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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 ...
Tom Copeland's user avatar
  • 10.5k
3 votes
0 answers
204 views

Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
Vincent Granville's user avatar
3 votes
1 answer
304 views

Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
CodeGolf's user avatar
  • 1,835
3 votes
1 answer
383 views

"Nice" functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point. Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...
Alexander Chervov's user avatar
3 votes
1 answer
142 views

How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?

Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$, $$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user avatar
3 votes
1 answer
251 views

Congruence modulo 2 for q-series

This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks. I would like to ask: QUESTION. Is this congruence true ...
T. Amdeberhan's user avatar
3 votes
0 answers
105 views

Recursive differences of Cantor set

Let $C$ be the Cantor ternary set obtained by repeatedly deleting the middle thirds of the interval $[0,1].$ Now define $$E_1=C$$ and $$E_{i+1}=\{ |x-y|,x \neq y\text{ and } \,x,y \in E_i \}$$ I ...
AgnostMystic's user avatar
3 votes
1 answer
274 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
Andrea Tauber's user avatar
3 votes
2 answers
309 views

Seeking proof to an asymptotics of a recursion or functional equation

My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion $$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...
Hans's user avatar
  • 2,239
3 votes
1 answer
557 views

Is there real or complex analytic function whose positive real zeros are the primes?

Related to this question Q1 Is there real or complex analytic function $f(x)$ such that its positive real zeros are the primes and it is given in closed form of compositions of already named ...
joro's user avatar
  • 25.4k
3 votes
1 answer
186 views

packing with special sets in high dimensional Euclidean space

Let $\lambda$ be Lebesgue measure on $[0,1]$. For $\mathbf{x}=(x_1,x_2,..,x_k)\in[0,1]^k$, define $$A(\mathbf{x}):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[...
Cuize Han's user avatar
3 votes
1 answer
220 views

Looking for non-polynomial functions: with the growth condition: $\phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)}$

I am for example(s) of an invertible Convex or concave function $\phi: [0,\infty)\to [0, \infty)$ such that $\phi(0)=0$ and there exists $\theta>0$ and for all $s\leq t$ we have \begin{align}\label{...
Guy Fsone's user avatar
  • 1,101
3 votes
1 answer
250 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 ...
CodeGolf's user avatar
  • 1,835
3 votes
1 answer
115 views

Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?

Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$. Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-...
Asaf Shachar's user avatar
  • 6,741
3 votes
2 answers
472 views

Regularity of lipschitz and derivable function

Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
Dattier's user avatar
  • 4,074
3 votes
1 answer
155 views

Smoothening a probability measure

Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define $$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\}, \ z\in{\mathbb ...
Seva's user avatar
  • 23k
3 votes
0 answers
166 views

Monotone version of one-dimensional Whitney extension theorem

Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...
Igor Belegradek's user avatar
3 votes
2 answers
487 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 ...
Manfred Weis's user avatar
  • 13.2k
3 votes
1 answer
496 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 ...
Anixx's user avatar
  • 10.1k
3 votes
1 answer
125 views

Relation between the local maxima and the local minima for approximating the generalized Laguerre polynomial

I have already asked my question in the link below: Minima approximation for Laguerre polynomials I have suggested to anyone to give me the approximations of the minima for the Laguerre polynomial, ...
Assinisa Hamidata's user avatar
3 votes
0 answers
290 views

Does there exist a supersmooth non-polynomial function?

Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
Joseph Van Name's user avatar
3 votes
1 answer
623 views

Forwards Feynman–Kac formula

This might be a simple question, but I'm having trouble with it. Consider the Cauchy problem with final condition. \begin{equation} \begin{cases} \frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
Paulo Rocha's user avatar
3 votes
0 answers
689 views

"Nicely" strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero". A set $X$ of reals is strong measure zero if, for any $f: \omega\...
Noah Schweber's user avatar
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 ...
Romain Gicquaud's user avatar
3 votes
3 answers
427 views

Quantitative analytic continuation estimate for a function small on a set of positive measure

The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here. In ...
Keefer Rowan's user avatar
3 votes
0 answers
1k views

On new (purely analytic) perspective towards theory of prime numbers

[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform. I myself am very skeptical about this but I want to know, from the experts' ...
bambi's user avatar
  • 375
3 votes
1 answer
201 views

"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients

In a lecture I once attended, I remember the speaker using a result of the following nature: $``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form $$P(n) ...
asrxiiviii's user avatar
3 votes
2 answers
447 views

Algebraic curve intersecting square-grid

Let us subdivide the unit square into square-grid cells with sidelength $w$. This will give us roughly $w^{-2}$ cells. Formally $$ g_{ij} = \{(wi, wj) + (x,y) : 0\leq x,y\leq w \},$$ for $i,j = 0,\...
Till's user avatar
  • 479
3 votes
3 answers
128 views

Detecting slow growth in a finite number of queries

The following question was asked at Can you solve this problem using a finite number of queries? : Let $g:[0,1]\to[0,1]$ be a continuous monotonically-increasing function. You can access $g$ using ...
Iosif Pinelis's user avatar
3 votes
0 answers
238 views

Move one element of finite set out from A in plane

Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
domotorp's user avatar
  • 18.9k
3 votes
1 answer
1k 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 $x\ge 0$...
Anand's user avatar
  • 1,649
3 votes
0 answers
232 views

When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?

In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder: How to characterize polynomials $f(x)$ with rational ...
Max Alekseyev's user avatar
3 votes
1 answer
296 views

Does this condition characterise intervals, among subsets of the real line?

For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$: $\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
Pietro Majer's user avatar
  • 60.5k
3 votes
1 answer
411 views

Continuation of a smooth function, whose every derivative is strictly monotonic

Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
H. Tomasz Grzybowski's user avatar
3 votes
1 answer
299 views

Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)

Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality \begin{align*} |f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n. \end{align*} For $n \ge 2$, can we ...
passerby51's user avatar
  • 1,731
3 votes
1 answer
496 views

"Square root" of multiplication operator on Sobolev space

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a non-negative, smooth, uniformly bounded function with uniformly bounded first derivative. Then $f$ defines a bounded operator on $L^2(\mathbb{R}^n)$ as ...
geometricK's user avatar
  • 1,903
3 votes
1 answer
193 views

Differentiability along hyperplanes

Definition. Let us say that a function $f\colon \mathbb R^d\to \mathbb R$ is differentiable along hyperplanes in the point $0\in \mathbb R^d$, if $f\circ \varphi\colon \mathbb R^{d-1}\to \mathbb R$ is ...
Jan Bohr's user avatar
  • 779
3 votes
0 answers
144 views

Noncrossing partitions in Hopf algebras/monoids via compositional inversion

Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
Tom Copeland's user avatar
  • 10.5k
3 votes
0 answers
306 views

Metric analogues of bounded variation

A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if $$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$ for some finite $V>0$, where the supremum is over all finite partitions $...
Aryeh Kontorovich's user avatar
3 votes
1 answer
442 views

Error of midpoint method for functions that are not twice-differentiable

All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...
James Propp's user avatar
  • 19.7k
3 votes
1 answer
139 views

Lower bound for coercive polynomials

For a polynomial $f \in \mathbb{R}[x_1, \cdots, x_n]$, we say that $f$ is coercive (see my earlier question: Real polynomials that go to infinity in all directions: how fast do they grow?) if $$\...
Stanley Yao Xiao's user avatar
3 votes
1 answer
231 views

Under which conditions the domain of the surjective function $f:[a,b]\times[c,d]\to[0,1]^{2}$ can be split s.t. the restrictions are bijective?

This is a follow-up question to this. Since it is not always possible to construct such partition, I would like to know if there are additional restrictions which we could impose so that the wanted ...
user avatar
3 votes
1 answer
322 views

Special version of Tonelli’s theorem

I am trying to prove this theorem. I have not found anything similar to it on the internet. Special version of Tonelli’s theorem Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
Mr. Proof's user avatar
  • 159
3 votes
1 answer
255 views

Is this constraint convex?

I have an optimization problem where the following constraint causes DCP Rule Error. $$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
Mojtaba's user avatar
  • 31
3 votes
1 answer
173 views

Weak Lebesgue spaces and an estimate for BV functions

Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that $$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$ holds for a.e. $...
Riku's user avatar
  • 839
3 votes
0 answers
154 views

Inequality involving convolution roots

I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is: increasing strictly convex on $(-\infty,0)$ strictly concave on $(0,+\infty)$ Let $\sigma>0$ ...
NancyBoy's user avatar
  • 393
3 votes
1 answer
248 views

"Lagrange inversion" around an extremum

Cross-posted from Math Stackexchange. In an older question to which I provided an answer it was asked how to compute a particular limit involving the roots of a transcedental function around its ...
K. Grammatikos's user avatar
3 votes
2 answers
203 views

Recovering a set from its projections in varying coordinate systems - a projection hull?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
M.G.'s user avatar
  • 7,127
3 votes
2 answers
293 views

On convergence of convex-concave functions

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that: $f_n$ is strictly convex on $(-\infty,x_n)$, $f_n$ is ...
Iosif Pinelis's user avatar
2 votes
0 answers
130 views

Smoothness of Radon transform

Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by $$ R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
dohmatob's user avatar
  • 6,853

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