# All Questions

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### Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following? When $C$ is the ...
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### X^2e + X^e + 1 is irreducible for e which is a power of 3? [on hold]

I am looking for a proof or reference to the fact that the trinomials of the form X^2e + x^e + 1 Are irreducible in GF(2) for e which is a power of 3. Please help! Lear
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### Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$. Two possible ways to compute $T_p$ mod $p$ seem to be: A) ...
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### Regular point of a map in algebraic geometry

What is the correct definition of a regular point of a map in algebraic geometry? More specifically, let $f:X\to Y$ be a map of varieties with $f(p) = q$, and let $Z=f^{-1}(q)$. Let $\hat{X}$ be the ...
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### bound on degree of certain polynomials

Consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $\Bbb R[x_1,\dots,x_n]$ of degree $d_i$ such that $$Z(f_i)\bigcap Z((p))\neq \emptyset$$ for the sphere $p$ passing through $\{0,1\}^n$. What is ...
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### Is elliptic curve point division defined over the field of real numbers?

An elliptic curve is defined over the field of real numbers: $y^2=x^3 + ax + b$ A point P and scalar n can be multiplied using a combination of point doubling and adding. What about point division? ...
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### degree of polynomials in nullstellensatz

If $K$ is algebraically closed field, then consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ for ...
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### Efficient computation of null space of large symbolic matrices?

Are there any computer algebra system/libraries that can compute the null space of a large symbolic matrix in parallel? This problem arises when finding invariant polynomials of a continuous linear ...
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### Equal-area projections of the hyperbolic plane [on hold]

I'm aware of the projections of the hyperbolic plane at http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane, but how would I project the hyperbolic plane onto a flat piece ...
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### Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
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### About Hausdorff characterization [on hold]

I am thinking about why only in complete metric space a set A is compact if and only if A is totally bounded and closed? Anyone can help me? Thanks a lot!
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### Langlands duality and multiplying cocharacters

Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation) of the Langlands dual group ...
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### Analogue of Baer's injectivity criterion for comodule algebras

Let $H$ be a Hopf algebra and $A$ an $H$-comodule algebra; denote by $M^H_A$ the category of right $(H,A)$-Hopf modules [i.e. $A$-module, $H$-comodules, everything is compatible with everything else]. ...
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### How to switch from the spectral density of the differential equation

I am modeling random process. It is described with the function of the spectral density, where $\alpha_x$ and $\beta_x$ are damping coefficient and the average frequency of the correlation function of ...
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### irreducible representation of a simple Lie group where each element has a fixed point

I was wondering if it possible that a simple Lie group $G$ could have a continuous irreducible representation on a finite-dimensional real vector space $V$ in which each element of $G$ has a non-zero ...
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### A question on the representation theory of finite group

By the Burnside theorem, we know that we can decompose the order of a group in to a sum of some integers' square, and these integers are the dimensions of the group's irreducible representations . ...
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### “Productively normal” space

If a set $S$ is endowed with the discrete topology $\mathcal{P}(S)$, then for every normal space $N$ the product $S\times N$ is normal. Question: can we endow a set $S$ with another Hausdorff ...
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### Mathematical induction understanding [on hold]

I need to proof that (k/k+1) + (1/(k+1)*(k+2)) = (k+1)/(k+2) can show me step by step how to proof that?
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### An obstruction to existence of invariant trivializations of central extensions in the presence of a group action

This is sort of a follow up question to this. My real question is a particular case of the following abstract situation. Let $G$, $\pi$ be groups and assume $\pi$ acts on $G$ by automorphisms. Assume ...
Let $H = (V, E)$ be a hypergraph, that is $V$ is a set and $E \subseteq \mathcal{P}(V)$. We say that $H$ is $T_1$ if for $v\neq w$ there are $e_v, e_w \in E$ such that $v\in e_v, w\notin e_v, w\in ... 0answers 33 views ### Derived tensor product and restriction Let$A$be a commutative ring, and let$B$be a commutative$A$-algebra. We have a restriction map$(-)_A : D(B) \to D(A)$which takes an object of the derived category of$B$-modules$M$, and ... 1answer 122 views ### “Almost” zeta function Given a sequence$(a_n)_{n\in\mathbb{N}}$with$a_n > 0$for all$n\in \mathbb{N}$and$\lim_{n\to\infty}a_n = 0$the series \begin{eqnarray} \zeta((a_n)_{n\in\mathbb{N}}) := \sum_{n=1}^\infty ... 1answer 101 views ### Systems of equations in Boolean Algebra I have to study systems of equations in a Boolean algebra, the matrix is$m\times n$with$m\neq n$. The Boolean algebra is actually the simplest one, it contains only$0$and$1$, let us denote it by ... 0answers 103 views ### reduction of elliptic curves to finite field [on hold] Let$E$be an elliptic curve which is defined over$\mathbb{Q}$and$p$be a prime number. I know we can reduced$E(\mathbb{Q})$to$E(\mathbb{F}_p)$, is there an algorithm to reduce$E(\mathbb{Q})$... 0answers 225 views ### Proving that a space is Hilbert Let$H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)be equipped with the norms \begin{align*} ... 0answers 123 views ### elliptic curves and tower of finite fields [on hold] LetE$be an elliptic curve which is defined over$\mathbb{F}_{p^n}$and$m< n$. Can we reduce$E(\mathbb{F}_{p^n})$to$E(\mathbb{F}_{p^m})$? Specially in the case where$m=1$? I mean, let$A$... 2answers 168 views ### Configuration space like subspace of sphere product For$k \geq 2, n \geq 1$let $$M^{n,k} = \{(x_1,\dots,x_k) \in S^n \times \dots \times S^n \ | \ x_1 + \dots + x_k = 0\}$$ This is a compact CW-complex and almost, but not quite, a manifold. I ... 1answer 45 views ### Existence of an equivariant Morse function Let$G$be a (finite) group and$M$a$G$manifold. Now I have a smooth real valued function$f: M\rightarrow R$with$f(x)=f(g(x)),\, \forall g\in G$. Now in general$f$will maybe not be a Morse ... 0answers 29 views ### Invariant subsets of a local action I have also asked this in MSE, but it seems to me that my question wasn't very well received there and I think someone in here will be able to answer it more quickly, hence this post. I don't ... 0answers 49 views ### Map between manifolds [on hold] Let$M,N \subset \mathbb{R}^3$be (not necessarily smooth) 2-manifolds without boundary. Let$f: M \rightarrow N$be a continuous function and suppose that$f$is injective. Let$x \in M$and let$U$... 1answer 643 views ### Von Neumann's consistency proof In the paper Zur Hilbertschen Beweistheorie, John Von Neumann has proposed a consistency proof for a fragment of first-order arithmetic (the fragment without induction and with the successor axioms ... 1answer 69 views ### Reductive subgroup and its derived subgroup with an irreducible represenation Could you please answer the following question: Let$V$be a faithful irreducible representation of a connected reductive group$H$defined over$\mathbb{R}$Is it true that the derived group of$H$, ... 0answers 66 views ### Actions on the mapping class groups on arcs Let$S$be an orientable punctured surface and denote by$MCG(S)$its$extended$mapping class group, i.e. the group of its homeomorphisms modulo isotopies fixing the punctures pointwise. ... 0answers 52 views ### Real number and axiom of continuity [on hold] I have just read Courant's Introduction to Calculus and Analysis. What makes me confusion is the section "Real Number and Nested Intervals". In the Postulate of Nested Intervals or the axiom of ... 0answers 53 views ### About some 'rigidity theorem' for the Kahler forms on projective bundles Let$E\to X$be a holomorphic vector bundle over a compact Kahler manifold$X$with Kahler form$\omega_{X}$. For a given hermitian metric on$E$, let$\omega_{E}$be the Chern form of the line bundle ... 0answers 98 views ### Irreducible representations of$Sp(4,\mathbb{F}_2)$I'm trying to construct the irreducible representations (over$\mathbb{C}$) of the finite group$Sp(4,\mathbb{F}_2)$. Using GAP, the character table is as follows: $$\left(\begin{matrix} 1 & 1 ... 0answers 80 views ### Solving for 2 numbers that both add and multiply to the same known [on hold] I started with the statement ab = a+b. I worked the solution for a and b when given ab (or a+b) and it is as follows.$$ \textrm{ If }x = ab \textrm{ and } x=a+b\\ a = \frac{x+\sqrt{x-4}\sqrt{x}}{2} ... 0answers 135 views ### Topology of the space of foliations on a 3-manifold Denote by$\mathcal{P} (M)$the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold$M$with the$C^{\infty}$topology, and by$\mathcal{F}(M)$... 0answers 91 views ### A question about the duality principle Suppose$X$and$Y$are finite sets and$K:X\times Y\to \mathbb R$is some function. We get an integral transform from the space of real functions on$X$to real functions on$Y$given by ... 0answers 71 views ### independent subset problems [on hold] I'm interested in the following which i suspect is probably a well studied problem. Given a set$N=\{1,2,...,n\}$and$M=\{1,2,...,m\}$consider a map $$f:N\rightarrow 2^{M}$$ (elements of$N$to ... 1answer 55 views ### Embedded Contact Homology and Manifold Decompositions Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in$\mathbb R\times Y$and "counting" them. In the symplectic ... 0answers 58 views ### A new method of solutions for partial linear differential equations [on hold] Recently,I read a book on partial differential equations,which says that the solution of second order linear equation of two differentiating variables and analytic coefficients can always be expressed ... 1answer 98 views ### totally disconnected sets and homeomorphisms For every totally disconnected perfect subset S in the plane one finds a homeomorphism of the plane onto itself mapping S onto the ternary Cantor set. This is an exercise in a book by Engelking and ... 0answers 58 views ### Identity of Bernoulli polynomials consider the Bernoulli polynomials defined by the generating function:$$\left(\prod_{i=1}^m \frac{a_i}{\left( e^{a_i}-1 \right)}\right)e^{xt}=\sum\limits_{n=0}^{\infty}B^{m}_n\left(x\vert ... 0answers 68 views ### A formal local triviality statement for smooth maps Let$f:X\to Y$be a smooth morphism of schemes of finite type over a field$k$, and suppose that$f(p) = q$. Let$Z = f^{-1}(q)$be the fiber of$f$. Let$\hat{X}$be the formal completion of$X$at ... 0answers 18 views ### What is the nilradical of$\mathfrak{gl}_n$? [migrated] I'm really embarrassed to ask but what is the nilradical of the Lie algebra$\mathfrak{gl}_n(\mathbb{C})$, i.e. the set of ad-nilpotent elements of$\mathfrak{gl}_n(\mathbb{C}) = ...
I am thinking a problem: given a subshift of finte type of $\{0,1\}^{\mathbb{N}}$ and $2>q>1$, where $q$ is a real number. Then how can we find the largest and smallest numbers of the projection ...