Algebraic independence of arithmetic gamma values and Carlitz zeta values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for F_q[theta] and provide complete algebraic independence results for them.Comment: 15 page

Adaptive Complex Contagions and Threshold Dynamical Systems

A broad range of nonlinear processes over networks are governed by threshold dynamics. So far, existing mathematical theory characterizing the behavior of such systems has largely been concerned with the case where the thresholds are static. In this paper we extend current theory of finite dynamical systems to cover dynamic thresholds. Three classes of parallel and sequential dynamic threshold systems are introduced and analyzed. Our main result, which is a complete characterization of their attractor structures, show that sequential systems may only have fixed points as limit sets whereas parallel systems may only have period orbits of size at most two as limit sets. The attractor states are characterized for general graphs and enumerated in the special case of paths and cycle graphs; a computational algorithm is outlined for determining the number of fixed points over a tree. We expect our results to be relevant for modeling a broad class of biological, behavioral and socio-technical systems where adaptive behavior is central.Comment: Submitted for publicatio

The topological glass in ring polymers

We study the dynamics of concentrated, long, semi-flexible, unknotted and unlinked ring polymers embedded in a gel by Monte Carlo simulation of a coarse-grained model. This involves the ansatz that the rings compactify into a duplex structure where they can be modelled as linear polymers. The classical polymer glass transition involves a rapid loss of microscopic freedom within the polymer molecule as the temperature is reduced toward Tg. Here we are interested in temperatures well above Tg where the polymers retain high microscopic mobility. We analyse the slowing of stress relaxation originating from inter-ring penetrations (threadings). For long polymers an extended network of quasi-topological penetrations forms. The longest relaxation time appears to depend exponentially on the ring polymer contour length, reminiscent of the usual exponential slowing (e.g., with temperature) in classical glasses. Finally, we discuss how this represents a universality class for glassy dynamics

Coding for Parallel Links to Maximize Expected Decodable-Message Value

Future communication scenarios for NASA spacecraft may involve multiple communication links and relay nodes, so that there is essentially a network in which there may be multiple paths from a sender to a destination. The availability of individual links may be uncertain. In this paper, scenarios are considered in which the goal is to maximize a payoff that assigns weight based on the worth of data and the probability of successful transmission. Ideally, the choice of what information to send over the various links will provide protection of high value data when many links are unavailable, yet result in communication of significant additional data when most links are available. Here the focus is on the simple network of multiple parallel links, where the links have known capacities and outage probabilities. Given a set of simple inter-link codes, linear programming can be used to find the optimal timesharing strategy among these codes. Some observations are made about the problem of determining all potentially useful codes, and techniques to assist in such determination are presented.Comment: 7 pages, submitted to ISIT200

Coding for Parallel Links to Maximize the Expected Value of Decodable Messages

When multiple parallel communication links are available, it is useful to consider link-utilization strategies that provide tradeoffs between reliability and throughput. Interesting cases arise when there are three or more available links. Under the model considered, the links have known probabilities of being in working order, and each link has a known capacity. The sender has a number of messages to send to the receiver. Each message has a size and a value (i.e., a worth or priority). Messages may be divided into pieces arbitrarily, and the value of each piece is proportional to its size. The goal is to choose combinations of messages to send on the links so that the expected value of the messages decodable by the receiver is maximized. There are three parts to the innovation: (1) Applying coding to parallel links under the model; (2) Linear programming formulation for finding the optimal combinations of messages to send on the links; and (3) Algorithms for assisting in finding feasible combinations of messages, as support for the linear programming formulation. There are similarities between this innovation and methods developed in the field of network coding. However, network coding has generally been concerned with either maximizing throughput in a fixed network, or robust communication of a fixed volume of data. In contrast, under this model, the throughput is expected to vary depending on the state of the network. Examples of error-correcting codes that are useful under this model but which are not needed under previous models have been found. This model can represent either a one-shot communication attempt, or a stream of communications. Under the one-shot model, message sizes and link capacities are quantities of information (e.g., measured in bits), while under the communications stream model, message sizes and link capacities are information rates (e.g., measured in bits/second). This work has the potential to increase the value of data returned from spacecraft under certain conditions

Proximity Full-Text Search with a Response Time Guarantee by Means of Additional Indexes

Full-text search engines are important tools for information retrieval. Term proximity is an important factor in relevance score measurement. In a proximity full-text search, we assume that a relevant document contains query terms near each other, especially if the query terms are frequently occurring words. A methodology for high-performance full-text query execution is discussed. We build additional indexes to achieve better efficiency. For a word that occurs in the text, we include in the indexes some information about nearby words. What types of additional indexes do we use? How do we use them? These questions are discussed in this work. We present the results of experiments showing that the average time of search query execution is 44-45 times less than that required when using ordinary inverted indexes. This is a pre-print of a contribution "Veretennikov A.B. Proximity Full-Text Search with a Response Time Guarantee by Means of Additional Indexes" published in "Arai K., Kapoor S., Bhatia R. (eds) Intelligent Systems and Applications. IntelliSys 2018. Advances in Intelligent Systems and Computing, vol 868" published by Springer, Cham. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-01054-6_66. The work was supported by Act 211 Government of the Russian Federation, contract no 02.A03.21.0006.Comment: Alexander B. Veretennikov. Chair of Calculation Mathematics and Computer Science, INSM. Ural Federal Universit