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We present a systematic formulation of scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition property similar to its linear analog's. We offer…
Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base…
We study the families of nonlinear modes described by the nonlinear Schr\"odinger equation with the PT-symmetric harmonic potential $x^2-2i\alpha x$. The found nonlinear modes display a number of interesting features. In particular, we have…
In this paper, we develop an algebraic approach to classifying contact symmetries of the second-order nonlinear evolution equations. Up to contact isomorphisms, all inequivalent PDEs admitting semi-simple algebras, solvable algebras of…
We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with aconstant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter,…
In this note we introduce a new technique to answer an issue posed in [7] concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper.
A particular form of non-linear $\sigma$-model, having a global gauge invariance, is studied. The detailed discussion on current algebra structures reveals the non-abelian nature of the invariance, with {\it{field dependent structure…
In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces
This article is based on the 5th Takagi Lectures delivered at the University of Tokyo in 2008. We discuss a hypothetical correspondence between holonomic D-modules on an algebraic variety X defined over a field of zero characteristic, and…
Let $u:A\to B$ be a morphism of noetherian local rings. We obtain smoothness criteria for algebras with differential bases, in the case of rings containing a field of characteristic $p>0.$ We also give smoothness criteria for reduced…
We look more closely at the higher nonabelian de Rham cohomology of a smooth projective variety or family of varieties that had been defined in some previous papers. We formalize using $n$-stacks the notion of shape underlying this…
We give a complete classification of non-loose Legendrian Hopf links in $L(p,q)$ generalizing a result of the author with Geiges and Onaran. The classification is for non-loose Hopf links for both zero and non-zero Giroux torsion in their…
Generally, natural scientific problems are so complicated that one has to establish some effective perturbation or nonperturbation theories with respect to some associated ideal models. In this Letter, a new theory that combines…
In the past decades, topological concepts have emerged to classify matter states beyond the Ginzburg-Landau symmetry breaking paradigm. The underlying global invariants are usually characterized by integers, such as Chern or winding…
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of…
This Chapter contains an overview of the effects of nonlinear interactions in selected problems of non-equilibrium statistical mechanics. Most of the emphasis is put on open setups, where energy is exchanged with the environment. With…
The paper deals with the existence of non-radial solutions for an $N$-coupled nonlinear elliptic system. In the repulsive regime with some structure conditions on the coupling and for each symmetric subspace of rotation symmetry, we prove…
This is the continuation of our previous work [5], where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp Hardy-Littlewood-Sobolev inequality. In this paper, we introduce some…
Binary multirelations can model alternating nondeterminism, for instance, in games or nondeterministically evolving systems interacting with an environment. Such systems can show partial or total functional behaviour at both levels of…
We introduce a "qualitative property" for Bott-Chern cohomology of complex non-K\"ahler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the…