V.9 - SUBDIVIDING SIMPLICES

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg, the aim of this paper is to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval.

This paper demonstrates convenient use of freely available Polmat library for computations with transfer function models of multidimensional systems within Matlab environment. Polmat functions rely on MuPAD, a computer algebra system, which has become the core engine for symbolic computations in Matlab. Basic manipulations with multivariate two-sided polynomials are illustrated, test of stability of spatially distributed systems and solution of linear (Diophantine) equations with multivariate polynomials are demonstrated by means of a few examples.

In this paper, the multivariate Bernstein polynomials defined on a simplex are viewed as sampling operators, and a generalization by allowing the sampling operators to take place at scattered sites is studied. Both stochastic and deterministic aspects are applied in the study. On the stochastic aspect, a Chebyshev type estimate for the sampling operators is established. On the deterministic aspect, combining the theory of uniform distribution and the discrepancy method, the rate of approximating continuous function and L p convergence for these operators are studied, respectively.

IT has a role to play in the current debate on climate change. The current discourse on IT and climate change views IT in a negative light, as a polluter. What remains unrecognised is the critical role of IT as a source of solutions to the climate change problem. We live in a massive, inter-connected Planet Earth Supply Chain. IT provides a range of tools to model, manage and optimise this supply chain.The University of Wollongong Carbon-Centric Computing Initiative (CCCI) seeks to seed a program of research that addresses the climate change problem with a range of computing technologies including (but not limited to): optimisation technologies, supply chain management technologies, business process management/process improvement technologies, grid computing (e.g., utility grid) and virtualisation technologies, ICT-enabled conferencing and collaboration technologies as well as ICT for knowledge sharing and network-centric advocacy.The contours of this new and exciting space for research and industry development are described in this report. The report provides insights into a set of representative points within this new space. It describes how existing web infrastructure could be leveraged to devise the optimising web a massive, globally inter-connected network of optimisers helping support decisions that would reduce the global carbon footprint. It describes how computer simulation models can provide the basis for sustainable manufacturing and environmental management in the enterprise. It describes how IT based techniques can help support supply chain optimisation audits to determine if and how value might be best derived from the judicious use of optimisation technology. It describes the critical role ICT-enabled collaboration technologies can play in reducing the carbon footprint. It also addresses the key role ICT-based knowledge sharing and network-centric advocacy can play in obtaining broader social engagement in this debate. The report addresses the policy dimension to these issues and the need for an industry-academia consortium to drive such an agenda forward.The DSL: The Decision Systems Lab has engaged in cutting-edge research in the areas of industrial optimisation, business process management, service-oriented computing, software engineering and the applications of artificial intelligence technology for over a decade. It prides itself in being able to effectively span the spectrum from basic to applied research, and has generated a range of high-impact insights and industry applications.ATUL: As the Activity Theory Usability Laboratory, ATUL was opened in 2001. Able to support both research and practice, ATUL was then equipped to conduct holistic and realistic usability evaluations of computer applications and websites. Since its opening ATUL has expanded to acquire an exciting range of tools in innovative areas of business analysis and training: team-building, group decision support, systems modeling, content analysis among these. Copyright 2009 Aditya Ghose, Helen Hasan and Trevor Spedding. No part of this article may be reproduced by any means without the written consent of the publisher.

Hydrodynamic models in coastal and estuarine systems are typically constrained by sparse bathymetry, boundary, and validation data, especially in regions where field campaigns are costly or impractical. Here we develop and test a fully satellite-driven framework for hydrodynamic modeling in South Africa’s Langebaan Lagoon without using any local in situ measurements. Bathymetry is derived by training multispectral Sentinel-2 reflectance against ICESat-2 ATL24 photon-derived depths using an XGBoost model optimized with Bayesian search. The final satellite derived bathymetry reproduces independent ATL24 points with RMSE = 0.45 m and R 2 = 0.97. This bathymetry was used in a depth-averaged Delft3D Flexible Mesh model driven at the open boundary by TPXO tidal harmonics and by ERA5 winds. We validate modeled water surface elevation against 16 SWOT low-rate (250 m, unsmoothed) passes in 2023. SWOT–model comparisons yield an overall RMSE of 0.11 m and R 2 = 0.61, with typical point differences &amp;lt;0.10 m ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> </mml:math> 5% of the 2 m tidal range), and showed consistent spatial gradients in water level from the offshore boundary, through Saldanha Bay, and into the lagoon. At the offshore boundary, TPXO and SWOT sea surface heights agree closely ( R 2 = 0.86). A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> </mml:math> 26 min phase lag, determined using a lag-correlation analysis, reduces the TPXO–SWOT RMSE from 0.18 m to 0.11 m, indicating that phase differences explain some of the mismatch, with remaining differences likely linked to non-tidal signals. Our results demonstrate that combining passive optical, photon-counting LiDAR, radar interferometry, and global tidal/atmospheric models enables robust, transferrable hydrodynamic modeling in data-scarce coastal systems, offering a cost-effective pathway for monitoring.

Known mathematical packages, MATLAB, Maple, Mathematica, MathCAD and others may get wrong, often plausible, the result of numerical solution of ODE systems with low, given by default, the requirements for mathematical accuracy of the results of the numerical solution of ODE systems. Since ODE system parameters obtained usually experimental, with a low mathematical precision, so the requirements for the precision of the results of mathematical solutions of ODE systems is low (for example, in MATLAB package required precision is 0.001). The article offers a basic set of tests to assess the range of applicability of the relevant solvers. The basic set of test problems for ODE solvers systems include linear ODE systems with the known analytic solution and nonlinear systems with known graphics solution. Presented comparative results of solutions for proposed problems using MATLAB solvers and manzhuk program from a library of standard mathematical programs SADEL (Sets of Algebraic and Differential Equations solvers Library), which has been designed for reliable and accurate solving of systems of linear algebraic equations (LAE) and ODE systems. The results can be used in mathematical modeling of dynamic systems described by ODE systems.

We study the polynomial reconstruction problem for low-degree multivariate polynomials over finite field $\mathbb{F}[2]$. In this problem, we are given a set of points $\mathbf{x}\in\{0,1\}^n$ and target values $f(\mathbf{x})\in\{0,1\}$ for each of these points, with the promise that there is a polynomial over $\mathbb{F}[2]$ of degree at most d that agrees with f at $1-\varepsilon$ fraction of the points. Our goal is to find a degree d polynomial that has good agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than $1-2^{-d}+\delta$ fraction of the points for any $\epsilon,\delta>0$. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, whereas the algorithm is allowed to find a polynomial of degree d. Previously the only known hardness of approximation (or even NP-completeness) was for the case when $d =1$, which follows from a celebrated result of Håstad [J. ACM, 48 (2001), pp. 798–859]. In the setting of Computational Learning, our result shows the hardness of nonproper agnostic learning of parities, where the learner is allowed a low-degree polynomial over $\mathbb{F}[2]$ as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results can be extended to multivariate polynomial reconstruction over any finite field.

A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well‐covered graphs are classified by the Wn property: For a positive integer n, a graph G belongs to class Wn if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of Wn graphs containing arbitrarily large independent sets. A characterization of Wn graphs in terms of well‐covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in Wn graphs.

All organisms are subject to large amounts of genetic and environmental variation and have evolved mechanisms that allow them to function well in spite of these challenges. This property is generally referred to as robustness. We start with the premise that phenotypes arise from dynamical systems and are therefore system properties. Phenotypes occur at all levels of the biological organizational hierarchy, from gene products, to biochemical pathways, to cells, tissues, organs, appendages, and whole bodies. Phenotypes at all these levels are subject to environmental and genetic challenges against which their form and function need to be protected. The mechanisms that can produce robustness are diverse and several different kinds often operate simultaneously. We focus, in particular, on homeostatic mechanisms that dynamically maintain form and function against varying environmental and genetic factors. Understanding how homeostatic mechanisms operate, how they reach their set point, and the nature of the set point pose difficult challenges. In developmental systems, homeostatic mechanisms make the progression of morphogenesis relatively insensitive to genetic and environmental variation so that the outcomes vary little, even in the presence of severe mutational and environmental stress. Accordingly, developmental systems give the appearance of being goal-oriented, but how the target phenotype is encoded is not known. We discuss why and how individual variation poses challenges for mathematical modeling of biological systems, and conclude with an explanation of how system population models are a useful method for incorporating individual variation into deterministic ordinary differential equation (ODE) models. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Physiology > Mammalian Physiology in Health and Disease Physiology > Organismal Responses to Environment Biological Mechanisms > Regulatory Biology.

Division algorithms for Bernstein polynomials

An identity for multivariate Bernstein polynomials

Two continuous formulations of the maximum independent set problem on a graph G=(V,E) are considered. Both cases involve the maximization of an n-variable polynomial over the n-dimensional hypercube, where n is the number of nodes in G. Two (polynomial) objective functions F(x) and H(x) are considered. Given any solution to x0 in the hypercube, we propose two polynomial-time algorithms based on these formulations, for finding maximal independent sets with cardinality greater than or equal to F(x0) and H(x0), respectively. A relation between the two approaches is studied and a more general statement for dominating sets is proved. Results of preliminary computational experiments for some of the DIMACS clique benchmark graphs are presented.

A fundamental question is whether one can maintain a maximum independent set in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of n disks of unit radius in the plane, we show that a 12-approximate maximum independent set can be maintained with worst-case update time $$O(\log n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension d , where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain $$O(1+\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -approximate maximum independent set in truly sublinear update time, under standard complexity assumptions. Our results build on two recent technical tools: (i) The MIX algorithm by Cardinal et al. (2021) that can smoothly transition from one independent set to another; hence it suffices to maintain a family of independent sets where the largest one is a constant-factor approximation of a maximum independent set. (ii) A dynamic nearest/farthest neighbor data structure for disks by Kaplan et al. (2020) and Liu (2022), which generalizes the dynamic convex hull data structure by Chan (2010), and allows us to quickly find a “replacement” disk (if any) when a disk in one of our independent sets is deleted.

Robust arithmetic for multivariate Bernstein-form polynomials

Computing critical points for invariant algebraic systems