Inclusions between Orlicz amalgam spaces on $\mathbb{R}^n$

On the weighted variable exponent amalgam space [formula omitted

Whenever one has to deal with problems involving rapidly increasing nonlinearities (e.g. of exponential type), Orlicz spaces are more appropriate than Lebesgue spaces. Since Orlicz spaces are ideal spaces, many statements of this section are just reformulations of the general results of Chapter 2, and therefore are cited mostly without proofs. However, in contrast to Lebesgue spaces, several new features occur in Orlicz spaces. For instance, the superposition operator may act from one Orlicz space into another and be bounded but not continuous, or continuous but not bounded. It turns out again that F is weakly continuous between Orlicz spaces if and only if f is affine in u . Further, the Lipschitz and Darbo conditions for F coincide, as in Lebesgue spaces, and one gets the same “degeneracy” phenomenon if the second space is “essentially smaller” than the first one. As in Lebesgue spaces, one can give conditions for differentiability and asymptotic linearity which are both necessary and sufficient. Again, f reduces to an affine function if F is differentiate from a “large” into a “small” space. On the other hand, the class of analytic superposition operators in Orlicz spaces is reasonably large; in contrast to Lebesgue spaces, a degeneration to polynomials occurs only under an additional restriction on the first space. Orlicz spaces As before, let Ω be an arbitrary set, M some σ-algebra of subsets of Ω, and µ, a σ-finite and countably additive measure on M ; together with µ we shall sometimes consider an equivalent normalized countably additive measure λ (see Section 1.1).

We derive some of the basic properties of weighted variable exponent Lebesgue spaces Lp(.)w (ℝn) and investigate embeddings of these spaces under some conditions. Also a new family of Wiener amalgam spaces W(Lp(.)w ;Lqv) is defined, where the local component is a weighted variable exponent Lebesgue space Lp(.)w (ℝn) and the global component is a weighted Lebesgue space Lqv (ℝn) : We investigate the properties of the spaces W(Lp(.)w ;Lqv): We also present new Hölder-type inequalities and embeddings for these spaces.

Let $\mathbb{A}$ be the affine group, $\Phi_1, \Phi_2$ be Young functions. We study the Orlicz amalgam spaces $W(L^{\Phi_1} (\mathbb{A}),L^{\Phi_2} (\mathbb{A}))$ defined on $\mathbb{A}$, where the local and global component spaces are the Orlicz spaces $L^{\Phi_1}(\mathbb{A})$ and $L^{\Phi_2}(\mathbb{A})$, respectively. We obtain an equivalent discrete norm on the amalgam space $W(L^{\Phi_1} (\mathbb{A}), L^{\Phi_2} (\mathbb{A}))$ using the constructions related to the affine group. Using the discrete norm we compute the dual space of $W(L^{\Phi_1} (\mathbb{A}),L^{\Phi_2} (\mathbb{A}))$. We also prove that the Orlicz amalgam space is a left $L^1(\mathbb{A})$-module with respect to convolution under certain conditions. Finally, we investigate some inclusion relations between the Orlicz amalgam spaces.

Let G be a locally compact group, Φ be a Young function. In this paper, we study the amalgam space W(LΦ,Y) defined on G, where the local component space is the Orlicz space LΦ and the global component is a Banach space Y containing the characteristic function of any compact subset of G. We obtain an equivalent norm on W(LΦ,Y). By using the equivalent norm, we investigate right translation invariance and completeness of the amalgam space W(LΦ,Y) with a non translation invariant space Y, in general. We also present an example of a non translation invariant space Y such that W(LΦ,Y) is right translation invariant in contrast to the literature.

Let \(L^{q ( x ) } (\mathbb{R} ) \) be variable exponent Lebesgue space and \(\ell^{ \{ q_{n} \} }\) be discrete analog of this space. In this work we define the amalgam spaces W(L p(x),L q(x)) and \(W ( L^{p ( x ) },\ell^{ \{ q_{n} \} } ) \), and discuss some basic properties of these spaces. Since the global components \(L^{q ( x ) } (\mathbb{R} ) \) and \(\ell^{ \{ q_{n} \} }\) are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent q(x) such that the sequence space \(\ell^{ \{ q_{n} \}}\) is the discrete space of \(L^{q ( x ) } (\mathbb{R} )\). By using this result we prove that \(W ( L^{p ( x ) },\ell ^{ \{ p_{n} \} } ) =L^{p ( x ) } (\mathbb{R} ) \). We also study the frame expansion in \(L^{p ( x ) } ( \mathbb{R} )\). At the end of this work we prove that the Hardy–Littlewood maximal operator from \(W ( L^{s ( x ) },\ell^{ \{t_{n} \} } ) \) into \(W ( L^{u ( x ) },\ell^{ \{v_{n} \} } ) \) is bounded under some assumptions.

The topological structure of the world air transportation network has been the subject of much research. However, to better understand the reality of air networks, one can consider the traffic, the number of passengers, or the distance between flights. This paper studies the weighted world air transportation network through the component structure, recently introduced in the network literature, by using the number of flights. The component structure is based on the community or multiple core-periphery structures and splits the network into local and global components. The local components capture the regional flights of these two mesoscopic structures (dense parts). The global components capture the inter-regional flights (links between the dense parts). We perform a comparative analysis of the world air transportation network and its components with their weighted counterparts. Moreover, we explore the strength and the s-core of these networks. Results display fewer local components well delimited and more global components covering the world than the unweighted world air transportation network. Centrality analysis reveals the difference between the top airports with high traffic and the top airports with high degrees. This difference is more pronounced in the global air network and the largest global component. Core analysis shows similitude between the s-core and the k-core for the local and global components, even though the latter includes more airports. For the world air network, the North and Central America-Caribbean airports dominate the s-core, whereas the European airports dominate the k-core.

Air transportation plays an essential role in the global economy. Therefore, there is a great deal of work to understand better the complex network formed by the links between the origins and destinations of flights. Some investigations show that the world air transportation network exhibits a community and a core-periphery structure. Although precious, these representations do not distinguish the inter-regional (global) web of connections from the regional (local) one. Therefore, we propose a new mesoscopic model called the component structure that decomposes the network into local and global components. Local components are the dense areas of the network, and global components are the nodes and links bridging the local components. As a case study, we consider the unweighted and undirected world air transportation network. Experiments show that it contains seven large local components and multiple small ones spatially well-defined. Moreover, it has a main global component covering the world. We perform an extensive comparative analysis of the structure of the components. Results demonstrate the non-homogeneous nature of the world air transportation network. The local components structure highlights regional differences, and the global component organization captures the efficiency of inter-regional travel. Centrality analysis of the components allows distinguishing airports centered on regional destinations from those focused on inter-regional exchanges. Core analysis is more accurate in the components than in the whole network where Europe dominates, blurring the rest of the world. Besides the world air transportation network, this paper demonstrates the potential of the component decomposition for modeling and analyzing the mesoscale structure of networks.

Global and local factors in earth surface systems

Introduction. Associations between chronic PM2.5 and mortality have been shown, although associations have varied. This variation may reflect confounding by long-term trends in PM2.5 and mortality. Methods. We assessed the impact of long-term time trends on the PM2.5-mortality association between 2000-2012. We did so using exposure measures based on monthly PM2.5 from 798 monitoring sites: (1) PM2.5 at each site (PMct) (2) PM2.5 centered by its mean site level, and (3) (γct), a new measure adjusting for annual PM2.5 trends. We used age-stratified log-linear Poisson models to assess the association between each PM2.5 measure and rate of mortality. We examined confounding by comparing PM2.5-associated rate ratios for data subsets based on every combination of follow-up period and baseline year. We also decomposed each PM2.5 measure into its local and global components and compared the difference in mortality rate ratios. Results We found statistically significant, positive associations between PM2.5 exposures and mortality. Centered PM2.5-associated mortality rate ratios were highest when data for the 13-year study period was examined, with a 5.5% increase in mortality rate per interquartile range (IQR), and decreased with shorter follow-up periods. When we adjusted for long-term PM2.5 time trends, we found lower rate ratios corresponding to a significant 2.7% increase in mortality rate per IQR increase in γct. Mortality rates associated with γct did not vary by follow-up period. When PMct was parsed into global and local components, their rate ratios differed, suggesting unmeasured confounding. This difference disappeared when γct was used as the exposure measure. Conclusions. We found associations between 1-year PM2.5 and mortality to be confounded by long-term time PM2.5 trends. This confounding was minimized using our year-adjusted PM2.5 measure, suggesting that future studies should adjust for long-term time trends in analyses of PM2.5 and mortality.

We analyze inflation rates in high-income OECD countries employing a multi-level factor structure that is estimated based on canonical correlation analysis (CCA) and sequential least squares (SLS). We show that inflation has a global component, mainly driven by G7 countries, explaining 77% of the variation on average, and a local component that accounts for substantial comovements in a subset of the countries. We demonstrate that this combination of global and local components has outstanding predictive ability, and can improve forecast performance significantly over a global-component-only specification for different policy horizons thus constituting a new benchmark for inflation forecasting.

The Orlicz spaces were introduced by Z.W. Birnbaum and W. Orlicz in 1931 as a natural generalization of the classical Lebesgue spaces. For this generalization the function �ݔ ௣ � entering in the definition of Lebesgue's space is replaced by a more general convex function Ф. Until today, Orlicz spaces have been studied by many authors from a different aspects. But, most of these studies used different definitions for the convex function Ф to define the Orlicz spaces. In this paper we will investigate the exact relationships between these different definitions of convex functions Ф and obtain the differences between the corresponding Orlicz spaces.

Local and global existence of solutions to a time-fractional wave equation with an exponential growth

INTRODUCTION: The task of approximation of complex nonlinear dependencies, especially in the case of short datasets, is important in various applied fields of medicine. Global approximation methods describe the generalized behavior of the model, while local methods explain the behavior of the model at specific data points. Global-local approximation combines both approaches, which makes such methods a powerful tool for processing short sets of medical data that can have both broad trends and local variations.OBJECTIVES: This paper aims to improve the method of sequential obtaining global and local components of the response surface to increase the accuracy of prediction in the case of short sets of medical data.METHODS: In this paper, the authors developed a new method that combined two ANNs: a non-iterative SGTM neural-like structure for obtaining the global component and GRNN as a powerful tool of local approximation in the case of short datasets.RESULTS: The authors have improved the method of global-local approximation due to the use of a General Regression Neural Network instead of RBF ANN for obtaining the local component, which ensured an increase in the accuracy of the body fat prediction task. The authors optimized the operation of the method and investigated the efficiency of the sequential obtaining global and local components of the response surface in comparison with the efficiency using a number of existing methods.CONCLUSION: The conducted experimental studies for solving the body fat prediction task showed the high efficiency of using the improved method in comparison with a number of existing methods, including ensemble methods.

Although oscillatory activity in the alpha band was traditionally associated with lack of alertness, more recent work has linked it to specific cognitive functions, including visual attention. The emerging method of rhythmic transcranial magnetic stimulation (TMS) allows causal interventional tests for the online impact on performance of TMS administered in short bursts at a particular frequency. TMS bursts at 10 Hz have recently been shown to have an impact on spatial visual attention, but any role in featural attention remains unclear. Here we used rhythmic TMS at 10 Hz to assess the impact on attending to global or local components of a hierarchical Navon-like stimulus (D. Navon (1977) Forest before trees: The precedence of global features in visual perception. Cognit. Psychol., 9, 353), in a paradigm recently used with TMS at other frequencies (V. Romei, J. Driver, P.G. Schyns & G. Thut. (2011) Rhythmic TMS over parietal cortex links distinct brain frequencies to global versus local visual processing. Curr. Biol., 2, 334-337). In separate groups, left or right posterior parietal sites were stimulated at 10 Hz just before presentation of the hierarchical stimulus. Participants had to identify either the local or global component in separate blocks. Right parietal 10 Hz stimulation (vs. sham) significantly impaired global processing without affecting local processing, while left parietal 10 Hz stimulation vs. sham impaired local processing with a minor trend to enhance global processing. These 10 Hz outcomes differed significantly from stimulation at other frequencies (i.e. 5 or 20 Hz) over the same site in other recent work with the same paradigm. These dissociations confirm differential roles of the two hemispheres in local vs. global processing, and reveal a frequency-specific role for stimulation in the alpha band for regulating feature-based visual attention.