Buffons Nadelproblem und verwandte probleme behandelt mit der Theorie der Gleichverteilung: Teil II

We denote with PC ^m the m-dimensional complex projective space, with U the unitary group acting on it with z _i(j=0, 1,..., m) the homogenous coordinates of a point [z] of PC ^m and assume that the z _i are normalized such that z ₀z₀ +...+z _mz_m=1. Furthermore we denote the U-invariant metric on PC ^m with d. We consider now a uniformly distributed sequence ([z] _k ; k=1,2,...) of points on PC ^m and study the sequence (d ^l([z] _k , [z]₀)), l0, [z]₀ a fixed point. We prove with the help of the theory of uniform distribution properties of this sequence. We consider furthermore a dual sequence suggested by the theory of H. Weyl and L. V. Ahlfors on meromorphic curves.     

Hamilton decompositions of cartesian products of multicycles

We define the multicycle C^(r)_m as a cycle on m vertices where each edge has multiplicity r. So C^(r)_m can be decomposed into r Hamilton cycles. We provide a complete answer to the following question: for which positive integers m, n, r, s with m, n ≥ 3 can the Cartesian product of two multicycles C^(r)_m x C^(s)_n be decomposed into r + s Hamilton cycles? We find some interesting characterizations of Hamilton cycles of C_m x C_n while answering the above question. © 1997 John Wiley & Sons, Inc.     

A remark on a maximal function over a Cantor set of directions

LetM _e ⁰ be the maximal operator over segments of length 1 with directions belonging to a Cantor set. It has been conjectured that this operator is bounded onL ². We consider a sequence of operators over finite sets of directions converging toM _e ⁰ . We improve the previous estimate for the (L ²,L ²)-norm of these particular operators. We also prove thatM _e ⁰ is bounded from some subsets ofL ² toL ². These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip. Partially supported by Spanish DGICYT grant no. PB90-0187.     

Index sets and parametric reductions

We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholakand the first author by proving the fundamental index sets associated with a computable set A, {e : W _e ≤ _q ^u A} for q∈ {m, T} are Σ₄ ⁰ complete. We also show hat FPT(≤ _q ⁿ ), that is {e : W _e computable and ≡ _q ⁿ ?}, is Σ₄ ⁰ complete. We also look at computable presentability of these classes. Received: 13 July 1996 / Revised version: 14 April 2000 / Published online: 18 May 2001     

Kaluza-Klein pistons with noncommutative extra dimensions

We calculate the scalar Casimir energy and Casimir force for an ℝ³ × N Kaluza-Klein piston setup in which the extra-dimensional space N contains a noncommutative two-dimensional sphere S _fz. We study the cases with T ^d ×S _fz and S _fz as the extra-dimensional spaces, where T ^d is the d-dimensional commutative torus, and examine the validity of the results and the regularization obtained in the piston setup in each case. We examine the Casimir energy with one-loop corrections for one piston chamber due to the self-interacting scalar field in the noncommutative geometry. We compute with some approximations. We compare the obtained results with the results of analogous computations for the Minkowski space-time M ^D . In conclusion, we discuss the stabilization of the extra-dimensional space in the piston setup.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Separating sets by Darboux Baire 1 and approximately continuous functions

We characterize sets A₀, A₁ for which there is a DB1 function f with [f = 0] = A₀ and [f = 1] = A₁. This characterization is a conjunction of necessary conditions for Darboux and for Baire 1 functions. We also characterize sets A^?, A⁺ for which there is a DB₁ function with [f < 0] = A^? and [f > 0] = A⁺. The same characterzations are provided for approximately continuous functions.     

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.     

Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle

In this article we consider the self-adjoint operator governing the propagation of elastic waves in a perturbed isotropic half space with a free boundary condition. We prove the limiting absorption principle in appropriate Hilbert spaces for this operator. We also prove decreasing properties for the eigenfunctions associated with strictly positive eigenvalues of this operator. The proofs are based on the limiting absorption principle for the self-adjoint operator governing the propagation of elastic waves in a homogeneous isotropic half space with a free boundary and on the so called division theorem for it. Both perturbations of R ₊² ={(x₁, x₂) ? R ²; x₂ > 0} and R ₊² = {(x₁, x₂, x₃) ? R ³; x₃ > 0} are considered.     

Almost Isometries of Balls

Let f be a bi-Lipschitz mapping of the Euclidean ball B ^_n into ℓ₂ with both Lipschitz constants close to one. We investigate the shape of f(B ^_n). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of ⁿ.     

Maximal three-independent subsets of {0, 1, 2} n

We consider a variant of the classical problem of finding the size of the largest cap in ther-dimensional projective geometry PG(r, 3) over the field IF₃ with 3 elements. We study the maximum sizef(n) of a subsetS of IF ₃ ⁿ with the property that the only solution to the equationx ₁+x₂+x₃=0 isx ₁=x₂=x₃. Letc _n=f(n)^1/n andc=sup{c ₁, c₂, ...}. We prove thatc>2.21, improving the previous lower bound of 2.1955 ...     

Van Geemen's families of lines on special quintic threefolds

We study the families of lines on the quintic threefolds of the pencilx ₀ ⁵ +x ₁ ⁵ +x ₂ ⁵ +x ₃ ⁵ +x ₄ ⁵ -5tx ₀ x ₁ x ₂ x ₃ x ₄=0. We show that on the generic threefold of the pencil there exists a 1-dimensional family of lines that is not a cone. Partially supported by funds M.P.I.     

Nerve complexes and moment-angle spaces of convex polytopes

We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope P to the nerve K _P of the cover of the boundary ∂P by facets. We develop a theory of nerve complexes and apply it to the moment-angle spaces Z _P of convex polytopes P. In the case of a polytope P with m facets, its moment-angle space Z _P is defined by the canonical embedding in the cone ℝ_≥ ^m . It is well-known that the space Z _P is homeomorphic to the polyhedral product (D ², S ¹)∂P* if the polytope P is simple. We show that the homotopy equivalence Z_P @ (D² ,S¹ )^K_P \mathcal{Z}_P \simeq (D^2 ,S^1 )^{K_P } holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known f-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, f-polynomials) coincide, while the new invariants are different.     

Families of multiresolution and wavelet spaces with optimal properties

Under suitable conditions, if the scaling functions ?₁ and ?₂ generate the multiresolutions V _(j)(?₁) and V _(j)(?₂), then their convolution ?₁*?₂also generates a multiresolution V _(j)(?₁*?₂) More over, if p is an appropriate convolution operator from l ₂ into itself and if ? is a scaling function generating the multiresolution V _(j)(?),then p*?is a scaling function generating the same multiresolution V _(j)(?)=V _(j)(p*?). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V _(j)(?ⁿ) and W _(j)(?ⁿ) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V ₍₀₎(?ⁿ)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W ₍₀₎(?ⁿ)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ ^b _nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.     

The braid group B4 is linear

We study a linear representation ρ:B _n ? GL _m (Z[q ^±1,t ^±1]) with m=n(n-1)/2. We will show that for n=4, this representation is faithful. We prove a relation with the new Charney length function. We formulate a conjecture implying that ρ is faithful for all n. Oblatum 15-VI-1999 & 24-II-2000?Published online: 18 September 2000     

Weighted Estimates for the Helmholtz Equation and Some Applications

We characterize the radial functionsVin ⁿfor which the a priori inequality u^1/2₂+k uV^1/2₂C V^−1/2(Δ+k²) u₂holds with constant independent ofk. The condition is forVto have the X-rays transform everywhere bounded. We apply these estimates to the well posedness of evolution Schrödinger equations with time dependent drift terms and to the restriction of the Fourier transform to Euclidean spheres.     

A Family of Partial Difference Sets with Denniston Parameters in Nonelementary Abelian 2-Groups

We construct a family of partial difference sets with Denniston parameters in the groupZ₄^t × Z₂^tby using Galois rings.     

Properties of the Flows Generated by Stochastic Equations with Reflection

We consider the properties of a random set ϕ _t (ℝ ₊ ^d ), where ϕ _t (x) is a solution of a stochastic differential equation in ℝ ₊ ^d with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ _t (ℝ ₊ ^d ) and prove that the Hausdorff dimension of the boundary ∂ϕ _t (ℝ ₊ ^d ) does not exceed d − 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.