Absolutely Continuous Research Articles

One-dimensional Schrödinger operator with decaying white noise potential

In this paper, we consider a random one-dimensional Schrödinger operator Hω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize Hω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that Hω is actually self-adjoint with probability one. When the white noise in Hω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of Hω. We shall further prove that the negative part of the spectrum of Hω is discrete, with no accumulation points other than 0.

Distorted copulas

In this paper, we deal with the problem of preserving the copula property by means of a transformation called distortion where automorphisms of the real unit interval are involved. A new methodological approach is followed by resorting to a special branch of group theory. The crucial role of absolute continuity of one-dimensional sections of a distorted copula is illustrated. Some necessary and sufficient conditions concerning the automorphisms which preserve the copula property for a large class of bivariate copulas are proved. Several examples are presented.

Rigidity and Absolute Continuity of Foliations of Anosov Endomorphisms

Rigidity and Absolute Continuity of Foliations of Anosov Endomorphisms

Absorbing Markov decision processes

In this paper, we study discrete-time absorbing Markov Decision Processes (MDP) with measurable state space and Borel action space with a given initial distribution. For such models, solutions to the characteristic equation that are not occupation measures may exist. Several necessary and sufficient conditions are provided to guarantee that any solution to the characteristic equation is an occupation measure. Under the so-called continuity-compactness conditions, we first show that a measure is precisely an occupation measure if and only if it satisfies the characteristic equation and an additional absolute continuity condition. Secondly, it is shown that the set of occupation measures is compact in the weak-strong topology if and only if the model is uniformly absorbing. Several examples are provided to illustrate our results.

About the absolute continuity of the Erdös measure for the golden ratio, tribonacci number, and second order Markov chains

Рассматривается степенной ряд в фиксированной точке $\rho \in (0.5,1)$, где случайные коэффициенты принимают значения $0$, $1$ и образуют стационарный, эргодический и апериодический процесс. Мера Эрдeша - это закон распределения такого ряда. Задача об абсолютной непрерывности меры Эрдeша сводится к вопросу, когда соответствующая скрытая марковская цепь является марковской цепью Перри. Для золотого сечения и 1-марковских цепей мы получаем необходимые и достаточные условия абсолютной непрерывности меры Эрдeша и, используя марковские цепи Блекуэлла, даем новое доказательство того, что необходимые условия, найденные ранее (Теория вероятн. и ее примен., 51:1 (2006), 5-21), являются достаточными. Для числа трибоначчи и 1-марковских цепей получено новое доказательство теоремы о сингулярности меры Эрдeша. Для числа трибоначчи и 2-марковских цепей найдено только два случая абсолютной непрерывности.

Dynamics analysis of a rigid-flexible-liquid coupled satellite antenna system via absolute nodal coordinate formulation curvature continuity constraints

Typically, Euler and Lagrangian approaches are primarily used in fluid dynamics and multibody system (MBS) dynamics models, respectively. Absolute nodal coordinate formulation (ANCF), as a Lagrangian nonlinear finite element method (FEM), can effectively handle large deformation problems and incorporate the arbitrary Lagrange-Eulerian (ALE) method for fluids turbulence and breaking. Therefore, we have established a rigid-flexible-liquid coupled satellite antenna system dynamics model via ANCF. Navier-Stokes equations and virtual work principle are applied to formulate the fluid-structure interaction (FSI) dynamics equations. A penalty function is used to describe liquid incompressibility. Bi-directional FSI effects are solved simultaneously and brick fluid element elastic force is validated with reference experiments. Moreover, we introduce curvature continuity constraints to make the liquid free surface continuous. Results indicate that curvature continuity constraints can smoother liquid free surface and eliminate the protrusions and depressions caused by assembling brick fluid elements. Furthermore, we achieve simulation results consistent with reference experiments by using fewer elements with curvature continuity constraints. In addition, due to bi-directional FSI effects, liquid sloshing will magnify satellite antenna vibrations and simultaneously oscillate the advancing tank.

Absolute Continuous Function of Two Variables in ℝ𝟐 Space

Absolute Continuous Function of Two Variables in ℝ𝟐 Space

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

AbstractLet $$\Omega \subset \mathbb {R}^{n+1}$$ Ω ⊂ R n + 1 , $$n\ge 2$$ n ≥ 2 , be an open set with Ahlfors–David regular boundary satisfying the corkscrew condition. When $$\Omega $$ Ω is quantitatively connected (i.e., it satisfies the Harnack chain condition) it is known that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures (i.e., its membership to the class $$A_\infty $$ A ∞ ) is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting, it is also known that these properties are stable under Fefferman–Kenig–Pipher’s perturbations. Nonetheless, it has been an open problem to show whether, without connectivity, the previous two conditions remain equivalent or whether there is a Fefferman–Kenig–Pipher perturbation theory. In this paper we settle the question of whether, for general real elliptic operators in sets without any connectivity assumption, quantitative absolute continuity of the elliptic measure (expressed via a corona decomposition for the elliptic measure) and Carleson measure type estimates for bounded null solutions are equivalent. We show that any of these conditions is also equivalent to the fact that the Green function is comparable, in the corona sense, to the distance to the boundary. Our characterization has profound consequences. First, we extend Fefferman–Kenig–Pipher’s perturbation result to sets which are not necessarily connected, and this is the first result in this general setting. Second, in the case of the Laplacian, and more generally for $$L^1$$ L 1 -Kenig–Pipher non-symmetric operators with variable coefficients, our conditions characterize the uniform rectifiability of the boundary. Last, our results generalize previous work in settings where quantitative connectivity is assumed.

Homeomorphisms between Finsler manifolds satisfying the Rohde condition

We study the main features of homeomorphisms between two Finsler manifolds satisfying the Rohde condition, i.e. the difference between the moduli of geodesic rings in the image and its preimage is uniformly bounded from below and above. Some equivalent conditions for such a characterization are established. Also we investigate the absolute continuity and the relations to the class of mappings with finite are distortion. As applications, the boundary correspondence results for homeomorphisms satisfying the Rohde condition between regular domains are also presented.

Lexicographic agreeing to disagree and perfect equilibrium

Aumann’s seminal agreement theorem deals with the impossibility for agents to acknowledge their distinct posterior beliefs. We consider agreeing to disagree in an extended framework with lexicographic probability systems. A weak agreement theorem in the sense of identical posteriors only at the first lexicographic level obtains. Somewhat surprisingly, a possibility result does emerge for the deeper levels. Agents can agree to disagree on their posteriors beyond the first lexicographic level. By means of mutual absolute continuity as an additional assumption, a strong agreement theorem with equal posteriors at every lexicographic level ensues. Subsequently, we turn to games and provide epistemic conditions for the classical solution concept of perfect equilibrium. Our lexicographic agreement theorems turn out to be pivotal in this endeavour. The hypotheses of mutual primary belief in caution, mutual primary belief in rationality, and common knowledge of conjectures characterize perfect equilibrium epistemically in our lexicographic framework.

Absolute continuity of non-homogeneous Gibbs measures of the Ising model on the Cayley tree

In this paper, for the Ising model on the Cayley tree of order , a sequence of boundary conditions is constructed depending on an initial value h which defines a Gibbs measure µ h . By investigating the dynamical behaviour of the renormalisation group map associated with the model, we prove that each measure µ h is equivalent to the disordered phase . This result shines a new light to the question closely related to the classical result by Kakutani which asserts that any two locally-equivalent probability product measures are either equivalent or mutually-singular.

Generalized Filippov solutions for systems with prescribed-time convergence

Dynamical systems with prescribed-time convergence sometimes feature a right-hand side exhibiting a singularity at the prescribed convergence time instant. In an open neighborhood of this singularity, classical absolutely continuous Filippov solutions may fail to exist, preventing indefinite continuation of such solutions. This note introduces a generalized Filippov solution definition based on the notion of generalized absolute continuity in the restricted sense. Conditions for the continuability of such generalized solutions are presented and it is shown, in particular, that generalized Filippov solutions of systems with an equilibrium that is attractive, in prescribed time or otherwise, can always be continued indefinitely. The results are demonstrated by applying them to a prescribed-time control design example for a perturbed second-order integrator chain

On copulas with a trapezoid support

Abstract A family of bivariate copulas given by: for v + 2 u < 2 v+2u\lt 2 , C ( u , v ) = F ( 2 F − 1 ( v ∕ 2 ) + F − 1 ( u ) ) C\left(u,v)=F\left(2{F}^{-1}\left(v/2)+{F}^{-1}\left(u)) , where F F is a strictly increasing cumulative distribution function of a symmetric, continuous random variable, and for v + 2 u ≥ 2 v+2u\ge 2 , C ( u , v ) = u + v − 1 C\left(u,v)=u+v-1 , is introduced. The basic properties and necessary conditions for absolute continuity of C C are discussed. Several examples are provided.

Absolute continuity in families of parametrised non-homogeneous self-similar measures

Absolute continuity in families of parametrised non-homogeneous self-similar measures

Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

On the Spectrum of Certain Hadamard Manifolds

We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the classes.

Remarks on absolute continuity of positive operators

Ando studied Lebesgue decomposition for (bounded) positive operators, where a notion known as the maximal absolutely continuous part plays a crucial role. Based on technique involving unbounded operators we study certain expressions for the maximal absolutely continuous part, Radon–Nikodym type results, behavior of the maximal absolutely continuous part under the tensor product operation and characterization of mutual absolute continuity and so on.

Overlapping Iterated Function Systems from the Perspective of Metric Number Theory

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the { 0 , 1 , 3 } \{0,1,3\} problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the { 0 , 1 , 3 } \{0,1,3\} problem has positive Lebesgue measure. For each t ∈ [ 0 , 1 ] t\in [0,1] we let Φ t \Phi _t be the iterated function system given by Φ t ≔ { ϕ 1 ( x ) = x 2 , ϕ 2 ( x ) = x + 1 2 , ϕ 3 ( x ) = x + t 2 , ϕ 4 ( x ) = x + 1 + t 2 } . \begin{equation*} \Phi _{t}≔\Big \{\phi _1(x)=\frac {x}{2},\phi _2(x)=\frac {x+1}{2},\phi _3(x)=\frac {x+t}{2},\phi _{4}(x)=\frac {x+1+t}{2}\Big \}. \end{equation*} We prove that either Φ t \Phi _t contains an exact overlap, or we observe Khintchine like behaviour. Our analysis shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures. Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

Fractional divergence-measure fields, Leibniz rule and Gauss–Green formula

AbstractGiven$$\alpha \in (0,1]$$α∈(0,1]and$$p\in [1,+\infty ]$$p∈[1,+∞], we define the space$${\mathcal {D}}{\mathcal {M}}^{\alpha ,p}({\mathbb {R}}^n)$$DMα,p(Rn)of$$L^p$$Lpvector fields whose$$\alpha $$α-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the$$\alpha $$α-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss–Green formula. The sharpness of our results is discussed via some explicit examples.

Anisotropic ball Campanato-type function spaces and their applications

Let A be a general expansive matrix and let X be a ball quasi-Banach function space on $${\mathbb {R}}^n$$ , whose certain power (namely its convexification) supports a Fefferman–Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy–Littlewood maximal operator. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both A and X, prove that these spaces are dual spaces of anisotropic Hardy spaces $$H_X^A({\mathbb {R}}^n)$$ associated with both A and X, and obtain various anisotropic Littlewood–Paley function characterizations of $$H_X^A({\mathbb {R}}^n)$$ . Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both A and X, further induce their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of the quasi-norm $$\Vert \cdot \Vert _X$$ under consideration, the authors embed X under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.