On Characterization of Absolute Geometries

Starting from a general absolute plane A = (P, L, α, ≡) in the sense of Karzel et al. (Einführung in die Geometrie, p. 96, 1973), Karzel and Marchi introduced the notion of a Lambert–Saccheri quadrangle (L-S quadrangle) in Karzel and Marchi (Le Matematiche LXI:27–36, 2006): A quadruple (a, b, c, d) of points of P, no three collinear, is a L-S quadrangle, if ${\overline{a,d}\bot\overline{a,b}\bot\overline{b,c}\bot\overline{c,d}}$ . Denoting the foot of a on the line ${\overline{c, d}}$ with ${a^{\prime}=\{a\bot\overline{c,d}\}\cap \overline{c,d}}$ , the L-S quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if ${a^{\prime}=d,\; a^{\prime}\,{\in}\, ]c,d[}$ or ${a^{\prime}\,{\notin}\, ]c,d[\cup \{d\}}$ respectively. Let LS be the set of all L-S quadrangles and LS _r , LS _h or LS _e the subset of all rectangles, hyperbolic or elliptic L-S quadrangles respectively. In Karzel and Marchi (Le Matematiche LXI:27–36, 2006) it was claimed that either LS =  LS _r or LS =  LS _h or LS =  LS _e . To this classification we add five further classifications of general absolute planes by using “distance” [defined in Karzel and Marchi (Discrete Math 308:220–230, 2008)] or the notions of “interior” and “exterior” angle, introduced in Karzel et al. (Resultate Math 51:61–71, 2007) and considering besides Lambert–Saccheri quadrangles, also triangles in particular right-angled triangles. For Lambert–Saccheri quadrangles (a, b, c, d) the relations between distances of the diagonal points (a, c) and (b, d) or between the “midpoint” ${o:=\overline{a,c}\cap\overline{b,d}}$ , and the corner points a, b, c, d give us possibilities for complete characterizations. Using triangles (a, b, c) and denoting by m and n the midpoints of (a, b) and (a, c) we classify the absolute planes by the relations between the distances |b, c| and 2|m, n|. All our main results are summarized at the end of the introduction.     

Suites dont la discrépance est comparable à un logarithme

We use the notion of self-similar sequences, introduced by the author in [1], to obtain in a natural way some sequences of points in the interval [0, 1] with the property: $$0< \ell \left( U \right)< L\left( U \right)< \infty $$ whereL(U) and ? (U) stand respectively for the lim sup and the lim inf of the ratioN·D _N ^* (U)/LogN. HereD _N ^* (U) is the star discrepancy of the sequenceU.     

Transfinite Extension to Q m-normality Theory

The theory of Q _m-normal families, m ∈ ?, was developed by P. Montel for the cases m = 0 (normal families) [5] and m = 1 (quasinormal families) [4] and later generalized by C.T. Chuang [2] for any m ≥ 0. In this paper, we extend the definition to an arbitrary ordinal number α as follows. Given E ? D, define the α-th derived set $E^{(\alpha)}_D$ of E with respect to D by $(E^{(\alpha-1)}_D)^{(1)}_D$ if α has an immediate predecessor and by ${\mathop \bigcap\limits_{\beta<\alpha}} E^{(\beta)}_D$ if α is a limit ordinal. Then a family ${\cal F}$ of meromorphic functions on a plane domain D is Q_α-normal if each sequence S of functions in ${\cal F}$ has a subsequence which converges locally χ-uniformaly on the domain DE, where E = E(S) ? D satisfies $E^{(\alpha)}_{D}=\emptyset$ . Inparticular, a Q ₀ -normal family is a normal family, and a Q ₁ -normal family is a quasi- normal family. We also give analogues to some basic results in Q_m-normality theory and extend Zalcman’s Lemma to Q _α -normal families where α is an infinite countable (enumerable) ordinal number.     

Discrepancy, separation and Riesz energy of finite point sets on the unit sphere

For $d \geqslant 2,$ we consider asymptotically equidistributed sequences of $\mathbb S^d$ codes, with an upper bound $\operatorname{\boldsymbol{\delta}}$ on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0?<?s?<?d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\,\Delta^{-s}\,N^{-s/d}\big),$ where N is the number of code points. For well separated sequences of spherical codes, this bound becomes $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\big).$ We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.     

Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

Construction of scrambled polynomial lattice rules over \mathbb{F}_{2} with small mean square weighted \mathcal{L}_{2} discrepancy

The \(\mathcal{L}_{2}\) discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Wo?niakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted \(\mathcal{L}_{2}\) discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field \(\mathbb{F}_{2}\) whose scrambled point sets have small mean square weighted \(\mathcal{L}_{2}\) discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of N ^?2+δ for all δ>0, where N denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol’ sequences.     

Quasi–Monte Carlo rules for numerical integration over the unit sphere {\mathbb{S}^2}

We study numerical integration on the unit sphere ${\mathbb{S}^2 \subseteq\mathbb{R}^3}$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a (0, m, 2)-net given in the unit square [0, 1]² to the sphere ${\mathbb{S}^2}$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J Sci Comput 18(2):595–609, 1997]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on ${\mathbb{S}^2}$ . And finally, we prove an upper bound on the spherical cap L ₂-discrepancy of order N ^?1/2(log N)^1/2 (where N denotes the number of points). This improves upon the bound on the spherical cap L ₂-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Commun Pure Appl Math 39(S, suppl):S149–S186, 1986] by a factor of ${\sqrt{\log N}}$ . Numerical results suggest that the (0, m, 2)-nets lifted to the sphere ${\mathbb{S}^2}$ have spherical cap L ₂-discrepancy converging with the optimal order of N ^?3/4.     

Fan triangulations of a hyperbolic plane of positive curvature

We study the families (?_λ) of normal partitions of a 3-(1)-contour F of a hyperbolic plane \(\hat H\) of positive curvature into simple 4-contours whose hyperbolic diagonal lines are parallel to the base of F. A 3-(1)-contour with a given partition from a family (?_λ) (or some its normal subpartition) is called a fan. We construct fan partitions P _e, P _h, and P _p of \(\hat H\) whose symmetry groups are generated by a shift along an elliptic (respectively, hyperbolic and parabolic) straight line. It is proved that the partitions P _h and P _p are normal. The partitions P _h and P _p whose cells are trihedrals present examples of the first triangulations of \(\hat H\) .     

Geométrie des suites de Kronecker

Let Θ be a element of the d-dimensional torus $\mathbb{T}$ ^d andτ the translationτ(x)=x + Θ. When d=1 there existe some partitions of $\mathbb{T}$ ¹ which are associated withτ. We prove the existence of partitions of $\mathbb{T}$ ^d which enjoyed the same kind of properties and whose elements (A _i ) _i≤n are convex polytopes. We also give a lower bound for the isotropic discrepancy of the sequence (nΘ) _nε?.     

Asymptotics of Discrete Riesz d-Polarization on Subsets of d-Dimensional Manifolds

We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as N?→?∞) of the N-point Riesz d-polarization constant for an infinite compact subset A of a d-dimensional C ¹-manifold embedded in ? ^m (d?≤?m). Moreover, if we assume further that the d-dimensional Hausdorff measure of A is positive, we show that any asymptotically optimal sequence of N-point configurations for the N-point d-polarization problem on A is asymptotically uniformly distributed with respect to \(\mathcal H_d|_A\) . These results also hold for finite unions of such sets A provided that their pairwise intersections have \(\mathcal H_d\) -measure zero.     

A dedekind finite borel set

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if ${B\subseteq 2^\omega}$ is a G _??? -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ^?? is the countable union of countable sets, then there exists an F _??? set ${C\subseteq 2^\omega}$ such that C is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite ${D\subseteq 2^\omega}$ which is F _??? .     

Radford’s Biproducts for Hopf Group-Coalgebras and Its Quasitriangular Structures

Let π be a group and H={H _α } _α∈π be a semi-Hopf π-coalgebra in the sense of Virelizier (J. Pure Appl. Algebra 171:75–122, 2002). Let H coact weakly on a coalgebra B and λ={λ _α,β :B?H _α ?H _β } be a family of k-linear maps. Then in this paper we first introduce the notion of a π-crossed coproduct $B\times_{\lambda }^{\pi}H=\{B\times_{\lambda }H_{\alpha }\}_{\alpha \in \pi }$ and find some sufficient and necessary conditions making it into a π-coalgebra, generalizing the main construction in Lin (Commun. Algebra 10:1–17, 1982). Secondly, we find a sufficient and necessary condition for $B_{\#^{\pi}}^{\times_{\lambda }^{\pi}} H$ , with the π-crossed coproduct $B\times_{\lambda }^{\pi}H$ and π-smash product B# ^π H to form a semi-Hopf π-coalgebra, if λ is convolution invertible dual 2-cocycle, which generalizes the well-known Radford’s biproduct in Radford (J. Algebra 92:322–347, 1985). Furthermore, we derive some sufficient conditions for $B_{\#^{\pi}}^{\times_{\lambda }^{\pi}} H$ to be a Hopf π-coalgebra. Finally, we construct a quasitriangular structure on the Hopf π-coalgebra $B\times_{\lambda }^{\pi}H$ (with the usual tensor product).     

Noetherian type in topological products

The cardinal invariant Noetherian type Nt(X) of a topological space X was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2:

1. There are spaces X and Y such that Nt(X×Y)< min{Nt(X), Nt(Y)}.
2. In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace.

The Noetherian type of the Cantor Cube of weight \({\aleph _\omega }\) with the countable box topology, \({({2^{{\aleph _\omega }}})_\delta }\) , is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of \({\aleph _\omega }\) . We discuss the influence of principles like \({\square _{{\aleph _\omega }}}\) and Chang’s conjecture for \({\aleph _\omega }\) on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an (?₄, ?₁)-sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ?₄ on the Noetherian type of \({({2^{{\aleph _\omega }}})_\delta }\) . The proof uses a method that was introduced by Shelah in 1993 [33].     

On invariant ccc σ-ideals on 2^{\mathbb{N}}

We study structural properties of the collection of all σ-ideals in the σ-algebra of Borel subsets of the Cantor group \(2^{\mathbb{N}}\) , especially those which satisfy the countable chain condition (ccc) and are translation invariant. We prove that the latter collection contains an uncountable family of pairwise orthogonal members and, as a consequence, a strictly decreasing sequence of length ω ₁. We also make some observations related to the σ-ideal I _ccc on \(2^{\mathbb{N}}\) , consisting of all Borel sets which belong to every translation invariant ccc σ-ideal on \(2^{\mathbb{N}}\) . In particular, improving earlier results of Rec?aw, Kraszewski and Komjáth, we show that:

• every subset of \(2^{\mathbb{N}}\) of cardinality less than can be covered by a set from I _ccc,
• there exists a set C∈I _ccc such that every countable subset Y of \(2^{\mathbb{N}}\) is contained in a translate of C.

     

Irregular discrepancy behavior of lacunary series II

In 1975 Philipp proved the following law of the iterated logarithm (LIL) for the discrepancy of lacunary series: let (n _k ) _{k ≥ 1} be a lacunary sequence of positive integers, i.e. a sequence satisfying the Hadamard gap condition n _k+1/n _k > q > 1. Then ${1/(4 \sqrt{2}) \leq \limsup_{N \to \infty} N D_N(n_k x) (2 N \log \log N)^{-1/2}\leq C_q}$ for almost all ${x \in (0,1)}$ in the sense of Lebesgue measure. The same result holds, if the “extremal discrepancy” D _N is replaced by the “star discrepancy” ${D_N^*}$ . It has been a long standing open problem whether the value of the limsup in the LIL has to be a constant almost everywhere or not. In a preceding paper we constructed a lacunary sequence of integers, for which the value of the limsup in the LIL for the star discrepancy is not a constant a.e. Now, using a refined version of our methods from this preceding paper, we finally construct a sequence for which also the value of the limsup in the LIL for the extremal discrepancy is not a constant a.e.     

Estimating the intensity of a random measure by histogram type estimators

The purpose of this paper is to estimate the intensity of some random measure N on a set ${\mathcal{X}}$ by a piecewise constant function on a finite partition of ${\mathcal{X}}$ . Given a (possibly large) family ${\mathcal{M}}$ of candidate partitions, we build a piecewise constant estimator (histogram) on each of them and then use the data to select one estimator in the family. Choosing the square of a Hellinger-type distance as our loss function, we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound. Moreover, the selection procedure leads to a final estimator satisfying some oracle-type inequality, with, as usual, a possible loss corresponding to the complexity of the family ${\mathcal{M}}$ . When this complexity is not too high, the selected estimator has a risk bounded, up to a universal constant, by the smallest risk bound obtained for the estimators in the family. For suitable choices of the family of partitions, we deduce uniform risk bounds over various classes of intensities. Our approach applies to the estimation of the intensity of an inhomogenous Poisson process, among other counting processes, or the estimation of the mean of a random vector with nonnegative components.     

The {k}-domatic number of a graph

For a positive integer k, a {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, . . . , k} such that for any vertex ${v\in V(G)}$ , the condition ${\sum_{u\in N[v]}f(u)\ge k}$ is fulfilled, where N[v] is the closed neighborhood of v. A {1}-dominating function is the same as ordinary domination. A set {f ₁, f ₂, . . . , f _d } of {k}-dominating functions on G with the property that ${\sum_{i=1}^df_i(v)\le k}$ for each ${v\in V(G)}$ , is called a {k}-dominating family (of functions) on G. The maximum number of functions in a {k}-dominating family on G is the {k}-domatic number of G, denoted by d _{k}(G). Note that d _{1}(G) is the classical domatic number d(G). In this paper we initiate the study of the {k}-domatic number in graphs and we present some bounds for d _{k}(G). Many of the known bounds of d(G) are immediate consequences of our results.     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

On the relation between entropy convergence rates and Baire category

Denoting by G the space of all invertible measure-preserving transformations on the unit interval, we show that the set of all transformations T satisfying the property that for all nontrivial finite measurable partitions α the entropy ${H(\alpha_T^n)}$ of the n-th refinements of α under T converges in the limit superior faster to ∞ than a given sublinear rate a _n is residual with respect to the weak topology on G. In slight modification of a well known result concerning positive-entropy transformations we also show that the set of all T for which ${H(\alpha_T^n)}$ converges in the limit inferior faster to ∞ than a given nontrivial rate (for all nontrivial α) is of first category in G.     

A characterisation of multiplicity

LetR be a ring with non-zero identity and unitary leftR-modules, while \(\mathcal{N}_R \) is the subcategory of NoetherianR-modules. Given a length functionL on \(\mathcal{N}_R \) and central elements α₁,...,α _n ofR we can define the multiplicity length functione _R (L|α₁,...α _n |) on \(\mathcal{N}_R \) with the same properties as the classical multiplicity. Here, we characterise multiplicity as the greatest length function which can be defined inductively in terms of a certain type of function on \(\mathcal{N}_R \) .