Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.     

On <Emphasis Type="Italic">W</Emphasis>-geometry of Toda systems

We describe W-geometry of two-dimensional Toda systems associated with the Lie algebra C _n . __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Integrals of open two-dimensional lattices

We present an explicit formula for integrals of the open two-dimensional Toda lattice of type A_n. This formula is applicable for various reductions of this lattice. As an illustration, we find integrals of the G₂ Toda lattice. We also reveal a connection between the open An Toda and Shabat-Yamilov lattices.     

On an integrable case of Kozlov-Treshchev Birkhoff integrable potentials

We establish, using a new approach, the integrability of a particular case in the Kozlov-Treshchev classification of Birkhoff integrable Hamiltonian systems. The technique used is a modification of the so called quadratic Lax pair for D _n Toda lattice combined with a method used by M. Ranada in proving the integrability of the Sklyanin case.      

The Hodge Structure on a Filtered Boolean Algebra

Let (B _n) be the order complex of the Boolean algebra and let B(n, k) be the part of (B _n) where all chains have a gap at most k between each set. We give an action of the symmetric group S _l on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S _n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k).     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].     

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a _n ,b _n )^T={f(?)} _n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.     

Counting lattice chains and Delannoy paths in higher dimensions

Lattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n₁,…,n_d, and let L denote the lattice of points (a₁,…,a_d)∈Z^d that satisfy 0≤a_i≤n_i for 1≤i≤d. We prove that the number of chains in L is given by where . We also show that the number of Delannoy paths in L equals Setting n_i=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension.     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Solvability of eight classes of nonlinear systems of difference equations

We have studied recently solvability and semi‐cycles of eight systems of difference equations of the following form: where a ∈ [0, + ∞), the sequences p_n, q_n, r_n, s_n are some of the sequences x_n and y_n, with positive initial values x_?j,y_?j, j = 1,2, in detail. This paper is devoted to the study of the other eight systems of the form. We show that these systems are also solvable in closed form and describe semi‐cycles of their solutions complementing our previous results on such systems of difference equations.     

Generalized Bäcklund–Darboux transformations for Coxeter–Toda flows from a cluster algebra perspective

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N _u,v in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group S _n and the corresponding networks N_u,v^°N_{u,v}^\circ in the annulus. Boundary measurements for N _u,v represent elements of the Coxeter double Bruhat cell G ^u,v ⊂GL _n . The Cartan subgroup H acts on G ^u,v by conjugation. The standard Poisson structure on the space of weights of N _u,v induces a Poisson structure on G ^u,v , and hence on the quotient G ^u,v /H, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for N_u,v^°N_{u,v}^\circ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for N _u,v . The corresponding Poisson bracket on the space of weights of N_u,v^°N_{u,v}^\circ induces a Poisson bracket on the certain space R_n {\mathcal{R}_n} of rational functions, which appeared previously in the context of Toda flows.     

Games of Chains and Cutsets in the Boolean Lattice II

Let 2 ^[n] denote the poset of all subsets of [n]={1,2,...,n} ordered by inclusion. Following Gutterman and Shahriari (Order 14, 1998, 321–325) we consider a game G _n (a,b,c). This is a game for two players. First, Player I constructs a independent maximal chains in 2 ^[n]. Player II will extend the collection to a+b independent maximal chains by finding another b independent maximal chains in 2 ^[n]. Finally, Player I will attempt to extend the collection further to a+b+c such chains. The last Player who is able to complete her move wins. In this paper, we complete the analysis of G _n (a,b,c) by considering its most difficult instance: when c=2 and a+b+2=n. We prove, the rather surprising result, that, for n7, Player I wins G _n (a,n–a–2,2) if and only if a3. As a consequence we get results about extending collections of independent maximal chains, and about cutsets (collections of subsets that intersect every maximal chain) of minimum possible width (the size of largest anti-chain).     

System of equations for stimulated combination scattering and the related double periodic <Emphasis Type="Italic">A</Emphasis><Stack><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Superscript>(1)</Superscript></Stack> Toda chains

We consider a system of equations describing stimulated combination scattering of light. We show that solutions of this system are expressed in terms of two solutions of the sine-Gordon equation that are related to each other by a Bäcklund transformation. We also show that this system is integrable and admits a Zakharov-Shabat pair. In the general case, the system of equations for the Bäcklund transformation of periodic A _n ⁽¹⁾ Toda chains is also shown to be integrable and to have a Zakharov-Shabat pair.     

On closeness of the sum of n subspaces of a Hilbert space

We give necessary and sufficient conditions for the sum of subspaces H ₁,…, H _n , n ≥ 2, of a Hilbert space H to be a subspace and present various properties of the n-tuples of subspaces with closed sum.     

Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere S ^r⊂R ^r+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d _n =dim P _n points, where P _n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S ² of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

     

Controlling multiparticle system on the line. II. Periodic case

As in [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772-4790] we consider classical system of interacting particles P₁,…,Pn on the line with only neighboring particles involved in interaction. On the contrast to [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772-4790] now periodic boundary conditions are imposed onto the system, i.e. P₁ and Pn are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its nonperiodic counterpart.Main result established is global controllability by means of single controlling force of the multiparticle system with a generic potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections 2, 3.     

The Arithmetical Hierarchy of Real Numbers

A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left (right) computable iff it is the supremum (infimum) of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable (multiple) sequences of rational numbers we introduce a non‐collapsing hierarchy {Σ_n, Π_n, Δ_n : n ∈ ℕ} of real numbers. We characterize the classes Σ₂, Π₂ and Δ₂ in various ways and give several interesting examples.     

Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks

We prove an isoperimetric inequality for wreath products of Markov chains with variable fibers. We use isoperimetric inequalities for wreath products to estimate the return probability of random walks on infinite groups and graphs, drift of random loops, the expected value E(exp(−tR _n )), where R _n is the number of distinct sites, visited up to the moment n, and, more generally, (where L _z,n is the number of visits of z up to the moment n and F(x, y) is some non-negative function).