General Kernel Research Articles

Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of C 0 C_{0} -semigroups ( V ( t ) ) t ⩾ 0 \left (\mathcal {V}(t)\right )_{t \geqslant 0} in L 1 ( T d × R d ) L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d}) governing conservative linear kinetic equations on the torus with general scattering kernel k ( v , v ′ ) \boldsymbol {k}(v,v’) and degenerate (i.e. not bounded away from zero) collision frequency σ ( v ) = ∫ R d k ( v ′ , v ) m ( d v ′ ) \sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v’,v)\boldsymbol {m}(\mathrm {d}v’) , (with m ( d v ) \boldsymbol {m}(\mathrm {d}v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N 0 N_{0} is the maximal integer s ⩾ 0 s \geqslant 0 such that 1 σ ( ⋅ ) ∫ R d k ( ⋅ , v ) σ − s ( v ) m ( d v ) ∈ L ∞ ( R d ) , \begin{equation*} \frac {1}{\sigma (\cdot )}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot ,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*} then, for initial datum f f such that ∫ T d × R d | f ( x , v ) | σ − N 0 ( v ) d x m ( d v ) > ∞ \displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) >\infty it holds ‖ V ( t ) f − ϱ f Ψ ‖ L 1 ( T d × R d ) = ε f ( t ) ( 1 + t ) N 0 − 1 , ϱ f ≔ ∫ R d f ( x , v ) d x m ( d v ) , \begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*} where Ψ \Psi is the unique invariant density of ( V ( t ) ) t ⩾ 0 \left (\mathcal {V}(t)\right )_{t \geqslant 0} and lim t → ∞ ε f ( t ) = 0 \lim _{t\to \infty }{\varepsilon }_{f}(t)=0 . We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of ( V ( t ) ) t ⩾ 0 \left (\mathcal {V}(t)\right )_{t \geqslant 0} and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp “subgeometric” convergence rate for Markov semigroups associated to general transition kernels.

Coagulation, non-associative algebras and binary trees

We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the coalescence term. The non-associative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binary-tree generating procedures, by: (i) grafting together the roots of all pairs of order-compatible trees at preceding orders, or (ii) attaching binary branches to all free branches of trees at the previous order. We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linear-loglinear in the number of spatial modes/nodes.

Stein–Weiss type inequality on the upper half space and its applications

In this paper, we establish some Stein–Weiss type inequalities with general kernels on the upper half space and study the extremal functions of the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of positive solutions of corresponding Euler–Lagrange system. As an application, we derive some Liouville type results for the Hartree type equations on the half space.

Equivalent conditions of a reverse Hilbert - type integral inequality in the whole plane and applications

In the present paper we establish a few equivalent conditions of a reverse Hilbert-type integral inequality with a general non-homogeneous kernel in the whole plane. In the form of applications, we deduce a few equivalent conditions of a reverse Hilbert-type integral inequality with a general homogeneous kernel in the whole plane. We also consider some particular cases and examples.

Novel generalized inequalities involving a general Hardy operator with multiple variables and general kernels on time scales

Novel generalized inequalities involving a general Hardy operator with multiple variables and general kernels on time scales

Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains.

Equivalent Statements of Two Multidimensional Hilbert-Type Integral Inequalities with Parameters

By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as H(||x||αλ1||y||βλ2)(λ1,λ2≠0) are given, which are some extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent conditions of the best value and several parameters related to the new inequalities are provided. Two corollaries regarding the kernel, represented as kλ(||x||αλ1,||y||βλ2)(λ1,λ2≠0), are given, and a few new inequalities for the particular parameters are obtained.

Approximating fractional calculus operators with general analytic kernel by Stancu variant of modified Bernstein–Kantorovich operators

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative.

Hierarchical team structure and multidimensional localization (or siloing) on networks

Knowledge silos emerge when structural properties of organizational interaction networks limit the diffusion of information. These structural barriers are known to take many forms at different scales—hubs in otherwise sparse organizations, large dense teams, or global core-periphery structure—but we lack an understanding of how these different structures interact and shape dynamics. Here we take a first theoretical step in bridging the gap between the mathematical literature on localization of spreading dynamics and the more applied literature on knowledge silos in organizational interaction networks. To do so, we introduce a new model that considers a layered structure of teams to unveil a new form of hierarchical localization (i.e. the localization of information at the top or center of an organization) and study its interplay with known phenomena of mesoscopic localization (i.e. the localization of information in large groups), k-core localization (i.e. around denser subgraphs) and hub localization (i.e. around high degree stars). We also include a complex contagion mechanism by considering a general infection kernel which can depend on hierarchical level (influence), degree (popularity), infectious neighbors (social reinforcement) or team size (importance). This very general model allows us to explore the multifaceted phenomenon of information siloing in complex organizational interaction networks and opens the door to new optimization problems to promote or hinder the emergence of different localization regimes.

Sparse Gaussian processes for solving nonlinear PDEs

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. We conjecture that a solution exists on a “condensed” subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the “condensed” subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost.Our contributions include formulating the nonlinear PDE problem as an optimization problem on a “condensed” subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels.

Data‐driven linear complexity low‐rank approximation of general kernel matrices: A geometric approach

SummaryA general, rectangular kernel matrix may be defined as where is a kernel function and where and are two sets of points. In this paper, we seek a low‐rank approximation to a kernel matrix where the sets of points and are large and are arbitrarily distributed, such as away from each other, “intermingled”, identical, and so forth. Such rectangular kernel matrices may arise, for example, in Gaussian process regression where corresponds to the training data and corresponds to the test data. In this case, the points are often high‐dimensional. Since the point sets are large, we must exploit the fact that the matrix arises from a kernel function, and avoid forming the matrix, and thus ruling out most algebraic techniques. In particular, we seek methods that can scale linearly or nearly linearly with respect to the size of data for a fixed approximation rank. The main idea in this paper is to geometrically select appropriate subsets of points to construct a low rank approximation. An analysis in this paper guides how this selection should be performed.

Parameterized Domain Mapping for Order Tracking of Rotating Machinery

Because of the periodicity of vibration signals from the cyclic motion of rotating machinery, many spectrum-based methods are widely applied. However, frequency distortion often occurs under non-stationary conditions, which weakens the effectiveness of spectrum-based methods. Order tracking can overcome the frequency distortion problem, but the instantaneous angular speed is hard to detect, especially under strong noise or with close-spaced frequencies. In this paper, we map the signal into a new domain called the pseudo time domain to eliminate the non-stationarity based on the mechanism of rotating machinery. A parameterized domain mapping function (PDMF) is used to represent the relationship between the time and pseudo time domains. Instead of conducting order tracking after detecting the instantaneous angular speed, we directly optimize the parameter set of the PDMF to construct an appropriate pseudo time domain. A new spectrum concentration indicator considering the continuity of the spectrum is built as the optimization objective. The Legendre polynomial, which shows good approximation property, is applied as the general kernel function of PDMF. Numerical and experimental verification shows good anti-noise performance and the ability to deal with complex and close-spaced frequencies under speed variation or speed fluctuation conditions.

On univariate fractional calculus with general bivariate analytic kernels

On univariate fractional calculus with general bivariate analytic kernels

Wind Power Interval Prediction Based on Robust Kernel Density Estimation

Wind power output has a high degree of randomness, so it is difficult to describe it accurately with some typical probability distribution. The extreme values influence the sample’s general non-parametric kernel density estimation method, and the estimated results are relatively conservative. A wind power interval prediction method based on robust kernel density estimation is proposed to improve the compactness and accuracy of interval prediction. In probability estimation, this method will assign a small weight to the extreme sample data to reduce its influence on the probability density estimation accuracy and has better robustness than the general kernel density estimation method. In addition, the bandwidth of robust kernel density estimation is a dynamic parameter, which can be adjusted with the sample data to improve the prediction accuracy and avoid over-conservative prediction.

General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

On bivariate fractional calculus with general univariate analytic kernels

We introduce a general bivariate fractional calculus, defined using a kernel based on an arbitrary univariate analytic function with an appropriate bivariate substitution. Various properties of the introduced general operators are established, including a series formula, function space mappings, and Fourier and Laplace transforms. A major result of this paper is a fractional Leibniz rule for the new operators, the derivation of which involves correcting a minor error in one of the classic textbooks on fractional calculus. We also solve some fractional differential equations using transform methods, revealing an interesting connection between bivariate type Mittag-Leffler functions.

Исключение интегрального члена в уравнениях одной эредитарной системы, связанной с задачей гидромагнитного динамо

В работе изучается двумерная система интегро-дифференциальных уравнений, которая является простейшей эредитарной моделью двумодового гидромагнитного динамо. Учет пространственной и временной нелокальности взаимодействий в динамо-системах сейчас активно исследуется. В маломодовых приближениях уравнений динамо можно рассматривать только временную нелокальность, т.е. эредитарность (память). Память в исследуемой системе реализована в виде обратной связи, распределенной по всем прошлым состояниям системы. Обратная связь представлена с помощью интегрального члена типа свертки от квадратичной комбинации фазовых переменных с ядром достаточно общего вида. Этот член моделирует подавление турбулентного генератора поля (α-эффекта) квадратичной формой от фазовых переменных. В реальных динамо-системах такое подавление обеспечивается силой Лоренца. Основной результат работы – доказательство возможности исключения интегрального члена для одного класса ядер. Такие ядра являются решениями однородного линейного дифференциального уравнения с постоянными коэффициентами. Доказано, что исследуемую интегро-дифференциальную систему можно заменить дифференциальной системой большей размерности с подходящими начальными условиями на дополнительные фазовые переменные. Если ядро является решением уравнения n-го порядка, то размерность системы может достигать 3n−2 и зависит от начальных условий, которым удовлетворяет ядро. В работе используются классические методы теории дифференциальных уравнений. Приводятся примеры динамических систем, возникающих при некоторых ядрах в результате исключения интегрального члена. Результаты работы можно использовать для верификации вычислительных алгоритмов и программных кодов, разработанных для решения интегро-дифференциальных уравнений.We study a two-dimensional system of integro-differential equations, which is the simplest hereditary model of a two-mode hydromagnetic dynamo. Accounting for the spatial and temporal nonlocality of interactions in dynamo systems is currently being actively studied. In the low-mode approximations of the dynamo equations, one can consider only temporal nonlocality, i.e. heredity (memory). Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by a convolution-type integral term of a quadratic combination of phase variables with a fairly general kernel. This term models the quenching of the turbulent field generator (α-effect) by a quadratic form in phase variables. In real dynamo systems, such quenchingn is provided by the Lorentz force. The main result of the work is a proof of the possibility of eliminating the integral term for one class of kernels. Such kernels are solutions of a homogeneous linear differential equation with constant coefficients. It is proved that the studed integro-differential system can be replaced by a higher-dimensional differential system with suitable initial conditions for additional phase variables. If the kernel is a solution to an n-order equation, then the dimension of the system can reach 3n−2 and depends on the initial conditions that the kernel satisfies. The work uses classical methods of the theory of differential equations. Examples are given of dynamical systems that arise for some kernels as a result of the elimination of the integral term. The results of the work can be used to verify computational algorithms and program codes developed for solving integro-differential equations.

Synchronization of uncertain general fractional unified chaotic systems via finite-time adaptive sliding mode control.

This paper employs two adaptive sliding mode control (ASMC) strategies to accomplish finite-time synchronization of uncertain general fractional unified chaotic systems (UGFUCSs) when uncertainty and external disturbance exist. First, general fractional unified chaotic system (GFUCS) is developed. GFUCS may be transitioned from general Lorenz system to general Chen system, and the general kernel function could compress and extend the time domain. Furthermore, two ASMC methods are applied to finite-time synchronization of UGFUCSs, where system states arrive at sliding surfaces in finite-time. The first ASMC approach utilizes three sliding mode controllers to achieve synchronization between chaotic systems, while the second ASMC method needs just one sliding mode controller to produce synchronization between chaotic systems. Finally, the effectiveness of the proposed ASMC approaches is verified using numerical simulations.

General Toeplitz kernels and -invariance

AbstractMotivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$ -invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

Rates of the Strong Uniform Consistency for the Kernel-Type Regression Function Estimators with General Kernels on Manifolds

Rates of the Strong Uniform Consistency for the Kernel-Type Regression Function Estimators with General Kernels on Manifolds