By Gheorghe Adam, Ján Buša, Michal Hnatič

This e-book constitutes the refereed post-proceedings of the overseas convention on Mathematical Modeling and Computational Physics, MMCP 2011, held in famous personá Lesná, Slovakia, in July 2011. The forty-one revised papers provided have been rigorously reviewed and chosen from various submissions. they're geared up in topical sections on mathematical modeling and techniques, numerical modeling and strategies, computational aid of the experiments, computing instruments, and optimization and simulation.

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Extra resources for Mathematical Modeling and Computational Science: International Conference, MMCP 2011, Stará Lesná, Slovakia, July 4-8, 2011, Revised Selected Papers

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Traditionally, the state of a qubit is denoted as |a , corresponding to the standard Dirac notation for quantum mechanical states (see [1]). Therefore, a quantum memory register is shown on a diagram visualizing the circuit as a column of states of the form |aj (j = 0, 1, . . , n − 1) from which “quantum wires” start. Although a quantum circuit doesn’t contain any wires as such, the term “wires” is merely used to show evolution of qubits acted on by various quantum gates. A number of gates and their arrangement in the circuit determines a quantum algorithm.

Then the smallest σ-ideal containing F˜ is said to be generated by F˜ If F˜ ⊂ R, and denoted by [F˜ ]. If for v, w ∈ M the equality w = t·θ ◦ v holds with θ ∈ Θ and t ∈ M we shall say that v divides w and write v | w. It is easy to see that this divisibility relation yields a partial order. Definition 6. Given a σ-ideal I and an admissible monomial ordering ˜ ⊂ I is its (difference) standard basis if [G] ˜ = I and subset G ˜ ) [ lm(˜ ( ∀ f˜ ∈ I )( ∃ g˜ ∈ G g ) | lm(f˜) ] . , a (6) If the standard basis is finite it is called Gr¨ obner basis.

In the linear case the algorithmic verification of the s-consistency is based on the completion of the initial differential system to involution and the construction of a Gr¨obner basis for the linear difference ideal generated by the FDA. It is important to emphasize that the involutivity of the linear differential system under consideration not only makes possible an algorithmic verification of the s-consistency but it is also necessary (cf. [25]) to the well-posedness of the Cauchy problem for the system what, if one believes in the extension of the LaxRichtmyer equivalence theorem to PDE systems, can guarantee the convergence for the s-consistent and stable FDA.

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