On chains of relatively saturated submodels of a model without the order property

1991 ◽  
Vol 56 (1) ◽  
pp. 124-128 ◽  
Author(s):  
Rami Grossberg
Keyword(s):  
Existence Theorems ◽  
Order Property ◽  
Similarity Type

AbstractLet M be a given model with similarity type L = L(M), and let L′ be any fragment of L∣L(M∣+,ω of cardinality ∣L(M)∣. We call N ≺ ML′-relatively saturated iff for every B ⊆ N of cardinality less than ∥N∥ every L′-type over B which is realized in M is realized in N. We discuss the existence of such submodels.The following are corollaries of the existence theorems.(1) If M is of cardinality at least ℶω1, and fails to have the ω order property, then there exists N ≺ M which is relatively saturated in M of cardinality ℶω1.(2) Assume GCH. Let ψ ∈ Lω1, ω, and let L′ ⊆ Lω1, ω be a countable fragment containing ψ. If ∃χ > ℵ0 such that I(χ, ψ) < 2χ, then for every M ⊨ ψ and every cardinal λ < ∥M∥ of uncountable cofinality, M has an L′-relatively saturated submodel of cardinality λ.

Classification and interpretation

1989 ◽  
Vol 54 (1) ◽  
pp. 138-159 ◽  
Author(s):  
Andreas Baudisch
Keyword(s):  
Order Property ◽  
Similarity Type ◽  
Complete Theories

AbstractLet S and T be countable complete theories. We assume that T is superstable without the dimensional order property, and S is interpretable in T in such a way that every model of S is coded in a model of T. We show that S does not have the dimensional order property, and we discuss the question of whether Depth(S) ≤ Depth(T). For Mekler's uniform interpretation of arbitrary theories S of finite similarity type into suitable theories TS of groups we show that Depth(S) ≤ Depth(TS) ≤ 1 + Depth(S).

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

Metaontology

2018 ◽  
Author(s):  
Uriah Kriegel
Keyword(s):  
Second Order ◽  
Order Property ◽  
First Order ◽  
First Approximation

Brentano’s theory of judgment serves as a springboard for his conception of reality, indeed for his ontology. It does so, indirectly, by inspiring a very specific metaontology. To a first approximation, ontology is concerned with what exists, metaontology with what it means to say that something exists. So understood, metaontology has been dominated by three views: (i) existence as a substantive first-order property that some things have and some do not, (ii) existence as a formal first-order property that everything has, and (iii) existence as a second-order property of existents’ distinctive properties. Brentano offers a fourth and completely different approach to existence talk, however, one which falls naturally out of his theory of judgment. The purpose of this chapter is to present and motivate Brentano’s approach.

scholarly journals On the boundary layer arising in the spin-up of a stratified fluid in a container with sloping walls

1997 ◽  
Vol 335 ◽  
pp. 233-259 ◽  
Author(s):  
P. W. DUCK ◽  
M. R. FOSTER ◽  
R. E. HEWITT
Keyword(s):  
Boundary Layer ◽  
Layer Structure ◽  
Outer Region ◽  
Stratified Fluid ◽  
Main Body ◽  
Similarity Type ◽  
Cone Apex ◽  
Uniform Solution ◽  
Steady Solutions ◽  
Finite Time Singularity

In this paper we consider the boundary layer that forms on the sloping walls of a rotating container (notably a conical container), filled with a stratified fluid, when flow conditions are changed abruptly from some initial (uniform) state. The structure of the solution valid away from the cone apex is derived, and it is shown that a similarity-type solution is appropriate. This system, which is inherently nonlinear in nature, is solved numerically for several flow regimes, and the results reveal a number of interesting and diverse features.In one case, a steady state is attained at large times inside the boundary layer. In a second case, a finite-time singularity occurs, which is fully analysed. A third scenario involves a double boundary-layer structure developing at large times, most significantly including an outer region that grows in thickness as the square-root of time.We also consider directly the nonlinear fully steady solutions to the problem, and map out in parameter space the likely ultimate flow behaviour. Intriguingly, we find cases where, when the rotation rate of the container is equal to that of the main body of the fluid, an alternative nonlinear state is preferred, rather than the trivial (uniform) solution.Finally, utilizing Laplace transforms, we re-investigate the linear initial-value problem for small differential spin-up studied by MacCready & Rhines (1991), recovering the growing-layer solution they found. However, in contrast to earlier work, we find a critical value of the buoyancy parameter beyond which the solution grows exponentially in time, consistent with our nonlinear results.

scholarly journals Mal’tsev products of varieties, I

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Tomasz Penza ◽  
Anna B. Romanowska
Keyword(s):  
Sufficient Condition ◽  
Similarity Type

AbstractWe investigate the Mal’tsev product $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W of two varieties $$\mathcal {V}$$ V and $$\mathcal {W}$$ W of the same similarity type. While such a product is usually a quasivariety, it is not necessarily a variety. We give an equational base for the variety generated by $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W in terms of identities satisfied in $$\mathcal {V}$$ V and $$\mathcal {W}$$ W . Then the main result provides a new sufficient condition for $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W to be a variety: If $$\mathcal {W}$$ W is an idempotent variety and there are terms f(x, y) and g(x, y) such that $$\mathcal {W}$$ W satisfies the identity $$f(x,y) = g(x,y)$$ f ( x , y ) = g ( x , y ) and $$\mathcal {V}$$ V satisfies the identities $$f(x,y) = x$$ f ( x , y ) = x and $$g(x,y) = y$$ g ( x , y ) = y , then $$\mathcal {V}\circ \mathcal {W}$$ V ∘ W is a variety. We provide a number of examples and applications of this result.

scholarly journals On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1219
Author(s):  
Marek T. Malinowski
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Theorems ◽  
Type Theorem ◽  
Integral Representations ◽  
Uniqueness Of The Solution ◽  
Nonempty Compact ◽  
Picard Type Theorem ◽  
Convex Subsets

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.

scholarly journals Solutions for parametric double phase Robin problems

2021 ◽  
Vol 121 (2) ◽  
pp. 159-170 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Boundary Condition ◽  
Existence Theorems ◽  
Robin Boundary Condition ◽  
Phase Problem ◽  
Double Phase ◽  
Robin Boundary ◽  
Double Phase Problem

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

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