Nonsymmetric Kernels Research Articles

Continuity and minimization of spectrum related with the two-component Novikov equation

This article is concerned with the spectral problem{dϕ1(x)=ϕ2(x)dh(x),dϕ2(x)=ϕ3(x)dg(x),dϕ3(x)=−λϕ1(x)dx, x∈(−1,1),ϕ2(−1)=0, ϕ3(−1)=0, ϕ3(1)=0,which is associated with the two-component Novikov equation. The coefficients g and h are functions of bounded variation. The first aim is to prove the continuous dependence of the n-th eigenvalue on the coefficients g and h with different topologies; in addition, we discuss the Fréchet differentiability of the algebraically simple eigenvalues on the coefficients g, h with respect to the norm topology of total variations. Secondly, utilizing the theory of oscillatory (non-symmetric) kernels, conditions on g, h under which the spectrum is nonnegative and algebraically simple are obtained. Finally, we solve the extremal problems of the lowest nonzero eigenvalue, when the norm ‖⋅‖V of the coefficients g, h is given. The latter can be seen as an application of our first two main results and the trace formulas.

Inversion formulas and their finite-dimensional analogs for multidimensional Volterra equations of the first kind

The paper focuses on solving one class of Volterra equations of the first kind, which is characterized by the variability of all integration limits. These equations were introduced in connection with the problem of identifying nonsymmetric kernels for constructing integral models of nonlinear dynamical systems of the “input-output” type in the form of Volterra polynomials. The case when the input perturbation of the system is a vector function of time is considered. To solve the identification problem, previously introduced test signals of duration h (mesh step) are used in the form of linear combinations of Heaviside functions with deviating arguments. The paper demonstrates a method for obtaining the desired solution, developing a method of steps for a one-dimensional case. The matching conditions providing the desired smoothness of the solution are established. The mesh analogs of the studied integral equations based on the formulas of middle rectangles are considered.

Numerical solution the Volterra dual integral equation of the first kind based on a method of Runge-Kutta type

The paper focuses on the non-classical Volterra dual equation of the first kind, to which the problem of identifying non-symmetric Volterra kernels is reduced. For this equation, we obtained difference analogs based on the application of the Runge-Kutta type method. Such equations arise in the problem of modeling nonlinear dynamical systems of black-box type, which employ the Volterra integro-power series, when the input perturbation is a vector function of time.

Permanental sequences related to a Markov chain example of Kolmogorov

Permanental sequences with non-symmetric kernels that are generalization of the potentials of a Markov chain with state space {0,1∕2,…,1∕n,…} and a single instantaneous state that was introduced by Kolmogorov, are studied. Depending on a parameter in the kernels we obtain an exact rate of divergence of the sequence at 0, an exact local modulus of continuity of the sequence at 0, or a precise bounded discontinuity for the sequence at 0.

Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques

Kernel analog forecasting (KAF), alternatively known as kernel principal component regression, is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data. This paper synthesizes descriptions of kernel methods and Koopman operator theory in order to provide a single consistent account of KAF. The framework presented here illuminates the property of the KAF method that, under measure-preserving and ergodic dynamics, it consistently approximates the conditional expectation of observables that are acted upon by the Koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization. More precisely, KAF yields optimal predictions, in the sense of minimal root mean square error with respect to the invariant measure, in the asymptotic limit of large data. The presented framework facilitates, moreover, the analysis of generalization error and quantification of uncertainty. Extensions of KAF to the construction of conditional variance and conditional probability functions, as well as to non-symmetric kernels, are also shown. Illustrations of various aspects of KAF are provided with applications to simple examples, namely a periodic flow on the circle and the chaotic Lorenz 63 system.

On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations

We obtain Schauder estimates for a class of concave fully nonlinear nonlocal parabolic equations of order $\sigma\in (0,2)$ with rough and non-symmetric kernels. As a application, we prove that the solution to a translation invariant equation with merely bounded data is $C^\sigma$ in $x$ variable and $\Lambda^1$ in $t$ variable, where $\Lambda^1$ is the Zygmund space.

Homogenization of Lévy-type Operators with Oscillating Coefficients

The paper deals with homogenization of Lévy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Lévy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem.

Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces

This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.

Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts

We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $\sigma\in (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.

Dini estimates for nonlocal fully nonlinear elliptic equations

We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order σ∈(0,2) with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined Cσ+α estimate in [9].

Permanental vectors with nonsymmetric kernels

A permanental vector with a symmetric kernel and index $2$ is a squared Gaussian vector. The definition of permanental vectors is a natural extension of the definition of squared Gaussian vectors to nonsymmetric kernels and to positive indexes. The only known permanental vectors either have a positive definite kernel or are infinitely divisible. Are there some others? We present a partial answer to this question.

An [formula omitted]-theory for a class of non-local elliptic equations related to nonsymmetric measurable kernels

We study the integro-differential operators L with kernels K(y)=a(y)J(y), where J(y) is rotationally invariant and J(y)dy is a Lévy measure on Rd (i.e. ∫Rd(1∧|y|2)J(y)dy<∞) and a(y) is an only measurable function with positive lower and upper bounds. Under few additional conditions on J(y), we prove the unique solvability of the equation Lu−λu=f in Lp-spaces and present some Lp-estimates of the solutions.

An analogue of the Robinson–Schensted–Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases

We prove a restriction of an analogue of the Robinson–Schensted–Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means that the pair of their right keys is such that one key is bounded by the Schützenberger evacuation of the other. This bijection is then used to obtain an expansion formula of the non-symmetric Cauchy kernel, over staircases or truncated staircases, in the basis of Demazure characters of type A, and the basis of Demazure atoms. The expansion implies Lascoux expansion formula, when specialized to staircases or truncated staircases, and make explicit, in the latter, the Young tableaux in the Demazure crystal by interpreting Demazure operators via elementary bubble sorting operators acting on weak compositions.

Regularity results for full nonlinear mixed type integraldifferential operators

We introduce a class of full nonlinear integral-differential equations with nonsymmetric positive kernels and study the regularity of their solutions. More precisely, we establish a nonlocal version of A-B-P estimate, a Harnack inequality, Holder regularity, and C 1, α of the solutions.

A Note on the Construction of a Sample Variance

Mukhopadhyay and Chattopadhyay (2013) proposed a new approach to construct unbiased estimators of by U-statistics starting with a class of non-symmetric initial kernels of degree m(> 2): But, surprisingly all symmetrized final estimators in the form of U-statistics reduced to when the parent distribution function (d.f.) F remained unknown. Now, we exhibit another class of non-symmetric initial kernels to come up with U-statistics of degree m(> 2) and unbiased for : Interestingly, the associated symmetrized final U-statistics again coincide with (Theorem 2.1), settling the open question from Remark 3.1 of Mukhopadhyay and Chattopadhyay (2013). DOI: http://dx.doi.org/10.4038/sljastats.v15i1.6795

Non-symmetric Cauchy kernels

Using an analogue of the Robinson-Schensted-Knuth (RSK) algorithm for semi-skyline augmented fillings, due to Sarah Mason, we exhibit expansions of non-symmetric Cauchy kernels $∏_(i,j)∈\eta (1-x_iy_j)^-1$, where the product is over all cell-coordinates $(i,j)$ of the stair-type partition shape $\eta$ , consisting of the cells in a NW-SE diagonal of a rectangle diagram and below it, containing the biggest stair shape. In the spirit of the classical Cauchy kernel expansion for rectangle shapes, this RSK variation provides an interpretation of the kernel for stair-type shapes as a family of pairs of semi-skyline augmented fillings whose key tableaux, determined by their shapes, lead to expansions as a sum of products of two families of key polynomials, the basis of Demazure characters of type A, and the Demazure atoms. A previous expansion of the Cauchy kernel in type A, for the stair shape was given by Alain Lascoux, based on the structure of double crystal graphs, and by Amy M. Fu and Alain Lascoux, relying on Demazure operators, which was also used to recover expansions for Ferrers shapes.

Regularity for solutions of nonlocal, nonsymmetric equations

We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear operators where the symmetric parts of the kernels have a fixed homogeneity σ and the skew symmetric parts have strictly smaller homogeneity τ. We prove a weak ABP estimate and C1,α regularity. Our estimates remain uniform as we take σ→2 and τ→1 so that this extends the regularity theory for elliptic differential equations with dependence on the gradient.

Regularity Results for Fully Nonlinear Integro-Differential Operators with Nonsymmetric Positive Kernels: Subcritical Case

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump L\`evy processes. If the index of the operator $\sigma$ is in $ (1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a H\"older regularity, and an interior $\rm C^{1,\alpha}$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index $\sigma$ of the operator is getting close to two, so that they can be regarded as a natural extension of regularity results for elliptic partial differential equations.

Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class \({\mathcal {S}^{\mathfrak {L}_0}}\) of uniformly elliptic nonlinear equations with 1 < σ < 2 (subcritical case) and to their subclass \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) with 0 < σ ≤ 1. We show that \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and C 1,α-regularity of the solutions by obtaining decay estimates of their level sets in each cases.

Non-symmetric Cauchy kernels for the classical groups

We give non-symmetric versions of the Cauchy kernel and Littlewood's kernels, corresponding to the types A, B, C and D, of the classical groups. Defining two families of key polynomials (one of them being the Demazure characters), we show that these new kernels are diagonal in the basis of key polynomials. We define scalar products such that the two families of key polynomials are adjoint to each other.