# All Questions

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### A game of stones

How I arrived at this question is a rather long story having to do with the honors calculus class I am teaching. At this point it's sheer curiosity on my part. Here is the game. ...
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### Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...
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### matrix theory understand the notion of transpose [on hold]

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ...
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In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ...
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### A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
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### Powers of orthogonal matrices is closed

This might be a basic question, nonetheless I cannot give a proof. Given an orthogonal matrix $A$ with eigendecomposition $A = Q \Lambda Q^{-1}$ with only non-real eigenvalues. Given also a diagonal ...
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### Weak topology and topology by semi-norm [on hold]

Wikipédia: -The weak topology on X is the initial topology with respect to X* (let's note it T') -If the field K has an absolute value , then the weak topology σ(X,F) is induced by the family of ...
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### State of the art in the theory of integer sequences

I was going through N.J.A. Sloane's 'Encyclopedia of Integer Sequences'. In it are discussed many tricks that are used to determine the recursive definition or explicit formula for a given sequence. ...
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### Strongly asymmetric graphs

Asymmetric graphs are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity. Let's call a graph $G$ strongly asymmetric ...
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### Continuity of a Functional

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is ...
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### Hausdorff topologies on Q

Is there any description known of the Hausdorff topologies on $\mathbb{Q}$ compatible with the group operations?
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### Rational curves and Serre's construction

Why rational curves, used in Serre's construction of vector bundles, usually corresponds to unstable bundle? I saw this affirmation in Richard Thomas's paper on an obstructed bundle on a CY threefold. ...
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### Examples of weird 'modular like mathematical space' that behaves like as if it is infinite until a threshold value is reached? [on hold]

(Might be a bit layman because I don't have rigorous math term to describe the concept) Generalize it to mathematical spaces, are there spaces which are sort of like a. Consists of multiple ...
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### Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem. Could any one give reference or a simple introduction about such result known in their ...
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### Schur's lemma for antiunitary operators on complex Hilbert spaces

Suppose to have a linear irreducible unitary representation $\rho:G\rightarrow U(H)$ on a complex Hilbert space $H$ with $G$ a generic group. Let $A$ be an $\textit{anti}$-linear operator such that ...
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### Expected value of minimum of an exponential function [on hold]

Find expected value of minimum of n random variables: x = (x1,x2,x3,..,xn) The distribution is an exponential function: ...
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### Continuous versions of tensors/ Tensors with infinite indices?

In linear algebra and general relativity, we knew that vectors can be represented by a linear combination of components and a basis $$\mathbf{V}=\sum_{i=1}^n A_i\mathbf{e_i}$$ Or in Einstein ...
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### decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $$\operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$ Let $j\in \mathbb N$ ...
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### infinitesimally commutative diagram [on hold]

Consider $f:X\rightarrow Y$, $g:Y\rightarrow Z$, $h:Y\rightarrow Z$ morphisms of intregal and separated $k$-schemes of finite type. We assume that at at point $x\in X$, $h(x)=g(f(x))$ the level of ...
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### Why is the finite extension field of the p-adic numbers $\mathbb{Q}_p$ spherically complete? [on hold]

Here by spherical completeness it is meant that given a non-empty nest of closed balls $\{B_\alpha|\alpha\in I\}$, that is, $\forall \alpha_1,\alpha_2\in I$ either $B_{\alpha_1}\subset B_{\alpha_2}$ ...
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### On a minimal algebraic number field which satisfies the principal ideal theorem

By an algebraic number field, we mean a finite extension field of the field of rational numbers. Let $k$ be an algebraic number field, we denote by $\mathcal{O}_k$ the ring of algebraic integers in ...
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### singular point of a complete intersection surface [migrated]

Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$. ...
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Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ... 0answers 27 views ### maximization of products of two trace function [on hold] consider the following optimization problem: \begin{array}{l} \mathop {\max }\limits_{\bf{X}} \,\,\,\operatorname{trace}\left( {{\bf{XA}}} \right)\operatorname{trace}\left( {{\bf{XB}}} \right)\\ ... 0answers 155 views ### Etale Slice Theorem I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ... 0answers 22 views ### Is triple point intersection 'generic' in Teichmuller space? Let$S$be a hyperbolic surface of finite type and$\alpha,\beta$be two closed curves. Consider$X$to be the set of all those points$\chi$in the Teichmuller space$\mathcal{T}(S)$of$S$such that ... 0answers 53 views ### Kontsevich integral for 2-bridge knots Are there any articles that explain a formula for Kontsevich integral of 2-bridge knots? 0answers 29 views ### Cell(J) vs Cof(J) in$\text{sSet}_{\text{Quillen}}$consider sSet equipped with its Quillen model structure$\text{sSet}_{\text{Quillen}}$, we know that a trivial cofibration is a retract of a transfinite composition of pushouts of horn inclusions. I ... 0answers 64 views ### Help me to proof Mobius-Euler equation [on hold] Can you help me to proof that $$\sum_{d | n}^{\, } \left ( \mu \left ( d \right ) \times \varphi \left ( d \right ) \right ) = 0\: for\: \mathbf{n}\geq 2, \mathbf{n}\: is\: even$$ where ... 0answers 29 views ### Poisson bivector on the product of two manifolds [migrated] Let$X, Y$be two manifolds. Let$(U, x_1, \ldots, x_n)$and$(V, y_1, \ldots, y_m)$local coordinates of$X, Y$respectively. A Poisson bivector on$Xis defined by \begin{align} \pi_X = \sum_{i,j} ... 0answers 30 views ### A question related to Bernoulli trial [on hold] I'm thinking a Bernoulli processX_1, X_2, X_3, ...$that stops when$n\left( X=0 \right)+2n\left( X=1 \right)\ge A$, where$n(X=0)$and$n(X=1)$are the number of 0 and 1 in the sequence ... 4answers 2k views ### Massive cancellations Let$A=\{a_1,\ldots,a_k\}$be a fixed, finite set of reals. Let$S_A(n)$be the set of all reals that are expressible as the sum of at most$2^n$terms, where each term is a product of at most$n$... 0answers 34 views ### Bound on change of function given bound on Hessian Suppose I have very some smooth function$F(x)$, and let$x_0 = \text{argmin}_x F(x)$. I would like to bound$F(x) - F(x_0)$from above, in terms of the gradient$\nabla f(x)$and the Hessian matrix ... 1answer 88 views ### Normals along a Sphere [on hold] Let$M \subset \mathbb{R}^d$be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points$x,y \in M$such that the normals$\angle(n_x, n_y) ...
By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the ...