Given an o-minimal expansion M of a real closed field R which is not polynomially bounded. Let P∞ denote the definable indefinitely Peano differentiable functions. If we further assume that M admits P∞ cell decomposition, each definable closed set A ⊂ Rn is the zero-set of a P∞ function f : Rn → R. This implies P∞ approximation of definable continuous functions and gluing of P∞ functions define...

It is known that the treewidth of a planar graph with a dominating set of size d is O( √ d) and this fact is used as the basis for several fixed parameter algorithms on planar graphs. An interesting question motivating our study is if similar bounds can be obtained for larger minor closed graph families. We say that a graph family F has the domination-treewidth property if there is some functio...

We say that a function f (with values in either R or C) is supported on a set A if it is zero on the complement of this set. The support of f , which we denote by supp(g), is the minimal closed set on which f is supported, equivalently, it is the closure of the set on which f is non-zero. We note that if f ∈ L1(R) is real-valued then the support of f̂ is symmetric around zero (since the real par...

The paper addresses the question of when the least-xed-point operator, in a cartesian closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a suucient condition on a cartesian closed full subcategory of the category of algebraic cpos for the characterisation to hold. The condition is quite mild, and...

A subset A of a space X is called regular open if A = IntA, and regular closed if X\A is regular open, or equivalently, if A= IntA. A is called semiopen [16] (resp., preopen [17], semi-preopen [3], b-open [4]) ifA⊂ IntA (resp.,A⊂ IntA,A⊂ IntA ,A⊂ IntA∪ IntA). The concept of a preopen set was introduced in [6] where the term locally dense was used and the concept of a semi-preopen set was introd...

Because antimatroid closure spaces satisfy the anti-exchange axiom, it is easy to show that they are uniquely generated. That is, the minimal set of elements determining a closed set is unique. A prime example is a discrete geometry in Euclidean space where closed sets are uniquely generated by their extreme points. But, many of the geometries arising in computer science, e.g. the world wide we...

Let R be a real closed field and n ≥ 2. We prove that: (1) for every finite subset F of R, the semialgebraic set R \ F is a polynomial image of R; and (2) for any independent linear forms l1, . . . , lr of R , the semialgebraic set {l1 > 0, . . . , lr > 0} ⊂ R is a polynomial image of R.

We study the minimization problem f (x) → min, x ∈ C, where f belongs to a complete metric space of convex functions and the set C is a countable intersection of a decreasing sequence of closed convex sets Ci in a reflexive Banach space. Let be the set of all f ∈ for which the solutions of the minimization problem over the set Ci converge strongly as i→∞ to the solution over the set C. In our r...

Let M be a compact Riemannian manifold, and let ∆ = ∆M denote the Laplace– Beltrami operator of M acting on L2(M). The eigenvalue spectrum E (M) consists of the eigenvalues of ∆ listed with their multiplicities. Two manifolds M1 and M2 are said to be isospectral if E (M1) = E (M2). Geometric and topological constraints are forced on isospectral manifolds; for example if the manifolds are hyperb...