Let $K$ be a totally real number field of degree $n$ . We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

On associe à toute structure de proto-bigèbre de Lie sur un espace vectoriel $F$ de dimension finie des structures d’algèbre de Lie d’homotopie définies respectivement sur la suspension de l’algèbre extérieure de $F$ et celle de son dual ${{F}^{*}}$ . Dans ces algèbres, tous les crochets $n$ -aires sont nuls pour $n\,\ge \,4$ du fait qu’ils proviennent d’une structure de proto-bigèbre de Lie. Plus généralement, on associe à un élément de degré impair de l’algèbre extérieure de la somme directe de $F$ et ${{F}^{*}}$ , une collection d’applications multilinéaires antisymétriques sur l’algèbre extérieure de $F$ (resp. ${{F}^{*}}$ ), qui vérifient les identités de Jacobi généralisées, définissant les algèbres de Lie d’homotopie, si l’élément donné est de carré nul pour le grand crochet de l’algèbre extérieure de la somme directe de $F$ et de ${{F}^{*}}$ .

In this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of such modules is given by spaces of jets of tensor fields.

We investigate large sieve inequalities such as

$$\frac{1}{m}\underset{j=1}{\overset{m}{\mathop{\sum }}}\,\,\psi \left( \log \,|P({{e}^{i\tau j}})| \right)\,\le \,\frac{C}{2\pi }\,\int_{0}^{2\pi }{\psi }\,\left( \log [e\,|P({{e}^{i\tau }})|] \right)\,d\tau ,$$

where $\psi$ is convex and increasing, $P$ is a polynomial or an exponential of a potential, and the constant $C$ depends on the degree of $P$ , and the distribution of the points $0\,\le \,{{\tau }_{1}}\,<\,{{\tau }_{2}}\,<\,\cdots \,<\,{{\tau }_{m}}\,\le \,2\pi$ . The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials.

We study the geometry of the set of closed extensions of index 0 of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

Let $G$ be a locally compact group and $\pi $ a representation of $G$ by weakly* continuous isometries acting in a dual Banach space $E$ . Given a probability measure $\mu $ on $G$ , we study the Choquet–Deny equation $\pi (\mu )x\,=\,x,\,x\,\in \,E$ . We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu $ . The relation between the space of solutions of the Choquet–Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the ${{W}^{*}}$ -crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space $E\,=\,B({{L}^{2}}(G))$ of bounded linear operators on ${{L}^{2}}(G)$ . Other general properties of the Choquet–Deny equation in a Banach space are also discussed.

Let $X$ be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the “forward” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$ . Thus we obtain for infinite $X$ that the $n$ -step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but small cycles are dense in $X$ , we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$ -step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.

In this paper, we characterize unitary representations of $\pi \text{ }:=\text{ }\pi {{\text{ }}_{1}}({{S}^{2}}\backslash \{{{s}_{1}},\ldots ,{{s}_{l}}\})$ whose generators ${{u}_{1}},\,\ldots ,\,{{u}_{l}}$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions ${{u}_{j}}\,=\,{{\sigma }_{j}}{{\sigma }_{j+1}}$ with ${{\sigma }_{l+1}}\,=\,{{\sigma }_{1}}$ . Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space ${{\mathcal{M}}_{C}}\,:\,=\,\text{Ho}{{\text{m}}_{C}}(\pi ,\,U(n))\,/\,U(n)$ . Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of ${{\mathcal{M}}_{C}}$ . To prove this, we use the quasi-Hamiltonian description of the symplectic structure of ${{\mathcal{M}}_{C}}$ and give conditions on an involution defined on a quasi-Hamiltonian $U$ -space $(M,\,\omega ,\,\mu :\,M\to \,U)$ for it to induce an anti-symplectic involution on the reduced space $M//U:=\,{{\mu }^{-1}}\,(\{1\})/U$ .

An ideal $I$ of a ring $R$ is called a radical ideal if $I\,=\,\mathcal{R}(R)$ where $\mathcal{R}$ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that if $R$ is a valuation domain, then a proper ideal $I$ of $R$ is a radical ideal if and only if $I$ is a distinguished ideal of $R$ (the latter property means that if $J$ and $K$ are ideals of $R$ such that $J\,\subset \,I\,\subset \,K$ then we cannot have $I/J\,\cong \,K/I$ as rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.