All Questions
1,579 questions with no upvoted or accepted answers
3
votes
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74
views
Discreteness of the higher inductive-inductive Cauchy real numbers in real cohesive homotopy type theory
We work in cohesive homotopy type theory with propositional resizing, so that there is only one type of Dedekind real numbers $\mathbb{R}$ up to equivalence, and Mike Shulman's axiom $\mathbb{R}\flat$,...
- 2,624
3
votes
0
answers
315
views
When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
- 43.2k
3
votes
0
answers
99
views
Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
3
votes
0
answers
84
views
A weighted $W^{2,p}$ estimates
Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have
$$
\|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
- 379
3
votes
0
answers
205
views
Uniform limit of pointwise limits of continuous functions
Let $X$ be topological spaces, $Y$ a metric space and $(f_n)_{n\in\mathbb{N}}$ a sequence of functions, with $f_n:X\rightarrow Y$ pointwise limit of continuous functions for each $n\in\mathbb{N}$. ...
- 2,286
3
votes
0
answers
185
views
Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix
The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
- 11.2k
3
votes
0
answers
204
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The inversion formula for the square root of a positive function
Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
- 4,058
3
votes
0
answers
289
views
Functional inverse of $z=1+w+\cdots+w^{n-1}$
Migrated from the MSE.
I am interested in the functional inverse of
$$
z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1.
$$
This function is strictly increasing on $w\geq0$ and thus admits an inverse.
By ...
3
votes
0
answers
137
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Is there a characterization of tempered functions whose Fourier transforms are also tempered functions?
A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions.
This MSE question asked ...
- 315
3
votes
0
answers
120
views
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2)$ then there is a subsequence such that $u_n(t)\rightharpoonup u(t)$ almost everywhere?
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?
I'm not sure if this question is trivial or not,...
- 153
3
votes
0
answers
119
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Question on the model completeness of the real field expanded by restricted Pfaffian functions
Currently I'm reading "Model completeness results for expansions of the ordered field
of real numbers by restricted Pfaffian functions and the exponential function" by Wilkie, and I do not ...
- 123
3
votes
0
answers
187
views
Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 21.2k
3
votes
0
answers
106
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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
0
answers
40
views
Non-existence of local generators for Sobolev tangent subundles
Let $U\subset\mathbb R^n$ be a bounded open set, a rank $r\le n$ measurable tangent subbundle $\mathcal V$ on $U$ is a map to the Grassmannian $\mathcal V:U\to Gr(r,\mathbb R^n)$ which is only defined ...
- 938
3
votes
0
answers
96
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
- 969
3
votes
0
answers
142
views
What are the possible asymptotics of the measure of a parametrised semialgebraic set?
Consider a family of semialgebraic sets $S_t \subset \mathbb{R}^d$ ($t \in [0,1]$) of the form
$$ S_t = \{ x \in \mathbb{R}^d \ : \ p_1(x,t) \geq 0,\ p_2(x,t) \geq 0, \dots, p_m(x,t) \geq 0 \} $$
...
- 1,914
3
votes
0
answers
56
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On Sobolev's inequality for weakly conformal maps
Suppose $u\in W^{2,p}(B^2,\mathbb{R}^n)$, $1<p<2$, is weakly conformal, that is
$$|u_x|=|u_y|,\quad u_x\cdot u_y=0$$
for almost every $(x,y)\in B^2$. Here $B^2$ is the unit open ball in $\mathbb{...
- 31
3
votes
0
answers
161
views
Chebyshev Equioscillation Theorem in presence of extra conditions
Let $P_\ell$ be polynomials of degree $\ell$. For $f \in C[0,1]$, define the minimax error $E_\ell(f) = \min_{p \in P_\ell} \max_{x \in [0,1]} |f(x) - p(x)|$. We know that for the above scenario the ...
- 431
3
votes
0
answers
84
views
Compact Sobolev embedding with boundary conditions
Let $X$ be some metric measure space on which Sobolev spaces can be defined in a reasonable way. In many cases, $H^1(X)$ is compactly embedded in $L^2(X)$ (e.g., if $X=\Omega$ is a bounded open set of ...
- 3,388
3
votes
0
answers
216
views
integration by parts on a Lipschitz domain as $\epsilon\to 0$
For a fixed, bounded, smooth domain $\Omega\subset \mathbb R^d$ and any $u\in W^{1,1}(\Omega)$ with trace $u|_{\partial\Omega}=g\in L^1(\partial\Omega)$ one can prove that
$$
\lim\limits_{\epsilon\to ...
- 5,380
3
votes
0
answers
121
views
Schatten norm estimate of spatially truncated resolvent of Laplacian
Consider the operator $-\Delta$ on $L^2(\mathbb R^d)$. In my studies I stumbled upon operators of the form
$$1_{\Gamma_n} (-\Delta -z)^{-1} 1_{\Gamma_m},$$
where $1_{\Gamma_m}$ denotes multiplication ...
- 91
3
votes
0
answers
79
views
Some exercise on the regularity of a summability method
I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
- 31
3
votes
0
answers
150
views
Is there any injective mapping from smooth functions on closed interval to smooth functions on circle? Motivated by signal processing
One advantage of Discrete Cosine Transform (DCT) over Discrete Fourier Transform (DFT) is that DCT maps any "continuous" signal defined on interval to a continuous one defined on circle.
I ...
- 1,615
3
votes
0
answers
234
views
Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $
What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
- 923
3
votes
0
answers
138
views
The mystery of the jumps of functions with the prescribed jumps: Eisenstein series and hidden symmetries(?)
Say that a function $f(t)$ “changes only by jumps” if $f(t) + \text{const} = C ∑_k j_k θ(t-t_k)$ for a certain constant $C$. Here $θ(t)$ is the Heaviside
step function which has a jump 1 at $t=0$ (it ...
- 455
3
votes
0
answers
117
views
Geometry of elements with prescribed multiplicity eigenvalues
Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...
- 1,061
3
votes
0
answers
154
views
Representation of a power of a quadratic form
Let $A=A^t$ be a non-singular symmetric matrix. For any multi-index $\gamma=(\gamma_1,\dots,\gamma_n)$ of degree $\vert \gamma\vert=\gamma_1+\dots+\gamma_n=2d$, let $b_\gamma=\frac{\partial}{\partial ...
3
votes
0
answers
84
views
Smallest matrix subspace containing many rank 1 matrices satisfying genericity condition
Let $X= \{A_i \in \mathbb{C}^{n \times n} : i \in \mathbb{N}\}$ be a set of rank 1 matrices with the property that for any $S \subset \mathbb{N}$ of size $n$, $\sum_{i \in S} A_i$ is invertible. What ...
- 539
3
votes
0
answers
90
views
Real analytic recursion
Fix an analytic function $f:\mathbb{R}\to\mathbb{R}$. Assume $f(x)>x$ for all $x\in \mathbb{R}$.
Is there an analytic function $g:\mathbb{R}\to\mathbb{R}$ such that $g(x+1)=f(g(x))$?
- 141
3
votes
0
answers
322
views
Heat equation damps backward heat equation?
In a previous question on mathoverflow, I was wondering about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
- 536
3
votes
0
answers
238
views
How to denote a partial derivative?
This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation .
When there are two ...
- 6,891
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' ...
- 375
3
votes
0
answers
46
views
Partial hypoellipticity
The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\...
- 4,145
3
votes
0
answers
467
views
Opposite of the curl operator and Biot-Savart kernel
Note: I just realized that using $\omega$ and $w$ might not have been the smartest choice of notation -- Sorry about that.
Let $\renewcommand{\div}{\operatorname{\div}}Q_0, Q_1$ be two real numbers, $...
- 1,326
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
0
answers
117
views
Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?
Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries.
Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\...
- 6,741
3
votes
0
answers
65
views
Elliptic equations in semi-infinite strips
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
- 4,145
3
votes
0
answers
125
views
Green operator of elliptic differential operator and radius of convergence
Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
- 1,383
3
votes
0
answers
222
views
Sets of finite perimeter: intersection with an half space
I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...
- 51
3
votes
0
answers
117
views
Optimal Poincaré constants under combined boundary and average conditions
Let $\Omega=[0,1]^2$ be the unit square, $\Gamma_1=[0,1]\times\{0;1\}$ its horizontal boundary and $\Gamma_2= \{0;1\}\times[0,1]$ its vertical boundary.
I would like to know the optimal Poincaré ...
- 53
3
votes
0
answers
119
views
Second derivative estimates
I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates.
In one of his papers, Lin proves the following result:
Let's consider a ...
- 59
3
votes
0
answers
235
views
Singular integral operator
I am working on a problem involving the Biot-Savart law in fluid dynamics. I found a theorem of singular integral which is intimately related to my research.
Assume that $K(x)$ is a classical Calderon-...
- 151
3
votes
0
answers
242
views
Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
- 561
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
3
votes
0
answers
155
views
Bass theorem on non-affine scheme
A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free.
Now, consider a connected noetherian scheme $...
- 3,472
3
votes
0
answers
126
views
Distance between two algebraic sets
We are in $M_n(\mathbb{R})$ equipped with the Frobenius norm $||A||^2=tr(AA^T)$.
Let $Z=\{(A,B)\in M_n(\mathbb{R})^2;A^2-AB-B^2=0\}$ and $T=O(n)^2$. It is easy to see that $Z\cap T=\emptyset$ and ...
- 3,741
3
votes
0
answers
56
views
Regular sequence of binary forms in a subspace
Let $A_d=\mathbb{C}[x_1,\dots,x_n]_d$ denote the vector space of homogeneous polynomials of degree d. And assume $U\subset A_d$ is a subspace of codimension $d-1$ such that there is no point in $\...
- 51
3
votes
0
answers
87
views
Is $|f^{-1}f(p)|$ constant on a conull set?
Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is an analytic map. Can we say that there is a conull set $U \subset \mathbb{R}^n$ (i.e. $\mathbb{R}^n \setminus U$ has measure zero) where $|f^{-1}f(...
- 276
3
votes
0
answers
238
views
Dominated convergence Theorem
I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...
- 31
3
votes
0
answers
241
views
Lower bound on coefficients in hermite transform of Tanh
I would like to understand $L^2\left(\mathbb{R}, \mu\right)$ approximation by polynomials of $\tanh$ (and more generally smooth functions), where $\mu$ is standard gaussian distribution. This leads to ...
