**-1**

votes

**0**answers

17 views

### Bundle with a symmetric bilinear form

Sorry if the question is not of high level!
Given a vector bundle $E$ of rank $r$ over a curve $X$, s.t there existe a symmetric bilinear form $$\psi:E\otimes E\rightarrow \mathcal O_X$$.
The ...

**0**

votes

**0**answers

15 views

### A question in matrix polynomial

Suppose
${A_j} \in {C^{n \times n}},(j = 0,1,2....m)$
${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a complex ...

**-1**

votes

**0**answers

30 views

### Faithfully flat ring extension

Let $R$ be a commutative ring with identity and $R [x]$ and $R[[x]]$ be polynomial ring and power series ring over $R$. Is $R[[x]]$ a faithfully flat ring extension for $R[x]$?

**0**

votes

**0**answers

6 views

### strong law of large number for semimartingale

I just want to know if for semimartingale X we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{<X>_{t}}=0$ or when it is possible. i know it is true for brownian motion.
Thanks

**0**

votes

**0**answers

33 views

### formula for sequence 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, [migrated]

There is a sequence with the values 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... (basically there are always four 0s followed by a 1, then it repeats).
Is there a function for this sequence?
Here are two ...

**3**

votes

**0**answers

55 views

### Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from
$$
xq+x^2q^4+x^3q^9+...+x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n};
$$
that is,
$$
\sum_{d|n}x^{\frac ...

**-4**

votes

**0**answers

16 views

### 2's complement subtraction conversion to decimal for checking [on hold]

I was having some problem when trying to perform a 2's complement subtraction. So the question is:
01110101
- 11010110
----------
Then I perform the following ...

**2**

votes

**0**answers

38 views

### Solution to $(A+x^2)e^x=B$ with Lambert W function

Is it possible to obtain a analytical solution for $(A+x^2)e^x=B$, where we want to solve for $x$ with $A,B$ as constants?

**-5**

votes

**0**answers

59 views

### Arun Bhandari,Master of philosophy in applied mathematicsm, Kathmandu University ,Nepal [on hold]

Greetings from Arun Bhandari, I am doing research in Numerical methods for nonlinear differential equations. Currently, I am working on He's Variational Iteration Method for this I need following ...

**1**

vote

**0**answers

15 views

### Stiefel-Whitney class of unordered configuration space

Let $S^n$ be the $n$-sphere. Then the unordered configuration space $B(S^m,2)=F(M,2)/\Sigma_2$ is the total space of a line bundle over $\mathbb{R}P^m$, i.e. we have a fibre bundle
$$
\mathbb{R}\to ...

**1**

vote

**0**answers

66 views

### tangent space of line bundles over projective space

Let a line bundle
$$
\eta:\mathbb{R}\to E(\eta)\to \mathbb{R}P^m.$$
I want to study the tangent bundle $TE(\eta)$.
Question 1. When $n$ is even, $\mathbb{R}P^m$ is non-orientable. Does this imply ...

**25**

votes

**5**answers

1k views

### Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g.,
$$z^3+z^2+2 z+3 \;.$$
Find its $n$ roots, and list them in order of their modulus:
$$-1.28, (0.14\pm 1.53 i)$$
Now form a new ...

**1**

vote

**1**answer

80 views

### Is the locus of points which have irreducible fibers constructible?

Suppose $X \rightarrow Y$ is a map of projective schemes over a field $k$. Is $\{y \in Y: \pi^{-1}(y) \text{ is irreducible}\}$ a constructible subset of $Y$?
Note: One cannot hope to do "better" ...

**2**

votes

**1**answer

30 views

### Avoiding the range of a bivariate function or Diophantine function

I have a bivariate integer function where x,y are positive integers in the function $f(x,y)=5+23x+7y+30xy$. The lattice points of this function, or its range, contain a large number of values. I'm ...

**3**

votes

**0**answers

137 views

### A conjecture like Cayley–Bacharach theorem

Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic ...

**5**

votes

**2**answers

344 views

### The sum of a series, continued

In this question the OP asks whether the sum
$$
f(q, \alpha) = \sum _{k=1}^{\infty } \frac{q^k \left(q^k-1\right)^\alpha}{(q;q)_k}
$$
is ever zero. An experiment with Mathematica indicates, to any ...

**-3**

votes

**0**answers

59 views

### What would be the impact - to the foundation of First Order Logic - of a sentence whose truth value is impossible to verify or know? [on hold]

Suppose there's a sentence F written in L(PA) that is undecidable in PA, and whose truth value is impossible to verify (know), then face value it seems both the formal systems T1 = PA + {F} and T2 = ...

**2**

votes

**3**answers

49 views

### Algorithm to determine isomorphism of 2 maximal planar graphs

I read on wikipedia that there are efficient algorithms to answer the question whether 2 (maximal) planar graphs F and G are isomorphic. However, after some (IMHO) substantial searching I don't seem ...

**0**

votes

**0**answers

21 views

### A Statement from Brauer and Nohel's book on stability of time-depending linear systems

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a 2x2 couterexample by Vinograd to the system $y'=A(t)y$ where $A(t)$= \begin{matrix} -1 -9 \cos^2 6t + ...

**1**

vote

**0**answers

37 views

### Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg_t = dB_t \circ g_t$$
where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...

**2**

votes

**1**answer

93 views

### Complete regularity in C*-algebras

It is clear that commutative C*-algebras correspond to locally compact Hausdorff spaces. And locally compact Hausdorff spaces are completely regular. Now, does the complete regularity statement have ...

**0**

votes

**0**answers

19 views

### Gauss Curvature Equation for hypersurface in Semi-Riemannian manifold [migrated]

We have the Gauss curvature equation:
$$\langle R(V,W)X,Y\rangle = \langle R'(V,W)X,Y\rangle - \langle II(V,X),II(W,Y)\rangle + \langle II(V,Y),II(W,X)\rangle$$
Here $M$ is an immersion in $N$. ...

**1**

vote

**0**answers

283 views

### Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...

**0**

votes

**1**answer

60 views

### Solving Shroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system [on hold]

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$
$r_1$ is the distance between the proton $1$ and the electron.
$r_2$ is the distance between the proton $2$ and the electron.
$R$ is the ...

**1**

vote

**1**answer

136 views

### Covering space theory, category theory [on hold]

Requiring covering spaces of a well-behaved connected topological space $X$ to be connected, let $\mathcal{Cov}(X)$ be the category of covering spaces of $X$ and maps over $X$ and maps over $X$. Can ...

**1**

vote

**0**answers

63 views

### Adjacency matrix, quivers

Let $Q$ be a quiver with finitely many edges and such that the underlying graph is connected. Let $I = \{1, \dots, n\}$ be the vertex set of $Q$, so we have $\mathbb{R}\{I\} \cong \mathbb{R}^n$.
For ...

**4**

votes

**0**answers

160 views

### Is a determinant 2x2 positive and increasing?

Let $X_1,X_2,X_3$ be a three discrete (integer and non-negative valued) random variables with local probabilities $a_k:=\mathbb{P}(X_1=k)$, $b_k:=\mathbb{P}(X_2=k)$, $c_k:=\mathbb{P}(X_3=k)$ and ...

**-3**

votes

**0**answers

26 views

### How to compute the direction of flattest ascent for a convex function [on hold]

Consider an infinitely differentiable convex function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ at the point $x_0$. So long as $x_0$ is not the minimum, it is well known that we can compute a unit vector ...

**0**

votes

**1**answer

63 views

### Why does optimization of a sum of two terms result in “neat” answers? [on hold]

This is a somewhat vague and philosophical question.
Consider the following three problems:
Problem 1:
Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$
...

**3**

votes

**1**answer

145 views

### characteristic classes of tangent bundle of 2-nd unordered configuration space

Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid ...

**7**

votes

**1**answer

263 views

### The sum of a series

Let $0< \alpha <1$ and $q>1.$
Consider the (alternating) series: $$
\sum_{k=1}^\infty
(-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$
Denote its sum by $f(q,\alpha).$
Prove (or ...

**0**

votes

**0**answers

27 views

### Problem regarding sum of a recursive sequence

Problem of the recursive sum is as follows.
Find the sum
$$\sum_{r=1}^n U_r$$
where
$$U_r = \frac{U_{r-1}M_r}{M_{r-1}(a+b M_r)}$$
and
$$U_1 = \frac{M_1}{a+b M_1} , \ \ \sum_{r=1}^{n} M_r = 1.$$
Here ...

**7**

votes

**1**answer

111 views

### Physical interpretation of the mellin transform variable?

I shall keep this to the point: Given a time domain signal say microphone recording of a conversation:
Laplace tranfrom of x is some function X(s) say defined in the complex plane. I like to think ...

**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 ...

**0**

votes

**0**answers

6 views

### About irreducible representation of symmetric group [on hold]

Consider the tensor space
$$\mathbb{C}^m\otimes \mathbb{C}^n\otimes\mathbb{C}^n\otimes\cdots\otimes\mathbb{C}^n$$ with $k$ factors.
The symmetric group $S_k$ on $k$ letters acts on this space (on ...

**11**

votes

**0**answers

144 views

### Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...

**0**

votes

**1**answer

50 views

### Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of:
$$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$
$$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ...

**1**

vote

**1**answer

69 views

### Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...

**-4**

votes

**0**answers

69 views

### Maximal ideals of R [x] [on hold]

Let $R$ be a commutative ring with identity. Is there any relation between maximal ideas of $R[x]$ and maximal ideas of $R$?

**0**

votes

**0**answers

56 views

### A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...

**9**

votes

**0**answers

107 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

**1**answer

55 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

**1**answer

221 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 ...

**4**

votes

**0**answers

30 views

### Current upper bound on length of addition chain

An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...

**4**

votes

**1**answer

96 views

### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...

**5**

votes

**0**answers

45 views

### Cutting a piece of cake that $n$ people value as exactly $w$

Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a ...

**20**

votes

**9**answers

1k views

### Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...

**0**

votes

**0**answers

33 views

### Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof.
Paper 1, page 1, line 10 says : Consider the topological image G of a ...

**1**

vote

**0**answers

38 views

### Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...

**3**

votes

**0**answers

62 views

### How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques.
For example, consider the equation $qn = mf$, where each of the variables ...