1. CMB 2014 (vol 57 pp. 277)
||On Mutually $m$-permutable Product of Smooth Groups|
Let $G$ be a
finite group and $H$, $K$ two subgroups of G. A group $G$ is said to
be a mutually m-permutable product of $H$ and $K$ if $G=HK$ and
every maximal subgroup of $H$ permutes with $K$ and every maximal
subgroup of $K$ permutes with $H$. In this paper, we investigate the
structure of a finite group which is a mutually m-permutable product
of two subgroups under the assumption that its maximal subgroups are
Keywords:permutable subgroups, $m$-permutable, smooth groups, subgroup lattices
Categories:20D10, 20D20, 20E15, 20F16
2. CMB 2011 (vol 55 pp. 697)
||Constructions of Uniformly Convex Functions|
We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.
Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm
Categories:52A41, 46G05, 46N10, 49J50, 90C25
3. CMB 2009 (vol 52 pp. 342)
||On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width |
The X-ray numbers of some classes of convex bodies are investigated.
In particular, we give a proof of the X-ray Conjecture as well as
of the Illumination Conjecture for almost smooth convex bodies
of any dimension and for convex bodies of constant width of
dimensions $3$, $4$, $5$ and $6$.
Keywords:almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, X-ray number, X-ray Conjecture
Categories:52A20, 52A37, 52C17, 52C35
4. CMB 2007 (vol 50 pp. 356)
||Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities |
In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$-Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Keywords:$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem
Categories:35J20, 35J60, 35J85
5. CMB 2005 (vol 48 pp. 455)
||On GÃ¢teaux Differentiability of Convex Functions in WCG Spaces |
It is shown, using the Borwein--Preiss variational principle
that for every continuous convex function $f$ on
a weakly compactly generated space $X$,
every $x_0\in X$ and every weakly compact convex symmetric set $K$ such that
there is a point of G\^ateaux differentiability of $f$ in $x_0+K$.
This extends a Klee's result for separable spaces.
Keywords:GÃ¢teaux smoothness, Borwein--Preiss variational principle,, weakly compactly generated spaces
6. CMB 1998 (vol 41 pp. 497)
||On the construction of HÃ¶lder and Proximal Subderivatives |
We construct Lipschitz functions such that for all $s>0$ they are
$s$-H\"older, and so proximally, subdifferentiable only on dyadic
rationals and nowhere else. As applications we construct Lipschitz
functions with prescribed H\"older and approximate subderivatives.
Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationals
Categories:49J52, 26A16, 26A24