The von Neumann Minimax Theorem and its relatives and a study of externality in on-line auctions

This work consists of a theoretical part and an experimental one. The first part provides a simple treatment of the celebrated von Neumann minimax theorem as formulated by Nikaid6 and Sion. It also discusses its relationships with fundamental theorems of convex analysis. The second part is about externality in sponsored search auctions. It shows that in these auctions, advertisers have externality effects on each other which influence their bidding behavior. It proposes Hal R.Varian model and shows how adding externality to this model will affect its properties. In order to have a better understanding of the interaction among advertisers in on-line auctions, it studies the structure of the Google advertisements networ.k and shows that it is a small-world scale-free network.

An Interactive Theorem Prover for First-Order Dynamic Logic

Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs. The method of proving correctness of properties of a computer program using the well-known Hoare Logic can be implemented by utilizing the robustness of dynamic logic. For a very broad range of languages and applications in program veri cation, a theorem prover named KIV (Karlsruhe Interactive Veri er) Theorem Prover has already been developed. But a high degree of automation and its complexity make it di cult to use it for educational purposes. My research work is motivated towards the design and implementation of a similar interactive theorem prover with educational use as its main design criteria. As the key purpose of this system is to serve as an educational tool, it is a self-explanatory system that explains every step of creating a derivation, i.e., proving a theorem. This deductive system is implemented in the platform-independent programming language Java. In addition, a very popular combination of a lexical analyzer generator, JFlex, and the parser generator BYacc/J for parsing formulas and programs has been used.

A Representation Theorem for Domains with Discrete and Continuous Variables.

This paper proves a new representation theorem for domains with both discrete and continuous variables. The result generalizes Debreu's well-known representation theorem on connected domains. A strengthening of the standard continuity axiom is used in order to guarantee the existence of a representation. A generalization of the main theorem and an application of the more general result are also presented.

Arrow's Theorem in Spatial Environments

In spatial environments, we consider social welfare functions satisfying Arrow's requirements. i.e., weak Pareto and independence of irrelevant alternatives. When the policy space os a one-dimensional continuum, such a welfare function is determined by a collection of 2n strictly quasi-concave preferences and a tie-breaking rule. As a corrollary, we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter's preference is strictly quasi-concave. When the policy space is multi-dimensional, we establish Arrow's impossibility theorem. Among others, we show that weak Pareto, independence of irrelevant alternatives, and non-dictatorship are inconsistent if the set of alternatives has a non-empty interior and it is compact and convex.

Harsanyi’s Social Aggregation Theorem : A Multi-Profile Approach with Variable-Population Extensions

This paper provides new versions of Harsanyi’s social aggregation theorem that are formulated in terms of prospects rather than lotteries. Strengthening an earlier result, fixed-population ex-ante utilitarianism is characterized in a multi-profile setting with fixed probabilities. In addition, we extend the social aggregation theorem to social-evaluation problems under uncertainty with a variable population and generalize our approach to uncertain alternatives, which consist of compound vectors of probability distributions and prospects.

A Generalization of a Theorem of Boyd and Lawton

Ce mémoire s’applique à étudier d’abord, dans la première partie, la mesure de Mahler des polynômes à une seule variable. Il commence en donnant des définitions et quelques résultats pertinents pour le calcul de telle hauteur. Il aborde aussi le sujet de la question de Lehmer, la conjecture la plus célèbre dans le domaine, donne quelques exemples et résultats ayant pour but de résoudre la question. Ensuite, il y a l’extension de la mesure de Mahler sur les polynômes à plusieurs variables, une démarche semblable au premier cas de la mesure de Mahler, et le sujet des points limites avec quelques exemples. Dans la seconde partie, on commence par donner des définitions concernant un ordre supérieur de la mesure de Mahler, et des généralisations en passant des polynômes simples aux polynômes à plusieurs variables. La question de Lehmer existe aussi dans le domaine de la mesure de Mahler supérieure, mais avec des réponses totalement différentes. À la fin, on arrive à notre objectif, qui sera la démonstration de la généralisation d’un théorème de Boyd-Lawton, ce dernier met en évidence une relation entre la mesure de Mahler des polynômes à plusieurs variables avec la limite de la mesure de Mahler des polynômes à une seule variable. Ce résultat a des conséquences en termes de la conjecture de Lehmer et sert à clarifier la relation entre les valeurs de la mesure de Mahler des polynômes à une variable et celles des polynômes à plusieurs variables, qui, en effet, sont très différentes en nature.

Studies on Thermodynamics and No-hair Theorem in Black hole Spacetime

Department of Physics, Cochin University of Science and Technology

Recurrence Quantification Analysis of System Signals for Detecting Tool and Chatter in Turning

In this thesis, the applications of the recurrence quantification analysis in metal cutting operation in a lathe, with specific objective to detect tool wear and chatter, are presented.This study is based on the discovery that process dynamics in a lathe is low dimensional chaotic. It implies that the machine dynamics is controllable using principles of chaos theory. This understanding is to revolutionize the feature extraction methodologies used in condition monitoring systems as conventional linear methods or models are incapable of capturing the critical and strange behaviors associated with the metal cutting process.As sensor based approaches provide an automated and cost effective way to monitor and control, an efficient feature extraction methodology based on nonlinear time series analysis is much more demanding. The task here is more complex when the information has to be deduced solely from sensor signals since traditional methods do not address the issue of how to treat noise present in real-world processes and its non-stationarity. In an effort to get over these two issues to the maximum possible, this thesis adopts the recurrence quantification analysis methodology in the study since this feature extraction technique is found to be robust against noise and stationarity in the signals.The work consists of two different sets of experiments in a lathe; set-I and set-2. The experiment, set-I, study the influence of tool wear on the RQA variables whereas the set-2 is carried out to identify the sensitive RQA variables to machine tool chatter followed by its validation in actual cutting. To obtain the bounds of the spectrum of the significant RQA variable values, in set-i, a fresh tool and a worn tool are used for cutting. The first part of the set-2 experiments uses a stepped shaft in order to create chatter at a known location. And the second part uses a conical section having a uniform taper along the axis for creating chatter to onset at some distance from the smaller end by gradually increasing the depth of cut while keeping the spindle speed and feed rate constant.The study concludes by revealing the dependence of certain RQA variables; percent determinism, percent recurrence and entropy, to tool wear and chatter unambiguously. The performances of the results establish this methodology to be viable for detection of tool wear and chatter in metal cutting operation in a lathe. The key reason is that the dynamics of the system under study have been nonlinear and the recurrence quantification analysis can characterize them adequately.This work establishes that principles and practice of machining can be considerably benefited and advanced from using nonlinear dynamics and chaos theory.

Analysis of System Signals Based on Cross Recurrence Method for System Dynamics Characterization

Natural systems are inherently non linear. Recurrent behaviours are typical of natural systems. Recurrence is a fundamental property of non linear dynamical systems which can be exploited to characterize the system behaviour effectively. Cross recurrence based analysis of sensor signals from non linear dynamical system is presented in this thesis. The mutual dependency among relatively independent components of a system is referred as coupling. The analysis is done for a mechanically coupled system specifically designed for conducting experiment. Further, cross recurrence method is extended to the actual machining process in a lathe to characterize the chatter during turning. The result is verified by permutation entropy method. Conventional linear methods or models are incapable of capturing the critical and strange behaviours associated with the dynamical process. Hence any effective feature extraction methodologies should invariably gather information thorough nonlinear time series analysis. The sensor signals from the dynamical system normally contain noise and non stationarity. In an effort to get over these two issues to the maximum possible extent, this work adopts the cross recurrence quantification analysis (CRQA) methodology since it is found to be robust against noise and stationarity in the signals. The study reveals that the CRQA is capable of characterizing even weak coupling among system signals. It also divulges the dependence of certain CRQA variables like percent determinism, percent recurrence and entropy to chatter unambiguously. The surrogate data test shows that the results obtained by CRQA are the true properties of the temporal evolution of the dynamics and contain a degree of deterministic structure. The results are verified using permutation entropy (PE) to detect the onset of chatter from the time series. The present study ascertains that this CRP based methodology is capable of recognizing the transition from regular cutting to the chatter cutting irrespective of the machining parameters or work piece material. The results establish this methodology to be feasible for detection of chatter in metal cutting operation in a lathe.

Modeling of ALFA Programs Using PVS Theorem Prover

In Safety critical software failure can have a high price. Such software should be free of errors before it is put into operation. Application of formal methods in the Software Development Life Cycle helps to ensure that the software for safety critical missions are ultra reliable. PVS theorem prover, a formal method tool, can be used for the formal verification of software in ADA Language for Flight Software Application (ALFA.). This paper describes the modeling of ALFA programs for PVS theorem prover. An ALFA2PVS translator is developed which automatically converts the software in ALFA to PVS specification. By this approach the software can be verified formally with respect to underflow/overflow errors and divide by zero conditions without the actual execution of the code.

Modeling of ALFA Programs Using PVS Theorem Prover

In Safety critical software failure can have a high price. Such software should be free of errors before it is put into operation. Application of formal methods in the Software Development Life Cycle helps to ensure that the software for safety critical missions are ultra reliable. PVS theorem prover, a formal method tool, can be used for the formal verification of software in ADA Language for Flight Software Application (ALFA.). This paper describes the modeling of ALFA programs for PVS theorem prover. An ALFA2PVS translator is developed which automatically converts the software in ALFA to PVS specification. By this approach the software can be verified formally with respect to underflow/overflow errors and divide by zero conditions without the actual execution of the code

Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

Solution properties of the de Branges differential recurrence equation

In this 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums thas was published by Askey and Gasper in 1976. The de Branges functions Tn/k(t) are defined as the solutions of a system of differential recurrence equations with suitably given initial values. The essential fact used in the proof of the Bieberbach and Milin conjectures is the statement Tn/k(t)<=0. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λn/k(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', Tn/k(t)=-kΛn/k(t), and the positivity results in both proofs Tn/k(t)<=0 are essentially the same. In this paper we study differential recurrence equations equivalent to de Branges' original ones and show that many solutions of these differential recurrence equations don't change sign so that the above inequality is not as surprising as expected. Furthermore, we present a multiparameterized hypergeometric family of solutions of the de Branges differential recurrence equations showing that solutions are not rare at all.

Femtosecond pump-probe study of the spreading and recurrence of a vibrational wave packet in Na_2

The motion of a vibrational wave packet in the bound A(^1 \summe^+_u) electronic state of the sodium dimer is detected in a femtosecond pump/probe molecular beam experiment. For short times harmonic motion is seen in the total ion yield of Na^+_2 as a function of delay time between the two laser pulses. The spreading of the wave packet results in the loss of the periodic variation of the ion signal. For longer delay times (47 ps) the wave packet regains its initial form which is reflected in the revival structure of the Na^+_2 signal. Time-dependent quantum calculations reproduce the measured effects.

Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen

Die q-Analysis ist eine spezielle Diskretisierung der Analysis auf einem Gitter, welches eine geometrische Folge darstellt, und findet insbesondere in der Quantenphysik eine breite Anwendung, ist aber auch in der Theorie der q-orthogonalen Polynome und speziellen Funktionen von großer Bedeutung. Die betrachteten mathematischen Objekte aus der q-Welt weisen meist eine recht komplizierte Struktur auf und es liegt daher nahe, sie mit Computeralgebrasystemen zu behandeln. In der vorliegenden Dissertation werden Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen vorgestellt. Alle Algorithmen sind in dem Maple-Package qFPS, welches integraler Bestandteil der Arbeit ist, implementiert. Nachdem in den ersten beiden Kapiteln Grundlagen geschaffen werden, werden im dritten Kapitel Algorithmen präsentiert, mit denen man zu einer q-holonomen Funktion q-holonome Rekursionsgleichungen durch Kenntnis derer q-Shifts aufstellen kann. Operationen mit q-holonomen Rekursionen werden ebenfalls behandelt. Im vierten Kapitel werden effiziente Methoden zur Bestimmung polynomialer, rationaler und q-hypergeometrischer Lösungen von q-holonomen Rekursionen beschrieben. Das fünfte Kapitel beschäftigt sich mit q-hypergeometrischen Potenzreihen bzgl. spezieller Polynombasen. Wir formulieren einen neuen Algorithmus, der zu einer q-holonomen Rekursionsgleichung einer q-hypergeometrischen Reihe mit nichttrivialem Entwicklungspunkt die entsprechende q-holonome Rekursionsgleichung für die Koeffizienten ermittelt. Ferner können wir einen neuen Algorithmus angeben, der umgekehrt zu einer q-holonomen Rekursionsgleichung für die Koeffizienten eine q-holonome Rekursionsgleichung der Reihe bestimmt und der nützlich ist, um q-holonome Rekursionen für bestimmte verallgemeinerte q-hypergeometrische Funktionen aufzustellen. Mit Formulierung des q-Taylorsatzes haben wir schließlich alle Zutaten zusammen, um das Hauptergebnis dieser Arbeit, das q-Analogon des FPS-Algorithmus zu erhalten. Wolfram Koepfs FPS-Algorithmus aus dem Jahre 1992 bestimmt zu einer gegebenen holonomen Funktion die entsprechende hypergeometrische Reihe. Wir erweitern den Algorithmus dahingehend, dass sogar Linearkombinationen q-hypergeometrischer Potenzreihen bestimmt werden können. ________________________________________________________________________________________________________________