Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

The mathematics of eigenvalue optimization

 Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread. Received: December 4, 2002 / Accepted: April 22, 2003 Published online: May 28, 2003 Key Words.  eigenvalue optimization – convexity – nonsmooth analysis – duality – semidefinite program – subdifferential – Clarke regular – chain rule – sensitivity – eigenvalue perturbation – partly smooth – spectral function – unitarily invariant norm – hyperbolic polynomial – stability – robust control – pseudospectrum – H _∞ norm Mathematics Subject Classification (2000): 90C30, 15A42, 65F15, 49K40     

Homoclinic and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle-focus and Saddle Under Reversible Condition

In this paper, we study the dynamical behavior for a 4-dimensional reversible system near its heteroclinic loop connecting a saddle-focus and a saddle. The existence of infinitely many reversible 1-homoclinic orbits to the saddle and 2-homoclinic orbits to the saddle-focus is shown. And it is also proved that, corresponding to each 1-homoclinic （resp. 2-homoclinic） orbit F, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1-periodic （resp. 2-periodic） and accumulate onto F. Moreover, each 2-homoclinic orbit may be also accumulated by a sequence of reversible 4-homoclinic orbits.     

Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

We compute the recently introduced Fan–Jarvis–Ruan–Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov–Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan–Jarvis–Ruan–Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau–Ginzburg/Calabi–Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental’s quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan–Jarvis–Ruan–Witten theory.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

A Bass–Heller–Swan formula for pseudoisotopies

The Bass–Heller–Swan formula is a basic calculational tool in pseudoisotopy K-theory. We describe the Nil-groups and the Bass–Heller–Swan splitting for the group of the pseudoisotopies of a closed manifold. We use the methods of controlled topology used in the Bass–Heller–Swan splitting in K-theory.     

K-subadditive and K-superadditive set-valued functions bounded on “large” sets

Aequationes mathematicae - We prove that every K–subadditive set–valued map weakly K–upper bounded on a “large” set (e.g. not null–finite, not...     

Skew quantum Murnaghan–Nakayama rule

We extend recent results of Assaf and McNamara on a skew Pieri rule and a skew Murnaghan–Nakayama rule to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf–McNamara’s original proof, and one via Lam–Lauve–Sotille’s skew Littlewood–Richardson rule. We end with some conjectures for skew rules for Hall–Littlewood polynomials.     

Asymptotic compactness of global trajectories generated by the Navier–Stokes–Poisson equations of a compressible fluid

In this paper, we consider the global behavior of weak solutions of Navier–Stokes–Poisson equations in time in a bounded domain–arbitrary forces. After proving the existence of bounded absorbing sets, we also obtain the conclusion on asymptotic compactness of global trajectories generated by the Navier–Stokes–Poisson equations of a compressible fluid. Supported by NSF(No:10531020) of China and the Program of 985 Innovation Engineering on Information in Xiamen University (2004–2007).     

Bifurcations of Fine 3-point-loop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to study the bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversal conditions and the non–twisted condition, the existence, coexistence and incoexistence of 2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and the number of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained. This work is supported by the National Natural Science Foundation of China (10371040), the Shanghai Priority Academic Disciplines and the Scientific Research Foundation of Linyi Teacher’s University     

Uniformly small Δ* Groups

It is established that in each uniformly small Δ*–group, there exists a closed system of representatives; moreover, it turns out that every narrow epimorphism onto a projective Δ*–group splits. (These properties are violated in a class of all Δ*–group). Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 24–39, January–February, 1999.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Renormalization and homogenization of fractional diffusion equations with random data

 This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations. Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002 Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60 Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler function – Stable distributions – Bessel potential – Riesz potential     

Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra B _n (q,r) by lifting bases for cell modules of B _n−1(q,r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke algebra of the symmetric group, acts triangularly. The triangular action of this Abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B–M–W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori–Hecke algebra of the symmetric group. Research supported by Japan Society for Promotion of Science.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

Quadratic forms for a 1-form on an isolated complete intersection singularity

We consider a holomorphic 1-form ω with an isolated zero on an isolated complete intersection singularity (V,0). We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair (V,ω). They generalize the Eisenbud–Levine–Khimshiashvili quadratic form defined for a smooth V. Partially supported by the DFG-programme 'Global methods in complex geometry' (Eb 102/4–3) grants RFBR–04–01–00762, NSh–1972.2003.1.     

On <Emphasis Type="Italic">γ</Emphasis>-Vectors Satisfying the Kruskal–Katona Inequalities

We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement of Gal’s conjecture, which asserts that such γ-vectors are nonnegative.     

On the complete integrability and linearization of a Burgers– Korteweg–de Vries-type nonlinear equation

In this work, on the basis of the Bogolyubov–Prykarpats’kyi gradient–holonomic algorithm for the investigation of the integrability of nonlinear dynamical systems on functional manifolds, the exact linearization of a Burgers–Korteweg–de Vries-type nonlinear dynamical system is established. As a result, we describe the linear structure of the space of solutions and show its relation to the convexity of certain functional subsets. The bi-Hamiltonian property of the Burgers–Korteweg–de Vries dynamical system is also established, and the infinite hierarchy of functionally independent invariants is constructed.     

Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal –regularity for the generator of an analytic semigroup in terms of the R–boundedness of the resolvent of A or the semigroup . Received July 19, 1999 / Revised July 13, 2000 / Published online February 5, 2001     

Strong Formulations of Robust Mixed 0–1 Programming

We introduce strong formulations for robust mixed 0–1 programming with uncertain objective coefficients. We focus on a polytopic uncertainty set described by a ``budget constraint' for allowed uncertainty in the objective coefficients. We show that for a robust 0–1 problem, there is an α–tight linear programming formulation with size polynomial in the size of an α–tight linear programming formulation for the nominal 0–1 problem. We give extensions to robust mixed 0–1 programming and present computational experiments with the proposed formulations.