On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.      

Compact solutions to the equation Tx = y in a weakly closed $$ \mathcal{T}(\mathcal{N}) $$-module

Given two vectors x, y in a Hilbert space and a weakly closed -module , we provide a necessary and sufficient condition for the existence of a compact operator T in satisfying Tx = y.     

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator defined on the whole L _(0,1)²(Ω), where is the multiplication by and is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the -Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and . Partially supported by the FWF grant P19147-N13.     

On ideals and quotients of A \mathcal{T} -algebras

Some results on A -algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of A -algebras are A -algebras or A -algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an A -algebra being an A -algebra.     

$${\mathcal{M}}$$ -decomposability and symmetric unimodal densities in one dimension

In this paper, we introduce the notion of -decomposability of probability density functions in one dimension. Using -decomposability, we derive an inequality that applies to all symmetric unimodal densities. Our inequality involves only the standard deviation of the densities concerned. The concept of -decomposability can be used as a non-parametric criterion for mode-finding and cluster analysis.     

On the size of the Jacobians of curves over finite fields

Abdract  Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field of q elements, we obtain the asymptotic formula q ^g+o(g) for the size of set of the -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.      

On a theorem of Cantor-Bernstein type for algebras

Freytes proved a theorem of Cantor-Bernstein type for algbras; he applied certain sequences of central elements of bounded lattices. The aim of the present paper is to extend the mentioned result to the case when the lattices under consideration need not be bounded; instead of sequences of central elements we deal with sequences of internal direct factors of lattices. This work was supported by Science and Technology Assistance Agency under the contract No APVT-51-032002.     

A combinatorial algorithm for the TDMA message scheduling problem

In this paper, we introduce a combinatorial algorithm for the message scheduling problem on Time Division Multiple Access (TDMA) networks. In TDMA networks, time is divided in to slots in which messages are scheduled. The frame length is defined as the total number of slots required for all stations to broadcast without message collisions. The objective is to provide a broadcast schedule of minimum frame length which also provides the maximum throughput. This problem is known to be -hard, thus efficient heuristics are needed to provide solutions to real-world instances. We present a two-phase algorithm which exploits the combinatorial structure of the problem in order to provide high quality solutions. The first phase finds a feasible frame length in which the throughput is maximized in phase two. Computational results are provided and compared with other heuristics in the literature as well as to the optimal solutions found using a commercial integer programming solver. Experiments on 63 benchmark instances show that the proposed method is able to provide optimal frame lengths for all cases with near optimal throughput.     

Analysis on adjoint non-recurrent property of nonlinear time series in random environment domain

By introducing a random interference into the typical of nonlinear time series model, this paper establishes a RENLAR model: . The author introduces the definition of adjoint non-recurrence, and utilizing general state space Markov chain theorem, we obtain some criteria for non-recurrence and adjoint non-recurrence of nonlinear time series models in random environment domain and analyze adjoint non-recurrence of some models by using these criteria. Research supported Science Foundation of China (10171009).     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

An algorithm for matching point sets using the l_1 norm

The problem is considered of matching two sets of points in , by translation and rotation. There are many applications, for example in geodesy, computer vision and in the assessment of manufactured parts. When the matching criterion is least squares, there is a well known solution process based on the singular value decomposition of an matrix. Here we consider the use of the norm, which may be more appropriate than least squares in the context of wild points in the data. An algorithm is developed, and is illustrated by some examples for the case .     

Generalized Continuous Posets and a New Cartesian Closed Category

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed. This research is supported by the National Natural Science Foundation of China, 10471035.     

Fixed points in {\mathcal{M}}-fuzzy metric spaces

In this paper, we give some common fixed point theorems for five mappings satisfying some conditions in -fuzzy metric spaces.     

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

Anomalous dimensions of Wilson operators in the N = 4\mathcal{N} = 4 supersymmetric Yang-Mills theory

We present results for the universal anomalous dimension γ_un(j) of Wilson twist-2 operators in the supersymmetric Yang-Mills theory in the first three orders of the perturbation theory. We obtain these expressions by extracting the most complicated terms from the corresponding anomalous dimensions in QCD. The result obtained agrees with the hypothesis of the integrability of the supersymmetric Yang-Mills theory in the context of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 249–262, February, 2007.     

Metric differentiability of Lipschitz maps defined on Wiener spaces

This note is devoted to the differentiability properties of -Lipschitz maps defined on abstract Wiener spaces and with values in metric spaces (Y,d _Y ). We prove the existence γ-a.e. of local seminorms in the Cameron-Martin space which allow to compute the metric derivative of the map. Second Author partially supported by DGES (Spain) MTM2006-03531.     

Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

Global error bounds for the extended vertical LCP

A new necessary and sufficient condition for the row -property is given. By using this new condition and a special row rearrangement, we provide two global error bounds for the extended vertical linear complementarity problem under the row -property, which extend the error bounds given in Chen and Xiang (Math. Program. 106:513–525, 2006) and Mathias and Pang (Linear Algebra Appl. 132:123–136, 1990) for the P-matrix linear complementarity problem, respectively. We show that one of the new error bounds is sharper than the other, and it can be computed easily for some special class of the row -property block matrix. Numerical examples are given to illustrate the error bounds. The work was in part supported by a Grant-in-Aid from Japan Society for the Promotion of Science, and the National Natural Science Foundation of China (10671010).     

The cubic Segre variety in PG(5, 2)

The Segre variety in PG(5, 2) is a 21-set of points which is shown to have a cubic equation Q(x) = 0. If T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing the cubic polynomial Q, then the associate U ^# of an r-flat is defined to be