Weakening arcs in tournaments

A weakening arc of an irreducible tournament is an arc whose reversal creates a reducible tournament. We consider properties of such arcs and derive recurrence relations for enumerating strong tournaments with no such arcs, one or more such arcs, and exactly one such arc. We also give some asymptotic results on the numbers of such tournaments, among other things. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 142–162, 2004     

No-arbitrage and optimal investment with possibly non-concave utilities: a measure theoretical approach

We consider a discrete-time financial market model with finite time horizon and investors with utility functions defined on the non-negative half-line. We allow these functions to be random, non-concave and non-smooth. We use a dynamic programming framework together with measurable selection arguments to establish both the characterisation of the no-arbitrage property for such markets and the existence of an optimal portfolio strategy for such investors.     

Analysis on an ODE arisen from studying the shape of a red blood cell

We study the existence problem of a special solution to the Helfrich functional such that it corresponds to a surface of the shape of a red blood cell. The Helfrich functional is also a perturbation of the Willmore functional involving some parameters with physical meanings. With the expected symmetry of the surface, it reduces to an analysis on an ODE with certain shape requirements. We discover a sufficient condition on the parameters which ensures the existence of such a special solution to the ODE.     

On the upper embedding of symmetric configurations with block size 3

We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.     

Elliptic dilation–contraction problems on manifolds with boundary. <Emphasis Type="Italic">C</Emphasis>*-theory

We study boundary value problems with dilations and contractions on manifolds with boundary. We construct a C*- algebra of such problems generated by zero-order operators. We compute the trajectory symbols of elements of this algebra, obtain an analog of the Shapiro–Lopatinskii condition for such problems, and prove the corresponding finiteness theorem.     

A Study of Composite Heuristic Algorithms

When faced with a difficult combinatorial problem whose optimization may be prohibitively expensive, researchers frequently turn to the study of fast heuristic algorithms in an effort to guarantee near-optimal results. An important aspect in the evaluation of such an algorithm is the determination of its worst-case behaviour, relative to an optimization rule. We here focus attention on schemes for combining two or more heuristics so as demonstrably to improve the worst case that can occur. We survey successful, illustrative examples of such composite algorithms, concentrating on problems of central significance from domains such as scheduling, packing and routeing. We also attempt to illuminate the conditions under which these strategies can be successful and why.     

On the Inverse Problem for Scattering of Electromagnetic Radiation by a Periodic Structure

We consider a smooth perturbation  δε( x , y , z )  of a constant background permittivity  ε=ε₀  that varies periodically with x , does not depend on y , and is supported on a finite-length interval in z . We investigate the theoretical and numerical determination of such perturbation from (several) fixed frequency y -invariant electromagnetic waves.
By varying the direction and frequency of the probing radiation a scattering matrix is defined. By using an invariant-imbedding technique we derive an operator Riccati equation for such scattering matrix. We obtain a theoretical uniqueness result for the problem of determining the perturbation from the scattering matrix.
We also investigate a numerical method for performing such reconstruction using multi-frequency information of the truncated scattering matrix. This relies on ideas of regularization and recursive linearization. Numerical experiments are presented validating such approach.     

Scalar and φ-Sectional Curvature of a Certain Type of Metric f-Structures

We consider a Riemannian manifold (M,g) equipped with an f-structure of constant rank with parallelizable kernel. We assume certain integrability conditions on such a manifold. We prove some inequalities involving the scalar and *-scalar curvature of g. We prove that the corresponding equalities characterize an -manifold, which is a generalization of a Sasakian manifold. We also give a method of constructing such structures on toroidal bundles. Dedicated to the memory of Professor Aldo Cossu Research supported by the Italian MIUR 60% and GNSAGA.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

On contact equivalence of Abel cubic differential equations of second order

We study the geometry of cubic Abel differential equations on one-dimensional real curve. We prove that such an equation is the kernel of some nonlinear differential operator. This operator is defined by a cubic on the Cartan distribution on the space of 1-jets. Based on this observation, we construct a contact-invariant {e}-structure associated with a non-degenerate Abel equation and obtain contact classification of such equations.     

Utility efficiency and its approximation

Summary We assume a multi-attribute decision making problem under uncertainty with partial information on the decision maker's preferences, by a vector utility function with two components and imprecision over their scaling constants. We propose an approximation set whose determination may be easier than the one of the utility efficient set and we consider an interactive procedure which uses such approximation to decision aid. We study some nesting and convergence properties based on the interactive reduction of the approximation set. Finally, we illustrate the procedure with a numerical example.     

Idempotent transformations of finite groups

We describe the action of idempotent transformations on finite groups. We show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group. This is done in terms of the first two homology groups. We prove for example that except special linear groups, such an orbit can have at most 7 elements. We also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite.     

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

Anderson-Cheeger limits of smooth Riemannian manifolds,and other Gromov-Hausdorff limits

We establish further regularity of the C^α and H^1,p limits of smooth, n-dimensional Riemannian manifolds with a lower bound on Ricci tensor and injectivity radius, and an upper bound on volume, first considered in [1]. We use this extra regularity to show that such a limit is a nonbranching geodesic space, as defined in [10], and to construct a variant of a geodesic flow for such a limit. We contrast the behavior of some slightly more singular limits.     

Abelian Hypercomplex 8-Dimensional Nilmanifolds

We study invariant Abelian hypercomplex structures on 8-dimensional nilpotent Lie groups. We prove that a group N admitting such a structure is either Abelian or an Abelian extension of a group of type H. We determine the Poincaré polynomials of the associated nilmanifolds and study the existence of symplectic and quaternionic structures on such spaces.     

Optimal control of a quasi-variational obstacle problem

We consider an optimal control where the state-control relation is given by a quasi-variational inequality, namely a generalized obstacle problem. We give an existence result for solutions to such a problem. The main tool is a stability result, based on the Mosco-convergence theory, that gives the weak closeness of the control-to-state operator. We end the paper with some examples.     

Global pointwise estimates for Green's matrix of second order elliptic systems

We establish global pointwise bounds for the Green's matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green's matrix. In the scalar case, such an estimate is a consequence of De Giorgi-Moser-Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In the vectorial case, one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.     

Step-like contrast structure for a quasilinear system of singularly perturbed differential equations with a zero characteristic number

We consider a system of singularly perturbed first-order differential equations with a zero characteristic number. The solution of such a problem is characterized by the presence of a contrast structure, that is, of an internal transition layer on a given interval. We prove the existence of an exact solution with a step-like contrast structure and construct its uniform asymptotic expansion. An example is given.     

Oblique derivative and interface problems on polygonal domains and networks

We Investigate oblique derivative problems associated to the Laplace operator on a polygon and we extend our study to "polygonal interface problems" which are an extension to networks of the prevlous ones. We focus on the non variational character of such problems. We obtain index formulae, a calculus of the dimension of the kernel, an expansion of the 'semi-variational" (or weak) solutions into regular and singular parts and formulae for the coefficients of the singularities In such expanslons.