Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

Zéros d’applications holomorphes deC n dansC n

It is known that, unlike the one dimensional case it is not possible to find an upper bound for the zeros of an entire map fromC ⁿ toC ⁿ ,n≥2, in terms of the growth of the map. However, if we only consider the “non-degenerate” zeros, that is, the zeros where the jacobian is not “too small”, it becomes possible. We give a new proof of this fact.      

Univalence Criteria Starting from the Method of Loewner Chains

The paper continues the work of Royster (Duke Math J 19:447–457, 1952), Mocanu [Mathematica (Cluj) 22(1):77–83, 1980; Mathematica (Cluj) 29:49–55, 1987], Cristea [Mathematica (Cluj) 36(2):137–144, 1994; Complex Var 42:333–345, 2000; Mathematica (Cluj) 43(1):23–34, 2001; Mathematica (Cluj), 2010, to appear; Teoria Topologica a Functiilor Analitice, Editura Universitatii Bucuresti, Romania, 1999] of extending univalence criteria for complex mappings to C ¹ mappings. We improve now the method of Loewner chains which is usually used in complex univalence theory for proving univalence criteria or for proving quasiconformal extensions of holomorphic mappings f : B → C ⁿ to C ⁿ . The results are surprisingly strong. We show that the usual results from the theory, like Becker’s univalence criteria remain true for C ¹ mappings and since we use a stronger form of Loewner’s theory, we obtain results which are stronger even for holomorphic mappings f : B → C ⁿ . In our main result (Theorem 4.1) we end the researches dedicated to quasiconformal extensions of K-quasiregular and holomorphic mappings f : B → C ⁿ to C ⁿ . We show that a C ¹ quasiconformal map f : B → C ⁿ can be extended to a quasiconformal map F : C ⁿ → C ⁿ , without any metric condition imposed to the map f.     

Quasi-parabolic analytic transformations of <Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>. Parabolic manifolds

In [Rong, F., Quasi-parabolic analytic transformations of C ⁿ , J. Math. Anal. Appl. 343 (2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of C ⁿ . Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.     

On binary linear <Emphasis Type="Italic">r</Emphasis>-identifying codes

A subspace C of the binary Hamming space F ⁿ of length n is called a linear r-identifying code if for all vectors of F ⁿ the intersections of C and closed r-radius neighbourhoods are nonempty and different. In this paper, we give lower bounds for such linear codes. For radius r =  2, we give some general constructions. We give many (optimal) constructions which were found by a computer search. New constructions improve some previously known upper bounds for r-identifying codes in the case where linearity is not assumed.     

Cohomology vanishing¶and a problem in approximation theory

For a simplicial subdivison Δ of a region in k ⁿ (k algebraically closed) and r∈N, there is a reflexive sheaf ? on P ⁿ , such that H ⁰(?(d)) is essentially the space of piecewise polynomial functions on Δ, of degree at most d, which meet with order of smoothness r along common faces. In [9], Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ℰ on P ⁿ in terms of the top two Chern classes and the generic splitting type of ℰ. We use a spectral sequence argument similar to that of [16] to characterize those Δ for which ? is actually a bundle (which is always the case for n= 2). In this situation we can obtain a formula for H ⁰(?(d)) which involves only local data; the results of [9] cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P ² and we discuss a possible connection between semi-stability and the conjectured answer to this open problem. Received: 9 April 2001     

Wavelet Filter Functions,the Matrix Completion Problem,and Projective Modules Over <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Bold">T</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective C(T ⁿ )-modules. Conversely, we show how cancellation properties for finitely generated projective modules over C(T ⁿ ) can often be used to prove the existence of continuous high pass filters, of the kind needed for multivariate wavelets, corresponding to a given continuous low-pass filter. However, we also give an example of a continuous low-pass filter for which it is impossible to find corresponding continuous high-pass filters. In this way we give another approach to the solution of the matrix completion problem for filters of the kind arising in wavelet theory.     

A new proof of the Abhyankar-Moh-Suzuki theorem

We give a new proof of the famous result that any two embeddings of the affine lineA ¹ inA ² are equivalent by an automorphism ofA ². We also prove that iff(X,Y) is a polynomial overC with one place at infinity, then for every C,f– also has one place at infinity. The proof uses basic results about algebraic surfaces.     

Test for properness using residue calculus

We will consider global problems in the ringK[X ₁, …,X _n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P ₁, …,P _n):K ⁿ →K ⁿ be a polynomial map such that (P ₁,…,P _n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P ₁, …,P _n) is a proper map by computing a finite number of residue symbols.     

LINEAR MAPPINGS THAT PRESERVE THE DERIVATIONAL STRUCTURE OF C*-ALGEBRAS

Abstract

Given a C*-algebra A and a suitable set of derivations on A, we consider the algebras A _n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C^(m)(U) → C ^(p) (V) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). As part of the theory developed we obtain a non-trivial extension of the Kleinecke-Shirokov theorem in the category of C*-algebras to unbounded partially defined *-derivations. This indicates the existence of a single mathematical principle governing both the non-increasibility of differentiability by continuous homomorphisms and the untenability of the Heisenberg Uncertainty Principle for bounded observables.     

Prescribing scalar curvature on Sn and related problems,part II: Existence and compactness

This is a sequel to [30], which studies the prescribing scalar curvature problem on Sⁿ. First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lower-dimensional cases n = 2, 3. Next we study the problem for n ≥ 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is β ϵ (n − 2, n). The key point there is that for the class of K mentioned above we have completed L^∞ apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is β = n − 2, the L^∞ estimates for solutions fail in general. In fact, two or more blowup points occur. On the other hand, we provide some existence and compactness results when the order of flatness at critical points of K(x) is β ϵ [n − 2,n). With this result, we can easily deduce that C^∞ scalar curvature functions are dense in C^1,α (0 < α < 1) norm among positive functions, although this is generally not true in the C² norm. We also give a simpler proof to a Sobolev-Aubin-type inequality established in [16]. Some of the results in this paper as well as that of [30] have been announced in [29]. © 1996 John Wiley & Sons, Inc.     

Polynomial diffeomorphisms of C2: V. critical points and Lyapunov exponents

There is an invariant measure μ, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C.In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to μ, analogous to the Brolin-Manning formula polynomial maps of C.Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these the critical points have a natural dynamical interpretation.     

Embedding a polytope in a lattice

We present a special similarity ofR ⁴ⁿ which maps lattice points into lattice points. Applying this similarity, we prove that if a (4n−1)-polytope is similar to a lattice polytope (a polytope whose vertices are all lattice points) inR ⁴ⁿ , then it is similar to a lattice polytope inR ⁴ⁿ⁻¹, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someR ^N if and only if it is similar to a lattice polytope inR ²ⁿ⁺¹, and if and only if sin²(<ABC) is rational for any three verticesA, B, C of the polytope.     

The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space M(n,m)\mathcal{M}(n,m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H_\mathbbCⁿ\mathbf{H}_{\mathbb{C}}^{n}, up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for M(n,m)\mathcal{M}(n,m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.     

Remarks on the Cauchy problem for first order hyperbolic systems with constant coefficient principal part

Abstract In this article we shall introduce the results obtained in [16], i.e., we shall give a necessary and sufficient condition that the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is C^∞ well-posed under the maximal rank condition (see the condition (R) below). We shall also give a simple sufficient condition without any assumptions on the rank. Keywords: Hyperbolic system, Cauchy problem, Constant coefficient principal part     

Examples for cross curvature flow on 3-manifolds

Recently, B. Chow and R.S. Hamilton [3] introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a S² bundle over a circle. We show that the global flow exists in both cases. However, on the former the flow diverges at time infinity, and on the latter the flow converges at time infinity. Mathematics Subject Classification (1991) 53C44     

Smooth Quasiregular Maps with Branching in R n

According to a theorem of Martio, Rickman and Väisälä, all nonconstant C^n/(n-2)-smooth quasiregular maps in Rⁿ, n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R³. We prove that the order of smoothness is sharp in R⁴. For each n≥5 we construct a C^1+ε(n)-smooth quasiregular map in Rⁿ with nonempty branch set.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Determination of a Domain in Complex Space by its Automorphism Group

We consider results, both in one complex variable and several, which show that the algebraic or geometric structure of the automorphism group of a domain z can determine that domain. The domains considered include Ω = B, the unit ball in C ⁿ , and Ω = C ⁿ . Various illustrative examples are provided.