0
votes
0answers
6 views

Frame-bundle reduction from spinor-bundle reduction

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...
1
vote
1answer
22 views

Tangent cone of a complete intersection

Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' ...
1
vote
0answers
31 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
2
votes
0answers
60 views

Is this diagram of sheaves actually Cartesian as claimed?

The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a ...
0
votes
0answers
35 views

the inverse of Trace theorem when $p = 2$

I can see there is an answer the inverse for the trace theorem and Image of the trace operator My question is that given $f\in H^{1/2}(\partial\Omega)$, is it possible to extended it into $\Omega$ ...
2
votes
0answers
89 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
-3
votes
0answers
24 views

construction of nonocommutative division rings [on hold]

I am studing "A first course in noncommutative rings by T.Y.Lam " Please introduce books or articles to better understand the contents of section 14 (noncommutative division rings) of this book. thank ...
-1
votes
0answers
14 views

Solving el Gammal given D can solve DDH [on hold]

I have a crypto final in 2 days and I am reviewing past finals but the prof does not give solutions. There is 1 question I cannot solve it and I spend a good 2-3 hours on it. Here is the question: ...
0
votes
0answers
31 views

What is the relation between the $K_0$ of a singular curve and its normalization?

Let $X$ be a singular curve over a field $k$. We define $K_0(X)$ to be the Grothendieck group of the category of coherent sheaves on $X$. For $X$ we have its normalization $\widetilde{X}$ and hence ...
5
votes
0answers
144 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
-2
votes
0answers
10 views

Random Walk Probability Including Drift [on hold]

What is the equation for the probability of a random walk with drift being equal to a specific value after n steps, given a specific standard deviation?
-2
votes
0answers
16 views

Determining odds of a slot machine given a payout value of the icon [on hold]

So most slot machines base the payout on the probability of the combination coming up. What I would like to do is flip that and set a payout and then have the probability based off of that if ...
2
votes
1answer
48 views

Density of polynomials which are soluble with respect to a set of primes

Suppose that $p$ is a prime, and $f(x)$ is a polynomial with integer coefficients and positive degree. Then there exists an integer $n_p$ such that $p | f(n_p)$ if and only if $f(x)$ has a linear ...
4
votes
0answers
71 views

On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
5
votes
0answers
122 views

Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
2
votes
1answer
165 views

Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...
0
votes
0answers
21 views

Euler transformation of pochhammer symbol

From the Euler transformation of Pochhammer symbol $$\sum_{n=0}^{\infty}\frac{(b)_n}{n!}a_nz^n=(1-z)^{-b}\sum_{n=0}^{\infty}\frac{(b)_n}{n!}\Delta^na_0(\frac{z}{1-z})^n$$ the following ...
1
vote
1answer
67 views

Counting primes powers $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is: $$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...
1
vote
1answer
98 views

Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...
-2
votes
0answers
40 views

How to solve this triple integral? [on hold]

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
0
votes
0answers
30 views

Is the following set of infinite absolutely convex combinations closed? [on hold]

Let X be an infinite dimensional Banach space and let $(x_n)$ be a weakly-null sequence in X. Let A:=$\{\sum_{n=1}^{\infty} a_nx_n: (a_n) \in B_{l_1}\}$, where $B_{l_1}$ is the closed unit ball of ...
0
votes
1answer
56 views

Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$. Ultimately, I'm interested in finding a ...
0
votes
0answers
17 views

Simple proof for a simple second-order approximation of a convex function

I am wondering if someone can prove (or outline the proof) for the following statement from p. 459 of Boyd and Vandenberge's Convex Optimization textbook. Consider a strongly convex function ...
0
votes
0answers
55 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
-1
votes
0answers
17 views

How to describe behavior of population system, given by system of ODEs? [on hold]

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
4
votes
1answer
130 views

Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...
1
vote
1answer
81 views

Explicit examples of Dehn presentations of hyperbolic groups

It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation. My question concerns something I'm struggling with since the first time I read the proof ...
2
votes
0answers
92 views

Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family of degree $2$ maps defined (for $t$ small and non zero) by $$u_t([X,Y]) := [X^2, t Y^2, XY].$$ Note that as $t$ goes to zero, ...
4
votes
0answers
60 views

Analysis of Nim-Like Game? [on hold]

There are a finite number of heaps, each with a finite number of counters. Two players take turns; on each move, they may remove exactly one counter from any heap, and also, if the heap is of size ...
0
votes
0answers
27 views

Contraction of simplicial presheaves

Let $X,Y$ be two simplicial presheaves on a small category $\mathcal{C}$, let $*$ be the final simplicial presheaf. Consider the category of simplicial presheaves equipped with its projective model ...
7
votes
0answers
99 views

What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...
-3
votes
0answers
52 views

assume subgroup $H$ of $G$ such that $N$ is also a subgoup of $H$, then $ P_{G/N}(H/N) = P_{G}(H)/N$ [on hold]

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
-3
votes
0answers
39 views

Graph Theory - k-connected graph [on hold]

I am trying to understand the concept of k-connected graphs in graph thoery. Reference books state that a graph G is k-connected if G is connected and if its vertex connectivity is greater than or ...
1
vote
0answers
43 views

Does there exist a projection (of a variety) birational onto its image and satisfying additional conditions?

Let $X \subset \mathbb P^n$ be an irreducible (projective) variety of dimension $k < n-1$. By Harris [Har, Lecture 18, page 224], the projection $\pi_p : \mathbb P^n - \{p\} \to \mathbb P^{n-1}$ ...
-1
votes
0answers
20 views

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system [on hold]

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system $$\frac{du_1}{dt}=u1(b_1-a_{11}u_1-a_{12}u_2)$$ ...
0
votes
0answers
64 views

Length of quotients and relations between $\ell(\mathrm{coker}\varphi),\ell(R/\det\varphi)$

Let $R$ be domain(not necessary local) with maximal ideal $\mathfrak{p}$ and $d \in R, d \neq 0$. $(R/(d))_{\mathfrak{p}} = 0 \iff (d) \not\subset \mathfrak{p}$(?). And if $ (d) \subset \mathfrak{p}$ ...
2
votes
0answers
187 views

random category theory

This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references ...
-4
votes
0answers
47 views

Distance Between two points [on hold]

Need help in solving my homework problem.How to find distance between 2 points when (4,5) and (12, 3) are given? I need to know the formula for finding it
-3
votes
0answers
61 views

how to compute the de Rham cohomology with compact support of a mobius strip [on hold]

I am having problem computing the de Rham cohomology with compact support of an open mobius strip,it's aquestion from Bott's book, and Bott said its cohomology is identically zero which can be ...
-3
votes
0answers
50 views

Schur multiplier [on hold]

Are there real world applications of Schur multiplier? I am interested in applications of topics specifically coming from Schur multiplier. for example, in biology, computer sience and other branch.
0
votes
0answers
86 views

Is there an algorithm that probably solves the Halting problem? [on hold]

Such an algorithm takes as input any program and returns a probability that it halts. In the limit of many programs, it must answer on average in the correct proportion. But im interested in other ...
1
vote
0answers
38 views

Geodesic equation and radial metric

Assume that $g(z)=f(|z|)$ is a radial metric on the unit disk in complex plane, where $f$ is a smooth real function. Is there any simple equation of geodesic lines w.r.t. metric $g$, e.g. ...
-1
votes
0answers
22 views

Is C^0 fine topology is finer that metric topology? [migrated]

Let C(E,F) be a set of continious maps between metric spaces E and F. Suppose we are given $C^0$ fine topology and a metric topology on C(E,F). We now that the fine topology is finer than compat-open ...
3
votes
1answer
99 views

Descending a monomorphism of stacks

The question is about Proposition 3.8.1 in Laumon and Moret-Bailly book on algebraic stacks. Let $S$ be a scheme and let $F: \mathscr{X} \rightarrow \mathscr{Y}$ be a morphism of $S$-stacks (for the ...
0
votes
0answers
24 views

Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...
2
votes
1answer
107 views

Gromov Geometric Structures and Killing fields

Let's fix some notations: $M$ will denote a real smooth, $m$-dimensional, manifold, $F^k(M)$ is the k-th order frame bundle on $M$ and $Gl^k(m)$ is the space of $k$-jets of diffeomorphisms of $\mathbb ...
0
votes
0answers
119 views

A question on (odd) perfect numbers

I have asked this question in MSE a few weeks back, but did not receive any responses. I have cross-posted it to MO, hoping that it is appropriate for this site. Let $\sigma(x)$ be the (classical) ...
0
votes
0answers
45 views

How to use Integrals to calculate the expected value of two-dimensional Gaussian distribution [on hold]

Given that I have the following joint density function (two-dimensional Gaussian): $f(u,v)= \frac{1}{1\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{-\frac{1}{2}Q(u,v)}$ where ...
3
votes
1answer
104 views

Average height of rational points on a curve

I am seeking a formalism to define the average height of the rational points on a curve. This is straightforward if the number of points is finite, but (to me) not straightforward when the rational ...
1
vote
0answers
33 views

how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written: $$ \frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1} $$ and so by symmetry: $$ \frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...

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