scholarly journals Generic elements of a Zariski-dense subgroup form an open subset

2017 ◽  
Vol 78 ◽  
pp. 299-314 ◽  
Author(s):  
G. Prasad ◽  
A. S. Rapinchuk
Keyword(s):  
Open Subset ◽  
Dense Subgroup ◽  
Generic Elements

Continuity of homotopies

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Open Subset ◽  
Small Neighborhood ◽  
Unit Vector ◽  
Zariski Topology ◽  
Simple Point ◽  
Smooth Variety ◽  
Additional Material ◽  
Generic Type ◽  
Zariski Open Subset

This chapter includes some additional material on homotopies. In particular, for a smooth variety V, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of V. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset U of V. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

Some properties of open subset intersection graph of a topological space

2021 ◽  
pp. 1-8
Author(s):  
R. A. Muneshwar ◽  
K. L. Bondar
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Intersection Graph

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

Finitely generic models of TUH, for certain model companionable theories T

1985 ◽  
Vol 50 (3) ◽  
pp. 604-610
Author(s):  
Francoise Point
Keyword(s):  
Discriminator Variety ◽  
Generic Models ◽  
Elementary Class ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Closed Element ◽  
Starting Point ◽  
Joint Embedding ◽  
Joint Embedding Property ◽  
Generic Elements

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

Some uniqueness results for singular elliptic problems

2015 ◽  
Vol 26 (03) ◽  
pp. 1550026 ◽  
Author(s):  
L. Caso ◽  
R. D'Ambrosio
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Weighted Sobolev Spaces ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Singular Data

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝn, n ≥ 2, not necessarily bounded or regular.

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
LUCK DARNIÈRE ◽  
EVA LEENKNEGT
Keyword(s):  
Algebraic Geometry ◽  
Open Subset ◽  
Cell Decomposition ◽  
Real Algebraic Geometry ◽  
Dimension Theory ◽  
Definable Set ◽  
Image Position ◽  
Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

scholarly journals Quasiconvexity and Density Topology

2014 ◽  
Vol 57 (1) ◽  
pp. 178-187 ◽  
Author(s):  
Patrick J. Rabier
Keyword(s):  
Open Subset ◽  
Measure Zero ◽  
Quasiconvex Functions ◽  
Density Topology ◽  
Approximate Continuity ◽  
The Common

AbstractWe prove that if f : ℝN → ℝ̄ is quasiconvex and U ⊂ ℝN is open in the density topology, then supU ƒ = ess supU f ; while infU ƒ = ess supU ƒ if and only if the equality holds when U = RN: The first (second) property is typical of lsc (usc) functions, and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of ƒ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

scholarly journals Examples of malformed subsets of a Riemann surface

1983 ◽  
Vol 24 (2) ◽  
pp. 101-106
Author(s):  
Moses Glasner
Keyword(s):  
Riemann Surface ◽  
Maximum Principle ◽  
Open Subset ◽  
Linear Mapping ◽  
The Maximum Principle ◽  
Hyperbolic Riemann Surface

Let R be a hyperbolic Riemann surface and W an open subset of R with ∂W piecewise analytic. Denote by the space of Dirichlet finite Tonelli functions on R and by π the harmonic projection of . Consider the relative HD–class on W, HD(W;∂W) = {u∈ │ u │ W∈HD(W) and u │ R\W = 0}. The extremization operation μis the linear mapping of HD(W;∂W) into HD(R) defined by μ. Since π preserves values of functions at the Royden harmonic boundary, the maximum principle implies that μis an order preserving injection and that Mμ is an isometry with respect to the supremum norms.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

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