We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

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.

The Stieltjes transformation is extended to generalised functions both by the direct approach and the method of adjoints, and the resulting extensions are correlated. Inversion formulae are developed, as is the application of fractional integration to these transforms. An integral transformation with a hypergeometric kernel is also briefly considered.

It is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup. This allows a number of existence theorems about non-congruence subgroups.

The level of a subgroup of the modular group can be defined in terms of the action on Q ∪ {∞}. We define a similar action to get information on congruence subgroups. In fact, we get a more powerful result, but this appears to be the most useful version.

A new method is developed for finding radially symmetric solutions of semilinear elliptic problems by phase space methods. The basic idea is to formulate a shooting argument from initial conditions at r = 0 which involves encoding the oscillation information about the trajectories under consideration. The main theorem is applied to a particular nonlinearity and produces a cascade of solutions wherein the multiplicity increases with the number of zeros.

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

Let X be a Banach space and Y a closed subspace. We introduce an intrinsic geometric property of Y—the k-ball sequence property—which is a weakening of the famous k-ball property due to Alfsen & Effros. We prove that Y satisfies the 2-ball sequence property if and only if Y has the Phelps uniqueness property U (i.e. every continuous linear functional g ∈Y* has a unique norm-preserving extension f ∈X*). We prove that Y is an ideal having property U if and only if Y satisfies the 3-ball sequence property, and in this case, Y satisfies the k-ball sequence property for all k.

The aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.

The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

Given a decreasing sequence of domains Ωn converging in measure to some domain Ω0, a sequence of subspaces V of a Hilbert space V is constructed in such a way that the convergence of the solutions of u −Δu = f on Ωn with Neumann Boundary Condition is given in terms of the convergence of the orthogonal projections Pn on Vn. Under dissipative assumptions, we can obtain continuation results for equations like u −Δu = f(x,u ∇u).

The main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles and such that certain combinations of derivatives of f have few non-real zeros.

In this paper we prove the existence of a weak solution of the Cauchy problem for the nonlinear Dirac equation in ℝ × ℝ

where X(r) is the characteristic function of a compact interval of ]0, + ∞[

We study the existence of multiple solutions for the problem

and we show that at least one of them is sign changing. Here Ω is a bounded domain in ℝN with N ⩾ 5 whose boundary is of class C2, ∂/∂ν is the outward normal derivative, and f ∈ LN/2(Ω) whose L2N/(N+2)(Ω) norm is sufficiently small.

In 1900 von Ihering directed attention to the presence in certain Lamellibranchs of a special muscle lying in the posterior portion of the ventral mantle edge, close to the inner end of the inhalent siphon. This muscle consisted of two strands, each running diagonally from an origin on one valve to be inserted on the other, and crossing one another in the mid-ventral line so as to form a muscular apparatus with the appearance of a St Andrew's cross. He regarded this cruciform muscle as a specially differentiated group of fibres belonging to the pallial edge, acting as an accessory adductor muscle, a point of view in which he has been followed by all subsequent observers. To this von Ihering added the speculation that it had been by some similar process of specialisation of marginal pallial muscle fibres that the two other true adductor muscles of Lamellibranchs had originated.

In this paper, we consider the following nonlinear Schrödinger equations with the critical Sobolev exponent and mixed nonlinearities:

\[\left\{\begin{aligned} & -\Delta u+\lambda u=t|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\ & u\in H^{1}(\mathbb{R}^{N}), \end{aligned}\right.\]

$N\geq 3$

$t>0$

$\lambda >0$

$2< q<2^{*}=\frac {2N}{N-2}$

1. (1) the above equation has two positive radial solutions for $N=3$, $2< q<4$ and $t>0$ sufficiently large, which gives a rigorous proof of the numerical conjecture in [J. Dávila, M. del Pino and I. Guerra. Non-uniqueness of positive ground states of non-linear Schrödinger equations. Proc. Lond. Math. Soc. 106 (2013), 318–344.];

2. (2) there exists $t_q^{*}>0$ for $2< q\leq 4$ such that the above equation has ground-states for $t\geq t_q^{*}$ in the case of $2< q<4$ and for $t>t_4^{*}$ in the case of $q=4$, while the above equation has no ground-states for $0< t< t_q^{*}$ for all $2< q\leq 4$, which, together with the well-known results on ground-states of the above equation, almost completely solve the existence of ground-states, except for $N=3$, $q=4$ and $t=t_4^{*}$.

$0<\overline {t}_{a,q}<+\infty$

$2< q<2+\frac {4}{N}$

$t>\overline {t}_{a,q}$

$\int _{\mathbb {R}^{N}}|u|^{2}{\rm d}x=a^{2}$

\[\left\{ \begin{aligned} & -\Delta u+\lambda u+(x_1^{2}+x_2^{2})u=|u|^{p-2}u\quad\text{in }\mathbb{R}^{3},\\ & u\in H^{1}(\mathbb{R}^{3}),\quad \int_{\mathbb{R}^{3}}|u|^{2}{\rm d}x=r^{2}, \end{aligned}\right.\]

$x=(x_1,x_2,x_3)\in \mathbb {R}^{3}$

$\frac {10}{3}< p<6$

$r>0$

$(u, \lambda )$

$\lambda$

$r>0$

The aim of this paper is to study a boundary time-optimal control problem for the heat equation in a two-dimensional ball. The main ingredient is the extension of a result concerning Müntz polynomials due to Borwein and Erdélyi that allows us to prove an observability inequality for the dynamical system's truncation to a finite number of modes. This result, combined with a well-known Lebeau–Robbiano argument used to show the null-controllability of parabolic type equations, enables us to deduce the existence, uniqueness and bang-bang properties for the boundary time-optimal control.

We prove some results on the global existence of smooth solutions for certain nonlinear parabolic systems of the form Ut + A(U)Ux = DUxx. Here U is a vector and A(U), D are matrices with D a constant, positive matrix. We show how to use our results to study the global continuous (or generalised) solutions to the corresponding nonlinear hyperbolic conservation laws and a conjecture is given.

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

We provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

We study the balanced Allen–Cahn problem in a singular perturbation setting. We are interested in the behaviour of clusters of layers, i.e. a family of solutions uε(x) with an increasing number of layers as ε → 0. In particular, we give a characterization of cluster of layers with asymptotically positive length by means of a limit energy function and, conversely, for a given admissible pattern, i.e. for a given a limit energy function, we construct a family of solutions with the corresponding behaviour.

The existence of solutions to equations in normed spaces is proved when the nonlinear part of the equation satisfies growth and asymptotic conditions, whether the linear part is invertible or not. For this, we use the coincidence degree theory developed by Mawhin. We apply our abstract results to boundary value problems for nonlinear vector ordinary differential equations. In particular, we consider the Picard boundary value problem at the first eigenvalue and the periodic boundary value problem at resonance. In both cases, the nonlinear term can be of superlinear type. Also, necessary and sufficient conditions of Landesman-Lazer type are obtained.