From Grushin to Heisenberg via an isoperimetric problem

The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections are solutions of an isoperimetric problem in the Grushin plane.     

Instanton contributions to theR-symmetry anomaly

In the N=2 supersymmetric Yang-Mills theory, the instanton contribution to the R-symmetry anomaly resulting from noninvariance of the collective coordinate measure is investigated. The one-instanton contribution is in agreement with the exact result. Multi-instanton contributions are investigated qualitatively. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115, No. 3, pp. 402–409, June, 1998.     

From Heisenberg uniqueness pairs to properties of the Helmholtz and Laplace equations

The aim of this paper is to establish uniqueness properties of solutions of the Helmholtz and Laplace equations. In particular, we show that if two solutions of such equations on a domain of ${\mathbb{R}}^d$ agree on two intersecting $d?1$-dimensional submanifolds in generic position, then they agree everywhere.     

Global strong solution to the Thirring model in critical space

We obtain a strong solution in charge critical space L²(R) of the Thirring system and Federbusch equations in one space dimension by using solution representation of the models. The uniqueness is obtained for the solution Ψ∈L^∞([0,T];L²(R)∩L⁴(R)). A decay of local charge and asymptotic behavior of the field can be shown directly.     

Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

Instanton moduli as a novel map from tori to K3-surfaces

A map is constructed from the moduli of hyper-Kähler tori to hyper-Kähler K3 surfaces which does not coincide with the Kummer map. The map takes a torus to the moduli space of SO(3) connections on a bundle with nontrivial first Stiefel-Whitney class and first Pontrjagin class equal to –4. This map is shown to intersect the Kummer moduli and also certain subvarieties of singular K3 surfaces. Our map is shown to satisfy the local Torelli theorem, and the K3-surfaces in its image are shown to carry a natural metric which is Calabi-Yau.     

Isomorphism Classes of Rings Related to Heisenberg Algebras

Abstract

In this paper we examine isomorphism classes of algebras which are generalizations of the enveloping algebra of the Heisenberg Lie algebra.     

KdV-Burgers-Kuramoto系统的渐近吸引子

研究了 KdV-Burgers-Kuramoto 方程的渐近吸引子,即利用正交分解法构造一个有限维解序列。首先用数学归纳法证明了该解序列不会远离方程的整体吸引子,接着证明解序列在长时间后无限趋于方程的整体吸引子,最后给出渐近吸引子的维数估计。     

Tau-function formulation for bright,dark soliton and breather solutions to the massive Thirring model

In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single-component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one- and two-soliton for bright, dark soliton and breather solutions are analyzed in details.     

A dynamical system approach to Heisenberg uniqueness pairs

A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials.     

修正的各向异性旋转链方程的整体解

本文研究一类修正的海森堡旋转链方程组整体光滑解的存在性和唯一性.     

On the Semistability of Instanton Sheaves Over Certain Projective Varieties

We show that instanton bundles of rank r ≤ 2n ? 1, defined as the cohomology of certain linear monads, on an n-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford–Takemoto. Furthermore, we show that rank r ≤ n linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.     

Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid

There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be non-unique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ωm to Ω=R×(−L,L), where is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ωm→Ω as m→+∞. That is, let A and Am be the global attractors of the fluid corresponding to Ω and Ωm, respectively, we establish that for any neighborhood O(A) of A, the global attractor Am enters O(A) if m is large enough.     

Homoclinical structure of the chaotic attractor

In the reference [Akhmet MU. Devaney chaos of a relay system. Commun Nonlinear Sci Numer Simulat 2009;14:1486–93.], a relay system was introduced, which admits a chaotic attractor with Devaney’s ingredients. Now, we prove that the attractor consists of homoclinic solutions. A simulation of the attractor is provided for a pendulum equation. Similar results for impulsive differential equations were announced in the plenary talk [Akhmet MU. Hyperbolic sets of impact systems. Dyn Contin Discrete Impuls Syst Ser A Math Anal 2008;15(Suppl. S1):1–2. Proceedings of the 5th international conference on impulsive and hybrid dynamical systems and applications, Beijin: Watan Press; 2008.].     

Extended Fisher-Kolmogorov系统的渐近吸引子

考虑了ExtendedFisher-Kolmogorov系统的解的长时间行为,构造了一个有限维解序列即该系统的渐近吸引子,证明了它在长时间后无限趋于方程的整体吸引子,并给出了渐近吸引子的维数估计.