The left-definite Legendre type boundary problem

The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionM _k [–] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation

The GHS inequality and the Riemann hypothesis

LetV(t) be the even function on (–, ) which is related to the Riemann xi-function by (x/2)=4 _– exp(ixt–V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.Communicated by Richard Varga.     

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form

Super spline spaces of smoothnessr and degreed≥3r+2

The problem of computing the dimension of spaces of splines whose elements are piecewise polynomials of degreed withr continuous derivatives globally has attracted a great deal of attention recently. We contribute to this theory by obtaining dimension formulae for certain spaces of super splines, including the case where varying amounts of additional smoothness is enforced at each vertex. We also explicitly construct minimally supported bases for the spaces. The main tool is the Bernstein-Bézier method.Communicated by Klaus Höllig.     

Normal ordering and quantum groups

The irreducible R-matrices associated with the quantum Liouville and sine-Gordon equations were classified by the su(2) index l, 2l integer. We find that the associated quantum field theories must have the following equal time operator product expansions in the lattice approximation

Oscillations of certain second order nonlinear differential equations

A second order nonlinear differential equation

Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at . Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].Communicated by Carl de Boor     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros

Sharp Remez-, Nikolskii-, and Markov-type inequalities are proved for functions of the form

Phenomenology of nonlocal cellular automata

Dynamical systems with nonlocal connections have potential applications to economic and biological systems. This paper studies the dynamics of nonlocal cellular automata. In particular, all two-state, three-input nonlocal cellular automata are classified according to the dynamical behavior starting from random initial configurations and random wirings, although it is observed that sometimes a rule can have different dynamical behaviors with different wirings. The nonlocal cellular automata rule space is studied using a mean-field parametrization which is ideal for the situation of random wiring. Nonlocal cellular automata can be considered as computers carrying out computation at the level of each component. Their computational abilities are studied from the point of view of whether they contain many basic logical gates. In particular, I ask the question of whether a three-input cellular automaton rule contains the three fundamental logical gates: two-input rules AND and OR, and one-input rule NOT. A particularly interesting edge-of-chaos nonlocal cellular automaton, the rule 184, is studied in detail. It is a system of coupled selectors or multiplexers. It is also part of the Fredkin's gate—a proposed fundamental gate for conservative computations. This rule exhibits irregular fluctuations of density, large coherent structures, and long transient times.