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3 2013

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On Language Equations with One-sided Concatenation

Baader F., Okhotin A.

Fundamenta Informaticae

2013

Vol. 126, nr 1

1--35

Language equations are equations where both the constants occurring in the equations and the solutions are formal languages. They have first been introduced in formal language theory, but are now also considered in other areas of computer science. In the present paper, we restrict the attention to language equations with one-sided concatenation, but in contrast to previous work on these equations, we allow not just union but all Boolean operations to be used when formulating them. In addition, we are not just interested in deciding solvability of such equations, but also in deciding other properties of the set of solutions, like its cardinality (finite, infinite, uncountable) and whether it contains least/greatest solutions. We show that all these decision problems are EXPTIME-complete.

Does there Exist an Algorithm which to Each Diophantine Equation Assigns an Integer which is Greater than the Modulus of Integer Solutions, if these Solutions form a Finite Set?

Tyszka A.

Fundamenta Informaticae

2013

Vol. 125, nr 1

85--99

Let En = {xi = 1; xi + xj = xk; xi · xj = xk : i; j; k ∈ {1,...,n}}. We conjecture that if a system S ⊆ En has only finitely many solutions in integers x1,...,xn, then each such solution (x1,...,xn) satisfies |x1|,...,|xn| ≤ 22n−1. Assuming the conjecture, we prove: (1) there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set, (2) if a set M Í \mathbbN is recursively enumerable but not recursive, then a finite-fold Diophantine representation of M does not exist.

Fundamental Solution for the Plane Problem in Magnetothermoelastic Diffusion Media

Kumar R., Chawla V.

Computational Methods in Science and Technology

2013

Vol. 19, No. 4

195--207

The aim of the present paper is to study the fundamental solution in orthotropic magneto- thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic magnetothermoelastic diffusion media is derived. On the basis of thegeneral solution, the fundamental solution for a steady point heat source in an infinite and a semiinfinite orthotropic magnetothermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, some special cases of interest are also deduced and compared with the previously obtained results. The resulting quantities are computed numerically for infinite and semi-infinite magnetothermoelastic material and presented graphically to depict the magnetic effect.