# All Questions

102,879 questions
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### Kazhdan–Lusztig polynomials in terms of Ext groups

Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest ...
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### Reading list in dynamical systems

So I’ve managed to gather from various sources, a plethora of books in dynamical systems. Now I’m wondering which of them to read, and in what order. So far these are the books I’ve found/been ...
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### Do disjoint manifolds get separated by embedded disks in higher Euclidean space?

Let $A,B$ two disjoint $p$ and $q$ manifolds embedded in $R^n$. Can we find always a PL-map $f:R^n \longrightarrow R^k$ such that $f(A)$ and $f(B)$ are contained in two separate disjoint embedded $k$-...
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### When volume ratio of concentric balls decays faster than Gaussian?

Let $\mu$ be a finite measure on $\mathbb{R}^n$. Let $B_1$ to be the unit Euclidean ball centered at 0 in $\mathbb{R}^n$. Therefore, for any $t>0$ and $\theta\in\mathbb{R}^n$, $tB_1+\theta$ is the ...
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### Under what conditions is this family normal?

Let $\mathcal{S} = \{s \in \mathbb{C}\,\mid\,|\Im(s)| < 1\}$ be a strip of the complex plane. Let $q(s,z)$ be a holomorphic function on $\mathcal{S} \times \mathbb{C}$. Letting $\mathcal{K}$ be a ...
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### Roots of $x^n-x^{n-1}-\cdots-x-1$

It is easy to see that $f(x)=x^n-x^{n-1}-\cdots-x-1$ has only one positive root $\alpha$ which lies in the interval $(1,2)$. But it is claimed that this root is a Pisot number, i.e., the other roots ...
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### Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but ...
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We have two convex bounded polytopes $P_1$ and $P_2$ where a. $P_2\subseteq P_1$ b. $\mathcal{V}(P_2)\cap\mathcal{V}(P_1)\neq\emptyset$. Is there a name for the polytope $P=\mbox{Conv}(\mathcal{V}(... 0answers 75 views ### Stable bundles on Curves Suppose E is an ample vector bundle on a curve. Is there a simple way to manufacture a (slope)stable vector bundle out of it, without destroying its ampleness? Sorry if this question is too vague or ... 2answers 447 views ### What is a really good book for complex variables? [on hold] I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was ... 0answers 105 views ### Would this definition of True(F, x) select all true sentences of F? [on hold] Would this definition of True(F, x) select all true sentences of F? Introduction to Mathematical logic Sixth edition Elliott Mendelson (2015):28 A wf C is said to be a consequence in S of a set Γ of ... 0answers 44 views ### In search of a proof for the Mayank-Israel determinants Let the$a_i$'s be some indeterminates. In this MO post, Mayank introduced the matrix $$M= \begin{pmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&... 0answers 53 views ### Divisibility Properties of Pisano Periods Let (F_n) the Fibonacci sequence and \pi(m) the Pisano period of m (i.e., the smallest period of F_n \pmod{m}). There are many proved results about \pi(m). For example, it is known that \pi(... 0answers 50 views ### How to obtain mathematical expectation with the vector as random variable? In my study, I wish to get the mathematical expectation for the term below. The vector \boldsymbol{z} \in \mathcal{C}^{N\times1} and \boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{... 0answers 73 views ### homotopy VS isotopy classes of embeddings Let X any compact set in R^n, not necessarily a manifold. Let f,g:X \longrightarrow M^k be two homotopic PL embeddings of X. When X is an m-manifold then the two embeddings are also ... 0answers 442 views ### Are we better in computing integrals than mathematicians of 19th century? When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ... 0answers 62 views ### Elementary constraints for the solutions of z^{n-2}y(y+z)=x^n? Related to FLT and this question. For natural n > 4 define the curve C_n : z^{n-2}y(y+z)=x^n. C_n has the trivial points with x=0 for all n. The answer in the linked question shows ... 0answers 109 views ### Can a non-trivial algebraic variety carry a vector bundle whose total space is affine space? Suppose X is an algebraic variety over \mathbb{C}, and let Y\to X be an algebraic vector bundle. Suppose Y is algebraically isomorphic to \mathbb{C}^n for some n. Does it follow that X ... 0answers 21 views ### Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality In my work I wish to obtain a lower bound for the term below, independent of the vector x. Here the expectation is taken over h, a standard random Gaussian vector of length n. The vector x is ... 0answers 60 views ### Solution existence theorems of polynomial system of equations Consider the following system a_i(1-\sum\limits_{z=1}^N(x_z\theta_z))=x_i(1-\theta_i) b_i(1-\sum\limits_{z=1}^N(y_z\phi_z))=y_i(1-\phi_i) where \theta_i=q^u(\sum\limits_{k=i+1}^N(y_kq_{ik})+\... 0answers 79 views ### Generalised CRT - How to compute the cokernel? Let R be a commutative ring of dimension one with minimal prime ideals P_1,\ldots,P_n. We have the canonical injective map$$\phi_n: R/(P_1 \cap \ldots \cap P_n) \to \prod_{i=1}^n R/P_i.$$My ... 0answers 35 views ### Symmetric and anti-symmetric parts of the covariant derivative of a connection The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program. Now we come to the question of higher derivatives. As usual in modern ... 1answer 253 views ### Realizing cohomology classes by submanifolds In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ... 0answers 84 views ### A typo in Jacquet-Langlands On pages 516-517 of "Automorphic forms on$GL(2)$", there is a list of expressions contributing to Selberg trace formula. In (IV), should there be an additional$\frac{1}{2}\$ multiple? In (VII), ...

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