A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given.

Let V be a countably generated right vector space over a division ring D. If D ≇ ℤ/2ℤ, ℤ/3ℤ, then for any γ ∈ EndD(V), there exists α ∈ AutD(V) such that γ+α, γ−α−1 ∈ AutD(V). This gives a generalization of [D. Zelinsky, Proc. Amer. Math. Soc. 5 (1954), 627–630, Theorem].

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Let K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K((X)) be the valued field of Laurent power series and let G = Aut(F/K). We prove that if L is a subfield of F, K ≠ L, such that L/K is a sub-extension of F/K and F/L is a Galois algebraic extension (L/K is Galois coalgebraic in F/K), then L is closed in F, F/L is a finite extension and Gal(F/L) is a finite cyclic group of G. We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F/K. Some other auxiliary results which are useful by their own are given.

Kummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

In this note, we give a homology-free proof that the non-abelian tensor product of two finite groups is finite. In addition, we provide an explicit proof that the non-abelian tensor product of two finite p-groups is a finite p-group.

We give sufficient conditions and necessary conditions for a Banach algebra, which is ℓ1-graded over a semi-lattice, to be biflat or biprojective. As an application we characterise biflat and biprojective discrete convolution algebras for commutative semi-groups.

For each positive integer n ≥ 2, there is a well-known regular orientable Hamiltonian embedding of Kn, n, and this generates a regular face 2-colourable triangular embedding of Kn, n, n. In the case n ≡ 0 (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of Kn, n. This paper presents an analysis of the face 2-colourable triangular embedding of Kn, n, n that results from this. The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.

Consider the p(x)-Laplacian–Dirichlet problem with sign-changing non-linearity of the form where Ω ⊂ ℝN is a bounded domain, p ∈ C0(Ω) and infx∈Ωp(x) > 1, m ∈ L∞(Ω) is non-negative, f : ℝ → ℝ is continuous and f(0) > 0, the coefficient a ∈ L∞(Ω) is sign-changing in (Ω). We give some sufficient conditions to assure the existence of a positive solution to the problem for sufficiently small λ > 0. Our results extend the corresponding results established in the p-Laplacian case to the p(x)-Laplacian case.

We study the eigenvalue problem = λ|u|q(x)−2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN with smooth boundary ∂Ω, λ is a positive real number, and p1,⋅ ⋅ ⋅, pN, q are continuous functions satisfying the following conditions: 2 ≤ pi(x) < N, 1 < q(x) for all x ∈ Ω, i ∈ {1,. . .,N}; there exist j, k ∈ {1,. . .,N}, j ≠ k, such that pj ≡ q in Ω, q is independent of xj and maxΩq < minΩpk. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that every λ ∈(λ1, ∞) is an eigenvalue, while no λ ∈ (0, λ0) can be an eigenvalue of the above problem.

Let Tn+1(R) be the algebra of all upper triangular n+1 by n+1 matrices over a 2-torsionfree commutative ring R with identity. In this paper, we give a complete description of the Jordan automorphisms of Tn+1(R), proving that every Jordan automorphism of Tn+1(R) can be written in a unique way as a product of a graph automorphism, an inner automorphism and a diagonal automorphism for n ≥ 1.

In this paper, using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semi-linear degenerate parabolic equation in an arbitrary domain, without restriction on the growth order of the polynomial type non-linearity and with a suitable exponential growth of the external force. The obtained results improve some recent ones for the non-autonomous reaction–diffusion equations.

We determine the extent to which the collection of Γ-Euler–Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the Γ-Euler–Satake characteristics corresponding to free or free abelian Γ and are not classified by those corresponding to any finite set of finitely generated discrete groups. These results demonstrate that the Γ-Euler–Satake characteristics corresponding to free abelian Γ constitute new invariants of orbifolds. Similarly, we show that such a classification is neither possible for non-orientable 2-orbifolds nor for non-effective 2-orbifolds using any collection of groups Γ.

Let n(≥ 3) be an odd integer. Let k:= be the imaginary quadratic field and k′:= the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

The solutions of a perturbed linear ordinary differential equation (ODE) system are studied. Provided that some integrability and oddness conditions are satisfied, we show that they are asymptotically equivalent at t = ±∞ to the solutions of the unperturbed one. This fact is used to determine the existence of almost periodic or pseudo-almost periodic solutions of the perturbed system.

We prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

Let R be a commutative ring with identity. We will say that an R-module M has Nakayama property, if IM = M, where I is an ideal of R, implies that there exists a ∈ R such that aM = 0 and a − 1 ∈ I. Nakayama's Lemma is a well-known result, which states that every finitely generated R-module has Nakayama property. In this paper, we will study Nakayama property for modules. It is proved that R is a perfect ring if and only if every R-module has Nakayama property (Theorem 4.9).

Type I domains are the domains of the self-adjoint operators determined by the weak formulation of formally self-adjoint differential expressions ℓ. This class of operators is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts agrees with the inner product 〈ℓu, v〉. A complete characterisation of the boundary conditions assumed by functions in these domains for second-order differential expressions is given in this paper. In the singular case, the boundary conditions are stated in terms of certain ‘boundary condition’ functions and in the regular case they are given in terms of classical function values.

In this paper, we investigate linear Weingarten hypersurfaces with two distinct principal curvatures in a real space form Mn+1(c), we obtain two rigidity results and give some characterization of the Riemannian product Sk(a) × Sn−k(), 1 ≤ k ≤ n − 1 in Mn+1(c)(c = 1), the Riemannian product Rk × Sn−k(a), 1 ≤ k ≤ n −1 in Mn+1(c)(c = 0) and the Riemannian product Hk(tanh2 ρ−1) × Sn−k(coth2 ρ−1), 1 ≤ k ≤ n −1 in Mn+1(c)(c = −1).

Let H be a Hopf algebra, A/B be an H-Galois extension. Let D(A) and D(B) be the derived categories of right A-modules and of right B-modules, respectively. An object M⋅ ∈ D(A) may be regarded as an object in D(B) via the restriction functor. We discuss the relations of the derived endomorphism rings EA(M⋅) = ⊕i∈ℤHomD(A)(M⋅, M⋅[i]) and EB(M⋅) = ⊕i∈ℤHomD(B)(M⋅, M⋅[i]). If H is a finite-dimensional semi-simple Hopf algebra, then EA(M⋅) is a graded sub-algebra of EB(M⋅). In particular, if M is a usual A-module, a necessary and sufficient condition for EB(M) to be an H*-Galois graded extension of EA(M) is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.