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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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

We consider the existence, uniqueness and nonlinear stability of stationary solutions for the three-dimensional compressible micropolar fluids with the external force of general form. More precisely, with the weighted L2 and L∞ estimates on solutions to the linearized problem, we show the existence and uniqueness of stationary solutions in some suitable function space by the contraction mapping principle. The stability of the steady flow is studied by an elementary energy method. [ABSTRACT FROM AUTHOR]

Lower strict monotonicity points and lower local uniform monotonicity points are considered in the case of Musielak–Orlicz function spaces L Φ endowed with the Mazur–Orlicz F-norm. The findings outlined in this study extend the scope of geometric characteristics observed in F-normed Orlicz spaces, as well as monotonicity properties within specific F-normed lattices. They are suitable for the Orlicz spaces of ordered continuous elements, specifically in relation to the Mazur–Orlicz F-norm. In addition, in this paper presents results that can be used to derive certain monotonicity properties in F-normed Musielak–Orlicz spaces. [ABSTRACT FROM AUTHOR]

In this article, we consider the correction of metric matrices in quasi-Newton methods (QNM) from the perspective of machine learning theory. Based on training information for estimating the matrix of the second derivatives of a function, we formulate a quality functional and minimize it by using gradient machine learning algorithms. We demonstrate that this approach leads us to the well-known ways of updating metric matrices used in QNM. The learning algorithm for finding metric matrices performs minimization along a system of directions, the orthogonality of which determines the convergence rate of the learning process. The degree of learning vectors' orthogonality can be increased both by choosing a QNM and by using additional orthogonalization methods. It has been shown theoretically that the orthogonality degree of learning vectors in the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is higher than in the Davidon–Fletcher–Powell (DFP) method, which determines the advantage of the BFGS method. In our paper, we discuss some orthogonalization techniques. One of them is to include iterations with orthogonalization or an exact one-dimensional descent. As a result, it is theoretically possible to detect the cumulative effect of reducing the optimization space on quadratic functions. Another way to increase the orthogonality degree of learning vectors at the initial stages of the QNM is a special choice of initial metric matrices. Our computational experiments on problems with a high degree of conditionality have confirmed the stated theoretical assumptions. [ABSTRACT FROM AUTHOR]

Let |$G$| be a real reductive algebraic group, and let |$H\subset G$| be an algebraic subgroup. It is known that the action of |$G$| on the space of functions on |$G/H$| is "tame" if this space is spherical. In particular, the multiplicities of the space |${\mathcal {S}}(G/H)$| of Schwartz functions on |$G/H$| are finite in this case. In this paper, we formulate and analyze a generalization of sphericity that implies finite multiplicities in |${\mathcal {S}}(G/H)$| for small enough irreducible representations of |$G$|⁠. [ABSTRACT FROM AUTHOR]

Soft robots require soft actuators for mobility and functionality. To integrate multiple functions in a limited space, researchers are developing multifunctional soft actuators with several functions integrated in one device. This study successfully demonstrates an electroluminescent (EL) soft actuator, a single device with both light‐emitting and actuating capabilities, using liquid dielectric and uniformly dispersed EL phosphor particles. EL phosphor particles are copper‐doped zinc sulfide (ZnS) particles doped with copper. The work is able to disperse the ZnS particles uniformly in the oil by coating the ZnS particles with silica, ensuring consistent luminescence across the device's active area. The performance of the soft actuator is notable, with a strain of 25% and a maximum EL intensity of 23 cd m−2 for a single actuator. When stacking multiple actuators, the intensity exceeds 29 cd m−2. The findings suggest that EL actuator devices will be valuable smart components for soft robots and interactive sensors. They can effectively represent external stimuli through color signals, expanding the capabilities of these systems. [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]