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Symbolic integration of polynomial functions over a linear polyhedron in Euclidean three-dimensional space

International Journal for Numerical Methods in Biomedical Engineering 1996年第8期12卷

作者： H. T. Rathod H. S. Govinda Rao S. V. Hiremath Department of Mathematics Central College Campus Bangalore University Bangalore 560 001 India Department of Mathematics Dr Ambedkar Institute of Technology Bangalore 560 056 India

The paper concerns analytical integration of polynomial functions over linear polyhedra in three-dimensional space. To the authors' knowledge this is a first presentation of the analytical integration of monomials over a tetrahedral solid in 3D space. A linear polyhedron can be obtained by decomposing it into a set of solid tetrahedrons, but the division of a linear polyhedral solid in 3D space into tetrahedra sometimes presents difficulties of visualization and could easily lead to errors in nodal numbering, etc We have taken this into account and also the linearity property of integration to derive a symbolic integration formula for linear hexahedra in 3D space. We have also used yet another fact that a hexahedron could be built up in two, and only two, distinct ways from five tetrahedral shaped elements These symbolic integration formulas are then followed by an illustrative numerical example for a rectangular prism element, which clearly verifies the formulas derived for the tetrahedron and hexahedron elements.

关键词： linear polyhedra symbolic integration polynomial functions monomials tetrahedron hexahedron

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polynomial Implicit Neural Framework for Promoting Shape Awareness in Generative Models

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INTERNATIONAL JOURNAL OF COMPUTER VISION 2025年第5期133卷 2967-2995页

作者： Nath, Utkarsh Singh, Rajhans Shukla, Ankita Kulkarni, Kuldeep Turaga, Pavan Arizona State Univ Sch Comp & Augmented Intelligence Tempe AZ 85281 USA Arizona State Univ Sch Elect Comp & Energy Engn Tempe AZ USA Arizona State Univ Sch Arts Media & Engn Geometr Media Lab Tempe AZ 85281 USA Univ Nevada Comp Sci & Engn Reno NV USA Adobe Res Bangalore Karnataka India

polynomial functions have been employed to represent shape-related information in 2D and 3D computer vision, even from the very early days of the field. In this paper, we present a framework using polynomial-type basis functions to promote shape awareness in contemporary generative architectures. The benefits of using a learnable form of polynomial basis functions as drop-in modules into generative architectures are several-including promoting shape awareness, a noticeable disentanglement of shape from texture, and high quality generation. To enable the architectures to have a small number of parameters, we further use implicit neural representations (INR) as the base architecture. Most INR architectures rely on sinusoidal positional encoding, which accounts for high-frequency information in data. However, the finite encoding size restricts the model's representational power. Higher representational power is critically needed to transition from representing a single given image to effectively representing large and diverse datasets. Our approach addresses this gap by representing an image with a polynomial function and eliminates the need for positional encodings. Therefore, to achieve a progressively higher degree of polynomial representation, we use element-wise multiplications between features and affine-transformed coordinate locations after every ReLU layer. The proposed method is evaluated qualitatively and quantitatively on large datasets such as ImageNet. The proposed Poly-INR model performs comparably to state-of-the-art generative models without any convolution, normalization, or self-attention layers, and with significantly fewer trainable parameters. With substantially fewer training parameters and higher representative power, our approach paves the way for broader adoption of INR models for generative modeling tasks in complex domains. The code is publicly available at https://***/Rajhans0/Poly_INR.

关键词： Generative models polynomial functions Shape awareness Implicit models

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Value sets of non-permutation polynomials over the residue class rings of integers

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RAMANUJAN JOURNAL 2025年第2期67卷 1-10页

作者： Shang, Shikui Shanghai Univ Dept Math Shanghai 200444 Peoples R China

In this paper, we study the value sets of non-permutation polynomial functions over the residue class ring Z/mZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}/m\mathbb {Z}$$\end{document}. When m=pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=p<^>r$$\end{document} is a power of some prime p, an upper bound is given for the size of the value set of a polynomial function which is not a permutation. We also show that this upper bound can be achieved by some integral polynomials. Finally, we generalize the results for any positive integer m with known prime decomposition.

关键词： polynomial functions Value set Residue ring of integers

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ON THE DIFFERENCE PROPERTY OF HIGHER ORDERS FOR DIFFERENTIABLE functions

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BANACH JOURNAL OF MATHEMATICAL ANALYSIS 2015年第3期9卷 85-97页

作者： Adam, Marcin Silesian Tech Univ Inst Math PL-44100 Gliwice Poland

In this paper, inspired by some results concerning the double difference property, we show that the class C-p(R, R) of p- times continuously differentiable functions has the difference property of p- th order, i.e. if a function f : R -> R is such that Delta(p)(h) f is an element of C-p (R x R,R), where Delta(p)(h) f is the p-th iterate of the well- known difference operator Delta(h)f(x) := f( x + h) - f( x), then there exists a polynomial function Gamma(p-1) : R -> R of ( p - 1)-th order such that f - Gamma(p-1) is an element of C-p( R, R). Moreover, some new equalities connected with the difference operator are also presented.

关键词： Difference property difference operator polynomial functions

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As Bad as it Gets: How Climate Damage functions Affect Growth and the Social Cost of Carbon (vol 72, pg 5, 2019)

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ENVIRONMENTAL & RESOURCE ECONOMICS 2019年第1期72卷 27-27页

作者： Bretschger, Lucas Pattakou, Aimilia Eidgenoss TH Zurich Zurich Switzerland

The original article displays figure errors caused by miscommunication during the proofing stage.

关键词： Climate damages Social cost of carbon Endogenous growth polynomial functions

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Finiteness of the nearring of congruence preserving and 0-preserving functions of an expanded group

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ACTA SCIENTIARUM MATHEMATICARUM 2018年第3-4期84卷 401-411页

作者： Peterson, Gary L. Scott, Stuart D. James Madison Univ Dept Math & Stat MSC 1911 Harrisonburg VA 22807 USA Univ Auckland Dept Math Auckland New Zealand

In this paper we shall obtain that the nearring C-0(V) of congruence preserving functions that are 0-preserving of a tame N-module V of a nearring N is finite when N is finite. As a consequence, C-0(V) of an expanded group < V, +, F > is finite when the nearring of 0-preserving polynomial functions P-0(V) of < V, +, F > is finite. We then go on to obtain further consequences of this result.

关键词： nearring expanded group tame module polynomial functions congruence preserving functions endomorphism nearring

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Upper bounds on algebraic immunity of boolean power functions

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13th International Workshop on Fast Software Encryption

作者： Nawaz, Yassir Gong, Guang Gupta, Kishan Chand Univ Waterloo Dept Elect & Comp Engn Waterloo ON N2L 3G1 Canada Univ Waterloo Ctr Appl Cryptog Res Waterloo ON N2L 3G1 Canada

ISBN: (纸本)3540365974

Algebraic attacks have received a lot of attention in studying security of symmetric ciphers. The function used in a symmetric cipher should have high algebraic immunity (A-T) to resist algebraic attacks. In this paper we are interested in finding A-T of Boolean power functions. We give an upper bound on the AI of any Boolean power function and a formula to find its corresponding low degree multiples. We prove that the upper bound on the A-T for Boolean power functions with Inverse, Kasami and Niho exponents are [root n] + [n/[root n]] - 2, [root n] + [n/[root n]] and [root n] + [n/[root n]] respectively. We also generalize this idea to Boolean polynomial functions. All existing algorithms to determine All and corresponding low degree multiples become too complex if the function has more than 25 variables. In our approach no algorithm is required. The A-I and low degree multiples can be obtained directly from the given formula.

关键词： algebraic attacks algebraic immunity inverse exponent Kasami exponent polynomial functions power functions Niho exponent

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On Types of Isolated KKT Points in polynomial Optimization

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Journal of Systems Science & Complexity 2023年第5期36卷 2186-2213页

作者： GUO Feng JIAO Liguo KIM Do Sang PHAM Tien-Son School of Mathematical Sciences Dalian University of TechnologyDalian 116024China Academy for Advanced Interdisciplinary Studies Northeast Normal UniversityChangchun 130024China Department of Applied Mathematics Pukyong National UniversityBusan 48513South Korea Department of Mathematics Dalat UniversityDalat 96300Vietnam.

Let f be a real polynomial function with n variables and S be a basic closed semialgebraic set in R^(n).In this paper,the authors are interested in the problem of identifying the type(local minimizer,maximizer or not extremum point)of a given isolated KKT point x^(*)of f over *** this end,the authors investigate some properties of the tangency variety of f on S at x^(*),by which the authors introduce the definition of faithful radius of f over S at x^(*).Then,the authors show that the type of x^(*)can be determined by the global extrema of f over the intersection of S and the Euclidean ball centered at x^(*)with a faithful ***,the authors propose an algorithm involving algebraic computations to compute a faithful radius of x*and determine its type.

关键词： Faithful radii KKT points polynomial functions tangency varieties types

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Simulation bounds for equivalence verification of polynomial datapaths using finite ring algebra

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IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 2008年第4期16卷 376-387页

作者： Shekhar, Namrata Kalla, Priyank Meredith, M. Brandon Enescu, Florian Univ Utah Dept Elect & Comp Engn Salt Lake City UT 84112 USA Georgia State Univ Dept Math & Stat Atlanta GA 30303 USA

This paper addresses simulation-based verification of high-level [algorithmic, behavioral, or register-transfer level (RTL)] descriptions of arithmetic datapaths that perform polynomial computations over finite word-length operands. Such designs are typically found in digital signal processing (DSP) for audio/video and multimedia applications;where the word-lengths of input and output Signals (bit-vectors) are predetermined and fixed according to the desired precision. Initial descriptions of such systems are usually specified as MATLAB/C code. These are then automatically translated into behavioral/RTL descriptions for subsequent hardware synthesis. In order to verify that the initial MATLAB/C model is bit-true equivalent to the translated RTL, how many simulation vectors need to be applied? This paper derives some important results that show that exhaustive simulation is not necessary to prove/disprove their equivalence. To derive these results, we model the datapath computations as polynomial functions over finite integer rings of the form Z(2)(rn), where m corresponds to the bit-vector word-length. Subsequently, by exploring some number theoretic and algebraic properties of these rings, we derive an upper bound on the number of simulation vectors required to prove equivalence or to identify bugs. Moreover, these vectors cannot be arbitrarily generated. We identify exactly those vectors that need to be simulated. Experiments are performed within practical computer-aided design (CAD) settings to demonstrate the validity and applicability of these results.

关键词： bit-vector arithmetic equivalence checking finite integer rings polynomial functions simulation-based verification

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False determinations of chaos in short noisy time series

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PHYSICA D-NONLINEAR PHENOMENA 2003年第1-2期180卷 115-127页

作者： Hunt, HW Antle, JM Paustian, K Colorado State Univ Nat Resource Ecol Lab Ft Collins CO 80523 USA Montana State Univ Dept Agr Econ & Econ Bozeman MT 59717 USA

A method (NEMG) proposed in 1992 for diagnosing chaos in noisy time series with 50 or fewer observations entails fitting the time series with an empirical function which predicts an observation in the series from previous observations, and then estimating the rate of divergence or convergence of two nearby trajectories of the function (sensitive dependence on initial conditions). Workers applying NENIG have concluded that chaos exists in some natural biological populations. However, there are insufficient published tests of the reliability of NEMG using time series from biologically realistic models with numerous (>4) state variables. Also, there are unresolved technical issues for NEMG: whether or not the data should be detrended (removal of linear or quadratic trends) before analysis, and how long a sampling interval should be used to define the time series. We addressed these issues by applying NEMG to time series of 50 observations generated by four different chaotic models, ranging from simple models of physical systems to a complex ecosystem model. We also analyzed output from non-chaotic versions of each model. Previously recommended guidelines for specifying parameters affecting NEMG diagnoses of chaos were often ineffective. For the models we examined, NEMG produced high incidences of false determinations of chaos in non-chaotic time series, and of the absence of chaos in chaotic time series. Thus the method appears to be unreliable for the circumstances we examined. (C) 2003 Elsevier Science B.V All rights reserved.

关键词： autocorrelation ecological models feedforward neural networks polynomial functions simplex optimization

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