# All Questions

102,940 questions
2answers
216 views

### Properties of signomial Functions in one variable

I am interested in functions of the form: \sum_{j=1}^\infty a_j x^{p_j} where p_j can be any non-negative real number. Wikipedia has informed me that this is a subset of the signomial functions, but ...
3answers
2k views

### Simple example of a ring which is normal but not CM

I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I'...
1answer
237 views

### Correlation measure between signals of different dimensions?

I have several temporal signals of different dimensions, for example the motion of a point throughout time which would be of dimension 3, and the value of a temperature sensor, of dimension 1. I ...
3answers
5k views

### Finite type/finite morphism

I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "...
1answer
176 views

### Homotopy type of stabilizers

Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y). My question is the following: is it ...
3answers
2k views

### K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?
3answers
8k views

### Is 8 the largest cube in fibonacci sequence?

Can you prove that 8 is the largest cube in fibonacci sequence?
2answers
812 views

### Are generalized cohomology theories a homotopy category of some category of invariants?

I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors ...
2answers
725 views

### Is there a free digraph associated to a graph?

A little bit of background: A graph G is, of course, a set of vertices V(G) and a multiset of edges, which are unordered pairs of (not necessarily distinct) vertices. We say that two vertices v_1, v_2 ...
4answers
1k views

### When is a scheme a zero-set of a section of a vector bundle?

Are there any general results on when a closed subscheme X of a quasi-projective smooth scheme M can be written as the zero-set of a section of a vector bundle E on M? To put it in a diagram: When is ...
5answers
1k views

### Pairs of shortest paths

It is known that the binomial coefficient $2n \choose n$ is equal to number of shortest lattice paths from $(0,0)$ to $(n,n)$. The Catalan number $\frac{1}{n+1} {2n\choose n}$is equal to the number of ...
1answer
319 views

### about Function of Random variables [closed]

Hello, I am studying random variables. Question is this: if rv X & a function g is known, what is the pdf of random variable Y = g(x)? in the textbook answer is explained as follows. P[y ≤ Y ≤...
3answers
2k views

### free homotopy groups — when do they exist?

Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,x), where ΩxX denotes the space of loops in X based at x, ...
2answers
129 views

### Multiplicative stability for integration [closed]

I remember back in undergraduate to ask myself this question : In the general case, I is an interval, \int_I fg =! \int_I f \int_I g (*) But how to describe the egality case, i.e find all couples (f,g)...
3answers
2k views

### Special cases for efficient enumeration of Hamiltonian paths on grid graphs?

While the general problem of detecting a Hamiltonian path or cycle on an undirected grid graph is known to be NP-complete, are there interesting special cases where efficient polynomial time ...
4answers
7k views

### Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
1answer
298 views

### Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed. Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...
1answer
421 views

### multidimensional multinomial density [closed]

I have data set X = {x_1, x_2, \ldots, x_N}, each x_i is a d-dimensional vector, where scalars are from some finite field (In practice they are categories, represented by integers from 1...C). If ...
2answers
1k views

### Can a singular Deligne-Mumford stack have a smooth coarse space?

Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...
5answers
929 views

### Estimating the number of clusters

For a collection of points in $\mathbb{R}^n$, is there a statistic that I can compute which will estimate the number of clusters with some level of confidence?
1answer
266 views

### Can you construct a mapping space from local data? (looking for reference)

I'd to know if/where there is a reference for the following construction. Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
7answers
5k views

### Quotients of Schemes by Free Group Actions

I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
3answers
871 views

### Categorifying the Reals via von Neumann Algebras?

So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...
5answers
2k views

### Point singularity of a Riemannian manifold with bounded curvature

Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold ...
7answers
1k views

### What is the smallest integer whose primality status is not known?

Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current ...
1answer
476 views

### How do you rotate a matrix to maximum sparsity?

Given a matrix M, I want to find an orthogonal matrix U that maximizes the number of entries that are zero in the product MU. How do I go about doing this?
3answers
2k views

### Unstable Vector Bundles

As a follow up to me other question, what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater ...
5answers
1k views

### Is there a tool for finding probability distributions given some samples?

I'm looking for a tool that does "probability distribution fitting" given a set of data points. Sort of like curve fitting, but tries to fit to standard density distributions. For example if I input ...
1answer
738 views

### Is there a Murnaghan-Nakayama Rule for GL(n,q)?

The Murnaghan-Nakayama rule for S_n is a combinatorial rule to compute the irreducible characters of the symmetric group. Is there a q-analogue of this rule for GL(n,q) to compute the irreducible ...
5answers
5k views

### Induction and Coinduction of Representations

I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...
2answers
1k views

### Pushforwards of Line Bundles and Stability

I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a ...
1answer
473 views

### Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
2answers
337 views

### Is the center of a free (as a module) algebra free?

A submodule of a free module need not be free (for instance, in the free Z[X]-module Z[X] the submodule generated by 2 and X is not free). But over a principal ideal domain, submodules of free modules ...
7answers
5k views

### What's so special about transcendental numbers? [closed]

It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? Are most numbers ...
2answers
5k views

### What is the geometric meaning of integral closure?

More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its ...
2answers
688 views

### Does projectiveness descend along field extensions?

Background: Properness is a much more robust notion than projectiveness. For example, properness descends along arbitrary fpqc covers (see, for example, Vistoli's Notes on Grothendieck topologies, ...
2answers
345 views

### High dimensional Steiner tree

Given n affinely independent points in n-1 dimensional Euclidean space, how is the minimum Steiner tree constructed? Or assuming that the topology of the Steiner tree is given, is there an easy way ...
2answers
317 views

### When do PROP-morphisms induce adjunctions?

If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where B-Alg(C)=tensor-...
4answers
474 views

### Software for rigorous optimization of real polynomials

I am looking for software that can find a global minimum of a polynomial function over a polyhedral domain (given by, say, some linear inequalities) in $\mathbb R^n$. The number of variables, $n$, is ...
4answers
1k views

### Moduli spaces of complex curves as algebraic varieties

Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a ...
2answers
6k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
5answers
802 views

### Models of the reals which have no unmeasurable sets

I recall being told -- at tea, once upon a time -- that there exist models of the real numbers which have no unmeasurable sets. This seems a bit bizarre; since any two models of the reals are ...
3answers
1k views

### Asymptotics of a hypergeometric series/Taylor series coefficient.

I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes: I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh ...
3answers
1k views

### What are some conserved quantities of Poisson brackets?

Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets. Let's say we are working on T^n x R^n (T^n ...
2answers
2k views

### Are curves with `fractional points' uniquely determined by their residual gerbes?

One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...
3answers
2k views

### Can we categorify the equation (1 - t)(1 + t + t^2 + …) = 1?

Polynomials in ℤ[t] are categorified by considering Euler characteristics of complexes of finite-dimensional graded vector spaces. Now, given a rational function that has a power series ...
8answers
2k views

### Euclidean volume of the unit ball of matrices under the matrix norm

The matrix norm for an n-by-n matrix A is defined as |A|=max(|Ax|) where x ranges over all vectors with |x|=1, and the norm on the vectors in R^n is the usual Euclidean one. This is also called the ...
3answers
688 views

### Is there a co-Hahn-Mazurkiewicz theorem for line-filling spaces?

A famous theorem on space-filling curves is the Hahn-Mazurkiewicz theorem: Let $X$ be a Hausdorff space, then there exists a surjective continuous map $[0,1] \to X$ if and only if $X$ is compact, ...
5answers
3k views

### Algorithm to Find all the Cycle Bases in a Graph

I am given a graph defined by vertexes and edges. I have to obtain all the cycle bases in a network. No coordinates will be given for the nodes. Here's a sketch that illustrates my point. Note that ...
3answers
830 views

### Freyd-Mitchell for triangulated categories?

Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a ...

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