We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has revealed essential aspects of the integrability of commutative soliton equations and the noncommutative extension is worth studying. We succeed in deriving va...

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax ope...

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions...

Clearly produced vowels exhibit longer duration and more extreme spectral properties than plain, conversational vowels. These features also characterize tense relative to lax vowels. This study explored the interaction of clear-speech and tensity effects by comparing clear and plain productions of three English tense-lax vowel pairs (/i-ɪ/, /ɑ-ʌ/, /u-ʊ/ in /kVd/ words). Both temporal and spectr...

We consider scalar conservation laws with convex flux and random initial data. The Hopf–Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative Markov process. Here we show that: (i) the Lax equations a...

The problem in phylogenetic tree construction, is to find the tree which best describes the relationship between a set of objects. We are mostly interested in the case when the objects are species or taxa. carrot whale chimpanzee human Figure 5.1: Phylogenetic tree of 4 species Phylogenetic Tree Construction is based on Cladistics, the systematic classification of groups of organisms on the bas...

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions...

We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg–deVries equation and show that it can be consistently extended multidimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series of q-discrete Painlevé equations, and Lax pairs for the reduced equations are found. A 2× 2 Lax pair is given f...

A propositional logic program P may be identified with a PfPf -coalgebra on the set of atomic propositions in the program. The corresponding C(PfPf )-coalgebra, where C(PfPf ) is the cofree comonad on PfPf , describes derivations by resolution. Using lax semantics, that correspondence may be extended to a class of first-order logic programs without existential variables. The resulting extension...

2.1 Examples of conservative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 The Godunov Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 The Lax-Friedrichs Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 The local Lax-Friedrichs Scheme . . . . . . . ....