On a Generalized Class of Minimax Inequalities

Some results on a generalized class of minimax inequalities based on the r–I–G–H-KKM mapping theorems in a G–H-space setting are presented. The r–I–G–H-KKM mappings represent a new class of KKM mappings in G–H-spaces as well as in the interval spaces.     

Interpolation of characteristic classes of singular hypersurfaces

We show that the Chern–Schwartz–MacPherson class of a hypersurface X in a nonsingular variety M ‘interpolates’ between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern–Schwartz–MacPherson class of such ‘nice’ hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern–Schwartz–MacPherson class of X in terms of its polar classes.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

Complexity of the min–max (regret) versions of min cut problems

This paper investigates the complexity of the min–max and min–max regret versions of the min s–t cut and min cut problems. Even if the underlying problems are closely related and both polynomial, the complexities of their min–max and min–max regret versions, for a constant number of scenarios, are quite contrasted since they are respectively strongly NP-hard and polynomial. However, for a non-constant number of scenarios, these versions become strongly NP-hard for both problems. In the interval scenario case, min–max versions are trivially polynomial. Moreover, for min–max regret versions, we obtain the same contrasted results as for a constant number of scenarios: min–max regret min s–t cut is strongly NP-hard whereas min–max regret min cut is polynomial.     

Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.     

The Maximal and Minimal Ranks of Some Expressions of Generalized Inverses of Matrices

In this paper, we first determine the maximal and minimal ranks of A — BXC with respect to X. Using those results, we then find the maximal and minimal ranks of the expressions AA^– — A^– A — BB^– A — AC^– C and B^– BA — ACC^– with respect to the choice of generalized inverses A^–, B^– and C^–. In particular, we consider the commutativity of A and A^–, A^k and A^–.The research of the author was supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Ultratertiary Quantization of Thermodynamics

We show that if the Dirac–Bogoliubov rule for replacing the bosonic creation and annihilation operators with the c-numbers is used, then the ultratertiary quantization allows obtaining the Bardeen–Cooper–Schrieffer–Bogoliubov formulas.     

Regularized Resolvent of Sums of Commuting Operators

For B ₁ and B ₂ commuting linear operators on a Banach space such that B ₁ generates a bounded strongly continuous semigroup and –B ₂ generates an exponentially decaying strongly continuous holomorphic semigroup, it is shown that (B ₁ – B ₂)^–1 B ₂ ^–r and (B ₁ – B ₂)^–1(–B ₁)^–r are bounded and everywhere defined, for any r > 0. Density of domains may also be removed. The results are applied to various abstract Cauchy problems.     

An Expansion Formula for the Askey–Wilson Function

The Askey–Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey–Wilson second order q-difference operator. The kernel is called the Askey–Wilson function. In this paper an explicit expansion formula for the Askey–Wilson function in terms of Askey–Wilson polynomials is proven. With this expansion formula at hand, the image under the Askey–Wilson function transform of an Askey–Wilson polynomial multiplied by an analogue of the Gaussian is computed explicitly. As a special case of these formulas a q-analogue (in one variable) of the Macdonald–Mehta integral is obtained, for which also two alternative, direct proofs are presented.     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems

Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems for scalar-valued measures on rings of sets that are in general not σ-rings are presented. As a consequence, the rings of subsets of N with density zero and uniform density zero are shown to have the Nikodym property. In addition, vector measure generalizations of the Vitali–Hahn–Saks theorem are given.     

Precise distribution properties of the van der Corput sequence and related sequences

The discrepancy is a quantitative measure for the irregularity of distribution of sequences in the unit interval. This article is devoted to the precise study of L_p–discrepancies of a special class of digital (0,1)–sequences containing especially the van der Corput sequence. We show that within this special class of digital (0,1)–sequences over ℤ₂ the van der Corput sequence is the worst distributed sequence with respect to L₂–discrepancy. Further we prove that the L_p–discrepancies of the van der Corput sequence satisfy a central limit theorem and we study the discrepancy function of (0,1)–sequences pointwise.     

A sharpening of the Johnson bound for binary linear codes and the nonexistence of linear codes with preparata parameters

We show fort>3 the nonexistence of binary [2 ^t –2, 2 ^t –2t–1, 5] codes.     

Riemann–Roch for Algebraic Stacks: I

In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

On the rigidity of super-Grassmannians

The rigidity of the complex super-Grassmannian Gr _m|n,k|l with 0<k<m, 0<l<n, supposing that (k, l) (1,n–1), (m–1, 1), (1,n–2), (m–2, 1), (2,n–1), (m–1, 2), is proved.Partially supported by Erwin Schrödinger International Institute for Mathematical Physics (Vienna, Austria)     

Decomposing the complete graph into cycles of many lengths

For all oddv 3 the complete graph onvK _v vertices can be decomposed intov – 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,...,v – 1. Also, for all oddv 7,K _v can be decomposed intov – 3 edge disjoint cycles whose lengths are 3, 4,...,v – 4,v – 2,v – 1,v. Research supported by Australian Research Council grant A49130102     

Characterisations of Flock Quadrangles

We characterise the Hermitian and Kantor flock generalized quadrangles of order (q ²,q), q even, (associated with the linear and Fisher–Thas–Walker flocks of a quadratic cone, and the Desarguesian and Betten–Walker translation planes) in terms of a self-dual subquadrangle. Equivalently, we show that a herd which contains a translation oval must be associated with the linear or Fisher–Thas–Walker flock. This result is a consequence of the determination of all functions which satisfy a certain absolute trace equation whose form is remarkably similar to that of an equation arising in recent studies of ovoids in three-dimensional projective space of finite order q.