Spaces of distributions on product metric spaces associated with operators

We investigate (uniform) mean ergodicity of weighted composition operators on the space of smooth functions and the space of distributions, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic with period 2. Our results are based on a characterization of mean ergodicity in terms of Cesàro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest.

In Section 18.2 we introduced the space of conormal distributions associated with a submanifold Y of a manifold X. This is a natural extension of the classical notion of multiple layer on Y. All such distributions have their wave front sets in the normal bundle of Y which is a conic Lagrangian manifold. In Section 25.1 we generalize the notion of conormal distribution by defining the space of Lagrangian distributions associated with an arbitrary conic Lagrangian Λ ⊂ T*(X)\0. This is the space of distributions u such that there is a fixed bound for the order of P1, ... P N u for any sequence of first order pseudo-differential operators P1,...,PN with principal symbols vanishing on Λ. This implies that WF(u) ⊂ Λ. Symbols can be defined for Lagrangian distributions in much the same way as for conormal distributions. The only essential difference is that the symbols obtained are half densities on the Lagrangian tensored with the Maslov bundle of Section 21.6.

Explicit representations of spaces of smooth functions and distributions

Jointing in bedded siliciclastic sedimentary successions has been mainly analyzed in terms of fractured layers thickness-normalized joint spacing and related probability distributions, commonly neglecting the influence of primary structures on the petrophysical properties, joint height patterns, spatial arrangements and best-fitted spacing distributions. Taking into account the depositional controls on mechanical and fracture stratigraphy is crucial for improving the predictive potential of geological modelling in exploration and exploitation of underground resources, gas injection and storage. Here, we present a high-resolution, structural-stratigraphic study carried out on a 72 m-thick succession of the Marnoso-arenacea foredeep turbidites, exposed in the Northern Apennines of Italy. The systematic integration of the structural and sedimentological characterization of this basin-plain turbiditic succession, coupled with petrophysical and geomechanical measurements, represents the fracture and mechanical stratigraphy database onto which we tested the standardly used fracture density parameters and the best-fitted probability distributions, i.e. normal, log-normal, gamma and exponential, applied to fracture spacing analysis. Our results document that cumulative analysis does not identify the variability of joint height patterns and spacing distributions in siliciclastic turbidites, which instead requires considering single sedimentary facies types. Specifically, sandstones are affected by facies bounded joints and gamma spacing distributions while clay-rich and poorly sorted facies show a higher abundance of top bounded and throughgoing joints, and are best fitted by log-normal spacing distributions. We suggest that the fracture and mechanical stratigraphy should be studied at the facies scale resolution to unravel the strong control of the sedimentary facies for improving the characterization of siliciclastic reservoirs.

Differential spectra in observables that resolve additional soft or collinear QCD emissions exhibit Sudakov double logarithms in the form of logarithmic plus distributions. Important examples are the total transverse momentum qT in color-singlet production, N -jettiness (with thrust or beam thrust as special cases), but also jet mass and more complicated jet substructure observables. The all-order logarithmic structure of such distributions is often fully encoded in differential equations, so-called (renormalization group) evolution equations. We introduce a well-defined technique of distributional scale setting, which allows one to treat logarithmic plus distributions like ordinary logarithms when solving these differential equations. In particular, this allows one (through canonical scale choices) to minimize logarithmic contributions in the boundary terms of the solution, and to obtain the full distributional logarithmic structure from the solution’s evolution kernel directly in distribution space. We apply this technique to the qT distribution, where the two-dimensional nature of convolutions leads to additional difficulties (compared to one-dimensional cases like thrust), and for which the resummation in distribution (or momentum) space has been a long-standing open question. For the first time, we show how to perform the RG evolution fully in momentum space, thereby directly resumming the logarithms [lnn(qT2/Q2)/qT2]+ appearing in the physical qT distribution. The resummation accuracy is then solely determined by the perturbative expansion of the associated anomalous dimensions.

This paper proposes two approaches to compositional semantics in distributional semantic spaces. Both approaches conceive the semantics of complex structures, such as phrases or sentences, as being other than the sum of its terms. Syntax is the plus used as a glue to compose words. The former kind of approach encodes information about syntactic dependencies directly into distributional spaces, the latter exploits compositional operators reflecting the syntactic role of words. We present a preliminary evaluation performed on GEMS 2011 "Compositional Semantics" dataset, with the aim of understanding the effects of these approaches when applied to simple word pairs of the kind Noun-Noun, Adjective-Noun and Verb-Noun. Experimental results corroborate our conjecture that exploiting syntax can lead to improved distributional models and compositional operators, and suggest new openings for future uses in real-application scenario.

An extension of the linear stability analysis for the prediction of multiple necking during dynamic extension of round bar

This paper studies some of the theoretical questions of large openings or gaps in a single stream of traffic. A gap in the traffic stream is defined as a headway between vehicles greater than or equal to some minimum size—say x. Several authors have studied the probability distribution of the wait that a randomly located observer must endure before he finds a gap. This paper, while briefly reviewing the solutions of this well-known problem, is primarily concerned with expressions for (i) the distribution of gap sizes, (ii) the distribution of spacings between vehicles and gaps, (iii) the mean and variance of intervehicle and intergap spacings, (iv) the stationary flow rates of gaps, and (v) the distribution of blocked and unblocked periods. It is assumed that the origin of measurements may be located (i) with the passing of a vehicle, (ii) at the beginning of a gap, or (iii) at random. It is also assumed that the distribution of intervehicle spacings are independently, but identically, distributed random variables.

About Kolmogorov's statistical theory of phase transformation

The Pathfinder core, collected in the South Eugene Island Block 330 field, offshore Louisiana, provides an outstanding sample of structures associated with a major growth fault that abuts a giant oil field and that is thought to have acted as a conduit for hydrocarbon migration into the producing reservoirs. Where cored, the growth-fault zone cuts semiconsolidated Pliocene-Pleistocene mudstone and is over 100 m wide. The fault zone in the core consists of three structural domains, each characterized by a distinct rock type, distribution of fault dips and dip azimuths, and distribution of spacing between adjacent faults and fractures. Although all of the domains contain oil-bearing sands, only faults and fractures in the deepest domain contain oil, even though the oil-barren fault domains contain numerous faults and fractures that are parallel to those containing oil in the deepest domain. The deepest domain is also distinguished from the other two domains by a greater degree of structural complexity and by a well-defined power-law distribution of fault and fracture spacings. Sediments in this domain behaved as competent rock with respect to fault and fracture spacing, whereas the departure from power-law distribution of fault and fracture spacing in the other two domains may reflect deformation of unconsolidated sediment. This departure from a power-law spacing distribution in the upper two domains, combined with stable isotope data that indicate low-temperature water-rock interaction within a gouge zone that separates these two fault domains, indicates that the faults in those domains may have been active only early in the history of the growth fault zone, when the sampled sediments were at shallow burial depths. Thus, these faults may predate oil migration. In contrast, the faults in the oil-bearing domain appear to have been active later in the fault zone’s history, when the sediments faulted as competent rock and when geologic and organic geochemical investigations indicate oil migrated into the Block 330 reservoirs. Even though oil is present in sands throughout the core, its restriction to faults and fractures in the youngest sampled portion of the fault zone implies that oil migrated only through that part of the fault that was active during the time when oil had access to it. The absence of oil in fractures or faults in the other, probably older, fault domains indicates that the oil was never sufficiently pressured to flow up the fault zone on its own, either by hydraulic fracture or by increased permeability as a result of decreased effective stress. Instead, fluid migration along faults and fractures in the Pathfinder core was enhanced by permeability created in response to relatively far-field stresses related to minibasin subsidence.

The bog mosses, Sphagnum , form a significant part of the total mass of plants in the world. Their rate of growth depends to a considerable extent on the supply of water to them, and different species occupy characteristic habitats which differ in their ability to supply water. We used the profiles of water content in almost undisturbed cores of two species to infer the size and distribution of spaces around the plants in an attempt to account for the observation that S. papillosum is usually found not far above the water table, while S. capillifolium is usually found on hummocks well above the water table. Profiles of water content were recorded non-destructively from 30 cm diameter cores of Sphagnum and underlying peat, with use of the absorbance of the soft gamma radiation of 241 Am. The distribution of water-fillable spaces of different size was inferred from profiles with the water table at different distances, to a maximum of 150 cm, below the surface. The larger spaces, which are the main path of water transport, are outside the plant cell walls: between leaves and between pendent branches and stems. The mean radius of such spaces around the hummock species S. capillifolium is smaller than that around S. papillosum . For a given depth of water table the water content of the apical tuft of branches, where growth occurs, is greater in the hummock species than it is in the lawn species. Of ecological importance is that, for a given water content in the apex, the water table is at a greater depth below the hummock species than it is below the lawn species. As the water table rises and falls, so the water content of both species shows hysteresis as large as the difference between them. The ecological significance of this and the need for measurements while water is flowing are discussed.

Tight formations usually have large area of coverage. There is large areal heterogeneity for reservoir petrophysics, rock facies, formation thickness, physical properties and oil saturations, in different reservoir blocks, in area between different wells, and also in different segments of a single well. Tight reservoirs have different scales of natural fracture development. Hydraulic fracturing also forms different scales of complex hydraulic fracture network, which has large macroscopic heterogeneity with prominent multi-scale features. Unconventional tight reservoirs consist of different scales of porous media from nano-sized to nano-micron to micron-sized. There is large difference for compositions and quantity distributions in pore spaces with different scales. Different scales of pores present discrete discontinuous distribution in space, with large heterogeneity in spatial distribution. There is large difference for geometric parameters and physical properties in different scales of porous media. Different scales of pores present large microscopic heterogeneity and multiple porous media features. All of these features make conventional reservoir modelling and numerical simulation theory and technology not applicable in this scenario. In this paper, a new multimedia modelling and simulation method for volume fracturing of horizontal wells in tight reservoirs is introduced in detail by using the unconventional reservoir modelling software Untog, taking a single well as an example.

The equivalence of the differential-equation and integral-equation approaches to the solution of the nonlinear traveling-wave amplifier problem is shown rigorously. The equations can be transformed one into the other without making any additional assumptions. The space-charge expression developed on the basis of considering the electron distribution in phase space is shown to give the same form for the space-charge weighting function as a space-charge expression based on the electron distribution in space. Efficiency calculations are compared for the two methods and the agreement is excellent. Corrections to earlier calculations are included. The effect of radial electric field variations due to the circuit is considered and it is shown that the efficiency for large streams is reduced in direct proportion to the square of the field reduction function.

Systems with distributions and viability theorem

In this work, we investigate, both theoretically and experimentally, the signal power distribution in the joint space (along the radio path), angle-of-arrival (AOA) and time-of-arrival (TOA) [or time delay, TD] domains on the framework of the statistical modeling approach that was originally proposed for the outdoor urban environment and now continuously is used for indoor/outdoor communication links. In the presented work, the proposed multi-parametric approach is adopted now for modeling the indoor propagation conditions, combined guiding effects of radio propagation along the crossing corridors in LOS conditions and multi-ray phenomena caused by the multiple scattering, reflection and diffraction by numerous obstructions surrounded both terminal antennas, the transmitter (Tx) and the receiver (Rx) inside rooms lining corridors. The joint spatial, AOA and TOA model for indoor crossing corridors, combined with rooms aligned each corridor, is derived using the modified statistical multi-parametric model. Similar formulas, as for outdoor communication links, are derived to obtain full information on signal 2D and 3D intensity distribution in different indoor/outdoor environments The accuracy of the proposed model was validated using the collected measurements carried out in specific indoor/outdoor scenarios occur in BenGurion University environment. The spatial, TOA and AOA measured data were obtained by special experimental campaigns using a highly directive sector-scanning narrow-beam antenna operating at frequency of approximately 2.5 GHz. A satisfactory agreement between theoretical prediction and results of measurements is obtained. The obtained results indicate that using the unified multi-parametric stochastic model, both 2D and 3G (accounting for buildings' overlay profile) allow predicting signal intensity distribution in joint spatial, AOA and TOA domains in different indoor/outdoor environments.