An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

A quantitative analysis of Oka’s lemma

In this paper, we will examine a strong form of Oka’s lemma which provides sufficient conditions for compact and subelliptic estimates for the -Neumann operator on Lipschitz domains. On smooth domains, the condition for subellipticity is equivalent to D’Angelo finite type and the condition for compactness is equivalent to Catlin’s condition (P). As an application, we will prove regularity for the -Neumann operator in the Sobolev space W ^s , , on C ² domains.     

Korn’s second inequality and geometric rigidity with mixed growth conditions

Geometric rigidity states that a gradient field which is $L^p$ -close to the set of proper rotations is necessarily $L^p$ -close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in $L^p+L^q$ and in $L^{p,q}$ interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn’s inequality to these spaces.     

A nonlinear Picone’s identity and its applications

The purpose of this note is to show a generalization to Picone’s identity in a nonlinear framework. The classical Picone’s identity turns out to be a particular case of our result. We show, as an application of our results, that the Morse index of the zero solution to a semilinear elliptic boundary value problem is 0 and also establish a linear relationship between the components of the solution of a nonlinear elliptic system.     

Amply regular graphs with Hoffman’s condition

It is known that, if the minimal eigenvalue of a graph is −2, then the graph satisfies Hoffman’s condition: for any generated complete bipartite subgraph K _1,3 (a 3-claw) with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to the vertices q ₁ and q ₂ is adjacent to p but not adjacent to q ₃. We prove the converse statement for amply regular graphs containing a 3-claw and satisfying the condition μ > 1.     

Strongly regular graphs with Hoffman’s condition

It is known that if the minimal eigenvalue of a graph is ?2, then the graph satisfies Hoffman’s condition; i.e., for any generated complete bipartite subgraph K _1,3 with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to two vertices from the second part is not adjacent to the third vertex and is adjacent to p. We prove the converse statement, formulated for strongly regular graphs containing a 3-claw and satisfying the condition gm > 1.     

Amply regular graphs with Hoffman’s condition

It is known that, if the minimal eigenvalue of a graph is ?2, then the graph satisfies Hoffman’s condition: for any generated complete bipartite subgraph K _1,3 (a 3-claw) with parts {p} and {q ₁, q ₂, q ₃}, any vertex distinct from p and adjacent to the vertices q ₁ and q ₂ is adjacent to p but not adjacent to q ₃. We prove the converse statement for amply regular graphs containing a 3-claw and satisfying the condition µ > 1.     

Refined geometric realization of a quantum group and Lusztig’s symmetries

In this paper, we shall give a refinement of the geometric realization of Lusztig’s algebra f given by Lusztig. This realization gives a geometric interpretation of the decomposition of f. As an application, we shall give geometric realizations of Lusztig’s symmetries on the whole quantum group.     

A geometric approach to Euler’s addition formula

Plane quartic curves given by equations of the form y ²=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form \(\int (1/\sqrt{P(x)})\,\mathrm{d}x\) with a quartic polynomial P can be derived directly from this addition law.     

Approximation of conic section by quartic Bzier curve with endpoints continuity condition

A new method for approximation of conic section by quartic B′ezier curve is presented, based on the quartic B′ezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme,and the effectiveness of this method is demonstrated by some numerical examples.     

A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations

Summary This paper is devoted to the numerical analysis of some finite volume discretizations of Darcys equations. We propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems. This leads to optimal a priori error estimates. In view of mesh adaptivity, we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way. Mathematics Subject Classification (2000):65G99, 65M06, 65M15, 65M60, 65P05This work was partially supported by Contract C03127/AEE2714 with the Laboratoire National dHydraulique of the Division Recherche et Développement of Électricité de France. We thank B. Gest and her research group for very interesting discussions on this subject.     

A counterexample to Herzog’s Conjecture on the number of involutions

In 1979, Herzog put forward the following conjecture: if two simple groups have the same number of involutions, then they are of the same order. We give a counterexample to this conjecture.     

Quasi-convex sets and size × curvature condition,application to nonlinear inversion

We define a family of sets of a Hilbert space (quasi-convex sets) on which a generalization of the usual theory of projection on convex sets can be defined (existence, uniqueness, and stability of the projection of all points of some neighborhood of the set). We then give a constructive sufficient condition, called the size × curvature condition, for a setD to be quasi-convex, which involves radii of curvatures of curves lying on the setD. Finally, we use the above result for the study of nonlinear least-squares problems, as they appear in parameter estimation, for which we give a sufficient condition ensuring existence, uniqueness, and stability.     

Periodic points of algebraic functions and Deuring’s class number formula

The Ramanujan Journal - In this paper, transformation formulas for the function $$\begin{aligned} A_{1}\left( z,s:\chi \right) =\sum \limits _{n=1}^{\infty }\sum \limits _{m=1} ^{\infty }\chi...