We investigate the global asymptotic stability for positive solutions to a class of general symmetric rational difference equations and prove that the unique positive equilibrium 1 of the general symmetric rational difference equations is globally asymptotically stable. As a special case of our result, we solve the conjecture raised by Berenhaut, Foley and Stević [‘The global attractivity of the rational difference equation yn=(yn−k+yn−m)/(1+yn−kyn−m)’, Appl. Math. Lett.20 (2007), 54–58].

Neuman and Reid describe a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' “SnapPea” program, they conjecture that the cusps are strongly geometrically isolated, a fact which we establish here. We also give a parameterisation of the Dehn Surgery Space of this orbifold which has amusing properties.

A family of n k-subsets of the integers modulo ν are said to be supplementary difference sets if developing them by addition modulo ν leads to a balanced incomplete block design, and to be minimal if no proper subfamily leads to a balanced incomplete block design when developed modulo ν. In other words, the family of supplementary difference sets is minimal precisely when it leads to a balanced incomplete block design which cannot be partitioned into a union of proper subdesigns, each consisting of complete cyclic sets of ν blocks. We discuss the conditions under which such a balanced incomplete block design can be partitioned in some non-cyclic fashion into a union of proper subdesigns.

The group of Poisson automorphisms of the coordinate ring of Poisson symplectic 2n-space is isomorphic to the algebraic torus (κ✶)n+1 and it confirms that the algebra constructed by K.L. Horton (2003) is a quantisation of the coordinate ring of Poisson symplectic 2n-space.

To explore the difficulties of classifying actions with the tracial Rokhlin property using K-theoretic data, we construct two $\mathbb{Z}_{2}$ actions $\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}$ on a simple unital AF algebra $A$ such that $\unicode[STIX]{x1D6FC}_{1}$ has the tracial Rokhlin property and $\unicode[STIX]{x1D6FC}_{2}$ does not, while $(\unicode[STIX]{x1D6FC}_{1})_{\ast }=(\unicode[STIX]{x1D6FC}_{2})_{\ast }$, where $(\unicode[STIX]{x1D6FC}_{i})_{\ast }$ is the induced map by $\unicode[STIX]{x1D6FC}_{i}$ acting on $K_{0}(A)$ for $i=1,2$.

In this paper we present a combinatorial proof of an identity involving the two kinds of Stirling numbers and the numbers of permutations with prescribed numbers of excedances and cycles. Several recurrence relations related to the numbers of excedances, fixed points and cycles are also obtained.

Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.

Let G be the free product of groups Gα, [Gα] the cartesian subgroup of G and k [Gα] the intersection of [G] with the k–th term of the lower central series for G. Then the k[Gα] form a descending chain of subgroups of ] and it is shown that if the intersection of all the subgroups in this chain is trivial then G and hence each Gα, is residually nilpotent. This answers a question of S. Moran.

A useful result for uniformly bounded sequences of functions in Orlicz spaces is generalised by means of a recent extension of Komlós' theorem. The same generalisation can also be proven differently, by means of Young measures.

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

We extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math. 74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in $\mathbb R^N$ and $\lambda _s(A)$, $\lambda _s(B)$ are the lowest eigenvalues of $(-\Delta )^s$, $0<s<1$, with Dirichlet boundary conditions, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$. Moreover, without the boundedness assumption on A and B, we show that for any $\varepsilon>0$, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$

The classes of finite groups with minimal sets of generators of fixed cardinalities, named ${\mathcal{B}}$-groups, and groups with the basis property, in which every subgroup is a ${\mathcal{B}}$-group, contain only $p$-groups and some $\{p,q\}$-groups. Moreover, abelian ${\mathcal{B}}$-groups are exactly $p$-groups. If only generators of prime power orders are considered, then an analogue of property ${\mathcal{B}}$ is denoted by ${\mathcal{B}}_{pp}$ and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic $q$-extensions of $p$-groups. In this paper we characterise all finite groups with the pp-basis property as products of $p$-groups and precisely described $\{p,q\}$-groups.

In this paper, a finite-difference method for the determination of an approximate solution of a fourth-order two-point boundary value problem is presented under the nonresonance condition. The solution of this linear problem can be used to find approximate solutions of a broad range of nonlinear problems in applications.

The notion of $\theta $ -congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $ -congruent over certain real number fields.

An upper bound is given of the dimension of certain spaces of cusp harmonic forms of arithmetic subgroups Γ of semisimple algebraic groups G in terms of the multiplicities of corresponding irreducible unitary representations of the group GR of real rational points of G in the space 0L2(GR/Γ) of cusp forms.

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

A generalisation of the notion of “sets of measure zero” for arbitrary Banach spaces is defined so that continuous convex functions are automatically Gateaux differentiable “almost everywhere”. It is then shown that this class of sets satisfies all the properties tht one expects of sets of measure zero. Moreover (in a certain large class of Banach spaces, at least) nonempty open sets are not of “measure zero”.

In this paper we show that given a σ-algebra Σ of subsets of a set Ω and a normed space X, then the normed space B(Σ, X), endowed with the usual supremum-norm, of the X-valued functions defined on Ω that are the uniform limit of a sequence of σ-simple X-valued functions on Ω is barrelled of class s if and only if X is barrelled of class s. This extends in the normed case the well known result obtained by Mendoza (1982) for barrelled spaces.

We compute the Alexander polynomial of a nonreduced nonirreducible complex projective plane curve with mutually coprime orders of vanishing along its irreducible components in terms of certain multiplier ideals.