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Product Code: Vol. 36, No. 4, 2010

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**David E. Dobbs,** Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300 (dobbs@math.utk.edu) and **Jay Shapiro, **Department of Mathematics, George Mason University, Fairfax, Virginia 22030-4444 (jshapiro@gmu.edu).

Patching together a minimal overring, pp. 985-995.

ABSTRACT. Let R be a (commutative integral) domain and M a maximal ideal of R. Let T(M) be a minimal ring extension of R_{M}. Our basic question is (*): does there exist a (necessarily minimal) ring extension T of R such that T_{M} is isomorphic to T(M) and T_{N} = R_{N} canonically for each prime ideal N of R that is distinct from M? The answer to (*) is affirmative if T(M) is not a domain. Several equivalences are given for an affirmative answer to (*) when T(M) is a domain, such as the existence of a in T(M) \ R_{M} such that M is the radical of (R:_Ra). If R is a Prüfer domain that has property (#), the answer to (*) is affirmative for all such data {M, T(M)}; the converse is false in general but holds for Prüfer domains each of whose maximal ideals is branched.

**J. Sichler,** Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 (sichler@cc.umanitoba.ca) and **V. Trnkova, **Mathematical Institute of Charles University, Sokolovska 83, 186 75, Praha 8, Czech Republic (trnkova@karlin.mff.cuni.cz).

On clones of infinitary algebras and their initial segments , pp. 997-1009.

ABSTRACT. We give the respective answers to the question of when the existence of all local clone isomorphisms implies the existence of a global clone isomorphism for term clones, polynomial clones and centralizer clones of universal algebras of an infinitary similarity type.

**Avendaño, Martín,** Math Department, Texas A&M University, College Station, TX 77843 (avendano@math.tamu.edu), and **Ibrahim, Ashraf,** Math Department, Texas A&M University, College Station, TX 77843 (aibrahim@math.tamu.edu).

Ultrametric root counting, pp. 1011-1022.

ABSTRACT. Let K be a complete non-archimedian field with respect to a discrete valuation, f be a polynomial with coefficients in K and non-zero discriminant, A the valuation ring of K, and M the maximal ideal of A. The first main result of this paper is a reformulation of Hensel's Lemma that connects the number of roots of f with the number of roots of its reduction modulo a power of M. We then define a condition - regularity - that yields a simple method to compute the exact number of roots of f in K. In particular, we show that regularity implies that the number of roots of f equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.

**G.A. Bagheri-Bardi, **Department of Mathematics, Persian Gulf University, Boushehr 75168, Iran (bagheri@pgu.ac.ir), **A.R. Medghalchi,** Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran 15614, Iran (medghalchi@saba.tmu.ac.ir), and **N. Spronk,** Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada (nspronk@uwaterloo.ca).

Operator-valued convolution algebras, pp. 1023-1036.

ABSTRACT. For any index sets I and J we consider the space of bounded I X J-matrices with entries in A. Under Schur multiplication, this space of matrices is itself a completely contractive Banach algebra. In particular, for any locally compact group, we obtain natural operator-valued Fourier-Stieltjes and measure algebras. We examine their properties in the context of abstract convolution algebras, which are defined via C*-bialgebras.

**Bennett, Grahame, ** Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A. __(bennettg@indiana.edu).__

Some forms of Majorization, pp. 1037-1066.

ABSTRACT. The Theory of Majorization provides easy proofs for many inequalities of the type

*wherein x and y are fixed N-tuples with positive entries and the estimate holds for all convex functions φ from (0,∞) into R. We show here that useful results can be obtained when other classes of functions are considered. *

**Jun Cao,** School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (caojun1860@mail.bnu.edu.cn), **Yu Liu, **Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, People's Republic of China (pkyuliu@yahoo.com.cn) and ** Dachun Yang (Corresponding author),** School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (dcyang@bnu.edu.cn).

Hardy spaces H_{L}^{1}(R^{n}) associated to Schrödinger Type Operators (-Δ)^{2}+V^{2}, pp. 1067-1095.

ABSTRACT. Let L=(-Δ)^{2}+V^{2} be the Schrödinger Type Operator in R^{n} with n≥ 5, where the nonnegative potential V belongs to the reverse Hölder class B_{q0}(R^{n}) with q_{0} ∈ (n/2, ∞). In this paper, the authors introduce the Hardy spaces H_{L}^{1}(R^{n}), which are defined in terms of radial maximal functions associated with the heat semigroup {e^{-tL}}_{t>0}, and establish their atomic decomposition characterizations. As applications, the authors show that the Hardy space H_{L}^{1}(R^{n}) coincides with the Hardy space H^{1}_{-Δ+V}(R^{n}) with equivalent norms. Moreover, the authors also prove that the higher order Riesz transform ∇^{2}(L^{-1/2}) is bounded from H_{L}^{1}(R^{n}) to the classical Hardy space H^{1}(R^{n}) and the corresponding fractional integral L^{-α/4} for α ∈ (0, n) maps H_{L}^{1}(R^{n}) continuously into the Lebesgue space L^{n/(n-α)}(R^{n}).

**Roch, Steffen,** Technical University Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany (roch@mathematik.tu-darmstadt.de).

Spatial discretization of Cuntz algebras, pp. 1097-1132.

ABSTRACT. The (abstract) Cuntz algebra is generated by non-unitary isometries and has therefore no intrinsic finiteness properties. To approximate its elements by finite-dimensional objects, we thus consider a spatial discretization of the Cuntz algebra by the finite section method. For we represent the Cuntz algebra as a (concrete) algebra of operators on a Hilbert space, choose a suitable basis, and associate with each operator in the Cuntz algebra the sequence of its finite sections with respect to the chosen basis. The goal of this paper is to examine the structure of the algebra which is generated by all sequences of this form. Our main results are the fractality of a suitable restriction of this algebra and a necessary and sufficient criterion for the stability of sequences in the restricted algebra. These results are employed to study spectral and pseudo-spectral approximations of elements of the Cuntz algebra.

**Botelho, Fernanda,** University of Memphis, Memphis, TN 38152 (mbotelho@memphis.edu) and and **Jamison, James,** University of Memphis, Memphis, TN 38152(jjamison@memphis.edu).

Algebraic reflexivity of tensor product spaces, pp. 1133-1137.

ABSTRACT. The isometry group of the tensor product of two symmetric sequences spaces, not isometric to a Hilbert space, is algebraically reflexive provided that the tensor product supports only dyadic surjective isometries.

**Young Sik Kim**, Department of Mathematics, Division of Natural Sciences, Yonsei University, Seoul 120-749, South Korea (yoskim@yonsei.ac.kr).

Fourier Feynman transforms and analytic Feynman integrals and convolutions of a Fourier transform μ of a measure on Wiener spaces., pp. 1139-1158.

ABSTRACT. We investigate the action of a Fourier transform on the Wiener space C_{0} [0,T].

**Luo, Qiu-Ming,** Chongqing Normal University, Chongqing 401331, People's Republic of China (luomath2007@163.com).

An explicit relationship between the generalized Apostol–Bernoulli and Apostol–Euler polynomials associated with λ–Stirling numbers of the second kind, pp. 1159-1171.

ABSTRACTIn this paper, we introduce the so–called λ–Stirling numbers of the second kind and research its some elementary properties. we give an explicit relationship between the generalized Apostol–Bernoulli and Apostol–Euler polynomials in terms of the λ–Stirling numbers of the second kind.

**Xiaoxiao,Wang,** East China Normal University, Shanghai,200241,P.R.China (xxwang39@126.com), **Xiaojun,Liu,** East China Normal University, Shanghai,200241, P.R.China (xiaojunliu2007@hotmail.com) and **Xuecheng,Pang,** East China Normal University, Shanghai,200241, P.R.China (xcpang@math.ecnu.edu.cn).

Normality of meromorphic functions concerning differential polynomials, pp. 1173-1184.

ABSTRACT. Let F be a family of functions meromorphic on a plane domain D, all of whose zeros have multiplicity at least k, where k≥2 be an integer, b, c be two nonzero finite complex numbers and 0&less; M<4|b| be a positive number. Also let a_{i}(z), b_{i}(z) be functions analytic on D, i=1,2,...,k. If for every f, we have (a) f(z)=0 implies |f^{(k)}(z)+a_{1}(z)f^{(k-1)}(z)+...+a_{k}(z)f(z)|≤M and f^{(k)}(z)+b_{1}(z)f^{(k-1)}(z)+...+b_{k}(z)f(z)=b if and only if f(z)=c. Then F is normal on D.

**Minc, Piotr, **Department of Mathematics and Statistics, Auburn University, AL 36849, USA (mincpio@auburn.edu).

An uncountable collection of dendroids mutually incomparable by continuous functions, pp. 1185-1205.

ABSTRACT. In this paper we answer a question of B. Knaster by constructing an uncountable collection of dendroids whose members are not comparable by continuous maps.

**Eiichi, Matsuhashi,** Department of Mathematics, Faculty of Engineering, Shimane University ,Matsue, Shimane 690-8504, Japan (matsuhashi@riko.shimane-u.ac.jp). and **Vesko Valov,** Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada (veskov@nipissingu.ca).

Krasinkiewicz spaces and parametric Krasinkiewicz maps, pp. 1207-1220.

ABSTRACT. We say that a metrizable space M is a Krasinkiewicz space if any map from a metrizable compactum X into M can be approximated by Krasinkiewicz maps (a map g : X → M is Krasinkiewicz provided every continuum in X is either contained in a fiber of g or contains a component of a fiber of g). In this paper we establish the following property of Krasinkiewicz spaces: Let f : X → Y be a perfect map between metrizable spaces and M a Krasinkiewicz complete ANR-space. If Y is a countable union of closed finite-dimensional subsets, then the function space C(X,M) with the source limitation topology contains a dense G_{δ}-subset of maps such that all restrictions g|f^{-1}(y), y in Y, are Krasinkiewicz maps. The same conclusion remains true if M is homeomorphic to a closed convex subset of a Banach space and Y is a C-space.

**Kenneth R. Kellum,** Department of Mathematics, San Jose State University, San Jose, California 95192 (kellum@math.sjsu.edu).

Functions that separate X × R, pp. 1221-1226.

ABSTRACT. Suppose f is a real-valued function defined on a connected topological space X where the complement of the graph of f is disconnected. We prove that f restricted to an open subset of X is continuous and the restriction of f to that open set separates the cross space. Also, we prove that if f has the Gibson property, that is, f(cl(U)) is contained in cl(f(U)) for each open set U, then f is continuous. Other conditions insuring the continuity of f are considered.

**Morayne, Michal,** Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wroclaw, Poland (Michal.Morayne@pwr.wroc.pl), and **Wojcik, Michal,** Department of Mathematics, University of Louisville, Louisville, USA (michal.ryszard.wojcik@gmail.com).

A nonseparably connected metric space as a dense connected graph, pp. 1227-1232.

ABSTRACT. We present a connected metric space that does not contain any nontrivial separable connected subspace. Our space is a dense connected graph of a function from the real line satisfying Cauchy's equation.

**Bankston, Paul, **Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53201 (paulb@mscs.mu.edu).

Not every co-existential map is confluent, pp. 1233-1242.

ABSTRACT. A continuous surjection between compacta is co-existential if it is the second of two maps whose composition is a standard ultracopower projection. Co-existential maps are always weakly confluent, and are even monotone when the range space is locally connected; so it is a natural question to ask whether they are always confluent. Here we give a negative answer. This is an interesting question, mainly because of the fact that most theorems about confluent maps have parallel versions for co-existential maps---notably, both kinds of maps preserve hereditary indecomposability. Where the known parallels break down is in the question of chainability. It is a celebrated open problem whether confluent maps preserve chainability, or even being a pseudo-arc; however, as has recently been shown, co-existential maps do indeed preserve both these properties.

**Proctor, C. Wayne,** Stephen F. Austin State University, Nacogdoches, TX 75962 (cproctor@sfasu.edu).

Continuously ray extendible continua and circle-like continua, pp. 1243-1323.

ABSTRACT. The class of all continuously ray extendible continua is shown to contain each non-planar circle-like continuum

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