# All Questions

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### 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 ...
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### 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 ...
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### 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 $R[x]$?
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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" ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 = ...
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### 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 ...
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### 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 + ...
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### 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 ...
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### 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 ...
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### 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 $N$. ...
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### 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 $R$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
### 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 ...
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 ...