# All Questions

**0**

votes

**0**answers

5 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

**0**answers

32 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 ...

**6**

votes

**1**answer

57 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

**0**answers

32 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 ...

**4**

votes

**0**answers

44 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

**0**answers

12 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

**2**answers

92 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 ...

**5**

votes

**5**answers

250 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 ...

**6**

votes

**1**answer

59 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

**0**answers

29 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

**2**answers

105 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

**0**answers

38 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**

votes

**0**answers

10 views

### Problem Formulation for SGD as Applied in Non-stationary Optimziation

This post is about how to formulate a time-varying (loss) problem. Basically I am looking for a target-tracking (time-varying) model which is amenable (error can be controlled) for stochastic ...

**1**

vote

**0**answers

28 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

**1**answer

59 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 $\mathbb{C}\Gamma$ ...

**0**

votes

**0**answers

61 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

**1**answer

178 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 ...

**9**

votes

**1**answer

298 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

**0**answers

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

**1**answer

113 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 ...

**20**

votes

**2**answers

410 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

**1**answer

95 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 ...

**1**

vote

**0**answers

71 views

### Gluing a cosheaf of spaces defined on a basis (“cosheafication”?)

Let $X$ be a locally ringed space and let $\mathcal{Z}: \mathcal{B} \to \mathsf{Lrs}_{/X}$ be a cosheaf of spaces over $X$ defined on a basis $\mathcal{B}$ of $X$ (subcategory of $Open(X)$). Suppose ...

**0**

votes

**0**answers

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

**0**answers

22 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

**1**answer

187 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

**0**answers

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

**2**answers

109 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 ...

**0**

votes

**0**answers

83 views

### Handbook Homogeneous Space Examples?

Homogeneous spaces are super common in differential geometry, with many classy tricks for linking them together. Is there anywhere available a list covering the well-known examples of the homogeneous ...

**2**

votes

**0**answers

60 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

**0**answers

108 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

**0**answers

65 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

**0**answers

99 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 ...

**0**

votes

**0**answers

32 views

### Equation of a curved line from a graph [on hold]

I am trying to calculate an equation to represent the graph attached to this question. It's an extract from a take-off performance graph used in aviation.
The second graph shows how it is used. The ...

**-2**

votes

**0**answers

32 views

### On Incidence structure of finite Projective plane [on hold]

Consider a finite projective plane $\mathcal{P}$ over a finite field $F_q$, $q$ a prime power. Is it possible to define a map $f:\mathcal{P}\times \mathcal{P}\rightarrow \mathcal{P}$ such that
(i) ...

**0**

votes

**0**answers

14 views

### Algebraic operations with memory hardness properties

In cryptography, there are password hash functions like scrypt and argon2 for which the fastest known algorithms employ large ...

**3**

votes

**1**answer

91 views

### Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets

Definitions.
By an $\mathbb{N}$-graded set, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the grading. These will simply be called graded sets hereafter.
If ...

**1**

vote

**0**answers

22 views

### Singularities of algebraic curves, and torsion in the cotangent space

The problem in the following :
given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...

**2**

votes

**0**answers

64 views

### How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of
$$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$
$$\partial_\nu u = \alpha \quad \text{on ...

**5**

votes

**3**answers

224 views

### Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for ...

**3**

votes

**0**answers

35 views

### Maximal TB number and slice genus relation of a knot in any 3-manifold

Lisca-Stipsicz say that if a knot $K\subset{S^{3}}$ satisfies $g_{s}(K)>{0}$ and $TB(K)=2g_{s}(K)-1$ then $S^{3}_r(K)$ carries positive, tight contact structures
for every ...

**-5**

votes

**0**answers

44 views

### Should I learn information technology? [on hold]

I am 18 y.o., and I want to go to university. I will study
Software Engineering (Computer science), but I do not know yet what I want to be exactly.
I love math, phisycs, psychology, philosophy, ...

**3**

votes

**1**answer

82 views

### Cells in affine Weyl groups

This may sound like a very general question, which pretty much reflects my ignorance on the subject.
In the case of Weyl groups $W$, there is a notion of left/right/double cells, which is roughly ...

**10**

votes

**0**answers

79 views

### Weak*-closure of finite rank operators on dual space

Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is ...

**4**

votes

**1**answer

87 views

### Continuous non-constant function with infinite intersections with horizontal line on a compact interval?

The title might be misleading, but whether such a function exists is what boggles me about the following problem:
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all ...

**1**

vote

**0**answers

39 views

### Wave-like equation with 1st order time derivative and non-constant coefficients

We start with the following recurrence relation for complex coefficients $c_{n,m}$:
$$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m-2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$
where ...

**6**

votes

**0**answers

105 views

### Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in ...

**4**

votes

**0**answers

94 views

### Another interpretation of the $16$ dimensional Severi Vairety

I asked about an interpretation of this variety here. There is another one that could be easier. Let $K$ be an algebraically closed field of characteristic $0$. We denote the set of terns of $3\times ...

**0**

votes

**0**answers

15 views

### The effective strategy for choosing epsilon_init of random initialization in neural networks [on hold]

gays. i meet some problem in a description about choosing epsilon_init for random initialization in neural networks.
enter image description here
i don't know why the epsilon_init is related to the ...

**-3**

votes

**0**answers

44 views

### The line graph of a complete graph [on hold]

Show that there exist a $\left\{P_{5},C_{4}\right\}$- decomposition of the graph $L(K_{9})$.