All Questions
23,892 questions
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
CommunityBot
- 1
0
votes
0
answers
34
views
Locally compact groupoid with range map restricted to isotropy groupoid is open
Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋
a locally compact space is such that isotropy subgroups of H are isomorphic to each other.
Can this be an example of a ...
4
votes
0
answers
140
views
Condition under a function is uniquely identifiable by the supremum values
Let $f(x),g(x)$ be two real-valued functions on $\mathbb{R}$ and $h(x,y)$ be a real-valued function on the plane. We can assume continuity (maybe piecewise differentiability also) of these functions. ...
- 271
1
vote
1
answer
65
views
Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 6,367
0
votes
1
answer
151
views
Inequality $(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$
Conjecture: Let $a_1, a_2, \cdots , a_n>0$ and $y \ge x $ then
$$(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$$
Equality iff $x=y$
Is the conjecture right? Have you ever seen this ...
- 328
3
votes
1
answer
172
views
Unique continuation from the boundary for inhomogeneous elliptic pde
Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
CommunityBot
- 1
11
votes
1
answer
950
views
Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
- 660
3
votes
0
answers
90
views
Sobolev embedding on a compact manifold without boundary
I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996).
In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
- 311
0
votes
2
answers
180
views
Inversion formula for discrete sine and cosine transforms
$\newcommand{\wh}[1]{{\widehat{#1}}}
\newcommand{\R}{{\mathbb{R}}}
$I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that these ...
- 128k
7
votes
1
answer
707
views
How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
CommunityBot
- 1
2
votes
1
answer
127
views
Strong Ditkin sets in the Fourier algebra
What is the definition of a Ditkin set (resp. a strong Ditkin set) for the Fourier algebra $A(G)$ of a locally compact (not necessarily abelian) group $G$?
More specifically, let $E$
be a closed ...
- 25.8k
0
votes
0
answers
49
views
Kadec-Klee property of an equivalent norm on a Hilbert space
Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm:
$$
\Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r ...
- 85
2
votes
0
answers
104
views
A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 1,759
1
vote
1
answer
70
views
Classifying lisse conformal vertex algebras using singularities of associated varieties
For the sake of keeping terminologies consistent, let me say that a conformal vertex algebra is a vertex algebra (VA) with a specified conformal vector, and a vertex operator algebra (VOA) is a ...
- 45.7k
3
votes
1
answer
232
views
Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
2
votes
1
answer
486
views
Common eigenvector of commuting unbounded operators
Let $T$, $S$ be two self-adjoint linear operators on a Hilbert space $\mathcal{H}$ with pure point spectrum.
Then $T$ and $S$ commute if and only if they have a complete set of common eigenvectors.
...
CommunityBot
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3
votes
0
answers
270
views
Categorical General Relativity
What are some good references for GR from a categorical point of view?
This is essentially just a big-list reference request.
I'm aware that the subject exists and can do some basic sleuthing to find ...
- 10.1k
7
votes
4
answers
560
views
Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 128k
6
votes
0
answers
162
views
Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
- 8,098
0
votes
0
answers
84
views
Question on approximation of norms
Suppose that $E\in Int[L_{p},L_{q}]$ for some $1<p<q<\infty$ and $E$ is $w$-concave with $1<w<\infty$. It is well-known that for each $r\geq w$, we have $E=L_{r}\odot F_{r}$ for some ...
- 45
1
vote
0
answers
67
views
regularity convolution of a $L^2$ function with $W^{1,1}$ function [closed]
Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution
$$u*v$$
Is it true that $w*u \in W^{1,2}(\mathbb R)$?
What regularity can we put on $w$ for this to be true?
- 73
9
votes
2
answers
418
views
Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 6,367
8
votes
1
answer
328
views
Standard reference for double category theory
Is there a ‘standard reference’ for double category theory?
Ideally something along the lines of CWM for $1$-category theory or Johnson and Yau’s book for $2$-category theory; some reference that ...
- 10.7k
4
votes
0
answers
165
views
Szegő's inequality
I know Erdős-Lax's inequality and a couple of proofs. It states that:
If $P(z)=\sum_{v=0}^{n} a_{v} z^{v}$ is a complex polynomial of degree $n$ having no zeros in $|z|<1$, then
$$
\max _{|z|=1}\...
- 2,829
1
vote
0
answers
122
views
Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
- 111
9
votes
1
answer
647
views
Explicit construction of a (the?) dual symmetric space
I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type.
...
CommunityBot
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1
vote
0
answers
65
views
Upsampling parameters in the Takahashi-Alexander model
Let me start by begging your forebearance; this question might at first glance appear to belong more on a forum for economics, but I hope by the end to convince you that there is mathematical content ...
- 721
6
votes
1
answer
464
views
Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
CommunityBot
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6
votes
1
answer
349
views
Where is the original theorem shooting a club to kill a Mahlo cardinal?
I just want to make sure that I have the correct reference for the original theorem of shooting a club of singular cardinals to make a Mahlo cardinal become a non-Mahlo inaccessible cardinal. I can't ...
- 17.7k
-2
votes
1
answer
210
views
Reference request on dynamics and hyperbolic dynamics (hyperbolicity in absence of periodic orbits)
I would appreciate if you introduce me a reference (paper or book) who address the concept of hyperbolic dynamics but with emphasis on absence of periodic orbits. a possible ...
- 6,581
2
votes
0
answers
210
views
Inverse problems and chaos theory
In the classical theory of inverse problems we want to recover an unknown $u \in U$ from its noisy measurements $y \in L^2$, where $U$ is a Banach space. In particular, we study the following problem:
...
- 760
7
votes
1
answer
243
views
Isoperimetric inequality, but $L_p$ norm
I would like to consider the isoperimetric problem of $L_p$ norm:
Given a region in $\mathbb R^2$ such that the boundary is a curve $C(x,y)$, where $\int_{C}(|\mathrm dx|^p+|\mathrm dy|^p)^{1/p}$ is a ...
- 109k
1
vote
0
answers
108
views
Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
- 869
6
votes
0
answers
141
views
Historical background of finding the roots of cubic equations using continued fractions
I came across an algebra problem book written in 1899 for students of Dar al-Fonun ([dɒːɾolfʊˈnuːn], meaning, "polytechnic college",) the only modern educational institute in Iran at the ...
- 63.9k
12
votes
1
answer
402
views
Boundedness of sequences and cardinality
Let $X$ be a set of sequences of real numbers that converge to zero with the property that for any unbounded sequence of real numbers $(y_n)$, there is a sequence $(x_n)$ in $X$ for which the ...
- 18.6k
3
votes
1
answer
186
views
Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 998
3
votes
0
answers
53
views
Bounds on Besov norms for mollification of a bounded Lipschitz function
Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
- 63.9k
4
votes
3
answers
311
views
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
- 1,759
1
vote
0
answers
165
views
Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
2
votes
1
answer
959
views
Do kernels provide a basis for a RKHS?
Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, \...
CommunityBot
- 1
3
votes
0
answers
79
views
Reference request for unitary Shimura varieties
Let $K$ be an imaginary quadratic field and let $X$ be the Shimura variety associated to the unitary group $U(m,n)$ over $K$ (after a suitable choice of PEL datum).
Is there a reference that explains ...
- 1,949
3
votes
1
answer
158
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 651
3
votes
0
answers
92
views
Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.7k
7
votes
1
answer
757
views
compression of a Turing machine run sequence
consider a Turing machine with a set of states $s_n$ and alphabet symbols $a_n$. now consider a "run sequence" generated from a starting input in the following sense. the run sequence is defined as ...
CommunityBot
- 1
3
votes
0
answers
132
views
Uniqueness for a nonlinear kinetic PDE-system with heat transfer coupling in one dimension
I am currently trying to understand the following article "Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit" (2021) by Favre, ...
- 185
11
votes
3
answers
949
views
Some "axiom of choice" and "dependent choice" issues
I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with ...
- 623
7
votes
1
answer
281
views
Norm in the minimal tensor product of C*-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
- 2,830
5
votes
3
answers
550
views
Completeness of the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ in $L^2[0,1]$
In the book R. M. Young: An introduction to non-harmonic Fourier series, I came across the following problem (page 18):
Problem. Show that the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+...
- 12.9k
6
votes
1
answer
154
views
Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero
Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $...
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