Space Of Functions Research Articles

Characterization of switchgrass (Panicum virgatum L.) PvKSL1 as a levopimaradiene/abietadiene-type diterpene synthase.

The diverse class of plant diterpenoid metabolites serves important functions in mediating growth, chemical defence, and ecological adaptation. In major monocot crops, such as maize (Zea mays), rice (Oryza sativa), and barley (Hordeum vulgare), diterpenoids function as core components of biotic and abiotic stress resilience. Switchgrass (Panicum virgatum) is a perennial grass valued as a stress-resilient biofuel model crop. Previously we identified an unusually large diterpene synthase family that produces both common and species-specific diterpenoids, several of which accumulate in response to abiotic stress. Here, we report discovery and functional characterization of a previously unrecognized monofunctional class I diterpene synthase (PvKSL1) via in vivo co-expression assays with different copalyl pyrophosphate (CPP) isomers, structural and mutagenesis studies, as well as genomic and transcriptomic analyses. In particular, PvKSL1 converts ent-CPP into ent-abietadiene, ent-palustradiene, ent-levopimaradiene, and ent-neoabietadiene via a 13-hydroxy-8(14)-ent-abietene intermediate. Notably, although featuring a distinct ent-stereochemistry, this product profile is near-identical to bifunctional (+)-levopimaradiene/abietadiene synthases occurring in conifer trees. PvKSL1 has three of four active site residues previously shown to control (+)-levopimaradiene/abietadiene synthase catalytic specificity. However, mutagenesis studies suggest a distinct catalytic mechanism in PvKSL1. Genome localization of PvKSL1 distant from other diterpene synthases, and its phylogenetic distinctiveness from known abietane-forming diterpene synthases, support an independent evolution of PvKSL1 activity. Albeit at low levels, PvKSL1 gene expression predominantly in roots suggests a role of diterpenoid formation in belowground tissue. Together, these findings expand the known chemical and functional space of diterpenoid metabolism in monocot crops.

A discretization-invariant extension and analysis of some deep operator networks

We present a generalized version of the discretization-invariant neural operator in Zhang et al. (2022) and prove that the network is a universal approximation in the operator sense. Moreover, by incorporating additional terms in the architecture, we establish a connection between this discretization-invariant neural operator network and those discussed in Chen and Chen (1995) and Lu et al. (2021). The discretization-invariance property of the operator network implies that different input functions can be sampled using various sensor locations within the same training and testing phases. Additionally, since the network learns a “basis” for the input and output function spaces, our approach enables the evaluation of input functions on different discretizations. To evaluate the performance of the proposed discretization-invariant neural operator, we focus on challenging examples from multiscale partial differential equations. Our experimental results indicate that the method achieves lower prediction errors compared to previous networks and benefits from its discretization-invariant property.

Critical local well-posedness of the nonlinear Schrödinger equation on the torus

In this paper, we study the local well-posedness of nonlinear Schrödinger equations on tori \mathbb{T}^{d} at the critical regularity. We focus on cases where the nonlinearity |u|^{a}u is nonalgebraic with small a>0 . We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering a , we prove a bilinear estimate for the Schrödinger operator on tori \mathbb{T}^{d} , which enhances previously known multilinear estimates. We design a function space adapted to the new bilinear estimate and a package of Strichartz estimates, which is not based on conventional atomic spaces.

Simultaneous predictive bands for functional time series using minimum entropy sets

Functional Time Series (FTS) are sequences of dependent random elements taking values on some functional space. Most of the research on this domain focuses on producing a predictor able to forecast the next function, having observed a part of the sequence. For this, the Autoregressive Hilbertian process is a suitable framework. Here, we address the problem of constructing simultaneous predictive confidence bands for a stationary FTS. The method is based on an entropy measure for stochastic processes. To construct predictive bands, we use a Reproducing Kernel Hilbert Space (RKHS) to represent the functions and a functional bootstrap procedure that allows us to estimate the prediction law and a Reproducing Kernel Hilbert Spaces (RKHS) to represent the functions, considering then the basis associated to the reproducing kernel. We then classify the points on the projected space according to those that belong to the minimum entropy set (MES) and those that do not. We map the minimum entropy set back to the functional space and construct a band using the regularity property of the RKHS. The proposed methodology is illustrated through artificial and real data sets.

Weighted approximations by sampling type operators: recent and new results

In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function $\rho$ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions.

On Kantorovich variant of Brass-Stancu operators

Abstract In this study, we deal with Kantorovich-type generalization of the Brass-Stancu operators. For the sequence of these operators, we study L p {L}^{p} -convergence and give some upper estimates for the L p {L}^{p} -norm of the approximation error via first-order averaged modulus of smoothness and the first-order K K -functional. Moreover, we show that the Kantorovich generalization of each Brass-Stancu operator satisfies variation detracting property and is bounded with respect to the norm of the space of functions of bounded variation on [ 0 , 1 ] \left[0,1] . Finally, we present graphical and numerical examples to compare the convergence of these operators to given functions under different parameters.

Testing covariance separability for continuous functional data

Analyzing the covariance structure of data is a fundamental task of statistics. While this task is simple for low‐dimensional observations, it becomes challenging for more intricate objects, such as multi‐variate functions. Here, the covariance can be so complex that just saving a non‐parametric estimate is impractical and structural assumptions are necessary to tame the model. One popular assumption for space‐time data is separability of the covariance into purely spatial and temporal factors. In this article, we present a new test for separability in the context of dependent functional time series. While most of the related work studies functional data in a Hilbert space of square integrable functions, we model the observations as objects in the space of continuous functions equipped with the supremum norm. We argue that this (mathematically challenging) setup enhances interpretability for users and is more in line with practical preprocessing. Our test statistic measures the maximal deviation between the estimated covariance kernel and a separable approximation. Critical values are obtained by a non‐standard multiplier bootstrap for dependent data. We prove the statistical validity of our approach and demonstrate its practicability in a simulation study and a data example.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

Exploration on the Application of Smart Home Technologies to Enhance the Quality Design of Elderly-Friendly Living space in Existing Residences

Drawing upon the progressively evolving theoretical frameworks and methodologies for housing adaptation tailored to the needs of the elderly, this study explores the enhancement of age-friendly living spaces through the integration of smart home technologies. A novel, multidimensional framework is introduced, integrating the elements of "elderly individuals, living areas, functional spaces, and technologies," fostering a comprehensive application of technology in elderly care. This approach aims to scrutinize the seamless integration of technological advancements into the daily lives of the elderly from a multidisciplinary perspective. The study delves into the effectiveness and adaptation strategies of smart home systems across three pivotal scenarios: daily living, health monitoring, and safety measures. It underscores the pivotal role of smart home technologies in elevating the well-being of the elderly. By implementing sophisticated smart solutions, such as voice commands and streamlined operations, these technologies empower seniors to effortlessly manage lighting, climate control, and security systems within their living spaces, fostering a customized, intelligent living ambiance. This initiative not only propels the application of smart home systems in the lives of the elderly but also ensures a safer, more comfortable, and autonomous lifestyle.

Sums of Generalized Weighted Composition Operators from Weighted Bergman Spaces Induced by Doubling Weights into Bloch-Type Spaces

The single generalized weighted composition operator Du,ψn on various spaces of analytic functions has been investigated for decades, i.e., Du,ψnf=u·(f(n)∘ψ), where f∈H(D). However, the study of the finite sum of generalized weighted composition operators with different orders, i.e., PU,ψkf=u0·f∘ψ+u1·f′∘ψ+⋯+uk·f(k)∘ψ, is far from complete. The boundedness, compactness and essential norm of sums of generalized weighted composition operators from weighted Bergman spaces with doubling weights into Bloch-type spaces are investigated. We show a rigidity property of PU,ψk. Specifically, the boundedness and compactness of the sum PU,ψk is equivalent to those of each Dun,ψn, 0≤n≤k.

Integrable Solutions and Continuous Dependence of a Nonlinear Singular Integral Inclusion of Fractional Orders and Applications

Let E be a reflexive Banach space. In this article we study the existence of integrable solutions in the space of all lesbesgue integrable functions on E, L1([0,T],E), of the nonlinear singular integral inclusion of fractional orders beneath the assumption that the multi-valued function G has Lipschitz selection in E. The main tool applied in this work is the Banach contraction fixed point theorem. Moreover, the paper explores a qualitative property associated with these solutions for the given problem such as the continuous dependence of the solutions on the set of selections S1G(τ,x(τ)). As an application, the existence of integrable solutions of the two nonlocal and weighted problems of the fractional differential inclusion is investigated. We additionally provide an example given as a numerical application to demonstrate the effectiveness and value of our results.

Sequence, Structure, and Functional Space of Drosophila De Novo Proteins.

During de novo emergence, new protein coding genes emerge from previously nongenic sequences. The de novo proteins they encode are dissimilar in composition and predicted biochemical properties to conserved proteins. However, functional de novo proteins indeed exist. Both identification of functional de novo proteins and their structural characterization are experimentally laborious. To identify functional and structured de novo proteins in silico, we applied recently developed machine learning based tools and found that most de novo proteins are indeed different from conserved proteins both in their structure and sequence. However, some de novo proteins are predicted to adopt known protein folds, participate in cellular reactions, and to form biomolecular condensates. Apart from broadening our understanding of de novo protein evolution, our study also provides a large set of testable hypotheses for focused experimental studies on structure and function of de novo proteins in Drosophila.

Spatiotemporal Evolution and Driving Forces of Production-Living-Ecological Space in Arid Ecological Transition Zone Based on Functional and Structural Perspectives: A Case Study of the Hexi Corridor

The rational allocation of land resources is crucial to ensuring human well-being, livelihood, and survival. The study of Production-Living-Ecological Space (PLES) provides new perspectives on land resource allocation. However, few studies have assessed the feasibility of PLES optimization in ecological transition zones. For this study, using the composite functional space classification method, a classification and functional utility scoring system were constructed. Various methods, including dynamic attitude, transfer matrix, and spatial autocorrelation, were employed to characterize the evolution of the quantity and quality of PLES in the Hexi Corridor. Moreover, the mechanisms driving these changes were explored using a geodetector. Our findings revealed that: (1) The distribution of Production-Ecological Space (PES) is higher in the west and south and lower in the east and north. Production-Living Space (PLS) is scattered. Ecological-Production Space (EPS) is mostly distributed in the south or west, whereas Ecological Space (ES) is mainly located in the north and west of the Hexi Corridor. (2) From 1980 to 2020, the area of PES and PLS increased by 2037.84 km2 and 673 km2, respectively; the area of EPS was relatively stable, and the area of ES decreased by 2523.06 km2. (3) The evolution of PLES quality indicated that the high functional utility area of PES and PLS was roughly the same as the expanded functional utility area, whereas the expanded functional utility area of EPS and ES is similar to the median functional utility area. (4) The spatiotemporal evolution of PLES is closely linked to natural, economic, and social factors.

Pseudo-differential operators on local Hardy spaces associated with ball quasi-Banach function spaces

Pseudo-differential operators on local Hardy spaces associated with ball quasi-Banach function spaces

Multimorphisms on pre-Riesz spaces of continuous functions

In the theory of vector lattices, mostly three classes of lattice ordered algebras are investigated, called f-algebras, d-algebras, and almost-f-algebras. Each class is endowed with a different type of positive bilinear map as the multiplication, namely with bi-orthomorphisms, bi-Riesz homomorphisms, and orthosymmetric maps, respectively. On the space C(K), with K a compact Hausdorff space, representations as weighted composition or integral operators are known. We give generalized definitions of the above mentioned maps in the setting of partially ordered vector spaces, where we deal with n-linear maps. We generalize the representations to the case of certain subspaces of some C(K) space, with K a compact Hausdorff space. With the representations at hand, we deduce basic properties of these maps. The results also cover the case of order unit spaces, which can be seen as order dense subspaces of some C(K) space via the functional representation.

On the convergence of Fourier representations and Schwartz distributions

While Fourier theory remains a cornerstone for analyzing and interpreting the spectral content of diverse signals, its limitations often surface in practical applications. Notably, popular signals like sinusoids, Dirac deltas, signums, and unit steps lack convergent Fourier representations within the conventional framework. This discrepancy necessitates distribution theory to build and interpret suitable representations for such signals. However, existing literature in signal processing and communication engineering often glosses over these intricacies, leaving researchers grappling with obscure concepts regarding the very existence of Fourier representations for non-conforming signals. This work bridges this critical gap by offering a comprehensive exploration of the conditions guaranteeing the existence of Fourier representations. We introduce a novel linear space – the Gauss–Schwartz (GS) function space – and its corresponding class of tempered superexponential (TSE) distributions. We demonstrate that the Fourier transform (FT) acts as an isomorphism on the GS space of test functions and, by duality, on TSE distributions. Crucially, the GS space proves to be minimal in the sense that its dual, encompassing TSE distributions, represents the largest possible linear space over which the FT can be defined through duality. This theoretical advancement signifies a twofold contribution. Firstly, it clarifies the existence and interpretation of Fourier representations for prevalent signals that evade conventional analysis. Secondly, the introduction of the GS-TSE framework expands the reach of the FT to encompass a broader spectrum of functions, enabling novel applications in diverse fields. Ultimately, this work paves the way for a more robust and accessible understanding of Fourier analysis, empowering researchers and practitioners to leverage its full potential in exploring and processing a wider range of signals.

Skew-product attractors of non-autonomous Caputo fractional differential equations.

A non-autonomous Caputo fractional differential equation (FDE) of order α∈(0,1) in Rd with a driving system on a compact base space P is shown to generate a skew-product semi-flow on Cα×P, where Cα is the space of continuous functions f:R+→Rd with a weighted norm giving uniform convergence on compact time subsets. This skew-product semi-flow is then shown to have a bounded and closed attractor when the vector field of the Caputo FDE satisfies a uniform dissipativity condition. It attracts bounded sets of constant initial functions f in here Cα. The properties and structure of this attractor in Cα×P are also discussed.

On the optimal rate for the convergence problem in mean field control

The goal of this work is to obtain (nearly) optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. Our main result is then to derive sharp rates of convergence in two distinct regimes. First, when the data is sufficiently regular, we obtain rates proportional to N−1/2, with N being the number of particles, and we verify that N−1/2 is indeed optimal in this setting. Second, when the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance, we obtain rates proportional to N−2/(3d+6). We do not expect this second estimate to be optimal, but it improves substantially on the existing literature. Moreover, we construct an example showing that the optimal rate is no faster than N−1/d, and we conjecture that the optimal rate should indeed be exactly N−1/d (at least when d≥3). The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. To make the whole easier, we limit ourselves to the periodic setting. We also provide some examples to show that our results are sharp up to some extent.

Does Consciousness Have Dimensions?

Whether consciousness is unidimensional with states defined along a single scale or it consists of multiple fundamental dimensions has been debated. Clinical assessment of consciousness distinguishes the content of consciousness (awareness) and the level of consciousness (wakefulness or arousal), which conflates firstperson phenomenal properties with third-person observable properties. The state of consciousness is more appropriately defined in terms of subjective level and content which are interdependent. On this account, the state of consciousness is exclusively defined by the experienced mental content, i.e.awareness, whereas behaviour and cognition are overt expressions of the state. Wakefulness and arousal are predisposing factors for specific forms of conscious experience. Nevertheless, a unidimensional representation of consciousness fails to account for the variety of qualitatively different experiences in both normal and altered states of consciousness. To overcome this problem, cognitive and abstract multidimensional models of consciousness have been proposed, but such dimensions are interdependent and lack axiomatic support. A novel multidimensional characterization of consciousness based on the brain's macroscale functional geometry provides an alternative, empirically grounded model whose dimensions are defined by neurofunctional rather than behavioural attributes. The state of consciousness is then represented as a point in this functional multidimensional space.