Communications in Mathematical Physics, Jun 1, 1978
We prove the existence of the limit Gibbs state for one-dimensional continuous quantum fermion sy... more We prove the existence of the limit Gibbs state for one-dimensional continuous quantum fermion systems with non-hard-core, non-negative, rapidly decreasing pair interaction potentials. Existence of the limit Gibbs state is also established for one-dimensional continuous quantum boson systems with pair interaction potentials as above which, in addition, increase sufficiently fast at small distances.
We analyse an analog of the entropy-power inequality for the weighted entropy. In particular, we ... more We analyse an analog of the entropy-power inequality for the weighted entropy. In particular, we discuss connections with weighted Lieb's splitting inequality and an Gaussian additive noise formula. Examples and counterexamples are given, for some classes of probability distributions.
This paper represents an extended version of an earlier note [10]. The concept of weighted entrop... more This paper represents an extended version of an earlier note [10]. The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy contextdependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy power inequality for the weighted entropy and discuss connections with weighted Lieb's splitting inequality. The concepts of rates of the weighted entropy and information are also discussed.
Following a series of works on capital growth investment, we analyse log-optimal portfolios where... more Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes 'weights' of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme. An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses. The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study.
Abstract B71: Stochastic model of contact inhibition and the proliferation of melanoma in situ
Clinical Cancer Research, 2018
Contact inhibition of proliferation can be described as the decrease of proliferation rates when ... more Contact inhibition of proliferation can be described as the decrease of proliferation rates when the cell density increases. Its loss is a key feature of tumor development. Cell density of human metastatic melanoma cell line (SK-MEL-147) and immortalized keratinocyte cells (HaCaT) was evaluated daily in culture experiments. Keratinocytes reach lower cell density than melanoma cells at confluence, indicating that the population growth arrest is associated with contact inhibition (KHaCaT = 1779.6 ± 130.5 cells/mm2; KSK-MEL-147 = 5043.5 ± 316.5 cells/mm2). We performed a coculture experiment to assess the proliferation restriction between both cell lines. After 8 days a spatial pattern was detected, characterized by the formation of clusters of melanoma cells surrounded by keratinocytes constraining their proliferation. Despite showing approximately the same proliferation rates when data were fitted with a logistic model (ρHaCaT = 1.1 ± 0.1 days-1; ρSK-MEL-147 = 1.1 ± 0.3 days-1), the ...
A number of inequalities for the weighted entropies is proposed, mirroring properties of a standa... more A number of inequalities for the weighted entropies is proposed, mirroring properties of a standard (Shannon) entropy and related quantities.
The aim of this paper is to analyze the weighted KyFan inequality proposed in [11]. A number of n... more The aim of this paper is to analyze the weighted KyFan inequality proposed in [11]. A number of numerical simulations involving the exponential weighted function is given. We show that in several cases and types of examples one can imply an improvement of the standard KyFan inequality.
We study spectral properties of a system of two quantum particles on an integer lattice with a bo... more We study spectral properties of a system of two quantum particles on an integer lattice with a bounded short-range two-body interaction, in an external random potential field $V(x,\omega)$ with independent, identically distributed values. The main result is that if the common probability density $f$ of random variables $V(x,\omega)$ is analytic in a strip around the real line and the amplitude constant $g$ is large enough (i.e. the system is at high disorder), then, with probability one, the spectrum of the two-particle lattice Schroedinger operator $H(\omega)$ (bosonic or fermionic) is pure point, and all eigen-functions decay exponentially. The proof given in this paper is based on a refinement of a multiscale analysis (MSA) scheme proposed by von Dreifus and Klein, adapted to incorporate lattice systems with interaction.
This chapter outlines physical origins and the development of rigorous mathematical methods of th... more This chapter outlines physical origins and the development of rigorous mathematical methods of the Anderson localization theory, describing unusual propagation properties of quantum particles (as well as electromagnetic and acoustic waves) in disordered media. While the main scope of the book is restricted to the analysis of Anderson localization in a strongly disordered environment, Chap. 1 gives the reader a broad perspective and indicates directions for possible future research in the area of multi-particle localization theory.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
This note addresses the issue of the proof of the entropy power inequality (EPI), an important to... more This note addresses the issue of the proof of the entropy power inequality (EPI), an important tool in the analysis of Gaussian channels of information transmission, proposed by Shannon. We analyse continuity properties of the mutual entropy of the input and output signals in an additive memoryless channel and show how this can be used for a correct proof of the entropy-power inequality.
Cambridge University Mathematical Tripos examination questions in IB Statistics (1992–1999)
Probability and Statistics by Example
Further Topics from Coding Theory
Information Theory and Coding by Example
Tables of random variables and probability distributions
Probability and Statistics by Example
Probability and Statistics by Example
Probability and Statistics are as much about intuition and problem solving as they are about theo... more Probability and Statistics are as much about intuition and problem solving as they are about theorem proving. Because of this, students can find it very difficult to make a successful transition from lectures to examinations to practice, since the problems involved can vary so much in nature. Since the subject is critical in many modern applications such as mathematical finance, quantitative management, telecommunications, signal processing, bioinformatics, as well as traditional ones such as insurance, social science andengineering, the authors have rectified deficiencies in traditional lecture-based methods by collecting together a wealth of exercises with complete solutions, adapted to needs and skills of students. Following on from the success of Probability and Statistics by Example: Basic Probability and Statistics, the authors here concentrate on random processes, particularly Markov processes, emphasising modelsrather than general constructions. Basic mathematical facts are supplied as and when they are needed andhistorical information is sprinkled throughout.
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