Nonlinear extended variational inequalities without differentiability: Applications and solution methods

We consider a class of nonlinear problems which is intermediate between equilibrium and variational inequality ones. Several classes of applications of such problems are described. Iterative methods are proposed for finding a solution. The methods are utilized without differentiability properties of the mappings and converge to a solution under weakened monotonicity type assumptions.     

Nonlinear programming without differentiability in banach spaces: necessary and sufficient constraint qualification

The problem under consideration is a maximization problem over a constraint set defined by a finite number of inequality and equality constraints over an arbitrary set in a reflexive Banach space. A generalization of the Kuhn-Tucker necessary conditions is developed where neither the objective function nor the constraint functions are required to be differentiable. A new constraint qualification is imposed in order to validate the optimality criteria. It is shown that this qualification is the weakest possible in the sense that it is necessary for the optimality criteria to hold at the point under investigation for all families of objective functions having a constrained local maximum at this point     

Nonlinear similarity analysis for epileptic seizures prediction

The prediction of epileptic seizures can promise a new diagnostic application and a novel approach for seizure control. This paper proposes an improved dynamical similarity measure to predict epileptic seizures in electroencephalographic (EEG). First, mutual information and Cao's method are employed to reconstruct a phase space of preprocessed EEG recordings by using the positive zero crossing method. Second, a Gaussian function replaces the Heavyside function within correlation integral at calculating a similarity index. The crisp boundary of the Heavyside function is eliminated because of the Gaussian function's smooth boundary. Third, an adaptive detection method based on the similarity index is proposed to predict the epileptic seizures. In light of test results of EEG recordings of rats, it is found that the new dynamical similarity index is insensitive to the selection of the radius value of Gaussian function and the size of segmented EEG recordings. Comparing with the dynamical similarity index proposed by Le Van Quyen et al. [Anticipating epileptic seizures in real time by a non-linear analysis of similarity between EEG recordings, NeuroReport 10 (1999) 2149–2155], the tests of twelve rats show the new dynamical similarity index is better to predict the epileptic seizures.     

Analyticity without differentiability

In this article we derive all salient properties of analytic functions, including the analytic version of the inverse function theorem, using only the most elementary convergence properties of series. Not even the notion of differentiability is required to do so. Instead, analytical arguments are replaced by combinatorial arguments exhibiting properties of formal power series. Along the way, we show how formal power series can be used to solve combinatorial problems and also derive some results in calculus with a minimum of analytical machinery.     

Automorphic forms and differentiability properties

We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.

     

Generalized holomorphic processes and differentiability

Holomorphic processes are studied when the parameter set is a general finite atomless separable measure space, using the differentiation operator and Malliavin calculus. Generalized martingales are connected with differentiable processes. In the two-parameter case, these methods give a new simpler proof of the characterization of holomorphic processes by power series expansion.This work has been done while D. Nualart was staying as a visitor at Bar-Ilan University.     

Asymptotic linearity and Hadamard differentiability

For a mapping between Banach spaces, two weaker variants of the usual notion of asymptotic linearity are defined and explored. It is shown that, under inversion through the unit sphere, they correspond to Hadamard and weak Hadamard differentiability at the origin of the inversion. Nemytskii operators from Sobolev spaces to Lebesgue spaces over R^N${\mathbb{R}}^N$ share these weaker properties but they are not asymptotically linear in the usual sense.     

Fractional programming without differentiability

The notion of quasi-differentiability is examined and related to fractional programming. Necessary and sufficient conditions are given and various other properties of quasi-differentiable functions are discussed. Differentiability is not assumed.This research was partially supported by N.R.C. Grants A7751 and A7675.     

Cone monotone mappings: Continuity and differentiability

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces, and we define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): K$K$-monotone dominated and cone-to-cone monotone mappings. K$K$-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and K$K$-monotone functions defined by Borwein, Burke and Lewis to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zají?ek, we show some relationships between these classes. Then, we show that every K$K$-monotone function f:X→R$f:X\to \mathbb{R}$, where X$X$ is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for K$K$-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning almost everywhere differentiability (also in metric and w^∗$w^{\ast }$-senses) of these mappings.     

Continuity and differentiability for weighted Sobolev spaces

Our aim in this paper is to discuss continuity and differentiability of functions in weighted Sobolev spaces in the limiting case of Sobolev's imbedding theorem.

     

Bump functions and differentiability in Banach spaces

We show that if a Banach space admits a continuous symmetrically Fréchet subdifferentiable bump function, then is an Asplund space.

     

Vector continuous-time programming without differentiability

In this work continuous-time programming problems of vector optimization are considered. Firstly, a nonconvex generalized Gordan’s transposition theorem is obtained. Then, the relationship with the associated weighting scalar problem is studied and saddle point optimality results are established. A scalar dual problem is introduced and duality theorems are given. No differentiability assumption is imposed.