Abstract:
A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the $n$-sphere then (i) their intersection is empty; (ii) the intersection of any proper subfamily of them is non-empty. In the case of caps (i) and (ii) are also sufficient for the family to be a covering of the sphere.

Abstract:
By using some lattice-like operations which constitute extensions of ones introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E. A. Ok.

Abstract:
By using the metric projection onto a closed self-dual cone of the Euclidean space, M. S. Gowda, R. Sznajder and J. Tao have defined generalized lattice operations, which in the particular case of the nonnegative orthant of a Cartesian reference system reduce to the lattice operations of the coordinate-wise ordering. The aim of the present note is twofold: to give a geometric characterization of the closed convex sets which are invariant with respect to these operations, and to relate this invariance property to the isotonicity of the metric projection onto these sets. As concrete examples the Lorentz cone and the nonnegative orthant are considered. Old and recent results on closed convex Euclidean sublattices due to D. M. Topkis, A. F. Veinott and to M. Queyranne and F. Tardella, respectively are obtained as particular cases. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac, H. Nishimura and E. A. Ok.

Abstract:
While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors of the present paper it has been shown that these lattice-like operators and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by metods for proving the existence of solutions of variational inequalities and methods for finding these solutions in a recursive way. It turns out, that the closed convex sets admitting isotone projections are exactly the sets which are invariant with respect to these lattice-like operations, called lattice-like sets. In this paper it is shown that the Jordan subalgebras are lattice-like sets, but the converse in general is not true. In the case of simple Euclidean Jordan algebras of rank at least three the lattice-like property is rather restrictive, e.g., there are no lattice-like proper closed convex sets with interior points.

Abstract:
The note describes the cones in the Euclidean space admitting isotonic metric projection with respect to the coordinate-wise ordering. As a consequence it is showed that the metric projection onto the regression cone (the cone defined by the general isotonic regression problem) admits a projection which is isotonic with respect to the coordinate-wise ordering.

Abstract:
The metric projection onto an order nonnegative cone from the metric projection onto the corresponding order cone is derived. Particularly, we can use Pool Adjacent Violators-type algorithms developed for projecting onto the monotone cone for projecting onto the monotone nonnegative cone too.

Abstract:
A wedge (i.e., a closed nonempty set in the Euclidean space stable under addition and multiplication with non-negative scalars) induces by a standard way a semi-order (a reflexive and transitive binary relation) in the space. The wedges admitting isotone metric projection with respect to the semi-order induced by them are characterized. The obtained result is used to show that the monotone wedge (called monotone cone in regression theory) admits isotone projection.

Abstract:
A very fast heuristic iterative method of projection on simplicial cones is presented. It consists in solving two linear systems at each step of the iteration. The extensive experiments indicate that the method furnishes the exact solution in more then 99.7 percent of the cases. The average number of steps is 5.67 (we have not found any examples which required more than 13 steps) and the relative number of steps with respect to the dimension decreases dramatically. Roughly speaking, for high enough dimensions the absolute number of steps is independent of the dimension.

Abstract:
Intracellular calcium level has a definite role in innate and adaptive immune signaling but its evolutionary aspects are not entirely clear yet. Very few information are accessible about calcium contents of invertebrate immunocytes, especially of celomocytes, the effector cells of earthworm immunity. Different basal and induced Ca2+ levels characterize the various celomocyte subgroups. Intracellular calcium is mostly located in the endoplasmic reticulum and celomocytes exert intracellular Ca2+ ATPase activity to maintain their calcium homeostasis. Immune molecules such as phytohemagglutinin and the chemoattractant fMLP caused the elevation of intracellular Ca2+ level in celomocytes. All the evidence suggests that Ca2+ influx may play a crucial role in the signal transduction of the earthworm’s innate immunity.