Formal Manipulation Research Articles

Computable identities in the algebra of formal matrices

In this paper, we want to give an explicit description of identities satisfied by matrices n × n over a field k of characteristic 0, in order to be able to compute with formal matrices (“forgetting” their representations with coefficients). We introduce a universal free algebra, where all formal manipulations are made. Using classical properties of identities in an algebra with trace, we reduce our problem to the study of identities among multilinear traces. These are closely linked with the action of the algebra of the symmetric group k[ S m ] on the mth tensor product of E = K n . By proving a theorem about the kernel of this action and its effective version, we can decompose all identities of matrices in an explicit way as linear combinations, substitutions, products or traces of the well-known Cayley-Hamilton identity. This leads to an algorithm of reduction to a canonical form, modulo the ideal of identities of matrices, in the free algebra. Moreover, this set of identities is in fact a prime ideal and a natural extension of the quotient by this ideal gives us a skew field, which is also the fraction ring of the algebra of generic matrices.

A formal approach to designing delay-insensitive circuits

A method for designing delay-insensitive circuits is presented based on a simple formalism. The communication behavior of a circuit with its environment is specified by a regular expression-like program. Based on formal manipulations this program is then transformed into a delay-insensitive network of basic elements realizing the specified circuit. The notion of delay-insensitivity is concisely formalized.

Specification of concurrent objects using auxiliary variables

The role of auxiliary variables in the specification of concurrent objects with multiple inputs and outputs is examined. Auxiliary variables needed in a specification are defined through logical assertions on the interface variables. Necessary conditions and generic rules for the definition of such auxiliary variables are presented. The method is illustrated by specifying a concurrent buffer. Based on this specification, a number of useful properties of concurrent buffers and systems composed of them are derived through formal manipulation.

Regularized string and flow equations

Improved derivations of the string and flow equations for single hermitian-matrix models are given, avoiding the use of formal manipulations previously adopted. Explicitly local and regularized expressions are written down for the scaling operators and their continuum limit is calculated in detail. It is shown that the fundamental property underlying the mathematical structure of the equations is the singularity of the large-N eigenvalue distribution at one of the ends of its support.

Baryons and ladders

By formal manipulation of the QCD functional integral we arrive at a relativistic low energy effective theory of nonlocal colour singlet mesons and baryons, which at tree level sums up ladders of effective glue exchange between constituent quarks.

Path Integral and Brownian Motion

A simple formulation of the path integral theory of Brownian motion is given on the basis of a stochastic differential equation. Besides a formal manipulation, a practical method is proposed to calculate the path integrals explicitly. To demonstrate the usefulness of the method, linear and nonlinear diffusion processes are treated and compared with analytical solutions.

Formal manipulation of modular software systems

Formal manipulation of modular software systems

The meaning of pure mathematics

There are two classical answers to the question what is the meaning of pure mathematics: (1) Formalism. Pure mathematics is a symbolic construction; it becomes meaningful only when it is applied to describe physical reality. (2) Platonism. Pure mathematics describes an ideal reality which exists independently of man. I cannot accept either of those answers. Formalism is unsatisfactory first because it ignores the structures imagined by mathematicians and underlying the symbols and second because many concepts of pure mathematics do not have any physical interpretation nor the slightest chance of getting one in the future (this is clear, e.g., about the cardinal 2!,, a paradoxical decompositions of the sphere, etc.). Platonism is unsatisfactory because it violates our instinctive drive to obey Ockham's principle of parsimony: entia non sunt multiplicanda praeter necessitatem. I propose a third answer which I call intentionalism. Intentionalism says that pure mathematics is a description of finite structures consisting of finitely many individually imagined objects. (The term intentionalism is chosen for its contrast with extensionalism which accepts actually infinite sets and leads naturally to Platonism. The difference between intentionalism and formalism is more subtle, see Remark 4.3 below.) The idea of intentionalism is not new but it is not generally accepted. I think that this is so because hitherto intentionalism had a difficulty with the interpretation of quantifiers. For example the axioms of arithmetic cannot be literally true about any collection of imagined individual integers since man can imagine only finitely many of them. The main result of this paper is a. solution of this problem. Quantifiers will be explained away as a formal trick which mathematicians can but do not have to use. I will try to describe how mathematicians think in reality and I believe that, when they do not mechanically perform formal manipulations, they think in the

Order-of-Magnitude Reasoning with Fuzzy Relations

Relative orders of magnitude of quantities (e.g. a is slightly larger than b, c is negligible in comparison with d, e is close to f) are a kind of partial knowledge which is often available in engineering problems. In this paper we study how to represent this type of information and how to reason with it. Order-of-magnitude relations are modelled by fuzzy relations. This kind of semantics enables us to acknowledge that i) the extent to which two numerical values satisfy such relations is often a matter of degree and is also context-dependent; ii) these relations are often weakly or strongly transitive rather than just transitive in the usual sense. Conclusions can be obtained by composing fuzzy relations representing relative-order-of-magnitude information together, or by propagating these relations through algebraic constraints. A proper choice of the form of the fuzzy relations reduces the composition operations in practice to arithmetic operations on fuzzy numbers which are easy to compute. Rules which enables a formal manipulation of the relations between variables, in agreement with the fuzzy semantics, are derived. This improves approaches to qualitative reasoning problems previously developed in Artificial Intelligence, in a way which better sticks to the semantics of the available information.

Reply to Jacquette

I believe he has missed the point of the Chinese Room argument. Though the point is really very simple, it does sometimes tend to get lost in all of the interpretations, criticisms, and intuitions which are discussed in these matters. Here it is: programs are purely syntactically defined. The program consists of formal symbol manipulations. But these formal symbol manipulations by themselves have no semantic content. Any semantic interpretation has to be from outside the system of the symbol manipulation. This is a strict logical consequence of the syntactical character of the programs and the fact that syntax is not sufficient for semantics. But in this respect programs differ from human minds, which have mental states with intrinsic mental or semantic content. That is the argument. It is that simple. In the abstract of the original BBS article and in various other places, I set it out as a derivation from axioms, but it can equally be understood discursively, as I stated it above. The Chinese Room Argument reminds us of these facts by reminding us that the man in the Chinese room can be the hardware implementing the formal syntactical steps in any program at all, and yet the 'system' of hardware and software still does not have the appropriate semantic content which is supposed to be associated with those steps. This will be the case, for example, when he is a native monolingual English speaker going through the program for understanding Chinese. Using obvious abbreviations, Strong Al says:

Principles of BRST quantization

The BRST quantization as a quantization method ised. It is pointed out that all formal manipulations are only valid if all inner products in the original state space are finite. This restriction is also shown to be necessary in order to obtain the correct results for trivial models. It is argued that the investigations of BRST invariant states of the form | M〉| G〉, where M and G denote matter and ghost states, are sufficient to determine the physics of the BRST invariant subspace. A prescription of how to solve Q| M〉| G〉 = 0 is given for a general class of models. Solutions are classified into canonical and noncanonical ones. The latter cannot be obtained in a canonical quantization of the corresponding gauge model.

Is relativity relevant for the electrical engineer?

For most of us Relativity evokes intricate formal manipulations, pleasing to the professional mathematician, but awe-inspiring, and even repulsive, to the average EE. And yet, Physics of the greatest importance hides behind the forest of tensors and fourdimensional symbols which confronts the reader of relativistic texts. Einstein's fundamental 1905 paper, "Zur Elektrodynamik Bewegter Korper", already contains, in its title, the reason why the Physics of Relativity may indeed be relevant to the Electrical Engineer [l]. The present paper endeavors to develop this idea by discussing, in very broad terms and without much rigor, a number of problems whch may be solved correctly and concisely by applying relativistic methods.

On the hidden supersymmetry in stochastic quantization.

It is shown that the hidden supersymmetry discovered in parabolic stochastic differential equations is a spurious artifact originating from formal manipulations which lead to incorrect results and contradictions. The contradictions are illustrated on simple physical examples. The correct formulas both for the conditional probability and for the generating functional are given in path-integral forms.

Excitation energy transfer dynamics in nonperiodic molecular systems: Nonequilibrium green's function description

The nonequilibrium Green's function (NGF) technique is extended to the derivation of a generalized master equation for the description of energy transfer dynamics within an ensemble of two-level molecules. The use of the localized representation of intramolecular excitations requires some formal manipulations different from the standard procedures within the NGF technique. Noting the possible fast oscillatory time dependence of the off-diagonal density matrix of intramolecular excitations, a non-Markovian version of the generalized master equation is derived from Dyson's equation for the NGF. The dynamics of the vibrational modes to which the intramolecular excitations are coupled have been splitted into a nonthermal (coherent) and thermal (incoherent) part. Such a splitting is most suitable for the consideration of the mechanism of self-localization of excitation energy. The presented approach gives the basis for a further study of the dynamics of Davydov solitons at finite temperatures.

On the existence of the chiral Schwinger model.

We give support to a speculation made in a recent paper that the (anomalous) chiral Schwinger model may exist as a well-defined unitary theory in the gauge-invariant sector, by calculating the vacuum expectation value of three representative gauge-invariant operators, using functional as well as operator methods. The calculations also demonstrate explicitly the absence of vacuum-polarization effects in this sector, as predicted recently on the basis of purely formal manipulations of the generating functional. As a consequence, we find a linearly rising qq\ifmmode\bar\else\textasciimacron\fi{} potential for all separations.

Formal manipulation of Forrester diagrams by graph grammars

A grammar that formally constructs the types of diagrams used in system dynamics is fully described. It belongs to the special kinds of grammars (attributed programmed graph grammars) that are applied to the construction of graphs and geometric figures. The flow diagrams used in system dynamics have been defined by 'attributed graphs' so that the approach could be applied. The grammar manipulates the graphs according to the requirements of the methodology. Basically, a method to achieve a formal program for interactive modeling upon any diagram, starting from the formal specification of the diagram, is presented. >

Systems semantics: principles, applications, and implementation

Systems semantics extends the denotational semantics of programming languages to a semantics for the description of arbitrary systems, including objects that are not computations in any sense. By defining different meaning functions, the same formal description may be used to denote different system properties, such as structure, behavior, component cost, and performance aspects (e.g., timing). The definition of these semantic functions also provides guidance in language design, in particular for the match between language constructs and the system concepts to be expressed. Aiming at compositionality ensures useful properties for formal manipulation. In this fashion, the meaning functions can be made sufficiently simple to serve not only as a direct implementation on a machine but also as rules for reasoning about systems in a transformational manner. As the applications show, however, compositionality can be ensured only through careful consideration of the characteristics of the flow of information inside the system. Two classes of application are discussed: Unidirectional systems, in particular digital systems without feedback (combinational) and with feedback (sequential), and a certain class of analog systems. Nonunidirectional systems, in particular two-port analog networks. The emphasis will be on the functional style of description and on formal reasoning (theorem proving, derivation of properties). Implementation and rapid prototyping strategies in various system description environments are also briefly discussed. These would permit the concepts of system semantics to be explored without the need for a complete implementation.

Center manifold reduction and normal form transformations in systems with additive noise

The center manifold reduction and normal form transformations may be applied stochastic differential equations in a manner analogous to the treatment of ordinary differential equations. Results are identical to those obtained from an analysis of Fokker-Planck equations, but the formal manipulations are much simpler.

Streamlined Darwin simulation of nonneutral plasmas

Efficient, new algorithms that require less formal manipulation than previous implementations have been formulated for the numerical solution of the Darwin model. These new procedures reduce the effort required to achieve some of the advantages that the Darwin model offers. Because the Courant-Friedrichs-Lewy stability limit for radiation modes is eliminated, the Darwin model has the advantage of a substantially larger time-step. Further, without radiation modes, simulation results are less sensitive to enhanced particle fluctation noise. We discuss methods for calculating the magnetic field that avoid formal vector decomposition and offer a new procedure for finding the inductive electric field. This procedure avoids vector decomposition of plasma source terms and circumvents some source gradient issues that slow convergence. As a consequence, the numerical effort required for each of the field time-steps is reduced, and more importantly, the need to specify several nonintuitive boundary conditions is eliminated.

Perturbation theory for point interactions in three dimensions

The formal manipulation with delta -potentials for modelling contact interactions is compared with the recent mathematical theory of this phenomenon.