Some important examples and explications of Grothendieck duality which should have been included in Chapter 5 of [C1] were overlooked. Several such topics are treated here. One question which can be asked about the trace map H(X,ΩX/k) → k for a proper smooth scheme X of pure dimension n over a field k is how it relates to the trace map for a smooth divisor. Generalizing a little, one can ask wh...

Esik and Maletti introduced the notion of a proper semiring and proved that some important (classes of) semirings – Noetherian semirings, natural numbers – are proper. Properness matters as the equivalence problem for weighted automata over a semiring which is proper and finitely and effectively presented is decidable. Milius generalised the notion of properness from a semiring to a functor. As...

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices (u,v), the contains a directed path (a walk) from u to v such that arcs adjacent on (on have distinct colors. The proper connection number pc→(D) (the proper-walk wc→(D)) minimum colours make (properly-walk connected). We prove pc→(Cn(S))≤2 every circulant Cn(S) with S⊆{1,…,n−1},|S|≥2 and 1∈S. Fu...

Uncertainty in spatial data attributes can produce unreliable spatial patterns in choropleth maps, but only a few studies have considered uncertainty in map classification processes. Unfortunately, a less desirable classification result often is generated by existing methods. For example, most observations are assigned to a single class while the remaining classes have a very small number of ob...

Let ω(·) denote the union of all ω-limit sets of a given map. As the main result of this paper we prove that, for given continuous interval maps f1, . . . , fm, the union of all ω-limit sets of the product map f1 × · · · × fm and the cartesian product of the sets ω(f1), . . . , ω(fm) coincide. This result enriches the theory of multidimensional permutation product maps, i.e., maps of the form F...

Due to the development in the field of network technology and multimedia applications, every minute thousands of messages which can be text, images, audios, videos are created and transmitted over wireless network. Improper delivery of the message may leads to the leakage of important information. So encryption is used to provide security. In last few years, variety of image encryption algorith...

In this paper we introduce a new method for image segmentation. It is based on a height map generated from the input image. The height map characterizes the image content in such a way that the application of the watershed concept provides a proper segmentation of the image. The height map enables the watershed method to provide better segmentation results on difficult images, e.g., images of n...

We show that if a locally compact group G acts properly on a locally compact σ-compact space X, then there is a family of G-invariant proper continuous finite-valued pseudometrics which induces the topology of X. If X is, furthermore, metrizable, then G acts properly on X if and only if there exists a G-invariant proper compatible metric on X.