Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas

Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given.

Calculations of The Change of State of a Gas By Integrating The Partial Derivatives

It will not be denied that the calculations of the change of state for a gas is highly important in most engineering applications. For determining the gas’s properties such as the pressure (P), the volume (V) and the temperature (T), engineers and scientists uses the Boyle’s, Charles’s and Gay-Lussac’s (B-C-G) law of P1V1/T1=P2V2/T2. Although the B-C-G law provides the accurate property values of a gas, it give no detailed information embedded in the process when a gas changes its state. In this study, the author theoretically carried out the integrations of the partial differentials when differentiating the B-C-G law, which has not been tried by anyone up to now. The integration results of this study were thoroughly compared with the experimentally measured data and it was confirmed that the integration methods suggested in this study accurately provides the differential properties on ΔP, ΔV and ΔT. In addition to it, through the stepwise analysis of the integration of the partial differentials, it revealed that the efficiency in the change of state of a gas inherently exists higher than the Carnot cycle, which is operating between the same conditions. Therefore, the results of this study can be lead to the conclusion that all changes of state of all materials inevitably accompanies an energy loss and it is a natural phenomenon.

Solving 2D boundary-value problems using discrete partial differential operators

Purpose Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions. Design/methodology/approach This paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann. Findings An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window. Research limitations/implications The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases. Practical implications This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings. Originality/value The presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

New Hermite-Hadamard type inequalities for m and (α, m)-convex functions on the coordinates via generalized fractional integrals

In this paper, we obtained a new Hermite-Hadamard type inequality for functions of two independent variables that are m-convex on the coordinates via some generalized Katugampola type fractional integrals. We also established a new identity involving the second order mixed partial derivatives of functions of two independent variables via the generalized Katugampola fractional integrals. Using the identity, we established some new Hermite-Hadamard type inequalities for functions whose second order mixed partial derivatives in absolute value at some powers are (α, m)-convex on the coordinates. Our results are extensions of some earlier results in the literature for functions of two variables.

On the Jacobian Construction

Показано, что как произведение дифференциалов независимых переменных, так и произведение частных производных функции многих (нескольких) переменных преобразуются как кососимметрическая форма с одним и тем же определителем Якоби при переходе от одной системы координат к другой. It is shown that both the product of independent variable differentials and the product of partial derivatives of a multivariable function can be rearranged as an antisymmetric form with the same Jacobian as we convert from one reference system to another one.  

Development a method for determining the coordinates of air objects by radars with the additional use of multilateration technology

This paper reports an experimental study aimed at confirming disruptions in the operation of ADS-B receivers. The experimental investigation into disruptions in the operation of ADS-B receivers involved the FlightAware Piaware receiver. Examples of the disrupted performance of ADS-B receivers are given. It was found that the experimentally detected disruptions in the operation of ADS-B receivers could lead to a decrease in the accuracy of determining the coordinates of air objects with the joint use of the radar and multilateration technology. A method for determining the coordinates of an air object by radar with additional use of multilateration technology has been devised. The method involves the following stages: entering initial data, the calculation of distances between the points of reception and the air object, the computation of the inconsistency vector, the calculation of the matrix of partial derivatives taking into consideration the estimates of the coordinates of an air object at the previous iteration, the computation of the correction, the calculation of the refined coordinates of the air object. Unlike those known ones, the improved method for determining the coordinates of an air object by a radar additionally uses multilateration technology. The accuracy of determining the air objects' coordinates by a radar with the additional use of multilateration technology was estimated. It was established that the additional application of multilateration technology would reduce the error in determining the coordinates of an air object by 1.58 to 2.39 times on average, compared to using only an autonomous radar

On PDE Characterization of Smooth Hierarchical Functions Computed by Neural Networks

Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that are computable by a given network. We study real, infinitely differentiable (smooth) hierarchical functions implemented by feedforward neural networks via composing simpler functions in two cases: (1) each constituent function of the composition has fewer in puts than the resulting function and (2) constituent functions are in the more specific yet prevalent form of a nonlinear univariate function (e.g., tanh) applied to a linear multivariate function. We establish that in each of these regimes, there exist nontrivial algebraic partial differential equations (PDEs) that are satisfied by the computed functions. These PDEs are purely in terms of the partial derivatives and are dependent only on the topology of the network. Conversely, we conjecture that such PDE constraints, once accompanied by appropriate nonsingularity conditions and perhaps certain inequalities involving partial derivatives, guarantee that the smooth function under consideration can be represented by the network. The conjecture is verified in numerous examples, including the case of tree architectures, which are of neuroscientific interest. Our approach is a step toward formulating an algebraic description of functional spaces associated with specific neural networks, and may provide useful new tools for constructing neural networks.