Consider the p(x)-Laplacian–Dirichlet problem with sign-changing non-linearity of the formwhere 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.

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).

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.

We present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.

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).

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.

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.

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 our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types A2, A3, B2, B3 and C3. In this paper, we consider the case of G2-type. We define certain analogues of Bernoulli polynomials of G2-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of G2-type. Next, we consider the meromorphic continuation of the zeta-function of G2-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.

We consider the following process for generating large random cubic graphs. Starting with a given graph, repeatedly add edges that join the midpoints of two randomly chosen edges. We show that the growing graph asymptotically almost surely has logarithmic diameter. This process is motivated by a particular type of peer-to-peer network. Our method extends to similar processes that generate regular graphs of higher degree.

The paper presents yet another way to reformulate the Four Colour Conjecture as a statement concerning conditional probabilities of certain events involving planar graphs.

Some of the results of § 5 of the cited paper are incorrect: in particular, the characterization of when an algebra is ultra-amenable, in terms of a diagonal like construction, is not proved; and Theorem 5.7 is stated wrongly. The rest of the paper is unaffected. We shall show in this corrigendum that Theorem 5.7 can be corrected and that the other results of § 5 are true if the algebra in question has a certain approximation property.

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d ≫ log n.

The bootstrap process has been extensively studied on [n]d when d is a fixed constant and 2 ⩽ r ⩽ d, and in these cases pc([n]d, r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined pc([2]d, 2) up to a constant factor, and Balogh, Bollobás and Morris determined pc([n]d, d) asymptotically if d ≥ (log log n)2+ϵ, and gave much sharper bounds for the hypercube.

Here we prove the following result. Let λ be the smallest positive root of the equationso λ ≈ 1.166. Thenif d is sufficiently large, and moreoveras d → ∞, for every function n = n(d) with d ≫ log n.

We characterize quasi-Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related to the third Gray identity and in the almost-Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames previously introduced by the second author. By using such frames we also give a simpler and shorter proof of a theorem of Goldberg. Furthermore, we study almost-Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi-Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

It is known that all k-homogeneous orthogonally additive polynomials P over C(K) are of the form

Thus, x ↦ xk factors all orthogonally additive polynomials through some linear form μ. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic function of bounded type f over C(K) is of the form

for some μ and holomorphic h : C (K) → L1(μ) of bounded type.

We compare two known definitions for a relative family of effective zero cycles: one based on traces of functions and one based on norms of functions. In characteristic zero we show that both definitions agree. In the general setting we show that the norm map on functions can be expanded to a norm functor between certain categories of line bundles, thereby giving a third approach to families of zero cycles.

We consider the spectra of the Laplacians of two sequences of fractal graphs in the context of the general theory introduced by Sabot in 2003. For the sequence of graphs associated with the pentagasket, we give a description of the eigenvalues in terms of the iteration of a map from (ℂ2)3 to itself. For the sequence of graphs introduced in a previous paper by the author, we show that the results found therein can be related to Sabot's theory.

We obtain an asymptotic formula for the number of pairs of closed orbits of a weak-mixing transitive Anosov flow whose homology classes have a fixed difference.

Let G be a finite group. If G has a cyclic Sylow 2-subgroup, then G has the same number of irreducible rational-valued characters as of rational conjugacy classes. These numbers need not be the same even if G has Klein Sylow 2-subgroups and a normal 2-complement.

We extend an affine-invariant inequality for vector polynomials established by Dendrinos and Wright to general rational functions. As a consequence we obtain sharp universal estimates for various problems in Euclidean harmonic analysis defined with respect to the so-called affine arc-length measure.