Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

On the Wl q-Regularity of Solutions to Systems of Differential Equations in the Case When the Equations Are Constructed from Discontinuous Functions

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C ^l -smooth solution to a system of lth-order nonlinear partial differential equations constructed from C ^l -smooth functions meet the local Hoelder condition with every exponent , 0<<1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W ^l _q,loc-solution, q>1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.     

Decay of global solutions of two nonlinear evolution equations

1.IntroductionTherehavebeenconsiderableliteratuxeonthedecayofsolutionstothebestialvalueproblemsforsomenonlinearevolutionequations[3,4,6,7,161.Undercertainassumptions,LZdecayandLoodecayofsolutionstotheseproblemswereestablished.Thereadersinterestedcanfindsuchworksinourreferences.OurillterestisfocusedonthedecayofsolutionsoftheinitialvalueproblemsfornonlinearBenjamin--OnthBurgers(BOB)l"'19--21]andSchlodinger-Burgers(SB)equationwhereHisHilberttransform,definedbyWewallttoshowthattheLZandLoon…     

On asymptotically best norms for powers of operators

In the study of the successive overrelaxation iterative method for solving large systems of linear equations, a frequently considered problem is the behavior of the norm of powers of the successive overrelaxation matrix,L _b ^m both as a function ofm and of the given norm. Our main result is a rather natural necessary and sufficient condition for the existence of a norm asymptotically best for a non-nilpotent matrixA. As a corollary of our main result, it is shown here that there isno asymptotically best norm for the successive overrelaxation matrixL _b ^m .Research supported in part by the Atomic Energy Commission under Grant AT(11-1)-2075.     

Explicit and Implicit Continuation Algorithms for Strongly Monotone Variational Inequalities with Box Constraints

In this paper we discuss the variational inequality problems VIP(X, F), where F is assumed to be a strongly monotone mapping from ⁿ to ⁿ, and the feasible set X = [l, u] has the form of box constraints. Based on the Chen-Harker-Kanzow smoothing functions, first we present an explicit continuation algorithm (ECA) for solving VIP(X, F). The ECA possesses main features as follows: (a) at each iteration, it yields a new iterative point by solving a system of equations in ^{(n + s)} with a parameter and nonsingular Jacobian matrix, where s = |{j: - < l _j < u _j < +}|, (b) it generates a sequence of iterative points in the interior of the feasible set X. Secondly we give an implicit continuation algorithm (ICA) for solving VIP(X,F), the prime character of the ICA is that it solves only one, rather than a series of, system of nonlinear equations to obtain a solution of VIP(X,F). The two proposed algorithms are shown to possess strongly global convergence. Finally, some preliminary numerical results of the two algorithms are reported.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

Polylogarithmic functional equations: A new category of results developed with the help of computer algebra (MACSYMA)

The interrelation of polylogarithmic functional equations and certain numerical results, known as ladders, is discussed, and leads to a consideration of three new, single-variable functional equations at the second order. Two of these families each contain six leading terms whose interrelationship constitutes a constraint on the integration process, but the third has only a single leading term with no such constraints. It is shown how this functional equation can be integrated to the third order, and the process reduced to an algorithm — actually a sequence of instructions — for incorporation into a computer program for symbolic manipulation. The procedure utilizes results from Kummer's equations to cancel out, in sequence, terms which do not vanish, or do vanish, with the variablez. Arguments are all of the form ±z ^p (1–z) ^q (1+z) ^r , and the process is algebraicized by using a (p,q,r,s) notation (withs=±1) to represent such terms. Application of the procedure leads to an integration to the fourth and fifth orders, the latter exhibiting 55 transcendental terms. The first step for the transition to the sixth order can also be achieved but the subsequent steps are frustrated by the restricted forms that the Kummer equations take at the fifth order — it is not possible to create the needed equations in a form which vanishes withz; this corresponding to the elimination of the (5) constant in the extension of the numerically determined ladders to the sixth and higher orders. The existence of the higher-order ladders strongly suggests functional equations af these orders, but the present process has not yet been successful in finding them. The new equations have, however, produced ladders that were inaccessible from Kummer's equations, and had heretofore been only obtainable numerically, up to the fifth order. The method which was developed should be capable of generalization to other systems of equations characterized by the appearance of arguments with recurrent factors. Some new feature, however, will need to be determined before the barrier to the sixth order can be breached.     

Acceleration Scheme for Parallel Projected Aggregation Methods for Solving Large Linear Systems

The Projected Aggregation methods generate the new point x ^k+1 as the projection of x ^k onto an aggregate hyperplane usually arising from linear combinations of the hyperplanes defined by the blocks. The aim of this paper is to improve the speed of convergence of a particular kind of them by projecting the directions given by the blocks onto the aggregate hyperplane defined in the last iteration. For that purpose we apply the scheme introduced in A new method for solving large sparse systems of linear equations using row projections [11], for a given block projection algorithm, to some new methods here introduced whose main features are related to the fact that the projections do not need to be accurately computed. Adaptative splitting schemes are applied which take into account the structure and conditioning of the matrix. It is proved that these new highly parallel algorithms improve the original convergence rate and present numerical results which show their computational efficiency.     

On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X, W ⁺, W ^–), where W ⁺ and W ^– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W ⁺, W ^–) ). We show that these equations imply the existence of a certain isomorphism between two algebras and associated with (X, W ⁺, W ^–) . When is the Bose-Mesner algebra of some association scheme, and is a duality of . These results had already been obtained in [15] when W ⁺, W ^– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.     

Optimal control problems for distributed parameter systems in Banach spaces

We consider the infinite-dimensional nonlinear programming problem of minimizing a real-valued functionf ₀ (u) defined in a metric spaceV subject to the constraintf(u) Y, wheref(u) is defined inV and takes values in a Banach spaceE and Y is a subset ofE. We derive and use a theorem of Kuhn-Tucker type to obtain Pontryagin's maximum principle for certain semilinear parabolic distributed parameter systems. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations inL ¹ andL spaces.This work was supported in part by the National Science Foundation under Grant DMS-9001793.