Joint signal representations of arbitrary variables extend the scope of joint time-frequency representations, and provide a useful description for a wide variety of nonstationary signal characteristics. Cohen's marginal-based theory for bilinear representations is canonical from a distributional viewpoint, whereas, from other perspectives, such as characterization of the eeect of unitary signal...

We generalize the respective “double recurrence” results of Bourgain and of the second author, which established for pairs of L∞ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sig...

New exact Grammian determinant solutions to two (3+1)-dimensional nonlinear evolution equations are derived. Extended set of sufficient conditions consisting of linear partial differential equations with variable-coefficients is presented. Moreover, a systematic analysis of linear partial differential equations is used for solving the representative linear systems. The bilinear Bäcklund transfo...

In this paper, semi-discretizations and full-discretization of the Leznov lattice are investigated via Hirota’s bilinear formalism. As a result, two integrable semi-discrete versions and one fully discrete version for the Leznov lattice are found. Bäcklund transformations, nonlinear superposition formulae and Lax pairs for these discrete versions are presented. Mathematics Subject Classificatio...

Some explicit traveling wave solutions to a Kolmogorov-PetrovskiiPiskunov equation are presented through two ansätze. By a Cole-Hopf transformation, this Kolmogorov-Petrovskii-Piskunov equation is also written as a bilinear equation and further two solutions to describe nonlinear interaction of traveling waves are generated. Bäcklund transformations of the linear form and some special cases are...

The class of tensor product cubic splines is considered in an optimization process for two-dimensional polyacrylamide gel electrophoresis (2D-PAGE) image alignment problems arising in proteomics studies. Numerical results are presented along with comparisons with previous results based on piecewise affine and bilinear transformations.

The Hirota bilinear formalism and soliton solutions for a generalized Volterra system is presented. Also, starting from the soliton solutions, we obtain a class of nonsingular rational solutions using the ”long wave limit” procedure of Ablowitz and Satsuma, and appropriate ”gauge” transformations. Their properties are also discussed and it is shown that these solutions interact elastically with...

A survey of the global linearization problem is presented. Known results are divided into two groups: results for general affine nonlinear systems and for bilinear systems. In the latter case stronger results are available. A comparision of various linearizing transformations is performed. Numerous illustrative examples are included.