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.     

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.      

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

A KAM phenomenon for singular holomorphic vector fields

Let X be a germ of holomorphic vector field at the origin of Cⁿ and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.     

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

In 1989, Kalai stated three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the “3 ^d -conjecture.” It is well known that the three conjectures hold in dimensions d≤3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d≥5.     

On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

Provability interpretations of modal logic

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev ^* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.     

On the minimal free resolution of finite sets in P N

Here we construct many possible free resolutions fors points inP ⁿ . In suitable ranges we construct configurations of points with “good” minimal free resolution and other configurations for which the difference with respect to a “good” resolution is prescribed in advance.     

On Rich Lines in Grids

In this paper we show that if one has a grid A×B, where A and B are sets of n real numbers, then there can be only very few “rich” lines in certain quite small families. Indeed, we show that if the family has lines taking on n ^ε distinct slopes, and where each line is parallel to n ^ε others (so, at least n ^2ε lines in total), then at least one of these lines must fail to be “rich”. This result immediately implies non-trivial sumproduct inequalities; though, our proof makes use of the Szemeredi-Trotter inequality, which Elekes used in his argument for lower bounds on |C+C|+|C.C|.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Nonholonomous geodesics as solutions to Euler-Lagrange integral equations and the differential of the exponential mapping

Equations of horizontal geodesics on a Riemannian (or pseudo-Riemannian) manifold with nonholonomous distribution are obtained using the Euler-Lagrange method in Pontryagin’s formulation. It is shown that if the distribution and the metric tensor of the distribution are C ^k -smooth, k ≥ 1, then any regular solution to the variational problem is C ^{k + 1}-smooth. The differential of the exponential mapping is obtained for nonholonomous distribution with the condition of cyclicity with respect to “vertical” coordinates. This differential is nonsingular provided that the distribution is strongly bracket generating.     

A Finitary Version of Gromov’s Polynomial Growth Theorem

We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is some R ₀ > exp(exp(Cd ^C )) for which the number of elements in a ball of radius R ₀ in a Cayley graph of G is bounded by R₀^d{R_0^d} , then G has a finite-index subgroup which is nilpotent (of step < C ^d ). An effective bound on the finite index is provided if “nilpotent” is replaced by “polycyclic”, thus yielding a non-trivial result for finite groups as well.     

Spectra of elements in the group ring of SU(2)

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S² as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. Received June 1, 1998 / final version received September 8, 1998     

Remark on the Regularities of Kato’s Solutions to Navier-Stokes Equations with Initial Data in L^d(R^d)

Motivated by the results of J. Y. Chemin in ＂J. Anal. Math., 77, 1999, 27- 50＂ and G. Furioli et al in ＂Revista Mat. Iberoamer., 16, 2002, 605-667＂, the author considers further regularities of the mild solutions to Navier-Stokes equation with initial data uo ∈ L^d（R^d）. In particular, it is proved that if u C ∈（[0, T^＊）; L^d（R^d）） is a mild solution of （NSv）, then u（t,x）- e^vt△uo ∈ L^∞（（0, T）;B2/4^1,∞）~∩L^1 （（0, T）; B2/4^3 ,∞） for any T 〈 T^＊.     

r-entropy,equipartition, and Ornstein’s isomorphism theorem inR n

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.     

The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization

We show the existence of a sequence (λ _n ) of scalars withλ _n =o(n) such that, for any symmetric compact convex bodyB ⊂R ⁿ , there is an affine transformationT satisfyingQ ⊂T(B) ⊂λ _n Q, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλ _n . We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l ₂ ^m →X andβ:X→l _∞ ^m with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum. Research performed while this author was visiting IHES. Supported in part by the NSF Grant DMS-8702058 and the Sloan Research Fellowship.     

Commutative lmc algebras with discrete spectrum

SupposeE is a topological algebra with non-empty spectrum Gel'fand spaceM(E) andE M(E)) the algebra of allC-valued continuous functions onM(E). EndowE M(E) with the topologies “c”, “e” of compact resp. equicontinuous convergence. ThenE M (E) characterizes all unital (commutative) semisimple complete lmc algebras with discrete spectrum, while all unital uniform complete lmc algebras with dispersed spectrum are of the formE(E)). The first result may fail if completeness is dropped. The second one fails if “e” is replaced by “c” even ifE is complete. The part of the work due to the 2nd author was carried out during her 3-month visit at the Institute of Mathematics, Univ. of Münster (Germany) in Spring of 1995. The warm hospitality from Professor Dr. G. Maltese and the financial support from a DAAD grant are thankfully acknowledged.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.