Let $G$ be a compact $p$-valued $p$-adic Lie group, and let $\varLambda(G)$ be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion $\varLambda(G)$-modules, up to pseudo-isomorphism, which are largely parallel to the classical theory when $G$ is abelian (except for basic differences which occur for those torsion modules which do not possess a non-zero global annihilator). We illustrate our general theory by concrete examples of such modules arising from the Iwasawa theory of elliptic curves without complex multiplication over the field generated by all of their $p$-power torsion points.

AMS 2000 Mathematics subject classification: Primary 11G05; 11R23; 16D70; 16E65; 16W70

We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point $Q(X)$ to any end $X\in\partial F_n$, given an action of $F_n$ on an $\bm{R}$-tree $T$ with trivial arc stabilizers; this point $Q(X)$ may be in $T$, or in its metric completion, or in its boundary.

AMS 2000 Mathematics subject classification: Primary 20F65; 20E05; 20E08

Nous conjecturons que la réduction modulo $p$ des représentations cristallines irréductibles de dimension 2 sur $\bar{\bm{Q}}_p$ de $\Gal(\bar{\bm{Q}}_p/\bm{Q}_p)$ peut être prédite par la réduction modulo $p$ de représentations $p$-adiques localement algébriques de $\GL_2(\bm{Q}_p)$. Nous explicitons quelques calculs de telles réductions confirmant cette conjecture. Cela suggère un lien arithmétique non trivial entre les deux types de représentations.

AMS 2000 Mathematics subject classification: Primary 11F33; 11F70; 11F80; 11F85

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus\{0,\mu_n,\infty\}$. Let $n$ be a prime number. We assume that for any $m\geq 1$ the numbers $Li_{m+1}(\xi_n^k)$ for $1\leq k\leq (n-1)/2$ are linearly independent over $\mathbb{Q}$ in $\mathbb{C}/(2\pi\ri)^{m+1}\mathbb{Q}$. Let $S=\{k_1,\cdots,k_q\}$ be a subset of $\{1,\dots,p-1\}$ such that if $k\in S$, then $p-k\in S$ and $(S+S)\cap S=\emptyset$ (the sum of two elements of $S$ is calculated $\mathrm{Mod}p$). Then we show that in the Lie algebra associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus \{0,\mu_n,\infty\}$ there are derivations $D^{k_1}_{m+1},\dots,D^{k_q}_{m+1}$ in each degree $m+1$ and these derivations are free generators of a free Lie subalgebra of this Lie algebra.

AMS 2000 Mathematics subject classification: Primary 11G55

For a function $f\in L^p(\mathbb{R}d)$, $d\ge 2$, let $A_tf(x)$ be the mean of $f$ over the sphere of radius $t$ centred at $x$. Given a set $E\subset(0,\infty)$ of dilations we prove various endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E}|A_tf(x)|$, under some regularity assumptions on $E$.

AMS 2000 Mathematics subject classification: Primary 42B25. Secondary 42B20

We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.

This paper proves the existence and uniqueness of a monotone increasing solution of the Painlevé 1 equation y″ = y2+x. The monotonicity of solution is then exploited to show stability of the plasma-sheath transition in a weakly ionizing plasma.

A generalization of the Keller–Segel model for chemotactic systems is studied. In this model there are several populations interacting via several sensitivity agents in a two-dimensional domain. The dynamics of the population is determined by a Fokker–Planck system of equations, coupled with a system of diffusion equations for the chemical agents. Conditions for global existence of solutions and equilibria are discussed, as well as the possible existence of time-periodic attractors. The analysis is based on a variational functional associated with the system.

A general formalism is described whereby some regular singular points are effectively removed and substantial simplifications ensue for a class of Fuchsian ordinary differential equations, and related confluent equations. These simplifications follow provided the exponents at the singular points satisfy certain relations; explicit, illustrative examples are constructed to demonstrate the ideas.

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

We prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in Hs(R), cannot be solved by an iterative scheme based on the Duhamel formula for s < −1 if (k, d) = (2, 0) and s < sc(k, d) = ½ − (2 − d)/(k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in Hs(R) for these equations and shows the particularity of the case (k, d) = (2, 0), for which we prove that the critical space Hsc(R) = H−3/2(R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

This paper is concerned with the existence of stationary solutions for some cross-diffusion systems with small parameters. Using a singular perturbation method, we prove the existence of stationary solutions with boundary and interior layers, which extends the results of Fife and Ito to the cross-diffusion cases.

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).

We consider the class of nonlinear eigenvalue problemswhere yp* = |y|p sgn y, pi > 0 and p0p1 … pn−1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.

We also consider the semilinear problemwhere f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω,and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.

Let Ai, i = 1, …, m, be a set of Ni × Ni−1 strictly totally positive (STP) matrices, with N0 = Nm = N. For a vector x = (x1, …, xN) ∈ RN and arbitrary p > 0, setWe consider the eigenvalue-eigenvector problemwhere p1 … pm−1 = r. We prove an analogue of the classical Gantmacher-Krein theorem for the eigenvalue-eigenvector structure of STP matrices in the case where pi ≥ 1 for each i, plus various extensions thereof.

In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

We obtain the existence and decay rates of the classical solution to the initial-value problem of a non-uniformly parabolic equation. Our method is to set up two equivalent sequences of the successive approximations. One converges to a weak solution of the initial-value problem; the other shows that the weak solution is the classical solution for t > 0. Moreover, we show how bounds of the derivatives to the classical solution depend explicitly on the interval with compact support in (0, ∞). Then we study decay rates of this classical solution.