0
votes
0answers
11 views

Proof - If one domain $D$ is contained in another domain $D'$, then $\lambda_n \leq \lambda_n'$

I would like to understand the proof of the theorem $4$ of the book page $326$. In fact, after a good while of trying to understand that proof, I am not even sure how it works. In being clear, does ...
0
votes
0answers
9 views

Is the space of holomorphic maps a manifold

To be more specific: Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...
3
votes
0answers
40 views

Groupoid cardinality and Egyptian fraction representations of 1

It is well-known that any rational number can be represented using a sum of distinct Egyptian fractions (that is, rational fractions of the form $1/n$ with $n\in\mathbb{N}$). This may be proven by ...
3
votes
0answers
13 views

On the complemented subspaces of $L_{p}(p>2)$

M.I. Kadec and A. Pe{\l}czy\'{n}ski proved that if $E$ is a subspace of $L_{p}(p>2)$ isomorphic to $l_{2}$, then $E$ is complemented in $L_{p}$. My question is: Is there a constant $C_{p}$ ...
0
votes
0answers
11 views

Complexity of truncated SVD of a sparse matrix

This is somewhat of a follow-up to this question: What is the time complexity of truncated SVD Let $A$ be an $m \times n$ matrix, where we assume $m < n$ without loss of generality. Suppose that ...
2
votes
0answers
24 views

Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots, x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl algebra. As usual $W$ ...
8
votes
0answers
43 views

Can you prove Givental's conjecture on wavefronts and the icosahedron?

In his remarkable book The Theory of Singularities and its Applications, Vladimir Arnol'd discussed a conjecture of A. B. Givental, which asserts that the symmetry group of the icosahedron is secretly ...
3
votes
0answers
32 views

Roughly equal number of swimmers in teams

$b^2$ swimmers are to be put into one of the teams $1,2,\dots,b$. A team $i$ has a value function $f_i$, so that if they get swimmer $k$, they get value $f_i(k)$. The value $f_i(k)$ is randomized ...
1
vote
0answers
12 views

Optimal control / Portoflio optimization: Maximize expected utility of total consumption

I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...
1
vote
0answers
10 views

Surjections to free commutative dgas

Consider the category $C$ of commutative dgas (unbounded in both degrees) over a ring $R$ (in practice the integers or the integers mod $p$). Let $F$ be the free functor from chain complexes to $C$ ...
5
votes
2answers
62 views

Graphs with prescribed numbers of k-cliques

Let $(a_1,a_2,\dots, a_n)$ be a sequence of non-negative integers. Q. When does there exists a simple graph $G$ such that its number of $k$-cliques is $a_k$ (that is $G$ has $a_1$ vertices, $a_2$ ...
0
votes
0answers
24 views

symmetry of a square - is it possible pure geometric approach in didactics? [on hold]

Consider a square : four points in a plane constructed with classical means (compass and straightedge). Since no point is different from others (no coordinates, no labels...) it seems that we can not ...
1
vote
1answer
65 views

Product of Bruhat Cells

Fix a $(B,N)$ pair (Tits system) of a semisimple Lie group $G$. Let $u$ and $v$ be two Weyl group elements such that $l(uv)=l(u)+l(v)$. It is known that $BuvB=(BuB)(BvB)$ (see for example Humphreys's ...
0
votes
0answers
25 views

stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g. $ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$ $ \nabla \cdot u = 0 \; \text{in} \; \Omega$ $ u = 0 \; \text{on} ...
0
votes
0answers
33 views

Capacity of two disks

Is there an explicit formula for the (logarithmic) capacity of a union of two disjoint disks? As far as I understand, one can assume without loss of generality that the disks have the same radii ...
0
votes
1answer
75 views

Normal bundle of a fiber of the family of curves

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: ...
3
votes
1answer
80 views

Convergence in trace

Let $A$ and $B$ be two self-adjoint, positive definite Compact operators on a Hilbert space $\mathcal{H}$. Further, let $A$ be trace class. Define $C_n \equiv AB(\frac{I}{n} + BAB)^{-1}$. Does ...
-1
votes
1answer
99 views

Prove this conjecture inequality 2 [on hold]

Let $n$ be postive integer,I conjecture $$(1+2n)^n\ge 1^n+2^n+4^n+6^n+\cdots+(2n)^n \tag{1}$$ This problem when I solve this equation $$(1+2n)^n=1^n+2^n+4^n+6^n+\cdots+(2n)^n\tag{2}$$ if this ...
0
votes
1answer
70 views

$Ext$ functor over a product of groups

Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups). Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$. Write $G = G_1 ...
1
vote
1answer
111 views

How many the distinct linear factors of $f(x)-f(y)$ can be for f in Q[x]?

Let $f \in \mathbb{Q}[x]$. Let $S(f)$ denote the number of distinct linear factors of $f(x)-f(y)$. $S(f)$ is bounded by $\deg(f)$. Q1 Is $S(f)$ bounded by constant? Q2 Is it possible ...
-2
votes
0answers
27 views

Representing a function's performance by varying 4+ parameters [on hold]

I have developped a function that implements the Genetic Algorithms for solving the Traveling Salesman problem. This function has many variables: static void AG(int numberIterations, int ...
1
vote
0answers
48 views

representability of a certain extension of group algebraic spaces

Let S be a scheme. Suppose we have sheaves in abelian groups $A,B,C$ over the big étale site of $S$. Suppose that $A$ and $C$ are representable by algebraic spaces in groups locally of finite type ...
4
votes
0answers
124 views

What is an example of a non-mixed $\ell$-adic sheaf?

$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and ...
-4
votes
0answers
35 views

Localization Technique on Earth [on hold]

i have a question that needs to be answered. I have an asignment from school for making a localizationtool. It works like this: i have 3 basestations. these basestations stand on the ground or on a ...
1
vote
0answers
26 views

Triange-free graph and its complement has Lovasz number > 3

I found an example by the method in the paper Explicit Ramsey graphs and orthonormal labelings by Noga Alon 1994. The graph is around $10^6$ vertices, anyone knows smaller graph which is Triangle-free ...
0
votes
1answer
140 views

On the quadratic reciprocity law? [on hold]

In the Quadratic Reciprocity Law $$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$. Is there any relation between $x$ and $y$ ...
2
votes
1answer
84 views

Transitivity of “being the zero locus of a section of a vector bundle”

Let $Z\subset Y\subset X$ be smooth projective $k$-varieties. Suppose $Z$ (resp. $Y$) is defined in $X$ by the vanishing of a section of a vector bundle $E$ (resp. $F$). Is it true that $Z\subset Y$ ...
1
vote
0answers
38 views

How many distinct sets of n collinear points are there in an evenly-spaced two-dimensional grid of m x m points?

I'm seeking the definition of some function $f(n,m)$ which evaluates to the number of distinct sets of $n$ collinear points which are selected from an evenly-spaced two-dimensional grid of $m \times ...
-3
votes
0answers
93 views

Is the Poincare duality still non-degenerate when restricted to subgroups of algebraic cycles? [on hold]

On smooth varieties over the complex numbers there are conjectures (1) the usual Hodge conjecture (generalized Hodge conj. of level 0) (2) generalized Hodge conjecture of level 1 (3) standard ...
0
votes
0answers
16 views

reBounds on the positive roots of a bivariate polynomial

It is well known that various real root isolation methods are based on computing, first, the bounds on the values of the positive real roots of a polynomial equation. For the univariate case such ...
-6
votes
0answers
111 views

Is Hodge conjecture a mist or simple fact? [on hold]

On smooth varieties over the complex numbers there are conjectures (1) the usual Hodge conjecture (generalized Hodge conj. of level 0) (2) generalized Hodge conjecture of level 1 (3) standard ...
-2
votes
0answers
29 views

Graph Isomorphism of Graphs that are not Locally Triangle-Free

What is the computational complexity of graph isomorphism problem for graph-class $\mathcal{G}$ that excludes graphs which are locally triangle-free ? Is this $\mathcal{G}$ class included in any ...
3
votes
0answers
58 views

Can I combine the category of Drinfeld modules and the category of the base O_S

I am learning about Drinfeld modules,T-modules,...They are said to be analogues of elliptic curves, abelian varieties,... Let K be a finite extension of k = Frac(A), and $O_K$ the integral closure of ...
0
votes
0answers
31 views

When can an analytical solution for the heat equation be obtained?

I am currently trying to model a system with a time varying heat flux. It seems most researchers are using FEM to obtain the heat distribution (solve the heat equation). When can the heat equation be ...
7
votes
0answers
42 views

Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds

Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...
1
vote
0answers
102 views

Decomposable elements in cohomology ring [on hold]

I feel this must be well known, but I don't have enough knowledge on the matter. So, if there is a literature on this, then please provide a reference. Let $E$ be a ring spectrum, and $X$ a pointed ...
2
votes
1answer
100 views

How many lines of exactly n points can be placed in a discrete, square grid of size m x m?

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of ...
-5
votes
0answers
29 views

The transpose of a matrix times its transpose is the matrix's transpose times itself [on hold]

Somebody please help me to show the steps to prove this is true: $R$ is a matrix and T is transpose: $(R^TR)^T = R^TR^{TT} = R^TR$
2
votes
0answers
72 views

Ideals of $L^1(G)$

I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
0
votes
0answers
55 views

An integral form of sum $\sum_{n\geq 0} \frac{f^{(n)}(0) g^{(n)}(0)}{(n!)^2}$ for two real analytic at 0 functions? Fourier|Taylor series parallels

Thinking of parallels between Fourier series and Taylor series, you might find out that the integral $\frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle$ ...
11
votes
1answer
280 views

Errata on Rezk's paper

I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in ...
2
votes
0answers
110 views

étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$. Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$. For ...
-1
votes
0answers
85 views

Positivity of holomorphic bisectional curvature and existence of rational curves

Motivation: Siu-Yau by the positive bisectional curvature assumption, produced a rational curve using the analyticity of a stable harmonic map. Also is there any example of a compact Kahler ...
15
votes
1answer
1k views

How much mathematics has been formally verified?

That's a vague question so allow me to tighten it up a bit. I recently noticed that there is a formal machine verified proof of the Central Limit Theorem (CLT) implemented with Isabelle. This ...
5
votes
0answers
67 views

Multiplication in universal enveloping algebra in terms of PBW isomorphism

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...
6
votes
1answer
131 views

Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$. Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1]. But perhaps that estimate is too crude. ...
11
votes
0answers
120 views

Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
2
votes
0answers
78 views

Gaussian Integrals and Pseudo-Anosov Maps

The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated. Here I take from: ...
3
votes
0answers
50 views

Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...
2
votes
0answers
80 views

An elementary question about metrics on the real plane [on hold]

Given the metric $d_p$ on the real plane, i.e. $$ d_p(x,y) = d_p((x_1, y_1), (x_2, y_2)) = [|x_1 - x_2|^p+ |y_1 - y_2|^p]^{1/p} $$ for which values of $p$ ($\geq 1$) is it true that the following ...

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