Joint decomposition of Boolean functions on the basis of a solution of logic equations

It is proposed that the apparatus of logic equations be used to achieve an exact solution of the problem of joint decomposition of a system of Boolean functions with respect to a two-block partitioning of the set of arguments.     

Modeling orientation fields of fingerprints with rational complex functions

In this paper, a novel model is proposed for the orientation field of fingerprints, which can be expressed as the argument of a rational complex function. It is suitable for all types of fingerprints. Experimental results show that it performs much better than the previous works.     

Sparse interpolation of multivariate rational functions

Consider the black box interpolation of a τ-sparse, n-variate rational function f, where τ is the maximum number of terms in either numerator or denominator. When numerator and denominator are at most of degree d, then the number of possible terms in f is O(dⁿ) and explodes exponentially as the number of variables increases. The complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore. It still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree. We remove the exponent n and introduce the sparsity τ in the complexity by reconstructing the auxiliary function’s coefficients via sparse multivariate interpolation.The approach is new and builds on the normalization of the rational function’s representation. Our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment. The latter is illustrated with several examples, running from exact finite field arithmetic to noisy floating point evaluations. In general, the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation. If the early termination Ben-Or/Tiwari algorithm is used, our method achieves rational interpolation in O(τd) black box evaluations and thus is sensitive to the sparsity of the multivariate f.     

Regional stabilisation of polynomial non-linear systems using rational Lyapunov functions

In this article, we propose a new non-linear stabilisation approach based on the popular linear parameter-varying control techniques. The regional state-feedback control problem of polynomial non-linear systems will be studied using rational Lyapunov functions of states. By bounding the variation rates of each state, the domain of attraction will be embedded in the region specified by the non-linear vector field. As a result, the state-feedback stabilisation conditions will be formulated as a set of polynomial matrix inequalities and can be solved efficiently by sum-of-squares programming. The resulting Lyapunov matrix and state-feedback gains are typically state-dependent rational matrix functions. This approach is also extended to a class of output-dependent non-linear systems where the stabilising output-feedback controller can be synthesised using rational Lyapunov functions of outputs. Finally, several examples will be used to demonstrate the proposed stabilisation approach and clarify the effect of various choices of Lyapunov function forms and state constraints.     

Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

The class of HDT0L sequences is closed with respect to rational functions

We show that the class of HDT0L sequences is closed with respect to total rational functions.     

Symbolic additive decomposition of rational functions

The problem of additive decomposition of rational functions minimizing the degree of the remainders denominator with the additional minimization of the degree of the remainders numerator is considered. An algorithm that solves this problem under certain additional assumptions is proposed. This algorithm is implemented in Maple.Translated from Programmirovanie, Vol. 31, No. 2, 2005.Original Russian Text Copyright © 2005 by Polyakov.     

Counting with rational generating functions

We examine two different ways of encoding a counting function: as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations. Examples of such counting functions include Ehrhart quasi-polynomials, vector partition functions, integer points in parametric polytopes, and projections of the integer points in parametric polytopes. For this last example, this algorithm provides the first known way to compute the explicit function in polynomial time. We rely heavily on results by Barvinok and Pommersheim [Barvinok, A., Pommersheim, J., 1999. An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97). In: Math. Sci. Res. Inst. Publ., vol. 38. Cambridge Univ. Press, Cambridge, pp. 91–147], and also by Verdoolaege et al. [Verdoolaege, S., Seghir, R., Beyls, K., Loechner, V., Bruynooghe, M., 2007. Counting integer points in parametric polytopes using Barvinok’s rational functions, Algorithmica 48 (1), 37–66].     

函数分段有理三次Bézier插值

根据函数的几何性质,对函数进行适当分段。定义了函数的分段三角形凸包,提出了一种控制顶点和权因子的确定方案。详细地讨论了函数的分段有理三次Bézier插值算法,定义了一种便于计算的新型误差。插值函数保持了原始函数的重要几何性质,如单调性、凹凸性、G1连续性。最后以数值实验结果表明了算法的有效性和可行性,该算法提供了函数近似表示的一条有效途径。     

On the revision of preferences and rational inference processes

Orderings and inference relations can be successfully used to model the behavior of a rational agent. This behavior is indeed represented either by a set of ordered pairs that reflect the agent's preferences, or by a rational inference relation that describes the agent's internal logics. In the finite case where we work, both structures admit a simple representation by means of logical chains. The problem of revising such inference processes arises when it appears necessary to modify the original model in order to take into account new facts about the agent's behavior. How is it then possible to perform the desired modification? We study here the possibilities offered by the technique of ‘chain revision’ which appears to be the easiest way to treat this kind of problem: the revision is performed through a simple modification of the logical chain attached to the agent's behavior, and the revision problem boils down to adding, retracting or modifying some of the links of the original chain. This perspective permits an effective treatment of the problems of both simple and multiple revision. The technique developed can also be used in some limiting cases, when the agent's inference process is only partially known, encoded by an incomplete set of preferences or a conditional knowledge base.     

Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.     

Representation of individual differences in rectangular proximity data through anti-Q matrix decomposition

Decomposition of a rectangular proximity matrix into a sum of equal-sized matrices, each constrained to display a certain order pattern, called an anti-Q form, can be interpreted as a less restrictive analogue of singular value decomposition. Both decomposition techniques share the ultimate goal of identifying a parsimonious representation of the original matrix in the form of an approximation through a small sum of components. The specific patterning of the extracted anti-Q matrices lends to subsequent analyses steps (by treating each anti-Q component as a separate proximity matrix), and representations as a discrete two-mode ultrametric or a continuous unidimensional unfolding. Because both models entail the same number of estimated weights, a direct comparison of their fit values can be carried out. Thus, for each extracted anti-Q matrix we can distinguish whether a categorical (discrete) or a dimensional structure provides the better representation. A generalization of anti-Q decomposition is proposed to a cube formed by rectangular proximity matrices observed from multiple data sources, and therefore, as a way of representing individual differences. In addressing this task within a ‘deviation-from-the-mean paradigm’, the individual proximity matrices are decomposed against a reference structure derived from the aggregate body of data. Assessment of overall agreement with the reference structure for each data source, as well as discrete and continuous representations fit to each extracted confirmatory anti-Q matrix, provide a detailed inter- and intra-individual analysis of the predominate characterization of the relationships between row and column objects. As an illustrative application, we analyze criterion-based rankings given by males and females for various contraceptive measures.     

On the limits of cache-oblivious rational permutations

Permuting a vector is a fundamental primitive which arises in many applications. In particular, rational permutations, which are defined by permutations of the bits of the binary representations of the vector indices, are widely used. Matrix transposition and bit-reversal are notable examples of rational permutations. In this paper we contribute a number of results regarding the execution of these permutations in cache hierarchies, with particular emphasis on the cache-oblivious setting. We first bound from below the work needed to execute a rational permutation with an optimal cache complexity. Then, we develop a cache-oblivious algorithm to perform any rational permutation, which exhibits optimal work and cache complexities under the tall cache assumption. We finally show that for certain families of rational permutations (including matrix transposition and bit reversal) no cache-oblivious algorithm can exhibit optimal cache complexity for all values of the cache parameters. This latter result specializes the one proved by Brodal and Fagerberg for general permutations to the case of rational permutations, and provides further evidence that the tall cache assumption is often necessary to attain cache optimality in the context of cache-oblivious algorithms.     

Rate of convergence of schmidt pairs and rational functions corresponding to best approximants of truncated hankel operators

The problem of approximating Hankel operators of infinite rank by finite-rank Hankel operators is considered. For efficiency, truncated infinite Hankel matrices ⁿ of are utilized. In this paper for any compact Hankel operator of the Wiener class, we derive the rate of l₂-convergence of the Schmidt pairs of ⁿ to the corresponding Schmidt pairs of . For a certain subclass of Hankel operators of the Wiener class, we also obtain the rate of l₁-convergence. In addition, an upper bound for the rate of uniform convergence of the rational symbols of best rank-k Hankel approximants of ⁿ to the corresponding rational symbol of the best rank-k Hankel approximant to asn is derived.Supported by SDIO/IST managed by the U.S. Army under Contract No. DAAL03-87-K-0025 and also supported by the National Science Foundation under Grant No. DMS 89-01345.     

On the scalar rational interpolation problem

A new technique is derived for determining a parametrization of all minimal complexity rational functionsa(x)/b(x) interpolating an arbitrary sequence of points. Complexity is measured in terms of max{deg(a), deg(b) +r } wherer is an arbitrary integer (so thatr=0 corresponds to the McMillan degree). Our construction uses Gröbner bases of submodules of the free module of rank 2 over the polynomial ring in one variable and extends previous work on the key equation of error control coding theory.     

State space realizations of rational interpolants with prescribed poles

We give a generic algorithm for computing rational interpolants with prescribed poles. The resulting rational function is expressed in the so-called Newton form. State space realizations for this expression of rational functions are given. Our main tool for finding state space realizations is Fuhrmann's shift realization theory from which we obtain concrete realizations by introducing suitable bases of the state space and expressing the abstract operators with respect to these bases in matrix form.     

On modeling with rational ringed surfaces

A surface in Euclidean space is called ringed (or cyclic) if there exists a one-parameter family of planes that intersects this surface in circles. Well-known examples of ringed surfaces are the surfaces of revolution, (not only rotational) quadrics, canal surfaces, or Darboux cyclides. This paper focuses on modeling with rational ringed surfaces, mainly for blending purposes. We will deal with the question of rationality of ringed surfaces and discuss the usefulness of the so called P-curves for constructing rational ringed-surface-blends. The method of constructing blending surfaces that satisfy certain prescribed constraints, e.g. a necessity to avoid some obstacles, will be presented. The designed approach can be easily modified also for computing $n$-way blends. In addition, we will study the contour curves on ringed surfaces and use them for computing approximate parameterizations of implicitly given blends by ringed surfaces. The designed techniques and their implementations are verified on several examples.     

Behaviors and symbols of rational matrices

Recently, we have defined rational differential equations, and derived conditions when two such equations have the same solution set. In this article, we study the same question, but present a different approach based on the new notion of symbols. Also we try to describe the relationship with the work “Behaviors defined by rational functions” of Willems and Yamamoto.