Net Convergence Research Articles

Semi-order Continuous Operators on Vector Spaces

In this manuscript, we will study \({\tilde{o}}\)-convergence in (partially) ordered vector spaces and we will study a kind of convergence in a vector space V. A vector space V is called semi-order vector space (in short semi-order space), if there exist an ordered vector space W and an operator T from V into W. In this way, we say that V is semi-order space with respect to \(\{W, T\}\). A net \(\{x_\alpha \}\subseteq V\) is said to be \({\{W,T\}}\)-order convergent to a vector \(x\in V\) (in short we write \(x_\alpha \xrightarrow {\{W, T\}}x\)), whenever there exists a net \(\{y_\beta \}\) in W satisfying \(y_\beta \downarrow 0\) in W and for each \(\beta \), there exists \(\alpha _0\) such that \(\pm (Tx_\alpha -Tx) \le y_\beta \) whenever \(\alpha \ge \alpha _0\). In this manuscript, we study and investigate some properties of \(\{W,T\}\)-convergent nets and its relationships with other order convergence in partially ordered vector spaces. Assume that \(V_1\) and \(V_2\) are semi-order spaces with respect to \(\{{W_1}, T_1\}\) and \(\{W_2, T_2\}\), respectively. An operator S from \(V_1\) into \(V_2\) is called semi-order continuous, if \(x_\alpha \xrightarrow {\{{W_1}, T_1\}}x\) implies \(Sx_\alpha \xrightarrow {\{W_2, T_2\}}Sx\) whenever \(\{x_\alpha \}\subseteq V_1\). We study some properties of this new classification of operators.

ON SEPARATE ORDER CONTINUITY OF ORTHOGONALLY ADDITIVE OPERATORS

Our main result asserts that, under some assumptions, the uniformly-to-order continuity of an order bounded orthogonally additive operator between vector lattices together with its horizontally-to-order continuity implies its order continuity (we say that a mapping f : E → F between vector lattices E and F is horizontally-to-order continuous provided f sends laterally increasing order convergent nets in E to order convergent nets in F, and f is uniformly-to-order continuous provided f sends uniformly convergent nets to order convergent nets).

Filter bornological convergence in topological vector spaces

The concept of ?-enlargement defined on metric spaces is generalized to the concept of Uenlargement by using neighborhoods U of the zero of the space on topological vector spaces. By using U-enlargement, we define the bornological convergence for nets of sets in topological vector spaces and we examine some of their properties. By using filters defined on natural numbers, we define the concept of filter bornological convergence on sequences of sets, which is a more general concept than the bornological convergence defined on topological vector spaces. We give similar results for the filter bornological convergence.

A Notion of Convergence in Fuzzy Partially Ordered Sets

The notion of sequential convergence in fuzzy partially ordered sets, under the name oF-convergence, is well known. Our aim in this paper is to introduce and study a notion of net convergence, with respect to the fuzzy order relation, named o-convergence, which generalizes the former notion and is also closer to our sense of the classic concept of "convergence". The main result of this article is that the two notions of convergence are identical in the area of complete F-lattices.

Impact of air-sea interaction during two contrasting monsoon seasons

The air-sea interaction processes and their relation to Indian summer monsoon rainfall via dynamic and thermodynamic components are vital. The present study examined the two contrasting monsoon years 2016 (normal) and 2017 (below normal) and highlighted the significance of air-sea interaction in the understanding the rainfall over central India. We investigate the difference in the sea surface height anomaly and propagation of oceanic Kelvin and Rossby wave over the tropical Indian Ocean and other atmospheric parameters during the contrasting monsoon years. The study gives information on how equatorial and coastal Kelvin/Rossby waves cause upwelling/downwelling and modulate convection and influence rainfall during 2016 and 2017. The high (low) sea surface height anomaly, sea surface temperature, and upper ocean heat content during 2016 (2017) lead to enhanced (subdued) convection in the Bay of Bengal and thereby contributing more rainfall to central India. During 2016 (2017), high (low) tropospheric temperature draws more (less) moisture through mid-tropospheric heat flux, and net convergence (divergence) is dominated over the central India that results in enhanced (diminish) convection. Also, the velocity potential determines the strengthening (weakening) of Walker circulation through associated atmospheric circulation and upper-level divergence (convergence) that become favorable for rainfall in the year 2016 (2017). This study will help in understanding the variations in the air-sea interaction processes over the tropical Indian Ocean and the Indian subcontinent, the results of which may improve the predictability of the rainfall.

Transformation and Upwelling of Bottom Water in Fracture Zone Valleys

AbstractClosing the overturning circulation of bottom water requires abyssal transformation to lighter densities and upwelling. Where and how buoyancy is gained and water is transported upward remain topics of debate, not least because the available observations generally show downward-increasing turbulence levels in the abyss, apparently implying mean vertical turbulent buoyancy-flux divergence (densification). Here, we synthesize available observations indicating that bottom water is made less dense and upwelled in fracture zone valleys on the flanks of slow-spreading midocean ridges, which cover more than one-half of the seafloor area in some regions. The fracture zones are filled almost completely with water flowing up-valley and gaining buoyancy. Locally, valley water is transformed to lighter densities both in thin boundary layers that are in contact with the seafloor, where the buoyancy flux must vanish to match the no-flux boundary condition, and in thicker layers associated with downward-decreasing turbulence levels below interior maxima associated with hydraulic overflows and critical-layer interactions. Integrated across the valley, the turbulent buoyancy fluxes show maxima near the sidewall crests, consistent with net convergence below, with little sensitivity of this pattern to the vertical structure of the turbulence profiles, which implies that buoyancy flux convergence in the layers with downward-decreasing turbulence levels dominates over the divergence elsewhere, accounting for the net transformation to lighter densities in fracture zone valleys. We conclude that fracture zone topography likely exerts a controlling influence on the transformation and upwelling of bottom water in many areas of the global ocean.

Statistical Convergence of Nets Through Directed Sets

The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with extensions to nets.

Interaction of Superimposed Megaripples and Dunes in a Tidally Energetic Environment

Jones, K.R. and Traykovski, P., 2019. Interaction of superimposed megaripples and dunes in a tidally energetic environment. Journal of Coastal Research, 35(5), 948–958. Coconut Creek (Florida), ISSN 0749-0208.The superposition of smaller bedforms on larger dunes have been observed in multiple energetic tidal environments. This study analyzes the interaction of two scales of superimposed bedforms at Wasque Shoals, an ebb delta located approximately 1 km SE of Martha's Vineyard, Massachusetts. Data from a fixed rotary sidescan sonar deployed on a submerged quadpod captured the interaction of megaripples, or bedforms with wavelengths on the order of 1 m, with the lee side of a migrating dune with a wavelength of approximately 100 m. The net convergence of megaripples on the tidally dominant lee face of the dune suggests that the megaripples serve as an intermediate step between grain-scale transport processes and the large-scale dune migration. This conceptual model is quantitatively tested given observations of seabed elevation changes and dune migration rates. With this novel data set, two scales of bedforms were resolved on subtidal timescales, facilitating new insights into the dynamics of superimposed bedforms in tidal environments.

On filter convergence of nets in uniform spaces

In this paper, we introduce F-convergent and Fst-fundamental nets in uniform spaces and study some their properties.

On a characterization of compact frames by nets

In this paper we analyze the property of compactness and almost compactness in frames using nets. By introducing and investigating net convergence in frames, we characterize almost compact frames and regular compact frames as frames where every net clusters. With the aid of the results derived, we also present here a characterization of minimal Hausdorff frames through nets.

THE UNIFORM BOUNDEDNESS PRINCIPLE IN FUZZIFYING TOPOLOGICAL LINEAR SPACES

The main purpose of this study is to discuss the uniform boundednessprinciple in fuzzifying topological linear spaces. At first theconcepts of uniformly boundedness principle and fuzzy equicontinuousfamily of linear operators are proposed, then the relations betweenfuzzy equicontinuous and uniformly bounded are studied, and with thehelp of net convergence, the characterization of fuzzyequicontinuous is proved. Finally, the famous theorem of the uniformboundedness principle is presented in fuzzifying topological linearspaces.

A note on convergence of nets of multifunctions

The purpose of this paper is to investigate some new types of continuous convergence and quasi-uniform convergence of nets of multifunctions. The main three theorems describe a general types of interrelationships between forms of convergence, the continuity of the limits of nets of multifunctions and the continuity of members of such nets.

Resolving high-frequency internal waves generated at an isolated coral atoll using an unstructured grid ocean model

We apply the unstructured grid hydrodynamic model SUNTANS to investigate the internal wave dynamics around Scott Reef, Western Australia, an isolated coral reef atoll located on the edge of the continental shelf in water depths of 500,m and more. The atoll is subject to strong semi-diurnal tidal forcing and consists of two relatively shallow lagoons separated by a 500 m deep, 2 km wide and 15 km long channel. We focus on the dynamics in this channel as the internal tide-driven flow and resulting mixing is thought to be a key mechanism controlling heat and nutrient fluxes into the reef lagoons. We use an unstructured grid to discretise the domain and capture both the complex topography and the range of internal wave length scales in the channel flow. The model internal wave field shows super-tidal frequency lee waves generated by the combination of the steep channel topography and strong tidal flow. We evaluate the model performance using observations of velocity and temperature from two through water-column moorings in the channel separating the two reefs. Three different global ocean state estimate datasets (global HYCOM, CSIRO Bluelink, CSIRO climatology atlas) were used to provide the model initial and boundary conditions, and the model outputs from each were evaluated against the field observations. The scenario incorporating the CSIRO Bluelink data performed best in terms of through-water column Murphy skill scores of water temperature and eastward velocity variability in the channel. The model captures the observed vertical structure of the tidal (M2) and super-tidal (M4) frequency temperature and velocity oscillations. The model also predicts the direction and magnitude of the M2 internal tide energy flux. An energy analysis reveals a net convergence of the M2 energy flux and a divergence of the M4 energy flux in the channel, indicating the channel is a region of either energy transfer to higher frequencies or energy loss to dissipation. This conclusion is supported by the mooring observations that reveal high frequency lee waves breaking on the turning phase of the tide.

On ms-Convergence of Nets and Filters

"In this paper I'm introduce and study another type of convergence in a minimal structures namely Minimal Structure Convergence (ms-convergence) of nets and filters by using the concept of ms-open sets". "Also I'm investigate some properties of these concepts". "As well as I'm used two functions on the basis of Minimal structure in various forms, one check transmission character compacting minimal structure and other check transmission two characters compacting minimal structure and compact from one side depending on the requirements of research".

Uniform topology on EQ-algebras

Abstract In this paper, we use filters of an EQ-algebra E to induce a uniform structure (E, 𝓚), and then the part 𝓚 induce a uniform topology 𝒯 in E. We prove that the pair (E, 𝒯) is a topological EQ-algebra, and some properties of (E, 𝒯) are investigated. In particular, we show that (E, 𝒯) is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, by using convergence of nets, the convergence of topological EQ-algebras is obtained.

On $g_{ij}$-closed bi-generalized topological spaces

In this paper, generalizations of adherence and convergence of nets and filters on a bi-GTS are introduced and studied. Several properties and interrelations among such adherence and convergence of nets and filters on a bi-GTS are discussed and characterized using graphs of functions. Finally, these results are applied to investigate the behaviour of a generalization of compactness, known as $g_{ij}$-$closedness$ of a bi-GTS.

Ideal convergence of nets of functions with values in uniform spaces

We consider the pointwise, uniform, quasi-uniform, and the almost uniform I-convergence for a net (fd)d?D of functions from a topological space X into a uniform space (Y,U), where I is an ideal on D. The purpose of the present paper is to provide ideal versions of some classical results and to extend these to nets of functions with values in uniform spaces. In particular, we define the notion of I-equicontinuous family of functions on which pointwise and uniform I-convergence coincide on compact sets. Generalizing the theorem of Arzel?, we give a necessary and sufficient condition for a net of continuous functions from a compact space into a uniform space to I-converge pointwise to a continuous function.

Simultaneous power factorization in modules over Banach algebras

Let A be a Banach algebra with a bounded left approximate identity {e_lambda }_{lambda in Lambda }, let pi be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that lim _{lambda }pi (e_lambda )s=s uniformly on S. If S is bounded, or if {e_lambda }_{lambda in Lambda } is commutative, then we show that there exist ain A and maps x_n: Srightarrow X for nge 1 such that s=pi (a^n)x_n(s) for all nge 1 and sin S. The properties of ain A and the maps x_n, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of mathrm {C}_0({mathbb R}) and unbounded S is included.

A completion construction for continuous dynamical systems

In this work we use the theory of exterior spaces to construct a~$\check{C}_{0}^\mathbf{r}$-completion and a $\check{C}_{0}^\mathbf{l}$-completion of a dynamical system. If $X$ is a~flow, we construct canonical maps $X\to \check{C}_{0}^\mathbf{lr(X)$ and $X\to \check{C}_{0}^{\mathbf{l}}(X)$ and when these maps are homeomorphisms we have the class of $\check{C}_{0}^{\mathbf{r}}$-complete and $\check{C}_{0}^{\mathbf{l}}$-complete flows, respectively. In this study we find out many relations between the topological properties of the completions and the dynamical properties of a given flow. In the case of a complete flow this gives interesting relations between the topological properties (separability properties, compactness, convergence of nets, etc.) and dynamical properties (periodic points, omega limits, attractors, repulsors, etc.).

Convergence in [Formula: see text]-quasicontinuous posets.

In this paper, we present one way to generalize {mathcal {S}}-convergence and {mathcal {GS}}-convergence of nets for arbitrary posets by use of the cut operator instead of joins. Some convergence theoretical characterizations of s_2-continuity and s_2-quasicontinuity of posets are given. The main results are: (1) a poset P is s_2-continuous if and only if the {mathcal {S}}-convergence in P is topological; (2) P is s_2-quasicontinuous if and only if the {mathcal {GS}}-convergence in P is topological.