We prove that the only solutions in coprime positive integers to the equationare (x, y, z)=(n!–2, 1, 1, n), n≥3.

Let be a sequence of real numbers satisfying for each k ≥ 0, where M ≥ 1 is a fixed number. We prove that, for any sequence of real numbers , there is a real number ξ such that for each k ≥ 0. Here, denotes the distance from to the nearest integer. This is a corollary derived from our main theorem, which is a more general matrix version of this statement with explicit constants.

Let K[x] be a polynomial algebra in a variable x over a commutative -algebra K, and Γ′ the monoid of K-algebra monomorphisms of K[x] of the type σ: x ↦ x+λ2x2 + . . . +λnxn, λi ∈ K, λn is a unit of K. It is proved that for each σ ∈ Γ′ there are only finitely many distinct decompositions σ = σ1. . .σs in Γ′. Moreover, each such decomposition is uniquely determined by the degrees of components: if σ = σ1. . . σs= τ1 . . . τs then σ1=τ1, λ. . ., σs=τs if and only if deg(σ1)=deg(τ1), . . ., deg(σs)=deg(τs). Explicit formulae are given for the components σi via the coefficients λj and the degrees deg(σk) (as an application of the inversion formula for polynomial automorphisms in several variables from [1]). In general, for a polynomial there are no formulae (in radicals) for its divisors (elementary Galois theory). Surprisingly, one can write such formulae where instead of the product of polynomials one considers their composition (as polynomial functions).

We describe a short and easy-to-analyse construction of constant-degree expanders. The construction relies on the replacement product, applied by Reingold, Vadhan and Wigderson (2002) to give an iterative construction of bounded-degree expanders. Here we give a simpler construction, which applies the replacement product (only twice!) to turn the Cayley expanders of Alon and Roichman (1994), whose degree is polylog n, into constant-degree expanders. This enables us to prove the required expansion using a simple new combinatorial analysis of the replacement product (instead of the spectral analysis used by Reingold, Vadhan and Wigderson).

It is shown that if C1 and C2 are maximal abelian self-adjoint subalgebras (masas) of C*-algebras A1 and A2, respectively, then the completion C1 ⊗ C2 of the algebraic tensor product C1 ⊙ C2 of C1 and C2 in any C*-tensor product A1 ⊗βA2 is maximal abelian provided that C1 has the extension property of Kadison and Singer and C2 contains an approximate identity for A2. Examples are given to show that this result can fail if the conditions on the two masas do not both hold. This gives an answer to a long-standing question, but leaves open some other interesting problems, one of which turns out to have a potentially intriguing implication for the Kadison-Singer extension problem.

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomized algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

This paper contributes to the classification problem of pq dimensional Hopf algebras H over an algebraically closed field k of characteristic 0, where p, q are odd primes. It is shown that such Hopf algebras H are semisimple for the pairs of odd primes (p, q)=(3,11),(3,13),(3,19),(5,17),(5,19),(5,23),(5,29),(7,17),(7,19),(7,23),(7,29),(11,29),(13,29).

In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered block-groups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev products NOS = EIPOI and NOS ∩ A = NPOI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated by all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing the equalities BG = EIEcom = NEcom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.

We study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspace KB generated by the Blaschke product B.

We show that for 0<α<1 and θ>−α, the Poisson–Dirichlet distribution with parameter (α, θ) is the unique reversible distribution of a rather natural fragmentation–coalescence process. This completes earlier results in the literature for certain split-and-merge transformations and the parameter α = 0.

We derive here the Friedland–Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in r-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three-dimensional monomer–dimer entropy.

In this short note we present a result of Perelman with detailed proof. The result states that if $g(t)$ is the Kähler Ricci flow on a compact, Kähler manifold $M$ with $c_1(M)>0$, the scalar curvature and diameter of $(M,g(t))$ stay uniformly bounded along the flow, for $t\in[0,\infty)$. We learned about this result and its proof from Grigori Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the Kähler Ricci flow.

Let K be an arbitrary number field, and let ρ : Gal(/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal(/ℚ) when n > 2.

This paper is a brief survey of recent results and some open problems related to linear groups of finite Morley rank, an area of research where Bruno Poizat's impact is very prominent. As a sign of respect to his strongly expressed views that mathematics has to be done, written and pulished only in the native tongue of the immediate author–the scribe, in effect–of the text, I insist on writing my paper in Russian, even if the results presented belong to a small but multilingual community of researchers of American, British, French, German, Kazakh, Russian, Turkish origin. To emphasise even further the linguistic subtleties involved, I use British spelling in the English fragments of my text.

We prove the existence of the global smooth solutions to $p$-systems with damping under weaker conditions than those given by Lin and Zheng. The analysis is based on several key a priori estimates, which are obtained via the maximum principle.

A model focusing on key components involved in tumour invasion is studied. Tumour cell migration is based on cell motility and haptotaxis, i.e., the directed migratory response of tumour cells up gradients of cell-adhesion molecules. Individual cell processes are modelled according to cell age and several tumour phenotypes are incorporated. Global existence and uniqueness of nonnegative solutions to the corresponding coupled system of nonlinear partial differential equations are shown.

A physically transparent and mathematically streamlined derivation is presented for a third-order nonlinear dynamical system that describes the curious chiral reversals of a celt (rattleback). The system is integrable, and its solutions are periodic, showing an infinite succession of spin reversals. Inclusion of linear dissipation allows any given number of reversals, and a typical celt's observed behaviour is well captured by tuning the dissipation parameters.

In this paper we prove that the best constant in the Sobolev trace embedding $H^1(\varOmega)\hookrightarrow L^q(\partial\varOmega)$ in a bounded smooth domain can be obtained as the limit as $\varepsilon\to0$ of the best constant of the usual Sobolev embedding $H^1(\varOmega) \hookrightarrow L^q(\omega_\varepsilon,\mathrm{d} x/\varepsilon)$, where $\omega_\varepsilon=\{x\in\varOmega:\mathrm{dist}(x,\partial\varOmega)<\varepsilon\}$ is a small neighbourhood of the boundary. We also analyse symmetry properties of extremals of the latter embedding when $\varOmega$ is a ball.

In both modern stochastic analysis and more traditional probability and statistics, one way of characterizing a static or dynamic probability distribution is through its quantile function. This paper is focused on obtaining a direct understanding of this function via the classical approach of establishing and then solving differential equations for the function. We establish ordinary differential equations and power series for the quantile functions of several common distributions. We then develop the partial differential equation for the evolution of the quantile function associated with the solution of a class of stochastic differential equations, by a transformation of the Fokker–Planck equation. We are able to utilize the static formulation to provide elementary time-dependent and equilibrium solutions.

Such a direct understanding is important because quantile functions find important uses in the simulation of physical and financial systems. The simplest way of simulating any non-uniform random variable is by applying its quantile function to uniform deviates. Modern methods of Monte–Carlo simulation, techniques based on low-discrepancy sequences and copula methods all call for the use of quantile functions of marginal distributions. We provide web resources for prototype implementations in computer code. These implementations may variously be used directly in live sampling models or in a high-precision benchmarking mode for developing fast rational approximations also for use in simulation.

This paper is contributed to the inhomogeneous semilinear elliptic equation

\begin{equation}\Delta u+K(|x|)u^{p}+f(x)=0\quad\text{in }\mathbb{R}^n, \tag{$*$} \label{*}\end{equation}

where

$$\Delta=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$$

is the Laplacian operator, $n\geq3$, $p>1$, $f(x)\geq0$ and $K(|x|)>0$ is a given locally Hölder continuous function in $\mathbb{R}^n\setminus\{0\}$. The existence, non-existence and decay properties of positive solutions for \eqref{*} are obtained under some assumptions on $f(x)$ and $K(|x|)$ satisfying the slow-decay condition, i.e. $K(|x|)\geq C|x|^{l}$ at infinity for some constants $C>0$ and $l>-2$. The decay properties of positive solutions for $(\ast)$ are also discussed for the critical decay case on $K(|x|)$ with $l=-2$.