Extensions completes d'une theorie forcing complete

We give two examples of a universal theoryT such that the forcing companionT ^F (as defined by A. Robinson for infinite forcing) has some model which is not the elementary equivalent of a generic model ofT. Our examples answer in a negative way a question posed by E. Fisher and A. Robinson. In the first example, forT ^F, there is an extensionT′ which is complete and forcing comcomplete (that is, (T′) ^F=T′) not generated by a generic model ofT. In the second exampleT ^F has a complete extensionT′ which is not forcing complete.      

Substructure lattices and almost minimal end extensions of models of Peano arithmetic

This paper concerns intermediate structure lattices Lt(??/??), where ?? is an almost minimal elementary end extension of the model ?? of Peano Arithmetic. For the purposes of this abstract only, let us say that ?? attains L if L ? Lt(??/??) for some almost minimal elementary end extension of ??. If T is a completion of PA and L is a finite lattice, then: (A) If some model of T attains L, then every countable model of T does. (B) If some rather classless, ?₁‐saturated model of T attains L, then every model of T does. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nearly Model Complete Theories

A theory T of a language L is 1-model complete (nearly model complete) iff for every formula ρ of L there is a formula ? (χ) of L which is a ??-formula (a Boolean combination of universal formulas) such that T ? ?x [??θ]. The main results of the paper give characterizations of nearly model complete theories and of 1-model complete theories. As a consequence we obtain that a theory T is nearly model complete iff whenever ?? is a model of T and ???₁??, then T ∪ Δ¹_?? is a complete L(A)-theory, where Δ¹_?? is the 1-diagram of ??. We also point out that our main results extend to (n + l)-model complete and nearly ra-model complete theories for all n > 0.     

Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field

The phase structure of the (2+1) Gross-Neveu model at nonzero temperatureT and in an external fieldH is studied in the leading order of the 1/N expansion. It is shown that for any fixed value ofT there exists a critical valueH _c of the magnetic field such that whenH>H _c the chiral invariance of the model is spontaneously broken. For fixedH, there exists a critical value of the temperatureT _c (H) such that forT>T _c (H) the original symmetry of the model is restored. The phase portrait of the model is constructed in theH-T plane.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 3–11, January, 1992.     

Recursive reconstruction on periodic trees

A periodic tree T_n consists of full n-level copies of a finite tree T. The tree T_n is labeled by random bits. The root label is chosen randomly, and the probability of two adjacent vertices to have the same label is 1−ϵ. This model simulates noisy propagation of a bit from the root, and has significance both in communication theory and in biology. Our aim is to find an algorithm which decides for every set of values of the boundary bits of T, if the root is more probable to be 0 or 1. We want to use this algorithm recursively to reconstruct the value of the root of T_n with a probability bounded away from ½ for all n. In this paper we find for all T, the values of ϵ for which such a reconstruction is possible. We then compare the ϵ values for recursive and nonrecursive algorithms. Finally, we discuss some problems concerning generalizations of this model. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 81–97, 1998     

Geometry of free cyclic submodules over ternions

Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T ². This set of points can be represented as a set of planes in the projective space over F ⁶. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality.     

Minimal models of theories of one function symbol

IfT ₁ is a theory of one function symbol andT ₁ has a minimal model which is not prime, thenT ₁ has 2^No non-isomorphic minimal models. This paper is part of the author’s doctoral dissertation prepared at the Hebrew University under the supervision of Professor Haim Gaifman.     

Probabilistic order level system with lead time when delay in payments are permissible

The probabilistic order level inventory model with lead time when shortages are permitted is developed whe supplier allows some credit periodT ^* for setting the accounts for the purchase quantity. The scheduling periodT is known. Mathematical models are derived for both the cases: i)T ^*≤T and ii)T ^*>T. Expressions are derived for average expected total cost of the system and for obtaining optimum order levelZ=Z ₀ under each case.     

An Efficient Procedure for Dynamic Lot-sizing Model with Demand Time Windows

We consider a dynamic lot-sizing model with demand time windows where n demands need to be scheduled in T production periods. For the case of backlogging allowed, an O(T ³) algorithm exists under the non-speculative cost structure. For the same model with somewhat general cost structure, we propose an efficient algorithm with O(max {T ², nT}) time complexity.     

Locally efficient estimation of regression parameters using current status data

In biostatistics applications interest often focuses on the estimation of the distribution of a time-variable T. If one only observes whether or not T exceeds an observed monitoring time C, then the data structure is called current status data, also known as interval censored data, case I. We consider this data structure extended to allow the presence of both time-independent covariates and time-dependent covariate processes that are observed until the monitoring time. We assume that the monitoring process satisfies coarsening at random.Our goal is to estimate the regression parameter β of the regression model T=Zβ+ε. The curse of dimensionality implies no globally efficient nonparametric estimator with good practical performance at moderate sample sizes exists. We present an estimator of the parameter β that attains the semiparametric efficiency bound if we correctly specify (a) a model for the monitoring mechanism and (b) a lower-dimensional model for the conditional distribution of T given the covariates. In addition, our estimator is robust to model misspecification. If only (a) is correctly specified, the estimator remains consistent and asymptotically normal. We conclude with a simulation experiment and a data analysis.     

Structure theorem for tournaments omitting N5

A finite tournament T is tight if the class of finite tournaments omitting T is well‐quasi‐ordered. We show here that a certain tournament N₅ on five vertices is tight. This is one of the main steps in an exact classification of the tight tournaments, as explained in [10]; the third and final step is carried out in [11]. The proof involves an encoding of the indecomposable tournaments omitting N₅ by a finite alphabet, followed by an application of Kruskal's Tree Theorem. This problem arises in model theory and in computational complexity in a more general form, which remains open: the problem is to give an effective criterion for a finite set {T₁,…,T_k} of finite tournaments to be tight in the sense that the class of all finite tournaments omitting each of T₁,…,T_k is well‐quasi‐ordered. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 165–192, 2003     

Embedded Picard–Vessiot extensions

We prove that if T is a theory of large, bounded, fields of characteristic 0 with almost quantifier elimination, and T_D is the model companion of T∪{“? is a derivation”}, then for any model (𝒰,?) of T_D, differential subfield K of 𝒰 such that C_K?T, and linear differential equation ?Y = AY over K, there is a Picard-Vessiot extension L of K for the equation with K≤L≤𝒰, i.e. L can be embedded in 𝒰 over K, as a differential field. Moreover such L is unique to isomorphism over K as a differential field. Likewise for the analogue for strongly normal extensions for logarithmic differential equations in the sense of Kolchin.     

Abstract Characteristic Function

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k_S(z,w) = (1 - z [`(w)])^-1{k_S(z,w) = (1 - z {\overline {w}})^{-1}} for |z|, |w| < 1, by means of (1/k _S )(T, T*) ≥ 0, we consider an arbitrary open connected domain Ω in \mathbb Cⁿ{{\mathbb {C}}^n}, a kernel k on Ω so that 1/k is a polynomial and a tuple T = (T ₁, T ₂, . . . , T _n ) of commuting bounded operators on a complex separable Hilbert space H{\mathcal H} such that (1/k)(T, T*) ≥ 0. Under some standard assumptions on k, it turns out that whether a characteristic function can be associated with T or not depends not only on T, but also on the kernel k. We give a necessary and sufficient condition. When this condition is satisfied, a functional model can be constructed. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples T.     

|T |+‐resplendent models and the Lascar group

In this paper we show that in every |T |⁺‐resplendent model N , for every A ? N such that |A | ≤ |T |, the group Autf(N/A ) of strong automorphisms is the least very normal subgroup of the group Aut(N/A ) and the quotient Aut(N/A )/Autf(N/A ) is the Lascar group over A . Then we generalize this result to every |T |⁺‐saturated and strongly |T |⁺‐homogeneous model. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two cardinal compactness

We prove that ifλ≧μ ^ℵo=μ≧|T | and if every finite subtheory ofT has a (λ, μ)-model (i.e. a model with a domain of powerλ, in which a distinguished predicate is interpreted as a set ofμ elements) thenT has such a model. There are generalizations forμ-like models (or, equivalently, to languages with generalized quantifiers).     

Simultaneous Distributions of Space-Time Wave Characteristics in a Gaussian Sea

A simultaneous pair (L, T) is defined, representing crest length and crest period of the same random wave, i.e. the length and duration of the half of the wave that contains the crest. The simultaneous density in the ergodic sense of (L, T) is derived and evaluated for a homogeneous Gaussian spatio-temporal random sea model. Other names of this type of distribution are Palm, intensity, and long-run distribution. The density is compared to the deterministic relation given by the dispersion relation for deep water: L=T ² g/. A two-dimensional sea model parametrized by time and a single space coordinate is used, based on a directional spectrum. Only the case of deep water is considered. Furthermore, the crest height associated with the (L, T)-pair is studied. The ergodic distribution of (L, T) for significant waves is compared to that for all waves. All results are verified through comparison to simulated observations, but no real data are considered.     

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

It is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σ_w(f(T)) = f(σ_w(T)), where σ(T) and σ_w(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T) + Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT + F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT + F.     

Joint and supremum distributions in the compound binomial model with Markovian environment

In this paper, we study the compound binomial model in Markovian environment, which is proposed by Cossette, et al. (2003). We obtain the recursive formula of the joint distributions of T, X(T − 1) and |X(T)| (i.e., the time of ruin, the surplus before ruin and the deficit at ruin) by the method of mass function of up-crossing zero points, as given by Liu and Zhao (2007). By using the same method, the recursive formula of supremum distribution is obtained. An example is included to illustrate the results of the model.     

Reduced products which are not saturated

Summary IfT is a complete theory of Boolean algebra, then we writeA ⊲_T B to denote that for every cardinal κ and every κ-regular filter over a setI such that the Boolean algebra 2 _F ^I of all subsets ofI reduced byF is a model ofT, the reduced powerA _F ^I isK ⁺-saturated wheneverB _F ^I isK ⁺-saturated. The relation ⊲_T generalizes the relation ◃ introduced by Keisler. As in the case of Keisler's ◃ it happens that ⊲_T’s are relations between complete theories, i.e. ifA≡B thenA ⊲_T B andB ⊲_T A. In this paper some examples of theories which are maximal (minimal) with respect to ⊲_T’s are provided and the relations ⊲_T are compared with each other. Presented by J. Mycielski