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

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0answers
25 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 ...
3
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0answers
129 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
0answers
17 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 ...
6
votes
1answer
227 views

Regularity of Hodge Laplacian on bounded domains

I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates $\lVert \omega \rVert_{W^{s+2,p}} \leq c \lVert f \rVert _{W^{s,p}}$, for ...
3
votes
0answers
33 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 ...
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0answers
124 views
+100

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 ...
5
votes
0answers
111 views

Semi-continuity of intersection numbers

I always trusted the following quite vague statement: If you have a family of divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say varying in ...
2
votes
2answers
293 views

Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields: we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...
1
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1answer
178 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 ...
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0answers
42 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 ...
1
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0answers
82 views

Second derivative estimates for gradient dipendent elliptic equations

Caffarelli in the article " Interior a priori estimates for solutions of Fully non linear Equations" Ann. Math 130,No.1, 1989 proved that a continuous viscosity solution $ u $ of a uniformly elliptic ...
0
votes
0answers
51 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 ...
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votes
2answers
97 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 ...
19
votes
1answer
724 views

Why can the general quintic be transformed to $w^5-5\beta w^3+10\beta^2w-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic, $$w^5-10\alpha w^3+45\alpha^2w-\alpha^2 = 0\tag1$$ This form is well-known for its connection to the symmetries of the ...
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0answers
63 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
2answers
649 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, ...
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votes
0answers
94 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, ...
3
votes
0answers
68 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 ...
16
votes
1answer
622 views

Explicit path in the unitary group of a $C^*$-algebra

For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...
11
votes
2answers
224 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
118
votes
46answers
39k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
20
votes
1answer
543 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? ...
1
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1answer
79 views

Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...
6
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1answer
220 views

Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation: $$ -\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi, $$ where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
228
votes
68answers
107k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
5
votes
0answers
67 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 ...
5
votes
0answers
155 views
+100

Topological entropy and periodic sequences of a subshift

Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$. ...
41
votes
3answers
2k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of ...
1
vote
0answers
117 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 ...
7
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1answer
110 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 ...
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0answers
89 views

Good reference for noncommutative $L^p$ spaces

I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
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votes
0answers
61 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 ...
16
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1answer
395 views

Axiomatizations of the real exponential field

According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an ...
8
votes
1answer
248 views

Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...
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1answer
149 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 $s:A\to{\mathbb C}$, ...
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0answers
46 views

Algorithm: Computing the intersection of two conics [on hold]

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 ...
0
votes
1answer
179 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
1answer
134 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 ...
5
votes
1answer
172 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 $. ...
0
votes
0answers
66 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 $\psi$. ...
5
votes
1answer
317 views

Reference request - Jacquet module and asymptotic of matrix coefficients

Hello, I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the ...
1
vote
2answers
44 views

List of tensor product spaces with uniform crossnorms

Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products $H:=\bigotimes_{j=1}^n H^{(j)}$ and ...
8
votes
1answer
162 views

Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
8
votes
3answers
342 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 ...
1
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1answer
373 views

Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
3
votes
2answers
184 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 ...
7
votes
1answer
268 views

Unique factorization of posets

Given two finite posets $P$ and $Q$, we can form the direct product poset $P \times Q$ whose elements are pairs $(p,q) \in P \times Q$ with $(p,q) \leq (p',q')$ if $p \leq p'$ and $q \leq q'$. Let us ...
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1answer
71 views

Reference for a local density theorem for binary vectors

I have the following theorem written on my whiteboard, but have misplaced the reference. I believe the probabilistic method may be involved in the proof. Any pointers appreciated. Theorem Let ...
1
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1answer
155 views

General Reference for surface singularities

Is there any "standard" reference for (rational) singularities on algebraic surfaces? I'm aware of Artin's papers and the one of Brieskorn (Rationale Singularitäten komplexer Flächen), but they seem ...
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135 views

Reference for algebraic manipulation of sheaves [closed]

I am currently playing with sheaves over families of algebraic varieties ($O_x$-modules) their torsion sub-sheaf, higher direct images and tensor products, I am looking for a good reference to learn ...