Restricted left principal ideal rings

A ring is an LD-ring ifR is left bounded, ifR/J is a left Artinian left principal ideal ring for every proper idealJ inR, and ifR has finite left Goldie dimension. IfR is non-Artinian thenR is an order in a simple Artinian ringS. The ideal theory of LD-rings is investigated, and we discuss some conditions under which an LD-ring is an hereditary ring, and some under which an LD-ring is a Noetherian, bounded, maximal Asano order. A central localization of an LD-ring is an LD-ring, and the center of some LD-rings is a Krull-domain. This research was supported in part by the National Science Foundation Grant GP 23861.     

The Turaev genus of an adequate knot

The Turaev genus of a knot is an obstruction to the knot being alternating. An adequate knot is a generalization of an alternating knot. A natural problem is a characterization of the Turaev genus of an adequate knot. In this paper, we show that the Turaev genus of an adequate knot is realized by the genus of the Turaev surface associated to an adequate diagram of the knot using the Khovanov homology. As a result, we obtain the additivity of the Turaev genus of adequate knots, and show that the Turaev genus of an adequate knot is “often” preserved under mutation. We also show that an n-semi-alternating knot is of Turaev genus n. This is the first examples of adequate knots of Turaev genus two or more.     

Archimedean ordered semigroups as ideal extensions

The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.     

Conditions on Normal Spaces of Finite Rank That Guarantee C(X) is SV

An f-ring A is an SV f-ring if for every minimal prime ?-ideal P of A, A/P is a valuation domain. A topological space X is an SV space if C(X) is an SV f-ring. For normal spaces, several conditions are shown to guarantee the space is an SV space. For example, a normal space of finite rank for which the closure of the set of points of rank greater than 1 is an F-subspace, is an SV space. For normal spaces of rank 2, a characterization of SV spaces is given.     

Images and Open Subspaces of SV Spaces

A topological space is finitely an F-space if its Stone–?ech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). This article studies the images under open continuous functions and the open subspaces of spaces that are finitely an F-space or are SV. It is shown that an open continuous image of a compact space that is finitely an F-space is finitely an F-space and an open continuous image of certain SV spaces is SV. Also, it is shown cozerosets, but not necessarily open sets, of SV spaces are SV spaces and a similar situation holds for spaces that are finitely an F-space.     

On shrinking arcs in metric spaces

By a sin (1/x)-curve is meant a metric continuum that is a 1-1 continuous image of the disjoint union of an arc and a semi-open interval that has the image of the arc as continuum of convergence. It is shown that ifM is a compact metric space,A ⊂M an arc, whileM/A is an arc havingA/A as an end-point, thenM is an arc, a triod, some sin (1/x)-curve, or some sin (1/x)-curve with an arc attached at one point, or some sin (1/x)-curve with two arcs attached. The case of shrinking finitely many arcs is also considered in an attaching theorem. Prepared under a NASA Research Grant No. NsG-568 at Kent State University.     

On the power and limitations of affine policies in two-stage adaptive optimization

We consider a two-stage adaptive linear optimization problem under right hand side uncertainty with a min–max objective and give a sharp characterization of the power and limitations of affine policies (where the second stage solution is an affine function of the right hand side uncertainty). In particular, we show that the worst-case cost of an optimal affine policy can be times the worst-case cost of an optimal fully-adaptable solution for any δ > 0, where m is the number of linear constraints. We also show that the worst-case cost of the best affine policy is times the optimal cost when the first-stage constraint matrix has non-negative coefficients. Moreover, if there are only k ≤ m uncertain parameters, we generalize the performance bound for affine policies to , which is particularly useful if only a few parameters are uncertain. We also provide an -approximation algorithm for the general case without any restriction on the constraint matrix but the solution is not an affine function of the uncertain parameters. We also give a tight characterization of the conditions under which an affine policy is optimal for the above model. In particular, we show that if the uncertainty set, is a simplex, then an affine policy is optimal. However, an affine policy is suboptimal even if is a convex combination of only (m + 3) extreme points (only two more extreme points than a simplex) and the worst-case cost of an optimal affine policy can be a factor (2 − δ) worse than the worst-case cost of an optimal fully-adaptable solution for any δ > 0.     

Relations Between Algebras and Their Subalgebras of Invariants

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points A^G is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then A^G is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is A^H.     

On Orderable Poly Engel Groups

It is known that an orderable n-Engel group is nilpotent. We show that an orderable group that is an extension of an n-Engel group by an n-Engel group is nilpotent-by-nilpotent. However, a finitely generated orderable poly n-Engel group need not be solvable in general.     

A dual control problem and application to marketing

The general approach to adaptive and dual control is to formulate an optimal stochastic control problem. However, for such an approach only mathematical representations of the solution are available which allow little insight into the structure of the optimal controller. Here, an alternative deterministic approach is presented based upon determining a control in which a disturbance attenuation function remains bounded for all allowable (L₂ functions) disturbances. The disturbance attenuation function is composed of the ratio of an L₂ function of the desired outputs over an L₂ function of the disturbance inputs. This disturbance attenuation problem is converted to a differential game. For this game, the optimal control law, in a closed-form, is obtained by performing a minmax operation with respect to a quadratic cost function subjected to a bilinear system. The resulting controller is time-varying and depends nonlinearly on the state and the parameter estimates vector, and on an associated Riccati-type matrix. We provide insights into the structure of the resulting dual controller and illustrate the method by two examples. One of the examples is an application to marketing, to set promotional spending of a company, considering that the effect of promotional effort on sales is unknown.     

Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs

A benzenoid graph is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of a side length one. A benzenoid graph G is elementary if every edge belongs to a 1-factor of G. A hexagon h of an elementary benzenoid graph is reducible, if the removal of boundary edges and vertices of h results in an elementary benzenoid graph. We characterize the reducible hexagons of an elementary benzenoid graph. The characterization is the basis for an algorithm which finds the sequence of reducible hexagons that decompose a graph of this class in O(n²) time. Moreover, we present an algorithm which decomposes an elementary benzenoid graph with at most one pericondensed component in linear time.     

Attractors of generalized IFSs that are not attractors of IFSs

Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m≥2$m\ge 2$, there exists a Cantor subset of the plane which is an attractor of some GIFS of order m  , but is not an attractor of a GIFS of order m−1$m-1$. In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS.     

On flatness of acts

We consider monoids $S=G\dot \cup I$ where G is a group and I is an ideal of S and show that if an S-act is principally weakly flat, (weakly) flat, torsion free or satisfies conditions (P) or (P_E) as an I¹-act, then it has these properties as an S-act. We also show that an S-act which is free, projective or strongly flat as an I¹-act may not generally have these properties as an S-act.     

The three-in-a-tree problem

We show that there is a polynomial time algorithm that, given three vertices of a graph, tests whether there is an induced subgraph that is a tree, containing the three vertices. (Indeed, there is an explicit construction of the cases when there is no such tree.) As a consequence, we show that there is a polynomial time algorithm to test whether a graph contains a “theta” as an induced subgraph (this was an open question of interest) and an alternative way to test whether a graph contains a “pyramid” (a fundamental step in checking whether a graph is perfect).     

David G. Luenberger,Linear and nonlinear programming, 2nd edition,Addison-Wesley,Reading (1984), xv + 492 pages, $35.50

This paper considers a stochastic version of the linear continuous type knapsack problem in which the cost coefficients are random variables. The problem is to find an optimal solution and an optimal probability level of the chance constraint. This problem P₀ is first transformed into a deterministic equivalent problem P. Then a subproblem with a positive parameter is introduced and a close relation between P and its subproblem is shown. Further, an auxiliary problem of the subproblem is introduced and a direct relation between P and the auxiliary problem is derived through a relation connecting the subproblem and its auxiliary problem. Fully utilizing these relations, an efficient algorithm is proposed that finds an optimal solution of P in at most O(n⁴) computational time where n is the number of decision variables. Finally, further research problems are discussed.     

Stochastic adaptive control for continuous-time linear systems with quadratic cost

An adaptive control problem is formulated and solved for a completely observed, continuous-time, linear stochastic system with an ergodic quadratic cost criterion. The linear transformationsA of the state,B of the control, andC of the noise are assumed to be unknown. Assuming only thatA is stable and that the pair (A, C) is controllable and using a diminishing excitation control that is asymptotically negligible for an ergodic, quadratic cost criterion it is shown that a family of least-squares estimates is strongly consistent. Furthermore, an adaptive control is given using switchings that is self-optimizing for an ergodic, quadratic cost criterion.This research was partially supported b y NSF Grants ECS-9102714, ECS-9113029, and DMS-9305936.     

APT环上幂等阵的对角化

设R是一阿贝尔环(R的所有幂等元都在中心里),A是R上的一幂等阵.本文证明了以下结果：(a)A相抵于一对角阵当且仅当A相似于一对角阵;(b)若R是一APT(阿贝尔投射平凡)环,则A在相似变换之下可唯一地化为对角形diag{e₁, ..., e_n｝,这里e_i整除e_i+1;(c)R是APT环当且仅当R/I是APT环,这里I是环R的一幂零理想.由(a),还证明了分离的阿贝尔正则环是APT环.     

Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not necessarily a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n ^1.5/log n)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n−1)×(n−1) integer grid, where n is the number of vertices in G.     

Armendariz Properties Relative to a Ring Endomorphism

For an endomorphism α of a ring R, we introduce the notion of an α-Armendariz ring to investigate the relative Armendariz properties. This concept extends the class of Armendariz rings and gives us an opportunity to study Armendariz rings in a general setting. It is obvious that every Armendariz ring is an α-Armendariz ring, but we shall give an example to show that there exists a right α-Armendariz ring which is not Armendariz. A number of properties of this version are established. It is shown that if I is a reduced ideal of a ring R such that R/I is a right α-Armendariz ring, then R is right α-Armendariz. For an endomorphism α of a ring R, we show that R is right α-Armendariz if and only if R[x] is right α-Armendariz. Moreover, a weak form of α-Armendariz rings is considered in the last section. We show that in general weak α-Armendariz rings need not be α-Armendariz.