‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

Five Kinds of Maximal Monotonicity

We discuss Gossez's type (D) maximal monotone multifunctions and the newer type (ED) subfamily (for which an analog of the Brøndsted-Rockafellar property holds). We then discuss the locally maximal monotone (= type (FP)) and maximal monotone locally (= type (FPV)) multifunctions of Fitzpatrick-Phelps and Fitzpatrick-Phelps-Verona-Verona. Finally, we discuss the strongly maximal monotone multifunctions. We prove that every maximal monotone multifunction of type (D) is locally maximal monotone, and every maximal monotone multifunction of type (ED) is both maximal monotone locally and strongly maximal monotone.     

On minimizing the jump number for interval orders

The purpose of this paper is to give an effective characterization of all interval orders which are greedy with respect to the jump number problem.This research (Math/1406/30) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

Local symmetries and conservation laws

Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of higher KdV equations are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that symmetry and conservation law are, in some sense, the dual conceptions which coincides in the self-dual case, namely, for Euler-Lagrange equations. Training examples are also given.Translated from the Russian by B. A. Kuperschmidt.     

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by transitions with which we can also define differential quotients of a map. Their various limits are called mutations of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.This work was motivated by evolution equations of tubes in visual servoing on one hand, mathematical morphology on the other, when the metric spaces are power spaces. This paper begins by listing some consequences of general theorems concerning mutational equations for tubes.     

Compressible flow induced by the transient motion of a wavemaker

The effect of fluid compressibility on the evolution of the pressure distribution and free surface elevation, following the initiation of a horizontal motion of a vertical wavemaker, is analysed. This effect is significant even in a liquid (like water) when the time scale of the motion is very short (e.g. impulsive motions).In the leading order the present problem is analogous to that of supersonic flow about a thin wing, thus the solution is represented by means of an appropriate supersonic source distribution. Closed-form results are obtained for the case of impulsive motion (i.e. a step function velocity). The pressure field corresponds to systems of double rarefaction and double compression waves traversing the fluid domain intermittently. Following the passage of a rarefaction (compression) wave, the free surface becomes locally concave (convex). The resulting free surface profile consists of an elongating wavetrain in front of a jet riding up the vertical wall.On the compressible time-scale the pressure and velocity fields approach a steady long-time limit. This limit corresponds to the short-time incompressible flow prevailing after the attenuation of the pressure waves. The spatial nonuniformity of the asymptotic expansion in the neighbourhood of the waterline is briefly discussed.     

A new example of ‘independence’ and ‘white noise’

Summary We examine the notion of free independence according to Voiculescu. This form of independence is used for defining free white noise or process with stationary and freely independent increments. We prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions. We realize the free analogues of the Wiener process and of the Poisson process as processes on the full Fock space ofL ² (—).     

Some New Transversality Properties

The theorems of Ceva and Menelaus are concerned with cyclic products of ratios of lengths of collinear segments of triangles or more general polygons. These segments have one endpoint at a vertex of the polygon and one at the intersection point of a side with a suitable line. To these classical results we have recently added a selftransversality theorem in which the suitable line is determined by two other vertices. Here we present additional transversality properties in which the suitable line is determined either by a vertex and the intersection point of two diagonals, or by the intersection points of two pairs of such diagonals. Unexpectedly it turns out that besides several infinite families of systematic cases there are also a few sporadic cases.     

Propriétés Analytiques du Spectre dans les Algèbres de Jordan-Banach

If f() is an analytic function from a domain D of the complex plane into a Jordan-Banach algebra we prove that Sp f() is an analytic multivalued function. From this derives the subharmonicity of Log (f()), where denotes the spectral radius. We apply these results to prove that a Jordan-Banach algebra A is associative if and only if the spectral radius is subadditive and submultiplicative on A and to prove that A/Rad A is isomorphic to the complex plane if and only if each element of A has only one point in its spectrum.

---

---

Le travail du permier auteur a été subventionné par le Conseil de recherches en sciences naturelles et en génie du Canada (subvention A 7668)     

Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem

We investigate the hot–spots property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the hot–spots property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.Supported in part by NSF Grant # 9700585-DMSSupported in part by NSF Grant # 0203961-DMS     

Langlands classification for real Lie groups with reductive Lie algebra

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations U^V, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G ₀ is finite, so the center of the connected semi-simple subgroup with Lie algebra [g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).     

Grope cobordism and feynman diagrams

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in [CT]. We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the class is used to organize gropes. This implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3. We also show that the class is not a useful organizing complexity in 4 dimensions since only the Arf invariant survives. In contrast, measuring gropes according to height does lead to very interesting 4-dimensional information [COT]. Finally, several low degree calculations are explained, in particular we show that S-equivalence is the same relation as grope cobordism based on the smallest tree with an internal vertex. Mathematics Subject Classification (2000):57M27The first author was partially supported by NSF VIGRE grant DMS-9983660. The second author was partially supported by NSF grant DMS-0072775 and the Max-Planck Gesellschaft.     

Stable Runge-Kutta integrations for differential systems with semi-stable equilibria

Summary. In this paper, the stability of Runge-Kutta methods for differential systems with semi-stable equilibria is analyzed. In fact, a way of circumventing the very severe unconditional stability barrier [3] is proposed. The usage of time-meshes with step-size ratios under some constant greater than one, as usual in standard numerical codes, is considered in order to recover stability for many implicit methods. The main result roughly says that any strongly A-stable Runge-Kutta method is stable for this kind of differential systems on this sort of meshes.Work supported by project BMF2001-2562Mathematical Subject Classifications (2000): 65L20     

On the approximation of discrete functions by fourier-hahn sums

— f: {0,1, ..., N–1}R. .     

On the Held-Karp relaxation for the asymmetric and symmetric traveling salesman problems

A long-standing conjecture in combinatorial optimization says that the integrality gap of the famous Held-Karp relaxation of the metric STSP (Symmetric Traveling Salesman Problem) is precisely 4/3. In this paper, we show that a slight strengthening of this conjecture implies a tight 4/3 integrality gap for a linear programming relaxation of the metric ATSP (Asymmetric Traveling Salesman Problem). Our main tools are a new characterization of the integrality gap for linear objective functions over polyhedra, and the isolation of hard-to-round solutions of the relaxations.     

d≤1 ∪d≥25 and constrained KP hierarchy from BRST invariance in thec≠3 topological algebra

The BRST invariance condition in a highest-weight representation of the topological ( twisted N=2) algebra captures the invariant content of two-dimensional gravity coupled to matter. The topological algebra allows reductions to either the DDK-dressed matter or the Kontsevich-Miwa-dressed matter related to Virasoro-constrained KP hierarchy. The standard DDK formulation is recovered by splitting the topological generators into c=–26 reparametrization ghosts + matter + Liouville, while a similar splitting involving c=–2 ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the Liouville differ also by their ghost numbers, which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge c3 provides a two-fold covering of the allowed regiond1 d25 of the matter central charge d via d=(c+1)(c+6)/(c–3). The Liouville field is identified as the ghost-free part of the topological U(1) current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 239–250, May, 1993.     

Elations and symmetric designs having projective subspaces

Consider symmetric 2-designs D which have an automorphism group G containing sufficiently many elations. This paper investigates the case where there are the least number of elations. Under certain assumptions it is shown that D has a projective subspace and in a particular case D _n,q or D is obtained from _n,q by a process called K-alteration.The results in this paper form part of a PhD thesis submitted by the author to the University of London. The author gratefully acknowledges the support of the Commonwealth Scholarship Commission.     

Invariant Subspaces for Positive Operators Acting on a Banach Space with Markushevich Basis

We introduce weak quasinilpotence for operators. Then, by substituting Markushevich basis and weak quasinilpotence at a nonzero vector for Schauder basis and quasinilpotence at a nonzero vector, respectively, we answer a question on the invariant subspaces of positive operators in [3].     

χ-Embeddings of ordered sets

We use the technique of -Embeddings to study retracts and varieties of ordered sets. We investigate the class of all ordered sets which are retract of every ordered set in which is -Embedded.     

Solving staircase linear programs by the simplex method, 1: Inversion

This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or staircase, structure. The present paper looks at inversion routines within the simplex method, particularly those for sparse triangular factorization of a basis by Gaussian elimination and for solution of triangular linear systems. The succeeding paper examines pricing routines. Both papers describe extensive (though preliminary) computational experience, and can point to some quite promising results.