A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory

The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower semicontinuous function defined on a Banach space, through an approach based on an abstract notion of subdifferential operator, and taking into account the “smoothness” of the Banach space. Here, we give (Theorem 1) an extension in a metric setting, based on the notion of slope from [11] and coercivity is considered in a generalized sense, inspired by [9]; our result allows to recover, for example, the coercivity result of [19], where a weakened version of the Palais-Smale condition is used. Our main tool (Proposition 1) is a consequence of Ekeland’s variational principle extending [12, Corollary 3.4], and deals with a function f which is, in some sense, the “uniform” Γ-limit of a sequence of functions.

Limit Cycles of Perturbations of a Class of Quadratic Hamiltonian Vector Fields

We prove that in quadratic perturbations of generic Hamiltonian vector fields with two saddle points and one center there can appear at most two limit cycles. This bound is exact.

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-Valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gˆateaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions.

Quadratic Mean Radius of a Polynomial in C(Z)

* Dedicated to the memory of Prof. N. Obreshkoff

Some Comments on Q-Irresolute and Quasi-Irresolute Functions

The aim of this paper is to continue the study of θ-irresolute and quasi-irresolute functions as well as to give an example of a function which is θ-irresolute but neither quasi-irresolute nor an R-map and thus give an answer to a question posed by Ganster, Noiri and Reilly. We prove that RS-compactness is preserved under open, quasi-irresolute surjections.

On the Quality of Decision Functions in Pattern Recognition

The problem of decision functions quality in pattern recognition is considered. An overview of the approaches to the solution of this problem is given. Within the Bayesian framework, we suggest an approach based on the Bayesian interval estimates of quality on a finite set of events.

Growing Neural Networks Using Nonconventional Activation Functions

In the paper, an ontogenic artificial neural network (ANNs) is proposed. The network uses orthogonal activation functions that allow significant reducing of computational complexity. Another advantage is numerical stability, because the system of activation functions is linearly independent by definition. A learning procedure for proposed ANN with guaranteed convergence to the global minimum of error function in the parameter space is developed. An algorithm for structure network structure adaptation is proposed. The algorithm allows adding or deleting a node in real-time without retraining of the network. Simulation results confirm the efficiency of the proposed approach.

Double-Wavelet Neuron Based on Analytical Activation Functions

In this paper a new double-wavelet neuron architecture obtained by modification of standard wavelet neuron, and its learning algorithm are proposed. The offered architecture allows to improve the approximation properties of wavelet neuron. Double-wavelet neuron and its learning algorithm are examined for forecasting non-stationary chaotic time series.

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35

On Integral Means for Fractional Calculus Operators of Multivalent Functions

2000 Mathematics Subject Classification: Primary 30C45, Secondary 26A33, 30C80

Mean-Periodic Functions Associated with the Jacobi-Dunkl Operator on R

2000 Mathematics Subject Classification: 34K99, 44A15, 44A35, 42A75, 42A63

Applications of the Owa-Srivastava Operator to the Class of K-Uniformly Convex Functions

2000 Mathematics Subject Classification: Primary 30C45, 26A33; Secondary 33C15

Spectrum of Functions for the Dunkl Transform on R^d

Mathematics Subject Classification: 42B10

Theorems on the Convergence of Series in Generalized Lommel-Wright Functions

Mathematics Subject Classification: 30B10, 30B30; 33C10, 33C20

Inclusion Properties for Certain Classes of Analytic Functions Involving a Family of Fractional Integral Operators

2000 Math. Subject Classification: 30C45