Skew Schubert polynomials

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.

     

关于亚纯函数导数亏量和的Ozawa问题

设σλ表示所有限级λ的亚纯函数构成的集合,R.Nevanlinna显示,当λ是正的非整数时,κ(λ)>0,其中设f为有限级λ的亚纯函数,Ozawa证明了存在正常数d=d(λ),满足1/2(5-(21~(1/2))≤d≤1/4,使我们曾将d的范围精确为1/4≤d≤4/13。本文中,我们得到一个更精确、更广泛的结论:设f是有限级λ的亚纯函数,则对任何自然数n,存在仅与n,λ有关的正常数d,满足2n(n+1)/(4n~2+7n+2)≤d≤4n(n+1)/(4n~2+6n+1+(16n~4+56n~3+60n~2+20n+1)~(1/2))使得     

Hyers–Ulam stability of linear differential operator with constant coefficients

Let P(z) be a polynomial of degree n with complex coefficients and consider the n–th order linear differential operator P(D). We show that the equation P(D)f = 0 has the Hyers–Ulam stability, if and only if the equation P(z) = 0 has no pure imaginary solution. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ON THE RATE OF CONVERGENCE OF BEZIER VARIANT OF SZASZ—DURRMEYER OPERATORS

In the present paper, we introduce Szasz-Durrmeyer-Bezier operators M.,.(f,x) , which generalize the Szasz-Durrmeyer operators. Here we obtain an estimate on the rate of convergence of Mn,a(f,x) for functions of bounded variation. Our result extends and improves that of Sahai and Prasad and Gupta and Pant.     

Orders on Braid Groups

It is shown that the braid group defies lattice ordering.     

On the Similarity between Volterra Operators and Transformation Operators for Integro-Differential Operators of Fractional Order

We establish the similarity between certain Volterra integral operators and the Riemann--Liouville fractional integration operator as well as the existence of a triangular transformation operator for integro-differential equations of fractional order. The results obtained are consistent with similar results for the case of integer order.     

On a Construction of Fast Direct Solvers

Fast direct solvers for the Poisson equation with homogeneous Dirichlet and Neumann boundary conditions on special triangles and tetrahedra are constructed. The domain given is extended by symmetrization or skew symmetrization onto a rectangle or a rectangular parallelepiped and a fast direct solver is used there. All extendable domains are found. Eigenproblems are also considered.     

Passive Runge–Kutta Methods—Properties, Parametric Representation, and Order Conditions

In this paper, a new class of Runge–Kutta methods is introduced. Some basic properties of this subgroup of algebraically stable methods are presented and a complete parametric representation is given. Necessary and sufficient order conditions for lower order methods as well as sufficient order conditions for higher order methods are derived yielding a significantly reduced number of conditions when compared with general Runge–Kutta methods. Design examples conclude this paper.     

The Boson Normal Ordering Problem and Generalized Bell Numbers

For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for $F [(a^\dag)^r a^s]$, with r, s positive integers, $F [(a, a^\dag]=1$, i.e., we provide exact and explicit expressions for its normal form $\mathcal{N} \{F [(a^\dag)^r a^s]\} = F [(a^\dag)^r a^s]$, where in $ \mathcal{N} (F) $ all a's are to the right. The solution involves integer sequences of numbers which, for $ r, s \geq 1 $, are generalizations of the conventional Bell and Stirling numbers whose values they assume for $ r=s=1 $. A complete theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas), recursion relations and generating functions. These last are special expectation values in boson coherent states.AMS Subject Classification: 81R05, 81R15, 81R30, 47N50.     

Existence of an Angular Density of Roots of Entire Functions of Order Zero

We obtain sufficient conditions for the set of roots of an entire function of order zero of strongly regular growth to possess an angular density.