# All Questions

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### Continuity of bounded sublinear or quasilinear operators

Let $\mathcal{T}$ be an operator defined on a linear space of complex-valued measurable functions on a measure space $(X,\mu)$ and taking values in the set of all complex-valued finite almost ...
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### On the use of the symmetric asymptotic upper density on $\mathbf Z$

The upper asymptotic density on $\mathbf Z$, viz. the function $${\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},$$ has a symmetric ...
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### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$. There is a candidate ...
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### u.p (unique product) group which is not Right ordered ($RO$)

I am looking for an example of a u.p (unique product) group which is not Right ordered. Almost any group I pick up (obviously torsion free, as u.p. group cannot have torsion element, so no use ...
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### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...
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### Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
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### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...
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### About properties of polynomials with common interlacing

Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order) Can ...
1answer
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### Segal maps for Segal precategories

A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...
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Let $k$ be an algebraically closed field. Consider the map $\phi:X:=\mathbb{A}^{1}\times\mathbb{A}^{\mathbb{N}}\rightarrow Y:=\mathbb{A}^{\mathbb{N}}$ given by $(\lambda,x(t))\mapsto (t-\lambda) ... 1answer 56 views ### Find the probability that the product of these numbers is a multiple of 3 [on hold] From the sequence of natural numbers randomly select a pair of numbers. Find the probability that the product of these numbers is a multiple of 3. 0answers 38 views ### What is the idea behind a projection operator?what does it do? [on hold] I need the idea behind this not the definitions of the examples can someone help? 1answer 77 views ### Computing a transversal of a subgroup$H$of$G$in expected$O(|G : H|^2 \log |G : H| + |H|)$time I have the book "Handbook of Computational Group Theory", by Derek Holt, and in it is a section on finding the transversal of a subgroup. Recall a transversal of a subgroup$H$of$G$is a single ... 0answers 6 views ### Laplacian Matrix for weighted Adjacency? [on hold] I have seen definitions for Laplacian matrix in many resources as follows : L = D − A where D and A are the degree and ... 0answers 161 views ### Excellent rings If A is an excellent commutative ring and G is a finite group of automorphisms of A, is the invariant subring A^G still excellent ? I think this is false -- because if not it would probably be written ... 1answer 48 views ### Evolution operator for a linear parabolic equation Let$A(t)$be a smooth family of positive definite operators on a Hilbert space$H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let$U(t):H\to H$be the evolution operator, i.e.,$U(0)=I$and ... 0answers 47 views ### TOPOLOGY DATA ANALYSIS [on hold] actually i am PHD student and my research in TOPOLOGY DATA ANALYSIS (TDA) what I dream about to understand this artcile cran.r-project.org/web/packages/TDA/vignettes/article.pdf may you help me: 1- MY ... 0answers 56 views ### Relation between Kahler form and Kahler potential [on hold] Let us consider an example. Take$\mathbb{C}^m$which is identified with$\mathbb{R}^{2m}$. Now, the Kahler form is given by $$\Omega = \frac{i}{2} \sum_{i=1}^m dz^i \wedge d\bar{z}^i$$ Now, how can ... 0answers 74 views ### The existence of proper schemes under complection Let$R$be a regular local ring,$\hat{R}$be its completion,$X$be a proper scheme over$\text{Spec}(\hat{R})$. In what case there exist a proper scheme$Y$over$\text{Spec}(R)$, such that$X$is ... 1answer 132 views ### Direct proof that$U$is an$E_\infty$-space An immediate consequence of Bott periodicity is that the infinite unitary space is an infinite loop space and so an$E_\infty$-space. I wonder if there is a direct proof (not using$U = \Omega^2 U$) ... 0answers 32 views ### why group completion of configuration space is the iterated suspension space In Lecture notes in mathematics Vol. 533, The homology of$C_{n+1}$-spaces,$n\geq 0$, F. Cohen, page 226, Corollary 3.3:$\alpha_{n+1}: C(\mathbb{R}^{n+1};X)\to \Omega^{n+1}\Sigma^{n+1}X$is a ... 0answers 54 views ### Endomorphism algebras of abelian surfaces with real multiplication Given an abelian variety$A$over a field$F$, one may consider the ring of endomorphisms$End(A)$, the ring of$F$-rational maps$A \to A$respecting the group structure on$A$. We may also consider ... 0answers 56 views ### Has solution of Brocard's Equation n!=m^2-1 [on hold] Brocard conjecture in 1904 that the only solution of are n=4,5,7. There are no other solutions with .(Berndt and Galway n.d).Another of Brocard’s conjecture is that there are at least four primes ... 0answers 36 views ### Sobolev-Poincaré inequality for curl-integrable functions Let$B=B(r)$denote a ball of radius$r$in$\Omega \subset \mathbb R^d$and $$u_B := \frac1{|B|}\int_B u \, dx.$$ The standard Sobolev-Poincaré inequality states that if$u \in W^{1,p}(\Omega)$, ... 1answer 63 views ### Infinite graphs with “similar” Hom-sets Let$G,H$be infinite simple undirected graphs with the property that for any graph$X$we have$|\text{Hom}(X,G)| = |\text{Hom}(X,H)|$. Does this imply that$G$is isomorphic to a subgraph of$H$, ... 0answers 95 views ### The multiplicative group generated by shifted primes I am asking for references about the following problem. In particular, it is still open? If not, what is the state of the art result? Problem 1. Let$\Gamma$be the multiplicative subgroup of ... 0answers 44 views ### Sparsifiers for 3-term arithmetic progressions Let$G$be a finite abelian group of odd order, let$D\subseteq G$, and$\epsilon \in (0,1)$. For$S\subseteq G$define $$\Lambda(S) = \frac{1}{|S||G|} \sum_{s\in S}\sum_{g\in G} ... 0answers 25 views ### Stochastic gradient descent interleaved with deterministic optimization I wish to solve \min_{x, y_k} \frac{1}{n} \sum_{k=1}^n f_k(x, y_k). where f_k are all smooth and convex. Using standard stochastic gradient descent (SGD), each iteration I sample a k from \{1, ... 2answers 128 views ### Algebraic groups “generated” by a Lie algebra element Here is a definition which I invented and which I would like to understand better. Let A be a complex affine algebraic group. Let X \in \mathfrak g be an element in its Lie algebra. We say ... 0answers 82 views ### Quickest and/or most elementary proof of “principal iff splits completely”? Let L be the Hilbert class field of a number field K, and let \mathfrak{p} be a prime ideal of K. Then \mathfrak{p} splits completely in L if and only if \mathfrak{p} is a principal ... 0answers 17 views ### Find paths in a graph that any 2 vertices can be reached through N of them Given a undirected weighted graph. I would like to find a finite set of paths (consecutive vertices and edges) each shorter than L any two vertices can be reached through at most N(in my case N=4) ... 1answer 86 views ### Iterated sumset inequalities in semigroups This question is motivated by the following well-known theorems: Thm (Plünnecke): If A is a finite nonempty subset of an abelian group, then for every n we have |A^n| \le ... 1answer 83 views ### iterated loop spaces and configuration spaces [on hold] In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map$$ \phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y) $$is defined. And a map$$ ... 0answers 24 views ### Symmetry conjecture for minimal dilatation pseudo Anosov mapping classes The conjecture is something like the following: The minimal dilatation among pseudo-Anosov mapping classes on a surface$S_{g,n}$is realized by$\rho\circ\omega$where$\omega$is supported on a ... 1answer 69 views ### References about spectral theory on Hyperbolic space Can anyone suggest me some books or papers that include details about spectral theory on Hyperbolic spaces or related topics such as scattering theory on Hyperbolic spaces? After some googling, I ... 0answers 24 views ### Meaning of k-connected directed graphs [on hold] Is there any existing definition for "k-connected directed graphs"? Any reference paper? 0answers 43 views ### If there is in a category$\mathcal{A}$finite products and equalizers then it has pullbacks [on hold] My homework consist in showing that "If there is in a category$\mathcal{A}$finite coproducts and coequalizers then it has pushouts" based on the proof that "If there is in a category$\mathcal{A}$... 0answers 27 views ### simplifying an equation that has infinitesimals [on hold] I'm trying to understand an equation with infinitesimal changes: 8*X*dX = d(4*X^2) I think this can be written$8X\Delta X = \Delta (4(X^{2}))\$ I'm guessing going from 8 outside the differential ...

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