All Questions
23,892 questions
1
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150
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What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,064
4
votes
1
answer
270
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Examples of discrete-space continuous-time dynamical systems
Something that I see occur repeatedly in my work is the need for formal notions of discrete-space continuous-time dynamics — these are generally realized as digital oscillators that are interact using ...
- 41
1
vote
1
answer
183
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Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
2
votes
0
answers
228
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Any rigorous construction of $\phi^4$ theories without the mass term in the Lagrangian? (revised)
There are various papers on rigorous construction of massive $\phi^4$ theories in $2$ or $3$ Euclidean dimensions.
In 2D, there are in fact more general results such as this one by Glimm, Jaffe and ...
- 3,477
2
votes
0
answers
90
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Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 93
4
votes
1
answer
195
views
Asymptotic spectrum of a complex Sturm-Liouville differential operator
Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by
$$
\mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x),
$$
with Neumann ...
- 333
4
votes
0
answers
132
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Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
11
votes
1
answer
883
views
Which paper is the "Taubes trick" from?
In symplectic geometry, the "Taubes trick" is an argument used to show that a moduli space $\mathcal{M}(J)$, depending on a parameter $J \in \mathcal{J}$, is cut out transversely for generic ...
- 243
0
votes
2
answers
222
views
Reference to get quickly to modern discrete probability theory
I've had some formal training in Analysis - Functional Analysis, Basic Operator Algebra - and I've started working on probability - specifically Combinatorial Statistical Mechanics and Spin-Glasses. ...
4
votes
2
answers
389
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Gaussian mixtures are dense in total variation?
Let $M_{TV}(\mathbb{R}^d)$ denote the set of probability measures on $\mathbb{R}^d$ with finite total variation norm which are absolutely continuous with respect to the Lebesgue measure.
By a Gaussian ...
- 5,405
8
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0
answers
244
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Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,462
2
votes
0
answers
76
views
Does a matrix ring over a ring satisfy the Koethe conjecture if the coefficient ring itself satisfies the Koethe conjecture?
I just want to know whether the following statement is true or false.
If $R$ is a ring satisfying the Koethe conjecture, then the matrix ring over $R$ also satisfies the Koethe conjecture.
Or is it ...
3
votes
0
answers
101
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Tuple rearrangement: a combinatoric problem emerging from the Hurwitz action on Coxeter groups
I am working on Artin Groups, so called Dual Artin groups and the conjecture that they are isomorphic. Tuples of $n$ group elements can be acted on by the braid group $B_n$ in a particular way called ...
- 31
4
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0
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128
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Errata for "Foliations and Geometric Structures" by Aurel Bejancu and Hani Reda Farran
I'm reading "Foliations and Geometric Structures" (2006) by Aurel Bejancu and Hani Reda Farran and have been looking for an errata sheet. Unfortunately Prof. Bejancu has passed away. I ...
1
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0
answers
104
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Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127
2
votes
1
answer
300
views
G-equivariant homotopy between G-spaces
I apologize for asking too many questions in a single post. I am not very conversant with equivariant homotopy theory. While discussing with some faculty I was told that certain fact is true. All ...
- 139
1
vote
1
answer
141
views
Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
- 69
9
votes
1
answer
304
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About the normal subgroups of Burnside groups
I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
- 91
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
1
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0
answers
71
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Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
- 109
11
votes
1
answer
500
views
Uncountable families of measurable sets with pairwise positive intersections
Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$.
Is there an ...
- 10.6k
3
votes
0
answers
122
views
Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,639
3
votes
0
answers
74
views
Reference for PDEs from system of SDEs
I'm working with a system of SDEs
\begin{align*}
dX_t &= b(X_t, t) + \sigma dB_t\\
dY_t &= c(X_t, Y_t, t) + \sigma dB_t.
\end{align*}
Here, the Brownian motion is the same.
I know that ...
5
votes
2
answers
217
views
Smooth toric variety which is a cube is a bott tower (reference request)
According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference.
Let $X$ be a toric variety. ...
- 156k
0
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0
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105
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Generalizing the property of linear independent set in infinite dimensional TVS
Given a infinite dimensional Hilbert space $H$, and a countable set of vectors $\{v_{i}\}_{i=1}^{\infty}$. I want to study the following property of $\{v_{i}\}_{i=1}^{\infty}$:
There exists sequences $...
- 523
2
votes
1
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104
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Looking for review of delay differential equations involving $f(x)$ and $f(x/k)$
A research problem unexpectedly leads me to a delay differential equation of the form
$$
f(x)=\alpha(f(x),f(x/2))\,f'(x)+\beta(f(x),f(x/2))\,f'(x/2)+\gamma(f(x),f(x/2))
$$
For special cases of $\alpha,...
- 3,065
3
votes
0
answers
140
views
Polynomial from degrees of Weyl group
Let $d_1, \dotsc d_n$ be the degrees (of fundamental invariants) of the Weyl group $W$ of a simple Lie group, (in the reflection representation; see table given on the Wikipedia page for their ...
9
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0
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163
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
3
votes
0
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267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
3
votes
1
answer
91
views
Conditional Expectation in Diffusion Process
Consider a $d$-dimensional diffusion process $\mathbf{X}=(\mathbf{X}_t)_{t\in [0,T]}=([X^1_t,...,X^d_t])_{t\in [0,T]}$ that is the unique strong solution of the following SDE:
$$\left\{\begin{matrix}
...
- 143
0
votes
0
answers
33
views
Non-positive definite solution for differential Riccati equation
Consider the matrix-valued differential Riccati equation (DRE):
$$
\dot P_t +PA+A^\top P+Q-(B^\top P+S)^\top (B^\top P+S)=0, \quad t\in [0,T];\quad P(T)=G,
$$
where all coefficients are continuous.
...
- 503
0
votes
0
answers
57
views
Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
1
vote
1
answer
100
views
Is Nelson-Symanzik positivity compatible with fermionic statistics?
Let $\{ S_n \}_{n =0}^\infty$ be a sequence of tempered distributions where $S_n \in \mathcal{S}'(\mathbb{R}^{nd})$ where $d \in \{2,3,4\}$ is fixed. Moreover, we put three additional conditions:
$...
- 3,477
1
vote
0
answers
38
views
About Carleson measures on the Hardy space on the bidisc
I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of.
...
8
votes
1
answer
634
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Availability of a copy the first volume of Segre's "Forme differenziali e loro integrali"
I am precisely referring to the following, first volume of the textbook/lecture notes/monograph written by Beniamino Segre in the fifties of the twentieth century (I own a copy of the second volume)
...
- 6,367
5
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112
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Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
2
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1
answer
124
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Choice of the eigenbasis for the Dirac operator on $S^d$
This question is a simplified version of my previous one. I think that adding a gauge potential complicates the problem too much.
Let us consider the Dirac operator $D$ on the $d$-sphere $S^d$ with ...
- 3,477
11
votes
3
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671
views
Merging single-sorted and multi-sorted theories
The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
- 63.1k
1
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1
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129
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Is every operator range a Baire space in the relative topology?
Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $...
- 483
5
votes
1
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202
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Independent stationary increment process but with finite propagation speed
Intuitively, standard Brownian motion has infinite propagation speed, as it has a non-zero probability of reaching any point in any arbitrarily short time. This is due to the fact that the probability ...
- 807
7
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1
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415
views
Is there a “Closure-of-Range Theorem” for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions:
$T(X)$ is $s$-closed; $T(X)$ is $...
- 60.5k
1
vote
0
answers
45
views
Existence of optimal entropic weights for empirical modeling
Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
- 111
4
votes
1
answer
162
views
Topology on $O_M$, the space of slowly increasing smooth functions?
A smooth function on $\mathbb{R}^n$ is called slowly increasing if each of its derivatives is polynomially bounded. It seems that the collection of such functions is denoted as $O_M$.
Obviously, $O_M$ ...
- 3,477
0
votes
0
answers
38
views
Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
0
votes
0
answers
68
views
Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.
If ...
1
vote
1
answer
91
views
Positive definite kernels on compact interval $[0,1]$
From How to prove that a kernel is positive definite? I learned that a function $f:[0,\infty)\to\mathbb{R}$ induces a positive definite kernel $K:\mathbb{R}^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ ...
- 216
10
votes
0
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287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
1
vote
0
answers
176
views
If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?
Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$.
Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
- 641
7
votes
0
answers
142
views
What is the forcing $\bf U$ from Bartoszyński-Judah?
In Set Theory - on the structure of the Real Line by Bartoszyński & Judah, a forcing notion $\bf U$ is mentioned on page 339, allegedly corresponding to $\rm{cof}(\cal N)$ as it has several ...
- 393