Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.

Neurolinguistic analysis: aspects of language and speech deviations in Palestinian Arabic aphasics

The characteristics of aphasics’ speech in various languages have been the core of numerous studies, but Arabic in general, and Palestinian Arabic in particular, is still a virgin field in this respect. However, it is of vital importance to have a clear picture of the specific aspects of Palestinian Arabic that might be affected in the speech of aphasics in order to establish screening, diagnosis and therapy programs based on a clinical linguistic database. Hence the central questions of this study are what are the main neurolinguistic features of the Palestinian aphasics’ speech at the phonetic-acoustic level and to what extent are the results similar or not to those obtained from other languages. In general, this study is a survey of the most prominent features of Palestinian Broca’s aphasics’ speech. The main acoustic parameters of vowels and consonants are analysed such as vowel duration, formant frequency, Voice Onset Time (VOT), intensity and frication duration. The deviant patterns among the Broca’s aphasics are displayed and compared with those of normal speakers. The nature of deficit, whether phonetic or phonological, is also discussed. Moreover, the coarticulatory characteristics and some prosodic patterns of Broca’s aphasics are addressed. Samples were collected from six Broca’s aphasics from the same local region. The acoustic analysis conducted on a range of consonant and vowel parameters displayed differences between the speech patterns of Broca’s aphasics and normal speakers. For example, impairments in voicing contrast between the voiced and voiceless stops were found in Broca’s aphasics. This feature does not exist for the fricatives produced by the Palestinian Broca’s aphasics and hence deviates from data obtained for aphasics’ speech from other languages. The Palestinian Broca’s aphasics displayed particular problems with the emphatic sounds. They exhibited deviant coarticulation patterns, another feature that is inconsistent with data obtained from studies from other languages. However, several other findings are in accordance with those reported from various other languages such as impairments in the VOT. The results are in accordance with the suggestions that speech production deficits in Broca’s aphasics are not related to phoneme selection but rather to articulatory implementation and some speech output impairments are related to timing and planning deficits.