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As an optimal one-dimensional reaction coordinate, the committor function not only describes the probability of a trajectory initiated at a phase space point first reaching the product state before reaching the reactant state but also preserves the kinetics when utilized to run a reduced dynamics model. However, calculating the committor function in high-dimensional systems poses significant challenges. In this paper, within the framework of milestoning, exact expressions for committor functions at two levels of coarse graining are given, including committor functions of phase space point to point (CFPP) and milestone to milestone (CFMM). When combined with transition kernels obtained from trajectory analysis, these expressions can be utilized to accurately and efficiently compute the committor functions. Furthermore, based on the calculated committor functions, an adaptive algorithm is developed to gradually refine the transition state region. Finally, two model examples are employed to assess the accuracy of these different formulations of committor functions. [ABSTRACT FROM AUTHOR]

Makerspaces are now a common feature in many public libraries in the United States. However, there has been little research into how these spaces function for adults with disabilities. In this study, qualitative data were collected from patrons with disabilities and public librarians through online focus groups, interviews, and an online survey to address this gap in knowledge. Findings speak to the needs of a significant subset of public library patrons, describe current practices in public library makerspaces, and suggest areas for implementing more inclusive practices. [ABSTRACT FROM AUTHOR]

A new implementation of the orbital-based two-component relativistic configuration interaction approach is reported and applied to calculations of the electronic g-shifts of three diatomic radicals: AlO, HgF, and PdH. The new implementation augments efficient routines for the calculation of nonrelativistic Hamiltonians with new vectorized routines for the calculation of the action of the one-electron spin–orbit operator and allows efficient calculations for the expansion of generalized active space type. The program makes full use of double group as well as time-reversal symmetry. Particle–hole reorganization of the operators is used to improve the efficiency for expansions with nearly fully occupied orbital spaces. The flexibility of the algorithm and program is used to investigate the convergence of electronic g-shifts for the three diatomic radicals as functions of the active space, states included in the orbital optimization, and excitation levels. It was possible to converge to the valence limits within a few percent using expansions containing up to quadruple excitations. However, when excitations from the core orbitals were added, it was not possible to demonstrate convergence to within a few percent with expansions containing at most 10 × 109 determinants. [ABSTRACT FROM AUTHOR]

The present paper reformulates and improves a previously proposed determinant-based second-order multi-reference perturbative formalism. Through a rather simple modification of the energy denominators, this formalism takes into account the interactions between the model space determinants, which are repeated in outer space. The method has been shown to be size-consistent when the model space is a complete active space, which is a severe limit. It is shown here that the completeness of the model space is not necessary to keep this property, provided that the zero-order function satisfies some conditions. For instance, size consistency may be obtained from truncated complete active spaces. It may even be satisfied from Singles and Doubles Configuration Interactions, provided that a coupled electron pair approximation is used in the definition of the model space wave function. The physical content of the method is illustrated by a series of model problems, showing its robustness. A major benefit of the fact that the perturbers are single determinants is the possibility to revise with full flexibility the model-space component of the wave function, i.e., to treat the feedback effect of the dynamic correlation on the valence component of the wave function. [ABSTRACT FROM AUTHOR]

We introduce Gaussian Process Regression (GPR) as an enhanced method of thermodynamic extrapolation and interpolation. The heteroscedastic GPR models that we introduce automatically weight provided information by its estimated uncertainty, allowing for the incorporation of highly uncertain, high-order derivative information. By the linearity of the derivative operator, GPR models naturally handle derivative information and, with appropriate likelihood models that incorporate heterogeneous uncertainties, are able to identify estimates of functions for which the provided observations and derivatives are inconsistent due to the sampling bias that is common in molecular simulations. Since we utilize kernels that form complete bases on the function space to be learned, the estimated uncertainty in the model takes into account that of the functional form itself, in contrast to polynomial interpolation, which explicitly assumes the functional form to be fixed. We apply GPR models to a variety of data sources and assess various active learning strategies, identifying when specific options will be most useful. Our active-learning data collection based on GPR models incorporating derivative information is finally applied to tracing vapor–liquid equilibrium for a single-component Lennard-Jones fluid, which we show represents a powerful generalization to previous extrapolation strategies and Gibbs–Duhem integration. A suite of tools implementing these methods is provided at https://github.com/usnistgov/thermo-extrap. [ABSTRACT FROM AUTHOR]

Recent work [Mirth et al., J. Chem. Phys. 154, 114114 (2021)] has demonstrated that sublevelset persistent homology provides a compact representation of the complex features of an energy landscape in 3 N-dimensions. This includes information about all transition paths between local minima (connected by critical points of index ≥ 1) and allows for differentiation of energy landscapes that may appear similar when considering only the lowest energy pathways (as tracked by other representations, such as disconnectivity graphs, using index 1 critical points). Using the additive nature of the conformational potential energy landscape of n-alkanes, it became apparent that some topological features—such as the number of sublevelset persistence bars—could be proven. This work expands the notion of predictable energy landscape topology to any additive intramolecular energy function on a product space, including the number of sublevelset persistent bars as well as the birth and death times of these topological features. This amounts to a rigorous methodology to predict the relative energies of all topological features of the conformational energy landscape in 3N dimensions (without the need for dimensionality reduction). This approach is demonstrated for branched alkanes of varying complexity and connectivity patterns. More generally, this result explains how the sublevelset persistent homology of an additive energy landscape can be computed from the individual terms comprising that landscape. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

We give exact formulas for the subdifferentials of perturbed distance functions in a normed space. Our method, seemingly novel and different from existing ones, is to turn the involved problem equivalently to a parametric optimization problem and then apply variational analysis technique to the optimal value function. In the convex setting, we obtain new representations for the subdifferential of perturbed distance functions, which do not depend on the relative position of the reference point with respect to the input set, and which are described directly via the input data. Our results complement those of Wang et al. [J. Global Optim. 2010;46:489–501] and of Li and Bounkhel [Nonlinear Anal. 2014;108:173–188], which were established by different methods. [ABSTRACT FROM AUTHOR]

In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko–Cantelli theorems, and estimates of weighted Poincaré–Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincaré–Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs. [ABSTRACT FROM AUTHOR]

This survey shows the way in which the Armenian mathematician Academician M.M. Djrbashian introduced the apparatus of fractional calculus in investigation of weighted classes and spaces of regular functions since his earliest work of 1945 (see [3-4] or Addendum to [22]). The investigations of M.M. Djrbashian in this topic reached their final point by his exhaustive factorization theory for functions meromorphic in the unit disc of the complex plane [11]. The contemporary development of M.M. Djrbashian's ideas can be found in the recent monograph [22]. The survey intends to complete the survey article "Mkhitar Djrbashian and his contribution to fractional calculus" [25], which described the contribution of M.M. Djrbashian mainly from the point of view of basic constructions of the fractional calculus, to the theory of fractional differential equations and integral transforms. [ABSTRACT FROM AUTHOR]

In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented. [ABSTRACT FROM AUTHOR]

It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any z ∈ U , in the set V z := { f ∈ V ∣ f ⁢ (z) = 1 } , if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality K ⁢ (z , w) = K ⁢ (w , z) ¯ does not have to be true without assumptions that K ⁢ (z , w) ≠ 0 , K ⁢ (w , z) ≠ 0 . We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory. [ABSTRACT FROM AUTHOR]

We consider the problem of approximating Nash equilibria of N functions f 1 , ... , f N of N variables. In particular, we show systems of the form u ̇ j (t) = − ∇ x j f j (u (t)) (j = 1 , ... , N) are well-posed and the large time limits of their solutions u (t) = (u 1 (t) , ... , u N (t)) are Nash equilibria for f 1 , ... , f N provided that these functions satisfy an appropriate monotonicity condition. To this end, we will invoke the theory of maximal monotone operators on a Hilbert space. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces. [ABSTRACT FROM AUTHOR]

Neglecting small fragments in space debris evolutionary models can lead to a significant underestimation of the collision risk for operational satellites. However, when scaling down to the millimetre range, the debris population grows to over a hundred million objects, making deterministic approaches too computationally expensive. On the contrary, probabilistic models provide a more efficient alternative, which however typically works under some simplifying assumptions on the dynamics, limiting their field of applicability. This work proposes an extension of the density-based collision risk models to any orbital dynamics and impact geometry. The impact rate with a target satellite is derived from a multi-dimensional phase space density function in orbital elements, which discretely varies over both phase space and time. The assumption of a bin-wise constant cloud density allows for the analytical transformation of the six-dimensional distribution in orbital elements into the three-dimensional spatial density function, guaranteeing an efficient and accurate evaluation of the fragments flux. The proposed method is applied to the assessment of the collision risk posed by occurred fragmentation events in different orbital regions on a high-risk target object. The effect on the impact rate of the modelling improvements, compared to previous probabilistic formulations, is discussed. • Density-based model used for the propagation of space debris clouds in different orbital regimes. • Piece-wise analytical method for the estimation of the in-orbit collision probability is derived. • Impact risk posed by occurred fragmentation events on a target spacecraft is assessed. • Density-based models allow the required computational cost to be independent of objects size. [ABSTRACT FROM AUTHOR]

This paper deals with the global approximation of the solutions of Boundary Value Problems (BVPs) of second order on the real line. We first reduce the BVP to an equivalent Fredholm integral equation of the second kind and then approximate its solution by a Nyström type method based on a suitable product quadrature rule. Such quadrature formula is based on a truncated interpolation process at the Hermite zeros. The stability and the convergence of the method as well as the well conditioning of the involved linear systems are studied in weighted spaces of continuous functions. Numerical tests confirming the theoretical error estimates are shown. [ABSTRACT FROM AUTHOR]

Quadrature formulas are often constructed to be exact on the space of functions that are easily integrated and that are in some sense similar to the integrand. This motivates us to explore how the known properties of the integrand can be used to improve the accuracy of certain quadrature rules. In the present paper, we propose an l -point Gauss-type quadrature rule G l into which the zeros and poles of the integrand outside the integration interval are incorporated. Formula G l proves to be exact for certain rational functions which have the same zeros and poles as the integrand. It converges, all its nodes are pairwise distinct and belong to the interior of the integration interval, and all its weights are positive. Theoretical results on the remainder term of G l suggest that formula G l is applicable both when the incorporated zeros and poles of the integrand are known exactly, as well as when they are known approximately. To practically and economically estimate the error of G l , some extensions that inherit the l nodes of G l are developed. They are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Numerical experiments confirm the accuracy of G l and its extensions. [ABSTRACT FROM AUTHOR]

Particle swarm optimization, one of the modern global optimization methods, is attracting widespread interest because it overcomes the difficulties of conventional inversion techniques, such as trapping at a local minimum and/or initial model dependence. The main characteristic of particle swarm optimization is the large search space of parameters, which in a sense allows the exploration of the entire objective function space if the input parameters are properly chosen. However, in the case of a high‐dimensional model space, the numerical instability of the solution may increase and lead to unrealistic models and misinterpretations due to the sampling problem of particle swarm optimization. Therefore, smoothness‐constrained regularization techniques used for the objective function or model reduction techniques are required to stabilize the solution. However, weighting and combining objective function terms is partly a subjective process, as the regularization parameter is generally chosen based on some kind of criteria of how the smoothing constraints affect the data misfits. This means that it cannot be completely predefined but needs to be adjusted during the inversion process, which begins with the response of an initial model. In this paper, a new modelling approach is proposed to obtain a smoothness‐constrained model from magnetotelluric data utilizing multi‐objective particle swarm optimization based on the Pareto optimality approach without using a regularization parameter and combining several objective function terms. The presented approach was verified on synthetic models and an application with field data set from the Çanakkale–Tuzla geothermal field in Turkey. Findings from these analyses confirm the usefulness of the method as a new approach for all constrained inversions of geophysical data without the need to combine the objective function terms weighted by a regularization parameter. [ABSTRACT FROM AUTHOR]

We present a result concerning the essential norm of linear operators mapping some general weighted‐type spaces of holomorphic functions on the open unit ball in ℂn$$ {\mathrm{\mathbb{C}}}&amp;#x0005E;n $$ to Banach spaces. As some applications, we compare essential norms of a product‐type operator of integral type and a weighted composition operator between some of the general weighted‐type spaces. [ABSTRACT FROM AUTHOR]

Machine learning (ML) is increasingly being used to guide biological discovery in biomedicine such as prioritizing promising small molecules in drug discovery. In those applications, ML models are used to predict the properties of biological systems, and researchers use these predictions to prioritize candidates as new biological hypotheses for downstream experimental validations. However, when applied to unseen situations, these models can be overconfident and produce a large number of false positives. One solution to address this issue is to quantify the model's prediction uncertainty and provide a set of hypotheses with a controlled false discovery rate (FDR) pre-specified by researchers. We propose CPEC, an ML framework for FDR-controlled biological discovery. We demonstrate its effectiveness using enzyme function annotation as a case study, simulating the discovery process of identifying the functions of less-characterized enzymes. CPEC integrates a deep learning model with a statistical tool known as conformal prediction, providing accurate and FDR-controlled function predictions for a given protein enzyme. Conformal prediction provides rigorous statistical guarantees to the predictive model and ensures that the expected FDR will not exceed a user-specified level with high probability. Evaluation experiments show that CPEC achieves reliable FDR control, better or comparable prediction performance at a lower FDR than existing methods, and accurate predictions for enzymes under-represented in the training data. We expect CPEC to be a useful tool for biological discovery applications where a high yield rate in validation experiments is desired but the experimental budget is limited. Author summary: Machine learning (ML) models are increasingly being applied as predictors to generate biological hypotheses and guide biological discovery. However, when applied to unseen situations, ML models can be overconfident and make enormous false positive predictions, resulting in the challenges for researchers to trade-off between high yield rates and limited budgets. One solution is to quantify the model's prediction uncertainty and generate predictions at a controlled false discovery rate (FDR) pre-specified by researchers. Here, we introduce CPEC, an ML framework designed for FDR-controlled biological discovery. Using enzyme function prediction as a case study, we simulate the process of function discovery for less-characterized enzymes. Leveraging a statistical framework, conformal prediction, CPEC provides rigorous statistical guarantees that the FDR of the model predictions will not surpass a user-specified level with high probability. Our results suggested that CPEC achieved reliable FDR control for enzymes under-represented in the training data. In the broader context of biological discovery applications, CPEC can be applied to generate high-confidence hypotheses and guide researchers to allocate experimental resources to the validation of hypotheses that are more likely to succeed. [ABSTRACT FROM AUTHOR]

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity. [ABSTRACT FROM AUTHOR]

Using binary function spaces, we give an example of a pseudocompact discretely selective topological group. We show that, under PFA, every compact space of countable tightness has a countable disjoint local π-base at every point. If X is a compact space of countable tightness and all non-empty open subsets of X are non-separable, then it is proved in ZFC that X possesses a countable disjoint local π-base at every point. We also establish that a Lindelöf Σ-space has the discrete shrinking property if and only if the outer π-character of any compact subset of X is uncountable. As a consequence, any non-metrizable topological group with a countable network has the discrete shrinking property. [ABSTRACT FROM AUTHOR]

The axiomatic theory of ordinary differential equations, owing to its simplicity, can provide a useful framework to describe various generalizations of dynamical systems. In this study, we consider how dynamical properties can be generalized to this setting. First, we study the generalizations of dynamical properties, such as invariance and limit sets and show that compact weakly invariant sets can be characterized in terms of the corresponding shift-invariant sets in the function space. This result enables us to consider weakly invariant sets in terms of shift-invariant sets. Next, we compare the axiomatic theory of ODE with other formalisms of generalized dynamical systems, namely generalized semiflow and multivalued semigroup. Finally, we introduce the notion of invariant measures and show that we can generalize classical results, such as Poincaré recurrence theorem. [ABSTRACT FROM AUTHOR]

In this paper, staggered discontinuous Galerkin (SDG) approximation for the Stokes problem is developed in a 2D square domain. The square domain is discretized by using rectangular meshes, which allow us easy identification of elements and hence a simple implementation of the SDG methods. By introducing proper function spaces and interpolations for the velocity vector field, its gradient, and the pressure, some required inf-sup conditions of our SDG method are shown. Then, thanks to the inf-sup conditions, the optimal convergence of our SDG approximation to the Stokes solution is obtained with respect to the polynomial order. Numerical results are included as well to validate the performance of our proposed SDG method for the Stokes problem. [ABSTRACT FROM AUTHOR]

We illustrate the angular and modal equipartitioning of elastic waves in scattering media using two-dimensional elastic radiative transfer equations. To solve these equations, we decompose the P and S specific intensities into direct and scattered components. We handle the direct component analytically, and derive integral equations for the scattered components of the P and S specific intensities. We construct a time-stepping algorithm with which we evolve the scattered components of the specific intensities numerically in time. We handle the advection of P and S energy analytically at the computational grid points and use numerical interpolation to deal with advection terms which do not lie on the grid points. We test this algorithm for a pure P source and a double couple, which radiates both P and S energy. We compare our numerical solutions against known approximations and find good agreement. We use this algorithm to illustrate the local behavior of equipartitioning over wave modes and angular directions. We find that both types of equipartitioning are a function of space and time, depending on the extent of scattering. This local behavior must be taken into account when studying diffusion and equipartitioning of elastic waves. [ABSTRACT FROM AUTHOR]

Unfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM. [ABSTRACT FROM AUTHOR]

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers. [ABSTRACT FROM AUTHOR]

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives. [ABSTRACT FROM AUTHOR]

The main aim of this work is to show the solvability of the homogeneous and inhomogeneous boundary conjugation problems for piecewise analytic functions in the Besov space, which is embedded into the class of continuous functions. [ABSTRACT FROM AUTHOR]

We study a "p-powered" version K n p ⁢ (F ⁢ (R)) of the well-known Bohr radius problem for the family F ⁢ (R) of holomorphic functions f : R → X satisfying ∥ f ∥ < ∞ , where ∥ ⋅ ∥ is a norm in the function space F ⁢ (R) , R ⊂ ℂ n is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 , we describe in full detail the asymptotic behavior of K n p ⁢ (F ⁢ (R)) , where F ⁢ (R) is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 픻 n ; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ (l t n) of the Minkowski space l t n . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results. [ABSTRACT FROM AUTHOR]

In this work, we present a new weighted method for proving the generalized Hyers-Ulam stability for nonlinear Volterra integral equations in modular spaces. Using the same technique, we also prove the generalized Hyers-Ulam stability for nonlinear functional equations under Δ2 conditions. Fixed-point theorems in modular spaces form the foundation of our main conclusions. [ABSTRACT FROM AUTHOR]

The combination of machine learning and topology optimization has preliminarily mitigated the enormous computational costs incurred during optimization processes. Its powerful design capabilities are warmly welcomed by many scholars, but this strategy still faces many problems, such as unclear structural boundaries and high dependence on big data. Here, we propose a new machine learning topology optimization paradigm that learns the mapping relationship between the initial and optimized Level set functions in the infinite-dimensional function space under the Fourier neural operator to truly parameterize the structural model and its mechanical information, thereby achieving purely data-driven topology optimization. A base model trained under the proposed approach could obtain a topological structure with clear and smooth structural boundaries within 0.4 seconds, making it faster than traditional methods by tens of times. In addition, a lightweight retraining strategy under the base model was proposed. Four additional numerical examples were used to demonstrate that this strategy requires only 100 or 200 new samples and 3-6 seconds of retraining time to attain prediction capabilities under different boundary conditions, different meshes (resolution), various initial level set functions, and multiple loads. In addition, we discussed all the experiments and found that most of the mechanical properties of the results were close to the ground truths, while some results were even better than those of the traditional method. Although we only used the maximum structural stiffness as an example, this method can be extended to different objective functions. • FNO-based topology optimization framework: faster and more powerful. • Purely data-driven TO of parameterized structures and their mechanical models. • A lightweight retraining strategy for attaining various predictive capabilities. • Instantly predicting topological structure with clear and smooth boundaries. • The predicted structure has high accuracy in terms of mechanical properties and volume. [ABSTRACT FROM AUTHOR]

This article studies typical 1-Lipschitz images of 푛-rectifiable metric spaces 퐸 into R m for m ≥ n . For example, if E ⊂ R k , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 H n -almost everywhere and, if m > n , preserves the Hausdorff measure of 퐸. In general, we provide sufficient conditions, in terms of the tangent norms of 퐸, for when a typical 1-Lipschitz map preserves the Hausdorff measure of 퐸, up to some constant multiple. Almost optimal results for strongly 푛-rectifiable metric spaces are obtained. On the other hand, for any norm | ⋅ | on R m , we show that, in the space of 1-Lipschitz functions from ([ − 1 , 1 ] n , | ⋅ | ∞) to (R m , | ⋅ |) , the H n -measure of a typical image is not bounded below by any Δ > 0 . [ABSTRACT FROM AUTHOR]

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension. [ABSTRACT FROM AUTHOR]

We prove new optimal C 1 , α regularity results for obstacle problems involving evolutionary p -Laplace type operators in the degenerate regime p > 2. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward p -paraboloids, up to the critical exponent, α ≤ 2 / (p − 2) , and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or C 1 , α intrinsic interpolative geometries. [ABSTRACT FROM AUTHOR]

In this paper, the sampling and reconstruction problems in function subspaces of L p (R n) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval's relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the L p -stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a L p -subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations. [ABSTRACT FROM AUTHOR]

In the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv(τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed. [ABSTRACT FROM AUTHOR]

We show that a differentiable function on the 2‐Wasserstein space is geodesically convex if and only if it is also convex along a larger class of curves which we call 'acceleration‐free'. In particular, the set of acceleration‐free curves includes all generalised geodesics. We also show that geodesic convexity can be characterised through first‐ and second‐order inequalities involving the Wasserstein gradient and the Wasserstein Hessian. Subsequently, such inequalities also characterise convexity along acceleration‐free curves. [ABSTRACT FROM AUTHOR]

We consider local solutions u of nonlinear elliptic systems of the type div A (x , D u) = div F in Ω ⊂ R n , where u : Ω → R N is in a weighted W loc 1 , p space, with p ≥ 2 , F is in a weighted W loc 1 , 2 space and x → A (x , ξ) has growth coefficients in the space of functions with bounded mean oscillation. We prove higher differentiability of u in the sense that the nonlinear expression of its gradient V μ (D u) : = (μ 2 + | D u | 2 ) p - 2 4 D u , with 0 < μ ≤ 1 , is weakly differentiable with D (V μ (D u)) in a weighted L loc 2 space. Moreover we derive some local Calderón–Zygmund estimates when the source term is not necessarily differentiable. Global estimates for a suitable Dirichlet problem are also available. [ABSTRACT FROM AUTHOR]

We study stochastic reaction–diffusion equation ∂tut(x)=12∂xx2ut(x)+b(ut(x))+Ẇt(x),t>0,x∈D$$\begin{equation*} \partial _tu_t(x)=\frac{1}{2} \partial ^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in {D} \end{equation*}$$where b$b$ is a generalized function in the Besov space Bq,∞β(R)$\mathcal {B}^\beta _{q,\infty }(\mathbb {R})$, D⊂R${D}\subset \mathbb {R}$ and Ẇ$\dot{W}$ is a space‐time white noise on R+×D$\mathbb {R}_+\times {D}$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β−1/q⩾−1$\beta -1/q\geqslant -1$, β>−1$\beta >-1$ and q∈[1,∞]$q\in [1,\infty ]$. This class includes equations with b$b$ being measures, in particular, b=δ0$b=\delta _0$ which corresponds to the skewed stochastic heat equation. For β−1/q>−3/2$\beta -1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020). [ABSTRACT FROM AUTHOR]

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch's theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach. [ABSTRACT FROM AUTHOR]

For a spherical dark matter halo with isotropic velocity distribution, the phase space distribution function (DF), the energy distribution, and the density profile form a set of self-consistent description of its equilibrium state, and knowing one is sufficient to determine the other two. The Navarro–Frenk–White density profile (NFW profile) is known to be a good approximation to the spherically averaged density distribution in simulated haloes. The DARKexp energy distribution is also known to compare well with the simulated energy distribution. We present a quantitative assessment of the NFW and DARKexp fits to the simulated DF and energy distribution for a wide range of haloes in a dark-matter-only simulation from the IllustrisTNG Project. As expected, we find that the NFW fits work well except at low energy when the density at small radii deviates from the NFW profile. Further, the NFW and DARKexp fits have comparable accuracy in the region where both fit well, but the DARKexp fits are better at low energy because they require matching of the central gravitational potential. We also find an approximate relation between the energy scale parametrizing the DARKexp energy distribution and that defined by the characteristic density and radius of the NFW profile. This relation may be linked to the relaxation process during halo formation. [ABSTRACT FROM AUTHOR]

Assume that we are given a filtration (Fn)$(\mathcal F_n)$ on a probability space (Ω,F,P)$(\Omega,\mathcal F,\mathbb {P})$ of the form that each Fn$\mathcal F_n$ is generated by the partition of one atom of Fn−1$\mathcal F_{n-1}$ into two atoms of Fn$\mathcal F_n$ having positive measure. Additionally, assume that we are given a finite‐dimensional linear space S$S$ of F$\mathcal F$‐measurable, bounded functions on Ω$\Omega$ so that on each atom A$A$ of any σ$\sigma$‐algebra Fn$\mathcal F_n$, all Lp$L^p$‐norms of functions in S$S$ are comparable independently of n$n$ or A$A$. Denote by Sn$S_n$ the space of functions that are given locally, on atoms of Fn$\mathcal F_n$, by functions in S$S$ and by Pn$P_n$ the orthoprojector (with respect to the inner product in L2(Ω)$L^2(\Omega)$) onto Sn$S_n$. Since S=span{1Ω}$S = \operatorname{span}\lbrace \mathbbm 1_\Omega \rbrace$ satisfies the above assumption and Pn$P_n$ is then the conditional expectation En$\mathbb {E}_n$ with respect to Fn$\mathcal F_n$, for such filtrations, martingales (Enf)$(\mathbb {E}_n f)$ are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences (Pn−Pn−1)f$(P_n - P_{n-1})f$ form an unconditionally convergent series and are democratic in Lp$L^p$ for 1

We study Orlicz functions that do not satisfy the \Delta _2-condition at zero. We prove that for every Orlicz function M such that \limsup _{t\to 0}M(t)/t^p \!>0 for some p\ge 1, there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero and such that \begin{equation*} \sup _{k\in \mathbb {N}}\frac {M(ct_k)}{M(t_k)} <\infty,\text { for all }c>1, \end{equation*} that is, M satisfies the \Delta _2 condition with respect to T. Consequently, we show that for each Orlicz function with lower Boyd index \alpha _M < \infty there exists an Orlicz function N such that: (a) there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero such that N satisfies the \Delta _2 condition with respect to T, and (b) the space h_N is isomorphic to a subspace of h_M generated by one vector. We apply this result to find the maximal possible order of Gâteaux differentiability of a continuous bump function on the Orlicz space h_M(\Gamma) for \Gamma uncountable. [ABSTRACT FROM AUTHOR]

This paper engages in critical debate with urban informality in interstitial urban spaces through the lens of micro-scalar spatial practices motivated by everyday needs and desires. The aim is to examine the generative potential of small-scale reappropriations to change the functions, meanings and governing policies of undervalued urban spaces. An empirical focus is taken on residential alleys in inner-city neighbourhoods of Melbourne, Australia. Remnants of 19th-century sanitation and drainage infrastructure, these alleys are now underdetermined spaces of manifold functions and meanings. Drawing from extensive fieldwork documentation and interviews, this study maps and interrogates the interplay of formal and informal spatial practices. Formal practices, driven by assertion of authority rather than vision for public space, operate like Bourdieu's habitus. Informal practices, driven by everyday needs and desires, have a teleoaffective dimension that can modify the social field in which these dispositions are formed and thereby alter habitus. [ABSTRACT FROM AUTHOR]

In this article we provide new insights into the well-posedness and maximal regularity of systems of abstract evolution equations, in the framework of periodic Lebesgue spaces of vector-valued functions. Our abstract model is flexible enough as to admit time-fractional derivatives in the sense of Liouville-Grünwald. We characterize the maximal regularity property solely in terms of R -boundedness of a block operator-valued symbol, and provide corresponding estimates. In addition, we show practical criteria that imply the R -boundedness part of the characterization. We apply these criteria to show that the Keller-Segel system, as well as a reactor model system, have L q − L p maximal regularity. [ABSTRACT FROM AUTHOR]

Let |$G$| be a connected reductive group over |$\overline{{\mathbb{F}}}_{q}$| and let |$\rho ^{\vee }:G^{\vee }\rightarrow \mathrm{GL}_{n}$| be an algebraic representation of the dual group |$G^{\vee }$|⁠. Assuming that |$G$| and |$\rho ^{\vee }$| are defined over |${\mathbb{F}}_{q}$|⁠ , Braverman and Kazhdan defined an operator on the space |${{\mathcal{C}}}(G({\mathbb{F}}_{q}))$| of complex valued functions on |$G({\mathbb{F}}_{q})$|⁠. In this paper we are interested in the case where |$G$| is either |$\mathrm{SL}_{2}$| or |$\mathrm{PGL}_{2}$| and |$\rho ^{\vee }$| is the symmetric square representation of |$G^{\vee }$|⁠. We construct a natural |$G\times G$| -equivariant embedding |$G\hookrightarrow{{\mathcal{G}}}={{\mathcal{G}}}_{\rho }$| and an involutive operator (Fourier transform) |${{\mathcal{F}}}^{{{\mathcal{G}}}}$| on the space of functions |${{\mathcal{C}}}({{\mathcal{G}}}({\mathbb{F}}_{q}))$| that extends Braverman-Kazhdan's operator. [ABSTRACT FROM AUTHOR]

The Orlicz function-defined sequence spaces of functions by relative uniform convergence of sequences related to p -absolutely summable spaces are a new concept that is introduced in this article. We look at its various attributes, such as solidity, completeness, and symmetry. We also look at a few insertional connections involving these spaces. [ABSTRACT FROM AUTHOR]

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order. [ABSTRACT FROM AUTHOR]

Let 0 < q ≤ ∞ , b be a slowly varying function and Φ : [ 0 , ∞) ⟶ [ 0 , ∞) be an increasing function with Φ (0) = 0 and lim r → ∞ Φ (r) = ∞ . In this paper, we introduce a new class of function spaces L Φ , q , b which unify and generalize the Lorentz-Karamata spaces with Φ (t) = t p and the Orlicz-Lorentz spaces with b ≡ 1 . Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob's maximal inequalities by using Hardy's inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38-39] and the condition q Φ - 1 < 1 / q in [24], respectively. [ABSTRACT FROM AUTHOR]