Net Convergence Research Articles

The order topology for a von Neumann algebra

The order topology $\tau_o(P)$ (resp. the sequential order topology $\tau_{os}(P)$) on a poset $P$ is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra $M$ we consider the following three posets: the self-adjoint part $M_{sa}$, the self-adjoint part of the unit ball $M_{sa}^1$, and the projection lattice $P(M)$. We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on $M$, and relate the properties of the order topology to the underlying operator-algebraic structure of $M$.

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.

Lateral continuity and orthogonally additive operators

We generalize the notion of a laterally convergent net from increasing nets to general ones and study the corresponding lateral continuity of maps. The main result asserts that, the lateral continuity of an orthogonally additive operator is equivalent to its continuity at zero. This theorem holds for operators that send laterally convergent nets to any type convergent nets (laterally, order or norm convergent).

On Filter $(\alpha)$-convergence and Exhaustiveness of Function Nets in Lattice Groups and Applications

We consider (strong uniform)continuity of thelimit of a pointwise convergent net of latticegroup-valued functions, (strong weak)ex\-hau\-sti\-ve\-ness and (strong)$(\alpha)$-con\-ver\-gen\-ce with respect to a pairof filters, which in the setting of nets aremore natural than the corresponding notionsformulated with respect to a single filter. Somecomparison results are givenbetween such concepts, inconnection with suitable properties of filters.Moreover, some modes of filter(strong uniform) continuity for lattice group-valuedfunctions are investigated, givingsome characterization.As an application, we getsome Ascoli-type theorem in an abstract setting,extending earlier results to the context of filter$(\alpha)$-con\-ver\-gen\-ce.Furthermore, we pose some open problems.

Some convergence results for partial maps

In this paper we analyze some aspects of a new notion of convergence for nets of partial maps, introduced in [8]. In particular, we show that the introduced bornological convergence reduces to a natural uniform convergence relative to the bornology when the partial maps have a common domain. We then provide a new notion of upper convergence, which looks much more manageable than the original one. We show that the two notions, though different in general cases, do agree for sequences of strongly uniformly continuous (relative to the bornology) partial maps. More generally, coincidence for nets is shown in case the target space of the maps is totally bounded. This last result is interesting in view of possible applications, since partial maps are usually utility functions, thus when dealing with general models, monotone transformations valued in [0; 1] give rise to the same utility functions.

On convergence of topological aggregation functions

The concept of information aggregation is used in all scientific areas. More metric properties than required are often used when working with the mathematical concept of aggregation. This often gives the advantage of simplicity, but can sometimes obscure clues that might lead to a new discovery. Imprecision in space irregularities can be addressed by relaxing the metrizability condition using aggregation in a topological space. We study the convergence of nets of topological aggregation functions and their theoretical properties, and, in particular, the sensitivity of topological aggregation to input errors. It is clear that the underlying topology plays a crucial role in determining the aggregation properties.

On the Proof of the Existence of Undominated Strategies in Normal Form Games

In the game theory literature, there are two versions of the proof of the well-known fact that in a normal form game of n persons with compact spaces of strategies and continuous utility functions, the sets of undominated strategies are nonempty. The older one, stated in the first edition of the well-known book by Herve Moulin, depends on certain, relatively nontrivial results from measure theory, metric topology, and mathematical analysis. The proof is valid only for metrizable topological spaces. The second, revised edition of the same book contains a simplified proof, which is, however, incorrect. The author implicitly assumes that any linearly ordered set contains a cofinal subsequence, which is certainly not true. In this paper we correct, simplify, and generalize the second proof of Moulin by its reformulation in terms of topological convergence of nets. This modified technique also yields a slightly better result than is stated in the original. The assertion now holds for almost compact spaces. The argument used is elementary and easily understandable to non-experts.

OBJECTS APPEAR SMALLER AS THEY RECEDE: HOW PROPER MOTIONS CAN DIRECTLY REVEAL THE COSMIC EXPANSION, PROVIDE GEOMETRIC DISTANCES, AND MEASURE THE HUBBLE CONSTANT

Objects and structures gravitationally decoupled from the Hubble expansion will appear to shrink in angular size as the universe expands. Observations of extragalactic proper motions can thus directly reveal the cosmic expansion. Relatively static structures such as galaxies or galaxy clusters can potentially be used to measure the Hubble constant, and test masses in large scale structures can measure the overdensity. Since recession velocities and angular separations can be precisely measured, apparent proper motions can also provide geometric distance measurements to static structures. The apparent fractional angular compression of static objects is 15 microarcseconds per year in the local universe; this motion is modulated by the overdensity in dynamic expansion-decoupled structures. We use the Titov et al. quasar proper motion catalog to examine the pairwise proper motion of a sparse network of test masses. Small-separation pairs (< 200 Mpc comoving) are too few to measure the expected effect, yielding an inconclusive 8.3 +/- 14.9 microarcsec/yr. Large-separation pairs (200-1500 Mpc) show no net convergence or divergence for z < 1, -2.7 +/- 3.7 microarcsec/yr, consistent with pure Hubble expansion and significantly inconsistent with static structures, as expected. For all pairs a "null test" gives -0.36 +/- 0.62 microarcsec/yr, consistent with Hubble expansion, and excludes a static locus at ~5-10 sigma significance for z ~ 0.5-2.0. The observed large-separation pairs provide a reference frame for small-separation pairs that will significantly deviate from the Hubble flow. The current limitation is the number of small-separation objects with precise astrometry, but Gaia will address this and will likely detect the cosmic recession.

Regional Energy and Water Cycles: Transports from Ocean to Land

Abstract The flows of energy and water from ocean to land are examined in the context of the land energy and water budgets, for land as a whole and for continents. Most atmospheric reanalyses have large errors of up to 15 W m−2 in the top-of-atmosphere (TOA) energy imbalance, and none include volcanic eruptions. The flow of energy from ocean to land is more reliable as it relies on analyzed wind, temperature, and moisture fields. It is examined for transports of the total, latent energy (LE), and dry static energy (DSE) to land as a whole and as zonal means. The net convergence of energy onto land is balanced by the loss of energy at TOA, measured by Clouds and the Earth’s Radiant Energy System (CERES), and again there are notable discrepancies. Only the ECMWF Interim Re-Analysis (ERA-I) is stable and plausible. Strong compensation between variations in LE and DSE transports onto land means that their sum is more stable over time, and the net transport of energy onto land is largely that associated with the hydrological cycle (LE). A more detailed examination is given of the energy and water budgets for Eurasia, North and South America, Australia, and Africa, making use of Gravity Recovery and Climate Experiment (GRACE) data for water storage on land and data on river discharge into the ocean. With ERA-I, the new land estimates for both water and energy are closer to achieving balances than in previous studies. As well as the annual means, the mean annual cycles are examined in detail along with uncertainty sampling estimates, but the main test used here is that of closure.

Fuzzifying topologies on the space of linear operators

We introduce the notion of fuzzifying topologies on the space of linear operators. The concepts of fuzzifying topologies of pointwise convergence, bounded convergence, and complete bounded convergence are presented. We prove a sufficient and necessary condition for which the fuzzifying topologies are compatible with the structure on the space of linear operators. The Hausdorff separation of a fuzzifying topology on the space of linear operators is discussed. Specially, the net convergence, fuzzy boundedness, and fuzzy continuity of evaluation mappings in the fuzzifying topological linear space of pointwise convergence are investigated.

Influences of Sea Surface Temperature Gradients and Surface Roughness Changes on the Motion of Surface Oil: A Simple Idealized Study

AbstractThe authors' modeling shows that changes in sea surface temperature (SST) gradients and surface roughness between oil-free water and oil slicks influence the motion of the slick. Physically significant changes occur in surface wind speed, surface wind divergence, wind stress curl, and Ekman transport mostly because of SST gradients and changes in surface roughness between the water and the slick. These remarkable changes might affect the speed and direction of surface oil. For example, the strongest surface wind divergence (convergence) occurring in the transition zones owing to the presence of an oil slick will induce an atmospheric secondary circulation over the oil region, which in turn might affect the surface oil movement. SST-related changes to wind stress curl and Ekman transport in the transition zones appear to increase approximately linearly with the magnitude of SST gradients. Both surface roughness difference and SST gradients give rise to a net convergence of Ekman transport for oil cover. The SST gradient could play a more important role than surface roughness in changes of Ekman transport when SST gradients are large enough (e.g., several degrees per 10 km). The resulting changes in Ekman transport also induce the changes of surface oil movement. Sensitivity experiments show that appropriate selections of modeled parameters and geostrophic winds do not change the conclusions. The results from this idealized study indicate that the feedbacks from the surface oil presence to the oil motion itself are not trivial and should be further investigated for consideration in future oil-tracking modeling systems.

Stability of active mantle upwelling revealed by net characteristics of plate tectonics

Viscous convection within the mantle is linked to tectonic plate motions and deforms Earth's surface across wide areas. Such close links between surface geology and deep mantle dynamics presumably operated throughout Earth's history, but are difficult to investigate for past times because the history of mantle flow is poorly known. Here we show that the time dependence of global-scale mantle flow can be deduced from the net behaviour of surface plate motions. In particular, we tracked the geographic locations of net convergence and divergence for harmonic degrees 1 and 2 by computing the dipole and quadrupole moments of plate motions from tectonic reconstructions extended back to the early Mesozoic era. For present-day plate motions, we find dipole convergence in eastern Asia and quadrupole divergence in both central Africa and the central Pacific. These orientations are nearly identical to the dipole and quadrupole orientations of underlying mantle flow, which indicates that these 'net characteristics' of plate motions reveal deeper flow patterns. The positions of quadrupole divergence have not moved significantly during the past 250 million years, which suggests long-term stability of mantle upwelling beneath Africa and the Pacific Ocean. These upwelling locations are positioned above two compositionally and seismologically distinct regions of the lowermost mantle, which may organize global mantle flow as they remain stationary over geologic time.

On fuzzy almost continuous convergence in fuzzy function spaces

In this paper, we study the fuzzy almost continuous convergence of fuzzy nets on the set FAC ( X , Y ) of all fuzzy almost continuous functions of a fuzzy topological space X into another Y . Also, we introduce the notions of fuzzy splitting and fuzzy jointly continuous topologies on the set FAC ( X , Y ) and study some of its basic properties.

Climate change simulated by full and mixed-layer ocean versions of CSIRO Mk3.5 and Mk3.0: Large-scale sensitivity

The CSIRO Mk3.5 coupled atmosphere-ocean model includes upgrades to atmospheric and oceanic processes that remove a cold bias of the earlier Mk3.0. The global mean warming over the 21st century from Mk3.5 is 3.1 K under the CMIP3 A1B scenario, some 25% larger than that from Mk3.0. Two mixed-layer ocean versions of Mk3.5 were constructed, and these are also more sensitive than Mk3.0. To elucidate these differences, a simple feedback analysis is extended to Mk3.5, using changes for doubled CO2 in each model version. The net feedback for the low-mid latitude region is the main driver of the sensitivity contrast. The clear-sky component is consistently larger in Mk3.5, as is the increase in specific humidity, even after standardizing by the global warming. Cloud forcing provides a small positive feedback, which is stronger in cases that had larger declines in low-layer cloud. The net positive feedback for the higher-latitude region is larger in the coupled Mk3.5 than Mk3.0, which had more stable Arctic sea ice. However, some contrasts differed among the versions. As for Mk3.0, the surface warming in the coupled Mk3.5 is suppressed over that from the MLO case. Over the ocean, the pattern of suppression is similar to the change in energy flux into the surface in the coupled model. There is also a gradient of equatorial warming in the Asia-Pacific region that relates to the change in net convergence of heat transport by ocean currents. The effect of this pattern on regional rainfall is a focus of Part 2 of the study.

Topological Wiener–Wintner theorems for amenable operator semigroups

AbstractInspired by topological Wiener–Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.

Ideal convergence and divergence of nets in (ℓ)-groups

In this paper we introduce the I- and I*-convergence and divergence of nets in (ℓ)-groups. We prove some theorems relating different types of convergence/divergence for nets in (ℓ)-group setting, in relation with ideals. We consider both order and (D)-convergence.By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived.We prove that I*-convergence/divergence implies I-convergence/divergence for every ideal, admissible for the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds. Finally we pose some open problems.

Temporal and spatial scale dependence of precipitation analysis over the tropical deep convective regime

Data from Goddard cumulus ensemble model experiment are used to study temporal and spatial scale dependence of tropical rainfall separation analysis based on cloud budget during Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment (TOGA COARE). The analysis shows that the calculations of model domain mean or time-mean grid-scale mean simulation data overestimate the rain rates of the two rainfall types associated with net condensation but they severely underestimate the rain rate of the rainfall type associated with net evaporation and hydrometeor convergence.

Uniform families of ergodic operator nets

We study mean ergodicity in amenable operator semigroups and establish the connection to the convergence of strong and weak ergodic nets. We then use these results in order to show the convergence of uniform families of ergodic nets that appear in topological Wiener–Wintner theorems.

On some convergence criteria for nets of positive operators on continuous function spaces

We establish a simple criterion concerning the convergence of nets (or generalized sequences) of positive linear operators on C(X), X compact, toward a positive linear operator. As a consequence we discuss several criteria about the asymptotic behavior of the iterates of Markov operators. Finally, we present some applications concerning the asymptotic behavior of the iterates of Bernstein–Schnabl operators on a convex compact subset (not necessarily associated with a positive projection) and the iterates of a special linear operator generalizing the so-called Cesàro operator on C([0,1]).

A Comparison of Inner and Outer Spiral Rainbands in a Numerically Simulated Tropical Cyclone

Abstract The simulated inner and outer spiral rainbands in a tropical cyclone are compared in this study. The inner rainbands are generally active immediately outside the eyewall in the rapid filamentation zone, while the outer rainbands are active in regions outside about 3 times the radius of maximum wind. The inner rainbands are characterized by the convectively coupled vortex Rossby waves. The movement of the outer rainbands follows the low-level vector winds associated with the azimuthally averaged low-level flow and the radially outward cross-band flow caused by the downdraft-induced cold pool in the boundary layer. Convective cells in outer rainbands are typical of convective systems and move cyclonically and radially outward (inward) at large (small) radii. Net upward vertical mass transports (VMTs) appear throughout the depth of the troposphere in the whole inner-rainband region, while net downward VMTs are found below 4-km height in the outer-rainband region. In the whole inner-rainband region, only a very shallow layer with net horizontal convergence appears below 2-km height, while a deep layer with net convergence is found below 7.5-km height with net divergence aloft in the outer-rainband region. The inner rainband shows two tangential wind maxima, respectively, located near the top of the inflow boundary layer and immediately below the upper-tropospheric outflow layer. A secondary horizontal wind maximum occurs at about 4-km height on the inner edge of the outer rainband. Distinct features of the upwind, middle, and downwind sectors of the outer rainband are also discussed.