It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.

In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.

Recently, wireless sensor network (WSN) has become a promising technology with a wide range of applications such as supply chain monitoring and environment surveillance. It is typically composed of multiple tiny devices equipped with limited sensing, computing and wireless communication capabilities. Design of such networks presents several technique challenges while dealing with various requirements and diverse constraints. Performance analysis and deployment techniquesare required to provide insight on design parameters and system behaviors. Based on network calculus, a deterministic analysis method is presented for evaluating the worst-case delay and buffer cost of sensor networks. To this end,traffic splitting and multiplexing models are proposed and their delay and buffer bounds are derived. These models can be used in combination to characterize complex traffic flowing scenarios. Furthermore, the method integrates a variable duty cycle to allow the sensor nodes to operate at low rates thus saving power. In an attempt to balance traffic load and improve resource utilization and performance,traffic splitting mechanisms are introduced for sensor networks with general topologies. To provide reliable data delivery in sensor networks, retransmission has been one of the most popular schemes. We propose an analytical method to evaluate the maximum data transmission delay and energy consumption of two types of retransmission schemes: hop-by-hop retransmission and end-to-end retransmission.In order to validate the tightness of the bounds obtained by the analysis method, the simulation results and analytical results are compared with various input traffic loads. The results show that the analytic bounds are correct and tight. Stochastic network calculus has been developed as a useful tool for Qualityof Service (QoS) analysis of wireless networks. We propose a stochastic servicecurve model for the Rayleigh fading channel and then provide formulas to derive the probabilistic delay and backlog bounds in the cases of deterministic and stochastic arrival curves. The simulation results verify that the tightness of the bounds are good. Moreover, a detailed mechanism for bandwidth estimation of random wireless channels is developed. The bandwidth is derived from the measurement of statistical backlogs based on probe packet trains. It is expressed by statistical service curves that are allowed to violate a service guarantee with a certain probability. The theoretic foundation and the detailed step-by-step procedure of the estimation method are presented. One fundamental application of WSNs is event detection in a Field of Interest(FoI), where a set of sensors are deployed to monitor any ongoing events. To satisfy a certain level of detection quality in such applications, it is desirable that events in the region can be detected by a required number of sensors. Hence, an important problem is how to conduct sensor deployment for achieving certain coverage requirements. In this thesis, a probabilistic event coverage analysis methodis proposed for evaluating the coverage performance of heterogeneous sensor networks with randomly deployed sensors and stochastic event occurrences. Moreover,we present a framework for analyzing node deployment schemes in terms of three performance metrics: coverage, lifetime, and cost. The method can be used to evaluate the benefits and trade-offs of different deployment schemes and thus provide guidelines for network designers. / <p>QC 20120906</p>

<p>This dissertation explores the ideational underpinnings of the rise of the west through a comparison of ancient Greek geometry, medieval Arabic algebra, and early modern European calculus. Blending insights from Thomas Kuhn, Michel Foucault, and William H. Sewell, I assert that there is an underlying logic, however clouded, to the unfolding of a given civilization, governed by a cultural episteme that delineates the boundaries of rational thought and the accepted domain of human endeavor. Amid a certain conceptual configuration, the rise of the west happens; under other circumstances, it does not. Mathematics, as an explicit exhibition of logic premised on culturally determined axioms, presents an outward manifestation of the lens through which a civilization surveys the world, and as such offers a window on the fundamental assumptions from which a civilization's trajectory proceeds. To identify the epistemological conditions favorable to the rise of the west, I focus specifically on three mathematical divergences that were integral to the development of calculus, namely analytic geometry, trigonometry, and the fundamental theorem of calculus. Through a comparative/historical analysis of original source documents in mathematics, I demonstrate that the logic in the earlier cases is fundamentally different from that of calculus, and furthermore, incompatible with the key developments that constitute the rise of the west. I then examine the conceptual similarities between calculus and several features of the rise of the west to articulate a description of the early modern episteme.</p> / Dissertation

In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.

Some of the applications in modern data networks are delay sensitive (e.g., video and voice). An end-to-end delay analysis is needed to estimate the required network resources of delay sensitive applications. The schedulers used in the network can impact the resulting delays to the applications. When multiple applications are multiplexed in a switch, a scheduler is used to determine the precedence of the arrivals from different applications. Computing the end-to-end delay and queue sizes in a network of schedulers is difficult and the existing solutions are limited to some special cases (e.g., specific type of traffic). The theory of Network Calculus employs the min-plus algebra to obtain performance bounds. Given an upper bound on the traffic arrival in any time interval and a lower bound on the available service (called the service curve) at a network element, upper bounds on the delay and queue size of the traffic in that network element can be obtained. An equivalent end-to-end service curve of a tandem of queues is the min-plus convolution of the service curves of all nodes along the path. A probabilistic end-to-end delay bound using network service curve scales with O(H logH) in the path length H. This improves the results of the conventional method of adding per-node delay bounds scaling with O(H^3). We have used and advanced Network Calculus for end-to-end delay analysis in a network of schedulers. We formulate a service curve description for a large class of schedulers which we call Delta-schedulers. We show that with this service curve, tight single node delay and backlog bounds can be achieved. In an end-to-end scenario, we formulate a new convolution theoii rem which considerably improves the end-to-end probabilistic delay bounds. We specify our probabilistic end-to-end delay and backlog bounds for exponentially bounded burstniess (EBB) traffic arrivals. We show that the end-to-end delay varies considerably by the type of schedulers along the path. Using these bounds, we also show that a if the number of flows increases, the queues inside a network can be analyzed in isolation and regardless of the network effect.

A small group of profit seeking publishers dominates the American textbook market and guides the learning of the majority of our nation’s calculus students. The College Board’s AP Calculus curriculum is a de facto national standard for this gateway course that is critically important to 21st century STEM careers. A multi-representational understanding of calculus is a central pillar of the AP curriculum. This dissertation asks whether this multi-representational vision is manifest in popular calculus textbooks. This dissertation began with a survey of all AP Calculus AB Examination free response items, 2002-2011, and found that students score worse on items characterized by numerical anchors or verbal targets. Based on previously elucidated models, a new cognitive model of five levels and six principles is developed for the purpose of calculus textbook task analysis. This model explicates complexity as a function of representational input and output. Eight popular secondary calculus textbooks were selected for study based on Amazon sales rank data. All verbally anchored mathematical tasks (n=555) from sections of those books concerning the mean value theorem and all AP Calculus AB prompts (n=226) were analyzed for cognitive complexity and representational diversity using the model. The textbook study found that calculus textbooks underrepresented the numerical anchor and verbal target. It found that the textbooks were both explicitly and implicitly less cognitively complex than the AP test. The article suggested that textbook tasks should be less dense, avoid cognitive attenuation, move away from the stand-alone item, juxtapose anchor representations, scaffold student solutions, incorporate previously considered overarching concepts and include more profound follow-up questions. To date there have been no studies of calculus textbook content based on established research on cognitive learning. Given the critical role that their calculus course plays in the lives of hundreds of thousands of students annually, it is incumbent upon the College Board to establish a textbook review process at the very least in the same vain as the teacher syllabus auditing process established in recent years.

Some of the applications in modern data networks are delay sensitive (e.g., video and voice). An end-to-end delay analysis is needed to estimate the required network resources of delay sensitive applications. The schedulers used in the network can impact the resulting delays to the applications. When multiple applications are multiplexed in a switch, a scheduler is used to determine the precedence of the arrivals from different applications. Computing the end-to-end delay and queue sizes in a network of schedulers is difficult and the existing solutions are limited to some special cases (e.g., specific type of traffic). The theory of Network Calculus employs the min-plus algebra to obtain performance bounds. Given an upper bound on the traffic arrival in any time interval and a lower bound on the available service (called the service curve) at a network element, upper bounds on the delay and queue size of the traffic in that network element can be obtained. An equivalent end-to-end service curve of a tandem of queues is the min-plus convolution of the service curves of all nodes along the path. A probabilistic end-to-end delay bound using network service curve scales with O(H logH) in the path length H. This improves the results of the conventional method of adding per-node delay bounds scaling with O(H^3). We have used and advanced Network Calculus for end-to-end delay analysis in a network of schedulers. We formulate a service curve description for a large class of schedulers which we call Delta-schedulers. We show that with this service curve, tight single node delay and backlog bounds can be achieved. In an end-to-end scenario, we formulate a new convolution theoii rem which considerably improves the end-to-end probabilistic delay bounds. We specify our probabilistic end-to-end delay and backlog bounds for exponentially bounded burstniess (EBB) traffic arrivals. We show that the end-to-end delay varies considerably by the type of schedulers along the path. Using these bounds, we also show that a if the number of flows increases, the queues inside a network can be analyzed in isolation and regardless of the network effect.

No description available.

This thesis consists of six chapters. In the first chapter, we review some basic definitions and concepts of fractional calculus. Then we introduce fractional difference equations involving the Riemann-Liouville operator of real number order between zero and one. In the second chapter, we apply the Brouwer fixed point and Contraction Mapping Theorems to prove that there exists a solution for up to the first order nabla fractional difference equation with an initial condition. In chapter three, we define a lower and an upper solution for up to the first order nabla fractional difference equation with an initial condition. Under certain assumptions we prove that a lower solution stays less than an upper solution. Some examples are given to illustrate our findings in this chapter. Then we give constructive proofs of existence of a solution by defining monotone sequences. In the fourth chapter, we derive a continuous form of the Mittag-Leffler function. Then we use successive approximations method to calculate a discrete form of the Mittag-Leffler function. In the fifth chapter, we focus on finding the model which fits best for the data of tumor growth for twenty-eight mice. The models contain either three parameters (Gompertz, Logistic) or four parameters (Weibull, Richards). For each model, we consider continuous, discrete, continuous fractional and discrete fractional forms. Nihan Acar who is a former graduate student in mathematics department has already worked on Gompertz and Logistic models [1]. Here we continue and work on Richards curve. The difference between Acar’s work and ours is the number of parameters in each model. Gompertz and Logistic models contain three parameters and an alpha parameter. The Richards model has four parameters and an alpha parameter. In addition, we use statistical computation techniques such as residual sum of squares and cross-validation to compare fitting and predictive performance of these models. In conclusion, we put three models together to conclude which model is fitting best for the data of tumor growth for twenty-eight mice. In the last chapter, we conclude this thesis and state our future work.