Topic
Homogeneous polynomial
About: Homogeneous polynomial is a research topic. Over the lifetime, 2047 publications have been published within this topic receiving 27657 citations.
Papers published on a yearly basis
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Papers
TL;DR: In this paper, the authors presented an analytic description of the complex transfer function superior to that given by minimization of the "weighted" sum of the squares of the errors in magnitude.
Abstract: Experimental frequency response data obtained from a linear dynamic system is processed to obtain the transfer function as a ratio of two frequency-dependent polynomials. The difference between the absolute magnitudes of the actual function and the polynomial ratio is the error considered. The polynomial coefficients are evaluated as the result of minimizing the sum of the squares of the above errors at the experimental points. The polynomial coefficients are computed numerically using an IBM 704 FORTRAN program. The magnitude and phase angle of the transfer function are evaluated at the various frequencies, using the computed polynomial ratio; and are compared with the observed data. The method presented here gives an analytic description of the complex transfer function superior to that given by minimization of the "weighted" sum of the squares of the errors in magnitude.
744 citations
TL;DR: In this article, two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem are presented, and they are shown to be orthogonal with respect to a positive definite inner product defined over the compact interval [ − 1, 1 ] or the half-line [ 0, ∞ ), respectively.
458 citations
9 Dec 2003
TL;DR: It is shown that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence the switches to be separated by a minimum dwell time is not constraint.
Abstract: We propose an algebraic geometric solution to the identification of a class of linear hybrid systems. We show that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence we do not constraint the switches to be separated by a minimum dwell time. The decoupling is obtained from the so-called hybrid decoupling constraint, which establishes a connection between linear hybrid system identification, polynomial factorization and hyperplane clustering. In essence, we represent the number of discrete states n as the degree of a homogeneous polynomial p and the model parameters as factors of p. We then show that one can estimate n from a rank constraint on the data, the coefficients of p from a linear system, and the model parameters from the derivatives of p. The solution is closed form if and only if n/spl les/4. Once the model parameters have been identified, the estimation of the hybrid state becomes a simpler problem. Although our algorithm is designed for noiseless data, we also present simulation results with noisy data.
357 citations
3 Jan 1991
TL;DR: The authors demonstrate that any functionf whose L -norm is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions.
Abstract: This work gives apolynomial time algorithm for learning decision trees with respect to the uniform distribution (This algorithm uses membership queries) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (ie, summation of a subset of the input variables over GF(2)) This paper shows how to learn in polynomial time any function that can be approximated (in norm L2) by a polynomially sparse function (ie, a function with only polynomially many nonzero Fourier coefficients) The authors demonstrate that any functionf whose L -norm (ie, the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions Moreover, it is shown that the functions with polynomial L -norm can be learned deterministically The algorithm can also exactly identify a decision tree of depth d in time polynomial in 2 a and n This result implies that trees of logarithmic depth can be identified in polynomial time
356 citations
TL;DR: In this paper, it was shown that the size of a Kakeya set is at least the dimension of the space of polynomials of degree q 2, which is q71'1 when q is large.
Abstract: The motivation for studying Kakeya sets over finite fields is to try to better understand the more complicated questions regarding Kakeya sets in W1. A Kakeya set K C Rn is a compact set containing a line segment of unit length in every direction. The famous Kakeya Conjecture states that such sets must have Hausdorff (or Minkowski) dimension equal to n. The importance of this conjecture is partially due to the connections it has to many problems in harmonic analysis, number theory and PDE. This conjecture was proved for n = 2 [Dav71] and is open for larger values of n (we refer the reader to the survey papers [Wol99, BouOO, TaoOl] for more information). It was first suggested by Wolff [Wol99] to study finite field Kakeya sets. It was asked in [Wol99] whether there exists a lower bound of the form Cn • qn on the size of such sets in Fn. The lower bound appearing in [Wol99] was of the form Cn c(n+2)/2. This bound was further improved in [RogOl, BKT04, MT04, Tao08] both for general n and for specific small values of n (e.g. for n = 3, 4). For general n, the most current best lower bound is the one obtained in [RogOl, MT04] (based on results from [KT99]) of Cn • q4n/7 . The main technique used to show this bound is an additive number theoretic lemma relating the sizes of different sum sets of the form A+rB, where A and B are fixed sets in Fn and r ranges over several different values in F (the idea to use additive number theory in the context of Kakeya sets is due to Bourgain [Bou99]). The next theorem, proven in Section 2, gives a near-optimal bound on the size of Kakeya sets. Roughly speaking, the proof follows by observing that any degree q 2 homogeneous polynomial in F[#i, . . . , xn] can be 'reconstructed' from its value on any Kakeya set K c¥n. This implies that the size of K is at least the dimension of the space of polynomials of degree q 2, which is « q71'1 (when q is large).
311 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 6 |
| 2024 | 10 |
| 2023 | 21 |
| 2022 | 80 |
| 2021 | 80 |
| 2020 | 67 |