# All Questions

**0**

votes

**0**answers

24 views

### Eigenvalues of tridiagonal matrix.

I need to find eigenvalues of following tri-diagonal matrix. $n_1,n_2,...,n_k$ are positive integers.
\begin{equation*}
T=\begin{bmatrix}
-n_1 & n_2 & 0 &.&.&0 & 0 \\
...

**-3**

votes

**0**answers

19 views

### Number of trials until completion?

You've got a discrete uniform distribution - what is the expected number of trials until each point is hit at least once?
I started my thinking with maybe a Geometric distribution representing each ...

**0**

votes

**0**answers

15 views

### Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that ...

**2**

votes

**1**answer

71 views

### Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, ...

**0**

votes

**0**answers

22 views

### Polynomials pulled back by momentum maps

Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$.
Assuming I can find ...

**0**

votes

**0**answers

28 views

### Algorithm: Computing the intersection of two conics

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conics curves. The curves are given by two equations of the form:
$a x^2 + b y^2 ...

**0**

votes

**1**answer

34 views

### Eigenvalues of cyclic tridiagonal matrix

Prove that following cyclic tridiagonal matrix has two zero eigenvalues if and only if $k=6r$ for any positive integer $r$.
\begin{equation*}
T_\lambda=\begin{bmatrix}
-n_1 & n_2 & 0 ...

**2**

votes

**0**answers

30 views

### Which simplicial objects are Čech nerves?

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and ...

**5**

votes

**0**answers

45 views

### Definition of discrete spectrum and continuous and basic properties

I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...

**1**

vote

**0**answers

49 views

### What explains the asymptotic and the pattern in this sequence related to Riemann zeta zeros?

As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form:
$$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$
where $x_n$ is unknown. Therefore I ...

**5**

votes

**0**answers

33 views

### Is there a subset of $\Sigma_n$ s.t. each pair of elements is once in each pair of positions?

Is there a subset $A \subset \Sigma_n$ such that for each pair $(x, y)$ and each pair $(i, j)$, there is exactly one permutation $\sigma \in A$ such that $\sigma(i) = x$ and $\sigma(j) = y$? Remark ...

**-1**

votes

**0**answers

29 views

### What will draw a shape for $L = \left\{ {\lambda \in \mathbb{C}:{s_4}(\lambda ) = {s_3}(\lambda )} \right\}$ [on hold]

Let $P(\lambda ) = \left( {\begin{array}{*{20}{c}}
{{\lambda ^2} - 1} & 0 \\
0 & {{\lambda ^2} - 2\lambda } \\
\end{array}} \right)$ and $\lambda \in \mathbb{C}$( $λ$ is a complex ...

**0**

votes

**0**answers

23 views

### What is difference between length of proof and length of its presentation in Peano Arithmetic?

In this paper http://www.sciencedirect.com/science/article/pii/0304397584901117 section three it seems to imply there is a difference between length of proof and length of its presentation in Peano ...

**0**

votes

**0**answers

28 views

### What is the known weakest axiom system has Löb's derivability conditions?

We know that Peano Arithmetic satisfies Löb's derivability conditions, which is required in the proof of Gödel's 2nd incompleteness theorem. Is this the best result? If not, is there any known weaker ...

**1**

vote

**1**answer

84 views

### The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...

**0**

votes

**0**answers

25 views

### Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...

**0**

votes

**0**answers

73 views

### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...

**0**

votes

**0**answers

23 views

### Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property.
Here is the definition of f.p.s.p.("map" means ...

**2**

votes

**2**answers

40 views

### Random processes with smooth paths

Is there any prototypical example of a Random process with smooth paths?
I imagine one can simple integrate each path of a Brownian motion and get a $C^{\frac32-\epsilon}$ path.
It's easy to ...

**1**

vote

**0**answers

77 views

### Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for a smooth ...

**-2**

votes

**0**answers

23 views

### Linear transformation from n-dimensinal vector space to Rn [on hold]

Suppose:
U is a real n-dimensional vector space, and B = {$u_1$, $u_2$,...,$u_n$} be a basis for U, let $T: U \to R^n$ be the linear transformation defined by $$
T(u) = [u]_B
$$
How to prove:
...

**2**

votes

**2**answers

51 views

### Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...

**-4**

votes

**0**answers

28 views

### Volume and Surface Area Algebra Rational Functions [on hold]

A frozen yogurt cone has a volume of 10 cubic inches. The surface area of a cone exluding the base is S = pi r sqrt(r^2+h^2), where r is the radius of the base and h is the height.
Find the ...

**1**

vote

**1**answer

111 views

### Completing class-sized Fields

Let's say that an ordered Field is a class (proper or not) which satisfies the axioms of ordered fields. We work in NBG set theory with global choice.
Let's say that an ordered Field is real closed ...

**2**

votes

**0**answers

42 views

### Two questions on the James $p$-space $J_{p}(1<p<\infty)$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that
...

**1**

vote

**0**answers

75 views

### First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...

**1**

vote

**0**answers

34 views

### Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations.
Let $A$ be an ...

**1**

vote

**0**answers

75 views

### Base change and geometrically generic reduced fiber

Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...

**3**

votes

**2**answers

189 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**3**

votes

**0**answers

82 views

### What is the integral cohomology of an Enriques surface over a finite field?

Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, ...

**0**

votes

**1**answer

150 views

### Field extension and nilpotent element

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...

**-3**

votes

**0**answers

143 views

### Maps from $S^3$ to $S^3$ [on hold]

As a physicist, I apologize for imprecise language.
I am interested in maps from $S^3$ to $S^3$ (identical to the group $SU(2)$). Since $S^3$ is threedimensional, there is some similarity to maps ...

**3**

votes

**1**answer

120 views

### Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...

**1**

vote

**0**answers

56 views

### Question about continuity in the “complete Skorohod Topology”?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process"
https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf
In one of the exercises, exercise 8.9 ...

**17**

votes

**1**answer

943 views

### Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path ...

**2**

votes

**0**answers

31 views

### Are rays in Carnot groups straight?

A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...

**-1**

votes

**0**answers

54 views

### An efficient algorithm for computing all semigroups of order n [on hold]

I attached two papers which give an algorithm for computing all semigroups of order n=3, and n=5.
I understood the first(table 3) and second(table 4) steps of algorithm, but I can't understand the ...

**4**

votes

**0**answers

45 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

**-2**

votes

**0**answers

32 views

### $k$-th diagonal element of an inverse matrix $(\textbf{H}^{\dagger}\textbf{H})^{-1}$ [on hold]

Let $\textbf{W}=\textbf{H}^{\dagger}\textbf{H}$ with $\{\cdot\}^{\dagger}$ being conjugate transpose operator. In some materials I have read so far, there is a common statement that the $k$-th ...

**0**

votes

**1**answer

67 views

### Surniversal spaces

Basic background
On one hand there is a complete result: $\,\ $for every non-negative integer $n$ there exists an $n$-dimensional compact metric space $M^n$ such that it contains a homeomorphic ...

**1**

vote

**0**answers

40 views

### On a sequence of integers

I recall a well-known theorem due to Minkowski.
Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...

**10**

votes

**2**answers

298 views

### From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...

**-3**

votes

**0**answers

48 views

### Variable modification of the functional equation for the Riemann Zeta-function

Be
$$f_x(s):=\frac{\zeta(x+1-s)}{\zeta(x+s)}-\frac{\Gamma(x+\frac{s}{2})}{\Gamma(x+\frac{1-s}{2})}\pi^{\frac{1}{2}-s}$$
with $x\in\mathbb{R}_0^+$, $s_k(x)$ and $s_{-k}(x)$ the zeros of $f_x(s)$ with ...

**1**

vote

**0**answers

24 views

### Perturbations on SVD decompostions

Given a symmetric matrix $A\in \mathbb{R}^{n\times n}$, with all the entries greater than zero $A_{i,j}>0$ with rank $k<n$, we can calculate its SVD decomposition:
$$
A = USU'
$$
from which we ...

**4**

votes

**1**answer

117 views

### Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...

**0**

votes

**0**answers

21 views

### How to prove prime avoidance for graded cases?

Let $R$ be a nonnegatively graded ring such that $R_0$ is local with infinite
residue field. Let $I,J_1,...,J_s$ be a homogeneous ideals of $R$ such that $I$ is not contained in $J_i$. Please prove ...

**1**

vote

**1**answer

95 views

### Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...

**0**

votes

**0**answers

58 views

### inverse of operator [migrated]

I want to calculate the inverse of the operator
$T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial ...

**0**

votes

**0**answers

35 views

### The exponential Diophantine equation $a^n-b^m=x^3+y^3$ for arbitrary large $n,m$

Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers.
Consider the exponential Diophantine equation
$$ a^n-b^m=x^3+y^3 \qquad (*) $$
Nontrivial solution ...

**3**

votes

**1**answer

115 views

### 4-th order diophantine equation

I met in many places the equation
$(a^4-b^4)(c^4-d^4)=\square$
It is well known that this was investigated by Euler.
But I was unable to find the general solution of this equation. Could you please ...