Space Of Functions Research Articles

On the Properties of Quasi-Banach Function Spaces

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.

Equilibrium States of Endomorphisms of $\mathbb{P}^{k}$: Spectral Stability and Limit Theorems

AbstractWe establish the existence of a spectral gap for the transfer operator induced on $\mathbb{P}^{k} = \mathbb{P}^{k} (\mathbb{C})$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

Completeness of the systems of Bessel functions of index $-5/2$

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

Relativity of Reductive Chain Complexes of Non-abelian Simplexes

Chain total double complexes with reductive differentials for non-abelian simplexes with associated spaces are considered. It is conjectured that corresponding relative cohomology is equivalent to the coset space of vanishing functionals over non-vanishing functionals related to differentials of complexes. The conjecture is supported by the theorem for the case of spaces of correlation functions and generalized connections on vertex operator algebra bundles.

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

On the running and the UV limit of Wilsonian renormalization group flows

Abstract In nonperturbative formulation of quantum field theory (QFT), the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation (RGE). Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.

The uniqueness and existence of solutions in a new complex function space for Kannan nonlinear dynamical systems

The uniqueness and existence of solutions in a new complex function space for Kannan nonlinear dynamical systems

Growth of hypercyclic functions: a continuous path between -frequent hypercyclicity and hypercyclicity

Abstract We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$ -frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$ -frequent hypercyclicity and hypercyclicity.

Through their eyes: Multi-subject brain decoding with simple alignment techniques

Abstract To-date, brain decoding literature has focused on single-subject studies, that is, reconstructing stimuli presented to a subject under fMRI acquisition from the fMRI activity of the same subject. The objective of this study is to introduce a generalization technique that enables the decoding of a subject’s brain based on fMRI activity of another subject, that is, cross-subject brain decoding. To this end, we also explore cross-subject data alignment techniques. Data alignment is the attempt to register different subjects in a common anatomical or functional space for further and more general analysis. We utilized the Natural Scenes Dataset, a comprehensive 7T fMRI experiment focused on vision of natural images. The dataset contains fMRI data from multiple subjects exposed to 9,841 images, where 982 images have been viewed by all subjects. Our method involved training a decoding model on one subject’s data, aligning new data from other subjects to this space, and testing the decoding on the second subject based on information aligned to the first subject. We also compared different techniques for fMRI data alignment, specifically ridge regression, hyper alignment, and anatomical alignment. We found that cross-subject brain decoding is possible, even with a small subset of the dataset, specifically, using the common data, which are around 10% of the total data, namely 982 images, with performances in decoding comparable to the ones achieved by single-subject decoding. Cross-subject decoding is still feasible using half or a quarter of this number of images with slightly lower performances. Ridge regression emerged as the best method for functional alignment in fine-grained information decoding, outperforming all other techniques. By aligning multiple subjects, we achieved high-quality brain decoding and a potential reduction in scan time by 90%. This substantial decrease in scan time could open up unprecedented opportunities for more efficient experiment execution and further advancements in the field, which commonly requires prohibitive (20 hours) scan time per subject.

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

Design logic of post-industrial landscape structure based on Metaphor-Deconstruction-Empowerment: A case study of the Truss Red Gallery in Hebei

Design logic of post-industrial landscape structure based on Metaphor-Deconstruction-Empowerment: A case study of the Truss Red Gallery in Hebei

Nice operators and nice spaces

In this paper we obtain a necessary and sufficient condition for an operator on a uniform algebra to be nice. We characterize nice operators on an expansive class of Banach spaces. Then as examples, we give a complete description of nice operators on the space of differentiable functions on compact perfect plane sets, Bloch and Zygmund spaces and the space of Lipschitz functions on intervals. We also prove that every surjective nice operators are isometries on some of these Banach spaces.

Fermion quadrature bases for Wigner functionals

Abstract A Grassmann functional phase space is formulated for the definition of fermionic Wigner functionals by identifying suitable fermionic operators that are analogues to boson quadrature operators. Instead of the Majorana operators, we use operators that are defined with relative spin transformations between the ladder operators. The eigenstates of these operators are shown to provide orthogonal bases, provided that the dual space is defined with the incorporation of a spin transformation. These bases then serve as quadrature bases in terms of which Wigner functionals are defined in a way equivalent to the bosonic case. As an application, we consider a two-level fermion system.

Electronic structure and optical spectral analysis of the MnO42- anion with consideration of site and Jahn-Teller distortion.

The ground state of 3d1 MnO42- was studied by density functional theory (DFT) and complete active space self-consistent field (CASSCF) methods in terms of a variety of molecular point group structures to ascertain the site and Jahn-Teller (JT) distortion effect. Modeling results from UB3LYP/6-31+G(d) calculations with natural bond orbital analysis show the four Mn-O bonds are coordinate covalent. The one-electron matrix elements from CASSCF(AILFT) (abinitio ligand field theory) with a second order perturbation treatment were used to calculate the parameters of the angular overlap model. These allowed the calculation of JT stabilization energies and first and second order JT coupling constants for MnO42- with C2v and D2d symmetry. Absorption spectra and excited state transition energies were calculated assuming the lattice distorted geometry was C2v with time-dependent DFT (TDDFT) using the unrestricted coulomb-attenuating UCAM-B3LYP density functional and with the spin-adapted spin-flip DFT using the UBH&HLYP density functional, both with the AUG-CC-PVTZ basis set. The mean absolute deviation of the calculations from experimental excited states for the ligand field and ligand-to-metal charge transfer (LMCT) bands was better than 0.1eV. Several new assignments for LMCT excited states were made on the basis of the TDDFT excitation results.

Sustainability of Historical Buildings Through Reuse Projects: A Case Study of the Kadıköy District in Istanbul

This research focuses on the preservation and adaptive reuse of historic structures. It emphasizes the importance of conserving cultural and historical heritage, suggesting that assigning new functions to these buildings contributes to sustainability. The study progresses through successful examples of adaptive reuse of historical buildings acquired by the Kadıköy Municipality, addressing spatial, cultural, and legal dimensions. It underscores the importance of preserving the architectural characteristics, spatial organizations, and surroundings of these buildings when assigning new functions. The research advocates for minimal intervention principles in adaptive reuse projects, ensuring the historical and cultural values of the buildings are considered. Furthermore, it highlights that these projects should not only focus on physical restoration but also faster social and cultural interaction. Adaptive reuse projects in the Kadıköy region have been developed in line with these principles, creating active and functional spaces that respond to community needs. These projects successfully demonstrate how the preservation and active use of historical buildings can contribute to regional and cultural sustainability. Finally, the research addresses the challenges encountered in the preservation and adaptive reuse of historical buildings in Turkey, discussing legal, economic, and technical issues, and offers suggestions for effective strategies in this field. The aim is to increase awareness and develop efficient strategies for the preservation of Turkey’s historical structures.

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Linearization of holomorphic Lipschitz functions

AbstractLet and be complex Banach spaces with denoting the open unit ball of . This paper studies various aspects of the holomorphic Lipschitz space , endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets of Lipschitz mappings and of bounded holomorphic mappings, from to . Thanks to the Dixmier–Ng theorem, is indeed a dual space, whose predual shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of . We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that contains a 1‐complemented subspace isometric to and that has the (metric) approximation property whenever has it. We also analyze when is a subspace of , and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics

While deep learning and data-driven modeling approaches based on deep neural networks (DNNs) have recently attracted increasing attention for solving partial differential equations, their practical applications to real-world scientific and engineering problems remain limited due to the relatively low accuracy and high computational costs. In this study, we present a differentiable programming-based hybrid meshfree approach within the field of computational mechanics. This approach seamlessly integrates neural network-based deep learning architectures with physics-based meshfree discretization techniques, thus referred to as the neural-integrated meshfree (NIM) method. To effectively approximate the solution, NIM employs a hybrid scheme called NeuroPU approximation, which combines continuous DNN representations with partition of unity (PU) basis functions associated with the underlying spatial discretization. This neural-numerical hybridization not only enhances the solution representation through functional space decomposition but also reduces both the size of DNN model and the need of automatic differentiation for spatial gradient computations, leading to a significant improvement in training efficiency. Under the NIM framework, we propose two truly meshfree solvers: the strong form-based NIM (S-NIM) and the local variational form-based NIM (V-NIM). In the S-NIM solver, the strong-form governing equation is directly considered in the loss function, while the V-NIM solver employs a local Petrov–Galerkin approach that allows the construction of variational residuals based on arbitrary overlapping subdomains. This ensures both the satisfaction of underlying physics and the preservation of the meshfree property. We perform extensive numerical experiments on both stationary and transient benchmark problems to assess the effectiveness of the proposed NIM methods in terms of accuracy, scalability, generalizability, and convergence properties. Moreover, comparative analysis with other physics-informed machine learning methods demonstrates that NIM, especially V-NIM, significantly enhances both accuracy and efficiency in end-to-end predictive capabilities.

An assessment of numerical and geometrical quality of bases on surface fitting on Powell–Sabin triangulations

It is well known that the problem of fitting a dataset by using a spline surface minimizing an energy functional can be carried out by solving a linear system. Such a linear system strongly depends on the underlying functional space and, particularly, on the basis considered. Some papers in the literature study the numerical behavior and processing of the above-mentioned linear systems in specific cases. The bases that have local support and constitute a partition of unity have been shown to be interesting in the frame of geometric problems. In this work, we investigate the numerical effects of considering these bases in the quadratic Powell–Sabin spline space. Specifically, we present a direct approach to explore different preconditioning strategies and assess whether the already known ‘good’ bases also possess favorable numerical properties. Additionally, we introduce an inverse optimization approach based on a nonlinear optimization model to identify new bases that exhibit both good geometric and numerical properties.

Extrapolation of Compactness on Banach Function Spaces

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.