Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)

The generating series of a radix-rational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radix-rational sequence with complex values we provide an asymptotic expansion for the sequence of its Ces\`aro means. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence; the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra

The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have been previously enumerated by length and perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find that the generating function is expressed in terms of a $q$-hypergeometric function, with an accumulation of poles towards the dominant singularity. This expression reveals an unusual asymptotic structure of the number of polygons of area $n$, where the critical exponent is the transcendental number $\log_23$ and and the amplitude involves tiny oscillations. Based on numerical data, we also expect similar phenomena to occur for 4-sided polygons. The asymptotic methodology involves an original combination of Mellin transform techniques and singularity analysis, which is of potential interest in a number of other asymptotic enumeration problems.Comment: 27 pages, 6 figure

Automatic sequences as good weights for ergodic theorems

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$.Comment: 31 page

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Zolotarev polynomials utilization in spectral analysis

Tato práce je zaměřena na vybrané problémy Zolotarevových polynomů a jejich vyuľití ke spektrální analýze. Pokud jde o Zolotarevovy polynomy, jsou popsány základní vlastnosti symetrických Zolotarevových polynomů včetně ortogonality. Rovněľ se provádí prozkoumání numerických vlastností algoritmů generujících dokonce Zolotarevovy polynomy. Pokud jde o aplikaci Zolotarevových polynomů na spektrální analýzu, je implementována aproximovaná diskrétní Zolotarevova transformace, která umoľňuje výpočet spektrogramu (zologramu) v reálném čase. Aproximovaná diskrétní zolotarevská transformace je navíc upravena tak, aby lépe fungovala při analýze tlumených exponenciálních signálů. A nakonec je navrľena nová diskrétní Zolotarevova transformace implementovaná plně v časové oblasti. Tato transformace také ukazuje, ľe některé rysy pozorované u aproximované diskrétní Zolotarevovy transformace jsou důsledkem pouľití Zolotarevových polynomů.This thesis is focused on selected problems of symmetrical Zolotarev polynomials and their use in spectral analysis. Basic properties of symmetrical Zolotarev polynomials including orthogonality are described. Also, the exploration of numerical properties of algorithms generating even Zolotarev polynomials is performed. As regards to the application of Zolotarev polynomials to spectral analysis the Approximated Discrete Zolotarev Transform is implemented so that it enables computing of zologram in real–time. Moreover, the Approximated Discrete Zolotarev Transform is modified to perform better in the analysis of damped exponential signals. And finally, a novel Discrete Zolotarev Transform implemented fully in the time domain is suggested. This transform also shows that some features observed using the Approximated Discrete Zolotarev Transform are a consequence of using Zolotarev polynomials

Fractal image compression and the self-affinity assumption : a stochastic signal modelling perspective

Bibliography: p. 208-225.Fractal image compression is a comparatively new technique which has gained considerable attention in the popular technical press, and more recently in the research literature. The most significant advantages claimed are high reconstruction quality at low coding rates, rapid decoding, and "resolution independence" in the sense that an encoded image may be decoded at a higher resolution than the original. While many of the claims published in the popular technical press are clearly extravagant, it appears from the rapidly growing body of published research that fractal image compression is capable of performance comparable with that of other techniques enjoying the benefit of a considerably more robust theoretical foundation. . So called because of the similarities between the form of image representation and a mechanism widely used in generating deterministic fractal images, fractal compression represents an image by the parameters of a set of affine transforms on image blocks under which the image is approximately invariant. Although the conditions imposed on these transforms may be shown to be sufficient to guarantee that an approximation of the original image can be reconstructed, there is no obvious theoretical reason to expect this to represent an efficient representation for image coding purposes. The usual analogy with vector quantisation, in which each image is considered to be represented in terms of code vectors extracted from the image itself is instructive, but transforms the fundamental problem into one of understanding why this construction results in an efficient codebook. The signal property required for such a codebook to be effective, termed "self-affinity", is poorly understood. A stochastic signal model based examination of this property is the primary contribution of this dissertation. The most significant findings (subject to some important restrictions} are that "self-affinity" is not a natural consequence of common statistical assumptions but requires particular conditions which are inadequately characterised by second order statistics, and that "natural" images are only marginally "self-affine", to the extent that fractal image compression is effective, but not more so than comparable standard vector quantisation techniques

Design and evaluation of the Hamal parallel computer

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2003."December 2002."Includes bibliographical references (p. 145-152).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Parallel shared-memory machines with hundreds or thousands of processor-memory nodes have been built; in the future we will see machines with millions or even billions of nodes. Associated with such large systems is a new set of design challenges. Many problems must be addressed by an architecture in order for it to be successful; of these, we focus on three in particular. First, a scalable memory system is required. Second, the network messaging protocol must be fault-tolerant. Third, the overheads of thread creation, thread management and synchronization must be extremely low. This thesis presents the complete system design for Hamal, a shared-memory architecture which addresses these concerns and is directly scalable to one million nodes. Virtual memory and distributed objects are implemented in a manner that requires neither inter-node synchronization nor the storage of globally coherent translations at each node. We develop a lightweight fault-tolerant messaging protocol that guarantees message delivery and idempotence across a discarding network. A number of hardware mechanisms provide efficient support for massive multithreading and fine-grained synchronization.(cont.) Experiments are conducted in simulation, using a trace-driven network simulator to investigate the messaging protocol and a cycle-accurate simulator to evaluate the Hamal architecture. We determine implementation parameters for the messaging protocol which optimize performance. A discarding network is easier to design and can be clocked at a higher rate, and we find that with this protocol its performance can approach that of a non-discarding network. Our simulations of Hamal demonstrate the effectiveness of its thread management and synchronization primitives. In particular, we find register-based synchronization to be an extremely efficient mechanism which can be used to implement a software barrier with a latency of only 523 cycles on a 512 node machine.by J.B. Grossman.Ph.D

Design and Evaluation of the Hamal Parallel Computer

Parallel shared-memory machines with hundreds or thousands of processor-memory nodes have been built; in the future we will see machines with millions or even billions of nodes. Associated with such large systems is a new set of design challenges. Many problems must be addressed by an architecture in order for it to be successful; of these, we focus on three in particular. First, a scalable memory system is required. Second, the network messaging protocol must be fault-tolerant. Third, the overheads of thread creation, thread management and synchronization must be extremely low. This thesis presents the complete system design for Hamal, a shared-memory architecture which addresses these concerns and is directly scalable to one million nodes. Virtual memory and distributed objects are implemented in a manner that requires neither inter-node synchronization nor the storage of globally coherent translations at each node. We develop a lightweight fault-tolerant messaging protocol that guarantees message delivery and idempotence across a discarding network. A number of hardware mechanisms provide efficient support for massive multithreading and fine-grained synchronization. Experiments are conducted in simulation, using a trace-driven network simulator to investigate the messaging protocol and a cycle-accurate simulator to evaluate the Hamal architecture. We determine implementation parameters for the messaging protocol which optimize performance. A discarding network is easier to design and can be clocked at a higher rate, and we find that with this protocol its performance can approach that of a non-discarding network. Our simulations of Hamal demonstrate the effectiveness of its thread management and synchronization primitives. In particular, we find register-based synchronization to be an extremely efficient mechanism which can be used to implement a software barrier with a latency of only 523 cycles on a 512 node machine