Serre-Taubes duality for pseudoholomorphic curves

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b₊>1 satisfy the duality Gr(α)=±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods, we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b₊=1 and b₁=0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.     

Computations of instanton invariants using Donaldson-Floer Theory

Summary We compute the Donaldson SU(2)-invariants of the double cover of ² branched over a smooth algebraic curve of degree eight. From this we deduce a formula for the relative invariants of the blow-up of the Gompf nucleusN ₂, and we show how this gives a blow-up formula for a class of 4-manifolds which includes essentially all the simply connected 4-manifolds known to have big diffeomorphism group. We apply the result on the nucleus also to prove a formula for the invariants of minimal simply connected elliptic surfaces which reduces the computation to the case of geometric genus one. In particular, we compute all the Donaldson invariants of minimal simply connected elliptic surfaces without multiple fibers. Our main tool is Donaldson-Floer theory.Oblatum IX-1993 & 26-IV-1994     

On the ring of invariants ofF_{2^n }^*

In [1] the first and last authors studied a decomposition ofH ^*(R P ^∞×…×R P ^∞;F ₂) into modules over the Steenrod algebra obtained from an action of the cyclic group . Here a minimal set of generators for the ring of invariants is characterized and counted by analyzing the associated ring of Laurent polynomials. A structure theorem for the ring of invariant Laurent polynomials is given and a ‘destabilisation cancels localisation’ theorem is obtained. The authors gratefully acknowledge the support of NSERC. 1980 Mathematics Subject classification, 13F20, 55. Keywords: Invariant theory, Steenrod algebra.     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

Harmonic morphisms from the classical compact semisimple Lie groups

In this article, we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be used to construct the first known examples of harmonic morphisms from the non-compact Lie groups , SU ^*(2n), , SO ^*(2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with their standard dual semi-Riemannian metrics.      

Karmarkar's linear programming algorithm and Newton's method

This paper describes a full-dimensional version of Karmarkar's linear programming algorithm, theprojective scaling algorithm, which is defined for any linear program in ⁿ having a bounded, full-dimensional polytope of feasible solutions. If such a linear program hasm inequality constraints, then it is equivalent under an injective affine mappingJ: ^{n m} to Karmarkar's original algorithm for a linear program in ^m havingm—n equality constraints andm inequality constraints. Karmarkar's original algorithm minimizes a potential functiong(x), and the projective scaling algorithm is equivalent to that version of Karmarkar's algorithm whose step size minimizes the potential function in the step direction.The projective scaling algorithm is shown to be a global Newton method for minimizing a logarithmic barrier function in a suitable coordinate system. The new coordinate system is obtained from the original coordinate system by a fixed projective transformationy = (x) which maps the hyperplaneH _opt ={x:c, x =c ₀} specified by the optimal value of the objective function to the hyperplane at infinity. The feasible solution set is mapped under to anunbounded polytope. Letf _LB(y) denote the logarithmic barrier function associated to them inequality constraints in the new coordinate system. It coincides up to an additive constant with Karmarkar's potential function in the new coordinate system. Theglobal Newton method iterate y ^* for a strictly convex functionf(y) defined on a suitable convex domain is that pointy ^* that minimizesf(y) on the search ray {y+ v _N(y): 0} wherev _N(y) =–(² f(y))^–1(f(y)) is the Newton's method vector. If {x ^(k)} are a set of projective scaling algorithm iterates in the original coordinate system andy ^(k) =(x ^(k)) then {y ^(k)} are a set of global Newton method iterates forf _LB(y) and conversely.Karmarkar's algorithm with step size chosen to minimize the potential function is known to converge at least at a linear rate. It is shown (by example) that this algorithm does not have a superlinear convergence rate.     

Defining Relations for the Algebra of Invariants of 2×2 Matrices

We obtain defining relations of the algebra of invariants of the classical subgroups of GL ₂(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL _m (C) and ideas coming from the theory of algebras with polynomial identities.     

The geometric and numerical properties of duality in projective algebraic geometry

In this paper we investigate some fundamental geometric and numerical properties ofduality for projective varieties inP _k ^N =P ^N . We take a point of view which in our opinion is somewhat moregeometric and lessalgebraic andnumerical than what has been customary in the literature, and find that this can some times yield simpler and more natural proofs, as well as yield additional insight into the situation. We first recall the standard definitions of thedual variety and theconormal scheme, introducing classical numerical invariants associated with duality. In section 2 we recall the well known duality properties these invariants have, and which was noted first byT. Urabe. In section 3 we investigate the connection between these invariants andChern classes in the singular case. In section 4 we give a treatment of the dual variety of a hyperplane section of X, and the dual procedure of taking the dual of a projection of X. This simplifies the proofs of some very interesting theorems due toR. Piene. Section 5 contains a new and simpler proof of a theorem ofA. Hefez and S. L. Kleiman. Section 6 contains some further results, geometric in nature.     

The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…     

Sylvester Waves in the Coxeter Groups

A new recursive procedure of the calculation of partition numbers function W(s, d ^m ) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s, d ^m ) and their periods (G) for the irreducible Coxeter groups and a list for the first twelve symmetric group _m are presented. A least common multiple (m) of the series of the natural numbers 1,2,...,m plays a role in the period ( _m ) of W(s, d ^m) in _m .     

Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L ²-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra (usually the full group C*-algebra C*() of the fundamental group =₁(M) of a manifold M).     

Prolongations of Vector Fields and the Invariants-by-Derivation Property

For any given vector field X defined on some open set M ², we characterize the prolongations X _n ^* of X to the nth jet space M ⁽ⁿ⁾, n1, such that a complete system of invariants for X _n ^* can be obtained by derivation of lower-order invariants. This leads to characterizations of C -symmetries and to new procedures for reducing the order of an ordinary differential equation.     

Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

Simultaneous reduction of a lattice basis and its reciprocal basis

Given a latticeL we are looking for a basisB=[b ₁, ...b _n ] ofL with the property that bothB and the associated basisB ^*=[b ₁ ^* , ...,b _n ^* ] of the reciprocal latticeL ^* consist of short vectors. For any such basisB with reciprocal basisB ^* let . Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)exp(c ₁·n ^1/3) for a constantc ₁ independent ofn. We improve this upper bound toS(B)exp(c ₂·(lnn)²) withc ₂ independent ofn.We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity . In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.     

Groups of spacetime transformations and symmetries of four-dimensional spacetime. I

The relativistic 4-interval (X-X ₍₀₎ ²=s ₂ ⁽⁰⁾ is interpreted as a 4-hyperboloid of radiuss ₍₀₎ and center at the pointX ⁽⁰⁾ that is formed by particles radiated isotropically from its center with velocities 0<1 whose positions in 4d spacetime are fixed at a proper times _(0)/c that is the same for all of them. Therefore, the 4-hyperboloid can be regarded as a mathematical model of an isotropically radiating source, and all transformations of the spacetime variables that leave its equation invariant have a physical meaning and determine the symmetry properties of 4d spacetime. These transformations form the group of motions of a rotating 4-hyperboloid. For constant radiuss ₍₀₎=const, its configuration space is the 8-dimensional bundleR(1,3)=R(1,3) (1,3), and the minimal group of motions isK=P O(1,3). It is shown that the well-known groupsP andO(1,3) are defined, respectively, only on the baseR(1,3) and only on the fiber (1,3) of the spaceR(1,3) and that the symmetry properties of 4d spacetime introduced by them are incomplete. The groupK extends the isotropy property of 4d spacetime to moving frames of reference. The group of spacetime transformations is extended to the case ofN bundles. It is shown that the new interpretation of the 4-interval makes it necessary to assume that the radiuss ₍₀₎ is variable. The groups of motion of a 4-hyperboloid of variable radius are constructed in the second part of the paper. They introduce new symmetry properties of 4d spacetime.D. V. Efremov Institute of Electrophysical Apparatus. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 458–475, September, 1994.     

Infrared Variables for the SU(3) Yang–Mills Field

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)³.     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05     

Fluctuation theory of media with pronounced spacetime inhomogeneity

A nonlinear determinstic thermodynamics is constructed for media with pronounced (r,t) inhomogeneity of the intensive variables or their derivatives. The balance equations in the theory are taken to be either the equations of an ideal fluid or an ideal fluid with heat conduction. The basic variables are taken to be (r,t) andT(r,t). The hypothesis of local equilibrium is represented in the form of the Gibbs-Duhem relation, the conjugate coordinates are (r,t) and (r,t), and the local potential isP(,T). The velocity potentialv _i (r,t) enters through the substantial derivative. A variational principle is formulated; in the case of an ideal fluid with heat conduction there arises naturally a local decrease of the entropy production:z ₂(t)=z ₀ ² exp(–2t/t), there0074 is the relaxation time.In memory of D. N. ZubarevJoint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 53–67, October, 1993.