An Algorithm for Reducibility of 3-arrangements

We consider a central hyperplane arrangement in a three-dimensional vector space. The definition of characteristic form to a hyperplane arrangement is given and we could make use of characteristic form to judge the reducibility of this arrangement. In addition, the relationship between the reducibility and freeness of a hyperplane arrangement is given     

Reducibility of hyperplane arrangements

Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.     

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: I. The Covariant Formalism

We present a covariant approach to the kinetic theory of quantum electrodynamic plasma in a strong electromagnetic field. The method is based on the relativistic von Neumann equation for the nonequilibrium statistical operator defined on spacelike hyperplanes in Minkowski space. We use the canonical quantization of the system on hyperplanes and a covariant generalization of the Coulomb gauge. The condensate mode associated with the mean electromagnetic field is separated from the photon degrees of freedom by a time-dependent unitary transformation of the dynamic variables and the nonequilibrium statistical operator. This allows using expansions of correlation functions and of the statistical operator in powers of the fine structure constant even in the presence of a strong electromagnetic field. We present a general scheme for deriving kinetic equations in the hyperplane formalism.     

Multiplier ideals of hyperplane arrangements

In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).

     

Linear connections of a framed distribution of hyperplane elements in conformal space

We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions $ \mathcal{M} We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions and in conformal space C _n , where is a distribution of hyperplane elements, and is a distribution of line elements. We consider invariant fields of pencils that are parallel with respect to the normal connection along any curve belonging to the distribution . Original Russian Text ? A.M. Matveeva, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 7, pp. 79–84.     

Valuations of glued near hexagons

Valuations of near polygons were introduced in [ 12 ] as an important tool for classifying dense near polygons. In the present article, we will introduce the class of the semi‐diagonal valuations. These valuations live in glued near hexagons. A glued near hexagon S can be coordinatized by a pair of admissible triples; such triples consist of a Steiner system , a group G, and a certain nice map . We will give a necessary and sufficient condition for the existence of semi‐diagonal valuations in in terms of these two admissible triples. For two classes of glued near hexagons, we will use this condition to determine all semi‐diagonal valuations. Each semi‐diagonal valuation will also give rise to a hyperplane of the glued near hexagon. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 35–48, 2007     

A new method for complete ranking of DMUs

So far numerous models have been proposed for ranking the efficient decision-making units (DMUs) in data envelopment analysis (DEA). But, the most shortcoming of these models is their two-stage orientation. That is, firstly we have to find efficient DMUs and then rank them. Another flaw of some of these models, like AP-model (A procedure for ranking efficient units in data envelopment analysis, Management Science, 39 (10) (1993) 1261–1264), is existence of a non-Archimedean number in their objective function. Besides, when there is more than one weak efficient unit (or non-extreme efficient unit) these models could not rank DMUs. In this paper, we employ hyperplanes of the production possibility set (PPS) and propose a new method for complete ranking of DMUs in DEA. The proposed approach is a one stage method which ranks all DMUs (efficient and inefficient). In addition to ranking, the proposed method determines the type of efficiency for each DMU, simultaneously. Numerical examples are given to show applicability of the proposed method.     

The Supersolvable Order of Hyperplanes of an Arrangement

This paper mainly gives a sufficient and necessary condition for an order of hyperplanes of a graphic arrangement being supersolvable. In addition, we give the relations between the set of supersolvable orders of hyperplanes and the set of quadratic orders of hyperplanes for a supersolvable arrangement.     

An Outer Approximation Algorithm Guaranteeing Feasibility of Solutions and Approximate Accuracy of Optimality

We treat a concave programming problem with a compact convex feasible set. Assuming the differentiability of the convex functions which define the feasible set, we propose two solution methods. Those methods utilize the convexity of the feasible set and the property of the normal cone to the feasible set at each point over the boundary. Based on the proposed two methods, we propose a solution algorithm. This algorithm takes advantages over classical methods: (1) the obtained approximate solution is always feasible, (2) the error of such approximate value can be evaluated properly for the optimal value of such problem, (3) the algorithm does not have any redundant iterations.     

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.