All Questions
739 questions
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 ...
- 1,835
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) ...
- 33
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 \...
- 60.5k
3
votes
2
answers
2k
views
Expected gradient vs. gradient of expectation
Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in ...
- 205
3
votes
0
answers
237
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
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
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 ...
3
votes
1
answer
80
views
prove spectral equivalence bounds for fractional power of matrices
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A ...
- 75
3
votes
1
answer
139
views
Lower bound for coercive polynomials
For a polynomial $f \in \mathbb{R}[x_1, \cdots, x_n]$, we say that $f$ is coercive (see my earlier question: Real polynomials that go to infinity in all directions: how fast do they grow?) if
$$\...
- 26.9k
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{...
- 159
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 $f\in C^1([0,1])$ ?
- 4,074
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 }[...
- 75
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, ...
3
votes
1
answer
84
views
Existence and uniqueness of an Euler-type ODE with varying parameters part 2
I am working on some non-local differential equations that appear in geometric analysis.
One of which I posted here and was answered by @WillieWong and @losifPinelis.
Consider this non-local ...
- 969
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-...
- 6,741
3
votes
3
answers
1k
views
a "reverse Hadamard inequality"
Is there an inequality of the form $|\det(A)| \geq F(v_1, \ldots, v_n)$ for a real $n\times n$-matrix $A$ with columns $v_i$, $F \geq 0$?
user19475
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 ...
- 7,127
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 ...
- 29.1k
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 ...
- 23k
3
votes
0
answers
106
views
The behavior of an integral related to the inward normal vector near a point of the boundary of a domain
Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit
$$
\lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz)
$$
where
$...
- 6,367
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 ...
- 151
3
votes
1
answer
158
views
Explicit eigenvalues of matrix?
Consider the matrix-valued operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
I am wondering if one can explicitly compute the eigenfunctions of that object on ...
- 536
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 $\sigma>0$ ...
- 393
3
votes
1
answer
248
views
Log-convexity of conditional variances
Let $K$ be a positive integer and $C$ be any $K \times K$ non-singular matrix. For positive real numbers $q_1, \dots, q_K$, define
$$\Sigma(q_1, \dots, q_K) = CC' + diag(\frac{1}{q_1}, \dots, \frac{1}...
- 765
3
votes
1
answer
201
views
"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients
In a lecture I once attended, I remember the speaker using a result of the following nature:
$``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form
$$P(n) ...
- 739
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 ...
- 3,098
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 ...
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 ...
- 10.1k
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 ...
- 19.7k
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 ...
- 24.9k
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 ...
- 128k
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 ...
- 34.3k
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,\...
- 479
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. $...
- 839
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 ...
user127351
3
votes
3
answers
546
views
Determining Roots of a Polynomial with Interval Estimates of Coefficients
Let $f$ be a monic univariate polynomial with real coefficients:
$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$
The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as $B=(b_{n-1},...,...
- 103
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 $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
3
votes
2
answers
265
views
Can one realize this as an ergodic process?
Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.
We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$
In other words: For ...
user135480
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 ...
- 779
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$...
- 1,649
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 ...
- 1,903
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 ...
- 1,731
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 ...
- 10.5k
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 ...
- 1,835
3
votes
1
answer
735
views
A similar Cauchy-Schwarz inequality with linear-algebra
Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction.
Assume that $A$ and $B$ are contractions such that
$I-AA^*$ and $I-BB^*$ are positive-...
- 4,280
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 $...
- 19k
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 ...
- 13.2k
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 825
2
votes
2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 201
2
votes
2
answers
218
views
Convergence for a non-linear second order difference equation
In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation
$$
u_0,u_1>0, \qquad \forall n\in \mathbb N, \; u_{n+2}=a\ln(1+u_n)+b\ln(1+u_{n+1})
$$
with ...
- 1,503