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All papers have been typed and translated trying to reproduce as much as possible their original aspect. We only used common fonts and the same style for the titles, the abstracts and the bibliography. [ABSTRACT FROM AUTHOR]

In this chapter we are going to prove a famous theorem due to Kodaira, which gives a characterization of which compact complex manifolds admit an embedding into complex projective space. In Sec. 1 we shall define Hodge manifolds as those which carry an integral (1, 1) form which is positive definite in local coordinates. We then give various examples of such manifolds. Kodaira's theorem asserts that a compact complex manifold is projective algebraic if and only if it is a Hodge manifold. This is a very useful theorem, as we shall see, since it is often easy to verify the criterion. Chow's theorem asserts that projective algebraic manifolds are indeed algebraic, i.e., defined by the zeros of homogeneous polynomials. Thus the combination of these two theorems allows one to reduce problems of analysis to ones of algebra (cf. Serre's famous paper [2] in which this program of comparison is carried out in great detail). [ABSTRACT FROM AUTHOR]

Throughout this paper we denote by Lp the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on Lp-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on Lp-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of Lp-spaces. We will discuss results related to the boundedness of positive linear operators on Lp-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from Lp to Lq. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur's criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on Lp-spaces and discuss as an example the weighted composition operators on Lp-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikišin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new. [ABSTRACT FROM AUTHOR]

We wrote a survey [18] on lattice ordered algebras five years ago. Why do we return to f-algebras once more? We hasten to say that there is only little overlap between the current paper and that previous survey.We have three purposes for the present paper. In our previous survey we remarked that one aspect that we did not discuss, while of some historical importance to the topic, is the theory of averaging operators. That theory has its roots in the nineteenth century and predates the rise of vector lattices. Positivity is a crucial tool in averaging, and positivity has been a fertile ground for the study of averaging-like operators. The fruits of positivity in averaging have recently (see [24]) started to appear in probability theory (to which averaging operators are close kin) and statistics. In the first section of our paper, we survey the literature for our selection of old theorems on averaging operators, at the same time providing some new perspectives and results as well. [ABSTRACT FROM AUTHOR]

For centuries celestial mechanics has been an exceptional rich source of problems and results in mathematics. To some extent this is still the case. Today one can discern, rather artificially, three problem fields. The first one is the study of classical problems like perturbed Kepler motion, orbits in the threebody problem, the theory of asteroids and comets, etc. The second one is a small but relatively important field in which the astrophysicists are interested; we are referring to systems with evolution like for instance changes caused by tidal effects or by exchange of mass. The third field is what one could call ‘mathematical celestial mechanics', a subject which is part of the theory of dynamical systems. The distinction between the fields is artificial. There is some interplay between the fields and hopefully, this will increase in the future. An interesting example of a study combining the first and the third field is the paper by Brjuno [41]. A typical example of an important mathematical paper which has found already some use in classical celestial mechanics is Moser's study on the geometrical interpretation of the Kepler problem [193]. Surveys of mathematical aspects of celestial mechanics have been given in [194] and [3]. [ABSTRACT FROM AUTHOR]

The main result of the present paper is a method to transform a matrix or operator which has a hierarchical semi-separable (HSS) representation into a URV (Moore-Penrose) representation in which the operators U and V represent collections of efficient orthogonal transformations and the block upper matrix R still has the HSS form. The paper starts with an introduction to HSS-forms and a survey of a recently derived multi resolution representation for such systems. It then embarks on the derivation of the main ingredients needed for a Moore-Penrose reduction of the system while keeping the HSS structure. The final result is presented as a sequence of efficient algorithmic steps, the efficiency resulting from the HSS structure that is preserved throughout. [ABSTRACT FROM AUTHOR]

The notion of class fields is usually attributed to Hilbert, but the concept was already in the mind of Kronecker and the term was used by Weber before the appearance of the fundamental papers of Hilbert. [ABSTRACT FROM AUTHOR]

Pairs of compact convex sets naturally arise in quasidifferential calculus as sub- and super-differentials of a quasidifferentiable function (see [1]). Since the sub- and superdifferential are not uniquely determined, minimal representations are of special importance. In this paper we show that the problem of finding minimal representatives for the elements of pairs of compact convex sets is a special case of the more general problem of determining minimal fractions in ordered commutative semigroups which satisfy the order cancellation law. All the material of this paper is taken from the recently published textbook on pairs of compact convex sets ([11]). [ABSTRACT FROM AUTHOR]

Two recent papers [1] and [2] have presented existence and uniqueness results for solutions of mixed variational inequality problems involving P-mappings and convex and separable but not necessarily differentiable functions where the feasible set is defined by box type constraints. In this paper we generalise these results for the case where the subspaces constituting the initial space are not real lines. [ABSTRACT FROM AUTHOR]

In this paper we survey some aspects of the theory of non-commutative Banach function spaces, that is, spaces of measurable operators associated with a semi- finite von Neumann algebra. These spaces are also known as non-commutative symmetric spaces. The theory of such spaces emerged as a common generalization of the theory of classical ("commutative") rearrangement invariant Banach function spaces (in the sense of W.A.J. Luxemburg and A.C. Zaanen) and of the theory of symmetrically normed ideals of bounded linear operators in Hilbert space (in the sense of I.C. Gohberg and M.G. Krein). These two cases may be considered as the two extremes of the theory: in the first case the underlying von Neumann algebra is the commutative algebra L∞ on some measure space (with integration as trace); in the second case the underlying von Neumann algebra is B (ℌ), the algebra of all bounded linear operators on a Hilbert space ℌ (with standard trace). Important special cases of these non-commutative spaces are the non-commutative Lp-spaces, which correspond in the commutative case with the usual Lp-spaces on a measure space, and in the setting of symmetrically normed operator ideals they correspond to the Schatten p-classes $$ \mathfrak{S}_p $$ . [ABSTRACT FROM AUTHOR]

This paper is a follow-up of K. Hrbacek's article showing how his approach can be pedagogically helpful when introducing analysis at pre-university level. Conceptual difficulties arise in elementary pedagogical approaches. In most cases it remains difficult to explain at pre-university level how the derivative is calculated at nonstandard values or how an internal function is defined. Hrbacek provides a modified version of IST [8] (rather Péraire's RIST) which seems to reduce all these difficulties. This system is briefly presented here in its pedagogical form with an application to the derivative. It must be understood as a state-of-the-art report. [ABSTRACT FROM AUTHOR]

The conclusion of a Radon-Nikodým theorem is that a measure μ can be represented as an integral with respect to a reference measure such that for all measurable sets A, μ(A) = ∫Afμ(x)dλ with a (Bochner or Lebesgue) integrable derivative or density fμ. The measure λ is usually a countably additive σ-finite measure on the given measure space and the measure μ is absolutely continuous with respect to λ. Different theorems have different range spaces for μ. which could be the real numbers, or Banach spaces with or without the Radon-Nikodým property. In this paper we generalize to derivatives of vector valued measures with respect a vector-valued reference measure. We present a Radon-Nikodým theorem for vector measures of bounded variation that are absolutely continuous with respect to another vector measure of bounded variation. While it is easy in settings such as μ << λ, where λ is Lebesgue measure on the interval [0,1] and μ is vector-valued to write down a nonstandard Radon-Nikodým derivative of the form ϕ : *[0,1] → fin(*E) by $$ \varphi _\mu (x) = \sum\nolimits_{i = 1}^H {\tfrac{{{}^*\mu (A_i )}} {{{}^*\lambda (A_i )}}1_{A_i } (x)} $$ a vector valued reference measure does not allow this approach, as the quotient of two vectors in different Banach spaces is undefined. Furthermore, generalizing to a vector valued control measure necessitates the use of a generalization of the Bartle integral, a bilinear vector integral. [ABSTRACT FROM AUTHOR]

In their important (but often overlooked) paper [1]. C. Ward Henson and Frank Wallenberg introduced the notion of S-measurability. and showed that S-measurable functions are "approximately standard" (in a sense made precise in the next section). [ABSTRACT FROM AUTHOR]

This work is based on Nelson's paper [1], where the central question was: under suitable regularity conditions, what is the form of the infinitesimal generator of a Markov semigroup? In the elementary approach using IST [2]. the idea is to replace the continuous state space, such as ℝ with a finite state space X possibly containing an unlimited number of points. The topology on X arises naturally from the probability theory. For x ε X, let $$ \mathcal{I}_x $$ be the set of all h ∈ $$ \mathcal{M} $$ vanishing at x where $$ \mathcal{M} $$ is the multiplier algebra of the domain $$ \mathcal{D} $$ of the infinitesimal generator. To describe the structure of the semigroup generator A, we want to split Ah(x)=∑y∈X\{x}a(x,y) h(y) so that the contribution of the external set Fx of the points far from x appears separately. A definition of the quantity αah(x)=∑y∈Fa(x,y) h(y) is given using the least upper bound of the sums on all internal sets W included in the external set F. This leads to the characterization of the global part of the infinitesimal generator. [ABSTRACT FROM AUTHOR]

Let (pθ)θ∈Θ be an exponential family in ℝk. After establishing nonstandard results about large deviations of the sample mean $$ \overline X $$, this paper defines the nonstandard likelihood ratio test of the null hypothesis H0 : θ ∈ hal($$ \widetilde\Theta _0 $$), where $$ \widetilde\Theta _0 $$ is a standard subset of Θ and hal($$ \widetilde\Theta _0 $$) its halo. If α is the level of the test, depending on whether lnα/n is infinitesimal or not we obtain different rejection criteria. We calculate risks of the first and second kinds (external probabilities) and prove that this test is more powerful than any "regular" nonstandard test based on $$ \overline X $$. [ABSTRACT FROM AUTHOR]

Neutrices are convex subgroups of the nonstandard real number system, most of them are external sets. They may also be viewed as modules over the external set of all limited numbers, as such non-noetherian. Because of the convexity and the invariance under some translations and multiplications, the external neutrices are appropriate models of orders of magnitude of numbers. Using their strong algebraic structure a calculus of external numbers has been developped. which includes solving of equations, and even an analysis, for the structure of external numbers has a property of completeness. This paper contains a further step, towards linear algebra and geometry. We show that in ℝ2 every neutrix is the direct sum of two neutrices of ℝ. The components may be chosen orthogonal. [ABSTRACT FROM AUTHOR]

Nonstandard Analysis suggests several ways in which the standard theories of distributions and other generalised functions could be reformulated. This paper reviews the contributions of José Sousa Pinto to this area up to his untimely death four years ago. Following the original presentation of nonstandard models for the Sebastião e Silva axiomatic treatment of distributions and ultradistributions he worked on a nonstandard theory of Sato hyperfunctions, using a simple ultrapower model of the hyperreals. (This in particular allows nonstandard representations for generalised distributions, such as those of Roumieu, Beurling, and so on.) He also considered a nonstandard theory for the generalised functions of Colombeau, and finally turned his attention to the hyperfinite representation of generalised functions, following the work of Kinoshita. [ABSTRACT FROM AUTHOR]

This appendix is an adaptation and extension of the paper [280]. [ABSTRACT FROM AUTHOR]

Several methods are currently available to simulate paths of the Brownian motion. In particular, paths of the BM can be simulated using the properties of the increments of the process like in the Euler scheme, or as the limit of a random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve series. In this paper we first propose a IFSM (Iterated Function Systems with Maps) operator whose fixed point is the trajectory of the BM. We then use this representation of the process to simulate its trajectories. The resulting simulated trajectories are self-affine, continuous and fractal by construction. This fact produces more realistic trajectories than other schemes in the sense that their geometry is closer to the one of the true BM's trajectories. The IFSM trajectory of the BM can then be used to generate more realistic solutions of stochastic differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with the process of crystallization.We first present two major models that describe this phenomenon either as a birth-and-growth process or in terms of a Johnson-Mehl random tessellation. Then, we estimate the parameters of these models and we establish the asymptotic law of the estimators for the geometrical aspect of this phenomenon. Simulations of these laws are also provided in some cases. [ABSTRACT FROM AUTHOR]

The paper investigates the problem of tracking a desired level of plasma glucose concentration. The model for the glucose-insulin system considered here, and recently published, belongs to the class of single-distributed delay models. The control law is obtained according to the feedback linearization theory. Both the cases of hyperglycemic and hypoglycemic patients have been considered. Simulations support theoretical results and show the physical reliability of the approach proposed. [ABSTRACT FROM AUTHOR]

In this paper we investigate two different models for minimizing the makespan of batch processes by mixed-integer linear programming models. Special emphasis is laid on a small number of binary variables and on valid constraints. After a reformulation of the objective function, it is for the first time possible to find optimal solutions for medium-sized benchmark problems. Furthermore, a powerful iterative construction heuristic for larger-sized problems is developed. [ABSTRACT FROM AUTHOR]

A subsoil cleanup technology is called bioventing: bacteria are used to biodegrade pollutants and air is injected to enhance their activity. In this paper a general mathematical model describing the physical phenomenon is presented. The model is based on the theory of fluid dynamics in porous media. A multi-component and multi-phase fluid is considered and the system of partial differential equations is coupled with a population bacteria equation. [ABSTRACT FROM AUTHOR]

In recent literature the authors have introduced a Delta formalism, á la Dirac, for the description of random closed sets of lower dimension with respect to the environment space ℝd. Mean densities can be introduced for expected measures associated with such sets, with respect to the usual Lebesgue measure. In this paper we offer a review of the main results; in particular approximating sequences for the quoted mean densities are provided, that are of interest in the concrete estimation of mean densities of fibre processes, surface processes, etc. For time dependent random closed sets, as the ones describing the evolution of birth-and-growth processes (of interest for many models in material science and in biomedicine), the Delta formalism provides a natural framework for deriving evolution equations for mean densities at any (integer) Hausdorff dimension, in terms of the relevant kinetic parameters. In this context connections with the concepts of hazard functions, and spherical contact functions are presented. [ABSTRACT FROM AUTHOR]

Butterfly wings are covered with a large number of colored scale cells. It is well known that there exist two different kinds of patterns in butterfly wings - the spacing pattern of scale cells and color pattern. The spacing pattern is cellular pattern in which scale cells form nearly parallel rows along the anteroposterior axis of the wing. On the other hand, the color pattern is mainly pigmentation pattern which is constructed as a finely-tiled mosaic of colored scale cells. In this paper, I present mathematical models together with numerical simulations for both the cellular spacing pattern and color pattern with experimental evidences. The relationship between color patterns of fore- and hind-wing are also discussed within the framework of the model. [ABSTRACT FROM AUTHOR]

‘Hierarchical Semi-separable' matrices (HSS matrices) form an important class of structured matrices for which matrix transformation algorithms that are linear in the number of equations (and a function of other structural parameters) can be given. In particular, a system of linear equations Ax = b can be solved with linear complexity in the size of the matrix, the overall complexity being linearly dependent on the defining data. Also, LU and ULV factorization can be executed ‘efficiently', meaning with a complexity linear in the size of the matrix. This paper gives a survey of the main results, including a proof for the formulas for LU-factorization that were originally given in the thesis of Lyon [1], the derivation of an explicit algorithm for ULV factorization and related Moore-Penrose inversion, a complexity analysis and a short account of the connection between the HSS and the SSS (sequentially semi-separable) case. A direct consequence of the computational theory is that from a mathematical point of view the HSS structure is ‘closed' for a number operations. The HSS complexity of a Moore-Penrose inverse equals the HSS complexity of the original, for a sum and a product of operators the HSS complexity is no more than the sum of the individual complexities. [ABSTRACT FROM AUTHOR]

In this paper we construct a class of homogeneous Hilbert modules over the disc algebra $$ \mathcal{A}(\mathbb{D}) $$ as quotients of certain natural modules over the function algebra $$ \mathcal{A}(\mathbb{D}^2 ) $$. These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class Bk($$ \mathbb{D} $$) and that they are mutually inequivalent, irreducible and homogeneous. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the truncated matricial moment problem on a finite closed interval in the case of an odd number of prescribed moments by using of the FMI method of V.P. Potapov. The solvability of this problem is characterized by the fact that two block Hankel matrices built from the data of the problem are nonnegative Hermitian (Theorem 1.3). An essential step to solve the problem under consideration is to derive an effective coupling identity between both block Hankel matrices (Proposition 2.5). In the case that these Hankel matrices are both positive Hermitian we parametrize the set of solutions via a linear fractional transformation the generating matrix-valued function of which is a matrix polynomial whereas the set of parameters consists of distinguished pairs of meromorphic matrix-valued functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Clifford algebra-valued admissible wavelets, which are associated to more than 2-dimensional Euclidean group with Dilations. We give an explicit characterization of the admissibility condition in terms of the Fourier transform, study the properties of this kind of wavelet transform, also give a family of admissible wavelets. [ABSTRACT FROM AUTHOR]

We show how fundamental ideas from signal processing, multiscale theory and wavelets may be applied to nonlinear dynamics. The problems from dynamics include iterated function systems (IFS), dynamical systems based on substitution such as the discrete systems built on rational functions of one complex variable and the corresponding Julia sets, and state spaces of subshifts in symbolic dynamics. Our paper serves to motivate and survey our recent results in this general area. Hence we leave out some proofs, but instead add a number of intuitive ideas which we hope will make the subject more accessible to researchers in operator theory and systems theory. [ABSTRACT FROM AUTHOR]

In earlier research multi-dimensional wavelets have been constructed in the framework of Clifford analysis. Clifford analysis, centered around the notion of monogenic functions, may be regarded as a direct and elegant generalization to higher dimension of the theory of the holomorphic functions in the complex plane. This Clifford wavelet theory might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one. In this paper we develop the idea of a metric dependent Clifford analysis leading to a so-called anisotropic Clifford wavelet theory featuring wavelet functions which are adaptable to preferential, not necessarily orthogonal, directions in the signals or textures to be analyzed. [ABSTRACT FROM AUTHOR]

In this paper we consider Jordan domains in real Euclidean spaces of higher dimension which have fractal boundaries. The case of decomposing a Hölder continuous multivector field on the boundary of such domains is obtained in closed form as sum of two Hölder continuous multivector fields harmonically extendable to the domain and to the complement of its closure respectively. The problem is studied making use of the Teodorescu transform and suitable extension of the multivector fields. Finally we establish equivalent condition on a Hölder continuous multivector field on the boundary to be the trace of a harmonic Hölder continuous multivector field on the domain. [ABSTRACT FROM AUTHOR]

In Chapters 4 and 7, we stressed that the crucial feature of Localization is shared by a number of basis constructions, most notably by those of wavelets and of certain classes of fractals. This includes basis constructions in Hilbert spaces built recursively on fractals and on state spaces in dynamics. The recursive approach to the more general basis constructions is a special case of a refined tool from probability which is based on martingales. (It should be contrasted to classical Fourier expansions, which are notoriously poorly localized.) [ABSTRACT FROM AUTHOR]

The paper mostly concerns applications of the generalized differentiation theory in variational analysis to Lipschitzian stability and metric regularity of variational systems in infinite-dimensional spaces. The main tools of our analysis involve coderivatives of set-valued mappings that turn out to be proper extensions of the adjoint derivative operator to nonsmooth and set-valued mappings. The involved coderivatives allow us to give complete dual characterizations of certain fundamental properties in variational analysis and optimization related to Lipschitzian stability and metric regularity. Based on these characterizations and extended coderivative calculus, we obtain efficient conditions for Lipschitzian stability of variational systems governed by parametric generalized equations and their specifications. [ABSTRACT FROM AUTHOR]

In the first section of this paper we study the Hölder-continuity of solutions of the Schrödinger degenerate equation $$ - \sum\limits_{i,j = 1}^n {\left( {a_{ij} u_{x_i } } \right)_{x_j } + cu = 0,} $$ assuming the potential c belonging to appropriate degenerate Morrey spaces. In the second section we obtain the existence and the uniqueness of the solution of a variational inequality associated to the degenerate operator $$ Lu = - \sum\limits_{i,j = 1}^n {\left( {a_{ij} \left( x \right)u_{x_i } + d_j u} \right)_{x_j } } + \sum\limits_{i = l}^n {b_i u_{x_i } + cu} $$ assuming the coefficients of the lower terms and the known term belonging to a suitable degenerate Stummel-Kato class. In both cases the weight w, which gives the degeneration, belongs to the Muckenoupt class A2. [ABSTRACT FROM AUTHOR]

This paper deals with results about projectors on the important class of closed, prox-regular sets in ℝn. These sets, which include all closed convex sets but also many nonconvex sets, have the property that their associated projection mappings are very well behaved, being locally single-valued and continuous among other good properties. We give elementary proofs of these properties of the projector, and for the case in which the projection is made onto a perturbed set we show that under suitable conditions the projector is jointly continuous in the perturbation variable and the variable expressing the point that is projected. We briefly describe an application to the extension of a normal-map construction from variational inequalities posed over polyhedral convex sets to variational conditions posed over sets that satisfy prox-regularity. [ABSTRACT FROM AUTHOR]

The Fenchel problem of level sets was formulated by Roberts and Varberg in their book titled "Convex functions" (1973, p. 271) is as follows: "What "nice" conditions on a nested family of convex sets will ensure that it is the family of level sets of a convex function?" The aim of the paper is to draw attention to this structural question of convex analysis and to survey some results in different directions. [ABSTRACT FROM AUTHOR]

In this paper, we survey the development of semismooth Newton methods for solving the shape-preserving interpolation problem, the option price problem, and the semi-infinite programming problem. [ABSTRACT FROM AUTHOR]

In this paper we illustrate the connections between generalized variational inequalities (GVI) and other mathematical models: optimization, complementarity, inclusions, dynamical systems. In particular, we analyse relationships between existence theorems of solutions of GVI and existence theorems of equilibrium points of inclusions and projected differential inclusions. [ABSTRACT FROM AUTHOR]

In this paper we treat problem arasing in semiconductor theory from a mathematical and numerical point of view, in particular we consider a boundary value problem with unknown interfaces arising by the determination of the depletion layer in the most basic semiconductor device namely the p-n junction diode. We present the numerical approximation of free boundary problem with double obstacle treated with quasi-variational inequalities. We deal with the L∞ convergence of the standard finite element approximation of the system of quasi-variational inequalities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate differential properties of a class of set-valued maps and gap functions involving vector variational inequalities. Relationship between their contingent derivatives are discussed. A formula computing contingent derivative of the gap functions is established. Optimality conditions of solutions for vector variational inequalities are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an augmented Lagrangian for nonlinear semidefinite programs. Some basic properties of the augmented Lagrangian such as differentiabilty, monotonicity and convexity, are discussed. Necessary and sufficient conditions for a strong duality property and an exact penalty representation in the framework of augmented Lagrangian are derived. Under certain conditions, it is shown that any limit point of a sequence of stationary points of augmented Lagrangian problems is a Karuh, Kuhn-Tucker (for short, KKT) point of the original semidefinite program. [ABSTRACT FROM AUTHOR]

In this paper Generalized Complementarity Problems are expressed in terms of suitable optimization problems and some optimality conditions are given. The infinite dimensional Lagrangean and Duality Theories play an important role in order to achieve the main result. [ABSTRACT FROM AUTHOR]

The paper presents a study of transport in urban areas served by a public transport system, as well as by private vehicles on which road pricing is imposed. It is supposed that the road pricing fare, the ticket price and the frequency of the lines of public transport are established by the Public Administration in such a way that the surplus of users of both the transport modes is maximised, under the conditions that the system is in equilibrium, the budget constraint of the company managing public transport is satisfied, and the private transport demand does not exceed a given threshold for environmental reasons. The theoretical model that has been devised leads to a problem of nonlinear programming, with an equilibrium constraint formulated as a fixed point problem. From an application of the model to an urban area it emerges that, if the proceeds of road pricing are used for financing public transport, the results of road pricing essentially depend on the proportion of demand that is captive to public transport, and on the level of congestion existing on the urban road network before the imposition of road pricing. [ABSTRACT FROM AUTHOR]

In this paper we study the solvability of a nonlinear Hammerstein integral equation by using a variational principle of B. Ricceri and methods of critical point theory. In particular we do not require any positivity assumption on the kernel of the equation. Our results can be applied to higher order elliptic boundary value problem with changing sign kernel. [ABSTRACT FROM AUTHOR]

In this paper we prove, in the framework of reflexive Banach spaces, that a linear and continuous functional f achieves its supremum on every small ε -uniform perturbation of a closed convex set C containing no lines, if and only if f belongs to the norm-interior of the barrier cone of C. This result is applied to prove that every closed convex subset C of a reflexive Banach space X which contains no lines is continuous if and only if every small ε -uniform perturbation of C does not allow non-attaining linear and continuous functionals. Finally, we define a new class of non-coercive variational inequalities and state a corresponding open problem. [ABSTRACT FROM AUTHOR]

In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called "oriented distance" function [14,15]. In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis is needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4]. [ABSTRACT FROM AUTHOR]

This notes concern the Navier-Stokes equations with gradient dependent viscosity and slip (or non-slip) type boundary conditions. Regularity up to the boundary still presents many open problems. In the sequel we present some regularity results for weak solutions to the Ladyzhenskaya model in the half space ℝ+n. See Theorems 3.1 and 3.2. Complete proofs of these results are done, and will appear in the forthcoming paper [6]. [ABSTRACT FROM AUTHOR]