Mathematics and Informatics in Natural Sciences and Engineering

Dedicated to the 80th Birthday of David Gordeziani

AMINSE 2017

**Philippe G. Ciarlet**, City University of Hong Kong.

*Nonlinear Estimates for Surfaces in Terms of Their Fundamental Forms, and Applications*

**Abstract:** It is well known that a surface can be recovered from its two fundamental forms if they satisfy the Gauss and Codazzi-Mainardi compatibility equations on a simply-connected domain, in which case the surface is uniquely determined only up to isometric equivalence.

It is less known that in this case the surface becomes a continuous function of its fundamental forms, again up to isometric equivalence, for various topologies, such as the Fréchet topology of continuously differentiable functions, or those corresponding to various Sobolev norms.

In this talk, we will review such continuity results obtained during the past fifteen years, with special emphasis on those that can be derived by means of nonlinear Korn inequalities on a surface.

We will also mention potential applications of such results, such as the intrinsic approach to nonlinear shell theory, where the unknowns are the fundamental forms of the deformed middle surface of a shell, or the numerical reconstruction of the Earth surface by means of the knowledge of its fundamental forms on a discrete grid.

**Philippe Destuynder**, Conservatoire National des Arts et Métiers.

*Few Results Concerning Non Destructive Testing of Structures*

**Abstract:** The non destructive testing (NDT) consists in detecting the existence, the shape and the position of a defect in a structure. In this talk the ultrasonic method is discussed in the framework of NDT. The use of a mathematical model is a necessity for solving this challenge . In fact, several problems should be tackled.

For instance for a given frequency, a defect can be hidden by the ultrasonic waves used. The goal of the mathematician is to give informations on these invisible defects in order to adjust the frequencies used.

Furthermore, only a part of a structure is equipped with sensors and therefore a lot of the signal reflected by the defect is lost (or can be lost). The use of a so-called harvester (based on a control method) seems to be a promising improvement in such a technology. In addition the problem set is an inverse one with many controllability difficulties. Finally a fast simulator would be required in order to operate in real time. This talk is mainly oriented towards a definition of the problems to be handled and partial results are given.

**Olga Gil-Medrano**, University of Valencia.

*Geometry of the Space of Curves. Application to a problem in Endodontics*

**Abstract:** In the first part of the talk we will review the geometry of spaces of maps with special emphasis in the spaces of curves. The second part will be devoted to describe an application to a problem proposed by the stomatology researchers B. Buenrostro and L. Forner concerning the root canal curvature of natural human teeth. We will report the results of our joint work in progress.

**Temur Kutsia**, RISC, Johannes Kepler University.

*Anti-Unification in Description Logic EL*

**Abstract:** In unification theory, anti-unification is a process for computing generalizations, i.e., to obtain more general objects from concrete ones. The objects are represented as logical expressions (terms, term sequences, clauses, etc.) and the generalization relation is specified (e.g., with respect to syntactic equality, equality modulo given equations, subsumption, etc.). The name anti-unification underlines its duality with unification that essentially computes common instances of the given general objects.

Current research on anti-unification is dominated by practically oriented topics. It is not surprising, because generalization computation, in one form or another, is a very important ingredient of various applications in reasoning, learning, information extraction, data compression, software development and analysis, etc.

In this talk, we present anti-unification for the description logic EL. Description logics are a family of knowledge representation languages, which provide logical formalism for ontologies and Semantic Web. Compared to the other description logics, EL has a limited expressive power, but it attracts significant interest due to its efficient inference (subsumption problem is polynomial) and successful application in biomedical informatics (large biomedical ontologies are defined in EL).

We introduced the notion of least general generalization in EL, which generalizes simultaneously the notions of least common subsumer and concept matching. The idea of generalization of two concepts is to detect maximal similarities between them, and to abstract over their differences uniformly. We show that generalization for EL is finitary, present an anti-unification algorithm, discuss its properties, and report on preliminary experimental evaluation.

This is a joint work with Boris Konev (University of Liverpool).

**Volodymyr Makarov**, Institute of Mathematics NAS Ukraine.

*Exact and Approximate Solutions of the Spectral Problems for the Differential Schrödinger Operator with a Polynomial Potential in R ^{k}, k≥2*

**Abstract:**
Spectral problems for the Schrödinger operator with polynomial potentials in R^{k}, k≥2 are considered in this paper. For the set of specifically given potentials we found in the exact form a few first eigenvalues of the given problem using a technique based on the combination of functional-discrete (FD-) method and Maple computer algebra system. In the case when the application of traditional FD-method to the given problem leads to a divergent scheme (the power of the polynomial potential of at least one of the independent variables exceeds 2) we propose a modification of FD-method, which is proved to be effective for the class of the problems under consideration. The obtained theoretical results are illustrated by numerical examples.

**Mircea Marin**, Computer Science Department, West University of Timisoara.

*Rule-Based Programming: The Mathematica Experience*

**Abstract:** Rule-based programming provides a common framework to model the two main processes of concern to computer science: Computation and Deduction. Both of them can be expressed as sequences of transformations that lead to a desired result or proof. Various kinds of systems of conditional rewrite rules emerged as suitable formalisms for the specification of stepwise transformations, whereas the sequences of transformations of interest are controlled by strategies.

The last two decades witnessed the advent of powerful formalisms, such as rewriting logic and rewriting calculi, that led to the emergence of rule-based languages and systems, like Elan, Maude, and Stratego. Their practical applications span from computations in various systems to deduction in theorem provers, program transformation, implementation of constraint solvers, access control policies.

In this talk I will describe our experience with the design of a calculus for rule-based programming called RhoLog, its implementation in Mathematica, and some practical applications.

**Hamlet Meladze**, N.Muskhelishvili Institute of Computational Mathematics, St.Andrew the First Called Georgian University.

*On Some Parallel Algorithms for Approximate Solution of Problems of Mathematical Physics*

**Abstract:** The present talk is devoted to the investigation of special decomposition methods for stationary and nonstationary problems of partial differential equations: the decomposition of the basic area or the basic operator of the initial problem. These methods are based on the reduction of the solution of initial problem to the solution of some more "simple" sub-problems and open the great possibilities in designing algorithms of parallel implementation and creation the program products for computers. We consider also the parallel version of the Schwarz alternating method, based on area decomposition. The independent problem is the solution of difference problems representing itself the system of linear or nonlinear algebraic equations. The parallel iterative methods for the numerical solution of nonlinear equations and systems of equations will be considered as well.

In the talk the primary attention will be inverted on the works, conducted in the Tbilisi State University and I.Vekua Institute of Applied Mathematics.

**Paolo Podio-Guidugli**, Accademia Nazionale dei Lincei.

*On the Modeling of Transport Phenomena in Continuum and Statistical Mechanics*

**Abstract:** The formulation of balance laws in continuum and statistical mechanics is expounded in forms that open the way to revise and review the correspondence instituted, in a manner proposed by Irving and Kirkwood and improved by Noll, between the basic balance laws of Cauchy continua and those of standard Hamiltonian systems of particles.

**Paolo Emilio Ricci**, International Telematic University UniNettuno.

*Computation of Spectral Characteristics for Charged Integral Equations*

**Abstract:** The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of charged Fredholm-Stieltjes integral equations, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, including approximation of the relevant eigenfunctions. Starting from the problem of a string charged by a finite number of cursors, a survey including the extensions to the 2D and 3D dimensional problems is presented.