<Emphasis Type="Italic">P</Emphasis>-orderings of finite subsets of Dedekind domains

If R is a Dedekind domain, P a prime ideal of R and S⊆R a finite subset then a P-ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101–127, 1997), is an ordering {a _i } _i=1 ^m of the elements of S with the property that, for each 1<i≤m, the choice of a _i minimizes the P-adic valuation of ∏ _j<i (s−a _j ) over elements s∈S. If S, S ^′ are two finite subsets of R of the same cardinality then a bijection φ:S→S ^′ is a P-ordering equivalence if it preserves P-orderings. In this paper we give upper and lower bounds for the number of distinct P-orderings a finite set can have in terms of its cardinality and give an upper bound on the number of P-ordering equivalence classes of a given cardinality.     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

Approximation in L p (R d ) from a space spanned by the scattered shifts of a radial basis function

A new multivariate approximation scheme on R ^d using scattered translates of the “shifted” surface spline function is developed. The scheme is shown to provide spectral L _p -approximation orders with 1 ≤ p ≤ ∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline interpolation method.     

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and _R M _S is an S-R-bimodule. In the main theorem of this paper we show that if T _S is a tilting S-module, then under certain homological conditions on the S-module M _S , one can extend T _S to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and T _S , and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M _S is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M _S has a finite projective resolution and Ext _S ⁿ (M _S , S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, Hom _S (M, S)).     

Smooth and strongly smooth points in symmetric spaces of measurable operators

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? S_E^×{f\in S_{E^{\times}}} supporting μ(x), or s(x ^*) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.     

A Dodds–Fremlin Property for Asplund and Radon–Nikodým Operators

Let E and F be Banach lattices and let S, T: E→ F be positive operators such that 0≤ S≤ T. It is shown that if T is a Radon–Nikodym operator, F has order continuous norm and E and F both have (Schaefer's) property (P), then S is a Radon–Nikodym operator; also, if T is an Asplund operator, E' has order continuous norm and E has property (P), then S is an Asplund operator.     

A note on two equivalence relations on inverse semigroups

Let S be an inverse semigroup. In Rezavand et al. (Semigroup Forum 77:300–305, 2008) and Munn (Proc. Glasgow Math. Assoc. 5:41–48, 1961) two equivalence relations are defined on  S. After describing these relations we show that they coincide.     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.     

Two cancellative commutative congruences and group diagrams

Remmers (Adv. Math. 36:283–296, 1980) uses group diagrams in the Euclidean plane to demonstrate how equality in a semigroup S “mirrors” that inside the group G sharing the same presentation with S, when S satisfies Adyan’s condition—no cycles in the left/right graphs of the semigroup’s presentation. Goldstein and Teymouri (Semigroup Forum 47:299–304, 1993) introduce a conjugacy equivalence relation for semigroups S. By closely examining the geometry of annular group diagrams in the plane, they show how their equivalence relation mirrors conjugacy inside G, for S satisfying Adyan’s. In this article we introduce two cancellative commutative congruences. Following their leads, we examine the geometry of group diagrams on closed surfaces of higher genera to demonstrate how these congruences mirror equality inside two naturally associated Abelian quotient groups G/[G,G] and G/G ², respectively. In these instances we can drop Adyan’s condition.     

Nonlinear Subdivision Schemes on Irregular Meshes

The present article deals with convergence and smoothness analysis of geometric, nonlinear subdivision schemes in the presence of extraordinary points. We discuss when the existence of a proximity condition between a linear scheme and its nonlinear analogue implies convergence of the nonlinear scheme (for dense enough input data). Furthermore, we obtain C ¹ smoothness of the nonlinear limit function in the vicinity of an extraordinary point over Reif’s characteristic parametrization. The results apply to the geometric analogues of well-known subdivision schemes such as Doo–Sabin or Catmull–Clark schemes.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

<Emphasis Type="Italic">G</Emphasis>-compactness and groups

Lascar described E _KP as a composition of E _L and the topological closure of E _L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G _L . We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories. The first author is supported by the Polish Goverment grant N N201 384134. The second author is supported by the Polish Goverment grant N201 032 32/2231.     

Smooth 4-folds which contain a P1-bundle¶as an ample divisor

We show that if a smooth projective 4-fold M contains an ample divisor A which is P ¹-bundle π :A→S over a smooth projective surface S, π is extended to a P ²-bundle π :S→S, unless $A$ is isomorphic to P ²×P ¹. Received: 28 September 1998 / Revised version: 16 August 1999     

Teichmüller Spaces and Negatively Curved Fiber Bundles

In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres ${\mathbb{S}^k}In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres \mathbbS^k{\mathbb{S}^k}, the forgetful map F_\mathbbS^k{F_{\mathbb{S}^k}} is not one-to-one. This result follows from Theorem A, which proves that the quotient map MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} is the space of negatively curved metrics on M and T  ^{sec < 0} (M) = MET  ^{sec < 0} (M)/ DIFF₀(M){\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)} is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M. In particular we conclude that T  ^{sec < 0} (M){\mathcal{T}^{\,\,sec <0 }(M)} is, in general, not connected. Two remarks: (1) the nontrivial elements in p_kMET  ^{sec < 0} (M){\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)} constructed in [FO3] have trivial image by the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; (2) the nonzero classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} constructed in [FO2] are not in the image of the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; the nontrivial classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} given here, besides coming from MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} and being harder to construct, have a different nature and genesis: the former classes – given in [FO2] – come from the existence of exotic spheres, while the latter classes – given here – arise from the non-triviality and structure of certain homotopy groups of the space of pseudo-isotopies of the circle \mathbbS¹{\mathbb{S}^1}. The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in Theorem B.     

Wiener–Hopf Operators on Spaces of Functions on $${\mathbb{R}}^{+}$$ with Values in a Hilbert Space

A Wiener–Hopf operator on a Banach space of functions on is a bounded operator T such that P ⁺ S _−a TS _a = T, a ≥ 0, where S _a is the operator of translation by a. We obtain a representation theorem for the Wiener–Hopf operators on a large class of functions on with values in a separable Hilbert space.      

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems

We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family has tension parameter ω. When ω = 0, it becomes the minimal length biorthogonal Coifman wavelet system (Wei et al., IEEE Trans Image Proc 7:1000–1013, 1998). Choosing ω away from zero, we can get better smoothness of the refinable functions at the expense of slightly larger support. Though the construction of the new biorthogonal wavelet systems, in fact, starts from a new class of quasi-interpolatory subdivision schemes, we find that the refinement masks accidently coincide with the ones by Cohen et al. (Comm Pure Appl Math 45:485–560, 1992, §6.C) (or Daubechies 1992, §8.3.5), which are designed for the purpose of generating biorthogonal wavelets close to orthonormal cases. However, the corresponding mathematical analysis is yet to be provided. In this study, we highlight the connection between the quasi-interpolatory subdivision schemes and the masks by Cohen, Daubechies and Feauveau, and then we study the fundamental properties of the new biorthogonal wavelet systems such as regularity, stability, linear independence and accuracy.