The real-analysis tag has no wiki summary.

**0**

votes

**0**answers

29 views

### is $x_{n}\ll \overline{x}_{n}^{2}$?

This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and ...

**3**

votes

**0**answers

41 views

### Counting norms on an infinite dimensional vector space

It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...

**3**

votes

**1**answer

168 views

### Characterization of a subset of [0,1] $II$

My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...

**1**

vote

**1**answer

92 views

### Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...

**1**

vote

**1**answer

161 views

### Characterization of a subset of $[0,1]$

Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property:
For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...

**-8**

votes

**0**answers

108 views

### Proof that $\sqrt2$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{2}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$.
Here is how I ...

**0**

votes

**0**answers

28 views

### Period doubling bifurcations [on hold]

In the bifurcation diagram, is it true that if the function $f_r(x)$ at $x^*$ (where $x^*$ is a fixed point where a bifurcation occurs) can be locally written as a smooth function then ...

**0**

votes

**2**answers

105 views

### Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...

**-3**

votes

**0**answers

74 views

### Question about continuity of Holder functions [closed]

Let the definition of the oscillation about a point be given as
$
\mathrm{osc}[f] (x) : = \lim\limits_{\epsilon \rightarrow 0} \sup_{[x -\epsilon, x + \epsilon]} f - \inf_{[x -\epsilon, x + ...

**4**

votes

**0**answers

137 views

### Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...

**0**

votes

**0**answers

66 views

### Ask for a good reference for the calculus involving singular continuous measure [migrated]

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here.
Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...

**0**

votes

**1**answer

298 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**0**

votes

**1**answer

49 views

### A function with one partial derivative Hölder continuos is Hölder continuos?

I'm having trouble finding a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$
1. $(t,x)\mapsto \partial_x^2 u(t,x)$ is $C^{0,\alpha}$;
2. $(t,x)\mapsto \partial_x ...

**0**

votes

**1**answer

115 views

### Is the span of those vectors dense in $\ell_2$?

For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} ...

**0**

votes

**0**answers

122 views

### When is there a polynomial transformation? [closed]

First part: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is polynomial ( that ...

**15**

votes

**1**answer

529 views

### Solving a non linear equation

I've been trying to prove that the following equation has a unique solution in interval 0 < x < 1 :
$$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$
Where K is a number (integer, if it ...

**3**

votes

**1**answer

381 views

### the existence of a real polynomial satisfying the following property

It is easy to verify that
$$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2}
\quad \quad 0<t\leq1$$
I want to ask if there exist a real polynomial $h(t)$ such ...

**2**

votes

**2**answers

92 views

### Question on the number of equilibria

Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or ...

**2**

votes

**1**answer

58 views

### Monotonicity of Trapezoid Approximations

Here's a numerical analysis question which may not be very important, especially in practice, but has been bugging me.
Suppose $f$ is a continuous function on an interval $[a,b]$. Let $T_n(f)$ be ...

**4**

votes

**3**answers

339 views

### Inequality of arithmetic, geometric and harmonic means

Let $a_1,\dots,a_n$ be positive numbers, does the following inequality holds?
$$\frac{a_1+a_2+\cdots+a_n}{n}-\sqrt[n]{a_1a_2\cdots a_n}\geq\sqrt[n]{a_1a_2\cdots ...

**2**

votes

**1**answer

120 views

### Inequality for the tail of normal distribution function

Let $ Ф(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt $ be the cumulative distribution function of the standard normal distribution.
Numerical calculations suggest the following ...

**-1**

votes

**0**answers

49 views

### Convergence of an integral containing the Fourier transform of compact supported function

Let $u\in {{L}^{1}}\left( \mathbb{R} \right)$, ${{\left\| u \right\|}_{{{L}^{1}}\left( \mathbb{R} \right)}}=1$, $u\ge 0$, $\operatorname{supp}\left( u \right)\subset \left[ -S,S \right]$. Here ...

**0**

votes

**2**answers

104 views

### Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians?

This question originates an engineering application.
There is a certain process that is presumed to be a sequence of diffusions and is usually modelled as a sum of Gaussians:
$$\Sigma_n ...

**2**

votes

**1**answer

168 views

### Integration of gaussian times absolute value of cosine

Is there a way to compute/estimate the following integral?
$\int_0^\infty e^{-(x/c)^2}\left|\cos{x}\right|dx$
where $c$ is a real constant. I would like to know if it is of order $e^{-c^2/4}$ like the ...

**0**

votes

**1**answer

52 views

### Extending point-wise bound to uniform bound

Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$
$$ C(t):= \sup_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$
Does it follow that $f$ is bounded?
Note ...

**3**

votes

**1**answer

104 views

### Separability of $R_+\times\mathcal{C}(R_+)$

Let $\mathcal{C}(R_+)$ be the space of continuous functions $f$ defined on $[0,+\infty)$ with $f(0)=0$. Denote by $\Omega$ the product of $R_+$ and $\mathcal{C}(R_+)$. Now endow $\Omega$ with the ...

**4**

votes

**1**answer

101 views

### Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...

**4**

votes

**0**answers

76 views

### Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...

**6**

votes

**0**answers

111 views

### Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that
$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...

**-1**

votes

**2**answers

114 views

### Is real analytic function good enough (see problem)? [closed]

Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then ...

**0**

votes

**0**answers

125 views

### Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...

**0**

votes

**0**answers

99 views

### how to solve f(f(x))=x^2+x [duplicate]

Now I just know the equation f(f(x))=x^2+x, how can I find the f(x)?
I have already tired many times,but I found it is difficult to solve it by any way I knew.So please help me solve the problem,and ...

**3**

votes

**3**answers

344 views

### Non-zero smooth functions vanishing on a Cantor set

It is easy to give examples of continuous functions $f:[0,1]\to \mathbb R_+\cup\{0\}$ non-zero but vanishing on a Cantor set (ex: Can Cantor set be the zero set of a continuous function?). It is ...

**0**

votes

**0**answers

11 views

### Hello, everyone, I want to ask you a question about a proof in the Terence Tao's Real Analysis notes [migrated]

everyone. I am using Terence Tao's Real Analysis notes to self learn Analysis 1. There is one thing in the proof of Theorem 27 (Least Upper Bound Property) in “week 2 note” that I don’t understand ...

**3**

votes

**0**answers

127 views

### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition
We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...

**7**

votes

**0**answers

132 views

### If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...

**0**

votes

**1**answer

203 views

### Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...

**-2**

votes

**1**answer

412 views

### Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...

**25**

votes

**9**answers

1k views

### Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of ...

**13**

votes

**4**answers

536 views

### Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...

**0**

votes

**0**answers

28 views

### Integrating over the Intersection of Convex Regions

Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?
The ...

**1**

vote

**2**answers

199 views

### Set of distinct real numbers such that all combination of sums are distinct

Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers.
For a given $p \in \mathbb N$, consider further the set
$$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, ...

**3**

votes

**1**answer

138 views

### Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...

**0**

votes

**0**answers

62 views

### what is the best estimation for the following

Suppose a continuous $2\pi$-periodic function $f:R\rightarrow R$ satisfies
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \mathrm{const}\frac{\delta}{\Big(\log\frac{1}{\delta}\Big)^{\gamma}}, \,\,\, ...

**4**

votes

**2**answers

200 views

### What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
...

**1**

vote

**0**answers

73 views

### Estimating convolutions of powers

I would like an asymptotic estimate of
$$
\sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}}
$$
that does not involve any infinite summation. In order to lighten the notation, I ...

**6**

votes

**1**answer

164 views

### Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...

**0**

votes

**1**answer

57 views

### Continuous real function on germs

Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider ...

**4**

votes

**1**answer

117 views

### On a.e. approximate differentiability of certain continuous real functions

I have the following question:
If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on ...

**2**

votes

**2**answers

105 views

### Name of a generalized version of semi-continuity

I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.
We will say a ...