Bilinear approach is applied to derive integrable multi-component generalizations of the socalled 1+1 dimensional special Toda lattice, the Volterra lattice, a simple differential-difference equation found by Adler, Moser, Weiss, Veselov and Shabat and another integrable lattice reduced from the discrete BKP equation. Their soliton solutions expressed by pfaffians and the corresponding bilinear...

Recent algorithms for sparse coding and independent component analysis (ICA) have demonstrated how localized features can be learned from natural images. However, these approaches do not take image transformations into account. As a result, they produce image codes that are redundant because the same feature is learned at multiple locations. We describe an algorithm for sparse coding based on a...

A real square matrix Q is a bilinear complementarity relation on a proper cone K in R if x ∈ K, s ∈ K∗, and 〈x, s〉 = 0⇒ xQs = 0, where K∗ is the dual of K [14]. The bilinearity rank of K is the dimension of the space of all bilinear complementarity relations on K. In this article, we continue the study initiated in [14] by Rudol et al. We show that bilinear complementarity relations are related...

In the two-Higgs-doublet model (THDM), generalized-CP transformations (φi → Xijφj where X is unitary) and unitary Higgs-family transformations (φi → Uijφj) have recently been examined in a series of papers. In terms of gauge-invariant bilinear functions of the Higgs fields φi, the Higgs-family transformations and the generalized-CP transformations possess a simple geometric description. Namely,...

A new general bilinear relationship is found between continuous and discrete generalized singular perturbation (GSP) reduced-order models. This result is applied to the problem of deriving discrete analogs of continuous singular perturbation and direct truncation model reduction and leads to a new definition of discrete "Nyquist" model reduction. Also, "unit circle" bilinear transformations are...

A new multivariate Toda hierarchy of nonlinear partial differential equations adapted to biorthogonal polynomials is discussed. This integrable associated with non-standard biorthogonality. Wave and Baker functions, linear equations, Lax Zakharov–Shabat KP type appropriate reductions, Darboux or spectral transformations, bilinear involving transformations are presented.

Hirota’s bilinear technique is applied to some integrable lattice systems related to the Bäcklund transformations of the 2DToda, Lotka-Volterra and relativistic LotkaVolterra lattice systems, which include the modified Lotka-Volterra lattice system, the modified relativistic Lotka-Volterra lattice system, and the generalized BlaszakMarciniak lattice systems. Determinant solutions are constructe...

We construct a Cantor set ̂ of limit-periodic Jacobi operators having the spectrum on the Julia set J of the quadratic map z ι-> z + E for large negative real numbers E. The density of states of each of these operators is equal to the unique equilibrium measure μ on J. The Jacobi operators in $ are defined over the von Neumann-Kakutani system, a group translation on the compact topological group...

This article presents, in a brief form, the recent results on application of the Inverse scattering method to some problems of Differential geometry. A connection between the theory of Integrable systems and Differential geometry is not a new concept. The Sine–Gordon equation was introduced in the theory of surfaces of constant negative curvature around 1860. Actually, it should be called the “...