# All Questions

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### 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 ...
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### 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 $N$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Stable cohomology operation, natural homomorphism

How do I see that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?
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### 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 $x$ ...
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### 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 ...
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### $E_{\infty}$ $R$-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 ...
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### 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 ...
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### 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 $6j$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 $f$ ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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 ...
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 ...