Solving the Selesnick-Burrus Filter Design Equations Using Computational Algebra and Algebraic Geometry

In a recent paper, I. Selesnick and C.S. Burrus developed a design method for maximally flat FIR low-pass digital filters with reduced group delay. Their approach leads to a system of polynomial equations depending on three integer design parameters $K,L,M$. In certain cases (their ``Region I''), Selesnick and Burrus were able to derive solutions using only linear algebra; for the remaining cases ("Region II''), they proposed using Gr\"obner bases. This paper introduces a different method, based on multipolynomial resultants, for analyzing and solving the Selesnick-Burrus design equations. The results of calculations are presented, and some patterns concerning the number of solutions as a function of the design parameters are proved.Comment: 34 pages, 2 .eps figure

One Theorem, Two Ways: A Case Study in Geometric Techniques

If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in same ratios: AH : HB = BΘ : ΘΓ = ΓK : KA, then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus\u27s original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic proof gets to the heart of the matter more efficiently, the synthetic proof does a better job of revealing hidden aspects of the geometric configuration. Moreover, as Pappus presents it, the synthetic proof provides a real element of surprise and a sense of discovering unexpected connections. We conclude with some general observations about synthetic versus algebraic techniques in geometry and in the teaching and learning of mathematics

A Mathematician Reads Plutarch: Plato\u27s Criticism of Geometers of His Time

This essay describes the author\u27s recent encounter with two well-known passages in Plutarch that touch on a crucial episode in the history of the Greek mathematics of the fourth century BCE involving various approaches to the problem of the duplication of the cube. One theme will be the way key sources for understanding the history of our subject sometimes come from texts that have much wider cultural contexts and resonances. Sensitivity to the history, to the mathematics, and to the language is necessary to tease out the meaning of such texts. However, in the past, historians of mathematics often interpreted these sources using the mathematics of their own times. Their sometimes anachronistic accounts have often been presented in the mainstream histories of mathematics to which mathematicians who do not read Greek must turn to learn about that history. With the original sources, the tidy and inevitable picture of the development of mathematics disappears and we are left with a much more interesting, if ultimately somewhat inconclusive, story

Determinants Associated to Zeta Matrices of Posets

We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive formula for this determinant in the case in which $P$ is a boolean algebra.Comment: 14 pages, AMS-Te

Retinoic acid inhibits the fixation of initial transformational damage in X-irradiated Balb/3T3 mouse fibroblasts in vitro

We have examined the effects of all-trans retinoic acid (RA) on confluent holding recovery (cell survival) and on the fixation of initial transformational damage expressed as the ultimate yield of transformed foci following X-irradiation of density-inhibited cultures of Balb/3T3 cells. Non-cytotoxic concentrations of RA suppressed both recovery of potentially lethal damage and neoplastic transformation in a dosedependent manner when added for 24 h during postirradiation confluent holding after a dose of 5 Gy. At 100 μM, RA inhibited the fixation of initial transformational damage by 80%. These findings are discussed in terms of the hypothesis that retinoids may allow a selective enhancement of the inactivation of certain irradiated tumor cells in vivo while reducing the risk of secondary malignancies in successfully treated patient