Local complete intersections in and Koszul syzygies

We study the syzygies of a codimension two ideal . Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus is generated by the Koszul syzygies iff is a local complete intersection. The proof uses a characterization of complete intersections due to Herzog. When is saturated, we relate our theorem to results of Weyman and Simis and Vasconcelos. We conclude with an example of how our theorem fails for four generated local complete intersections in and we discuss generalizations to higher dimensions.

     

Small complete caps in PG(N,q), q even

A new family of small complete caps in PG(N,q), q even, is constructed. Apart from small values of either N or q, it provides an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N,q): for N even, the leading term is replaced by with , for N odd the leading term is replaced by with . © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 420–436, 2007     

Diffeomorphism classification of smooth weighted complete intersections

Xn(d1, . . . , dr-1, dr; w) and Xn(e1, . . . , er-1, dr; w) are two complex odd-dimensional smooth weighted complete intersections defined in a smooth weighted hypersurface Xn+r-1(dr; w). We prove that they are diffeomorphic if and only if they have the same total degree d, the Pontrjagin classes and the Euler characteristic, under the following assumptions: the weights w = (ω0, . . . , ωn+r) are pairwise relatively prime and odd, νp(d/dr) ≥ 2n+1/ 2(p-1) + 1 for all primes p with p(p-1) ≤ n + 1, where νp(d/dr) satisfies d/dr =Ⅱp prime pνp (d/dr).     

On the probability distribution of data at points in real complete intersection varieties

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multi-variate polynomial equations and convergence radius of Newton's operator for under-determined system of multi-variate polynomial equations.     

Quadratic forms for a 1-form on an isolated complete intersection singularity

We consider a holomorphic 1-form ω with an isolated zero on an isolated complete intersection singularity (V,0). We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair (V,ω). They generalize the Eisenbud–Levine–Khimshiashvili quadratic form defined for a smooth V. Partially supported by the DFG-programme 'Global methods in complex geometry' (Eb 102/4–3) grants RFBR–04–01–00762, NSh–1972.2003.1.     

New inductive constructions of complete caps in PG(N,q), q even

Some new families of small complete caps in PG(N, q), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N, q), N≥4, for all q≥2³. In particular, substantial improvements are obtained for infinite values of q square, including q=2^2Cm, C≥5, m≥3; for q=2^Cm, C≥5, m≥9, with C, m odd; and for all q≤2¹⁸. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010     

Common fixed points for generalized contractive multivalued operators in complete metric spaces

In this paper,we discuss some new fixed point theorems for a pair of multivalued operators which satisfy generalized contractive condition. Our results are the extension and improvement of corresponding results of single-valued operators. Finally, some examples are given to illustrate the usability of our results.     

Fixed points of multivalued operators in ordered metric spaces with applications

The purpose of this paper is to present some new (hybrid) fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space. An example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results.     

Limit laws for the intersection local time of stable processes in u2

We prove renormalization results for self‐intersections of a stable process of index β in the plane. If 1 < β ≤4÷3 we show that a suitably renormalized version of the intersection local time converges in law to a Brownian motion     

Conjugate points in length spaces

In this paper we extend the concept of a conjugate point in a Riemannian manifold to geodesic spaces. In particular, we introduce symmetric conjugate points and ultimate conjugate points and relate these notions to prior notions developed for more restricted classes of spaces. We generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. We close with applications of these new kinds of conjugate points to CBA(κ) spaces: proving both known and new theorems. In particular we prove a Rauch comparison theorem, a Relative Rauch Comparison Theorem, the fact that there are no ultimate conjugate points less than π apart in a CBA(1) space and a few facts concerning closed geodesics. This paper is written to be accessible to students and includes open problems.     

Brownian intersection local times: Exponential moments and law of large masses

Consider independent Brownian motions in , each running up to its first exit time from an open domain , and their intersection local time as a measure on . We give a sharp criterion for the finiteness of exponential moments,

where are nonnegative, bounded functions with compact support in . We also derive a law of large numbers for intersection local time conditioned to have large total mass.

     

Chains of function spaces over Euclidian spaces and local fields

The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho&#168;lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a “chain of function spaces“ over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields.     

Boundedly complete weak-Cauchy basic sequences in Banach spaces with the PCP

It is proved that every non-trivial weak-Cauchy sequence in a Banach space with the PCP (the Point of Continuity Property) has a boundedly complete basic subsequence. The following result, due independently to S. Bellenot and C. Finet, is then deduced as a corollary. If a Banach space X has separable dual and the PCP, then every non-trivial weak-Cauchy sequence in X has a subsequence spanning an order-one quasi-reflexive space.     

K6‐minors in triangulations and complete quadrangulations

In this paper, we shall prove that a projective‐planar (resp., toroidal) triangulation G has K₆ as a minor if and only if G has no quadrangulation isomorphic to K₄ (resp., K₅ ) as a subgraph. As an application of the theorems, we can prove that Hadwiger's conjecture is true for projective‐planar and toroidal triangulations. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 302‐312, 2009     

Maximal points with respect to cone dominance in Banach spaces and their existence

Given a convex cone in a Banach spaceV, an examination of the cone maximal points of a setX inV (with respect to the cone dominance induced by ) with respect to their characterization and existence is undertaken. The totality of cone maximal points ofX is called the conical frontier ofX. Comparisons of the conical frontiers of related sets and corresponding to related cones are made. By relaxing the compactness requirements of the underlying setX and by assuming some cone-related weaker forms of compactness, existence theorems for cone maximal points are developed. These theorems are believed to be generalizations of the existing results in one way or another.Maximizing points onX of certain linear functionals in the dual cone * of provide natural examples of cone maximal points. Properties characterizing a maximizing point of a linear functional in *, including the generalized version of Geoffrion's characterization of proper efficiency, are compiled and proved to be valid characterizations. Functionals in * with special properties are studied. Existence theorems are also obtained for the maximizing points of these functionals.The author is indebted to Professor James V. Whittaker for helpful discussions and comments and to Professors P. S. Bullen, C. W. Clark, B. N. Moyls, and F. Y. M. Wan for their encouragement and support.     

Minimal 1-saturating sets and complete caps in binary projective spaces

In binary projective spaces PG(v,2), minimal 1-saturating sets, including sets with inner lines and complete caps, are considered. A number of constructions of the minimal 1-saturating sets are described. They give infinite families of sets with inner lines and complete caps in spaces with increasing dimension. Some constructions produce sets with an interesting symmetrical structure connected with inner lines, polygons, and orbits of stabilizer groups. As an example we note an 11-set in PG(4,2) called “Pentagon with center”. The complete classification of minimal 1-saturating sets in small geometries is obtained by computer and is connected with the constructions described.     

Coprime packedness and set theoretic complete intersections of ideals in polynomial rings

A ring is said to be coprimely packed if whenever is an ideal of and is a set of maximal ideals of with , then for some . Let be a ring and be the localization of at its set of monic polynomials. We prove that if is a Noetherian normal domain, then the ring is coprimely packed if and only if is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of is a set theoretic complete intersection. A similar result is also proved when is either a Noetherian arithmetical ring or a Bézout domain of dimension one.