The essential coexistence phenomenon in Hamiltonian dynamics

We construct an example of a Hamiltonian flow $f^t$ on a four-dimensional smooth manifold $\mathcal {M}$ which after being restricted to an energy surface $\mathcal {M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$ -invariant subset $U\subset \mathcal {M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except for the direction of the flow) and is a Bernoulli flow while, on the boundary $\partial U$ , which has positive volume, all Lyapunov exponents of the system are zero.

The Unexpected Rapid Intensification of Tropical Cyclones in Moderate Vertical Wind Shear. Part III: Outflow–Environment Interaction

Interactions between the upper-level outflow of a sheared, rapidly intensifying tropical cyclone (TC) and the background environmental flow in an idealized model are presented. The most important finding is that the divergent outflow from convection localized by the tilt of the vortex serves to divert the background environmental flow around the TC, thus reducing the local vertical wind shear. We show that this effect can be understood from basic theoretical arguments related to Bernoulli flow around an obstacle. In the simulation discussed, the environmental flow diversion by the outflow is limited to 2 km below the tropopause in the 12–14-km (250–150 hPa) layer. Synthetic water vapor satellite imagery confirms the presence of upshear arcs in the cloud field, matching satellite observations. These arcs, which exist in the same layer as the outflow, are caused by slow-moving wave features and serve as visual markers of the outflow–environment interface. The blocking effect where the outflow and the environmental winds meet creates a dynamic high pressure whose pressure gradient extends nearly 1000 km upwind, thus causing the environmental winds to slow down, to converge, and to sink. We discuss these results with respect to the first part of this three-part study, and apply them to another atypical rapid intensification hurricane: Matthew (2016).

Regenerative processes for Poisson zero polytopes

Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

Moving From V&V to V&V&C in Nuclear Thermal-Hydraulics

V&V constitutes a powerful framework to demonstrate the capability of computational tools in several technological areas. Passing V&V requirements is a needed step before applications. Let’s focus hereafter to the area of (transient) Nuclear Thermal-hydraulic (NTH) and let’s identify V1 and V2 as acronyms for Verification and Validation, respectively. Now, V1 is performed within NTH according to the best available techniques and may not suffer of important deficiencies if compared with other technological areas. This is not the case of V2. Three inherent limitations shall be mentioned in the case of Validation in NTH: 1. Validation implies comparison with experimental data: available experimental data cover a (very) small fraction of the parameter range space expected in applications of the codes; this can be easily seen if one considers data in large diameter pipe, high velocity and high pressure or high power and power density. Noticeably, the scaling issue must be addressed in the framework of V2 which may result in controversial findings. 2. Water is at the center of the attention: the physical properties of water are known to a reasonable extent as well as large variations in values of quantities like density or various derivatives are expected within the range of variation of pressure inside application fields. Although not needed for current validation purposes (e.g. validation ranges may not include a situation of critical pressure and large heat flux) physically inconsistent values predicted by empirical correlations outside validation ranges, shall not be tolerated. 3. Occurrence of complex situations like transition from two-phase critical flow to ‘Bernoulli-flow’ (e.g. towards the end of blow-down) and from film boiling to nucleate boiling, possibly crossing the minimum film boiling temperature (e.g. during reflood). Therefore, whatever can be mentioned as classical V2 is not or cannot be performed in NTH. So, the idea of the present paper is to add a component to the V&V. This component, or step in the process, is called ‘Consistency with Reality’, or with the expected phenomenological evidence. The new component may need to be characterized in some cases and is indicated by the letter ‘C’. Then, the V&V becomes V&V&C. The purpose of the paper is to clarify the motivations at the bases of the V&V&C.

Introducing V&V&C in Nuclear Thermal-Hydraulics

V&V constitutes a powerful framework to demonstrate the capability of computational tools in several technological areas. Passing V&V requirements is a needed step before applications. Let’s focus hereafter to the area of (transient) Nuclear Thermal-hydraulic (NTH) and let’s identify V1 and V2 as acronyms for Verification and Validation, respectively. Now, V1 is performed within NTH according to the best available techniques and may not suffer of important deficiencies if compared with other technological areas. This is not the case of V2. Three inherent limitations shall be mentioned in the case of Validation in NTH: 1. Validation implies comparison with experimental data: available experimental data cover a (very) small fraction of the parameter range space expected in applications of the codes; this can be easily seen if one considers data in large diameter pipe, high velocity and high pressure or high power and power density. Noticeably, the scaling issue must be addressed in the framework of V2 which may result in controversial findings. 2. Water is at the center of the attention: the physical properties of water are known to a reasonable extent as well as large variations in values of quantities like density or various derivatives are expected within the range of variation of pressure inside application fields. Although not needed for current validation purposes (e.g. validation ranges may not include a situation of critical pressure and large heat flux) physically inconsistent values predicted by empirical correlations outside validation ranges, shall not be tolerated. 3. Occurrence of complex situations like transition from two-phase critical flow to ‘Bernoulli-flow’ (e.g. towards the end of blow-down) and from film boiling to nucleate boiling, possibly crossing the minimum film boiling temperature (e.g. during reflood). Therefore, whatever can be mentioned as classical V2 is not or cannot be performed in NTH. So, the idea of the present paper is to add a component to the V&V. This component, or step in the process, is called ‘Consistency with Reality’, or with the expected phenomenological evidence. The new component may need to be characterized in some cases and is indicated by the letter ‘C’. Then, the V&V becomes V&V&C. The purpose of the paper is to clarify the motivations at the bases of the V&V&C.

STIT tessellations are Bernoulli and standard

Let (Yt:t>0) be a STIT tessellation process and a>1. We prove that the random sequence (anYan:n∈ℤ) is a non-anticipating factor of a Bernoulli shift. We deduce that the continuous time process (atYat:t∈ℝ) is a Bernoulli flow. We use the techniques and results in Martínez and Nagel [Ergodic description of STIT tessellations. Stochastics 84(1) (2012), 113–134]. We also show that the filtration associated to the non-anticipating factor is standard in Vershik’s sense.

Ergodicity of hard spheres in a box

We prove that the system of two hard balls in a $\nu$-dimensional ($\nu\ge 2$) rectangular box is ergodic and, therefore, actually it is a Bernoulli flow.

α-equivalence: a refinement of Kakutani equivalence

For a fixed irrational α > 0 we say that probability measure-preserving transformationsSandTare α-equivalent if they can be realized as cross-sections in a common flow such that the return time functions on the cross-sections both take values in {1, 1 +α} and have equal integrals. Similarly we call two flowsFandGα-equivalent ifFhas a cross-sectionSandGhas a cross-sectionTisomorphic toSand again both the return time functions take values in {1, 1 + α} and have equal integrals. The integer kα(S), equal to the least positivesuchsuch that exp2πikα-1belongs to the point spectrum ofS, is an invariant of α-equivalence.We obtain a characterization of a-equivalence as a particular type of restricted orbit equivalence and use this to prove that within the class of loosely Bernoulli mapska(S) together with the entropyh(S) are complete invariants of α-equivalence. There is a corresponding a-equivalence theorem for flows which has as a consequence, for example, that up to an obvious entropy restriction, any weakly mixing cross-section of a loosely Bernoulli flow can also be realized as a cross-section with a {1,1 + α}-valued return time function.For the proof of the α-equivalence theorem we develop a relative Kakutani equivalence theorem for compact group extensions which is of interest in its own right. Finally, an example of Fieldsteel and Rudolph is used to show that in generalkα(S) is not a complete invariant of α-equivalence within a given even Kakutani equivalence class.