This tag is used if a reference is needed in a paper or textbook on a specific result.

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1
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1answer
25 views

Differences of consecutive ordered fractional parts

Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...
3
votes
0answers
39 views

Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence: $$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$ and this ...
1
vote
0answers
27 views

Regarding a result of I.Vekua on integral equations of first kind

Suppose $k\in L^2([0,1]\times[0,1])$ is a symmetric kernel. Presumably the following is true(under the assumption that $k$ is weakly singular): Either $$\displaystyle\int_0^1f(y)k(x,y)dy=0$$or ...
3
votes
1answer
123 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
2
votes
0answers
68 views

Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...
7
votes
2answers
248 views

When is a sequence the sum of two Beatty sequences?

In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that $$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$ for every positive integer $n$? ...
1
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0answers
33 views

Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems. Are there any papers or books that ...
0
votes
0answers
13 views

reference help on regular singular points of differential equations

I'm looking for books on systems of holomorphic partial differential equations with regular singular points. I know the book 'Équations différentielles à points singuliers réguliers' by Deligne, but I ...
11
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0answers
425 views

“To operate the machine, it is not necessary to raise the bonnet.”

The quotation in the title is attributed to Frank Adams and appears in several places: In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not ...
1
vote
0answers
143 views

Can triangulated categories be “approximated by countable subcategories” (that are triangulated but not full!)?

For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain ...
1
vote
0answers
37 views

Inverse Laplace Transforms of Exponential Form

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
2
votes
2answers
144 views

Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...
3
votes
0answers
112 views

When there exists some “cone” of a morphism of (ind-representable) cohomological functors?

I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups. The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...
4
votes
1answer
173 views

continuity of the Boltzmann entropy in the Wasserstein metric

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ ...
1
vote
0answers
87 views

References for modular curves over finite fields [on hold]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
-3
votes
0answers
223 views

Mathematical theories of changes - except from calculus? [closed]

Unfortunately motion is regarded as displacement in geometry: By a motion or displacement in the general sense is not meant a change of position of a single point or any bounded figure, but a ...
1
vote
0answers
38 views

Are spherical harmonics uniformly bounded?

The spherical harmonics are given by $$Y^m_l(\phi,\theta):=N^m_l e^{im\theta}P^m_l(\cos \phi) $$ where $P^m_l$ are the associated Legendre Polynomials and $N^m_l$ is the normalisation. From ...
0
votes
1answer
130 views

Marcel Berger's “Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes.”

I would appreciate any reference that contains either a translation or proof of the main theorem in this paper. Thank you in advanced.
0
votes
0answers
86 views

Definition of Hardy spaces [closed]

Classically, Hardy spaces $H^p$on the disk are introduced as set of functions analytic on $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$, which has bounded $H^p$ norm: $$ \|f\|_{H^p} = \sup_{0\leq r ...
4
votes
2answers
268 views

Understanding how to construct Bruhat-Tits buildings for non-split groups by Galois descent

Is there any way to get on top of the procedure for constructing Bruhat-Tits buildings for non-split groups over a non-archimedean local field $k$, by Galois descent, other than reading both the ...
3
votes
1answer
173 views

Transcendence of $\Gamma(1/3), \Gamma(1/4)$

This is a re-post from MSE as I did not get even a single comment there. Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
0
votes
0answers
95 views

book for help on problems with noetherian rings

Can you please introduce to me a book which would help me to prove the two following problems? In a noetherian ring, every integrally closed ideal is unmixed. Let $R$ be a noetherian ring, $P$ a ...
2
votes
0answers
50 views

Stationary distribution for time-inhomogeneous Markov process

I have a two state, discrete time, time-inhomogeneous Markov process with transition matrix defined by $$T_i=\begin{pmatrix} 1-p_i\alpha & p_i\alpha \\ p_i\beta& 1-p_i\beta \end{pmatrix}$$ ...
1
vote
1answer
249 views

Van Kampen colimits

nLab uses the following definition of van Kampen colimits --- a colimit in a category $\mathbb{C}$ is called van Kampen iff it is preserved by the internal indexing functor $\mathbb{C}/(-) \colon ...
2
votes
1answer
63 views

Domain of fractional powers of operators

Let $A$ and $B$ be non-negative ($(A x, x) \geq 0$ for all $x \in \mathcal{D}(A)$, similarly for $B$) densely defined self-adjoint operators on a Hilbert space $H$. Then the spectral theorem defines ...
1
vote
1answer
111 views

Unitary irreps of the Poincare group in dimension <4

It is well-known that long ago, Wigner classified the unitary irreducible representations of the Poincare group in dimension 4. I am looking for a convenient reference describing all unitary ...
-5
votes
0answers
63 views

Is this a legitimately correct equation, once I assign some values to the letters? [closed]

Could this be a legitimate equation with the proper brackets or do I need to write this equation differently with perhaps different brackets? SDP(F + E + [(P²)(P)] + I = H³ I plan to assign values to ...
10
votes
0answers
231 views

Reference request: a combinatoric result [closed]

When I tried to construct a counterexample in my research, I encountered the following result, which should be true. Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = ...
6
votes
3answers
671 views

Ambidexterity and Quantization

Here the nlab says about Hopkins-Lurie's ambidexterity paper: "The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in ...
1
vote
1answer
151 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
1
vote
0answers
32 views

Parametric Sard-Smale theorem - when is the generic set open?

I am learning Morse/Floer Theory and in my work on my master's thesis I want to apply the parametric Sard-Smale theorem. I.e. I consider a Banach bundle $\mathcal{L} \to \mathcal{M}\times \mathcal{G}$ ...
3
votes
1answer
143 views

Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets

Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...
0
votes
1answer
170 views

Axiomatic explanation of why the volume of a parallelepiped is equal to the area of its base times its height [closed]

I asked this in MSE, it flashed and disappeared. Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope ...
5
votes
1answer
230 views

If a topological space has vanishing $n$th homology for every possible homology theory, does it have vanishing $n$th homotopy?

I don't have any strong preference as to whether or not the homology theories are required to be ordinary. Also, if this does not hold in general, does it hold for some nice category of spaces, like ...
5
votes
0answers
150 views

Does there exist a smooth version of Cohen's factorization theorem?

A famous factorization theorem of Cohen states that for any locally compact group $G$, $$L^1(G)=L^1(G)*L^1(G).$$ I want to know if analogous results exist for the class of smooth functions when $G$ is ...
4
votes
1answer
145 views

Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$

Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X ...
1
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0answers
51 views

Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
3
votes
1answer
78 views

Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...
1
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0answers
47 views

Convergence to equilibrium for time in-homogeneous diffusions

Consider the long time behavior for a time in-homogeneous diffusion such as $$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$ where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent ...
2
votes
0answers
111 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
8
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0answers
154 views

Simplices and cubes

Question: What is the first appearance in the literature of one of the following statements: The result of intersecting a simplex with a cell of the dual subdivision is a cube There ...
3
votes
1answer
231 views

Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.) Suppose we have an algebraic group $G$ ...
0
votes
0answers
60 views

The following ODE global existence theorem reference?

There is an ODE existence theorem of the form: Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function. Suppose that there is a constant $c$ such that if $y$ is a ...
6
votes
1answer
113 views

Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed: Suppose $P, ...
1
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0answers
132 views

Mathematical journals (maybe in the past) with regular competitions? [closed]

This question from SEM but no answer : In Yōko Ogawa's "The Housekeeper and the Professor," there is a character who regularly takes part in contests published in a fictitious mathematical journal ...
0
votes
1answer
91 views

Borel subsets of Polish groups

Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...
1
vote
0answers
35 views

Is there a polynomial time algorithm for Poly-trees (Oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
15
votes
1answer
937 views

The list of problems for Grothendieck's thesis

Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...
5
votes
1answer
179 views

Proof of a soft version of Moschovakis's lemma

The following fact, which I've heard being called "soft version of Moschovakis's lemma" (see top answer here) is the following: Under AD, if there is a surjection $\Bbb R\rightarrow\alpha$, then ...
1
vote
0answers
92 views

Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$

This question is related to my previous question (here). Let $P_\lambda$(q,t) be the Macdonald polynomials with partition $\lambda$. Let $\Lambda$ denote the ring of symmetric functions over the ...