Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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21
votes
0answers
210 views

How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...
20
votes
0answers
334 views

How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$

How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$ Edit. Let $\mu$ be the irrationality measure ...
18
votes
0answers
197 views

Can $ f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $f^{-1}(y) = \{a_y,b_y\} $ be continuous?

This is the problem we want to solve: Can $f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $ f^{-1}(y) = \{a_y,b_y\}, a_y \neq b_y $ be continuous? ...
18
votes
0answers
382 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
17
votes
0answers
430 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{...
15
votes
0answers
138 views

Function defined as a limit

Q. If $f(x)=\lim\limits_{n\to\infty}\dfrac{\log(2+x)-x^{2n}\sin(x)}{1+x^{2n}}$, then explain why the function does not vanish anywhere in the interval $[0,\pi/2]$, although $f(0)$ and $f(\pi/2)$ ...
14
votes
0answers
200 views

Is this function nowhere analytic?

One usually sees $f(x):=\exp\frac{-1}{x^2}$ as an example of a $C^\infty$ function that is not analytic, having one point of non-analyticity (the point $0$). The Fabius function is a canonical ...
13
votes
0answers
163 views

Uniform continuity, uniform convergence, and translation

Let $f:\mathbb R \to \mathbb R$ be a continuous function. Define $f_n:\mathbb R \to \mathbb R$ by $$ f_n(x) := f(x+1/n). $$ Suppose that $(f_n)_{n=1}^\infty$ converges uniformly to $f$. Does it follow ...
13
votes
0answers
165 views

Regularity of PDEs, general question (Example: Kolmogorov equation)

Let us consider a differential operator $L$ and the PDE $$Lu=0$$ with some boundary conditions (assume Neumann conditions). For example, Kolmogorov's equation $$(1) \quad Lu(t,x) = \partial_t u(t,x) + ...
13
votes
0answers
585 views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
12
votes
0answers
252 views

Finding an example of nonhomeomorphic closed connected sets

Question: Find two closed, connected subsets in $\mathbb{R}^2$, $A$ and $B$, such that $A$ is not homeomorphic to $B$, but there is a continuous bijection $f:A \rightarrow B$ and a continuous ...
12
votes
0answers
126 views

References about Iterating integration, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$

Are there any references that discuss Iterating integration in general, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$, conditions in which they converge, some special values, some special tricks to ...
12
votes
0answers
91 views

A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that $\...
12
votes
0answers
414 views

Evaluate $\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} dx$

I found two different approaches, both is giving the same answer. Fubini: $$ \begin{align} \int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\ &...
11
votes
0answers
91 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\...
11
votes
0answers
426 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
11
votes
0answers
656 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
10
votes
0answers
119 views

Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don't even converge, for any $c > 0$, so the series ...
10
votes
0answers
180 views

Limits of series, proof of the convergence of two sequences

I have two sequences $x_i$ and $y_i$ defined by their expressions : $$x_i-x_{i+1}=y_i-y_{i+1}=\sqrt{x_{i+1}y_{i+1}}$$ I have to prove that $xy(x-y)=0$. I tried this : I have $$x_i=x^{2^{i-1}}\prod_{j=...
10
votes
0answers
141 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really stuck ...
10
votes
0answers
244 views

Integer values of the Riemann Zeta function

When $s>1$ is real, the Riemann zeta function $\zeta(s)$ takes all finite positive value $> 1$. I am studying the values of $s$ for which $\zeta(s)$ is a positive integer. I have the following ...
10
votes
0answers
312 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
9
votes
0answers
197 views

Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$

Let $a$, $b$ and $c$ be non-negative numbers such that $(a+b+c)^2(a^2+b^2+c^2)=27$. Prove that: $$\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$$ A big problem here around $(a,b,c)=(1.6185...,...
9
votes
0answers
219 views

Brezis Exercise 3.27 Extension.

Let $E$ be a separable Banach space with norm $\|\cdot\|$. The dual norm on $E^*$ is also denoted $\|\cdot\|$. Let $(a_n) \subset B_E$ be a dense subset of $B_E$ with respect to the strong ...
9
votes
0answers
111 views

Floor function and convergence of the sequence

Sequence $\{a(n)\}$ of real numbers is such that $\forall\space\lambda\in(1,2)$ sequence $a(\lfloor{\lambda}^n\rfloor)$ has a finite limit. Does it follow that $\{a(n)\}$ is convergent?
9
votes
0answers
182 views

Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
9
votes
0answers
129 views

Norms and pointwise convergence

It is known that There is no norm $\|.\|$ on the space $E$ of continuous real-valued functions on an interval, say $[0,1]$ such that $f_n \to f$ for $\|.\|$ if and only if $f_n$ converges pointwise ...
9
votes
0answers
169 views

Finding an integer $n$ such that $\sin(n)$ is close to 1

Given some $\epsilon>0$, is there an efficient way to find an integer $n$ such that $$1-\sin(n)<\epsilon$$ We all know there is always one (and many), and so I can test all $n$ from $0$ until ...
9
votes
0answers
373 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
9
votes
0answers
432 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let $\mathcal{D}(\mathbb{R}^n)$...
9
votes
0answers
488 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
9
votes
0answers
219 views

Invariant functions on the space of finite sequences of reals

Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation. 1. For each $\mathbf s\in S$ we ...
8
votes
0answers
81 views

On the convergence of a more complex iterated radical

After exploring Ramanujan's famed $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}} $$ and $$4=\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}},$$ both of which can be expressed more generally by $$x+n+a=\sqrt{ax+(n+a)^2+x\...
8
votes
0answers
100 views

Are any tools or techniques available to solve the “placement of safety points” problem?

Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ (This as an asymmetric version of the Hausdorff distance.) Here's ...
8
votes
0answers
87 views

Which functions are irrelevant?

Call a bijection $f : \mathbb{N} \rightarrow \mathbb{N}$ irrelevant over $\mathbb{R}$ iff for all sequences $a : \mathbb{N} \rightarrow \mathbb{R}$, if $$\sum_{i=0}^\infty a_n$$ exists, call its value ...
8
votes
0answers
144 views

prove uniqueness from orthogonality relation

The problem: we have two functions $f(x), g(x)\in C^{1}[-\pi, \pi]$, and we know that \begin{align} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \cos\left(ky\right) \sin\left(k\left\lvert y-z\right\rvert\...
8
votes
0answers
180 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
8
votes
0answers
69 views

Evaluate the following integral involving $\sin \pi x$

Let $F: \Bbb{R} \to \Bbb{R}$ be defined by $$F(s)=\begin{cases}1, & \text{if }s\ge \dfrac12 \\[0.2cm]0, & \text{if }s< \dfrac12 \end{cases}$$ I need to evaluate $$\int^{1}_{0} F(\sin \pi ...
8
votes
0answers
387 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h \...
8
votes
0answers
118 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
7
votes
0answers
51 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}udy. $$ But for the mean-value property of the Heat equation, ...
7
votes
0answers
148 views

Sard's Theorem Using Integration?

How exactly do we clean up this heuristic proof of Sard's theorem, from Schwarz' `Differential Topology for Physicists' using the Jacobian $J(f)(x)$: "A heuristic justification for Sard's theorem ...
7
votes
0answers
91 views

Proving continuity and monotonicity of $t\mapsto t^x, t>0$ with minimal assumptions.

I'm trying to prove that The function $t\mapsto t^x,\, x\in \Bbb R,\, t>0$ is continuous and monotonic. Suppose $+, \cdot\,:\Bbb R^2\to \Bbb R$ (addition and multiplication) have already ...
7
votes
0answers
70 views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega \...
7
votes
0answers
175 views

How do i evaluate this sum $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2n!}$?

How do I evaluate this sum: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2n!}$$ Note: The series converges by the ratio test. I have tried to use this sum:$$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}= \...
7
votes
0answers
189 views

The study of the properties of the solution of Heat equation.

I am studying the following basic heat equation. (All notations follows Evans book) \begin{align} u_t -\Delta u =0 &\text{ in }\mathbb R^2\times(0,\infty)\\ u=u_0 &\text{ on }\mathbb R^2\times\...
7
votes
0answers
305 views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids using complex numbers in the proof. Existence of complex roots will ...
7
votes
0answers
106 views

Let $(X,\Sigma, \mu)$ be a measure space. If $f$ is integrable and $\int_E {f}\,{d\mu} = 0$ for all $E\in\Sigma$ then $f=0$ almost everywhere.

My attempt: I called: $A = \{ x\in X : f(x) \ne 0 \}$ $B_n = \{ x\in X: f(x) \gt \frac{1}{n} \}$ $C_n = \{ x\in X:f(x) \lt -\frac{1}{n} \}$ $B = \bigcup_{n=1}^{\infty}{B_n}$ $C = \bigcup_{n=1}^{\...
7
votes
0answers
247 views

An alternative proof of Cauchy's Mean Value Theorem

Let's focus on the following version of Cauchy's Mean Value Theorem: Cauchy's Mean Value Theorem: Let $f, g$ be functions defined on closed interval $[a, b]$ such that 1) Both $f, g$ are ...
7
votes
0answers
91 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...