Interactive decision software and computer graphics for architectural and space planning

We describe the development and successful implementation of a decision support system now being used by several leading firms in the architecture and space planning industries. The system, which we call SPDS (spatial programming design system) has the following characteristics: (i) user-friendly convenience features permitting architects and space planners to operate the system without being experienced programmers; (ii) interactive capabilities allowing the user to control and to manipulate relevant parameters, orchestrating conditions to which his or her intuition provides valuable input; (iii) informative and understandable graphics, providing visual displays of interconnections that the computer itself treats in a more abstract methematical form; (iv) convenient ways to change configurations, and to carry out ‘what if’ analyses calling on the system’s decision support capabilities; (v) a collection of new methods, invisible to the user, capable of generating good solutions to the mathematical programming problems that underlie each major design component. These new methods succeed in generating high quality solutions to a collection of complex discrete, highly nonlinear problems. While these problems could only be solved in hours, or not at all, with previously existing software, the new methods obtain answers in seconds to minutes on a minicomputer. Major users, including Dalton, Dalton, Newport, and Marshal Erdwin, report numerous advantages of the system over traditional architectural design methods.     

Hyperidentities and solid varieties

Let V be a variety of type τ. A type τ hyperidentity of V is an identity of V which also holds in an additional stronger sense: for every substitution of terms of the variety (of appropriate arity) for the operation symbols in the identity, the resulting equation holds as an identity of the variety. Such identities were first introduced by Walter Taylor in [27] in 1981. A variety is called solid if all its identities also hold as hyperidentities. For example, the semigroup variety of rectangular bands is a solid variety. For any fixed type τ, the collection of all solid varieties of type τ forms a complete lattice which is a sublattice of the lattice L(τ) of all varieties of type τ. In this paper we give an overview of the study of hyperidentities and solid varieties, particularly for varieties of semigroups, culminating in the construction of an infinite collection of solid varieties of arbitrary type. This paper is dedicated to Walter Taylor. Received July 16, 2005; accepted in final form January 3, 2006. This paper is an expanded version of a talk presented at the Conference on Algebras, Lattices and Varieties in Honour of Walter Taylor, in Boulder Colorado, August 2004. The author’s research is supported by NSERC of Canada.     

Tools of the mind: Techniques and methods for intellectual work: V. STIBIC North-Holland,Amsterdam, 1982, xiii + 297 pages,Dfl.85.00

Many location problems may be separated into a series of interrelated macro, meso and micro decision-making states. The macro scale decision determines the type, capacity and number of facilities, the meso scale decision determines the location and allocation of facilities and the micro scale decision determines such considerations as routing and scheduling of service vehicles. This paper concerns the first two levels of decision-making.The present paper demonstrates the use of two models: (i) an analytical model that uses continuum approximations and methods of calculus to determine the number of facilities, the capacity and the approximate location of each that minimizes the sum of the transportation and facility costs for a slowly varying demand rate, and (ii) a traditional location-allocation model that determines more exactly the resulting locations and allocations. These two approaches have specific requirements in terms of data input, cost of data collection and cost of solution and, consequently, yield unique insights and benefits for practising planners. The strengths and weaknesses of the two models are complementary. This thesis is developed with an analysis of the Calgary, Alberta refuse collection and disposal system.     

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Summary. Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well. Received January 10, 1995     

Error analysis of upwind‐discretizations for the steady‐state incompressible Navier–Stokes equations

Within the framework of finite element methods, the paper investigates a general approximation technique for the nonlinear convective term of the Navier–Stokes equations. The approach is based on an upwind method of finite volume type. It is proved that the discrete convective term satisfies a well‐known collection of sufficient conditions for convergence of the finite element solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A modified nearly exact method for solving low-rank trust region subproblem

In this paper, we first discuss how the nearly exact (NE) method proposed by Moré and Sorensen [14] for solving trust region (TR) subproblems can be modified to solve large-scale “low-rank” TR subproblems efficiently. Our modified algorithm completely avoids computation of Cholesky factorizations by instead relying primarily on the Sherman–Morrison–Woodbury formula for computing inverses of “diagonal plus low-rank” type matrices. We also implement a specific version of the modified log-barrier (MLB) algorithm proposed by Polyak [17] where the generated log-barrier subproblems are solved by a trust region method. The corresponding direction finding TR subproblems are of the low-rank type and are then solved by our modified NE method. We finally discuss the computational results of our implementation of the MLB method and its comparison with a version of LANCELOT [5] based on a collection extracted from CUTEr [12] of nonlinear programming problems with simple bound constraints.      

Non-linear operators on sets of measures

Summary If M[ℬ, U(C, C)] is the collection of U(C, C)-valued (non-linear) set functions defined on the Borel subsetsℬ of the compact Hausdorff space S, one may define operators on M[ℬ, U(C, C)] which are ? of the Hammerstein type ?. We initiate a study of a concept analogous to the second dual of a space of continuous functions by inquiring as to what representation theorems one may obtain for these operators. A ? Lebesgue type ? decomposition theorem for elements of M[ℬ, U(C, C)] is obtained. A ? density ? theorem is also obtained for the space M[ℬ, U(C, C)]. Entrata in Redazione il 6 marzo 1974.     

A non-parametric competing risks model for manpower planning

One of the most important variables for manpower planners is duration until a specified event occurs. This is frequently the completed length of service until leaving a job, but may also include such variables as length of service in a grade until promotion, or length of a spell of withdrawal from the labour force. In this paper we develop non-parametric maximum likelihood estimators for the survivor functions of length of stay in a grade until leaving for a number of different possible destinations. Since the data are statistically incomplete, including right censored and left truncated durations, as well as complete durations, we must modify the competing risks theory in the biostatistical literature to take such incompleteness into account. Right censored durations arise when the individual is still in the grade when data collection ceases and left truncated durations when the individual is already in service when data collection commences. The competing risks model is fitted to data for Northern Ireland nursing service and used to predict staff flows between grades. We may thus estimate future movements within the system and predict the future manpower stocks.     

Stiefel—Whitney currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v₁,…,v_m, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ₂-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle. More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle. In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps. These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer’s theorem is offered in an appendix. This approach is simpler and is via sheaf theory.     

Convergence and computation of simultaneous rational quadrature formulas

We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

Towards the planning of actions in complex environments

This paper investigates the requirements for planning systems that must plan actions in complex environments. First the general characteristics of planning systems are described which are then compared to the properties of classical planners. It appears that most of these planning systems are not capable of planning actions in non trivial environments. Therefore an alternative approach is proposed where planning is no longer independent of the execution of the plan. Essential to this process is the ability of the system to observe its environment and adapt its behaviour according to sensory data. These characteristics make the system an open system contrary to traditional planners that are characterised by a closed system architecture.     

Hierarchic control

Distributed control is applied to a system modelled by a parabolic evolution equation. One considers situations where there are two cost (objective) functions. One possible way is to cut the control into 2 parts, one being thought of as “the leader” and the other one as “the follower”. This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper. Dedicated to the memory of Professor K G Ramanathan     

Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an “alternative theorem”. A version of a Green’s functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green’s functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green’s functional. The author passed away in 2006 prior to publication of the article.     

Interactive decision software and computer graphics for architectural and space planning

We describe the development and successful implementation of a decision support system now being used by several leading firms in the architecture and space planning industries. The system, which we call SPDS (spatialprogrammingdesignsystem) has the following characteristics: (i) user-friendly convenience features permitting architects and space planners to operate the system without being experienced programmers; (ii) interactive capabilities allowing the user to control and to manipulate relevant parameters, orchestrating conditions to which his or her intuition provides valuable input; (iii) informative and understandable graphics, providing visual displays of interconnections that the computer itself treats in a more abstract methematical form; (iv) convenient ways to change configurations, and to carry out what if analyses calling on the system's decision support capabilities; (v) a collection of new methods, invisible to the user, capable of generating good solutions to the mathematical programming problems that underlie each major design component. These new methods succeed in generating high quality solutions to a collection of complex discrete, highly nonlinear problems. While these problems could only be solved in hours, or not at all, with previously existing software, the new methods obtain answers in seconds to minutes on a minicomputer. Major users, including Dalton, Dalton, Newport, and Marshal Erdwin, report numerous advantages of the system over traditional architectural design methods.     

The entropies of renewal systems

Renewal systems are symbolic dynamical systems originally introduced by Adler. IfW is a finite set of words over a finite alphabetA, then the renewal system generated byW is the subshiftX _W ⊂A ^Z formed by bi-infinite concatenations of words fromW. Motivated by Adler’s question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy. We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems. The authors were supported in part by NFS grants DMS-8706284, DMS-8814159 and DMS-8820716.     

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of “volume” type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic “matrix” material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a “naive” continuous model. The present paper continues along these lines, exploring “normal boundary component controllability” criterion for determining achievable configurations for the controlled system in the two-dimensional case. Connections with conformal mapping lead to affirmative results for approximate controllability in this sense and Fourier series techniques provide exact controllability results for the case wherein the domain of the uncontrolled system is a two-dimensional disk.      

Weak first-order interior estimates and Hessian equations

Development of the modern theory of fully nonlinear, second-order partial differential equations has amazingly enriched the classical collection of ideas and methods. In this paper, we construct first-order interior a priori estimates of new type for solutions of Hessian equations. These estmates are applied to present the most transparent version of Krylov’s method and its tight connection with fully nonlinear equations. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 55–66.     

Generalized symmetric spaces,Yu. P. Solovyov’s formula,and the generalized Hermitian geometry

We collect some basic results on canonical affinor structures of classical type on generalized symmetric spaces. Yu. P. Solovyov’s stimulating influence on this topic during its initial stages is illustrated. Using special canonical f-structures on homogeneous k-symmetric spaces, we also present a new collection of homogeneous Hermitian f-manifolds. Dedicated to the memory of Yu. P. Solovyov Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 43–60, 2007.