $f$, $$\operatorname{Graph}(f) := \{\,(y, x) \in \mathbb R^2\mid y = f(x), x \in [a, b]\,\}$$ is a Jordan ...
2
votes
1answer
33 views

Fundamental solution for a parabolic PDE with costant coefficents

as it is well known, the fundamental solution of the heat equation is the function $G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}}$, for all $t>0,x\in\mathbb{R}^n$. I wonder if exists (and ...
0
votes
0answers
27 views

Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
0
votes
0answers
26 views

Continuity of translation property [duplicate]

Let $u \in L^{p}(U)$ where $1 \leq p \lt \infty$ & $U \subseteq \mathbb R^{n}$ . Define : $F : \mathbb R^{n} \to L^{p}(U) $ by $ F(y) := u(x+y)$ . Prove that: as a function of $y$ ; $F(y) $ is ...
1
vote
1answer
61 views

Asymptotic expansion of integral with hyperbolic functions

Consider the integral given by $$f(r)=\int_{0}^{\tanh(r)} \arccos\left(\frac{\sigma}{\sinh(r)\sqrt{1-\sigma^2}}\right)\cdot \frac{1}{\sqrt{\sigma^2+a^2}}d\sigma,$$ where $a>0$. I am wondering ...
-1
votes
0answers
58 views

Find a function that maps real numbers to real numbers, that is monotone, differentiable, and has a derivative that is positive for all X.

Let $T\colon \mathbb R\to\mathbb R$ be monotone and differentiable, with a positive derivative. Suppose $f: \mathbb{R} \to \mathbb{R}$ is also differentiable. I need to find the derivative of $c(x) = ...
3
votes
2answers
120 views

Why doesn't this work for Rudin Exercise 3.8

The problem is 3.8 exercise in baby Rudin: If $ \sum{a_n} $ converges and $\{b_n\}$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. Why can't I just do this?: Let $M$ be an ...
1
vote
0answers
31 views

continuous, compactly supported function with zero not in the image of the support

Rudin defines (Def. 2.9) the support $K$ of a complex function $f$ on a topological space $X$ to bet the closure of the set $\{x:f(x)\neq 0\}$. He then claims that if $X$ is not compact and $f$ is a ...
0
votes
0answers
26 views

Statement about the discrete (metric) space, and both an open and closed ball.

I have the following statement from my notes: "Let $(X,d)$ be the discrete space i.e. any non-empty set with the discrete metric ($d_d(x,y)=1$ for all $x\neq y$). Then, amazingly, $B_1(x)=\{x\}$, a ...
0
votes
1answer
37 views

Dense $G_{\delta}$ set implies comeagre set

Suppose that $X$ is a metric space. Show that if $D$ is a dense $G_{\delta}$ set, then $D$ is comeagre, that is, countable intersection of dense sets. My attempt: Let $D=\bigcap_{n \in ...
0
votes
1answer
43 views

Expressing $\cos(\varphi x)$ as a function of $x\sin\varphi,x\cos\varphi$

Let $\varphi,x\in\mathbb{R}$. I wonder if one can explicitly express $\cos(\varphi x)$ as a function of the variables $x\sin\varphi$ and $x\cos\varphi$. Suppose we denote ...
1
vote
1answer
26 views

Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
0
votes
0answers
35 views

How to prove the convergence of a sequence satisfying $a_{k+1} \leq c_{1} a_{k} + \frac{c_{2}}{a_{k}} +1$?

Assume a positive sequence $\left\{ a_{k} \right\}$ satisfying \begin{equation} a_{k+1} \leq c_{1} a_{k} + \frac{c_{2}}{a_{k}} +1, k \in \mathbb{N} \end{equation} where $c_{1},c_{2},a_{1} > 0$. ...
4
votes
1answer
86 views

How to prove there exist distinct $a_{i}$ such $f'(a_{1})f'(a_{2})f'(a_{3})\cdots f'(a_{n})=1$

Let $f$ be a continuous map from $[0,1]$ to $R$ that is differentiable on $(0,1)$,with $f(0)=0,f(1)=1$, show that for each postive integer $n$ there exist distinct numbers $a_{1},a_{2},\cdots,a_{n}\in ...
3
votes
0answers
36 views

“composition” of “pointwise convergent sequences of functions”

Intuitively, if $f_n\to f$ as $n\to\infty$ and $g^{(n)}_i\to f_n$ as $i\to\infty$, can we get $g_j\to f$ as $j\to\infty$? Formally, Let $\{f_n\}_n$ be a sequence of functions from $\mathbb{R}^d$ ...
2
votes
0answers
20 views
+50

Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence: Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...
0
votes
0answers
23 views

Existence and uniqueness of a pde solution

I have the PDE system: $\frac{\delta}{\delta t}u(t,r)=-\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)$ $\frac{\delta}{\delta t}v(t,r)=\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)-v(t,r)$ $x(0,r)=\rho(r), ...
0
votes
2answers
23 views

What can I say about the constant of a Lipschitz condition for a scaled norm?

Let's say $X$ is a vector space with inner product $\langle \cdot,\cdot\rangle$ and induced norm $\|\cdot\|$. Then for a scalar $\theta > 0$ we define $\langle \cdot,\cdot\rangle_{\theta} := ...
1
vote
0answers
36 views

why does a nonempty convex set have a nonempty interior

Suppose $A \neq \emptyset$ is a convex set in $R^n$, or generally a metric space, then is there any neat proof to show that $A$ has nonempty interior? Thanks for the help.
1
vote
0answers
42 views

On the size of rational numbers and Irrational numbers. [duplicate]

Being a high school student, It's obvious to me that there are both an infinite number of rational and irrational numbers. However I don't really see if there is more rational than irrational, ...
3
votes
1answer
35 views

Smooth maps preserve dimension

I stumbled over a useful consequence, that is apparently wrong for only continuous maps. Imagine $A \subset \mathbb{R}^{n-1}$ is a compact set and $F : \mathbb{R}^{n-1} \rightarrow S^{n}$ a smooth ...
0
votes
1answer
62 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
1
vote
1answer
28 views

What is a martingale array - its definition and importance?

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.
1
vote
1answer
13 views

Lipschitz maps on locally compact groups

Suppose $G$ is a locally compact second countable group. This means that there exists a proper (closed bounded sets are compact) left invariant ($d(gx,gy) = d(x,y) \ \forall g,x,y \in G$) metric on ...
1
vote
4answers
60 views

Is there a way to show the following equality without induction

I wanted to show the following equality without using induction: $$ \sum_{k=2}^n \frac{1}{k(k-1)} = \frac{n-1}{n} $$ Any hint on how to do it?
1
vote
1answer
39 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
0
votes
1answer
75 views

Induction proof $\sum_{k=1}^{n}\frac{1}{\sqrt{k}} \gt \sqrt{n}$

I want to show the following statement through induction for n>1: $$ \sum_{k=1}^{n}\frac{1}{\sqrt{k}} \gt \sqrt{n} $$ The induction step isn't completly working out. I should show that the last term ...
0
votes
1answer
42 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
0
votes
1answer
23 views

Are those uses of sum and product notations correct?

I wanted to use th eproduct and um notations to describe the following sequences. Are those correct? $$ 6+12+18+24=\sum_{i=1}^4 6i \\ x_{1}-x_{2}+x_{3}-x_{4}+...+x_{7}-x_{8} = \sum_{i=1}^8 ...
1
vote
0answers
18 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
0
votes
1answer
20 views

$ \exists f:\mathbb C \setminus D \to \mathbb C$ is bounded one-one holomorphic, how?

We note that there cannot exist bounded one-one holomorphic map $f:\mathbb C \setminus \{0\} \to \mathbb C.$ Put $D=\{z\in \mathbb C: |z|\leq1\}$ (closed disk). My Question: How to show there ...
1
vote
1answer
18 views

$\lim_{N \rightarrow \infty}\dfrac{1}{N^d}|E \cap \dfrac{1}{N}\mathbb{Z^d}|$ may not exist

In this page, under Remark $1$, the limit $$\lim_{N \rightarrow \infty}\dfrac{1}{N^d}|E \cap \dfrac{1}{N}\mathbb{Z^d}|$$ may not exist. Question: For $d=1$, that is, in $\mathbb{R}$, what is the ...
1
vote
0answers
8 views

baire $1$ function

Here is a new definition of Baire Class one function. Suppose that $X$ is a complete separable metric space. A function $f:X \rightarrow \mathbb{R}$ is said to be Baire class one if for any ...