Narrow Results

We consider the rational dynamical quantum group $E_y({𝔤𝔩}_2)$ and introduce an $E_y({𝔤𝔩}_2)$-module structure on ${\oplus }_{k=0}^n{H}_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }{(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))}^{\prime }$, where $H_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }{(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))}^{\prime }$ is the equivariant cohomology algebra $H_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))$ of the cotangent bundle of the Grassmannian ${\mathrm{\text{Gr}}}_k({ℂ}^n)$ with coefficients extended by a suitable ring of rational functions in an additional variable $\lambda$. We consider the dynamical Gelfand–Zetlin algebra which is a commutative algebra of difference operators in $\lambda$. We show that the action of the Gelfand–Zetlin algebra on $H_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }{(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))}^{\prime }$ is the natural action of the algebra $H_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))\otimes ℂ[{\delta }^{\pm 1}]$ on $H_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }{(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))}^{\prime }$, where $\delta :\zeta (\lambda )\to \zeta (\lambda +y)$ is the shift operator.

The $E_y({𝔤𝔩}_2)$-module structure on ${\oplus }_{k=0}^n{H}_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }{(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))}^{\prime }$ is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov [Quantum Groups and quantum cohomology, preprint (2012) 1–276, arXiv:1211.1287]. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [G. Felder, V. Tarasov and A. Varchenko, Solutions of the elliptic QKZB equations and Bethe ansatz I, in Topics in Singularity Theory V. I. Arnold’s 60th Anniversary Collection, Advances in the Mathematical Sciences, AMS Translations, Series 2, Vol. 180 (AMS, 1997), pp. 45–76.] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in $H_{{\mathrm{\text{GL}}}_n\times {ℂ}^{\times }}^{\ast }{(T^{\ast }{\mathrm{\text{Gr}}}_k({ℂ}^n))}^{\prime }$ induced by the weight functions are dynamical variants of Chern–Schwartz–MacPherson classes of Schubert cells.

P. Jarratt has developed a family of fourth-order optimal methods. He suggested two members of the family. The dynamics of one of those was discussed previously. Here we show that the family can be written using a weight function and analyze all members of the family to find the best performer.

In this paper stochastic relations and closure results for weighted and transformed distributions including the proportional hazards model are presented. Exponential approximations to the class of increasing failure rate (IFR) and decreasing failure rate (DFR) weighted distributions including transformed distributions with monotone weight functions are obtained. These include approximations via the proportional hazards and length-biased exponential distributions. Also, bounds and moment-type inequalities for weighted life distributions including proportional hazards models and some applications are presented.

In this paper, we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules, we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom–Tsfasman weight.