Given a domain $\Omega $ of class $C^{k,1}$, $k\in \Bbb N $, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of $\Omega $ in the sense that $(\partial- {\partial x_n})\alpha (x',0)= - N(x')$ and that still is of class $C^{k,1}$. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to $k$ on domains of class $C^{k,1}$. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.
The meadow spittlebug genus Philaenus (Auchenorrhyncha: Aphrophoridae) is known to display marked colour polymorphism. This study presents the results of a karyotype analysis of P. arslani from Lebanon using conventional chromosome staining, C-banding, fluorescent banding using base-specific fluorochromes (CMA3 and DAPI) and AgNOR-staining. This species has 2n = 18 + neo-XY, and differs from P. spumarius both in the number of chromosomes and sex chromosome system. During meiosis, the neo-XY bivalent is clearly heteromorphic being the largest in the complement. Furthermore, sex chromosomes show marked differences in C-banding pattern. The NOR-bearing chromosomes are the first and one of the middle-sized pairs of autosomes. NORs are G-C rich. Furthermore, some blocks of constitutive heterochromatin on the sex chromosomes are also G-C rich. All other C-bands are DAPI or DAPI/ CMA3 positive, thus containing A-T rich DNA. The significant difference in the karyotype of P. arslani and P. spumarius indicates chromosomal transformations during the evolution of the genus Philaenus.
Hagler and the first named author introduced a class of hereditarily $l_1$ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily $l_p$ Banach spaces for $1\leq p<\infty $. Here we use these spaces to introduce a new class of hereditarily $l_p(c_0)$ Banach spaces analogous of the space of Popov. In particular, for $p=1$ the spaces are further examples of hereditarily $l_1$ Banach spaces failing the Schur property.
In [5] and [10], statistical-conservative and $\sigma $-conservative matrices were characterized. In this note we have determined a class of statistical and $\sigma $-conservative matrices studying some inequalities which are analogous to Knopp’s Core Theorem.
In this paper, we study the limit properties of countable nonhomogeneous Markov chains in the generalized gambling system by means of constructing compatible distributions and martingales. By allowing random selection functions to take values in arbitrary intervals, the concept of random selection is generalized. As corollaries, some strong limit theorems and the asymptotic equipartition property (AEP) theorems for countable nonhomogeneous Markov chains in the generalized gambling system are established. Some results obtained are extended.