# Tagged Questions

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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Definition Let $E$ and $F$ be two Banach spaces together with a plain subset ...
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### A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
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### Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
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### Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$. We fix the value of one variable $q_i \in (0,1)$, then maximize ...
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### a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...
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### Theorem with an example [closed]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
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### Has anyone seen this series?

I come across the following infinite series. $$\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for t>0 and a>0}.$$ In particular, I am interested in the case where $a=1/4$. ...
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### Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
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### How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0)$$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence $$a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),$$ with $a(1,1)=1$ and ...
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### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$ ...
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### Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by ...
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### The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
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### Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points and a positive integer $n$, is there ...
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I am interested in nonnegative solutions of $-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $u=0$ on $\partial \Omega$. Or instead the equation $-\Delta u + ... 3answers 513 views ### Relating the roots of polynomials to the solution sets of certain functional equations Consider a functional equation of the following form: $$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ... 2answers 1k views ### Is a function with nowhere vanishing derivatives analytic? My question is the following: Let f\in C^\infty(a,b), such that f^{(n)}(x)\ne 0, for every n\in\mathbb N, and every x\in (a,b). Does that imply that f is real analytic? EDIT. According to a ... 2answers 942 views ### Generalization of Darboux's Theorem Darboux's Theorem. If f:[a,b]\to\mathbb R is differentiable and f'(a)<\xi<f'(b), then there exists a c\in (a,b), such that \,f'(c)=\xi. Does any of the following generalizations Let ... 1answer 223 views ### Alternative proof of Lojasiewicz inequality is there a "brute force proof" of the Lojasiewicz inequality? By "brute force" i mean without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e. standard ... 1answer 97 views ### Compactly supported smooth function with Laplace transform bounded on a cone My question is if it is possible to find a compactly supported smooth function \varphi:\mathbf{R}\to \mathbf{R} s.t. the following integration \int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt stays ... 1answer 66 views ### Expressions in “continued” monotone functions Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction Now take a look at this question: ... 1answer 282 views ### Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem? From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ... 1answer 202 views ### A special case of the Divergence theorem I am interested in the following statement: Let F be a vector field in \mathbb{R}^n that is C^1-smooth in a domain U, continuous up to the boundary \partial U, and vanishing on ... 0answers 78 views ### Representing quasianalytic functions in several variables For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ... 1answer 410 views ### Is the mapping f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n} surjective? Is the mapping$$ f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n} $$surjective? If not, what is its image? If yes, what can be said about ... 0answers 81 views ### Does the difference quotient of an absolut cont. funct. converge in L^1? Assume that \mu is a finite Radon measure on the real line and f is integrable wrt. \mu. Define F(x)=\int_{]\infty;t]}f(y)d\mu(y) Is the following statement true? The functions ... 2answers 198 views ### specific improper integral involving erf I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:$$ \int_{1}^{\infty} ... 2answers 293 views ### Is it possible to have the set$f^{-1}(\lbrace x \rbrace)$perfect for every$x$? There are examples of functions$f \colon [0,1] \longrightarrow [0,1]$such that for any$\alpha $,$f^{-1}(\lbrace \alpha \rbrace)$is uncountable. My favorite example is$$f(r) = \limsup_n ... 0answers 201 views ### decreasing rearrangements: why the asymmetry of measure-preserving maps? Ryff proved in 1970 that the decreasing rearrangement$f^*$of a, say, continuous function$f:[0,1]\to\mathbb{R}$admits a measure preserving map$\phi$such that$f=f^*\circ\phi$. In general it is ... 1answer 76 views ### An inequality involving multi-index I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For$x \in \mathbb{R}^{n}$and$\alpha = ...
Where to find a proof of theorem which says that: if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesque measure or on the set of second category with Baire propert ...