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**7**

votes

**0**answers

281 views

### About the first decimal of $\sqrt {n!}$

Do we have :
$$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$
Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.

**0**

votes

**0**answers

68 views

### variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...

**1**

vote

**1**answer

194 views

### A calculus question related to the nonnegative definite functions

I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge ...

**4**

votes

**1**answer

446 views

### Elementary Proof of the Uniqueness of Smooth Structures on R

Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm ...

**-3**

votes

**1**answer

235 views

### Hilbert space vector representation for data in a metric space. Where am i wrong in this experiment?

Consider the function space $M$ such that all its elements are of bounded variation, square integrable and of unit norm. An equivalence class is defined over this set as, $f \sim g$ iff for some ...

**1**

vote

**1**answer

223 views

### Nepero game (by Yacov Perelman)

I have already posted this question time before on stackexchange, but didn't receive a definitive solution.
So this is the game: consider a positive integer number $n$ and divide it in a finite ...

**1**

vote

**0**answers

112 views

### Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?

Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ ...

**0**

votes

**1**answer

59 views

### Precompactness of a sequence of convex functions

Suppose we have a bounded convex open set $\Omega$ in $\mathbf{R}^n$,and a sequence of convex functions $P_n$ such that $||P_n||_{L^2(\Omega)}\leq C\forall n$.Is it possible to find a subsequence ...

**0**

votes

**0**answers

86 views

### Asymptotic Expansion of Double integral

Crosspost from math.stackexchange. Have a look at the great answers there, even though they do not quite answer the question completely.
Define
$$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} ...

**10**

votes

**1**answer

443 views

### Metric $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$

I'm wondering where the relative probabilistic distance was first studied:
$$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$
where $\overline A$ is the complement of $A$.
A web search ...

**-1**

votes

**1**answer

157 views

### What is the advantage of the knowledge of jumps for approximating a function with trigonometric polynomials?

Let $f:(a,b)\to\mathbb{R}$ be square integrable, bounded variation and piecewise continuous function. Let the points of jump be $\{x_i/a<x_i<b,i = 1,2,3,...n\}$. The goal is to approximate the ...

**0**

votes

**0**answers

41 views

### Uniform continuity in the first variable on compacts w.r.t the other variables for a continuous function

It is a notational doubt.
I have seen the following notation in a book for the continuous function $f$:
$f(.,.):\Bbb R^d \times \Bbb R^d \to \Bbb R$ is uniformly continuous in the first argument on ...

**3**

votes

**2**answers

349 views

### Looking for some function

Is there a continuous function $F: R\to R$ such that $F$ is a surjection but not an injection, $F(Q)\subset Q$ and the restriction $F: Q\to Q$ is an injection, but not a surjection. Here $Q$ denotes ...

**1**

vote

**0**answers

230 views

### On the remainder term in Taylor's formula [closed]

(1) What are the main differences, in terms of "usefulness" while solving problems (even at research level), among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula? Could ...

**12**

votes

**2**answers

1k views

### New research and re-discovering classic results in “basic” real analysis

Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, ...

**5**

votes

**0**answers

39 views

### Cusp point and straightness of a smooth curve.

I have a smooth curve of length $L$ with a single cusp point $P$ occuring at length $s = L_P$. Let the curve in arc length parametrization be $\alpha_t(s) \equiv (X_t(s),Y_t(s)) $. They are actually a ...

**1**

vote

**0**answers

55 views

### Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation
$$
g(t) = \int_0^t K_n(t,s)w_n(s) ds
$$
where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...

**8**

votes

**4**answers

720 views

### What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?

My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples.
Let me start the discussion with ...

**11**

votes

**1**answer

286 views

### Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
...

**3**

votes

**0**answers

427 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 ...

**2**

votes

**0**answers

72 views

### Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]:
Let $K$ be a real ...

**-2**

votes

**1**answer

169 views

### Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
...

**0**

votes

**0**answers

73 views

### Baire sets in locally compact Hausdorff spaces

I posed this on 14 Dec. at http://math.stackexchange.com/questions/1067751/baire-sets-in-locally-compact-hausdorff-spaces, but there has been no response:
(This is a follow-up to ...

**3**

votes

**0**answers

458 views

### Question on a proof by Solonnikov,Ladyzhenskaya,Ural'tseva

I have already asked this question on Mathematics SE, because I suppose that it is not research level. But I haven't got an answer, possibly here someone can answer.
Let $G(t,x)$ be the fundamental ...

**1**

vote

**0**answers

60 views

### Schoenberg correspondence on $L^p$

Schoenberg correspondence states that $\psi: \mathbb R^d\longrightarrow \mathbb C$ is conditionally positive definite and hermitian if and only if $e^{t\psi}$ is positive definite for each $t>0$. ...

**2**

votes

**1**answer

192 views

### Fourier transform of $sin(\frac{1}{x})$ for $x > 0 (x > 1)$

Please, give me the cue: does exist analytical representation of Fourier Transform of $sin(\frac{1}{x})$ for$ x>0$ (or $x>1$). Maybe exist an approximation of $FT(sin(\frac{1}{x}))$ by Bessel ...

**0**

votes

**0**answers

81 views

### Implicit function theorem on boundary points

I have the following examples:
(1) $xy-1=0$ with $x\ge 0$. By the implicit function theorem, we can solve when $x\in(0, \infty)$. Here on the boundary we have $y=\frac{1}{x}\rightarrow \infty$ as ...

**0**

votes

**1**answer

116 views

### Surjectivity of “nice maps” from local properties

What tools are available from real algebraic geometry, analysis and
topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$
from local properties and maybe function values?
...

**1**

vote

**2**answers

79 views

### Finding conditions to guarantee existence of solutions to IVP [closed]

Consider the following IVP:
$x'=f(t,x)$ and $x(0)=x_0$, where $x\in \mathbb{R}^n$ and $t\in \mathbb{R}$.
Suppose that for all $(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$.
In order for the ...

**2**

votes

**1**answer

287 views

### Banach space of discontinuous functions(Killing continuous functions)

Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
...

**0**

votes

**1**answer

113 views

### Prove a function, defined by integration of a harmonic function, is log-convex [closed]

Let $u$ be a harmonic function and we define
$$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$
The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function ...

**0**

votes

**0**answers

29 views

### continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following:
Does a continuous and concave function
\begin{eqnarray*}
f: N_{\mathbb{Q}} \to \mathbb{R}
...

**5**

votes

**0**answers

194 views

### A generalization of Jensen's Inequality

Jensen's inequality is well known as
$$E\big[f(X)\big]\le f\big(E[X]\big)$$
where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also ...

**-2**

votes

**1**answer

95 views

### Upper and lower limits [closed]

Find the following limits:
(1) $\limsup_{n\to\infty } \sin (n!) $
(2) $\liminf_{n\to\infty } \sin (n!) $
(3) $\limsup_{n\to\infty } \cos (n!) $
(4) $\liminf_{n\to\infty } \cos (n!) .$

**2**

votes

**1**answer

117 views

### Question regarding to approximate continuity

Given $u\in BV(R^N)$, we say $u$ is approximate continuous at $x$ and the approximation limit is $l\in R$ if
$$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$
for all ...

**4**

votes

**0**answers

64 views

### A monoid-structure on pairs of interlacing polynomials

Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace ...

**0**

votes

**1**answer

141 views

### The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads.
Let's rewrite the product in a way ...

**1**

vote

**0**answers

161 views

### Separability of the space $C(C[0, 1], \mathbb{R})$

Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...

**7**

votes

**2**answers

568 views

### A generalized mean-value theorem

I'm pretty sure that if the function $f$ is continuous on $[x_1,x_3]$ and twice-differentiable on $(x_1,x_3)$, with $x_1 < x_2 < x_3$, then there must exist $x$ in $(x_1,x_3)$ for which $f''(x)$ ...

**22**

votes

**1**answer

387 views

### Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...

**9**

votes

**4**answers

320 views

### Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...

**2**

votes

**0**answers

112 views

### Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point.
I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...

**3**

votes

**1**answer

269 views

### Does this function have any exponential growth?

Has anyone seen any function of the following type?
$$
g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant ...

**0**

votes

**0**answers

87 views

### Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$
$$
...

**0**

votes

**0**answers

92 views

### Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...

**1**

vote

**1**answer

126 views

### Absolutely continuous functions

it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality
$$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$
for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...

**0**

votes

**1**answer

219 views

### Continuity of a Functional

A certain functional $T$ is defined as:
$$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
The result that above functional is ...

**0**

votes

**0**answers

90 views

### Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?

By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the ...

**-2**

votes

**2**answers

56 views

### Systems of ODEs that fulfill a matrix relationship at steady state [closed]

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...

**2**

votes

**1**answer

87 views

### Do the sequences with divergent associated $\zeta$-function form a vector space?

Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication.
...