In this paper we introduce a modified version of Aleksandrov Theorem on
non-discrete Hausdorff locally compact groups. This also provides us a method to construct
Cantor type sets in any positive left Haar measure subset.
First we see that Ando’s inequality A ∗ B ≥ (A#B) ∗ (A#B) gives a
characterization of the geometric operator mean #, where ∗ is the Hadamard product.
Extending this, we discuss inequalities for operator means and the Hadamard product.
Moreover we show monotone convergence theorems including such inequalities.
A class of immigration superprocesses with dependent spatial motion for deterministic
immigration rate is considered, and we discuss a convergence problem for the rescaled
processes. When the immigration rate converges to a non-vanishing deterministic one, then
we can prove that under a suitable scaling, the rescaled immigration superprocesses associated
with SDSM converge to a class of immigration superprocesses associated with coalescing spatial
motion in the sense of probability distribution on the space of measure-valued continuous
paths. This scaled limit not only provides with a new class of superprocesses but also gives a
new type of limit theorem.
In this paper we consider the ruin probability for a storage process. This
process has two phases, the inflow phase and the outflow one, and the switchover of
which is controlled by a certain storage level. In these phases the storage increases
or decreases at each rate dependent on the present level. Furthermore the large scale
demand for the system may happen in both phases according to Poisson process. The
limiting probability distribution for the storage level and the ruin probability are given
by the solution of a system of renewal equations.
In this paper we show that the class of PC-algebras and the class of
B-algebras with condition (D) are Smarandache disjoint, and show that an algebra
(X; ∗, 0) is a Coxeter algebra if and only if it is a PC-algebra with (N). Moreover, we
show that there is no non-trivial quadratic PC-algebras on a field with |X| ≥ 3.
The notion of BCK/BCI-bialgebras and sub-bialgebras is introduced, and
related properties are investigated. A characterization of X = pI(X1) pI(X2) is
provided.
The notion of essences in subtraction algebras is introduced, and many
properties are investigated. Relations among subalgebras, ideals and essences are given.
Homomorphic image and inverse image are considered.
The Smarandache structure of generalized BCK-algebras is considered.
Several examples of a qS-gBCK-algebra are provided. The notion of SΩ-ideals and
qSΩ-ideals is introduced, and related properties are investigated.