The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an operatorvalued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold.

In this paper, we present a bi-objective Virtual-force local search particle swarm optimization (BVFPSO) algorithm to improve the placement of sensors in wireless sensor networks while it simultaneously increases the coverage rate and preserves the battery energy of the sensors. Mostly, sensor nodes in a wireless sensor network are first randomly deployed in the target area, and their deployment should be then modified such that some objective functions are obtained. In the proposed BVFPSO algorithm, PSO is used as the basic meta-heuristic algorithm and the Virtual-force operator is used as the local search. As far as we know, this is the first time that a bi-objective PSO algorithm has been combined with a Virtual force operator to improve the coverage rate of sensors while preserving their battery energy. The results of the simulations on some initial random deployments with the different numbers of sensors show that the BVFPSO algorithm by combining two objectives and using Virtual-force local search is enabled to achieve a more efficient deployment in comparison to the competitive algorithms PSO, GA, FRED and VFA with providing simultaneously maximum coverage rate and the minimum energy consumption.

Introduction Devices that allow using the functionality of natural hand movements are of the greatest interest. The purpose of this study was to select areas of research at the intersection of several fields of science –,biomechanics and cybernetics to develop scientific and technical approaches to track the movements of the operator’, s fingers and form feedback tactile and force communication received from the control object to achieve a new level of accuracy in work with Virtual and with real objects by converting Virtual contact action into physical. Methods of force feedback were implemented according which they can be divided into two groups: active and passive feedback. The main technologies used to implement various functionalities of the Virtual reality glove were identified: measuring hand positions and feedback generation. The main advantages of the planned development were also identified: the ability to digitize up to 16 finger joints, tactile and force feedback, and moderate cost, a benchmark for the mass market. on R commander 3. 4. 1 was used with the Kruskal-Wallis h-criterion and Pearson’, s χ, ²,-test. Conclusion The development of the design of the glove will be implemented using a kinematic scheme based on the kinematics of the human hand, considering the requirements of aesthetics and ergonomics of the solution. As a sensor system, it is planned to use digital Hall sensors with a capacity of up to 14 bits, high speed and, at the same time, small dimensions, allowing them to be installed directly inside the swivel joints.

In this investigation, using Opoola differential operator ($D^{m}(\mu,\beta,t)f(z)$), a new integral operator: $I_{t,\beta,\mu}^{m,\sigma}(f_{1},...,f_{n})(z): A^{n}\rightarrow A$  is defined in the unit disk, $U=\left\lbrace z\in C:\left|z\right|<1\right\rbrace$; and we investigated the Univalence conditions of this generalized operator. Finally, a number of corollaries and remarks which show the extension of our results are presented.