The classical algebra Λ of symmetric functions has a remarkable deformation Λ, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions sμ, where μ ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in Z(gl(n)), the center of th...

In this paper, we consider shifted tridiagonal matrices. We prove that the standard algorithm to compute the LU factorization in this situation is mixed forward-backward stable and, therefore, componentwise forward stable. Moreover, we give a formula to compute the corresponding condition number in O(n) flops. Mathematics Subject Classification (2000). 65F35, 65F50, 15A12, 15A23, 65G50.

Bijective proofs of the hook formulas for the number of ordinary standard Young tableaux and for the number of shifted standard Young tableaux are given. They are formulated in a uniform manner, and in fact prove q-analogues of the ordinary and shifted hook formulas. The proofs proceed by combining the ordinary, respectively shifted, Hillman–Grassl algorithm and Stanley's (P, ω)-partition theor...

This paper is devoted to the adaptive Morley element algorithms for a biharmonic eigenvalue problem in [Formula: see text] ([Formula: see text]). We combine the Morley element method with the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to propose multigrid discretization schemes in an adaptive fashion. We establish an inequality on ...

Let G be any of the complex classical groups GL(n) , SO(2n + 1) , Sp(2n) , O(2n) , let g denote the Lie algebra of G , and let Z(g) denote the subalgebra of G -invariants in the universal enveloping algebra U(g) . We derive a Taylortype expansion for finite-dimensional characters of G (the binomial formula) and use it to specify a distinguished linear basis in Z(g) . The eigenvalues of the basi...

We consider the numerical stability of the second barycentric formula for evaluation at points in [0, 2π ] of trigonometric interpolants in an odd number of equispaced points in that interval. We show that, contrary to the prevailing view, which claims that this formula is always stable, it actually possesses a subtle instability that seems not to have been noticed before. This instability can ...

We give an exact formula for the L2 discrepancy of two-dimensional digitally shifted Hammersley point sets in base b. This formula shows that for certain bases b and certain shifts the L2 discrepancy is of best possible order with respect to the general lower bound due to Roth. Hence, for the first time, it is proved that, for a thin, but infinite subsequence of bases b starting with 5, 19, 71,...

Quartz tuning forks are being increasingly employed as sensors in non-contact atomic force microscopy especially in the "qPlus" design. In this study a new and easily applicable setup has been used to determine the static spring constant at several positions along the prong of the tuning fork. The results show a significant deviation from values calculated with the beam formula. In order to und...

Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometric functions received a brief presentation on the rectangular coordinate system. Next, they participated in a computer-interactive matching-to-sample procedure that trained formula-to-formula and formula-to-graph relations. Then, they were exposed to 40 novel formula-to-graph tests and 10 novel graph-to-formul...