We investigate the joint moments of the 2k-th power of the characteristic polynomial of random unitary matrices with the 2h-th power of the derivative of this same polynomial. We prove that for a fixed h, the moments ...
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We investigate the joint moments of the 2k-th power of the characteristic polynomial of random unitary matrices with the 2h-th power of the derivative of this same polynomial. We prove that for a fixed h, the moments are given by rational functions of k, up to a well-known factor that already arises when h = 0. We fully describe the denominator in those rational functions (this had already been done by Hughes experimentally), and define the numerators through various formulas, mostly sums over partitions. We also use this to formulate conjectures on joint moments of the zeta function and its derivatives, or even the same questions for the Hardy function, if we use a "real" version of characteristic polynomials. Our methods should easily be applied to other similar problems, for instance with higher derivatives of characteristic polynomials. More data and computer programs are available as expanded content.
提出了基于AFS(Axiomatic Fuzzy Set)理论的模糊聚类分析算法(FCA_AFS),并且给出了聚类有效性指标。该指标能够判断合理的聚类数,而且能给出达到最高准确率的参数值。与其他算法比较:FCA_AFS算法主要通过模糊概念及其逻辑运算求出描述每类特征的模糊集,然后用这些具有确切语义的模糊集来确定每个样本归属的类。规避了其他模糊聚类算法涉及的复杂优化问题,同时不需要事先给出聚类数。在著名数据集-Iris、Wine、Wisconsin Breast Cancer的应用说明该算法实用、有效。
We consider monic (with higher coefficient 1) polynomials of fixed degree n over an arbitrary finite field GF(q), where q >= 2 is a prime number or a power of a prime number. It is assumed that on the set F-n = {f(...
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We consider monic (with higher coefficient 1) polynomials of fixed degree n over an arbitrary finite field GF(q), where q >= 2 is a prime number or a power of a prime number. It is assumed that on the set F-n = {f(n)} of all q(n) such polynomials the uniform measure is defined which assigns the probability q n to each polynomial. For an arbitrary polynomial f(n) is an element of F-n, its local structure K-n = K(f(n)) is defined as the set of multiplicities of all irreducible factors in the canonical decomposition of fn, and various structural characteristics of a polynomial (its exact and asymptotic as n -> infinity distributions)
We have constructed one-hidden-layer neural networks capable of approximating polynomials and their derivatives simultaneously. Generally, optimizing neural network parameters to be trained at later steps of the BP tr...
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We have constructed one-hidden-layer neural networks capable of approximating polynomials and their derivatives simultaneously. Generally, optimizing neural network parameters to be trained at later steps of the BP training is more difficult than optimizing those to be trained at the first step. Taking into account this fact, we suppressed the number of parameters of the former type. We measure degree of approximation in both the uniform norm on compact sets and the LP-norm on the whole space with respect to probability measures.
A real polynomial P(X-1,..., X-n) sign represents f : A(n) -> {0, 1} if for every (a(1),..., a(n)) is an element of A(n), the sign of P(a(1),..., a(n)) equals (-1) (f(a1,..., an)). Such sign representations are wel...
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A real polynomial P(X-1,..., X-n) sign represents f : A(n) -> {0, 1} if for every (a(1),..., a(n)) is an element of A(n), the sign of P(a(1),..., a(n)) equals (-1) (f(a1,..., an)). Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. The work in this area aims to determine the minimum degree and sparsity possible for a polynomial that sign represents a function f. While the degree of such polynomials is relatively well-understood, far less is known about their sparsity. Known bounds apply only to the cases where A = {0, 1} or A = {-1, +1}. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set A. We show that sign representing parity over {0,..., m-1}(n) with the degree in each variable at most m-1 requires sparsity at least m(n). We show that a tradeoff exists between sparsity and degree, by exhibiting a sign representation that has higher degree but lower sparsity. We show a lower bound of n(m-2) + 1 on the sparsity of polynomials of any degree representing parity over {0,..., m-1}(n). We prove exact bounds on the sparsity of such polynomials for any two element subset A. The main tool used is Descartes' Rule of Signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. As an application, we use bounds on sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with a Threshold Gate at the top.
Starting from previous results concerning determinants and permanents of (0, 1) circulant matrices, and using theory on finite fields with characteristic 2, we first see (denoting by A the n x n (0, 1) circulant matri...
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Starting from previous results concerning determinants and permanents of (0, 1) circulant matrices, and using theory on finite fields with characteristic 2, we first see (denoting by A the n x n (0, 1) circulant matrix whose 1's in the first row are exactly in positions i(1), i(2),..., i(k)) that Per(A) is even iff gcd(x(i1) + x(i2) +... + x(ik), 1 + x(n)) not equal 1 in Z(2)[x]. We then derive some consequences for specific classes of primes n: in particular, when n is a prime Such that the group Z(n)* is generated by the residue class 2, we see that for k not equal n n and k odd the value Per(A) is always odd. Similarly, for n > 7 and n prime with the group Of Squares in Z(n)* generated by the class 2, we see that for k = 3 the value Per(A) is always odd. Contrary to such cases, we find that when n is a Mersenne prime greater than 3, there are a significant number of circulants A of the above form with k odd and Per(A) even.
Social Choice theory (SCT, see [2] for an introduction) studies social aggregation problems, i.e., the problem of aggregating individual choices, preferences, opinions, judgments, etc. into a group choice, preference,...
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This short survey of moment invariants points out interesting clues. Moments offer a sound theoretical framework for solving the generic problems encountered in many imaging applications. The diverse families of ortho...
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This short survey of moment invariants points out interesting clues. Moments offer a sound theoretical framework for solving the generic problems encountered in many imaging applications. The diverse families of orthogonal moments provide the flexibility that may be required to face a particular target. However, they have to satisfy the time computation constraints that are inherent in many applications. Suk and Flusser proposed a solution of simultaneously dealing with affine transformation and blur (with centrosymmetric PSF) for pattern recognition, template matching, and image registration. Flusser and Zivota suggested a set of combined moments that are invariant to both rotation and blurring. Based on complex moments, Liu and Zhang derived a subset of moment features that are not affected by image blurring and geometric transformation such as translation, scale, and rotation. All these works, however, point out the problems related to the number of invariants to be selected, the choice of the size of the region of interest where moments are computed, and the dependence with object features (i.e., symmetry).
This expanded new edition presents a thorough and up-to-date introduction to the study of linear algebra Linear algebra, Third Edition provides a unified introduction to linear algebra while reinforcing and emphasizin...
ISBN:
(纸本)9780470178843
This expanded new edition presents a thorough and up-to-date introduction to the study of linear algebra Linear algebra, Third Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, the book successfully helps readers to understand not only how to implement a technique, but why its use is important. The book outlines an analytical, algebraic, and geometric discussion of the provided definitions, theorems, and proofs. For each concept, an abstract foundation is presented together with its computational output, and this parallel structure clearly and immediately illustrates the relationship between the theory and its appropriate applications. The Third Edition also features: * A new chapter on generalized eigenvectors and chain bases with coverage of the Jordan form and the Cayley-Hamilton theorem * A new chapter on numerical techniques, including a discussion of the condition number * A new section on Hermitian symmetric and unitary matrices * An exploration of computational approaches to finding eigenvalues, such as the forward iteration, reverse iteration, and the QR method * Additional exercises that consist of application, numerical, and conceptual questions as well as true-false questions Illuminating applications of linear algebra are provided throughout most parts of the book along with self-study questions that allow the reader to replicate the treatments independently of the book. Each chapter concludes with a summary of key points, and most topics are accompanied by a "computer Projects" section, which contains worked-out exercises that utilize the most up-to-date version of MATLAB(r). A related Web site features Maple translations of these exercises as well as additional supplemental material. Linear algebra, Third Edition is an excellent undergraduate-level textbook for c
The proceedings contain 16 papers. The topics discussed include: interpolation of the double discrete logarithm;finite Dedekind sums;transitive q-Ary functions over finite fields, or finite sets: counts, properties an...
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ISBN:
(纸本)3540694986
The proceedings contain 16 papers. The topics discussed include: interpolation of the double discrete logarithm;finite Dedekind sums;transitive q-Ary functions over finite fields, or finite sets: counts, properties and applications;fast point multiplication on elliptic curves without precomputation;efficient finite fields in the Maxima computeralgebra system;subquadratic space complexity multiplication over binary fields with Dickson polynomial representation;digit-serial structures for the shifted polynomial basis multiplication over binary extension fields;some theorems on planar mappings;on the number of two-weight cyclic codes with composite parity-check polynomials;on field size and success probability in network coding;and Montgomery ladder for all Genus 2 curves in characteristic 2.
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