Let D be a non empty set and let E be a complex-membered set. One can verify that every element of D→̇E is complex-valued. Let D be a non empty set, let E be a complex-membered set, and let F1, F2 be elements of D→̇E. Then F1 + F2 is an element of D→̇C. Then F1 − F2 is an element of D→̇C. Then F1 · F2 is an element of D→̇C. Then F1/F2 is an element of D→̇C. Let D be a non empty set, let E be a comple...

In Dellunde et al. (J. Symbolic Logic 67(3) (2002) 997–1015), we determined the complete theory Te of modules of separably closed %elds of characteristic p and imperfection degree e, e∈! ∪ {∞}. Here, for 0 = e∈!, we describe the closed set of the Ziegler spectrum corresponding to Te. Further, we establish a correspondence between certain submodules and n-types and we investigate several notions...

We study the problem of listing all closed sets of a closure operator σ that is a partial function on the power set of some finite ground set E, i.e., σ : F → F with F ⊆ P(E). A very simple divide-and-conquer algorithm is analyzed that correctly solves this problem if and only if the domain of the closure operator is a strongly accessible set system. Strong accessibility is a strict relaxation ...

7a-set and let G = R E} be the Hahn group" with real coefficients and "exponents" in E: i.e., let G be the set of all mappings g of E into R whose support, s(g), is either empty or an anti-well-ordered subset of E, where the addition is point-wise addition and the order is lexicographic. Let Ga = 9g e G: Is(g) <Ka}. Let Eo C E and let H = Ig EG:g(e) e Z if e EEo}. Finally, let Ha = H n Ga. Then...

Let E ⊂ R be a closed set of Hausdorff dimension α. We prove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions. Mathematics Subject Classification: 28A78, 42A32, 42A38, 42A45, 11B25.

A subset E of the d-dimensional torus Td is called a set of uniqueness, or U -set, if every multiple trigonometric series spherically converging to 0 outside E vanishes identically. We show that all countable sets are U -sets and also that HJ sets are U -sets for every J . In particular, C × Td−1, where C is the Cantor set, is an H1 set and hence a U -set. We will say that E is a UA-set if ever...

Closed classes of three-valued logic generated by periodic symmetric funtions that equal 1 in tuples from {1, 2} and equal 0 on the rest tuples are considered. Criteria for bases existence and finite bases existence for these classes is obtained. The problem of the bases existence for some families of closed classes of the three-valued logic functions is considered in the paper. E. Post [1] (se...

A graph G=(V; E) is a tolerance graph if there is a set I={Iv | v∈V} of closed real intervals and a set ={ v | v∈V} of positive real numbers such that (x; y)∈E ⇔ |Ix∩ Iy|¿min{ x; y}. We show that every tolerance graph has a polynomial size integer representation. It follows that tolerance graph recognition is in NP. c © 2004 Elsevier B.V. All rights reserved.

Game Graphs The model is based on a graph G = (S,E), consisting of a set S of states and a set E of directed edges between those states, the state transitions. We assume that every state has at least one outgoing edge. Figure 1(a) contains an example of such a graph representing a closed system where all indeterminism comes from the system itself (rather than from interaction with an outside en...

The main purpose of this paper is to study normed hypervector spaces. We generalize some definitions such as basis, convexity, operator norm, closed set, Cauchy sequences, and continuity in such spaces and prove some theorems about them.