Neighborly combinatorial 3-manifolds with 9 vertices

A complete classification is given for neighborly combinatorial 3-manifolds with 9 vertices. It is found that there are 51 types, only one of which is not a sphere.     

Tolerances in self-cohering antenna arrays of arbitrary geometry

In a conventional array, the location of each element must be known to an accuracy of aboutlambda/10for phasing and scanning purposes. A larger tolerance on the element location results in the loss of gain and deterioration of the radiation pattern. In a self-cohering phased array, the phasing of the array elements is accomplished closed-loop and adaptively; therefore, no information on the element location is necessary. On the other hand, to scan the adaptively formed beam about the direction of the received wave, which has to be performed open-loop, one requires information on the element locations. In this paper, basic relations pertaining to the open-loop scanning of adaptive antenna arrays of arbitrary geometry are derived for both far-field and near-field scanning problems, and the transition between the two cases and the required accuracy in the knowledge of the same distance are explored. It is shown that the tolerance requirement on the element locations is the conventional tolerance divided by the maximum scan angle. In large systems designed for target imaging, it is shown that the tolerance on element location can be two or more orders of magnitude larger than that in a conventional array.     

The Global Dimension of the full Transformation Monoid (with an Appendix by V. Mazorchuk and B. Steinberg)

The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid \(\mathfrak {T}_{n}\) (the monoid of all self-maps of an n-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by Ponizovski?, Putcha and Ringel between 1987 and 2000. From their work, one can deduce that the global dimension of \(\mathbb {C}\mathfrak {T}_{n}\) is n?1 for n = 1, 2, 3, 4. We prove in this paper that the global dimension is n?1 for all n ≥ 1 and, moreover, we provide an explicit minimal projective resolution of the trivial module of length n?1. In an appendix with V. Mazorchuk we compute the indecomposable tilting modules of \(\mathbb {C}\mathfrak T_{n}\) with respect to Putcha’s quasi-hereditary structure and the Ringel dual (up to Morita equivalence).     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

Riccati equations and Lie series

Lie series and a special matrix notation for first-order differential operators are used to show that the Lie group properties of matrix Riccati equations arise in a natural way. The Lie series notation makes it evident that the solutions of a matrix Riccati equation are curves in a group of nonlinear transformations that is a generalization of the linear fractional transformations familiar from the classical complex analysis. It is easy to obtain a linear representation of the Lie algebra of the nonlinear group of transformations and then this linearization leads directly to the standard linearization of the matrix Riccati equations. We note that the matrix Riccati equations considered here are of the general rectangular type.     

On observing the states of uncertain dynamical systems

For the implementation of a stable asymptotic reduced-order Luenberger observer, which is not sensitive to unknown disturbances entering the system, it is shown that specific columns of the measurement matrixC must be different from zero.This research was supported by the Technion VPR Fund, L. Rogow Aeronautical Research Fund.The author wishes to thank E. P. Ryan, G. Leitmann, and S. Gutman for helpful discussions.     

Continue,quit, restart probability model

We discuss a new applied probability model: there is a system whose evolution is described by a Markov chain (MC) with known transition matrix on a discrete state space and at each moment of a discrete time a decision maker can apply one of three possible actions: continue, quit, and restart MC in one of a finite number of fixed “restarting” points. Such a model is a generalization of a model due to Katehakis and Veinott (Math. Oper. Res. 12:262, 1987), where a restart to a unique point was allowed without any fee and quit action was absent. Both models are related to Gittins index and to another index defined in a Whittle family of stopping retirement problems. We propose a transparent recursive finite algorithm to solve our model by performing O(n³) operations.     

A Preference Order Dynamic Program for a Knapsack Problem with Stochastic Rewards

In this paper, we extend the knapsack problem to include more realistic situations by treating the rewards (or values) associated with each item included in the solution as random variables with distributions that are known (or may be estimated) rather than known integers, as in the usual formulation. We propose a dynamic programming solution methodology where the usual real-valued return function is replaced by a preference ordering on the distributions of returns from the items selected. In addition to extending previous solutions to the knapsack problem, we demonstrate the selection of a preference ordering criterion and illustrate the conditions required of the ordering to guarantee optimality of the procedure. A sample problem is shown to demonstrate the algorithm, and results of computational experience with 459 problems of varying sizes and parameters are presented.