Class Of Generalized Functions Research Articles

Multistability analysis of octonion-valued neural networks with time-varying delays

In this article, the multistability analysis is studied for n-dimensional octonion valued neural networks (OVNNs) with time-varying delays for a general class of activation functions. Firstly, OVNNs are decomposed into eight real-valued systems, and then based on geometrical properties of activation functions, 38n disjoint regions are constructed in On. Then, by using the inequality technique, several sufficient conditions are obtained to ensure the existence of 38n equilibrium points of the system, each of which is located in one of the regions, and 28n of them are locally exponentially stable. Moreover positively invariant sets are also estimated in this scientific contribution. Two numerical examples are provided to illustrate the effectiveness of the obtained results. Especially, the numerical Example 2 demonstrates that the designed OVNNs work efficiently on storing and retrieving the truecolor images.

An Optimal Control Framework for Online Job Scheduling with General Cost Functions

The paper “An Optimal Control Framework for Online Job Scheduling with General Cost Functions,” by Etesami devises online speed-augmented competitive algorithms for minimizing the generalized completion time on a single and multiple unrelated machines for a very general class of cost functions. To that end, a novel optimal control formulation for the offline version of the problem is developed. Such a formulation allows using tools from optimal control theory, such as the minimum principle and Hamilton-Jacobi-Bellman equation, to set the dual variables as close as possible to the optimal dual variables and leverage them to design primal-dual online algorithms as an iterative application of the offline problem. The analysis can achieve state-of-the-art competitive ratios for several special cases and provide new competitive ratios which are the first in their settings. In particular, the analysis offers a principled method of estimating dual variables in a general setting of online job scheduling.

Frequency Response and Transfer Functions of Large Self-Similar Networks

Abstract Large-scale dynamical systems, no matter whether possessing interconnected appearances, are frequently modeled as networks. For instance, graphs, multi-agent systems, and materials' intricate behaviors are often treated as networked dynamical systems. However, only a few studies have approached the problem in the frequency domain, mostly due to the complexity of evaluating their frequency response. That gap is filled by this paper, which proposes algorithms computing a general class of self-similar networks' frequency response and transfer functions, no matter they are finite or infinite, damaged or undamaged. In addition, this paper shows that for infinite self-similar networks, even when they are damaged, fractional-order and irrational dynamics naturally come into sight. Most importantly, this paper illustrates that for a network under different operating conditions, its frequency response would form a set of neighboring plants, which sets the basis of applying robust control methods to dynamic networks.

Cost-Aware Deployment of Check-In Nodes in Complex Networks

It is challenging to deploy check-in nodes optimally in a complex network so as to perform specific check-in like services, e.g., fuel supplements etc., but crucial in many real-world applications. In this article, we propose and study the new optimization problem of placing check-in nodes with the minimum cost, i.e., the problem of finding the minimum-cost check-in nodes (MCCN), which is of great interests for real-world application situations, but even more difficult. We motivate the new algorithms through three typical worse cases by the often-used greedy-type algorithms, i.e., One-to-Many, Many-to-One, and Duplicate-Overrides. With this respect, the proposed novel optimization algorithms utilize a novel metric of contribution density for selecting check-in nodes iteratively, which successfully avoid the occurrence of the two worse cases of One-to-Many and Many-to-One. We also introduce an extra backward extraction step in one of new algorithms, which overcomes the crucial worse case of “duplicate overrides” and largely improves the algorithmic performance in solving the introduced optimization problem of MCCN. Meanwhile, we extend the new contribution density metric to a more general class of functions and study their effectiveness to eliminate the two worse cases of One-to-Many and Many-to-One; also, a detailed analysis on complexity and performance of the proposed algorithms is presented to show their numerical efficiency and accuracy. Extensive experiments over two classical artificial networks, i.e., BA network and ER network, and ten real-world networks, under two typical cost setups, show the proposed algorithms significantly outperform the four state-of-the-art algorithms. We also demonstrate that our proposed algorithms are much reliable and robust with different experiment settings of deployment cost and network-type and scale.

Triple Integral involving the Product of the Logarithmic and Bessel Functions expressed in terms of the Lerch Function

The aim of the present document is to evaluate a triple integral involving a general class of logarithmic, special function and exponential function. Importance of our resultss lies in the fact that they involve the Bessel function of the First Kind, whichis used in a wide range of areas spanning Science and Engineering. Further we establish some special cases

Model-based clustering of high-dimensional longitudinal data via regularization.

We propose a model-based clustering method for high-dimensional longitudinal data via regularization in this paper. This study was motivated by the Trial of Activity in Adolescent Girls (TAAG), which aimed to examine multilevel factors related to the change of physical activity by following up a cohort of 783 girls over 10 years from adolescence to early adulthood. Our goal is to identify the intrinsic grouping of subjects with similar patterns of physical activity trajectories and the most relevant predictors within each group. The previous analyses conducted clustering and variable selection in two steps, while our new method can perform the tasks simultaneously. Within each cluster, a linear mixed-effects model (LMM) is fitted with a doubly penalized likelihood to induce sparsity for parameter estimation and effect selection. The large-sample joint properties are established, allowing the dimensions of both fixed and random effects to increase at an exponential rate of the sample size, with a general class of penalty functions. Assuming subjects are drawn from a Gaussian mixture distribution, model effects and cluster labels are estimated via a coordinate descent algorithm nested inside the Expectation-Maximization (EM) algorithm. Bayesian Information Criterion (BIC) is used to determine the optimal number of clusters and the values of tuning parameters. Our numerical studies show that the new method has satisfactory performance and is able to accommodate complex data with multilevel and/or longitudinaleffects.

Analysis of a continuously stirred two tank reactor cascade with Haldane kinetics

Biological reactors are employed in industrial applications to break down organic waste. We view the cascade of two open loop continuously stirred tank reactors with Haldane growth function as chemostats with bacterial inputs. A function of some of the reactor parameters is derived, the sign of which determines the maximum number of critical points a reactor can have. This allows us to determine the parameter combinations which ensure a reactor has only a single critical point for all bacterial removal rates (dilution rate plus death rate). Where a simple condition on the above function is confirmed to hold, if the first reactor in a cascade only has a single critical point for all bacterial removal rates then, the next reactor will also only have a single critical point for all bacterial removal rates. A global stability result is also given for some of these cases. A simple proof is given for the local stability of critical points of a reactor with a general class of bacterial growth functions, bacteria and substrate input, and a death rate. For the special case where the first reactor has zero bacteria input, we compare a two reactor cascade with a single reactor under various conditions, long and short residence times, and different death rates. This follows the pattern of similar papers that considered cascades using the Monod and Contois growth functions.

Stable learning laws design for long short-term memory identifier for uncertain discrete systems via control Lyapunov functions

This study introduces a method for designing stable learning laws of Long Short-Term Memory (LSTM) networks working as a non-parametric identifier of nonlinear systems with uncertain models. The strategy applies the concept of stability for discrete-time systems in the sense of Lyapunov to prove that origin is a practical stable equilibrium point for the identification error. The laws consider a general class of sigmoidal functions placed at the different gates of a LSTM structure (long and short memory). The design of the learning laws uses a matrix inequality framework to obtain the rate gains associated with the evolution of the weights. Numerical results show the designed learning laws for the non-parametric identifier based on a LSTM approximation tested on two classes of nonlinear systems: the first one describes the ozone-based degradation of organic contaminants, and the second one represents the dynamics of a Van Der Poll oscillator. The LSTM identifier is compared against a classical Lyapunov-based recurrent neural network. This comparison demonstrates how the proposed algorithm approximates the trajectories of both systems with a smaller mean squared error, which serves as an indicator of the benefits obtained with these new learning laws.

Geodesic locally starshaped G-invex sets and geodesic semilocal preinvex functions

In this paper, we introduce a new class of sets and functions, namely geodesic locally starshaped G-invex sets and geodesic semilocal preinvex functions on Riemannian manifolds. We study the properties of these classes of sets and functions and derive certain characterizations. The results of the paper extend and unify several known results from literature to a more general class of functions as well as in more general space setting

Pointwise Convergence of the Fractional Schrödinger Equation in $\mathbb{R}^2$

We investigate the pointwise convergence of the solution to the fractional Schrödinger equation in $\mathbb{R}^2$. By establishing $H^s(\mathbb{R}^2) - L^3(\mathbb{R}^2)$ estimates for the associated maximal operator provided that $s > 1/3$, we improve the previous result obtained by Miao, Yang, and Zheng [19]. Our estimates extend the refined Strichartz estimates obtained by Du, Guth, and Li [10] to a general class of elliptic functions.

Competing risks regression models with covariates-adjusted censoring weight under the generalized case-cohort design.

A generalized case-cohort design has been used when measuring exposures is expensive and events are not rare in the full cohort. This design collects expensive exposure information from a (stratified) randomly selected subset from the full cohort, called the subcohort, and a fraction of cases outside the subcohort. For the full cohort study with competing risks, He et al. (Scand J Stat 43:103-122, 2016) studied the non-stratified proportional subdistribution hazards model with covariate-dependent censoring to directly evaluate covariate effects on the cumulative incidence function. In this paper, we propose a stratified proportional subdistribution hazards model with covariate-adjusted censoring weights for competing risks data under the generalized case-cohort design. We consider a general class of weight functions to account for the generalized case-cohort design. Then, we derive the optimal weight function which minimizes the asymptotic variance of parameter estimates within the general class of weight functions. The proposed estimator is shown to be consistent and asymptotically normally distributed. The simulation studies show (i) the proposed estimator with covariate-adjusted weight is unbiased when the censoring distribution depends on covariates; and (ii) the proposed estimator with the optimal weight function gains parameter estimation efficiency. We apply the proposed method to stem cell transplantation and diabetes data sets.

Logarithm and program testing

Randomized property-based testing has gained much attention recently, but most frameworks stop short at polymorphic properties. Although Bernardy et al. have developed a theory to reduce a wide range of polymorphic properties to monomorphic ones, it relies upon ad-hoc embedding-projection pairs to massage the types into a particular form. This paper skips the embedding-projection pairs and presents a mechanical monomorphization for a general class of polymorphic functions, a step towards automatic testing for polymorphic properties. The calculation of suitable types for monomorphization turns out to be logarithm .

Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

<abstract><p>In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 > 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>).</p></abstract>

Collective modes in anisotropic systems

We study collective modes in anisotropic plasmas of quarks and gluons using a quasi-particle picture and a hard loop approximation. We use a general class of anisotropic distribution functions, and we consider chirally asymmetric systems. We introduce a complete tensor basis to decompose the gluon polarization tensor into a set of nine scalar functions. We derive and solve the corresponding dispersion equations. Imaginary modes are particularly important because of their potential influence on plasma dynamics. We explore in detail their dependence on the chiral chemical potential and the parameters that characterise the anisotropy of the system. We show that our generalized distributions produce dispersion relations that are much richer in structure than those obtained with a simple one parameter deformation of an isotropic distribution. In addition, the size and domain of the imaginary solutions are enhanced, relative to those obtained with a one parameter deformation. Finally, we show that the influence of even a very small chiral chemical potential is significantly magnified when anisotropy is present.

ON TIME SERIES MODEL ORDER SELECTION OF NON-STATIONARY AND NON-NORMAL DATA STRUCTURE

The study is aimed at identifying the orders of Time Series Models in Non-Stationarity Non-normal data structure from Uniform distributions with a view to determining the best Autoregressive/ Moving Average orders from time series models (ARMA and ARIMA) under different underline distributions when Non-Stationarity assumption is assumed. The data is generated from Autoregressive (AR) linear of second orders of general classes of Autoregressive functions. The generation of the data used for this simulation study is non-stationary cases and nonnormal. The data were simulated for both response variables and error terms from non-normal Distribution. The result shows that the values of the penalty function: AIC, BIC, HQIC and FPE of the order selection increases with increase in sample size but decreases with increasing order. It was observed that at both lower (20, 40) and larger (160,180 and 200) sample sizes, models with smallest orders. Similarly, the selection process is tied to the principle of parsimony i.e smaller orders are selected, but vary with the variation in the distribution of the series. The study recommends the need to develop a methodology for model selection combining objective and subjective techniques.

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schrödinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender et al (2014 J. Phys. A: Math. Theor. 47 235204), separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e. for y′(x) = J ν (xy), we show that the eigenvalues asymptotically grow as 241/42 n 1/4. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e. for y′(x) = 1/Γ(−xy), we show that the nth eigenvalue grows factorially fast as , where r k is the kth root of the digamma function.

Teichmüller space theory and classical problems of geometric function theory

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmuller spaces. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions

PE‐EDD: An efficient peer‐effect‐based financial fraud detection approach in publicly traded China firms

Financial fraud arises from the exaggeration of business interests, and an accurate detection or prediction is a useful tool for both corporate management and capital market systems. A collection of computer technologies has been made on this problem so far, and one of the most important solutions is unsupervised learning algorithms. Among them, most approaches work by analysing the internal relations in financial data and finding a new description of non-fraud firms. However, current studies focus a lot on the geometry attribute of financial data, while overlooking the obvious behaviour patterns and peer effects among firms. This has limited the accuracy of representation and furthermore the detection performance. In this work, a very general class of functions is allowed to represent firms, constraining them by peer effects between firms and presenting an error-distribution-based financial fraud firm detection approach. Experimental results have shown great performance of the proposed approach.

Directional distance functions and social welfare: Some axiomatic and dual properties

The paper introduces in social choice theory a directional distance function (DDF) that quantifies the level of inefficiency of a given allocation with respect to the utility possibilities frontier. The paper shows that the DDF has a simple geometric interpretation and involves a complete transitive preference relation. If the uility possibilities set is not convex, it is shown that the Rawls welfare function is dual (in a certain sense) to the DDF. Finally a more general class of distance functions is introduced that is shown to be closely related to a large class of welfare functions widely used in the study of economic inequality.

Teichmuller space theory and classical problems of geometric function theory

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions.