9
votes
0answers
259 views
+200

Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
-1
votes
0answers
209 views

Is there any significance in Heegner numbers (or class number 1) representation symmetry?

$\mathrm{A003173}(n) = 1+((1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1})/(2\sqrt{3})$ for n = 1,2,3,4. $\mathrm{A003173}(n) = 19+24((1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6)})/(2\sqrt{3})$ for n = ...
0
votes
0answers
14 views

Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D: $$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g ...
0
votes
0answers
45 views

Necessary condition for decouplings for surfaces in $\mathbb{R}^4$

I'm currently studying the paper Decouplings for surfaces in $\mathbb{R}^4$ written by Bourgain and Demeter. This paper is available in here. As an example of nondegenerate $2$-dimensional surfaces ...
19
votes
4answers
1k views

Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
-2
votes
0answers
30 views

Correct definition of submodularity

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...
-1
votes
0answers
29 views

Norm of a linear operator in a tight frame

My question certainly has a simple answer, but I am not sure about how to formalize my thoughts; to put it simply, I am looking for the norm of a linear operator that is a composition of 2 linear ...
4
votes
2answers
556 views

What is the mathematical significance of the IHES logo?

The logo of the IHES http://www.ihes.fr/jsp/site/Portal.jsp (upper left) is lovely, but what exactly does represent mathematically? (There's a slightly larger version at ...
9
votes
3answers
237 views

Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R). The idea is that each non-trivial ...
-1
votes
0answers
47 views

QR(pivot) vs SVD for low rank approximation

Define the low rank problem as finding the approximation of matrix $A$, $B$: where we want to minimize rank($B$), and we want the $2$-norm of the residue of $A-B$ to be less than $\epsilon$. Could ...
1
vote
2answers
73 views

Transformations whose product with a given rotation are ergodic

I am interested in the ergodic (invertible) transformations $T$ such that $T\times R_\theta$ is ergodic where $R_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all ...
0
votes
0answers
25 views

Connection between maximal regularity and optimal control [on hold]

I would like to know the connection between maximal regularity and optimal control of some equations. I do not find any related topic on internet. Anyone can help me for this question ?
4
votes
3answers
321 views

When can the “homotopy exact sequence” of etale fundamental groups for a smooth curve fail to be exact?

Suppose you have a smooth proper curve $f : \overline{X}\rightarrow S$ over an arbitrary locally noetherian scheme $S$ with a section $e : S\rightarrow \overline{X}$. Let $X := \overline{X} - e(S)$. ...
0
votes
0answers
17 views

Bounding the error of the optimal solution from an approximated objective

Let $f$ be a smooth convex function defined in a bounded region $X$ and its Hessian is bounded $m I \le |\frac{\partial^2 f(x)}{\partial x^2}| \le M I$ for some $M>m>0$. Let $x^*=argmin_{x\in X} ...
0
votes
0answers
12 views

Approximation preserving reductions [on hold]

I've seen in the following document https://hal.archives-ouvertes.fr/hal-00958028/document A definition of the $\leq_{S}$ reduction defined specifically for minimisation problems at the bottom of ...
2
votes
0answers
82 views

Laplace-Beltrami of the Gauss map

Let $M$ be a surface in $\mathbb{R}^3$ given by a regular chart, say $X:M \longrightarrow \mathbb{R}^3$, with its first fundamental form $g$, Gauss map $N$, Gaussian curvature $K$ and mean curvature ...
2
votes
1answer
86 views

Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic, linear operators, which says the following: Defintion: Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...
5
votes
0answers
140 views

Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic: (1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...
1
vote
0answers
34 views

Bounds on $\sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)}$

During our search of real rooted entire function approximations to Riemann $\Xi$ function, we need to calculate the upper and lower bounds of $$f_m(x):=\sum_{j=1}^m ...
2
votes
1answer
238 views

A question about generating set of groups and epimorphism

Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an ...
180
votes
109answers
47k views

What are some examples of colorful language in serious mathematics papers? [closed]

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ...
0
votes
0answers
41 views

Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...
-1
votes
0answers
12 views

affecting the final result of function depending on external factor [on hold]

Suppose I have a function f(x) = x / (x+y) whose range is in the interval [0,1] and there is an external factor say a, such that a is in the interval [0,1], moreover, a predefined threshold t: when ...
1
vote
0answers
13 views

How can I filter the effects of a variable from a correlation matrix?

I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...
0
votes
0answers
24 views

Minimum weight odd cycle with certain edge pairs forbidden

Given a weighted graph $G=(V,E)$ and several disjoint sets $S_1, \dots, S_t \subset E$ of edges, is there a polynomial-time algorithm to find a minimum weight odd cycle which does not contain more ...
1
vote
0answers
11 views

Maximization of the difference of a monotone submodular function and a linear function with a cardinality constraint

Maximizing a monotone submodular function with a cardinality constraint can be solved by using a simple greedy heuristic. However, if the submodular function is non-monotone, the greedy heuristic can ...
-1
votes
0answers
44 views

Monoidal product over database table entries

We have seen from Spivak that database schemas are categories, having tables as objects and relationships as morphisms. I am wondering if we can have a monoidal product over objects, that is, over ...
15
votes
0answers
323 views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
5
votes
4answers
370 views

How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$?

I'm actually interested in a slightly smaller quantity, but I'm willing to accept the following simplification, especially if there are small error terms. Let's start with $ n \gt 1$, Euler's totient ...
10
votes
0answers
206 views

Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence

This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ ...
116
votes
94answers
64k views

Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wiki rules apply: one ...
1
vote
1answer
228 views

On compact, orientable 3-manifolds with non-empty boundary

I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave: Let $M$ be a compact, orientable 3 manifold with ...
0
votes
0answers
46 views

Locally nilpotent derivations on noncommutative rings

I am interested on LNDs on noncommutative rings, specially noncommutative polynomial rings. I have found only some stuff related to the Weyl algebras and Lie algebras. Anyone can tell me a ...
0
votes
0answers
21 views

Vanishing ideal of a finite set of points does not have expected amount of cones in Gröbner fan

I am reading the paper A Gröbner fan method for biochemical network modeling. In Chapter 4.3 (i.stack.imgur.com/h2O8B.png) they calculate the vanishing ideal of some tuples (input points of Series ...
15
votes
3answers
496 views

Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$. Probable primes are the union of the primes and base two pseudoprimes. This definition is much ...
5
votes
1answer
120 views

Is the (super-)symmetric power of the exterior algebra free?

Let $V$ be a vector space over $k$ of dimension $m$. (I'm only interested in the case $k=\mathbb{Q}$.) Let $R:=\Lambda^*V$ be the exterior algebra. It carries the structure of a supercommutative ring: ...
0
votes
0answers
40 views

Regularity result for Neumann problem

I have two questions. On Elliptic regularity for the Neumann problem, the OP asked whether the test function $v$ must be of mean value zero. However, isn't it true that we only need $f$ is of mean ...
0
votes
0answers
51 views

Term for meshes bounding a non-degenerate tetrahedral mesh in 3D?

Is there a term in the literature referring to triangle meshes that are the closed boundary of some non-self-intersecting, non-degenerate tetrahedral mesh embedded in $R^3$? This class of triangle ...
91
votes
17answers
9k views

How does one justify funding for mathematics research?

G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be ...
12
votes
2answers
486 views

Every finitely generated flat module over a ring with finitely many minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, ...
20
votes
1answer
650 views

Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$

When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following: Problem. We have a surface of a cube $n\times n \times n$ such that each ...
57
votes
41answers
9k views

Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider ...
0
votes
1answer
46 views

Is spectral properties a general term for condition number?

I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of ...
167
votes
41answers
62k views

A single paper everyone should read? [closed]

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues. Do ...
6
votes
2answers
221 views

“Database” of simplicial polytopes/spheres

Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep ...
-1
votes
0answers
10 views

Binary Nonlinear Optimization Problem Transform to Continuous Form

Hi I have a mixed binary NLP problem \begin{equation} min\: f(x) \end{equation} s.t. $$ g_i(x)<=0 $$ $$ h_j(x)=0 $$ $$ x_k=1\text{ or }0 $$ I know the optimal of f(x) is approximately 0, so can I ...
4
votes
1answer
175 views

How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
95
votes
43answers
18k views

Examples of eventual counterexamples

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make. where ...
2
votes
1answer
120 views

A canonical representative in Morita equivalence class

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. If $A$ is semisimple, then $A$ is Morita equivalence with a commutative algebra, that is $A \backsim K^n$ where $n$ is ...
0
votes
2answers
70 views

Index Reduction of Differential Algebraic Equations by Hand

I dont really understand how to reduce the index of DAEs ? Does Reducing the index of DAE result in an ODE ? How would I reduce the index of the DAE by Hand ? Say I have : $$ \begin{matrix} ...

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