<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.     

On the E-Epigraph of an E-Convex Function

In Ref 1, Yang shows that some of the results obtained in Ref. 2 on E-convex programming are incorrect, but does not prove that the results which make the connection between an E-convex function and its E-epigraph are incorrect. In this note, we show that the results obtained in Ref. 2 concerning the characterization of an E-convex function f in terms of its E-epigraph are incorrect. Afterward, some characterizations of E-convex functions using a different notion of epigraph are given.     

Patch-generated Frames and Projectable Hulls

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal ℓ-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal ℓ-ideals of A. For Bernhard Banaschewski, on the occasion of his 80th birthday.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Dimension in algebraic frames

In an algebraic frame L the dimension, dim(L), is defined, as in classical ideal theory, to be the maximum of the lengths n of chains of primes p ₀ < p ₁ < ... < p _n , if such a maximum exists, and ∞ otherwise. A notion of “dominance” is then defined among the compact elements of L, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of L, including the frames dL and zL of d-elements and z-elements, respectively. The more concrete illustrations regarding the frame convex ℓ-subgroups of a lattice-ordered group and its various natural frame quotients occupy the second half of this exposition. For example, it is shown that if A is a commutative semiprime f-ring with finite ℓ-dimension then A must be hyperarchimedean. The d-dimension of an ℓ-group is invariant under formation of direct products, whereas ℓ-dimension is not. r-dimension of a commutative semiprime f-ring is either 0 or infinite, but this fails if nilpotent elements are present. sp-dimension coincides with classical Krull dimension in commutative semiprime f-rings with bounded inversion.     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Invertible sequences of bounded linear operators

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.     

Nonstable <Emphasis Type="Italic">K</Emphasis>-theory for Graph Algebras

We compute the monoid V(L _K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L _K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L _K (E) and the lattice of order-ideals of V(L _K (E)). When K is the field of complex numbers, the algebra is a dense subalgebra of the graph C ^*-algebra C ^*(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property. The first author was partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM2002-01390, the second and the third by the DGI and European Regional Development Fund, jointly, through Project MTM2004-00149 and by PAI III grant FQM-298 of the Junta de Andalucía. Also, the first and third authors are partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.     

Construction of markov process with markov resolvent onL P (E, m)

LetE be a Hausdorff topological space and let β(E) be its Borel σ-field. Letm be a σ-finite measure on (E, β(E)). A necessary and sufficient condition for a Markov resolvent onL ^P (E,m) to be associated with anm-tightm-special standard process (with state spaceE) is given. Furthermore some new examples which do not belong to the framework of Dirichlet space are also given. Project supported by the Youth Foundation of Chinese Academy of Sciences.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Some complete metrics on spaces of fuzzy subsets

The classes of the L_p,∞- and L_p-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L_p,∞-metric there exists some L_p-metric which induces the same topology. This paper deals with the completion of the L_p,∞- and L_p-metrics. We can also show that the relationship between the classes of L_p,∞- and L_p-metrics still holds for the obtained respective classes of their completions.     

A note on the left and right translations in L-topological groups

In this note, we show by a counterexample that a proposition on L-homeomorphism property of the left and right translations in L-topological groups given by Bayoumi [On initial and final L-topological groups, Fuzzy Sets and Systems 156 (2005) 43–54] is incorrect, and offer its modified version.     

Two-sample rank tests based on exceeding observations

Simple rank statistics are used to test that two samples come from the same distribution. Šidák’s E-test (Apl. Mat. 22 (1977), 166–175) is based on the number of observations from one sample that exceed all observations from the other sample. A similar test statistic is defined in Ann. Inst. Stat. Math. 52 (1970), 255–266. We study asymptotic behavior of the moments of both statistics.     

An acceleration property of theE-algorithm for alternate sequences

We prove a convergence acceleration result by theE-algorithm for sequences whose error has an asymptotic expansion on the scale of comparison for which a determinantal relation holds. This result is generalized to the vector case. Moreover we prove a result which contains an acceleration property for columns and diagonals of theE array. This result is applied to some alternating series.     

A Method of Deducing L-Polyhedra for n-Lattices

We suggest a method for selecting an L-simplex in an L-polyhedron of an n-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an L-simplex and by considering a simplex selected from an L-polyhedron, we present a new method for describing all types of L-polyhedra in lattices of given dimension n. We apply the method to deduce all types of L-polyhedra in n-dimensional lattices for n=2,3,4, which are already known from previous results.     

Local c- and E-optimal Designs for Exponential Regression Models

In this paper we investigate local E- and c-optimal designs for exponential regression models of the form . We establish a numerical method for the construction of efficient and local optimal designs, which is based on two results. First, we consider for fixed k the limit μ _i → γ (i = 1, ... , k) and show that the optimal designs converge weakly to the optimal designs in a heteroscedastic polynomial regression model. It is then demonstrated that in this model the optimal designs can be easily determined by standard numerical software. Secondly, it is proved that the support points and weights of the local optimal designs in the exponential regression model are analytic functions of the nonlinear parameters μ ₁, ... , μ _k . This result is used for the numerical calculation of the local E-optimal designs by means of a Taylor expansion for any vector (μ ₁, ... , μ _k ). It is also demonstrated that in the models under consideration E-optimal designs are usually more efficient for estimating individual parameters than D-optimal designs.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

Frames and bases in tensor products of Hilbert spaces and Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

In this article, we study tensor product of Hilbert C*-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F, and we get more results. For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K.