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Mixed trigonometric-polynomials (MTPs) are functions of the form f (x , sin ⁡ x , cos ⁡ x) where f is a trivariate polynomial with rational coefficients, and the argument x ranges over the reals. In this paper, an algorithm "isolating" all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [ μ − , μ + ] while the other consists of countably many roots in R ﹨ [ μ − , μ + ]. For the roots in [ μ − , μ + ] , the algorithm returns isolating intervals and corresponding multiplicities while for those greater than μ + , it returns finitely many mutually disjoint small intervals I i ⊂ [ − π , π ] , integers c i > 0 and multisets of root multiplicity { m j , i } j = 1 c i such that any root > μ + is in the set (∪ i ∪ k ∈ N (I i + 2 k π)) and any interval I i + 2 k π ⊂ (μ + , ∞) contains exactly c i distinct roots with multiplicities m 1 , i ,... , m c i , i , respectively. The efficiency of the algorithm is shown by experiments. The method used to isolate the roots in [ μ − , μ + ] is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether there is any root, or whether there are infinitely many roots in unbounded intervals of the form (− ∞ , a) or (a , ∞) with a ∈ Q. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the biorthogonal matrix extension problem with symmetry and its application to construction of biorthogonal multiwavelets. Given a pair of biorthogonal matrices , the biorthogonal matrix extension problem is to find a pair of extension matrices of Laurent polynomials with symmetry such that the submatrix of the first r rows of is the given matrix , respectively; and are biorthogonal satisfying ; and and have the same compatible symmetry. We satisfactorily solve the biorthogonal matrix extension problem with symmetry and provide a step-by-step algorithm for constructing the desired pair of extension matrices from the given pair of matrices . Moreover, our results cover the case for paraunitary matrix extension with symmetry (i.e., the given pair satisfies ). Matrix extension plays an important role in many areas such as wavelet analysis, electronic engineering, system sciences, and so on. As an application of our general results on biorthogonal matrix extension with symmetry, we obtain a satisfactory algorithm for constructing univariate biorthogonal multiwavelets with symmetry for any dilation factor from a given pair of biorthogonal -refinable function vectors with symmetry. Correspondingly, pairs of -dual filter banks with the perfect reconstruction property and with symmetry can be derived by applying our algorithm to a given pair of -dual low-pass filters with symmetry. Several examples of symmetric biorthogonal multiwavelets are provided to illustrate our results in this paper. [Copyright &y& Elsevier]

The efficient evaluation of multivariate polynomials at many points is an important operation for polynomial system solving. Kedlaya and Umans have recently devised a theoretically efficient algorithm for this task when the coefficients are integers or when they lie in a finite field. In this paper, we assume that the set of points where we need to evaluate is fixed and "sufficiently generic". Under these restrictions, we present a quasi-optimal algorithm for multi-point evaluation over general fields. We also present a quasi-optimal algorithm for the opposite interpolation task. [ABSTRACT FROM AUTHOR]

Abstract: Let be a graph and let and be two integer-valued functions defined on such that for every . Let be vertex-disjoint subgraphs of with (). In this paper, we prove that every -graph contains a subgraph such that has a -factorization orthogonal to (), where and are positive integers with . We also prove that there are polynomial-time algorithms for finding the desired subgraph . [ABSTRACT FROM AUTHOR]