Search

The aim of this concise paper is to present an extension of the local inverse theorem of polynomial type operators, recently considered by the same authors in [l-4] for nonlinear systems analysis. This extension gives a local inversion of certain operators which are represented by generalized power series. The homogeneous operator terms (i.e., powers) of the series are defined algebraically via multilinear mappings. In this generality, theory of polynomial operators and power series is previously introduced by, e.g., Michal [5] and Cartan [6], by using a somewhat different presentation. However, the results concerning the inversion are not presented anywhere, as far as the authors know. Among others, a general recursive representation is obtained which gives the homogeneous operators of the inverted power series successively with increasing orders. The main theorem is presented here without proof because it can be performed by a pedantic modification of the corresponding one in [l, 31.

We consider constitutive relations of the form W = W( B ,…, C ) where W is a scalar polynomial function of the components of B,…,C which is invariant under the group A of transformations describing the symmetry of the material considered. We make no restrictions as to the number or kind of quantities B,…,C appearing as arguments of W. We proceed by listing the elements of the integrity basis for the polynomial W( B ,…, C ) invariant under A which are multilinear in the components of B,…,C. The non-linear elements of the integrity basis are readily obtained once the multilinear elements are given. We give complete results for the cases where the material considered is a single crystal belonging to one of the triclinic, monoclinic, rhombic, tetragonal or hexagonal crystal systems (except for the crystal D6h).

A predesign cost-estimation function for quite arbitrary buildings is derived utilizing historical cost data. The assumed function is multilinear and its coefficients are determined by means of the least-squares estimation technique. The selection and measurement of the six independent variables of the cost function, the collection of limited but proper historical cost data, and the use of mathematical tools both as a means in arriving at the desired function and as an end in explaining the value and impact of estimation hypotheses are the real contribution of this work. The derived predesign cost-estimation function is subjected to an error analysis and tested with two actual buildings not belonging in the sample data.

This paper considers certain linear and multilinear aspects of the classical isomorph rejection problem. Specifically, elements of a finite set S are considered as elements of a finite dimensional vector space. Some classical and more recent formulations of Bumside's Lemma are stated and proved in this linear algebraic setting. It is observed that the specialization of S as a finite function space yields a multilinear structure on the underlying vector space. It is remarked that the multilinear structure may be exploited to achieve Po1ya-like counting theorems

Introduction. A function of two variables h = F(f, g), where h, f, and g are all elements of Hilbert Space may be termed a bilinear transformation if it is linear in f and linear in g. A more formal definition is given in ?1. While a complete treatment of bilinear transformations would obviously require a very lengthy discussion, we wish to point out in this paper that many of the methods used in the study of linear transformations are applicable to them, with, of course, certain modifications. Many elementary notions can be extended and corresponding results obtained. For certain classes of bilinear transformations, there is even a "canonical resolution" (cf. ?5, Theorem 7). Bilinear transformations have appeared in the work of Kerner. t While the first Frechet differential is a linear transformation, the second is bilinear, and it is this connection which was studied by Kerner. We shall show the relationship between bilinear transformations and rings of operators.: Mazur and Orlicz have pointed out the relationship between bilinear (and multilinear) transformations and polynomial transformations (cf. [5 1, p. 59). Polynomial transformations have also been studied by Banach (cf. [2]). We shall have occasion to use some of their results. There is a very simple relationship between bilinear transformations and trilinear forms. For instance, if F(f, g) is a bilinear transformation, then

The theory of three terminal RC networks is given a new unified formulation based on five relatively prime polynomials. A set of necessary and sufficient mutually independent conditions, which are equivalent to the residue theorem, coefficient conditions, and RC admittance and impedance conditions, is given. These conditions are stated by means of the zero distributions of the five polynomials and a multilinear relation among them.

Thurstone's latent-distance method was used to find the coordinates of fifty-five societies in a three-dimensional space that has level of sociocultural development as its principal axis. By assuming that the most probable evolutionary precursor of any society was of a type represented by the society at a lower level of development that showed the greatest similarity to it, societies were linked to form a branched evolutionary “tree.” Its structure supported the postulate of multilinear sociocultural evolution and suggested its sequence and dynamics in three main lines of development up to the level of early civilization.

Let V be a vector space over a field F of zero characteristic, which is acted upon by the symmetric group. Systems of generators for V are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree 2 n + 1 2n + 1 which holds in the matrix ring F n ( n > 2 ) {F_n}(n > 2) is a consequence of the standard identity s 2 n {s_{2n}} . The notions of rigid and semirigid sequences of matrices are defined and treated.

It is well known (see, e.g., R. E. Kalman, Proc. Nat. Acad. Sci. (USA), December, 1965) that pattern-recognition linear machine properties of a (finite-dimensional linear dynamical system) may be identified with the input/state map polynomials}→ {polynomials mod characteris-tic polynomial}. In a similar way, the pattern-recogn ition action of a multilinear machine is describable by projections into certain complicated equivalence classes constructed on the input space. The results reported here, obtained very recently, give a firm mathematical foundation to earlier attempts by Norbert Wiener and many others to analyze non-linear system problems using generalized Volterra kernels and Hermite expansions.

For vector spaces over certain algebraic fields, homogeneous functionals which are derivable from multilinear forms are characterized by an identity involving two-dimensional subspaces. The quadratic case is the “parallelogram law” which distinguishes Hilbert spaces among Banach spaces.

A multilinear system with r input lines may be realized as an r -layer network, with the j th layer containing one linear system for each j -element subset of {1,...,r} with any machine receiving inputs from those machines, and those system inputs corresponding to subsets of its own subset.

In the study of inequalities, the cases of equality are often the most difficult and interesting part. The case of equality is, in some sense, a measure of the tightness of the inequality. In this paper, we generalize two inequalities of Brualdi and Newman [1, Theorems 3, 4], but the instances of equality are probably more interesting because of the variety of cases which can occur.Let A = (aij) be an n × n matrix. Define the permanent of A byWe say that A is row stochastic if all entries are non-negative and all row sums are 1. In [1], several inequalities involving permanents of row stochastic matrices were proved. In two of these results, the case of equality was not determined. We will generalize both of these results to a class of functions which includes the permanent, and determine all cases of equality.

Systems of linear relationships have received an increasing amount of attention in recent years, partly in their own right as mathematical entities, but probably more so because of their frequent occurrence in physical situations. Multilinear systems, however, involving linear combinations of linear expressions, seem to have been almost completely neglected. Yet many physical situations are more adequately described by multilinear than by linear equations. Examples arise in the topological synthesis of electrical networks, and in sensitivity and stability studies of active networks. There is therefore a need for mathematical development in this area. To the end of initiating this, two theorems, based on the representation of a multilinear system as a multi-weighted sum of vectors belonging to some linear space, are presented in this paper, together with an example of their application in the stability analysis of linear systems.

The inelastic flexural behaviour of reinforced concrete beams is described, as usual, by a trilinear (or, in general, multilinear) moment–rotation relationship. The analysis of beams and frames subject to given external actions is formulated as a ‘linear complementary problem’ and, in two dual ways, as a ‘quadratic programming problem’. These problems are solved by means of classical and recent algorithms in use in operations research. By the procedures pointed out it is also possible to calculate the distribution of plastic deformations along the beams. Two methods are proposed for the evaluation of the safety factor against local failure due to limited rotation capacity: the solution technique of the direct one essentially consists of an application of the simplex method for linear programming. Finally, a procedure is established for determining, allowing for the irreversibility of plastic rotations, the structural response to non-proportional loading histories which consist of individually proportional stages. Illustrative examples are given.

A typology of evolutionary theories, derived from Steward's (1955) uni lincar, universal, and multilinear evolution, and from Sahlins' (1960) spe cific and general evolution, is developed in this article. Two attributes, the unit of analysis and the number of sequences, are found to be common to all five theories. The attributes are used to construct the typology. It is shown that unilinear and general evolution are the same theory, while universal, multilinear, and specific evolution are distinct theories.

n/a

A subtensor product of a family of modules is defined by using a subdirect product of the family of modules considered as sets. A tensor product of modules can be decomposed into a direct sum of subtensor products of the modules. Subtensor products of graded modules and graded algebras are also studied. As an application of these, a certain subtensor product of a family (not necessarily finite) of anticommutative algebras is shown to be a coproduct of this family in the category of unitary anticommutative algebras, and it can be imbedded as a direct summand into a tensor product of the family as modules. 1. Subtensor product of modules. Let (A),ei be a family of modules over a commutative ring R with unit, and U a subset of the cartesian product FIKI Ma as sets. For an R-module N, a mapping 9: U-+N will be called a multilinear mapping of U into N if for any (x)ei, (Y)ei and (Za)~ei in U such that x#=AyXfl+tzfl, where A and ,u are in R, for one fi EI, and x =yL=za for all = E I with cxof3, t(0a).sI) = A99((YcJcei) + P99((Zca)acXi) holds. A multilinear mapping of FLaei M. into N is a usual multilinear mapping; and a restriction mapping of this to U is an example of a multilinear mapping in our sense. By a similar construction to that of a tensor product of modules [1, Theorem 37, p. 87], the following theorem can be proved. THEOREM 1. Forany U ' lTLei M., there exist an R-module A and a multilinear mapping a of U into A such that for any R-module N andfor any multilinear mapping 9 of U into N there exists a unique linear mappingf of A into N such thatfo o= 9?. PROOF. Let F be a free R-module with U as its basis; and J be the submodule of F generated by the elements (X e)l a(y )aeI[-(Z zc)i,, where Presented to the Society, January 21, 1972; received by the editors September 28, 1971 and, in revised form, February 28, 1972. AMS (MOS) subject classifications (1969). Primary 1580; Secondary 1610, 1590.

n/a

The multilinear extension of an n-person game v is a function defined on the n-cube IN which is linear in each variable and which coincides with v at the conrners of the cube, satisfying f(x) = v({i ∣ xi = 1}). Multilinear extensions are useful as a help in computing the values of large games, and give a generalization of the Shapley value.

A number of different authors have studied classes of nonassociative algebras or rings satisfying a multilinear identity in three variables (see Bibliography). In this paper we are interested in the structure of those rings which satisfy a multilinear identity in three variables, but which do not fall into certain of the well-known classes of such rings (the latter turn out to be the exceptional cases of the general theory). In order to rule out certain less interesting cases, we shall, however, find it convenient to restrict our attention to those identities which are satisfied by at least one algebra with unity element. We show in ?1, that, up to quasi-equivalence, every such identity either implies one of a certain one-parameter class of identities, or implies one of seven other identities, all but one of which have been treated in the literature. Rings satisfying x2x = xx2 and an identity from our one-parameter class are treated in another paper appearing in this issue [6]. Some remarks on the remaining seven identities are to be found in ?2.

For a multilinear differential operator which satisfies the necessary conditions, we establish a method of constructing a smooth function which maps no points to the zero operator.

Although the notion of a covariant v-valued v-functor, exemplified by the v-valued hom functors a(A, -) on a v-category a, has been recognized by Eilenberg and Kelly, in their comprehensive foundational treatise [EK, esp. pp. 454, ff.] on closed and monoidal categories, for general closed categories v, those authors pointedly renounce consideration of that notion's contravariant counterpart until v is at least symmetric, and carefully refrain from even mentioning the two analogous possibilities for general (not necessarily closed) monoidal categories v.

This paper consists of new results continuing the series of papers on optimal design theory by Kiefer (1959), (1960), (1961), Kiefer and Wolfowitz (1959), (1960), Farrell, Kiefer and Walbran (1965) and Karlin and Studden (1966a). After disposing of the necessary preliminaries in Section 1, we show in Section 2 that in several classes of problems an optimal design for estimating all the parameters is supported only on certain points of symmetry. This is applied to the problem (introduced by Scheffe (1958)) of multilinear regression on the simplex. In Section 3 we consider optimality when nuisance parameters are present. A new sufficient condition for optimality is given. A corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality, and we prove the natural invariance theorem involving this condition. These results are applied to the problem of multilinear regression on the simplex when estimating only some of the parameters. Section 4 consists primarily of a number of bounds on the efficiency of designs; these are summarized at the beginning of that section.

A method of predicting the stress distribution in a multilayered circular cylinder subjected to combined bending and axial loading and constructed of materials represented by multilinear stressstrain relationships is presented. Each layer may have a different stress-strain characteristic and may include perfectly plastic behavior as well as different properties in tension and compression, The analysis assumes that the strain developed at any cross section is a linear function of the distance from the neutral axis. For such a strain condition, the stresses developed in the cross section can be calculated from the assumed stressstrain relationships. The resultant axial and bending loads reacted by each layer may be determined by integrating these stresses over the cross-sectional area of each layer. The results of the analysis are presented either as interaction curves rel'ating all combinations of applied loads which will produce a specified stress condition or as a predicted load-strain relationship for the cylinder subjected to increasing load at a fixed ratio of bending moment and axial load. The analysis is compared with experimental data for layered filament-wound S glass-epoxy cylinders of various helical wrap angle combinations.

We study a method to construct multilinear invariant forms from a given set of unitary irreducible representations of the groupSLn,C. We use a global technique based on the theory of Gel’fand and Neumark.