Simple Algebraic Structure Research Articles

Well-Ordered Steiner Triple Systems and 1-Perfect Partitions of the n-cube

Binary 1-perfect codes which give rise to partitions of the n-cube are presented. The 1-perfect partitions are characterized as homomorphic images of simple algebraic structures on Fn and are constructed starting from a particular case of a structure defined in Fn . A special property (so-called well-ordering) of STS(n) is given in such a way that for this kind of STS it is possible to define the algebraic structure we need in Fn and to construct 1-perfect partitions of the n-cube. These 1-perfect partitions give us a kind of 1-perfect code for which it is easy to do the coding and decoding. Furthermore, there exists a syndrome which allows us to perform error correction. We present systematic codes of length n=15 and we give examples of how to do the coding, decoding, and error correction.

Exact results related to the periodic Anderson model in D » 1 dimensions

We present exact results related to the T = 0 phase diagram of the periodic Anderson model in the strong coupling U =infinity limit for D = 2,3 dimensions. In a restricted parameter region, a ground state with simple algebraic structure was deduced in both dimensions, and its region of existence was characterized by sufficient conditions. The quasiparticle creation operator that builds up the ground state wave function does not satisfy the standard fermionic anticommutation relations.

Integrable structure of the new Calogero models

We show that the integrability of the dynamical system recently proposed by Calogero and characterized by the Hamiltonian H=∑j,kNpjpk{λ+μ cos[ν(qj−qk)]} is due to a simple algebraic structure. It is shown that this system is canonically equivalent to the classical Heisenberg magnet with long-range interactions. Our method shows clearly how these types of systems can be generalized and provides the mechanism of integrability of a large class of similar systems.

AN ALGEBRA FOR SLOPE-MONOTONE CLOSED CURVES

The paper proposes a new operation for a class of slope-monotone closed curves. This operation is closely related to the Minkowski sum in that the Minkowski sum can be considered as the binary operation for figures bounded by those closed curves. In this sense, the proposed algebra is a generalization of the Minkowski sum. Conventionally, the inverse of the Minkowski sum can be defined naturally for convex figures, but the generalization of the inverse to nonconvex figures results in a complicated algebraic structure. This paper, on the other hand, gives a natural definition of the inverse for nonconvex figures, and thus constructs a simple algebraic structure.

The almost PV behavior of some far from PV algebraic integers

This paper studies divisibility properties of sequences defined inductively by a 1 = 1, a n+1 = sa n + t⌊θ a m ⌋, where s,t are integers, and θ is a quadratic irrationality. Under appropriate hypotheses (especially that s + tθ be a PV-number) it is proved that the highest power of Δ that divides a n , where Δ is the discriminant of θ, tends to infinity. This is noteworthy in that truncation would normally be expected to destroy any simple algebraic structure. Moreover, we establish related results that imply the a n are not uniformly distributed modulo Δ in cases where the smaller conjugate of s + tθ exceeds 1 in modulus (the non-PV case).

Exact ground-state energy of the periodic Anderson model in d=1 and extended Emery models in d=1,2 for special parameter values.

We generalize an approach, which was recently introduced by Brandt and Giesekus to calculate the exact ground-state energy for strongly interacting particles on special perovskitelike lattices, to the periodic Anderson model in the dimension d=1 and to extended Emery models in d=1,2 on regular lattices for arbitrary spin degeneracy. For these models we calculate the exact ground-state energy for a restricted parameter regime in the strong-coupling limit. The ground-state energy shows a simple algebraic structure. We also present an eigenfunction of the Hamiltonian with the ground-state energy as its corresponding eigenvalue.

CONLIN: An efficient dual optimizer based on convex approximation concepts

The Convex Linearization method (CONLIN) exhibits many interesting features and it is applicable to a broad class of structural optimization problems. The method employs mixed design variables (either direct or reciprocal) in order to get first order, conservative approximations to the objective function and to the constraints. The primary optimization problem is therefore replaced with a sequence of explicit approximate problems having a simple algebraic structure. The explicit subproblems are convex and separable, and they can be solved efficiently by using a dual method approach. In this paper, a special purpose dual optimizer is proposed to solve the explicit subproblem generated by the CONLIN strategy. The maximum of the dual function is sought in a sequence of dual subspaces of variable dimensionality. The primary dual problem is itself replaced with a sequence of approximate quadratic subproblems with non-negativity constraints on the dual variables. Because each quadratic subproblem is restricted to the current subspace of non zero dual variables, its dimensionality is usually reasonably small. Clearly, the Hessian matrix does not need to be inverted (it can in fact be singular), and no line search process is necessary. An important advantage of the proposed maximization method lies in the fact that most of the computational effort in the iterative process is performed with reduced sets of primal variables and dual variables. Furthermore, an appropriate active set strategy has been devised, that yields a highly reliable dual optimizer.

Error Propagation for Large Errors

An essential facet of a risk assessment is the correct evaluation of uncertainties inherent in the numerical results. If the calculation is based on an explicit algebraic expression, an analytical treatment of error propagation is possible, usually as an approximation valid for small errors. In many instances, however, the errors are large and uncertain. It is the purpose of this paper to demonstrate that despite large errors, an analytical treatment is possible in many instances. These cases can be identified by an analysis of the algebraic structure and a detailed examination of the errors in input parameters and mathematical models. From a general formula, explicit formulas for some simple algebraic structures that occur often in risk assessments are derived and applied to practical problems.

Complete semi-Thue systems for abelian groups

We show that there are infinitely many examples of finite semi-Thue systems with decidable word problem and which have no finite equivalent noetherian and confluent (=complete) reduction system. These examples have a very simple algebraic structure: they are abelian groups. More precisely, we develop necessary and sufficient conditions for the existence of complete semi-Thue systems for presentations of finitely generated abelian groups.

Does Accidental Degeneracy Imply a Symmetry Group?

The question whether an accidental degeneracy in quantum mechanical system always implies the internal symmetry group of the system is probed by means of the simple 'model of the three·dimensional harmonic oscillator with a constant spin-orbit potential: H = (1/ 2)(p2+ r2)+ ;\6' L. For the fixed values of ; = += 1, this model has an interesting degeneracy of remarkable regularity; in particular, the ground state is infinitely degenerate, consisting of all the nodeless j = 1 ± 1/ 2 states. In order to systematically seek the symmetry group, we first investigate the full dynamical group of the system. To our surprise, we find that the· dynamical group of our system for .an arbitrary value of ; is the graded 5p(6R) - a supergroup extension of the dynamical group 5P(6R) of the three-dimensional harmonic oscillator. We find the graded 5U(3) as its subgroup, which is a supergroup extension of the well-known symmetry group 5U(3) of the harmonic oscillator. It is further shown that there exists a natural group chain, gr5P(6R)~5P(2R) XgrO(3), which corresponds to the group chain 5p(6R)~5p(2R)X 0(3), where x denotes the direct­ product. Next, we examine whether any subgroup of the full dynamical group constitutes the symmetry group of the system responsible for the accidental degeneracy. It is shown that there is no such a symmetry group in the system and that, instead of the symmetry group, there exists a special, simple algebraic structure which is essentially responsible for the accidental degeneracy.

Design of exclusive or sum-of-products (ESP) logic arrays with universal tests for detecting stuck-at and bridging faults

The detection problem of bridging faults in AND-EXOR arrays is considered in this paper in a new framework. These AND-EXOR arrays are different from the arrays based on the so-called Reed-Muller canonic (RMC) expansion of functions. The multiple stuck-at fault detection test set in such arrays as already derived by Pradhan[1] has been utilized to detect bridging faults. One most important advantage of this test set is that it is independent of the function realized and it has a simple algebraic structure and hence can be generated easily. As this conventional test set is insufficient to detect all bridging faults, we propose a technique of augmenting the network with some additional observation points which take care of otherwise undetectable bridging faults.

Relativistic wave equation for the harmonically confined quark-antiquark system

A covariant wave equation with simple algebraic structure is proposed to described the binary quark-antiquark system. The harmonic confinement mechanisms built in the equation leads to the linear square-mass spectrum with adjustable hyperfine splitting.

Automorphic forms for Schottky groups

Publisher Summary The chapter discusses a (revised and expanded) version of a lecture delivered at the Los Alamos Workshop on Mathematics in June 1974 under the title A Glimpse into Complex Analysis. It discusses automorphic forms for Schottky groups. Schottky groups stand out, among other Kleinian groups, by a particularly simple algebraic and geometric structure. Every finitely generated free Kleinian group all of which nontrivial elements are loxodromic is a Schottky group. One can always construct standard fundamental regions bounded by analytic curves. There are Schottky groups for which no standard fundamental region, no matter what generators one chooses, is bounded by circles as every finitely generated subgroup of a Schottky group is a Schottky group.