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

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33 views

Question on viscosity solution through stochastic differential equations

I have learned that for the equation $\partial_tu+a(u)\partial_xu=0$, the entropy solution could be obtained as the limit of the equation $\partial_tu+a(u)\partial_xu=\epsilon u_{xx}$ with ...
2
votes
0answers
81 views

Comprehensive survey on mathematical modelling of neural networks: from the basic ideas to contemporary research topics

I am looking for a comprehensive survey (paper(s) or book(s)) on mathematical modelling of neural networks (both artificial and biological). It should start from the very basic concepts of modelling ...
12
votes
3answers
487 views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
0
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0answers
36 views

System of integral equations

Let $K_1,K_2,K_3,K_4$ be integral operators. I'm interested in the following system of integral equations. $$\begin{cases} g_1 = K_1f_1 + K_2f_2 \\ g_2 = K_3f_1 + K_4f_2 \end{cases}$$ I'm ...
4
votes
1answer
122 views

Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, ...
2
votes
1answer
129 views

Lifting one parameter subgroups of algebraic groups

Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one ...
3
votes
0answers
107 views

Growth of the number of generators in hyperbolic groups

Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index. I would like to know if one ...
0
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2answers
104 views

Comparing the Rational Approximability of Infinite Continued Fractions

It is known, that $\phi := \frac{sqrt(5)-1}{2}$, is the number, that is hardest to approximate by rationals (cf e.g. the section properties of the golden ratio $\phi$ here: ...
2
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3answers
246 views

Growth of $r_{2}(n)$

Let $n$ be a positive integer. From Jacobi's two-square theorem we know that the number $r_{2}(n)$ of representations of $n$ as a sum of two squares is given by $$ r_{2}(n)=4(d_{1}(n)-d_{3}(n)), $$ ...
10
votes
2answers
592 views

Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?

This may be inappropriate for MO, but here goes: if I have understood the statement of the Erdős–Rado theorem correctly, then it contains as a special case the following result: if $\mu$ is ...
6
votes
2answers
319 views

Generalization of Sylvester-Gallai theorem

The Sylvester-Gallai theorem states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single ...
5
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1answer
140 views

Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
0
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0answers
20 views

about the weak comparison principle for p - Laplace equation [on hold]

Let $\Omega$ an open bounded domain in $R^n$ with smooth boundary. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$ and $u \in W^{1,p}(\Omega) $ with $\Delta_p u = 0$ in $\Omega$ and $u - ...
9
votes
4answers
733 views

Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...
2
votes
2answers
154 views

Tensor product over a monoid in a monoidal category

nLab article on tensor product says: "Given two objects in a monoidal category $(C,\otimes)$ with a right and left action, respectively, of some monoid $A$, their tensor product over $A$ is the ...
1
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0answers
40 views

Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem $$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...
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0answers
53 views

The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$: $$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$ This is A001923 in the OEIS. I don't have much experience with ...
5
votes
2answers
373 views

Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement. Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...
2
votes
3answers
728 views

Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...
2
votes
0answers
42 views

Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse. Does there ...
15
votes
1answer
887 views

Joyal's letter to Grothendieck

Mostly out of curiosity: Where do I find Joyal's letter to Grothendieck in which he defines a model structure on simplicial sheaves? The question was already asked in this MO post, but that ...
4
votes
1answer
290 views

Results about moduli of surfaces

There are early successes of the moduli theory - the construction and compactification of the moduli spaces of curves $\overline{\mathcal{M}}_g$ . I want to study about the moduli of algebraic ...
3
votes
1answer
128 views

invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results; For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1) (1) There exist ...
4
votes
1answer
417 views

Beginners Guide to Cartan for Beginners [closed]

I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance. Question: I am seeking ...
2
votes
1answer
99 views

Hopf structures on “pictorial” descriptions of permutations

There is a well-known Hopf structure on permutations, due to Malvenuto and Reutenauer. Here, the F basis elements are permutations in one-line notation, the product of two permutations is their ...
0
votes
1answer
85 views

References: Solutions of the Bethe Ansatz Equations [closed]

Could someone show me good references to find solutions of the Bethe Ansatz Equations, for simple cases (using algebraic geometry or others interfaces with mathematics)?
5
votes
1answer
139 views

When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$ Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...
11
votes
2answers
305 views

A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ? I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
3
votes
1answer
207 views

A More Advanced Version of Aluffi's Chapter 0

This is a crosspost of this question from MSE. Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce ...
21
votes
2answers
595 views

Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...
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0answers
39 views

finding dominating cycles in $2K_2$-free graphs

A cycle $C$ in a connected graph $G$ is called dominating if its complement $V(G)-V(C)$ is an independent set. H.J. Veldman proved in 1983 (Disc. Math. v.43, 281-96) a general result that in ...
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1answer
300 views

Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...
0
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0answers
52 views

Reference : Special case of Banach-valued function integration by parts

Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...
10
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0answers
265 views

Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in ...
2
votes
1answer
80 views

Any two bivariate algebraically dependent polynomials are always in the same ring generated by some bivariate polynomial?

If $f(x,y)$ and $g(x,y)$ are two algebraically dependent polynomials over some field $k$, is it true that there exists a bivariate polynomial $p(x,y)$ such that both $f(x,y)$ and $g(x,y)$ are in the ...
2
votes
1answer
85 views

System of linear first order PDE with constant coefficients

recently in my researches I've come across the following operator $$L\left(\begin{array}{c} a_1\\ \vdots\\ a_n \end{array}\right)=M_1\left(\begin{array}{c} ...
11
votes
4answers
579 views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
27
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1answer
657 views

Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations, where we can use forcing to prove the existence of finite objects with some ...
1
vote
1answer
65 views

Smooth unit vector field on a tetrahedron to interpolate vertex constraints

For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$ consider the (energy) functional ...
1
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0answers
59 views

Better version of “Monotonicity methods in Hilbert spaces and some applications to nonlinear PDEs..”

I am asking whether any one knows of a better source for the text Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations by H. Brezis which I ...
1
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0answers
41 views

Reason for the Choice of Line Parameters in the Radon Transform

Why are the lines, over which the integrals in a Radon Transform are calculated, apparently always parameterized as $L(t,\phi,\alpha) := ...
2
votes
0answers
120 views

Good Pre-Calculus book? [closed]

I was reading this article and the author mentioned I should come here and get some advice. I'm 17, currently taking Pre-Calc in high schooling doing really good, but I feel like I'm not getting the ...
2
votes
0answers
112 views

Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
3
votes
1answer
209 views

Reference request for non-commutative analogues of exterior algebras

I am reading Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (preprint available on web page). Here is an extract of the paper: Cohen called $A^R_n$ "a standard tool used in ...
0
votes
2answers
263 views

A sum-of-determinants identity [closed]

I posed a terser version of this question on math.stackexchange.com and after 24 hours I get only a comment on a detail of notation and neither votes nor answers. Suppose ...
1
vote
0answers
47 views

Regularity of Schrödinger Resolvent

The following problem keeps bothering me: Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on ...
3
votes
1answer
74 views

What is Known About the Complexity of Calculating Minimal Surface Polyhedra?

I am currently ruminating about ways of generalizing Minimum Spanning Trees to Minimum Spanning "Hypertrees", where the cost is associated with simplex volumes and, where certain topological ...
2
votes
0answers
148 views

Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...
5
votes
1answer
82 views

Continuity of taking collapse maps

Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
0
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1answer
101 views

Surjectivity of “nice maps” from local properties

What tools are available from real algebraic geometry, analysis and topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$ from local properties and maybe function values? ...