Class Of Generalized Functions Research Articles

OSCILLATORY PROPERTIES OF REAL FUNCTIONS WITH WEAKLY BOUNDED MELLIN TRANSFORM

We show that the error term of the counting function of sums of two squares has oscillations of logarithmic frequency and size x1/2(log x)−3/2−ε. We do that by showing a theorem on oscillations for error terms of a rather general class of real functions. If the Mellin transform of f(x) has a singularity at ρ=β+iγ, γ≠0, with the principal summand of the form (s−ρ)−b(log(s−ρ))ch(s), h(ρ)≠0, we obtain oscillations of logarithmic frequency and size xβ(log x)b−1−ε. We also apply this result to other arithmetical functions related to sums of two squares.

Sharp coefficient estimates for a certain general class of Spirallike functions by means of differential subordination

In the present paper, the authors derive several sharp estimates for the Taylor-Maclaurin coefficients of functions in a certain general class S?(A, B) of spirallike functions in the open unit disk U, which is defined here by using the principle of differential subordination The results presented here would generalize those given in the earlier work of R. J. Libera.

Peer Effects in Endogenous Networks

This paper presents a model of strategic network formation with local complementarities in effort levels and positive local externalities. Results are obtained for a general class of payoff functions, which subsumes the linear-quadratic specification frequently used in the literature. We assume homogeneous agents and characterize equilibria for two-sided and one-sided link formation. (Pairwise) Nash equilibrium networks are shown to be nested split graphs, which are a strict subset of core-periphery networks. The relevance of the convexity of the value function in obtaining these structures is highlighted. In equilibrium more central agents exert more effort and obtain higher gross payoffs. However, net of linking cost, central agents may obtain strictly lower net payoffs. Under additional assumptions on payoffs we show that the only efficient network structures are also nested split graphs. These findings are relevant for a wide range of social and economic phenomena, such as educational attainment, criminal activity, labor market participation, and R&D expenditures of firms.

Improvements on Non-quadratic Stabilization of Takagi-Sugeno Models via Line-Integral Lyapunov Functions

Abstract This paper introduces two different generalizations on non-quadratic stabilization of Takagi-Sugeno models via line-integral Lyapunov functions. Inspired in recent developments which have proved to relax the conservativeness in the quadratic framework via suitable matrix transformations using a more general class of Lyapunov functions, the results here provided outperform the existing ones for line-integral Lyapunov functions with a linear matrix inequalities (LMI) form instead of bilinear matrix inequalities (BMI) in second order systems. Examples are provided to illustrate the advantages of the proposed technique over the existing approach.

On the distribution of auto-correlation value of balanced Boolean functions

In this paper, we study the lower bound on the sum-of-square indicator of balanced Boolean functions obtained by Son, et al. in 1998, and give a sufficient and necessary condition under which balanced Boolean functions achieve this lower bound. We introduce a new general class of balanced Boolean functions in $n$ variables $(n\geq 4)$ with optimal auto-correlation distribution, and we study two sub-classes more explicitely. Finally, we study the sets of Boolean functions having a same auto-correlation distribution, and derive a lower bound on the number of elements in such set.

A New Class of Functions for Measuring Solution Integrality in the Feasibility Pump Approach

Mixed integer optimization is a powerful tool for modeling many optimization problems arising from real-world applications. Finding a first feasible solution represents the first step for several mixed integer programming (MIP) solvers. The feasibility pump is a heuristic for finding feasible solutions to mixed integer linear programming (MILP) problems which is effective even when dealing with hard MIP instances. In this work, we start by interpreting the feasibility pump as a Frank--Wolfe method applied to a nonsmooth concave merit function. Then we define a general class of functions that can be included in the feasibility pump scheme for measuring solution integrality, and we identify some merit functions belonging to this class. We further extend our approach by dynamically combining two different merit functions. Finally, we define a new version of the feasibility pump algorithm, which includes the original version of the feasibility pump as a special case, and we present computational results on binary MILP problems showing the effectiveness of our approach.

Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation

We consider a singular version with state constraints of the stochastic target problems studied in Soner and Touzi (SIAM J. Control Optim. 41:404–424, 2002; J. Eur. Math. Soc. 4:201–236, 2002) and more recently Bouchard et al. (SIAM J. Control Optim. 48:3123–3150, 2009), among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly fits the market models with proportional transaction costs and the order book liquidation issues. Our main result is a direct PDE characterization of the associated pricing function. As an example application, we discuss the valuation of VWAP-guaranteed-type book liquidation contracts, for a general class of risk functions.

A Schur transformation for functions in a general class of domains

In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting.

Deterrence in Contests

This paper explores the role of deterrence in contests. As a general rule, we show that for a deterrence strategy to be played by rational agents, it is necessary that the contest be destructive. We show for a very general class of functions that pure strategy deterrence equilibria where contestants deter one another do not exist. A corollary of this finding is that under fairly general conditions, agents should always be expected to engage in contests. Applied to international relations, our results imply that war is always a potential outcome despite deterrence attempts.

On Optimal Stopping Problems for Matrix-Exponential Jump-Diffusion Processes

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

On Optimal Stopping Problems for Matrix-Exponential Jump-Diffusion Processes

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

Convex spacelike hypersurfaces of constant curvature in de Sitter space

We show that for a very general and natural class of curvature functions (for example the curvature quotients$(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurfacein de Sitter space satisfying$f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinityhas at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.

On the equivalence of posterior inference based on retrospective and prospective likelihoods: application to a case‐control study of colorectal cancer

The paper develops a class of priors that leads to equivalent posterior inference for odds ratio parameters based on prospective and retrospective models for categorical response data. The results are applicable to both unmatched and matched case-control studies. The results hold for a general class of link functions for categorical response. The proposed method can accommodate multiple and possibly ordered disease states. The results are applied to the analysis of discrete subtypes in an ongoing case-control study of colorectal cancer. A simulation study illustrates the need for carefully considering prior choices in Bayesian analysis of data collected under retrospective design.

On the convergence to the multiple subfractional Wiener–Itô integral

In this paper, we construct a family of continuous stochastic processes that converges in law to the multiple Wiener–Itô integrals with respect to the subfractional Brownian motion with H>12 for the integrand f in a rather general class of functions. We mainly use Donsker and Stroock approximations and the techniques of the multiple Wiener–Itô integral with respect to the Wiener process.

Nonsmooth optimization via quasi-Newton methods

We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.

Bayesian prediction of future number of failures based on finite mixture of general class of distributions

This paper is concerned with the Bayesian prediction problem of the number of components which will fail in a future time interval. The failure times are distributed according to a finite mixture of a general class of distributions. Type-I censored sample from this nonhomogeneous population and a general class of prior density functions are used. A one-sample scheme is used to predict the number of failures in a future time interval. An example of a finite mixture of k exponential components is given to illustrate our results.

Efficient importance sampling function design for sequential Monte Carlo PHD filter

In this paper, we propose a novel implementation of the probability hypothesis density (PHD) filter based on the sequential Monte Carlo (SMC) method called SMC-PHD filter. The SMC-PHD filter is analogous to the sequential importance sampling which generates samples using an importance sampling (IS) function. Even though this filter permits general class of IS density function, many previous implementations have simply used the state transition density function. However, this approach leads to a degeneracy problem and renders the filter inefficient. Thus, we propose a novel IS function for the SMC-PHD filter using a combination of an unscented information filter and a gating technique. Further, we use measurement-driven birth target intensities because they are more efficient and accurate than selecting birth targets selected using arbitrary or expected mean target states. The performance of the SMC-PHD filter with the proposed IS function was subsequently evaluated through a simulation and it was shown to outperform the standard SMC-PHD filter and recently proposed auxiliary PHD filter.

Multiperiodicity Evoked by Periodic External Inputs in Cohen-Grossberg-type BAM Networks with Discrete and Distributed Delays

In this paper, by using contraction mapping theorem, analysis approach and decomposition of solution space, the multiperiodicity issue is discussed for Cohen-Grossberg-type (CG-type) bidirectional associative memory networks (BAMNs) with discrete and distributed delays and a general class of activation functions, where the general class of activation functions consist of nondecreasing functions with saturations including piecewise linear functions with two corner points and standard activation functions as their special cases. It is shown that for any saturation region, if there is a periodic orbit located in it, it must be locally exponentially stable. Then, based on this result, some conditions are derived for ascertaining the (n + m)-neuron CG-type BAMNs can have 2 locally exponentially stable limit cycles located in two saturation regions respectively which are symmetrical. Also, taking account of different saturation regions, results about 2(p + q) (p ≤ m, q ≤ n − 1), 2min{n,m} locally exponentially stable limit cycles can be obtained, where n is the number of the neurons in one layer, m is the number of the neurons in the other layer. Meanwhile, for every locally exponentially stable limit cycle given, the corresponding saturation region can be expressed concretely. Finally, three examples are given to illustrate the effectiveness of the obtained results.

An FPTAS for optimizing a class of low-rank functions over a polytope

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.

On the Direct Versus Indirect Taxation Controversy

The well-known Atkinson and Stiglitz (1976, section 5) theorem on the superiority of nonlinear directtaxation over nonlinear indirect taxation is proved to hold under the second order approach, even on in-tervals where the monotonicity constraint is binding. this confirms for the nonlinear case previous find-ings by Kaplow (2006) and Laroque (2005), who considered a joint combination of linear indirect taxesand, possibly suboptimal, nonlinear income taxation. the intuition behind is that, when indirect taxationis not restricted to linearity, its elimination is only justified if the income taxation scheme becomes opti-mally chosen, as happens in Atkinson and Stiglitz (1976).In addition, our analysis allows for a more general class of sub-utility functions than the one in the Atkin-son and Stiglitz theorem, which includes work done or income further to consumption goods and servic-es. Finally, the possibility that optimal allocations fail to be differentiable, or continuous, is also contem-plated unlike in the standard literature.