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
votes
3answers
291 views

What mathematical analysis book should I read (research, Putnam, personal enrichment)? [closed]

S.E advisers, I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the fields of computation theory and cryptography. Recently, it became important matter that ...
0
votes
0answers
18 views

derivative of indicator function composed with a relaxation of a heaviside function

I want to compute the following: $|\partial_t I_{\{H(u(x),t) \geq \mu\}}|,$ where $I$ represents the indicator function and $H(u(x),t)$ is actually a smoothly relaxation of another indicator ...
-1
votes
3answers
95 views

How to prove this inequality $(\frac{n+1}{e})^{n} < n! < e(\frac{n+1}{e})^{n+1}$? [closed]

$\Bigl(\frac{n+1}{e}\Bigr)^{n} < n! < e\Bigl(\cfrac{n+1}{e}\Bigr)^{n+1}$
4
votes
1answer
49 views

Integrability of $(x+y) ^{-3}$.

I'm asked to determine for what positive values of $\alpha$ is $(x+y)^{-3}$ integrable in the region where $0<x<1$ and $0<y<x^\alpha$. I've found that the function is integrable when ...
2
votes
1answer
57 views

Solving the functional equation $2f(x)-f(1/x)=3x$

If $$2f(x)-f(1/x)=3x$$ how would I find $f(x)$? I have tried various linear and other functions but I do not know how to start this
0
votes
1answer
39 views

A misconception about cauchy's criterion for existence of limit of a function

Let us consider a function F(x). Let's say that the limit of a function exists at a point $P$ and $Q$, but let's say the limit of that function doesn't exist at a point $Q$, $Q>P$ and $Q-P= D$. ...
1
vote
0answers
19 views

Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite ...
1
vote
1answer
40 views

Limit of continuous convex functions.

Let $\left( \, f_n(x)\, \right)$, $x \in \mathbb{R}$, be convex continuous increasing functions. Let $$f(x) := \lim\limits_{n \rightarrow \infty} f_n(x)$$ and assume that the limit exists for every ...
1
vote
2answers
22 views

Monotonicity of this series

Given the function $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=e^{x-1}$ we have the series $(x_n)_{n\ge1}$ where: $x_1=2$ and $x_{n+1}=f(x_n), n\ge1$ Find the monotonicity of the series and compute ...
2
votes
1answer
32 views

Existence of an entire function with certain property

Let $\{a_n\}$ and $\{b_n\}$ be two sequence of complex numbers such that $|a_n|\to\infty$ as $n\to\infty$. Prove that there exists an entire function $f:\Bbb C\to\Bbb C$ (i.e. $f$ is complex ...
-3
votes
2answers
97 views

Can you give me your opinion on Courant, Rudin, Apostol, Bartle or Zakon for Analysis? [closed]

Experts, can you give me your opinion on each book? I need a rigorous and complete book. Thanks for the advice.
2
votes
2answers
33 views

Pullback of $1$-form in coordinates

Let $$\theta(p) = \sum_{i=1}^n f_i(p) \, dx_i$$ be a $1$-form in local coordinates. then we define $F^*(\omega(p))(X_1,\ldots,X_n) = \omega(F(p))(DF(p)(X_1),\ldots,DF(p)(X_n))$ as the pullback of a ...
8
votes
3answers
160 views

Rudin assumes $(x^a)^b=x^{ab}$(for real $a$ and $b$) without proof?

I am currently self studying Baby Rudin and I'm having some problems with his proof of Theorem 3.20(a) on page 58. I have read all previous chapters and I can't find any mention of real exponents ...
2
votes
1answer
31 views

subadditivity for sets and monotonic function

Let $f : \Sigma \rightarrow \mathbb{R}^+$ be a function, where $\Sigma$ is the space of finite subsets of $\mathbb{Z}^d$. Assume that, if $A_1, A_2 \subset \mathbb{Z}^d$ are two disjoint finite sets, ...
2
votes
0answers
48 views

Density of subset with nonlocal boundary condition

I am having difficulty proving that $E=\bigcap_{n\geq 0} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a dense subset of: $F=\{f\in C^2 (\mathbb{R}) : ...
2
votes
1answer
58 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
2
votes
0answers
41 views

Extensions of $C^k$ functions to the boundary [closed]

Assume $\Omega \subset \mathbb R{^n} $ is an open connected smooth domain. I have some propositions that I guess they are correct , but I want to be confident. If $f\in C_0^k(\Omega) $ then $f\in ...
1
vote
1answer
31 views

Clever way to simplify sum?

Is there a clever way to rewrite the sum $$\sum_{i=2}^{n} (x_i-x_{i-1})\left(\frac{(x_i-x_{i-1})}{2}-x_i \right) ?$$ I haven't been able to come up with anything useful thus far.
0
votes
0answers
49 views

Prove exponential $e^f$ is of class $C^\infty$

Let $E$ a Banach space, $F=L(E,E)$ of linear and continuous functions. Define $f^0={\rm id}_E$, $f^n=f\circ\cdots\circ f$, $n$ times. Put $\exp(f)=\sum_{n=0}^\infty \frac{f^n}{n!}$. How to show the ...
1
vote
2answers
69 views

porous sets: why measure zero?

We call a measurable set $E\subset\mathbb R^N$ porous if every ball $B_r(x)$ contains a smaller ball $B_{cr}(y)$ for some $c\in(0,1)$ such that $$ B_{cr}(y)\subset B_r(x)\setminus E. $$ So I've read ...
0
votes
1answer
40 views

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma

Here's Theorem 4.3-2 (i.e. the Hahn Banach theorem for normed spaces): Let $f$ be a bounded linear functional defined on a subspace $Z$ of a normed space $X$. Then there exists a bounbed linear ...
1
vote
0answers
24 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
1
vote
1answer
48 views

Any real valued continuous function from a closed bounded set in $\mathbb R^2$ is bounded and attains its bounds

Without using the idea of compact or sequential compactness , can we prove that if $A$ is a closed bounded set in $\mathbb R^2$ , then any continuous function $f:A \to \mathbb R$ is bounded and ...
0
votes
1answer
58 views

Derivative of $g(x)=(f(x))(x)$?

Let $E,F$ Banach spaces, $U$ an open set, $0\in U$. Let $f:U\to L(E,F)$ of class $C^1$ in $U$, and let $g(x)=(f(x))(x)$, for all $x\in U$. Problem have two parts: a) Show $g$ is of class $C^1$ ...
0
votes
0answers
29 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
0
votes
1answer
39 views

How to find the roots of a 2 variable polynomial of 2nd degree?

The following polynomial is just an example: $$(3-3y)(x^2-y)$$ and is what does it mean to find the critical points of this polynomial? These are the maxima minima. Are they always concerned with ...
0
votes
1answer
11 views

Convergence of the real part of the integral associated with a characteristic function

Let $I_n (t) = \int (e^{itu} -1)\frac{1+ u^2}{u^2}\, dG_n(u)$ be such that $$ I_n(t) \to \log f(t) $$ where $f(t)$ is the characteristic function of an infinitely divisible law. Why is it that $$ ...
4
votes
1answer
46 views

Compute the derivatives of $\frac{d^{2\ell}}{dx^{2\ell}}\tanh(x)^{2k}$ in $x=0$

I would like to compute the derivatives $\frac{d^{2\ell}}{dx^{2\ell}}_{\vert x=0}\tanh(x)^{2k}$ at $x=0$ where $k,\ell\in \mathbb{N}$ positive integers with $\ell\geq k$. I am not sure how to attack ...
3
votes
1answer
44 views

Locally constant property

Suppose f is positive and Schwartz function. Fix $N>0$ and $A>0$. Suppose that for any $x \in [-N,N]$, $$A \leq \int_{-N}^{N}f(x-z)dz$$ Then do the inequality $$A \leq C_{r} ...
1
vote
2answers
22 views

Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function is surjective when it's codomain is restricted to it's image ...
1
vote
4answers
60 views

Is this a Norm?

How is the following formula calculated, assuming $a$ and $b$ are $n$-dimensional vectors? $\parallel \overrightarrow{a} - \overrightarrow{b}\parallel^2$
1
vote
0answers
21 views

Curvature shortening flow of embedded curves

QUESTION: I'm not sure how they proved part c in particular. Note that theorem 2.1 refers to Huiskan's distance comparison principle for evolving curves. I don't see why a separating boundary curve ...
34
votes
3answers
2k views

Addition is to Integration as Multiplication is to ______

Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors" ...
0
votes
0answers
39 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F ...
2
votes
2answers
88 views

mapping is bijective if $\lambda=? $

We consider the mapping $f$ from $\mathbb{C}$ in itself defined by~: $$ f:z\longmapsto Z=f(z)=\dfrac{z+i\lambda \bar{z}}{1+i\lambda}$$ with $\bar{z}$ is the conjugates of $z$ and $\lambda > ...
-1
votes
0answers
21 views

uniform convergence on compact spaces and differentiation

Prove or disprove: Let $\{f_n\} $ be a complex analytic function defined on an open set $U $ in $\mathbb {C} $. Suppose $f_n\to f$ uniformly on every compact subset of $U $. Then, $f'_n\to f' $ ...
0
votes
1answer
66 views

Sum of primes at minimal $\gt t!$

$$2+3+5+17+97+599\cdots a_t \gt t!$$ What does that mean? Well it is a sum that follows specific rules. For one, the number of terms in the sequence is $t$. Similarly, $a_t$ represents the $t$'th ...
2
votes
0answers
35 views

How to speak on limit of sequence categorically? [duplicate]

I was thinking on ways to define limit of a sequence (over the reals, or over a metric space, or even better, over a general topological space) using the categorical limit (final or inicial object of ...
5
votes
4answers
354 views

Find the roots of the summed polynomial

Find the roots of: $$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$ I got that: $$\frac{1 - x^8}{1-x} - x^6 - x = 0$$ But that doesnt make it any easier.
1
vote
1answer
22 views

Proof Verification Regarding Uniform Continuity

Assume that $g$ is defined on an open interval $(a, c)$ and it is known to be uniformly continuous on $(a, b]$ and $[b, c)$, where $a < b < c$. Prove that g is uniformly continuous on $(a, c)$. ...
2
votes
2answers
124 views

Can a vector field be conservative if its domain is not a star domain?

Can a vector field be conservative if its domain is not a star domain? I was trying to figure out whether the vector field $$\vec{f}(\vec{x}):=\frac{1}{\lvert \lvert \vec{x} \rvert \rvert} ...
0
votes
1answer
34 views

Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of ...
2
votes
1answer
23 views

How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
0
votes
2answers
75 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
5
votes
2answers
149 views

Chess rating calculating algorithm

In competitive chess tournaments, there is a complex rating system that evaluates your rating based on how you do well you do playing games. I am referring to the FIDE system not USCF. Are there FIDE ...
3
votes
1answer
48 views

Frechet derivative of squared norm $\|x\|^2$

I got this from Analysis II, H. Amann, J. Escher, p. 152. Their definition of the derivative of a map $f$ between Banach spaces $E,F$ over the field $\mathbb{K}$ is a bounded linear operator ...
0
votes
0answers
23 views

New variables of function.

Let be $z=z(x,y)$ function of the independent variables $x$ and $y$. Given to the new variables $(u,v)$ subject $$x = u + \log v$$ $$y = v - \log u.$$ Express $\frac{\partial z}{\partial x}, ...
1
vote
1answer
20 views

Support of convolution

Assume $u \in L^1(\mathbb{R}^n)$ and $\mathrm{ess\,supp}(u) \subset U,$ where $U$ is a bounded open set. Now we compute the convolution of $u$ with a function $\eta \in C(\mathbb{R}^n)$ with ...
0
votes
1answer
22 views

How to prove this identity $H=M-\dfrac{\sigma^2}{M}?$

Let $$M=\dfrac{\displaystyle \sum_{i=1}^{m}x_{i}}{m},H=\dfrac{m}{\displaystyle\sum_{i=1}^{m}\dfrac{1}{x_{i}}},\sigma^2=\dfrac{\displaystyle\sum_{i=1}^{m}(x_{i}-M)^2}{m}$$ show that ...
5
votes
2answers
140 views

Infinite sum of elements in a finite field

This is a bit of a curiosity that intrigues me. Let $p$ be a prime and consider the sum of reciprocals of squares divisible by $p$. This is just $$ \dfrac{1}{p^2}\sum_{n=1}^\infty \dfrac{1}{n^2} = ...