44 research outputs found
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 . 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
Resultant optimization of the three-dimensional intersection problem
Resultant approach is here employed to optimize the dimensionless space angles to solve in a closed form the over-determined three-dimensional intersection problem. The advantages of the resultant optimization approach are the non-requirement of the approximate initial starting values, non iterative and does not rely on linearization during its operation, save for the nonlinear variance-covariance/error propagation to generate the weight matrix. Resultant method, a branch of abstract algebra, is employed to compute the combinatorial scatters, which are then optimized to offer a closed form solution. Using the test network Stuttgart Central as an example, it is demonstrated that the resultant optimization approach can be applied as an alternative approach to conventional methods such as least squares for point positioning within the over-determined intersection framework, especially when the approximate starting values for linearization and iterative approaches are not known as may happen in Photogrammetry, Machine Vision or in Robotics
Exact matrix formula for the unmixed resultant in three variables
We give the first exact determinantal formula for the resultant of an unmixed
sparse system of four Laurent polynomials in three variables with arbitrary
support. This follows earlier work by the author on exact formulas for
bivariate systems and also uses the exterior algebra techniques of Eisenbud and
Schreyer. Along the way we will prove an interesting new vanishing theorem for
the sheaf cohomology of divisors on toric varieties. This will allow us to
describe some supports in four or more variables for which determinantal
formulas for the resultant exist.Comment: 24 pages, 2 figures, Cohomology vanishing theorem generalized with
new combinatorial proof. Can state some cases of exact resultant formulas in
higher dimensio
Algorithmic Boundedness-From-Below Conditions for Generic Scalar Potentials
Checking that a scalar potential is bounded from below (BFB) is an ubiquitous
and notoriously difficult task in many models with extended scalar sectors.
Exact analytic BFB conditions are known only in simple cases. In this work, we
present a novel approach to algorithmically establish the BFB conditions for
any polynomial scalar potential. The method relies on elements of multivariate
algebra, in particular, on resultants and on the spectral theory of tensors,
which is being developed by the mathematical community. We give first a
pedagogical introduction to this approach, illustrate it with elementary
examples, and then present the working Mathematica implementation publicly
available at GitHub. Due to the rapidly increasing complexity of the problem,
we have not yet produced ready-to-use analytical BFB conditions for new
multi-scalar cases. But we are confident that the present implementation can be
dramatically improved and may eventually lead to such results.Comment: 27 pages, 2 figures; v2: added reference