The entropy condition and the admissibility of shocks

The author proposed (Trans. Amer. Math. Soc.199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author (J. Differential Equations18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand [Uspehi Mat. Nauk14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller [Comm. Pure Appl. Math.23 (1970), 867–884] proved that a shock satisfies Lax's shock inequalities [cf. Comm. Pure Appl. Math.14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.     

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On Gronwall-Bellman-Bihari-type integral inequalities in several variables with retardation

Presented are some new nonlinear integral inequalities of the Gronwall-Bellman-Bihari type in n-independent variables with delay which extend recent results of C. C. Yeh and M.-H. Shin [J. Math. Anal. Appl.86, (1982), 157–167], C. C. Yeh [J. Math. Anal. Appl.87, (1982), 311–321], and A. I. Zahariev and D. D. Bainor [J. Math. Anal. Appl.89, (1981), 147–149]. Some applications of the results are included.     

A sharp Sobolev inequality on Riemannian manifoldsUne inégalité de Sobolev optimale sur les varietés riemanniennes

We outline our results in [11] concerning some sharp Sobolev inequalities on Riemannian manifolds. Our inequalities emphasize the role of scalar curvature in this context. To cite this article: Y.Y. Li, T. Ricciardi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 519–524.     

Random nonlinear Hammerstein equations

We prove existence theorems for random differential equations defined in a separable reflexive Banach space. These theorems are proved through the use of theory of random analysis established in [X. Z. Yuan, Random nonlinear mappings of monotone type, J. Math. Anal. Appl. 19] which differs from the other means, for example in [R. Kannan and H. Salehi, Random nonlinear equations and monotonic nonlinearities, J. Math. Anal. Appl. 57 (1977), 234–256; D. Kravvaritis, Existence theorems for nonlinear random equations and inequalities, J. Math. Anal. Appl. 86 (1982), 61–73; D. A. Kandilakis and N. S. Papageorgious, On the existence of solutions for random differential inclusions in a Banach space, J. Math. Anal. Appl. 126 (1987), 11–23].     

On discrete inequalities of Wirtinger's type

Discrete inequalities of Wirtinger's type are considered. Constants in the obtained inequalities are the best ones. In the special case the inequalities (1) and (2) are obtained. They are proved by K. Fan, O. Taussky, and J. Todd: Discrete analogs of inequalities of Wirtinger, Montash. Math.59 (1955), 73–79.     

Corrigendum to the Note “Symplectic capacities of toric manifolds and combinatorial inequalities” [C. R. Acad. Sci. Paris,Ser. I 334 (10) (2002) 889–892]

In this Note we correct some results in Lu, Symplectic capacities of toric manifolds and combinatorial inequalities [C. R. Acad. Sci. Paris, Ser. I 334 (10) (2002) 889–892] on (pseudo) symplectic capacities for toric manifolds. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A Bennett concentration inequality and its application to suprema of empirical processesUne inégalité de concentration de type Bennett et son application aux maxima de processus empiriques

We introduce new concentration inequalities for functions on product spaces. They allow to obtain a Bennett type deviation bound for suprema of empirical processes indexed by upper bounded functions. The result is an improvement on Rio's version [6] of Talagrand's inequality [7] for equidistributed variables. To cite this article: O. Bousquet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 495–500.     

Isoperimetry and symmetrization for Sobolev spaces on metric spaces

Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, Poincaré inequalities, as well as metric versions of the Pólya–Szegö and Faber–Krahn principles. To cite this article: J. Martín, M. Milman, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Some Remarks on Korn Inequalities

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks(see[Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I,350, 2012, 861–865].     

Estimates for Wallis’ ratio and related functions

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function.     

On some conformally invariant fully nonlinear equationsSur certaines équations completement nonlinéaires invariantes par transformation conforme

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.     

Non-Liouville numbers and a Theorem of Hörmander

It is shown in this paper that Theorem 1 of [G. H. Meisters, “Translation-invariant linear forms and a formula for the Dirac measure,” J. Functional Analysis 8 (1971), 173–188] can be deduced from a very general result of Lars Hörmander, namely, Theorem 1 of “Generators for some rings of analytic functions” [Bull. Amer. Math. Soc.73 (1967), 943–949]. However, Hörmander's theorem is evidently not applicable in several other cases where Meisters'-type results have been obtained (e.g., Theorem 1 of G.H. Meisters and Wolfgang M. Schmidt, “Translation-invariant linear forms on L²(G) for compact abelian groups G,” J. Functional Analysis11 (1972), 407–424).     

Efficiency and Cramér-Rao type lnequalities for convex loss functions

A general method for obtaining inequalities of Cramér-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cramér-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts (“Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,”) and of Blyth (Ann. Statist.2, 464–473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.     

Weak convergence theorems for strict pseudo-contractions in Banach spaces

In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257–1265(2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results(Lemma 3.4, Theorems 3.5–3.6, 3.8–3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages(2010)] do not hold in Lpfor p 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include Lpfor p ≥ 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above.     

Variational Inequalities: an Elementary Approach

We will deal with the following problem: Let M be an n×n matrix with real entries. Under which conditions the family of inequalities: x∈? ⁿ ;x?0;M·x?0has non–trivial solutions? We will prove that a sufficient condition is given by m_i,j+m_j,i?0 (1?i,j?n); from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.     

Spaces of lower semicontinuous set-valued maps

We give a negative answer to Problem 2 posed by R. A. McCoy in his paper [McCoy R. A.: Spaces of lower semicontinuous set-valued maps II, Math. Slovaca 60 (2010), 541–570]. Some topological properties of the space L ^?(X) introduced in [McCOY R. A.: Spaces of lower semicontinuous set-valued maps I, Math. Slovaca 60 (2010), 521–540] equipped with the Vietoris topology are also investigated.     

On linear forms with algebraic coefficients and diophantine equations

In my paper, [Man. Math.18 (1976), Satz 1.1] I proved a result on simultaneous diophantine inequalities for p-adic linear forms with algebraic coefficients. In this paper I shall generalize this result and give a necessary and sufficient criterion for the estimation of a product of complex and p-adic linear forms with algebraic coefficients, implying a theorem of Schmidt, [Math. Ann.191 (1971), Satz 1]. Using this estimate I shall obtain the p-adic generalization of Schmidt's theorems on diophantine equations of norm form type [Ann. of Math.96 (1972)].     

The asymmetric travelling salesman problem and a reformulation of the Miller–Tucker–Zemlin constraints

In this paper we compare the linear programming (LP) relaxations of several old and new formulations for the asymmetric travelling salesman problem (ATSP). The main result of this paper is the derivation of a compact formulation whose LP relaxation is characterized by a set of circuit inequalities given by Grotschel and Padberg (In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A., Shmoys, D.B. (Eds.), The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985). The new compact model is an improved and disaggregated version of a well-known model for the ATSP based on the subtour elimination constraints (Miller et al., Journal of ACM 7 (1960) 326–329). The circuit inequalities are weaker than the subtour elimination constraints given by Dantzig et al. However, each one of these circuit inequalities can be lifted into several different facet defining inequalities which are not dominated by the subtour elimination inequalities. We show that some of the inequalities involved in the previously mentioned compact formulation can be lifted in such a way that, by projection, we obtain a small subset of the so-called D_k^← and D_k^→ inequalities. This shows that the LP relaxation of our strongest model is not dominated by the LP relaxation of the model presented by Dantzig et al. (Operations Research 2 (1954) 393–410). The new models motivate a new classification of formulations for the ATSP.     

Concentration and isoperimetry are equivalent assuming curvature lower bound

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry–Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. Our method is entirely geometric, continuing the approach of Gromov, Buser and Morgan. To cite this article: E. Milman, C. R. Acad. Sci. Paris, Ser. I 347 (2009).