Space Of Functions Research Articles

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.

Analysis of Functional Properties of Fuzzy \(\tilde{L}\) \(^p\Omega\) Spaces

This paper proposes a study of fuzzy \(\tilde{L}\)p (Ω) spaces, incorporating functions with triangular fuzzy coefficients. These fuzzy functional spaces provide a better adaptation to fuzzy or imprecise functions. We establish the theoretical foundations of these spaces by examining key functional properties, such as the fuzzy scalar product and the fuzzy norm. To do this, we checked the bilinearity, symmetry, positivity, homogeneity and triangular inequality in a fuzzy environment and in the presence of functions whose coefficients are triangular fuzzy numbers, by the \(\alpha\) -cut Dubois and Prade approach. The aim of this paper is to address observations identified in the existing literature, where some functional properties of \(\tilde{L}\)p (Ω) fuzzy spaces are often addressed in a too general manner, without specifying the types of fuzzy functions. This study aims to provide a more detailed and rigorous analysis, thus enriching mathematical understanding and paving the way for practical applications in diverse fields such as: fuzzy differential equations, artificial intelligence, information processing, and decision making in uncertain environments.

Compactness of averaging operators on Banach function spaces

Compactness of averaging operators on Banach function spaces

Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point

AbstractFor a sample $$X_1, X_2,\ldots X_N$$ X 1 , X 2 , … X N of independent identically distributed copies of a log-logistically distributed random variable X the maximum likelihood estimation is analysed in detail if a left-truncation point $$x_L>0$$ x L > 0 is introduced. Due to scaling properties it is sufficient to investigate the case $$x_L=1$$ x L = 1 . Here the corresponding maximum likelihood equations for a normalised sample (i.e. a sample divided by $$x_L$$ x L ) do not always possess a solution. A simple criterion guarantees the existence of a solution: Let $$\mathbb {E}(\cdot )$$ E ( · ) denote the expectation induced by the normalised sample and denote by $$\beta _0=\mathbb {E}(\ln {X})^{-1}$$ β 0 = E ( ln X ) - 1 , the inverse value of expectation of the logarithm of the sampled random variable X (which is greater than $$x_L=1$$ x L = 1 ). If this value $$\beta _0$$ β 0 is bigger than a certain positive number $$\beta _C$$ β C then a solution of the maximum likelihood equation exists. Here the number $$\beta _C$$ β C is the unique solution of a moment equation,$$\mathbb {E}(X^{-\beta _C})=\frac{1}{2}$$ E ( X - β C ) = 1 2 . In the case of existence a profile likelihood function can be constructed and the optimisation problem is reduced to one dimension leading to a robust numerical algorithm. When the maximum likelihood equations do not admit a solution for certain data samples, it is shown that the Pareto distribution is the $$L^1$$ L 1 -limit of the degenerated left-truncated log-logistic distribution, where $$L^1(\mathbb {R}^+)$$ L 1 ( R + ) is the usual Banach space of functions whose absolute value is Lebesgue-integrable. A large sample analysis showing consistency and asymptotic normality complements our analysis. Finally, two applications to real world data are presented.

Adaptive Sampling for Signals Associated with the Special Affine Fourier Transform

Adaptive sampling is inspired by the working principle of biological neurons, which converts the amplitude information of the received signal into a time sequence through a time encoding machine (TEM). The article mainly studies uniform sampling and adaptive sampling of non-bandlimited signals in two kinds of function spaces associated with the special affine Fourier transform (SAFT). Firstly, based on the stability characterization of the basis functions, we prove the existence of orthogonal projection operators in function spaces associated with the canonical convolution and the SAFT-convolution, respectively. Secondly, uniform sampling theorems are established for two kinds of signals living in the proposed function spaces. Finally, adaptive sampling schemes based on the crossing TEM and the integrate-and-fire TEM are studied. Moreover, the corresponding iterative reconstruction algorithms with exponential convergence are provided.

Solid bases and functorial constructions for (p-)Banach spaces of analytic functions

Abstract Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials $\{z^n\}_{n\geq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of $\mathcal{X}^\mathrm{max}$ (and $\mathcal{X}^\mathrm{min}$ ), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.

Inverse initial‐value problems for time fractional diffusion equations in fractional Sobolev spaces

AbstractWe study the time fractional diffusion equation , , in a bounded domain with an elliptic operator and a locally Lipschitz nonlinearity on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time , but at a final time , that is, is given instead of . The problem is, therefore, called an inverse initial‐value problem. We first establish the well‐posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of . Second, an susceptible‐infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial ‐estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and estimates for heat semigroup.

An integral RB operator

We introduce the novel concept of integral Read–Bajraktarević (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read–Bajraktarević (RB) operator and we derive conditions for the fixed point of the iRB operator to belong to certain function spaces.

Linear Statistics of Determinantal Point Processes and Norm Representations

Abstract We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.

Perception of the Vegetation Elements of Urban Green Spaces with a Focus on Flower Beds.

Urban vegetation plays a crucial role in meeting the challenges posed by rapid urbanization and climate change. The presence of plants and green spaces in urban areas provides a variety of environmental, social, and economic benefits. Understanding how users perceive ornamental plants in public green spaces and what their preferences are for certain vegetation elements is extremely important for planning and designing functional and aesthetically interesting urban landscapes. Although landscape experts sometimes use their creativity to create new trends, it is important not to ignore the attitudes and preferences of the public, who sometimes have a different opinion from that of the experts. The aim of the study was to determine the perceptions and preferences of the public and landscape experts for different vegetation elements and the differences in attitudes between these two groups. The study was conducted in Croatia in April 2012 using an online survey (n = 348). The results showed that trees were the most preferred vegetation element and that the public preferred flower beds and lawns to a greater extent than the professionals. All respondents perceived vegetation elements as volumes (trees, shrubs, and hedges) and plains (flower beds and lawns). In addition, respondents perceived two basic types of flower beds according to the features that characterize them: conventional and sustainable. The results show that users perceive the functional and spatial characteristics of the different vegetation elements, which is very important for the design of functional and sustainable urban green spaces.

Dynamical mixed boundary-transmission problems of the generalized thermo-electro-magneto-elasticity theory for composed structures

Abstract In the present paper we investigate three-dimensional dynamical mixed boundary-transmission problems for composed body consisting of two adjacent anisotropic elastic components having a common interface surface. The two contacting elastic components are subject to different mathematical models: Green–Lindsay’s model of generalized thermo-electro-magneto-elasticity in one component and Green–Lindsay’s model of generalized thermo-elasticity in the other one. We assume that the composed elastic structure under consideration contains an interfacial crack. We prove the uniqueness and existence theorems in appropriate function spaces.

An inner product on Adelic measures with applications to the Arakelov–Zhang pairing

We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of K K . The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov–Zhang pairing from arithmetic dynamics. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with a height on the space of rational functions with fixed degree. As a consequence, we show that the Arakelov–Zhang pairing of two rational maps f f and g g is, when holding g g fixed, commensurate with the height of f f .

Exploring Traditional Nepalese Building Techniques and Timber’s Role Structural Integrity

This paper aims to explore traditional Nepalese building construction techniques and the significant role of timber in enhancing the structural integrity and longevity of these buildings. With a long history of utilizing materials that are readily available locally, particularly wood, clay, stone, Nepal a country renowned for its rich architectural legacy has built buildings employing these resources. The study delves into the various traditional building techniques employed in Nepal, including the intricate craftsmanship and cultural significance associated with them. Additionally, it investigates the unique properties of timber that make it an ideal choice for construction in the Nepalese context. The finding talks about the construction of buildings various techniques which are employed to assemble structures. This includes information, on building materials, functional space allocation and the terminologies used in building construction. These terminologies hold value as they are derived from our tradition, in building construction technology. The findings of this research can serve as a valuable resource for architects, engineers, conservationists, and policymakers involved in the restoration and sustainable development of traditional Nepalese buildings.

Research on The Application of Interactive Design in Subway Public Art

China's urbanization is developing rapidly, and subway space, as an important part of urban construction, is receiving more and more attention from society and the government. The subway has alleviated the city's traffic congestion problem to the greatest extent, and has made rational use of the city's public space system. Public art in underground space has also become a window for urban cultural propaganda. Underground space is transforming from a functional space to a humanistic art space. In recent years, China's subway public art has developed rapidly and presented a diverse state, but there are still some common problems, such as the passive and single communication between space art and passengers, the lack of a certain degree of artistic publicity, and the failure to better meet people's current needs for humanistic art.

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When u0∈Ḃp,qα−2p(R+n)∩Ḃn+22,n+221−4n+2(R+n)∩Ḃp,11+np(R+n) is given, we will find the weak solutions for the Eq. (1.1) in the function space Cb[0,∞;Ḃp,qα−2p(R+n)∩Cb(0,∞;Ḃn+221−4n+2(R+n))∩L∞(0,∞;Ẇ∞1(R+n)), n+2<p<∞,1≤q≤∞,1+n+2p<α<2. We show the existence of weak solutions in the anisotropic Besov spaces Ḃp,qα,α2(R+n×(0,∞)) (see Theorem 1.2) and we show the embedding Ḃp,qα,α2R+n×(0,∞)⊂Cb[0,∞;Ḃp,qα−2p(R+n) (see Lemma 2.8). For the global existence of solutions, we assume that the extra stress tensor S is represented by S(A)=F(A)A, where F satisfies the assumption (A). Note that S1, S2 and S3 introduced in (1.2) satisfy our assumptions.