Homogeneous Kernel Research Articles

Data-Driven Learning of the Generalized Langevin Equation with State-Dependent Memory.

We present a data-driven method to learn stochastic reduced models of complex systems that retain a state-dependent memory beyond the standard generalized Langevin equation with a homogeneous kernel. The constructed model naturally encodes the heterogeneous energy dissipation by jointly learning a set of state features and the non-Markovian coupling among the features. Numerical results demonstrate the limitation of the standard generalized Langevin equation and the essential role of the broadly overlooked state-dependency nature in predicting molecule kinetics related to conformation relaxation and transition.

Fractional operators with homogeneous kernel on the Calderón product of Morrey spaces

AbstractWe investigate fractional operators with homogeneous kernel in Morrey spaces. In particular, we prove that fractional integral operators and fractional maximal operators with homogeneous kernel are bounded from the Calderón product of Morrey spaces to certain Morrey spaces. Our results can be seen as a generalization of a recent result on the relation between the boundedness of (classical) fractional operators and interpolation of Morrey spaces. What is new about this paper is not only the passage from the classical fractional integral operators to the rough integral operators. Even the case of fractional integral operators, handled in earlier papers, is significantly simplified.

Compressed Fast Multipole Representations for Homogeneous 3-D Kernels

For homogeneous kernels, the memory requirements associated with H2 representations of integral equation matrices can be reduced by incorporating translational invariance. Starting with a non-translationally invariant H2 representation, this can be accomplished using a left/right iterative algorithm. In this paper, it is shown that a similar algorithm can also be used to compress an existing fast multipole method (FMM). It is observed that the iterative compression converges faster when used to compress an FMM than when it is applied to an H2 representation. Resulting savings in floating-point operations are indicated, and extensions of the reported method are discussed.

Compactness of commutators of fractional integral operators on ball Banach function spaces

<abstract><p>Let $ 0 < \alpha < n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.</p></abstract>

Discrete Local Fractional Hilbert-type Inequalities

The main objective of this paper is a study of some new discrete local fractional Hilbert-type inequalities. We apply our general results to homogeneous kernels. Also, the obtained results have the best possible constants.

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.

Raman Multi-Omic Snapshots of Koshihikari Rice Kernels Reveal Important Molecular Diversities with Potential Benefits in Healthcare.

This study exploits quantitative algorithms of Raman spectroscopy to assess, at the molecular scale, the nutritional quality of individual kernels of the Japanese short-grain rice cultivar Koshihikari in terms of amylose-to-amylopectin ratio, fractions of phenylalanine and tryptophan aromatic amino acid residues, protein-to-carbohydrate ratio, and fractions of protein secondary structures. Statistical assessments on a large number of rice kernels reveal wide distributions of the above nutritional parameters over nominally homogeneous kernel batches. This demonstrates that genetic classifications cannot catch omic fluctuations, which are strongly influenced by a number of extrinsic factors, including the location of individual grass plants within the same rice field and the level of kernel maturation. The possibility of collecting nearly real-time Raman "multi-omic snapshots" of individual rice kernels allows for the automatic (low-cost) differentiation of groups of kernels with restricted nutritional characteristics that could be used in the formulation of functional foods for specific diseases and in positively modulating the intestinal microbiota for protection against bacterial infection and cancer prevention.

A new reverse half-discrete Mulholland-type inequality with a nonhomogeneous kernel

In this paper, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h(v(x)ln n) and the best possible constant factor is obtained by using the weight functions and the technique of real analysis. The equivalent reverses are considered. As corollaries, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form k_{lambda }(v(x),ln n). A few particular cases are provided. Our new reverse half-discrete Mulholland-type inequality which has a nonhomogeneous kernel is more general than in the previous homogeneous kernel work. The harmonized integration will have more applications.

Multilinear singular integrals with homogeneous kernels near $$L^1$$

Multilinear singular integrals with homogeneous kernels near $$L^1$$

An Equivalent Form Related to a Hilbert-Type Integral Inequality

In the present paper, we establish an equivalent form related to a Hilbert-type integral inequality with a non-homogeneous kernel and a best possible constant factor. We also consider the case of homogeneous kernel as well as certain operator expressions.

On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients

On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients

On a more accurate half-discrete multidimensional Hilbert-type inequality involving one derivative function of m-order

By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard’s inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as frac{1}{(x + Vert k - xi Vert _{alpha} )^{lambda + m}} (x,lambda > 0) involving one derivative function of m-order is given. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms. the operator expressions and some particular inequalities are obtained.

A New Hardy–Hilbert-Type Integral Inequality Involving One Multiple Upper Limit Function and One Derivative Function of Higher Order

Using weight functions and parameters, as well as applying real analytic techniques, we derive a new Hardy–Hilbert-type integral inequality with the homogeneous kernel 1(x+y)λ+n involving one multiple upper limit function and one derivative function of higher order. Certain equivalent statements of the optimal constant factor related to some parameters are considered. A few particular inequalities and the case of reverses are also provided.

On a reverse Hardy–Hilbert-type integral inequality involving derivative functions of higher order

By means of the weight functions, the idea of introducing parameters and the technique of real analysis related to the beta and gamma functions, a new reverse Hardy–Hilbert-type integral inequality with the homogeneous kernel as frac{1}{(x + y)^{lambda + m + n}} (lambda > 0) involving two derivative functions of higher order is given. As applications, the equivalent statements of the best possible constant factor related to several parameters are considered, and some particular inequalities are obtained.

The invariant solution of Smoluchowski coagulation equation with homogeneous kernels based on one parameter group transformation

The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.

A note on the fractional Hardy inequality

We give a direct proof of fractional Hardy inequality by means of Littlewood–Paley decomposition and properties of singular homogeneous kernels of degree -d. A refinement when q>2 is proved.

Some Local Fractional Hilbert-Type Inequalities

The main purpose of this paper is to prove some new local fractional Hilbert-type inequalities. Our general results are applicable to homogeneous kernels. Furthermore, the best possible constants in terms of local fractional hypergeometric function are obtained. The obtained results prove that the employed method is very simple and effective for treating various kinds of local fractional Hilbert-type inequalities.

A More Accurate Half-Discrete Multidimensional Hilbert-Type Inequality Involving One Multiple Upper Limit Function

By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard’s inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as 1(x+||k−ξ||α)λ(x,λ>0) involving one multiple upper limit function is given, which is a new application of Hilbert-type inequalities. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms the operator expressions and some particular inequalities are obtained.

Equivalent Conditions of the Reverse Hardy-Type Integral Inequalities

Hardy-type integral inequalities play a prominent role in the study of analytic inequalities, which are essential in mathematical analysis and its various applications, such as in the study of symmetry and asymmetry phenomena. In this paper, employing methods of real analysis and using weight functions, we investigate some equivalent conditions of two kinds of reverse Hardy-type integral inequalities with a particular non-homogeneous kernel. A few equivalent conditions of two kinds of reverse Hardy-type integral inequalities with a particular homogeneous kernel are deduced in the form of applications.

A Hardy–Hilbert-type integral inequality involving two multiple upper-limit functions

By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality with the homogeneous kernel frac{1}{(x + y)^{lambda}} (lambda > 0) involving two multiple upper-limit functions is obtained. The equivalent statements of the best possible constant factor related to the beta and gamma functions are considered. As applications, the equivalent forms and the case of a nonhomogeneous kernel are deduced. Some particular inequalities and the operator expressions are provided.