| 1. |
- Engström, Christopher, 1987-, et al.
(författare)
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PageRank, a Look at Small Changes in a Line of Nodes and the Complete Graph
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 223-247
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Bokkapitel (refereegranskat)abstract
- In this article we will look at the PageRank algorithm used as part of the ranking process of different Internet pages in search engines by for example Google. This article has its main focus in the understanding of the behavior of PageRank as the system dynamically changes either by contracting or expanding such as when adding or subtracting nodes or links or groups of nodes or links. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others. We will look at PageRank as the solution of a linear system of equations and do our examination in both the ordinary normalized version of PageRank as well as the non-normalized version found by solving corresponding linear system. We will show that using two different methods we can find explicit formulas for the PageRank of some simple link structures.
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| 2. |
- Engström, Christopher, 1987-, et al.
(författare)
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PageRank, Connecting a Line of Nodes with a Complete Graph
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
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Bokkapitel (refereegranskat)abstract
- The focus of this article is the PageRank algorithm originally defined by S. Brin and L. Page as the stationary distribution of a certain random walk on a graph used to rank homepages on the Internet. We will attempt to get a better understanding of how PageRank changes after you make some changes to the graph such as adding or removing edge between otherwise disjoint subgraphs. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others and different ways to combine the two. Both the ordinary normalized version of PageRank as well as a non-normalized version of PageRank found by solving corresponding linear system will be considered. We will see that it is possible to find explicit formulas for the PageRank in some simple link structures and using these formulas take a more in-depth look at the behavior of the ranking as the system changes.
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| 3. |
- Guariglia, Emanuel, et al.
(författare)
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Fractional-Wavelet Analysis of Positive definite Distributions and Wavelets on D’(C)
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 337-353
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Bokkapitel (refereegranskat)abstract
- In the following chapter we describe a wavelet expansion theory for positivedefinite distributions over the real line and define a fractional derivative operator for complex functions in the distribution sense. In order to obtain a characterisation of the complex fractional derivative through the distribution theory, the Ortigueira-Caputo fractional derivative operator is rewritten as a convolution product according to the fractional calculus of real distributions. In particular, the fractional derivative of the Gabor-Morlet wavelet is computed together with its plots and main properties.
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| 4. |
- Hellström, Lars, 1974-
(författare)
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Valued Custom Skew Fields with Generalised PBW Property from Power Series Construction
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 33-55
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Bokkapitel (refereegranskat)abstract
- This chapter describes a construction of associative algebras that, despite starting from a commutation relation that the user may customize quite considerably, still manages to produce algebras with a number of useful properties: they have a Poincaré–Birkhoff–Witt type basis, they are equipped with a norm (actually an ultranorm) that is trivial to compute for basis elements, they are topologically complete, and they satisfy their given commutation relation. In addition, parameters can be chosen so that the algebras will in fact turn out to be skew fields and the norms become valuations. The construction is basically that of a power series algebra with given commutation relation, stated to be effective enough that the other properties can be derived. What is worked out in detail here is the case of algebras with two generators, but only the analysis of the commutation relation is specific for that case.
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| 5. |
- Kaijser, Sten, et al.
(författare)
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Lp-Boundedness of Two Singular Integral Operators of Convolution Type
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
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Bokkapitel (refereegranskat)abstract
- We investigate boundedness properties of two singular integral operators defined on Lp-spaces (1 < p < ∞) on the real line, both as convolution operators on Lp(R) and on the spaces Lp(w), where w(x) = 1/2cosh πx/2. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for p = 2 and weak boundedness for p = 1, and then using interpolation to obtain boundedness for 1 < p ≤ 2. To obtain boundedness also for 2 ≤ p < ∞, we use duality in the translation invariant case, while the weighted case is partly based on the expositions on the conjugate function operator in [7].π/2
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| 6. |
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| 7. |
- Nazir, Talat, 1982-, et al.
(författare)
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Linear and Nonlinear Classifiers of Data with Support Vector Machines and Generalized Support Vector Machines
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 377-396
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Bokkapitel (refereegranskat)abstract
- The support vector machine for linear and nonlinear classification of datais studied. The notion of generalized support vector machine for data classifications is used. The problem of generalized support vector machine is shown to be equivalent to the problem of generalized variational inequality and various results for the existence of solutions are established. Moreover, examples supporting the results are provided.
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| 8. |
- Nazir, Talat, 1982-, et al.
(författare)
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Linear Classification of data with Support Vector Machines and Generalized Support Vector Machines
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
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Bokkapitel (refereegranskat)abstract
- In this paper, we study the support vector machine and introduced the notion of generalized support vector machine for classification of data. We showthat the problem of generalized support vector machine is equivalent to the problem of generalized variational inequality and establish various results for the existence of solutions. Moreover, we provide various examples to support our results.
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| 9. |
- Richter, Johan, 1986-
(författare)
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Centralizers and Pseudo-Degree Functions
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 65-73
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Bokkapitel (refereegranskat)abstract
- This paper generalizes a proof of certain results by Hellström and Silvestrov on centralizers in graded algebras. We study centralizers in certain algebras with valuations. We prove that the centralizer of an element in these algebras is a free module over a certain ring. Under further assumptions we obtain that the centralizer is also commutative.
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| 10. |
- Richter, Johan, 1986-, et al.
(författare)
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Computing Burchnall–Chaundy Polynomials with Determinants
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 57-63
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Bokkapitel (refereegranskat)abstract
- In this expository paper we discuss a way of computing the Burchnall-Chaundy polynomial of two commuting differential operators using a determinant.We describe how the algorithm can be generalized to general Ore extensions, andwhich properties of the algorithm that are preserved.
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