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26. CMB 2014 (vol 57 pp. 814)

Hou, Ruchen
On Global Dimensions of Tree Type Finite Dimensional Algebras
A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k-$algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum of possible global dimensions is $n-1$.

Keywords:global dimension, tree type finite dimensional $k-$algebra, quiver
Categories:16D40, 16E10, , 16G20

27. CMB Online first

Pollack, Paul; Vandehey, Joseph
Some normal numbers generated by arithmetic functions
Let $g \geq 2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\phi$ denote Euler's totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number \[ 0. f(1) f(2) f(3) \dots \] obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and Erdős in 1946 to prove the $10$-normality of $0.235711131719\ldots$.

Keywords:normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's number
Categories:11K16, 11A63, 11N25, 11N37

28. CMB 2014 (vol 57 pp. 884)

Xu, Yong; Zhang, Xinjian
$m$-embedded Subgroups and $p$-nilpotency of Finite Groups
Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and $G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$ or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G = AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.

Keywords:finite group, $p$-nilpotent group, $m$-embedded subgroup
Categories:20D10, 20D15

29. CMB 2014 (vol 57 pp. 749)

Cavalieri, Renzo; Marcus, Steffen
Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers
We describe double Hurwitz numbers as intersection numbers on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$. Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the $\psi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera $0$ and $1$).

Keywords:double Hurwitz numbers, wall crossings, moduli spaces, ELSV formula
Category:14N35

30. CMB Online first

Tikuisis, Aaron Peter; Toms, Andrew
On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras
We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple $\mathrm{C}^*$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named author, this shows that slow dimension growth coincides with $\mathcal Z$-stability, for approximately subhomogeneous algebras.

Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growth
Categories:46L35, 46L05, 46L80, 47L40, 46L85

31. CMB 2014 (vol 57 pp. 673)

Ahmadi, S. Ruhallah; Gilligan, Bruce
Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two
If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$ acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification $\widehat{G}$ of $G$ on $X$ except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of $X$.

Keywords:homogeneous complex manifold, non-compact dimension two, complexification
Category:32M10

32. CMB Online first

Boulabiar, Karim
Characters on $C( X)$
The precise condition on a completely regular space $X$ for every character on $C(X) $ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

Keywords:characters, realcompact, evaluation, real-valued continuous functions
Categories:54C30, 46E25

33. CMB 2014 (vol 57 pp. 708)

Brannan, Michael
Strong Asymptotic Freeness for Free Orthogonal Quantum Groups
It is known that the normalized standard generators of the free orthogonal quantum group $O_N^+$ converge in distribution to a free semicircular system as $N \to \infty$. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator norm of any non-commutative polynomial in the normalized standard generators of $O_N^+$ converges as $N \to \infty$ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm equivalence for non-commutative polynomials in free semicircular systems.

Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay
Categories:46L54, 20G42, 46L65

34. CMB 2014 (vol 57 pp. 834)

Koh, Doowon
Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields
We study $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

Keywords:finite fields, radial functions, restriction operators
Categories:42B05, 43A32, 43A15

35. CMB Online first

Khamsi, M. A.
Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces
In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that $T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space $(M,d)$. In particular we prove that under suitable conditions, the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings.

Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic space
Categories:47H09, 46B20, 47H10, 47E10

36. CMB 2014 (vol 57 pp. 648)

Tang, Juping; Miao, Long
On the ${\mathcal F}{\Phi}$-Hypercentre of Finite Groups
Let $G$ be a finite group, $\mathcal F$ a class of groups. Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre of $G$ which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A subgroup $H$ is called $\mathcal M$-supplemented in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$ of $H$. The main purpose of this paper is to prove: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that $1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$ is $\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal U}{\Phi}}(G)$.

Keywords:${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formation
Categories:20D10, 20D20

37. CMB 2014 (vol 57 pp. 477)

Eghbali, Majid
On Set Theoretically and Cohomologically Complete Intersection Ideals
Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities \[ \operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq \operatorname{ara}(\mathfrak a) \leq l(\mathfrak a) \leq \mu(\mathfrak a) \] are known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.

Keywords:set-theoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of ideals
Categories:13D45, 13C14

38. CMB 2014 (vol 57 pp. 579)

Larson, Paul; Tall, Franklin D.
On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of $\omega_1$ is hereditarily paracompact.

Keywords:locally compact, hereditarily normal, paracompact, Axiom R, PFA$^{++}$
Categories:54D35, 54D15, 54D20, 54D45, 03E65, 03E35

39. CMB 2014 (vol 57 pp. 803)

Gabriyelyan, S. S.
Free Locally Convex Spaces and the $k$-space Property
Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$.

Keywords:free locally convex space, $k$-space, countable tightness
Categories:46A03, 54D50, 54A25

40. CMB 2014 (vol 57 pp. 485)

Franc, Cameron; Mason, Geoffrey
Fourier Coefficients of Vector-valued Modular Forms of Dimension $2$
We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$-dimensional vector-valued modular form on $\operatorname{SL}_2(\mathbb{Z})$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

Keywords:vector-valued modular form, modular group, bounded denominators
Categories:11F41, 11G99

41. CMB Online first

Chung, Jaeyoung
On an Exponential Functional Inequality and its Distributional Version
Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb R$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality $ \Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G, $ where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$.

Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability
Categories:46F99, 39B82

42. CMB 2014 (vol 57 pp. 780)

Erzakova, Nina A.
Measures of Noncompactness in Regular Spaces
Previous results by the author on the connection between three of measures of non-compactness obtained for $L_p$, are extended to regular spaces of measurable functions. An example of advantage in some cases one of them in comparison with another is given. Geometric characteristics of regular spaces are determined. New theorems for $(k,\beta)$-boundedness of partially additive operators are proved.

Keywords:measure of non-compactness, condensing map, partially additive operator, regular space, ideal space
Categories:47H08, 46E30, 47H99, 47G10

43. CMB Online first

Ghenciu, Ioana
Limited Sets and Bibasic Sequences
Bibasic sequences are used to study relative weak compactness and relative norm compactness of limited sets.

Keywords:limited sets, $L$-sets, bibasic sequences, the Dunford-Pettis property
Categories:46B20, 46B28, 28B05

44. CMB Online first

Daniilidis, A.; Drusvyatskiy, D.; Lewis, A. S.
Orbits of Geometric Descent
We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points.

Keywords:differential inclusion, quasiconvex function, self-contracted curve, sweeping process
Categories:34A60, 49J99

45. CMB Online first

Kamalov, F.
Property T and Amenable Transformation Group $C^*$-algebras
It is well known that a discrete group which is both amenable and has Kazhdan's Property T must be finite. In this note we generalize the above statement to the case of transformation groups. We show that if $G$ is a discrete amenable group acting on a compact Hausdorff space $X$, then the transformation group $C^*$-algebra $C^*(X, G)$ has Property T if and only if both $X$ and $G$ are finite. Our approach does not rely on the use of tracial states on $C^*(X, G)$.

Keywords:Property T, $C^*$-algebras, transformation group, amenable
Categories:46L55, 46L05

46. CMB 2014 (vol 57 pp. 721)

Bruillard, Paul; Galindo, César; Hong, Seung-Moon; Kashina, Yevgenia; Naidu, Deepak; Natale, Sonia; Plavnik, Julia Yael; Rowell, Eric C.
Classification of Integral Modular Categories of Frobenius--Perron Dimension $pq^4$ and $p^2q^2$
We classify integral modular categories of dimension $pq^4$ and $p^2q^2$, where $p$ and $q$ are distinct primes. We show that such categories are always group-theoretical except for categories of dimension $4q^2$. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension $4q^2$ is equivalent to either one of these well-known examples or is of dimension $36$ and is twist-equivalent to fusion categories arising from a certain quantum group.

Keywords:modular categories, fusion categories
Category:18D10

47. CMB 2014 (vol 57 pp. 431)

Tagami, Keiji
The Rasmussen Invariant, Four-genus and Three-genus of an Almost Positive Knot Are Equal
An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, $4$-genus and $3$-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of $4$-genus one.

Keywords:almost positive knot, four-genus, Rasmussen invariant
Categories:57M27, 57M25

48. 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
Category:16W30

49. CMB 2014 (vol 57 pp. 765)

da Silva, Rosângela Maria; Tenenblat, Keti
Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids.

Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers space
Categories:53A10, 53B40

50. CMB 2014 (vol 57 pp. 231)

Bagherian, J.
On the Multiplicities of Characters in Table Algebras
In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset which is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

Keywords:table algebra, faithful module, multiplicity of character
Categories:20C99, 16G30
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