# Tagged Questions

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

**3**

votes

**0**answers

114 views

### Reference for the Hodge polynomial or the Hodge Characteristic

What is the first work that studies, refers to, or mentions the Hodge characteristic?
The Hodge polynomial is the unique ring homomorphism
$$
P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to ...

**4**

votes

**1**answer

89 views

### Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge ...

**14**

votes

**1**answer

297 views

### Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...

**9**

votes

**2**answers

236 views

### Finite objects for which isomorphism is NP-hard or harder?

Are there finite objects for which deciding isomorphism
is NP-hard or harder?
Graphs and groups are not solutions.
Searching the web didn't return answer for me.
Partial result based on Chow's ...

**1**

vote

**1**answer

176 views

### Deformation Quantization

I am a beginner and I want to learn about deformation quantization. Please suggest me with which book or notes, I should start?

**3**

votes

**1**answer

132 views

### Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...

**1**

vote

**0**answers

54 views

### History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -
$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall ...

**1**

vote

**0**answers

45 views

### Reference: Heat Kernel for Siegel Upper Half plane

Is there a ready reference for explicit computation of the heat kernel for Siegel upper half space $\mathbb{H}_n=\{Z=X+iY\in \mathrm{Mat}_n(\mathbb{C}) \vert Y>0\} $? I could find it for general ...

**0**

votes

**0**answers

162 views

### Matrix with roots of unity entries

For given prime p, i am interested in the norms of $p \times p$ matrices which have roots of unity entries, $M_{k,l} ∈ \{1,ζ,…,ζ^{p−1}\}$ with $ζ=\exp(2πI/p)$. Are there any studies of the norms of ...

**5**

votes

**0**answers

147 views

### When a proper smooth fibration is isotrivial?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two ...

**4**

votes

**1**answer

165 views

### (Etale) fundamental group of quotient singularity $\mathbb{C}^n/G$

I don't know much about (algebraic/etale) fundamental groups, so sorry if this question sounds stupid. I am interested in quotient singularities (quotients $X$ of $\mathbb{C}^n$ by a finite subgroup ...

**3**

votes

**0**answers

51 views

### Reference for classification of positive involutions

An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if ...

**0**

votes

**0**answers

162 views

### SDP based heuristics for graph coloring

This is a question about semidefinite programming heuristics for graph vertex coloring based on the Lovasz theta number such as "Approximate Graph Coloring" by Karger, Motwani and Sudan or "A ...

**7**

votes

**4**answers

440 views

### General Relativity and Differential Geometry intuitions of Second Bianchi Identity

In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-
$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$
It is said that ...

**0**

votes

**0**answers

42 views

### Largest instance of highly nonlinear benchmark functions (e.g. Rastrigin function)

What is the largest instance size (number of variables) ever numerically solved for highly nonlinear (continuous, not combinatorial) optimization benchmarks functions, such as Rastrigin, Schwefel or ...

**7**

votes

**1**answer

149 views

### $C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a ...

**3**

votes

**2**answers

277 views

### A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...

**5**

votes

**3**answers

342 views

### Elementary reference for the isometry group of $\mathbb{RP}^2$

Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ".
It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an ...

**1**

vote

**0**answers

97 views

### Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every ...

**1**

vote

**1**answer

85 views

### Weighted counting of circular codes

Given a circular code $X$ (for example: $X=\{ w,b \}$) with generating function $u(z)=\sum\limits_{k=0}^{\infty}{u_k z^k}$ (in this example : $u(z)=2z$), the generating function ...

**4**

votes

**2**answers

698 views

### Divergent Series as a topic of research

About a year ago, while studying real analysis, I got very much interested in divergent series. I discussed possible research topics related to divergent series with my teachers but couldn't find any. ...

**2**

votes

**0**answers

115 views

### $f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other ...

**1**

vote

**0**answers

59 views

### A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...

**0**

votes

**0**answers

49 views

### Convexity condition of matrices

In studying the viscoelastic theory of elastodynamics, I encounter a problem on the convexity condition of matrix functions. It has been known that for the energy function $E=E(v,F) = \frac{1}{2} v^2 ...

**1**

vote

**0**answers

130 views

### On the cardinality of the set of right-truncatable primes

We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ ...

**4**

votes

**1**answer

78 views

### Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...

**6**

votes

**0**answers

164 views

### Semi-continuity of intersection numbers

I always trusted the following quite vague statement:
If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...

**0**

votes

**0**answers

79 views

### Continuity of solutions of nonlinear elliptic PDEs

Consider the nonlinear 2nd order elliptic PDE $$\sum_{i, j} a_{ij}(x, t) \partial_i\partial_j u + \sum_k b_k(x, t) \partial_k u + c u = F(u), \quad x \in \mathbb{R}^n, t \in [0, \infty).$$
Here ...

**1**

vote

**0**answers

58 views

### Fluid dynamics of a rotating liquid droplet

I'm looking for an analytical solution of the Navier-Stokes equation with the following boundary conditions: a liquid is held inside a spherical shell, which is rotating at a constant rate, and is ...

**4**

votes

**0**answers

94 views

### Cofibrations in the model structures for non-negative graded (commutative) DG algebras

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}_{k}^{+}$ denote the category of non-negative graded ...

**7**

votes

**2**answers

711 views

### Complexity of Turing Machine behavior

If one looks at the code for a Turing Machine (TM) with
$q$ states and, let's say, $2$ symbols, they all look
pretty much the same:
A list of $5$-tuples:
$$
< state, symbol{-}read, ...

**6**

votes

**2**answers

115 views

### Relationship between Multiplicative Ergodic Theorems

One version of Oseledets' Multiplicative Ergodic Theorem states that if $\sigma$ is an ergodic measure-preserving transformation of a space $(\Omega,\mathbb P)$ and if $A\colon\Omega\to GL(d,\mathbb ...

**3**

votes

**0**answers

74 views

### Bott-type vanishing results for the weighted Grassmannian wGr(2,5)

If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted ...

**1**

vote

**1**answer

205 views

### Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...

**0**

votes

**1**answer

199 views

### Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true.
Since posting the question, ...

**20**

votes

**1**answer

685 views

### Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?

$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic.
Is this folklore, or is it credited to someone? ...

**5**

votes

**0**answers

93 views

### Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of ...

**1**

vote

**0**answers

154 views

### When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto ...

**0**

votes

**0**answers

168 views

### Milnor numbers and mixed multiplicities

section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration ...

**4**

votes

**0**answers

245 views

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**1**

vote

**0**answers

49 views

### Algorithm: Computing the intersection of two conics [closed]

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conics curves. The curves are given by two equations of the form:
$a x^2 + b y^2 ...

**2**

votes

**1**answer

158 views

### The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...

**3**

votes

**0**answers

177 views

### First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...

**0**

votes

**1**answer

183 views

### Field extension and nilpotent element

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...

**3**

votes

**1**answer

144 views

### 4-th order diophantine equation

I met in many places the equation
$(a^4-b^4)(c^4-d^4)=\square$
It is well known that this was investigated by Euler.
But I was unable to find the general solution of this equation. Could you please ...

**0**

votes

**0**answers

77 views

### Fastest algorithm to compute isogeny

Let $E/GF(p)$ and $E'/GF(p')$ are two isogenous elliptic curves($\#E=\#E'$). We know that there exist the map
$$\psi : E \to E'$$
Suppose that we haven't any information about degree of $\psi$.
...

**4**

votes

**2**answers

195 views

### Equivalence of different cohomology groups

Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups:
(1) singular cohomology ...

**6**

votes

**1**answer

116 views

### Two notions of bundle of C* algebras

One can find in the literature two notions of $C^*$-algebra over a topological space $X$.
The first is as the data of an open surjection $ \pi: B \rightarrow X$ together with the structure of a ...

**5**

votes

**1**answer

179 views

### Injectivity/Surjectivity of $F_A :=\frac{d}{dt} +A(t) $ for a hyperbolic path of matrices $A(t)$ on $H^1 $

I am looking for a reference to the following problem:
Given a hyperbolic (no purely imaginary eigenvalues), continuous path of matrices $A(t)$ in $\mathbb{R}$ with hyperolic limits at $\pm \infty $.
...

**9**

votes

**3**answers

552 views

### References for Yang-Mills Theory

We are looking to run a working seminar about the Yang-Mills story. We hope that our seminars is of interest to analysts (working with curvatures and Ricci flows on Riemannian manifolds), the ...