We investigate the maximal numberMk of offspring amongst all individuals in a critical Galton–Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index −α for 1 < α < 2, then k−1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Uryson, initially ω1-compact and non-compact space X of size ω2 having the following property: for every open (or closed) set A in X we have |A| ≤ ω1 or |X \A| ≤ ω1.

We consider Toeplitz operators with symbols enjoying a uniform radial limit on Segal–Bargmann type spaces. We show that such an operator is compact if and only if the limiting function vanishes on the unit sphere. The structure of the C∗-algebra generated by Toeplitz operators whose symbols admit continuous uniform radial limits is also analyzed.

We investigate the planar quasi-septic non-analytic systems which have a center-focus equilibrium at the origin and whose angular speed is constant. The system could be changed into an analytic system by two transformations, with the help of computer algebra system MATHEMATICA, the conditions of uniform isochronous center are obtained. Keywords—Non-analytic; Center–focus problem; Lyapunov const...

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

Data and computation alignment is an important part of compiling sequential programs to architectures with non uniform memory access times In this paper we show that elementary matrix methods can be used to determine communication free alignment of code and data We also solve the problem of replicating data to eliminate communication Our matrix based approach leads to algorithms which work well...

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any scheme, and D(G) ring kG, we show self-extension algebra trivial representation a algebra, each D(G)-representation V extensions from form module over aforementioned algebra. As corollary, find all categories \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual \({{\,\ma...

We study the range transformations 0p(/fD, Reß) and Op(/fD, B) for Banach function algebras A and B. As a special instance, the harmonicity of functions in Op(/fD, Re A) for a nontrivial function algebra A is established and is compared with previous investigations of Op( An, A) and Op((Re/l);, (Re/1)) for an interval /. In §2 we present some results on Op(AD, B) and use them to show that funct...