Explicit Function Research Articles

NUMERICAL COMPARISON OF THREE LOCAL MULTIVARIATE GAUSSIAN-BASED INTERPOLATION SCHEMES

This study introduces the Local Explicit Radial Basis Function scheme (LE-RBF), alongside an investigation of its effectiveness compared with two other Gaussian RBF-based methods: the Modified Radial Basis Function Shepard's method (MRBFS) and the Local Radial Basis Function Karel's algorithm (KAREL), for interpolation problems. The research focuses on three challenging applications: Franke’s function in two and three dimensions, and grayscale image reconstruction. All three schemes, operating in a local interpolation manner, are assessed across multiple criteria, including accuracy, CPU time and storage, and sensitivity to parameters and node sizes. The numerical experiments conducted reveal that while all three schemes yield reasonably good results in most scenarios, the proposed LE-RBF method stands out for its higher accuracy, reduced sensitivity to nodes and shape choices, and lower CPU time and storage requirements. Notably, LE-RBF demonstrates superior performance in various node densities and shape parameter-selecting strategies, especially when combined with specific strategies like the Carlson shape at 142×142 nodes. Its efficiency in processing time further highlights its practicality for complex applications. The study concludes that LE-RBF, with its robust performance and flexibility, presents a promising avenue for future application in diverse scientific and engineering tasks.

Tackling nonlinear price impact with linear strategies

AbstractEmpirical studies in various contexts find that the price impact of large trades approximately follows a power law with exponent between 0.4 and 0.7. Yet, tractable formulas for the portfolios that trade off predictive trading signals, risk, and trading costs in an optimal manner are only available for quadratic costs corresponding to linear price impact. In this paper, we show that the resulting linear strategies allow to achieve virtually optimal performance also for realistic nonlinear price impact, if the “effective” quadratic cost parameter is chosen appropriately. To wit, for a wide range of risk levels, this leads to performance losses below 2% compared to a numerical algorithm proposed by Kolm and Ritter, run at very high accuracy. The effective quadratic cost depends on the portfolio risk and concavity of the impact function, but can be computed without any sophisticated numerics by simply maximizing an explicit scalar function.

Across-horizon propagation and infinite time-dilation: An independent approach to Hawking radiation

The conformal scalar propagator as an explicit function of the Schwarzschild black-hole space-time has been established in the preceding three projects. The present project examines the precise relation which that propagator has to the amplitude the Schwarzschild black hole emit a scalar particle in a particular mode of positive energy. It is thereby established that the effect of infinite time-dilation on the event horizon results in both, the thermal radiation in the background of a static exterior space-time geometry and the detraction from thermality if energy-conservation is properly imposed as a constraint on scalar propagation. The derivation of the latter signifies an equivalent approach to the result of Parikh and Wilczek from the semi-classical perspective of the distant observer. From that perspective for that matter, Hawking radiation emerges in both contexts as the exclusive consequence of infinite time-dilation on the event horizon. These results are consistent with Hawking radiation as the exclusive consequence of the causal structure of the Schwarzschild black-hole space-time both, in a static exterior geometry and in such dynamical exterior geometry as energy-conservation signifies. The extension of the thermal case to other spin-fields and black-hole geometries is discussed.

Integrability and Inverse Scattering Transform of the Modified Benjamin-Ono Equation

Abstract This paper presents a B{"a}cklund transformation, the Lax representation, and conserved quantities for the modified Benjamin--Ono equation. The initial problem of the modified Benjamin--Ono equation on the line was studied by the inverse scattering transform method, presenting a nonlocal Riemann--Hilbert problem in the inverse problem to reconstruct the explicit potential function. Further, the exact $N$-soliton solutions and long--time asymptotic behavior are provided, respectively. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms. As a revised version, the mBO equation incorporates the solution features of the BO equation, and its solutions are presented in logarithmic form, which seeks to describe related physical phenomena more accurately. Our results will contribute to further exploration of physical and mathematical problems related to the mBO equation.

Localizability of the approximation method

We use the approximation method of Razborov to analyze the locality barrier which arose from the investigation of the hardness magnification approach to complexity lower bounds. Adapting a limitation of the approximation method obtained by Razborov, we show that in many cases it is not possible to combine the approximation method with typical (localizable) hardness magnification theorems to derive strong circuit lower bounds. In particular, one cannot use the approximation method to derive an extremely strong constant-depth circuit lower bound and then magnify it to an NC1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NC}^{1}$$\\end{document} lower bound for an explicit function. To prove this, we show that lower bounds obtained by the approximation method are in many cases localizable in the sense that they imply lower bounds for circuits which are allowed to use arbitrarily powerful oracles with small fan-in.

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that | E ( G )| ≥ α | V ( G )| 2 for some fixed α > 0. While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given ε > 0, returns an integer g such that G has an embedding in a surface of genus g , and this is ε-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G) \) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G) \) . The running time of the algorithm is O ( f (ε) n 2 ), where f (·) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time O ( f 1 (ε) n 2 ). Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [25]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [17] and to Fox, Lovász, and Zhao [14, 15]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ε-approximation ν ( G ) for the maximum number of edge-disjoint triangles in G . The value ν ( G ) can be computed by solving a linear program whose size is bounded by certain value f 2 (ε) depending only on ε. After solving the linear program, the genus can be approximated (see Corollary). The proof of this result is long and will be of independent interest in topological graph theory.

An explicit multilevel power turbulent wall function based on the de-thresholding Douglas–Peucker algorithm

Wall functions are extensively applied in engineering simulations with turbulence. They facilitate a significant increase in the scale of the grids next to the wall, which in turn reduces the total number of grids needed. This optimization enhances computational efficiency, making the simulation process more effective and streamlined. However, the current commonly used wall functions, such as the Spalding wall function, are an implicit expression that needs to be solved iteratively, which affects the computational efficiency, and the multilayer segmented wall function is not smoothly articulated, which affects the fidelity. In this study, based on flat plate direct numerical simulation (DNS) data, combined with structural ensemble dynamics theory, the de-thresholding Douglas–Peucker algorithm is introduced to construct an explicit wall function expression in the form of multilevel power exponential concatenated multiplication. The comparison of the new wall function against DNS data reveals that it demonstrates superior fitting accuracy in contrast to the traditional ones, and eliminates the need for manual calibration, reduces subjective influence, and enhances reliability. The numerical simulation outcomes for the flat plate boundary layer and a series of airfoils showcase the new wall function's exceptional accuracy, which not only meets but also surpasses the demanding standards of engineering practice.

Brayton Cycle Analysis of the Effect of Wake Ingestion on Aircraft Range

The effect of wake ingestion on the propulsive efficiency of turbojet and turbofan engines has been examined to determine the effect of wake ingestion on jet aircraft range. In a new approach, the Brayton turbine cycle was analyzed to show how wake ingestion affects the thrust and specific fuel consumption of jet engines. It is argued that it is not necessary to add an expression for propulsive efficiency to the Breguet range equation, because the thrust-specific fuel consumption is an explicit function of the inlet Mach number. It is concluded that wake ingestion does not increase the aircraft’s maximum range because it reduces the engine’s maximum thrust.

Co-design of an unmanned cable shovel for structural and control integrated optimization: A highly heterogeneous constrained multi-objective optimization algorithm

Unmanned cable shovel is an intelligence-based large machine used for excavating in open-pit coal mines, with significant implications for the security and development of energy resources. The co-design of unmanned cable shovels, considering elements from both structural and control systems, has demonstrated superior performance in obtaining optimal solutions compared to the traditional approaches that focus on individual systems. However, the inherent complexities of diverse systems often lead to heterogeneous function evaluations, posing challenges for the existing optimization algorithms. To address highly heterogeneous multi-objective optimization problems characterized by surrogate-approximated expensive functions and explicit inexpensive functions, this study proposes a highly heterogeneous constrained multi-objective optimization (HHCMO) algorithm. Leveraging NSGA-III as an evolutionary optimizer and the Kriging model for approximating expensive functions, HHCMO strategically addresses the challenges posed by highly heterogeneous objective evaluations. A Monte Carlo sampling-based expected hypervolume improvement (EHVI) criterion, employing a new nearest point approximation method in a unified sampling space, effectively handles the impact of heterogeneous objective evaluations on the sequential infill of the Kriging model. Thorough evaluations on both unconstrained and constrained multi-objective benchmark problems validate the correctness of HHCMO and its superiority relative to state-of-the-art algorithms. Finally, the effectiveness of the proposed method is successfully demonstrated in an engineering scenario of an unmanned cable shovel, where it adeptly handles expensive functions from the structural system and inexpensive functions from the control system.

Recursive identification of kernel-based hammerstein model for online energy efficiency estimation of large-scale chemical plants

Accurate energy efficiency prediction is crucial for decision-making in large-scale chemical production, especially considering energy management and environmental concerns. The complexity of chemical processes, characterized by strong nonlinearities, dynamics, and uncertainties, challenges traditional prediction methods. This paper introduces KHSM (Kernel-based Hammerstein Subspace Model), which models nonlinear static and linear dynamic processes in series by embedding kernel function-based nonlinear mapping into subspace modeling for energy efficiency estimation. KHSM identifies nonlinear process models without explicit nonlinear mapping functions, constructing dynamic models that encompass both energy input and product output variables. An adaptive KHSM variant dynamically updates the energy efficiency model, improving accuracy by iteratively selecting and excluding samples. Comparative analysis shows KHSM outperforms existing methods, achieving a coefficient of determination (R2) closest to 1 and reducing the root-mean-square error (RMSE) by up to an order of magnitude, with the smallest mean and standard deviation. Additionally, modeling efficiency increases by an order of magnitude. The developed model enables ongoing energy efficiency estimation and proactive identification of efficiency trends. Validation through mathematical and real-world ethylene process cases demonstrates KHSM's efficacy and applicability. These estimates facilitate informed adjustments to energy management strategies and optimization of production approaches in energy-intensive chemical processes.

Mesenchymal Vangl1 and Vangl2 facilitate airway elongation and widening independently of the planar cell polarity complex.

Adult mammalian lungs exhibit a fractal pattern, as each successive generation of airways is a fraction of the size of the parental branch. Achieving this structure likely requires precise control of airway length and diameter, as the embryonic airways initially lack the fractal scaling observed in the adult. In monolayers and tubes, directional growth can be regulated by the planar cell polarity (PCP) complex. Here, we characterized the roles of PCP complex components in airway initiation, elongation and widening during branching morphogenesis of the lung. Using tissue-specific knockout mice, we surprisingly found that branching morphogenesis proceeds independently of PCP complex function in the lung epithelium. Instead, we found a previously unreported Celsr1-independent role for the PCP complex components Vangl1 and Vangl2 in the pulmonary mesenchyme, where they are required for branch initiation, elongation and widening. Our data thus reveal an explicit function for Vangl1 and Vangl2 that is independent of the core PCP complex, suggesting a functional diversification of PCP complex components in vertebrate development. These data also reveal an essential role for the embryonic mesenchyme in generating the fractal structure of airways in the mature lung.

Loops in anti de Sitter space

We discuss general one and two-loop banana diagrams and one-loop diagrams with external lines with arbitrary masses on the anti de Sitter spacetime by using methods of AdS quantum field theory in the dimensional regularization approach. The banana diagrams explicitly computed in this paper are indeed the necessary ingredients for the evaluation of the two-loop effective potential of the Standard Model and can be used to extend the flat space results in presence of a negative cosmological constant. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. In the two-loop case we show the explicit calculation is possible thanks to a remarkable discrete Källén-Lehmann formula which we found and proved sometimes ago and whose domain of applicability we extend in the present paper.

Energetic Information from Information-Theoretic Approach in Density Functional Theory as Quantitative Measures of Physicochemical Properties.

The Hohenberg-Kohn theorem of density functional theory (DFT) stipulates that energy is a universal functional of electron density in the ground state, so energy can be thought of having encoded essential information for the density. Based on this, we recently proposed to quantify energetic information within the framework of information-theoretic approach (ITA) of DFT (J. Chem. Phys. 2022, 157, 101103). In this study, we systematically apply energetic information to a variety of chemical phenomena to validate the use of energetic information as quantitative measures of physicochemical properties. To that end, we employed six ITA quantities such as Shannon entropy and Fisher information for five energetic densities, yielding twenty-six viable energetic information quantities. Then, they are applied to correlate with physicochemical properties of molecular systems, including chemical bonding, conformational stability, intermolecular interactions, acidity, aromaticity, cooperativity, electrophilicity, nucleophilicity, and reactivity. Our results show that different quantities of energetic information often behave differently for different properties but a few of them, such as Shannon entropy of the total kinetic energy density and information gain of the Pauli energy density, stand out and strongly correlate with several properties across different categories of molecular systems. These results suggest that they can be employed as quantitative measures of physicochemical properties. This work not only enriches the body of our knowledge about the relationship between energy and information, but also provides scores of newly introduced explicit density functionals to quantify physicochemical properties, which can serve as robust features for building machine learning models in future studies.

Loops in de Sitter space

We discuss general one and two-loops banana diagrams with arbitrary masses on the de Sitter spacetime by using direct methods of dS quantum field theory in the dimensional regularization approach. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. For the two-loop case we show that the calculation is made easy thanks to a remarkable Källén-Lehmann formula that has been in the literature for a while. We discuss the divergent cases at d = 3 using a contiguity formula for generalized hypergeometric functions and we extract the dominant term at d = 4 proving a general formula to deal with a divergent hypergeometric series.

Many-Body Function Corrected Neural Network with Atomic Attention (MBNN-att) for Molecular Property Prediction.

Recent years have seen a surge of machine learning (ML) in chemistry for predicting chemical properties, but a low-cost, general-purpose, and high-performance model, desirable to be accessible on central processing unit (CPU) devices, remains not available. For this purpose, here we introduce an atomic attention mechanism into many-body function corrected neural network (MBNN), namely, MBNN-att ML model, to predict both the extensive and intensive properties of molecules and materials. The MBNN-att uses explicit function descriptors as the inputs for the atom-based feed-forward neural network (NN). The output of the NN is designed to be a vector to implement the multihead self-attention mechanism. This vector is split into two parts: the atomic attention weight part and the many-body-function part. The final property is obtained by summing the products of each atomic attention weight and the corresponding many-body function. We show that MBNN-att performs well on all QM9 properties, i.e., errors on all properties, below chemical accuracy, and, in particular, achieves the top performance for the energy-related extensive properties. By systematically comparing with other explicit-function-type descriptor ML models and the graph representation ML models, we demonstrate that the many-body-function framework and atomic attention mechanism are key ingredients for the high performance and the good transferability of MBNN-att in molecular property prediction.

Imaging quality enhancement in photon-counting single-pixel imaging via an ADMM-based deep unfolding network in small animal fluorescence imaging

Photon-counting single-pixel imaging (SPI) can image under low-light conditions with high-sensitivity detection. However, the imaging quality of these systems will degrade due to the undersampling and intrinsic photon-noise in practical applications. Here, we propose a deep unfolding network based on the Bayesian maximum a posterior (MAP) estimation and alternating direction method of multipliers (ADMM) algorithm. The reconstruction framework adopts a learnable denoiser by convolutional neural network (CNN) instead of explicit function with hand-crafted prior. Our method enhances the imaging quality compared to traditional methods and data-driven CNN under different photon-noise levels at a low sampling rate of 8%. Using our method, the sensitivity of photon-counting SPI prototype system for fluorescence imaging can reach 7.4 pmol/ml. In-vivo imaging of a mouse bearing tumor demonstrates an 8-times imaging efficiency improvement.

On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

In this article we solve the following problem: “The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing.” We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation.

A novel formulation of wind velocity spectrum incorporating rainfall influence

Wind-rain interaction constitutes a common physical phenomenon in nature. Nevertheless, in fluid mechanics and wind engineering, whether and how rainfall influences wind remains an unresolved challenge. This study relies on a substantial dataset obtained through on-site measurement. Initially, three distinct rainfall categories—light rain, moderate rain, and heavy rain—are established based on the magnitude of vertical wind velocity. Notably, significant disparities in the high-frequency constituents of the fluctuating wind velocity spectrum are discerned across various rainfall levels. Subsequently, a rigorous quantitative analysis explores the relationship between the high-frequency cumulative energy proportion of the fluctuating wind velocity spectrum and rainfall intensity, leading to formulations that delineate the high-frequency cumulative energy proportion in the alongwind, acrosswind, and vertical wind directions as explicit functions of rainfall intensity. Moreover, a comprehensive unified expression models the three-dimensional fluctuating wind velocity spectrum. This expression incorporates parameters encompassing the relative scales of fluctuation and mean components, the extent of conversion between high and low-frequency energy, and the scaling laws governing the inertial subrange. Additionally, explicit functional dependencies for these parameters concerning variations in rainfall intensity are provided. The novel formulation of the fluctuating wind velocity spectrum proposed in this study bears substantial scientific significance, as it facilitates a deeper understanding of wind-rain interaction and offers valuable insights for conducting intricate environmental flow field simulations.

A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains

The large deviations at various levels that are explicit for Markov jump processes satisfying detailed balance are revisited in terms of the supersymmetric quantum Hamiltonian H that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density p^(C) of the configurations C seen during a trajectory over the large time window [0,T] , and rewrite the explicit Donsker–Varadhan rate function as the matrix element I[2][p^(.)]=⟨p^|H|p^⟩ involving the square-root ket |p^⟩ . (The analog formula is also discussed for reversible diffusion processes as a comparison.) We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density p^(C) and the empirical local activities a^(C,C′) characterizing the density of jumps between two configurations (C,C′) . Finally, the explicit rate function for the joint probability of the empirical density p^(C) and of the empirical total activity A^ that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements ⟨p^|H|p^⟩ and ⟨p^|Hoff|p^⟩ , where Hoff represents the off-diagonal part of the supersymmetric Hamiltonian H. This general framework is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian H corresponds to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites. It is then useful to introduce the quantum density matrix ρ^=|p^⟩⟨p^| associated with the empirical density p^(.) in order to rewrite the various rate functions in terms of reduced density matrices involving only two or three neighboring sites.

Second-order reliability analysis of an energy pile with CPT data

As pile foundations and ground source heat producers, energy piles are becoming more and more popular. Even after much research, engineers find their design to be challenging. An effective way for using the Second-Order Reliability method (SORM) in foundation engineering scenarios including explicit limit state functions is shown in this article. The suggested SORM process is simple to implement in a spreadsheet and is based on an approximate paraboloid fitted to the limit state surface close to the design point. Using well-established closed-form SORM formulae, the failure probability is automatically determined using the primary curvatures of the limit state surface and the reliability index. An energy pile foundation engineering design examples are analyzed using the proposed method and discussed. Comparison with FORM, SORM and Genetic Programming (GP)-based FORM and SORM are made. In the case of determination of group capacity of pile and thermal load the input random variables are qc0, qc1, qc2, fs and E. The reliability index and probability of failure value are found to be 9.34 for FORM, 9.55 for SORM, 9.24 for GP-based FORM, and 9.45 for GP-based SORM. The probability of failure value for FORM 4.60E-21, SORM 3.24E-21, GP-based FORM 1.20E-20, and GP-based SORM is 8.50E-21. The relative percentage error for the average second-order approximation corresponding to FORM is also calculated, and found that Pf2 is 29.48 % in the case of simple SORM and 29.15 % in the case of GP-based SORM. The Breitung provides the most accurate estimation, and the Cai_Elishakoff provides the least accurate estimate.