Mathematical Physics, Analysis and Geometry最新文献

We consider multi-component Kadomtsev-Petviashvili hierarchy of type C (the multi-component CKP hierarchy) originally defined with the help of matrix pseudo-differential operators via the Lax-Sato formalism. Starting from the bilinear relation for the wave functions, we prove existence of the tau-function for the multi-component CKP hierarchy and provide a formula which expresses the wave functions through the tau-function. We also find how this tau-function is related to the tau-function of the multi-component Kadomtsev-Petviashvili hierarchy. The tau-function of the multi-component CKP hierarchy satisfies an integral relation which, unlike the integral relation for the latter tau-function, is no longer bilinear but has a more complicated form.

We provide a concrete characterization of the poly-analytic planar automorphic functions, a special class of non analytic planar automorphic functions with respect to the Appell–Humbert automorphy factor, arising as images of the holomorphic ones by means of the creation differential operator. This is closely connected to the spectral theory of the magnetic Laplacian on the complex plane.

We introduce (delta ) type vertex conditions for beam operators, the fourth-order differential operator, on finite, compact and connected metric graphs. Our study the effect of certain geometrical alterations (graph surgery) of the graph on their spectra. Results are obtained for a class of vertex conditions which can be seen as an analogue of (delta )-conditions for graphs Laplacian. There are a number of possible candidates of (delta ) type conditions for beam operators. We develop surgery principles and record the monotonicity properties of their spectrum, keeping in view the possibility that vertex conditions may change within the same class after certain graph alterations. We also demonstrate the applications of surgery principles by obtaining several lower and upper estimates on the eigenvalues.

A Lie algebra morphism triple is a triple ((mathfrak {g}, mathfrak {h}, phi )) consisting of two Lie algebras (mathfrak {g}, mathfrak {h}) and a Lie algebra homomorphism (phi : mathfrak {g} rightarrow mathfrak {h}). We define representations and cohomology of Lie algebra morphism triples. As applications of our cohomology, we study some aspects of deformations, abelian extensions of Lie algebra morphism triples and classify skeletal sh Lie algebra morphism triples. Finally, we consider the cohomology of Lie group morphism triples and find a relation with the cohomology of Lie algebra morphism triples.

We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve (y=x^n) with (n ge 2). We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of n, proving the extended 16th Hilbert problem in this case. In particular, we show that for (n=2) this bound can be reached.

In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data.

One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology classes on the moduli spaces of curves satisfying certain natural splitting properties. Typically, these PDEs have an infinite expansion in the dispersive parameter, which happens because they involve contributions from the moduli spaces of curves of arbitrarily large genus. In this paper, for each rank (Nge 2), we present a family of F-CohFTs without unit, for which the equations of the associated integrable system have a finite expansion in the dispersive parameter. For (N=2), we explicitly compute the primary flows of this integrable system.

In the present paper, we consider a p-adic Ising model on a Cayley tree. The existence of non-periodic p-adic generalized Gibbs measures of this model is investigated. In particular, we construct p-adic analogue of the Bleher–Ganikhodjaev construction and generalize some constructive methods. Moreover, the boundedness of obtained measures are established, which yields the occurrence of a phase transition.

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars–Schneider system. We find its commutation representation in the form of the Manakov triple and study properties of the spectral curve. By studying more general elliptic solutions (elliptic families), we also suggest an extension of the deformed Ruijsenaars–Schneider system to a field theory.