Artist Study: The Compositional Style of Jazz Guitarist Nathen Page

For this project, I have composed three and arranged two compositions for jazz quartet in the style of Page. The featured instrumentation will be guitar, piano, drums, and bass, which is the same instrumentation that Page had used almost exclusively since he first formed his own group. In preparation for writing my compositions and arrangements, I first had to learn Page’s compositions and arrangements by transcribing them from his recordings. In presenting my compositions/arrangements, I will first present the Page composition that my work will be derived from, along with a short written explanation of the song. Then I will present my own work, along with an explanation of how exactly I derived it from the preceding Page composition

Temperature of nonextensive system: Tsallis entropy as Clausius entropy

The problem of temperature in nonextensive statistical mechanics is studied. Considering the first law of thermodynamics and a "quasi-reversible process", it is shown that the Tsallis entropy becomes the Clausius entropy if the inverse of the Lagrange multiplier, $beta$, associated with the constraint on the internal energy is regarded as the temperature. This temperature is different from the previously proposed "physical temperature" defined through the assumption of divisibility of the total system into independent subsystems. A general discussion is also made about the role of Boltzmann's constant in generalized statistical mechanics based on an entropy, which, under the assumption of independence, is nonadditive.Comment: 14 pages, no figure

A step beyond Tsallis and Renyi entropies

Tsallis and R\'{e}nyi entropy measures are two possible different generalizations of the Boltzmann-Gibbs entropy (or Shannon's information) but are not generalizations of each others. It is however the Sharma-Mittal measure, which was already defined in 1975 (B.D. Sharma, D.P. Mittal, J.Math.Sci \textbf{10}, 28) and which received attention only recently as an application in statistical mechanics (T.D. Frank & A. Daffertshofer, Physica A \textbf{285}, 351 & T.D. Frank, A.R. Plastino, Eur. Phys. J., B \textbf{30}, 543-549) that provides one possible unification. We will show how this generalization that unifies R\'{e}nyi and Tsallis entropy in a coherent picture naturally comes into being if the q-formalism of generalized logarithm and exponential functions is used, how together with Sharma-Mittal's measure another possible extension emerges which however does not obey a pseudo-additive law and lacks of other properties relevant for a generalized thermostatistics, and how the relation between all these information measures is best understood when described in terms of a particular logarithmic Kolmogorov-Nagumo average

Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.Comment: LaTeX, 22 pages, 2 figure

Stability of Tsallis antropy and instabilities of Renyi and normalized Tsallis entropies: A basis for q-exponential distributions

The q-exponential distributions, which are generalizations of the Zipf-Mandelbrot power-law distribution, are frequently encountered in complex systems at their stationary states. From the viewpoint of the principle of maximum entropy, they can apparently be derived from three different generalized entropies: the Renyi entropy, the Tsallis entropy, and the normalized Tsallis entropy. Accordingly, mere fittings of observed data by the q-exponential distributions do not lead to identification of the correct physical entropy. Here, stabilities of these entropies, i.e., their behaviors under arbitrary small deformation of a distribution, are examined. It is shown that, among the three, the Tsallis entropy is stable and can provide an entropic basis for the q-exponential distributions, whereas the others are unstable and cannot represent any experimentally observable quantities.Comment: 20 pages, no figures, the disappeared "primes" on the distributions are added. Also, Eq. (65) is correcte

Statistical mechanical foundations of power-law distributions

The foundations of the Boltzmann-Gibbs (BG) distributions for describing equilibrium statistical mechanics of systems are examined. Broadly, they fall into: (i) probabilistic paaroaches based on the principle of equal a priori probability (counting technique and method of steepest descents), law of large numbers, or the state density considerations and (ii) a variational scheme -- maximum entropy principle (due to Gibbs and Jaynes) subject to certain constraints. A minimum set of requirements on each of these methods are briefly pointed out: in the first approach, the function space and the counting algorithm while in the second, "additivity" property of the entropy with respect to the composition of statistically independent systems. In the past few decades, a large number of systems, which are not necessarily in thermodynamic equilibrium (such as glasses, for example), have been found to display power-law distributions, which are not describable by the above-mentioned methods. In this paper, parallel to all the inquiries underlying the BG program described above are given in a brief form. In particular, in the probabilistic derivations, one employs a different function space and one gives up "additivity" in the variational scheme with a different form for the entropy. The requirement of stability makes the entropy choice to be that proposed by Tsallis. From this a generalized thermodynamic description of the system in a quasi-equilibrium state is derived. A brief account of a unified consistent formalism associated with systems obeying power-law distributions precursor to the exponential form associated with thermodynamic equilibrium of systems is presented here.Comment: 19 pages, no figures. Invited talk at Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, Santa Fe, USA, November 6-9, 200

Distributivity and deformation of the reals from Tsallis entropy

We propose a one-parameter family \ $\mathbb{R}_q$ \ of deformations of the reals, which is motivated by the generalized additivity of the Tsallis entropy. We introduce a generalized multiplication which is distributive with respect to the generalized addition of the Tsallis entropy. These operations establish a one-parameter family of field isomorphisms \ $\tau_q$ \ between \ $\mathbb{R}$ \ and \ $\mathbb{R}_q$ \ through which an absolute value on \ $\mathbb{R}_q$ \ is introduced. This turns out to be a quasisymmetric map, whose metric and measure-theoretical implications are pointed out.Comment: 16 pages, Standard LaTeX2e, To be published in Physica

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy $S_{q}=k\frac{1-\sum\limits_{i=1}^{W} p_{i} ^{q}}{1-q} (q\in \Re)$ ($S_{1} \equiv S_{BG}=-k\sum\limits_{i=1}^{W}p_{i} \ln p_{i}$). More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices $q$ in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation $q_{stat}+q_{sens}=2$ is numerically satisfied, $q_{stat}$ and $q_{sens}$ being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.Comment: Invited paper to appear in special issue of Eur. Phys. J. B dedicated to econophysics, edited by T. Di Matteo and T. Aste. 7 page

Towards information theory for q-nonextensive statistics without q-deformed distributions

In this paper we extend our recent results [Physica A340 (2004)110] on q-nonextensive statistics with non-Tsallis entropies. In particular, we combine an axiomatics of Renyi with the q-deformed version of Khinchin axioms to obtain the entropy which accounts both for systems with embedded self-similarity and q-nonextensivity. We find that this entropy can be uniquely solved in terms of a one-parameter family of information measures. The corresponding entropy maximizer is expressible via a special function known under the name of the Lambert W-function. We analyze the corresponding "high" and "low-temperature" asymptotics and make some remarks on the possible applications.Comment: Presented at Next2005, uses Elsevier LaTeX macros, revised version with minor changes, accepted to Physica

A smart polymer for sequence-selective binding, pulldown, and release of DNA targets

Selective isolation of DNA is crucial for applications in biology, bionanotechnology, clinical diagnostics and forensics. We herein report a smart methanol-responsive polymer (MeRPy) that can be programmed to bind and separate single- as well as double-stranded DNA targets. Captured targets are quickly isolated and released back into solution by denaturation (sequence-agnostic) or toehold-mediated strand displacement (sequence-selective). The latter mode allows 99.8% efficient removal of unwanted sequences and 79% recovery of highly pure target sequences. We applied MeRPy for the depletion of insulin, glucagon, and transthyretin cDNA from clinical next-generation sequencing (NGS) libraries. This step improved the data quality for low-abundance transcripts in expression profiles of pancreatic tissues. Its low cost, scalability, high stability and ease of use make MeRPy suitable for diverse applications in research and clinical laboratories, including enhancement of NGS libraries, extraction of DNA from biological samples, preparative-scale DNA isolations, and sorting of DNA-labeled non-nucleic acid targets. © 2020, The Author(s)