We obtain a removal lemma for systems of linear equations over the circle group, using a similar result for finite fields due to Král′, Serra and Vena, and we discuss some applications.

The famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and if

is the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

In this paper, we investigate the existence of global weak solutions to an integrable two-component Camassa–Holm shallow-water system, provided the initial data u0(x) and ρ0(x) have end states u± and ρ±, respectively. By perturbing the Cauchy problem of the system around rarefaction waves of the well-known Burgers equation, we obtain a global weak solution for the system under the assumptions u− ≤ u+ and ρ− ≤ ρ+.

Given a sequence of matrices (Am)m∈ℕ whose Lyapunov exponents are limits, we show that this asymptotic behaviour is reproduced by the sequences xm+1 = Amxm + fm(xm) for any sufficiently small perturbations fm. We also consider the general case of exponential rates ecρm for an arbitrary increasing sequence ρm. Our approach is based on Lyapunov's theory of regularity.

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers $n$ with $8\nmid n$, $n\not\equiv 2~(\text{mod} ~3)$ and $n\not\equiv 14~(\text{mod} ~16)$ are the sum of two squares and three biquadrates.

In this paper, a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier–Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids; thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model verifies a dissipative entropy inequality up to a second-order term. Moreover, we propose a correction of the model – by taking into account the second-order extension for the pressure – that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first test, we compare the numerical results with the viscous bilayer shallow water model proposed in Narbona-Reina et al. (Comput. Model. Eng. Sci., 2009, Vol. 43, pp. 27–71). In the second test, the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test, the influence of the friction coefficient on the coastal area affected by the pollutant is studied.

Let $p\gt 2$ be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod $p$ representations of the absolute Galois group of ${ \mathbb{Q} }_{p} $. We also state a conjectural generalization to $n$-dimensional representations of the absolute Galois group of an arbitrary finite extension of ${ \mathbb{Q} }_{p} $, and give a conditional proof of this conjecture, subject to a certain $R= \mathbb{T} $-type theorem together with a strong version of the weight part of Serre’s conjecture for rank $n$ unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate representations.

We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graph H. Maker aims to occupy a given target graph G and Breaker tries to prevent Maker from achieving his goal. We show that for every d there is a constant c = c(d) with the property that for every graph G on n vertices of maximum degree d there is a graph H on at most cn edges such that Maker has a strategy to occupy a copy of G in the game on H.

This is a result about a game-theoretic variant of the size Ramsey number. For a given graph G, $\hat{r}'(G)$ is defined as the smallest number M for which there exists a graph H with M edges such that Maker has a strategy to occupy a copy of G in the game on H. In this language, our result yields that for every connected graph G of constant maximum degree, $\hat{r}'(G) = \Theta(n)$.

Moreover, we can also use our method to settle the corresponding extremal number for universal graphs: for a constant d and for the class ${\cal G}_{n}$ of n-vertex graphs of maximum degree d, $s({\cal G}_{n})$ denotes the minimum number such that there exists a graph H with M edges where, for everyG ∈ ${\cal G}_{n}$, Maker has a strategy to build a copy of G in the game on H. We obtain that $s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})$.

Given a set $A\subset\mathbb{Z}_{N}$, we say that a function $f\colon A \to \mathbb{Z}_{N}$ is a Freiman homomorphism if f(a)+f(b)=f(c)+f(d) whenever a,b,c,d ∈ A satisfy a+b=c+d. This notion was introduced by Freiman in the 1970s, and plays an important role in the field of additive combinatorics. We say that A is linear if the only Freiman homomorphisms are functions of the form f(x) = ax+b.

Suppose the elements of A are chosen independently at random, each with probability p. We shall look at the following question: For which values of p=p(N) is A linear with high probability as N → ∞? We show that if p=(2logN − ω(N))1/3N−2/3, where ω(N) → ∞ as N → ∞, then A is not linear with high probability, whereas if p=N−1/2+ε for any ε>0 then A is linear with high probability.

This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability of taking other values.

We generalize Reimer's Inequality [6] (a.k.a. the BKR Inequality or the van den Berg–Kesten Conjecture [1]) to the setting of finite distributive lattices.

It is shown that the set of conservation laws for the nonlinear system of equations describing plane steady potential barotropic flow of gas is given by the set of conservation laws for the linear Chaplygin system. All the conservation laws of zero order for the Chaplygin system are found. These include both known and new nonlinear conservation laws. It is found that the number of conservation laws of the first order is not more than three, assuming that the laws do not depend on the velocity potential and are not non-obvious ones. The components of these conservation laws are quadratic with respect to the stream function and its derivatives. All the Chaplygin functions are found, for which the Chaplygin system has three non-obvious conservation laws of the first order that are independent of velocity potential. All such non-obvious first-order conservation laws are found.