All Questions
5,909 questions
0
votes
2
answers
235
views
An inequality on length of two curves [closed]
I am looking for a proof, reference, comment of an inequality as follows:
If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that:
$f(a)=g(a)$ and $f(b)=g(b)$
$(...
- 30.1k
0
votes
1
answer
190
views
Is $f(x)$ is more curvature than $g(x)$ then length of $f(x)$ seem longer than length of $g(x)$?
In my obsevation:
If $f(x)$ and $g(x)$ be two continuous funcions, and they have derivative and second derivative are also continuous in interval $[a, b]$. If $f(x)$ is more curvature than $g(x)$ in ...
- 128k
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 60.6k
4
votes
1
answer
357
views
Lipschitz function admits Whitney stratification
I've been reading Topological Aspects of Nonsmooth Optimization by Vladimir Shikhman.
There I have found the following observation that:
Lipschitz functions $f : \mathbb{R}^n \to \mathbb{R}$ admit
...
- 28k
1
vote
0
answers
211
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
39
votes
8
answers
13k
views
Can Cantor set be the zero set of a continuous function?
More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth?
Some days ago I discovered that in this proof ...
- 4,729
8
votes
1
answer
486
views
An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...
- 1,991
5
votes
1
answer
171
views
Invariant subspace in infinite dimensions
Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
- 12.6k
1
vote
1
answer
737
views
$L^2$ function in Schwartz space?
Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...
- 5,169
5
votes
1
answer
332
views
Convergence of a sequence by iteration
Let $F:\mathbb R^d\to\mathbb R$ be a convex function. Assume that $F$ has a uniformly bounded gradient, $|\sup_{x\in\mathbb R^d}\nabla F(x)|<+\infty$. Define the sequence as follows: Take an ...
- 17.2k
7
votes
0
answers
266
views
When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?
Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
3
votes
1
answer
233
views
Bounds on the positive roots of a bivariate polynomial
It is well known that various real root isolation methods are based on computing, first, the bounds on the values of the positive real roots of a polynomial equation. For the univariate case such ...
- 701
4
votes
2
answers
126
views
Extension of a certain type of (very) smooth functions to a larger interval
Let $[a,b]\subset[-1,1]$ and $f\in C^{\infty}([a,b],\mathbb{R})$ a function which satisfies for all $n\in\mathbb{N}$ that $\sup_{x\in[a,b]}|f^{(n)}(x)|\leq n!$.
Does there exist a function $g\in C^{\...
- 5,343
1
vote
1
answer
870
views
Borel $\sigma$-algebra on the space of Hölder continuous functions
Let
$(M,d)$ be a separable metric space
$E$ be a $\mathbb R$-Banach space
$\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...
- 29.8k
5
votes
0
answers
83
views
A subadditive bijection on the positive reals
I posed some time ago this question on MSE, which I am proposing also here since we got no definitive answer.
Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)...
- 1,511
5
votes
3
answers
668
views
Continuity and sequential continuity of a linear functional
Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...
- 60.6k
4
votes
3
answers
211
views
Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$
I want to prove that function
\begin{equation}
f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt
\end{equation}
is quasi-concave. One approach is to obtain the closed form ...
- 128k
13
votes
1
answer
1k
views
How continuous can a bijection between line and plane be?
Is there a bijection $f$ from $[0, 1]$ to $[0, 1]^2$ such that the set of points of discontinuity of $f$ has measure zero? If not, could it be dense/comeager?
- 4,713
6
votes
1
answer
397
views
iterated limit sets of a countable subset of real numbers
Let $A\subset \mathbb{R}$ be a closed subset, and $A'$ be the sets of limit points. We know that if $A$ is a countable set, $A'$ is a proper subset of $A$. Is it possible to find a subset( closed and ...
- 221
13
votes
1
answer
638
views
A question on the sine function
The Fejer-Jackson-Gronwall inequality involving the sine function is as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
Here I ask the ...
- 43.7k
2
votes
1
answer
448
views
$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0 ?$ with $\alpha \ge 1$ and $n=1, 2,\cdots$
Could You give a poof, comment or reference for the inequality as follows:
$$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$
for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha \ge 1$
See ...
CommunityBot
- 1
4
votes
1
answer
388
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
CommunityBot
- 1
2
votes
0
answers
116
views
Growth estimates for polynomials with natural coefficients
Suppose $p(x)$ is a degree $m$ polynomial whose coefficients are natural numbers. Suppose further that we have $p(1)=n$, $p(2)\leq nm$ and $p(3)=n^2$, and assume that $m\leq \log n$. So $p$ only grows ...
- 21
1
vote
0
answers
156
views
Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
- 105
3
votes
0
answers
169
views
Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?
Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$?
Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
-6
votes
1
answer
141
views
Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]
Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
...
- 128k
3
votes
1
answer
1k
views
Reference request: interpolation of Hölder spaces
On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space ...
- 2,670
5
votes
1
answer
795
views
How to define transfinite derivatives of a function?
There are all manners of theories generalizing the notion of derivative. Amongst them is the fractional calculus, a rich theory which gives a sense to the derivation and integration of non-integer (i....
7
votes
1
answer
683
views
The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
- 188k
1
vote
0
answers
85
views
Conditional expectation with respect to a topological factor map
Let $\pi: (X,T) \to (Y,S)$ be a factor map of minimal topological dynamical systems, and let $U \subseteq X$ be open, non-empty.
Question: Does there exist a $T$-invariant, Borel probability measure $...
- 859
10
votes
1
answer
328
views
Asymptotic behavior of an integral depending on an integer
A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where
$$
f(n) := \...
- 105k
6
votes
2
answers
618
views
Solution of an equation with Jacobi theta function
I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework).
It has to do with the real solutions $x$ to the ...
- 2,784
10
votes
0
answers
845
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
CommunityBot
- 1
5
votes
1
answer
243
views
Integral involving Gamma function: density of Kendall-Ressel family of distributions
I came across the following function when reading the famous paper of Letac and Mora "Natural exponential families with cubic variance functions", i.e.,
$$f(x) = \frac{x^x e^{-x}}{\Gamma(x+2)}$$ for $...
- 984
10
votes
1
answer
496
views
Does this Osgood-like condition imply continuity?
Let us consider a bounded, Borel function $F\colon \mathbb R^d \to \mathbb R^d$. Assume it satisfies the following
Osgood-like condition:
$$\tag{O}
\boxed{\vert \langle F(x) - F(y), x-y \rangle\...
- 61.9k
3
votes
1
answer
285
views
Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$
Definitions: Let $X$ be a Polish space (separable completely metrizable topological space).
A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...
- 42k
1
vote
1
answer
155
views
Proving a sum to be sublinear in growth
Suppose that $\alpha \in (0,1)$. The goal is to prove that the following sum is of $o(T)$ (or, if possible, give a more accurate growth rate, e.g. $O(T^{1-\alpha})$ or something like that):
$$ \sum_{t=...
CommunityBot
- 1
2
votes
1
answer
246
views
Does this sequence contain a nonnegative number?
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
- 109k
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 5,169
4
votes
1
answer
184
views
Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...
- 4,361
0
votes
1
answer
166
views
Corner integrals of $\exp$
I feel that there is a good chance to apply certain integrals of
$\ \exp(-(\sum_{k=1}^n x_k))\ $ over corners (see below) to the analytic number theory. I have obtained two formulas to start with ...
- 17.2k
1
vote
1
answer
399
views
I want to disprove an equality involving a double integral
I want to show that the following equality does not hold:
\begin{equation}\label{at3}
\frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
CommunityBot
- 1
3
votes
1
answer
2k
views
How "compact" are sets of finite measure?
Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover.
Now consider the following situation:
Everything I say in the following is with respect to the ...
- 54.2k
7
votes
1
answer
407
views
A conjecture on writing a function as a sum of uncountably many points
Define the sum of the non-negative numbers $\{r_s \mid s \in S\}$ $S$ uncountable to be
$$\sup _{D \subseteq S} \sum _{d \in D} r_d$$
($D$ being finite), which exists if this supremum is finite.
...
- 128k
2
votes
2
answers
304
views
Continuous monotone real functions of several real variables
Let $O$ be an open bounded connected set in $R^n$ and K its boundary. Given a continuous real function $f$ defined on $K$, I would like to extend $f$ to a continuous real function $g$ (i.e. $g$ ...
CommunityBot
- 1
8
votes
1
answer
460
views
Real-rootedness of some polynomials
Denote the unsigned Stirling numbers of the first kind by $s(n,j)$.
Question. Is it true that the polynomials
$$P_n(x)=\sum_{j\geq0}s(n,j)\binom{x}j$$
have only real roots?
Note. Obviously, the ...
CommunityBot
- 1
1
vote
0
answers
136
views
Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x↦x^{-x}$
The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...
4
votes
1
answer
282
views
A weak fragment of analysis?
Question: Is there a decidable theory sufficient to formulate and prove (many) theorems of classical analysis?
What I have in mind is a theory with two kinds of objects, reals (which are introduced ...
- 6,891
11
votes
1
answer
704
views
Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$
We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions.
One can generalize the definition above by taking pointwise limit of ...
- 4,727