Trigonometric Fmn-Transform of Multi-Variable Functions and Its Application to the Partial Differential Equations and Image Processing

In this study, we focus on the extension of the trigonometric F m-transform technique for functions with one-variable in order to improve its approximation properties at the end points of [a,b] and then generalize the extended trigonometric Fm -transform technique to functions with more variables. The approximation and convergence properties of the direct and inverse multi-variable extended trigonometric Fm -transforms are discussed. The applicability of multi-variable trigonometric F m -transforms to approximate multi-variable functions are illustrated by some examples. Moreover, some direct formulas for the multi-variable extended trigonometric Fm -transforms of partial derivatives of multi-variable functions are obtained and they are applied to solving the Cauchy problem of the transport equation. Also, the application of multi-variable extended trigonometric Fm -transforms for image compression is described. Some examples for the validity of the obtained results about the partial differential equations and image compression are given. The results are compared with some existence ones in the literature.

Convergence properties of generalized Lupaş-Kantorovich operators

In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $\rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.

Hybrid Selection Based Multi/Many-Objective Evolutionary Algorithm

In the last decade, numerous Multi/Many-Objective Evolutionary Algorithms (MOEAs) have been proposed to handle Multi/Many-Objective Problems (MOPs) with challenges such as discontinuous Pareto Front (PF), degenerate PF, etc. MOEAs in the literature can be broadly divided into three categories based on the selection strategy employed such as dominance, decomposition, and indicator-based MOEAs. Each category of MOEAs have their advantages and disadvantages when solving MOPs with diverse characteristics. In this work, we propose a Hybrid Selection based MOEA, referred to as HS-MOEA, which is a simple yet effective hybridization of dominance, decomposition and indicator-based concepts. In other words, we propose a new environmental selection strategy where the Pareto-dominance, reference vectors and an indicator are combined to effectively balance the diversity and convergence properties of MOEA during the evolution. The superior performance of HS-MOEA compared to the state-of-the-art MOEAs is demonstrated through experimental simulations on DTLZ and WFG test suites with up to 10 objectives.

The Mathematics of Smoothed Particle Hydrodynamics (SPH) Consistency

Since its inception Smoothed Particle Hydrodynamics (SPH) has been widely employed as a numerical tool in different areas of science, engineering, and more recently in the animation of fluids for computer graphics applications. Although SPH is still in the process of experiencing continual theoretical and technical developments, the method has been improved over the years to overcome some shortcomings and deficiencies. Its widespread success is due to its simplicity, ease of implementation, and robustness in modeling complex systems. However, despite recent progress in consolidating its theoretical foundations, a long-standing key aspect of SPH is related to the loss of particle consistency, which affects its accuracy and convergence properties. In this paper, an overview of the mathematical aspects of the SPH consistency is presented with a focus on the most recent developments.

On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

Stochastic approximation cut algorithm for inference in modularized Bayesian models

Bayesian modelling enables us to accommodate complex forms of data and make a comprehensive inference, but the effect of partial misspecification of the model is a concern. One approach in this setting is to modularize the model and prevent feedback from suspect modules, using a cut model. After observing data, this leads to the cut distribution which normally does not have a closed form. Previous studies have proposed algorithms to sample from this distribution, but these algorithms have unclear theoretical convergence properties. To address this, we propose a new algorithm called the stochastic approximation cut (SACut) algorithm as an alternative. The algorithm is divided into two parallel chains. The main chain targets an approximation to the cut distribution; the auxiliary chain is used to form an adaptive proposal distribution for the main chain. We prove convergence of the samples drawn by the proposed algorithm and present the exact limit. Although SACut is biased, since the main chain does not target the exact cut distribution, we prove this bias can be reduced geometrically by increasing a user-chosen tuning parameter. In addition, parallel computing can be easily adopted for SACut, which greatly reduces computation time.

Asymptotic properties of Urysohn type generalized sampling operators

The concern of this study is to continue the investigation of convergence properties of Urysohn type generalized sampling operators, which are defined by the author in [Dolomites Res. Notes Approx. 2021, 14 (2), 58-67]. In details, the paper centers around to investigation of the asymptotic properties together with some Voronovskaya-type theorems for the linear and nonlinear counterpart of Urysohn type generalized sampling operators.

An adaptive gBOIN design with shrinkage boundaries for phase I dose-finding trials

Background With the emergence of molecularly targeted agents and immunotherapies, the landscape of phase I trials in oncology has been changed. Though these new therapeutic agents are very likely induce multiple low- or moderate-grade toxicities instead of DLT, most of the existing phase I trial designs account for the binary toxicity outcomes. Motivated by a pediatric phase I trial of solid tumor with a continuous outcome, we propose an adaptive generalized Bayesian optimal interval design with shrinkage boundaries, gBOINS, which can account for continuous, toxicity grades endpoints and regard the conventional binary endpoint as a special case. Result The proposed gBOINS design enjoys convergence properties, e.g., the induced interval shrinks to the toxicity target and the recommended dose converges to the true maximum tolerated dose with increased sample size. Conclusion The proposed gBOINS design is transparent and simple to implement. We show that the gBOINS design has the desirable finite property of coherence and large-sample property of consistency. Numerical studies show that the proposed gBOINS design yields good performance and is comparable with or superior to the competing design.

On Bregman-Type Distances and Their Associated Projection Mappings

We investigate convergence properties of Bregman distances induced by convex representations of maximally monotone operators. We also introduce and study the projection mappings associated with such distances.