Homogeneous Kernel Research Articles

A Hilbert-type integral inequality in the whole plane with the homogeneous kernel

A Hilbert-type integral inequality in the whole plane with the homogeneous kernel

Endpoint Estimates for Generalized Commutators of Hardy Operators onH1Space

We study theH1-boundedness of the generalized commutators of Hardy operator with a homogeneous kernel as follows:ℋΩ,A,βmf(x)=(1/|x|n-β)∫|y|<|x|(‍Ω(x-y)/|x-y|m-1)Rm(A;x,y)f(y)dy, whereRm(A;x,y)=A(x)-∑|α|<m‍(1/α!)DαA(y)(x-y)αwithm∈Z+,0≤β<nandΩ∈Lip1(Sn-1). We prove that, whenm≥1,ℋΩ,A,βmis not bounded fromH1toLn/(n-β)unlessℋΩ,A,βm≡0. Finally, we prove thatℋΩ,A,βmis bounded fromH1toLn/(n-β),∞withm≥1.

Study of Multiple-Kernel Relevance Vector Machine Based on Kernel Alignment

The relevance vector machine (RVM) was a Bayesian framework for learning sparse regression models and classifiers, it used single kernel function to map training data from low dimension sample space to high dimension feature space. The prediction accuracy and generalization of traditional single-kernel RVM (sRVM) were not ideal both in classification and regression, so we constructed homogeneous and heterogeneous multiple kernels function (MKF) by kernel function combination in which we testified the validity of basic kernel function (BKF) and its parameters we employed by kernel alignment (KA), then we acquired optimized multiple-kernel RVM (mRVM). Experiment results on LIBSVM datasets not only indicate that both homogeneous and heterogeneous multiple-kernel RVM we constructed possess lower error rate in classification and smaller root mean square (RMS) in regression than single-kernel RVM, but also prove the effectiveness of kernel alignment.

A Hilbert-Type Integral Inequality with the Homogeneous Kernel of Degree-3 and a Best Constant Factor

By establishing the weight function, we present a new Hilbert-type inequality with the integral in whole plane and with a best constant factor, and its kernel is a homogeneous form of degree-3, and also we put forward its equivalent form.

On a multidimensional version of the Hilbert type inequality

The main objective of this paper is a study of some new multidimensional Hilbert type inequalities with a general homogeneous kernel. We derive a pair of equivalent inequalities, and also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered.

Dynamic scaling in magnetophoretic separation

Using numerical solutions of the Langevin equations, to simulate a magnetic separation of colloidal suspensions under an external magnetic-field gradient, we show the existence of a regime of dynamic scaling. We find that the separation time depends strongly on the size (or length) of aggregates taking place in the separation process. In the irreversible aggregation regime, we show that linear chains of particles steadily grow and their average size, S, increases with time as a power low. The chain-size (s) distribution-function (cs) approaches (for long times) the following scaling form cs(t)∝s−2f(s/S). The exponent, z, of the mean chain size S(t)∝tz is compatible with Smoluchowski's coagulation equation, with a homogeneous kernel, even if the separation is a process with particle migration. We also show that the regime corresponding to the dynamic scaling is limited in time due to the separation. Furthermore, we find a characteristic time, which we identify as a similarity variable, and we relate it to the separation time.

Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in Lp(ℝn), 1 < p < +∞. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the C*-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator K-theory.

Classification of Fruits Using Computer Vision and a Multiclass Support Vector Machine

Automatic classification of fruits via computer vision is still a complicated task due to the various properties of numerous types of fruits. We propose a novel classification method based on a multi-class kernel support vector machine (kSVM) with the desirable goal of accurate and fast classification of fruits. First, fruit images were acquired by a digital camera, and then the background of each image was removed by a split-and-merge algorithm; Second, the color histogram, texture and shape features of each fruit image were extracted to compose a feature space; Third, principal component analysis (PCA) was used to reduce the dimensions of feature space; Finally, three kinds of multi-class SVMs were constructed, i.e., Winner-Takes-All SVM, Max-Wins-Voting SVM, and Directed Acyclic Graph SVM. Meanwhile, three kinds of kernels were chosen, i.e., linear kernel, Homogeneous Polynomial kernel, and Gaussian Radial Basis kernel; finally, the SVMs were trained using 5-fold stratified cross validation with the reduced feature vectors as input. The experimental results demonstrated that the Max-Wins-Voting SVM with Gaussian Radial Basis kernel achieves the best classification accuracy of 88.2%. For computation time, the Directed Acyclic Graph SVMs performs swiftest.

Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect

A reaction–diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel.

Weighted Trudinger inequality associated with rough multilinear fractional type operators

Let I Ω , α Θ Open image in new window be the multilinear fractional type operator defined by I Ω , α Θ ( f → ) ( x ) = ∫ R n Ω ( y ) ∏ j = 1 m f j ( x − θ j y ) | y | ( α − n ) d y Open image in new window. In this paper, we study the weighted estimates for the Trudinger inequality associated to I Ω , α Θ Open image in new window with rough homogeneous kernels, which improve some known results significantly. A similar Trudinger inequality holds for another type of fractional integral defined by I ¯ Ω , α ( f → ) ( x ) = ∫ ( R n ) m ∏ j = 1 m | f j ( y j ) | | Ω j ( x − y j ) | | ( x − y 1 , x − y 2 , … , x − y m ) | m n − α d y → Open image in new window, where d y → = d y 1 ⋯ d y m Open image in new window.

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space Lp(ℝn), 1 < p < ∞, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.

Comparison of ray‐ and adjoint‐based sensitivity kernels for body‐wave seismic tomography

We compare finite‐frequency sensitivity kernels computed from ray‐theoretical wavefields (“banana‐doughnut” kernels) and from full waveform computations (often called “adjoint” kernels) in order to evaluate resolution, accuracy and computational cost. We focus here on body‐wave seismic tomography at regional and local scales. Our results show that: (1) for homogeneous reference media, ray‐based and adjoint kernels agree except for the expected differences in the regions close to the source and the receiver, where near‐field contributions are neglected in the ray‐based kernels; (2) for a smooth 3D background velocity model, the differences in predicted delay times for the two methods are generally well below 10 % of the delay for P waves, though as much as 20 % for S‐waves, suggesting that extra care should be taken when performing S‐wave tomography with ray‐based “banana doughnut” kernels.

On Opial's type inequalities for an integral operator with homogeneous kernel

In this article, we establish some new Opial's type integral inequalities for an integral operator with homogeneous kernel. The results in special cases yield some of the interrelated results and provide new estimates on inequalities of this type. MS (2000) Subject Classification: 26D15.

On a Hilbert-type inequality with a homogeneous kernel in ℝ2 and its equivalent form

By using the way of weight functions and the technique of real analysis, a new integral inequality with a homogeneous kernel and the best constant factor in ℝ2 is given. The equivalent form and the reverses are considered.

Double SO(2, 1)-integrals and formulas for Whittaker functions

With the help of some double integral bilinear functionals with homogeneous kernels defined on a pair of representation spaces of the group SO(2, 1) we obtain some functional relations for Whittaker functions and calculate the sum of one series of Gauss hypergeometric functions converging to a Whittaker function.

Classical non-mass-preserving solutions of coagulation equations

In this paper we construct classical solutions of a family of coagulation equations with homogeneous kernels that exhibit the behaviour known as gelation. This behaviour consists in the loss of mass due to the fact that some of the particles can become infinitely large in finite time.

A refined discrete Hilbert inequality obtained via the Hermite–Hadamard inequality

In this paper we establish a general method for improving the discrete Hilbert-type inequalities via the Hermite–Hadamard inequality. The general result is then applied to homogeneous kernels. Moreover, some particular homogeneous kernels are considered, and a whole series of refinements of some recent results known from the literature are yielded. In addition some particular non-homogeneous cases are also considered.

A New Hilbert-type Integral Inequality with the Homogeneous Kernel of 1/2 Degree Form and the Integral in Whole Plane

A New Hilbert-type Integral Inequality with the Homogeneous Kernel of 1/2 Degree Form and the Integral in Whole Plane

ON SOME REFINEMENTS AND CONVERSES OF MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES

AbstractThe main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

Efficient Additive Kernels via Explicit Feature Maps

Large scale nonlinear support vector machines (SVMs) can be approximated by linear ones using a suitable feature map. The linear SVMs are in general much faster to learn and evaluate (test) than the original nonlinear SVMs. This work introduces explicit feature maps for the additive class of kernels, such as the intersection, Hellinger's, and χ2 kernels, commonly used in computer vision, and enables their use in large scale problems. In particular, we: 1) provide explicit feature maps for all additive homogeneous kernels along with closed form expression for all common kernels; 2) derive corresponding approximate finite-dimensional feature maps based on a spectral analysis; and 3) quantify the error of the approximation, showing that the error is independent of the data dimension and decays exponentially fast with the approximation order for selected kernels such as χ2. We demonstrate that the approximations have indistinguishable performance from the full kernels yet greatly reduce the train/test times of SVMs. We also compare with two other approximation methods: Nystrom's approximation of Perronnin et al., which is data dependent, and the explicit map of Maji and Berg for the intersection kernel, which, as in the case of our approximations, is data independent. The approximations are evaluated on a number of standard data sets, including Caltech-101, Daimler-Chrysler pedestrians, and INRIA pedestrians.