Transformation Semigroup

The explosive nature of cholera epidemic over the years in different parts of the world has been a subject of interest to scientists in proffering interventions towards controlling its spread. Over the years many models has been created by the following people Capaso and Pavari-Fontana (1973), Codeco (2001), Hartley, Tien (2009), Mukandivare (2009) etc. In the present study, we modify the Cholera model proposed by Mukandivare incorporating three (3) containment options such as vaccination, Therapeutic treatment and water treatment and solved the system analytically using Homotopy Perturbation Method. The results shows that with improved use of vaccination, therapy and proper sanitation we have a more healthy population. This research is therefore recommended to modelers who desire to know how homotopy perturbation methods works. The computations were done and further analyzed mathematically using a computer symbolic package MAPLE 13.

In this paper, we are interested in parallels to the classical notions of special subsets in R defined in the generalized Cantor and Baire spaces (2 κ and κ κ). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ-sets, γ-sets, sets with Menger, Rothberger or Hurewicz property, but also of some less-know classes like X-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller and Laver-null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems.

In this paper we study the class of s-topological groups and a wider class of S-topological groups which are defined by using semi-open sets and semi-continuity introduced by N. Levine. It is shown that these groups form a generalization of topological groups, and that they are different from several distinct notions of semitopological groups which appear in the literature. Counterexamples are given to strengthen these concepts. Some basic results and applications of sand S-topological groups are presented. Similarities with and differences from topological groups are investigated.

Food systems lie squarely at the intersection of several overarching goals of the UN and member states, as embodied in the Sustainable Development Goals, including eliminating poverty, hunger, and malnutrition in all its forms, achieving good health and well-being, while promoting environmental sustainability. The need for radical transformation of current food systems is inescapable if the world is to achieve one, let alone all, of these goals. Meeting this challenge will inevitably be disruptive to current food systems, carry costs, and be politically onerous. But the projected benefits far outweigh these difficulties. This commentary spells out the complexity of issues that need to be tackled to design and implement food systems that improve diets, nutrition, and health in an equitable fashion, while simultaneously respecting planetary boundaries. Six critical domains are identified that must be addressed for the successful transformation of food systems: 1) reinvent agriculture,...

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

In this paper we introduce the definition of (k, l)-universal transversal property, which is a refinement of the definition of k-universal transversal property, which in turn is a refinement of the classic definition of k-homogeneity for permutation groups. In particular, a group possesses the (2, n)-universal transversal property if and only if it is primitive; it possesses the (2, 2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k, l)universal transversal property, for k ≥ 3. Then we apply this result for studying regular semigroups of partial transformations.

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklóssy and the first author, we show that this statement fails for countably compact regular spaces, and even for ω-bounded regular spaces. In fact, there are κ-bounded counterexamples for every infinite cardinal κ. The proof makes essential use of the so-called strong colorings that were invented by the second author.

introduced certain types of separation axioms which are Z-regular space, ZT3 space, semi-Z-regular space and semi-ZT3 space, by using boundary points and semi boundary points. The relationships among them and relationships with each of T3, T1 and semi-T1 axioms have been given.

For a set X, an equivalence relation Ω on X, and a cross-section R of the partition X/Ω, consider the following subsemigroup of the semigroup T (X) of full transformations on X: T (X, Ω, R) = {a 2 T (X) : Ra µ R and (x, y) 2 Ω) (xa, ya) 2 Ω}. The semigroup T (X, Ω, R) is the centralizer of the idempotent transformation with kernel Ω and image R. We prove that the automorphisms of T (X, Ω, R) are the inner automorphisms induced by the units of T (X, Ω, R) and that the automorphism group of T (X, Ω, R) is isomorphic to the group of units of T (X, Ω, R).

A deterministic mathematical model of cholera infection incorporating health education campaign, vaccination of susceptible humans, treatment of infected human and water sanitation is developed. It is shown that the solution of the model uniquely exist, it is positive and bounded in a certain region. The disease-free equilibrium (DFE) state of the model was determined and used to compute the basic reproduction number  as a threshold for effective disease management. The result from stability analysis for the disease-free equilibrium state (DFEs) shows that it is locally as well as globally asymptotically stable whenever the basic reproduction number  is less than unity (). The results obtained from the sensitivity index of   show that the control parameters of public health education campaign, vaccination of susceptible individuals, treatment of infected humans and water sanitation are crucial parameters to cholera management. Numerical simulations show that, expanded and improved v...

introduced certain types of separation axioms which are Z-regular space, ZT3 space, semi-Z-regular space and semi-ZT3 space, by using boundary points and semi boundary points. The relationships among them and relationships with each of T3, T1 and semi-T1 axioms have been given.

Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a, b} of vertices such that a = b and ab = ba. Denote by T (X) the semigroup of full transformations on a finite set X. Let J be any ideal of T (X) such that J is different from the ideal of constant transformations on X. We prove that if |X| ≥ 4, then, with a few exceptions, the diameter of G(J) is 5. On the other hand, we prove that for every positive integer n, there exists a semigroup S such that the diameter of G(S) is n. We also study the left paths in G(S), that is, paths a1 − a2 − • • • − am such that a1 = am and a1ai = amai for all i ∈ {1,. .. , m}. We prove that for every positive integer n ≥ 2, except n = 3, there exists a semigroup whose shortest left path has length n. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.

This work aims at understanding the structure of process algebras via holonomy decomposition. In that connection, the work studies the skeleton of the transformation semigroup which is obtained from the natural transition relation between the processes of process ...

Let G be a group of cardinality κ > ℵ 0 endowed with a topology T such that |U | = κ for every non-empty U ∈ T and T has a base of cardinality κ. We prove that G could be factorized G = AB (i.e. each g ∈ G has unique representation g = ab, a ∈ A, b ∈ B) into dense subsets A, B, |A| = |B| = κ. We do not know if this statement holds for κ = ℵ 0 even if G is a topological group.

Topologies τ 1 and τ 2 on a set X are called T 1-complementary if τ 1 ∩ τ 2 = {X \ F : F ⊆ X is finite} ∪ {∅} and τ 1 ∪ τ 2 is a subbase for the discrete topology on X. Topological spaces (X, τ X) and (Y, τ Y) are called T 1-complementary provided that there exists a bijection f : X → Y such that τ X and {f −1 (U) : U ∈ τ Y } are T 1-complementary topologies on X. We provide an example of a compact Hausdorff space of size 2 c which is T 1-complementary to itself (c denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff space of size c that is T 1-complementary to itself is both consistent with and independent of ZFC. On the other hand, we construct in ZFC a countably compact Tikhonov space of size c which is T 1-complementary to itself and a compact Hausdorff space of size c which is T 1-complementary to a countably compact Tikhonov space. The last two examples have the smallest possible size: It is consistent with ZFC that c is the smallest cardinality of an infinite set admitting two Hausdorff T 1-complementary topologies [8]. Our results provide complete solutions to Problems 160 and 161 (both posed by S. Watson [14]) from Open Problems in Topology (North-Holland, 1990).

Let X be a finite set such that |X| = n, and let k < n/2. A group is k-homogeneous if it has only one orbit on the sets of size k. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate k-homogeneous groups (for k ≥ 1); in particular we show that 2-homogeneous groups are 2-generated. We also describe the orbits of k-homogenous groups on partitions with n − k parts, classify the 3-homogeneous groups G whose orbits on (n − 3)-partitions are invariant under the normalizer of G in S n , and describe the normalizers of 2-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

We investigate semigroup topologies on the full transformation monoid T(X) of an infinite set X. We show that the standard pointwise topology is the weakest Hausdorff semigroup topology on T(X), show that the pointwise topology is the unique Hausdorff semigroup topology on T(X) that induces the pointwise topology on the group of all permutations of X, and construct |X| distinct Hausdorff semigroup topologies on T(X). In the case where X is countable, we prove that the pointwise topology is the only Polish semigroup topology on T(X). We also show that every separable semigroup topology on T(X) is perfect, describe the compact sets in an arbitrary Hausdorff semigroup topology on T(X), and show that there are no locally compact perfect Hausdorff semigroup topologies on T(X) when |X| has uncountable cofinality.

This work aims at understanding the structure of process algebras via holonomy decomposition. In that connection, the work studies the skeleton of the transformation semigroup which is obtained from the natural transition relation between the processes of process algebra, and observes that it is of height one.

The explosive nature of cholera epidemic over the years in different parts of the world has been a subject of interest to scientists in proffering interventions towards controlling its spread. Over the years many models has been created by the following people Capaso and Pavari – Fontana (1973), Codeco (2001), Hartley, Tien (2009), Mukandivare (2009) etc. In the present study, we modify the Cholera model proposed by Mukandivare incorporating three (3) containment options such as vaccination, Therapeutic treatment and water treatment and solved the system analytically using Homotopy Perturbation Method. The results shows that with improved use of vaccination, therapy and proper sanitation we have a more healthy population. This research is therefore recommended to modelers who desire to know how homotopy perturbation methods works. The computations were done and further analyzed mathematically using a computer symbolic package MAPLE 13.

Homotopy analysis method (HAM) is a well-known device for solving a wide range of functional equations such as partial and ordinary differential equations, integral equations and so on. In this article, we adopt homotopy analysis method to obtain the solutions of a system of first order nonlinear ordinary differential equations arising from transmission dynamics of cholera disease. The method provides a direct scheme for solving the problem without linearization, perturbation, massive computation and transformation. Numerical simulations are carried out and graphical results are presented to illustrate the solutions. The results obtained confirm the power and accuracy of homotopy analysis method in solving differential equations.

In this paper, we propose a general SIR mathematical model for infectious diseases dynamics for a given constant population. These mathematical models are described by nonlinear first order differential equations. We first find the analytical solution which is obtained using Homotopy Perturbation Method (HPM). The results of the numerical simulation show that the disease has the tendency of dying out over a period of time. (Keywords: SIR infectious disease model, epidemic model, homotopy perturbation method, HPM)

The explosive nature of cholera epidemic over the years in different parts of the world has been a subject of interest to scientists in proffering interventions towards controlling its spread. Over the years many models has been created by the following people Capaso and Pavari – Fontana (1973), Codeco (2001), Hartley, Tien (2009), Mukandivare (2009) etc. In the present study, we modify the Cholera model proposed by Mukandivare incorporating three (3) containment options such as vaccination, Therapeutic treatment and water treatment and solved the system analytically using Homotopy Perturbation Method. The results shows that with improved use of vaccination, therapy and proper sanitation we have a more healthy population. This research is therefore recommended to modelers who desire to know how homotopy perturbation methods works. The computations were done and further analyzed mathematically using a computer symbolic package MAPLE 13.