An Optimal Way of Moving a Sequence of Points onto a Curve in Two Dimensions

Let (t), 0 t T, be a smooth curve and let _i , i = 1, 2, , n, be a sequence of points in two dimensions. An algorithm is given that calculates the parameters t_i, i = 1, 2, , n, that minimize the function max{ _i – (t_i) ₂ : i = 1, 2, , n } subject to the constraints 0 t₁ t₂ t_n T. Further, the final value of the objective function is best lexicographically, when the distances _i – (t_i)₂, i = 1, 2, , n, are sorted into decreasing order. The algorithm finds the global solution to this calculation. Usually the magnitude of the total work is only about n when the number of data points is large. The efficiency comes from techniques that use bounds on the final values of the parameters to split the original problem into calculations that have fewer variables. The splitting techniques are analysed, the algorithm is described, and some numerical results are presented and discussed.     

On Sparse Parity Check Matrices

We consider the extremal problem to determine the maximal number of columns of a 0-1 matrix with rows and at most ones in each column such that each columns are linearly independent modulo . For fixed integers and , we shall prove the probabilistic lower bound = ; for a power of , we prove the upper bound which matches the lower bound for infinitely many values of . We give some explicit constructions.     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group onto a space of right cosets of w.r.t. some subgroup of . Examples are constructed showing that the condition of being convex for in is insufficient for this. A necessary and sufficient condition (in terms of a subgroup and a positive cone of ) is specified under which an order of can be induced onto . Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups for which is partially ordered for every convex subgroup , and properties of the class of groups such that is partially ordered for every partial right order on and every subgroup that is convex under .     

Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

A New Critical Behavior for Nonlinear Wave Equations

We study the inhomogeneous semilinear wave equations on with initial values and ,where is a noncompact, complete manifold. We founda new critical behavior in the following sense. There exists ap* > 0. When 1 < p p*, the above problem hasno global solution for any nonnegative not identicallyzero and for any and ; when the problem has a global solution for some and some and . If , which is equipped with the Euclideanmetric, then . If we show that belongs to the blow upcase. Although homogeneous semilinear wave equations are known to exhibit acritical behavior for a long time, this seems to be the first result oninhomogeneous ones.     

Transversal Twistor Spaces of Foliations

The transversal twistor space of a foliation of an even codimension is the bundle of the complex structures of the fibers of the transversalbundle of . On there exists a foliation by covering spaces of the leaves of , and any Bottconnection of produces an ordered pair of transversal almost complex structures of . The existence of a Bott connection which yields a structure ₁ that is projectable to the space of leaves isequivalent to the fact that is a transversallyprojective foliation. A Bott connection which yields a projectablestructure ₂ exists iff isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, ₁is projectable as well. ₂ is never integrable.The essential integrability condition of ₁ isthe flatness of the transversal projective structure of .     

A Canonical Directly Infinite Ring

Let be the set of nonnegative integers and the ring of integers. Let be the ring of N × N matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of all matrices in with only a finite number of nonzero entries and the subring of generated by the identity matrix. Regular elements are also described. We characterize all ideals of , show that all ideals are finitely generated and that not all ideals of are principal. Some general ring theoretic properties of are also established.     

On the Best Ranges for A p + and RH r +

In this paper we study the relationship between one-sided reverse Holder classes and the classes. We find the best possible range of to which an weight belongs, in terms of the constant. Conversely, we also find the best range of to which a weight belongs, in terms of the constant. Similar problems for , and are solved using factorization.     

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra , one may obtain a space of vector fields on Euclidean space such that and are isomorphic when is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of is a Borel subalgebra . Furthermore, one may adjoin to the vector fields in multiplication operators to obtain an -parameter family of distinct presentations of as spaces of differential operators, where is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of can be presented in this way. All of this is carried out explicitly for arbitrary . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.     

Joint Approximations of Distributions in Banach Spaces

For a given homogeneous elliptic partial differential operator with constant complex coefficients, two Banach spaces and of distributions in , and compact sets and in , we study joint approximations in the norms of the spaces and (the spaces of Whitney jet-distributions) by the solutions of the equation in neighborhoods of the set . We obtain a localization theorem, which, under certain conditions, allows one to reduce the above-cited approximation problem to the corresponding separate problems in each of the spaces.     

A Constructive Proof of the Generalized Gelfand Isomorphism

Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius -homomorphism. For , this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let X be a compact Hausdorff space, the th symmetric power of X, and the algebra of continuous complex-valued functions on X with the sup-norm; then the evaluation map defined by the formula identifies the space with the space of all Frobenius -homomorphisms of the algebra into with the weak topology.     

Extremal Decomposition Problems in the Space of Riemann Surfaces

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface with a prescribed set of distinguished points to the case of the Teichmüller space corresponding to under quasiconformal homeomorphisms f. For the functional of our problem on extremal decomposition of a surface , we consider a function expressing the dependence of the extremal value of on a point . Differentiation formulas for the function are derived. These formulas are different and depend on the genus g of the surface . The case where the function is pluriharmonic is considered. Bibliography: 8 titles.     

Computing Eigenvalues of Lommel-Type Equations by the Sampling Method

Using the semiclassical approximation for where and as , we shall show that theeigenvalues are the zeros of a certain function in a Paley–Wienerspace, which allows us to use the Whittaker–Shannon–Kotelnikovsampling theorem to approximate the eigenvalues. We show how thedistribution of the eigenvalues depends on the asymptotics of thecoefficients and as . After abrief discussion on the truncation error, numerical examples are provided.