Disassociated indiscernibles

This work addresses a basic question by Kunen: how many normal measures can there be on the least measurable cardinal? Starting with a measurable cardinal κ of Mitchell order less than two () we define a Prikry type forcing which turns the number of normal measures over κ to any while making κ the first measurable.     

Stability theory, permutations of indiscernibles, and embedded finite models

We show that the expressive power of first-order logic over finite models embedded in a model is determined by stability-theoretic properties of . In particular, we show that if is stable, then every class of finite structures that can be defined by embedding the structures in , can be defined in pure first-order logic. We also show that if does not have the independence property, then any class of finite structures that can be defined by embedding the structures in , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let be a set of indiscernibles in a model and suppose is elementarily equivalent to where is -saturated. If is stable and is saturated, then every permutation of extends to an automorphism of and the theory of is stable. Let be a sequence of -indiscernibles in a model , which does not have the independence property, and suppose is elementarily equivalent to where is a complete dense linear order and is -saturated. Then -types over are order-definable and if is -saturated, every order preserving permutation of can be extended to a back-and-forth system.

     

A minimal prime model with an infinite set of indiscernibles

We show the existence of a prime minimal model containing an infinite set of indiscernibles. We then find a sentence ofL _ω1ω which is categorical in ω₁ but whose model of power ω₁ is not (ω₁,L _ω1ω-homogeneous. This answers a question posed by Keisler.     

Simple monadic theories and indiscernibles

Aiming for applications in monadic second‐order model theory, we study first‐order theories without definable pairing functions. Our main results concern forking‐properties of sequences of indiscernibles. These turn out to be very well‐behaved for the theories under consideration (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

A Banach space with few operators

Assuming the axiom (of set theory)V=L (explained below), we construct a Banach space with density character ℵ₁ such that every (linear bounded) operatorT fromB toB has the forma I+T ₁, whereI is the identity, andT ₁ has a separable range. The axiomV=L means that all the sets in the universe are in the classL of sets constructible from ordinals; in a sense this is the minimal universe. In fact, we make use of just one consequence of this axiom, ℵ₁ proved by Jensen, which is widely used by mathematical logicians.     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

Asymptotic theory for varying coefficient regression models with dependent data

The varying coefficient models (VCMs) are extremely important tools in the statistical literature and are widely used in many subject areas for data modeling and exploration. In linear VCMs, typically the errors are assumed to be independent. However, in many situations, especially in spatial or spatiotemporal settings, this is not a viable assumption. In this article, we consider nonparametric VCMs with a general dependent error structure which allows for both spatially autoregressive and spatial moving average models as special cases. We investigate asymptotic properties of local polynomial estimators of the model components. Specifically, we show that the estimates of the unknown functions and their derivatives are consistent and asymptotically normally distributed. We show that the rate of convergence and the asymptotic covariance matrix depend on the error dependence structure and we derive the explicit formula for the convergence results.     

A superclass of Edge-Path-Tree graphs with few cliques

Edge-Path-Tree (EPT) graphs are intersection graphs of EPT matrices that is matrices whose columns are incidence vectors of edge-sets of paths in a given tree. EPT graphs have polynomially many cliques [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinational Theory Series B 38 (1985) 8–22; C.L. Monma, V.K. Wey, Intersection graphs of paths in a tree, Journal of Combinational Theory Series B 41 (1986) 141–181]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time. We extend this result to a proper superclass of EPT graphs.     

A note on ELECTRE TRI-nB with few limiting profiles

4OR - ELECTRE TRI-nB is a method designed to sort alternatives evaluated on several attributes into ordered categories. It is an extension of ELECTRE TRI-B that uses several limiting profiles,...     

A monotonicity analysis theory for the dependent eigenvalues of the vibrating systems

This paper presents and discusses the monotonicity analysis theory for the generalized eigenvalues of the nonlinear structural eigensystems. The analysis is based on investigating the mass and stiffness matrices which are associated with both the mixed and exact finite element models. These models can be distinguished by the shape functions derived from the choice of displacement field which plays a crucial role in the accuracy and efficiency of the solution. The main emphasis of this contribution is the derivation and the investigation of this analysis for large scale eigenproblems.     

positivity in time dependent linear transport theory

We present methods using positive semigroups and perturbation theory in the application to the linear Boltzmann equation. Besides being a review, this paper also presents generalizations of known results and develops known methods in a more abstract setting.In Section 1 we present spectral properties of the semigroup operatorsW _a(t) of the absorption semigroup and its generatorT _a. In Section 2 we treat the full semigroup (W(t);t0) as a perturbation of the absorption semigroup. We discuss part of the problems (perturbation arguments and existence of eigenvalues) which have to be solved in order to obtain statements about the large time behaviour ofW(·). In Section 3 we discuss irreducibility ofW(·).In four appendices we present abstract methods used in Sections 1, 2 and 3.     

Scheduling with few changes

In this work we consider scheduling problems where a sequence of assignments from products to machines – or from tasks to operators, or from workers to resources – has to be determined, with the goal of minimizing the costs (=money, manpower, and/or time) that are incurred by the interplay between those assignments. To account for the different practical requirements (e.g. few changes between different products/tasks on the same machine/operator, few production disruptions, or few changes of the same worker between different resources), we employ different objective functions that are all based on elementary combinatorial properties of the schedule matrix. We propose simple and efficient algorithms to solve the corresponding optimization problems, and provide hardness results where such algorithms most likely do not exist.     

A class of Banach spaces with few non-strictly singular operators

We construct a family (X_γ) of reflexive Banach spaces with long (countable as well as uncountable) transfinite bases but with no unconditional basic sequences. The method we introduce to achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to precisely describe the corresponding spaces on non-strictly singular operators. For example, for every pair of countable ordinals γ,β, we are able to decompose every bounded linear operator from X_γ to X_β as the sum of a diagonal operator and an strictly singular operator. We also show that every finite-dimensional subspace of any member X_γ of our class can be moved by and (4+?)-isomorphism to essentially any region of any other member X_δ or our class. Finally, we find subspaces X of X_γ such that the operator space L(X,X_γ) is quite rich but any bounded operator T from X into X is a strictly singular pertubation of a scalar multiple of the identity.     

Time dependent neutron transport theory in multigroup-P L-approximation

Summary A method is developed for solving the time dependent neutron transport equation in multigroupP _L approximation for one-dimensional geometries. The partial differential equations in time and space are solved by means of a power series expansion in the spatial variable. The resulting ordinary differential equations are solved up to theNth order, and the last spatial coefficients are used to satisfy the boundary conditions. Integration of the purely time dependent differential equations is carried out by means of Lie series.Numerical oscillations, appearing for high ordersN, are avoided by subdividing each zone into smaller subzones. Even and odd spatial moments must be developed in opposite directions in each subzone, and the stationary solution representing the initial condition for the time dependent calculation must be developed in the same manner.Results of two calculations in spherical geometry are presented. One is the start-up of a small experimental reactor usingP ₃ theory, the other is a demonstration of neutron waves inP ₁ theory.

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Zusammenfassung Es wird eine Methode zur Lösung der zeitabhängigen Neutronentransportgleichung in der Multigruppen-P _L-Näherung für eindimensionale Geometrien entwickelt. Für die Lösung der partiellen Differentialgleichungen in Ort und Zeit wird eine Potenzreihe im Ort angesetzt. Das sich ergebende gewöhnliche Differentialgleichungssystem wird bis zu einer gewählten OrdnungN erfüllt, und die letzten Koeffizienten der Ortsentwicklungen dienen zur Befriedigung der Randbedingungen. Die Integration in der Zeit erfolgt dann mit der Lie-Reihen-Methode.Zur Vermeidung der Oszillationen, die bei hohen OrdnungenN auftreten, werden die einzelnen Zonen in kleinere Abschnitte unterteilt. Die geraden und ungeraden Momente müssen in den Abschnitten in entgegengesetzter Richtung entwickelt werden. Die stationäre Lösung, die als Anfangsbedingung für die zeitabhängige Rechnung dient, muss mit demselben Entwicklungsschema ermittelt werden.Die Methode wird angewendet zur Lösung derP ₁- undP ₃-Gleichungen in sphärischen Reaktoren. Zwei Beispiele werden damit berechnet: Das Anfahren eines kleinen Versuchsreaktors inP ₃-Näherung und die Nachweisung von Neutronenwellen inP ₁-Theorie.