On Exact Sufficient Oscillation Conditions for Solutions of Linear Differential and Difference Equations of the First Order With Aftereffect

We obtain new unimprovable effective oscillation conditions for all solutions of linear first-order differential and difference equations with several delays. We show that known results of the kind are consequences of the new results. We reveal the reasons for the impossibility to obtain oscillation conditions for equations with several delays, as sharp as the conditions for the equation with one delay, in the case when only known approaches are used.     

Global Optimality Conditions for Some Classes of Optimization Problems

We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity.     

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.     

Linearizability of planar polynomial Hamiltonian systems

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous non-linearities of degrees 5, 6 and 7. We also present some sufficient conditions for systems with nonhomogeneous nonlinearities of degrees two, three and five.     

Approximate deconvolution boundary conditions for large eddy simulation

We introduce new boundary conditions for large eddy simulation. These boundary conditions are based on an approximate deconvolution approach. They are computationally efficient and general, which makes them appropriate for the numerical simulation of turbulent flows with time-dependent boundary conditions. Numerical results are presented to demonstrate the new boundary conditions in a simplified linear setting.     

Conversion of second-class constraints and resolving the zero-curvature conditions in the geometric quantization theory

In the approach to geometric quantization based on the conversion of second-class constraints, we resolve the corresponding nonlinear zero-curvature conditions for the extended symplectic potential. From the zero-curvature conditions, we deduce new linear equations for the extended symplectic potential. We show that solutions of the new linear equations also satisfy the zero-curvature condition. We present a functional solution of these new linear equations and obtain the corresponding path integral representation. We investigate the general case of a phase superspace where boson and fermion coordinates are present on an equal basis.     

Necessary and Sufficient Conditions for Local Invariance for Semilinear Differential Inclusions

We establish new necessary and sufficient conditions for local invariance referring to semilinear differential inclusions, conditions expressed in terms of a new suitable defined tangency concept. An application to the study of the Hamilton-Jacobi-Bellman equation is given.     

On measure-compact operators

We obtain new sufficient solvability conditions for equations with measure-compact operators comparable with partially additive ones. We also prove new conditions under which these operators are locally condensing with respect to the Hausdorff measure of noncompactness. As an application of the obtained results we prove one property of bifurcation points which occur, in particular, in nonlinear mechanics.     

Functionals with sign-definite derivative for the stabilization of delay systems

We present new conditions for the uniform asymptotic stability of equilibria in delay systems. These conditions are based on Lyapunov functionals that have negative definite derivatives along the trajectories of the system only on some part of the phase space.By using these conditions, we establish new optimal stabilization tests, which admit the use of a performance functional whose weight functional is not negative definite on the entire phase space.We introduce a new notion of smallness for perturbations in delay systems and present stabilization tests by the first approximation.     

A dual dynamic programming for minimax optimal control problems governed by parabolic equation

In the article, minimax optimal control problems governed by parabolic equations are considered. We apply a new dual dynamic programming approach to derive sufficient optimality conditions for such problems. The idea is to move all the notions from a state space to a dual space and to obtain a new verification theorem providing the conditions, which should be satisfied by a solution of the dual partial differential equation of dynamic programming. We also give sufficient optimality conditions for the existence of an optimal dual feedback control and some approximation of the problem considered, which seems to be very useful from a practical point of view.     

Modified Numerov method for solving system of second-order boundary-value problems

We introduce and discuss a new numerical method for solving system of second order boundary value problems, where the solution is required to satisfy some extra continuity conditions on the subintervals in addition to the usual boundary conditions. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.     

Beyond Canonical DC-Optimization: The Single Reverse Polar Problem

We propose a novel generalization of the Canonical Difference of Convex problem (CDC), and we study the convergence of outer approximation algorithms for its solution, which use an approximated oracle for checking the global optimality conditions. Although the approximated optimality conditions are similar to those of CDC, this new class of problems is shown to significantly differ from its special case. Indeed, outer approximation approaches for CDC need be substantially modified in order to cope with the more general problem, bringing to new algorithms. We develop a hierarchy of conditions that guarantee global convergence, and we build three different cutting plane algorithms relying on them.     

Uniqueness theorems for (sub-)harmonic functions with applications to operator theory

We obtain uniqueness theorems for harmonic and subharmonic functionsof a new type. They lead to new analytic extension criteriaand new conditions for stability of operator semigroups in Banachspaces with Fourier type.     

On Transience Conditions for Markov Chains and Random Walks

We prove a new transience criterion for Markov chains on an arbitrary state space and give a corollary for real-valued chains. We show by example that in the case of a homogeneous random walk with infinite mean the proposed sufficient conditions are close to those necessary. We give a new proof of the well-known criterion for finiteness of the supremum of a random walk.     

Necessary and sufficient conditions for the existence of shock curves of hyperbolic 1-conservation laws

We present three conditions for the existence of shock curves of hyperbolic 1-conservation laws. If at least two of the conditions hold, then shock curves exist uniquely in the classical sense. We present an example (Example 2, Section 4) for which two of the conditions do not hold and a new kind of singularity occurs. In Remark 1 we present two other conditions similar to the above three conditions which also imply the existence of shock curves.     

Slant and pseudo-slant submanifolds in LCS-manifolds

We show new results on when a pseudo-slant submanifold is a LCS-manifold. Necessary and sufficient conditions for a submanifold to be pseudo-slant are given. We obtain necessary and sufficient conditions for the integrability of distributions which are involved in the definition of the pseudo-slant submanifold. We characterize the pseudoslant product and give necessary and sufficient conditions for a pseudo-slant submanifold to be the pseudo-slant product. Also we give an example of a slant submanifold in an LCS-manifold to illustrate the subject.     

A reconsideration on convergence of three-step iterations for asymptotically nonexpansive mappings

We illustrate that the control conditions of the main convergence theorems of Yao and Noor [Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput. in press] are incorrect. We also provide new control conditions which are complementary to Nilsrakoo and Saejung’s results [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput. 181 (2006) 1026–1034].     

Weaker convergence conditions for the secant method

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.     

A canonical duality approach for the solution of affine quasi-variational inequalities

We propose a new formulation of the Karush–Kunt–Tucker conditions of a particular class of quasi-variational inequalities. In order to reformulate the problem we use the Fisher–Burmeister complementarity function and canonical duality theory. We establish the conditions for a critical point of the new formulation to be a solution of the original quasi-variational inequality showing the potentiality of such approach in solving this class of problems. We test the obtained theoretical results with a simple heuristic that is demonstrated on several problems coming from the academy and various engineering applications.     

Starlike and close-to-convex functions defined by differential inequalities

We present new sufficient conditions for an analytic function to be close-to-convex and starlike, respectively. These conditions are easy to apply and can be used to obtain functions in these classes.