A mathematical theory of the value of information

An algorithm is given for finding where the variables of a program are active or live. While the algorithm is modeled after that of Kennedy, it is based on the flow graph straightening procedure of Earnest, Balke, and Anderson, rather than on Cocke-Allen intervals. Thus it can be applied to any program, without appeal to any additional mechanism such as node splitting.     

Processor sharing: A survey of the mathematical theory

During last few decades the Egalitarian Processor Sharing (EPS) has gained a prominent role in applied probability, especially, in queueing theory and its computer applications. While the EPS paradigm emerged in 1967 as an idealization of round-robin (RR) scheduling algorithm in time-sharing computer systems, it has recently capture renewed interest as a powerful concept for modeling WEB servers. This paper summarizes the most important results concerning the exact solutions for the M/GI/1 queue with egalitarian processor sharing. The material is drawn, mainly, from recent authors’ papers which are supplemented, in small degree, by other related results. Many of the further results are established under the direct influence of our earlier papers. Our main purpose is to give a survey of state-of-the-art with regard to main achievements of the contemporary theory of the M/GI/1 queueing system with processor sharing. The focus is on the methods and techniques of exact and asymptotic analysis of this queueing system. In contrast to the standard surveys, the abridged proofs (or their ideas) of some key theorems and corollaries are included in the paper. We outline recent developments in exact analysis of the M/GI/1-EPS queue with further emphasis on time-dependent (transient) probability distributions of the main characteristics. In particular, the present paper includes the results on the joint time-dependent distribution of the sojourn time of a job arriving at time t with the service demand (length) u, and of the number of jobs at time t- in the M/GI/1 queue with egalitarian processor sharing, which obtained in form of multiple transforms. We also show how the non-stationary solutions can be used to obtain known and new results which allow to predict the behaviour of the EPS queue and to yield additional insights into its new unexpected properties. We also discuss a number of limit theorems arising under the study of the processor sharing queues.     

A theory of the motion fields of curves

This article reports a study of the motion field generated by moving 3-D curves that are observed by a camera. We first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend.We then go ahead to study the motion field of a general curve. We first study the general case of a curve moving nonrigidly and introduce the notion of isometric motion. In order to do this, we introduce the notion of spatiotemporal surface and study its differential properties up to the second order. We show that, contrary to what is commonly believed, the full motion field of the curve (i.e., the component tangent to the curve) cannot be recovered from this surface. We also give the equations that characterize the spatio-temporal surface completely up to a rigid transformation. Those equations are the expressions of the first and second fundamental forms and the Gauss and Codazzi-Mainardi equations. We then relate those differential expressions computed on the spatio-temporal surface to quantities that can be computed from the images intensities. The actual values depend upon the choice of the edge detector.We then show that the hypothesis of a rigid 3-D motion allows in general to recover the structure and the motion of the curve, in fact without explicitly computing the tangential motion field, at the cost of introducing the three-dimensional accelerations. We first study the motion field generated by the simplest kind of rigid 3-D curves, namely lines. This study is illuminating in that it paves the way for the study of general rigid curves and because of the useful results which are obtained. We then extend the results obtained in the case of lines to the case of general curves and show that at each point of the image curve two equations can be written relating the kinematic screw of the moving 3-D curve and its time derivative to quantities defined in the study of the general nonrigid motion that can be measured from the spatio-temporal surface and therefore from the image. This shows that the structure and the motion of the curve can be recovered from six image points only, without establishing any point correspondences.Finally we study the cooperation between motion and stereo in the framework of this theory. The use of two cameras instead of one allows us to get rid of the three-dimensional accelerations and the relations between the two spatio-temporal surfaces of the same rigidly moving 3-D curve can be used to help disambiguate stereo correspondences.     

A new stage in mathematical teletraffic theory

History of the development of theory of queues and mathematical teletraffic theory achieved at the MSU and PFUR are outlined. Basic theorems and algorithms for calculation of probability measures for multiservice loss networks with unicast connections, multicast connections, as well as with both types of connections are presented. A Triple Play network analysis is determined to be the key task for the future.     

A mathematical analysis of the cumulus pricing scheme

One important task in current and future communication networks is to define a suitable pricing scheme. It is then preferable to formulate a mathematical model, so that parameters will be optimized and important properties such as fairness or truthful anticipated load revelation (or incentive compatibility) will be verified. In this paper we study a simple and promising scheme called the cumulus pricing scheme, which can address service differentiation and scalability among other issues. Based on a mathematical model, we determine values for optimizing the provider’s revenue, which happens under the constraint that each user has an incentive to reveal its anticipated load. This has led to a small variation of the initial model from the literature as in the modelling, cumulus points are translated into financial terms, and measurements induce a cost as well.     

Probabilistic motion estimation based on temporal coherence

We develop a theory for the temporal integration of visual motion motivated by psychophysical experiments. The theory proposes that input data are temporally grouped and used to predict and estimate the motion flows in the image sequence. This temporal grouping can be considered a generalization of the data association techniques that engineers use to study motion sequences. Our temporal grouping theory is expressed in terms of the Bayesian generalization of standard Kalman filtering. To implement the theory, we derive a parallel network that shares some properties of cortical networks. Computer simulations of this network demonstrate that our theory qualitatively accounts for psychophysical experiments on motion occlusion and motion outliers. In deriving our theory, we assumed spatial factorizability of the probability distributions and made the approximation of updating the marginal distributions of velocity at each point. This allowed us to perform local computations and simplified our implementation. We argue that these approximations are suitable for the stimuli we are considering (for which spatial coherence effects are negligible).     

A mathematical theory of evolution by heterochrony in colonial invertebrates

This work provides a foundation for a quantitative dynamical theory of heterochronic processes in The Evolution of Colonial Animals including Bryozoans, Siphonophores and Social Insects. These are environmentally induced changes in the time-sequencing of growth and development of a colonial individual (e.g. Ant colony, Portuguese Man-of-War, etc.) which can, via the Finsler differential geometric theory of Wagner transformations, produce alterations in social interactions. This mathematical theory predicts the usual Paleontological Allometric relationships between fossil morphospecies and their ancestors. Emphasis is on the extensive Bryozoan fossil record because of its great importance in establishing The Theory of Punctuated Evolution of N. Eldredge and S. J. Gould. But, facultative neoteny, as occurs in Salamanders is also described by this model, as is behavioral switching in Ants.Partially supported by NSERC A-7667.     

A mathematical analysis of improved EigenAnt algorithm

As a variant of Ant Colony Optimization, the EigenAnt algorithm finds the shortest path between a source node and a destination node based on negative feedback in the form of selective pheromone removal that occurs only on the path which is actually chosen for each trip. EigenAnt algorithm also could change quickly to reflect to the dynamic variety of initial pheromone concentrations and path length etc. However, in general, the solution of EigenAnt algorithm is not always convergent. In this paper, we propose an improved EigenAnt (iEigenAnt) algorithm in terms of both negative and positive feedback; that is, selective pheromone updates are decided by smart ants or stupid ones, which depends whether the amount of the pheromone at the selected path increases or not. The system modelled by our algorithm has a unique equilibrium as the shortest path. Besides, using mathematical analysis, we demonstrate that the equilibrium is global asymptotically stable, i.e., stable and convergent. Finally, we also implement the iEigenAnt algorithm under four different cases and apply it on travelling salesman problem problem, the simulation result shows that our iEigenAnt algorithm is faster convergent and more effective compared to the original EigenAnt algorithm, and some combinatorial optimisation problems can be effectively solved based on our iEigenAnt algorithm.     

Silhouette coherence for camera calibration under circular motion

We present a new approach to camera calibration as a part of a complete and practical system to recover digital copies of sculpture from uncalibrated image sequences taken under turntable motion. In this paper, we introduce the concept of the silhouette coherence of a set of silhouettes generated by a 3D object. We show how the maximization of the silhouette coherence can be exploited to recover the camera poses and focal length. Silhouette coherence can be considered as a generalization of the well-known epipolar tangency constraint for calculating motion from silhouettes or outlines alone. Further, silhouette coherence exploits all the geometric information encoded in the silhouette (not just at epipolar tangency points) and can be used in many practical situations where point correspondences or outer epipolar tangents are unavailable. We present an algorithm for exploiting silhouette coherence to efficiently and reliably estimate camera motion. We use this algorithm to reconstruct very high quality 3D models from uncalibrated circular motion sequences, even when epipolar tangency points are not available or the silhouettes are truncated. The algorithm has been integrated into a practical system and has been tested on more than 50 uncalibrated sequences to produce high quality photo-realistic models. Three illustrative examples are included in this paper. The algorithm is also evaluated quantitatively by comparing it to a state-of-the-art system that exploits only epipolar tangents     

A mathematical analysis of the vibration characteristics of the human tibia

The frequencies and damping coefficients of a vibrating human tibia are studied in this paper, by modelling the tibia as a truncated conical structure. The experimentally observed principal mechanical properties of osseous tissues, viz. anisotropy, nonhomogeneity and material damping have been duly accounted for in the analysis. The applicability of the model has been illustrated through numerical computation of the derived analytical expressions. The computed values are presented in tabular form.     

One problem of mathematical optimal control theory

The necessary conditions obtained by the author in a previous study for the extremum of a functional that depends on the control function on the compact set and on a -measurable subset in (where is the ordinary Lebesgue measure) subject to functional equality and inequality constraints are generalized here to an arbitrary continuous Radon measure.Translated from Kibernetika, No. 4, pp. 69–73, July–August, 1990.     

Towards a theory of mathematical operational semantics

Turi and Plotkin gave a precise mathematical formulation of a notion of structural operational semantics in their paper “Towards a mathematical operational semantics.” Starting from that definition and at the level of generality of that definition, we give a mathematical formulation of some of the basic constructions one makes with structural operational semantics. In particular, given a single-step operational semantics, as is the spirit of their work, one composes transitions and considers streams of transitions in order to study the dynamics induced by the operational semantics. In all their leading examples, it is obvious that one can do that and it is obvious how to do it. But if their definition is to be taken seriously, one needs to be able to make such constructions at the level of generality of their definition rather than case-by-case. So this paper does so for several of the basic constructions associated with structural operational semantics, in particular those required in order to speak of a stream of transitions and hence of dynamics.     

Some mathematical questions of theory of information transmission

A survey talk given by M.S. Pinsker at the Summer Seminar on Information Theory and Statistical Methods of Automatic Control, Prague, May 25–June 4, 1965, was published in Kybernetika journal of the Czech Academy of Sciences (1966, vol. 2, no. 2, pp. 117–146). By the courtesy of the editors of Kybernetika, we are allowed to publish an abridged version of the talk, which concisely and clearly exposes final mathematical results that underlie Shannon’s information theory. The content of the talk is deeply intertwined with the author’s works. The talk is almost unknown to Russian readers, and with this publication we fill this gap, thereby paying tribute to our outstanding scientist and friend.     

基于数学建模理论的问题研究与分析

数学模型是架于数学与实际问题之间的桥梁,在数学发展的进程中无时无刻不留下数学模型的印记。数学模型是一种数学的思维方法,用数学语言和方法来抽象简化实际问题,以便于实际问题的解决。数学建模不仅是应用数学解决实际问题的重要工具,而且是揭示基本自然规律,产生新的数学思想和方法的重要途经。文章用实例介绍了数学建模理论及其应用。     

A mathematical model for transient analysis of computer systems

The study of computer system dynamic behavior is a prerequisite for the design and implementation of automatic mechanisms for performance control. An analytic technique for modelling the transient behavior of computer systems is presented and a suitable method for modelling job dynamics is given.

The system model is discussed from the viewpoint of transient analysis with particular reference to bottleneck identification and to bottleneck migration analysis.

A mathematical study is given together with a numerical algorithm. The model is validated on the basis of suitable experimental results.