## The proposed continuous-time problem is solved in a finite-dimensional function space spanned by Bernstein polynomials, which converts the problem into a solvable mixed-integer linear programming problem.

Continuous-Time Look-Ahead Optimization of Energy Storage in Real-Time Balancing and Regulation Markets

## A function space-based method is developed to solve the proposed model, which converts the continuous-time problem into a mixed-integer linear programming problem with finite dimensional decision space.

Scheduling and Pricing of Energy Generation and Storage in Power Systems

## The first aim of this work is to introduce a new function space called the measurable set-valued Borel functions space and new topological space called the confine measurable set-valued Borel function topology on the set of the measurable set-valued Borel functions.

Measurable SB-Functions Space And Confine MŞB-Function Topology

## Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$.

Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

## Let X be a ball Banach function space on R.

Compactness Characterizations of Commutators on Ball Banach Function Spaces

## These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power H α in the sense that similar estimates might fail with the classical Besov spaces.

Regularity estimates for the Cauchy problem to a parabolic equation associated to fractional harmonic oscillators

## Appropriate function spaces and derivation of adjoint operators are investigated.

All-at-once formulation meets the Bayesian approach: A study of two prototypical linear inverse problems

## In this paper, we introduce the notion of ρ-attractive elements in modular function spaces.

$rho$-Attractive Elements in Modular Function Spaces

## In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.

Multistep-type construction of fixed point for multivalued ρ-quasi-contractive-like maps in modular function spaces

## In these operators, σ not only features the sequences of operators but also features the Korovkin function set { 1 , σ , σ 2 } $\lbrace 1,\sigma ,\sigma ^{2} \rbrace$ in a weighted function space such that the operators fix exactly two functions from the set.

On approximation properties of a new construction of Baskakov operators

## Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate.

Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness

## As applications, we obtain that the above operators are bounded on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces and on the vector-valued function spaces.

Weighted estimates for Marcinkiewicz integrals with applications to angular integrability

## The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting \`a la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Triebel-Lizorkin spaces.

An intersection representation for a class of anisotropic vector-valued function spaces

## In this paper we study Toeplitz and Cesàro-type operators on holomorphic function spaces on a homogeneous Siegel domain of Type II.

Toeplitz and Cesàro-type operators on homogeneous Siegel domains

## We study several connected problems of holomorphic function spaces on homogeneous Siegel domains.

Holomorphic function spaces on homogeneous Siegel domains

## The proposed continuous-time problem is solved in a finite-dimensional function space spanned by Bernstein polynomials, which converts the problem into a solvable mixed-integer linear programming problem.

Continuous-Time Look-Ahead Optimization of Energy Storage in Real-Time Balancing and Regulation Markets

## A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup.

Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

## One of the main advantages of our proposal are both its simplicity and its ability to scale up (in objective function space).

An Empirical Study on the Use of the S-energy Performance Indicator in Mating Restriction Schemes for Multi-Objective Optimizers

## In addition, this approach identifies a diverse set of solutions in the multi-objective function space which can be challenging to estimate with single-objective formulations.

Multi-Objective Evolutionary Algorithm for PET Image Reconstruction: Concept

## As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space $$A^{2}({\mathbb {D}})$$ , Fock space $$F^{2}({\mathbb {C}})$$ etc.

Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces

## The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces.

Path Components of the Space of (Weighted) Composition Operators on Bergman Spaces

## As an application, we give a characterization for the boundedness of the Volterra integral operator $$J_{g}$$ from $$\mathcal{L}_{p,\lambda}$$ to general function spaces $$F(p,p-1-\lambda,s)$$.

Volterra Integral Operators from Campanato Spaces into General Function Spaces

## Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces D p p−� to general function spaces are also investigated.

BCM volume 64 issue 3 Cover and Back matter

## The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence.

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

## It is shown that the marginal function of the considered control system is lower semi-continuous and the optimal states operator generates a continuous branch in a suitable function space.

Optimal Boundary Control of the Boussinesq Approximation for Polymeric Fluids

## Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces.

Solving a well-posed fractional initial value problem by a complex approach

## In this paper, we obtain some inequalities about commutators of a rough p -adic fractional Hardy-type operator on Herz-type spaces when the symbol functions belong to two different function spaces.

The boundedness of commutators of rough p-adic fractional Hardy type operators on Herz-type spaces

## Here, we present a bio-inspired global stochastic optimisation method applicable in Hilbert function spaces.

Global optimisation in Hilbert spaces using the survival of the fittest algorithm

## The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

On Pointwise Product Vector Measure Duality

## Using operator-valued Ċα -Fourier multiplier results on vector-valued Hölder continuous function spaces and the Carleman transform, we characterize the Cα -well-posedness of second order degenerate differential equations with infinite delay: (Mu)′′(t) = Au(t)+ ∫ t −∞ a(t− s)Au(s)ds+ f (t) and (Mu′)′(t) = Au(t)+ ∫ t −∞ a(t−s)Au(s)ds+ f (t) on R, where A : D(A)→ X and M : D(M)→ X are closed linear operators in a complex Banach space X , and a ∈ L(R+)∩L(R+; tα dt).

Well-posedness of second order degenerate differential equations with infinite delay in Hölder continuous function spaces

## In this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$ -Fourier multiplier results on vector-valued Holder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$ -well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$ , ( $t\in \mathbb{R}$ ), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$ , $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

## In the variational forms we introduce, the solution space is defined as a subspace V of the graph space associated with the differential operator in question, whereas the test function space L is a tuple of L 2 spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.

Well-posed variational formulations of Friedrichs-type systems

## Scalar and vector basis functions for the phase space $$\Omega$$ (realizations of a turbulent flow) and the test function space $$\mathcal{N}_p$$ (argument functions of the characteristic functional) plus analytic functions, for the purpose of testing numerically the convergence properties of the bases, are constructed using cylindrical coordinates suitable for the periodic flow through straight pipes with circular cross section.

Appendix C: Scalar and Vector Bases for Periodic Pipe Flow

## We establish the well-posedness of the MHD boundary layer system in Gevrey function space without any structural assumption.

Well-posedness of the MHD Boundary Layer System in Gevrey Function Space without Structural Assumption

## We first establish the existence and uniqueness of solution in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ , then with the definition modulus of continuity, we show that the solution of Euler system is continuously dependent of the initial data $$v_{0}$$ in $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$.

Isentropic Approximation and Gevrey Regularity for the Full Compressible Euler Equations in $${\mathbb {R}}^{N}$$

## We construct the relevant function space, which is significantly constrained due to the extended Steinmann relations, up to weight 13 in coproduct form, and up to weight 12 as an explicit polylogarithmic representation.

Hexagon bootstrap in the double scaling limit

## Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kreĭn-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

## The proper function spaces and assumptions are proposed to discuss the existence of mild solutions.

Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises

## A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (<jats:italic>C</jats:italic>).

Schrödinger–Newton equations in dimension two via a Pohozaev–Trudinger log-weighted inequality

## Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parametrization of a low-dimensional transition manifold in a certain function space.

Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds

## To unveil the inverse and composition properties of the IF operators, certain function spaces with their characterizations are presented and analyzed.

Novel improved fractional operators and their scientific applications

## In this current manuscript, some general classes of weighted analytic function spaces in a unit disc are defined and studied.

Estimates on Some General Classes of Holomorphic Function Spaces

## In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces.

Global solvability of 2D MHD boundary layer equations in analytic function spaces

## The Volterra operator is defined on an arbitrary normed function space F d that is continuously embedded in the space of square integrable functions defined on the unit d -cube.

Tractability for Volterra problems of the second kind with convolution kernels

## This paper will introduce you to some properties of normed function spaces with many groups variables field of Analysis and it helps me appreciate how normed Lebesgue–Morrey space with many groups of variables that build and studied new normed spaces nowadays.

Some Characterization of the Function Space Type of Lizorkin−Triebel−Morrey

## This includes the basic definitions and properties of classical function spaces and distributions, the Fourier transform and the definition of Hadamard's finite part integrals which, in fact, represent the natural regularization of homogeneous distributions and of the hypersingular boundary integral operators.

Representation Formulae, Local Coordinates and Direct Boundary Integral Equations

## We characterize the reducing subspaces of S 2 ⁎ S 1 or S 1 S 2 ⁎ as wandering subspaces with additional structures, and give a unified way to describe the reducing subspaces of Toeplitz operators induced by non-analytic monomial on weighted Hardy spaces of several variables, which including many classical function spaces, such as weighted Bergman space and Dirichlet space over the polydisk.

Reducing subspaces for the product of a forward and a backward operator-weighted shifts

## In this paper, based on generalized Herz-type function spaces K̇ q (θ) were introduced by Y.

Some results concerning localization property of generalized Herz, Herz-type Besov spaces and Herz-type Triebel-Lizorkin spaces

Pekka Koskela

## We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

On the dynamic slip boundary condition for Navier–Stokes-like problems

## The classes of acceptable variations are described by convex cones in corresponding function spaces.

On generalized solvability of variable nonlinear integral equations on cones

## These results find theoretical support in the application of the concept of scoring function space.

Computational Prediction of Binding Affinity for CDK2-ligand Complexes. A Protein Target for Cancer Drug Discovery.

## OBJECTIVE Our purpose here is to review some the of methods used to calculate the electrostatic energy of protein-drug complexes and explore the capacity of these approaches for the generation of new computational tools for drug discovery using the abstraction of scoring function space.

Electrostatic Potential Energy in Protein-Drug Complexes.

## On harmonic function spaces, we define shift operators using zonal harmonics and partial derivatives, and develop their basic properties.

Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality

## We define the Kelvin-Mobius transform of a function harmonic on the unit ball of R n and determine harmonic function spaces that are invariant under this transform.

Kelvin-Möbius-Invariant Harmonic Function Spaces on the Real Unit Ball

## We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.

A Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

## In this paper we investigated a relationship between the analytic generalized Fourier–Feynman transform associated with Gaussian process and the function space integral for exponential type functionals on the function space $$C_{a,b}[0,T]$$ C a , b [ 0 , T ].

Relationship Between the Analytic Generalized Fourier–Feynman Transform and the Function Space Integral

## For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup.

Well-posedness and asymptotic behaviour of a wave equation with non-monotone memory kernel

## We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

On the dynamic slip boundary condition for Navier–Stokes-like problems

## A function space solution method is proposed to reduce the dimensionality of the continuous-time problem by modeling the parameter and decision trajectories in a function space formed by Bernstein polynomials, which converts the continuous-time problem into a linear programming problem.

Continuous-time Look-Ahead Scheduling of Energy Storage in Regulation Markets

## A scalable and computationally efficient function space solution method is proposed that converts the continuous-time problems into mixed integer linear programming problems with finite-dimensional decision space.

Continuous-Time Locational Marginal Price of Electricity

## The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form  du dt +Au = F (t, ut), t ≥ s, us(θ) = φ(θ), ∀θ ∈ (−∞, 0], s ∈ R, where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line.

ADMISSIBLE INERTIAL MANIFOLDS FOR INFINITE DELAY EVOLUTION EQUATIONS

## VC Dimension is calculated on a class of least ϵ identifiable function space defined over the weight space identified by the processing state of the network.

On the Stability and Generalization of Neural Networks with VC Dimension and Fuzzy Feature Encoders

## The Function Space Optimization (FSO) method, recently developed by Feigl et al.

Catchment to model space mapping – learning transfer functions from data by symbolic regression

## Function Space Optimization (FSO) ist eine symbolische Regressionsmethode, die automatisiert Transferfunktionen aus Daten schätzen kann.

Regionalisierung hydrologischer Modelle mit Function Space Optimization

## We consider a semigroup acting on the function space L based a measure space.

The pointwise existence and properties of heat kernel

## In the variational forms we introduce, the solution space is defined as a subspace V of the graph space associated with the differential operator in question, whereas the test function space L is a tuple of L 2 spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.

Well-posed variational formulations of Friedrichs-type systems

## The Volterra operator is defined on an arbitrary normed function space F d that is continuously embedded in the space of square integrable functions defined on the unit d -cube.

Tractability for Volterra problems of the second kind with convolution kernels

## In the special case of Kothe-Bochner spaces of measurable vector-valued functions our main result asserts that every dominated operator T : E ( X ) → F from a Kothe-Bochner space E ( X ) to an order continuous Banach function space F has a unique representation T = T N + T H where T N is the narrow part and T H is the atomic part of T.

The Kalton and Rosenthal type decomposition of operators in Köthe-Bochner spaces

## We characterize cofinally complete metric spaces in terms of some function space topologies as well.

Cofinal completeness vis-á-vis hyperspaces

## The admissibility of the function space topology and KKM-Theorem have played important role in proving the results.

A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

10.21203/RS.3.RS-317340/V1

## In this paper, we first construct the function space of ( L,M )-fuzzy Q-convergence spaces to show the Cartesian-closedness of the category ( L,M )- QC of ( L,M )-fuzzy Q-convergence spaces.

Cartesian-closedness and Subcategories of (L,M)-fuzzy Q-convergence Spaces

10.1016/J.TOPOL.2021.107750

## Examples in function spaces are given.

Sub-posets in ω and the strong Pytkeev⁎ property

## In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in function spaces.

A global universality of two-layer neural networks with ReLU activations

10.1016/J.TOPOL.2021.107729

## These properties provide us with necessary and sufficient conditions for an isomorphic classification of function spaces C p ⁎ ( X ) , where X is any countable metric space of scattered height less than or equal to ω.

On the lp⁎-equivalence of metric spaces

10.22034/CMDE.2021.39351.1725

## ‎The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space‎.

Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems

10.1016/j.jcp.2020.109770

## Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method.

An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

10.1007/s11854-021-0155-6

## Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry.

Krein Reproducing Kernel Modules in Clifford Analysis

10.1007/s13226-021-00140-6

## Also, some examples of function spaces satisfying the given conditions are considered.

Reflexive operators on analytic function spaces

## In this paper we investigate weight cubature formula in function spaces of S.

Weight optimal order of convergence cubature formulas in Sobolev space

10.20944/preprints202010.0626.v1

## We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.

A Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

10.1002/cbic.202100401

## In this work, the sequence-function space of glycoside hydrolase family 94 (GH94) was explored in detail, using a combined approach of phylogenetic analysis and sequence similarity networks.

Exploration of GH94 sequence space for enzyme discovery reveals a novel glucosylgalactose phosphorylase specificity.

## ) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty)$ such that $f^*\chi_{[0,1]}\in E$ and $f^*\chi_{[1,\infty)}\in L^2$.

Rosenthal’s space revisited

10.1007/s00332-021-09729-x

## The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.

Vortex Patches Choreography for Active Scalar Equations

10.1007/S11749-021-00755-1

## We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain.

Regularity and approximation of Gaussian random fields evolving temporally over compact two-point homogeneous spaces

10.1007/s40314-021-01622-3

## Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms.

On fractional calculus with analytic kernels with respect to functions

10.1093/oso/9780198868781.001.0001

## It is an extensive study of metric spaces, including the core topics of completeness, compactness, and function spaces, with a good number of applications.

Fundamentals of Mathematical Analysis

10.1103/PHYSREVA.103.012413

## In this paper we incorporate derivatives of the Hamiltonian into the VQE and develop some hybrid quantum-classical algorithms, which explore both Hamiltonian and wave-function spaces for solving quantum chemistry problems more efficiently.

Hybrid quantum-classical algorithms for solving quantum chemistry in Hamiltonian–wave-function space

## As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$.

Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups

10.1016/j.cmpb.2020.105902

## A wavelet filter is used to acquire the informative matrix of each picture and decrease the dimensionality of the function space in the suggested method.

Optimal Deep Belief Network with Opposition based Pity Beetle Algorithm for Lung Cancer Classification: A DBNOPBA Approach

10.1007/S11785-021-01106-6

## The first approach is universal, in the sense that it applies in general Banach spaces; the technical conditions are typically easy to verify in sequence spaces, but are more complicated in function spaces.

On Approximate Operator Representations of Sequences in Banach Spaces

10.3389/fmolb.2021.668184

## He was a mentor and supervisor to many other leading scientists who continued his quest to characterise structure and function space.

Tracing Evolution Through Protein Structures: Nature Captured in a Few Thousand Folds

10.1016/J.JMAA.2021.125278

## The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space.

On positive linear functionals and operators associated with generalized means

10.1101/2021.01.26.428348

## Prospective statistical models developed on the datasets to efficiently explore the sequence-function space will guide towards the intelligent design of proteins and peptides through deep directed evolution.

Deep Directed Evolution of Solid Binding Peptides for Quantitative Big-data Generation

10.1007/s00006-021-01158-z

## The aim of this work is to show how the Splitting Lemma and the Representation Formula intrinsically determine a fiber bundle over the space of quaternionic slice regular functions and as a consequence, several properties of this function space are interpreted in terms of sections, pullbacks and isomorphism of fiber bundles.

A Fiber Bundle over the Quaternionic Slice Regular Functions

## In this paper, we investigate the strict inclusion relation associated with intersections and unions of a general family of function spaces.

Intersections and unions of a general family of function spaces

10.1088/1742-6596/1999/1/012136

## By studying and solving the direct problem for the heat equation in composite materials, we can determine the function spaces and solve the inverse problem.

Development the Regularization Computing Method for Solving Boundary Value Problem to Heat Equation in the Composite Materials

10.1287/MOOR.2020.1095

## Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability.

Extension of Monotonic Functions and Representation of Preferences

10.1016/J.JCP.2020.109863

## The function space, initial and boundary conditions are carefully chosen so that it fixes the relative orientation and displacement, and we follow a gradient flow to let the interface find its optimal structure.

Computing interface with quasiperiodicity

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