# All Questions

**0**

votes

**0**answers

8 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 $S^3$?

**0**

votes

**0**answers

13 views

### The weak-star closure of closed left ideals corresponded 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 ...

**0**

votes

**0**answers

5 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 ...

**0**

votes

**0**answers

13 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 ...

**-4**

votes

**0**answers

21 views

**1**

vote

**1**answer

43 views

### When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable

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

34 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 ...

**-3**

votes

**0**answers

38 views

### How to evaluate this equality [on hold]

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$
where $\cdot$ is the ...

**6**

votes

**1**answer

134 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 ...

**6**

votes

**1**answer

252 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

97 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 ...

**13**

votes

**2**answers

244 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

74 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

62 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

12 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

19 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 | ...

**9**

votes

**1**answer

161 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

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

96 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

56 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 ...

**1**

vote

**0**answers

45 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

104 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

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

56 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

94 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

30 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 ...

**2**

votes

**1**answer

81 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

19 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

59 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 ...

**4**

votes

**3**answers

211 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 ...

**2**

votes

**0**answers

24 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

73 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

73 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 ...

**3**

votes

**1**answer

85 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

38 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 ...

**5**

votes

**0**answers

96 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

90 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

13 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

43 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 $L(K_{9})$.

**4**

votes

**2**answers

93 views

### Bounding and dominating numbers ${\frak b}, {\frak d}$ via ultrafilters

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ and suppose that ${\cal U}$ is a free ultrafilter on $\omega$. We write $f \leq_{\cal U} g$ if $$\{n\in\omega: f(n) \leq ...

**9**

votes

**1**answer

108 views

### Calculation of the integral related to the gravitational shock wave

The following integral
$$\int\limits_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$
can be found in the paper
Tevian Dray and Gerard 't Hooft, The ...

**3**

votes

**1**answer

154 views

### Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...

**1**

vote

**0**answers

42 views

### Difference in the Four Color Theorem [on hold]

How is proving that any planar graph with maximum degree of four has a four coloring, different from proving the four color theorem? If they are different, then how would one prove it?

**-2**

votes

**0**answers

47 views

### Adherent value of sin(sqrt(n)) [on hold]

I have been struggling against this question for several hours and I really need some help. It is from my undergraduate real analysis course. Your time and help is greatly appreciated.
I got struck ...

**2**

votes

**0**answers

49 views

### Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...

**1**

vote

**1**answer

64 views

### Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?
It seems like it should be easy ...

**1**

vote

**1**answer

52 views

### Is there a generalization for the discrete fourier transform whereby eigenvalues are other roots of unity?

The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions.
https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors
Is ...