Class Of Generalized Functions Research Articles

Improved estimation of a function of scale parameter of a doubly censored exponential distribution

In the present article, we have studied the estimation of the reciprocal of scale parameter , that is, hazard rate of a two parameter exponential distribution based on a doubly censored sample. This estimation problem has been investigated under a general class of bowl-shaped scale invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of scale equivariant estimator is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for three special loss functions. A simulation study is conducted to compare the risk performance of the proposed estimators. Finally, we analyze a real data set.

Analytic functions in the unit ball of bounded L-index in joint variables and of bounded 𝐿-index in direction: a connection between these classes

Abstract We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variables and has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2 → C.

The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices

We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to the large AdS radius limit of gapped theories in AdS2. In this regime we demonstrate analytically that optimal bounds on OPE coefficients lead to extremal solutions to crossing arising from integrable field theories placed in large AdS2. In the process, we uncover a close connection between asymptotic extremal functionals and S-matrices of integrable field theories in flat space and explain how 2D S-matrix bootstrap results can be derived from the 1D conformal bootstrap equations. These points illustrate that our formalism is capable of capturing non-trivial solutions of CFT crossing.

Estimating the common hazard rate of several exponential distributions

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.

Sensitivity Properties of Parametric Nonconvex Evolution Inclusions with Application to Optimal Control Problems

The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first order involving a general class of nonconvex functions. Using recent results on the stability of the subdifferentials, with respect to the Gamma convergence, of the associated sequence of subsmooth or semiconvex functions, we give some continuity properties of the solution set associated to these problems. The particular case of the parametric sweeping process involving uniformly subsmooth or uniformly prox-regular sets is studied in details. As an application, we study the sensitivity analysis of the generalized Bolza/Mayer problem governed by a nonsmooth dynamic of a sweeping process type.

Existence of tripled fixed points and solution of functional integral equations through a measure of noncompactness

In this paper, we propose fixed point results through the notion of a measure of noncompactness and give a generalization of a Darbo’s fixed point theorem. We also prove some new tripled fixed point results via a measure of noncompactness for a more general class of functions. Our results generalize and extend some comparable results in the literature. Further, we apply the obtained fixed point theorems to prove the existence of solutions for a general system of non-linear functional integral equations. In the end, an example is given to illustrate the validity of our results.

Slices of parameter spaces of generalized Nevanlinna functions

In the early 1980's, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps.Here, we extend these ideas to transcendental functions.In [16], it was shown that for the tangent family, $ \{ \lambda \tan z \} $, the hyperbolic components meet at a parameter $ \lambda^* $ such that $ f_{ \lambda^*}^n( \lambda^*i) = \infty $ for some $ n $. The behavior there reflects the dynamic behavior of $ \lambda^* \tan z $ at infinity. In Part 1. we show that this duality extends to a more general class of transcendental meromorphic functions $ \{f_{\lambda}\} $ for which infinity is not an asymptotic value. In particular, we show that in 'dynamically natural' one-dimensional slices of parameter space, there are 'hyperbolic-like' components $ \Omega $ with a unique distinguished boundary point such that for $ \lambda \in \Omega $, the dynamics of $ f_\lambda $ reflect the behavior of $ f_\lambda $ at infinity. Our main result is that every parameter point $ \lambda $ in such a slice for which the iterate of the asymptotic value of $ f_\lambda $ is a pole is such a distinguished boundary point.In the second part of the paper, we apply this result to the families $ \lambda \tan^p z^q $, $ p, q \in \mathbb Z^+ $, to prove that all hyperbolic components of period greater than $ 1 $ are bounded.

On Complete Stability of Recurrent Neural Networks With Time-Varying Delays and General Piecewise Linear Activation Functions.

This paper addresses the problem of complete stability of delayed recurrent neural networks with a general class of piecewise linear activation functions. By applying an appropriate partition of the state space and iterating the defined bounding functions, some sufficient conditions are obtained to ensure that an n -neuron neural network is completely stable with exactly ∏i=1n(2Ki-1) equilibrium points, among which ∏i=1nKi equilibrium points are locally exponentially stable and the others are unstable, where Ki (i=1,…,n) are non-negative integers which depend jointly on activation functions and parameters of neural networks. The results of this paper include the existing works on the stability analysis of recurrent neural networks with piecewise linear functions as special cases and hence can be considered as the improvement and extension of the existing stability results in the literature. A numerical example is provided to illustrate the derived theoretical results.

Nowhere differentiable hairs for entire maps

Devaney and Krych (Ergod Theory Dyn Syst 4:35–52, 1984) showed that for the exponential family $$\lambda e^z$$ , where $$0<\lambda <1/e$$ , the Julia set consists of uncountably many pairwise disjoint simple curves tending to $$\infty $$ , which they called hairs. Viana proved that these hairs are smooth. Baranski as well as Rottenfuser, Ruckert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana’s result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable.

Self-similar solutions of curvature flows in warped products

In this paper we study self-similar solutions in warped products satisfying F−F=g¯(λ(r)∂r,ν), where F is a nonnegative constant and F is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such F. We also obtain a similar uniqueness result in hyperbolic space H3 for Gauss curvature F and F≥1.

Restricted strong convexity implies weak submodularity

We connect high-dimensional subset selection and submodular maximization. Our results extend the work of Das and Kempe [In ICML (2011) 1057–1064] from the setting of linear regression to arbitrary objective functions. For greedy feature selection, this connection allows us to obtain strong multiplicative performance bounds on several methods without statistical modeling assumptions. We also derive recovery guarantees of this form under standard assumptions. Our work shows that greedy algorithms perform within a constant factor from the best possible subset-selection solution for a broad class of general objective functions. Our methods allow a direct control over the number of obtained features as opposed to regularization parameters that only implicitly control sparsity. Our proof technique uses the concept of weak submodularity initially defined by Das and Kempe. We draw a connection between convex analysis and submodular set function theory which may be of independent interest for other statistical learning applications that have combinatorial structure.

Density estimates and central limit theorem for the functional of fractional SDEs

In this paper, we consider a general class of functionals of stochastic differential equations driven by fractional Brownian motion. For this class, we obtain Gaussian estimates for the density and a quantitative central limit theorem. The main tools of the paper are the techniques of Malliavin calculus.

A Note on Strategic Stability of Cooperative Solutions for Multistage Games

The problem of strategic stability of cooperative solutions for multistage games is studied. The sufficient conditions related to discount factors are presented, which guarantee the existence of Nash or strong Nash equilibria in such games and therefore guarantee the strategic stability of cooperative solutions. The deviating payoffs of players are given directly, which are related closely to these conditions and avoid the loss of super-additivity of a class of general characteristic functions. As an illustration, Nash and strong Nash equilibria are found for the repeated infinite stage Prisoner’s dilemma game.

An extended family of nonconforming quasi-Wilson elements for solving elasticity problem

This contribution (part two) focuses on numerical implementation and efficiency aspects of an extended family of nonconforming quasi-Wilson elements type. Such a class of nonconforming elements has been introduced recently in Achchab et al. (2015). Here, based on a rectangular mesh, it is used for the approximate solution of a planar elasticity problem. It is shown that this family passes the patch-test, and that by suitably adding some orthogonality conditions, on a general class of enrichment functions, we can derive higher order consistency error estimates. Our general theoretical results, see Theorems 4.1 and 4.2, unify, simplify and extend a number of existing works on the improvement of the order of consistency error. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamé parameter μ, shear modulus λ and locking free when the Poisson parameter ν approaches close to 0.5.

Higher Genus Modular Graph Functions, String Invariants, and their Exact Asymptotics

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter t. For arbitrary genus, the Arakelov–Green function and the Kawazumi–Zhang invariant degenerate to a Laurent polynomial in t of degree (1, 1) in the limit \({t\to\infty}\) . For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree (w, w) in t, where w + 2 is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in t, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in t of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.

On the Efficiency of Nash Equilibria in Aggregative Charging Games

Several works have recently suggested to model the problem of coordinating the charging needs of a fleet of electric vehicles as a game, and have proposed distributed algorithms to coordinate the vehicles towards a Nash equilibrium of such game. However, Nash equilibria have been shown to posses desirable system-level properties only in simplified cases. In this work, we use the concept of price of anarchy to analyze the inefficiency of Nash equilibria when compared to the social optimum solution. More precisely, we show that i) for linear price functions depending on all the charging instants, the price of anarchy converges to one as the population of vehicles grows; ii) for price functions that depend only on the instantaneous demand, the price of anarchy converges to one if the price function takes the form of a positive pure monomial; iii) for general classes of price functions, the asymptotic price of anarchy can be bounded. For finite populations, we additionaly provide a bound on the price of anarchy as a function of the number vehicles in the system. We support the theoretical findings by means of numerical simulations.

Geodesic semilocal E-convex functions on Riemannian manifolds

In this paper, we have introduce a new class of sets and functions, called geodesic local E-convex sets, geodesic local E-convex functions and geodesic semilocal E-convex functions on Riemannian manifolds and study their properties. 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.

Multistability and instability of competitive neural networks with non-monotonic piecewise linear activation functions

This paper addresses the issue of multistability for competitive neural networks. First, a general class of continuous non-monotonic piecewise linear activation functions is introduced. Then, based on the fixed point theorem, the contraction mapping theorem and the eigenvalue properties of strict diagonal dominance matrix, it is shown that under some conditions, such n-neuron competitive neural networks have exactly 5n equilibrium points, among which 3n equilibrium points are locally exponentially stable and the others are unstable. Moreover, it is revealed that the neural networks with non-monotonic piecewise linear activation functions introduced in this paper can have greater storage capacity than the ones with Mexican-hat-type activation function and nondecreasing saturated activation function. In addition, unlike most existing multistability results of neural networks with nondecreasing activation functions, the location of those obtained 3n locally stable equilibrium points in this paper is more flexible. Finally, a numerical example is provided to illustrate and validate the theoretical findings via comprehensive computer simulations.

Robust decentralized control design for aircraft engines: A fractional type

A new decentralized control for aircraft engines is proposed. In the proposed control approach, aircraft engines are considered as uncertain large-scale systems composed of interconnected uncertain subsystems. For each subsystem, the time-varying uncertainty, including parameter disturbances and interconnections in/between subsystems, is depicted by a class of general nonlinear functions. A fractional robust decentralized control with two parts, the nominal one and the fractional one, is presented. The nominal control guarantees the asymptotical stability of the engine system without uncertainty. The fractional part aims at overcoming the influences of uncertainty. Compared to the previous studies, the presented control provides not only an extra flexibility for the system performance tuning by the fraction-type gain but also a facility for the control input calculation. The proposed control approach is applied to a turbofan engine with two subsystems. The computer simulation shows that, in the flight envelope, the fractional control not only guarantees the closed-loop system uniform boundedness and ultimate uniform boundedness but also shows good economy.

General overlap functions

As a generalization of bivariate overlap functions, which measure the degree of overlapping (intersection for non-crisp sets) of n different classes, in this paper we introduce the concept of general overlap functions. We characterize the class of general overlap functions and include some construction methods by means of different aggregation and bivariate overlap functions. Finally, we apply general overlap functions to define a new matching degree in a classification problem. We deduce that the global behaviour of these functions is slightly better than some other methods in the literature.