We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or −1 and if ⊂ ℤ/pℤ is not too large (with respect to p), then | + t ⋅ | is significantly larger than 2|| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | + t ⋅ | ≥ 2.08 ||−2.

Plünnecke's inequality is a standard tool for obtaining estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be attained.

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.

We show that there are graphs with n vertices containing no K5,5 which have about n7/4 edges, thus proving that ex(n, K5,5) ≥ (1 + o(1))n7/4. This bound gives an asymptotic improvement to the known lower bounds on ex(n, Kt, s) for t = 5 when 5 ≤ s ≤ 12, and t = 6 when 6 ≤ s ≤ 8.

The present paper deals with the following system:

where p and p* are positive numbers satisfying 1/p + 1/p* = 1, and ϕq(z) = |z|q−2z for q = p or q = p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the non-integrable case (e(t), f(t)) ≢ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

The harmonic mapping problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in the calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents a considerable challenge and leads to several interesting open questions.

We consider a time-varying inclusion in a thermal conductor specimen. In particular, the thermal conductivity is a variable function depending on space and time with a jump of discontinuity along the interface of the unknown anomalous region. Provided with some a priori information on the conductivity and its support, we study the continuous dependence of the inclusion from infinitely many thermal measurements taken on an open portion of the boundary of our specimen. We prove a rate of continuity of logarithmic type showing, in addition, its optimality.

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

We study travelling-front solutions of pyramidal shapes in the Allen–Cahn equation in ℝN with N ≥ 3. It is well known that two-dimensional V-form travelling fronts and three-dimensional pyramidal travelling fronts exist and are stable. The aim of this paper is to show that for N ≥ 4 there exist N-dimensional pyramidal travelling fronts. We construct a supersolution and a subsolution, and find a pyramidal travelling-front solution between them. For the construction of a supersolution we use a multi-scale method.

We use the extended multiple Riccati equations expansion method to construct a series of double-soliton-like solutions, double triangular function solutions and complexiton soliton solutions for nonlinear Jaulent-Miodek PDEs. With the help of symbolic computation using Maple and Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combinations of trigonometric periodic functions and hyperbolic function solutions, various combinations of trigonometric periodic functions and rational function solutions, and various combinations of hyperbolic functions and rational function solutions.

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial-value problem, and these solutions need to be supplemented with further admissibility conditions. This paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns.

Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on , mainly for the case where n = 2 and M = . But there are very few results on the non-homogeneous polynomial vector fields of degree n on . This paper attempts to rectify this slightly.

By using the symmetric mountain pass theorem, we establish some new existence criteria to guarantee that the second-order Hamiltonian systems ü(t) − L(t)u(t) + ∇W(t,u(t)) = 0 have infinitely many homoclinic orbits, where t ∈ ℝ, u ∈ ℝN, L ∈ C(ℝ, ℝN × N) and W ∈ C1(ℝ × ℝN, ℝ) are not periodic in t. Our results generalize and improve some existing results in the literature by relaxing the conditions on the potential function W(t, x).

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.

A characterization is obtained on weight function u so that is bounded for 1 < p < ∞ and 0 < q < ∞, where T are integral operators and related maximal operators, and for 0 < p, q < ∞, where T are geometric mean operators and related geometric maximal operators. The equivalence of such weighted inequalities for these operators are established.

A fully nonlinear, degenerate parabolic equation arising in the theory of damage mechanics is shown to be well-posed. Its solutions blow up in finite time and, under suitable conditions on the initial configuration, the blow-up set, corresponding to the portion of the material which breaks at the blow-up time, is an interval of nonzero measure. In a special but physically relevant case the problem reduces to the study of the blow-up set of solutions of the quasilinear equation