Doubly stochastic measures with prescribed support

Summary Denote the set of doubly stochastic measures on the unit square X ×X that are supported on the graphs of measurable maps L,HXX by (L, H). Conditions are given that imply that (L, H) is a singleton. Since (L, H) is in any event a (possibly empty) extremal subset of the set of all doubly stochastic measures on X×X, our results are intimately related to the problem of describing the supports of the extreme points of .     

Improvement of the Nash-Tognoli theorem

Let be a smooth closed manifold in ⁿ. The Nash-Tognoli theorem says that M can be arbitrarily well approximated (in the C^r-topology with r < ) in ⁿ by a nonsingular real algebraic set under the condition that dim <(n-1)/2 There is a familiar conjecture, going back at least to Nash, that the restriction on dim in the Nash-Tognoli theorem is unnecessary. However, up to now in unstable dimensions [i.e., for dim (n-1)/2 ] the possibility of approximating was known only for orientable of codimension (in ⁿ) 1 or 2. The goal of the paper is to prove the following theorem, relaxing the restriction on dim in the Nash-Tognoli theorem to dim M<(2n-1)/3. If is a smooth closed manifold in IK and dim M<(2n–1)/3, then can be arbitrarily well approximated in ⁿ by a nonsingular real algebraic set.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 66–71, 1982.     

Flowchart machines

Let denote a conventional flowchart. Any algorithm can be represented by a flowchart. If action nodes in call then is a recursive flowchart. We show how to decompose arbitrary non-self-modifying programs into structure and atomic parts. We specifically give the synthesis procedure for a controller . can serve as the only sequencer in an execution of . If is recursive then is a pushdown machine, otherwise is a finite state machine. The next-state functionf and the output functiong of represent respectively all of the structure-, i.e. the programmer-oriented-, and all of the atomic-, i.e. the data-oriented-, parts of .f defines the flow or pattern of computations andg the actual transformations or operations on data. Thus we construct and analyze programs by constructing and analyzing their sequencers .     

Sharp determinants

We introduce a sharp trace Tr^# and a sharp determinant Det^#(1-z) for an algebra of operators acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of . The relationship with weighted zeta functions of interval maps and Milnor-Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators.Oblatum 8-V-1995 & IX-1995On leave from CNRS, UMR 128, ENS Lyon, France     

Semimodular lattices with isomorphic graphs

Two discrete modular lattice and have isomorphic graphs if and only if is of the form A × and is of the form A × for some lattices A and and . We prove that for discrete semimodular lattices and this latter condition holds if and only if and have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.     

The geometry of the moduli space of CP2 instantons

Summary The moduli space of self-dual connections (instantons) of instanton numberk=1 overCP ² is topologically a cone onCP ². The complement ^* of the vertex of this cone carries a natural Riemannian metric (the L ² metric). We find an explicit formula for this metric and deduce certain consequences. In particular, (i) the boundary is a totally geodesic submanifold of the completion, (ii) is positively curved at , and (iii) is negatively curved in certain directions near the vertex of the cone.Partially supported by N.S.F. grant DMS-8405661     

On the bigℳ in the affine scaling algorithm

When we apply the affine scaling algorithm to a linear program, we usually construct an artificial linear program having an interior feasible solution from which the algorithm starts. The artificial linear program involves a positive number called the big. Theoretically, there exists an ^* such that the original problem to be solved is equivalent to the artificial linear program if > ^*. Practically, however, such an ^* is unknown and a safe estimate of is often too large. This paper proposes a method of updating to a suitable value during the iteration of the affine scaling algorithm. As becomes large, the method gives information on infeasibility of the original problem or its dual.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Supported by Grant-in-Aids for Co-Operative Research (03832017) of the Japan Ministry of Education, Science and Culture.     

A canonical form of a system of microdifferential equations with non-involutory characteristics and branching of singularities

We study a system of microdifferential (=pseudodifferential) equations. We assume that the characteristic varietyV of takes the formV=V ₁V₂; hereV ₁andV ₂are regular involutory submanifolds and intersect normally, andV ₁V₂is non-involutory and 1-codimensional both inV ₁and inV ₂. We also assume that has regular singularities alongV. Then we give a canonical form of in the complex domain. This enables us to investigate the branching of supports of microfunction solutions of when is hyperbolic.     

A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space

We study the spaces and and _Lip of smooth (resp. non-degenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions. We prove that and _Lip are infinite dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite dimensional version of the Kirwan, Kempf-Ness Theorem [Kir84], [KN78], [Nes84] relating symplectic quotients to holomorphic quotients, applied to the action ofPSL ₂() on the free loop space ofS ².Oblatum 15-X-1994 & 5-VII-1995This research was supported in part by NSF grant DMS-92-05154.This research was partially supported by AFOSR grant F49620-92-J-0093.     

A little theorem of the bigℳ in interior point algorithms

When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an artificial problem having a known interior feasible solution from which we start the algorithm. The artificial problem involves a constant (or constants) which we need to choose large enough to ensure the equivalence between the artificial problem and the original problem. Theoretically, we can always assign a positive number of the order O(2 ^L ) to in linear cases, whereL denotes the input size of the problem. Practically, however, such a large number is impossible to implement on computers. If we choose too large, we may have numerical instability and/or computational inefficiency, while the artificial problem with not large enough will never lead to any solution of the original problem. To solve this difficulty, this paper presents a little theorem of the big, which will enable us to find whether is not large enough, and to update during the iterations of the algorithm even if we start with a smaller. Applications of the theorem are given to a polynomial-time potential reduction algorithm for positive semi-definite linear complementarity problems, and to an artificial self-dual linear program which has a close relation with the primal—dual interior point algorithm using Lustig's limiting feasible direction vector.     

Quantum Poisson processes and dilations of dynamical semigroups

Summary The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM _n with stationary faithful state . If (, ) is the quantum measure space in question ( a von Neumann algebra and a faithful normal weight), then the semigroupe ^tL of transition operators on (M _n , ) has generator whereu is an arbitrary unitary element of the centraliser of (M _n ,).Supported by the Netherlands Organization for Scientific Research NWO     

Allgemeine Zerlegungstheorie

The notion -disjoint decomposition is introduced as a common generalization of the elementary geometric decomposition of polyhedras and the disjoint decomposition of sets: Let Rø be a set, G a subgroup of the symmetric group S_R and 2^R with ø and ¦¦>1; then A, B 2^R are called -disjoint if ø. is the only set of contained in A B, and then A B A B is called the disjoint union. If fullfills the axioms (0, I, II, III) of §2, then can be provided in a natural way with certain binary relations. With respect to the decomposition equivalence can be turned in a partially ordered commutative semigroup with identity (cf. Satz §5).

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Dedicated to Prof.R.Artzy on the occasion of his 80. birthday     

Some coinductive graphs

Summary LetT be a universal theory of graphs such that Mod(T) is closed under disjoint unions. Let _T be a disjoint union _i such that each _i is a finite model ofT and every finite isomorphism type in Mod(T) is represented in{ _i i<3}. We investigate under what conditions onT, Th( _T ) is a coinductive theory, where a theory is called coinductive if it can be axiomatizated by -sentences. We also characterize coinductive graphs which have quantifier-free rank 1.     

On obtaining a stationary process isomorphic to a given process with a desired distribution

Let _e denote the set of distributions of all stationary, ergodic, aperiodic processes with a given finite state space, and let the metric on _e be Ornstein's process distance. Suppose is a subset of _e which is a in the weak topology and for which (µ _n ,)0 whenever { _n } is a sequence from _e converging weakly to a positive entropy measure in . It is shown that ifX is a stationary ergodic aperiodic process with entropy rate less than the entropy of one of the distributions in , thenX is isomorphic to a process whose distribution lies in . As special cases, one obtains the invulnerable source coding theorem of information theory and also the Grillenberger-Krengel theorem on the existence of a generator whose process has a desired marginal distribution.Research of author supported by NSF Grant ECS-78-21335.     

Functorial semantics and HSP type theorems

We show that if ] is the category of models for a theory in the sense of Linton over an arbitrary base category, then a full subcategory of ] is closed under Homomorphic images that split in the underlying category, under -Subobjects for a class of monomorphisms and under Products if and only if it is an intersection of a nest of subcategories, each determined from the preceding by a class of Horns, in which the crucial arrow lies in the class of epimorphisms orthogonal to .Dedicated to the memory of Evelyn Nelson, Alan Day and Alan MeklerPresented by F. E. J. Linton.This research has been supported by grants from the NSERC of Canada and the FCAR du Québec.     

Stability of the extreme point set of a polyhedron

This paper is focused on the stability properties of the extreme point set of a polyhedron. We consider a polyhedral setX(A,b) which is defined by a linear system of equality and inequality constraintsAxb, where the matrixA and the right-hand sideb are subject to perturbations. The extreme point setE(X(A,b)) of the polyhedronX(A,b) defines a multivalued map :(A,b)E(X(A,b)). In the paper, characterization of continuity and Lipschitz continuity of the map is obtained. Boundedness of the setX(A,b) is not assumed It is shown that lower Lipschitz continuity is equivalent to the lower semicontinuity of the map and to the Robinson and Mangasarian-Fromovitz constraint qualifications. Upper Lipschitz continuity is proved to be equivalent to the upper semicontinuity of the map . It appears that the upper semicontinuity of the map implies the lower semicontinuity of this map. Some examples of using the conditions obtained are provided.The author wishes to thank Dr. N. M. Novikova, Dr. S. K. Zavriev, and anonymous referees for their helpful comments and advice. The research described in this publication was made possible in part by Grant NJCU100 from the International Science Foundation and Russian Government, and by the Euler Grant, Deutsche Mathematiker Vereinigung.     

A characterization of hadamard designs with SL(2,q) acting transitively

Given a hyperoval in a projective plane of even orderq, we can associate a Hadamard 2-design. In the case when is the Desarguesian plane P_2,q ,q=2 ^h ,h>1 and is a regular hyperoval (conic and its nucleus) then a design (q) is obtained. (q) has a point transitive automorphism group isomorphic to PSL(2,q)( SL(2,q)). We classify the designs (q) and P_2h–1,2 (the projective space of dimension 2h–1 overF ₂) among all the designsH with the same parameters as (q) admitting an automorphism groupGSL(2,q) acting transitively the points ofH. We also describe how all such designsH may be constructed and discuss the problem of when two such designs are isomorphic.This research was supported by Science and Engineering Research Council Grant GR/G 03359.     

Sur un mouvement particulier d'un profil muni d'une cavité fermée par un couvercle élastique et remplie de liquide

Resumé L'auteur considère le mouvement d'un profil sans pointe pesant, possédant une cavité fermée par un couvercle élastique et contenant un liquide parfait, dans un fluide parfait incompressible, en mouvement irrotationnel, au repos à l'infini. Dans la première partie, il forme les équations du mouvement du système, démontre l'existence de mouvements de translation horizontale uniforme du profil avec repos relatif du liquide et du couvercle et établit que l'un d'eux, noté mouvement _o est stable.Dans le deuxiéme partie, il étudie les petits mouvements du système voisins du mouvement _o.

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The motion of a heavy profile without sharp edge, with a cavity closed by an elastic cover and containing an in viscid liquid in an inviscid incompressible fluid in irrotational motion (at rest at infinity) is investigated.In the first part the equations of motion of the system are given. The existence of motions of horizontal uniform translation of the profile with liquid and cover at rest is proved. The stability of one of them, denoted by motion _o, is established. The second part deals with the study of the small oscillations of the system in the vicinity of the motion _o.

     

On the tautological ring of ℳ g

We prove that any product of tautological classes of _g of degreed (in the Chow ring of _g with rational coefficients) vanishes ford>g–2 and is proportional to the class of the hyperelliptic locus in degreeg–2.Oblatum 28-I-1195 & 15-IV-1995In memory of Nicolaas H. Kuiper (1920–1994)     

Maximum monoreflections

It is shown that, in a category with a specified class of monics and under some mild hypothesis,there is a monoreflection maximum among those whose reflection maps lie in . Thus, for example, any variety, and most SP-classes in a variety, have both amaximum monoreflection and amaximum essential reflection (which might be the same, but frequently aren't, and which might be the identity functor, but frequently aren't). And, for example, under some mild hypotheses, beneath each completion lies a maximum monoreflection, so that, for example, any category of rings has amaximum functorial ring of quotients.