1
vote
1answer
40 views

characteristic classes of symmetric product

Given a (real or almost complex) manifold $M$, Let the symmetric product be $$ B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2 $$ where $$ \Delta=\{(m,m)\mid m\in M \}. $$ Then $B(M,2)$ is a (real ...
0
votes
1answer
24 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
0
votes
0answers
6 views

Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of: $$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$ $$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ...
2
votes
0answers
14 views

Current upper bound on length of addition chain

An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...
0
votes
0answers
6 views

Dynamics of the distribution of prime factorization types in increasing intervals

I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...
3
votes
1answer
104 views

characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal $$ ...
-2
votes
0answers
14 views

constant rate of change

When downloading a large file, Travis noticed that the estimated time remaining to complete the download decreased by 35 seconds for each additional megabyte downloaded. When he started the download ...
0
votes
0answers
6 views

When is the identity Hilbert-Schmidt between weighted Sobolev spaces?

Set $w(x) = (1 + |x|)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space $$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := ...
20
votes
5answers
789 views

Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g., $$z^3+z^2+2 z+3 \;.$$ Find its $n$ roots, and list them in order of their modulus: $$-1.28, (0.14\pm 1.53 i)$$ Now form a new ...
0
votes
0answers
31 views

Is the map $\exp_x(\sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification. Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...
0
votes
0answers
12 views

norm of a matrix that have integral operator as its entries

what is the norm of a matrix that have integral operator as its entries? for example $$\bordermatrix{\text{corner}&c_1&c_2&\ldots &c_n\cr & A_{11} & A_{12}\cr ...
4
votes
0answers
27 views

Cutting a piece of cake that $n$ people value as exactly $w$

Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a ...
2
votes
0answers
29 views

How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation $qn = mf$, where each of the variables ...
0
votes
0answers
15 views

Help with notations from 2D to 3D FFT representations as 1D FFT

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my ...
4
votes
1answer
91 views

Partitioning ${\cal P}([[1,n]])$

In an analysis of the Jacobi method for the computation of the spectrum of a Hermitian matrix, I face the following problem. Denote ${\cal P}_2(n)$ the set of doubletons $\{a,b\}$ in ...
4
votes
1answer
84 views

Stable cohomology operation, natural homomorphism

How do I see that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?
19
votes
0answers
268 views

Relaxed Collatz 3x+1 conjecture

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ ...
2
votes
0answers
22 views

How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular? It would be helpful is someone can share some specific ...
5
votes
0answers
65 views

$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
-5
votes
0answers
24 views

Integration of an inverse trig function [on hold]

Hey guys I have this inverse trig function that needs to be integrated however there are certain aspects of it that throw me off. The function is (sin^-1(x^2))^2 The portions that are throwing me ...
3
votes
0answers
68 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ ...
3
votes
2answers
68 views

Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...
3
votes
1answer
57 views

Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...
5
votes
1answer
110 views

The sum of a series

Let $0< \alpha <1$ and $q>1.$ Consider the (alternating) series: $$ \sum_{k=1}^\infty (-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$ Denote its sum by $f(q,\alpha).$ Prove (or ...
0
votes
0answers
57 views

Minimal number for sums and differences of primes

Let $\mathbb{N}$ denote the set of positive integers. For any set $X$, let ${\cal P}_{\text{fin}}(X)$ be the set of finite subsets of $X$, and let $\mathbb{P}$ be the set of prime numbers in ...
3
votes
0answers
37 views

Polynomial constraints triggered by irreducibility

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic: $$af^2 + bf + c = 0$$ If we're working in a ring, ...
0
votes
0answers
49 views

good book on differential forms for engineers? [on hold]

I find the language of differential forms heavy in formailisms. Most books around are written for mathematicians and/or physicist which have a style slightly inaccesible for engineers. I understand ...
1
vote
0answers
52 views

The self intersection class of exceptional divisor of 3-fold blown up along a curve

Suppose $X$ is a smooth complete variety of dimension $3$, let $\sigma\colon\widetilde{X}\to X$ the blow-up along smooth curve $C\subset X$, let $\sigma^{-1}(C)=E$ be the exceptional divisor, let $f$ ...
3
votes
0answers
67 views

Is a successor to a successor of the trivial group topology totally bounded?

Is there an example of an abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
5
votes
3answers
176 views

Generating (or availability of) large strongly regular graphs

Are there collections of already generated large strongly regular graphs available to download? By large I mean $n >= 200$ where $n$ is the number of vertices. I found Ted Spence's page on srgs, ...
-3
votes
0answers
126 views

A question about Category Theory [on hold]

The Review of Symbolic Logic for June 2015 contains an article by Michael Ernst, in which it is proved that Unlimited Category Theory (as defined by S. Feferman) is inconsistent. This seems to me to ...
8
votes
0answers
113 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...
1
vote
0answers
42 views

Graph Theory for Dummies Book [migrated]

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course ...
1
vote
2answers
87 views

Asymptotic of the sum of squared primes [on hold]

I have a rather simple question of number theory which I can't seem to be able to find a good reference for. I am not a specialist and I don't really know where to look. I would like to show that the ...
1
vote
0answers
47 views

Normal Subgroups of $U_n(q)$

What is known about normal subgroups of $U_n(q)$, the group of upper triangular matrices with entries in the finite field $\mathbb{F}_q$ and ones on the diagonal?
3
votes
1answer
74 views

Method to construct a bipartite graph G' with 2n vertices from a graph G

I have seen mentioned in a talk an operation that takes a graph $G=(V,E)$ and constructs a new bipartite graph $G'=(V',E')$ such that $V' = V\times \{0,1\}$ and $E'=\{((i,1),(j,0)) : (i,j)\in E\} \cup ...
-4
votes
0answers
43 views

Uncountable set and uncountable accumulations [on hold]

Show uncountable set of real numbers has uncountable accumulation points in itself. So far I have proved that we can assume that this set is bounded and there is a accumulation point not necessary ...
-2
votes
0answers
65 views

Degree of map into Lie group representation

Suppose $M$ is a smooth manifold with unit volume and that $G$ is a compact Lie group of the same dimension. Given a smooth map $\phi: M\rightarrow G$, we can compute the degree of $\phi$ as: ...
7
votes
3answers
579 views

Category theory for Algebraic Geometry

How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
-4
votes
0answers
36 views

Algebra II exercise (help) [on hold]

Can anyone please help me with this exercise for my exam? It says: 1)Given S={(x₁,x₂,x₃) ∈ R^3 : x₁ + x₂ -2x₃ = 0}. a)Prove that S is a subspace. b)For each of the matrices A shown, check if ...
53
votes
46answers
7k views

Important formulas in Combinatorics

Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...
4
votes
1answer
147 views

Physical strength of a link [on hold]

Assuming that we construct a link/chain using a collection of knots. Is there a way to measure the physical strength of this chain?
10
votes
2answers
688 views

Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?
2
votes
1answer
50 views

Set of density matrices

A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$ In Quantum Mechanics it is natural to look at a group action $\Phi: ...
8
votes
2answers
212 views

Free $k[x_1, \dots, x_n]^{S_n}$-module?

Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} ...
2
votes
1answer
53 views

Effective Realization of GCD of middle binomials?

So, it is well-known that $$ \gcd \left(\binom{m}{k}\mid 1\leq k\lt m \right) = e^{\Lambda(m)}$$ which can incidentally be sparsified for prime $p$ $$ \gcd ...
1
vote
0answers
20 views

Automorphism groups of symmetric cones

Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone $C$ is a real Lie group $Aut(C)$. What is the associated class of Lie groups $Aut(C)$ for each of ...
4
votes
1answer
90 views

cohomology ring of infinite iterated loop space

What is the cohomology ring $$ H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)? $$ I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...
3
votes
3answers
102 views

Simplifying a Taylor polynomial that involves Stirling numbers of the second kind

I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form: $$\sum_{n=1}^k ...
3
votes
2answers
184 views

bar construction and loop space cohomology

Let $X$ be a topological space. Then its chain complex $C_{*}(X)$ is naturally a coalgebra (as explained here Does homology have a coproduct?). In particular if $X$ is simply connected we have that ...

15 30 50 per page