On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

Ockham’s Calculus of Strict Implication

In his main work Summa Logicae written around 1323, William of Ockham developed a system of propositional modal logic which contains almost all theorems of a modern calculus of strict implication. This calculus is formally reconstructed here with the help of modern symbols for the operators of conjunction, disjunction, implication, negation, possibility, and necessity.     

Non Deterministic Classical Logic: The λμ++ ‐calculus

In this paper, we present an extension of λμ‐calculus called λμ⁺⁺‐calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel‐or.     

A calculus for ideal triangulations of three‐manifolds with embedded arcs

Refining the notion of an ideal triangulation of a compact three‐manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,α), where M is a three‐manifold and α is a collection of properly embedded arcs. We also show that certain well‐understood combinatorial moves are sufficient to relate to each other any two refined triangulations representing the same (M,α). Our proof does not assume the Matveev–Piergallini calculus for ideal triangulations, and actually easily implies this calculus. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Specialization of Loop Rules of a Sequent Calculus of Intuitionistic Temporal Logic with Time Gaps

The paper deals with the loop-rule problem in the first-order intuitionistic temporal logic sequent calculus LBJ. The calculus LBJT is intended for the specialization of the antecedent implication rule. The invertibility of some of the LBJT rules and the syntactic admissibility of the structural rules and the cut rule in LBJT, as well as the equivalence of LBJ and LBJT, are proved. The calculus LBJT2 is intended for the specialization of the antecedent universal quantifier and antecedent box rules. The decidability of LBJT2 is proved.     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

On contraction and the modal fragment

We observe that removing contraction from a standard sequent calculus for first‐order predicate logic preserves completeness for the modal fragment. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit algebras with the Leibniz–Mellin translation product

A sub‐calculus of the calculus (“algebra”) of all complete conormal symbols arising in the edge pseudodifferential calculus is constructed. This calculus of complete conormal symbols is suitable for constructing sub‐calculi of the general edge pseudodifferential calculus, for which the edge‐degenerate pseudodifferential operators involved map conormal asymptotics of distributions near the edges in a prescribed manner. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ideal density‐dependent incompressible viscoelastic flow in the critical Besov spaces

In this paper, we consider the well‐posedness issue for the density‐dependent incompressible viscoelastic fluids of the Oldroyd model for the ideal case in space dimension greater than 2. We obtain the local well‐posedness of this model under the assumption that the initial density is bounded away from zero in the critical Besov spaces by means of the Littlewood‐Paley theory and Bony's paradifferential calculus. In particular, we obtain a Beale‐Kato‐Majida–type regularity criterion.     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

Finite‐time stability of fractional‐order stochastic singular systems with time delay and white noise

In this article, the finite‐time stochastic stability of fractional‐order singular systems with time delay and white noise is investigated. First the existence and uniqueness of solution for the considered system is derived using the basic fractional calculus theory. Then based on the Gronwall's approach and stochastic analysis technique, the sufficient condition for the finite‐time stability criterion is developed. Finally, a numerical example is presented to verify the obtained theory. © 2016 Wiley Periodicals, Inc. Complexity 21: 370–379, 2016     

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half‐space

We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.     

A note on the discrete Cauchy‐Kovalevskaya extension

In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

New operations in intuitionistic calculus

The question is studied of the possibility of extending the intuitionistic propositional calculus by adding single-valued one-place operations which are not expressible in terms of conjunction, disjunction, implication, or negation. There is constructed a system of correct calculi with new single-valued operations which is isomorphic to the family of proper extensions of the intuitionistic propositional calculus.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 23–28, July, 1977.The author wishes to thank L. L. Maksimova for his help with this paper.     

Universal Classes of MV‐chains with Applications to Many‐valued Logics

In this paper we characterize, classify and axiomatize all universal classes of MV‐chains. Moreover, we accomplish analogous characterization, classification and axiomatization for congruence distributive quasivarieties of MV‐algebras. Finally, we apply those results to study some finitary extensions of the Łukasiewicz infinite valued propositional calculus.     

Fractional modeling and control of a complex nonlinear energy supply‐demand system

This article deals with the fractional‐order modeling of a complex four‐dimensional energy supply‐demand system (FOESDS). First, the fractional calculus techniques are adopted to describe the dynamics of the energy supply‐demand system. Then the complex behavior of the proposed fractional‐order FOESDS is studied using numerical simulations. It is shown that the FOESDS can exhibit stable, chaotic, and unstable states. When it exhibits chaos, the FOESDS's strange attractors are plotted to validate the chaotic behavior of the system. Moreover, we calculate the maximal Lyapunov exponents of the system to confirm the existence of chaos. Accordingly, to stabilize the system, a finite‐time active fractional‐order controller is proposed. The effects of model uncertainties and external disturbances are also taken into account. An estimation of the stabilization time is given. Based on the latest version of the fractional Lyapunov stability theory, the finite‐time stability and robustness of the proposed method are proved. Finally, two illustrative examples are provided to illustrate the usefulness and applicability of the proposed control scheme. © 2014 Wiley Periodicals, Inc. Complexity 20: 74–86, 2015     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.