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3 votes
0 answers
89 views

A special type of differential equations

Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state. Here is a simplified ...
  • 992
0 votes
1 answer
88 views

Does an affine building associated to a group satisfy the axioms of building?

Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
  • 429
0 votes
0 answers
59 views

Matrix algebra as sub algebra of von Neumann algebra

Let $\mathcal M$ be a infinite dimensional von Neumann algebra which is not abelian. Suppose it is known that $\mathcal M$ does not contain type $I_n$ factors von Neumann subalgefbras for all $n\geq N....
2 votes
1 answer
269 views

Points on curves of genus 3

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find a point ...
  • 219
1 vote
0 answers
74 views

Proof that $\max_i(d_i^{(n)})/ \sqrt{n}$ tends to $0$ when $\sum_i (d_i^{(n)})^2 / n$ converges

Let $d_i^{(n)} \in \mathbb{N}$ be a sequence of integer valued sequences such that \begin{equation} \lim_{n\rightarrow\infty} \sum_{i=1}^n \frac{(d_i^{(n)})^2}{n} = C < \infty \end{equation} Is it ...
0 votes
0 answers
79 views

Truncated circle method and partitions

Let $p(n)$ be the unrestricted partition function. Then it is known that its generating function $F(z)$ is expressible as an infinite product $$ F(z)=1+\sum_{n\ge1}p(n)z^n=\prod_{k\ge1}(1-z^k)^{-1}. $$...
  • 444
1 vote
1 answer
68 views
+100

Popular algorithms (stopping rules) with output - a prefix of a permutation

What are some popular settings, when we look at the elements of a randomly generated permutation one by one, and we use certain stopping rule which, as a result gives us a prefix of the observed ...
  • 123
0 votes
0 answers
81 views

Hardy's inequality proof using Doob's inequalities

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
  • 135
-4 votes
0 answers
34 views

Given function in MathCAD [closed]

Sorry for my bad english and i have never used this site. I'm studying to be an air conditioning engineer, but I'm terrible at math. I watched dozens of videos on this Given function, but I still didn’...
13 votes
0 answers
275 views

Does the Cheeger constant satisfy a heat-type equation?

It was shown by Hamilton in the 1990s that the isoperimetric ratio $C_H$ on the $2$-sphere improves along the Ricci flow. A way to prove this is to use the fact that if $(M^2, g(t))$ is a solution of ...
6 votes
0 answers
121 views

3-term recurrence relation including integral or differential operator for polynomials

Sequences of polynomials with a 3-term recurrence relations are well known for orthogonal polynomials. Do recurrence relations using differential or integral operators also appear in some theories? I ...
6 votes
0 answers
63 views

m-point-homogeneous, but not (m+1)-point-homogeneous

It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large). In other words, if $A\subset Q$ ...
1 vote
1 answer
164 views

Two unknowns: one vector, one scalar, one equation

I would like to know if this equation is solvable for $a$ and $\alpha$: \begin{equation} \Sigma = \Gamma + a \left( \alpha 1^\top + 1\alpha^\top \right) +a^2 b \end{equation} $\Sigma$ & $\Gamma$ ...
1 vote
0 answers
124 views
+100

Proving a sign rule for $f_{2n}$

If $t_{1},...,t_{n}$ are real numbers, consider the set of indexed linear operators $T(t_{1}),...,T(t_{n})$ on a Hilbert space $\mathcal{H}$ and define its ordering by: $$\pi[T(t_{1})\cdots T(t_{n})] :...
15 votes
1 answer
331 views

Conjecture on sum over permutations of products of Catalan numbers

Context In a recent paper involving entanglement in linear optics, we came across some summations involving Catalan numbers and permutations. In particular, these sums arise when doing integration ...
1 vote
0 answers
67 views

Exists $G$-equivariant embedding with faithful representation of $G$?

Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...
  • 911
-5 votes
0 answers
104 views

Does every smooth manifold admit a smooth bijection with Euclidean space? [closed]

Consider $M$, a connected $m$ dimensional smooth manifold. Does there exist a smooth bijection $X: \mathbf{R}^m\to M$? $X^{-1}$ may not be smooth so $M$ doesn't have to be diffeomorphic to $\mathbf{R}^...
  • 343
1 vote
1 answer
92 views
+50

Special function: Pulse peak modified with a power term

PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
  • 657
0 votes
0 answers
37 views

Uniqueness of graph embeddings on surfaces for highly connected graphs

A theorem of Whitney says that if $i_1$ and $i_2$ are two embeddings of a 3-connected planar graph into $S^2$, then there exists a homeomorphism $h : S^2 \to S^2$ such that $h \circ i_1 = i_2$. Is ...
  • 607
2 votes
0 answers
77 views

Examples of the use of forcing to build up models of stronger theories?

I'm very new to the subject of forcing. I always got the impression that with forcing we begin with say a model $M$ of a theory $\sf T+I$ and produce another model $M[G]$ that is also a model of $\sf ...
1 vote
0 answers
34 views

under what conditions do the two normal conditional expectations coincide

Let $M$ ba a von Neumann algebra and $N$ be the von Neumann subalgebra of $M$. Suppose there are two normal states $\rho_1$ and $\rho_2$ on $M$. If $\sigma_t^{\rho_i}(N)=N, i=1,2, \forall t\in \Bbb R$,...
0 votes
0 answers
51 views

Ito-Tanaka formula for SPDEs

I found a Tanaka's formula proved by Briand, et al in (Stochastic Processes and their Application, 108 (2003) 109-129), that is, Let $\left\{K_t\right\}_{t \in[0, T]}$ and $\left\{H_t\right\}_{t \in[0,...
  • 1
4 votes
1 answer
155 views

Non-trivial examples of selective coideals of $\omega$

$\newcommand{\H}{\mathcal{H}}$ $\newcommand{\A}{\mathcal{A}}$ Recall that a coideal $\H$ over $\omega$ is selective if for every $\{A_n : n < \omega\} \subseteq \H$, where $i < j \implies A_i \...
2 votes
1 answer
81 views

Does this condition on $f$ imply essential boundedness on compacts?

Let $f: \mathbb R \to \mathbb R$ be a nonnegative measurable function, and $\{q_n\}$ some enumeration of the rational numbers. Suppose for every $0 < r < 1$ it holds that $$\sum_{n = 0}^\infty r^...
  • 1,674
4 votes
0 answers
138 views

Rational solutions to Catalan's equation

Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation $$ x^{a}-y^{b}=1. $$ for $a, b > 1$ and $x, y > 0$ is $x = 3,...
-4 votes
0 answers
52 views

Find the integer solution to when $15 \cdot 2^n + 1$ is a perfect square [closed]

I tried to solve it using the $\text{mod}$ but the only reasonable observation made by me was that $n≥3$ except when $n=0$ using ($\text{mod}$ $8$), which did not help, unfortunately. Then I tried to ...
1 vote
0 answers
38 views

Lax pair and cubic nonlinear Schrödinger equation

Motivation: I'm trying to understand the Section 4 in the this section 4 in paper" Low regularity conservation laws for integrable PDE by Killip-Visan-Zhang ö It reads as follows: many completely ...
1 vote
1 answer
131 views

Concrete sheaves

On the nLab, given a local $S$-topos $E$, a concrete sheaf is defined as an object that is separated with respect to the local isomorphisms (the morphisms that are inverted by the global sections ...
  • 161
1 vote
0 answers
42 views

Size of minimal generating set of a module generated by columns of a diagonal matrix with extra structure

Let $R$ be a commutative ring with unit. Let $A \in R^{k \times k}$ be a diagonal matrix such that $A_{11} | A_{22} | \dots | A_{rr}$ for some $r \leq k$ and are non-zero, while $A_{ii}=0$ for all $i &...
0 votes
0 answers
41 views

Complemented subalgebra in a free Lie ring

A Lie ring is a triple $(G,+, [\ ,\ ]),$ where $(G,+)$ is an abelian group and $ [\ ,\ ]$ is a bilinear map satisfying $[x,x]=0$ $[\ ,\ ]$ is bilinear $[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,...
  • 101
2 votes
0 answers
35 views

Topological rings with a final topology

Given a family of ring homomorphisms $ \phi_i : X \rightarrow Y_i $ where each $ Y_i $ is a topological ring and consider the initial topology on $ X $, i.e. the coarest topology such that each map is ...
1 vote
0 answers
38 views

Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
0 votes
0 answers
26 views

Use Green's theorem to find a line integral for a open curve [migrated]

An area $R$ in the $xy$-plane is bounded by two curves $ C_1 $ and $ C_2 $. The two curves form a closed curve $C$ where the positive direction of rotation is anti-clockwise. The two curves can be ...
1 vote
1 answer
40 views

Is it true that any semisimple Jordan algebra has the unit element?

I found this theorem in Minnesota notes of Koecher [Thm. 9 p.70]: Thm. Any semisimple Jordan algebra has a unit element. During the text Koecher do not state that the Jordan algebra has to be finite ...
  • 263
0 votes
0 answers
29 views

On boundedness of fractional integral with respect to the fractional Laplacian

Let $\alpha \in [0,1]$ and $p> 1$. Consider the smooth function $\varphi (x)= \sqrt{x^2+1}$, for all $x\in \mathbb{R}$, is it true that \begin{equation} \begin{split} \int_{\mathbb{T}^2}\varphi ^{p-...
  • 1
6 votes
0 answers
75 views
+50

Learning roadmap for admissible representations of $\widehat{\mathfrak{g}}$ (affine Lie algebras)

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbf{C}$. A priori one might expect the representation theory of the affine Lie algebra $\widehat{\mathfrak{g}}$ (the Lie ...
  • 4,214
2 votes
1 answer
153 views

Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)

Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f_+:\mathsf{D}^b(\mathcal{D}_X)\to \mathsf{...
  • 1,401
1 vote
0 answers
46 views

Dyson's lemma implies index is small (in proving Roth's theorem)

I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
9 votes
1 answer
515 views

The paper behind an olympiad problem

In IMO Shortlist 2013, there is a number theory problem: Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $...
5 votes
1 answer
155 views

What are the solutions in numbers of $xyz \mid x^n + y^n + z^n$, $x,y,z$ globally coprime

What are globally coprime integers $x,y,z\in \mathbb Z^*$ such that $xyz$ divide $x^n + y^n + z^n$? I have no other motivation for that problem but its inherent beauty and interest. Note that it can ...
  • 637
-3 votes
0 answers
21 views

Integration and Gaussian Elimination questions using an Expression [closed]

I am learning Integration and Gaussian Elimination and have reached this question which has left me completely stumped. I don't know where to begin and am looking for any advice on how to solve it. ...
  • 1
1 vote
1 answer
99 views

Proof of lower bound on variance

I'm reading through the paper Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube. However, I'm having a difficult time understanding the following proof: Lemma 2.1 ...
  • 13
11 votes
1 answer
667 views

Is $e^{{e^{\ \dots\ }}^n}$ ever an integer?

Let $n$ be a positive integer. It is clear that $e^n$ is not integer because $e$ is transcendental (not algebraic). Now for each positive integer $k$ let $F^k(n)$ denote the $k$-fold composition of $F(...
  • 2,764
1 vote
0 answers
94 views

Will the least class satisfying Scott set theory interpret AC and CH?

I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality, in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. ...
1 vote
0 answers
30 views

Automorphic representations on non-cyclic covering groups

The theory of theta functions can be interpreted as automorphic representations on metaplectic groups (2-fold covering groups of $\mathrm{Sp}_{2}$, or $\mathrm{GL}_2$), and there's also a notion of $n$...
  • 1,341
4 votes
1 answer
106 views

CW structure for $\mathrm{BSp}(n,\mathbb{C})$ and $\mathrm{BPSp}(n,\mathbb{C})$ in degrees $4i$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\USp{USp}\DeclareMathOperator\BSp{BSp}\DeclareMathOperator\BUSp{BUSp}\DeclareMathOperator\BPSp{BPSp}$Let $\USp(n,\mathbb{C})...
  • 315
0 votes
0 answers
107 views

Young tableaux — irreps correspondence for simple complex Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally introduced to study the irreducible representations of finite symmetric groups $S_n$ ...
2 votes
1 answer
52 views

Equivalence of Hilbert space norm associated to the harmonic oscillator and a sum of Sobolev and weighted $L^2$ norms

I have seen an equivalence claimed in a few places, but I do not know of a reference that actually proves it with details and it has been a while since I took graduate courses on all this. Apologies ...
0 votes
0 answers
55 views

Can an upper bound for $r_{0}(n)$ be reached from a duality principle about the distinct primes $n$ "defines"?

Under Goldbach's conjecture, denote by $r_{0}(n)$ the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$, so that $k_{0}(n)$ ...
0 votes
0 answers
233 views

How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...

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