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1

The Complexity of Synthesis of b-Bounded Petri Nets

Tredup Ronny

Fundamenta Informaticae

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2021

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Vol. 183, nr 1-2

125--167

EN

For a fixed type of Petri nets τ, τ-SYNTHESIS is the task of finding for a given transition system A a Petri net N of type τ(τ-net, for short) whose reachability graph is isomorphic to A if there is one. The decision version of this search problem is called τ-SOLVABILITY. If an input A allows a positive decision, then it is called τ-solvable and a sought net N τ-solves A. As a well known fact, A is τ-solvable if and only if it has the so-called τ-event state separation property (τ-ESSP, for short) and the τ-state separation property (τ-SSP, for short). The question whether A has the τ-ESSP or the τ-SSP defines also decision problems. In this paper, for all b ∈ ℕ, we completely characterize the computational complexity of τ-SOLVABILITY, τ-ESSP and τ-SSP for the types of pure b-bounded Place/Transition-nets, the b-bounded Place/Transitionnets and their corresponding ℤb+1-extensions.

2

The progress of science from a computational point of view: the drive towards ever higher solvability

Marciszewski Witold

Foundations of Computing and Decision Sciences

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2019

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Vol. 44, No. 1

11--26

EN

This essay's content is rendered by the titles of the successive sections. 1. Effective solvability versus intuitive solvability. — 2. Decidability, i.e. effective solvability, in predicate logic. The speedup phenomenon — 3. Contributions of the second-order logic to the problems of solvability — 4. The infinite progress of science in the light of Turing's idea of the oracle. The term "oracle" is a technical counterpart of the notion of mathematical intuition. A more detailed summary can be obtained through juxtaposing the textboxes labelled with letters A...F. Conclusion: in the progress of science an essential role is played by the feedback between intellectual intuitions (intuitive solvability) and algorithmic procedures (effective solvability).

3

Symplectic singularities and solvable Hamiltonian mappings

Fukuda T., Janeczko S.

Demonstratio Mathematica

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2015

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Vol. 48, nr 2

118--146

EN

We study singularities of smooth mappings (…) of R2n into symplectic space (…) by their isotropic liftings to the corresponding symplectic tangent bundle (…). Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over (…) endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one.

4

On the solvability of Dirichlet problem for the weighted p-Laplacian

Szlachtowska E.

Opuscula Mathematica

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2012

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Vol. 32, no. 4

775-781

EN

The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted ρ-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.

5

On the initial value problem for two-dimensional systems of linear functional differential equations with monotone operators

Sremr J.

Fasciculi Mathematici

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2007

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Nr 37

87-108

EN

We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators.

6

A priori solution estimates for nonlinear vector integral equations

Gil` M. I.

Journal of Applied Analysis

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2003

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Vol. 9, nr 2

187-200

EN

Nonlinear vector integral equations are considered. Solution estimates and solvability conditions are derived. Applications to the periodic boundary value problem are also discussed. Under some restrictions our results improve the well-known ones. The main tool in the paper is the recent estimates for the resolvent of Hilbert-Schmidt operators.