Multi-operator scaling random fields

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling property. Actually, they locally look like operator scaling random fields, whose order is allowed to vary along the sample paths. We also give an upper bound of their modulus of continuity. Their pointwise Hölder exponents may also vary with the position x and their anisotropic behavior is driven by a matrix which may also depend on x.     

A local support theorem for the radiation fields on asymptotically euclidean manifolds

We prove a local support theorem for the radiation fields on asymptotically euclidean manifolds which generalizes the local support theorem for the Radon transform.     

局部环上典型群的生成问题

回顾了域上典型群的生成问题所得成果,对局部环上典型群生成问题的研究,作了一个展示,并按展示的范式,给出了辛群生成定理的证明.     

Stopping for two-dimensional stochastic processes

The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R²₊. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

A New Approach to Investigation of Carnot�CCarath��odory Geometry

We develop a new approach to studying the geometry of Carnot–Carathéodory spaces under minimal assumptions on the smoothness of basis vector fields. We obtain quantitative comparison estimates for the local geometries of two different local Carnot groups, as well as of a local Carnot group and the original space. As corollaries, we deduce some results that are well-known and basic for the “smooth” case: the generalized triangle inequality for d _∞, the local approximation theorem for the quasimetric d _∞, the Rashevskiǐ–Chow theorem, the ball-box theorem, and so on.     

On the normal approximation for random fields via martingale methods

We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random sequences by Maxwell and Woodroofe. Our approach is based on new results for triangular arrays of martingale differences, which have interest in themselves. We provide as applications new results for linear random fields and nonlinear random fields of Volterra-type.     

Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations

We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.     

Oriented Chow groups, Hermitian K-theory and the Gersten conjecture

We show that the oriented Chow groups of Barge–Morel appear in the E ₂-term of the coniveau spectral sequence for Hermitian K-theory. This includes a localization theorem and the Gersten conjecture (over infinite base fields) for Hermitian K-theory. We also discuss the conjectural relationship between oriented and higher oriented Chow groups and Levine’s homotopy coniveau spectral sequence when applied to Hermitian K-theory.     

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.     

A family of quintic cyclic fields with even class number parameterized by rational points on an elliptic curve

We give a family of quintic cyclic fields with even class number parametrized by rational points on an elliptic curve associated with Emma Lehmer's quintic polynomial. Further, we use the arithmetic of elliptic curves and the Chebotarev density theorem to show that there are infinitely many such fields.     

Riemann-integration and a new proof of the Bichteler–Dellacherie theorem

We give a new proof of the celebrated Bichteler–Dellacherie theorem, which states that a process S$S$ is a good integrator if and only if it is the sum of a local martingale and a finite-variation process. As a corollary, we obtain a characterization of semimartingales along the lines of classical Riemann integrability.     

A Note on Class Field Theory for Two-Dimensional Local Rings

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K ₂-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.     

Lusztig's isomorphism theorem for cellular algebras

In this paper, we first introduce a notion of semisimple system with parameters, then we establish Lusztig's isomorphism theorem for any cellular semisimple system with parameters. As an application, we obtain Lusztig's isomorphism theorem for Ariki-Koike algebras, cyclotomic q-Schur algebras and Birman-Murakami-Wenzl algebras. Second, using the results for certain Ariki-Koike algebras, we prove an analogue of Lusztig's isomorphism theorem for the cyclotomic Hecke algebras of type G(p,p,n) (which are not known to be cellular in general). These generalize earlier results of [G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981) 490-498.] on such isomorphisms for Iwahori-Hecke algebras associated to finite Weyl groups.     

A note on a refined version of Anderson-Brownawell-Papanikolas criterion

We give a refinement of the linear independence criterion over function fields developed by Anderson, Brownawell and Papanikolas [Greg W. Anderson, W. Dale Brownawell, Matthew A. Papanikolas, Determination of the algebraic relations among special Γ-values in positive characteristic, Ann. of Math. 160 (2004) 237-313]. As a consequence, a function field analogue of the Siegel-Shidlovskii theorem is derived.     

A sampling theorem for multivariate stationary processes

The notion of sampling for second-order q-variate processes is defined. It is shown that if the components of a q-variate process (not necessarily stationary) admits a sampling theorem with some sample spacing, then the process itself admits a sampling theorem with the same sample spacing. A sampling theorem for q-variate stationary processes, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloy's work. This sampling theorem is used to show that if a q-variate stationary process admits a sampling theorem, then each of its components will admit a sampling theorem too.     

Liouville's theorem and the maximum modulus principle for a system of complex vector fields

This work is concerned with Liouville's theorem and the maximum principle for the homogeneous solutions of systems of complex vector fields . Necessary and sufficient conditions are provedfor tube structures to have the Liouville property. Maximum principles are proved for a general system of complex vector fields which are integrable. As an application, in the case of vector fields, we get new characterizations of the local solvability property (P) of Nirenberg andTreves. Another application concerns a solvability condition (P_n-1) introduced by P.Cordaro and J. Hounie in differential complexes associated to locally integrable structures.     

Self-dual normal integral bases for infinite unramified extensions

We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.     

A note on the Mordell-Weil rank modulo n

Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.     

A central limit theorem for stationary random fields

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form _Xk=g(_εk−s,s∈Z^d)$X_k=g({\epsilon }_{k-s},s\in {\mathbb{Z}}^d)$, k∈Z^d$k\in {\mathbb{Z}}^d$, where _(εi)i∈Zd${\left({\epsilon }_i\right)}_{i\in {\mathbb{Z}}^d}$ are iid random variables and g$g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.