401. CMB 2007 (vol 50 pp. 474)
 Zhou, Jiazu

On Willmore's Inequality for Submanifolds
Let $M$ be an $m$ dimensional submanifold in the Euclidean space
${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain
some low geometric estimates of the total square mean curvature
$\int_M H^2 d\sigma$. The low bounds are geometric invariants
involving the volume of $M$, the total scalar curvature of $M$,
the Euler characteristic and the circumscribed ball of $M$.
Keywords:submanifold, mean curvature, kinematic formul, scalar curvature Categories:52A22, 53C65, 51C16 

402. CMB 2007 (vol 50 pp. 447)
 Śniatycki, Jędrzej

Generalizations of Frobenius' Theorem on Manifolds and Subcartesian Spaces
Let $\mathcal{F}$ be a family of vector fields on a manifold or a
subcartesian space spanning a distribution $D$. We prove that an orbit $O$
of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$
and it has constant rank on $O$. This result implies Frobenius' theorem, and
its various generalizations, on manifolds as well as on subcartesian spaces.
Keywords:differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theorem Categories:58A30, 58A40 

403. CMB 2007 (vol 50 pp. 434)
 Õzarslan, M. Ali; Duman, Oktay

MKZ Type Operators Providing a Better Estimation on $[1/2,1)$
In the present paper, we introduce a modification of the MeyerK\"{o}nig and
Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and
$f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation
than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with
respect to the modulus of continuity and the Lipschitz class functionals.
Furthermore, we present the $r$th order generalization of our operators and
study their approximation properties.
Keywords:MeyerKÃ¶nig and Zeller operators, Korovkin type approximation theorem, modulus of continuity, Lipschitz class functionals Categories:41A25, 41A36 

404. CMB 2007 (vol 50 pp. 206)
 Golasiński, Marek; Gonçalves, Daciberg Lima

Spherical Space Forms: Homotopy Types and SelfEquivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$
Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$dimensional
$CW$complex of the homotopy type of an $n$sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$actions on all $CW$complexes $X(2dn1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn1)/\mu$ associated with
free and cellular $G$actions $\mu$ on $X(2dn1)$ are determined as
well.
Keywords:automorphism group, $CW$complex, free and cellular $G$action, group of self homotopy equivalences, LyndonHochschildSerre spectral sequence, special (linear) group, spherical space form Categories:55M35, 55P15, 20E22, 20F28, 57S17 

405. CMB 2007 (vol 50 pp. 284)
 McIntosh, Richard J.

Second Order Mock Theta Functions
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $q<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
Keywords:$q$series, mock theta function, Mordell integral Categories:11B65, 33D15 

406. CMB 2007 (vol 50 pp. 234)
 Kuo, Wentang

A Remark on a Modular Analogue of the SatoTate Conjecture
The original SatoTate Conjecture concerns the angle distribution
of the eigenvalues arising from nonCM elliptic curves. In this paper,
we formulate a modular analogue of the SatoTate Conjecture and prove
that the angles arising from nonCM holomorphic Hecke
eigenforms with nontrivial central characters are not distributed
with respect to the SateTate measure
for nonCM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.
Keywords:$L$functions, Elliptic curves, SatoTate Categories:11F03, 11F25 

407. CMB 2007 (vol 50 pp. 85)
 Han, Deguang

Classification of Finite GroupFrames and SuperFrames
Given a finite group $G$, we examine the classification of all
frame representations of $G$ and the classification of all
$G$frames, \emph{i.e.,} frames induced by group representations of $G$.
We show that the exact number of equivalence classes of $G$frames
and the exact number of frame representations can be explicitly
calculated. We also discuss how to calculate the largest number
$L$ such that there exists an $L$tuple of strongly disjoint
$G$frames.
Keywords:frames, groupframes, frame representations, disjoint frames Categories:42C15, 46C05, 47B10 

408. CMB 2007 (vol 50 pp. 149)
 Śliwa, Wiesław

On Quotients of NonArchimedean KÃ¶the Spaces
We show that there exists a nonarchimedean
Fr\'echetMontel space $W$ with a basis and with a continuous norm
such that any nonarchimedean Fr\'echet space of countable type is isomorphic
to a quotient of $W$. We also prove that any nonarchimedean nuclear
Fr\'echet space is isomorphic to a quotient of some nonarchimedean nuclear
Fr\'echet space with a basis and with a continuous norm.
Keywords:Nonarchimedean KÃ¶the spaces, nuclear FrÃ©chet spaces, pseudobases Categories:46S10, 46A45 

409. CMB 2007 (vol 50 pp. 113)
 Li, ZhenYang; Zhang, Xi

Hermitian Harmonic Maps into Convex Balls
In this paper, we consider Hermitian harmonic maps from
Hermitian manifolds into convex balls. We prove that there exist
no nontrivial Hermitian harmonic maps from closed Hermitian
manifolds into convex balls, and we use the heat flow method to
solve the Dirichlet problem for Hermitian harmonic maps when the
domain is a compact Hermitian manifold with nonempty boundary.
Keywords:Hermitian harmonic map, Hermitian manifold, convex ball Categories:58E15, 53C07 

410. CMB 2007 (vol 50 pp. 11)
 Borwein, David; Borwein, Jonathan

van der Pol Expansions of LSeries
We provide concise series representations for various
Lseries integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying Lseries.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, Lseries Categories:11M35, 11M41, 30B50 

411. CMB 2006 (vol 49 pp. 536)
412. CMB 2006 (vol 49 pp. 624)
 Teragaito, Masakazu

On NonIntegral Dehn Surgeries Creating NonOrientable Surfaces
For a nontrivial knot in the $3$sphere,
only integral Dehn surgery can create a closed $3$manifold containing a projective plane.
If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true.
In contrast to these, we show that nonintegral surgery on a hyperbolic knot
can create a closed nonorientable surface of any genus greater than two.
Keywords:knot, Dehn surgery, nonorientable surface Category:57M25 

413. CMB 2006 (vol 49 pp. 560)
 Luijk, Ronald van

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain nonKummer, singular
K3 surface
are dense. We will also compute the entire NÃ©ronSeveri group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ronSeveri group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 

414. CMB 2006 (vol 49 pp. 481)
 Browkin, J.; Brzeziński, J.

On Sequences of Squares with Constant Second Differences
The aim of this paper is to study sequences of integers
for which the second differences between their squares are
constant. We show that there are infinitely many nontrivial
monotone sextuples having this property and discuss some related
problems.
Keywords:sequence of squares, second difference, elliptic curve Categories:11B83, 11Y85, 11D09 

415. CMB 2006 (vol 49 pp. 371)
 Floricel, Remus

Inner $E_0$Semigroups on Infinite Factors
This paper is concerned with the structure of
inner $E_0$semigroups. We show that any inner
$E_0$semigroup acting on an infinite factor
$M$ is completely determined by a continuous
tensor product system of Hilbert spaces in
$M$ and that the product system associated
with an inner $E_0$semigroup is a complete cocycle conjugacy invariant.
Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacy Categories:46L40, 46L55 

416. CMB 2006 (vol 49 pp. 472)
417. CMB 2006 (vol 49 pp. 358)
 Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed

On the Principal Eigencurve of the $p$Laplacian: Stability Phenomena
We show that each point of the principal eigencurve of the
nonlinear problem
$$
\Delta_{p}u\lambda m(x)u^{p2}u=\muu^{p2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.
Keywords:$p$Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability Categories:35P30, 35P60, 35J70 

418. CMB 2006 (vol 49 pp. 203)
 Çömez, Doğan

The Ergodic Hilbert Transform for Admissible Processes
It is shown that the ergodic Hilbert transform
exists for a class of bounded symmetric admissible processes
relative to invertible measure preserving transformations. This
generalizes the wellknown result on the existence of the ergodic
Hilbert transform.
Keywords:Hilbert transform, admissible processes Categories:28D05, 37A99 

419. CMB 2006 (vol 49 pp. 281)
 Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.

Correction to a Theorem on Total Positivity
A wellknown theorem states that if $f(z)$ generates a PF$_r$
sequence then $1/f(z)$ generates a PF$_r$ sequence. We give two
counterexamples
which show that this is not true, and give a correct version of the theorem.
In the infinite limit the result is sound: if $f(z)$ generates a PF
sequence then $1/f(z)$ generates a PF sequence.
Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur function Categories:15A48, 15A45, 15A57, 05E05 

420. CMB 2006 (vol 49 pp. 270)
421. CMB 2006 (vol 49 pp. 265)
 Nicholson, W. K.; Zhou, Y.

Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial
If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in
C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$clean if every element is
the sum of a unit and a root of $g(x)$. If $V$ is a vector space of
countable dimension over a division ring $D,$ they showed that
$\end {}_{D}V$ is
$g(x)$clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=xx^{2}$
this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan.
In this paper we remove the countable condition, and in fact prove that
$\Mend {}_{R}M$ is $g(x)$clean for any semisimple module $M$ over an arbitrary
ring $R$ provided that $g(x)\in (xa)(xb)C[x]$ where $a,b\in C$ and both $b$
and $ba$ are units in $R$.
Keywords:Clean rings, linear transformations, endomorphism rings Categories:16S50, 16E50 

422. CMB 2006 (vol 49 pp. 256)
 Neelon, Tejinder

A BernsteinWalsh Type Inequality and Applications
A BernsteinWalsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) BochnakSiciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) ZornLelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) AbhyankarMohSathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:BernsteinWalsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 

423. CMB 2006 (vol 49 pp. 247)
424. CMB 2006 (vol 49 pp. 213)
425. CMB 2006 (vol 49 pp. 185)
 Averkov, Gennadiy

On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the RogersShephard inequality we obtain that
$\binom{2d}{d}{}^{1} \le \alpha(B)/V(B) \le 2^{d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a crosspolytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified BanachMazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, BanachMazur compactum, (modified) BanachMazur distance, volume ratio Categories:52A40, 46B20 
