Kernel Of Operator Research Articles

Image edge detection with a photonic spiking VCSEL-neuron.

We report both experimentally and in theory on the detection of edge features in digital images with an artificial optical spiking neuron based on a vertical-cavity surface-emitting laser (VCSEL). The latter delivers fast (< 100 ps) neuron-like optical spikes in response to optical inputs pre-processed using convolution techniques; hence representing image feature information with a spiking data output directly in the optical domain. The proposed technique is able to detect target edges of different directionalities in digital images by applying individual kernel operators and can achieve complete image edge detection using gradient magnitude. Importantly, the neuromorphic (brain-like) spiking edge detection of this work uses commercially sourced VCSELs exhibiting responses at sub-nanosecond rates (many orders of magnitude faster than biological neurons) and operating at the important telecom wavelength of 1300 nm; hence making our approach compatible with optical communication and data-centre technologies.

Fast-diffusion limit for reaction-diffusion equations with multiplicative noise

In this paper we consider a class of a stochastic partial differential equations with multiplicative noise and consider the limit of fast diffusion or other fast elliptic operator. We show that one can approximate solutions of the stochastic partial differential equations by the solution of a suitable stochastic ordinary differential equations on the kernel of the linear operator. We focus on equations with polynomial nonlinearities and give applications to Fisher–Kolmogorov–Petrovsky–Piscounov equation and the Ginzburg–Landau equation.

On the deformation of coated functionally graded structures with delamination

The paper considers problems for extensional inhomogeneous layered structures with functionally graded coatings in the presence of delamination. An approach that allows, from a single point of view, to study the problem of equilibrium of a strip non-uniform along the vertical coordinate with a delaminated coating and the problem of steady-state oscillations of a cylindrical waveguide with a ring crack is developed. Using the Fourier transform, the problems for the strip and waveguide are reduced to canonical systems of differential equations with variable coefficients with respect to transformants of the stress tensor components and displacement vector. Based on the formulation of several auxiliary Cauchy problems, a scheme for constructing a system of integral equations with hypersingular kernels with respect to jumps of displacements on the banks of delamination is presented. The asymptotic behavior of the Fourier symbols of kernels of integral operators for large values of the transformation parameter is investigated. A computational scheme for solving the system of integral equations based on the boundary element method is developed. The results of computational experiments on the determination of the disclosure functions of the delamination banks for a strip and a waveguide in the presence of a discontinuity in the elastic moduli at the boundary of the main body and the coating are presented.

Hardware design of real-valued NSCT transform for biomedical video compression

Abstract Number based Non-Subsampled Contourlet Transform (NSCT) diminishes the difficulty of the transformation operation kernel in biomedical video compression by dispensing the requirement for floating number mathematics. Yet, the dynamic scope of INSCT is huge and in this manner, equipment cost is more. In this manuscript, an innovative transformation operation kernel for biomedical video compression is projected which utilizes a novel set of complex and real- esteemed -NSCT coefficients. The projected NSCT decreases the equipment cost, area utilized and propagation time by diminishing the complication with intermediary information duration. Proposed transformation can be implemented by digit parallel multiplications and serial shift operations. Thus the itinerant fault can be decreased by standardize this numeral information and shift quantity for Biomedical video compression. Intermediary information duration and area, power consumption of the proposed method is fewer.

Kernels of Trace Operators and Boundary Value Problems in Field Theory

We propose an approach to formulation and solution of a series of boundary value problems in field theory based on the use of kernels of the trace operators and functionals in Sobolev spaces. The boundary conditions are nonlocal and can contain the main first order operators (gradient, curl, and divergence) of field theory. The case of plane vector fields is separately studied.

The Harmonicity of Slice Regular Functions

In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over {mathbb {H}}. Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over {mathbb {H}} (analogous to an holomorphic function over {mathbb {C}}) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over {mathbb {H}}? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over {mathbb {H}} and a Jensen’s Formula for semi-regular ones.

Застосування колокаційно-ітеративного методу до нелінійних інтегро-функціональних рівнянь

The article investigates the application of the collocation-iterative method to one type of nonlinear integro-functional equation. The conditions that guarantee the existence of a single solution of such an equation are given. The basic algorithm of the method is given and the conditions under which this method will be convergent are indicated. In substantiating this method, the fact is used that the original nonlinear integro-functional equation can be reduced to a nonlinear integral equation, the kernels of integral operators of which are written explicitly. Another important point in the justification is that the collocation-iterative method can be interpreted as a partial case of the projection-iterative method. The application of the latter to different types of nonlinear operator equations has been systematically investigated in the works of A. Y. Luchkа and his students. A significant difference between the method studied in this work is that at each step of the iteration it is necessary to solve systems of nonlinear algebraic or transcendental equations, which is the main technical complexity of this process. But the problem of finding solutions of such systems is simpler than solving the original nonlinear integro-functional equation. The article shows that the method of successive approximations and the method of collocation of the solution of the initial nonlinear equation can be considered as partial cases of the collocation-iterative method. In addition to the basic algorithm of the method, its computational algorithm is given, which is more convenient for direct calculations than the method itself and which, if necessary, can be successfully implemented on a computer by creating an appropriate program.

Expansion of a size disaggregation profile library for particulate matter emissions processing from three generic profiles to 36 source-type-specific profiles

ABSTRACT This study describes a significant upgrade to the particulate matter (PM) size disaggregation profile library used for preparing emissions files for the GEM-MACH (Global Environmental Multiscale-Modelling Air-quality and CHemistry) chemical transport model (CTM). This model uses a sectional (bin) approach to represent the PM size distribution, where one configuration employs 12 size bins to disaggregate PM2.5 and PM10 inventory emissions into the first 10 bins ranging from 0.01 to 10.24 μm in diameter. For the size disaggregation step, a small library of three generic PM size disaggregation profiles is currently applied for three broad source categories (area, mobile, and point). However, as might be expected, these generic profiles are not always representative: for example, emissions from two very different area sources ‒ paved road dust and residential wood combustion ‒ are disaggregated using the same generic size distribution profile. In order to improve the current small PM size disaggregation profile library, a comprehensive literature review was conducted: over 100 relevant publications were identified and PM size distribution profiles for 36 different emission source types were selected and compiled. These 36 source-type-specific PM size distribution profiles were then combined based on process type with corresponding PM speciation profiles to create a library of chemically speciated and size-resolved PM disaggregation profiles. This library can now be used by the SMOKE (Sparse Matrix Operator Kernel Emissions) emissions processing system for the 12-bin version of GEM-MACH to perform PM chemical speciation and size allocation in one step. The size-profile data collected and compiled in this study may also be used for emissions processing for other CTMs with a size-resolved representation of PM. Details of the compilation of the 36 PM size disaggregation profiles are discussed, and the differences in processed PM emissions based on the current and updated PM size disaggregation profile libraries are shown. Implications: A new and expanded particulate matter (PM) size disaggregation profile library covering 36 emission source types has been developed based on an extensive literature review. Its use can produce significant changes in the size allocation of bulk PM inventory emissions processed for input to size-resolved PM chemical transport models. Such models are used to predict atmospheric visibility and to simulate the interactions of aerosol particles with atmospheric radiation and with clouds. The use of more accurate, size-resolved primary PM emissions by these models should improve their predictive skill for atmospheric PM processes affecting air quality, meteorology, and climate.

Towards efficient tile low-rank GEMM computation on sunway many-core processors

Tile low-rank general matrix multiplication (TLR GEMM) is a novel method of matrix multiplication on large data-sparse matrices, which can significantly reduce storage footprint and arithmetic complexity under given accuracy. To implement high-performance TLR GEMM on Sunway many-core processor, the following challenges remain to be addressed: 1) design an efficient parallel scheme; 2) provide an efficient kernel library of math functions commonly used in TLR GEMM. This paper proposes swTLR GEMM, an efficient implementation of TLR GEMM. We assign LR GEMM computation to a single computing processing element (CPE) and use grouped task queue to process different data tiles of the TLR matrix. Moreover, we implement an efficient kernel library (swLR Kernels) for low-rank matrix operations. To scale to massive (CGs), we organize the CGs into the CG grid and partition the matrices into blocks accordingly. We also apply Cannon’s algorithm to enable efficient communication when processing the matrix blocks across CGs simultaneously. The experiment results show that the DGEMM kernel in swLR Kernels achieves 102 $$\times$$ speedup on average. In terms of overall performance, swTLR GEMM-LLD and swTLR GEMM-LLL achieve 91 $$\times$$ and 20.1 $$\times$$ speedup on average, respectively. In addition, our implementation of swTLR GEMM exhibits good scalability when running on 1,024 CGs of Sunway processors (66,560 cores in total).

Nuclear operators on Banach function spaces

Let X be a Banach space and E be a perfect Banach function space over a finite measure space (Omega ,Sigma ,lambda ) such that L^infty subset Esubset L^1. Let E' denote the Köthe dual of E and tau (E,E') stand for the natural Mackey topology on E. It is shown that every nuclear operator T:Erightarrow X between the locally convex space (E,tau (E,E')) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator T:L^infty rightarrow X between the locally convex space (L^infty ,tau (L^infty ,L^1)) and a Banach space X is nuclear if and only if its representing measure m_T:Sigma rightarrow X has the Radon-Nikodym property and |m_T|(Omega )=Vert TVert _{nuc} (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on L^infty are nuclear. Moreover, it is shown that every nuclear operator T:L^infty rightarrow X admits a factorization through some Orlicz space L^varphi , that is, T=Scirc i_infty , where S:L^varphi rightarrow X is a Bochner representable and compact operator and i_infty :L^infty rightarrow L^varphi is the inclusion map.

Existence and bifurcation of non-constant positive steady states for a tumor–immune model

Models incorporating diffusion terms into tumor and immune cells interaction systems become more realistic to understand the scenarios with better compatibility. In this paper, we consider a tumor–immune model with diffusion and nonlinear functional response and investigate the effect of diffusion on the existence of non-constant positive steady states and the steady-state bifurcations. By using the Leray–Schauder degree theory, the Crandall–Rabinowitz bifurcation theory, the space decomposition and implicit function theorem, we mainly obtain two results: (a) for the fixed diffusion rate of tumor cells, the system exists non-constant positive steady states when the diffusion rate of effector cells is large; (b) taking the diffusion rate of effector cells as a parameter, no matter the kernel of a certain operator is one- or two-dimensional, the system can always produce the steady-state bifurcation solutions. Moreover, some numerical simulations are presented to support and illustrate our theoretical analysis.

Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings

In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first discuss factorization of their Dirichlet heat kernels in metric measure spaces. Then we establish explicit estimates of the Dirichlet heat kernels under critical killings in C1,1 open subsets of Rd or in Rd∖{0}. The decay rates of our explicit estimates come from the values of the multiplicative constants in the killing potentials. Our method also provides an alternative and unified proof of the main results of [18–20].

Slice Dirac operator over octonions

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

Semigroups Generated by Volterra Integro-Differential Equations

We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of one-parameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation. The existence of a contraction $$C_0 $$ -semigroup is proved under certain assumptions about the kernels of integral operators. Examples of exponential and fractional-exponential (Rabotnov functions) kernels of integral operators satisfying the above assumptions are given.

Solving Problems of Vibrational Processes of Isotropically Homogeneous Elastic Plates

The relevance of the topic is determined by the search for the transcendental frequency equations that are reduced to algebraic and the influence of both boundary conditions at the edges of a rectangular plate and geometric and mechanical characters on the inherent oscillatory frequencies of rectangular flat elements is considered and the previous results for a rectangular plate whose material satisfies a viscoelastic model of Maxwell is generalized. When studying oscillatory processes in a solid deformable body, it is advisable to take the kernel of viscoelastic operators regularly, since only such operators describe the instantaneous elasticity and then the viscous flow, which is characteristic of deformable solids. Integro-differential equations with regular kernels are known to be equivalent to partial differential equations.

The Asymptotic Behavior of Solutions of a Fractional Integro-differential Equation

In this paper, we study the asymptotic behavior of solutions for an initial value problem with a nonlinearfractional integro-differential equation. Most of the existing results in the literature assume the continuity of theinvolved kernel. We consider here a kernel that is not necessarily continuous, namely, the kernel of the RiemannLiouville fractional integral operator that might be singular. We determine certain sufficient conditions underwhich the solutions, in an appropriate underlying space, behave eventually like power functions. For this purpose,we establish and generalize some well-known integral inequalities with some crucial estimates. Our findings aresupported by examples and numerical calculations.

Toeplitz kernels and the backward shift

In this paper we study the kernels of Toeplitz operators on both the scalar and the vector-valued Hardy space for 1<p<∞. We show existence of a minimal kernel for any element of the vector-valued Hardy space and we determine a symbol for the corresponding Toeplitz operator. In the scalar case we give an explicit description of a maximal function for a given Toeplitz kernel which has been decomposed in to a certain form. In the vectorial case we show not all Toeplitz kernels have a maximal function and in the case of p=2 we find the exact conditions for when a Toeplitz kernel has a maximal function. For both the scalar and vector-valued Hardy space we study the minimal Toeplitz kernel containing multiple elements of the Hardy space, which in turn allows us to deduce an equivalent condition for a function in the Smirnov class to be cyclic for the backward shift.

Information potential for some probability density functions

This paper is related to the information theoretic learning methodology, whose goal is to quantify global scalar descriptors (e.g., entropy) of a given probability density function (PDF). In this context, the core concept is the information potential (IP) S[s](x):=∫Rps(t,x)dt,s > 0 of a PDF p(t, x) depending on a parameter x; it is naturally related to the Rényi and Tsallis entropies. We present several such PDF, viewed also as kernels of integral operators, for which a precise relation exists between S[2](x) and the variance Var[p(t, x)]. For these PDF we determine explicitly the IP and the Shannon entropy. As an application to Information Theoretic Learning we determine two essential indices used in this theory: the expected value E[log p(t, x)] and the variance Var[log p(t, x)]. The latter is an index of the intrinsic shape of p(t, x) having more statistical power than kurtosis. For a sequence of B-spline functions, considered as kernels of Steklov operators and also as PDF, we investigate the sequence of IP and its asymptotic behaviour. Another special sequence of PDF consists of kernels of Kantorovich modifications of the classical Bernstein operators. Convexity properties and bounds of the associated IP, useful in Information Theoretic Learning, are discussed. Several examples and numerical computations illustrate the general results.

NPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius δ goes to zero, and to the fractional Laplacian as δ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to δ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, δ and α, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order α(y), which is more suitable for modeling wall-bounded turbulence, e.g. turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as δ. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

Convergence of sampling Kantorovich operators in modular spaces with applications

In the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such as Musielak–Orlicz and Orlicz spaces. As a consequence of these results we obtain convergence theorems in the classical and weighted versions of the L^p and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented.