Non-classical Systems Research Articles

Topographical analyses of homonuclear multiple bonds between main group elements

Recent experiments have resulted in the completion of the series of Group 14 and Group 15 element double-bond systems, R(n)E=ER(n) (E = C - Pb, n = 2; E = N - Bi, n = 1). Furthermore, new families of multiple-bonded species have been discovered, such as the radical anion [RSnSnR](-) , the close ion pairs [RE(mu2Na)2ER] (E = Ge,Sn), and a digallyne [RGa(mu2Na)2GaR] for which a Ga=Ga triple bond was formulated. Some of these compounds show classical multiple bond features (i.e. the dipnictogens RE=ER, E=N-Bi) in the sense that planar structures with short E-E distances are observed. However, many (i.e. R2E=ER2, E = Si - Pb) do not behave as expected for compounds with multiple bonds. They have trans bent structures, show enormous variation in their E-E distances, and some dissociate easily under E-E bond cleavage in solution. These properties raised doubts as to whether these compounds can be formulated as multiple-bonded systems. Using the electron localization function (ELF) it is possible to clearly show the topographical similarities between classical and nonclassical multiple bonds; ELF divides these systems into unslipped (classical) and slipped (nonclassical) systems. ELF can also be employed to confirm the nonexistence of multiple bonds. Therefore, topographical analyses using ELF are useful to categorize a bonding system. In particular, the bonds in the heavier Group 14 double systems and the Ga-Ga bond in digallyne are clearly shown by this method as slipped double and triple bonds, respectively.

On the intuitionistic force of classical search

The combinatorics of classical propositional logic lies at the heart of both local and global methods of proof-search enabling the achievement of least-commitment search. Extension of such methods to the predicate calculus, or to non-classical systems, presents us with the problem of recovering this least-commitment principle in the context of non-invertible rules. One successful approach is to view the non-classical logic as a perturbation on search in classical logic and characterize when a least-commitment (classical) search yields sufficient evidence for provability in the (non-classical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a type-theoretic view of this approach for the case in which the non-classical logic is intuitionistic. We develop a system of realizers (proof-objects) for sequents in classical propositional logic (the types) by extending Parigot's λμ-calculus, a system of realizers for classical free deduction ( cf. natural deduction). Our treatment of disjunction exploits directly the multiple-conclusioned form of LK as opposed to the single-conclusioned form of LJ. Consequently, it requires the addition of another binding operator, called ν, to λμ. This choice is motivated by our concern to reflect the properties of classical proof-search in the system of realizers. Using this framework, we illustrate the sense in which intuitionistic search can be viewed as a perturbation on classical search. As an application, we develop a proof procedure based on the natural extension of the notion of uniform proof to the multiple-conclusioned classical sequent calculus Harrop fragment of intuitionistic logic. This paper develops the proof-theoretic aspects of the approach.

Is there a zande logic?

The issue of what consequences to draw from the existence of non-classical logical systems has been the subject of an interesting debate across a diversity of fields. In this paper the matter of alternative logics is considered with reference to a specific belief system and its propositions :the Azande are said to maintain beliefs about witchcraft which, when expressed propositionally, appear to be inconsistent. When the Azande have been presented with such inconsistencies, they either fail to see them as such or else accept them as non-problematical. Is our knowledge of logical truths a relative and culturally determined phenomenon, or is there some (transcendent) criterion that allows us to adjudicate between alternative logical systems? The authors propose an approach for resolving disputes about the status of Azande reasoning which assumes a paraconsistent framework, thus providing a new perspective on this debate

A Quasi-Decoupling Approach for Nonclassical Linear Systems in State Space

By adopting the orthogonal transformations provided by the generalized real Schur decomposition, it is shown that every nonclassical linear system in state space can be transformed into block upper triangular form, to which the quasi-decoupling solution can be progressively carried out by solving the either first or second-order component equations with the “back substitution.” The distinct characteristics of generalized eigenvalue problems from those of standard ones are discussed. Favorable properties of the proposed method include: no inverting of any system matrix, indiscriminate applicability to both defective and nondefective systems, the simultaneous decoupling of the adjoint problem, and numerical stability. Illustrative examples are also presented.

Non-stationary states in radical polymerisation with size-dependent termination: Studies by the group-termination procedure—4. The effects of chain-transfer

The occurrence of chain-transfer introduces complications into calculations involving non-classical systems, both stationary and non-stationary. A procedure for treating the latter is described and applied in this paper. It resembles that formerly developed for systems without transfer and depends on setting up simultaneous differential equations for the time variation of r ̄ , the mean radical size, and R·, the total radical concentration. Changes in the overall reactivity of the group of radicals with changes in size-distribution are accommodated by a factor f f , which plays a role similar to that of the factor f in the earlier treatment. The procedure is applied to the non-stationary phases arising when the rate of chain-initiation in a stationary reaction is abruptly increased or decreased. The time-variations of r ̄ and R· are calculated, together with the resulting pre- and after-effects ( ΔM pre, ΔM aft, respectively). Some new features in the time-variations mentioned have been found, notably the existence of reversal points (even in classical systems with transfer). The occurrence of transfer reduces ΔM pre and ΔM aft. Plots of log( ΔM pre) or log( ΔM aft)) vs log( k p M) or log k to are very nearly—but not exactly—linear in the systems examined. Factors influencing the mean slopes of these lines are discussed. Semi-quantitative equations are developed relating ΔM pre (or ΔM aft) to the reaction parameters. They may be used to estimate certain ratios of k p M and k to from pre- (or after-) affects measured experimentally under prescribed conditions.

The Inheritance of Phenotypes: an Adaptation to Fluctuating Environments

We discuss simple models for the evolution of rates of spontaneous and induced heritable phenotypic variations in a periodically fluctuating environment with a cycle length between two and 100 generations. For the simplest case, the optimal spontaneous transition rate between two states is approximately 1/ n(where nis the cycle length). It is also shown that selection for the optimal transition rate under these conditions is surprisingly strong. When nis small, this means that the heritable variations are produced by non-classical inheritance systems, including non-DNA inheritance systems. Thus, it is predicted that in genes controlling adaptation to such environments, non-classical genetic effects are likely to be observed. We argue that the evolution of spontaneous and induced heritable transitions played an important role in the evolution of ontogenies of both unicellular and multicellular organisms. The existence of a machinery for producing induced heritable phenotypic variations introduces a “Lamarckian” factor into evolution.

Exact stationary response solutions of nonlinear dynamic systems

Exact stationary solutions are constructed for nonlinear dynamic systems subjected to stochastic excitation. The results are then applied to both classical and nonclassical stochastic vibration systems.

Non-stationary states in free-radical polymerizations with size-dependent termination. Studies by the group-termination procedure—3. Further consideration of after-and pre-effects and evaluation of absolute coefficients

In earlier studies of the kinetic quantities in non-stationary stages of non-classical free-radical polymerization a relation between the total pre- and after-effects ( ΔM pre, ΔM af, respectively) and the rate coefficients k p k to was proposed. This arose from the results of numerical computation. Further work shows that the relation is derivable from the basic kinetic equations and is therefore of general validity. We have been able to formulate a new quantity Λ which is a function of the ratio of the final and initial rates of initiation (I 2, I 1, respectively) (not the individual values) and is independent of the rate coefficients. It is useful for the evaluation of absolute rate coefficients in non-classical systems and the procedure involved is explained. Values of Λ are presented in the form of contour lines on a β vs I 2/I 1 diagram. A method for calculating Λ based on empirical relations between radical concentrations and mean radical sizes is outlined. A special case has been found [ β = 0.25, see equation (1)] for which analytical solutions can be obtained. The derived expressions for radical concentrations and ΔM pre and ΔM aft are presented. As in earlier papers in this series concerned with non-stationary states we assume chain transfer to be insignificant.

Closure to “Discussion of ‘Complex Modes and Solvability of Nonclassical Linear Systems’” (1993, ASME J. Appl Mech., 60, pp. 787–788)

Closure to “Discussion of ‘Complex Modes and Solvability of Nonclassical Linear Systems’” (1993, ASME J. Appl Mech., 60, pp. 787–788)

Complex Modes and Solvability of Nonclassical Linear Systems

The purpose of this article is to give necessary and sufficient conditions under which nonclassical linear systems can be decoupled or become solvable.

Non-stationary states in free-radical polymerizations with size-dependent termination. Studies by the group-termination procedure—2. Pre-effects and systems with intermittent initiation

Calculations on non-classical systems of the type mentioned in the title have been carried out using the treatment described in Part I. As before chain transfer has been neglected. It is shown that the properties of the total pre-effect ΔM pre are analogous to those previously deduced for the after-effect. Thus, for given rates of chain-starting in the initial and final stationary states, the following proportionalities are found: ΔM pre ∝(k p M) (1+β) (1−β) k to (1−β) , ΔM pre ∝k p MR 1p 2· and ΔM pre∝ k p MR 2p 2·. R 1 · and R 2· are the initial and final radical concentrations, β is the geometric mean exponent in the relation between termination coefficient and radical size and other symbols have their conventional meanings. Non-classical polymerizations with intermittent initiation of the “rotating sector” type (light and dark rates of initiation I 2p, I 2a, respectively) have been treated by the same procedure. Plots of the ratio ω/ω s ( ω = mean rate of polymerization and ω s = rate with continuous full rate of initiation) against t l / τ have been constructed ( t l = duration of “light” period, τ = mean life-time of chains ) for β = 0.1, β = 0.139 and β = 0.2. It is shown that, for given β and with I 2 a = 0, ω/ω s is a single-valued function of t l / τ. The resulting curves, although resembling those for classical systems, are displaced from the latter; moreover, members of different families (different β) are not superposable. Systems with intermittent initiation together with a constant background (thermal) initiation and β = 0.1 and β = 0.2 have been investigated similarly. Applications to the evaluation of absolute rate coefficients are discussed.

Modal Synthesis Method for General Dynamic Systems

A modal synthesis method is presented to calculate the lower eigenproperties of a general dynamic system divided into subsystems. The left and right eigenproperties of the subsystems associated with their equations of motion expressed in the state form are used. To account for the truncated higher eigenproperties of a subsystem, the force derivative approach formulated for general dynamic systems is used. The reduced order equations of the subsystems are coupled by enforcing displacement and force compatibility conditions at the interface. The approach is quite general, so it can be used with nonclassical damped systems as well as with systems with asymmetric matrices, such as gyroscopic and other circulatory systems. The numerical examples of: (1) A piping system carrying rotating parts and provided with discrete dampers; and (2) a large rotor‐disk‐bearing system are presented to demonstrate the effectiveness of the proposed approach. It is shown that lower eigenproperties of a system can be accurately c...

A New Class of Narrow-Band Organic Polymers with a Ferromagnetic Ground State

Abstract A new topological method predicting the number of nonbonding molecular orbitals (NBMO's) in organic molecules is elaborated. The method is a generalization of the Longuet-Higgins theorem and extends significantly the number of non-classical (non-Kekule) π -electron systems to alternant and non-alternant hydrocarbons and their heteroanalogues. The polymers of this type possess a band of degenerate NBMO's. A study of the spin correlation in the degenerate band shows that this group of nonclassical systems represents a new class of high-spin polymers, some of which probably with a ferromagnetic ground state.

Structure and properties of nonclassical polymers. VII. Polymers with heteroatoms

AbstractIt is shown that the Coulson–Rushbrooke theorem for odd alternant hydrocarbons [15] can be generalized for some classes of nonclassical systems containing heteroatoms. The energy spectrum of such conjugated polymers containing heteroatomic groups, in particular a nitroxide group >NO, is analogous to that studied [1–4] in homonuclear alternant nonclassical polymers, whose ground state is characterized by maximum spin multiplicity. Their band structure is characterized by a wide energy gap, in the middle of which there is a band of degenerate or nearly degenerate quantum states.The presence of a narrow band of degenerate quantum states, the Hund rule, and the alternant character of the polymers of this group determine the maximum multiplicity of their ground state and may give rise to the creation of magnetic order in such compounds, if interchain correlations occur. This is in line with the investigations in Refs. 1–11.

The modal status of antinomies.

What is the modal status of antinomies? Classical modal logic provides no interesting answer to this question because it lets antinomies turn all wellformed formulas (including all modal formulas) into theorems. In the present note we propose two nonclassical modal systems which do not suffer from this defect. Both systems are obtained by supplementing the semantics of Asenjo's and Tamburino's antinomic propositional logic L (see [1], familiarity with which will be assumed in this article) with a very natural-sounding truth condition for modal formulas. The surprising result is that antinomies are in any case both necessary and impossible: according to the second system we propose, they are both non-necessary and possible as well. It may be doubted whether these results are in accord with our intuitions. However, it should be remembered that our intuitions were formed during centuries of classical slumber; acquiring the right intuitions in antinomic thinking may simply be a matter of time.

An extension of the Coulson-Rushbrooke theorem

It is shown that the Coulson-Rushbrooke theorem for odd alternant hydrocarbons and polymers can be extended to some classes of non-classical (non-Kekulé) systems containing heteroatoms. The extension is only applicable to non-bonding molecular orbitals. The band structure of such polymers is analogous to that of odd alternant non-classical polymers, whose ground state is characterized by maximum spin multiplicity.

Sobre sistemas formales no clásicos

Resumen

Positivity of the K-entropy on non-abelian K-flows

A non-commutative extension of certain aspects of classical probability theory is presented in such a manner that the notion of Kolmogorov entropy can be extended to a large class of non-classical dynamical systems. In particular, the generalized K-entropy so defined is shown to be strictly positive on the class of non-abelian K-flows.

Provability as a deontic notion

The purpose of this paper is to mark a significant difference between classical and several non-classical prepositional calculi. The argument presupposes familiarity with Kripke/Hintikka semantics for modal logic. The non-classical systems are Hintikka's logic of belief and alethic modal systems which have Kripke/Hintikka semantics.

On nonclassical canonical systems

Generalizations in the canonical theory of dynamics are made; at first transformations which augment the number of canonical variables, and secondly differential transformations of the independent variable are outlined. This is applied to the perturbed two-body problem. The results are canonical systems using independent variables other than time. This leads to ‘Delaunay-similar’ sets of 8 canonical elements when the Jacobian equation is separable. The application of the theory to the KS-transformation yields a completely regular canonical system in a 10-dimensional phase-space, using the eccentric anomaly as independent variable. Subsequently sets of 10 regular canonical elements are introduced.