# All Questions

**-4**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**4**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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 ...