Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities

For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

On Solvability of the Integro-Differential Equations

In this paper, we study the existence of solution to integro-differential equations in the space of Lebesgue-integrable on un-bounded interval after transformed to nonlinear integral functional equation, the used tool is the fixed point theorem due to Schauder with weak measure of non compactness, due to De-Blasi. In addition, we give an example which satisfies the conditions of our existence theorem.

PARAMETRIC SYNTHESIS OF A MOVING OBJECT STABILIZER

The problem of choosing the variable parameters of a stabilizer of an object which minimize an additive quadratic integral functional reflecting the complex of requirements for a closed stabilization system is considered. To solve the problem a combined method of parametric synthesis of the stabilizer, which is a sequential combination of the Sobol grid method and the Nelder-Mead method, is proposed. At the first stage of the method by applying the Sobolev grid method a working point of the closed system in the pace of its variable parameters is transferred into a neighborhood of the quality functional global minimum point. Then at the second stage the Nelder-Mead method is used to relocated the working point into a small neighborhood of the global minimum. The method proposed comprises a particular algorithm for choosing the weight coefficient of the additive quality functional as well as makes use of the stabilization object state vector main coordinates, which provide the most adequate description of its dynamic features. The properties of a mathematical model of controlled system with discontinuous stabilization process control are studied numerically. The analysis of the plots in the dynamical system state phase space indicates non-spiral approach of the system to its equilibrium state. The synthesized control is realized in the form of a sequence of switchovers.

On the route construction in changing environments using solutions of the eikonal equation

The article deals with the vehicle routing problem in an environment with dynamically changing properties. The problem is relevant in current conditions when the delivery cost has a steady upward trend and is often comparable to the cost of the product itself. A central feature of the study is that the optimality criterion is the minimum delivery time, but not the distance traveled. The optical-geometric approach developed by the authors, based on the analogy between the propagation of light in an optically inhomogeneous medium and the minimization of the integral functional, is used as a research tool. We use exact and approximate solutions of the eikonal equations to describe wave fronts. Two original numerical algorithms for route construction are proposed and implemented as software. A computational experiment is performed that justified the effectiveness of the proposed model-algorithmic tools.

APPLICATION OF THE ”WAVE“ METHOD TO SOLVE THE PROBLEMS OF OPTIMIZATION OF PASSENGER AND CARGO TRANSPORTATION IN MEGACITIES TO IMPROVE THEIR TRANSPORT NETWORK AND INFRASTRUCTURE. Conceptual basis, mathematical framework

The paper proposes a new approach to solving optimization problems arising in engineering and transport logistics in designing and construction of roads (in particular, in megacities, near large transport hubs, near state borders) for cargo and passenger transportation and implementation of international trade. The fundamental problems of modern engineering logistics - the problem of optimal location (transport hubs) and the problem of identification and segmentation of logistics, transport and logistics zones are considered. These problems are solved using methods of variational calculus, in particular, the so-called "wave" method based on the Fermat principle existing in physical optics, which is based on the analogy between finding the global extremum of the integral functional and the propagation of light in an optically heterogeneous medium. A numerical method for the above technique has been developed programatically. The idea of the "wave method/approach belongs to V.V. Bashurov, who proposed to use the methods of geometrical and physical optics to investigate applied safety problems and some related issues. The essence of the "wave method" is that initially the safety problem is reduced to the search for the global minimum of a nonlinear functional. In turn, the minimization problem is solved by constructing the trajectory of motion of the front of the "light wave" moving in an optically inhomogeneous medium. Finding the minimum of a functional is a classical problem of variational calculus, for the solution of which a significant mathematical apparatus has been developed. However, most of known methods effectively determine only local extrema. "Wave" method allows to solve the problem of finding a global extremum with greater efficiency. This paper proposes a conceptual framework and scientifically justified modification of this "wave" method for solving optimization problems arising in engineering and transport logistics, including the problem of optimal location of the transport hub, transport and logistics center (warehouse) and the problem of optimal identification and segmentation of logistical zones (metropolitan areas, large transport hubs).

A Study Based on Controlled Drug Release by 3D Printed Aloe-Vera Bi-Layer Tablet

Aloe-Vera or Aloe barbadensis (botanical name) is a plant with many medicinal properties and have great importance in Ayurveda. Its leaves are succulent, erect, forming a thick rosette. The internal translucent pulp of Aloe-Vera is bound to a waxy crust or cuticle, and its vascular tissues transport minerals as well as water from the soil. Aloe Vera is being used as a major skin rejuvenating product, although it has varied medicinal properties also. In the present study, an attempt to make a method to create bi-layer tablets of Aloe-Vera, utilizing 3D printing techniques is presented. The method created doesnt affect the integral functional characteristics of the tablet. The method here contains creating an immediate release and sustained release tablet for making the Aloe-Vera to be used directly by the person for its numerous health effects. The tablet is designed so to be consumed by vegans as well since it is completely herbal.

Using Organotypic Tissue Slices to Investigate the Microenvironment of Pancreatic Cancer: Pharmacotyping and Beyond

Organotypic tissue slices prepared from patient tumors are a semi-intact ex vivo preparation that recapitulates many aspects of the tumor microenvironment (TME). While connections to the vasculature and nervous system are severed, the integral functional elements of the tumor remain intact for many days during the slice culture. During this window of time, the slice platforms offer a suite of molecular, biomechanical and functional tools to investigate PDAC biology. In this review, we first briefly discuss the development of pancreatic tissue slices as a model system. Next, we touch upon using slices as an orthogonal approach to study the TME as compared to other established 3D models, such as organoids. Distinct from most other models, the pancreatic slices contain autologous immune and other stromal cells. Taking advantage of the existing immune cells within the slices, we will discuss the breakthrough studies which investigate the immune compartment in the pancreas slices. These studies will provide an important framework for future investigations seeking to exploit or reprogram the TME for cancer therapy.

On Well-Posedness of Some Constrained Variational Problems

By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by defining the set of approximating solutions for the class of variational problems under study, we establish several results on well-posedness.

Generalized proportional fractional integral functional bounds in Minkowski’s inequalities

In this research paper, we improve some fractional integral inequalities of Minkowski-type. Precisely, we use a proportional fractional integral operator with respect to another strictly increasing continuous function ψ. The functions used in this work are bounded by two positive functions to get reverse Minkowski inequalities in a new sense. Moreover, we introduce new fractional integral inequalities which have a close relationship to the reverse Minkowski-type inequalities via ψ-proportional fractional integral, then with the help of this fractional integral operator, we discuss some new special cases of reverse Minkowski-type inequalities through this work. An open issue is covered in the conclusion section to extend the current findings to be more general.

Well Posedness of New Optimization Problems with Variational Inequality Constraints

In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by introducing the set of approximating solutions for the considered class of constrained optimization problems, we established some characterization results on well posedness. Furthermore, to illustrate the theoretical developments included in this paper, we present some examples.