In this paper we study the oscillatory behaviour of solutions of equations of the forms , where pi, and τi, for i = 1, 2,…, n, are positive constants and p is an arbitary constant. Three necessary and sufficient conditions for all solutions of (A) and (B) respectively to be oscillatory are established. These results extend and generalise previous results in a new approach.

We study the fractional parabolic Lichnerowicz equation

$$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$

$p>0$

$ 0<s<1 $

$p>1$

$0<p\leq 1$

$$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$

$p>0$

$0<s<1$

A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra. Any tree vith the descending chain condition has a valuation in the natural numbers, considered as a commutative BCK-algebra; the valuation is the height-function. Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure. The finite trees with at most one atom and height at most n are precisely the finitely generated subdirectly irreducible (simple) algebras in the subvariety of commutative BCK-algebras which satisfy the identity (En): xyn = xyn+1. Due to congruence-distributivity, it is then possible to describe the associated lattice of subvarieties.

We prove that most group algebras of free products have left zero divisors that are not right zero divisors.

A graph Γ is said to be a G-locally primitive graph, for G ≥ Aut Γ, if for every vertex, α, the stabiliser Gα induces a primitive permutation group on Γ (α) the set of vertices adjacent to α. In 1978 Richard Weiss conjectured that there exists a function f: ℕ →ℕ such that for any finite connected vertex-transitive G-locally primitive graph of valency d and a vertex α of the graph, |Gα| ≥ f(d). The purpose of this paper is to prove that, in the case Soc(G) = Sz(q), the conjecture is true.

A consideration of the separation properties of pre-neighbourhood lattices, leads to the definition and characterisation of T1-neighbourhood lattices in terms of the properties of the neighbourhood mapping, independently of points. It is then shown that if net convergence is defined in neighbourhood lattices as a consequence of replacing ‘point’ by ‘set’ in topological convergence, then the limits of convergent nets are unique. The relationship between continuity and convergence is established with the proof of the statement that a residuated function between conditionally complete T1-neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. If P(X) denotes the power set of X, then the observation that a filter in a topological space (X, T) is a net in P(X) leads to a discussion of the net convergence of a filter as a special case of net convergence. Particular attention is paid to maximal filters, Fréchet filters and to the filter generated by the limit elements of a net. Further, if the ‘filter’ convergence of a filter F in a topological space (X, T) is given by , if η(x) ⊆ F, then the relationship between ‘filter’ convergence and the net convergence of a filter in P(X) is established. Finally, it is proved that, in the neighbourhood system ‘lifted’ from a topological space to P(P(X)), the continuous image of a filter which converges to a singleton set is a convergent filter with the appropriate image set as the limit.

The isodiametric problem in the Euclidean plane is solved for bounded convex sets, which are symmetric about the origin, and which contain no interior non-zero point of an arbitrary lattice L.

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$.

In this note we consider Hermite-Fejér interpolation at the zeros of Jacobi polynomials and with additional boundary conditions. For the associated Hermite-Fejér type operators and special values of α, β it was proved by the first author in recent papers that one has uniform convergence on the whole interval [−1,1]. The second author could show by introducing the concept of asymptotic positivity how to get the known convergence results for the classical Hermite-Fejér interpolation operators. In the present paper we show, using a slightly modified Bohman-Korovkin theorem for asymptotically positive functionals, that the Hermite-Fejér type interpolation polynomials , converge pointwise to f for arbitrary α, β > −1. The convergence is uniform on [−1 + δ,1 − δ].

We give the generating function of split $(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split $n$-colour partitions. We derive a combinatorial relation between the number of restricted split $n$-colour partitions and the function $\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with $d(a)$ copies of each part $a$ and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’, Indian J. Pure Appl. Math 22(9) (1991), 737–743].

In this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

Let B be the Bloch space of all those functions f holomorphic in the open unit disc D of the complex plane satisfying . We establish sufficient conditions for the boundedness of functions f belonging to B satisfying a certain uniform radial boundedness condition, and, by introducing a wide class of subsets E of ∂D, which we call negligible sets for boundedness, we show that if f ∈ B and there is a constant K > 0 such that , then f is bounded in D. Hence a significant extension of a theorem of Goolsby is obtained.

Let ${\mathcal{H}}ol(B_{d})$ denote the space of holomorphic functions on the unit ball $B_{d}$ of $\mathbb{C}^{d}$, $d\geq 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in {\mathcal{H}}ol(B_{d})$ such that the standard integral means $M_{p}(f,r)$ and $w(r)$ are equivalent for any $p$ with $0<p\leq \infty$. We also obtain similar results related to volume integral means.

In this paper, some existence and uniqueness results for generalized solutions to a periodic-Dirichlet problem for semilinear wave equations are given, using a global inverse function theorem. These results extend those known in the literature.

We prove that each Polish space is homeomorphic to the ω–limit set of a bounded solution of an autonomous equation x′ = f (x) in l2 × c0, in which f is Lipschitz continuous on bounded sets.

The unsolved problem of whether there exists a positive constant $c$ such that the number $k(G)$ of conjugacy classes in any finite group $G$ satisfies $k(G) \geq c \log _{2}|G|$ has attracted attention for many years. Deriving bounds on $k(G)$ from (that is, reducing the problem to) lower bounds on $k(N)$ and $k(G/N)$, $N\trianglelefteq G$, plays a critical role. Recently Keller proved the best lower bound known for solvable groups:

\[ k(G)\gt c_{0} \frac {\log _{2}|G|} {\log _{2} \log _{2} |G|}\quad (|G|\geq 4) \]

$k(G/N) \geq \beta [G : N]^{\alpha }$

$k(G/N) \geq \beta (\log [G : N])^{t}$

$G$

$N$

$k(N)$

$k(G)$

We investigate how varying the parameters of t-(ν, κ, λ) designs affects the sizes of smallest defining sets. In particular, we consider the effect of varying each of the parameters t, ν and λ. We establish a number of new bounds for the sizes of smallest defining sets and find the size of smallest defining sets for an infinite family of designs. We also show how one of our results can be applied to the problem of finding critical sets of Latin squares.

In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n\times n$ matrices. Some Hoffman–Wielandt-type relative perturbation bounds are proved using the Clifford algebra technique. We also extend a result for diagonalisable matrices which improves a relative perturbation bound for single matrices.