Finite Members Research Articles

An approach for improved modal control of multi-input multi-output discrete time systems†

Abstract It is shown that the state feedback gains for a given multi-input multi-output discrete time system can be computed by matching a finite model output sequence members to prespecified ones. Relationship of the pole positioning control schemes to such matching is first examined and it is shown that better control is possible by adopting the method proposed in this paper. The results are illustrated through a numerical example.

The spectrum of a lattice-primal algebra

Let L = 〈L; ⩽〉 be a finite non-trivial lattice; we denote by L ∗ = 〈L;F〉 any algebra whose term functions of positive rank are exactly the operations on L isotone with respect to ⩽. Every such algebra L ∗ is called lattice-primal. For any finite algebra, A, Spec( A) is the set of cardinalities of the finite members of the equational class generated by A. We conjecture that for L directly irreducible there is a positive integer m (depending on L) such that if i ∈ Spec( L ∗) , then with finitely many exceptions j ∈ Spec( L ∗) for every positive integer j congruent to i modulo m. In this paper the conjecture is proved for L = M n, the (n + 2) -element lattice with n atoms; in particular, for L = M n, m = (n 2-2n) and the congruence classes which occur are exactly the quadratic residues modulo n-2.

Algebras with minimal spectrum

The spectrum of a class of algebras, ~ , is the set of cardinalities of the finite members of ~K; Spec (~K) denotes the spectrum of ~s If ~K is a variety (equational class), then Spec (~) is a multiplicative monoid of positive integers. Le t 92 be a finite non-trivial algebra and let ~s be the variety generated by 92; we say that 92 has minimal spectrum if Spec (~s = {1921 ~ [ n 0}; the aim of this paper is to characterize all algebras with minimal spectrum. Closely connected with this problem is the concept of the fine spectrum of ~ ; this was introduced and studied by Walter Taylor in [16]. The fine spectrum of ~K is the function fx such that fx(n) is the number of isomorphism classes of algebras in ~ of cardinality n. We will abbreviate fxr by f~; if 92 is finite, then fga(1) = 1 and f~(n) is finite. Note that if 92 has minimal spectrum, then fa(n)= 0 if n is not a power of [92 I. In [16] Taylor showed that there are exactly 9 2-e lement algebras with minimal spectrum (of course, up to polynomial equivalence). Moreover , he showed that for 7 of them, fga(2 "~) = 1 while for the other 2, fa(2 m) = 2 for m > 0. This property carries over to every algebra with minimal spectrum. Examples of the first kind are ~2, the 2-element boolean algebra, and @2, the 2-element group. An example of the second kind is ~2"=({0, 1};+, 1), the 2-element group in which the non-identity element is the value of a nuUary operation. The two isomorphism classes of cardinality 2 ~ are the one in which the added nullary operat ion is not the identity and the one in which it is the identity. For instance, if we factor ((~2") 2 by the congruence induced by the diagonal subgroup, then in the quotient the nullary is the identity. All 9 algebras are closely related to either ~2 or (~2. In particular, each example generates a congruence permutable variety. This property carries over to every algebra with minimal spectrum.

Transient Response of Long Structural Members

The dynamic response of long structural members has usually been approached using wave propagation methodology. With such methods an arbitrary structural member, with variations in geometry and material and complicated loading, is a challenge to analyze for transient response. A simple procedure is proposed here which reduces the long member analysis to that of a finite length member analysis. This permits a long member analysis to that of a finite length member analysis. This permits a long member with arbitrary loading, material, and geometry to be studied using the integration techniques commonly employed for finite members.

The Variety of Modular Lattices is Not Generated by its Finite Members

The Variety of Modular Lattices is Not Generated by its Finite Members

Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces

We study Fourier transforms of distributions on a symmetric space X . Eguchi et al. [1] characterized the image of E ′( X )-distributions of compact support under the Fourier transform. We give a simpler proof of Eguchi's result and characterize the size of the singular support for the K -finite members of E ′( X ). We apply this Paley-Wiener type theorem to invariant differential equations on X .

The variety of modular lattices is not generated by its finite members

This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given.

2x o varieties of Heyting algebras not generated by their finite members

2x o varieties of Heyting algebras not generated by their finite members

Free Ortholattices

It has been known for some time but does not seem to be anywhere in the literature that the variety of all ortholattices is generated by its finite members (see (4.2) of this paper). This is well known to imply that the word problem for free ortholattices is solvable. On the other hand, it is also known that the solution obtained this way is of no practical use. The main purpose of this paper is to present a workable solution.

Varieties of Orthomodular Lattices

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],

The axial contact of finite elastic cylinders with application to thermal contact resistance

Predictions of the macroscopic thermal contact resistance have been severely restricted because the macroscopic contact area between finite members could not be determined. A new method of solution to this contact problem in elasticity is developed which is applicable to a wide variety of geometries and boundary conditions. A physical lumped-parameter model is employed from which the finite difference equations in terms of displacement are derived. Calculations using this method indicate that large errors in the prediction of the thermal contact resistance can result if solutions for bodies of infinite extent are employed for finite regions of interest. Especially large errors may occur if the members are thin. The calculations indicate that the maximum deviation from flatness is insufficient for an accurate prediction of the macroscopic contact resistance. The form of the large scale surface geometry must also be considered.