Things That are Right with the Traditional Square of Opposition

The truth conditions that Aristotle attributes to the propositions making up the traditional square of opposition have as a consequence that a particular affirmative proposition such as ‘Some A is not B’ is true if there are no Bs. Although a different convention than the modern one, this assumption remained part of centuries of work in logic that was coherent and logically fruitful.      

The Oblique Derivative Problem for the Laplace Equation in a Plain Domain

The oblique derivative problem for the Laplace equation is studied in a planar multiply connected domain. The boundary condition has a form where is the unit normal vector, is the unit tangential vector and is a fixed real number. If is a Hölderian function and the corresponding domain has Ljapunov boundary then the classical problem is studied. If on the boundary and the domain has a locally Lipschitz boundary then a solution, which fulfils the boundary condition in the sense of a nontangential limit, is studied. If is a real measure on the boundary and the domain has bounded cyclic variation then a solution in a sense of distributions is studied. The solution is looked for in a form of a linear combination of a single layer potential and an angular potential.     

Finite dimensional convex structures II: the invariants

Various relations between the dimension and the classical invariants of a topological convex structure have been obtained, leading to an equivalence between Helly's and Carathéodory's theorem, and to the closedness of the hull of compact sets in finite-dimensional convexities. It is also shown that the Radon number of an n-dimensional binary convexity is in most cases equal to the Radon number of the n-cube, and a natural condition is presented under which the invariants are equal to dimension plus one.     

The word problem for involutive residuated lattices and related structures

It will be shown that the word problem is undecidable for involutive residuated lattices, for finite involutive residuated lattices and certain related structures like residuated lattices. The proof relies on the fact that the monoid reduct of a group can be embedded as a monoid into a distributive involutive residuated lattice. Thus, results about groups by P. S. Novikov and W. W. Boone and about finite groups by A. M. Slobodskoi can be used. Furthermore, for any non-trivial lattice variety , the word problem is undecidable for those involutive residuated lattices and finite involutive residuated lattices whose lattice reducts belong to . In particular, the word problem is undecidable for modular and distributive involutive residuated lattices. The author would like to thank the Deutsche Telekom Stiftung for financial support. Received: 10 November 2005     

Resonance expansions of propagators in the presence of potential barriers

We show that the Schrödinger propagator can be expanded in terms of resonances at energy levels at which a barrier separates the interaction region from infinity. The expansions hold for all times with errors small in the semi-classical parameter. As a byproduct we obtain a result on the approximation of clusters of resonant states by clusters of eigenfunctions of a self-adjoint reference operator.     

Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations

We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ²ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Mock tridiagonal systems

We introduce the notion of a mock tridiagonal system. This is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain nonvanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism.     

Decay for nonlinear Klein-Gordon equations

We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid. In these spaces of fractional order we obtain a weighted Sobolev embedding and a nonlinear estimate. Using these, we establish the decay estimate of the solution for large time provided the power of nonlinearity is greater than a critical value.     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003     

Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

Natural bounded concentrators

We give the first known direct construction for linear families of bounded concentrators. The construction is explicit and the results are simple natural bounded concentrators. Let be the field withq elements,g(x)∈F _q [x] of degree greater than or equal to 2, and . LetI _nputs=H/A,O _utputs=H/B, and draw an edge betweenaA andbB iffaA∩bB≠ϕ. We prove that for everyq≥5 this graph is an concentrator. Part of this research was done while the author was at the department of Computer Science, The University of British Columbia, Vancouver, B.C., Canada.     

Matching binary convexities

We showed in an earlier paper that the Radon number of an n-dimensional binary convexity equals the Radon number of the n-cube, except for a well-determined sequence of dimensions, in which case the Radon number may be one unit larger. Examples of the latter are obtained in every predicted dimension. The basic tool is a matching procedure which works for binary convexities.     

Hidden lemmas in the early history of infinite series

Euler, Fourier, Poisson and Cauchy appear to have used, in a more or less implicit form, some facts on infinitely small quantities. Attempting to state and prove several lemmata, I shall discuss their relationships to interchanges of limits in series and integrals. Early methods of summation for divergent series and integrals, including a conjecture of Poisson, are discussed.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.