Ordinary Algebra Research Articles

Emergy and co-emergy

We introduce a method of calculating emergy that requires only ordinary algebra without any reliance on special rules to account for co-production. This is accomplished by using an intermediate computation “co-emergy”, and treating co-production as a problem of scale. In addition, we compare emergy calculations using inputs to the system with emergy calculations using what was used up in the system. It is shown that this can lead to slightly different results. We show how these methods can be used to compute emergy in systems at steady state, with imports and exports and with changes in stocks. These techniques allow direct comparison of competitive species, industries, or technologies using standard methods of linear algebra. It also enables us to include the efficiencies of various processes explicitly, which can help in the formulation and testing of conjectures about the relationships between emergy and local and system-wide efficiencies.

Beyond Fock space in three-dimensional semiclassical gravity

Quantization of relativistic point particles coupled to three-dimensional Einstein gravity naturally leads to field theories living on the Lorentz group in their momentum representation. The Lie group structure of momentum space can be traced back to the classical phase space of the particles coupled to topological gravity. In this work we show how the non-trivial structure of momentum space leads to an unusual description of Fock space. The latter is reflected in a deformed algebra of creation and annihilation operators which reduces to the ordinary algebra when momentum space ‘flattens’ to Minkowski space in the limit in which the three-dimensional Newton’s constant vanishes. The construction is covariant under the action of relativistic symmetries acting on the Lorentz group-momentum space. This shows how it is possible to build a Fock space on a group manifold momentum space in a way consistent with the underlying (deformed) relativistic symmetries.

Fault diagnosis of timed discrete event systems using Dioid Algebra

This paper deals with the fault diagnosis problem in a concurrent Timed Discrete Event System (TDES). In a TDES, concurrency leads to more complexity in the diagnoser and appears where, at a certain time, some user must choose among several resources. To cope with this problem, a new model-based diagnoser is proposed in this paper. This diagnoser uses Durational Graph (DG), a main subclass of timed automata for representing the time evolution of the TDES. The proposed diagnoser predicts all possible timed-event trajectories that may be generated by the DG. This prediction procedure is complicated for nondeterministic DG’s that are obtained for concurrent TDES’s. To solve this problem, a new Dioid Algebra, Union-Plus Algebra is introduced in this paper. Based on this Algebra, a reachability matrix is defined for a DG that plays an essential role in predicting the time behavior of TDES. By using reachability matrix, a prediction procedure is carried on via an effective equation set that is similar to linear system state equations in ordinary algebra. These results provide a suitable framework for designing an observer-based diagnoser that is illustrated by an example.

Understanding Measurement in Light of Its Origins

During the course of history, the natural sciences have seen the development of increasingly convenient short-hand symbolic devices for denoting physical quantities. These devices ultimately took the form of physical algebra. However, the convenience of algebra arguably came at a cost – a loss of the clarity of direct insights by Euclid, Galileo, and Newton into natural quantitative relations. Physical algebra is frequently interpreted as ordinary algebra; i.e., it is interpreted as though symbols denote (a) numbers and operations on numbers, as opposed to (b) physical quantities and quantitative relations. The paper revisits the way in which Newton understood and expressed physical definitions and laws. Accordingly, it reviews a compact form of notation that has been used to denote both: (a) ratios of physical quantities; and (b) compound ratios, involving two or more kinds of quantity. The purpose is to show that it is consistent with historical developments to regard physical algebra as a device for denoting relations among ratios. Understood in the historical context, the objective of measurement is to establish that a physical quantity stands in a specific ratio to another quantity of the same kind. To clarify the meaning of measurement in terms of the historical origins of physics carries basic implications for the way in which measurement is understood and approached. Possible implications for the social sciences are considered.

ON COREPRESENTATION OF HOPF π-COALGEBRAS

Let π be a group and H be a Hopf π-coalgebra. We investigate the criterion for the existence of a total integral of π-H-comodule algebra A in the setting of Hopf π-coalgebras, and prove that there exists a total integral θ = {θα : Hα → A} if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of H, which generalizes the result invented by Doi in the ordinary Hopf algebra setting.

SPECIAL AND EXCEPTIONAL JORDAN DIALGEBRAS

In this paper, we study the class of Jordan dialgebras (also called quasi-Jordan algebras). We develop an approach for reducing problems on dialgebras to the case of ordinary algebras. It is shown that straightforward generalizations of the classical Cohn's, Shirshov's, and Macdonald's Theorems do not hold for dialgebras. However, we prove dialgebraic analogues of these statements. Also, we study multilinear special identities which hold in all special Jordan algebras and do not hold in all Jordan algebras. We find a natural correspondence between special identities for ordinary algebras and dialgebras.

Division Algebras with Radicable Multiplicative Groups

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≅ ℤ2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups D* is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and is divisible.

Structurable Superalgebras of Classical Type

This article classifies simple structurable superalgebras with nontrivial involution over algebraically closed fields of characteristic 0 whose Kantor Lie superalgebra is of classical (i.e., non-Cartan) type. These are either structurable superalgebras of a hermitian superform or two examples constructed from cubic forms. It also makes extensive use of the Grassmann envelope to transfer constructions and results from ordinary algebras to superalgebras.

Perturbative quantum field theory via vertex algebras

In this paper, we explain how perturbative quantum field theory can be formulated in terms of (a version of) vertex algebras. Our starting point is the Wilson–Zimmermann operator product expansion (OPE). Following ideas of a previous paper (S. Hollands, e-print arXiv:0802.2198), we consider a consistency (essentially associativity) condition satisfied by the coefficients in this expansion. We observe that the information in the OPE coefficients can be repackaged straightforwardly into “vertex operators” and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras. We develop a general theory of perturbations of the algebras that we encounter, similar in nature to the Hochschild cohomology describing the deformation theory of ordinary algebras. The main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction Lagrangian (such as λφ4) in practice, using the consistency condition and the corresponding nonlinear field equation. We derive graphical rules, which display the vertex operators (i.e., OPE coefficients) in terms of certain multiple series of hypergeometric type.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees ⩽6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees ⩽6, and demonstrate the existence of further new identities in degree 7. Our proofs depend on computer algebra; we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem.

Lifting and Restricting Recollement Data

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in "Parametrizing recollement data", allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a 'right bounded' derived category of a differential graded(=dg) category to be a recollement of 'right bounded' derived categories of dg categories. In Theorem 2 we consider the particular case in which those dg categories are just ordinary algebras.

Angular momentum coupling in NMR: a new view of rotation matrix products via Racah algebra

Choosing the rotation angle-rotation axis parameters instead of the conventional Euler angle parameters {α, β, γ} to parametrise rotations, and a spherical tensor operator basis, Happer was the first to propose a novel multipole operator expansion of the rotation operator [W. Happer, Ann. Phys. (N.Y.) 48, 579 (1968)]. As Happer pointed out, such an expansion in irreducible spherical tensor operators readily lends itself to the methods of Racah algebra, a feature we exploit to simplify the calculus of rotation matrix products. Working with a Clebsch–Gordan coefficient expansion [W. Happer, ibid. cit.; M.S. Marinov, Yad. Fiz. 5, 943 (1967)] of the corresponding rotation matrices , and closure relations for these (2J + 1)-dimensional irreducible representations of the rotation group, we state, and prove for the first time, simple recoupling coefficient expansions for rotation matrix products. With an eye towards applications in NMR, where multiple rotations in spin or coordinate space often require the efficient evaluation of such rotation matrix products, we focus on expansions defining the matrices representing the result of two or three consecutive rotations. We show that the coefficients of these expansions are defined by 6 − j symbols and bipolar spherical harmonics (two rotations), or by 9 − j symbols and tripolar spherical harmonics (three rotations). We show that these recoupling coefficient expansions, in conjunction with Racah algebra, are the cornerstone of an unusually simple and direct method of (1) proving the Euler–Rodrigues (quaternion) parameter composition rules for two and three consecutive rotations and (2) constructing rotational invariants expressed in Weyl-invariant form. These methods are distinguished by their independence on the usual algebraic approaches, either those of operator algebra, or ordinary algebra.

Perfect IFG-Formulas

IFG logic is a variant of the independence-friendly logic of Hintikka and Sandu. We answer the question: “Which IFG-formulas are equivalent to ordinary first-order formulas?” We use the answer to prove the ordinary cylindric set algebra over a structure can be embedded into a reduct of the IFG-cylindric set algebra over the structure.

Exploring M-theory symmetries through D = 11 supergravity

We study the reduction of the Free Differential Algebra (FDA) of D = 11 supergravity to an ordinary algebra. Such a reduction permits to discover as underlying symmetry of trivial backgrounds an Inönü-Wigner contraction of Sezgin's M-Algebra. Considering non trivial configurations, we also find evidences for symmetries reproducing the lowest levels of E11. This result suggests that the E11 symmetries, which act on non perturbative states as well, are already encoded in the D = 11 FDA, i.e. in the classical supergravity theory.

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct.

On the underlyingE11symmetry of theD= 11 free differential algebra

We study the reduction of the Free Differential Algebra (FDA) of D=11 supergravity to an ordinary algebra. We show that in flat background and with vanishing three-form field strength, the corresponding minimal FDA can be reduced to an Inonu-Wigner contraction of Sezgin's M-Algebra. We also prove that in flat background but with a non trivial three-form field strength, the bosonic FDA can be reduced to the lowest levels of E11. This result suggests that the E11 symmetries, which act on perturbative states as well, are already encoded in the D=11 FDA and are made explicit when the theory is formulated on a enlarged group manifold.

A Normal Form and the Term-linearity of Affine Spaces over GF(3)

An algebra is called term-linear if every term is equal to a linear term or a constant and every n-ary linear term is essential. The term-linearity is a generalization of the linearity of polynomials over ordinary linear algebras or vector spaces. In this paper, as a sequel to a previous paper, we present an explicit normal form for terms over an affine space over GF(3) by binary trees, which shows that the word problem for affine spaces over GF(3) is solvable. Using this normal form, we count the number of all essentially n-ary linear terms and apply the Csákány formula to show that an affine space over GF(3) is term-linear.

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R T )-Lie coalgebra ΓT through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L T , [, ], ΔT, R T ) is an (H,R T )-Lie bialgebra of an ordinary Lie algebra.

Fuzzy transforms: Theory and applications

The technique of direct and inverse fuzzy (F-)transforms of three different types is introduced and approximating properties of the inverse F-transforms are described. All three types of the direct F-transform are transformations from a function space to a finite dimensional vector space. The first (ordinary) F-transform is constructed on the basis of the ordinary algebra of reals, while the other two types of the F-transform are constructed on the basis of residuated lattice. The core idea of the technique of F-transforms is a fuzzy partition of a universe into fuzzy subsets (factors, clusters, granules etc.). We claim that for a sufficient representation of a function defined on this universe, we may consider its average values over fuzzy subsets from the partition. Thus, a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values. A number of theorems establishing best approximation properties of the inverse F-transforms are proved. In fact, three types of the inverse F-transform are the best approximations in average, from below, and from above respectively. As one of many possible applications, we present a method of image compression and reconstruction on the basis of the F-transform.

Congruence preserving functions of Wilke’s tree algebras

As a framework for characterizing families of regular languages of binary trees, Wilke introduced a formalism for defining binary trees that uses six many-sorted operations involving letters, trees and contexts. In this paper a completeness property of these operations is studied. It is shown that all functions involving letters, binary trees and binary contexts which preserve congruence relations of the free tree algebra over an alphabet, are generated by Wilke’s functions, if the alphabet contains at least seven letters. That is to say, the free tree algebra over an alphabet with at least seven letters is affine-complete. The proof yields also a version of the theorem for ordinary one-sorted term algebras: congruence preserving functions on contexts and members of a term algebra are substitution functions, provided that the signature consists of constant and binary function symbols only, and contains at least seven symbols of each rank. Moreover, term algebras over signatures with at least seven constant symbols are affine-complete.