¡v in this paper we consider a strongly regular relation ƒá on hypermodules so that the quotient is amodule (with abelian group) over a fundamental commutative ring. also, we state necessary and sufficientconditions so that the relation ƒá is transitive, and finally we prove that ƒá is transitive on hypermodules.

Kim argues that weak and global supervenience are too weak to guarantee any sort of dependency. Of the three original forms of supervenience, strong, weak, and global, each commonly wielded across all branches of philosophy, two are thus cast aside as uninteresting or useless. His arguments, however, fail to appreciate the strength of weak and global supervenience. I investigate what weak and g...

Kim argues that weak and global supervenience are too weak to guarantee any sort of dependency. Of the three original forms of supervenience, strong, weak, and global, each commonly wielded across all branches of philosophy, two are thus cast aside as uninteresting or useless. His arguments, however, fail to appreciate the strength of weak and global supervenience. I investigate what weak and g...

Template dependencies (TD’s) are a class of data dependencies that include multivalued and join dependencies and embedded versions of these. A collection of techniques, examples and results about TD’s are presented. The principal results are: 1) Finite implication (implication over relations with a finite number of tuples) is distinct from unrestricted implication for TD’s. 2) There are, for TD...

In this paper, we investigate the global behavior of the difference equation of order three xn+1 = α+ xn−2 xn , n = 0, 1, . . . where the parameters α, k ∈ (0,∞) and the initial values x−2, x−1 and x0 are arbitrary positive real numbers.

Proof. (1) These are the well known classical Plücker relations. For completeness’ sake we include a proof. Let P ∈ ΛV and consider the induced linear mapping ♯P : Λ V ∗ → V . Its image, W , is contained in each linear subspace U of V with P ∈ ΛU . Thus W is the minimal subspace with this property. P is decomposable if and only if dimW = s, and this is the case if and only if w ∧ P = 0 for each...

In this paper, existence criteria for single and multiple positive solutions of periodic boundary value problems for first order difference equations of the form