**-1**

votes

**0**answers

34 views

### Irreducible linear set of quadratics over $\Bbb F_p$

Given $f(x)\in\Bbb F_p[x],\gamma\in\Bbb F_p$ with $\mathsf{deg}(f(x))=2$.
Denote $S(f(x),\gamma)$ to be set of polynomials of form $$f(x)+x(x-\gamma)\beta$$ where $\beta\in\Bbb F_p$.
Denote ...

**0**

votes

**1**answer

8 views

### Generating (or availability of) large strongly regular graphs

Are there collections of already generated large strongly regular graphs available to download? By large I mean $n >= 200$ where $n$ is the number of vertices. I found Ted Spence's page on srgs, ...

**0**

votes

**0**answers

8 views

### Conditions for existence of Penrose diagrams

A Penrose diagram (also known as a conformal diagram or Carter-Penrose diagram) is a technique for visualizing the causal (light-cone) structure of a 3+1-dimensional manifold. Usually the diagram is ...

**-3**

votes

**0**answers

16 views

### summation of non-linear function [on hold]

Can anyone have idea for dealing with the two following series summations
∑_(i=1)^n▒1/(a+bx_i )=c
∑_(i=1)^n▒x_i/(a+bx_i )=d
I need to find the values of 'a' and 'b'; 'c' and 'd' are known. x_i is ...

**4**

votes

**0**answers

23 views

### Arithmetic Points are Dense on a Hida Family

I am reading the paper "Constancy of the Adjoint L-invariant" by H. Hida (http://www.math.ucla.edu/~hida/ConstP.pdf).
Correct me if I'm wrong, but I've read/heard that the arithmetic points $p \in ...

**2**

votes

**0**answers

19 views

### The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

Recall the notion of a "nearly holomorphic modular form" introduced by Shimura:
A function $f : \mathfrak h \to \mathbb C$ is said to be nearly
holomorphic of level $\Gamma_1(N)$, weight $k$ and ...

**2**

votes

**2**answers

54 views

### Asymptotic expression for $j$ which satisfies $\binom{n}{j}/j! \sim k$ as $n\to\infty$

Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$
The asymptotic expression for ...

**1**

vote

**0**answers

60 views

### Reference request: Ebin

I'm after the paper The manifold of Riemanian metrics by D. Ebin. A link to the reference is:
http://www.ams.org/mathscinet-getitem?mr=0267604
The paper seems to be very hard to track down. Can ...

**-3**

votes

**0**answers

29 views

### Check if the equality holds [on hold]

I have the following problem. For orthogonal $8\times 8$ matrix $M$ ($M\cdot M^{T} = 1$) check if the following equality holds
$$
U = M^{T} \cdot \left( \begin{array}{cc} 1_{3\times 3} & 0\\
...

**8**

votes

**2**answers

169 views

### Free $k[x_1, \dots, x_n]^{S_n}$-module?

Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} ...

**15**

votes

**0**answers

130 views

### Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.
$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...

**-3**

votes

**0**answers

9 views

### Generating Random Variates from CDF [on hold]

Suppose I am given a CDF of a distribution, given by F(x) ∝ (x + x^2 + x^4 + x^7).
How do I generate random variates from this distribution ?
Thanks in advance.

**3**

votes

**2**answers

39 views

### Simplifying a Taylor polynomial that involves Stirling numbers of the second kind

I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form:
$$\sum_{n=1}^k ...

**6**

votes

**2**answers

386 views

### Category theory for Algebraic Geometry

How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?

**1**

vote

**0**answers

17 views

### General Markov Chains on same Probability Space?

A Markov chain $(X_i)_{i\in \mathbb{N}}$ on a measurable space $(E,\Sigma)$ is (see e.g. Revuz or Meyn/Tweedie) constructed on the following probabilty space.
$$ \Omega = \{ (x_l)_{l \in \mathbb{N}} ...

**1**

vote

**2**answers

132 views

### indecomposable decomposition for a commutative ring

Let $R$ be a commutative ring with identity. We say that $R$ has an indecomposible decomposition if it can be wrighten as a finite direct sum of indecomposiable rings.
Is there any characterization ...

**3**

votes

**0**answers

23 views

### “Spherelike” embedding for smooth Fano polytope

Let $\mathbb{R}^n$ be a vector space with Euclidean inner product and $\mathbb{Z}^n \subset \mathbb{R}^n$ be a lattice. Let $P$ be a full dimension convex lattice polytope in $\mathbb{R}^n$ containing ...

**3**

votes

**0**answers

45 views

### Undecidability of the existential theory

Do you know if I can find the proof that the existential theory of $\mathbb{Z}$ with the structure of addition , divisibility and the relation $(\exists s \in \mathbb{Z})m=np^s$ is undecidable, ...

**1**

vote

**0**answers

18 views

### A lower bound for orthogonal partial circulant matrices

Let us call an $m$ by $n$ matrix with $m<n$ a partial circulant matrix it is the first $m$ rows of some square circulant matrix.
Consider partial circulant matrices whose elements are either $-1$ ...

**2**

votes

**1**answer

23 views

### Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points

I am currently working on a problem where I have to minimize a $m$-strongly convex function
$$f ~: ~\mathbb{R}^n \rightarrow \mathbb{R}^+$$
over a bounded integer lattice,
$$L = \mathbb{Z}^n \cap ...

**9**

votes

**2**answers

137 views

### Curvature of a finite metric space

I am sorry to ask a very vague question, but:
What are good ways to define the curvature of a finite metric space?
The best way I can think of is: the curvature of a finite metric space $M$
is ...

**2**

votes

**0**answers

26 views

### Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...

**8**

votes

**1**answer

533 views

### Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?

**5**

votes

**0**answers

31 views

### What's the relation between half-twists and star structures on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...

**2**

votes

**0**answers

33 views

### Polynomials with positive coefficients passing through fixed points/range of Vandermonde matrices

I'll give two equivalent statements of the setup, then give my questions.
Fix integers $M \leq N$ and define the Vandermonde-like matrix $V_{M,N}[i,j] = (1 - \frac{i}{M})^{j-1}$ for $i \in ...

**-4**

votes

**0**answers

27 views

### Multivariate Calculus: Switching the Order of Integration [on hold]

This is confusing the heck out of me... I am asked to switch the order of integration of the following function:
$$
\int_{0}^{1} \int_{0}^{1-x} \int_{0}^{x+y} f[x,y,z] dz dy dx
$$
The order of ...

**2**

votes

**0**answers

58 views

### Deformation of finite coverings between smooth projective varieties

Assume that we have a finite covering $$f \colon X \longrightarrow Y,$$
where $X$ and $Y$ are smooth, complex projective varieties of dimension $n$. Therefore we obtain a splitting $$f_* \mathscr{O}_X ...

**1**

vote

**0**answers

48 views

### Representing trigonometric functions in a form of rational functions

Here I introduced a non-Archimedean numerical system in which the real numbers are extended by elements $\omega_-$, $\tau=\omega_-+1/2$, $\omega_+=\omega_-+1$ in such a way that standard parts of ...

**2**

votes

**0**answers

45 views

### Positive existential theory of $(\mathbb{Z}; +, |_n)$

I am reading a paper and there is the following theorem:
Let $n$ be a fixed integer, and $n >1$.
Denote divisibility in $\mathbb{Z}[\frac{1}{n}]$ by $|_n$, thus for
all $x, y \in ...

**1**

vote

**0**answers

65 views

### vector spaces with uncountable dimension and a nice basis

Unlike the reals considered as a vector space over the rationals, I know of a number of nice examples of vector spaces with uncountable dimension that have a nice basis.
For example, the space of ...

**6**

votes

**1**answer

402 views

### Segal's 1999 Stanford lecture notes on TQFT, where to find them?

I am trying to track down a copy of Graeme Segal's 1999 lecture notes on topological field theory. These are sometimes referred to as the "Stanford lectures" or something similar.
For many years ...

**-4**

votes

**0**answers

88 views

### Are polls good approximations [on hold]

Let $X$ be a finite set and $A\subseteq X$ and $m$ be a natural number satisfying $m\le |X|$ and $\epsilon$ be a small positive number.
I'm interested to know if one selects a random $Y\subseteq X$ ...

**53**

votes

**46**answers

7k views

### Important formulas in Combinatorics

Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...

**1**

vote

**0**answers

64 views

### Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves:
$\frac{d U_s}{ds} = (a + w(s)b)U_s$
for some given $a,b \in \mathfrak{su}(4)$ and a smooth real, bounded function $w: [0,T] ...

**4**

votes

**2**answers

39 views

### convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...

**0**

votes

**0**answers

25 views

### What is the Segre class of a generating line of a cone

Suppose $U=\textrm{Proj}\ k[X,Y,Z,W]/(XY-Z^2)$ is the projective closure of an affine cone, let $V$ be a generating line of the cone $V=V(Y,Z)$, how do we calculate the Segre class $s(V,U)$?
(We can ...

**0**

votes

**0**answers

16 views

### Quantification of the extent of periodicity in a time series using fractal analyses

Suppose I have a time series which is almost periodic. If I were to segment each of the visually most evident periods i.e. say of the longest period, I would find a strong mean cross-correlation among ...

**0**

votes

**0**answers

34 views

### Properties of a specific antichain of a lattice formed by the cartesian product of finite ordered sets

Let $X$ be a poset of all $n$-tuples, $x = (x_1, x_2, ..., x_n)$, where $0 \leq x_i \leq m_i - 1$ for $i = 1, ..., n$ together with the relation $x \prec y$ defined so that for $y=(y_1,y_2, ..., ...

**-3**

votes

**0**answers

56 views

### Eigenvalues of a random matrix [on hold]

For test cases i generated a random real uniform distributed matrix with
entries from the intervall $[0,1]$.
Here is the MATLAB Code i used
...

**2**

votes

**2**answers

163 views

### Connectedness of moduli of vector bundles

Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? ...

**3**

votes

**1**answer

125 views

### Example of measure of non-compactness

I can't understand the following example of measure of non-compactness, which was given in this article.
Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...

**5**

votes

**1**answer

42 views

### Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...

**4**

votes

**0**answers

79 views

### How “small” can an ordinal be made by forcing?

I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ...

**5**

votes

**1**answer

358 views

### A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...

**2**

votes

**1**answer

107 views

### Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...

**-1**

votes

**0**answers

30 views

### Obtaining z-transform of a multivariate nonlinear difference equation [on hold]

I need to obtain the z-transform of difference equations that are as follows:
My problem however is multivariate and looks like this:
x[k+1]=ay[k]+ ((x[k])^2)(y[k]) ...

**0**

votes

**1**answer

25 views

### floating point representation via the perspective of TTE/computable analysis

Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ...

**7**

votes

**1**answer

176 views

### Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer.
Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$
denote the sum of divisors of $n$. Recall that we have ...

**20**

votes

**2**answers

848 views

### What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...

**2**

votes

**2**answers

125 views

### Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 ...