The Equation x<Superscript>i</Superscript> = y<Superscript>j</Superscript>z<Superscript>k</Superscript> in a Free Semigroup

The equation xⁱ = y^jz^k has only periodic solutions in a free semigroup. This result was first proven by Lyndon and Schützenberger. We present a very short proof of this classical result. Moreover, we establish that the power of two or more of a primitive word cannot be factorized into conjugates of a different word.     

Two-dimensional quotients of C<Superscript>n</Superscript> are isomorphic to C<Superscript>2</Superscript>/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C² modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.     

A Scheduling Problem<Superscript>*</Superscript>

This note treats the optimization problem of minimizing the completion time of jobs in a job-shop with no wait in process. Techniques are presented for restricted cases of the problem.     

Product Development Plans<Superscript>†</Superscript>

A decision as to which of several possible products should be developed involves at least two different phases:(a) The technical phase, during which attempts are made to improve some aspects of the product.(b) The commercial phase, during which the product is produced and hopefully sold.Both phases are uncertain and this paper tries to provide explicit treatment of the combined uncertainty in the two phases.The expected value of return is maximized subject to budget constraints and chance-constraints on total return and/or pay-back requirements.The distributions consist of spikes and Poisson distributions and a formula is developed to compute compound probabilities. Dynamic programming is used for the numerical solution.An application of the method in a company is described.     

Defence Manpower Studies<Superscript>†</Superscript>

The Defence Manpower Studies Unit (DMSU) has been created to co-ordinate and develop manpower planning and research for the Armed Forces. Some of its objectives and modes of operation are described.An important characteristic of the Unit is that it seeks to be both an operational research unit and an integral part of management. The arguments for this dual role are based on experience and particularly on the experience of the Naval Manpower Department. The relevant history of this Department is described, and examples of its tasks given.The Armed Forces, taken together, absorb a substantial part of the national manpower resources at the recruiting ages, but also contribute to national trained manpower. The DMSU, therefore, is paying considerable attention to macro-scale aspects. Some of the work of the new Unit is relevant to the work of other government departments, universities, and industry. Examples are given.     

S<Superscript>1</Superscript>-Valued Sobolev maps

We describe the structure of the space W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) , where 0 < s < ∞ and 1 ≤ p < ∞. According to the values of s, p, and n, maps in W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) can either be characterised by their phases or by a couple (singular set, phase).     

Pseudo-self-affine tilings in ℝ<Superscript>d</Superscript>

It is proved that every pseudo-self-affine tiling in ?^d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoï tessellations is a corollary. Previously, these results were obtained in the planar case, jointly with Priebe Frank. The new approach is based on the theory of graph-directed iterated function systems and substitution Delone sets developed by Lagarias and Wang. Bibliography: 18 titles.     

Forecasting Traffic on STOL<Superscript>*</Superscript>

A subsidiary of Air Canada will be operating an experimental STOL (Short Take-Off and Landing) service between Montreal and Ottawa from downtown airports. A model was developed to forecast traffic and evaluate marketing strategies for the proposed service. Traditional forecasting methods relying on historical data are of little use as STOL will offer a level of service outside the range of existing experience. The model therefore uses attitudinal data on such things as comfort in conjunction with data on more concrete variables such as price. The data came from a market research survey.     

Homogenization of the parabolic cauchy problem in the Sobolev class H<Superscript>1</Superscript>(R<Superscript>d</Superscript>)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in R^d is studied. The coefficients are assumed to be periodic in R^d with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(R^d) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H ¹(R^d) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H ¹(R^d) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.     

The Katchalski-Lewis Transversal Problem in R<Superscript>n</Superscript>

Let F be a family of disjoint translates of a compact convex set in the plane. In 1980 Katchalski and Lewis showed that there exists a constant k, independent of F, such that if each three members of F are met by a line, then a "large" subfamily G ⊂ F, with |F\G| ≤ k, is met by a line. In this paper we obtain a higher-dimensional analogue containing the Katchalski-Lewis result. Also we give two constructions of families of pairwise disjoint translates of the unit ball in R³ which answer some related questions.     

Convexification of Nonsmooth Monotone Functions<Superscript>1</Superscript>

We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations. Communicated by X. Q. Yang This research was partially supported by the National Natural Science Foundation of China under Grants 70671064 and 60473097 and by the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.     

Laminar currents in ℙ<Superscript>2</Superscript>

 In this paper we study laminar currents in ℙ². Given a sequence of irreducible algebraic curves (C _n ) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ², and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ², as well as for all bimeromorphic maps of projective surfaces. Received: 24 September 2001 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50     

Dynamic Pricing via Dynamic Programming<Superscript>1</Superscript>

This article specifies an efficient numerical scheme for computing optimal dynamic prices in a setting where the demand in a given period depends on the price in that period, cumulative sales up to the current period, and remaining market potential. The problem is studied in a deterministic and monopolistic context with a general form of the demand function. While traditional approaches produce closed-form equations that are difficult to solve due to the boundary conditions, we specify a computationally tractable numerical procedure by converting the problem to an initial-value problem based on a dynamic programming formulation. We find also that the optimal price dynamics preserves certain properties over the planning horizon: the unit revenue is linearly proportional to the demand elasticity of price; the unit revenue is constant over time when the demand elasticity is constant; and the sales rate is constant over time when the demand elasticity is linear in the price. ¹We acknowledge professor robert e. kalaba for initiating this work and suggesting solution methods.     

Topological hypercovers and <Superscript>1</Superscript>-realizations

We show that if U_* is a hypercover of a topological space X then the natural map hocolim U_* X is a weak equivalence. This fact is used to construct topological realization functors for the ¹-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.Mathematics Subject Classification (2000):55U35, 14F20, 14F42The second author was supported by an NSF Postdoctoral Research Fellowship     

State spaces of <Emphasis Type="Italic">JB</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-triples<Superscript>*</Superscript>

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^*-triple, and up to complete isometry, of one-sided ideals in C^*-algebras.Mathematics Subject Classification (2000):17C65, 46L07Both authors are supported by NSF grant DMS-0101153