First-order Operators Research Articles

Intensional Functions

Functions in functional languages have a single elimination form — application — and cannot be compared, hashed, or subjected to other non-application operations. These operations can be approximated via defunctionalization: functions are replaced with first-order data and calls are replaced with invocations of a dispatch function. Operations such as comparison may then be implemented for these first-order data to approximate e.g. deduplication of continuations in algorithms such as unbounded searches. Unfortunately, this encoding is tedious, imposes a maintenance burden, and obfuscates the affected code. We introduce an alternative in intensional functions , a language feature which supports the definition of non-application operations in terms of a function’s definition site and closure-captured values. First-order data operations may be defined on intensional functions without burdensome code transformation. We give an operational semantics and type system and prove their formal properties. We further define intensional monads , whose Kleisli arrows are intensional functions, enabling monadic values to be similarly subjected to additional operations.

Another Evidence to not Employ Customized Masked Hardware

As a well-studied countermeasure against side-channel analysis attacks, there is a general interest in applying masking to different cryptographic functions executed on different platforms. On the one hand, despite their high performance, masked hardware implementations are dedicated to specific algorithms, making them inflexible. On the other hand, applying masking on software involves serious challenges including significant overhead in terms of efficiency and difficulties to maintain theoretical security guarantees in practice. As a result, a line of research has been devoted to enable masked operations in flexible platforms (i.e., microprocessors) by including some masked modules in their hardware, hence a combination of flexibility and performance. In such scenarios, RISC-V is a natural choice as hardware can be adjusted to the extended instruction set. One such attempt presented at CHES 2021 is known as SCARV, which extends the Instruction Set Architecture (ISA) of a RISC-V core with a rich number of first-order masked operations on both Boolean and arithmetic masked operands. In this work, we conduct a comprehensive analysis of SCARV. Instead of relying on empirical measurements to demonstrate security, we utilize tool-assisted evaluations. Through these evaluations, we identified a couple of design flaws that lead to leakage in the masked implementations hosted by the corresponding processor. These flaws are primarily due to the lack of composability of cascaded components. While heuristic and ad-hoc design principles can result in secure, small, and efficient designs, they lack formal security proofs, which may lead to security flaws, like those we report here. Consequently, this work serves as a motivation for using composable masked modules and tool-assisted evaluations when constructing complex circuits.

Generalized Hom–Lie–Rinehart algebras

The first-order Hom-differential operators on a commutative Hom-associative algebra with unit are introduced. We describe Hom–Lie algebra associated with a module of first-order Hom-differential operators. We give the notion of a generalized Hom–Lie–Rinehart algebra and we show the characterization of generalized Hom–Lie–Rinehart algebras in terms of Hom–Gerstenhaber–Jacobi algebras. We prove the necessary and sufficient condition to obtain a generalized Hom–Lie–Rinehart algebra structure on a Hom–Lie algebroid.

Implicit-explicit Runge-Kutta for radiation hydrodynamics I: Gray diffusion

Radiation hydrodynamics are a challenging multiscale and multiphysics set of equations. To capture the relevant physics of interest, one typically must time step on the hydrodynamics timescale, making explicit integration the obvious choice. On the other hand, the coupled radiation equations have a scaling such that implicit integration is effectively necessary in non-relativistic regimes. A first-order Lie-Trotter-like operator split is the most common time integration scheme used in practice, alternating between an explicit hydrodynamics step and an implicit radiation solve and energy deposition step. However, such a scheme is limited to first-order accuracy, and nonlinear coupling between the radiation and hydrodynamics equations makes a more general additive partitioning of the equations non-trivial. Here, we develop a new formulation and partitioning of radiation hydrodynamics with gray diffusion that allows us to apply (linearly) implicit-explicit Runge-Kutta time integration schemes. We prove conservation of total energy in the new framework, and demonstrate 2nd-order convergence in time on multiple radiative shock problems, achieving error 3–5 orders of magnitude smaller than the first-order Lie-Trotter operator split at the hydrodynamic CFL, even when Lie-Trotter applies a 3rd-order TVD Runge-Kutta scheme to the hydrodynamics equations.

Normal Extensions of Differential Operators for Degenerate First-order

Normal Extensions of Differential Operators for Degenerate First-order

Grey Duane model for reliability growth prediction under small sample uncertain failure data

Traditional reliability growth models require failure data to adhere to a specific distribution, which greatly limits the applicability in data-scarce scenarios. Drawing from the characteristic in modeling small sample and poor information of grey system theory, this study introduces a novel grey Duane reliability growth prediction model, tailored for analyzing uncertainty in limited failure data. The characteristics and adaptability of the proposed grey Duane model (GDM) are analyzed and compared with two traditional reliability growth prediction models. Utilizing a first-order accumulation generation operation, a differential function is established to estimate the unknown parameters in GDM. The proposed GDM enables synchronized predictions of failure time, failure number, and instantaneous mean time between failures. To validate the applicability of GDM in reliability growth management, an industrial case in particular electronic equipment during an aircraft’s flight-testing process is applied, whose results demonstrate its superior predictive accuracy compared to alternative models.

Learning Distinct Relationship in Package Recommendation With Graph Attention Networks

Recommendation systems have been widely developed and extensively used in various websites and platforms to promote products or services to interested users. However, in quite a few sale scenarios, the platform has the necessity to display users a series of items, which is called package recommendation. There is very little research in this area. This article develops a novel and realistic package recommendation system named package graph attention network (PGAT) based on graph neural network. PGAT integrates users, items, and packages to build a unified heterogeneous graph and treat them as a whole. PGAT incorporates an attention mechanism in the first-order neighborhood aggregation operation, which can differentiate the weight of different neighbor nodes to the center node. By performing graph attention and graph convolution operations on the tripartite graph, PGAT can learn node embeddings more expressively and address the problem of data sparse to a large extent. Extensive experiment results on two real-world datasets validate the outstanding performance of PGAT, which is superior to the state-of-the-art baselines by 0.77%–10.12%.

On Some Class of Normal Differential Operators for First Order

In this work, we construct the minimal and maximal operators generated by linear differential-operator expression for first order in the Hilbert space of vector-functions on finite symmetric interval. Then, deficiency indices of the minimal operator will be calculated and the space of boundary values of this operator will be constructed. By using of Calkin-Gorbachuk method, the general representation of all normal extensions of the formally normal minimal operator in terms of boundary values will also be established. Moreover we explore the spectrum structure of these extensions.

Maxwell operator in a cylinder

The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such an operator is represented as a sum of a countable set of matrix differential operators of first order acting in L 2 ( R ) . Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: (1) in the case of coefficients stabilizing at infinity; and (2) in the case of periodic coefficients.

On the Big Hartley transform

In this paper, we introduce a new type of singular first-order differential-difference operator of Dunkl type on the real line. This operator is obtained as a limiting case from both the first-order Dunkl-type operators corresponding to Bannai-Ito and Big 1-Jacobi orthogonal polynomials. We provide an explicit expression for the eigenfunction of this operator in terms of Bessel functions. The obtained kernel is called the Big Hartley function, which is a generalization of the usual Hartley kernel and the little Hartley function studied in Bouzeffour [The generalized Hartley transform. Integral Transforms Spec Funct. 2014;25(3):230–239]. Additionally, we present a new product formula for the little Hartley function, which is related to the Kingman-Bessel hypergroup and the Rosler-Dunkl signed hypergroup. Finally, we investigate inversion formulae for the transforms of both the little Hartley function and the big Hartley function.

An efficient autofocus method for microscope based on the improved first-order derivative Gaussian filtering operator

This paper introduces an improved filtering operator based on the first-order derivative Gaussian filtering operator. As the defocus distance decreases, the edges of an image transition from being smooth to sharp. Using the operator proposed in this study, a parameter can be obtained to quantify the degree of edge sharpness, where a smaller value indicates sharper edges. By using the characteristic, we can judge whether the object is near the focal plane. Combining this information, we proposed two focus method. Both of them can avoid the problem of the hill climbing algorithm locating at local extremum points and one of them can locate the watershed between “coarse search” and “fine search” before the sharpness value crosses the peak value, so as to realize the transition from “coarse search” to “fine search” before crossing the peak value and improve focusing efficiency. This is different from previous passive focusing strategies. The final experiment shows that the proposed solutions in this paper is effective. However, the methods proposed in this study are only applicable to incoherent optical microscopy imaging systems and are not suitable for coherent light source microscopy imaging systems. Besides, the prerequisite for the successful implementation of the two proposed focusing strategies in this study is that the target exhibits sharp edges when it is within the focal plane.

Multi-dimensional visual data completion via weighted hybrid graph-Laplacian

Graph Laplacian has been extensively studied as a priori to assist in visual completion of partially observed entries. However, a simple treatment using first-order operation may be suboptimal since the inherent smooth property within the visual data is ill-considered. To relieve this issue, in this paper, we propose an improved priori, namely weighted hybrid graph Laplacian (WHGL), to better characterize the spatially hierarchical knowledge hidden in the multi-dimensional data. On the whole, two individual layers are presented to encode the structured properties of the given data. In the first layer, the underlying adjacency matrix of the local and nonlocal relationships within the data is composed to capture the pixel-level structure information. Global smoothness is considered as the second layer to reveal the piecewise correlations of different visual elements. Mathematically, we generalize the graph Laplacian in the first- and second-order to construct the respective layers. Finally, a re-weighting strategy is used to mitigate the discontinuity of the reconstructed data under extreme cases, e.g. 95% voxels missing. With an initial step borrowed from some off-the-shelf methods, our model can be solved under limited iterations by the Preconditioned Conjugate Gradients algorithm. The experimental results demonstrate the superiority of the proposed method.

Recursive solution of initial value problems with temporal discretization

We construct a continuous domain, as a model of interval analysis, for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.

A new Poisson-type equation applicable to the three-dimensional non-hydrostatic model in the framework of the discontinuous Galerkin method

In this study, a new Poisson-type equation for non-hydrostatic pressure in terrain-following σ-coordinates is proposed. The resulting equation consists of a standard elliptic operator and other first-order operators for non-hydrostatic pressure. Compared with the original Poisson-type equation, the treatment of the boundary condition at the lateral boundary can be simplified. Numerical tests indicate that the reformulated Poisson-type equation shows better convergence properties than the original equation when the quadrature-free nodal discontinuous Galerkin method is adopted for numerical discretization. Additionally, for the standing wave case, the numerical error of the three-dimensional non-hydrostatic model with the reformulated Poisson equation is smaller than that with the original equation, the numerical results of the solitary wave case indicate that the dissipation of the three-dimensional non-hydrostatic model is small, and transformation of wave over a circular shoal indicates that the developed model can properly simulate combined wave refraction and diffraction.

A comparison of linear consistent correction methods for first-order SPH derivatives

A well-known issue with the widely used Smoothed Particle Hydrodynamics (SPH) method is the neighborhood deficiency. Near the surface, the SPH interpolant fails to accurately capture the underlying fields due to a lack of neighboring particles. These errors may introduce ghost forces or other visual artifacts into the simulation. In this work we investigate three different popular methods to correct the first-order spatial derivative SPH operators up to linear accuracy, namely the Kernel Gradient Correction (KGC), Moving Least Squares (MLS) and Reproducing Kernel Particle Method (RKPM). We provide a thorough, theoretical comparison in which we remark strong resemblance between the aforementioned methods. We support this by an analysis using synthetic test scenarios. Additionally, we apply the correction methods in simulations with boundary handling, viscosity, surface tension, vorticity and elastic solids to showcase the reduction or elimination of common numerical artifacts like ghost forces. Lastly, we show that incorporating the correction algorithms in a state-of-the-art SPH solver only incurs a negligible reduction in computational performance.

Hydroelastic analysis of a floating bridge under spatially inhomogeneous waves, with emphasis on the effect of drift force modeling

The pontoon-supported floating bridge is an alternative concept for crossing a fjord with a width of up to 5 km. Due to the huge span, the first two natural periods of the bridge are more than 60 s, indicating that the corresponding modes can be excited by slowly-varying drift forces on the pontoons. A simplified approach for determining the drift forces is to assume that the pontoons are independent of the floating bridge, i.e., fixed or freely floating. However, the first-order motions of the pontoons, which are restricted by the deformation of the floating bridge, have a significant effect on the drift forces. To evaluate the uncertainties implied by the simplification of drift forces, in this study, a second-order beam-connected-discrete-modules (BCDM) hydroelastic method is adopted to assess the effect of different drift force models on the floating bridge. In this method, the first-order response amplitude operators (RAOs) of bridge-restricted pontoons are first solved in the frequency-domain. Based on the RAOs, the mean drift forces for each pontoon are then determined by using potential theory. Considering the spatial inhomogeneity of the wave field, the time series of the first- and second-order wave forces are generated by using the linear transfer functions and Newman’s approximation, respectively. Then, the method is applied to investigate the hydroelastic responses of a straight side-anchored floating bridge for the crossing site of Bjørnafjord. The results show that the horizontal displacement is very sensitive to the different force models implied by various pontoon boundary conditions, i.e., free, fixed, and bridge-restricted. The drift forces on the bridge-restricted pontoons are approximately 10%–20% larger than those for the fixed pontoons. Moreover, wave inhomogeneity results in increased vertical displacements and weak axis bending moments.

Isotropic Two-Dimensional Differentiation Based on Dual Dynamic Volume Holograms

We study the use of two dynamic thick holograms to realize isotropic two-dimensional (2D) differentiation under Bragg diffraction. Acousto-optic modulators (AOMs) are used as dynamic volume holograms. Using a single volume hologram, we can accomplish a first-order derivative operation, corresponding to selective edge extraction of an image. Since the AOM is a 1D spatial light modulator, filtering of the image only occurs along the direction of the sound propagation. To achieve 2D image processing, two AOMs are used within a Mach–Zehnder interferometer (MZI). By aligning one AOM along the x-direction on the upper arm of the interferometer and another AOM along the y-direction on the lower arm, we accomplish the sum of two first-derivative operations, leading to isotropic edge extraction. We have performed both computer simulations and optical experiments to verify the proposed idea. The system provides additional operations in optical computing using AOMs as dynamic holograms.

Generalized multistep Steffensen iterative method. Solving the model of a photomultiplier device

It is well known that the Steffensen-type methods approximate the derivative appearing in Newton's scheme by means of the first-order divided difference operator. The generalized multistep Steffensen iterative method consists of composing the method with itself m times. Specifically, the divided difference is held constant for every m steps before it is updated. In this work, we introduce a modification to this method, in order to accelerate the convergence order. In the proposed, scheme we compute the divided differences in first and second step and use the divided difference from the second step in the following m−1 steps. We perform an exhaustive study of the computational efficiency of this scheme and also introduce memory to this method to speed up convergence without performing new functional evaluations. Finally, some numerical examples are studied to verify the usefulness of these algorithms.

Nonlinear Uncertainty Estimator-Based Robust Control for PMSM Servo Mechanisms With Prescribed Performance

Active nonlinear uncertainties compensation is crucial for realizing high-precision servo control. However, the rejection of the unknown uncertainties is still a challenging problem. In view of these issues, this paper proposes a novel nonlinear uncertainty estimator based time-varying sliding mode control scheme for servo systems with prescribed performance. A nonlinear uncertainty estimator, which employs first-order low-pass filter operations with only one tuning parameter, is constructed to deal with unknown nonlinearities, while the NDE estimation error and uncertainties of the servo system are handled by the robust integral of the sign of the error (RISE) feedbacks. Meanwhile, a modified prescribed performance function (PPF) with a faster convergence rate is incorporated into the control design to restricted the tracking errors within the predefined asymptotic boundaries. By combining PPF and NDE, a time-varying sliding mode controller is developed to eliminate the reaching phase and improve the control performance. The validity and feasibility of the presented control scheme are verified with simulations and experiments based on a motor driving system.

Fibered universal algebra for first-order logics

We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing fibered algebraic semantics for general first-order logics. This framework includes a natural notion of substitution, which allows first-order logics to be considered as structural closure operators just as propositional logics are in abstract algebraic logic. We then establish an extension of the homomorphism theorem from universal algebra for generalized prop-categories and characterize two natural closure operators on the prop-categorical semantics. The first closes a class of structures (which are interpreted as morphisms of prop-categories) under the satisfaction of their common first-order theory and the second closes a class of prop-categories under their associated first-order consequence. It turns out that these closure operators have characterizations that closely mirror Birkhoff's characterization of the closure of a class of algebras under the satisfaction of their common equational theory and Blok and Jónsson's characterization of closure under equational consequence, respectively. These algebraic characterizations of the first-order closure operators are unique to the prop-categorical semantics. They do not have analogues, for example, in the Tarskian semantics for classical first-order logic. The prop-categories we consider are much more general than traditional intuitionistic prop-categories or triposes (i.e., topos representing indexed partially ordered sets). Nonetheless, to the best of our knowledge, our results are new, even when restricted to these special classes of prop-categories.