‘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.     

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.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Inductive Construction of Repletion

The aim of this paper is to give a purely logical construction of repletion, i.e. the reflection of an arbitrary set to a replete one. Replete sets within constructive logic were introduced independently by M. Hyland and P. Taylor as the most restrictive but sufficiently general notion of predomain suitable for the purposes of denotational semantics à la Scott.For any set A its repletion R(A) appears as an inductively defined subset of S²(A) (A S) S which can be expressed within the internal language of a model of type theory. More explicitly, R(A) is the least subset of S²(A) containing all point filters and closed under a class of generalised limit processes. Improvements of our construction arise from several results saying that it suffices for the purpose of repletion to consider more restrictive classes of generalised limit processes.     

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 (—).     

Generic continuity of restricted weak upper semi-continuous set-valued mappings

Set-valued mappings from a topological space into subsets of a Banach space which satisfy a restricted form of weak upper semi-continuity, have particularly noteworthy properties. We establish a selection theorem for certain set-valued mappings from a (-) unfavourable topological space into subsets of a Banach space and as a consequence derive the property that restricted weak upper semi-continuous set-valued mappings which satisfy a minimality condition, from a (-) unfavourable topological space into subsets of a Banach space are single-valued and norm upper semi-continuous at the points of a residual subset of their domain.     

An extension of Willard’s Finite Basis Theorem: Congruence meet-semidistributive varieties of finite critical depth

Ross Willard proved that every congruence meet-semidistributive variety of algebras that has a finite residual bound and a finite signature can be axiomatized by some finite set of equations. We offer here a simplification of Willards proof, avoiding its use of Ramseys Theorem. This simplification also extends Willards theorem by replacing the finite residual bound with a weaker condition.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived February 26, 2004; accepted in final form August 2, 2004.     

Fractionally colouring total graphs

Bchzad and Vizing have conjectured that given any simple graph of maximum degree , one can colour its edges and vertices with +2 colours so that no two adjacent vertices, or two incident edges, or an edge and either of its ends receive the same colour. We show that for any simple graphG, V(G)E(G) can be fractionally coloured with +2 colours.     

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.     

Central Limit Theorems for dependent variables. I

Summary This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak mixing condition called -mixing. Roughly speaking, an array of real r.v.s is said to be -mixing if linear combinations of its past and future are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, -mixing and -mixing) are special cases of -mixing. Linear processes are shown to be -mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are nearly second order stationary.     

A probabilistic framework for the continuous observation of a quantum process

This paper treats quantum measurement within von Neumann's abstract framework. Specifically, observation is defined as a fixed self-adjoint operator with countable spectrum and nondegenerate eigenstates. Suppose scenarios for the observation of a quantum process over time are expanded by adding extra observations at time points interspersed among those of a previous scenario. If each observation leads to a mixture of eigenstates rather than a pure state, then the naturally defined joint probability measures on observed results are not consistent as scenarios vary. Nevertheless, we characterize the limiting subprobability measure when the times of observation become infinitely dense in any finite interval. This limiting measure corresponds to a continuous-time sub-stochastic process which decays with exponential rate out of any initial state and never reappears in any other state. Thus the process loses probability exponentially over time, and this loss occurs equally fast in the case of nonselective observation as for selective observation.Previous treatments of this problem have concentrated on the special case when Zeno's Paradox is in force, i.e. the rate of decay out of any state is zero and the process is immobilized by continuous observation. This situation exists, for instance, when the initial state is in the domain of the generator for the unitary group underlying the quantum process.     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube*

Gomorys and Chvátals cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The Chvátal rank of the polyhedron is the number of rounds needed to obtain all valid inequalities. It is well known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is 2-dimensional, and if its integer hull is a 0/1-polytope.We show that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, we prove that the rank of every polytope contained in the n-dimensional 0/1-cube is at most n ² (1+log n). Moreover, we also demonstrate that the rank of any polytope in the 0/1-cube whose integer hull is defined by inequalities with constant coefficients is O(n).Finally, we provide a family of polytopes contained in the 0/1-cube whose Chvátal rank is at least (1 + ) n, for some > 0.* An extended abstract of this paper appeared in the Proceedings of the 7th International Conference on Integer Programming and Combinatorial Optimization [20].     

Some Geometric Aspects of Integrability of Differential Equations in Two Independent Variables

The relation between scalar evolution equations which are the integrability condition of sl(2,R)-valued linear problems with parameter (kinematic integrability) and those which possess recursion operators (formal integrability) is studied: using that kinematically integrable equations describe one-parameter families of pseudo-spherical surfaces and vice versa, it is shown that every second order formally integrable evolution equation is kinematically integrable, and that this result cannot be extended as proven to the third-order case.Conservation laws of kinematically integrable equations obtained from their underlying pseudo-spherical structure are compared with the ones one finds from the Riccati equation version of their associated linear problems. Symmetries (generalized/nonlocal) for these equations are also studied, by considering infinitesimal deformations of the associated pseudo-spherical surfaces.Finally, conservation laws for equations describing pseudo-spherical surfaces immersed in a flat three-space are found, and the class of equations describing Calapso–Guichard surfaces is introduced.     

Decompositions of partially ordered sets into chains and antichains of given size

Every partially ordered set P on at least (1+o(1))n ³ elements can be decomposed into subposets of size n that are almost chains or antichains. This lower bound on P is asymptotically best possible. Similar results are presented for other types of combinatorial structures.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant 1-3-86-264.     

Sharp parameter ranges in the uniform anti-maximum principle for second-order ordinary differential operators

We consider the equation (pu)-qu+wu = f in (0,1) subject to homogenous boundary conditions at x = 0 and x = 1, e.g., u(0) = u(1) = 0. Let ₁ be the first eigenvalue of the corresponding Sturm-Liouville problem. If f 0 but 0 then it is known that there exists > 0 (independent on f) such that for (₁, ₁ + ] any solution u must be negative. This so-called uniform anti-maximum principle (UAMP) goes back to Clément, Peletier [4]. In this paper we establish the sharp values of for which (UAMP) holds. The same phenomenon, including sharp values of , can be shown for the radially symmetric p-Laplacian on balls and annuli in ⁿ provided 1 n < p. The results are illustrated by explicitly computed examples.     

Linearity properties of Shimura varieties, II

Let A=A_{g, 1, n} denote the moduli scheme over Z[1/N] of p.p. g-dimensional abelian varieties with a level n structure; its generic fibre can be described as a Shimura variety. We study its Shimura subvarieties. If x A is an ordinary moduli point in characteristic p, then we formulate a local linearity property in terms of the Serre–Tate group structure on the formal deformation space (= formal completion of A at x). We prove that an irreducible algebraic subvariety of A is a Shimura subvariety if, locally at an ordinary point x, it is formally linear. We show that there is a close connection to a differential-geometrical linearity property in characteristic 0.We apply our results to the study of Oort's conjecture on subvarieties Z A with a dense collection of CM-points. We give a reformulation of this conjecture, and we prove it in a special case.     

Simplex method and groups generated by reflections

A detailed account of the simplex method applied to a class of polyhedra (called Coxeter complexes) is given. The specific geometric properties of these polyhedra enable their use as a testing-ground for comparison of various linear programming algorithms. Applications to sorting problems are given.     

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirkas theorem on the extension of functions holomorphic in a neighbourhood of (F)(D×D) – where D is the open unit disc and (F) is the graph of a continuous D-valued function F – to the bidisc. We extend holomorphic functions by applying the Kontinuitätssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirkas theorem.Mathematics Subject classification (2000): 32D15Work supported by NSF Grant DMS-0072237.