Our object of study is the arithmetic of the differential modules (l) (l ∈ ℕ – {0}), associated by Dwork's theory to a homogeneous polynomial f (λ,X) with coefficients in a number field. Our main result is that (1) is a differential module of type G, c'est-à-dire, a module those solutions are G-functions. For the proof we distinguish two cases: the regular one and the non regular one.

Our method gives us an effective upper bound for the global radius of (l), which doesn't depend on “l” but only on the polynomial f (λ,X). This upper bound is interesting because it gives an explicit estimate for the coefficients of the solutions of (l).

In the regular case we know there is an isomorphism of differential modules between (1) and a certain De Rham cohomology group, endowed with the Gauss-Manin connection, c'est-à-dire, our module “comes from geometry”. Therefore our main result is a particular case of André's theorem which assert that at least in the regular case, all modules coming from geometry are of type G.

It is shown that the free orthomodular word problem is solvable. Since the free orthomodular lattice L0 on countable many generators has, as a partial subalgebra, every finite partial orthomodular lattice P, which is contained in some orthomodular lattice as a partial subalgebra, it is sufficient to prove Evans embedding property for these P only. The construction of the finite orthomodular lattice containing P as a partial subalgebra has and can be done outside of L0. It uses the coatom extension for ortholattices.

In Problem 1 of her book Varieties of groups, Hanna Neumann asked whether a fully invariant subsemigroup of a free group of infinite rank is necessarily a subgroup. This note presents an example which shows that the answer is negative.

We show

(i) that the Rayleigh-Ritz method is a practical procedure for obtaining approximations to the velocity potential for compressible flows, and

(ii) how to calculate an estimate of the error in such approximations.

We discuss permutation properties of a specific kind of binomials over finite fields. As a result, we complete Cavior's classification of binomial octic permutation polynomials over Fq with q odd.

An oscillation criterion is established for a class of functional differential equations including the generalized Emden-Fowler equation

as a special case. The deviating arguments involved may be retarded or advanced or otherwise. The result extends and improves known fundamental oscillation criteria for superlinear differential equations with retarded arguments.

We sharpen a result of Byers on the existence of periodic points for some continuous expanding maps of the interval and generalize it to some classes of continuous maps of the interval which are not necessarily expanding. We then use these results to construct one-parameter families of continuous maps of the interval which have a bifurcation form fixed points directly to period 3 points together with a series of reverse bifurcations from period 3 points back to fixed points. Consequently, our results also provide examples of one-parameter families of continuous maps of the interval whose topological entropy jumps form zero to some positive number and then changes back to zero as the parameter varies.

Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X, there exists a subspace A⊆X with property P such that X=St(A,𝒰), where St(A,𝒰)=⋃ {U∈𝒰:U∩A≠0̸}. In this paper we construct an example of a pseudocompact Tychonoff space that is not star Lindelöf, which gives a negative answer to Alas et al. [‘Countability and star covering properties’, Topology Appl.158 (2011), 620–626, Question 3].

We use Zalcman’s lemma to study a uniqueness question for meromorphic functions where certain associated nonlinear differential polynomials share a nonzero value. The results in this paper extend Theorem 1 in Yang and Hua [‘Uniqueness and value-sharing of meromorphic functions’, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406] and Theorem 1 in Fang [‘Uniqueness and value sharing of entire functions’, Comput. Math. Appl. 44 (2002), 823–831]. Our reasoning in this paper also corrects a defect in the reasoning in the proof of Theorem 4 in Bhoosnurmath and Dyavanal [‘Uniqueness and value sharing of meromorphic functions’, Comput. Math. Appl. 53 (2007), 1191–1205].

A linear complementarity problem, involving a given square matrix and vector, is generalised by including an element of the subdifferential of a convex function. The existence of a solution to this nonlinear complementarity problem is shown, under various conditions on the matrix. An application to convex nonlinear nondifferentiable programs is presented.

For a locally compact group G and an arbitrary subset J of [1,∞], we introduce ILJ(G) as a subspace of ⋂ p∈JLp(G) with some norm to make it a Banach space. Then, for some special choice of J, we investigate some topological and algebraic properties of ILJ(G) as a Banach algebra under a convolution product.

The main purpose of this paper is to give a new, elementary proof of Flanigan’s theorem, which says that a given ring A has a maximal essential extension ME(A) if and only if the two-sided annihilator of A is zero. Moreover, we discuss the problem of description of ME(A) for a given right ideal A of a ring with an identity.

Let k be a finite field of characteristic 2, and let G be the three dimensional special linear group over k. The principal indecomposable modules of G over k are constructed from tensor products of the irreducible modules, and formulae for their dimensions are given.

In this paper, we consider the Diophantine equations

$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$

$q,p\geq 2$

$\gcd (F_{n},F_{m})=1$

$F_{k}$

$q=2$

$q$

$q\equiv 3\;(\text{mod}\;4),3<q<1087$

$q=3$

We give an existence result for a fully nonlinear system consisting of a parabolic equation strongly coupled with an elliptic one. It models in particular miscible displacement in porous media. To this aim, we adapt the tools of Ladyzenskaja, Solon-nikov and Uralćeva [27, 28] to the coupled nonlinear setting. Under some reasonable assumptions on the data, we state the existence of semi-classical solutions for the problem. We also give an existence result of weak solutions for a degenerate form of the problem.

If H and M are right R-modules, H is M-injective if every R-homonorphism N → H, N a right R-submodule of M, can be extended to an R-homonorphism from M to H.

H is strongly M-injective if H is injective for inclusions whose cokernels are isomorphic to factor modules of M.

For the case of abelian groups H and M, one settles the questions “when is H M-injective” and “when is H strongly M-injective”. The latter can be characterized in terms of the vanishing of Ext. Results for general module categories are also given.