0
votes
0answers
3 views

derived automorphisms of K3 surfaces of picard rank one

I am aware of work of Bayer-Bridgeland, which describes G=Aut(D(K3)) for a K3 of Picard rank one. Is it possible to use their result to give an explicit presentation of G, in terms of generators and ...
0
votes
0answers
12 views

New (Almost General) Primality Tests Based On Gaussian and Eisenstein Integers

After a few years on trying to prove deterministicability. I decided I'd pass of my observations of these algorithms as my limited knowledge of Number Theory has constrained me. It would really be ...
0
votes
0answers
14 views

Binomial Distribution and Proof relating to Factorials

I am studying probability and statistics at my university but haven't had a solid math course in awhile(mostly forget algebra dealing with factorials)thus I am stuck with the following proof. ...
6
votes
0answers
54 views

Reference Request for Hilbert Schemes

I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...
2
votes
0answers
34 views

Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$? Is it enough to consider the injection via Hochschild: $Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$? In ...
0
votes
0answers
19 views

How to find the number of possible sub rectangles touching the edge of a larger one? [on hold]

http://i.imgur.com/iUDIeMG.png If there exists a rectangular matrix of order M by N then how to find the number of ways to pick a sub-rectangle matrix of any size which is a multiple of 1 square ...
3
votes
0answers
50 views

Graph spectra and topology

This is a somewhat vague question, but I'm wondering if there has been any research into connections between the spectrum of a graph and some notion of the "topology" of that graph. To give an ...
3
votes
2answers
64 views

Basic question about polytope duals

The following must be well known. Is there a beginning or midlevel text where the answer is discussed? Thanks. Along with a polytope one has the notion of its dual which is officially defined via ...
-4
votes
0answers
17 views

first order linear systems differential equations [on hold]

This is an easy question on one dimension but when moving into a system of equations I can’t find the exact solution using matrixes and vectors. given $\dot {\vec x} =A{\vec x}+\vec b$ (where A is a ...
18
votes
2answers
309 views

Recent observation of gravitational waves [on hold]

It was exciting to hear that LIGO detected the merging of two black holes one billion light-years away. One of the black holes had 36 times the mass of the sun, and the other 29. After the merging the ...
6
votes
1answer
71 views

Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$. ...
-4
votes
0answers
31 views

lebegue integral

we suppose that $f:R^+\rightarrow R^+$ is a positive lebegue integrable function and for every $\epsilon >0$, $\int_0^\epsilon f(s)ds >0$ .if we have $\sqrt[n]{\int_0^a f(s)ds} <1$ ...
1
vote
0answers
25 views

For a division ring $D$, does $[D:C_D(a)]_{right}$ vary when $D$ is enlarged?

In a commutative field $K$, the Zariski dimension of an algebraic subset of $K^n$ over $K$ does not vary if one enlarges $K$ if I understood well. In particular, for two Zariski-closed vector spaces ...
3
votes
0answers
43 views

Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695). The particular question is about ...
0
votes
0answers
76 views

Express $ \int_0^1 \frac{dz}{\sqrt{x(x^2 - 1)(x - \lambda)}} $ as hypergeometric function

How do we express the following as hypergeometric function? Let $\lambda > 1$: $$ \int_0^1 \frac{dz}{\sqrt{x(x^2 - 1)(x - \lambda)}} $$ Is this still of the ${}_2F_1$ type? How to find the ...
-1
votes
0answers
53 views

Maschke Theorem descomposition [on hold]

When I was doing an exercise to illustrate to myself the decomposition of the famous Maschke´s Theorem, I realized I didn't understand how was the decomposition stated in the theorem. This is the ...
1
vote
0answers
60 views

Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following: Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...
-3
votes
0answers
68 views

Ramsey theory and logic [on hold]

i need literature which contains formal proof of finite Ramsey theorem in PA, possibly, available on- line.
0
votes
0answers
12 views

combination of field and particle methods for fluid dynamics

in numerical fluid dynamics there are field methods like finite-volume, finite-element, etc. and particle methods like Smoothed-Particle-Hydrodynamics – SPH and others. Both approaches have advantages ...
2
votes
0answers
38 views

Resolution of indeterminacies for a map to Grassmannian of planes

Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
1
vote
0answers
9 views

Least Width of Planar Unimodal Curves with Unit Diameter

I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...
1
vote
0answers
75 views

Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension. In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
12
votes
1answer
384 views

What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?
0
votes
0answers
39 views

On Lie theory with special functions. [on hold]

I research in Lie theory with special functions. But I saw a lot of research on the use of lie theory in hyper-geometric and hermit and other .. Is there a new kind of functions, not considered ...
6
votes
1answer
120 views

Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...
-5
votes
0answers
13 views

When comparing different bars on a bar chart, can you use percentage difference/change? [on hold]

I wanted to know if you can use percentage difference for discontinuous data
1
vote
2answers
163 views

Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$

Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has ...
7
votes
8answers
625 views

What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just ...
7
votes
1answer
97 views

Why do we want $p$-permutation modules in splendid equivalences?

First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...
1
vote
0answers
39 views

How do the heat kernel coefficients depend on the curvature tensor

this is a crosspost, the same question was asked first here: http://math.stackexchange.com/questions/1640092/polynomial-in-the-components-of-the-curvature-tensor Since I have not received any answers ...
5
votes
2answers
137 views

Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?
2
votes
0answers
48 views

The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment. Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...
1
vote
0answers
35 views

Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory. Currently I am working through Murodock's book: Pertubations: Theory and Methods. But I am rest assure ...
1
vote
1answer
72 views

When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [on hold]

Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...
1
vote
0answers
86 views

Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [on hold]

This question follows from the information provided below. Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...
6
votes
1answer
198 views

Is there a consistent arithmetically definable extension of PA that proves its own consistency?

I asked this on stackexchange with no answer. The negation would be the obvious generalization of Gödel's second incompleteness from r.e. extensions of PA to any arithmetically definable extension of ...
10
votes
1answer
323 views

“Set theory” founded on lists rather than sets

On a computer, sets are often represented rather "indirectly / implicitly", e.g. in terms of some properties that they or their members satisfy. But some sets can be represented more "directly / ...
-3
votes
0answers
21 views

minimizing sum of distances [on hold]

I have three points A(-3.5, 0), B(2,0), C(0.3). Looking for D(0,d) such that AD + BD + CD is minimal. Fermat does not work here due to D lying on the y-axis. I thought I could just minimize the sum ...
1
vote
1answer
118 views

Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad ...
21
votes
2answers
515 views

Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element $$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$ ...
2
votes
1answer
99 views

About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve

I was looking for some information related to the values of the characteristic polynomial $\chi(t)$ of the Frobenius of a Jacobian of a hyperelliptic curve $C$ of genus 2 over $\mathbb{F}_q$ and in ...
0
votes
0answers
18 views

Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem : $ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $ where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices ...
0
votes
0answers
24 views

Gradient Descent with L2 Norm Regularization [on hold]

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $ w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t) $ $p(\mathbf{y} = 1 | ...
11
votes
1answer
207 views

Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures

I'm new here. I hope to do it right! I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations. Let us take a smooth complex variety $X$ and a ...
1
vote
0answers
21 views

Divergence of a second order tensor [on hold]

I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct. Suppose you would like to compute the components of a ...
2
votes
2answers
116 views

Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...
2
votes
0answers
64 views

Classifications of Lie bialgebras

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets ...
1
vote
0answers
110 views

if 0→A→A⊕B→B→0 is an exact sequence of finitely generated modules over a commutative Noetherian ring, then the exact sequence does split [duplicate]

Here, Martin Brandenburg says it is not true that "Every short exact sequence of the form $0 \to A \to A \oplus B \to B \to 0$ splits." Then Mohan says in comments that "As a positive result, If ...
1
vote
0answers
69 views

to what extent is a reductive group hyperbolic?

The group $SL(2,F)$ where $F$ is a local nonArchemidian field is hyperbolic. Various generalizations of the notion of hyperbolicity have been studied in the literature (I've seen terms like ...
3
votes
0answers
103 views

For which fields are the 1-dimensional algebraic groups known?

Given an algebraically closed field, or even a perfect one, a connected 1-dimensional algebraic group $G$ over the field $K$ is isomorphic to either $\mathbf G_a$ or $\mathbf G_m$. For which fields ...

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