‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

We present a new probabilistic method for solving systems of polynomial equations and inequations. Our algorithm computes the equidimensional decomposition of the Zariski closure of the solution set of such systems. Each equidimensional component is encoded by a generic fiber, that is a finite set of points obtained from the intersection of the component with a generic transverse affine subspac...

A portfolio credit derivative is a contingent claim on the aggregate loss of a portfolio of credit sensitive securities. We develop an economically motivated and computationally tractable top down valuation framework in which portfolio loss follows an affine point process. The magnitude of each loss is random and defaults are governed by an intensity that is driven by affine jump diffusion risk...

We show that almost all metric–affine theories of gravity yield Einstein equations with a non–null cosmological constant Λ. Under certain circumstances and for any dimension, it is also possible to incorporate a Weyl vector field Wμ and therefore the presence of an anisotropy. The viability of these field equations is discussed in view of recent astrophysical observations.

We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff’s rod, where they provide a unified geometric interpretation.

‎Let $R$ be a ring‎ ‎with the Jacobson radical $J(R)$ and let $picolon Rto R/J(R)$ be‎ the canonical map‎. ‎Then $pi$ induces an order preserving group homomorphism‎ ‎$K_0picolon K_0(R)to K_0(R/J(R))$ and an‎ ‎affine continuous map $S(K_0pi)$ between the state space $St(R/J(R))$ and the‎ ‎state space $St(R).$‎ ‎In this paper‎, ‎we consider the natural affine map $S(K_0pi).$ We give a condition ...

In this paper, we establish global W 2,p estimates for solutions to the linearizedMonge–Ampère equations under natural assumptions on the domain, Monge– Ampère measures and boundary data. Our estimates are affine invariant analogues of the global W 2,p estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global W 2,p estimates for th...

Preface Roughly speaking, a deterministic extractor is a function that 'extracts' almost perfect random bits from a 'weak random source'-a distribution that contains some entropy but is far from being truly random. In this book we explicitly construct deterministic extractors and related objects for various types of sources. A basic theme in this book is a methodology of recycling randomness th...

in this paper, some basic results concerning strict, nonstrict inequalities, local existence theorem and differential inequalities  have been proved for an ivp of first order hybrid  random differential equations with the linear perturbation of second type. a comparison theorem is proved and  applied to prove the uniqueness of random solution for the considered perturbed random differential equ...

For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct generic solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We ...