All Questions
739 questions
4
votes
1
answer
150
views
Quantitative analytic continuation estimate for functions small except on a small set
This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
- 324
4
votes
0
answers
151
views
Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$
Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define
$$
\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \},
$$
i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
- 1,245
4
votes
1
answer
642
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 697
4
votes
1
answer
134
views
On partial absolute continuity
$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
- 128k
4
votes
2
answers
345
views
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 queries of two kinds:
Given $x\in[0,1]$, return $g(x)$.
Given $y\in[0,1]$, return $g^{-1}(y)$.
Given ...
- 3,549
4
votes
1
answer
223
views
Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
- 43.2k
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,467
4
votes
2
answers
846
views
Generalized Jordan theorem and winding number
By the generalized Jordan theorem any continuous injective map
$S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It ...
- 1,875
4
votes
2
answers
977
views
Articles with examples of Darboux functions without fixed points
A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \...
- 2,222
4
votes
0
answers
219
views
Is every supersmooth function a local polynomial?
This question is a follow up question to this question that I recently asked.
A $C^{\infty}$ function $f:(c,d)\rightarrow\mathbb{R}$ shall be called a local polynomial if whenever $f:(c,d)\rightarrow\...
4
votes
1
answer
109
views
Dividing a spherical cap into three equal wedges
Background: Optimal ways to cut an orange.
In this problem, we have a spherical orange, and we do not wish to eat its central column which is modelled as a cylinder. Part of the procedure involves an ...
- 1,459
4
votes
1
answer
228
views
Haar-null union of dense subsets
Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...
- 63
4
votes
2
answers
446
views
About Euclidean distances
$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$.
Let $d$ denote the Euclidean distance in $\R^n$.
Do then ...
- 128k
4
votes
1
answer
222
views
A continuous bi-Lipschitz shrinking of a domain into a compact subset
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is:
(1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...
- 401
4
votes
2
answers
548
views
Convergence of a sequence
Let $x_0=1$ and
$$x_{k+1} = (1-a_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x_k}\right)$$
where $a_n$ is a known sequence satisfying that $a_k\in(0,1)$ for all $k$ and $a_k\to 0$ as $k\to\infty$. How to ...
- 311
4
votes
1
answer
153
views
Linear maps preserved by algebra automorphisms
Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field.
Let $s_A:A\to F$ be an $F$-linear map.
Now consider an arbitrary field extension $K/F$, and define $s_{A\...
- 1,766
4
votes
2
answers
4k
views
Pointwise convergence for continuous functions
Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set $f(x)=\...
- 16.2k
4
votes
1
answer
249
views
Does this functional admit an absolute minimizer?
This is a close relative of the following problem.
Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
- 6,213
4
votes
1
answer
369
views
Proving two inequalities involving the gamma and digamma functions
I'm having trouble proving the following inequality:
$$\forall p>1 \quad \forall m\geq 0 \quad \dfrac{m^2\Gamma(\dfrac{2m}{p})\Gamma(\dfrac{2m}{q})}{\Gamma(\dfrac{2m+2}{p})\Gamma(\dfrac{2m+2}{q})}\...
4
votes
1
answer
277
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
4
votes
1
answer
668
views
Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
- 825
4
votes
1
answer
251
views
Superadditivity of the lower density
Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) $\mu^\star(\...
- 1,511
4
votes
1
answer
202
views
Subalgebras of singular matrices (less naive version)
Is it true that, for any subalgebra $\cal S$
of the algebra of linear operators in a finite-dimensional vector space $V$ over a field,
$$
\bigcap_{A\in\cal S}\ker A=\{0\}\hbox{ and }
\bigcup_{A\in\cal ...
- 3,932
4
votes
1
answer
149
views
An algebraic inequality in three real variables
Is it true that
$$(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}
+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-...
- 128k
4
votes
1
answer
204
views
Is there a density theorem for Banach measure?
Fix a finitely additive measure $\mu$ on $\mathbb R^2$ that is consistent with the Lebesgue measure. Does Lebesgue's density theorem hold for such a $\mu$, i.e., is it true that for every $A$ we have $...
- 19k
4
votes
1
answer
321
views
Can planar set contain even many vertices of every unit equilateral triangle?
Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle?
I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
- 19k
4
votes
1
answer
143
views
Mean value of a function associated with continued fractions
Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?
- 3,443
4
votes
1
answer
1k
views
seeking proofs: infinite series inequalities
Question. Numerically, the following is convincing. However, is there a proof?
$$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4
<\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\...
- 43.2k
4
votes
1
answer
875
views
comparing norms of tensor product of two Hilbert spaces
Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...
- 41
4
votes
1
answer
377
views
Differential inequalities under which a flat function must be identically zero
Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $.
Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
- 356
4
votes
2
answers
394
views
Is this projection on the boundary of a convex Lipschitz?
Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, and $u\in\mathbb{R}^n\setminus\{0\}$ such that $u$ does not belong to the asymptotic cone of $C$ and is nowhere tangent to $\partial C$. ...
- 449
4
votes
0
answers
281
views
Dual space of ${\rm Lip}_0(\mathbb R^d)$
This question comes to me when I read this paper : https://arxiv.org/pdf/1702.06049.pdf
Let ${\rm Lip}_0(\mathbb R^d)$ be the space of Lipschitz functions $F$ on $\mathbb R^d$ with $F(0)=0$. Then is $...
user128095
4
votes
1
answer
92
views
Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary).
Does there exist a sequence of ...
- 6,741
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
4
votes
1
answer
379
views
A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
4
votes
1
answer
279
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
4
votes
2
answers
3k
views
Chain rule for fractional laplacian
Does anyone know a formula of chain rule for fractional laplacian?
say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...
- 41
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
"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
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
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
$$\...
- 26.9k
3
votes
0
answers
237
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
3
votes
0
answers
290
views
Does there exist a supersmooth non-polynomial function?
Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
3
votes
1
answer
201
views
"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients
In a lecture I once attended, I remember the speaker using a result of the following nature:
$``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form
$$P(n) ...
- 739
3
votes
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
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{...
- 1,101
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
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
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
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$...
- 1,649
3
votes
1
answer
415
views
Inverse of block matrix
Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$
Consider the block matrix
$$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$
...
- 192