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.     

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     

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     

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 .     

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.     

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 .     

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     

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 regularity problem for generalized harmonic maps into homogeneous spaces

Let be a Riemannian surface and be a standard sphere, or more generally a Riemannian manifold on which a Lie group,, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu W _loc ^1,1 (, ). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu W ^1,p(, ) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural -regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ). To prove this -regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality.     

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.     

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 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.     

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.     

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.     

Strong Exceptional Sequences Provided by Quivers

Let Q be a finite quiver without oriented cycles. Denote by U (Q) the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext _(Q) ^l (U, U) = 0 for all l > 0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived categories of finitely generated right k Q-modules and that of coherent sheaves on (Q) are related via the full and faithful functor – _kQ ^L U.     

Geometrical Aspects of Hilbert C*-modules

The aim of the present paper is to solve some major open problems of Hilbert C^*-module theory by applying various aspects of multiplier C^*-theory. The key result is the equivalence established between positive invertible quasi-multipliers of the C^*-algebra of compact operators on a Hilbert C^*-module {, ., } and A-valued inner products on , inducing an equivalent norm to the given one. The problem of unitary isomorphism of C^*-valued inner products on a Hilbert C^*-module is considered and new criteria are formulated. Countably generated Hilbert C^*-modules turn out to be unitarily isomorphic if they are isomorphic as Banach C^*-modules. The property of bounded module operators on Hilbert C^*-modules of being compact and/or adjointable is unambiguously connected to operators with respect to any choice of the C^*-valued inner product on a fixed Hilbert C^*-module if every bounded module operator possesses an adjoint operator on the module. Every bounded module operator on a given full Hilbert C^*-module turns out to be adjointable if the Hilbert C^*-module is orthogonally complementary. Moreover, if the unit ball of the Hilbert C^*-module is complete with respect to a certain locally convex topology, then these two properties are shown to be equivalent to self-duality.     

Intersecting families of convex cover order m

Let S=cl(int S) be a subset of the plane, Q the set of points of local non-convexity of S, Qfinite, with p a point in (bdry S ker S) Q. If denotes the collection of allmaximal families of wedges of S having convex cover order m, m1, then the members of are characterized in the following manner: For W _o a wedge of S, W _o is in if and only if for every w _o in W _o and every collection of points w ₁, ...w _m in S, then [w _i, w_j] S for at least one pair i, j, 0i<jm.AMS(MOS) Subject Classification 1970: Primary 52A10, 52A40.     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

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.     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.