Using spatial dynamics, we prove a Hopf bifurcation theorem for viscous Lax shocks in viscous conservation laws. The bifurcating viscous shocks are unique (up to time and space translation), exponentially localized in space, periodic in time, and their speed satisfies the Rankine–Hugoniot condition. We also prove an ”exchange of spectral stability” result for superand subcritical bifurcations, ...

Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor − ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of −⊗A B and comodules (or coalgebras) of − ⊗A C...

We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on Wasserstein space. prove that are symplectic submersions in geometry. Based this transformation, energy splitting inequalities provided.

These results were discovered in an attempt to embed real projective spaces. The well-known Hopf fiberings of Pu and Pi give the following representations of these spaces: Pi is a fiber space over S4 with P% as fiber, and Pu is a fiber space over S8 with P7 as fiber. Since any embedding of Pz in R6 has a trivial normal bundle [ó], Theorem 1.2 applies for Pi. Theorem 1.1 applies for Pu and, henc...

Using Hirasawa-Murasugi’s classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on S. We also construct L-space knots with arbitrarily larg...

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener–Hopf functional equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The key step in the so...

We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke R-matrix. Their behaviour with respect to Hecke sum of R-matrices is studied. Given a Hecke R-matrix in n-dimensional vector space, we construct a Hecke R-matrix in 2n-dimensional vector space commuting with a differential. The notion o...

In this paper, we study the duals of some finite dimensional pointed Hopf algebras working over an algebraically closed field k of characteristic 0. In particular, we study pointed Hopf algebras with coradical k[Γ] for Γ a finite abelian group, and with associated graded Hopf algebra of the form B(V )#k[Γ] where B(V ) is the Nichols algebra of V = ⊕iV χi gi ∈ k[Γ] k[Γ]YD. As a corollary to a ge...

Two coalgebra structures are used in quantum field theory. The first one is the coalgebra part of a Hopf algebra leading to quantization. The second one is a co-module co-algebra over the first Hopf algebra and it is used to define connected chronological products and renormalization. Paper written for the Encyclopaedia of Mathematics. Co-algebra is a pervasive structure of quantum field theory...