A study of barred preferential arrangements with applications to numerical approximation in electric circuits
- Authors: Nkonkobe, Sithembele
- Date: 2015
- Subjects: Electric circuits , Numerical calculations , Sequences (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5433 , http://hdl.handle.net/10962/d1020394
- Description: In 1854 Cayley proposed an interesting sequence 1,1,3,13,75,541,... in connection with analytical forms called trees. Since then there has been various combinatorial interpretations of the sequence. The sequence has been interpreted as the number of preferential arrangements of members of a set with n elements. Alternatively the sequence has been interpreted as the number of ordered partitions; the outcomes in races in which ties are allowed or geometrically the number of vertices, edges and faces of simplicial objects. An interesting application of the sequence is found in combination locks. The idea of a preferential arrangement has been extended to a wider combinatorial object called barred preferential arrangement with multiple bars. In this thesis we study barred preferential arrangements combinatorially with application to resistance of certain electrical circuits. In the process we derive some results on cyclic properties of the last digit of the number of barred preferential arrangements. An algorithm in python has been developed to find the number of barred preferential arrangements.
- Full Text:
- Date Issued: 2015
- Authors: Nkonkobe, Sithembele
- Date: 2015
- Subjects: Electric circuits , Numerical calculations , Sequences (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5433 , http://hdl.handle.net/10962/d1020394
- Description: In 1854 Cayley proposed an interesting sequence 1,1,3,13,75,541,... in connection with analytical forms called trees. Since then there has been various combinatorial interpretations of the sequence. The sequence has been interpreted as the number of preferential arrangements of members of a set with n elements. Alternatively the sequence has been interpreted as the number of ordered partitions; the outcomes in races in which ties are allowed or geometrically the number of vertices, edges and faces of simplicial objects. An interesting application of the sequence is found in combination locks. The idea of a preferential arrangement has been extended to a wider combinatorial object called barred preferential arrangement with multiple bars. In this thesis we study barred preferential arrangements combinatorially with application to resistance of certain electrical circuits. In the process we derive some results on cyclic properties of the last digit of the number of barred preferential arrangements. An algorithm in python has been developed to find the number of barred preferential arrangements.
- Full Text:
- Date Issued: 2015
Some general convergence theorems on fixed points
- Authors: Panicker, Rekha Manoj
- Date: 2014
- Subjects: Fixed point theory , Convergence , Coincidence theory (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5426 , http://hdl.handle.net/10962/d1013112
- Description: In this thesis, we first obtain coincidence and common fixed point theorems for a pair of generalized non-expansive type mappings in a normed space. Then we discuss two types of convergence theorems, namely, the convergence of Mann iteration procedures and the convergence and stability of fixed points. In addition, we discuss the viscosity approximations generated by (ψ ,ϕ)-weakly contractive mappings and a sequence of non-expansive mappings and then establish Browder and Halpern type convergence theorems on Banach spaces. With regard to iteration procedures, we obtain a result on the convergence of Mann iteration for generalized non-expansive type mappings in a Banach space which satisfies Opial's condition. And, in the case of stability of fixed points, we obtain a number of stability results for the sequence of (ψ,ϕ)- weakly contractive mappings and the sequence of their corresponding fixed points in metric and 2-metric spaces. We also present a generalization of Fraser and Nadler type stability theorems in 2-metric spaces involving a sequence of metrics.
- Full Text:
- Date Issued: 2014
- Authors: Panicker, Rekha Manoj
- Date: 2014
- Subjects: Fixed point theory , Convergence , Coincidence theory (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5426 , http://hdl.handle.net/10962/d1013112
- Description: In this thesis, we first obtain coincidence and common fixed point theorems for a pair of generalized non-expansive type mappings in a normed space. Then we discuss two types of convergence theorems, namely, the convergence of Mann iteration procedures and the convergence and stability of fixed points. In addition, we discuss the viscosity approximations generated by (ψ ,ϕ)-weakly contractive mappings and a sequence of non-expansive mappings and then establish Browder and Halpern type convergence theorems on Banach spaces. With regard to iteration procedures, we obtain a result on the convergence of Mann iteration for generalized non-expansive type mappings in a Banach space which satisfies Opial's condition. And, in the case of stability of fixed points, we obtain a number of stability results for the sequence of (ψ,ϕ)- weakly contractive mappings and the sequence of their corresponding fixed points in metric and 2-metric spaces. We also present a generalization of Fraser and Nadler type stability theorems in 2-metric spaces involving a sequence of metrics.
- Full Text:
- Date Issued: 2014
A study of maximum and minimum operators with applications to piecewise linear payoff functions
- Authors: Seedat, Ebrahim
- Date: 2013
- Subjects: Options (Finance) Piecewise linear topology Geometry, Affine Riesz spaces Lattice theory Algebra, Boolean Pricing , Max and min operators
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:931 , http://hdl.handle.net/10962/d1001457
- Description: The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives
- Full Text:
- Date Issued: 2013
- Authors: Seedat, Ebrahim
- Date: 2013
- Subjects: Options (Finance) Piecewise linear topology Geometry, Affine Riesz spaces Lattice theory Algebra, Boolean Pricing , Max and min operators
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:931 , http://hdl.handle.net/10962/d1001457
- Description: The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives
- Full Text:
- Date Issued: 2013
Fixed points of single-valued and multi-valued mappings with applications
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
- Full Text:
- Date Issued: 2013
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
- Full Text:
- Date Issued: 2013
Qualitative and quantitative properties of solutions of ordinary differential equations
- Authors: Ogundare, Babatunde Sunday
- Date: 2009
- Subjects: Differential equations , Lyapunov functions , Chebyshev polynomials , Algorithms
- Language: English
- Type: Thesis , Doctoral , PhD (Applied Mathematics)
- Identifier: vital:11588 , http://hdl.handle.net/10353/244 , Differential equations , Lyapunov functions , Chebyshev polynomials , Algorithms
- Description: This thesis is concerned with the qualitative and quantitative properties of solutions of certain classes of ordinary di erential equations (ODEs); in particular linear boundary value problems of second order ODE's and non-linear ODEs of order at most four. The Lyapunov's second method of special functions called Lyapunov functions are employed extensively in this thesis. We construct suitable complete Lyapunov functions to discuss the qualitative properties of solutions to certain classes of non-linear ordinary di erential equations considered. Though there is no unique way of constructing Lyapunov functions, We adopt Cartwright's method to construct complete Lyapunov functions that are required in this thesis. Su cient conditions were established to discuss the qualitative properties such as boundedness, convergence, periodicity and stability of the classes of equations of our focus. Another aspect of this thesis is on the quantitative properties of solutions. New scheme based on interpolation and collocation is derived for solving initial value problem of ODEs. This scheme is derived from the general method of deriving the spline functions. Also by exploiting the Trigonometric identity property of the Chebyshev polynomials, We develop a new scheme for approximating the solutions of two-point boundary value problems. These schemes are user-friendly, easy to develop algorithm (computer program) and execute. They compare favorably with known standard methods used in solving the classes of problems they were derived for
- Full Text:
- Date Issued: 2009
- Authors: Ogundare, Babatunde Sunday
- Date: 2009
- Subjects: Differential equations , Lyapunov functions , Chebyshev polynomials , Algorithms
- Language: English
- Type: Thesis , Doctoral , PhD (Applied Mathematics)
- Identifier: vital:11588 , http://hdl.handle.net/10353/244 , Differential equations , Lyapunov functions , Chebyshev polynomials , Algorithms
- Description: This thesis is concerned with the qualitative and quantitative properties of solutions of certain classes of ordinary di erential equations (ODEs); in particular linear boundary value problems of second order ODE's and non-linear ODEs of order at most four. The Lyapunov's second method of special functions called Lyapunov functions are employed extensively in this thesis. We construct suitable complete Lyapunov functions to discuss the qualitative properties of solutions to certain classes of non-linear ordinary di erential equations considered. Though there is no unique way of constructing Lyapunov functions, We adopt Cartwright's method to construct complete Lyapunov functions that are required in this thesis. Su cient conditions were established to discuss the qualitative properties such as boundedness, convergence, periodicity and stability of the classes of equations of our focus. Another aspect of this thesis is on the quantitative properties of solutions. New scheme based on interpolation and collocation is derived for solving initial value problem of ODEs. This scheme is derived from the general method of deriving the spline functions. Also by exploiting the Trigonometric identity property of the Chebyshev polynomials, We develop a new scheme for approximating the solutions of two-point boundary value problems. These schemes are user-friendly, easy to develop algorithm (computer program) and execute. They compare favorably with known standard methods used in solving the classes of problems they were derived for
- Full Text:
- Date Issued: 2009
Studies of equivalent fuzzy subgroups of finite abelian p-Groups of rank two and their subgroup lattices
- Authors: Ngcibi, Sakhile Leonard
- Date: 2006
- Subjects: Abelian groups Fuzzy sets Finite groups Group theory Polynomials
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5416 , http://hdl.handle.net/10962/d1005230
- Description: We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.
- Full Text:
- Date Issued: 2006
- Authors: Ngcibi, Sakhile Leonard
- Date: 2006
- Subjects: Abelian groups Fuzzy sets Finite groups Group theory Polynomials
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5416 , http://hdl.handle.net/10962/d1005230
- Description: We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.
- Full Text:
- Date Issued: 2006
- «
- ‹
- 1
- ›
- »