| 1. |
- Abola, Benard
(författare)
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Perturbed Markov Chains with Damping Component and Information Networks
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis brings together three thematic topics, PageRank of evolving tree graphs, stopping criteria for ranks and perturbed Markov chains with damping component. The commonality in these topics is their focus on ranking problems in information networks. In the fields of science and engineering, information networks are interesting from both practical and theoretical perspectives. The fascinating property of networks is their applicability in analysing broad spectrum of problems and well established mathematical objects. One of the most common algorithms in networks' analysis is PageRank. It was developed for web pages’ ranking and now serves as a tool for identifying important vertices as well as studying characteristics of real-world systems in several areas of applications. Despite numerous successes of the algorithm in real life, the analysis of information networks is still challenging. Specifically, when the system experiences changes in vertices /edges or it is not strongly connected or when a damping stochastic matrix and a damping factor are added to an information matrix. For these reasons, extending existing or developing methods to understand such complex networks is necessary.Chapter 2 of this thesis focuses on information networks with no bidirectional interaction. They are commonly encountered in ecological systems, number theory and security systems. We consider certain specific changes in a network and describe how the corresponding information matrix can be updated as well as PageRank scores. Specifically, we consider the graph partitioned into levels of vertices and describe how PageRank is updated as the network evolves.In Chapter 3, we review different stopping criteria used in solving a linear system of equations and investigate each stopping criterion against some classical iterative methods. Also, we explore whether clustering algorithms may be used as stopping criteria.Chapter 4 focuses on perturbed Markov chains commonly used for the description of information networks. In such models, the transition matrix of an information Markov chain is usually regularised and approximated by a stochastic (Google type) matrix. Stationary distribution of the stochastic matrix is equivalent to PageRank, which is very important for ranking of vertices in information networks. Determining stationary probabilities and related characteristics of singularly perturbed Markov chains is complicated; leave alone the choice of regularisation parameter. We use the procedure of artificial regeneration for the perturbed Markov chain with the matrix of transition probabilities and coupling methods. We obtain ergodic theorems, in the form of asymptotic relations. We also derive explicit upper bounds for the rate of convergence in ergodic relations. Finally, we illustrate these results with numerical examples.
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| 2. |
- Al-Shujary, Ahmed, 1980-
(författare)
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Kähler-Poisson Algebras
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we introduce Kähler-Poisson algebras and study their basic properties. The motivation comes from differential geometry, where one can show that the Riemannian geometry of an almost Kähler manifold can be formulated in terms of the Poisson algebra of smooth functions on the manifold. It turns out that one can identify an algebraic condition in the Poisson algebra (together with a metric) implying that most geometric objects can be given a purely algebraic formulation. This leads to the definition of a Kähler-Poisson algebra, which consists of a Poisson algebra and a metric fulfilling an algebraic condition. We show that every Kähler- Poisson algebra admits a unique Levi-Civita connection on its module of inner derivations and, furthermore, that the corresponding curvature operator has all the classical symmetries. Moreover, we present a construction procedure which allows one to associate a Kähler-Poisson algebra to a large class of Poisson algebras. From a more algebraic perspective, we introduce basic notions, such as morphisms and subalgebras, as well as direct sums and tensor products. Finally, we initiate a study of the moduli space of Kähler-Poisson algebras; i.e for a given Poisson algebra, one considers classes of metrics giving rise to non-isomorphic Kähler-Poisson algebras. As it turns out, even the simple case of a Poisson algebra generated by two variables gives rise to a nontrivial classification problem.
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| 3. |
- Seleka Biganda, Pitos, 1981-
(författare)
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Analytical and Iterative Methods of Computing PageRank of Networks
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is about variants of PageRank, methods of PageRank computation and perturbation analysis of a PageRank vector as a stationary distribution of a kind of perturbed Markov chain model. Chapter 2 of this thesis gives closed form formulae for ordinary and lazy PageRanks for some specific simple line graphs. Different cases of changes made to the simple line graph are considered and for each case, a corresponding formula for each of the two variants of PageRank is provided.Chapter 3 is dedicated to the exploration of relationships that exist between three known variants of PageRank: ordinary PageRank, lazy PageRank and random walk with backstep PageRank in terms of their convergence and consistency in rank scores for different graph structures with reference to PageRank parameters, the damping factor c and backstep parameter β. In Chapter 4, we discuss numerical methods used in solving the PageRank problem as a linear system and evaluate some stopping criteria that can be employed in such methods. Finally, in Chapter 5, we address the PageRank problem as a first order perturbed Markov chain problem and study the perturbation analysis for stationary distributions of Markov chains with damping component. We illustrate our results on asymptotic perturbation analysis by using different computational examples.
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| 4. |
- Silvestrov, Dmitrii, 1947-, et al.
(författare)
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Perturbation analysis for stationary distributions of markov chains with damping component
- 2020
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Ingår i: Algebraic Structures and Applications. - Cham : Springer Nature. - 9783030418496 ; , s. 903-933
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Bokkapitel (refereegranskat)abstract
- Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix P0 of an information Markov chain is usually approximated by matrix Pε = (1 - ε) P0 + ε D, where D is a so-called damping stochastic matrix with identical rows and all positive elements, while ε is a damping (perturbation) parameter. We perform a detailed perturbation analysis for stationary distributions of such Markov chains, in particular get effective explicit series representations for the corresponding stationary distributions πε, upper bounds for the deviation |πε- π0 |, and asymptotic expansions for πε with respect to the perturbation parameter ε.
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| 5. |
- Abramov, Viktor, et al.
(författare)
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3-Hom-Lie Algebras Based on σ-Derivation and Involution
- 2020
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Ingår i: Advances in Applied Clifford Algebras. - : Springer. - 0188-7009 .- 1661-4909. ; 30:3
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Tidskriftsartikel (refereegranskat)abstract
- We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using σ-derivation and involution, and we obtain two 3-Hom-Lie algebras.
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| 6. |
- Armakan, Abdoreza, et al.
(författare)
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Enveloping algebras of certain types of color hom-Lie algebras
- 2020
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Ingår i: Algebraic Structures and Applications. - Cham : Springer Nature. - 9783030418496 - 9783030418502 ; , s. 257-284
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Bokkapitel (refereegranskat)abstract
- In this paper the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free hom-associative color algebra on a hom-module is described for a certain type of color hom-Lie algebras and is applied to obtain the universal enveloping algebra of those hom-Lie color algebras. Finally, this construction is applied to obtain the extension of the well-known Poincaré–Birkhoff–Witt theorem for Lie algebras to the enveloping algebra of the certain types of color hom-Lie algebra such that some power of the twisting map is the identity map.
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| 7. |
- Bakayoko, I., et al.
(författare)
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Multiplicative n-hom-lie color algebras
- 2020
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Ingår i: Algebraic Structures and Applications. - Cham : Springer Nature. - 9783030418496 ; , s. 159-187
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Bokkapitel (refereegranskat)abstract
- The purpose of this paper is to generalize some results on n-Lie algebras and n-Hom-Lie algebras to n-Hom-Lie color algebras. Then we introduce and give some constructions of n-Hom-Lie color algebras.
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| 8. |
- Biganda, Pitos, 1981-, et al.
(författare)
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Exploring The Relationship Between Ordinary PageRank, Lazy PageRank and Random Walk with Backstep PageRank for Different Graph Structures
- 2020
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Ingår i: Data Analysis and Applications 3. - : John Wiley & Sons, Ltd. - 9781786305343 - 9781119721871 ; , s. 53-73
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Bokkapitel (refereegranskat)abstract
- PageRank is an algorithm for ranking web pages. It is the first and best known webgraph-based algorithm in the Google search engine. The algorithm is simple, robust and reliable to measure the importance of web pages. This chapter presents a comparative review of three variants of PageRank, namely ordinary PageRank (introduced by Brin and Page as a measure of importance of a web page), lazy PageRank and random walk with backstep PageRank. It compares the variants in terms of their convergence and consistency in rank scores for different graph structures with reference to PageRank’s parameters, damping factor and backstep parameter. The chapter also shows that ordinary PageRank can be formulated from the other two variants by some proportionality relationships.
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| 9. |
- Canhanga, Betuel, et al.
(författare)
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Advanced Monte Carlo pricing of european options in a market model with two stochastic volatilities
- 2020
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Ingår i: Algebraic Structures and Applications. - Cham : Springer Nature. - 9783030418496 ; , s. 857-874
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Bokkapitel (refereegranskat)abstract
- We consider a market model with four correlated factors and two stochastic volatilities, one of which is rapid-changing, while another one is slow-changing in time. An advanced Monte Carlo method based on the theory of cubature in Wiener space is used to find the no-arbitrage price of the European call option in the above model.
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| 10. |
- Cantuba, Rafael Reno, et al.
(författare)
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Lie polynomial characterization problems
- 2020
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Ingår i: Algebraic Structures and Applications. - Cham : Springer Nature. - 9783030418496 ; , s. 593-601
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Bokkapitel (refereegranskat)abstract
- We present a review of some results about Lie polynomials in finitely-generated associative algebras with defining relations that involve deformed commutation relations. Such algebras have arisen from various areas such as in the theory of quantum groups, of q-oscillators, of q-deformed Heisenberg algebras, of orthogonal polynomials, and even from algebraic combinatorics. The q-deformed Heisenberg-Weyl relation is so far the most successful setting for a Lie polynomial characterization problem. Both algebraic and operator-theoretic approaches have been found. We also discuss some partial results for other algebras related to quantum groups.
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