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The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151-160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of m subsets of a universe of size n is on the order of max {log n, √log m}. On the other hand, building upon work of Matousek [Proc. Amer. Math. Soc., 141 (2013), pp. 451-460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308-313] showed that this gap is always bounded up to constants by √log(m)log(n). This is tight when m is polynomial in n but leaves open the case of large m. We show that the bound of Jiang and Reis is tight for nearly the entire range of m. Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization issue for a multidimensional wave equation with localized Kelvin--Voigt damping on a cuboidal domain, in which the damping region does not satisfy the geometric control condition (GCC). The variable damping coefficient is assumed to be degenerate near the interface. We prove that the system is polynomially stable with a decay rate depending on the degree of the degeneration γ. A relationship between the decay order and γ is identified. In particular, this decay rate is consistent with the optimal one for the corresponding system with constant damping coefficient (i.e., γ = 0) obtained in [K. Yu and Z.-J. Han, SIAM J. Control Optim., 59 (2021), pp. 1973--1988]. Moreover, it is the first result on the decay rates of the solutions to multidimensional wave equations with localized degenerate Kelvin--Voigt damping when GCC is not satisfied. [ABSTRACT FROM AUTHOR]

n the article "A Rational QZ Method" by D. Camps, K. Meerbergen, and R. Vande-bril [SIAM J. Matrix Anal. Appl., 40 (2019), pp. 943--972], we introduced rational QZ (RQZ) methods. Our theoretical examinations revealed that the convergence of the RQZ method is governed by rational subspace iteration, thereby generalizing the classical QZ method, whose convergence relieson polynomial subspace iteration. Moreover the RQZ method operates on a pencil more general than Hessenberg---upper triangular, namely, a Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. However, the RQZ method can only be made competitive to advanced QZ implementations by using crucial add-ons such as small bulge multi shift sweeps, aggressive early deflation, and optimal packing. In this paper we develop these techniques for the RQZ method.In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that the presented method is competitive in terms of speed and accuracy. [ABSTRACT FROM AUTHOR]

We study numerical integration of smooth functions defined over the |$s$| -dimensional unit cube. A recent work by Dick et al. (2019 , Richardson extrapolation of polynomial lattice rules. SIAM J. Numer. Anal. , 57 , 44–69) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of convergence for numerical integration, and can be constructed by the fast component-by-component search algorithm with smaller computational costs as compared to interlaced polynomial lattice rules. In this paper we prove that, instead of polynomial lattice point sets, truncated higher-order digital nets and sequences can be used within the same algorithmic framework to explicitly construct good quadrature rules achieving the almost optimal rate of convergence. The major advantage of our new approach compared to original higher-order digital nets is that we can significantly reduce the precision of points, i.e. the number of digits necessary to describe each quadrature node. This finding has a practically useful implication when either the number of points or the smoothness parameter is so large that original higher-order digital nets require more than the available finite-precision floating-point representations. [ABSTRACT FROM AUTHOR]

A convex optimization problem in conic form involves minimizing a linear functional over the intersection of a convex cone and an affine subspace. In some cases, it is possible to replace a conic formulation using a certain cone, with a “lifted” conic formulation using another cone that is higher-dimensional, but simpler, in some sense. One situation in which this can be computationally advantageous is when the higher-dimensional cone is a Cartesian product of many “low-complexity” cones, such as second-order cones, or small positive semidefinite cones. This paper studies obstructions to a convex cone having a lifted representation with such a product structure. The main result says that whenever a convex cone has a certain neighborliness property, then it does not have a lifted representation using a finite product of cones, each of which has only short chains of faces. This is a generalization of recent work of Averkov [SIAM J. Appl. Alg. Geom., 3 (2019), pp. 128–151], which considers only lifted representations using products of positive semidefinite cones of bounded size. Among the consequences of the main result is that various cones related to nonnegative polynomials do not have lifted representations using products of “low-complexity” cones, such as smooth cones, the exponential cone, and cones defined by hyperbolic polynomials of low degree. [ABSTRACT FROM AUTHOR]

It is shown in [SIAM J. Sci. Comput., 39 (2017), pp. B424-B441] that free-form curves used in computer-aided geometric design can usually be represented as the solutions of linear differential systems and that points and derivatives on the curves can be evaluated dynamically by solving the differential systems numerically. In this paper we present an even more robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transformation of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a precomputed matrix. Like former dynamic evaluation algorithms, the newly proposed approach needs only arithmetic operations for evaluating exponential polynomial curves and surfaces. Unlike conventional numerical methods that solve a linear differential system, the new method can give robust and accurate evaluation results for any chosen parameter steps. The basis transformation technique also enables dynamic evaluation of polynomial curves with changing parameter steps using a constant matrix, which reduces time costs significantly than computing each point individually by classical algorithms. [ABSTRACT FROM AUTHOR]

A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [SIAM J. Numer. Anal., 51 (2013), pp. 1634-1654] for the Laplace operator. The main tool is the computation of an explicit upper bound to the error of the Galerkin projection. The error is split into two parts. One part is controlled by a hypercircle principle and an auxiliary eigenvalue problem. The second part requires a perturbation argument for the right-hand side replaced by a suitable piecewise polynomial. The latter error is controlled through the use of the commuting quasi-interpolation by Falk and Winther and computational bounds on its stability constant. This situation is different from the Laplace operator where such a perturbation is easily controlled through local Poincaré inequalities. The practical viability of the approach is demonstrated in test cases for two and three space dimensions. [ABSTRACT FROM AUTHOR]

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O (1. 1389 n) for graphs with maximum degree 3 and in time O (1. 2356 n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O (2. 2356 n) as well as an exponential-space O (1. 2330 n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström's compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties. [ABSTRACT FROM AUTHOR]

A numerical semigroup S is cyclotomic if its semigroup polynomial P S is a product of cyclotomic polynomials. The number of irreducible factors of P S (with multiplicity) is the polynomial length ℓ (S) of S. We show that a cyclotomic numerical semigroup is complete intersection if ℓ (S) ≤ 2 . This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between ℓ (S) and the embedding dimension of S. [ABSTRACT FROM AUTHOR]

We study Leja sequences on the unit disc and resulting mapped sequences to conventional domain, ellipses and real intervals, for the problem of relaxation of Richardson iteration. Using simple considerations, we establish upper and lower estimates for the growth of associated Newton polynomials and the so-called residual polynomials. These results broaden the understanding of such sequences and add to results established in Calvetti et al. (Numer. Math. 67(1), 21–40, 1994), Calvetti and Reichel (Numerical Algorithms 11(1-4), 79–98, 1996; J. Comput. Appl. Math. 71(2), 267–286 1996), Fischer and Reichel (Numer. Math. 54(2), 225–242, 1989), Nachtigal et al. (SIAM Journal on Matrix Analysis and Applications 13(3), 796–825, 1992), Reichel (SIAM J. Numer. Anal. 25(6), 1359–1368, 1988), Tal-Ezer (J. Sci. Comput. 4(1), 25–60, 1989). We also propose adaptive strategies for the selection of relaxation parameters that build on fast strategies (Balgama et al., Electron. Trans. Numer. Anal. 7, 124–140, 1998; Reichel, Linear Algebra and its Applications 154-156(C), 389–414, 1991) which are known to be suitable for the problem. In particular a Leja ordering is enforced on the mapped sequences. New strategies for conveniently confining the spectrums of certain matrices arising in PDEs discretization is described and demonstrate the flexibility of the Richardson iteration. [ABSTRACT FROM AUTHOR]

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control, and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in Guglielmi et al. (SIAM J. Numer. Anal. 55, 1456–1482, 2017), which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover, we show that the algorithm becomes faster by replacing the eigenvalues by the singular values. [ABSTRACT FROM AUTHOR]

The widely available laboratory spectrometers detect targets at spectral regions restricted to visible and near-infrared (VNIR). The spectral response of soils in this region is predominantly featureless and obstructs the exploitation of absorption features as diagnostic criterion. In this study, polynomial based modelling was developed as an alternative method of estimating soil organic matter (OM) from VNIR spectral region. Forty-one core samples, collected from Lop Buri, Thailand, were subjected to chemical and radiometric analysis. Computations were made across four categories of synthesized bandwidths. The selection procedure identified bands at 960, 1100 and 520 nm as OM sensitive. The widening interval of bandwidth has corresponded with diminishing predictive power, termed 'bandwidth decay effect'. The use of polynomial models and their validations showed a higher performance than the analysis made with multiple regressions analysis. The polynomial based approach offers a fresh opportunity for modelling other non-photoactive soil nutrient parameters. Furthermore, it may form the basis for integration of spectrometers and satellite sensors, aimed at mapping of non-vegetated soils. [ABSTRACT FROM AUTHOR]