In this paper we study linear fractional relations defined in the following way. Let i, 'i, i = 1,2, be Banach spaces. We denote the space of bounded linear operators by . Let T ε ( 1 ⊕ 2, '1 ⊕ '2). To each such operator there corresponds a 2 × 2 operator matrix of the form(*) where T ij ε ( j , 'i . For each such T we define a set-valued map G T from ( 1, 2) into the set of closed affine subspaces of ('1, '2) by

The map G T is called a linear fractional relation.

The paper is devoted to the following two problems.

• • Characterization of operator matrices of the form (*) for which the set G T(K) is non-empty for each K in some open ball of the space (1,2).

• • Characterizations of quadruples (1, 2, '1, '2) of Banach spaces such that linear fractional relations defined for such spaces satisfy the natural analogue of the Liouville theorem “a bounded entire function is constant”.

In this note, we show that if we write ⌊en!⌋ = s(n)u(n)2, where s(n) is square-free then has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the mth power-free part of s(n) as n ranges from 1 to N, where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊en!⌋ is a square.

Let E be a Banach space such that its dual E* is separable. We show that there exists a hypercyclic bounded operator T on E such that its adjoint T* is also hypercyclic on E*. We also exhibit a new kind of dual hypercyclic operator. Thus answers affirmatively two of the questions raised by Henrik Petersson in a recent paper.

In this paper, we give a characterization of Clifford tori and in a unit sphere S n+1 (1). Our results extend the results due to Cheng and Yau [4], and Wang and Xia [11].

In this paper, we show that the semilinear elliptic systems of the form(0.1) possess at least one positive solution pair (u, v) ∈ H 1 0(Ω) × H 1 0(Ω), where Ω is a smooth bounded domain in , f(x,t) and g(x, t) are continuous functions on and asymptotically linear at infinity.

We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {M n }∞ n = 0 is a sequence of positive numbers such that M 0 = 1 and (m + n)!/M m+n ≤ (m!/M m )(n!/M n ) for m, n ∈ N. Let d = lim sup(n!/Mn )1/n and Xd = {z ∈ C : dist(z, X) ≤ d}. We show that, under certain conditions on X, every f ∈ D(X, M) has an analytic extension to X d . Let DP [D R ]) be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of D P is , the polynomial convex hull of X d , and the maximal ideal space of D R is X d . Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.

Using the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects of flow pulsation, and remain considerably simpler and more efficient than computationally solving three-dimensional Navier–Stokes type models. Using methods from gradient flows, we demonstrate that our unsteady evolution equation for the stream function has a unique solution. The solution is continuous with respect to variations in the model physical data and will decay exponentially to a steady-state distribution if the data do not change with time. In the event that density differences between the fluids are small and that the fluids have a yield stress, then if the flow rate is decreased suddenly to zero, the stream function (hence velocity) decays to zero in a finite time. We verify these decay properties, using a numerical solution. We then use the numerical solution to study the effects of pulsating the flow rate on a typical displacement.

Amortization of a shock in an electro-rheological shock absorber is carried out in the motion of a piston in an electrorheological fluid. The drag force acting on the piston is regulated by varying the voltage applied to electrodes. A model of an electrorheological shock absorber is constructed. A problem on shock absorber reduces to the solution of a coupled problem for motion equation of the piston and non-linear equations of fluid flow in an unknown domain that varies with the time. A method of semi-discretization for approximate solution of the coupled problem is considered. Results on the existence and on the uniqueness of the solution of the coupled problem are obtained. Convergence of approximate solutions to the exact solution is proved. Numerical simulation of the operation of the shock absorber is performed.

We generalise the notion of fractal interpolation functions (FIFs) to allow data sets of the formwhere I=[0,1]n. We introduce recurrent iterated function systems whose attractors G are graphs of continuous functions f:I→, which interpolate the data. We show that the proposed constructions generalise the previously existed ones on . We also present some relations between FIFs and the Laplace partial differential equation with Dirichlet boundary conditions. Finally, the fractal dimensions of a class of FIFs are derived and some methods for the construction of functions of class Cp using recurrent iterated function systems are presented.

The quasi-steady power-law Stokes flow of a mixture of incompressible fluids with shear-dependent viscosity is studied. The fluids are immiscible and have constant densities. Existence results are presented for both the no-slip and the no-stick boundary value conditions. Use is made of Schauder's fixed-point theorem, compactness arguments, and DiPerna–Lions renormalized solutions.

This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property that the accuracy improves as the frequency of oscillations increases. This asymptotic expansion is closely related to the method of stationary phase, but presented in a way that allows the derivation of an alternate approximation method that has similar asymptotic behaviour, but with significantly greater accuracy. This approximation method does not require moments.

Riesz transforms $R_\mu$ related to the Bessel operators

$$\Delta_\mu=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2}$$

are studied in this work. We develop for $R_\mu$ a theory that runs parallel to that for the Euclidean Hilbert transform. It is proved that $R_\mu$ is actually a Calderón–Zygmund singular integral operator. Also, $R_\mu$ is seen to be the boundary value of the appropriate harmonic extension for this context. Finally, we analyse weighted inequalities involving $R_\mu$.

We study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the classical (local) problem, there exist travelling-wave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits.

We investigate the existence and properties of the Jost solution associated with the differential equation $-y''+q(x)y=\lambda y$, $x\geq0$, for a class of real- or complex-valued slowly decaying potentials $q$. In particular, it is shown how the traditional condition $q\in L(\mathbb{R}^{+})$ for the existence of the Jost solution can be replaced by $q'\in L(\mathbb{R}^{+})$ for a class of potentials considered here. We also examine the asymptotics of the Titchmarsh–Weyl function for a class of real- or complex-valued slowly decaying potentials and the form of the spectral density for a class of real-valued slowly decaying potentials.

Let $D$ be a bounded, finitely connected domain in $\mathbb{C}$ without isolated points in the boundary and let $f$ be a continuous function on $bD$. Let $\tilde{f}$ be a continuous extension of $f$ to $\bar{D}$. We prove that $f$ extends holomorphically through $D$ if and only if the degree of $\tilde{f}+h$ is non-negative for every holomorphic function $h$ on $D$ such that $\tilde{f}+h$ is bounded away from $0$ near $bD$.

We prove that the integral of the product of two functions over a symmetric set in $\mathbb{S}^1\times\mathbb{S}^1$, defined as $E=\{(x,y)\in\mathbb{S}^1\times\mathbb{S}^1:d(\sigma_1(x),\sigma_2(y))\leq\alpha\}$ (where $\sigma_1$, $\sigma_2$ are diffeomorphisms of $\mathbb{S}^1$ with certain properties and $d$ is the geodesic distance on $\mathbb{S}^1$), increases when we pass to their symmetric decreasing rearrangement. We also give a characterization of the diffeomorphisms $\sigma_1$, $\sigma_2$ for which the rearrangement inequality holds. As a consequence, we obtain the result for the integral of the function $\varPsi(f(x),g(y))$ (where $\varPsi$ is a supermodular function) with a kernel given as $k[d(\sigma_1(x),\sigma_2(y))]$, with $k$ decreasing.

A new method for asymptotic summation of linear systems of difference equations is proposed and studied. It is based on the introduction of a certain summation equation that pinpoints sufficient conditions for asymptotic summation. These conditions serve as a framework from which new and old theorems follow. In particular the analogues of the fundamental theorems of Levinson and Hartman and Wintner are shown to follow from one and the same framework. Examples are given that are not amenable to other techniques.

Travelling wavefronts for a system of two reaction–diffusion equations are studied. The existence of a family of wavefronts (one for each wave speed) as well as the existence of the minimal speed (in the case that the Lewis number is greater than $1$) are proved. Asymptotic formulae for the wavefronts are established. New results are obtained when applying the main theorems to an isothermal autocatalytic chemical reaction system.

The main purpose of this paper is to study the mean value properties of certain Hardy sums over a short interval by using the mean-value theorems of the Dirichlet $L$-functions. Our main result is a mean-value formula for these sums.