Electronic Edition Vol. 42, No. 2, 2016

Editors:  D. Bao (San Francisco, SFSU), D. Blecher (Houston), Bernhard G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station),  M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice).
Managing Editor: K. Kaiser (Houston)


Houston Journal of Mathematics


Hetzel, Andrew J., Tennessee Tech University, Cookeville, TN 38505 (ahetzel@tntech.edu), and Lawson, Ashley M., Tennessee Tech University, Cookeville, TN 38505.
Finite superideal domains, pp. 381-395.
ABSTRACT. We develop the basic theory of integral domains R such that for each nonzero ideal I of R, there exist only finitely many ideals J of R for which I is a subset of J, termed “finite superideal domains” (FSD's). It is shown that the class of FSD's properly encompasses both the class of Dedekind domains and the class of Cohen-Kaplansky domains (CK-domains). By characterizing FSD's in terms of special principal ideal rings (SPIR's) and finite rings, we demonstrate that the property of being an FSD is stable under the formation of localizations. We also characterize when the classical D+M construction yields an FSD and provide both a necessary condition and a sufficient condition for a monoid domain to be an FSD.

Carmen Constantin, Quantum Group, Department of Computer Science, University of Oxford (i.m.carmen@gmail.com) and Andreas Döring, Institute for Theoretical Physics, Department of Physics, Friedrich-Alexander Universität Erlangen-Nürnberg,Schlossplatz 4, 91054 Erlangen, Germany (andreas.doering@fau.de)
Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices, pp. 397-409.
ABSTRACT. Harding and Navara have shown that an orthomodular lattice L is determined up to isomorphism by the poset BL of its Boolean subalgebras. Here, we give a reasonably direct order-theoretic construction of an atomic orthomodular lattice L from the poset BL. This relates to an open problem presented by Harding and Navara. A further motivation for this line of work comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=PH, the projection lattice of the algebra BH of bounded operators on Hilbert space.

Mendes, Carla, Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal (cmendes@math.uminho.pt).
Subdirectly irreducible double Km,n-algebras, pp. 411-425.
ABSTRACT. In this paper we study subdirectly irreducible double Km,n-algebras. In particular, we prove that every finitely subdirectly irreducible double Km,n-algebra is subdirectly irreducible and we characterize subdirectly irreducible double Km,n-algebras by determining their lattice of congruences.

Chang, Gyu Whan, Department of Mathematics Education, Incheon National University, Incheon 22012, Korea (whan@inu.ac.kr), Houston, Evan, University of North Carolina at Charlotte, Charlotte, NC 28223 (eghousto@uncc.edu), and Park, Mi Hee, Chung-Ang University, Seoul 156-756, Korea (mhpark@cau.ac.kr).
Star operations on strong Mori domains, pp. 427-446 
ABSTRACT. Let D be a strong Mori domain, S(D) the set of star operations on D, and Sw(D) the subset consisting of those star operations at least as large as the w-operation. We give a product formula for the cardinality of Sw(D), which allows a reduction to the t-local case, we characterize when the cardinality of Sw(D) is 2 or 3, and we prove that Sw(D[X]) and Sw(D) have the same cardinality when the former is finite.

Le Giang, Department of Mathematics, Hanoi National University of Education, 136-Xuan Thuy, Cau Giay, Hanoi, VietNam (legiang01@yahoo.com).
An explicit estimate on multiplicity truncation in the degenerated second main theorem, pp. 447-462.

ABSTRACT. Chen, Ru and Yan proved a weak Cartan-type Second Main Theorem for an algebraically non-degenerate holomorphic curve from the complex plane into a complex projective variety intersecting hypersurfaces in subgeneral position that included truncated counting functions. We improve their results by giving an explicit estimate for the level of truncation.

Xiaoqing Lu, Department of Mathematics, Nanjing University, 22 Hankou Road, 210093,Nanjing, P.R.China (luxiaoqing1984@126.com), and Liangwen Liao, Department of Mathematics, Nanjing University, 22 Hankou Road, 210093, Nanjing, P.R. China. (maliao@nju.edu.cn).
On entire solutions of a certain type of nonlinear differential equations, pp. 463-477.
ABSTRACT. By utilizing Nevanlinna's value distribution theory of meromorphic functions,we prove that a certain type of nonlinear differential equations has no transcendental entire solutions.

Morales, CA, Instituto de Matematica, UFRJ, 68530, 21945-970, Rio de Janeiro, Brazil (morales@impa.br).
Finiteness of periods for diffeomorphisms, pp. 479-497.
ABSTRACT. We give a sufficient condition for the finiteness of periods of a diffeomorphism. It is based on the growth of the derivative at the periodic points of nearby diffeomorphisms.

Kentaro Saji, Department of Mathematics, Graduate School of Science, Kobe University, Rokko, Nada, Kobe 657-8501, Japan (saji@math.kobe-u.ac.jp).
Isotopy of Morin singularities
, pp. 499-519.
ABSTRACT. We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other well-known low-dimensional singularities. We also give an application to stable perturbations of simple equi-dimensional map-germs.

Royal, Jennifer, University of Georgia, Athens, GA 30602 (jroyal@uga.edu).
Regularity of geodesics in sets of positive reach
, pp. 521-535.
ABSTRACT. The question of regularity of geodesics is a central theme in geometry, having been addressed fully in the context of smooth Riemannian manifolds with and without boundary. In this paper, we examine geodesics in regular sets of positive reach (PR sets) in Euclidean space, and we prove that any geodesic in such a set has Lipschitz continuous velocity. To prove the result, we first show that the velocity of a geodesic has essentially bounded variation; we then use the geometric structure of the associated tangent and normal cones to conclude that the first derivative is Lipschitz continuous.

Jameson Cahill, Department of Mathematical Sciences, New Mexico State University, 1290 Frenger Mall, MSC 3MB/Science Hall 236, Las Cruces, NM 88003-8001 (jameson.cahill@gmail.com), Peter G. Casazza, Department of Mathematics, University of Missouri, Columbia, MO 65211-4100 (Casazzap@missouri.edu), Jesse Peterson, Department of Mathematics and Statistics, Air Force Institute of Technology, WPAFB, OH 45433-7765 (jesse.peterson@afit.edu), and Lindsey Woodland, (lmwvh4@gmail.com).
Phase retrieval by projections, pp. 537-558.
The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. Much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps towards answering these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with 2M-1 subspaces of arbitrary rank in a real M-dimensional vector space and 4M-3 subspaces of arbitrary rank in a complex M-dimensional vector space. Finally, we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

Andreas Döring, Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany (andreas.doering@fau.de), and Harding, John, New Mexico State University, Las Cruces, NM 88003, USA (jharding@nmsu.edu).
Abelian subalgebras and the Jordan structure of a von Neumann algebra, pp. 559-568.
ABSTRACT. For von Neumann algebras M, N not isomorphic to C + C and without type I_2 summands, we show that for an order-isomorphism f from the poset AbSub M of abelian von Neumann subalgebras of M to AbSub N, there is a unique Jordan *-isomorphism g from M to N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of a von Neumann algebra not isomorphic to C + C and without type I_2 summands is determined by the poset of its abelian subalgebras. This result has implications in recent approaches to foundational issues in quantum mechanics.

George Costakis, Department of Mathematics, University of Crete, GR-700 13 Heraklion, Crete, Greece (costakis@math.uoc.gr).
Two problems on power-regular operators, pp. 569-575.
ABSTRACT. We answer two questions raised by A. Atzmon in  Power regular operators, Trans. Amer. Math. Soc. 347 (1995), 3101-3109. In particular, we show that there exists an operator T, acting on a complex infinite dimensional separable Hilbert space H, whose spectrum is contained either in the real line or in a circle and it is not power-regular, that is, for some h∈H\{ 0} the sequence ( ∥Tn∥h1/n ) does not converge. Our second result shows that the sum of two commuting power-regular operators is not necessarily power-regular. The spectrum of a Hilbert space hypercyclic operator plays a key role in the proofs.
Chávez-Domínguez, Javier Alejandro, The University of Texas at Austin, Austin, TX 78249 (jachavezd@math.utexas.edu).
The Chevet-Saphar tensor norms for operator spaces, pp. 577-596.
We introduce the operator space versions of the Chevet-Saphar tensor norms, and show that they share many properties with their Banach-space counterparts. Most importantly, these tensor norms are in operator space duality with the completely p-summing maps of G. Pisier. Our approach, based on the algebraic idea of a tensor contraction, complements previous results on duality for completely p-summing maps due to E. Effros and Z.-J. Ruan, and M. Junge. We also obtain an operator-space version of the Chevet-Persson-Saphar inequalities and give several applications of it, including a completely isomorphic characterization of quotients of subspaces of ultrapowers of the Schatten p-class.

Mehdi Rostami
, Faculty of Mathematical and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, 15914 Tehran, Iran (mross@aut.ac.ir) and Nico Spronk, Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada (nspronk@uwaterloo.ca).
Convolutions on the Haagerup tensor products of Fourier algebras, pp. 597-611.
ABSTRACT. The authors study ranges of maps of convolutions and notice that a certain  operator algebra has an unexpected set of spectral synthesis.

J.D. Rossi, University of Buenos Aires, Argentina (jrossi@dm.uba.ar) and N. Saintier, University of Buenos Aires and University of Gral Sarmiento, Argentina (nsaintie@dm.uba.ar).
On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions , pp. 613-635.
ABSTRACT. We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ(Ut)=λ(U)+O(t). Although λ(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative.

Michel Smith, Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849 (smith01@auburn.edu) and Scott Varagona,  Department of Biology, Chemistry and Mathematics, University of Montevallo, Montevallo, Alabama 35115 (svaragona@montevallo.edu).
Generalized inverse limits with N-type bonding functions, pp. 637-657.
ABSTRACT.  We construct a specific type of inverse limit with set valued bonding functions all but one of whose coordinate spaces are finite non-Hausdorff spaces. The resulting space is a metric chainable indecomposable Hausdorff continuum which is then used to show that a wide range of inverse limits with upper semi-continuous bonding functions are all homeomorphic to well known and studied chainable indecomposable metric continua.

Alas, Ofelia T., Universidade de São Paulo, 05314-970 São Paulo, Brazil (alas@ime.usp.br)Junqueira, Lucia R., Universidade de São Paulo, 05314-970 São Paulo, Brazil (lucia@ime.usp.br) Tkachuk, Vladimir V., Universidad Autónoma Metropolitana, 09340, Mexico City, Mexico (vova@xanum.uam.mx), and  Wilson, Richard G., Universidad Autónoma Metropolitana, 09340, Mexico City, Mexico (vova@xanum.uam.mx).
Discrete reflexivity in squares, pp. 659-673.

ABSTRACT.We establish that first countability is discretely reflexive in the class of pseudocompact spaces of character at most ω1. Given a property P from the list {σ-compactness, zero-dimensionality, analyticity}, it is shown that a space X with a countable network has P if and only if the closure of D has P for any discrete set D⊂ X2. If X is a Lindelöf Σ-space and the closure of D is countable for every discrete subspace D ⊂ X2, then X is countable. However, under CH, there exists an example of an uncountable space X such that the closure of D is countable for any discrete D⊂ X2. We also prove that for any space X of countable π-weight, there exists a discrete D⊂ X2 such that ΔX={(x,x): x∈ X} ⊂ cl(D). Therefore, if X has countable π-weight, then for every closed-hereditary property P , if X2 is discretely P, then the space X has P.

Dimov, Georgi D., University of Sofia, 1164 Sofia, Bulgaria (gdimov@fmi.uni-sofia.bg), and Ivanova, Elza P., University of Sofia, 1164 Sofia, Bulgaria (elza@fmi.uni-sofia.bg).
Yet another duality theorem for locally compact spaces, pp. 675-700.
ABSTRACT. In 1962, H. de Vries (Compact Spaces and Compactifications, an Algebraic Approach, Van Gorcum, The Netherlands) proved a duality theorem for the category HC of compact Hausdorff spaces and continuous maps. In 2010 this duality theorem was extended by G. Dimov (A de Vries-type duality theorem for the category of locally compact spaces and continuous maps - I, Acta Math. Hungarica, 129, 314-349) to the category HLC of locally compact Hausdorff spaces and continuous maps. The composition of the morphisms of the dual categories, obtained in the papers cited above, differs from the set-theoretic one. Here we obtain a new category MDHLC dual to the category HLC; the composition of the MDHLC-morphisms is a more natural one but the MDHLC-morphisms are multi-valued maps.

Michał Ryszard Wójcik, Institute of Geography and Regional Development, University of Wrocław, pl. Uniwersytecki 1, 50-137 Wrocław, Poland  (michal.wojcik@uni.wroc.pl).
The generalized Aron-Maestre comb, pp. 701-707
ABSTRACT. This note is inspired by the connected, not separably connected metric space constructed by Aron and Maestre in 2003. It is a comb-like space embedded in a Banach space. The construction explicitly uses a Vitali set and takes up two printed pages. We present an isometric half-a-page version, with no use of the axiom of choice, as a subspace of the plane whose topology is finer than the Euclidean topology but coarser than the river metric topology, making it easy to visualize. Using a Vitali set we show it to be absolutely F sigma. We also generalize this construction using topological formulations and arguments instead of the tricks which make it so short to write, which is still shorter than the original. We make no external references.

Themba, Dube, Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003 Pretoria, SOUTH AFRICA (dubeta@unisa.ac.za) and Jissy, Nsonde-Nsayi, Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003 Pretoria, SOUTH AFRICA (jissy@aims.ac.za).
Another ring-theoretic characterization of boundary spaces, pp. 709-722.
ABSTRACT. A Tychonoff space X is called a boundary space if the boundary of every zero-set of X is contained in a zero-set with empty interior. These spaces were studied in [F. Azarpanah and M. Karavan, On nonregular ideals and z-ideals in C(X), Czechoslovak Math. J. 55 (2005), 397-407.]. They characterized them as those X for which every prime ideal of C(X) that consists entirely of zero-divisors is a d-ideal. In this note we give another ring-theoretic characterization. Call a ring A a boundary ring if, for every a
∈A, the ideal M(a) + Ann(a) contains a non-divisor of zero, where M(a) designates the intersection of all maximal ideals of A containing a, and Ann(a) is the annihilator of a. We show that X is a boundary space if and only if C(X) is a boundary ring. We also show that if X×Y is z-embedded in βX×βY, then X and Y are boundary spaces if X×Y is a boundary space.