# All Questions

2 views

### $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$

In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals), an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$. [See Comm. Rings by Kaplansky, ...
31 views

### Why calculus textbooks do not include the natural integration constants in the tables of integrals?

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
23 views

### Proof that $G(3)\le 7$ [migrated]

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers. Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...
14 views

### An ideal that contained in finitely many maximal ideals but all of its elements contained in infinitely many maximal ideals [migrated]

Is it possible that an ideal I in an integral domain D is contained in only finitely many maximal ideals but each element of I is contained in infinitely many maximal ideals? I am quit sure that it is ...
17 views

139 views

### Continuous relations? [on hold]

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
60 views

### Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...
123 views

### Approaching convex and discrete geometry from other disciplines

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...
82 views

### Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
114 views

### Plucker embedding and tautological/universal quotient bundle

Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
107 views

### A lifting problem

Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles. For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...
90 views

### Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$. There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...
49 views

### Reference for Elliptic PDE on $\mathbb{R}^d$

Could anyone suggest a textbook, article, or lecture notes that covers elliptic PDE theory (existence, uniqueness, regularity) on all of $\mathbb{R}^d$, as opposed to the Dirichlet or Neumann problem ...
27 views

### Eigendecomposition of analytic Hermitian matrix-valued functions of several variables

If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and ...
55 views

### Quantifier problem of equations in physics [on hold]

Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. To assert that static energy equals mass ...
121 views

### Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers $$A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3 = \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .$$ In an email, physicist Alan Sokal ...
69 views

### Group action induced on homology under change of coefficients

Let $M$ be a closed manifold equipped with a (continuous or smooth) $\mathbb{Z}_2$-action such that - for simplicity - both $H_{k-1}(M;\mathbb{Z})$ and $H_{k+1}(M;\mathbb{Z})$ are zero for some ...
26 views

### Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform Probability ...
100 views

### Operator norm vs spectral radius for positive matrices

I believe the following statement should be true but somehow I don't see an argument: For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
52 views

### Semidirect products with braid groups and type $F_\infty$

Let $F$ be a group which is strongly type $F_\infty$ in the sense that every subgroup is of type $F_\infty$. Here, type $F_\infty$ means that the group admits a classifying space with compact skeleta. ...
90 views

### Without Skolem–Mahler–Lech Theorem? [on hold]

Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$. Is there a "simple way" to prove this ...
16 views

### Mixed Integer Quadratic Programming using Opti Toolbox in MATLAB [on hold]

I wish to solve a mixed integer quadratic program with linear constraints using OPTI toolbox in MATLAB. I want some of my decision variables to be continuous and some decision variables to be binary. ...
30 views

### Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...
48 views

### Find base of kernel with as many 0 as possible

I have a 400x132 rectangular matrix with only 0 and 1. I am looking for the linear combinations of the columns of the matrix that sum to 0. For example C1 + C2 - C3 = 0. I want to find the linear ...
47 views

### Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate ...
37 views

### Linear transformation of a polyhedral cone

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ with H- and V-representations $C = \{x : A x \le 0 \} = \{R y : y \ge 0\}.$ The pair $(A,R)$ is referred to as a double description (DD) pair of the ...
124 views

### concepts in fields other than physics and computer science which touch on concepts that are fundamental in pure mathematics [on hold]

[ Contextual Information: Is mathematics held back by it's relative lack of ties to disciplines other than physics and computer science? Are there areas of mathematics which have gone underexplored ...
86 views

### How does the Atiyah-Singer index theorem in a relative setting related to “ringed spaces and pseudocoherent complexes of finite tor-dimension”?

I come across the following paragraph from the article Reminiscences of Grothendieck and His School, here is from the part of the interview by Luc Illusie,: " I was indeed looking for an ...
Let A, B be k-algebra (k is a field). What is free product of k-algebra A and k-algebra B? ($A \ast_{k}B$)
### natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$?
is there any natural map from natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$? where $G$ is a discrete group, $BG$ is the classifying space, and $\underline E G$ is the ...