Class Of Generalized Functions Research Articles

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.

On Krylov’s estimates for optional semimartingales

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

A Note on the Refined Strichartz Estimates and Maximal Extension Operator

There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain-Demeter's $l^2$ decoupling theorem and induction on scales. In the second part, we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main methods are sharp $L^2$ estimates by Du and Zhang, and the bilinear method by Wolff and Tao.

On the generalized fractional Laplace transform

Abstract In the present paper a generalization of the Laplace transform is introduced and studied. Its inversion formula is also obtained. As an application, we obtain the generalized fractional Laplace transform of a general class of functions and a product of the Fox’s H- function and general class of functions. The results obtained are of general nature and capable of yielding a large number of known or new results as special cases. For illustration, some special cases involving important special functions are mentioned.

Effect of spin–phonon coupling on quantum correlation in the spin-1 XY model

The XY model coupled with an external system as a phonon system may be studied in terms of its own variables only. Hence, the effect of phonons external in such system can be included in a general class of functionals of the coordinates of the system only. The specific form of this influence functional representing the effect of definite and random forces of a linear dissipative system and combination of these are analyzed in detail. Thus the effect of the spin–phonon interaction on entanglement is investigated where we obtain a small effect of the spin–phonon coupling on von Neumann entropy.

On Some Classes with Norms of Meromorphic Function Spaces Defined by General Spherical Derivatives

The main concerned target of this article is to define and study some concerned classes of meromorphic function spaces using the general spherical derivatives. The general Besov-type classes of meromorphic functions as well as the general normal functions are considered intensively and both are compared deeply with each other. Specifically, multiple results concerning general meromorphic-type classes as well as non-normal classes are obtained by the help of general spherical derivatives. The concerned results are proved by constructing some specific mild conditions on the sequences of points belonging to the concerned meromorphic-type classes. The obtained results generalize and improve the corresponding previous results in some concerned respects. The concerned proofs and methods are simply presented.

Determining the competition outcome in the chemostat: General response functions and delayed growth

Competitive exclusion has been proven to be true mainly for competition in the chemostat with monotone response functions. For the case with nonmonotone response functions, strong restrictions are imposed on the property of response functions in the literature to ensure the competitive exclusion. In this work, by constructing a novel Liapunov functional, we show that only one species can survive when n species are competing for a single essential resource in the chemostat. The conditions are quite generic and our result applies to chemostat competition models with differential removal rates, delayed growth, and a large class of general response functions including the simplified Holling type IV response functions.

Stability results for second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay and a very general class of memory kernel functions. Under some suitable conditions and by introducing a suitable Lyapunov functional, we state and prove explicit and general stability results of the energy. We give also some applications to illustrate our results. This work generalizes the previous results without time delay term to those with time-varying delay.

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most rn from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for G(n,rn) in the thermodynamic limit with nrnd= const., and also in the dense limit with nrnd→∞, rn→0. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the traveling salesman, spanning tree, matching, bipartite matching and bipartite traveling salesman problems, for a general class of weight functions with at most polynomial growth of order d−ε, under thermodynamic scaling of the distance parameter.

Portfolio selection with drawdown constraint on consumption: a generalization model

In this study, we generalize the results of Arun (The Merton problem with a drawdown constraint on consumption. Working paper, 2013) on the optimal consumption and investment problem of an infinitely lived agent who does not accept her consumption falling below a fixed proportion of her historically highest level, the so-called drawdown constraint on consumption. We extend the results to a general class of utility functions. We use the martingale method to study the dual problem, which involves the choice of a maximum consumption process. The dual problem can be formulated as a two-dimensional singular control problem, with the free boundary depending on a state variable of the maximum process. We establish the duality theorem and provide semi-closed form solutions regarding the optimal strategies. To highlight our methodology, we present some special cases of utility functions that do not allow for the dimension reduction considered in Arun (2013).