Showing papers on "Affine transformation published in 1986"

Quantum $R$ matrix for the generalized Toda system

Abstract: We report the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.

Complete affine hypersurfaces. Part I. The completeness of affine metrics

Abstract: On etudie les proprietes des hypersurfaces invariantes sous le groupe des transformations affines unimodulaires

On some affine isoperimetric inequalities

Abstract: In [15] it was shown that a certain intermediary inequality can be combined with the Blaschke-Santalό inequality to obtain a general version of the affine isoperimetric inequality (of affine differential geometry) and, in turn, that the equality conditions of this intermediary inequality can be used to obtain the Blaschke-Santalό inequality if one starts with this general version of the affine isoperimetric inequality. It was shown in [16] that another intermediary inequality can be combined with the Petty projection inequality to obtain a general version of the Busemann-Petty centroid inequality and, in turn, the equality conditions of this intermediary inequality can be used to obtain the Petty projection inequality if one starts with the general version of the Busemann-Petty centroid inequality. The two situations are remarkably similar. The similarity between the Blaschke-Santalό inequality and the Petty projection inequality is striking. However, no similar analogy appears to exist between the affine isoperimetric inequality and the Busemann-Petty centroid inequality. One of the objects of this article is to show that such an analogy does exist. The setting for this article is Euclidean w-dimensional space, R\" (n > 2). We use Jf\" to denote the space of convex bodies (compact, convex sets with nonempty interiors) in R, endowed with the topology induced by the Hausdorff metric. The support function of a convex body K will be denoted by hκ\\ i.e.,

Affine semigroups and Cohen-Macaulay rings generated by monomials

Abstract: We give a criterion for an arbitrary ring generated by monomials to be Cohen-Macaulay in terms of certain numerical and topological properties of the additive semigroup generated by the exponents of the monomials. As a consequence, the Cohen-Macaulayness of such a ring is dependent upon the characteristic of the ground field. Introduction. Let N denote the set of nonnegative integers. By an affine semigroup we mean a finitely generated submonoid S of the additive monoid N n, where n is some positive integer. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t l , ... , tn] generated by the monomials IX = lil ••• t:n, x = (Xl"'" Xn) E S. Obviously, every subring of k[t l , ... , In] generated by a finite set of monomials is the semigroup ring of the affine semigroup in N n generated by the exponents of the monomials. So one has, up to isomorphisms, a one-to-one correspondence between affine semigroups and affine varieties which are given parametrically by finite sets of monomials. When illustrating problems of algebraic geometry one almost inevitably tends to choose varieties of this type. Even when dealing with a quite general variety, either its singularities or a certain blowup may well be defined in local coordinates by monomials [15, 19]. Moreover, one can also use rings generated by monomials to study solutions of linear equations in nonnegative integers or, equivalently, invariants of a torus acting linearly on a polynomial ring [11, 26]. Therefore, a criterion for such a ring to be Cohen-Macaulay in terms of the associated semigroup would be very useful. It should be mentioned that the first example of a non-Coh~n­ Macaulay domain (in modem language), given by F. S. Macaulay at the beginning of this century [17, p. 98], was the ring k[/t, Ith, Il/~, Ii] and that analyzing this example, Grobner [7] already posed the problem of classifying rings generated by monomials of the same degree with respect to their Cohen-Macaulayness. The first step toward such a criterion was taken by Hochster [11], who succeeded in characterizing normal rings generated by monomials in terms of the associated semigroups and showed that they are always Cohen-Macaulay. Although this result was motivated by a conjecture on rings of invariants of reductive linear algebraic groups, which was later settled [12], its proof deserved much attention. It suggested Received by the editors July 22, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 13RI0; Secondary 14M05. 145 ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 146 N. V. TRUNG AND L. T. HOA the use of topological techniques in studying rings generated by monomials. The next step was a criterion given by Goto et al. for the case that the ring has a system of parameters consisting of monomials [5]. These results inspired many other works and were re-proved many times by different techniques, such as rational resolution [15], Hilbert functions of graded algebras [25], homology of polyhedral complexes [13,26], and the Hodge algebra [10]. In both steps, one used the coincidence of an affine semigroup S with one of its extensions to indicate the Cohen-Macaulayness of k[S]. Inspired by this phenomenon, Goto and Watanabe [6] defined a suitable extension Sf of S (see below) and claimed that Sf = S is a necessary and sufficient condition for k[S] to be CohenMacaulay. Let Z and Q denote the sets of integers and rational numbers, respectively. Consider the elements of S as points in the space Qn. Let G denote the additive group in zn generated by S and put r = rankzG. Let Cs denote the convex rational polyhedral cone spanned by S in Qn. Then Cs is r-dimensional. Suppose that Fl , ... , Fm are the (r I)-dimensional faces of Cs. Let Sj denote the set of elements x E G such that x + yES for some element yES n F;, i = 1, ... , m. Then they define Sf = n?_lSj' In this paper, we shall see that the condition Sf = S is not sufficient for the Cohen-Macaulayness of k[S], and that one has to add some topological condition on the convex cone Cs to get a correct criterion. To formulate this we need some more notation. Let [1, m] denote the set of the integers 1, ... , m. For every subset J of [1, m], set GJ = n Sj \ U Sj' i~J JEJ and let 'lTJ be the simplicial complex of nonempty subsets I of J with the property n j E IS n F; *" (0). Note that one calls 'lTJ acyclic if the reduced homology group Hi'ITJ ; k) vanishes for all q ~ O. MAIN THEOREM. Let S be an arbitrary affine semigroup. Then k[S] is a CohenMacaulay (resp. Gorenstein) ring if and only if the following conditions are satisfied: (i) Sf = S (resp. there exists an element x E G such that every element of G[l,m] is the difference of x by some element of S). (ii) For every nonempty proper subset J of [1, m], GJ = 0 or 'lTJ is acyclic. It will follow from some property of the Cousin complex of k[S] and from an explicit description of all local cohomology modules of k[Sf] in terms of GJ and 'lTJ • We will also give some simple methods for checking the above conditions. As a consequence, one immediately gets the abovementioned results of Hochster and Goto et al. Another application is a criterion for the Rees algebra (blowing-up) of a ring generated by monomials to be Cohen-Macaulay (resp. Gorenstein). In particular, as in the work of Reisner on polynomial rings modulo ideals generated by square-free monomials [20] where a similar link to topology is given, we will show that the Cohen-Macaulayness of k[S] is dependent upon the characteristic of the License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AFFINE SEMIGROUPS AND COHEN-MACAULAY RINGS 147 field k. Moreover, using the main theorem, we have been able to solve Grabner's problem for some particular cases [31] (cf. [21, 29, and 30)). We would like to mention that Stanley [26] also obtained a similar result on modules associated with solutions of systems of linear diophantine equations, which overlaps with ours only in Hochster's normal case. This paper is organized in five sections. §1 gives a counterexample to the result of Goto and Watanabe via some consideration on the Rees algebras of affine semigroup rings. In §2, S' will be related to the Cousin complex of k[S] in order to show that S' = S if k[S] is a Cohen-Macaulay ring. §3 deals with the local cohomology modules of k[S']. Criteria for k[S] to be Cohen-Macaulay (resp. Gorenstein) are given in §4. There we will also deal with the Buchsbaumness of k[S]. The aim of §5 is to construct an affine semigroup ring whose local cohomology modules are just the reduced homology groups of a given finite simplicial complex. All notations introduced above will be used throughout. Moreover, if A and B are subsets of zn, G(A) denotes the additive group generated by A in zn, and A ± B is the set of elements a ± b with a E A and b E B. If x, y, ... are elements of zn, we will denote their components by Xi'Yi"'" i = 1, ... , n, respectively. For unexplained notations and standard facts in commutative algebra, algebraic topology, and local cohomology, we refer the reader to [18, 24, and 8]. ACKNOWLEDGMENT. The authors would like to thank S. Ikeda for pointing out that our earlier conclusion on the Cohen-Macaulayness of the Rees algebras of Cohen-Macaulay rings generated by monomials is false (see §1). This led us to check the result of [6, II]. Thanks are also due to S. Goto for encouraging our study, and to L. Robbiano for some useful suggestions. 1. Counterexamples to the result of Goto and Watanabe. Let S be an arbitrary affine semigroup in N n. Set S = {x E G; PX E S for some p > O}, S(i) = {x E S; Xi = A}, i = 1, ... ,n. Then we call S standard if the following conditions are satisfied: (1) S = G n N n , (2) S(i) =F S(j) for i =F j, (3) rankZG(S(i») = r 1, i = 1, ... , n. Geometrically, these conditions mean that Cs has exactly n (r 1 )-dimensional faces lying on the hyperplane Xi = 0, i = 1, ... , n. In this case, we may assume that S(i) = S n F; and Si = S S(i)' Goto and Watanabe [6, Theorem 3.3.3] claimed that if S is standard, then k[S] is a Cohen-Macaulay ring if and only if S' = S. We shall see that this is false. First, we have to remark that every affine semigroup can be transformed isomorphically onto a standard one by the following technique which is due to Hochster [11, p. 323]. HOCHSTER'S TRANSFORMATION. Let W denote the vector space generated by S in Qn. Then one can find m linear functionals 11"'" 1m from W to Q corresponding with Fl , ..• , Fm such that Cs = {x E W; li{x} ~ Oforalli}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 148 N. V. TRUNG AND L. T. HOA See also [15, p. 6]. Let T denote the linear transformation which sends every element x E W to the element (ll(X), ... , Im(x» E Qm. By replacing Ii by a suitable positive integer multiple, one can assume that T(S) ~ N m. Hochster has shown that T(S) is isomorphic to S and that T(S) = G(T(S)) n N m. Obviously, T also induces an isomorphism between the semigroups S n F; and T(S){iJ' i = 1, ... , m. Since S n F; =1= S n Fj for i =1= j, and rankzG(S n F;) = r 1, we can conclude that T(S) is a standard affine semigroup in N m. Moreover, since T(S)i = T(S) T(S)(i) = T(S S n F;) = T(Si)' we also have T(S)' = T(S'). According to this transformation, one can omit the assumption on the standardness of S in the claim of Goto and Watanabe. The counterexample will also be constructed in the nonstandard case by using the following observation. Let A: Qn -+ Q be a linear functional such that A(S) ~ N, and, if xES and A(X) = 0, then x = O. Then we call the affine semigroup E>,.:= {(x, p) E N n+1 ; xES and A(X) > p} a blowing-up extension of S. The name stems from the

A general moment-invariants/attributed-graph method for three-dimensional object recognition from a single image

Abstract: A consistent development of general moment invariants of affine transformations for two-dimensional image functions is presented. Based on this development, a new general moment-invariants/attributed-graph (MIAG) method is presented for the identification of three-dimensional objects from a single observed image using a model-matching approach. The three-dimensional location and orientation parameters of the object are also obtained as a byproduct of the matching procedure. The scheme presented allows the observed object to be partially Occluded. For identification purposes, a three-dimensional object is represented by an attributed graph describing the geometrical structure and shape of the surface bounding the object. In such a description, two-dimensional general moment invariants of the rigid planar patches (RPP) constituting the object faces are used as attributes or feature vectors which are invariant under three-dimensional motion. With this representation, the identification problem becomes a subgraph isomorphism problem between the observed image and a library model. An algorithm is presented for this matching process, and the results are illustrated by computer simulations.

Cell Graphs: A Provable Correct Method for the Storage of Geometry *

Abstract: Several methods have been proposed for the storage of geometric properties in Geographic Information Systems, but many are based on the storage of metric data (coordinates) and analytical geometry. Because of the well-known limitations of implementations of the algebra of computer real numbers, such as incidence and inclusion during affine transformations. We propose an approach in which topological relations are separately recorded and independent of metric positions. The method is based on the use of simplices, which are the simplest polyhedrons of each dimension. The zero- dimensional simplex is the point, the one-dimensional one line, etc. In order to allow for non-straight lines as connections between points, we actually use cells, which are the homeomorphic image of simplices. In order to store topological relations, we use two completeness principles: Completeness of incidence and completeness of inclusion. We can show that in such a geometric configuration topology is invariant to affine transformations, independently of the method selected for recording metric information. For formal treatment we form a multi-sorted algebra (abstract data types). The axioms for this algebra must be selected such that the above-mentioned principles are maintained as invariants. We rely on an arbitrary method to learn about the topological relations initially. This "oracle" may use the calculation of a distance and a threshold, query the user or decide randomly, but it cannot influence the consistancy of the resulting geometry, as it is consulted only if the same information was not previously available and thus cannot lead to an inconsistent situation. Reasonable performance is expected, as this method imposes a 'neighborhood' structure on the data. All operations use and change only data of objects in immediate proximity. Databases suitable for handling spatial data should permit clustering of data by proximity.

Stochastic Teams with Nonclassical Information Revisited: When is an Affine Law Optimal?

Abstract: In this paper we consider a parameterized family of two-stage stochastic control problems with nonclassical information patterns, which includes the famous 1968 counterexample of Witsenhausen. We show that the parameter region can be partitioned into two regions, in one of which the optimal solution is linear whereas in the other it is inherently nonlinear. In the latter, the best piecewise-constant solution does not always outperform the best linear solution, whereas a linear plus piecewise-constant policy leads to a uniformly better performance in that region. Extensive numerical computations complement the study.

Affine manifolds and orbits of algebraic groups

Abstract: This paper is the sequel to The radiance obstruction and parallel forms on a;ffne manifolds (lYans. Amer. Math. Soc. 286 (1984), 629 649) which introduced a new family of secondary characteristic classes for affine structures on manifolds. The present paper utilizes the representation of these classes in Lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold Mn to the action on Rn Of the algebraic hull A(r) of the affine holonomy group r c Aff(Rn). A main technical result of the paper is that if M has a nonzero cohomology class represented by a parallel k-form, then every orbit of A(r) has dimension > k. When M is compact, then A(r) acts transitively provided that M is complete or has parallel volume; the converse holds when r is nilpotent. A 4-dimensional subgroup of Aff(R3) is exhibited which does not contain the holonomy group of any compact affine 3-manifold. When M has solvable holonomy and is complete, then M must have parallel volume. Conversely, if M has parallel volume and is of the homotopy type of a solvmanifold, then M is complete. If M is a compact homogeneous affine manifold or if M possesses a rational Riemannian metric, then it is shown that the conditions of parallel volume and completeness are equivalent. This paper is the sequel to our previous paper [22]. In that paper we exploited certain characteristic classes (exterior powers of the radiance obstruction) to obtain relationships between various properties of affine manifolds. Our results supported the conjecture (first made by L. Markus [36]) that a compact affine manifold is complete if and only if it has parallel volume. We shall refer to this as the main conjecture. Let M be a compact affine manifold with developing map dev: M > E and affine holonomy representation h: 7r > h(7r) = r c Aff(E). Here 1r is the group of deck transformations of the universal cover M, E is the vector space Rn, and Aff(E) is the group of affine automorphisms of E. The map dev is a locally affine immersion which is equivariant respecting h. The linear holonomy homb morphism is the composition A: 1r > GL(E) of h with the natural homomorphism L: Aff(E) > GL(E). The obstruction to r fixing a point in E is a 1-dimensional cohomology class CM E H1(M; Ex) with coefficients in E twisted by A. It comes from a universal class in the group cohomology H1 (Aff(E); EL) which contains the crossed homomorphism u: Aff(E) ) E, g g(0). Received by the editors December 12, 1984 and, in revised form, May 29, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R99, 53C05; Secondary 53C10, 55R25. 1 Research partially supported by fellowships and grants from the National Science Foundation. (a)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Mathematical Programming: An Introduction to Optimization

Abstract: 1. An Introduction to Mathematical Programming 2. Subspaces, Matrices, Affine Sets, Cones, Convex Sets, and the Linear Programming Problem 3. The Primal Simplex Procedure 4. Duality and the Linear Complementarity Problem 5. Other Simplex Procedures 6. Network Programming 7. Convex and Concave Functions 8. Optimality Conditions 9. Search Techniques for Unconstrained Optimization Problems 10. Penalty Function Methods

The regular local noninteracting control problem for nonlinear control systems

Abstract: We study the Noninteracting Control Problem for affine nonlinear control systems under the assumption that the number of scalar inputs equals the number of vector outputs. Our purpose is to find a static state feedback law for the system which achieves noninteraction. Using the recently developed differential geometric approach to nonlinear systems theory and working under a set of regularity assumptions, we give necessary and sufficient conditions for the local solvability of the problem. This work extends earlier results in the “geometric approach” for linear systems.

On Affine Incentives for Dynamic Decision Problems

Abstract: Construction of optimal incentive strategies for continuous time two-person game problems described by integral convex cost criteria is considered. The strategies are affine in the data available and they are represented by means of Stieltjes measures.

The regular local noninteracting control problem for nonlinear control systems

Abstract: We study the Noninteracting Control Problem for affine nonlinear control systems under the assumption that the number of scalar inputs equals the number of vector outputs. Our purpose is to find a static state feedback law for the system which achieves noninteraction. Using the recently developed differential geometric approach to nonlinear systems theory and working under a set of regularity assumptions, we give necessary and sufficient conditions for the local solvability of the problem. This work extends earlier results in the “geometric approach” for linear systems.

Image rotating system by an arbitrary angle

Abstract: A system for rotating an image by an arbitrary angle which is effective in image processing techniques using terminal equipment such as a work station, and will not distort a rotated image even with an increasing rotation angle θ. Skew transformation is implemented for respective skew angles in horizontal and vertical directions corresponding to a desired rotation angle three times alternately, so that affine transformation for image rotation may be replaced by triple skew transformations. Thus, arbitrary two-dimensional image data is precisely rotated by any desired angle at high speeds without resorting to approximation of the arithmetic equation.

A Higher Dimensional Mordell Conjecture

Abstract: Faltings’ long awaited proof of the Mordell conjecture completes, roughly speaking, the question of whether a given curve has only finitely many integral or rational points. Indeed, if a complete curve has genus g ≥ 2, then it has finitely many rational points; any affine curve whose projective closure is a curve of genus at least two will, a fortiori, have only finitely many integral points. A curve of genus 1 is an elliptic curve; it will have infinitely many rational points over a sufficiently large ground field, but no affine subvariety has an infinite number of integral points. Finally, a curve of genus zero is, after a base change, the projective line, which has an infinite number of rational points; affine sub-varieties omitting at most two points will have infinitely many integral points over a sufficiently large ring; but affine sub-varieties omitting at least three points will have only finitely many integral points. Thus the answer to the finiteness question is given entirely by the structure of the curve over the complex numbers.

Periodic solutions of affine stochastic differential equations

Abstract: We discuss the problem of the existence of periodic and stationary solutions of affine stochastic differential equations. We prove that under a controllability condition the system has a periodic solution if and only if the linear part is eyponentially stable in mean square. It is also shown that the controllability assumption is necessary for the existence of a “unique” weakly periodic solution with nondegenerate covariance.

Self-affine fractal sets, II: Length and surface dimensions

Abstract: For a self-similar curve, one is able to estimate the fractal dimension by “walking a divider. It is shown that -for self-affine curves, this procedure yields a local and a global value, both doubly anomalous. Other problems raised by length and area measurement -for fractals are investigated.

Some remarks on affine rings

Abstract: Various topics on affine rings are considered, such as the relationship between Gelfand-Kirillov dimension and Krull dimension, and when a "locally affine" algebra is affine. The dimension result is applied to study prime ideals in fixed rings of finite groups, and in identity components of group-graded rings. 0. Introduction. We study affine rings (i.e., rings finitely generated as algebras over a central subring) and group actions on them. Of particular interest is the relationship between the Gelfand-Kirillov (GK) dimension and the classical Krull dimension. Our first theorem shows for affine prime PI rings over a field that the GK dimension of a prime subalgebra is the same as its Krull dimension. This result is then applied to give a rapid proof of a result of Alev [1], on prime ideals in fixed rings. Later we turn to group-graded affine rings deriving results similar to those previously obtained for group actions; the notion of equivalence is introduced for prime ideals in the identity component, and an analog of Alev's theorem is proved. We also consider when a prime PI algebra which is "locally affine" must be affine. Some of the results in this paper (including Theorem 1) were announced at the NATO A.S.I. in Ring Theory, Antwerp, 1983 (see [7]). 1. Prime subrings of PI rings. We begin by stating a form of the Artin-Tate lemma, which will be used throughout this paper. LEMMA 1. Let R c S be algebras over a commutative Noetherian ring C, such that S is a finite module over R and affine over C. If R is contained in the center of S, then R is C-affine. Consequently R is Noetherian and S is a Noetherian R-module. PROOF. The usual commutative argument works in this situation; see also Lemma 27 of [11]. Our first theorem extends a result of Malliavin [4] on affine PI algebras to prime subrings of such rings, which are not necessarily affine. For any algebra A over a field k, we let cl(A) denote the classical Krull dimension of A, and GK(A) denote the Gelfand-Kirillov dimension of A. THEOREM 1. Let A be a prime PI algebra which is affine over a field k. If B is a prime subalgebra of A, then GK(B) = cl(B). Received by the editors March 25, 1985 and, in revised form, August 19, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A38, 16A72. 'Both authors acknowledge support from the NSF. 01986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page