Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

L p theory for semilinear nonlocal problems with measure of noncompactness in separable Banach spaces

We study the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in a separable Banach space X. We derive conditions in terms of the Hausdorff measure of noncompactness under which mild solutions exist in L^p(0, b; X). For illustration, a partial integral differential system is worked out. Dedicated to Felix Browder on his 80th birthday     

Boundary Harnack Principle for <Emphasis Type="Italic">p</Emphasis>-harmonic Functions in Smooth Euclidean Domains

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C ^1,1-domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. H. A. was partially supported by Grant-in-Aid for (B) (2) (No. 15340046) Japan Society for the Promotion of Science. N. S. was partially supported by NSF grant DMS-0355027. X. Z. was partially supported by the Taft foundation.     

The optimal convergence of the h–p version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains

In the framework of the Jacobi-weighted Besov spaces, we analyze the lower and upper bounds of errors in the h–p version of boundary element solutions on quasiuniform meshes for elliptic problems on polygons. Both lower bound and upper bound are optimal in h and p, and they are of the same order. The optimal convergence of the h–p version of boundary element method with quasiuniform meshes is proved, which includes the optimal rates for h version with quasiuniform meshes and the p version with quasiuniform degrees as two special cases. Dedicated to Professor Charles Micchelli on the occasion of his sixtieth birthday Mathematics subject classification (2000) 65N38. Benqi Guo: The work of this author was supported by NSERC of Canada under Grant OGP0046726 and was complete during visiting Newton Institute for Mathematical Sciences, Cambridge University for participating in special program “Computational Challenges in PDEs” in 2003. Norbert Heuer: This author is supported by Fondecyt project No. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis. Current address: Mathematical Sciences, Brunel University, Uxbridge, U.K.     

Existence and uniqueness results for a nonlinear stationary system

We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves L⊃ coupling terms we use the techniques of renormalized solutions for elliptic equations with L⊃ data. We also prove partial uniqueness results.AMS Subject Classification: 32D05, 74D99.     

The Cauchy Problem for Second-Order Elliptic Systems on the Plane

Regular solutions to second-order elliptic systems on the plane are representable in terms of A-analytic functions satisfying an operator equation of the Beltrami type. We prove Carleman-type formulas for reconstruction of solutions from data on a part of the boundary of the domain. We use these formulas for solving the Cauchy problems for the system of Lame equations, the Navier–Stokes system, and the system of equations of elasticity with resilience.     

Non-commutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.     

Multiplicity and vanishing lemmas for differential and <Emphasis Type="Italic">q</Emphasis>-difference equations in the Siegel–Shidlovsky theory

We present a general multiplicity estimate for linear forms in solutions of various types of functional equations, which extends the zero estimates used in some recent works on the Siegel–Shidlovsky theorem and its q-analogues. We also present a dual version of this estimate, as well as a new interpretation of Siegel’s theorem itself in terms of periods of Deligne’s irregular Hodge theory.     

Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

ComplexK-theory ofBSL3(Z)

ComplexK-theory ofBSL₃(Z) andBSt₃(Z) are computed. We first study a Brown-Peterson (BP) version of Soulé's arguments. Then we consider complexK-theory by using a Conner-Floyd type theorem.     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

A Hybrid Heuristic for the p-Median Problem

Given n customers and a set F of m potential facilities, the p-median problem consists in finding a subset of F with p facilities such that the cost of serving all customers is minimized. This is a well-known NP-complete problem with important applications in location science and classification (clustering). We present a multistart hybrid heuristic that combines elements of several traditional metaheuristics to find near-optimal solutions to this problem. Empirical results on instances from the literature attest the robustness of the algorithm, which performs at least as well as other methods, and often better in terms of both running time and solution quality. In all cases the solutions obtained by our method were within 0.1% of the best known upper bounds.     

Population monotonic solutions on convex games

The Dutta-Ray solution and the Shapley value are two well-known examples of population-monotonic solutions on the domain of convex games. We provide a new formula for the Dutta-Ray solution from which population-monotonicity immediately follows. Then we define a new family of population-monotonic solutions, which we refer to as “sequential Dutta-Ray solutions.” We also show that it is possible to construct several symmetric and population-monotonic solutions by using the solutions in this family. Received September 1998/Revised version: December 1999     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

Strong converse inequalities for Baskakov operators

We give a strong converse inequality of type B in terms of unified K-functional K_λ^α( f,t²)(0λ1, 0<α<2) for Baskakov operators.     

W 2,p -a priori estimates for the emergent Poincaré Problem

We derive W ^2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.      

The hp-Version of the Boundary Element Method in ℝ3 The Basic Approximation Results

This paper deals with the basic approximation properties of the h–p version of the boundary element method (BEM) in ℝ³. We extend the results on the exponential convergence of the h–p version of the boundary element method on geometric meshes from problems in polygonal domains to problems in polyhedral domains. In 2D elliptic boundary value problems the solutions have only corner singularities whereas in 3D problems they contain additional edge and corner-edge singularities. The solutions of the corresponding boundary integral equations inherit those singularities. The detailed investigations in our analysis take care of the various types of those singularities. While edge singularities can be analysed using standard one-dimensional approximation results the corner-edge singularities demand a new analysis. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.