Let X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.

Let $\mathscr{C}$ be a $(d+2)$ -angulated category with d-suspension functor $\Sigma^d$ . Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$ -angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander–Reiten $(d+2)$ -angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ is d-Auslander–Reiten translation. These results generalize work by Bondal–Kapranov and Reiten–Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$ , the quotient category $\mathscr{C}/\mathscr{X}$ is a $(d+2)$ -angulated category if and only if $(\mathscr{C},\mathscr{C})$ is an $\mathscr{X}$ -mutation pair, and if and only if $\tau_d\mathscr{X} =\mathscr{X}$ .

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic 4-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well-known invariants of homology spheres that obstruct acyclic bounding 4-manifolds and then use a method based on equivariant Yang–Mills moduli spaces to rule out extensions of the actions.

The fractional part of the sequence {αnk}, where α is an irrational real number and k is an integer, was first studied early this century, initiated by the work of Hardy, Littlewood and Weyl. It seems very natural to consider the subsequence {αpk}, where p denotes a prime variable. The pioneering work in this direction was conducted by Vinogradov [13,14]. Improvements have since been made by Vaughan [12], Ghosh [4], Harman [6,7,8] and Jia [11]. The best results to date have been obtained by Harman for k = 1 [9], by Baker and Harman for 2 ≤ k ≤ 12 [1], and by Harman for larger k [8]. In the following work, we shall adopt a sieve technique developed by Harman in [6] to show the following.

We prove the longstanding conjecture that the 3-Moufang condition for generalized quadrangles is equivalent to the Moufang condition. We mention some other characterizations of Moufang quadrangles that follow from this result. We also provide a short proof of Tent's recent result that every half Moufang quadrangle is necessarily a Moufang quadrangle.

We introduce in this paper the concepts of rings characterized by minimal one-sided ideals and concern ourselves with rings containing an injective maximal left ideal. Some known results for idempotent reflexive rings and left HI rings can be extended to left MC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with non-zero socle and extend some known results for strongly regular rings.

We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci.126 (2016), 591–612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from θ-covers, and consider related problems.

Let X be a compact Hausdorff space, let C(X) denote the algebra of all continuous functions on X, let B be a Banach algebra, and let θ: → C(X) → B be a (possibly discontinuous) homomorphism with dense range. A classical theorem by W. G. Bade and P. C. Curtis ([2, Theorem 4.3]) describes in great detail the structure of θ we shall refer to this result as the Bade–Curtis theorem. Before we give a brief sketch of this theorem, we fix some notation. For Y ⊂ X let I(Y) and J(Y) denote the ideals of all functions in C(X) that vanish on Y and on a neighborhood of Y respectively; if Y = {x} for some x ɛ X, we write mx and Jx for I(Y) and J(Y) respectively. According to the Bade–Curtis theorem there is a finite set {x1,…, xn) ⊂ X, the so-called singularity set of θ, such that θ | ({x1, …, xn}) is continuous. As a consequence, the restriction of θ to the dense subalgebra of C (X) consisting of all those functions which are constant near each Xj (j = 1,…, n) is continuous, and extends to a continuous homomorphism θcont: C(X)→ B. Let θsing: = θ – θcont. Then θsing | I({x1,…, xn}) is a homomorphism onto a dense subalgebra of rad (B). θcont, and θsing are called the continuous and the singular part of θ respectively. Moreover, there are linear maps : C(X)⊒ B such that

(i)

(ii) is a homomorphism, and

Condition (iii) forces the homomorphisms to map into rad(B); such homomorphisms are called radical homomorphisms.

We will compute the subgroup permutability degree of the projective special linear groups PSL(2,pn).

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results. Let Proj (G, α) denote the set of irreducible projective characters of a group G with cocyle α. In previous papers (for exampe [2], [4], and [6]) numerous authors have considered the situation when Proj(G, α) = 1 or 2; such groups are said to be of α-central type or of 2α-central type, respectively. In particular in [4, Theorem A] the author showed that if Proj(G, α) = {ξ1, ξ2}, then ξ1(1)=ξ2(1). This result has recently been independently confirmed in [8, Corollary C].

Three different sets of equations connecting the sums of angles in an n-dimensional simplex have been given by Sommerville [7], Höhn [5], and Peschl [6]. The equivalence of the first two sets of equations has been proved by Sprott [7].

In the present note it is shown that results are simplified if we consider averages instead of sums, and that the averages form a sequence which is self-reciprocal with respect to the transformation

The equivalence of the sets of equations is then easily proved by symbolic methods.

If we investigate symmetry of an infra-nilmanifold M, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groups E with finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms on M, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group of E.

In an earlier paper [1] on groups which are the products of two finite cyclic groups with trivial intersection, certain permutations, called “semi-special”, played a certain role. The permutation π of the numbers 1, 2,…, n is semi-special if† πn=n, and if, for every y ε [n],

is again a permutation, namely a power (depending on y) of π.

Let X be a non-locally convex F-space (complete metric linear space) whose dual X′ separates the points of X. Then it is known that X possesses a closed subspace N which fails to be weakly closed (see [3]), or, equivalently, such that the quotient space X/N does not have a point separating dual. However the question has also been raised by Duren, Romberg and Shields [2] of whether X possesses a proper closed weakly dense (PCWD) subspace N, or, equivalently a closed subspace N such that X/N has trivial dual. In [2], the space Hp (0<p<1) was shown to have a PCWD subspace; later in [9], Shapiro showed that ℓp (0<p<1) and certain spaces of analytic function have PCWD subspaces. Our first result in this note is that every separable non-locally convex F-space with separating dual has a PCWD subspace.

In this paper the author continues the search for a suitable integral transform that can be applied to certain boundary value problems involving the Helmholtz equation and the condition of radiation. The transform in question must be capable of eliminating the r-dependence appearing in the partial differential equation

Dr V. O. Ennola has pointed out that there is a mistake in this paper. The inequality for dox at the foot of page 76 does not imply the stated inequality between the second and third lines on page 77. This error vitiates the entire argument, and I have not so far been able to put it right.

A preliminary attempt is made to place the theory of completions of boolean algebras and of partially ordered sets in a wider context. The theory and construction of injective hulls in abelian categories is generalised and it is demonstrated that any variety with enough injectives admits injective hulls. Then the methods developed are applied to a non-algebraic bicategory, that of ordered sets.

be a given series and let

With Flett [4], we say that the series is summable , q real, if

where Summability |C, k, 0|1 is identical with absolute Cesàro summability (C, k), or summability |C, k|, as defined by Fekete [3].

A series ∑an is said to be summable (C, — 1) to s if it converges to s and nan = o(1) [8]. It is well known that this definition is equivalent to tn→s (n→∞), where tn = sn + nan, sn = ao + … + an. The series is summable | C, — 1 | to s if the sequence t = {tn} is of bounded variation (t ∈ B.V.), i.e. ∑ |; ▲tn |; = ∑ | tn - tn-1 | < ∞, and ∑ ▲tn = lim tn = s. An equivalent condition is ∑ | an |; < ∞, ∑an = s and ∑ | ▲(nan) | < ∞. For, suppose that ∑an = s | C, - 1 |. Since {sn} is the sequence of (C, 1)-means of {tn} and since | C, 0 | ⊂ | C, 1 |, we have ∑ | an | < ∞ and ∑an = s whence ∑ | ▲(nan) | < ∞. Conversely, ∑ | an | < ∞, ∑an = s and ∑ | ▲(nan) | < ∞ imply t ∈ B.V. and ∑▲n = s + lim nan. But lim nan = 0, since ∑ | an | < ∞.

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε ℋU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the ℋW(p + 1/2) operator  by A = |A|1/2U |A|l/2 Let  = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.