Strongly 2-dimensional theories

1988 ◽  
Vol 53 (3) ◽  
pp. 931-936
Author(s):  
Akito Tsuboi
Keyword(s):  
Simple Structure ◽  
Stable Theory ◽  
Morley Rank ◽  
Dimensional Theory ◽  
Finite Sequences ◽  
Strongly Regular ◽  
Definition Of ◽  
Condition B

In [8], we have shown the equivalence of almost strong minimality and strong unidimensionality. More precisely, we proved:Theorem [8]. Let T be a countable stable theory. Then the following two conditions are equivalent:(i) T is almost strongly minimal;(ii) T can be extended to a theory such that any two nonalgebraic types are not almost orthogonal.In the present paper, we define the notion of strong 2-dimensionality (of T). We show that if T is strongly 2-dimensional then T is ω-stable and its model has a simple structure. Roughly speaking, in a model of a strongly 2-dimensional theory, one of the following holds: (a) every element is in acl (δi is strongly minimal), or (b) every element is in acl (δ is strongly regular). Shelah's definition of 2-dimensionality does not imply even superstability. (See Exercise 5.5 in [6, Chapter V, §5].) We show also that condition (a) above implies strong 2-dimensionality of T. However condition (b) does not imply strong 2-dimensionality in general.Our notations and conventions are standard. T is always countable and stable. We work in . A,B,… are used to denote small subsets of . , … are used to denote finite sequences of elements in . δ, φ,… are used to denote formulas (with parameters), p, q, … are used to denote types (with parameters). The fact that p is a nonforking (forking) extension of q is denoted by p ⊃nfq(p ⊃fq). If p is stationary, p∣A denotes the type in S(A) which is parallel to p. (or ) denotes the set of realizations of p (or δ). The Morley rank of p is denoted by RM(p).

The Pn-Integral

1975 ◽  
Vol 18 (4) ◽  
pp. 493-497 ◽  
Author(s):  
George Cross
Keyword(s):  
The Difference ◽  
Definition Of ◽  
Condition B

In the definition of the Pn-integral [2] there is a difficulty with the condition Bn-2 ([2], p. 150) since it is not linear on the set of major and minor functions. As a result, the proof of Lemma 5.1 [2] fails since the difference Q(x)—q(x) need not satisfy the conditions of Theorem 4.2, [2].

FREE VIBRATION OF AN ORTHOTROPIC HOLLOW BODY OF REVOLUTION

2005 ◽  
Vol 05 (02) ◽  
pp. 299-312
Author(s):  
D. REDEKOP
Keyword(s):  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Constant Thickness ◽  
Body Of Revolution ◽  
Orthotropic Cylinder ◽  
Dimensional Theory ◽  
Hollow Cylinders ◽  
Definition Of

A method is developed to determine the natural frequencies of vibration of an orthotropic hollow body of revolution of constant thickness but of arbitrary smooth meridian. Equations are derived using the linear three-dimensional theory of elasticity, and a numerical solution is obtained using the differential quadrature method. The geometric generality of the solution is attained by delaying definition of local geometric parameters until the solution stage. Validation is by comparison with previously published results, including results for a hollow orthotropic cylinder. Sample results are given for orthotropic hollow cylinders and spherical segments, and conclusions are drawn.

scholarly journals ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

2017 ◽  
Vol 23 (2) ◽  
pp. 32-43
Author(s):  
M. V. Kukushkin
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Fractional Differentiation ◽  
Qualitative Properties ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Integral Equality ◽  
Definition Of

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

Efficient Solving Method of Common-Mode Interference in Balanced Bus

2013 ◽  
Vol 340 ◽  
pp. 418-421
Author(s):  
Xiao Ming Chang ◽  
Chun Geng Gao
Keyword(s):  
Simple Structure ◽  
Low Cost ◽  
Cost Effective ◽  
Communication Line ◽  
Effective Solution ◽  
Common Mode ◽  
Mathematical Formula ◽  
Common Mode Voltage ◽  
Definition Of ◽  
Mode Interference

To analyze the causes of common-mode interference from the definition of common-mode voltage, and the combination of graphics using mathematical formula image illustrates the size of common-mode voltage. At the same time, to analyze the method and defects of common-mode interference in the traditional two-line balanced communication line. On this basis, a cost-effective solution circuit is proposed. The results prove that the circuit has the advantages of low cost, simple structure, convenient features.

Strongly Regular Graphs Derived from Combinatorial Designs

1970 ◽  
Vol 22 (3) ◽  
pp. 597-614 ◽  
Author(s):  
J. M. Goethals ◽  
J. J. Seidel
Keyword(s):  
Strongly Regular Graph ◽  
Hadamard Matrices ◽  
Latin Squares ◽  
Discrete Mathematics ◽  
Strongly Regular Graphs ◽  
Block Designs ◽  
Regular Graphs ◽  
Geometric Configurations ◽  
Strongly Regular ◽  
Definition Of

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.

scholarly journals Introduction to Liouville Numbers

2017 ◽  
Vol 25 (1) ◽  
pp. 39-48
Author(s):  
Adam Grabowski ◽  
Artur Korniłowicz
Keyword(s):  
Positive Integer ◽  
Algebraic Number ◽  
Finite Sequence ◽  
Rational Numbers ◽  
Liouville Number ◽  
Liouville Numbers ◽  
Constant Sequence ◽  
Transcendental Numbers ◽  
Finite Sequences ◽  
Definition Of

Summary The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https://en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.

Countable models of nonmultidimensional ℵ0-stable theories

1983 ◽  
Vol 48 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Elisabeth Bouscaren ◽  
Daniel Lascar
Keyword(s):  
Finite Number ◽  
Homogeneous Model ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Stable Theory ◽  
Strong Type ◽  
First Case ◽  
Skolem Functions ◽  
Finite Sequences ◽  
Homogeneous Models

In this paper T will always be a countable ℵ0-stable theory, and in this introduction a model of T will mean a countable model.One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M, if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E, definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L1. In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize.The motivation of this research is to distinguish two classes of ℵ0-Stable theories:(1) theories such that all models are almost homogeneous;(2) theories with 2ℵ0 nonalmost homogeneous models.The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.

There is no sharp transitivity on q6 when q is a type of Morley rank 2

1992 ◽  
Vol 57 (4) ◽  
pp. 1198-1212 ◽  
Author(s):  
Ursula Gropp
Keyword(s):  
Group Action ◽  
Group Actions ◽  
Transitive Group ◽  
Closed Field ◽  
Stable Theory ◽  
Morley Rank ◽  
Independent Sequence ◽  
Generic Type ◽  
Rank 2 ◽  
Sharply Transitive

In this paper we study transitive group actions.:G × X → X, definable in an ω-stable theory, where G is a connected group and X a set of Morley rank 2, with respect to sharp transitivity on qα. Here q is the generic type of X (X is of degree 1 by Proposition (1)), for ordinals α > 0, qα is the αth power of q, i.e. (aβ)β < α, ⊨ qα iff (aβ)β < α is an independent sequence (in the sense of forking) of realizations of q, and G is defined to be sharply transitive on qα iff for all (aβ)β < α, (bβ)β < α ⊨ qα there is one and only one g ∈ G with g.aβ = bβ for all β < α. The question studied here is: For which powers α of q are there group actions subject to the above conditions with G sharply transitive on qα?In §1 we will see that for group actions satisfying the above conditions, G can be sharply transitive only on finite powers of q. Moreover, if G is sharply transitive on qn for some n ≥ 2, then the action of the stabilizer Ga on a certain subset Y of X satisfies the conditions above with Ga being sharply transitive on qm−1, where q′ is the generic type of Y (Proposition (8)). Thus, there would be a complete answer to the question if one could find some n < ω such that there is no group action as above with G sharply transitive on qn, but for n – 1 there is. In this paper we prove that such n exists and that it is either 5 or 6. More precisely, in §2 we prove that there is no group action satisfying the above conditions with G sharply transitive on q6. This is the main result of this paper. It is not known to the author whether the same also holds for q5 instead of q6. However, it does not hold for q4, as is seen in §3. There we give an example provided from projective geometry, for a group action satisfying the above conditions with G sharply transitive on q4; for G we choose PGL(3, K) and for X the projective plane over K, where K is some algebraically closed field.

scholarly journals Nur für Schwindelfreie? Eine Geographie politischer Praktiken nach Hannah Arendt

2020 ◽  
Vol 75 (2) ◽  
pp. 81-91
Author(s):  
Florian Dünckmann
Keyword(s):  
Social Life ◽  
Political Action ◽  
Human Condition ◽  
The Political ◽  
Spatial Structures ◽  
Point Of Interest ◽  
Definition Of ◽  
Social Orders ◽  
The Web ◽  
Condition B

Abstract. Times of crisis often call the legitimacy of existing social orders into question. These practices of dispute and debate that question, challenge, or affirm the rules that govern our social life are what constitutes the realm of the political. This article fathoms the potential of a Political Geography that makes political practices its main point of interest. Arendt's political philosophy provides the foundation for a geography of political practices that asks about (a) the way in which the possibility and necessity of the political is tied to the spatiality of our human condition, (b) the relation of political practices to spatial structures and their production of particular places, spaces and scales, and (c) the role which materiality plays in stabilizing, constraining and shaping political practices. Combining insights from Arendt's concept of political action with recent ideas of practice theories, a definition of political practices that relies on three characteristics – reflexivity, perspectivity, and expressivity – is introduced. I will argue that these metapragmatic practices, although they distinguish themselves from pragmatic practices, nevertheless, always remain embedded in and related to the web of our everyday doings.

scholarly journals The Determination of nonlinear second-order diffraction forces acting on a ship based on three-dimensional theory

2021 ◽  
pp. 20-27
Author(s):  
В.Ю. Семенова ◽  
Д.А. Альбаев
Keyword(s):  
Free Surface ◽  
Three Dimensional ◽  
Second Order ◽  
Complete Agreement ◽  
Dimensional Theory ◽  
Nonlinear Diffraction ◽  
Definition Of ◽  
Limited Depth ◽  
Good Agreement

В статье рассматривается определение нелинейных дифракционных сил второго порядка, на основании применения трехмерной потенциальной теории. Для их определения необходимо вычисление потенциалов второго порядка малости. Представленное решение в отечественной практике является новым. Решение задачи осуществляется на основании методов малого параметра и интегральных уравнений с учетом нелинейного граничного условия на свободной поверхностью жидкости. В работе показана возможность расчета интегралов по свободной поверхности напрямую за счет их сходимости на бесконечном удалении от судна. Нелинейные дифракционные силы и моменты определяются в работе с использованием различных функций Грина: для бесконечно-глубокой жидкости и жидкости ограниченной глубины, когда H→∞. Полученные результаты практически полностью согласуются между собой. Приводятся результаты расчетов дифракционных сил и моментов для четырех разных судов. Расчеты представлены в сравнении с расчетами по двумерной теории, выполненными также для случая бесконечно глубокой жидкости и жидкости ограниченной глубины при больших значениях отношения глубины к осадке H/T. Показано хорошее согласование результатов между собой. Показана возможность расчета нелинейных дифракционных сил на произвольных курсовых углах. The article discusses the definition of nonlinear diffraction forces of the second order, based on the application of three-dimensional potential theory. To determine them, it is necessary to calculate the potentials of the second order of smallness. The presented solution is new in domestic practice. The problem is solved on the basis of small parameter methods and integral equations taking into account the nonlinear boundary condition on the free surface of the liquid. The paper shows the possibility of calculating the integrals over the free surface directly due to their convergence at an infinite distance from the ship. Nonlinear diffraction forces and moments are determined in the work using various Green's functions: for an infinitely deep fluid and a fluid of limited depth when H → ∞. The results obtained are in almost complete agreement with each other. The results of calculations of diffraction forces and moments for four different ships are presented. The calculations are presented in comparison with the calculations according to the two-dimensional theory, performed also for the case of an infinitely deep liquid and a liquid of limited depth at large values of ratio H / T. Good agreement of the results with each other is shown. The possibility of calculating nonlinear diffraction forces at arbitrary heading angles is shown.

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