二阶线性微分方程的乘子可积性

在偏微分方程Riemann解法和微分方程裂变思想的启发下,引入了微分方程乘子函数(解)和乘子解法的概念,系统地讨论了二阶线性微分方程的乘子可积性.得到了二阶线性微分方程乘子可积的条件以及Riceati方程可积的充分必要条件,并分别给出了二阶线性微分方程和Riccati方程在乘子解下的通积分.     

Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-type Spaces

The problem of the boundedness of the Riesz potential I _α , 0 < α < n, in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted L _p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, these sufficient conditions coincide with the necessary ones. V. Guliyev’s research was partially supported by the grant of the Azerbaijan-U. S. Bilateral Grants Program II (project ANSF Award / AZM1-3110-BA-08) and the Turkish Scientific and Technological Research Council (TUBITAK, programme 2221, no. 220.01-619-4889).     

Log-Sigmoid Multipliers Method in Constrained Optimization

In this paper we introduced and analyzed the Log-Sigmoid (LS) multipliers method for constrained optimization. The LS method is to the recently developed smoothing technique as augmented Lagrangian to the penalty method or modified barrier to classical barrier methods. At the same time the LS method has some specific properties, which make it substantially different from other nonquadratic augmented Lagrangian techniques.We established convergence of the LS type penalty method under very mild assumptions on the input data and estimated the rate of convergence of the LS multipliers method under the standard second order optimality condition for both exact and nonexact minimization.Some important properties of the dual function and the dual problem, which are based on the LS Lagrangian, were discovered and the primal–dual LS method was introduced.     

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coeffcients in Banach-valued Lp-spaces are studied.To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations,the suffcient conditions are founded.By using these facts,the unique continuation properties are established.In the application part,the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.     

On the Regularity Condition for Vector Programming Problems

In this work, we use a notion of approximation derived from Jourani and Thibault [13] to ascertain optimality conditions analogous to those that established but applicable to larger class of vector valued objective mappings and constraint set-valued mappings. To this end, we introduce an appropriate regularity condition to help us discern the Karush-Kuhn-Tucker multipliers.     

On congruences and ideals of partially ordered quasigroups

Some results concerning congruence relations on partially ordered quasigroups (especially, Riesz quasigroups) and ideals of partially ordered loops are presented. These results generalize the assertions which were proved by Fuchs in [5] for partially ordered groups and Riesz groups.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

Weighted norm inequalities,off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace–Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.      

Calibration Invariance of the MaxEnt Distribution in the Maximum Entropy Principle

The maximum entropy principle consists of two steps: The first step is to find the distribution which maximizes entropy under given constraints. The second step is to calculate the corresponding thermodynamic quantities. The second part is determined by Lagrange multipliers’ relation to the measurable physical quantities as temperature or Helmholtz free energy/free entropy. We show that for a given MaxEnt distribution, the whole class of entropies and constraints leads to the same distribution but generally different thermodynamics. Two simple classes of transformations that preserve the MaxEnt distributions are studied: The first case is a transform of the entropy to an arbitrary increasing function of that entropy. The second case is the transform of the energetic constraint to a combination of the normalization and energetic constraints. We derive group transformations of the Lagrange multipliers corresponding to these transformations and determine their connections to thermodynamic quantities. For each case, we provide a simple example of this transformation.     

Exponential stability of uniform Euler–Bernoulli beams with non-collocated boundary controllers

We study the stability of a robot system composed of two Euler–Bernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical spectra of the system are distributed in the complex left-half plane and there is a sequence of the generalized eigenfunctions that forms a Riesz basis in the energy space. Since there exist at most finitely many spectral points of the system in the right half-plane, to obtain the exponential stability, we show that one can choose suitable feedback gains such that all eigenvalues of the system are located in the left half-plane. Hence the Riesz basis property ensures that the system is exponentially stable. Finally we give some simulation for spectra of the system.