0
votes
0answers
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
0answers
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
0answers
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
0answers
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 ...
1
vote
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
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
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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 ...

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