Kernel methods provide an attractive framework for aggregating and learning from ranking data, and so understanding the fundamental properties of kernels over permutations is a question of broad interest. We provide a detailed analysis of the Fourier spectra of the standard Kendall and Mallows kernels, and a new class of polynomial-type kernels. We prove that the Kendall kernel has exactly two ...

where 1 < p < N+2 N−2 , N ≥ 3. It is well-known that this equation has a unique positive radial solution. The existence of sign-changing radial solutions with exactly k nodes is also known. However the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial solution is unique when p is close to N+2 N−2 . Moreover, those solutions are non-degenerate, i.e., the...

Abstract. This paper is concerned with the stability analysis of the discretized collocation method for the second-kind Volterra integral equation with degenerate kernel. A fixed-order recurrence relation with variable coefficients is derived, and local stability conditions are given independent of the discretization. Local stability and stability with respect to an isolated perturbation of som...

In this paper, we derive a simple and efficient matrix formulation using Laguerre polynomials to solve the singular integral equation with degenerate kernel. This method is based on replacement of unknown function by truncated series well known expansion functions. leads system algebraic equations coefficients. Thus, solving equation, coefficients are obtained. Some numerical examples included ...

In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with KZ monodromy constructed by Varagnolo–Vasserot. prove centraliser property our and give a characterisation its kernel. establish these results family algebras, called quiver which includes as special ...

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ Rm×n, the kernel problem requires a positive vector in the kernel of A, and the image problem requires a positive vector in the image of A. Both algorithms iterate between simple first order steps and rescaling steps. These rescalings steps improve natural geometric potentials in the domain ...

We consider Bayesian inference in sequential latent variable models in general, and in nonlinear state space models in particular (i.e., state smoothing). We work with sequential Monte Carlo (SMC) algorithms, which provide a powerful inference framework for addressing this problem. However, for certain challenging and common model classes the state-of-the-art algorithms still struggle. The work...