The complex Frobenius theorem for rough involutive structures

We establish a version of the complex Frobenius theorem in the context of a complex subbundle of the complexified tangent bundle of a manifold having minimal regularity. If the subbundle defines the structure of a Levi-flat CR-manifold, it suffices that be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on recent work of the authors.

     

A vanishing result for strictly p-convex domains

In view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for \(q\) -complete domains in \(\mathbb C ^{n}\) , we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly \(p\) -convex domains in \(\mathbb R ^n\) in the sense of Harvey and Lawson (The foundations of \(p\) -convexity and \(p\) -plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the \({L}^2\) -techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965).     

The Wiener-Ikehara theorem by complex analysis

The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman's contour integration method can be adapted to establish the Wiener-Ikehara theorem. A simple special case suffices for the PNT. But what about the twin-prime problem?

     

Central limit theorem for complex measures

We prove two central limit theorems for real identically distribution random variables where the distribution is a complex-valued Borel measure . The results involve the weak convergence of the suitably scaledn-fold convolution of certain complex atomic or absolutely continuous measures of spectral radius 1 to ahypergaussian measure whose Fourier-Stieltjes transform is exp(–² for a positive integer . The proof uses a generalization of the method of characteristic functions. Counter-examples are given to rather more general statements that had appeared previously in the literature. This research arose in connection with problems related to general tauberian theorems on the line for complexvalued summability methods which are discussed at the end of the paper. Some interesting examples are given generalizing well-known theorems related to Euler and Borel summability.     

The boundary of a Busemann space

Let be a proper Busemann space. Then there is a well defined boundary, , for . Moreover, if is (Gromov) hyperbolic (resp. non-positively curved), then this boundary is homeomorphic to the hyperbolic (resp. non-positively curved) boundary.

     

The Bloch theorem in several complex variables

The background theory for the Bloch theorem is generalized to several complex variables. This work involves study of the Bergman kernel functions in order to extend work of Landau and Bonk. The main conclusion is an estimate for Bloch’s constant for mappings of domains of the first classical type. In the special case of then-dimensional ball, the estimate of Bloch’s constant coincides with that of Liu.     

A Gauss-Bonnet-Chern theorem for complex Finsler manifolds

Using Chern’s method of transgression and the currents, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds (M, F). This result extends the classical Gauss-Bonnet-Chern theorem for Hermitian manifolds. Furthermore, a simplified version of the Gauss-Bonnet-Chern theorem is obtained in the case of complex Berwald 1-manifolds.     

A min-max theorem for complex symmetric matrices

We optimize the form Re x^tTx to obtain the singular values of a complex symmetric matrix T. We prove that for ,

     

On the finite-increment theorem for complex polynomials

For an arbitrary polynomial P of degree at most n and any points z ₁ and z ₂ on the complex plane, we establish estimates of the form $$ \left| {P(z_1 ) - P(z_2 )} \right| \geqslant d_n \left| {P'(z_1 )} \right|\left| {z_1 - \zeta } \right| $$ , where ζ is one of the roots of the equation P(z) = P(z ₂), and d _n is a positive constant depending only on the number n.     

A generalized localization theorem and geometric inequalities for convex bodies

In this article, we generalize a localization theorem of Lovász and Simonovits [Random walks in a convex body and an improved volume algorithm, Random Struct. Algorithms 4-4 (1993) 359-412] which is an important tool to prove dimension-free functional inequalities for log-concave measures. In a previous paper [Fradelizi and Guédon, The extreme points of subsets of s-concave probabilities and a geometric localization theorem, Discrete Comput. Geom. 31 (2004) 327-335], we proved that the localization may be deduced from a suitable application of Krein-Milman's theorem to a subset of log-concave probabilities satisfying one linear constraint and from the determination of the extreme points of its convex hull. Here, we generalize this result to more constraints, give some necessary conditions satisfied by such extreme points and explain how it may be understood as a generalized localization theorem. Finally, using this new localization theorem, we solve an open question on the comparison of the volume of sections of non-symmetric convex bodies in Rⁿ by hyperplanes. A surprising feature of the result is that the extremal case in this geometric inequality is reached by an unusual convex set that we manage to identify.     

A duality theorem for complex quadratic programming

A duality theory for complex quadratic programming over polyhedral cones is developed, following Dorn, by using linear duality theory.This research was partly supported by the National Science Foundation, Project No. GP-7550, and by the US Army Research Office, Durham, North Carolina, Contract No. DA-31-124-ARO-D-322. The authors are indebted to the referee for his helpful suggestions.     

The second main theorem for algebroid functions of several complex variables

The work of the first author was partially supported by the National Natural Science Foundation of China and the second one's by U.P.G.C. of Hong Kong     

Contact problems for bodies with complex coatings

The paper deals with various statements and mathematical models of contact and contact-wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.     

Refinements of Milnor's fibration theorem for complex singularities

Let X be an analytic subset of an open neighbourhood U of the origin in Cn. Let be holomorphic and set V=f−1(0). Let Bε be a ball in U of sufficiently small radius ε>0, centred at . We show that f has an associated canonical pencil of real analytic hypersurfaces Xθ, with axis V, which leads to a fibration Φ of the whole space (X∩Bε)?V over S¹. Its restriction to (X∩Sε)?V is the usual Milnor fibration , while its restriction to the Milnor tube f−1(∂Dη)∩Bε is the Milnor-Lê fibration of f. Each element of the pencil Xθ meets transversally the boundary sphere Sε=∂Bε, and the intersection is the union of the link of f and two homeomorphic fibres of ? over antipodal points in the circle. Furthermore, the space obtained by the real blow up of the ideal (Re(f),Im(f)) is a fibre bundle over RP¹ with the Xθ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.