Quasicomponents in topos theory: the hyperpure, complete spread factorization

2007 ◽  
Vol 142 (1) ◽  
pp. 47-62
Author(s):  
MARTA BUNGE ◽  
JONATHON FUNK
Keyword(s):  
Connected Domain ◽  
Existence And Uniqueness ◽  
Topos Theory ◽  
Local Connectedness ◽  
Special Case ◽  
Geometric Morphism ◽  
Geometric Morphisms

AbstractWe establish the existence and uniqueness of a factorization for geometric morphisms that generalizes the pure, complete spread factorization for geometric morphisms with a locally connected domain. A complete spread with locally connected domain over a topos is a geometric counterpart of a Lawvere distribution on the topos, and the factorization itself is of the comprehensive type. The new factorization removes the topologically restrictive local connectedness requirement by working with quasicomponents in topos theory. In the special case when the codomain topos of the geometric morphism coincides with the base topos, the factorization gives the locale of quasicomponents of the domain topos, or its ‘0-dimensional’ reflection.

scholarly journals A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

scholarly journals On the Bean critical-state model in superconductivity

1996 ◽  
Vol 7 (3) ◽  
pp. 237-247 ◽  
Author(s):  
L. Prigozhin
Keyword(s):  
Variational Inequalities ◽  
Critical State ◽  
Existence And Uniqueness ◽  
Critical State Model ◽  
Type Ii ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Uniqueness Of The Solution ◽  
Type Ii Superconductivity ◽  
Special Case

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

THE ROLE OF OXYGEN IN TISSUE MAINTENANCE: MATHEMATICAL MODELING AND QUALITATIVE ANALYSIS

2008 ◽  
Vol 18 (08) ◽  
pp. 1409-1441 ◽  
Author(s):  
AVNER FRIEDMAN ◽  
BEI HU
Keyword(s):  
Mathematical Modeling ◽  
Free Boundary ◽  
Qualitative Analysis ◽  
Phase I ◽  
Stationary Solution ◽  
Oxygen Concentration ◽  
Existence And Uniqueness ◽  
Symmetric Case ◽  
Special Case

The cells in a tissue occupying a region Ωt are divided according to their cycling phase. The density pi of cells in phase i depends on the spatial variable x, the time t, and the time si since the cells entered in phase i. The pi(x, t, si) and the oxygen concentration w(x, t) satisfy a system of PDEs in Ωt, and the boundary of Ωt is a free boundary. We denote by [Formula: see text] the oxygen concentration on the free boundary and consider the radially symmetric case, so that Ωt = {r < R(t)}. We prove that R(t) is always bounded; furthermore, if [Formula: see text] is small, then R(t) → 0 as t → ∞, and if [Formula: see text] is large, then R(t) ≥ c > 0 for all t. Finally, we prove the existence and uniqueness of a stationary solution in a special case.

Simply Connected Limits

1990 ◽  
Vol 42 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Left Adjoint ◽  
Topos Theory ◽  
Essential Ingredient ◽  
Original Definition ◽  
The Subject ◽  
Definition Of ◽  
Simply Connected ◽  
Geometric Morphisms ◽  
Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

scholarly journals Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
T. E. Govindan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Special Case ◽  
Partial Functional Differential Equation

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

A SPECIAL CASE OF THE GENERALIZED TRICOMI PROBLEM FOR THE LAVRENTIEV–BITSADZE EQUATION

2011 ◽  
Vol 08 (01) ◽  
pp. 9-19 ◽  
Author(s):  
DIAN HU
Keyword(s):  
Boundary Condition ◽  
Existence And Uniqueness ◽  
Tricomi Problem ◽  
Hyperbolic Region ◽  
Special Case ◽  
Uniqueness Of A Solution

We study the generalized Tricomi problem for the Lavrentiev–Bitsadze equation in a sector, when the boundary condition prescribed in the hyperbolic region is far away from the characteristic. The existence and uniqueness of a solution to this problem is proven and further estimates of interest are established.

scholarly journals Metric Characterizations ofα-Well-Posedness for a System of Mixed Quasivariational-Like Inequalities in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
L. C. Ceng ◽  
Y. C. Lin
Keyword(s):  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Equilibrium Problems ◽  
Uniqueness Of Solution ◽  
Quasi Equilibrium ◽  
Well Posedness ◽  
Special Case

The purpose of this paper is to investigate the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept ofα-well-posedness to the system of mixed quasivariational-like inequalities, which includes symmetric quasi-equilibrium problems as a special case. Second, we establish some metric characterizations ofα-well-posedness for the system of mixed quasivariational-like inequalities. Under some suitable conditions, we prove that theα-well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. The corresponding concept ofα-well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. The results presented in this paper generalize and improve some known results in the literature.

scholarly journals Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

2019 ◽  
Vol 2019 (751) ◽  
pp. 27-89 ◽  
Author(s):  
Robert J. Berman ◽  
Sebastien Boucksom ◽  
Philippe Eyssidieux ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Ricci Flow ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Einstein Metrics ◽  
Discrete Version ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics ◽  
Special Case

AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

Essential geometric morphisms between toposes over finite sets

1980 ◽  
Vol 87 (1) ◽  
pp. 21-24
Author(s):  
John Haigh
Keyword(s):  
Finite Sets ◽  
Generating Set ◽  
Cogenerating Set ◽  
Geometric Morphism ◽  
Geometric Morphisms

We show that if {Gi}J ε I is a generating set for an (elementary) topos ℰ then {P(Gi)}iεI is a cogenerating set for x2130;. From this we show that if topos ℰ contains an object G whose subobjects generate ℰ, then ΩG is a cogenerator for ℰ. Let denote the topos of finite sets and functions. We also show that if ℰ1 is a topos and ℰ2 is a bounded -topos then every geometric morphism ℰ1 → ℰ2 is essential.

CHUA SYSTEMS WITH DISCONTINUITIES

1999 ◽  
Vol 09 (04) ◽  
pp. 591-616 ◽  
Author(s):  
C.-H. LAMARQUE ◽  
O. JANIN ◽  
J. AWREJCEWICZ
Keyword(s):  
Lyapunov Exponents ◽  
Existence And Uniqueness ◽  
Special Class ◽  
Equilibrium Solution ◽  
Phase Portraits ◽  
Global Behavior ◽  
Bifurcation Diagrams ◽  
Trivial Equilibrium ◽  
Analytical Expressions ◽  
Special Case

We present a special class of mechanical systems that could be written as Chua circuits with discontinuities. We recall the general frame for the study of such models. Results of existence and uniqueness are given. Then numerical results obtained via piecewise analytical expressions are presented. We discuss some bifurcation diagrams, phase portraits. Chaos is characterized by computing Lyapunov exponents. We analyze the global behavior in a special case where discontinuity stabilizes the trivial equilibrium solution.

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