5
votes
2answers
183 views

Existence of finite set of points in the revolving circles

Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ ...
4
votes
1answer
191 views

Do discrete groups with property $(T)$ have “modest” subgroup growth?

I saw it conjectured at http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above ...
3
votes
1answer
200 views

Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...
9
votes
3answers
360 views

Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...
44
votes
4answers
3k views

History of “without loss of generality”

"Without loss of generality" is a standard in the mathematical lexicon, and I am writing to ask if anyone knows where the expression was popularized. (The idea has been around since antiquity, I'm ...
103
votes
13answers
22k views

Knuth's intuition that Goldbach might be unprovable

Knuth's intuition that Goldbach's conjecture (every even number greater than 2 can be written as a sum of two primes) might be one of the statements that can neither be proved nor disproved really ...
7
votes
2answers
413 views

Hausdorff dimension of R x X

In general, the Hausdorff dimension of a product is at least the sum of the dimensions of the two spaces. Does equality hold if one space is Euclidian? So let $X$ be a metric space and let ...
77
votes
9answers
6k views

Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason! Is ...
18
votes
7answers
3k views

The Chern-Simons/Wess-Zumino-Witten correspondence

I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. I guess in the condensed matter ...
14
votes
4answers
1k views

Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, ...
1
vote
0answers
125 views

Ext and cup products and subvarieties

I am trying to understand Remark 11.3 in Huybrechts's amazing book on derived categories (FM transforms in AG). He starts with smooth projective varieties $j\colon Y \subset X$ and aims to describe ...
8
votes
1answer
236 views

Gromov-Hausdorff convergence for non-compact metric spaces

Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent? $\forall r > 0: \bar{B}_r(p_i) ...
27
votes
6answers
2k views

Are all polynomial inequalities deducible from the trivial inequality?

I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...
0
votes
1answer
282 views

Largest eigenvalue of the sum of hermitian matricies [closed]

Is there an expression for the largest eigenvalue of the sum of two hermitian matricies in terms of the spectrum of the same matricies?
19
votes
11answers
3k views

When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...
4
votes
1answer
517 views

equivalence between katz and classical modular forms

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\PP}{\mathbb{P}}$ $\newcommand{\QQ}{\mathbb{Q}}$ $\newcommand{\hH}{\mathcal{H}}$ $\newcommand{\eE}{\mathcal{E}}$ ...
4
votes
1answer
372 views

Computing cotangent complex

I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given? In my precise situation, ...
14
votes
4answers
895 views

rational function identity

I just had to make use of an elementary rational function identity (below). The proof is a straightforward exercise, but that isn't the point. First, "my" identity is almost surely not original, but ...
15
votes
2answers
5k views

Truth of the Poisson summation formula

The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable ...
3
votes
2answers
357 views

On well-formedness of weighted projective spaces and a Hurwitz theorem calculation

This question has two parts: A calculation that is giving me a lot of troubles, and a theoretical one on weighted projective spaces. 1) I want to find the genus of the curve $C_7 \subset ...
4
votes
3answers
2k views

Finding a recursion for a sum of Legendre polynomials

The polynomial $a_n(x):=P_n(x)-\frac{n-1}{n}P_{n-2}(x)$ where $P_n(x)$ is a Legendre polynomial came up while I was investigating methods for estimating the error in Gaussian quadrature. I am ...
12
votes
2answers
389 views

Classifying spaces for enriched categories

Is there a standard construction of a classifying space $BC$ for a category $C$ which is enriched which takes into account the enrichment? This is of course vague... The simplest example I can think ...
10
votes
3answers
1k views

The first complete proof of the Kronecker-Weber theorem

While the Kronecker-Weber theorem —that every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field— is always attributed to, well, Leopold Kronecker and Heinrich Martin Weber, ...
8
votes
1answer
522 views

Is this pleasing polynomial irreducible?

Let: $f(x)=x^n+2x^{n-1}+3x^{n-2}+4x^{n-3}+\ldots + (n-1)x^2+nx+(n+1)$. Is $f(x)$ irreducible? In light of the answers to this question, I now know that this is true when $n+1$ is prime. What about ...
7
votes
4answers
3k views

When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
12
votes
0answers
327 views

Nilpotent pro-$p$ groups

Is it true that every finitely generated (topologically) torsion free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
8
votes
1answer
563 views

what was Hilbert's geometric construction in his 17th problem?

Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
1
vote
2answers
775 views

Is the isomorphism class of a fixed cardinality a set?

Is the isomorphism class of a fixed cardinality a set(not a proper class)? Or a fixed ordinality for that matter? By "isomorphism" I mean just bijection for cardinals and order preserving bijection ...
5
votes
0answers
58 views

A “universally non Hypercomplete” $\infty$-topos?

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected ...
1
vote
0answers
29 views

Minkowski spacetime in Newman Penrose formalism

I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere: I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's ...
3
votes
1answer
91 views

Classifying space for homology endomorphisms supported on a graph?

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the ...
4
votes
0answers
92 views

Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...
22
votes
5answers
1k views

Errata database?

Some authors do a really great job by collecting errors and comments to their books and putting a list on their websites. I wonder if there is some (perhaps wiki-style) website where errata are ...
4
votes
3answers
2k views

n-dimensional “cross product” reference request

I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to. Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ ...
14
votes
3answers
1k views

Relation between motivic homotopy category and the derived category of motives

What's the relation between the pointed motivic homotopy category $\mathcal{H}_*(k)$ and the derived category of motives $\mathbf{DM}^-_{eff}(k)$ besides the representability of motivic cohomology in ...
8
votes
1answer
572 views

Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there. Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...
0
votes
0answers
66 views

A question on theorem 1.1 of Fritz John ultrahyperbolic pde

I have the following paper: Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, pages 300-322 doi:10.1215/S0012-7094-38-00423-5 Now ...
2
votes
1answer
100 views

Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form $$\begin{bmatrix} \pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\ A_{2n} & ...
6
votes
2answers
151 views

Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?

Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...
2
votes
1answer
96 views

Do the full subcategories have a simple structure in higher category theory?

Let $C \in Cat$ be an $(\infty,1)$-category. Let $P$ be the partially ordered subset of full subcategories of $C$. Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer ...
0
votes
0answers
19 views

A book on discriminant analysis

Can anyone suggest a good book on discriminant analysis - comprehensible and detailed? (Kendall and Stuart write about the subject too concisely.) Thanks in advance.
10
votes
6answers
960 views

Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said: Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...
0
votes
0answers
93 views

Can someone explain some of the steps in this paper clearly?

I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$ Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...
19
votes
2answers
1k views

History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to ...
2
votes
1answer
129 views

Is the top interval of a finite distributive lattice an hypercube lattice?

Let $(L,\wedge,\vee)$ be a finite distributive lattice. Let $M$ be the (unique) maximum element. An element $a \in L$ is called maximal if $a \le a' < M$ implies $a = a'$. Let $b = ...
0
votes
0answers
42 views

The property reservation conditions in the functional iteration process

Given a integral equation: $$K(y,p)=\int_{a}^{b}f(x,y)K(x,p)dx$$ Using the iteration method,choosing an arbitrary inition function $K^{0}(x,p)$: $$K^{1}(y,p)=\int_{a}^{b}f(x,y)K^{0}(x,p)dx$$ ...
1
vote
1answer
86 views

extension of holomorphic mappings

In 1971 Phillip.A.Griffith proved that Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact ...
1
vote
1answer
115 views

Are most random variables trivially sub-gaussian? [closed]

I'm trying to understand sub-gaussian RVs to see if they could be relevant to my work. The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform ...
2
votes
2answers
146 views

Divisibility among discriminants

Let $f(x)$ be an algebraic function over the field $\mathcal F$ of algebraic numbers over $\mathbb{Q}$. Suppose that $r \in \mathcal F$. Does the discriminant of $f(r)$ divide the discriminant of ...
3
votes
2answers
260 views

the space of continuous maps between 3-manifolds

Let $X$ be a connected hyperbolic 3-manifold (without boundary), $S^3$ the 3-sphere and $Map(X,S^3)$ the space of continuous maps between $X$ and $S^3$. Question: Is the space $Map(X,S^3)$ connected ...

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