A Stokesian fluid in motion along a porous medium saturated by the same fluid is modelled by the Beavers—Joseph—Saffman boundary-value problem to generalized Forchheimer—Stokes—Fourier systems: what we call the Beavers—Joseph—Saffman (BJS) problem. The model has nonlinear character given by the temperature dependence of physical parameters such as the viscosity, the permeability, the thermal conductivity and the thermal expansion. The paper is concerned with the study of the steady-state and the time-dependent regimes via the Galerkin and the Faedo—Galerkin techniques, respectively.

The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.

The period function of Liénard systems ẋ = −y + F(x), ẏ = g(x) associated with a centre is considered and sufficient conditions are presented for the period function to be strictly and globally monotone increasing and to have at least one critical point. Examples of applications and remarks on related work are also given.

We study the Helmholtz equation with electromagnetic-type perturbations, in the exterior of a domain, in dimension n ≥ 3. Using multiplier techniques in the style of Morawetz, we prove a family of a priori estimates from which the limiting absorption principle follows. Moreover, we give some standard applications to cases with an absence of embedded eigenvalues and zero resonances, under explicit conditions on the potentials.

In this paper we study the maximum displacement for linear probing hashing. We use the standard probabilistic model together with the insertion policy known as First-Come-(First-Served). The results are of asymptotic nature and focus on dense hash tables. That is, the number of occupied cells n and the size of the hash table m tend to infinity with ratio n/m → 1. We present distributions and moments for the size of the maximum displacement, as well as for the number of items with displacement larger than some critical value. This is done via process convergence of the (appropriately normalized) length of the largest block of consecutive occupied cells, when the total number of occupied cells n varies.

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi’s golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_j\otimes F^n, s)$. This is the Rankin–Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane +2 with a step set that is a subset of

\[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\]

The analysis of randomized search heuristics on classes of functions is fundamental to the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in bounding the expected optimization time of a simple evolutionary algorithm, called (1+1) EA, on the class of linear functions. We improve the previously best known bound in this setting from (1.39+o(1))en ln n to en ln n+O(n) in expectation and with high probability, which is tight up to lower-order terms. Moreover, upper and lower bounds for arbitrary mutation probabilities p are derived, which imply expected polynomial optimization time as long as p = O((ln n)/n) and p = Ω(n−C) for a constant C > 0, and which are tight if p = c/n for a constant c > 0. As a consequence, the standard mutation probability p = 1/n is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. Furthermore, the algorithm turns out to be surprisingly robust since the large neighbourhood explored by the mutation operator does not disrupt the search.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

A graph is called universal for a given graph class (or, equivalently, -universal) if it contains a copy of every graph in as a subgraph. The construction of sparse universal graphs for various classes has received a considerable amount of attention. There is particular interest in tight -universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from . Arguably, the most studied case is that when is some class of trees. In this work, we are interested in (n,Δ), the class of all n-vertex trees with maximum degree at most Δ. We show that every n-vertex graph satisfying certain natural expansion properties is (n,Δ)-universal. Our methods also apply to the case when Δ is some function of n. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constant c such that the random graph G(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for (n,O(1)). Moreover, a corresponding result holds for the random regular graph of degree cn2/3log2n. Another interesting consequence is the existence of locally sparse n-vertex (n,Δ)-universal graphs. For example, we show that one can (randomly) construct n-vertex (n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose (n,Δ)-universal graphs with merely O(n) edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly (n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) × (2k + 1) array A on the symbols (1,. . .,2k+1) with at most c(2k+1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1) × (2k+1) Latin square S on the symbols 1,. . .,2k+1 such that, for each i,j ∈ {1,. . .,2k+1}, the symbol in position (i,j) of S does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integer n, if each cell in an n × n array B is assigned a set of m ≤ ρ n symbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability that B is avoidable tends to 1 as n → ∞.

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for ‘moments’ of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.