In this paper, we establish the generalized discriminant theorem governing uniqueness of symmetric polynomials in n variables of least total degree dn, k which vanish when any k + 1 of the variables are equal.

In this paper, a class of generalized implicit inclusion problems is introduced, which can be regarded as a generalization of variational inequality problems, equilibrium problems, optimization problems and inclusion problems. Some existence results of solutions for such problems are obtained on noncompact subsets of Hausdorff topological vector spaces using the famous FKKM theorem. As applications, some existence results for vector equilibrium problems and vector variational inequalities on noncompact sets of Hausdorff topological vector spaces are given.

A Banach space X has the asymptotic-norming property if and only if the Lebesgue-Bochner function space Lp (μ, X) has the asumptotic-norming property for p with 1 < p < ∞. It follows that a Banach space X is Hahn-Banach smooth if and only if Lp (μ, X) is Hahn-Banach smooth for p with 1 < p < ∞. We also show that for p with 1 < p < ∞, (1) if X has the compact Mazur intersection property then so does Lp(μ, X); (2) if the measure μ is not purely atomic, then the space Lp(μ, X) has the Mazur intersection property if and only if X is an Asplund space and has the Mazur intersection property.

A foliation on a Riemannian manifold (M, g) is harmonic if all the leaves are minimal submanifolds. We give a new characterisation of the harmonicity of a foliation on (M, g) by the harmonicity of an associated bundle map of (T M, gC), where gC is the complete lift metric of g to the tangent bundle as introduced by Yano and Ishihara.

We characterise in terms of free topological groups those Banach-Lie algebras with finite-dimensional centre coming from Lie groups.

It is shown that there exists a metric under which a diffeomorphism f on a Riemannian manifold M becomes an isometry, provided that the dynamical system generated by f is of characteristic 0± and all its orbits are closed. Furthermore, it is shown that the foliation given by the suspension of f is parallel in this case.

A power-associative ring A is called a p-ring provided there exists a prime p so that for every x in A, xp = x and px = 0. It is shown that if A is such a ring with p ≠ 2, then A is isomorphic to a subdirect sum of copies of GF(p), the Galois field with p elements.

It is proved that, for any m given distinct real numbers a1, …, am, there exist transcendental entire functions f(z) at most of order m for which all the values

are rational integers.

In this paper I prove that if a semigroup S is stable then ∧L(S) and ∧R(S) (the Rhodes expansions), and ∧+(SA) (the iteration of those expansions) are also stable. I also prove that if S is stable and has a J-depth function then these expansions also have a J-depth functon. More generally, if X →→ S is a J*-surmorphism and if S is stable and has a J-depth function then X has a J-depth function. All these results are needed for the structure theory of semigroups which are stable and have a J-depth function.

The techniques used were originally developed by the author to prove that ∧+(SA) is finite if S is finite (later Rhodes found a much more direct proof of that result).

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

We give conditions under which pointwise multiplication is a continuous bounded operation on kth order Sobolev-Orlicz spaces. This result is used to derive a sufficient condition under which the superposition operator is a continuous bounded operator on these spaces.

For an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising in the theory of stochastic processes and other areas.

A multi-dimensional analogue of a well known inequality of Littlewood and Paley is obtained on the ball.

We give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.

Let G = (X, U) be a digraph of order n. P(G) denotes the minimal cardinal of a path-partition of the arcs of G.

Oystein Ore, Theory of graphs (Amer. Math. Soc, Providence, Rhode Island, 1962) has proved that , where . We say that G satisfies Q if the preceeding inequality is an equality.

We give some properties of the digraphs satisfying Q, and in particular the case where G is strongly connected. Then we prove that P(G) ≤ [n2/4], and that this result is the best possible. Next we exhibit the existence of digraphs with circuits such that P(G) = [n2/4].

Finally we prove that if G is a strongly connected digraph of order n which satisfies Q, then there exists a strongly connected digraph H of order n + 1 having G as a sub-digraph and satisfying Q, too.

Let R be a ring. A right R-module M is called an SC-module if every M-singular right R-module is continuous. The purpose of this note is to give some characterisations of SC-modules.

A number of Banach space properties have been shown to imply the weak fixed point property. If the dual of a Banach space were to possess some of these properties then the original space can been shown to satisfy related conditions.

We consider the sum $\sum 1/\gamma $ , where $\gamma $ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$ , and examine its behaviour as $T \to \infty $ . We show that, after subtracting a smooth approximation $({1}/{4\pi }) \log ^2(T/2\pi ),$ the sum tends to a limit $H \approx -0.0171594$ , which can be expressed as an integral. We calculate H to high accuracy, using a method which has error $O((\log T)/T^2)$ . Our results improve on earlier results by Hassani [‘Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function’, Appl. Math. E-Notes16 (2016), 109–116] and other authors.

Let A = {ai} be a finite set of integers and let p, m denote the orders of A + A = {ai+aj} and A – A = {ai–aj} respectively. J.H. Conway conjectured that p ≤ m always and that p = m only if A is symmetric about 0. This conjecture has since been disproved; here we make several other observations on the values of p and m.