http://ir.library.sust.edu:8080/xmlui/handle/123456789/63 2026-05-08T01:15:27Z 2026-05-08T01:15:27Z Uddin, Mohammad Salah http://ir.library.sust.edu:8080/xmlui/handle/sust/247 2025-08-19T05:35:29Z 2021-11-01T00:00:00Z

A Study of Recognition and Enumeration for Decomposable Ordered Sets Uddin, Mohammad Salah In this research, we consider mainly two repeatedly studied combinatorial problems in the theory of posets (partially ordered sets). The first one, known as the Recognition Problem, is to recognize those classes of posets that satisfy some common structural properties. Here, we give the recognitions of the most computationally tractable classes of posets, namely the decomposable posets. Well-known classes of the decomposable posets are the classes of P -graphs, P - series, and series-parallel posets. Also, we introduce the notions of the classes of factorable posets and composite posets. Due to many computational aspects of the incidence matrices, they have classical applications in recognizing various classes of posets and graphs. For this, we introduce the notion of poset matrix, an incidence matrix, to represent finite posets. Here, we define the order relation in a square (0,1)-matrix and give an association of this matrix to the posets. We show that every poset matrix can be relabeled to an upper (equivalently, lower) triangular matrix that represents a unique poset up to isomorphism. We introduce the notions of the ordinal sum, ordinal product, and composition of matrices. We establish the algebraic interpretations of the direct sum, ordinal sum, Kronecker product, ordinal product, and composition of matrices in the case of poset matrices. We give the matrix recognitions of the classes of P - graphs, P -series, series-parallel posets, factorable posets, and composite posets by using the poset matrix. Finally, we give a matrix recognition of the class of all decomposable posets that generalizes most of the above results regarding the matrix recognitions of posets. viiThe second problem, known as the Enumeration Problem, is to count the number of pairwise nonisomorphic posets with a certain number of elements be- longing to a particular class of posets. Among several methods for the enumera- tion of posets, here, we consider the exact enumeration method. The algorithmic methods for the enumerations of some classes of decomposable posets considered in the previous researches are of type generate-one and count-one. As a result, the running time of these algorithms grows more rapidly even though the posets are significantly small in size. It is mainly due to the recursions in generating pairwise nonisomorphic posets that make these algorithms highly time-complex. Therefore, it was always a great challenge to give polynomial-time algorithms to make some enumeration process time-efficient. Since the generating methods for the decomposable posets seem to consist of the recursive constructions, algorith- mic methods for the enumerations of such posets were ignored by some authors. Here, we give an exact enumeration method for the unlabeled P -series and series- parallel posets by using the poset matrix. In both cases, we give the enumeration of the unlabeled disconnected posets according to the number of connected direct terms of the posets. In the case of the unlabeled connected series-parallel posets, we give the enumeration of the posets according to the number of ordinal terms that are either the singleton or disconnected posets. For these, we use the results regarding the matrix recognitions of the classes of P -series and series-parallel posets. We also give some algorithms to determine the parameters involved in the enumeration formulae, and finally, compute the number of unlabeled posets. We show that these enumeration algorithms have polynomial-time complexities. Moreover, we implement these enumeration algorithms into the computer and obtain the numbers of unlabeled P -series up to 75 elements and the numbers of unlabeled series-parallel posets up to 33 elements. A thesis for the degree of Doctor of Philosophy in Mathematics : "A Study of Recognition and Enumeration for Decomposable Ordered Sets".

2021-11-01T00:00:00Z Ahamad, Razwan http://ir.library.sust.edu:8080/xmlui/handle/sust/237 2025-07-30T05:33:12Z 2017-11-01T00:00:00Z

Development of appropriate structured mathematical models for fish population dynamics along with efficient numerical schemes for the solutions of the developed and other models of engineering problems Ahamad, Razwan This thesis is mainly concentrated to obtain numerical solutions of real world problems encountered in continuum mechanics and other branches of sciences like development, management and production oriented agricultural sectors. Appropriate mathematical models are well established for all continuum mechanics problems and hence in such instances a faster algorithm or a technique is the only requirement for obtaining numerical solutions. On the other hand, though the dynamic model approach is a widely applied technique to a number of environmental management and sustainability issues like fisheries management problems still needed to be developed. Therefore, primarily the thesis intends to develop appropriate mathematical models for such real problems and then stresses to obtain their numerical solutions. More specifically, we intended firstly to develop appropriate mathematical model in order to calculate fish population. As an outcome, we finally present the model by a system of hyperbolic partial differential equations with linear and nonlinear boundary conditions for the calculation of fish population. Secondly, an appropriate model for mathematical estimation of fish production performances is developed for the calculation of fish sizes in different time span depending on initial sizes. Then, as an important integral component, computer codes in FORTRAN that employs Finite Volume Method are developed for obtaining numerical solutions of such models. Afterward, other codes in MATLAB are developed for analyzing and graphical presentation of computed data. Substantiation of the outcomes of the developed models is then established by comparing the computed results with the experimental data. The versatility and popularity of Finite Element Method (FEM) is well known. The main and important time consuming step in FEM is the formation of all element matrices. Generally, all the elements in global space are transformed into respective contiguous elements in local space by use of isoparametric/ Abstract   subparametric/ superparametric transformation. For such transformations only all the components of element matrices become integrals of rational functions for the popular quadrilateral as well as for the curved triangular finite elements. The Gaussian quadrature schemes, used in most cases for its simplicity cannot evaluate such rational integrals as it can evaluate the integrals of polynomials of degree/ order (2n-1) with n Gaussian points. For the desired accuracy of the evaluations, more and more Gaussian points are used and eventually that increases the computing time. Therefore, it is an important task to make a proper balance between the accuracy and efficiency of evaluations of numerous rational integrals. So, the thesis concentrates to develop the faster technique by reducing steps of various stages of usual FEM solution procedure for obtaining numerical solutions of numerous boundary value problems governed by hyperbolic, elliptic partial differential equations. For doing so, it stresses to present faster closed form formulae needed to form exactly all types of element matrices for solving such two dimensional boundary value problems encountered in the realm of science and engineering. Since, the faces of finite volume are finite elements so all the formulae are applicable in both FEM, FVM methods. Computer codes compatible with the formulations are also developed accordingly. The efficiency and accuracy of the technique is then demonstrated through application of the formulae in order to obtain the solutions of test problems. Thus, in brief the Thesis includes: (1) appropriate mathematical model for the calculation of fish population, (2) appropriate mathematical model for the calculation of fish population performances (size of fishes), (3) computer codes employing suitable numerical methods (FVM) for obtaining best approximations of solutions of the developed models, and (4) computer codes based on the developed technique for exact computing all the element matrices efficiently in order to solve numerous two dimensional boundary value problems. All the relevant concepts, mathematical tools, devised; modified; improved algorithms, other related topics and the gradual development of the Thesis work are elaborately described in 7 (seven) chapters. A thesis submitted for the degree of Doctor of Philosophy: "Development of appropriate structured mathematical models for fish population dynamics along with efficient numerical schemes for the solutions of the developed and other models of engineering problems".

2017-11-01T00:00:00Z Nag, Chandrani http://ir.library.sust.edu:8080/xmlui/handle/sust/228 2025-07-24T05:11:37Z 2016-03-01T00:00:00Z

A Study on P-algebras and Its Underlying Lattices Nag, Chandrani In this thesis we study p-algebras and its underlying lattices. An algebra L = ⟨L, ∧, ∨, ∗ , 0, 1⟩ is called a p-algebra if (i) ⟨L, ∧, ∨, 0, 1⟩ is a bounded lattice, and (ii) for all x, y ∈ L, x ∧ x ∗ = 0 and x ∧ y = 0 implies y 6 x ∗ . The lattice ⟨L, ∧, ∨⟩ is called the underlying lattice of L. There are many research works on p-algebras where the underlying lattices are distributive. In this thesis we study p-algebras in general. The underlying lattice of a p-algebra is 0-distributive but not every un- derlying lattice of a p-algebra is 0-modular or 1-distributive. We study some classes of ideals of 0-distributive lattice for study of p-algebras. We charac- terize some subclasses of p-algebras. We introduce a notion of DM-algebras which is a nice subclass of p-algebras. P-ideals and p-filters of a p-algebra play an important role to study p- algebras. I ∗ (L), the set of all p-ideals of a p-algebra L, itself form a com- plete distributive p-algebra which is isomorphic to the complete distributive p-algebra formed by the set of all p-filters of L. We study congruences, particularly, kernel ideals of a p-algebra. We show that an ideal of a p-algebra is a kernel ideal if and only if it is a p-ideal. We also characterize co-kernel filters of a p-algebra. Finally, we show that if f : L → M is an epimorphism of p-algebras, then there is an epimorphism f ∗ : I ∗ (L) → I ∗ (M) if and only if ker f is a principal ideal. We also show that for any p-algebra L we have I ∗ (L) ∼ = I(S(L)). A thesis for the degree of Doctor of Philosophy in Mathematics "A Study on P-algebras and Its Underlying Lattices".

2016-03-01T00:00:00Z