Tait showed in 1878 that the Four Color Theorem is equivalent to being able to three-color the edges of any planar, three-regular, two-edge connected graph. Not surprisingly, this equivalent problem proved to be equally difficult. We consider the problem of fractional colorings, which resemble ordinary colorings but allow for some degree of cheating. Happily, it is known that every planar three-regular, two-edge connected graph is fractionally three-edge colorable. Is there an analogue to Tait’s Theorem which would allow us to derive the Fractional Four Color Theorem from this edge-coloring result?

In recent years, the field of Fractional Brownian motion, Fractional Gaussian noise and long-range dependent processes has gained growing interest. Fractional Brownian motion is of great interest for example in telecommunications, hydrology and the generation of artificial landscapes. In fact, Fractional Brownian motion is a basic continuous process through which we show that it is neither a semimartingale nor a Markov process. In this work, we will focus on the path properties of Fractional Brownian motion and will try to check the absence of the property of a semimartingale. The concept of volatility will be dealt with in this work as a phenomenon in finance. Moreover, some statistical method like R/S analysis will be presented. By using these statistical tools we examine the volatility of shares and we demonstrate empirically that there are in fact shares which exhibit a fractal structure different from that of Brownian motion.

<p>In recent years, the field of Fractional Brownian motion, Fractional Gaussian noise and long-range dependent processes has gained growing interest. Fractional Brownian motion is of great interest for example in telecommunications, hydrology and the generation of artificial landscapes. In fact, Fractional Brownian motion is a basic continuous process through which we show that it is neither a semimartingale nor a Markov process. In this work, we will focus on the path properties of Fractional Brownian motion and will try to check the absence of the property of a semimartingale. The concept of volatility will be dealt with in this work as a phenomenon in finance. Moreover, some statistical method like R/S analysis will be presented. By using these statistical tools we examine the volatility of shares and we demonstrate empirically that there are in fact shares which exhibit a fractal structure different from that of Brownian motion.</p>

Thesis (Ph. D.)--Illinois State University, 1995. / Title from title page screen, viewed May 2, 2006. Dissertation Committee: John A. Dossey (chair), Stephen H. Friedberg, Beverly S. Rich, Kenneth Strand, Jane O. Swafford. Includes bibliographical references (leaves 93-97) and abstract. Also available in print.

In this thesis we focus on the investigation of the relation between the the chip-firing and fractional coloring. Since chi_{f}(G)=inf {n/k : G is homomorphic to K(n,k)}, we find that G has an (n,k)-periodic sequence for some configuration if and only if G is homomorphic to K(n,k). Then we study the periodic configurations for the Kneser graphs. Finally, we try to evaluate the number of chips of the periodic configurations for K(n,k).

The loop bandwidth of a fractional-N PLL is a desirable parameter for many applications. A wide bandwidth allows a significant attenuation of phase noise arising from the VCO. A good VCO typically requires a high Q LC oscillator. It is difficult to build an on-chip inductor with a high Q factor. In addition, a good VCO also requires a lot of power. Both these design challenges are relaxed with a wide loop bandwidth PLL. However a wide loop bandwidth reduces the effective oversampling ratio (OSR) between the update rate and loop bandwidth and makes quantization noise from the ΔΣ modulator a much bigger noise contributor. A wide band loop also makes the noise and linearity performance of the phase detector more significant. The key to successful implementation of a wideband fractional-N synthesizer is in managing jitter and spurious performance. In this dissertation we present a new PLL architecture for bandwidth extension or phase noise reduction. By using clock squaring buffers with built-in offsets, multiple clock edges are extracted from a single cycle of a sinusoidal reference and used for phase updates, effectively forming a reference frequency multiplier. A higher update rate enables a higher OSR which allows for better quantization noise shaping and makes a wideband fractional-N PLL possible. However since the proposed reference multiplier utilizes the magnitude information from a sinusoidal reference to obtain phases, the derived new edges tend to cluster around the zero-crossings and form an irregular clock. This presents a challenge in lock acquisition. We have demonstrated for the first time that an irregular clock can be used to lock a PLL. The irregularity of the reference clock is taken into account in the divider by adding a cyclic divide pattern along with the ΔΣ control bits, this forces the loop to locally match the incoming patterns and achieve lock. Theoretically this new architecture allows for a 6x increase in loop BW or a 24dB improvement in phase noise. One potential issue associated with the proposed approach is the degraded spurious performance due to PVT variations, which lead to unintended mismatches between the irregular period and the divider pattern. A calibration scheme was invented to overcome this issue. In simulation, the calibration scheme was shown to lower the spurs down to inherent spurs level, of which the total energy is much less than the integrated phase noise. A test chip for proof of concept is presented and measurements are carefully analyzed. / text

Thesis (Ph.D.)--University of Tulsa, 1974. / Bibliography: leaves 40-44.

Partial differential equations (PDEs) are used to describe multi-dimensional physical phenomena. However, some of these phenomena are described by a more general class of systems called fractional systems. Indeed, fractional calculus has emerged as a new tool for modeling complex phenomena thanks to the memory and hereditary properties of fraction derivatives. In this thesis, we explore a class of controllers and estimators that respond to some control and estimation challenges for both PDE and FPDE. We first propose a backstepping controller for the flow control of a first-order hyperbolic PDE modeling the heat transfer in parabolic solar collectors. While backstepping is a well-established method for boundary controlled PDEs, the process is less straightforward for in-domain controllers. One of the main contributions of this thesis is the development of a new integral transformation-based control algorithm for the study of reference tracking problems and observer designs for fractional PDEs using the extended backstepping approach. The main challenge consists of the proof of stability of the fractional target system, which utilizes either an alternative Lyapunov method for time FPDE or a fundamental solution for the error system for reference tracking, and observer design of space FPDE. Examples of applications involving reference tracking of FPDEs are gas production in fractured media and solute transport in porous media. The designed controllers, require knowledge of some system’s parameters or the state. However, these quantities may be not measurable, especially, for space-evolving PDEs. Therefore, we propose a non-asymptotic and robust estimation algorithm based on the so-called modulating functions. Unlike the observers-based methods, the proposed algorithm has the advantage that it converges in a finite time. This algorithm is extended for the state estimation of linear and non-linear PDEs with general non-linearity. This algorithm is also used for the estimation of parameters and disturbances for FPDEs. This thesis aims to design an integral transformation-based algorithm for the control and estimation of PDEs and FDEs. This transformation is defined through a suitably designed function that transforms the identification problem into an algebraic system for non-asymptotic estimation purposes. It also maps unstable systems to stable systems to achieve control goals.

As an extension of the geometric Brownian motion, a geometric fractional Brownian motion (GFBM) is considered as a stock-price model. The modeled GFBM is compared with empirical Chinese stock prices. Comparisons are performed by considering logarithmic-return densities, autocovariance functions, spectral densities and trajectories. Since logarithmic-return densities of GFBM stock prices are Gaussian and empirical stock logarithmic-returns typically are far from Gaussian, a GFBM model may not be the most suitable stock price model.

This thesis deals with developing Galerkin based solution strategies for several important classes of differential equations involving derivatives and integrals of various fractional orders. Fractional order calculus finds use in several areas of science and engineering. The use of fractional derivatives may arise purely from the mathematical viewpoint, as in controller design, or it may arise from the underlying physics of the material, as in the damping behavior of viscoelastic materials. The physical origins of the fractional damping motivated us to study viscoelastic behavior of disordered materials at three levels. At the first level, we review two first principles models of rubber viscoelasticity. This leads us to study, at the next two levels, two simple disordered systems. The study of these two simplified systems prompted us towards an infinite dimensional system which is mathematically equivalent to a fractional order derivative or integral. This infinite dimensional system forms the starting point for our Galerkin projection based approximation scheme. In a simplified study of disordered viscoelastic materials, we show that the networks of springs and dash-pots can lead to fractional power law relaxation if the damping coefficients of the dash-pots follow a certain type of random distribution. Similar results are obtained when we consider a more simplified model, which involves a random system coefficient matrix. Fractional order derivatives and integrals are infinite dimensional operators and non-local in time: the history of the state variable is needed to evaluate such operators. This non-local nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Following this, we identify eight important classes of fractional differential equations (FDEs) and fractional integrodifferential equations (FIEs), and develop separate Galerkin based solution strategies for each of them. Distinction between these classes arises from the fact that both Riemann-Liouville as well as Caputo type derivatives used in this work do not, in general, follow either the law of exponents or the commutative property. Criteria used to identify these classes include; the initial conditions used, order of the highest derivative, integer or fractional order highest derivative, single or multiterm fractional derivatives and integrals. A key feature of our approximation scheme is the development of differential algebraic equations (DAEs) when the highest order derivative is fractional or the equation involves fractional integrals only. To demonstrate the effectiveness of our approximation scheme, we compare the numerical results with analytical solutions, when available, or with suitably developed series solutions. Our approximation scheme matches analytical/series solutions very well for all classes considered.