Showing papers on "Affine transformation published in 1981"

Nonlinear regulation: The piecewise linear approach

Abstract: This paper approaches nonlinear control problems through the use of (discrete-time) piecewise linear systems. These are systems whose next-state and output maps are both described by PL maps, i.e., by maps which are affine on each of the components of a finite polyhedral partition. Various results on state and output feedback, observers, and inverses, standard for linear systems, are proved for PL systems. Many of these results are then used in the study of more general (both discrete- and continuous-time) systems, using suitable approximations.

Affine manifolds with nilpotent holonomy

Abstract: An affine manifold is a differentiable manifold together with an atlas of coordinate charts whose coordinate changes extend to affine automorphisms of Euclidean space. These charts are called atline coordinates. A map between affine manifolds is called affine it its expression in affine coordinates is the restriction of an aftine map between vector spaces. Thus we form the category of affine manifolds and affine m_nps. Let M be a connected affine manifold of dimension n-> 1, locally isomorphic to the vector space E. Its universal covering/~/ inheri ts a unique affine structure for which the covering projection/~/--~ M is an aifine immersion. The group ~r of deck transformations acts on /V/by afline automorphisms. It is well known that there is an affine immersion D :/~5/--~ E, called the developing map. This follows, for example, from Chevalley's Monodromy Theorem; a proof is outlined in Section 2. Such an immersion is unique up to composition with an atiine automorphism of E. Thus for every g ~ "n" there is a unique affine automorphism a(g) of E such that D o g =ct(g)oD. The resulting homomorphism a : ~r --~ Aft (E) from 7r into the group of affine automorphisms of E is called the affine holonomy representation. It is unique up to inner automorphisms of Aft (E). The composition A : 7r ~ G L (E) is called the linear holonomy representation. The affine structure on M is completely determined by the pair (D, a) . M is called complete when D is a homeomorphism. This is equivalent to geodesic completeness of the connection on M (in which parallel transport is locally defined by affine charts as ordinary parallel transport in E). It is notorious that compactness does not imply completeness. The main results of this paper are about aftine manifolds whose affine holonomy groups a(Tr) are nilpotent. An important class of such manifolds are the affine nilmanifolds 7r\G. Here 7r is a discrete subgroup of a simply connected nilpoint Lie group G. It is assumed that G has a left-invariant afline structure; the space of right cosets of r then inherits an affine structure.

Obtaining surface orientation from texels under perspective projection

Abstract: A new method to obtain surface orientation of textural plane under perspective projection is described. Firstly, we derive a 2-D Affine transformation which approximates the distortion of texel patterns under perspective projection. Then we present a method to obtain the vanishing line of the plane from the area of texels in the picture. It can obtain a vanishing point from an arbitrary pair of texels; when n texels exist, C2, vanishing points can be obtained. An algorithm to solve the 2-D Affine matrix between two patterns is also presented. It reinforces the method to be applicable to a texture which is constituted from more than two kinds of texel patterns. Experiments are performed by using artificially generated textures.

A spectral sequence for the K-theory of affine glued schemes

Abstract: An affine scheme is ‘glued’ if it is the colimit of a finite diagram of affine schemes. We first develop several recognition criteria for determining when an affine scheme is glued. Under mild hypotheses, for example, glued schemes are seminormal. We then investigate the K-theory of glued schemes and develop an Atiyah-Hirzebruch type spectral sequence which converges to the Karoubi-Villamayor K-theory of the glued scheme. This allows us to compute K0 of some interesting rings and generalize a number of previous results in the literature.

A Characterization of Well-Posed Optimal Control Systems

Abstract: A constrained optimal regulator problem is considered. Continuous dependence of the optimal control on the desired trajectory (Hadamard well-posedness) or convergence toward the optimal control of any minimizing sequence (Tykhonov well-posedness) are proved when the dynamics are affine (linear plus constant). Dense well-posedness is obtained in the non-affine case. A necessary and sufficient condition for well-posedness for all desired trajectories is shown to be the affine structure of the plant.

Two examples of affine manifolds

Abstract: An affine manifold is a manifold with a distinguished system of affine coordinates, namely, an open covering by charts which map homeomorphically onto open sets in an affine space E such that on overlapping charts the homeomorphisms differ by an affine automorphism of E. Some, but certainly not all, affine manifolds arise as quotients Ω/Γ of a domain in E by a discrete group Γ of affine transformations acting properly and freely. In that case we identify Ω with a covering space of the affine manifold. If Ω—E, then we say the affine manifold is complete. In general, however, there is only a local homeomorphism of the universal covering into E, which is equivariant with respect to a certain affine representation of the fundamental group. The image of this representation is a certain subgroup of the affine group on E, is called the affine holonomy and is well defined up to conjugacy in the affine group. See Fried, Goldman, and Hirsch.

Hessian Manifolds and Convexity

Abstract: Let M be a flat affine manifold, that is, M admits open charts (Ui, x i 1 ,..., x i n such that M =U Ui and whose coordinate changes are all affine functions. Such local coordinate systems { i n ,...,x i n } will be called affine local coordinate systems. Throughout this note the local expressions for geometric concepts on M will be given in terms of affine local coordinate systems. A Riemannian metric g on M is said to be Hessian if for each point p∈M there exists a C∞-function (⌽ defined on a neighborhood of p such that gij=∂2o/∂xi ∂xj. Such a function ⌽ is called a primitive of g on a neighborhood of p. Using the flat affine structure we define the exterior differentiation dl for tensor bundle valued forms on M. Let g be the cotangent bundle valued 1-form on M corresponding to a Riemannian metric g on M. Then we know that g is Hessian if and only if g0 is dl-closed. A flat affine manifold provided with a Hessian metric is called a Hessian manifold[4], [5]. Koszul dealt with the case where g0 is dl-exact [1], [2], [3].

The dynamic linear exponential Gaussian team problem

Abstract: The dynamic team problem for a linear system with Gaussian noise, exponential of a quadratic performance index, and one-step delayed sharing information pattern is considered. It is shown, via dynamic programming, that the multistage problem can be decomposed into a series of static team problems. Moreover, the optimal policy of the i th team member at time k is an affine function of both the one-step predicted Kalman filter estimate and the i th team member's observation at time k . Efficient algorithms are available for determining the gains of this affine controller. This model and solution are applied to an approximate resource allocation problem associated with a defense network, and a numerical example is discussed.

Controlling a constrained linear system to an affine target

Abstract: We consider the problem of steering the state of a linear system to an affine target when the admissible controls are required to satisfy magnitude constraints. A necessary and sufficient condition for the existence of an admissible control which steers the system to the target from a specified initial condition is presented, as well as a necessary condition and a sufficient condition for global controllability to the target. The conditions are similar to those available in the literature in that they involve a finite dimensional search. Here, however, the affine nature of the target is exploited to reduce the dimensionality of the resulting optimization problem. Hence, the results are more easily applied than those developed in [13].

A Generalization of Bieberbach's Theorem.

Abstract: In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be separated into two conditions. First, M has an affine structure a distinguished covering by coordinate charts, whose coordinate changes are affine. Second, M has a Riemannian metric whose coefficients in the affine charts are constants. In this paper we relax the second condition. A polynomial Riemannian metric on the affine manifold M is a Riemannian metric whose local expression in affine coordinates has the form ~gi~(x)dxidx ~ where the gij are polynomial functions on Euclidean space E. By letting the gij be rational functions on E, we arrive at the more general notion of a rational Riemannian metric. (It is not assumed that these expressions define Riemannian metrics on all of E.) The object of this paper is to determine which affine manifolds have polynomial Riemannian metrics and to give examples of affine manifold having rational Riemannian metrics.

Constructions for resolvable and related designs

Abstract: Methods are given for constructing block designs, using resolvable designs. These constructions yield methods for generating resolvable and affine designs and also affine designs with affine duals. The latter are transversal designs or semi-regular group divisible designs with λ1=0 whose duals are also designs of the same type and parameters. The paper is a survey of some old and some recent constructions.

Quasicoherent sheaves over affine Hensel schemes

Abstract: The following two theorems concerning affine Hensel schemes are proved. Theorem A. Every quasi-coherent sheaf over an affine Hensel scheme is generated by its global sections. Theorem B. Hp(X, F) = 0 for all positive p and all quasi-coherent sheaves F over an affine Hensel scheme X. Introduction. The first one to consider the \"Henselian structure\" of an algebraic variety along a closed subvariety was Hironaka [17], but the theory of Henselian schemes was developed systematically a few years later by Kurke in his doctoral thesis, now included in the book [18] by Kurke, Pfister and Roczen. Some results were obtained independently in [11] and [12], while morphisms and fiber products were studied by Mora [19]. Henselian schemes are similar to formal schemes ([22], [7]) and provide a good notion of \"algebraic tubular neighborhood\" of a subvariety, which has the advantage, with respect to the widely used formal neighborhoods, to be \"closer\" to the algebraic situation; this idea, included in the above paper by Hironaka, was developed by Cox [3], [4]. In this paper we show that quasi-coherent sheaves over an affine Hensel scheme behave as they are expected to; namely, they are generated by global sections (Theorem A, see 1.11) and their cohomology is trivial (Theorem B, see 1.12). These results were announced in [14]; applications are given by Roczen [21]. The paper is divided into 4 sections. In the first one we recall some facts from the theory of Hensel couples, and we give the main results, along with some obvious corollaries. We give also, as a consequence of Theorem A, a particular case of the fundamental theorem of affine morphisms (see 1.22). §2 contains some technical results. We study the canonical homomorphism : hAf ®A hAg -+hAfg, where h denotes Henselization, and (/, g) = (1). The main facts are Theorem 2.4 ( is surjective). Our proofs are based on some nice properties of the absolutely integrally closed rings (already used by M. Artin [1]), and on a result of Gruson [13] concerning etale coverings over Hensel couples. Received by the editors May 8, 1980 and, in revised form, November 25, 1980. 1980 Mathematics Subject Classification. Primary 14A20, 14F20, 13J15; Secondary 14B25, 14E20, 14F05, 13B20. © 1981 American Mathematical Society 0002-9947/81/0000-05 56/$06.2 5 445 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 446 SILVIO GRECO AND ROSARIO STRANO In §3 we study Ker . This allows us to prove Theorem A. Note that by [12, §6, Theorem 1], Theorem A implies Gruson's Theorem [13]. The proof of Theorem B is given in §4, as a consequence of Theorem A. The authors wish to thank M. Hochster and D. Buchsbaum for some helpful discussions on the subject of this paper. 1. Preliminaries and main results. We recall some known facts concerning Henselian couples and Henselian schemes, and we state the main results of this paper, along with some corollaries. A. Hensel couples and Henselization. 1.1. A couple (A, a) consists of a ring A (commutative with 1) and of an ideal a a A. A morphism of couples /: (A,a) —>(B, b) is a ring homomorphism /: A -» B such that fia) c b. 1.2. An N-polynomial over the couple (A, a) is a monic polynomial a0 + axX + ' • • + X\" G A[X] such that a0 G a, and a, is a unit modulo a. The couple (A, a) is said to be a Hensel couple (shortly //-couple) if (i) a c rad A, (ii) every A/-polynomial has a root in a. A local ring A with maximal ideal m is Henselian if and only if (A, m) is a Hensel couple [20, p. 76, Proposition 3]. For more details on Hensel couples we refer to [10], [20], [18]. Here we list some properties we shall use freely throughout this paper. See [10] for indications on the proofs. 1.3. (A, a) is an //-couple if and only if a c red A, and for any finite A -algebra B the canonical map B —> B/aB induces a bijection between the sets of idempotents. 1.4. If (A, a) is an //-couple and B is an A -algebra integral over A, then (B, q_B) is an //-couple. 1.5. Let (A, a) be a couple and let b c a be an ideal. Then (A, a) is an //-couple if and only if (A/b, a/b) and (A, b) are //-couples. It follows that (A, a) is an //-couple if and only if (A, Va ) is such, if and only if (Ated, gAted) is such. 1.6. To every couple (A, a) one can attach an //-couple (B, b) along with a morphism (A, a) —> (B, b) such that for any //-couple (B', b') the canonical map Hom[(B, b), (B', b')] -* Hom[(^, a), (B', b')] is bijective. This couple is called the Henselization of (A, a) and is denoted by h(A, a). We often write hA in place of B, and we call it the Henselization of A with respect to a. If C is an A -algebra we often write hC for the Henselization of C with respect to aC. We summarize some properties of the Henselization we shall need later: 1.7. Let (A, a) be a couple. Then: (i) h(A, a) exists and is unique up to canonical isomorphism. (ii) hA/ahA = A/a and the a-adic completions of A and hA coincide. (iii) h(A, a) is the direct limit of the set of all local etale (L.E.) neighborhoods of (A, a) (see [20, Theorem 2, Chapter XI]). In particular hA is a direct limit of etale A -algebras, and depends only on Va . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use QUASI-COHERENT SHEAVES 447 (iv) hA is y4-flat, and is faithfully flat if and only if a c rad A. (v) hA =h(Ax+a). Hence the kernel of the canonical map A -^>hA coincides with the kernel of A -* A, +a. Thus if a ¥= A and A is a domain, then A -^>hA is injective. (vi) If A is noetheriah (resp. normal, regular, excellent) the same holds for hA. (vii) If (A, a) = lim (A,., qj, then h(A,q)= limA(v4,., qj. (viii) If B is an /I-algebra, integral over B, then hB = B <8>AhA. In particular h(A/I) = (hA)/IhA for every ideal / c A. B. Henselian schemes. 1.8. Let (A, a) be a Hensel couple and put X = Spec^4/a. For each/ G /I put Xj = D(f) n X, and S^(A^) =^4/This defines a presheaf of rings over X, which is actually a sheaf (whence T(X, 6X) = A). More generally to any A -module M one can associate the presheaf M defined by M(Xf) =hAj ®A M. It turns out that Af is a sheaf over X (the above claims are proved in [18, 7.1.3]; another proof is sketched in [11]). 1.9. The ringed space (X, 6X) is called the Henselian spectrum of (A, a) and is denoted by Sph(v4, a) or Sph A if a is understood. An affine Henselian scheme is a ringed space isomorphic to Sph(A,a) for some //\"-couple (A, a). A Henselian scheme is defined accordingly, in the obvious way. An important example of Hensehan scheme is the Henselization of a scheme along a closed subscheme (see [17], [11], [18]). 1.10. Let X = Sph(A, a) be an affine Henselian scheme. Then: (i) If x G X corresponds top G Spec A, then 6Xx =hAp (see 1.7(vii)). (ii) Sph(/1, a) depends only on Va (see 1.7(iii)). (iii) The functor M h» M, from (A-modules) to (0^-modules), is exact and fully faithful, and commutes with direct limits. Hence M is always quasi-coherent, and is coherent if A is noetherian and M is finitely generated (apply 1.7(iv) and (vi)). C. Main results of this paper and corollaries. 1.11. Theorem (Theorem A). Let X be an affine Hensel scheme, and let ^ be a quasi-coherent &x-module. Then (i) ?F = M where M = T(X, f), or equivalently (ii) ?F is generated by its global sections. 1.12. Theorem (Theorem B). Let X, <5 be as in 1.11. Then HP(X, <») = Ofor all /? > 0. The proofs of 1.11 and 1.12 will be given in §§3 and 4 respectively. Here we give some corollaries. 1.13. Corollary. If X = Sph(A,a) is an affine Henselian scheme, then the functor M h> M is an equivalence between the categories of A-modules and of quasi-coherent 6x-modules. If A is noetherian it induces an equivalence between the categories of finitely generated A-modules and of coherent 6x-modules. Proof. Immediate from 1.11 and 1.10(iii). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 448 SILVIO GRECO AND ROSARIO STRANO 1.14. Corollary. Let X = Sph(A,a) be an affine Henselian scheme, with A noetherian. Then any quasi-coherent 6x-module is the direct limit of the family of its coherent submodules. Proof. Apply 1.13 and l.lO(iii). 1.15. Corollary. Let X be a Hensel scheme, and let 0 -»<% -» § —» % -> 0 be an exact sequence of 6x-modules. If any two of them are quasi-coherent, so is the third. Proof. It follows from 1.11, by the same argument used for ordinary schemes (see [7, 1.4.7]). 1.16. Corollary. Let X be a Henselian scheme, and let % be a quasi-coherent 6x-module. Let % be an affine covering of X. Then for allp > Owe have. Hp(X, W) = //'(%, f) = Hp(X, f) Proof. It follows from 1.12, by general facts on cohomology (see e.g. [6]). 1.17. Remark. Theorems 1.11 and 1.12 are well known for ordinary schemes (see [7], [9]). Moreover they are true for coherent sheaves over a noetherian affine formal scheme ([7, 10.10.2] for 1.11; [15, Proposition 4.1] for 1.12). A general theory of quasi-coherent sheaves over a formal scheme is not known, and very likely it cannot be as well behaved as in the Henselian case. Application of \"Theorem B\" to the equivalence of singularities is given by Roczen [21]. D. Application to integral morphisms. An important fact in the theory of ordinary or formal schemes is that if X —» Y is an affine morphism, and Y is affine, then X is also affine. We shall prove this fact for a class of morphisms of Henselian schemes; so far we are not able to prove the general case. We recall first some facts on morphisms of Henselian schemes. For details see [19]. 1.18. Let A\"bea Henselian scheme. An ideal of definition of A' is a quasi-coherent ideal j cfl^ with the following property: there is an affine open covering Uj = Sph(.4,, a,) of X such that the ideals T(Ut, $) and a, of At have the same radical for all /. If $ is an ideal of definition of X then (X, &x/$) is an ordinary scheme having X as underlying topological space. One can show that there is a unique maximal idea

Utilization of the affine transformation of thermoanalytical curves for the determination of kinetic parameters

Abstract: A method is suggested for the affine transformation of thermoanalytical curves, by means of which their comparison with one another becomes feasible. It is demonstrated that the results obtained by the traditional methods of non-isothermal kinetics depend on the heating rate, whereas the results attained by affine transformation are independent of the heating rate. They are consistent with the results obtained by Merzhanov's dmethod, which is also a non-aprioristic method.

Recursive constructions for skew resolutions in affine geometries

Abstract: A resolutionR inAG(n, q) is defined to be a partition of the lines into classesR1,R2, ...,Rt (t=(qn−1)/(q−1)) such that each point of the geometry is incident with precisely one line of each classRl, 1≤i≤t. Of course, the equivalence relation of parallelism defines a resolution in any affine geometry. A resolutionR is said to be a skew resolution provided noRi, 1≤i≤t, contains two parallel lines. Skew resolutions are useful for producing packings of lines in projective spaces and doubly resolvable block designs. Skew resolutions are known to exist inAG(n, q),n=2t−1,i≥2,q a prime power. The entire spectrum is unknown. In this paper, we give two recursive constructions for skew resolutions. These constructions produce skew resolutions inAG(n, q) for infinietly many new values ofn.

On Hessian Structures on an Affine Manifold

Abstract: On a smooth manifold, an affine connection whose torsion and curvature vanish identically is called an affine structure. A smooth manifold provided with an affine structure is called an affine manifold. Let M be an affine manifold with an affine structure D. The co-variant differentiation by D will be also denoted by D. A Riemannian metric h on M is called a hessian metric if for each point x∈M there exist a neighborhood U of x and a smooth function ⌽ on U such that g = D2⌽ on U [5]. In this note we shall give an example of an affine manifold which does not admit any hessian metric and then determine the structure of A-Lie algebras which admit hessian metrics. For these purposes, we shall also establish a vanishing theorem of a certain cohomology group. The author would like to thank Professor H. Shima who introduced him to the problem discussed here.

A Column Generation Technique for the Computation of Stationary Points

Abstract: In two recent papers, Eaves showed that Lemke's algorithm can be used to compute a stationary point of an affine function over a polyhedral set. This paper proposes an alternative method which is based on parametric principal pivoting. The proposed method involves solving systems of linear equations and parametric linear subprograms over the given polyhedral set. An obvious advantage of the method is that any special structure of the polyhedral set can be exploited profitably in the solution of the subprograms.

Lagrangean functions and affine minorants

Abstract: We give hypotheses, valid in reflexive Banach spaces (such as L p for ∞>p>1 or Hilbert spaces), for a certain modification of the ordinary lagrangean to close the duality gap, in convex programs with (possibly) infinitely many constraint functions.

Products of Axial Affinities and Products of Central Collineations

Abstract: In order to obtain information about a mapping, it is often advantageous to factorize it into mappings of a special nature. We shall announce a number of results dealing with the factorization of affinities and projectivities into axial affinities and central collineations, respectively. These mappings are as simple as possible, since they leave all points of a hyperplane fixed. We shall distinguish different types of axial affinities such as shears, affine reflections, and affine hyperreflections, and of central collineations such as elations, projective reflections, and projective hyperreflections. We shall be interested in factorizations with a minimal number of factors for each mapping and also in finding upper bounds for the number of factors needed to express all mappings in a certain group.