Search

Given a ring endowed with a ring order, we provide sufficient conditions for the order topology induced by the ring order to become a ring topology (analogous results for module orders are consequently derived). Finally, the notions of Radon and regular measures are transported to the scope of module-valued measures through module orders. Classical characterizations of these measures are obtained as well as the hereditariness of regularity for conditional ring-valued measures. [ABSTRACT FROM AUTHOR]

We propose a new strategy for curve reconstruction in an image through an off-the-grid variational framework, inspired by spike reconstruction in the literature. We introduce a new functional CROC on the space of 2-dimensional Radon measures with finite divergence denoted V, and we establish several theoretical tools through the definition of a certificate. Our main contribution lies in the sharp characterization of the extreme points of the unit ball of the V -norm: there are exact measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise characterization of our functional minimizers and further opening a promising avenue for the algorithmic implementation. [ABSTRACT FROM AUTHOR]

We study existence and uniqueness of Radon measure-valued solutions for a class of nonlinear elliptic equations in inhomogeneous media. Solutions are constructed by a regularization procedure which relies on a standard approximation of the measure data and satisfy both a persistence property and a compatibility condition prescribing the structure of their concentrated and diffuse parts with respect to a suitable capacity. [ABSTRACT FROM AUTHOR]

A Markov process is constructed in an explicit way for an infinite system of entities arriving in and departing from a habitat X, which is a locally compact Polish space with a positive Radon measure χ. Along with its location x ∈ X, each particle is characterized by age α ≥ 0 - time since arriving. As the state space one takes the set of marked configurations ..., equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator L, expressed through the measure χ and a departure rate m(x, α) ≥ 0, and acting on bounded continuous functions F : ... → ℝ. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero. [ABSTRACT FROM AUTHOR]

An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space , the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case. [ABSTRACT FROM AUTHOR]

We define a local (mean curvature flow) entropy for Radon measures in Rn or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen-Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen-Cahn equation has unit density for all future time. [ABSTRACT FROM AUTHOR]

In this paper, we present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case V ϲ C+(X) is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case V ϲ C+(X), is presented in [3], Chapter 2, subparagraph 2.5. [ABSTRACT FROM AUTHOR]

The main result of this paper, Theorem 1.4, establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a self-adjoint reproducing kernel, the system of kernels sampled at the points of a random configuration is complete in the range of the kernel. A key step in the proof, Lemma 1.9, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.10 is a new local property for kernels of conditional point processes. In Theorem 1.6 we prove the triviality of the tail -algebra for determinantal point processes governed by self-adjoint kernels. [ABSTRACT FROM AUTHOR]

We study the existence of measure-valued solutions for a class of degenerate elliptic equations with measure data. The notion of solution is natural, since it is obtained by a regularization procedure which also relies on a standard approximation of the datum μ. We provide partial uniqueness results and qualitative properties of the constructed solutions concerning, in particular, the structure of their diffuse part with respect to the harmonic-capacity.

We study the formation of singularities for the problem { u t = φ (u) x x + ε ψ (u) t x x in Ω × (0 , T) φ (u) + ε ψ (u) t = 0 in ∂ Ω × (0 , T) u = u 0 ≥ 0 in Ω × { 0 } , where ϵ and T are positive constants, Ω a bounded interval, u 0 a nonnegative Radon measure on Ω, φ a nonmonotone and nonnegative function with φ (0) = φ (∞) = 0 , and ψ an increasing bounded function. We show that if u 0 is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures. [ABSTRACT FROM AUTHOR]

The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set. [ABSTRACT FROM AUTHOR]

We look at continuum solutions in optimisation problems associated to linear inverse problems $ y = Ax $ with non-negativity constraint $ x \geq 0 $. We focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which cover a wide range of common noise statistics such as Gaussian and Poisson. Considering $ x $ as a Radon measure over the domain on which the reconstruction is taking place, we show a general singularity result. In the high noise regime corresponding to $ y \notin\{A x \mid x \geq 0\} $ and under a key assumption on the divergence as well as on the operator $ A $, any optimiser has a singular part with respect to the Lebesgue measure. We hence provide an explanation as to why any possible algorithm successfully solving the optimisation problem will lead to undesirably spiky-looking images when the image resolution gets finer, a phenomenon well documented in the literature. We illustrate these results with several numerical examples inspired by medical imaging. [ABSTRACT FROM AUTHOR]

Copyright of Ricerche di Matematica is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Weighted spaces of continuous functions are introduced such that Weyl fractional integrals with orders from any finite nonnegative interval define equicontinuous sets of continuous linear endomorphisms for which the semigroup law of fractional orders is valid. The result is obtained from studying continuity and boundedness of convolution as a bilinear operation on general weighted spaces of continuous functions and measures. [ABSTRACT FROM AUTHOR]

Let Q = (0 , T) × Ω , where Ω is a bounded open subset of R d. We consider the parabolic p -capacity on Q naturally associated with the usual p -Laplacian. Droniou, Porretta, and Prignet have shown that if a bounded Radon measure μ on Q is diffuse, i.e. charges no set of zero p -capacity, p > 1 , then it is of the form μ = f + div (G) + g t for some f ∈ L 1 (Q) , G ∈ (L p ′ (Q)) d and g ∈ L p (0 , T ; W 0 1 , p (Ω) ∩ L 2 (Ω)). We show the converse of this result: if p > 1 , then each bounded Radon measure μ on Q admitting such a decomposition is diffuse. [ABSTRACT FROM AUTHOR]

Abstract We prove a global Calderón–Zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right-hand side is a bounded Radon measure. Minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove the existence of an initial trace T u for any positive solution u to the semilinear fractional diffusion equation (H) ∂ t u + (− Δ) s u + f (t , x , u) = 0 in (0 , + ∞) × R N , where N ≥ 1 , the operator (− Δ) s with s ∈ (0 , 1) is the fractional Laplacian, f : R + × R N × R + → R is a Caratheodory function satisfying f (t , x , u) u ≥ 0 for all (t , x , u) ∈ R + × R N × R + and R + = [ 0 , + ∞). We define the regular set of the trace T u as an open subset of R u ⊂ R N carrying a nonnegative Radon measure ν u such that lim t → 0 ⁡ ∫ R u u (t , x) ζ (x) d x = ∫ R u ζ d ν u , ∀ ζ ∈ C 0 2 (R u) , and the singular set S u = R N ∖ R u as the set points a such that lim sup t → 0 ∫ B ρ (a) u (t , x) d x = + ∞ for any ρ > 0. We also study the reverse problem of constructing a positive solution to (H) with a given initial trace (S , ν) , where S ⊂ R N is a closed set and ν is a positive Radon measure on R = R N ∖ S and develop the case f (t , x , u) = t β u p with β > − 1 and p > 1. [ABSTRACT FROM AUTHOR]

Abstract Realistic models of traffic flow are nonlinear and involve nonlocal effects in balance laws. Flow characteristics of different types of vehicles, such as cars and trucks, need to be described differently. Two alternatives are used here, $$L^p$$ L p -valued Lebesgue measurable density functions and signed Radon measures. The resulting solution spaces are metric spaces that do not have a linear structure, so the usual convenient methods of functional analysis are no longer applicable. Instead ideas from mutational analysis will be used, in particular the method of Euler compactness will be applied to establish the well-posedness of the nonlocal balance laws. This involves the concatenation of solutions of piecewise linear systems on successive time subintervals obtained by freezing the nonlinear nonlocal coefficients to their values at the start of each subinterval. Various compactness criteria lead to a convergent subsequence. Careful estimates of the linear systems are needed to implement this program.

We introduce and analyze a new, nonlinear fourth-order regularization of forwardbackward parabolic equations. In one space dimension, under general assumptions on the potentials, which include those of Perona--Malik type, we prove existence of Radon measure-valued solutions under both natural and essential boundary conditions. If the decay at infinity of the nonlinearities is sufficiently fast, we also exhibit examples of local solutions whose atomic part arises and/or persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations). [ABSTRACT FROM AUTHOR]

Let 1 < p < 1 and let Ω be a bounded domain of RN (N ≥ 1). In this paper, we consider a class of second order quasilinear elliptic operators A in Ω including the p-Laplace operator Δp. First we establish various type of Kato's inequalities for A when Au is a Radon measure. Then we prove the inverse maximum principle and describe the strong maximum principle. For this purpose it is crucial to introduce a notion of admissible class for the operator A and use it effectively. [ABSTRACT FROM AUTHOR]

Abstract In this article we consider weak solutions of the Euler- α equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than O ((t log ⁡ t) 1 / 4). This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and p -th power integrable, p > 2 , due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati. [ABSTRACT FROM AUTHOR]

The tail measure of a regularly varying stationary time series has been recently introduced. It is used in this contribution to reconsider certain properties of the tail process and establish new ones. A new formulation of the time change formula is used to establish identities, some of which were indirectly known and some of which are new. [ABSTRACT FROM AUTHOR]

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative solutions, under a quadratic lower bound on the Ricci curvature. On the other hand, we prove that any weak solution of the porous medium equation necessarily takes on as initial datum a finite Radon measure. In addition, we obtain some results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions, and properties of potentials of positive Radon measures. Those results are new and of independent interest, and are crucial for our approach. [ABSTRACT FROM AUTHOR]

Highlights • Near-surface soil gas and groundwater data can vary naturally and significantly. • Multiple approaches should be employed when assessing near-surface site data. • Air temperature modeling of soil gas data can assist in natural trend evaluation. • Identification of key groundwater parameters can assist in CO 2 exposure monitoring. • Modeling near-surface site data enhances monitoring efficiency and effectiveness. Abstract Near-surface soil gas and groundwater measurements can be helpful tools in assuaging concerns of potential out-of-zone migration of CO 2 from a geologic storage unit into the overlying near-surface environment. These data, therefore, help to build confidence with local stakeholders and regulators that stored CO 2 is not impacting surface/near-surface environments. Routine monitoring of soil gas concentrations in the vadose zone can be used to show a lack of change or effect. However, both air temperature modeling and the Romanak et al. (2012) process-based approach should be applied when soil gas data are evaluated, as increased CO 2 concentrations can occur naturally from changes in the soil environment. Laboratory testing of groundwater and formation rock (drill cuttings) samples, exposed to varying concentrations of CO 2 under in situ temperature and pressure conditions, yield valuable information with respect to water chemistry changes that could occur from a potential out-of-zone migration. Key field-measured groundwater monitoring parameters that change significantly in response to low levels of CO 2 are pH (rapid decrease), alkalinity (increase), and conductivity (increase). Empirical models that predict soil gas concentrations using routinely measured climatic data such as air temperature, as well as models that predict the magnitude and duration of potential CO 2 exposure in groundwater, should be employed as components of a broad surface and subsurface monitoring program. [ABSTRACT FROM AUTHOR]

Let dμ(x1, . . . , xn) = dμ1(x1) . . . dμn(xn) be a product measure which is not necessarily doubling in Rn (only assuming dμi is doubling on R for i = 2, . . . , n), and Mndμ be the strong maximal function defined by ... [ABSTRACT FROM AUTHOR]

Highlights • A survey on a sample of 154 buildings was carried out. • The survey considers radon measures before and after energy retrofit. • A statistical analysis of results was accomplished. • Results underline the need of considering energy saving and indoor air quality at the same time. Abstract Energy regulation, policy and targets enhance energy retrofit in buildings with a wide distribution in Europe and Switzerland. These actions are mainly aimed at reducing heat dispersion through the envelope. The interventions affect the permeability of the envelope influencing indoor air quality. Focusing on radon concentration, this study reports the results of a survey on 154 buildings measuring the radon concentrations before and after energy remediation. The buildings were located in the southern part of Switzerland (Canton Ticino), a region with measurements of radon concentration in more than half of the buildings (over 55,000 building in 2018), within a population of approximately 355,000. These figures make this region an area with an exceptionally high number of radon measurements, performed in 2005–10 upon mandate of the local public health authorities. The survey reveals the increasing of radon concentrations, in particular where windows were replaced with more performant ones. Results underline the need of considering energy saving and indoor air quality at the same time, in the frameworks of orienting public and private investment towards improving long-term public health. Adequate techniques for improving ventilation could be very helpful to that end. [ABSTRACT FROM AUTHOR]

Abstract: The two‐weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures σ and w on R. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two‐weight problem by Lacey, Sawyer, Shen, and Uriarte‐Tuero. Our characterization is in terms of Sawyer‐type testing conditions and a variant of the two‐weight A 2 condition, where σ and w are integrated over complementary intervals only. A key novelty of the proof is a two‐weight inequality for the Poisson integral with ‘holes’. [ABSTRACT FROM AUTHOR]

We analyse the behaviour of solutions of the linear heat equation in R d for initial data in the classes M ε ( R d ) of Radon measures with ∫ R d e − ε | x | 2 d | u 0 | < ∞ . We show that these classes are optimal for local and global existence of non-negative solutions: in particular M 0 ( R d ) : = ∩ ε > 0 M ε ( R d ) consists of those initial data for which a solution of the heat equation can be given for all time using the heat kernel representation formula. We prove existence, uniqueness, and regularity results for such initial data, which can grow rapidly at infinity, and then show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t → ∞ and that one can prescribe the behaviour of u ( 0 , t ) to be any real-analytic function γ ( t ) on [ 0 , ∞ ) . [ABSTRACT FROM AUTHOR]

Regarding the some useful advantages of spherical coordinates for some special problems, in this paper, based on Radon measure properties, we present a new and basic solution method for general shape optimization problems defined in spherical coordinates. Indeed, our goal is to determine a bounded shape located over the (x, y)-plane, such that its projection in the (x, y)-plane and its volume is given and also it minimizes some given surface integral. To solve these kinds of problems, we somehow extend the embedding process in Radon measures space. First, the problem is converted into an infinite-dimensional linear programming one. Then, using approximation scheme and a special way for discretization in spherical region, this problem is reduced to a finite-dimensional linear programming one. Finally, the solution of this new problem is used to construct a nearly optimal smooth surface by applying an outlier detection algorithm and curve fitting. More than reducing the complexity, this approach in comparison with the other methods has some other advantages: linear treatment for even nonlinear problems, and the minimization is global and does not depend on initial shape and mesh design. Numerical examples are also given to demonstrate the effectiveness of the new method, especially for classical and obstacle problems. [ABSTRACT FROM AUTHOR]

In this paper, we aim to characterize the cylindrical Wigner measures associated to regular quantum states in the Weyl C*-algebra of canonical commutation relations. In particular, we provide conditions at the quantum level sufficient to prove the concentration of all the corresponding cylindrical Wigner measures as Radon measures on suitable topological vector spaces. The analysis is motivated by variational and dynamical problems in the semiclassical study of bosonic quantum field theories. [ABSTRACT FROM AUTHOR]

We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD2) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that GSBD2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD2 in dimension two. Moreover, for all d ≥ 2 we show GSBD2 (Ω) ⊂ (GBV (Ω; R))d and the embedding SBD2 (Ω) ∩ L∞(Ω; Rd) ,→ SBV (Ω; Rd) into the space of special functions of bounded variation (SBV ). Finally, we present a Korn-Poincaré inequality for functions with small jump sets in arbitrary space dimension. [ABSTRACT FROM AUTHOR]

It has been shown that all Radon probability measures on R are Mycielski-regular, as well as Lebesgue measure on the unit cube and certain self-similar measures. In this paper, these results are extended to Gibbs measures on cookie-cutter sets. [ABSTRACT FROM AUTHOR]

Abstract: We establish the boundedness and weak boundedness of the maximal operator and generalized fractional integral operators on generalized Morrey spaces over metric measure spaces ( X , d , μ ) without the assumption of the growth condition on μ. The results are generalization and improvement of some known results. We also give the vector‐valued boundedness. Moreover we prove the independence of the choice of the parameter in the definition of generalized Morrey spaces by using the geometrically doubling condition in the sense of Hytönen. [ABSTRACT FROM AUTHOR]

We prove the existence of parabolic initial boundary value problems of the type ut - div(a∈(t, x, u∈, ▽u∈)) = μ∈ in Q := (0, T) x Ω, μ∈ = 0 on (0, T) x ∂Ω, u∈(0) = u0, ∈ in Ω, with respect to suitable convergence of the nonlinear operators a∈ and of the measure data μ∈. As a consequence, we obtain the existence of a renormalized solution for a general class of nonlinear parabolic equations with right-hand side measure. [ABSTRACT FROM AUTHOR]

In this paper, we present a dispersive regularization approach to construct a global N -peakon weak solution to the modified Camassa-Holm equation (mCH) in one dimension. In particular, we perform a double mollification for the system of ODEs describing trajectories of N - peakon solutions and obtain N smoothed peakons without collisions. Though the smoothed peakons do not give a solution to the mCH equation, the weak consistency allows us to take the smoothing parameter to zero and the limiting function is a global N -peakon weak solution. The trajectories of the peakons in the constructed solution are globally Lipschitz continuous and do not cross each other. When N = 2, the solution is a sticky peakon weak solution. At last, using the N -peakon solutions and through a mean field limit process, we obtain global weak solutions for general initial data m0 in Radon measure space. [ABSTRACT FROM AUTHOR]

This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the distance as a geodesic distance over the space of measures (ii) a static “Kantorovich” formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures. Both formulations are convex optimization problems, and the ability to switch from one to the other depending on the targeted application is a crucial property of our models. Of particular interest is the Wasserstein–Fisher–Rao metric recently introduced independently by [7,15] . Defined initially through a dynamic formulation, it belongs to this class of metrics and hence automatically benefits from a static Kantorovich formulation. [ABSTRACT FROM AUTHOR]

We start from the question raised by Williams (Proc. Am. Math. Soc. 2016): Must a second countable, locally compact, transitive groupoid G have open range map? If the answer is positive, the topology of G is in fact locally transitive (in the sense of [Seda, 1976]). We prove that even if the answer is negative, we can replace the original topology of G with a local transitive topology so that the topologies of the r-fibres are not affected. The new topology is locally compact Hausdorff but not necessary second countable. However its full C*-algebra (introduced in [Renault, 1980]) is still isomorphic to C*(H)⊗K(L2(μ)), where H is the isotropy group at a unit u and μ is a positive Radon measure on the unit space. We also present a few remarks concerning the Haar systems on locally compact groupoids and for every locally compact groupoid having paracompact unit space and second countable r-fibres, we prove the existence of a pre-Haar system bounded on the compact sets. [ABSTRACT FROM AUTHOR]

The group of diffeomorphisms of a compact manifold endowed with the L 2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein–Fisher–Rao distance is a natural extension of the classical L 2 -Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the H div right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa–Holm (CH) equation in one dimension. Geometrically, we present an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L 2 type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On S 1 , solutions to the standard CH thus give radially 1-homogeneous solutions of the incompressible Euler equation on R 2 which preserves a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler–Arnold equation for the H div right-invariant metric are length minimizing geodesics for sufficiently short times. [ABSTRACT FROM AUTHOR]

The Bost-Connes system is a C*-dynamical system related to class field theory. The purpose of this paper is to construct Bost-Connes systems for local fields of characteristic zero. The notable phenomenon in the case of local fields is the absence of the phase transition. We also investigate the relation between the global and local Bost-Connes systems. [ABSTRACT FROM AUTHOR]