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76. CMB 2012 (vol 57 pp. 113)

Madras, Neal
A Lower Bound for the End-to-End Distance of Self-Avoiding Walk
For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists.

Keywords:self-avoiding walk, critical exponent
Categories:82B41, 60D05, 60K35

77. CMB 2012 (vol 56 pp. 844)

Shparlinski, Igor E.
On the Average Number of Square-Free Values of Polynomials
We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ for some fixed $\varepsilon\gt 0$. Individual results of this kind for polynomials of degree $k \gt 3$, due to A. Granville (1998), are only known under the $ABC$-conjecture.

Keywords:polynomials, square-free numbers
Category:11N32

78. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities
Categories:47A63, 15A45

79. CMB 2012 (vol 56 pp. 850)

Teragaito, Masakazu
Left-orderability and Exceptional Dehn Surgery on Twist Knots
We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a $3$-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.

Keywords:left-ordering, twist knot, Dehn surgery
Categories:57M25, 06F15

80. CMB 2012 (vol 56 pp. 466)

Aulaskari, Rauno; Rättyä, Jouni
Inclusion Relations for New Function Spaces on Riemann Surfaces
We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, BMOA or the recently defined $Q_p$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.

Keywords:Bloch space, BMOA, $Q_p$, Green's function, hyperbolic Riemann surface
Categories:30F35, 30H25, 30H30

81. CMB 2012 (vol 56 pp. 503)

Bu, Qingying
Weak Sequential Completeness of $\mathcal K(X,Y)$
For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either $X^\ast$ or $Y$ has the bounded compact approximation property.

Keywords:weak sequential completeness, reflexivity, compact operator space
Categories:46B25, 46B28

82. CMB 2012 (vol 56 pp. 723)

Bérczes, Attila; Luca, Florian
On the Sum of Digits of Numerators of Bernoulli Numbers
Let $b\gt 1$ be an integer. We prove that for almost all $n$, the sum of the digits in base $b$ of the numerator of the Bernoulli number $B_{2n}$ exceeds $c\log n$, where $c:=c(b)\gt 0$ is some constant depending on $b$.

Keywords:Bernoulli numbers, sums of digits
Category:11B68

83. CMB 2012 (vol 56 pp. 520)

Elbasraoui, Abdelkrim; Sebbar, Abdellah
Equivariant Forms: Structure and Geometry
In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of $\operatorname{SL}_2(\mathbb{Z})$ by means of the cross-ratio, the weight 2 modular forms, the quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

Keywords:equivariant forms, modular forms, Schwarz derivative, cross-ratio, differential forms
Category:11F11

84. CMB 2012 (vol 56 pp. 769)

Lahiri, Indrajit; Kaish, Imrul
A Non-zero Value Shared by an Entire Function and its Linear Differential Polynomials
In this paper we study uniqueness of entire functions sharing a non-zero finite value with linear differential polynomials and address a result of W. Wang and P. Li.

Keywords:entire function, linear differential polynomial, value sharing
Category:30D35

85. CMB 2012 (vol 56 pp. 491)

Bahmanpour, Kamal
A Note on Homological Dimensions of Artinian Local Cohomology Modules
Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring (with identity), $M$ be a non-zero finitely generated $R$-module. In this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that $ \operatorname{{\rm injdim_{_{R_{\frak p}}}}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and ${\rm fd}_{R_{\p}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ are bounded from above by $ \operatorname{{\rm injdim_{_{R}}}} H^i_{\frak m}(M)$ and $ {\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.

Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology
Category:13D45

86. CMB 2012 (vol 56 pp. 544)

Gauthier, P. M.
Universally Overconvergent Power Series via the Riemann Zeta-function
The Riemann zeta-function is employed to generate universally overconvergent power series.

Keywords:overconvergence, zeta-function
Categories:30K05, 11M06

87. CMB 2011 (vol 56 pp. 306)

Pérez, Juan de Dios; Suh, Young Jin
Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel
We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition.

Keywords:complex projective space, real hypersurface, structure Jacobi operator
Categories:53C15, 53C40

88. CMB 2011 (vol 56 pp. 593)

Liu, Congwen; Zhou, Lifang
On the $p$-norm of an Integral Operator in the Half Plane
We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

Keywords:Bergman projection, integral operator, $L^p$-norm, the upper half plane
Categories:47B38, 47G10, 32A36

89. CMB 2011 (vol 56 pp. 317)

Dorais, François G.
A Note on Conjectures of F. Galvin and R. Rado
In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H \subseteq G$ with size $\aleph_1$. Todorčević has shown that Rado's Conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's Conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal.

Keywords:Galvin conjecture, Rado conjecture, perfect graph, comparability graph, chordal graph, clique-cover number, chromatic number
Categories:03E05, 03E35, 03E55

90. CMB 2011 (vol 56 pp. 225)

Agashe, Amod
On the Notion of Visibility of Torsors
Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$. Let $x$ be an element of $H^1(\mathbf{Q},A)$. Then there are at least two definitions of $x$ being visible in $J$: one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$ to a subvariety of $J$, and the other asks that $x$ be in the kernel of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we clarify the relation between the two definitions.

Keywords:torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group
Categories:11G35, 14G25

91. CMB 2011 (vol 56 pp. 184)

Shen, Zhongmin
On Some Non-Riemannian Quantities in Finsler Geometry
In this paper we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the S-curvature. We show some relationships among the flag curvature, the S-curvature, and the new non-Riemannian quantity.

Keywords:Finsler metric, S-curvature, non-Riemannian quantity
Categories:53C60, 53B40

92. CMB 2011 (vol 56 pp. 39)

Ben Amara, Jamel
Comparison Theorem for Conjugate Points of a Fourth-order Linear Differential Equation
In 1961, J. Barrett showed that if the first conjugate point $\eta_1(a)$ exists for the differential equation $(r(x)y'')''= p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first systems-conjugate point $\widehat\eta_1(a)$. The aim of this note is to extend this result to the general equation with middle term $(q(x)y')'$ without further restriction on $q(x)$, other than continuity.

Keywords:fourth-order linear differential equation, conjugate points, system-conjugate points, subwronskians
Categories:47E05, 34B05, 34C10

93. CMB 2011 (vol 56 pp. 412)

Sanders, T.
Structure in Sets with Logarithmic Doubling
Suppose that $G$ is an abelian group, $A \subset G$ is finite with $|A+A| \leq K|A|$ and $\eta \in (0,1]$ is a parameter. Our main result is that there is a set $\mathcal{L}$ such that \begin{equation*} |A \cap \operatorname{Span}(\mathcal{L})| \geq K^{-O_\eta(1)}|A| \quad\text{and}\quad |\mathcal{L}| = O(K^\eta\log |A|). \end{equation*} We include an application of this result to a generalisation of the Roth--Meshulam theorem due to Liu and Spencer.

Keywords:Fourier analysis, Freiman's theorem, capset problem
Category:11B25

94. CMB 2011 (vol 56 pp. 510)

Dubickas, Artūras
Linear Forms in Monic Integer Polynomials
We prove a necessary and sufficient condition on the list of nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form $u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in \mathbb{Z}[x]$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials $f_1,\dots,f_k$ can be chosen to be irreducible in $\mathbb{Z}[x]$.

Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion
Categories:11R09, 11C08, 11B83

95. CMB 2011 (vol 56 pp. 395)

Oancea, D.
Coessential Abelianization Morphisms in the Category of Groups
An epimorphism $\phi\colon G\to H$ of groups, where $G$ has rank $n$, is called coessential if every (ordered) generating $n$-tuple of $H$ can be lifted along $\phi$ to a generating $n$-tuple for $G$. We discuss this property in the context of the category of groups, and establish a criterion for such a group $G$ to have the property that its abelianization epimorphism $G\to G/[G,G]$, where $[G,G]$ is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews--Curtis conjecture.

Keywords:coessential epimorphism, Nielsen transformations, Andrew-Curtis transformations
Categories:20F05, 20F99, 20J15

96. CMB 2011 (vol 56 pp. 500)

Browning, T. D.
The Lang--Weil Estimate for Cubic Hypersurfaces
An improved estimate is provided for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

Keywords:cubic hypersurface, rational points, finite fields
Categories:11G25, 14G15

97. CMB 2011 (vol 56 pp. 442)

Zelenyuk, Yevhen
Closed Left Ideal Decompositions of $U(G)$
Let $G$ be an infinite discrete group and let $\beta G$ be the Stone--Čech compactification of $G$. We take the points of $ėta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U(G)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show that if $|G|$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the finest decomposition of $U(G)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U(G)$ is Hausdorff.

Keywords:Stone--Čech compactification, uniform ultrafilter, closed left ideal, decomposition
Categories:22A15, 54H20, 22A30, 54D80

98. CMB 2011 (vol 56 pp. 326)

Erdoğan, M. Burak; Oberlin, Daniel M.
Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$
We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms of some fractal measures.

Keywords:Fourier transforms of fractal measures, Fourier restriction
Categories:42B10, 28A12

99. CMB 2011 (vol 56 pp. 272)

Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

Keywords:super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

100. CMB 2011 (vol 56 pp. 258)

Chandoul, A.; Jellali, M.; Mkaouar, M.
The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
Dufresnoy and Pisot characterized the smallest Pisot number of degree $n \geq 3$ by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot's result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element (SPE) of degree $n$ in the field of formal power series over a finite field is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$ is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$ (as a finite total ordered set). We prove that the sequence of SPEs of degree $n$ is decreasing and converges to $\alpha X.$ Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

Keywords:Pisot element, continued fraction, Laurent series, finite fields
Categories:11A55, 11D45, 11D72, 11J61, 11J66
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