Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and let C ⁿ denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C ⁿ →C ⁿ and for holomorphic automorphisms of C ⁿ on discrete subsets of C ⁿ.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C ⁿ.For each closed complex submanifold (or subvariety) M ⊂ C ⁿ of complex dimension m < n we construct a domain Ω ⊂C ⁿ containing M and a biholomorphic map F: Ω → C ⁿ onto C ⁿ with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C ^n−m →C ⁿ at infinitely many points. If m = n − 1, we construct F as above such that C ⁿ ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C ^m →C ^m−1 such that the complement C ^m+1 ∖F(C ^m )is hyperbolic.     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

The torsion of a submanifold of a Riemannian manifold

For a submanifoldM ⁿ of a Riemannian manifoldM ^q, the concept of a torsion bivector at the point x M ⁿ for given one- and two-dimensional directions fromT _x M ⁿ is introduced using only the first and second fundamental forms ofM ⁿ. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM ⁿ is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM ⁿ into some totally geodesicM ⁿ⁺¹ inM ^q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM ² inM ^q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

Curvature Estimates for Stable Minimal Hypersurfaces in R4 and R5

We obtain curvature estimates for certain stable minimalhypersurfaces in R ⁴ and R ⁵without using volume bounds. It follows that if M is acomplete stable minimal hypersurface in R ⁴ orR ⁵, then M is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.     

Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity

We consider the global attractor for the weakly damped forced KdV equation in Sobolev spaces [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}for s < 0. Under the assumption that the external forcing term belongs to [(L)\dot]²(T),{\dot{L}^2({\mathbf T}),} we prove the existence of the global attractor in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})} for −1/2 ≤ s < 0, which is identical to the one in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and thus is compact in H ³(T). The argument is a combination of the I-method and decomposing the solution into two parts, one of which is uniformly bounded in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and the other decays exponentially in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}.     

Quantified propositional calculus and a second-order theory for NC1

Let H be a proof system for quantified propositional calculus (QPC). We define the Σ^q_j-witnessing problem for H to be: given a prenex Σ^q_j-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σ^q₁-witnessing problems for the systems G^*₁and G₁ are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G^*₀ and G₀, in which cuts are restricted to quantifier-free formulas, and prove that the Σ^q_j-witnessing problem for each is complete for NC¹. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G^*₀ and proving that G₀-proofs are TC⁰-recognizable. We also introduce QPC systems for TC⁰ and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC¹ of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σ^b ₀-CA for uniform NC¹. We describe simple translations of VNC¹ proofs of all bounded theorems to polynomial size families of G^*₀ proofs. From this and the above theorem we get alternative proofs of the NC¹ witnessing theorems for VNC¹ and AID.This research was carried while this author was a student at the University of Toronto.     

Frobenius problem for semigroups \mathsf{S}(d_{1},d_{2},d_{3})

We find the matrix representation of the set Δ(d ³), where d ³=(d ₁,d ₂,d ₃), of integers that are unrepresentable by d ₁,d ₂,d ₃ and develop a diagrammatic procedure for calculating the generating function Φ(d ³;z) for the set Δ(d ³). We find the Frobenius number F(d ³), the genus G(d ³), and the Hilbert series H(d ³;z) of a graded subring for nonsymmetric and symmetric semigroups and enhance the lower bounds of F(d ³) for symmetric and nonsymmetric semigroups.      

Mapping Zr into Zs with Maximal Contraction

Given any bijection f: Z ^r →f: Z ^s with s≥ r , easy volume comparisons show that there must be a universal constant K>0 (depending only on r and s ) and infinitely many pairs of points x,y∈ Z ^r such that || f(x)-f(y)|| > K|| x-y|| ^r/s . This puts a bound on how much contraction can be achieved for any such bijection. We show that, conversely, for any s≥ r there is a bijection f: Z ^r →Z ^s and a constant C>0 such that for all x,y∈ Z ^r we have || f(x)-f(y)|| <C|| x-y|| ^r/s . Phrased differently there is a bijection f: Z ^r →Z ^s which shrinks the distance between the images of any two points as much as possible, up to a constant factor. This generalizes a construction in fractal image processing and answers in the affirmative a question of Michael Freedman. Received May 15, 1996.     

Almost periodic measures on the torus

Given a skew product flow (T,T ²) on the two torus, we construct a family of flows onT ³ parametrized by elements of the circleT. We show that under a certain condition on (T,T ²) almost every flow in this family is strictly ergodic. This is used to characterize minimal subsets of the flow (T,P(T ²)) induced byT on the space of probability measures onT ². Using a result of M. Herman, we give an example to show that this characterization does not hold for everyT. To the memory of Shlomo Horowitz     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

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Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

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Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno     

Two composition methods for solving certain systems of linear equations

Summary This note is concerned with the following problem: Given a systemA·x=b of linear equations and knowing that certains of its subsystemsA ¹·x ¹=b ¹, ...,A ^m ·x ^m =b ^m can be solved uniquely what can be said about the regularity ofA and how to find the solutionx fromx ¹, ...,x ^m ? This question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively [7, 13, 15].Work performed under NATO Research Grant 027-81     

Diffeomorphic types of complements of nice point arrangements in CP<Superscript><Emphasis Type="Italic">l</Emphasis></Superscript>

We use a new method to study arrangement in CP ^l , define a class of nice point arrangements and show that if two nice point arrangements have the same combinatorics, then their complements are diffeomorphic to each other. In particular, the moduli space of nice point arrangements with same combinatorics in CP ^l is connected. It generalizes the result on point arrangements in CP ³ to point arrangements in CP ^l for any l.     

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.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

Existence and uniqueness of complete constant mean curvature surfaces at infinity ofH 3

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H ³.It is shown that analogs of these equivalences exist for surfaces in S _∞ ² ,the bounding ideal sphere of H ³,leading to a notion of constant mean curvature at infinity of H ³.A parametrization of all complete constant mean curvature surfaces at infinity of H ³ is given by holomorphic quadratic differentials on Ĉ,C, and D.     

On Self-Associated Sets of Points in Small Projective Spaces

We study moduli of “self-associated” sets of points in P ⁿ for small n. In particular, we show that for n = 5 a general such set arises as a hyperplane section of the Lagrangian Grassmanian LG(5,10) ? P ¹⁵. For n = 6, a general such set arises as a hyperplane section of the Grassmanian G(2,6) ? P ¹⁴. We also make a conjecture for the next case n = 7. Our results are analogues of Mukai's characterization of general canonically embedded curves in P ⁶ and P ⁷, respectively.     

Two Extensions to Finsler's Recurring Theorem

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R ⁿ : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R ⁿ → R ^k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y R ⁿ for some λ∈ R ^k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R ⁿ by a real vector space X . Accepted 22 February 1998     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

The Incompressible Limits of Viscous Polytropic Fluids with Zero Thermal Conductivity Coefficient

ABSTRACT

We study the singular limit of viscous polytropic fluids without thermal conductivity as the Mach number tends to zero. A uniform existence result for the Cauchy problem in R ³ is proved under the assumption that the initial data belongs uniformly to H ^k (R ³) with k = 2, 3 and is well-prepared in H ¹ (R ³).