Let S=k[x~ ..... x,] be a polynomial ring over an infinite field k, and let 1 be a homogeneous ideal of S.An algorithm for computing the (first) syzygies of I is due independently to Spear [Spe77] and Schreyer [Sch80]: One chooses an ordering on the monomials of S, and then constructs a monomial ideal in(1) generated by the lead terms of all elements of I, in(l) can be viewed as the limit of 1 under the action of a 1-parameter subgroup of GL(n) on the Hilbert scheme [Bay 82], so in(1) occurs as the special fiber of a flat family whose general fiber is isomorphic to I. It follows from a well-known criterion for flatness [Art 76] that each syzygy of in(l) can be lifted to a syzygy of I; the set of syzygies thus obtained can be trimmed to give a complete set of minimal syzygies of I.The rnonomial ideal in(I) was first studied The following problem arises in using this syzygy algorithm in practice: in(l) can have minimal generators and syzygies in degrees higher than any minimal generator or syzygy of I. In this situation, computations in these higher degrees are unnecessary; one should compute the generators and syzygies of in(1) in only those degrees necessary to find all minimal syzygies of 1.In order to modify the syzygy algorithm to take advantage of this observation, one would like a criterion for determining when all minimal syzygies of I have been found. This problem appears to be intractable at present. However, the question of bounding the degrees of the minimal jth syzygies of I, for all j, is tractable. Recall that I is defined to be m-regular if the jth syzygy module of I is generated in degrees
Abstract.This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called ^-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of ^-trajectories and /1-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and ^-trajectories. It shows that projective transformations map ^-trajectories into f-trajectories. It presents Karmarkar's interpretation of /1-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 = ^2"_x dx¡dxj/xf restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of /1-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm.These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that /"-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesies (actually distinguished chords) of a geometry isometric to a Hubert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The /1-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesies of a geometry isometric to Euclidean geometry.
Available research analysing the playing structure in kids tennis on different scaled courts identifies a severe problem at the transition from the U9 (Orange stage) to the U10 (Green stage), which can mainly be reasoned by the enlargement of the court from a small-sided field to the full-sized court. Aware of this problem, an intermediate stage, called Lime Court (stage), between Orange and Green was introduced in Austria. The study at hand aims to compare the playing structure between the Green and the Lime Court in kids' tennis aged 9-10 years (U10). Twelve videos from matches on Lime in 2013 were analysed and compared to the results found in 2014. The playing structure was defined by 18 performance parameters. The differences in the mean values as well as one-way ANOVA were calculated between the groups. The results found in the study lead to the conclusion that the Lime Court enables children to play more similar to elite players than the Green Court. Thus, Lime closes the existing gap between the Tennis10s stage Orange and Green and should be used for 10-year old tennis players in order to properly develop their playing skills.
Abstract.Karmarkar's projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called P-trajectories, as well as a related set of trajectories, called A-trajectories. The /1-trajectories arise from another linear programming algorithm, the affine scaling algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively.These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canonical form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center.The .4-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible solutions is bounded, they are the complete set of geodesies of a Riemannian geometry isometric to Euclidean geometry. Each /1-trajectory is part of a real algebraic curve.Each P-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The P-trajectory through 0 in Legendre transform coordinates, called the central P-trajectory, is part of a straight line, and is contained in the ^-trajectory through 0 , called the central A-trajectory.Each /"-trajectory is part of a real algebraic curve. The central ^-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form (c, x) = p . It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central .4-trajectory, which is also the central P-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar's algorithm.-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point Xoo of a central ^-trajectory on the boundary of the feasible solution polytope P is the center of the unique face of P containing Xoo in its relative interior.Received by the editors October 8, 1986 and, in revised form, June 9, 1987 and March 25, 1988. 1980 Mathematics Subject Classification (1985. Primary 90C05; Secondary 52A40, 34A34.The first author was partially supported by ONR contract N00014-87-K0214. The central trajectory of a combined primal-dual linear program has a simple set of polynomial relations determining it as an algebraic curve. These relations are a relaxed form of the complementary slackness ...
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