The structure of harmonizable isotropic random fields

The class of harmonizable processes and fields are a natural extension of the class of stationary processes and fields. Random fields admit an additional property called isotropy. The classical spectral and covariance representations for stationary isotropic random fields are extended to the harmonizable isotropic case. A classification of these fields is obtained based upon the smoothness properties of their covariances. In contrast to the stationary case, it is also shown that there exist non-trivial harmonizable isotropic fields which satisfy the Laplace operator in the L ²-sense     

Halbordnungen und maximale Teilkörper in gelochten algebraisch-abgeschlossenen Körpern

In a former notice the author has shown that the so-calledh-closed fields (coming from halfordered fields) can be characterized in a similar way than the real closed fields (coming from ordered fields). But the case of characteristic 2 had been excluded. In this notice now a new characterization ofh-closed fields is given which includes fields of characteristic 2. Indeed, they are all maximal subfields of an algebraically closed field not containing a given element.     

Witt equivalence of function fields of curves over local fields

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13 G?adki, P., Marshall, M. Witt equivalence of function fields over global fields. Trans. Am. Math. Soc., electronically published on April 11, 2017, doi: https://doi.org/10.1090/tran/6898 (to appear in print).[Crossref] , [Google Scholar]] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.     

Axiomatisations simples des theories des corps de rolle

Using Becker's results we obtain here a simple first order axiomatization, looking like those by Artin-Schreier and also written in the language of fields, for the theory of Rolle fields (i.e. fields with the Rolle's property for every order). In fields having a finite number of orders, we characterize Rolle fields as those which are pythagorean at level 4 and do not admit any algebraic extension of odd degree. Then we give an axiomatization for Rolle fields having exactly 2ⁿ orders (n≥0); in fact, for n=0 we recover an axiomatization of the theory of real-closed fields and for n=1 we get exactly an axiomatization given for the theory of chain-closed fields by the author in [G1]. Finally we prove that a Rolle field with exactly 2ⁿ orders is the intersection of n+1 real closures of the field.      

Two-parameter Lévy processes: ItÔ formula,semigroups, and generators

We consider random Lévy fields, i.e., stationary fields continuous in probability and having independent increments. We prove that the trajectories of such fields have at most one jump on every line parallel to the axes. We derive an expression for the ItÔ change of variables for Lévy fields. We also consider semigroups generated by Lévy fields and their generators.Published in Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 7, pp. 952–961, July, 1995.     

Harmonicity of vector fields on four-dimensional generalized symmetric spaces

Let (M = G/H;g)denote a four-dimensional pseudo-Riemannian generalized symmetric space and g = m + h the corresponding decomposition of the Lie algebra g of G. We completely determine the harmonicity properties of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.     

Lokal-Global-Prinzipien für die Brauergruppe

This article looks at Local-Global-Principles for the Brauer group, modeled after the celebrated theorem of Hasse-Brauer-Noether for the Brauer group of a number field. Pop introduced a property for fields, which holds especially for real closed andp-adically closed fields and yields a Local-Global-Principle for function fields of one variable over such fields. Then he used model theoretical means to generalize these results to arbitrary extensions of transcendental degree one over real closed andp-adically closed fields. This paper achieves this in a more elementary manner. Another result are examples of fields where the Local-Global-Principle is violated.

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This article was processed by the author using the IATEX style filecljour1 from Springer-Verlag.     

Approximation by ridge function fields over compact sets

We study the approximation of a continuous function field over a compact set T by a continuous field of ridge approximants over T, named ridge function fields. We first give general density results about function fields and show how they apply to ridge function fields. We next discuss the parameterization of sets of ridge function fields and give additional density results for a class of continuous ridge function fields that admits a weak parameterization. Finally, we discuss the construction of the elements in that class.     

一类独立随机场的强定律的收敛率

§ 1　IntroductionDefinition1 .[1 ] A field{ Xi,i∈Nd} is called negatively associated(NA) if for every pair ofdisjoint subsets T1 ,T2 of Nd,Cov(f1 (Xi,i∈ T1 ) ,f2 (Xj,j∈ T2 ) )≤ 0 ,whenever f1 and f2 are coordinatewise increasing.Definition2 .[1 ] A field{ Xi,i∈Nd} is calledρ* -mixing ifρ* (s) =sup{ (ρ(S,T) ;S,T N,dist(S,T)≥ s}→ 0 (s→∞ ) ,whereρ(S,T) =sup{ |E(f -Ef) (g -Eg) |/‖ f -Ef‖2 ‖ g -Eg‖2 ,f∈ L2 (σ(S) ) ,g∈ L2 (σ(T) ) } .Definition 3.[1 ] A field { Xi…     

The global dynamics of a class of vector fields in ℝ<Superscript>3</Superscript>

In this paper, we find a bridge connecting a class of vector fields in ℝ³ with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of a class of vector fields in ℝ³. All the possible nonwandering sets of this class of vector fields fall into three classes: (i) singularities; (ii) closed orbits; (iii) graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.     

A generalization of Voronoi's unit algorithm I

By means of a new method of mapping an algebraic number field into a euclidean space Voronoi's unit algorithm is generalized to all algebraic number fields and it is proved that the generalized Voronoi algorithm computes the fundamental units of all algebraic number fields of unit rank 1, i.e., of the real quadratic fields, of the complex cubic fields, and of the totally complex quartic fields.     

The uniform companion for large differential fields of characteristic 0

We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic , which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.

     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

Topological differential fields

We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields (requiring no interaction between the derivation and the other primitives of the language). We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax-Kochen-Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations (respectively several orders).     

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

Spectral representation of intrinsically stationary fields

Transferring the concept of processes with weakly stationary increments to arbitrary locally compact Abelian groups two closely related notions arise: while intrinsically stationary random fields can be seen as a direct analog of intrinsic random functions of order k$k$ applied by G. Matheron in geostatistics, stationarizable random fields arise as a natural analog of definitizable functions in harmonic analysis. We concentrate on intrinsically stationary random fields related to finite-dimensional, translation-invariant function spaces, establish an orthogonal decomposition of random fields of this type, and present spectral representations for intrinsically stationary as well as stationarizable random fields using orthogonal vector measures.     

Quantifier elimination for henselian fields relative to additive and multiplicative congruences

Several classes of henselian valued fields admit quantifier elimination relative to structures which reflect the additive and multiplicative congruences of the field. Value groups and residue fields may be viewed as reduts of these structures. A general theorem is given using the theory of tame extensions of henselian fields. Special cases like the case ofp-adically closed fields and the case of henselian fields of residue characteristic 0 are discussed. This work was completed during a visit in the Institute for Advanced Studies at the Hebrew University of Jerusalem.     

Function Fields and Elementary Equivalence

Continuing work of Duret, we treat the relation between isomorphismand elementary equivalence of function fields over algebraicallyclosed fields. For function fields of curves, these are ‘usually’the same, but in characteristic zero, for elliptic curves withcomplex multiplication, a weak variant of elementary equivalenceof their function fields corresponds to isomorphism of the endomorphismrings of the curves, not to isomorphism of the curves themselves.1991 Mathematics Subject Classification 14H52, 11U09, 03C52.     

ON THE COMPUTATION OF TRACE FORMS OF ALGEBRAS WITH INVOLUTION

In this paper, we compute explicitely the isomorphism class of the trace of k-algebras with involution (of any kind) for some special base fields, especially for non formally real global fields, euclidean fields and the field of rational numbers.