A nice labelling for tree-like event structures of degree 3

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the index of an event structure of degree 3 is bounded by a linear function of the height. The main theorem of the paper states that event structures of degree 3 whose causality order is a tree have a nice labelling with 3 colors. We exemplify how to use this theorem to construct upper bounds for the index of other event structures of degree 3.     

Trading order for degree in creative telescoping

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms.     

Counting Hexagonal Patches and Independent Sets in Circle Graphs

A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the enumeration of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem. It is also shown how to subsequently generate hexagonal patches.     

Smolyak's Construction of Cubature Formulas of Arbitrary Trigonometric Degree

We study cubature formulas for d-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degree in dimension d, we need about function values if d is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare different methods to obtain formulas with high trigonometric degree. Received: April 8, 1998     

Selective Degree Elevation for Multi‐Sided Bézier Patches

This paper presents a method to selectively elevate the degree of an S‐Patch of arbitrary dimension. We consider not only S‐Patches with 2D domains but 3D and higher‐dimensional domains as well, of which volumetric cage deformations are a subset. We show how to selectively insert control points of a higher degree patch into a lower degree patch while maintaining the polynomial reproduction order of the original patch. This process allows the user to elevate the degree of only one portion of the patch to add new degrees of freedom or maintain continuity with adjacent patches without elevating the degree of the entire patch, which could create far more degrees of freedom than necessary. Finally we show an application to cage‐based deformations where we increase the number of control points by elevating the degree of a subset of cage faces. The result is a cage deformation with higher degree triangular Bézier functions on a subset of cage faces but no interior control points.     

Multiple View Geometry of General Algebraic Curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus.We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d – 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.     

Identities for the Associator in Alternative Algebras

The associator is an alternating trilinear product for any alternative algebra. We study this trilinear product in three related algebras: the associator in a free alternative algebra, the associator in the Cayley algebra, and the ternary cross product on four-dimensional space. This last example is isomorphic to the ternary subalgebra of the Cayley algebra which is spanned by the non-quaternion basis elements. We determine the identities of degree  ≤  7 satisfied by these three ternary algebras. We discover two new identities in degree 7 satisfied by the associator in every alternative algebra and five new identities in degree 7 satisfied by the associator in the Cayley algebra. For the ternary cross product we recover the ternary derivation identity in degree 5 introduced by Filippov.     

Structural Properties of Recurrent Neural Networks

In this article we research the impact of the adaptive learning process of recurrent neural networks (RNN) on the structural properties of the derived graphs. A trained fully connected RNN can be converted to a graph by defining edges between pairs od nodes having significant weights. We measured structural properties of the derived graphs, such as characteristic path lengths, clustering coefficients and degree distributions. The results imply that a trained RNN has significantly larger clustering coefficient than a random network with a comparable connectivity. Besides, the degree distributions show existence of nodes with a large degree or hubs, typical for scale-free networks. We also show analytically and experimentally that this type of degree distribution has increased entropy.     

Relativized Degree Spectra

We present a relativized version of the notion of a degree spectrumof a structure with respect to finitely many abstract structures.We study the connection to the notion of joint spectrum. Weprove that some properties of the degree spectrum as a minimalpair theorem and the existence of quasi-minimal degrees aretrue for the relative spectrum.     

基于粗隶属函数的粗糙集模型的推广

通过粗隶属函数,将粗糙集理论与模糊集理论联系起来,建立一种粗糙集理论与模糊集理论间的关系。把粗隶属函数视为论域上的一个特殊模糊集,用它的!-截集和强"-截集的概念,将经典粗糙集模型进行推广,提出基于等价关系的隶属度粗糙集模型,验证一些有用的性质,并证明该模型比Pawlak粗糙集模型具有更好的精度。最后将基于等价关系的隶属度粗糙集模型拓展到基于一般二元关系的广义隶属度粗糙集模型,并给出其相应的性质。     

Prioritized aggregation operators based on the priority degrees in multicriteria decision-making

We consider the multicriteria decision-making (MCDM) problems where there exists a prioritization relationship over the criteria. We introduce the concept of the priority degree. Then we give three kinds of prioritized aggregation operators based on the priority degrees: the prioritized averaging operator with the priority degrees, the prioritized scoring operator with the priority degrees, and the prioritized ordered weighted averaging operator with the priority degrees. Some desired properties of these prioritized aggregation operators are also investigated. The priority degree plays an important role in the prioritized MCDM problems. We also investigate how to select a proper priority degree according to the giving decision information. By using an illustrative example, we show that the prioritized aggregation operators based on the priority degrees provide the decision-makers more choices and they are more flexible in the process of decision-making.     

一类H布尔函数的代数次数、相关免疫性与代数免疫性的关系

以布尔函数的导数和自定义的e-导数为研究工具,研究了一类特定Hamming重量的H布尔函数的代数次数、代数免疫性、相关免疫性之间的关联问题.得出H布尔函数的组成部分e-导数的代数次数决定了H布尔函数的代数次数;H布尔函数的e-导数与H布尔函数的代数免疫阶的大小紧密关联;H布尔函数的e-导数可将H布尔函数的代数免疫性、零化子、相关免疫性、代数次数联系到一起等.同时,导出了公式法和级联法两类求解H布尔函数最低代数次数零化子的不同方法.     

A new parameterization of stable polynomials

In this note, we develop a new characterization of stable polynomials. Specifically, given n positive, ordered numbers (frequencies), we develop a procedure for constructing a stable degree n monic polynomial with real coefficients. This construction can be viewed as a mapping from the space of n ordered frequencies to the space of stable degree n monic polynomials. The mapping is one-one and onto, thereby giving a complete parameterization of all stable, degree n monic polynomials. We show how the result can be used to generate parameterizations of stabilizing fixed-order proper controllers for unity feedback systems. We apply these results in the development of stability margin lower bounds for systems with parameter uncertainty.     

Constructing lattice rules based on weighted degree of exactness and worst case error

Recall that an integration rule is said to have a trigonometric degree of exactness m if it integrates exactly all trigonometric polynomials of degree ≤ m. In this paper we focus on high dimensions, say, d ? 6. We introduce three notions of weighted degree of exactness, where we use weights to characterize the anisotropicness of the integrand with respect to successive coordinate directions. Unlike in the classical unweighted setting, the minimal number of integration points needed to achieve a prescribed weighted degree of exactness no longer grows exponentially with d provided that the weights decay sufficiently fast. We present a component-by-component algorithm for the construction of a rank-1 lattice rule such that (i) it has a prescribed weighted degree of exactness, and (ii) its worst case error achieves the optimal rate of convergence in a weighted Korobov space. Then we introduce a modified, more practical, version of this algorithm which maximizes the weighted degree of exactness in each step of the construction. Both algorithms are illustrated by numerical results.     

Fractional Path Coloring in Bounded Degree Trees with Applications

This paper studies the natural linear programming relaxation of the path coloring problem. We prove constructively that finding an optimal fractional path coloring is Fixed Parameter Tractable (FPT), with the degree of the tree as parameter: the fractional coloring of paths in a bounded degree trees can be done in a time which is linear in the size of the tree, quadratic in the load of the set of paths, while exponential in the degree of the tree. We give an algorithm based on the generation of an efficient polynomial size linear program. Our algorithm is able to explore in polynomial time the exponential number of different fractional colorings, thanks to the notion of trace of a coloring that we introduce. We further give an upper bound on the cost of such a coloring in binary trees and extend this algorithm to bounded degree graphs with bounded treewidth. Finally, we also show some relationships between the integral and fractional problems, and derive a 1+5/3e≈1.61—approximation algorithm for the path coloring problem in bounded degree trees, improving on existing results. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (wdm) optical networks.     

Near optimal decentralised control with a pre-specified degree of stability

In this paper we develop a method for computing near optimal decentralised control with a pre-specified degree of stability for large scale, linear, interconnected dynamical systems. All the calculations in the new method are performed off-line using a three level hierarchical structure. We provide a condition the satisfaction of which ensures that the system has a pre-specified degree of stability. We also show that the control developed using the new method is more stable than the optimal decentralised control obtained by neglecting all interactions between the subsystems.     

On convolutions of algebraic curves

We focus on the investigation of relations between plane algebraic curves and their convolution. Since the convolution of irreducible algebraic curves is not necessarily irreducible, an upper bound for the number of components is given. Then, a formula expressing the convolution degree using the algebraic degree and the genus of the curve is derived. In addition, a detailed analysis of the so-called special and degenerated components is discussed. We also present some special results for curves with low convolution degree and for rational curves, and use our results to investigate the relation with the theory of the classical offsets and Pythagorean Hodograph (PH) curves presented in Arrondo et al. (1997).     

Efficient striping techniques for variable bit rate continuous media file servers

The performance of striped disk arrays is governed by two parameters: the stripe unit size and the degree of striping. In this paper, we describe techniques for determining the stripe unit size and degree of striping for disk arrays storing variable bit rate continuous media data. We present an analytical model to determine the optimal stripe unit size in redundant and non-redundant disk arrays. We then use the model to study the effect of various system parameters on the optimal stripe unit size. To determine the degree of striping, we first demonstrate that striping a continuous media stream across all disks in the array causes the number of clients supported to increase sub-linearly with increase in the number of disks. To overcome this limitation, we propose a technique that partitions a disk array and stripes each media stream across a single partition. We then propose an analytical model to determine the optimal partition size and maximize the number of clients supported by the array.     

Recursion theory on processes

A general definition of a process is given. Recursive processes and relative recursion for processes are defined and the degree theory of processes is studied. We construct a process of minimal degree     

Semi‐sharp Creases on Subdivision Curves and Surfaces

We explore a method for generalising Pixar semi‐sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high‐degree univariate examples and explore analogous methods for subdivision surfaces.