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Search: All articles in the CMB digital archive with keyword UCT

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1. CMB Online first

Jiang, Chunlan
Reduction to dimension two of local spectrum for $AH$ algebra with ideal property
A $C^{*}$-algebra $A$ has the ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.

Keywords:AH algebra, reduction, local spectrum, ideal property

2. CMB Online first

Liu, Li; Weng, Peixuan
Globally asymptotic stability of a delayed integro-differential equation with nonlocal diffusion
We study a population model with nonlocal diffusion, which is a delayed integro-differential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As application, an example for a population model with age structure is discussed at the end of the article.

Keywords:integro-differential equation, nonlocal diffusion, equilibrium, globally asymptotic stability, population model with age structure
Categories:45J05, 35K57, 92D25

3. CMB 2016 (vol 60 pp. 217)

Wang, Yuanyi
Condition $C'_{\wedge}$ of Operator Spaces
In this paper, we study condition $C'_{\wedge}$ which is a projective tensor product analogue of condition $C'$. We show that the finite-dimensional OLLP operator spaces have condition $C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.

Keywords:operator space, local theory, tensor product

4. CMB 2016 (vol 59 pp. 813)

Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of Their generalized Tanaka-Webster Lie Derivative
On a real hypersurface $M$ in a non-flat complex space form there exist the Levi-Civita and the k-th generalized Tanaka-Webster connections. The aim of the present paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operator with respect to the Levi-Civita connections coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in directions of any vecro field orthogonal to the structure vector field.

Keywords:$k$-th generalized Tanaka-Webster connection, non-flat complex space form, real hypersurface, Lie derivative, structure Jacobi operator
Categories:53C15, 53B25

5. CMB 2016 (vol 59 pp. 705)

Chen, Yichao; Yin, Xuluo
The Thickness of the Cartesian Product of Two Graphs
The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G.$ A $t$-minimal graph is a graph of thickness $t$ which contains no proper subgraph of thickness $t.$ In this paper, upper and lower bounds are obtained for the thickness, $t(G\Box H)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t(G)$ and $t(H)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph $K_{4k,4k},$ the thickness of the Cartesian product of two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also given, for $n\neq 4k+1$.

Keywords:planar graph, thickness, Cartesian product, $t$-minimal graph, complete bipartite graph

6. CMB 2016 (vol 59 pp. 326)

Jiang, Chunlan; Shi, Rui
On the Uniqueness of Jordan Canonical Form Decompositions of Operators by $K$-theoretical Data
In this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in $\mathcal {L}(\mathcal {H})$. A complete criterion for similarity for this class of operators in terms of $K$-theory for Banach algebras is given.

Keywords:strongly irreducible operator, similarity invariant, reduction theory of von Neumann algebras, $K$-theory
Categories:47A15, 47C15, 47A65

7. CMB 2016 (vol 59 pp. 279)

Dimca, Alexandru
The Poincaré-Deligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined
Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré-Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

Keywords:line arrangement, Milnor fiber, monodromy, mixed Hodge structures
Categories:32S22, 32S35, 32S25, 32S55

8. CMB 2016 (vol 59 pp. 311)

Ilten, Nathan; Teitler, Zach
Product Ranks of the $3\times 3$ Determinant and Permanent
We show that the product rank of the $3 \times 3$ determinant $\det_3$ is $5$, and the product rank of the $3 \times 3$ permanent $\operatorname{perm}_3$ is $4$. As a corollary, we obtain that the tensor rank of $\det_3$ is $5$ and the tensor rank of $\operatorname{perm}_3$ is $4$. We show moreover that the border product rank of $\operatorname{perm}_n$ is larger than $n$ for any $n\geq 3$.

Keywords:product rank, tensor rank, determinant, permanent, Fano schemes
Categories:15A21, 15A69, 14M12, 14N15

9. CMB 2016 (vol 59 pp. 363)

Li, Dan; Ma, Wanbiao
Dynamical Analysis of a Stage-Structured Model for Lyme Disease with Two Delays
In this paper, a nonlinear stage-structured model for Lyme disease is considered. The model is a system of differential equations with two time delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$, then there exists a unique positive equilibrium whose local asymptotical stability and the existence of Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Some numerical simulations are performed to confirm the correctness of theoretical analysis. At last, some conclusions are given.

Keywords:Lyme disease, stage-structure, time delay, Lyapunov functional stability Hopf bifurcation.

10. CMB 2015 (vol 58 pp. 877)

Zaatra, Mohamed
Generating Some Symmetric Semi-classical Orthogonal Polynomials
We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relation
Categories:33C45, 42C05

11. CMB 2015 (vol 58 pp. 858)

Williams, Kenneth S.
Ternary Quadratic Forms and Eta Quotients
Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly 10 eta quotients \[ f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0, \] such that the Fourier coefficients $c(n)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$ we have $c(n)=0$ for all $n$ in each of the arithmetic progressions $\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k \geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.

Keywords:Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formula
Categories:11F20, 11E20, 11E25

12. CMB 2015 (vol 58 pp. 548)

Lü, Guangshi; Sankaranarayanan, Ayyadurai
Higher Moments of Fourier Coefficients of Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL(2, \mathbb{Z})$. Let $\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in S_{k_3}(\Gamma)$ respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$, $\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted by the arithmetic function $\lambda_f(n)$.

Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$-function, Perron's formula
Categories:11F30, 11F66

13. CMB 2015 (vol 58 pp. 730)

Efrat, Ido; Matzri, Eliyahu
Vanishing of Massey Products and Brauer Groups
Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the mod-$p$ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináċ, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields
Categories:16K50, 11R34, 12G05, 12E30

14. CMB 2015 (vol 58 pp. 374)

Szabó, Gábor
A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in $E$-Theory
Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$-algebra and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action. Provided that $\alpha$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in $E$-theory passes from $A$ to the crossed product $\mathrm{C}^*$-algebra $A\rtimes_\alpha G$ and the fixed point algebra $A^\alpha$. This extends a similar result by Gardella for $KK$-theory in the case of unital $\mathrm{C}^*$-algebras, but with a shorter and less technical proof. For circle actions on separable, unital $\mathrm{C}^*$-algebras with the continuous Rokhlin property, we establish a connection between the $E$-theory equivalence class of $A$ and that of its fixed point algebra $A^\alpha$.

Keywords:Rokhlin property, UCT, KK-theory, E-theory, circle actions
Categories:46L55, 19K35

15. CMB 2015 (vol 58 pp. 281)

Kalus, Matthias
On the Relation of Real and Complex Lie Supergroups
A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complex Harish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed.

Keywords:Lie supergroup, almost complex structure, Harish-Chandra pair, universal complexification
Categories:32C11, 58A50

16. CMB 2014 (vol 58 pp. 91)

Hasegawa, Kei
Essential Commutants of Semicrossed Products
Let $\alpha\colon G\curvearrowright M$ be a spatial action of countable abelian group on a "spatial" von Neumann algebra $M$ and $S$ be its unital subsemigroup with $G=S^{-1}S$. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten $p$-class or the compact operators, of the w$^*$-semicrossed product of $M$ by $S$ when $M'$ contains no non-zero compact operators. We also prove a weaker result when $M$ is a von Neumann algebra on a finite dimensional Hilbert space and $(G,S)=(\mathbb{Z},\mathbb{Z}_+)$, which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.

Keywords:essential commutant, semicrossed product
Categories:47L65, 47A55

17. CMB 2014 (vol 57 pp. 264)

Dai, Li; Dong, Jingcheng
On Semisimple Hopf Algebras of Dimension $pq^n$
Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $pq^n$ over an algebraically closed field of characteristic $0$. This paper proves that $H$ must possess one of the following structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group algebra of group $G$ of order $p$, and $R$ is a semisimple Yetter--Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.

Keywords:semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld double

18. CMB 2013 (vol 57 pp. 375)

López, S. C.; Muntaner-Batle, ; Rius-Font,
A Problem on Edge-magic Labelings of Cycles
Kotzig and Rosa defined in 1970 the concept of edge-magic labelings as follows: let $G$ be a simple $(p,q)$-graph (that is, a graph of order $p$ and size $q$ without loops or multiple edges). A bijective function $f:V(G)\cup E(G)\rightarrow \{1,2,\ldots,p+q\}$ is an edge-magic labeling of $G$ if $f(u)+f(uv)+f(v)=k$, for all $uv\in E(G)$. A graph that admits an edge-magic labeling is called an edge-magic graph, and $k$ is called the magic sum of the labeling. An old conjecture of Godbold and Slater sets that all possible theoretical magic sums are attained for each cycle of order $n\ge 7$. Motivated by this conjecture, we prove that for all $n_0\in \mathbb{N}$, there exists $n\in \mathbb{N}$, such that the cycle $C_n$ admits at least $n_0$ edge-magic labelings with at least $n_0$ mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle $C_n$, depending on the sum of the exponents of the odd primes appearing in the prime factorization of $n$.

Keywords:edge-magic, valence, $\otimes_h$-product

19. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
Assouad-Nagata Dimension of Wreath Products of Groups
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.

Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groups
Categories:54F45, 55M10, 54C65

20. CMB 2013 (vol 57 pp. 401)

Perrone, Domenico
Curvature of $K$-contact Semi-Riemannian Manifolds
In this paper we characterize $K$-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat $K$-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature $\kappa=\varepsilon$, where $\varepsilon =\pm 1$ denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a $K$-contact Lorentzian manifold.

Keywords:contact semi-Riemannian structures, $K$-contact structures, conformally flat manifolds, Einstein Lorentzian-Sasaki manifolds
Categories:53C50, 53C25, 53B30

21. CMB 2013 (vol 57 pp. 821)

Jeong, Imsoon; Kim, Seonhui; Suh, Young Jin
Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator
In this paper we give a characterization of a real hypersurface of Type~$(A)$ in complex two-plane Grassmannians ${ { {G_2({\mathbb C}^{m+2})} } }$, which means a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in ${G_2({\mathbb C}^{m+2})}$, by the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.

Keywords:real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, Reeb parallel, structure Jacobi operator
Categories:53C40, 53C15

22. CMB 2013 (vol 57 pp. 598)

Lu, Yufeng; Yang, Dachun; Yuan, Wen
Interpolation of Morrey Spaces on Metric Measure Spaces
In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasi-metric measure spaces, which generalizes some known results on ${\mathbb R}^n$.

Keywords:complex interpolation, Morrey space, Gagliardo interpolation, Calderón product, quasi-metric measure space
Categories:46B70, 46E30

23. CMB 2012 (vol 57 pp. 97)

Levy, Jason
Rationality and the Jordan-Gatti-Viniberghi decomposition
We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan-Kac-Vinberg decompositions of a rational vector all lie in a single rational orbit.

Keywords:reductive group, $G$-module, Jordan decomposition, orbit closure, rationality
Categories:20G15, 14L24

24. CMB 2012 (vol 57 pp. 326)

Ivanov, S. V.; Mikhailov, Roman
On Zero-divisors in Group Rings of Groups with Torsion
Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

Keywords:Burnside groups, free products of groups, group rings, zero-divisors
Categories:20C07, 20E06, 20F05, , 20F50

25. CMB 2012 (vol 57 pp. 80)

Khemphet, Anchalee; Peters, Justin R.
Semicrossed Products of the Disk Algebra and the Jacobson Radical
We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

Keywords:semicrossed product, disk algebra, Jacobson radical
Categories:47L65, 47L20, 30J10, 30H50
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