A note on the generalized symmetries of lattice hierarchies

We show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential-delay equations. All members of these hierarchies of lattice and differential-delay equations commute. It is thus seen that differential-delay hierarchies provide a broader context within which to place lattice hierarchies.     

Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure

Theoretical and Mathematical Physics - We recently constructed a series of integrable discrete autonomous equations on a quadratic lattice with a nonstandard structure of higher symmetries. Here,...     

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Classification of generalized symmetries of the Yang-Mills fields with a semi-simple structure group

A complete classification of generalized (or local) symmetries of the Yang-Mills equations on four dimensional Minkowski space with a semi-simple structure group is carried out. It is shown that any generalized symmetry, up to a generalized gauge symmetry, agrees with a first order symmetry on solutions of the Yang-Mills equations. Let be the decomposition of the Lie algebra of the structure group into simple ideals. First order symmetries for -valued Yang-Mills fields are found to consist of gauge symmetries, conformal symmetries for -valued Yang-Mills fields, 1?m?n, and their images under a complex structure of .     

Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation

In this paper, we consider a class of generalized diffusion equations which are of great interest in mathematical physics. For some of these equations that model fast diffusion, nonclassical and nonclassical potential symmetries are derived. These symmetries allow us to increase the number of solutions. These solutions are unobtainable neither from classical nor from classical potential symmetries. Copyright © 2007 John Wiley & Sons, Ltd.     

On generalized symmetries and structure of modular categories

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.     

广义KdV—Burgers方程的势对称和不变解

用微分形式的吴方法讨论了广义KdV—Burgers方程不同系数情况下的势对称,并且利用这些对称求得了相应的不变解,这些解对进一步研究广义KdV—Burgers方程所描述的物理现象具有重要意义．     

Discrete analysis on a radial lattice

A discrete model for analytic functions is constructed using lattice points of the complex plane arranged in radial form. The discrete analytic functions are defined as solutions of a finite-difference approximation to the polar Cauchy-Riemann equations. The resulting discrete powerz⁽ⁿ⁾ (an analogue of zⁿ) has a simple algebraic form (a direct analogue of ?ⁿ exp{inθ}) and has some surprising properties. For example every discrete polynomial ∑₀^ma_nz⁽ⁿ⁾ has a factorization in terms of the zeros of its classical counterpart ∑₀^ma_nzⁿ every discrete entire function has a power series representation ∑ a_nz⁽ⁿ⁾.     

Classification of discrete systems on a square lattice

We consider the classification up to a M?bius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A₁, A₂, and A₃ linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice.     

Summation of closed diagrams on a square lattice

S. M. Kirov Ural Polytechnic Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 2, pp. 305–310, May, 1990.     

Discrete optimization on a multivariable boolean lattice

Consider the problem of finding the minimum value of a scalar objective function whose arguments are theN components of 2 ^N vector elements partially ordered as a Boolean lattice. If the function is strictly decreasing along any shortest path from the minimum point to its logical complement, then the minimum can be located precisely after sequential measurement of the objective function atN + 1 points. This result suggests a new line of research on discrete optimization problems.This paper was presented at the 7th Mathematical Programming Symposium, The Hague, The Netherlands.This research was supported in part by U.S. Office of Saline Water Grant No. 699.     

The symmetries of a nerve conduction equation

The infinitesimal operators of the symmetry group of a special nonlinear nerve conduction equation are determined.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

Equilibria with nonstandard prices in vector lattice overlapping generations economies

The paper investigates overlapping generations (OLG) economies in vector lattices framework. Agents' preferences are assumed uniformly proper, though they may be nontransitive and incomplete. Existence is stated for the “equilibrium with nonstandard prices,” a notion that may be looked upon as a particular (or generalized, in other aspect) case of known “compensated equilibria” of OLG-economies. The difference is that compensated values are described via explicit formula given in nonstandard analysis terms. This approach enables more clear economic interpretation and shows some new properties of compensated values, such as their linearity over agents' endowments. Also it allows easy to prove the existence of equilibria under classical additional assumptions on agents' endowments.