Singularities ofJ-holomorphic curves in almost complex 4-manifolds

This note concerns the structure of singularities of mapsf from a neighborhood of {0} in the complex plane ? to an almost complex manifold (V, J), which areJ-holomorphic in the sense thatdf oi =J odf and are singular (i.e.,df = 0) at {0}. The main result is that whenV has dimension 4, the topology of these singularities is the same as in the case whenJ is integrable. Thus, if the image Imf =C is not multiply-covered, there is a neighborhoodU of the pointx = f(0), such that the pair (U, U ∩C) is homeomorphic to the cone overS ³,K _x whereK _x is an algebraic knot in S³ that depends only on the germC atx.     

Poisson Traces and D-Modules on Poisson Varieties

To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism ${\phi : X \to Y}To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism f: X ? Y{\phi : X \to Y} and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_f(X,N){M_\phi(X,N)} on X, and prove that it is holonomic if X has finitely many symplectic leaves, f{\phi} is finite, and N is coherent.     

New hierarchy of integrable system bi-Hamiltonian structure and constrained flows

We have considered the hierarchy of integrable systems associated with the unstable nonlinear Schrodinger equation. The spectral gradient approach and the trace identity are used to derive the bi-Hamiltonian structure of the system. The bi-Hamiltonian property and the square eigenfunctions determined via the spectral gradient approach are then used to construct constrained flows, which is also proved to be derivable from a rational Lax operator. This new Lax operator of the constrained flows is seen to generate the classical r-matrix. Lastly it is also explicitly demonstrated that the different integrals of motion of the constrained flows Poisson commute.     

Determination of the limits for multivariate rational functions

The purpose of this paper is to solve the problem of determining the limits of multivariate rational functions.It is essential to decide whether or not limxˉ→0f g=0 for two non-zero polynomials f,g∈R[x1,...,xn]with f(0,...,0)=g(0,...,0)=0.For two such polynomials f and g,we establish two necessary and sufcient conditions for the rational functionf g to have its limit 0 at the origin.Based on these theoretic results,we present an algorithm for deciding whether or not lim(x1,...,xn)→(0,...,0)f g=0,where f,g∈R[x1,...,xn]are two non-zero polynomials.The design of our algorithm involves two existing algorithms:one for computing the rational univariate representations of a complete chain of polynomials,another for catching strictly critical points in a real algebraic variety.The two algorithms are based on the well-known Wu’s method.With the aid of the computer algebraic system Maple,our algorithm has been made into a general program.In the final section of this paper,several examples are given to illustrate the efectiveness of our algorithm.     

Jointly ergodic measure-preserving transformations

The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that ifT ₁,T ₂, …,T _s are invertible commuting measure-preserving transformations of a probability space (X, ?, μ) then 1 $$\frac{1}{{N - M}}\sum\limits_{n = M}^{N - 1} {T{}_1^n } f_1 .T_2^n f_2 .....T_s^n f_s \xrightarrow[{N - M \to \propto }]{{I^2 (X)}}(\int_X {f1d\mu )} (\int_X {f2d\mu )...(\int_X {fsd\mu )} } $$ for anyf ₁,f ₂, …,f _s∈L ^x (X, ?, μ) iffT ₁×T ₂×…×T _s and all the transformationsT _iT_j ¹,i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.     

The extended f-vectors of 4-polytopes

For P a d-dimensional convex polytope and S = {i₁,…, i_s} ⊂ {0, 1,…, d−1}, let f_s(P) be the number of chains of faces Ø ⊂ F₁ ⊂ F₂ ⊂ … ⊂ F_s ⊂ P with dim F_j = i_j. By the generalized Dehn-Sommerville equations the dimension of the affine span of the extended f-vectors (f_s(P))_S ⊂ {0,1,2,3} as P ranges over all 4-polytopes is 4, and the extended f-vectors are determined by the values of f₀, f₁, f₂ and f₀₂. Six linear and four nonlinear inequalities on extended f-vectors of 4-polytopes are given. The consequences for the basic f-vector, (f₀, f₁, f₂, f₃), are derived. These include the inequality, 4f₂ ⩾ 3f₀ − 10 + 7f₃, conjectured by Barnette.     

Poisson structures on Hilbert schemes of points¶of a surface and integrable systems

In this paper we prove that if S is a Poisson surface, i.e., a smooth algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville [B1] and S. Mukai [M2] in the symplectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the simple case S=ℙ² we obtain by this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke in [V1] and, more generally, in [V2]. Received: 9 March 1998 / Revised version: 19 June 1998     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

Superpositions for nonlinear operators II. Generalized affine operators

From a previously given type of operators having strong superpositions, the C_f1,f2 classes, a generalization of affine transformations is obtained. The resulting operators have invariant quasilinear means. In addition, the operators have strong superpositions which are abelian semigroup operations with an idempotent property. It is natural, in this case, to define scalar operations on pairs of scalars and pairs of vectors on the domain and range spaces. Properties of this algebraic structure and its similarity to the superposition rules for color sensations are shown.     

Minimizing makespan in re-entrant permutation flow-shops

A re-entrant flow-shop (RFS) describes situations in which every job must be processed on machines in the order of M₁, M₂, …, M _m , M₁, M₂, …,M _m , …, and M₁, M₂, …,M _m . In this case, every job can be decomposed into L levels and each level starts on M₁, and finishes on M _m . In a RFS case, if the job ordering is the same on any machine at each level, then it is said that no passing is allowed since any job is not allowed to pass any previous job. The RFS scheduling problem where no passing is allowed is called the re-entrant permutation flow-shop (RPFS) problem. This paper proposes three extended mixed BIP formulations and six extended effective heuristics for solving RPFS scheduling problems to minimize makespan.     

Coxeter–Petrie Complexes of Regular Maps

Coxeter–Petrie complexes naturally arise as thin diagram geometries whose rank 3 residues contain all of the dual forms of a regular algebraic map M. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. Jones and Thornton have shown that these involutory duality operations generate the symmetric groupS₃ , giving in all six dual forms, and whose source is the outer automorphism group of the infinite triangle group generated by involutions s₁, s₂, s₃, subject to the additional relation s₁s₃ =  s₃s₁. In fact, this outer automorphism group is parametrized by the permutations of the three commuting involutions s₁,s₃ , s₁s₃. These involutions together with the involutions₂ can be taken to define the nodes of a Coxeter diagram of shape D₄(with the involution s₂at the central node), and when the original map M is regular, there is a natural extension from M to a thin Coxeter complex of rank 4 all of whose rank 3 residues are isomorphic to the various dual forms of M. These are fully explicated in case the original algebraic map is a Platonic map.     

Bilinear and quadratic variants on the Littlewood-Offord problem

If f(x ₁, …, x _n ) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a ₁, …,a _n , what is the maximum number of sums of the form ±a ₁ ± a ₂ ± … ± a _n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n ^?1 are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n ^?1/2. In both cases the results are nearly tight.     

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.     

On the representation of functions as a sum of derivatives

We show that there is an integrable function g of two variables which cannot be represented as a sum g = f₀ + ∂f₁ + ∂₂f₂, where f₀,f₁,f₂ are functions with integrable gradient.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Bifurcation analysis of the Zhukovskii-Volterra system via bi-Hamiltonian approach

The main goal of this paper consists of bifurcation analysis of classical integrable Zhukovskii-Volterra system. We use the fact that the ZV system is bi-Hamiltonian and apply new techniques [1] for analysis of singularities of bi-Hamiltonian systems, which can be formulated as follows: the structure of singularities of a bi-Hamiltonian system is determined by that of the corresponding compatible Poisson brackets.     

Non-simply connected copes

Let (W⁴,?W⁴) be a 4-manifold. Let f₁,f₂,…,f_k:(D²,?D²)→ (W⁴,?W⁴) be transverse immersions that have spherical duals $\text{α}{}_1\text{,α}{}_2\text{,…,α}{}_{\mathrm{k}}\text{:S}{}^2\text{→}\text{W}\text{?}$. Then there are open disjoint subsets V₁, V₂,…,V_k of W, such that for each 1?i?k, (a) ?V_i=V₁∩?W and ?V_i is an open regular neighborhood of f_i(?D²) in ?W, and (b) (V_i,?V_i,f_i(?D²)) is proper homotopy equivalent to (M, ?M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.     

Hua operators and Poisson transform for bounded symmetric domains

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)^n/r/²|h(z,u)^n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n?m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image.     

On the form of homomorphisms into the differential group Ls 1 and their extensibility

Let L_s ¹ (s ∈ ?) be the s-th differential group, that is the set {(x1,…,x_s): x₁ ≠ 0, x_n ∈ K, n =1,2,…,s} (K ∈ {?,?}) together with the group operation which describes the chain rules (up to order s) for C^s-functions with fixed point 0. We consider homomorphisms Φ_s, Φ_s = (f₁,…,f_s) from an abelian group (G,+) into L_s ¹ such that f₁ = 1, f₂ = … = f_p+2 = 0, 0_p+2 ≠ 0 for a fixed, but arbitrary p ≥ 0 such that p + 2 ≤ s (then f_p+2 is necessarily a homomorphism from (G, +) to (K, +). Let l ∈ ? or l = ∞. We present a criterion for the extensibility of Φ_s to a homomorphism Φ_s+l from (G, +) to L_s+1 ¹ (L_∞ ¹, if l = ∞), by proving that such an extension (continuation) exists iff the component functions f_n of Φ_s with s - p ≤ n ≤ min(s - p + l - 1,s) are certain polynomials in f_P+2 (see Theorem 1). We also formulate the problem in the language of truncated formal power series in one indeterminate X over K. The somewhat easier situation f ₁ ≠ 1 will be studied in a separate paper.