In this survey article we consider equations related to the minimal surface equation div Tu = 0, where Tu = ∇u √1+|∇u|2 , ∇u is the gradient of u, and derive some structural inequalities related to the vector function Tu. These structural inequalities give rise to striking uniqueness properties of the solutions.

In the past quarter century white noise theory has made great progress. We are further given many interesting and important questions which should be discussed in line with white noise analysis. On the other hand, some of the new directions are before our eyes and they require new analytic methods to be investigated, by generalizing the method established so far. It is now the time to remind the original idea of our stochastic analysis and to think of further directions of this theory which are going to appear in front of us. While we are reviewing the results, we have recognized the foundation of the theory should be refined. This will be done also in Part I. The plan of Part II is as follows. From what have been established, there naturally arise several new topics to be discussed. This is, in a sense, a continuation of Part I. On the other hand, having suggested by many significant examples and problems in applications, we come to a framework of the new white noise theory which is going to be constructed. A significant development is to grow out of L2-theory and to establish an analysis on a wider class of suitable functionals of sample paths. This approach will be proposed.

We consider the existence and nonexistence of global solutions of the following initial-boundary value problem for a system of nonlinear wave equations in a domain Ω × [0, T): ½ utt − Δu +m21 u = −4λ(u + αv)3 − 2βuv2, vtt − Δv +m22 v = −4αλ(u + αv)3 − 2βu2v, where Ω is a bounded domain in R3 with a smooth boundary. Some sufficient conditions on the given parameters λ, α and β for the global existence and blow-up are imposed. The estimates for the lifespan of solutions is given.

We show that the compactness of the difference of two cosine functions can be characterized by that of the difference of their resolvents. We also make some comparisons between such results for C0-semigroups and cosine functions.

The authors consider four-point boundary value problem for a fourth order ordinary differential equations of the form (φp(u00(t)))00 (E) = a(t)f(u(t)), t ∈ (0, 1), with one of the following boundary conditions (B1) u(0) − λu0(η) = u0(1) = 0, u000(0) = α1u000(ξ), u00(1) = β1u00(ξ), or (B2) u(1) + λu0(η) = u0(0) = 0, u000(0) = α1u000(ξ), u00(1) = β1u00(ξ). They impose growth conditions on f which guarantee existence of at least two positive solutions for the problems (E) − (B1) and (E) − (B2).

In this paper we initiate the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator algebras.

With tame transformation method (TTM), T. Moh invented a cryp- tosystem. The success of the system relies on the construction of the Qk component. Some constructions of Qk components are known but most of the Qk components have square terms. There are some possible risks for square terms. In this paper, we give a systematic construction of square-free Qk components to avoid the possible attacks.

Given an infinite-dimensional Banach space X, we introduce the daugavetian index of X, daug(X), as the greatest constant m > 0 such that kId + Tk > 1 +mkTk for all T ∈ K(X). We givetwo characterizations of this index and we estimate it in some examples. We show that the daugavetian index of a c0-, l1- or l∞-sum of Banach spaces is the infimum index of the summands. Finally, we calculate the daugavetian index of some vector-valued function spaces: daug ¡ C(K,X) ¢ ¡ resp. daug ¡ L1(µ,X) ¢ , daug ¡ L∞(µ,X) ¢¢ is the maximum of daug (X) and daug (C(K)) resp. daug (L1(µ)), daug(L∞(µ)).

We prove the generalized Hyers-Ulam-Rassias stability of general- ized Jensen’s equations in Banach modules over a unital C∗-algebra associ- ated with its unitary group. It is applied to show the stability of generalized Jensen’s equations in a Hilbert module over a unital C∗-algebra associated with its unitary group.

A graph G is decomposable into subgraphs G1, G2, . . . , Gn of G if no Gi (i = 1, 2, . . . , n) has isolated vertices and the edge set E(G) can be partitioned into the subsets E(G1), E(G2), . . . , E(Gn). If Gi ∼= P4 for all i, then G is called P4-decomposable. In this paper, we show the P4-decomposability of some classes of graphs, and prove in particular that a complete r-partite graph is P4-decomposable if and only if its size is a multiple of 3. We also give an example of a 2-connected graph of size 3k which is not P4-decomposable, disproving a conjecture of Chartrand.

In this note the authors establish several results concerning the uniform convergence of series in vector-valued sequence spaces. Corollaries include sufficient conditions for the weak sequential completeness of β-duals of sequence spaces, versions of the Uniform Boundedness Theorem and the Banach-Steinhaus Theorem for elements of operator-valued β-duals, and a characterization of weakly convergent sequences in β-duals. A further appli- cation establishes a vector-valued version of the Hahn-Schur Lemma.