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 \`a toute structure de proto-big\`ebre de Lie sur un espace
vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie
d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre
ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres,
tous les crochets $n$-aires sont nuls pour $n \geq 4$ du fait qu'ils
proviennent d'une structure de proto-big\`ebre de Lie. Plus
g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de
l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une
collection d'applications multilin\'eaires antisym\'etriques sur
l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les
identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres
de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour
le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de
$F$ et de~$F^*$.
To any proto-Lie algebra structure on a finite-dimensional vector
space~$F$, we associate homotopy Lie algebra structures defined on
the suspension of the exterior algebra of $F$ and that of its dual
$F^*$, respectively. In these algebras, all $n$-ary brackets for $n
\geq 4$ vanish because the brackets are defined by the proto-Lie
algebra structure. More generally, to any element of odd degree in
the exterior algebra of the direct sum of $F$ and $F^*$, we associate
a set of multilinear skew-symmetric mappings on the suspension of the
exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized
Jacobi identities, defining the homotopy Lie algebras, if the given
element is of square zero with respect to the big bracket of the
exterior algebra of the direct sum of $F$ and~$F^*$.
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
\[
\frac{1}{m}\sum_{j=1}^{m}\psi ( \log | P( e^{i\tau
_{j}}) | ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left(
\log [ e| P( e^{i\tau }) | ] \right) \,d\tau
,
\]
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\leq \tau _{1}<\tau _{2}<\dots<\tau
_{m}\leq 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 old results 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 $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. 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
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is
non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasi-Hamiltonian $U$-space
$(M, \w, \mu\from M \to U)$ 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
$I={\mathcalR}(R)$ 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.