On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

ACM Computing Classification System (1998): G.1.1, G.1.2.

Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)

Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively, with n > m, three new, and easy to understand methods — along with the more efficient variants of the last two of them — are presented for the computation of their subresultant polynomial remainder sequence (prs). All three methods evaluate a single determinant (subresultant) of an appropriate sub-matrix of sylvester1, Sylvester’s widely known and used matrix of 1840 of dimension (m + n) × (m + n), in order to compute the correct sign of each polynomial in the sequence and — except for the second method — to force its coefficients to become subresultants. Of interest is the fact that only the first method uses pseudo remainders. The second method uses regular remainders and performs operations in Q[x], whereas the third one triangularizes sylvester2, Sylvester’s little known and hardly ever used matrix of 1853 of dimension 2n × 2n. All methods mentioned in this paper (along with their supporting functions) have been implemented in Sympy and can be downloaded from the link http://inf-server.inf.uth.gr/~akritas/publications/subresultants.py

On some Extremal Problems of Landau

2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.

Complex Analogues of the Rolle's Theorem

2000 Mathematics Subject Classification: 30C10.

A Unified Group-Theoretic Method on Improper Partial Semi-Bilateral Generating Functions

2000 Mathematics Subject Classification: 33A65, 33C20.

The Lower Estimate for Bernstein Operator

MSC 2010: 41A10, 41A15, 41A25, 41A36

Extention of Apolarity and Grace Theorem

MSC 2010: 30C10

Hermite Series with Polar Singularities

MSC 2010: 33C45, 40G05

Class Number Two for Real Quadratic Fields of Richaud-Degert Type

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.

Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants

MSC 2010: 11B83, 05A19, 33C45

The Direct and Inverse Spectral Problems for some Banded Matrices

2000 Mathematics Subject Classification: 15A29.

Zolotarev's proof of Gauss reciprocity and Jacobi symbols

2000 Mathematics Subject Classification: Primary 11A15.

Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group

2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.

Equiconvergence and equisummability of Jacobi series

2010 Mathematics Subject Classification: 33C45, 40G05.

Non-manipulable domains for the Borda count

We characterize the preference domains on which the Borda count satises Arrow's "independence of irrelevant alternatives" condition. Under a weak richness condition, these domains are obtained by xing one preference ordering and including all its cyclic permutations ("Condorcet cycles"). We then ask on which domains the Borda count is non-manipulable. It turns out that it is non-manipulable on a broader class of domains when combined with appropriately chosen tie-breaking rules. On the other hand, we also prove that the rich domains on which the Borda count is non-manipulable for all possible tie-breaking rules are again the cyclic permutation domains.