Products in the category of -manifolds

We prove that the category of -manifolds has all finite products. Further, we show that a -manifold (resp., a -morphism) can be reconstructed from its algebra of global -functions (resp., from its algebra morphism between global -function algebras). These results are of importance in the study of Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure sheafs of two -manifolds is not a sheaf. We rely on a number of results on (pre)sheaves of topological algebras, which we establish in the appendix.     

Generalized Solvable Structures and First Integrals for ODEs Admitting an Symmetry Algebra

The notion of solvable structure is generalized in order to exploit the presence of an algebra of symmetries for a kth-order ordinary differential equation with k > 3. In this setting, the knowledge of a generalized solvable structure for allows us to reduce to a family of second-order linear ordinary differential equations depending on k ? 3 parameters. Examples of explicit integration of fourth and fifth order equations are provided in order to illustrate the procedure.     

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t)+A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on , then p takes a quadratic form of x. 2) If A = f (t)I + D with D^T = ?D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.     

Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides

The evolution equations mentioned in the title of this paper read as follows:

where ? is the “discrete-time” independent variable taking integer values (? = 0, 1, 2,?…?), xn ≡ xn(?) are the 2 dependent variables, , and the 2 functions P⁽ⁿ⁾(x₁, x₂), n = 1, 2, are 2 polynomials in the 2 dependent variables x₁(?) and x₂(?). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t—in which case x_n ≡ x_n(t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts.     

Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,

which are algebraically solvable. Here l is the “discrete-time” independent variable taking integer values (l = 0, 1, 2, . . .), x_n ≡ x_n(l) are 2 dependent variables, and are the corresponding 2 updated variables. In a previous paper the 2 functions F⁽ⁿ⁾(x₁, x₂), n = 1, 2, were defined as follows: F⁽ⁿ⁾(x₁, x₂) = P₂ (x_n, x_n+1), n = 1, 2 mod[2], with P₂(x₁, x₂) a specific second-degree homogeneous polynomial in the 2 (indistinguishable!) dependent variables x₁(l) and x₂(l). In the present paper we further clarify some aspects of that model and we present its extension to the case when a specific homogeneous function of arbitrary (integer) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x₁(l) and x₂(l).     

Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlevé equations are then obtained by deautonomising the various mappings thus obtained. We apply the restoration method to two challenging examples, one of which does not lead to a QRT mapping at the autonomous limit but we verify that even in that case our method is indeed still applicable. For one of the equations we derive we also show how, starting from a form where the independent variable advances one step at a time, we can obtain versions that correspond to multiple-step evolutions.     

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

If G is a finite Coxeter group, then symplectic reflection algebra H := H_1,η (G) has Lie algebra of inner derivations and can be decomposed under spin: H = H₀ ⊕ H_1/2 ⊕ H₁ ⊕ H_3/2 ⊕ …?We show that if the ideals of all the vectors from the kernel of degenerate bilinear forms B_i(x, y) := sp_i (x · y), where sp_i are (super)traces on H, do exist, then if and only if .     

Differential Operators,Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract

With each second-order differential equation Z in the evolution space J ¹(M _n+1) we associate, using the natural f(3, ?1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J ¹(M _n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J ¹(M _n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincaré–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.     

On (2)-relative cohomology of the Lie algebra of vector fields and differential operators

Abstract

Let Vect(?) be the Lie algebra of smooth vector fields on ?. The space of symbols Pol(T*?) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(?)-module that becomes trivial once the action is restricted to (2) ? Vect(?). The deformations of Pol(T*?), which become trivial once the action is restricted to (2) and such that the Vect(?)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D _λ,μ = Homdiff(F _λ, F _μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).     

Free-field realizations of the W,N-algebra

In this paper, we will construct free-field realizations of the W_,N algebra associated to an -valued differential operator

where is a Frobenius algebra with the uint I_n.     

A convenient expression of the time-derivative,of arbitrary order k,of the zero zn(t) of a time-dependent polynomial pN(z;t) of arbitrary degree N in z,and solvable dynamical systems

Let p_N (z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let z_n ≡ z_n (t) be its N zeros: . In this paper we report a convenient expression of the k-th time-derivative of the zero z_n (t). This formula plays a key role in the identification of classes of solvable dynamical systems describing the motion of point-particles moving in the complex z-plane while nonlinearly interacting among themselves; one such example, featuring many arbitrary parameters, is reported, including its variation describing the motion of many particles moving in the real Cartesian xy-plane and interacting among themselves via rotation-invariant Newtonian equations of motion (”accelerations equal forces”).     

Hamiltonian Structures and Integrability of Frobenius Algebra-Valued (n,m)th KdV Hierarchy

We introduce Frobenius algebra ?-valued (n, m)^th KdV hierarchy and construct its bi-Hamiltonian structures by employing ?-valued pseudo-differential operators. As an illustrative example, the (1, 1)^th -valued case is analyzed in detail. Its Hamiltonian structures and recursion operator are derived. Infinitely many symmetries, conservation laws and explicit flow equations are also obtained.