Expand all Collapse all | Results 76 - 100 of 421 |
76. CMB 2012 (vol 57 pp. 113)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie $\mathbb{D}$-parallel |
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)
On the $p$-norm of an Integral Operator in the Half Plane We give a partial answer to a conjecture of DostaniÄ 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)
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 $\aleph_1$. TodorÄeviÄ 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)
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)
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)
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)
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)
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)
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)
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)
Closed Left Ideal Decompositions of $U(G)$ Let $G$ be an infinite discrete group and let $\beta G$ be the
Stone--Äech compactification of $G$. We take the points of $Äta
G$ 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--Äech compactification, uniform ultrafilter, closed left ideal, decomposition Categories:22A15, 54H20, 22A30, 54D80 |
98. CMB 2011 (vol 56 pp. 326)
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)
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate |
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 FrÃ©chet
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 FrÃ©chet differentiability, representation Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50 |
100. CMB 2011 (vol 56 pp. 258)
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 |