Class Of Generalized Functions Research Articles

An Iterative Coordinate Descent Algorithm for High-Dimensional Nonconvex Penalized Quantile Regression

We propose and study a new iterative coordinate descent algorithm (QICD) for solving nonconvex penalized quantile regression in high dimension. By permitting different subsets of covariates to be relevant for modeling the response variable at different quantiles, nonconvex penalized quantile regression provides a flexible approach for modeling high-dimensional data with heterogeneity. Although its theory has been investigated recently, its computation remains highly challenging when p is large due to the nonsmoothness of the quantile loss function and the nonconvexity of the penalty function. Existing coordinate descent algorithms for penalized least-squares regression cannot be directly applied. We establish the convergence property of the proposed algorithm under some regularity conditions for a general class of nonconvex penalty functions including popular choices such as SCAD (smoothly clipped absolute deviation) and MCP (minimax concave penalty). Our Monte Carlo study confirms that QICD substantially improves the computational speed in the p ≫ n setting. We illustrate the application by analyzing a microarray dataset.

On the “Preconditioning” Function Used in Planewave DFT Calculations and its Generalization

AbstractThe Teter, Payne, and Allan “preconditioning” function plays a significant role in planewave DFT calculations. This function is often called the TPA preconditioner. We present a detailed study of this “preconditioning” function. We develop a general formula that can readily generate a class of “preconditioning” functions. These functions have higher order approximation accuracy and fulfill the two essential “preconditioning” purposes as required in planewave DFT calculations. Our general class of functions are expected to have applications in other areas.

On a general theorem connecting Laplace transform and generalized Weyl fractional integral operator

The main aim of the present paper is to establish a general theorem which asserts an interesting relationship between the Laplace transform and generalized Weyl fractional integral operator involving general class of functions. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to the sequence, one can easily evaluate generalized Weyl fractional integral operator of special functions of several variables. An illustration for generalized Lauricella function is mentioned. Several interesting special cases of the general theorem are also mentioned.

Firm location and monopolistic competition

AbstractThis paper investigates the equilibrium spatial distributions of firms in a circular city under monopolistic competition. We show that the uniform distribution constitutes an equilibrium for a very general class of transport cost functions. Moreover, we establish that symmetric partial agglomeration arises as an equilibrium if and only if the transport cost function is linear and the number of locations is even. We also find that all rotationally symmetric distributions with continuous support are spatial equilibria if the transport costs are linear. This multiplicity of equilibria however disappears in the case of strictly concave transport costs: the uniform distribution is then the unique spatial equilibrium distribution with continuous support.

Bernstein type inequalities for rational functions

In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions.

Generalized fractional integrals of product of two H-functions and a general class of polynomials

The purpose of this paper is to compute two unified fractional integrals involving the product of two H-functions, a general class of polynomials and Appell function . These integrals are further applied in proving two theorems on Saigo–Maeda fractional integral operators. Some consequent results and special cases are also pointed out in the concluding section.

Blow-up phenomena for compressible Euler equations with non-vacuum initial data

In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding blowup results of the 1-dimensional case in non-radial symmetry are also included.

Worst-Case Approach to Strategic Optimal Portfolio Selection Under Transaction Costs and Trading Limits

We study a worst-case scenario approach to the problem of strategic portfolio selection in presence of transaction costs and trading limits under uncertain stochastic process of market parameters. Unlike classic stochastic programming, the approach is model-free, solution of the arising Bellman-Isaacs equation can be easily found numerically under some general assumptions. All results hold for a general class of utility functions and several risky assets. For a special case of proportional transaction costs and CRRA utility, we present a numerical scheme which allows to reduce the dimension of the Bellman-Isaacs equation by a number of risky assets.

Robust investment–reinsurance optimization with multiscale stochastic volatility

This paper investigates the investment and reinsurance problem in the presence of stochastic volatility for an ambiguity-averse insurer (AAI) with a general concave utility function. The AAI concerns about model uncertainty and seeks for an optimal robust decision. We consider a Brownian motion with drift for the surplus of the AAI who invests in a risky asset following a multiscale stochastic volatility (SV) model. We formulate the robust optimal investment and reinsurance problem for a general class of utility functions under a general SV model. Applying perturbation techniques to the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation associated with our problem, we derive an investment–reinsurance strategy that well approximates the optimal strategy of the robust optimization problem under a multiscale SV model. We also provide a practical strategy that requires no tracking of volatility factors. Numerical study is conducted to demonstrate the practical use of theoretical results and to draw economic interpretations from the robust decision rules.

Dynamical Behaviors of Multiple Equilibria in Competitive Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions.

This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such n -neuron competitive neural networks can have 5(n) equilibria, among which 3(n) equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the 3(n) locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

Objective function to obtain multiple representative waveforms for a novel helical CT scan protocol.

To develop objective functions for selecting multiple representative respiratory waveforms. A specific application considered is to reduce the number of swiping scans in a novel helical CT scan protocol to harvest efficiency and dose reduction benefit. The authors consider a general class of potential objective functions consisting of weighted norms on pointwise profile differentials. The authors utilize the Lagrangian approach and derive proper conditions on the formulation based on first and second order optimality conditions. The derived objective functions are applied to clinically acquired respiratory trajectories for swipe subset selection to verify the validity and generality of the proposed rationale. An end-to-end 4DCT reconstruction comparison is performed using a swipe subset of data corresponding to 3 out of the full 25 waveforms to assess the consequence in image quality and dose. Their results show that maximizing the proposed objective function with the suggested parameters yields maximal spread of trajectories among the selected subset. 4DCT Reconstruction using the chosen subset of data indicates the potential for further dose reduction by about 5 to 10 folds without significant sacrifice in image quality. Experimental results also support further generalization to include slice prioritization. The authors have derived a formulation that is both simple and general as a metric to quantify the spread of a set of respiratory trajectories, which can be used for subset selection with potential computation and dose reduction benefit when applied to a newly developed helical 4DCT scan protocol.

A Stochastic Flows Approach for Asset Allocation with Hidden Economic Environment

An optimal asset allocation problem for a quite general class of utility functions is discussed in a simple two-state Markovian regime-switching model, where the appreciation rate of a risky share changes over time according to the state of a hidden economy. As usual, standard filtering theory is used to transform a financial model with hidden information into one with complete information, where a martingale approach is applied to discuss the optimal asset allocation problem. Using a martingale representation coupled with stochastic flows of diffeomorphisms for the filtering equation, the integrand in the martingale representation is identified which gives rise to an optimal portfolio strategy under some differentiability conditions.

Fundamental aspects of shape optimization in the context of isogeometric analysis

We develop a mathematical foundation for shape optimization problems under state equation constraints where both state and control are discretized by B-splines or NURBS. In other words, we use isogeometric analysis (IGA) for solving the partial differential equation and a nodal approach to change domains where control points take the place of nodes and where thus a quite general class of functions for representing optimal shapes and their boundaries becomes available. The minimization problem is solved by a gradient descent method where the shape gradient will be defined in isogeometric terms. This gradient is obtained following two schemes, optimize first–discretize then and, reversely, discretize first–optimize then. We show that for isogeometric analysis, the two schemes yield the same discrete system. Moreover, we also formulate shape optimization with respect to NURBS in the optimize first ansatz which amounts to finding optimal control points and weights simultaneously. Numerical tests illustrate the theory.

Fekete-Szego coefficient for a general class of spirallike functions in unit disk

The aim of this paper is to give sharp bound for the Fekete-Szego coecient functional of normalized analytic functions f(z) dened on the open unit disk in the complex plane are derived for the general class of spirallike functions in unit disk.

Initial coefficient bounds for a general class of bi-univalent functions

Recently, Srivastava et al. [22] reviewed the study of coefficient problems for bi-univalent functions. Inspired by the pioneering work of Srivastava et al. [22], there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions (see, for example, [1, 3, 6, 7, 27, 29],). Motivated essentially by the aforementioned works, in this paper we propose to investigate the coefficient estimates for a general class of analytic and bi-univalent functions. Also, we obtain estimates on the coefficients |a2| and |a3| for functions in this new class. Further, we discuss some interesting remarks, corollaries and applications of the results presented here.

Asymptotics of Suprema of Weighted Gaussian Fields with Applications to Kernel Density Estimators

For centered stationary Gaussian fields $X(s), s\in {\bf R}^d, d>1,$ we show that properly centered and normalized $\sup_{s\in {\bf R}^d}w(s/T)|X(s)|$ with a nonnegative weight function $w$ converges in distribution to a double exponential law as $T\to\infty$, provided that $w$ and the covariance function of $X$ satisfy certain smoothness and regularity conditions. This limit theorem extends the results for compact sets without weight functions obtained earlier in [V. Piterbarg, Asymptotics Methods in the Theory of Gaussian Processes and Fields, Amer. Math. Soc., Providence, RI, 1996]. This new result for Gaussian fields is then applied to obtain necessary and sufficient conditions for the properly centered and normalized sequence $\sup_{x\in {\bf R}^d}|\Psi(x)(\widehat{f}_n(x)-\e\,\widehat{f}_n(x))|$ to converge in distribution to a double exponential law under natural smoothness conditions, where $\widehat{f}_n$ denotes the kernel density estimator of the bounded continuous density $f$ on ${\bf R}^d$ based on the sample of size $n$, and $\Psi$ is a positive continuous function such that $\sup_{x\in {\bf R}^d}|\Psi(x) f(x)^\beta|<\infty$ for some $\beta\in (0, 1/2)$. This paper extends results of the paper [E. Giné, V. Koltchinskii, and L. Sakhanenko, Probab. Theory Related Fields, 130 (2004), pp. 167--198] to the case of $d>1$. It also extends the results of [E. Rio, Probab. Theory Related Fields, 98 (1994), pp. 21--45] for $\Psi(\cdot)=f^{-1/2}(\cdot)I_S(\cdot)$ with some compact set $S$ in ${\bf R}^d$ to a general class of functions $\Psi$. An example of an application completes this work.

Renormalization of correlations in a quasiperiodically forced two-level system for a general class of modulation function

We provide a renormalization analysis of correlations in a quasi-periodically forced two-level system in a time dependent field with periodic kicks whose amplitude is given by a general class of discontinuous modulation function. For certain intensities of modulation, we give a complete understanding of the autocorrelation function. Furthermore, once the locations of the discontinuities of the modulation function are known, aperiodic orbits lead to correlations on renormalization strange sets which are determined by two specified features of the modulation function of which there are only a finite number of variations.

Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach

This paper introduces a framework for robustifying convex, law invariant risk measures. The robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure, which represent the investors' beliefs about the distribution of future asset losses. It is shown that under mild conditions, the infinite dimensional optimization problem of finding the worst-case risk can be solved analytically and closed-form expressions for the robust risk measures are obtained. Using these results, robust versions of several risk measures including the standard deviation, the Conditional Value-at-Risk, and the general class of distortion functionals are derived. The resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems, and a numerical study shows that in most cases they perform significantly better out-of-sample than their nonrobust variants in terms of risk, expected losses, and turnover.

Coalition configurations and share functions

Albizuri and Aurrekoetxea (Soc Choice Welf 26:571–596, 2006a) and Albizuri et al. (Games Econ Behav 57:1–17, 2006b) defined values for games in which the players are organized into an a priori coalition configuration. In games with coalition configuration, we suppose that players organize themselves into coalitions that are not necessarily disjoint. A player can belong to more than one a priori coalition. In this paper we redefine coalition configuration values by using the concept of share function, as introduced by van der Laan and van den Brink (Theory Decis 53:61–86, 2002). A share function assigns to every player in a game its share in the worth to be distributed. We also define and characterize a general class of share function for games with coalition configuration which contains among other values those introduced by Albizuri et al.

Aggregate Stability and Balanced-Budget Rules

It has been shown that under perfect competition and a Cobb-Douglas production function, a basic real business cycle model may exhibit indeterminacy and sunspot fluctuations when income tax rates are determined by a balanced-budget rule (BBR). This paper introduces in an otherwise standard real business cycle model a more general and data-coherent class of production functions, namely, a constant elasticity of substitution production function. We show that the degree of substitutability between production factors is a key ingredient to understanding the (de)stabilizing properties of a BBR. Then we calibrate the model consistently with the empirical evidence; that is, we set the elasticity of substitution between labor and capital below unity. We show that compared to the Cobb-Douglas case, the likelihood of indeterminacy under a BBR is greatly reduced in the U.S., the EU, and the UK.