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 $l=-2$.

We present some transference results for a convolution operator with kernel $K$ which is bounded from $L^{p_0}(w_0)$ into $L^{p_1}(w_1)$. Our results are a natural extension of the classical results of Coifman and Weiss and their further extensions. We shall give several applications in the setting of restriction of Fourier multipliers.

We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.

We investigate the behaviour of the eigenvalues of a self-adjoint Sturm–Liouville problem with a separated boundary condition when the interval of the problem shrinks to an end point. It is shown that all the eigenvalues, except possibly the first, approach $+\infty$. The choices of the boundary condition are found for which the first eigenvalue tends to $+\infty$, independent of the coefficient functions, and the same is done for the $-\infty$ limit. For the remaining choices of the boundary condition, several types of condition on the coefficient functions are given, so that the first eigenvalue has a finite or infinite limit and, when the limit is finite, an explicit expression for the limit is obtained. Moreover, numerous examples are presented to illustrate these results, and a construction is given to perturb the finite-limit case to the no-limit case.

By using the moving plane method combined with integral inequalities and Hardy's inequality, some new Liouville-type theorems for semilinear polyharmonic equations in $\mathbb{R}^N$ and in $\mathbb{R}^N_+$ are proved.

The global entropy solution to the radial motion of isothermal self-gravitating gases is studied. A transformation of variables is applied. The global approximate solutions are constructed by a generalized shock-capturing scheme. The source terms include an integral which is non-local and is carefully estimated. The uniform estimate of the approximate solutions is obtained. The convergence of the approximate solutions is proved by the compensated compactness method. The global existence of weak entropy solution with large $L^{\infty}$ initial data is established. The solution allows a possible vacuum state.

We consider the Cauchy problem for quasilinear parabolic equations $u_t=\Delta\phi(u)+f(u)$, with the bounded non-negative initial data $u_0(x)$ ($u_0(x)\not\equiv0$), where $f(\xi)$ is a positive function in $\xi>0$ satisfying a blow-up condition $\int_1^{\infty}1/f(\xi)\,\mathrm{d}\xi<\infty$. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation $\mathrm{d} v/\mathrm{d} t=f(v)$ with the initial data $\|u_0\|_{L^{\infty}(\mathbb{R}^N)}>0$. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on $u_0$ for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on $u_0$ for blow-up with the least blow-up time, provided that $f(\xi)$ grows more rapidly than $\phi(\xi)$.

The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography.

In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4), 897–936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.

Let $H$ be a torsion-free compact $p$-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra $\varLambda_H$ of $H$ has trivial centre and use this result to classify the prime $c$-ideals in the Iwasawa algebra $\varLambda_G$ of $G:=H\times\mathbb{Z}_p$. We also show that a finitely generated torsion $\varLambda_G$-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are Noetherian whenever R is; this question arose naturally with Bernstein's fundamental work for R = ℂ, in which case he proved this Noetherian property. In a first step, we prove that Noetherianity would follow from a generalization of the so-called second adjointness property between parabolic functors, also due to Bernstein for complex representations. Then, to attack this second adjointness, we introduce and study ‘parahoric functors’ between representations of groups of integral points of smooth integral models of G and of their ‘Levi’ subgroups. Applying our general study to Bruhat-Tits parahoric models, we get second adjointness for minimal parabolic groups. For non-minimal parabolic subgroups, we have to restrict to classical and linear groups, and use smooth models associated with Bushnell-Kutzko and Stevens semi-simple characters. The same strategy should apply to ‘tame’ groups, using Yu's smooth models and generic characters.