Space Of Functions Research Articles

Multiplication Operators on Generalized Orlicz Spaces Associated to Banach Function Spaces

Multiplication Operators on Generalized Orlicz Spaces Associated to Banach Function Spaces

The global loss of avian functional and phylogenetic diversity from anthropogenic extinctions.

Humans have been driving a global erosion of species richness for millennia, but the consequences of past extinctions for other dimensions of biodiversity-functional and phylogenetic diversity-are poorly understood. In this work, we show that, since the Late Pleistocene, the extinction of 610 bird species has caused a disproportionate loss of the global avian functional space along with ~3 billion years of unique evolutionary history. For island endemics, proportional losses have been even greater. Projected future extinctions of more than 1000 species over the next two centuries will incur further substantial reductions in functional and phylogenetic diversity. These results highlight the severe consequences of the ongoing biodiversity crisis and the urgent need to identify the ecological functions being lost through extinction.

Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

Abstract In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa’s type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on ℙ k {\mathbb{P}^{k}} discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order 𝒪 ⁢ ( N - t + h k ) {\mathcal{O}(N^{-t}+h^{k})} for a smooth solution which admits partial derivatives of order k + 1 {k+1} and t with respect to the spatial and angular variables, respectively. For k = 0 {k=0} (piecewise constant finite elements) we also obtain a convergence result of order 𝒪 ⁢ ( N - t + h 1 2 ) {\mathcal{O}(N^{-t}+h^{\frac{1}{2}})} . Numerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the L 2 {L^{2}} -norm, typically of one more order, 𝒪 ⁢ ( N - t + h k + 1 ) {\mathcal{O}(N^{-t}+h^{k+1})} .

Banach lattices with upper p-estimates: free and injective objects

AbstractWe study the free Banach lattice $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) together with a finite dimensional reduction, it is shown that the spaces $$\ell ^{p,\infty }(n)$$ ℓ p , ∞ ( n ) witness the universal property of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] isomorphically. As a consequence, we obtain a functional representation for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] , answering a question from Oikhberg et al. [Free Banach lattices. J Eur Math Soc (JEMS), 2024]. More generally, our proof allows us to identify the norm of any free Banach lattice over E associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] . Notably, we prove that the norm for $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] cannot be isometrically witnessed by $$L^{p,\infty }(\mu )$$ L p , ∞ ( μ ) and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper p-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $$\ell ^p$$ ℓ p allows us to give an alternative proof of the subspace problem for free p-convex Banach lattices. On the other hand, we prove that $$\ell ^{p,\infty }$$ ℓ p , ∞ is not injective in the class of Banach lattices with upper p-estimates, elucidating one of many difficulties arising in the study of $$\textrm{FBL}^{(p,\infty )}[E]$$ FBL ( p , ∞ ) [ E ] .

A study on various generalizations of generalized centers (GC) in Banach spaces

In [Generalized centers of finite sets in Banach spaces. Acta Math Univ Comenian (N.S.). 1997;66(1):83–115], Veselý developed the idea of generalized centers for finite sets in Banach spaces. In this work, we explore the concept of restricted F -center property for a triplet ( X , Y , F ( X ) ) , where Y is a subspace of a Banach space X and F ( X ) is the family of finite subsets of X. In addition, we generalize the analysis to include all closed, bounded subsets of X. Similar to how Lindenstrauss characterized n .2 . I . P . , we characterize n . X . I . P . . So, it is possible to figure out that Y has n . X . I . P . in X for all natural numbers n if and only if rad Y ( F ) = rad X ( F ) for all finite subsets F of Y. It then turns out that, for all continuous, monotone functions f, the f-radii viz. rad Y f ( F ) , rad X f ( F ) are same whenever the generalized radii viz. rad Y ( F ) , rad X ( F ) are also same, for all finite subsets F of Y. We establish a variety of characterizations of central subspaces of Banach spaces. With reference to an appropriate subfamily of closed and bounded subsets, it appears that a number of function spaces and subspaces exhibit the restricted weighted Chebyshev center property.

Evaluating the spatiotemporal land ecological changes in the Yangtze-to-Huaihe Water Diversion Project area.

As a fundamental component of human existence, land is inextricably linked to human development, and its ecological functions are closely associated with multiple sustainable development goals. This paper presents a framework for constructing and optimizing ecological function space, with the Yangtze-to-Huaihe Water Diversion Project area serving as a case study. A comprehensive land ecological index system is established, encompassing natural foundation, land degradation, land production, ecological structure, and ecological protection. An identity-discrepancy-contrary connection method is employed to investigate changes in regional land ecological functions before (2013) and during (2017, 2020, and 2022) the project's construction based on remote sensing data. The results indicated that the mean values of the land ecological index for each period were 0.1883, 0.1981, 0.2253, and 0.1370, respectively. The study calculated the connection, differences, and contradictions in the land ecological impacts across the counties, revealing a gradual decrease in differences and a growing prominence of contradictions. The land ecology of the Yangtze-to-Huaihe Water Diversion Project area is affected by the project construction, particularly within the construction area, showing an overall improvement. Most counties exhibited a trend of ecological improvement compared to the land ecology before the project's construction. However, after the project implementation, most districts demonstrated a trend of ecological deterioration. As the distance from the construction canal increases, the characteristics of each section and stage vary, generally exhibiting an exponential decrease in the land ecological index. The study highlighted the significance of enhancing the land ecological pattern, improving water quality, increasing water supply along the project, and alleviating groundwater overexploitation. The study can serve as a reference for land ecological protection and restoration in water transfer areas and river basins worldwide.

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

A Composition Formula for the Modified Analytic Function Space Fourier–Feynman Transform

Composition formula is one of the most important research topics in functional analysis theory. Various relationships can be obtained using the composition formula. Since the generalized Brownian motion process used in this paper has a non-zero mean function, there are many restrictions on obtaining a composition formula for the modified analytic function space Fourier–Feynman transform. This paper contains an idea of how the composition Formula (9) below is established for the modified analytic function space Fourier–Feynman transform on function space. Using this idea, we are able to solve a problem that had never been solved before.

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Exploring the Coordination of Park Green Spaces and Urban Functional Areas through Multi-Source Data: A Spatial Analysis in Fuzhou, China

The coordinated development of park green spaces (PGS)with urban functional areas (UFA) has a direct impact on the operational efficiency of cities and the quality of life of residents. Therefore, an in-depth exploration of the coupling patterns and influencing factors between PGS and UFA is fundamental for efficient collaboration and the creation of high-quality living environments. This study focuses on the street units of Fuzhou’s central urban area, utilizing multi-source data such as land use, points of interest (POI), and OpenStreetMap (OSM) methods, including kernel density analysis, standard deviational ellipse, coupling coordination degree model, and geographical detectors, are employed to systematically analyze the spatial distribution patterns of PGS and UFA, as well as their coupling coordination relationships. The findings reveal that (1) both PGS and various UFA have higher densities in the city center, with a concentric decrease towards the periphery. PGS are primarily concentrated in the city center, exhibiting a monocentric distribution, while UFA display planar, polycentric, or axial distribution patterns. (2) The spatial distribution centers of both PGS and UFA are skewed towards the southwest of the city center, with PGS being relatively evenly distributed and showing minimal deviation from UFA. (3) The dominant type of coupling coordination between PGS and various UFA is “Close to dissonance”, displaying a spatial pattern of “high in the center, low on the east-west and north-south wings”. Socioeconomic factors are the primary driving force influencing the coupling coordination degree, while population and transportation conditions are secondary factors. This research provides a scientific basis for urban planning and assists planners in more precisely coordinating the development of parks, green spaces, and various functional spaces in urban spatial layouts, thereby promoting sustainable urban development.

Regularity and uniqueness of solutions for compressible non-newtonian MHD equations with variable electrical conductivity and energy effects

This paper examines the existence, uniqueness, and regularity of solutions for a system of compressible non-Newtonian magnetohydrodynamics (MHD) equations with variable electrical conductivity and considering the energy equation. We establish the necessary conditions on fluid properties, external forces, and initial and boundary conditions to ensure the well-posedness of the system. Our results demonstrate the existence of regular solutions in appropriate functional spaces, the uniqueness of these solutions, and their higher regularity under smooth initial and boundary conditions. These findings provide a mathematical baseline for the study of complex interactions in compressible non-Newtonian MHD flows, and particularly the findings can be considered to support more accurate numerical simulations concerning a physical situation as well as deeper analytical solutions.

Truncated smooth function spaces

We introduce truncated Besov and Triebel–Lizorkin function spaces and study their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allow us to improve several known results in functional analysis and PDE’s. In particular, we obtain a full solution to the trace/extension problem in the critical case as well as sharp Sobolev-type embeddings with critical smoothness parameters.

The role of third parties in shaping transboundary cooperation of the Sixaola river basin (Costa Rica and Panama): an analysis through States’ involvement, technocratic turn, and reterritorialization

Transboundary water management in the Sixaola basin, shared by Costa Rica and Panama, has historically involved significant contributions from thrid - third and not thrid - parties. This study investigates their evolving roles and the implications for governance in this context. We conducted a comprehensive analysis of non-State actors interventions over time, focusing on changes in strategies, priorities, and their impact on transboundary basin governance. Our findings indicate that while non-state actor initiatives have attracted attention and financial resources, they have also introduced significant governance challenges, leading to inconsistencies in basin development. The analysis reveals that limited governmental involvement and the technocratic turn in development strategies have raised questions about basin development and its tangible impacts. Additionally, reterritorialization efforts highlight a local tendency to better align existing functional spaces with local concerns. This study contributes to the literature on transboundary basin management by providing insights into the multifaceted role of non-state actors in the Sixaola basin. It explores a case where governments remain distant in defining governance systems, highlighting the added value of focusing on non-state actors to grasp the complexity of hydropolitics and discussing the politics of river basin development.

Research on the Modular Design Method and Application of Prefabricated Residential Buildings

As one of the key ways to realize the industrialization and green development of construction, prefabricated construction is conducive to saving resources and energy and improving labor productivity and quality. Aiming to solve the problem of the lack of standardization in the design of prefabricated residential buildings, which leads to the components not being universally used and the industrial characteristics not being fully embodied, while excessive standardization leads to a lack of personalization and flexibility, the modular design theory is applied to the standardized design of prefabricated residential buildings in this study. The application route of modular design theory in the standardized design is constructed, that is, “system decomposition—module design—module combination”. Taking residential buildings within a height of 54 m as an example, each basic functional module is standardized and combined into standard plans. At the same time, the functional space module design based on modular coordination and the module combination design based on the trinity of “modulus, pattern, and mode” are discussed. This research is of great significance for giving full play to the comprehensive benefits of prefabricated concrete structures in quality improvement, cost reduction, and rapid assembly.

The classical and quantum particle on a flag manifold

Abstract In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace–Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace–Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.

Portfolio problem for the α−hypergeometric stochastic volatility model with consumption

This paper solves the finite horizon optimal consumption-investment problem for an investor that trades in the α−hypergeometric stochastic volatility model, with preferences based on a power utility function. To find the optimal strategy, we follow the dynamic programming approach. First, we perform a suitable change of variables in order to fully transform the non-linear Hamilton-Jacobi-Bellman (HJB) equation into a semilinear one. Next, we apply the Girsanov theorem to deduce an implicit Feynman-Kac formula depending upon the volatility process. The operator defined by the Feynman-Kac representation is shown to be a contraction operator on a designed function space. Finally, the Banach fixed point theorem yields the existence of a solution to the HJB equation in the designed space. Moreover, we apply a verification theorem to guarantee that the solution of the HJB equation coincides with the value function.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

Fixed points of regular set-valued mappings in quasi-metric spaces

Metric fixed point theory is becoming increasingly significant across various fields, including data science and iterative methods for solving optimization problems. This paper aims to introduce new fixed point theorems for set-valued mappings under novel regularity conditions, such as orbital regularity and orbital pseudo-Lipschitzness. Instead of traditional metric spaces, we adopt the framework of quasi-metric spaces, motivated by the need to address problems in spaces that are not necessarily metric, such as function spaces of homogeneous type. We also explore the stability of the set of fixed points under variations of the set-valued mapping. Additionally, we provide estimates for the distances from a given point to the set of fixed points and between two sets of fixed points. Building on these findings, we extend the discussion to similar problems involving fixed, coincidence, and cyclic/double fixed points within this framework. Our results generalize recent findings from the literature, including those in Ait Mansour M, Bahraoui MA, El Bekkali A. [Metric regularity and Lyusternik-Graves theorem via approximate fixed points of set-valued maps in noncomplete metric spaces. Set-Valued Var Anal. 2022;30(1):233–256. doi: 10.1007/s11228-020-00553-1], Dontchev AL, Rockafellar RT. [Implicit functions and solution mappings. a view from variational analysis. Dordrecht: Springer; 2009. Springer Monographs in Mathematics], Ioffe AD. [Variational analysis of regular mappings. Springer, Cham; 2017. Springer Monographs in Mathematics; theory and applications. doi: 10.1007/978-3-319-64277-2], Lim TC. [On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J Math Anal Appl. 1985;110(2):436–441. doi: 10.1016/0022-247X(85)90306-3] and Tron NH. [Coincidence and fixed points of set-valued mappings via regularity in metric spaces. Set-Valued Var Anal. 2023;31(2):22. Paper No. 17. doi: 10.1007/s11228-023-00680-5].

On Soft Topological Groups and Soft Function Spaces

In this paper, the definition of a soft topological group based on the concept of soft points is introduced. Let [Formula: see text] denote the collection of all soft continuous functions between soft topological spaces [Formula: see text] and [Formula: see text]. We investigate the structures of abstract groups, and soft groups within the context of the soft function space. Furthermore, the soft topological group structure of the group [Formula: see text] under soft topologies is explored.

Compactness results for a Dirichlet energy of nonlocal gradient with applications

AbstractWe prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.