Lecture Notes in Economics and Mathematical Systems

(1) Models of mean field, Hamilton-Jacobi-Bellman Equations and numerical methods. Yves Achdou. (2) First-order Hamilton-Jacobi Equations and Applications. Guy Barles. (3) Basic properties of viscosity solutionsand some aspects of weak KAM theory. Hitoshi Ishii. (4) Idempotent/Tropical Analysis and the Hamilton-Jacobi-Bellman Equations. Grigory L. Litvinov. (5) Homogenization and Approximation for Hamilton-Jacobi equations. Panagiotis E. Souganidis.

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Hamilton-Jacobi equations: approximations, numerical analysis and applications

The complicated nature of calculations in general relativity was one of the driving forces in the early development of computer algebra (CA). CA has become widely used in gravity research (GR) and its use can be expected to grow further. Here the general nature of computer algebra is discussed, along with some aspects of CA system design; features particular to GR’s requirements are considered; information on packages for CA in GR is provided, both for those packages currently available and for their predecessors; and applications of CA in GR are outlined.

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Computer algebra in gravity research

Pulse width influence in fast frequency measurements using rational approximations

Weighted labelled transition systems are LTSs whose transitions are given weights drawn from a commutative monoid. WLTSs subsume a wide range of LTSs, providing a general notion of strong (weighted) bisimulation. In this paper we extend this framework towards other behavioural equivalences, by considering semirings of weights. Taking advantage of this extra structure, we introduce a general notion of weak weighted bisimulation. We show that weak weighted bisimulation coincides with the usual weak bisimulations in the cases of non-deterministic and fully-probabilistic systems; moreover, it naturally provides a definition of weak bisimulation also for kinds of LTSs where this notion is currently missing (such as, stochastic systems). Finally, we provide a categorical account of the coalgebraic construction of weak weighted bisimulation; this construction points out how to port our approach to other equivalences based on different notion of observability.

Weak bisimulations for labelled transition systems weighted over semirings

This isaut]Grigory L. Litvinovaut]Victor P. Maslovaut]Anatoly Ya. Rodionovaut]Andrei N. Sobolevskii a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.

Universal Algorithms, Mathematics of Semirings and Parallel Computations

We will describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding proce- dure is based on the continued fraction expansion of real numbers. Results of computer experiments are given in order to compare efficiencyand accuracy of different types of approximate arithmetics and rounding procedures. Relations with universal algorithms and generic programming are briefly discussed.

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Approximate rational arithmetics and arbitrary precision computations for universal algorithms

This paper is a survey on universal algorithms for solving the matrix Bellman equations over semirings and especially tropical and idempotent semirings. However, original algorithms are also presented. Some applications and software implementations are discussed.

Universal algorithms for solving the matrix Bellman equations over semirings

A new incremental algorithm for data compression is presented. For a sequence of input symbols algorithm incrementally constructs a p-adic integer number as an output. Decoding process starts with less significant part of a p-adic integer and incrementally reconstructs a sequence of input symbols. Algorithm is based on certain features of p-adic numbers and p-adic norm. p-adic coding algorithm may be considered as of generalization a popular compression technique - arithmetic coding algorithms. It is shown that for p = 2 the algorithm works as integer variant of arithmetic coding; for a special class of models it gives exactly the same codes as Huffman's algorithm, for another special model and a specific alphabet it gives Golomb-Rice codes.

P-adic arithmetic coding

An implementation of an object-oriented CA system for reducing tensor expression to the canonical form is described. The specific feature of this system is the use of operations on mathematical systems, such as a direct and semidirect product of groups, to increase the efficiency of computations. The system is implemented in Java.

Anatoly Ya. Rodionov

Papers

Universal Algorithms, Mathematics of Semirings and Parallel Computations

Approximate rational arithmetics and arbitrary precision computations for universal algorithms

Universal algorithms for solving the matrix Bellman equations over semirings

P-adic arithmetic coding

The use of group and algebraic properties of tensor expressions in an object-oriented CA system