Yosida Complementarity Problem with Yosida Variational Inequality Problem and Yosida Proximal Operator Equation Involving XOR-Operation

Due to the importance of Yosida approximation operator, we generalized the variational inequality problem and its equivalent problems by using Yosida approximation operator. The aim of this work is to introduce and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. We prove an existence result together with convergence analysis for Yosida proximal operator equation involving XOR-operation. For this purpose, we establish an algorithm based on fixed point formulation. Our approach is based on a proximal operator technique involving a subdifferential operator. As an application of our main result, we provide a numerical example using the MATLAB program R2018a. Comparing different iterations, a computational table is assembled and some graphs are plotted to show the convergence of iterative sequences for different initial values.

On a Volterra Integral Equation with Delay, via w-Distances

The paper deals with a Volterra integral equation with delay. In order to apply the w-weak generalized contraction theorem for the study of existence and uniqueness of solutions, we rewrite the equation as a fixed point problem. The assumptions take into account the support of w-distance and the complexity of the delay equation. Gronwall-type theorem and comparison theorem are also discussed using a weak Picard operator technique. In the end, an example is provided to support our results.

Reliability of Focused Assessment With Sonography for Trauma in the Diagnosis of Blunt Torso Trauma

Objective: Focused assessment with sonography for trauma (FAST) is a widely used imaging technique for the diagnosis of blunt abdominal trauma and has its limitations and advantages. A meta-analysis was completed to evaluate the reliability of FAST, in the diagnosis of blunt torso trauma. Materials and Methods: A search was completed with Google Scholar, PubMed, National Center for Biotechnology Information (NCBI), MEDLINE, and Medscape databases, from 1993 up to 2020. Eligible studies were included for information about FAST examination of abdominal trauma. The animal research was excluded from this review process. The eligible studies were first categorized and then data analysis was performed, according to specific pathologic conditions. This literature review retrieved studies’ sample size, application, sensitivity, and a specificity of diagnosis using FAST for abdominal trauma. Results: In total, 100 articles were identified through the database search. Besides, five articles were identified through other sources. Then, screening was performed, and as such, 46 published studies were included that had a qualitative synthesis. Conclusion: FAST has a high sensitivity and specificity in the diagnosis of blunt abdominal trauma. However, a large range of sensitivity and specificity of FAST in the evaluation of torso trauma reveal that sonography is operator, technique, and equipment dependent.

Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains

Electron pumping through energy-gapped systems is restricted for vanishing local density of states at the Fermi level. In this paper, we propose a topological Su–Schrieffer–Heeger (SSH) chain between unbiased leads as an effective electron pump. We analyze the electron transport properties of topologically trivial and nontrivial systems in the presence of external time-dependent forces in the form of one-Gaussian or two-Gaussian perturbations (train impulses). We have found that the topologically trivial chain stands for much better charge pump than other normal or nontrivial chains. It is important that, during the perturbation, electrons are pumped through the mid-gap temporary states or through the induced sidebands states outside the energy gap. We also analyze the local density of states dynamics during the quench transition between different topological phases of the SSH chain. It turns out that after the quench, the edge topological states migrate through other sites and can temporarily exist in a topologically trivial part of the system. The tight-binding Hamiltonian and the evolution operator technique are used in our calculations.

SHAPING OPERATOR TECHNIQUE FOR MODELLING CYCLOSTATIONARY RANDOM PROCESSES

The paper proposes the shaping operator technique for modelling cyclostationary random process (CSRP) which belong to the class of second-order processes being described by the parametric pulse train model. Generating CSRP is expressed as the propagation result of the specially chosen elementary CSRP through the linear time-invariant system with a known impulse response. The paper describes the approach to forming the elementary CSRP based on the train made of Dirac delta-function uniformly following in time domain with constant step and defining the desired structural periodicity of the process being modelled. The analytical expressions for the main characteristics used for describing cyclostationary properties of the elementary process, including two-dimensional, cyclic and spectral correlation functions, were obtained. In addition to the case of the elementary CSRP with correlated weighting coefficients of delta-functions, the case of statistically independent coefficients was considered. It was shown that utilization of the spectral correlation function (SCF) for describing the cyclic property reveals the explicit analytical relation between characteristics of the elementary and modelled CSRPs provided the filter frequency response is known. The paper presents the comparative example which describes the modelling of two CSRPs: one of them was chosen as the train of rectangle pulses which is pulse amplitude modulated by stationary random time series, while the other is considered as a signal formed by the method of direct sequence spread spectrum. The chosen short-length Barker sequence as the code allowed performing visual comparison between absolute value of SCF components taken at the cyclic frequencies multiple of the chip frequency and cyclic frequencies multiple of the symbol frequency. The future development of the methods proposed in the paper opens the road to improving the performance of modern radar and telecommunication systems by means of utilizing cyclic frequencies which are non-random parameters describing the structural properties of signals under processing.

A degenerate Gaussian weight connected with Painlevé equations and Heun equations

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight [Formula: see text] where [Formula: see text] and [Formula: see text]. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto [Formula: see text] form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the [Formula: see text]-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case [Formula: see text], we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight [Formula: see text], which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].