Injectivity and adjoint extremals

The following investigation deals with injectivity results for general diffeomorphisms as well as for Haar's transformation. We apply these results to adjoint variational problems introduced by Haar [6]. In particular, we prove a generalization of Krust's theorem for two-dimensional adjoint extremals. Received March 20, 1998 / Accepted April 24, 1998     

Dual conditions characterizing optimality for convex multi-objective programs

Received June 13, 1995 / Revised version received February 6, 1998 Published online August 18, 1998     

On the Convergence of Newton’s Method for Polynomial Equations and Applications in Radiative Transfer

 Newton’s method is used to approximate a locally unique zero of a polynomial operator F of degree in Banach space. So far, convergence conditions have been found for Newton’s method based on the Newton-Kantorovich hypothesis that uses Lipschitz-type conditions and information only on the first Fréchet-derivative of F. Here we provide a new semilocal convergence theorem for Newton’s method that uses information on all Fréchet-derivatives of F except the first. This way, we obtain sufficient convergence conditions different from the Newton-Kantorovich hypothesis. Our results are extended to include the case when F is a nonlinear operator whose kth Fréchet-derivative satisfies a H?lder continuity condition. An example is provided to show that our conditions hold where all previous ones fail. Moreover, some applications of our results to the solution of polynomial systems and differential equations are suggested. Furthermore, our results apply to solve a nonlinear integral equation appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field. Received 9 December 1997 in revised form 30 March 1998     

Improved Hessian approximations for the limited memory BFGS method

This paper considers simple modifications of the limited memory BFGS (L-BFGS) method for large scale optimization. It describes algorithms in which alternating ways of re-using a given set of stored difference vectors are outlined. The proposed algorithms resemble the L-BFGS method, except that the initial Hessian approximation is defined implicitly like the L-BFGS Hessian in terms of some stored vectors rather than the usual choice of a multiple of the unit matrix. Numerical experiments show that the new algorithms yield desirable improvement over the L-BFGS method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Homogeneous variational complexes and bicomplexes

We present a family of complexes playing the same rôle, for homogeneous variational problems, that the horizontal parts of the variational bicomplex play for variational problems on a fibred manifold. We show that, modulo certain pullbacks, each of these complexes (apart from the first one) is globally exact. All the complexes may be embedded in bicomplexes, and we show that, again modulo pullbacks, the latter are locally exact. The edge sequence is an important part of such a bicomplex, and may be used for the study of homogeneous variational problems.     

Chromatographic retention prediction and octanol-water partition coefficient determination of monobasic weak acidic compounds in ion-suppression reversed-phase liquid chromatography using acids as ion-suppressors

Although simple acids, replacing buffers, have been widely applied to suppress the ionization of weakly ionizable acidic analytes in reversed-phase liquid chromatography (RPLC), none of the previously reported works focused on the systematic studies about the retention behavior of the acidic solutes in this ion-suppression RPLC mode. The subject of this paper was therefore to investigate the retention behavior of monobasic weak acidic compounds using acetic, perchloric and phosphoric acids as the ion-suppressors. The apparent octanol-water partition coefficient (K^″ow) was proposed to calibrate the octanol-water partition coefficient (Kow) of these weak acidic compounds, which resulted in a better linear correlation with log kw, the logarithm of the hypothetical retention factor corresponding to neat aqueous fraction of hydroorganic mobile phase. This log K^″ow−log kw linear correlation was successfully validated by the results of monocarboxylic acids and monohydrating phenols, and moreover by the results under diverse experimental conditions for the same solutes. This straightforward relationship not only can be used to effectively predict the retention values of weak acidic solutes combined with Snyder-Soczewinski equation, but also can offer a promising medium for directly measuring Kow data of these compounds via Collander equation. In addition, the influence of the different ion-suppressors on the retention of weak acidic compounds was also compared in this RPLC mode.     

Notes on a class of one-dimensional Landau-Brazovsky models

In the paper, a class of one-dimensional Landau–Brazovsky models is investigated. We present a sufficient condition under which the corresponding functional achieves its minimum. Moreover, a nonexistence result for nontrivial critical points is given. Project supported by the Doctoral Programme Foundation of the Ministry of Education of China.     

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.     

Decomposing the essential spectrum

We use C^*-algebra theory to provide a new method of decomposing the essential spectra of self-adjoint and non-self-adjoint Schrödinger operators in one or more space dimensions.     

A new class of transport distances between measures

We introduce a new class of distances between nonnegative Radon measures in . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. J. Dolbeault and B. Nazaret have been partially supported by the ANR project IFO. The second author has also been partially supported by the ANR project OTARIE. G. Savaré has been partially supported by grants of M.I.U.R., PRIN ’06. Part of this research was carried out while the third author was visiting professor at Ceremade, Université Paris-Dauphine, whose hospitality and support are also gratefully acknowledged.