In this paper we analyze the behavior of the Gamma function at its critical points and points of discontinuity on the negative side of the x-axis. We will also explain the bluntness of the gamma function on this negative side.

No description available.

The present study examines the role of turbulence on the incipient motion of sediment. For this purpose, well-controlled experiments are performed at the laboratory in a tilting flume. In these tests glass beads of the same size and density are used as the testing material to isolate the role of turbulence. State of the art equipment are used during the course of this study. Specifically, a 3-D Laser Doppler Velocimetry system is employed to measure the instantaneous velocity components at different points near the vicinity of a ball while the ball motion is monitored with a video camera. An image analysis program is developed here to analyze the motion of the particles within a test area. To examine the importance of the different stress components in the entrainment of sediment, five tests of different packing configuration are performed. Specifically three different roughness regimes are examined namely, the isolated, the wake interference, and the skimming flow. The results reveal that the instantaneous normal stress in the streamwise direction is the most dominant component of the instantaneous stress tensor. The backbone of this study is the development of a methodology to link the effects of turbulence with the commencement of sediment motion. It is considered that the metastable bursting cycle (i.e. sweeps, ejections, inward and outward interactions) is responsible for the sediment entrainment. And that the sediment entrainment, if any, occurs within a bursting period. The main concept behind the determination of the critical conditions is that the probability of the entrainment of sediment (effect) is equal to the probability of occurrence of these highly energetic turbulent events that have magnitude greater than the critical (cause). The probability of sediment entrainment is computed by means of the image analysis tool. The balance of moments is obtained here to determine the minimum moment that is required for the commencement of sediment motion. The balance of moments yields the deduction of a new variable that is used to describe the probability of occurrence of the different turbulent events. This variable is the summation of the instantaneous normal stresses in the streamwise and vertical direction. It is shown here that a two-parameter gamma density function describes quite well the statistical behavior of this variable. The results that are obtained from the existing model suggest that the present methodology can adequately describe the commencement of sediment motion. It is shown here that the traditionally used shear stress term uw may not be the appropriate measure for the determination of the critical conditions. / Ph. D.

The study of coherent states (CS) for a quantum mechanical system has received a lot of attention. The definition, applications, generalizations of such states have been the subject of work by researchers. A common starting point of all these approaches is the observation of properties of the original CS for the harmonic oscillator. It is well-known that they are described equivalently as (a) eigenstates of the usual annihilation operator, (b) from a displacement operator acting on a fundamental state and (c) as minimum uncertainty states. What we observe in the different generalizations proposed is that the preceding definitions are no longer equivalent and only some of the properties of the harmonic oscillator CS are preserved. In this thesis we propose to study a new class of coherent states and its properties. We note that in one example our CS coincide with the ones proposed by Glauber where a set of three requirements for such states has been imposed. The set of our generalized coherent states remains invariant under the corresponding time evolution and this property is called temporal stability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we get all coherent states by acting on one of these states ['fiducial vector'] with operators. They are highly non-classical states, in the sense that in general, their Bargmann functions have zeros which are related to negative regions of their Wigner functions. Examples of these coherent states with Bargmann function that involve the Gamma and also the Riemann ⲝ functions are represented. The zeros of these Bargmann functions and the paths of the zeros during time evolution are also studied.

No description available.

Η παρούσα διπλωματική εργασία έχει στόχο τη μελέτη της συνάρτησης Ζήτα του Riemann μέσω της Μιγαδικής ανάλυσης δηλαδή ως μία επέκταση αυτής από την ευθεία των πραγματικών αριθμών στο μιγαδικό επίπεδο. Η σύνδεση της συνάρτησης αυτής με τους πρώτους αριθμούς, η διάσημη υπόθεση Riemann, η συναρτησιακή εξίσωση, η αναλυτικότητά της εκτός σημείου είναι μερικά αποτελέσματα της μελέτης. Το αρχικό βήμα της εργασίας πριν από την μελέτη της συνάρτησης Ζήτα είναι μια εκτενής αναφορά στην συνάρτηση Γάμμα ως επεκτεινόμενη στο Μιγαδικό επίπεδο, την αναλυτικότητά της και κάποιων βασικών ιδιοτήτων της. Κάτι τέτοιο ήταν αναγκαίο, διότι η συνάρτηση Γάμμα αποτελεί ένα εργαλείο για τη μελέτη της συνάρτησης Ζήτα. / --

The study of coherent states (CS) for a quantum mechanical system has received a lot of attention. The definition, applications, generalizations of such states have been the subject of work by researchers. A common starting point of all these approaches is the observation of properties of the original CS for the harmonic oscillator. It is well-known that they are described equivalently as (a) eigenstates of the usual annihilation operator, (b) from a displacement operator acting on a fundamental state and (c) as minimum uncertainty states. What we observe in the different generalizations proposed is that the preceding definitions are no longer equivalent and only some of the properties of the harmonic oscillator CS are preserved. In this thesis we propose to study a new class of coherent states and its properties. We note that in one example our CS coincide with the ones proposed by Glauber where a set of three requirements for such states has been imposed. The set of our generalized coherent states remains invariant under the corresponding time evolution and this property is called temporal stability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we get all coherent states by acting on one of these states [¿fiducial vector¿] with operators. They are highly non-classical states, in the sense that in general, their Bargmann functions have zeros which are related to negative regions of their Wigner functions. Examples of these coherent states with Bargmann function that involve the Gamma and also the Riemann ¿ functions are represented. The zeros of these Bargmann functions and the paths of the zeros during time evolution are also studied. / Libyan Cultural Affairs

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

We consider the regularization of some equivariant Euler classes of certain infinite-dimensional vector bundles over a finite-dimensional manifold M using the framework of zeta-regularized products [35, 53, 59]. An example of such a regularization is the Atiyah–Witten regularization of the T-equivariant Euler class of the normal bundle v(TM) of M in the free loop space LM [2]. In this thesis, we propose a new regularization procedure — W-regularization — which can be shown to reduce to the Atiyah–Witten regularization when applied to the case of v(TM). This new regularization yields a new multiplicative genus (in the sense of Hirzebruch [26]) — the ^Γ-genus — when applied to the more general case of a complex spin vector bundle of complex rank ≥ 2 over M, as opposed to the case of the complexification of TM for the Atiyah–Witten regularization. Some of its properties are investigated and some tantalizing connections to other areas of mathematics are also discussed. We also consider the application of W-regularization to the regularization of T²- equivariant Euler classes associated to the case of the double free loop space LLM. We find that the theory of zeta-regularized products, as set out by Jorgenson–Lang [35], Quine et al [53] and Voros [59], amongst others, provides a good framework for comparing the regularizations that have been considered so far. In particular, it reveals relations between some of the genera that appeared in elliptic cohomology, allowing us to clarify and prove an assertion of Liu [44] on the ˆΘ-genus, as well as to recover the Witten genus. The ^Γ₂-genus, a new genus generated by a function based on Barnes’ double gamma function [5, 6], is also derived in a similar way to the ^Γ-genus. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

This paper consists of a discussion of the properties and applications of certain improper integrals, namely the gamma function and the beta function. There are also specific examples of application of these functions in certain fields of applied science.