We introduce the class of rational Kripke models and study symbolic model checking of the basic tense logic Kt and some extensions of it in models from that class. Rational Kripke models are based on (generally infinite) rational graphs, with vertices labeled by the words in some regular language and transitions recognized by asynchronous two-head finite automata, also known as rational transdu...

Let a, b, c, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in P defined by ax + by + cz + dw = 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic

On a rationally connected variety we would like to use rational curves to obtain a similar result. Complete intersection curves are essentially never rational. (For instance, if X ⊂ P is a hypersurface then a general complete intersection with hyperplanes is rational iff X is a hyperplane or a quadric.) Therefore we have to proceed in a quite different way. Let X be a smooth, projective, ration...

We consider indeterminate rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna type parameterization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove that the growth at the isolated singu...

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them is one less than a perfect square. Recently Gibbs constructed several examples of rational Diophantine sextuples with positive elements. In this note, we construct examples of rational Diophantine sextuples with mixed signs. Indeed, we show that such examples exist for all possible combinatio...

Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of padic rational integer, Qp denotes the ring of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N {0}. Let Cpn {ζ | ζpn 1} be the cyclic g...

We find the primitive integer solutions to x + y = z. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and...

We denote the real numbers by R and the positive real numbers by R+. Similarly we denote the rational numbers by Q and the positive rational numbers by Q+. We denote the natural numbers (including 0) by N. For all a and b in R with a < b, we denote the open interval between a and b by (a, b) and the closed interval by [a, b]. If S is a set, S∗ denotes the set of all finite sequences consisting ...

The behavior of a linear time-invariant differential system is defined as the set of solutions of a system of linear constantcoefficient differential equations. However, these behaviors can be represented in many other ways, for example, as the set of solutions of a system of equations involving a differential operator in a matrix of rational functions, rather than in a matrix of polynomials. T...

Manin’s Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s Conjecture is a thin set.