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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5 views

Functions linearly independent and linearly independent gradients?

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
2
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0answers
11 views

finding polynomials to approximate a multivariable function

Let $U := B_1(0) \subseteq \mathbb{R}^2$, with $B_1(0) := \{(x, y) \in \mathbb{R}^2,\space \|(x, y)\| _1 < 1\}$. Now consider the function: $$g: U \to \mathbb{R}^2, (x, y) \mapsto ...
1
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1answer
17 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
3
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1answer
21 views

diffeomorphism inbetween two subsets of $\mathbb{R}^2$

Consider the function $$f: \mathbb{R}^2 \to \mathbb{R}^2, \space\space f(x, y) := \pmatrix{x(1-y) \cr x y}$$ Now first, why is $f$ continuously differentiable? Then, I want to prove that $f$ ...
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2answers
16 views

second derivative of the composition of two multivariable functions

Let $U \subset \mathbb{R}^n$ be open, and let $\gamma: \mathbb{R} \to U$ and $f: U \to \mathbb{R}$ be to functions that are differentiable at least twice. I want to show that $\frac{d^2}{dt^2}(f ...
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0answers
12 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
2
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9answers
2k views

After switching a lamp on and off infinitely many times in one minute, is it on or off? [duplicate]

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
1
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1answer
42 views

An open interval as a union of closed intervals

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I ...
2
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1answer
39 views

Prove that $Df(p)=f(p)T$ where $T(q)=\int_{0}^1q$

Let $E=\mathcal{C}[0,1]$ provide with $\|\cdot\|_\infty$ norm. Let $f:E\to \mathbb{R}$, given by $f(p)=e^{\int_0^1 p}$. I need to prove that $f$ is differentiable. My approach: Let ...
2
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0answers
55 views

A few questions on the later chapters in Principles of Mathematical Analysis by Walter Rudin (3rd Edition)

I am currently reading Principles of Mathematical Analysis by Walter Rudin (3rd Edition). I am enjoying the book and it's terseness, which isn't an issue for me. What I do have a problem with is that ...
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1answer
40 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,\frac{3R}{4}))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,\frac{3R}{4})^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le ...
7
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1answer
65 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
3
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1answer
34 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
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1answer
37 views

Finding Laurent series of a function $f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$ [on hold]

How do i transform this function into Laurent series $$f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$$ where $ \frac{1}{3} < |z| < 1 $.
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0answers
16 views

combine analysis and artificial intelligence

I'm sorry if I ask this question at the wrong place, but I don't know a better one. I am a Master's student and I am really interested in analysis, but I also want to get into AI. Does anyone know a ...
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0answers
41 views

Is the field of real algebraic numbers a complete field?

Let $\mathbb{R}_{alg}$ be the field of real algebraic numbers. Is there exist a metric $|\cdot|$ for which $(\mathbb{R}_{alg}, |\cdot|)$ is a complete field (i.e. any Cauchy sequence converges in ...
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1answer
25 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
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1answer
19 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
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0answers
23 views

Continuously differentiable operator

if i consider the operator $A$ defined on $H^1_0$ by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s),~~t\leq s\\s(1-t),~~s\leq t\end{cases}$$ What is the expretion of $A'u$ ...
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0answers
25 views

Approximating functions such that the left- and right limit exists everywhere

Every continuous function $f : \mathbb R \to \mathbb R$ could be uniformly approximated by step functions. For a proof consider an interval $[a,b]$, then $f$ is bounded on this compact interval, i.e. ...
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2answers
62 views

find $\lim_n\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+…+\frac{a^\frac nn}{n+\frac 1n}$ using Riemann integral

Here is the question: prove that $S_n=\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+...+\frac{a^\frac nn}{n+\frac 1n}$ is convergent for $a>0$ then find its limit. My attempt: If we accept ...
2
votes
3answers
281 views

Non linear Differential Equation

Let $\Omega:=\{(x_1,x_2) \subset \mathbb{R}^2 | x_2>0\}$. I want to solve the differential equation $$\begin{pmatrix} \dot{x_1} \\\dot{x_2} \end{pmatrix}=\begin{pmatrix}x_2^2-x_1^2 ...
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3answers
143 views

Show the following set is connected

For any $x \in \Bbb R^n$ how do I show that the set $B_x := \{{kx\mid k \in \Bbb R}$} is connected. It should also be concluded that $\Bbb R^n$ is connected. I was thinking of starting by assuming ...
2
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0answers
29 views

non-analytic smooth function

Given the function $$f(z) = \sum_{k \in \mathbb{Z}} \exp\left(-\sqrt{|k|}\right) z^{k}$$ Defined on the unit circle $S^1$. That $f$ is smooth is not hard to see. Why is there not a $\Phi$ on an open ...
0
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1answer
32 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
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0answers
19 views

If $\{x_n\}$ and $\{y_n\}$ are two bounded sequences then prove that $\liminf x_n -\liminf y_n \leq \liminf(x_n - y_n)$ [on hold]

Please help me in finding the proof of this theorem.I am in a fix If $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ are two bounded sequences then prove that $$\liminf_{n\to\infty}\,x_n - ...
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3answers
94 views

Prove that the following function is $C^{\infty}$ [duplicate]

Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}}, & \text{if $|x|<1$} \\ 0, & \text{if $|x| \ge 1$} \end{cases}$$ is $C^{\infty}$ I found this problem ...
2
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0answers
20 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
2
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1answer
29 views

If a function is enclosed by lower and upper sums, does its limit w.r.t. partitions equals the integral

Let $a < b$ and denote by $\mathcal P[a,b]$ the set of all finite partitions of the compact interval $[a,b]$, i.e. all sets of the form $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. ...
1
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1answer
31 views

Why periodic functions form a dense subset in $ C[a,b]$ with $L^2$ norm?

Let's consider the linear space $C[a,b]$ but with $L^2$ norm $$ \|f\|=(\int_a^b |f(t)|^2dt)^{\frac{1}{2}} $$ How to prove that the subspace $$ V=\{f\in C[a,b]: f(a)=f(b)\} $$ is dense in this normed ...
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0answers
20 views

Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
2
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0answers
48 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
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0answers
24 views

How do I formulate a specific formula for a sequence?

I have three arrays, for instance s = [1:2], j = [1:20] and b = [1:8], and I am trying to build a single row. The problem that I actually have is that I need to find a formula f(s,j,b) such that ...
2
votes
1answer
37 views

Slopes of curves from complex derivative [on hold]

Show that the slopes of the level curves$$u(x,y)=\text{constant} \ \ \text{and} \ \ v(x,y)=\text{constant}$$ are respectively given by $$\cot(\arg(f'(z))) \ \ \text{and} \ \ -\tan(\arg(f'(z)))$$ If ...
3
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2answers
109 views

Show that $\frac{(x^2 + y^2 )}{4} \leq e^{x+y-2}$

Show that \begin{equation} \frac{x^2 + y^2}{4} \leq e^{x+y-2} \end{equation} is true for $x,y \geq 0$. As far, I have prove that \begin{equation} x^2 + y^2 \leq e^{x}e^{y}\leq e^{x+y} ...
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0answers
91 views

Determine the minimum of $\int_0^\infty\left|x^3+ax^2+bx+c\right|e^{-x}dx$

This question appeared on a graduate preliminary exam in real analysis. Determine $$\min_{a,b,c\in\mathbb{R}} \int_0^\infty\left|x^3+ax^2+bx+c\right|e^{-x}dx.$$
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1answer
24 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
2
votes
1answer
28 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...
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0answers
24 views

Approximation Lemma for Riemann-integrable functions

In the following let $f : [a,b] \to \mathbb R$ be bounded functions. For a regulated function, the integral could be written as the limit $$ \int_a^b f(x) dx = \lim_{n\to \infty} \int_a^b ...
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2answers
38 views

Bounded function on compact interval that is not Lebesgue integrable

Is there an example of a bounded function $f : [a,b] \to \mathbb R$ which is not Lebesgue integrable?
3
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0answers
65 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
0
votes
1answer
56 views

Evaluate $ \int_{\varepsilon}^1 \sin\left( \frac{1}{x} \right) dx$

Let $0 < \varepsilon < 1$, how to solve the integral: $$ \int_{\varepsilon}^1 \sin\left( \frac{1}{x} \right) dx $$
2
votes
2answers
50 views

Prove that a classical solution of $-\langle\nabla,A\nabla u\rangle=f$ is also a weak one

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable and $A(x)$ be symmetric, for all $x\in\Omega$ $u\in C^2(\Omega)$ with $A\nabla ...
0
votes
0answers
46 views

Riemann-integrable iff pointwise limit of step functions

Is the following true? Let $f : [a,b] \to \mathbb R$ be a bounded function: The function $f$ is Riemann-integrable if and only if there exists a sequence of step functions $\varphi_n$ converging ...
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0answers
19 views

Develop a concept of weak solvability for $-\langle\nabla,A\nabla u\rangle=f$

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable with $A(x)$ is symmetric, for all $x\in\Omega$ $\exists c_1,c_2>0$ with ...
0
votes
1answer
25 views

Integration problem of a modified 'standard integral'

Consider $\int\limits_0^\infty ye^{-y}e^{-xy}dy$ I can use the fact that $\int\limits_0^\infty u^ne^{-u}=n!$ Clearly, $\int\limits_0^\infty ye^{-y}e^{-xy}dy=\int\limits_0^\infty ye^{-y(1+x)}dy$. ...
2
votes
0answers
27 views

Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
3
votes
1answer
36 views

Compact linear operator

Today in lecture we were told that for a linear compact operator $T$ on an infinite-dimensional Hilbert space with infinite-dimensional range, we have that $\ker(T)^{\perp}$ is infinite-dimensional, ...
0
votes
0answers
18 views

A problem about a family of mesurable fuctions

Let $(X,\mathcal{M},\mu)$ a measure set such that $\mu(X)<\infty$ and $\mathcal{F}$ a family of $\mu$-measurable functions. Let $E(f,t)=\{x\in X\mid f(x)\geq t\}$ with $f\in\mathcal{F}$. If ...
0
votes
2answers
30 views

If a positive term series is less or equal to a positive real number for any finite n, will $S_n$ still bound by the same number for $n \to +\infty$?

For example, if $S_n = \sum\limits_{k = 0}^n a_k$ $\leqslant$ $R$ for any finite positive integer $n$ where $R$ is a fixed real number, will $Sn$ still bound by $R$ for $n \to +\infty$? if so, how to ...