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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1answer
80 views

Ways to represent functions of bounded variation as the difference of two monotone functions

I learnt (it is called Jordan's representation theorem) that every function $f : [a,b] \to \mathbb R$ with bounded variation could be represented as the difference $f = g - h$ of two monotone ...
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1answer
343 views

Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
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1answer
661 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi (...
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1answer
85 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
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1answer
66 views

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates $$ \left\{ \begin{array}{c} u = yz\sin(x)\\ v = y^2 - x\\ w = xz \end{array} \right. $$ Determine the ranks ...
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3answers
95 views

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
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0answers
36 views

Calderón reproducing formula : $\int_{0}^{\infty}\int_{R^d}|\phi_{t}(x-y)||(\phi_t*f)(y)|\frac{dt}{t}dy<\infty$

Suppose that $\int f=0$, $f \in L^2$ and $f$ has a compact support. Let $\phi$ be radial, and such that $\mathrm{supp}(\phi) \in B(0,1)$. Plus, assume that $\int_{R^+} |\hat{\phi}(t\xi)|^2t^{-1}dt$=1 ...
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1answer
45 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
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1answer
65 views

Solution of differential equations with discontinuity

Suppose that we have scalar differential equation \begin{equation} \dot{x}(t)=u(t) \end{equation} Here $u(t)$ is a piecewise constant function with discontinuity. If the points of discontinuity is ...
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1answer
64 views

Definition of a metric-nonnegativity condition

There is a question in my mind which seems to be silly but I am desperately wanting the answer. Why a metric is defined from $X\times X$ to $\mathbb R$ and not to the set of nonnegative reals? I ...
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1answer
39 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
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1answer
26 views

Proving a particular subset of a Hilbert space is a subspace

I have a small question please, how to prove that this set: $F=\lbrace h\in H, \langle f''(u)h,h\rangle <0\rbrace$ is a sub space of the Hilbert space $H$, where $f''(u)$ is a self-adjoint ...
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0answers
40 views

Spectral decomposition of a Hilbert space

I have this proof, but I don't understand how they do the spectral decomposition of $H$ into $H_-$and $H_+$? Please help me. Thank you.
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4answers
312 views

Is there a bijection from a bounded open interval of $\mathbb{Q}$ onto $\mathbb{Q}$?

It is easy to create a bijection between two bounded open intervals of $\mathbb{R}$, such as: $$ \begin{align} f : (a,b) &\to (\alpha,\beta) \\ x &\mapsto \alpha+(x-a)(\beta-\alpha). \end{...
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2answers
41 views

Proof by Induction: $(1+x)^n \le 1+(2^n-1)x$

I have to prove the following by induction: $$(1+x)^n \le 1+(2^n-1)x$$ for $n \ge 1$ and $0 \le x \le 1$. I start by showing that it's true for $n=1$ and assume it is true for one $n$. $$(1+x)^{n+...
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0answers
144 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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1answer
109 views

Some properties of points with Lebesgue density equal to $1$.

I am studying Evans-Gariepy book and in corollary 1 of section 3.1.2, he prove that if $f:\mathbb{R}^N\to\mathbb{R}^M$ is locally Lipschitz and $$Z=\{x:\ f(x)=0\},$$ then $Df(x)=0$ a.e. $x\in Z$. He ...
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0answers
61 views

Clarification of a theorem from Chang's Methods in Nonlinear Analysis

The following theorem is taken from Chang's Methods in Nonlinear Analysis. It has a complete proof; however, I have some trouble understanding it (for example, I don't see what $K(f_{\sigma_i})$ means)...
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0answers
52 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
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1answer
121 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{...
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1answer
85 views

Norm of the multiplication operator $f\mapsto (x\mapsto xf(x))$ on $L^2[a,b]$ [duplicate]

We have a linear operator $T : L^2[a,b] \rightarrow L^2[a,b]$ (with $|a| \le |b|$), $f \mapsto (x \mapsto xf(x))$ Now I shall determine what $\Vert T\Vert$ is. We clearly have $\Vert x \mapsto xf(x)...
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5answers
154 views

Find all real numbers $x$ such that $2+\sqrt{|x|}=x$

I'm revising for an exam and my lecture notes and online sheets etc. from my lecturers are very unhelpful, and I've completely forgotten how to do this type of question. So any help would be greatly ...
4
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0answers
33 views

Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
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2answers
114 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
0
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1answer
142 views

Few Questions about analysis in Rudins book

I have been looking at intro to real analysis. I am using the text book "Principals of Mathematical Analysis, third edition" by Walter Rudin. I have some questions about things I found confusing and ...
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2answers
55 views

Why is this a quotient map

Is there a direct way to see that $p \times id : [0,1]^2 \rightarrow S^1 \times [0,1]$ is a quotient map with $(p \times id)(x,y) = (e^{ix},y)$? By direct way, I mean is there an obvious argument why ...
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1answer
35 views

Bounding the $\ell^{1}$ norm given the $\ell^{2}$ norm

Suppose $x = (x_{1}, x_{2}, \ldots) \in \ell^{2}$. If $\sum_{n = 1}^{\infty}n|x_{n}|^{2} \leq 1$, is it possible to bound $\sum_{n = 1}^{\infty}|x_{n}|$?
4
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3answers
144 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
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2answers
140 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow \mathbb{T}\...
5
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0answers
608 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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2answers
37 views

$ \Bbb B( \Bbb R^{n+m} ) = \Bbb B( \Bbb R^{n} ) \times \Bbb B( \Bbb R^{m} ) $

Let $ \Bbb B( \Bbb R^{n} ) $ denote a Borel algebra on $ \Bbb R^n $. Why is it true, that: $ \Bbb B( \Bbb R^{n+m} ) = \Bbb B( \Bbb R^{n} ) \times \Bbb B( \Bbb R^{m} ) $ I think, that "$ \supset$" ...
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1answer
123 views

Lyapunov stability and finite time convergence

I have two questions: Problem 1: Let $V (x)$ be the Lyapunov function candidate with $x \in \mathbf{R}$, and the time derivative of $V(x)$ is given by $\dot{V} (x) \le - x ( x - \alpha (t) )$ where ...
2
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1answer
92 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
6
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1answer
96 views

Which negative integer powers of 2 belong to the Cantor Set?

Consider the Cantor set $C$, and negative integer powers $2^{-k}$. Clearly, for $k=1$, $2^{-1} \notin C$ since $1/2 \in (1/3, 2/3)$, the first deleted open interval. It is known that $1/4 = 2^{-2}...
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1answer
57 views

Which technique of integration should use to solve the question?

Quote from the paper I read: Given $F=(1-\lambda)f$ + $\lambda zf'$, we find that $f(z)= \tfrac{1}{\lambda} z^{1- \tfrac{1}{\lambda}}\int_0^zF(t)t^{\tfrac{1}{\lambda}-2}dt.$ My questions is Which ...
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7answers
525 views

If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges

Let $\displaystyle \sum a_n$ be convergent series of positive terms and set $\displaystyle b_n=\sum_{k=n}^{\infty}a_n$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges. I could see ...
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3answers
107 views

What is $\frac{d^n}{dx^n} \frac{e^{\lambda x}}{x}$?

I was wondering whether there is an explicit way to say what the derivative of $\dfrac{d^n}{dx^n} \dfrac{e^{\lambda x}}{x}$ for $n \in \mathbb{N}_0$is, where we assume that $\lambda \neq 0$.
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3answers
53 views

Length of Difference Curve

Let $\varphi : [a,b] \to \mathbb R^n$ be a curve, and for some partition $\pi = \{ t_0 = a, t_1, \ldots, t_m = b \}$ of $[a,b]$ set $$ l(\pi, \varphi) = \sum_{i=1}^m \| \varphi(t_i) - \varphi(t_{i-1})...
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0answers
62 views

Rules to evaluate Suprema and Infima

Note: All functions considered are supposed to be bounded. Nowhere I found rules to evaluate suprema and infima. Obviously, $$ c \cdot \sup_{x\in \mathbb R} f(x) = \sup_{x\in\mathbb R} cf(x) \quad\...
2
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1answer
77 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} k^...
3
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1answer
52 views

$1/r^2\int_{\mathbb{S}_r}u-u(x)$ converging to $\Delta u(x)$?a

When reading some papers on PDEs, the following shows up several times: For a $C^{\infty}$ function $u$, $\frac{\int_{\mathbb{S}_r(x)}u-u(x)}{r^2}$ converges to $1/2n\Delta u(x)$ uniformly on ...
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0answers
26 views

Integration of nonlinear and linear ODEs

\begin{equation} \frac{dc_1}{d\tau}= \alpha I(1-c_{0}) + c_{1} (-K_{F} - K_{D}-K_{N} s_{0}-K_{P}(1-q_{0}))+ c_{0}(-K_{N} s_{1}+K_{P}q_{1}), \nonumber \end{equation} \begin{equation} \frac{ds_1}{d\...
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0answers
47 views

Inequality among trigonometric sums of normal random variables

This is an inequality used in a proof which I do not know how to prove. $$\left(\sum_{k = 2^j +1}^{2^{j+1}} \frac{\sin(k\pi t)}{k}G_k\right)^2 \leq \left|\sum_{k = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi t}}...
2
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1answer
134 views

Why define the Lebesgue-Integral just for measurable functions?

Usually, the Lebesgue integral, for example on Wikipedia, is defined for non-negative measureable functions as $$ \int_E f \, d\mu := \sup\left\{ \int_E s \, d\mu : 0 \le s \le f, s \text{ simple } \...
0
votes
1answer
33 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
2
votes
2answers
71 views

Limits in complex plane.

Let $z$ and $z_n$ be complex numbers and assume $z_n \rightarrow z$. It it true that $$\lim\limits_{n\to \infty}\left(1+\frac{z_n}{n}\right)^n= \lim\limits_{n\to \infty}\left(1+\frac{z}{n}\right)^n ?...
1
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1answer
68 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
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1answer
32 views

Formula of Squaring Sums / Integrals

I'm trying to find a proof for the identities (which I use quite often) $$\left ( \int_{a}^{\infty}f(x)\,dx \right )^2=\int_{a}^{\infty}\int_{a}^{\infty}f(x, y)\,dx\,dy$$ and similarly for the series ...
1
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1answer
30 views

Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?

We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
1
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1answer
50 views

Problem with constructing a smooth function with given properties

I wish to construct a function $f:\mathbb R \rightarrow \mathbb R$ of class $C^\infty (\mathbb R)$ with the folowing properties: $f(x)=0$ for $|x|\leq 1$ $f(x)=x$ for $|x| \geq 2$, $|f(x)| \leq |x|$...