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502
Eigenvalues and invariants of tensors
, 2007
"... A tensor is represented by a supermatrix under a coordinate system. In this paper, we define Eeigenvalues and Eeigenvectors for tensors and supermatrices. By the resultant theory, we define the Echaracteristic polynomial of a tensor. An Eeigenvalue of a tensor is a root of the Echaracteristic p ..."
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Cited by 53 (22 self)
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A tensor is represented by a supermatrix under a coordinate system. In this paper, we define Eeigenvalues and Eeigenvectors for tensors and supermatrices. By the resultant theory, we define the Echaracteristic polynomial of a tensor. An Eeigenvalue of a tensor is a root of the Echaracteristic polynomial. In the regular case, a complex number is an Eeigenvalue if and only if it is a root of the Echaracteristic polynomial. We convert the Echaracteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under coordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order ndimensional tensors is a function of m and n. We denote it by d(m,n) and show that d(1, n) = 1, d(2, n) = n, d(m,2) = m for m 3 and d(m,n) mn−1 + · · · + m for m,n 3. We also define the rank of a tensor. All real eigenvectors associated with nonzero Eeigenvalues are in a subspace with dimension equal to its rank.
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
"... Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asym ..."
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Cited by 47 (17 self)
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Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morsetheoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have
On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points
 J. Symbolic Computation
, 2000
"... Techniques from algebraic geometry and commutative algebra are adopted to establish sufficient polynomial conditions for the validity of implicitization by the method of moving quadrics both for rectangular tensor product surfaces of bidegree (m; n) and for triangular surfaces of total degree n in ..."
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Cited by 47 (4 self)
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Techniques from algebraic geometry and commutative algebra are adopted to establish sufficient polynomial conditions for the validity of implicitization by the method of moving quadrics both for rectangular tensor product surfaces of bidegree (m; n) and for triangular surfaces of total degree n in the absence of base points. 1 Introduction Several years ago, Tom Sederberg introduced a new technique for finding the implicit equation of a rational surface [Sederberg & Chen 1995]. The classical method for finding the implicit equation of a rational parametric surface x = x(s; t) w(s; t) ; y = y(s; t) w(s; t) ; z = z(s; t) w(s; t) is to compute the bivariate resultant of the three polynomials: x(s; t) \Gamma x \Delta w(s; t); y(s; t) \Gamma y \Delta w(s; t); z(s; t) \Gamma z \Delta w(s; t): Unfortunately for many applications, the resultant of these three polynomials vanishes identically when the surface has base points  that is, parameter values (s 0 ; t 0 ) for which x(s 0 ;...
Asymptotic Behaviour of the Degree of Regularity of SemiRegular Polynomial Systems
 In MEGA’05, 2005. Eighth International Symposium on Effective Methods in Algebraic Geometry
"... We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] ..."
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Cited by 46 (25 self)
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We compute the asymptotic expansion of the degree of regularity for overdetermined semiregular sequences of algebraic equations. This degree implies bounds for the generic complexity of Gröbner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryptographic community. 1 Motivations and Results The worstcase complexity of Gröbner bases has been the object of extensive studies. In the most general case, it is well known after work by Mayr and Meyer that the complexity is doubly exponential in the number of variables. For subclasses of polynomial systems, the complexity may be much smaller. Of particular importance is the class of regular sequences of polynomials. There, it is known that after a generic linear change of variables the complexity of the computation for the degreereverselexicographic order is simply exponential in the number of variables. Moreover, in characteristic 0, these systems are generic. Our goal is to give similar complexity bounds for overdetermined systems, for a class of systems that we
LengthBased Attacks for Certain Group Based Encryption Rewriting Systems
, 2002
"... In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a ..."
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Cited by 45 (1 self)
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In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group.
Camera pose and calibration from 4 or 5 known 3D points
 In Proc. 7th Int. Conf. on Computer Vision
, 1999
"... We describe two direct quasilinear methods for camera pose (absolute orientation) and calibration from a single image of 4 or 5 known 3D points. They generalize the 6 point ‘Direct Linear Transform ’ method by incorporating partial prior camera knowledge, while still allowing some unknown calibratio ..."
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Cited by 42 (0 self)
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We describe two direct quasilinear methods for camera pose (absolute orientation) and calibration from a single image of 4 or 5 known 3D points. They generalize the 6 point ‘Direct Linear Transform ’ method by incorporating partial prior camera knowledge, while still allowing some unknown calibration parameters to be recovered. Only linear algebra is required, the solution is unique in nondegenerate cases, and additional points can be included for improved stability. Both methods fail for coplanar points, but we give an experimental eigendecomposition based one that handles both planar and nonplanar cases. Our methods use recent polynomial solving technology, and we give a brief summary of this. One of our aims was to try to understand the numerical behaviour of modern polynomial solvers on some relatively simple test cases, with a view to other vision applications.
Decomposition plans for geometric constraint systems
 J. Symbolic Computation
, 2001
"... A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past ..."
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Cited by 41 (1 self)
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A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)planning problem as well as several performance measures by which DRplanning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best and worstchoice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DRplanners that use two wellknown types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DRplanning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.
A minimal solution to the autocalibration of radial distortion
, 2007
"... Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special ..."
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Cited by 39 (12 self)
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Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this paper we provide a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation in nine variables. We simplify the system by eliminating six of these variables. Then, we solve the system by finding eigenvectors of an action matrix of a suitably chosen polynomial. We show how to construct the action matrix without computing complete Gröbner basis, which provides an efficient and robust solver. The quality of the solver is demonstrated on synthetic and real data. 1.
Algebraic methods in discrete analogs of the Kakeya problem
, 2008
"... Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect noncoplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line inters ..."
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Cited by 39 (2 self)
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Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect noncoplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N 3). Both our proofs are adaptations of Dvir’s argument for the finite field Kakeya problem. 1.
Higher order positive semidefinite diffusion tensor imaging
 SIAM J. Imaging Sci
"... Due to the wellknown limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) is used to characterize nonGaussian diffusion processes. One approach to analyze HARDI data is to model the apparent diffusion coefficient (ADC) with higher order diffusion tensors ..."
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Cited by 39 (24 self)
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Due to the wellknown limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) is used to characterize nonGaussian diffusion processes. One approach to analyze HARDI data is to model the apparent diffusion coefficient (ADC) with higher order diffusion tensors (HODT). The diffusivity function is positive semidefinite. In the literature, some methods have been proposed to preserve positive semidefiniteness of second order and fourth order diffusion tensors. None of them can work for arbitrary high order diffusion tensors. In this paper, we propose a comprehensive model to approximate the ADC profile by a positive semidefinite diffusion tensor of either second or higher order. We call this model PSDT (positive semidefinite diffusion tensor). PSDT is a convex optimization problem with a convex quadratic objective function constrained by the nonnegativity requirement on the smallest Zeigenvalue of the diffusivity function. The smallest Zeigenvalue is a computable measure of the extent of positive definiteness of the diffusivity function. We also propose some other invariants for the ADC profile analysis. Experiment results show that higher order tensors could improve the estimation of anisotropic diffusion and the PSDT model can