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Inspired by the construction of the Cuntz semigroup for a C ⁎ -algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C ⁎ -algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimension functions. [ABSTRACT FROM AUTHOR]

Ahmad, Bhat and Sheikh characterized composite wavelets based on results of affine and quasi affine frames. We continued their study and provided the characterization of nonuniform composite wavelets based on results of affine and quasi affine frames. Moreover all the nonuniform composite wavelets associated with AB-MRA are characterized onL²(Λ). [ABSTRACT FROM AUTHOR]

We prove that for any dimension function h with h ≺ xd and for any countable set of linear patterns, there exists a compact set E with Hh (E) > 0 avoiding all the given patterns. We also give several applications and recover results of Keleti, Maga, and Máthé. [ABSTRACT FROM AUTHOR]

We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L -modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L -modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, F Γ -modules) via a new colimit model for M ⊗ F Γ L. [ABSTRACT FROM AUTHOR]

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then a noncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given noncommutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a given noncommutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases. [ABSTRACT FROM AUTHOR]

When G is a row-finite digraph, we classify all finite-dimensional modules of the Leavitt path algebra L(G) via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph G. The category of (unital) L(G)-modules is equivalent to a full subcategory of quiver representations of G. However, the category of finite-dimensional representations of L(G) is tame in contrast to the finitedimensional quiver representations of G, which are almost always wild. [ABSTRACT FROM AUTHOR]

Dimension functions play a significant role in the study of wavelets and have been attracting many waveletters' interest. In recent years, the study of wavelet dimension functions has seen great achievements, but the study of Parseval frame wavelet (PFW) dimension functions has not. Bownik, Rzeszotnik and Speegle in 2001 and Arambašić, Bakić and Rajić in 2007 characterized [ABSTRACT FROM AUTHOR]

Let X be a Polish space. We prove that the generic compact set K ⊆ X (in the sense of Baire category) is either finite or there is a continuous gauge function h such that 0 < Hh(K) < ∞, where Hh denotes the h-Hausdorff measure. This answers a question of Cabrelli, Darji, and Molter. Moreover, for every weak contraction f : K → X we have Hh (K ⋂ f(K)) = 0. This is a measure theoretic analogue of a result of Elekes. [ABSTRACT FROM AUTHOR]

Abstract: For in , a subset of is called a Furstenberg set of type or -set if for each direction in the unit circle there is a line segment in the direction of such that the Hausdorff dimension of the set is greater than or equal to . In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for , , we construct a set of Hausdorff dimension not greater than . Since in a previous work we showed that is a lower bound for the Hausdorff dimension of any , with the present construction, the value is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions . [Copyright &y& Elsevier]

Generalizing the result of Bownik and Speegle [Approximation Theory X: Wavelets, Splines and Applications, Vanderbilt University Press, pp. 63-85, 2002], we provide plenty of non-MSF A-wavelets with the help of a given A-wavelet set. Further, by showing that the dimension function of the non-MSF A-wavelet constructed through an A-wavelet set W coincides with the dimension function of W, we conclude that the non-MSF Awavelet and the A-wavelet set through which it is constructed possess the same nature as far as the multiresolution analysis is concerned. Some examples of non-MSF d-wavelets and non-MSF A-wavelets are also provided. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function. [ABSTRACT FROM AUTHOR]

In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ε (0, 1], we will say that a set E ⊂ ℝ² is an Fαβ-set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each direction e in L, there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is equal to or greater than α. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E) ≥ max Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. for any E ε Fαβ. In particular we are able to extend previously known results to the "endpoint" α = 0 case. [ABSTRACT FROM AUTHOR]

In this article, we obtain two classes of non-MSF wavelets by considering two different classes of six-interval wavelet sets provided by Arcozzi, Behera and Madan (J. Geom. Anal.13 (2003) 557-579). Out of these classes one is countable, the non-MSF wavelets of which are non-MRA and the other one is uncountable, the non-MSF wavelets of which are MRA. The set of all non-MSF MRA wavelets of the latter class is shown to be pathconnected. [ABSTRACT FROM AUTHOR]

For a natural number n, selecting a 2n-interval symmetric wavelet set by making use of a result of Arcozzi, Behera and Madan [J. Geom. Anal. 13 (2003), 557-579], we construct a family of symmetric wavelet sets accumulating at the origin. Such a family of wavelet sets is also obtained from a family of symmetric six-interval wavelet sets provided by them in the same paper. Three-interval wavelet sets are employed in having a family of wavelet sets accumulating at the origin which are non-symmetric. Further, we obtain a larger class of H²-wavelet sets accumulating at the origin, which include the one given by Behera in [Bull. Polish Acad. Sci. Math. 52 (2004), 169-178]. Finally, non-MSF non-MRA wavelets are obtained through the selected family of 2n-interval symmetric wavelet sets. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For , a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment in the direction of e for which . It is well known that , and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if , there always exists such that (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint . [Copyright &y& Elsevier]

Abstract: Given a non-degenerate Peano continuum , a dimension function defined on the family of compact subsets of , and a subset , we recognize the topological structure of the system , where is the hyperspace of non-empty compact subsets of and is the subspace of , consisting of non-empty compact subsets with . [Copyright &y& Elsevier]

Abstract: In this paper we study structural properties of shift–modulation invariant (SMI) spaces, also called Gabor subspaces, or Weyl–Heisenberg subspaces, in the case when shift and modulation lattices are rationally dependent. We prove the characterization of SMI spaces in terms of range functions analogous to the well-known description of shift-invariant spaces [C. de Boor, R. DeVore, A. Ron, The structure of finitely generated shift-invariant spaces in , J. Funct. Anal. 119 (1994) 37–78; M. Bownik, The structure of shift-invariant subspaces of , J. Funct. Anal. 177 (2000) 282–309; H. Helson, Lectures on Invariant Subspaces, Academic Press, New York/London, 1964]. We also give a simple characterization of frames and Riesz sequences in terms on their behavior of the fibers of the range function. Next, we prove several orthogonal decomposition results of SMI spaces into simpler blocks, called principal SMI spaces. Then, this is used to characterize operators invariant under both shifts and modulations in terms of families of linear maps acting on the fibers of the range function. We also introduce the fundamental concept of the dimension function for SMI spaces. As a result, this leads to the classification of unitarily equivalent SMI spaces in terms of their dimension functions. Finally, we show several results illustrating our fiberization techniques to characterize dual Gabor frames. [Copyright &y& Elsevier]

Abstract: A generalization of Mallat''s classic multiresolution analysis (MRA), based on the theory of spectral pairs, was considered in two papers by Gabardo and Nashed. In this nonstandard setting, the translation set is no longer a subgroup or a translate of a subgroup of , but is a spectrum associated with a one-dimensional spectral pair. In this paper, we continue the study based on this nonstandard setting and give the characterization for nonuniform wavelets associated with a nonuniform MRA. These characterizations are consistent with both the known necessary and sufficient conditions for the existence of nonuniform MRA wavelets and the known characterization for standard dyadic wavelets associated with an MRA. [Copyright &y& Elsevier]

New dimension functions for topological spaces are introduced in the spirit of Nagata’s approach. Expressions for the new functions in terms of covering dimension include the Bruijning—Nagata and Hashimoto—Hattori formulas. [ABSTRACT FROM AUTHOR]

We investigate Riesz wavelets in the context of generalized multiresolution analysis (GMRA). In particular, we show that Zalik''s class of Riesz wavelets obtained by an MRA is the same as the class of biorthogonal wavelets associated with an MRA. [Copyright &y& Elsevier]

The dimension function Dψ of a band-limited wavelet ψ is bounded by n if its Fourier transform is supported in [−(2n+2/3)π,(2n+2/3)π]. For each n∈N and for each &epsiv;, 0<&epsiv;<δ=δ(n), we construct a wavelet ψ with supp ψ⊆[−(2n+2/3)π,(2n+2/3)π+&epsiv;] such that Dψ>n on a set of positive measure, which proves that [−(2n+2/3)π,(2n+2/3)π] is the largest symmetric interval for estimating the dimension function by n. This construction also provides a family of (uncountably many) wavelet sets each consisting of infinite number of intervals. [Copyright &y& Elsevier]

We obtain some properties that Riesz wavelets and the corresponding scaling functions should satisfy in order that the Riesz wavelets be associated with multiresolution analyses (MRAs). They are given in terms of the low/high-pass filters and in terms of the Fourier transform by using the newly obtained necessary and sufficient condition for the sum of two principal shift-invariant subspaces to be closed. The properties are used to improve the characterizations of Riesz wavelets associated with MRAs previously obtained by some of the authors. [Copyright &y& Elsevier]

This note corrects three typographical errors in the paper Wavelet Sets Accumulating at the Origin, Real Anal. Exchange 35(2), 463-478. The corrections are listed according to the page of the original article. [ABSTRACT FROM AUTHOR]