**1**

vote

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**5**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**46**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**2**answers

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 ...