Basic Types of Coarse-Graining

We consider two basic types of coarse-graining: the Ehrenfests' coarse-graining and its extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining based on uncertainty of dynamical models and Epsilon-motions (orbits). Non-technical discussion of basic notions and main coarse-graining theorems are presented: the theorem about entropy overproduction for the Ehrenfests' coarse-graining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for Epsilon-motions of general dynamical systems including structurally unstable systems. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfests' coarse-graining. General theory of reversible regularization and filtering semigroups in kinetics is presented, both for linear and non-linear filters. We obtain explicit expressions and entropic stability conditions for filtered equations. A brief discussion of coarse-graining by rounding and by small noise is also presented.Comment: 60 pgs, 11 figs., includes new analysis of coarse-graining by filtering. A talk given at the research workshop: "Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena," University of Leicester, UK, August 24-26, 200

Digital Filters

The new technology advances provide that a great number of system signals can be easily measured with a low cost. The main problem is that usually only a fraction of the signal is useful for different purposes, for example maintenance, DVD-recorders, computers, electric/electronic circuits, econometric, optimization, etc. Digital filters are the most versatile, practical and effective methods for extracting the information necessary from the signal. They can be dynamic, so they can be automatically or manually adjusted to the external and internal conditions. Presented in this book are the most advanced digital filters including different case studies and the most relevant literature

Spin-polarized Quantum Transport in Mesoscopic Conductors: Computational Concepts and Physical Phenomena

Mesoscopic conductors are electronic systems of sizes in between nano- and micrometers, and often of reduced dimensionality. In the phase-coherent regime at low temperatures, the conductance of these devices is governed by quantum interference effects, such as the Aharonov-Bohm effect and conductance fluctuations as prominent examples. While first measurements of quantum charge transport date back to the 1980s, spin phenomena in mesoscopic transport have moved only recently into the focus of attention, as one branch of the field of spintronics. The interplay between quantum coherence with confinement-, disorder- or interaction-effects gives rise to a variety of unexpected spin phenomena in mesoscopic conductors and allows moreover to control and engineer the spin of the charge carriers: spin interference is often the basis for spin-valves, -filters, -switches or -pumps. Their underlying mechanisms may gain relevance on the way to possible future semiconductor-based spin devices. A quantitative theoretical understanding of spin-dependent mesoscopic transport calls for developing efficient and flexible numerical algorithms, including matrix-reordering techniques within Green function approaches, which we will explain, review and employ.Comment: To appear in the Encyclopedia of Complexity and System Scienc

Passband flattened binary-tree structured add-drop multiplexers using sion waveguide technology

When writing this introduction I saw the following press release on the Internet: “Nielsen//Netratings reports a record half billion people worldwide now have home internet access‿. The number of home users grew worldwide with 5 % over the last quarter of 2001. The growth was nearly doubled compared to Q3 2001. The growth in Europe was 4.9%, almost equal to the world growth. One in three households in Europe/Middle East and Africa have Internet access, compared with over half in the US. The Netherlands has 52 % of the households connected to the Internet and 82 % of the computers is connected to the Internet. Another press release also fromNielsen//Netratings was titled as “Broadband Usage Outpaces Narrowband for the first time.‿ 1.19 billion of the total 2.3 billion hours was spent by broadband surfers online in January 2002 in the US. The broadband time spent in January 2002 was 64 % higher than in January 2001. Nearly 21.9 million surfers (in the US) at-home accessed the Internet via broadband connection in January 2002 compared to 13.1 million in January 2001, a boost of 67% in one year time. So there is an unstoppable march towards broadband. (See www.nielsen-netratings.com) This demand can be fulfilled with the tremendous bandwidth of the optical fiber of 30 THz (1420-1670 nm). It is not possible to directly address this complete band, since the current maximum speed of the electronics and modulators is 40-100 GHZ. Wavelength division multiplexing (WDM) is used to divide the band in multiple sub bands. The spacing between the sub band channels is defined by the ITU grid. Common spacings between channels are 12.5, 25, 50, 100 and 200 GHz. The device that combines these channels onto one fiber is called a Multiplexer (Mux) and the device that does the opposite, spatial separation of frequency channels onto different fibers, is called a demultiplexer (Demux). When Mux and Demux are combined it is possible to select only one (or more) channel to be dropped or added and leaving the remaining channels undisturbed. Such a device is called an Add-drop multiplexer(ADM). Optical transmission systems 3.28 Tbit/s over a few hundred of kilometers[Nielsen 2000] or 2 Tbit/s over almost ten thousand kilometers [Yamada 2002] have already be reported

The Partition Lattice in Many Guises

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-element

Applied Signal Processing

Being an inter-disciplinary subject, Signal Processing has application in almost all scientific fields. Applied Signal Processing tries to link between the analog and digital signal processing domains. Since the digital signal processing techniques have evolved from its analog counterpart, this book begins by explaining the fundamental concepts in analog signal processing and then progresses towards the digital signal processing. This will help the reader to gain a general overview of the whole subject and establish links between the various fundamental concepts. While the focus of this book is on the fundamentals of signal processing, the understanding of these topics greatly enhances the confident use as well as further development of the design and analysis of digital systems for various engineering and medical applications. Applied Signal Processing also prepares readers to further their knowledge in advanced topics within the field of signal processing