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.

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.

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 $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$

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.

It has been shown that the existence of a (left) approximate identity in the tensor product A ⊗ B of Banach algebras A and B, where α is an admissible algebra norm on A ⊗ B, implies the existence of approximate identities in A and B. The question has been raised as to whether the boundedness of the approximate identity in A ⊗αB implies the boundedness of the approximate identities in A and B. This paper answers the question affirmatively with a being the greatest cross-norm.

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.

S(R) is the semigroup, under composition, of all continuous selfmaps of the space R of real numbers. We show that the primes of S(R) are precisely those continuous selfmaps which are surjective and have exactly two local extrema. Additional results are then derived from this. For example, if f is any surjective continuous selfmap of R with n ≥ 2 local extrema, then there exist homeomorphisms from R onto R such that m ≤ 1 + n/2 and

where P is the polynomial defined by P(x) = x3 − x. It follows from this that the homeomorphisms together with the polynomial P generate a dense subsemigroup of S(R) where the topology on S(R) is the compact-open topology.

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.

Our main result shows that a conformal mapping of hyperbolic n-space into another Riemannian manifold with scalar curvature bounded above by −n(n − 1) is necessarily distance decreasing. This is a generalisation of Ahlfors' version of the Schwarz-Pick lemma to Riemannian Geometry.

Lin introduced the partition function $\text {PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\text {PDO}_t(n)$, and conjectured certain congruences modulo $3^{k+2}$ for $k\geq 0$. He proved the conjecture for $k=0$ and $k=1$ [‘The number of tagged parts over the partitions with designated summands’, J. Number Theory 184 (2018), 216–234]. We prove the conjecture for $k=2$. We also study the lacunarity of $\text {PDO}_t(n)$ modulo arbitrary powers of 2 and 3. Using nilpotency of Hecke operators, we prove that there exists an infinite family of congruences modulo any power of 2 satisfied by $\text {PDO}_t(n)$.

A Gaussian measure is introduced on the space of Hilbert space valued tempered distributions. It is used to define a Hilbert space valued Q-Wiener process and a white noise process with a nuclear covariance operator Q. The proposed construction is used for solving operator-differential equations with additive noise with the operator coefficient generating an n-times integrated exponentially bounded semigroup.

Sharp gradient estimates are derived for positive eigenfunctions on complete Riemannian manifolds with Ricci curvature bounded below.

We consider a Helgason-type Fourier transform on SL2(ℝ) and prove various results on L1-harmonic analysis on the full group analogous to those on symmetric spaces.

Let QT = ω x (0, T), where ω is a bounded domain in ℝn (n ≥ 3) having the cone property and T is a positive real number; let Y be a nonempty, closed connected and locally connected subset of ℝh; let f be a real-valued function defined in QT × ℝh × ℝnh × Y; let ℒ be a linear, second order, parabolic operator. In this paper we establish the existence of strong solutions (n + 2 ≤ p < + ∞) to the implicit parabolic differential equation

with the homogeneus Cauchy-Dirichlet conditions where u = (u1, u2, …, uh), Dxu = (Dxu1, Dxu2, …, Dxuh), Lu = (ℒu1, ℒu2, … ℒuh).

A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if $\langle Q,g\rangle $ is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such that $G/Q$ is Černikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Černikov normal subgroup N such that $G/N$ is quasihamiltonian. This result should be compared with theorems of Černikov and Schlette stating that if a group G is Černikov over its centre, then G is abelian-by-finite and its commutator subgroup is Černikov.

A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion R → T is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.