Frequent Questions
18,060 questions
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Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 131
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93
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On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ...
- 5,979
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213
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For which $f \in \mathbb{C}[x,y][T]$, all irreducible elements of $\mathbb{C}[x,y]$ remain irreducible in $\mathbb{C}[x,y,T]/(f)$
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$.
Denote: $A=\mathbb{C}[x,y]$ and $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}[x,y][...
- 2,837
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101
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Does the tensor product of mollifiers work for $L^{p,q}$ spaces?
Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively.
For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \...
- 3,477
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294
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Exponential Convexity
$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all ...
- 153
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128
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How small the radical of $xyz(x+y+z)$ can be infinitely?
This is an open problem.
Let $x,y,z$ be coprime integers (not necessarily pairwise coprime)
and no proper subset sum of $\{x,y,z,-(x+y+z)\}$ is zero.
For a quadruple $(x,y,z,-(x+y+z))$ define the ...
- 25.4k
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votes
1
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96
views
What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?
In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result:
I don't understand the significance of this result. If I don't misinterpret the assertion, ...
- 167
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93
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Continuity of generalised Legendre transform
Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider ...
- 463
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236
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Is this a contraction mapping for small $T$?
Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$
$$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$
For $T>0$, let $\mathcal H_T:=\{h:[0,T]\...
- 1,334
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195
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Dirac's theorem and the 1-factorization conjecture
Let $G=(V,E)$ be a simple, undirected graph. A matching is a subset $M\subseteq E$ such that all members of $M$ are pairwise disjoint; moreover we call $M$ perfect if $\bigcup M = V$.
The 1-...
- 48.9k
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113
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On question on quadratic forms in four variables
Let $F$ be a non-singular quadratic form in four variables and let $w: \mathbb{R}^4 \to \mathbb{R}$ be a non-negative compactly supported function satisfying certain suitable conditions. Set $$N(F,w,m)...
- 429
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565
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directional derivative along geodesic flow of vector field
A rather elementary question for the differential geometers. Let $M$ be a Riemannian manifold, let $X\colon M \to TM$ be a vector field, and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" of the ...
- 11
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69
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Is the right-hand term of the dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x,t),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z,t),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, ...
- 109
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250
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Proof of the Davies-Gaffney estimate in elliptic pdes?
I'd like a reference to a proof of the Davies-Gaffney estimate; which is an off-diagonal decay result. See for instance assumption H2 in the paper "Hardy Spaces associated to non-negative self-adjoint ...
- 391
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112
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Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
- 1,334
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147
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Additivity of genus of a graph [duplicate]
Let $G$ be a finite simple undirected graph. Suppose there exist subgraph $G_1,G_2,\dots,G_n$ of $G$, such that $E(G_i)\cap E(G_j) = \emptyset$ and $|V(G_i)\cap V(G_j)| \leq 2$, for $i\neq j$. Then, ...
- 309
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148
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Is this reflection schema equivalent to second order Bernays reflection?
This posting comes as a possible salvage to this earlier presented reflective theory which was proved inconsistent. Attesting the particulars of the underlying language in reflection axioms.
Working ...
- 11.3k
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102
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Limit of minimizers of a class of functionals
Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ ...
- 1,219
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489
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Solving a parabolic PDE with boundary conditions given over ranges
How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given over ranges?
Here is an example: How to solve the equation $u_{xx}+u_{yy}-\alpha^{2}u_{t}...
- 183
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2
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278
views
Solvability of two-variable quadratic equations with a parameter
(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...
- 7,182
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326
views
Prime numbers property. A Merten's third theorem like sequence
Here is a question I have asked on Math Stack Exchange https://math.stackexchange.com/questions/2290917/prime-numbers-property-mertens-theorem-related-sequence , that I would like this community to ...
- 251
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2k
views
Are Chow groups a birational invariant?
Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a ...
- 179
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341
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Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
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1
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127
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asymptotic of ratio between two summations (l1 / l2 norm)
Let $B$ as a $n\times n$ matrix where
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\theta_j), 1\leq i<j\leq n$$ and other entries equals to $0$, and $$\theta=[\theta_1,\cdots,\theta_n]\...
- 405
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182
views
Expansion of continued fraction using recursion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where
$$
G(j)=1-\frac{f(j)x}{G(j+1)}
$$
Here we have
$$
G(...
- 4,938
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258
views
How to compare convergence of two equations for the arctangent function?
I want to compare two equations for the arctangent function:
$$\tag{1}\arctan\left(x\right)=\sum\limits_{n=0}^{\infty}{\frac{{2^{2n}}{\left(n! \right)}^{2}}{\left(2n+1\right)!}\frac{x^{2n+1}}{{\left(1+...
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175
views
Asymptotic of ratio between l1 / l2 norm of a structured vector
As suggested in this discussion, I would like to inquire about the following question:
Consider a matrix B of size $n\times n$ defined as:
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\...
- 405
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votes
2
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230
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Characterization of the generator of a Lévy process using martingale problems
Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
- 167
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1
answer
475
views
uniqueness for Poisson equation in R^d with mildly regular data
I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
- 5,380
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14
answers
2k
views
Priming for the primes [closed]
I have to confess that most often my eyes begin to glaze over when someone starts discussing the prime numbers. However, my ears have perked up at times over the primes--maybe first when I learned of ...
-1
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1
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370
views
What's the probability of two independent events in time domain?
Suppose there are two independent events A and B. The probability that A or ...
- 29
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votes
1
answer
261
views
Is Proper Class Choice equivalent to Global Choice?
Working in "MK-Regularity-Limitation of Size + Replacement for sets", call it the Base theory, let's coin the following axiom:
Axiom of Super-Choice:$$\forall \ relation \ R \ \exists F \...
- 11.3k
-1
votes
1
answer
143
views
Does transforming a periodic function imply periodicity
Let $f(x,y)$ be a periodic function for every fixed $y = \beta$ with respect to $x$ in the domain $x\in \mathbb{R}$ and consider this transform of $f$:
\begin{equation}
f^\star(\alpha,\beta ) = \sum_{...
-1
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1
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315
views
Bounds for the number of points on projective hyperelliptic curves over finite fields
Let $C$ be projective hyperelliptic curve over finite field $K$.
What are bounds for the number of points $\#C(K)$?
The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are
not smooth ...
- 25.4k
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1
answer
280
views
A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
- 149
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1
answer
507
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loop homology product for oriented compact manifolds with boundary
This is my first steeps in string topology and please forgive the basic level of my questions: I reformulate my question
Chas and Sullivan define the loop homology product for closed (=compact with ...
- 189
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1
answer
218
views
Function on quadratic numbers
Let $\mathbb{N}$ denote the set of the positive integers. We consider the following function $f:\mathbb{N}\times \mathbb{N}\to \mathbb{Q}$: $$f(a,b)=\frac{a^2+b^2}{1+ab} \text{ for all } a,b\in\mathbb{...
- 48.9k
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answer
573
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is the existence of an inaccessible cardinal stronger than just CON(ZFC)? [closed]
is it even stronger than that ZFC has a transtitive model?
- 21
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395
views
Odd & even permutations and unit fractions
One more motivated by recent questions of Zhi-Wei Sun.
Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$.
Is it true that, for every $n \ge 8$, there is at least one even permutation $\...
- 4,589
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1
answer
179
views
Categories that admit all finite products but not all finite coproducts
What are examples for categories that admit all finite products but not all finite coproducts?
(See also this question: Categories that admit all products but not all coproducts .)
-1
votes
1
answer
163
views
Alternate property of H^2(T, Z) [closed]
Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
- 563
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1
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519
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Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
-1
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1
answer
342
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Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
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votes
3
answers
312
views
Binomial Coefficients sum [closed]
Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$
has a closed formula on $a$ and $b$ (and on what it is, in case it does)?
It is supposed that $b \le a$.
- 369
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votes
2
answers
225
views
Precise asymptotic of diophantine approximation
I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that
$$
\left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2}
$$
for infinitely many choices of $p$ and $q$...
- 459
-1
votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362
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votes
1
answer
402
views
Ubiquity beyond infinity, transitive closure and the recursion theorem?
I am considering a Principle of Ubiquity, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:
For $\alpha(y,z)$ a first order condition so ...
- 3,574
-1
votes
1
answer
108
views
Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?
In this paper, if we denote the $k$ largest eigenvalues by $\lambda_N,\lambda_{n-1},··· ,\lambda_{N-k+1}, $ then for Gaussian ensembles the joint distribution function of rescaled eigenvalues has the ...
- 288
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votes
2
answers
263
views
Selection problem in a collection of non-empty sets
Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties?
$a\in {\cal F} \implies |a|\geq 2$,
$...
- 48.9k
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votes
1
answer
204
views
Cauchy reduction formula with measure (a variation)
The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
- 141