A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.      

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

Completeness of the Description of an Equilibrium Canonical Ensemble by a Two-Particle Partition Function

We show that in the equilibrium classical canonical ensemble of particles with pair interaction, the full Gibbs partition function can be uniquely expressed in terms of the two-particle partition function. This implies that for a fixed number N of particles in the equilibrium system and a fixed volume V and temperature T, the two-particle partition function fully describes the Gibbs partition as well as the N-particle system in question. The Gibbs partition can be represented as a power series in the two-particle partition function. As an example, we give the linear term of this expansion. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 123–132, October, 2005.     

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

We consider a classical system of n charged particles in an external confining potential in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n^?1 (mean‐field scaling). By a suitable splitting of the Hamiltonian, we extract the next‐to‐leading‐order term in the ground state energy beyond the mean‐field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new “renormalized energy” functional providing a way to compute the total Coulomb energy of a jellium (i.e., an infinite set of point charges screened by a uniform neutralizing background) in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next‐to‐leading‐order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations, and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than 2 by an alternative approach. © 2016 Wiley Periodicals, Inc.     

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.     

Canonical transformations of the extended phase space and integrable systems

We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.     

Gauge theories as matrix models

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.     

Construction of Canonical Difference Schemes for Hamiltonian Formalism via Generating Functions

This paper discusses the relationship between canonical maps and generating functions and gives the general Hamilton-Jacobi theory for time-independent Hamiltonian systems. Based on this theory, the general method — the generating function method — of the construction of difference schemes for Hamiltonian systems is considered. The transition of such difference schemes from one time-step to the next is canonical. So they are called the canonical difference schemes. The well known Euler centered scheme is a canonical difference scheme. Its higher order canonical generalisations and other families of canonical difference schemes are given. The construction method proposed in the paper is also applicable to time-dependent Hamiltonian systems.     

Canonical partition theorems for parameter sets

A canonical (i.e., unrestricted) version of the partition theorem for k-parameter sets of Graham and Rothschild (Trans. Amer. Math. Soc.159 (1971), 257–291) is proven. Some applications, e.g., canonical versions, of the Rado-Folkman-Sanders theorem and of the partition theorem for finite Boolean algebras are given. Also the Erdös-Rado canonization theorem (J. London Math. Soc.25 (1950), 249–255) turns out to be an immediate corollary.     

Hamiltonian Structures of Fermionic Two-Dimensional Toda Lattice Hierarchies

By exhibiting the corresponding Lax-pair representations, we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies, which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows describing these hierarchies in the framework of the Hamiltonian formalism and constructing their first two Hamiltonian structures. We obtain the first Hamiltonian structure for both bosonic and fermionic Lax operators and the second Hamiltonian structure only for bosonic Lax operators. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 90–102, January, 2006.     

Canonical Euler-Lagrange equations and Jacobi's theorem on regular surfaces

In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobi's theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.     

Canonical and monophonic convexities in hypergraphs

Known properties of “canonical connections” from database theory and of “closed sets” from statistics implicitly define a hypergraph convexity, here called canonical convexity (c-convexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski-Krein-Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (or m-convexity), and prove that: (1) m-convexity is coarser than c-convexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3) m-convex hulls can be computed in the same efficient way as c-convex hulls.     

Twistors, 4-symmetric spaces and integrable systems

An order four automorphism of a Lie algebra gives rise to an integrable system introduced by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. As applications, we find that surfaces with holomorphic mean curvature in 4-dimensional real or complex space forms constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in 4-dimensional Hermitian symmetric spaces (this last providing a conceptual explanation of a result of Hélein-Romon).     

一类Schr(o)dinger算子的谱范围

针对半直线上可积势对应的Schr(o)dinger算子,研究A-函数和谱之间的关系,利用复分析中保形变换的方法给出了谱测度关于A-函数的局部表示,进而得到算子的谱范围,该结论说明了这一类Schr(o)dinger算子是下半有界的.     

Some Formulas for the Canonical Kernel Function

In this paper we present some expressions for the canonical kernel function of a bounded symmetric domain. We show how it is related both to the reproducing kernel of a holomorphic discrete series representation and to the contravariant form on highest weight modules over semisimple Lie algebras of Hermitian type.     

Canonical frame of a curve on a conformal plane

We show how differential geometry of smooth curves on the conformal plane can be studied by Élie Cartan’s method of exterior forms and moving frames. We find the canonical form of the derivation equations of a curve (which is not a circle) in the case of a semi-isotropic frame. We give a new proof of the theorem that states that curves of constant (in particular, zero) conformal curvature are loxodromes. We integrate the system of structure equations of the isotropy subgroup of a point.     

Combinatorial expansions of conformal blocks

A representation of Nekrasov partition functions in terms of a nontrivial two-dimensional conformal field theory was recently suggested. For a nonzero value of the deformation parameter ∈ = ∈ ₁ + ∈ ₂ , the instanton partition function is identified with a conformal block of the Liouville theory with the central charge c = 1 + 6∈ ² /∈ ₁ ∈ ₂ . The converse of this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with nondegenerate Virasoro representations, has a nontrivial decomposition into a sum over Young diagrams that differs from the natural decomposition studied in conformal field theory. We provide some details about this new nontrivial correspondence in the simplest case of the four-point correlation functions.     

Bundle bispectrality for matrix differential equations

We consider the fundamental solutions of a wide class of first order systems with polynomial dependence on the spectral parameter and rational matrix potentials. Such matrix potentials are rational solutions of a large class of integrable nonlinear equations, which play an important role in different mathematical physics problems. The concept of bispectrality, which was originally introduced by Grünbaum, is extended in a natural way for the systems under consideration and their bispectrality is derived via the representation of the fundamental solutions. This bispectrality is preserved under the flows of the corresponding integrable nonlinear equations. For the case of Dirac type (canonical) systems the complete characterization of the bispectral potentials under consideration is obtained in terms of the system's spectral function.     

A Fresh Perspective on Canonical Extensions for Bounded Lattices

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Plo??ica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphisms the Plo??ica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Plo??ica’s paper. This leads to a construction of canonical extension valid for all bounded lattices, which is shown to be functorial, with the property that the canonical extension functor decomposes as the composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors.     

Quantum Twist Maps and Dual Canonical Bases

In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the “reverse” lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.