Sheva-sheva-sheva: Large creatures

We develop the theory of the forcing with trees and creatures for an inaccessible λ continuing Rosłanowski and Shelah [15], [17]. To make a real use of these forcing notions (that is to iterate them without collapsing cardinals) we need suitable iteration theorems, and those are proved as well. (In this aspect we continue Rosłanowski and Shelah [16] and Shelah [20], [21].)     

Small idempotent clones I

G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning p _n-sequences and free spectra of algebras to the topic Small idempotent clones (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in p _n-sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids (G, ·) with p ₂(G, ·) 2 (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26, 27], [25], [28, 30, 31, 32] and [34].     

On ultraproducts of Boolean algebras and irr

1. Consistent inequality [We prove the consistency of irrirr(B_i)/D where D is an ultrafilter on and each B_i is a Boolean algebra and irr(B) is the maximal size of irredundant subsets of a Boolean algebra B, see full definition in the text. This solves the last problem, 35, of this form from Monk's list of problems in [M2]. The solution applies to many other properties, e.g. Souslinity.] 2. Consistency for small cardinals [We get similar results with =₁ (easily we cannot have it for =₀) and Boolean algebras B_i (i<) of cardinality .] This article continues Magidor Shelah [MgSh:433] and Shelah Spinas [ShSi:677], but does not rely on them: see [M2] for the background. I would like to thank Alice Leonhardt for the beautiful typing. This research was partially supported by the Israel Science Foundation. Publication 703     

Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems {ie133-01} where {ie133-2},u is a complex valued function and φ is a real valued function. When (c ₁,c ₂) = (-1, 2) the system (*) is called DSI equation in the inverse scattering literature. Our purpose in this paper is to prove the local existence of a unique solution to (*) in the Sobolev spaceH ²(R ²) without the smallness condition on the data which were assumed in previous works [7], [17], [19], [26], Our result is a positive answer to Question 7 in [24].     

Iterated souslin forcing,the principles ⋄(E) and a generalisation of the axiom SAD

The axiom SAD was introduced in our paper with Avraham and Shelah [1]. It is a Martin’s Axiom type of principle, having some of the consequences of MA plus , but nonetheless provably consistent with GCH. In [1] it was shown to be consistent (with GCH) and used to demonstrate the consistency with CH of some known consequences of MA + . In particular, SAD implies the negation of Jensen’s ⋄ principle. In this paper we present a generalisation of SAD, let us call it SAD(E), whereE will be an arbitrary stationary subset ofω ₁, and show that although SAD(E) implies the negation of ⋄(E), it is consistent with ⋄. SAD(E) resembles the axiom SA of Shelah, described in our survey article [2], and indeed is a sort of blending of the two principles SA and SAD. (In particular, Shelah proved that SA is consistent with ⋄ but implies the failure of some ⋄(E).) Our proof (of the consistency of SAD(E) with ⋄) will be of interest to forcing enthusiasts, since it shows that iterated Souslin forcingcan distinguish between different stationary sets (it was previously thought that this was not the case), and can indeed be used to establish the non-provability of the principles ⋄(E) from ⋄ alone.     

Tree indiscernibilities,revisited

We give definitions that distinguish between two notions of indiscernibility for a set ${\{a_{\eta} \mid \eta \in ^{\omega>}\omega\}}$ that saw original use in Shelah [Classification theory and the number of non-isomorphic models (revised edition). North-Holland, Amsterdam, 1990], which we name s- and str?indiscernibility. Using these definitions and detailed proofs, we prove s- and str-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP₁ or TP₂ that has not seen explication in the literature. In the Appendix, we exposit the proofs of Shelah [Classification theory and the number of non-isomorphic models (revised edition). North-Holland, Amsterdam, 1990, App. 2.6, 2.7], expanding on the details.     

The Spectra of General Differential Operators in the Direct Sum Spaces

In this paper, the general ordinary quasi-differential expression M _pof n-th order with complex coefficients and its formal adjoint M _p ⁺ on any finite number of intervals I _p =(a _p ,b _p ),p= 1,...,N, are considered in the setting of the direct sums of L _wp ² (a _p ,b _p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.     

Optimal controllability in viscoelasticity of rate type

Recently, the control engineers interest turns upon material with structural damping, such as hysteretic effects in elastoplastic systems [27] and flexible space structures [30] and in particular with damping due to the viscoelastic nature of the material, in the sense of Kelvin, [7], [12]. The reason for this renewed interest is the possibility of constructing finite rank compensators with small spillover in the observation [1], [5], [11], [25]. However, a control theory for the infinite system does not seem to exist in the literature. It is shown that the trajectories, initially in a certain subspace, can be steered to rest in any time T > 0 using distributed load controls

1 The reesults are partially contained in [19].

. Furthermore, and perhaps more interesting, the L₂ (0, T, H)-norm of the control decreases as T increases. This gives rise to the problem of time-optimal-controllability and the correlation to the problem of minimum norm controllability. It is shown that the time-optimal control is characterized by a weak “bang-bang” principle.     

Large normal ideals concentrating on a fixed small cardinality

The property on the filter in Definition 1, a kind of large cardinal property, suffices for the proof in Liu Shelah [LiSh484] and is proved consistent as required there (see Conclusion 6). A natural property which looks better, not only is not obtained here, but is shown to be false (in Claim 7). On earlier related theorems see Gitik Shelah [GiSh310]. On such games see e.g. [Je], [Sh-b], [Sh-f]. Received June 27, 1994     

The model theory of m‐ordered differential fields

In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields (more precisely domains) with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable (see also [8]). In this paper we study the case where the fields are expanded with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language L = {+, –, ·, D, <₁, …, <_m, 1, 0}. We denote this model companion by CODF_m and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover (just by putting m = 1) the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4]. Finally, we prove that it is possible to describe the completions of CODF_m and to obtain quantifier elimination in a slightly enriched (infinite) language. This generalizes van den Dries' results in the “derivation free” case. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the order of the antipode of a pointed Hopf algebra

Let k be a field and let H denote a pointed Hopf k-algebra with antipode S. We are interested in determining the order of S. Building on the work done by Taft and Wilson in [7], we define an invariant for H, denoted m_H, and prove that the value of this invariant is connected to the order of S. In the case where char k?=?0, it is shown that if S has finite order then it is either the identity or has order 2?m_H. If in addition H is assumed to be coradically graded, it is shown that the order of S is finite if and only if m_H is finite. We also consider the case where char k?=?p?>?0, generalizing the results of [7] to the infinite-dimensional setting.     

Values of rational functions on non-hilbertian fields and a question of Weissauer

We answer in the negative a question raised by Fried and Jarden, asking whether the quotient field of a unique factorization domain with infinitely many primes is necessarily hilbertian. This implies a negative answer to a related question of Weissauer. Our constructions are simple and take place inside the field of algebraic numbers. Simultaneously we investigate the relation of hilbertianity of a fieldK with the structure of the value sets of rational functions onK: we construct a non-hilbertian subfieldK of such that, given anyf ₁ ,…,f _h ∈K(x), each of degree ≥2, the union ∪ _z=1 ^h f _z(K) does not containK. See e.g. [FrJ], [L1], [L2], [Sch], [Se1], or [Se2] for the classical theory of hilbertian fields.     

Addendum to “A definable nonstandard enlargement”

?o?'s theorem for bounded D ‐ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [4], is established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universality at the edge of the spectrum for unitary,orthogonal, and symplectic ensembles of random matrices

We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e^?V(x) where V is a polynomial, V(x) = κ_2mx^2m + · · ·, κ_2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel‐Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.     

S. G Mikhlin (1908 – 1990)

Our concern is to find a representation theorem for operators in B(c(X), c(Y)) where X and Y are Banach spaces with Y containing an isomorphic copy of c₀. Cass and Gao [1] obtained a representation theorem that always applies if Y does not contain an isomorphic copy of c₀. Maddox [3], Melvin - Melvin [4], and Robinson [5] consider operators in B(c(X), c(Y)) that are given by matrices. In this paper we show that Cass's and Gao's result in [1] can be extended, when Y contains an isomorphic copy of c₀, to certain operators that we call represent able. In addition, we show that when Y contains an isomorphic copy of co there are always operators that fall outside the scope of our representation theorem. Light is also cast on a theorem given in Maddox [3, Theorem 4.2] and [5, Theorem IV].     

On voronoi diagrams in theL p-metric in ℝD

We prove upper bounds on the number ofL _p-spheres passing throughD+1 points in general position in ℝ”, and on the sum of the Betti numbers of the intersection of bisectors in theL _p-metric, wherep is an even positive integer. The bounds found do not depend onp. Our result implies that the complexity of Voronoi diagrams (for point sites in general position) in theL _p-metric is bounded for increasingp. The proof for this upper bound involves the techniques of Milnor [12] and Thom [16] for finding a bound on the sum of the Betti numbers of algebraic varieties, but instead of the usual degree of polynomials we use their additive complexity, and apply results of Benedetti and Risler [2], [13]. Furthermore, we prove that inD dimensions and for evenp the number ofL _p-spheres passing throughD+1 points in general position is odd. In particular, combined with results of [8], [9], our results clarify the structure of Voronoi diagrams based on theL _p-metric (with evenp) in three dimensions. For the proof we use the theory of degree of continuous mappings in ℝ^D, which is a tool widely applied in nonlinear analysis [14]. This work was partially supported by Deutsche Forschungsgemeinschaft, Grant K1 655/2-1. A preliminary version of this paper was presented at the 11th Annual Symposium on Theoretical Aspects of Computer Science, France, 1994.     

Fixed point results for multivalued contractions on ordered gauge spaces

The purpose of this article is to present fixed point results for multivalued E _≤-contractions on ordered complete gauge space. Our theorems generalize and extend some recent results given in M. Frigon [7], S. Reich [12], I.A. Rus and A. Petruşel [15] and I.A. Rus et al. [16].      

Direct products of automorphism groups of graphs

In this article we study the product action of the direct product of automorphism groups of graphs. We generalize the results of Watkins [J. Combin Theory 11 (1971), 95–104], Nowitz and Watkins [Monatsh. Math. 76 (1972), 168–171] and W. Imrich [Israel J. Math. 11 (1972), 258–264], and we show that except for an infinite family of groups S_n × S_n, n≥2 and three other groups D₄ × S₂, D₄ × D₄ and S₄ × S₂ × S₂, the direct product of automorphism groups of two graphs is itself the automorphism group of a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 26–36, 2009     

Note onk-point separation measurement

Summary Glick [1] introduced the notion of a separation measurement and showed that for a setf ₁,…,f _k of densities, is aK-point separation measurement. This notion is some generalization of Matusita's distance (affinity) of densitiesf ₁,f ₂, …,f _k, and its interesting applications were shown in Matusita [2], [3]. In this paper we given some statistical remarks on a separation measurement.