-4
votes
0answers
56 views

sum of the series and decimal system [closed]

This is a lighter version of the questions that I asked yesterday. If you have the sums $f(n) = 1^{29} + 2^{29} + 3^{29} + \cdots + (10^n)^{29}$ and $g(n) = 1^{2} + 2^{2} + 3^{2} + \cdots + ...
3
votes
0answers
42 views

Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...
0
votes
0answers
38 views

Conditional probability of dependent random variables

Let $ X \sim f_X(x), Y \sim f_Y(y) $ are two dependent random variables and their corresponding PDFs. I want to find a probability $$ P(Y\ge 0 | X+Y\ge 0) .$$ If these variables were independent I'd ...
0
votes
0answers
29 views

Number of real singular points

Let $f\in\mathbb{R}[X,Y]$ be a real polynomial in two variables. Are there bounds on the number of singular points of $f$ which take into account that $f$ might have high degree but be rather sparse? ...
2
votes
1answer
27 views

Information on special matrices similar to Jacobi matrices

Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...
-1
votes
0answers
96 views

field of constants of a curve [on hold]

I'm trying to gain some intuition about the field of constants of a curve. If $C$ is over a field $k$, then it is defined as the set of elements of $k(C)$ algebraic over $k$. If I understood ...
5
votes
2answers
139 views

What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?

Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot. Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$? Are there any ...
1
vote
0answers
12 views

Bifurcations in flows on two dimensional torus

I want to have a research about bifurcations which are appeared in flows on two dimensional torus. Especially bifurcations that can not be seen in flows of $\mathbb{R}^2$. Can anyone introduce me ...
4
votes
1answer
176 views

continuity of the Boltzmann entropy in the Wasserstein metric

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ ...
2
votes
0answers
33 views

Is this infinite family of non-trivial snarks resulting from the first Celmins-Swart?

Non-trivial snark is cubic graph with chromatic index $4$, girth at least $5$ and doesn't to contain three edges whose deletion results in a disconnected graph, each of whose components is nontrivial. ...
0
votes
0answers
42 views

Characterisation of vector fields solution to a simple equation

This question is complementary to another question I asked on math.stackexchange. I believe it is more subtle than it seems - it will become clearer when I provide more context - and probably hides ...
0
votes
0answers
69 views

Existence of non-trivial characters on compact abelian group [closed]

Does for every compact (compact metric) abelian group $(G, \odot )$ there exist a non-trivial homomorphism $\varphi : (G, \odot ) \to (\mathbb{C} , \cdot ) $ such that $|\varphi (g) |=1$ for all ...
2
votes
1answer
150 views

An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold $\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
11
votes
1answer
203 views

Travelling Salesman Problem: Can the nearest neighbor algorithm be $n$ times longer than the optimal solution?

This is inspired by a recent question. Given a positive integer $n\in\mathbb{N}$, is there a setting of finitely many points and a designated "starting point" $s$ in $\mathbb{R}^2$ such that the ...
5
votes
1answer
270 views

Computing an eigencuspform in $S_2(\Gamma_0(1776))$

Consider $$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$ the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then ...
0
votes
0answers
129 views

Is anything known about a ternary equivalent of groups?

Group theory studies the properties of algebraic structures that combine a set of elements with a binary operation. Different structures such as Monoids, Semigroups, Groups, Rings, Fields etc demand ...
35
votes
2answers
1k views

Arctangents of odd powers of the golden ratio

While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of ...
6
votes
0answers
77 views

Optimal strategy for game of 'online sorting' into a poset

Consider a single-player game played with an arbitrary finite poset, and a random number generator with a known distribution: Each turn, the RNG produces a number, and the player must assign that ...
1
vote
1answer
80 views

Elliptic pde with bilaplacian; boundary conditions.

I am interested in the solvability of $$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ ...
0
votes
0answers
21 views

property of orthonormal systems and sequences in Hilbert space [closed]

Problem: Let $H$ be a separable Hilbert space and {$e_n$} a complete orthonormal system of $H$. Prove that, if {$y_k$} is a bounded sequence in $H$, the condition $\lim_{k→∞} (e_n , y_k ) = 0$ for ...
0
votes
0answers
124 views

Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence $$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$ splits? I assume $K $ to be a number field and ...
2
votes
0answers
54 views

Tensor product of bounded analytic functions

I asked this question on math.SE, but couldn't get an answer. Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. ...
-2
votes
0answers
68 views

sum of the series and infinity [closed]

If you have the sums $f(n) = 1^{59} + 2^{59} + 3^{59} + \cdots + (10^n)^{59}$ and $g(n) = 1^{5} + 2^{5} + 3^{5} + \cdots + (10^n)^{5}$, for large enough $n$, $f(n)$ is approximately $\frac{1}{60} ...
2
votes
0answers
54 views

Choice of framing in Gravitational Chern Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
0
votes
0answers
14 views

Bayes' Rule where the probabilities are taken as conditional [migrated]

I'm encountering some difficulty beginning statistics work with a basic Bayes' Rule problem. You can see the problem and answer on page 16 here, but I've explained it below. ...
5
votes
2answers
65 views

Discrete optimization problem

Suppose you had $N$ many fixed points $X_1, X_2, ..., X_N$ in some Euclidean space $R^d$ and from these coordinates you had to choose $n$ many of them ($n \leq N$ also being fixed) to form a subset ...
4
votes
1answer
46 views

Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$

Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem? The only thing I can find is ...
-4
votes
0answers
50 views

Let p be prime, is there a divisor d of p-1 or p+1 with gcd(d!,p+d)=1 such that p+d is prime ? [closed]

My previous question was incomplete. Please accept my apologies.
5
votes
2answers
265 views

Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$ by $$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$ The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...
14
votes
4answers
682 views

Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
0
votes
0answers
12 views

heavy subgraph searching result in pseudopatterns in tensor [closed]

I encounter problem while trying to find heavy subgraph in tensor. I'm trying to maximize H(x,y)=1/2 summation a(ijk)x(i)x(j)y(k) Why do I only find pseudopatterns in heavy subgraph searching in ...
1
vote
0answers
88 views

References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
4
votes
0answers
99 views

Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L

I am hoping someone can estimate the number of primes that come up this way: take a number $L,$ then let $$ C = \operatorname{lcm} (1,2,3,\ldots,L). $$ We know that $C$ has quite a lot of divisors; ...
-1
votes
0answers
16 views

How should strongly correlated covariates for logistic regression be treated? [closed]

I have to build a logistic regression for multiple covariates (predictor variables), two of which are strongly correlated. How should they be treated? Am I to exclude one of them from the regression? ...
2
votes
2answers
155 views

Permutation covering of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$. We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
3
votes
1answer
103 views

Stationary distribution of last passage percolation

Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ...
-2
votes
0answers
32 views

Finding the equation of a curve from two given points [closed]

" A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve. "
3
votes
0answers
83 views

Can I find the gap between the two least eigenvalues of this special matrix A(t)?‎

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse ‎matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal ...
0
votes
0answers
201 views

Whitehead group [closed]

Whitehead group (WG) is known for some groups (e.g. free abelian group, cyclic group, Braid group etc). For example: The Whitehead group of the trivial group is trivial. The Whitehead group of a ...
-3
votes
0answers
58 views

Messages on rotating wheels [closed]

The question is: Would it be possible display a message (image, logo, text, ...) on a rotating wheel so that it would become readable once rotating at a certain speed, knowing that our brain will ...
0
votes
1answer
39 views

Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of: $$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ...
0
votes
1answer
100 views

rationality of residues of differentials

Let $C$ be a smooth curve over a field $k$, $\overline{C}$ the smooth compactification and $S=\overline{C} \setminus C$. We think of $S$ as a reduced divisor defined over $k$. Take the sheaf of ...
1
vote
0answers
35 views

Independence of inverse system to define continuous cohomology for profinite groups

I have a problem concerning cohomology of profinite groups as it is defined e.g. in Gille's and Szamuely's "Central Simple Algebras and Galois Cohomology" on page 86. For a profinite group ...
1
vote
0answers
29 views

Boundary conditions of PDE from SV model with stochastic interest rate

The PDE for the American put option price $P(S,\sigma ,r,t)$ is \begin{align*} 0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\ +& \frac{1}{2}P_{SS}S^2\sigma ^2 + ...
1
vote
0answers
98 views

Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...
-3
votes
0answers
224 views

Mathematical theories of changes - except from calculus? [closed]

Unfortunately motion is regarded as displacement in geometry: By a motion or displacement in the general sense is not meant a change of position of a single point or any bounded figure, but a ...
-3
votes
0answers
78 views

Prove bijection beetween sets [closed]

Prove that if the set $ X $ is infinite, and a subset $ Y $ is finit, there is a bijection $ X \setminus Y \to Y $. It seems a simple task, but no ideas yet. At first I thought that between these ...
7
votes
2answers
850 views

How to prove that this equation has only one solution?

I can't find a way to prove that the following equation has only one solution : $$ X = \frac{2^Q - 1}{2^{P+Q} - 3^P} $$ with $X,P,Q$ integers $> 0$. One trivial solution is $X = 1, P = 1, Q = ...
1
vote
1answer
76 views

integral closure of m-primary ideals

I need help with this excercise Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that ...
1
vote
2answers
93 views

Are spherical harmonics uniformly bounded?

The spherical harmonics are given by $$Y^m_l(\phi,\theta):=N^m_l e^{im\theta}P^m_l(\cos \phi) $$ where $P^m_l$ are the associated Legendre Polynomials and $N^m_l$ is the normalisation. From ...

15 30 50 per page