2
votes
0answers
45 views

I discovered something, what do I do?

I've recently made a mathematical discovery and being still in high school I have no idea how to publish my discovery or who to tell. I told my teachers but that got me nowhere.
0
votes
0answers
6 views

convex optimization

Attached below is a convex problem. I just start learning this and is kind of confused of this question. I notice that the dom of W is convex, tr(WQ) is convex, the composition of convex functions ...
-3
votes
0answers
10 views

Coming up with a function or a single graph, given its characteristics (pre-calculus) [on hold]

Give an example of a function or a single graph which has the following characteristics: (a) Hole at (3,-1). ...
1
vote
0answers
28 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 ...
2
votes
0answers
24 views

Resource needed on Lerch's transcendent

I am looking for resources in english which discuss basic properties of the Lerch's transcendent function. The Lerch Transcendant is defined by: ...
2
votes
3answers
29 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, ...
4
votes
1answer
34 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 ...
-3
votes
0answers
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 ...
2
votes
0answers
23 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 ...
-3
votes
0answers
33 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\\ ...
1
vote
0answers
72 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 ...
1
vote
0answers
22 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}} ...
3
votes
0answers
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 ...
1
vote
0answers
25 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 $-1$ ...
3
votes
0answers
51 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, ...
15
votes
0answers
139 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 ...
2
votes
0answers
29 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 ...
2
votes
3answers
72 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 ...
3
votes
2answers
42 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 ...
2
votes
0answers
34 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 ...
2
votes
1answer
25 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 ...
5
votes
0answers
37 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 ...
-4
votes
0answers
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 ...
0
votes
0answers
30 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 ...
1
vote
0answers
49 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
0answers
46 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 ...
2
votes
0answers
65 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 ...
2
votes
0answers
82 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 ...
0
votes
0answers
36 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, ..., ...
8
votes
2answers
174 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}} ...
9
votes
2answers
140 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 ...
8
votes
1answer
579 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?
0
votes
0answers
54 views

Chern classes of ideal sheaf of locally complete intersection

Let $C\varsubsetneq X$ be a reducible locally complete intersection closed pure $1$-dimensional subscheme in a smooth projective variety of dimension $3$ over a algebraically closed field $k$ of ...
0
votes
0answers
17 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 ...
-3
votes
0answers
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
0answers
57 views

cohomology ring of base-point-preserving maps on the 3-sphere

I find that $\text{Map}_*(S^3;S^3)=\Omega^3S^3$. I want to find the cohomology ring of $H^*(\Omega^3S^3;\mathbb{Z}_2)$. In the paper On configuration spaces, their homology, and Lie groups, I find ...
5
votes
0answers
81 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 ...
-3
votes
0answers
35 views

eigenvalues of cycle and its complement [on hold]

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
1
vote
2answers
156 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 ...
2
votes
0answers
83 views

Can such categorical notion of action be formalized?

I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the ...
3
votes
1answer
130 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 ...
6
votes
2answers
420 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?
3
votes
2answers
112 views

bar construction and loop space cohomology

Let $X$ be a topological space. Then its chain complex $C_{*}(X)$ is naturally a coalgebra (as explained here Does homology have a coproduct?). In particular if $X$ is simply connected we have that ...
5
votes
1answer
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 ...
6
votes
1answer
408 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
0answers
25 views

What is the upper bound for training Linear separable set with Perceptron, Rosenblatt rule? [on hold]

I have the following neural networks problem and couldnt find any answer on the web. Any hints would help. I am not looking for a complete solution, just some pointing in the right direction. ...
2
votes
1answer
108 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 ...
4
votes
1answer
80 views

unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then $$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence $ (*) $ $$ ...
3
votes
0answers
46 views

Are all transversely oriented foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the ...
-2
votes
0answers
79 views

Fundamental Group of SL_2 [on hold]

I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ...

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