We consider the Dirac equation in 3+1 dimensions with spherical symmetry and coupling to 1/r singular vector potential. An approximate analytic solution for all angular momenta is obtained. The approximation is made for the 1/r orbital term in the Dirac equation itself not for the traditional and more singular 1/r 2 term in the resulting second order differential equation. Consequently, the validity of the solution is for a wider energy spectrum. As examples, we consider the Hulthén and Eckart potentials.

This paper provides equational semantics for Dung's argumentation networks. The network nodes get numerical values in [0,1], and are supposed to satisfy certain equations. The solutions to these equations correspond to the ?extensions? of the network. This approach is very general and includes the Caminada labelling as a special case, as well as many other so-called network extensions, support systems, higher level attacks, Boolean networks, dependence on time, and much more. The equational approach has its conceptual roots in (...) the nineteenth century following the algebraic equational approach to logic by George Boole, Louis Couturat, and Ernst Schroeder. (shrink)

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a 0+a 1tan ξ+a 2tan 2 ξ, which (...) indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method. (shrink)

Structural equations have become increasingly popular in recent years as tools for understanding causation. But standard structural equations approaches to causation face deep problems. The most philosophically interesting of these consists in their failure to incorporate a distinction between default states of an object or system, and deviations therefrom. Exploring this problem, and how to fix it, helps to illuminate the central role this distinction plays in our causal thinking.

This paper uses duality to elaborate Slutzky equations of risks in quasi-linear decision models extended by independent background risks. Wealth, substitution and total effects are characterized in terms of mean-variance preferences. It is shown that both Pratt and Zeckhauser’s proper risk aversion and Kimball’s standard risk aversion are sufficient for negative substitution effects.

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, $W\gamma _{ik} = - G_{ik} + \tfrac{1}{2}g_{ik} g^{mn} G_{mn} - k(T_{ik} - \tfrac{1}{2}g_{ik} g^{mn} T_{mn} )$ together with the covariant Hilbert-gauge condition, $(\gamma _i^h - \tfrac{1}{2}\delta _i^k g^{mn} \gamma _{mn} )_{;k} = 0$ where W denotes the covariant wave operator and G ik the Einstein tensor of the metric g ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave (...) solutions γ ik(z)with g ik Z,i Z,k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,Δp(Δg*)2(Δx)3≥h hG/ c 3 concerning the total nonlinear gravitational field g *ik =g k +γ k. (shrink)

This paper provides a philosophical analysis of the Price equation and its role in evolutionary theory. Traditional models in population genetics postulate simplifying assumptions in order to make the models mathematically tractable. On the contrary, the Price equation implies a very specific way of theorizing, starting with assumptions that we think are true and then deriving from them the mathematical rules of the system. I argue that the Price equation is a generalization-sketch, whose main purpose is to provide a unifying (...) framework for researchers, helping them to develop specific models. The Price equation plays this role because, like other scientific principles, shows features as abstractness, unification and invariance. By underwriting this special role for the Price equation some recent disputes about it could be diverted. (shrink)

Locus equations contain an economical set of hidden (i.e., not directly observable in the data) parameters of speech that provide an elegant way of characterizing the ubiquitous context-dependent behaviors exhibited in speech acoustics. These hidden parameters can be effectively exploited to constrain the huge set of context-dependent speech model parameters currently in use in modern, mainstream speech recognition technology.

A nonlocal equation of motion with damping is derived by means of a Mori-Zwanzig renormalization process. The treatment is analogous to that of Mori in deriving the Langevin equation. For the case of electrodynamics, a local approximation yields the Lorentz equation; a relativistic generalization gives the Lorentz-Dirac equation. No self-acceleration or self-mass difficulties occur in the classical treatment, although runaway solutions are not eliminated. The nonrelativistic quantum case does not exhibit runaways, however, provided one remains within a weak damping approximation. (...) The correspondence limit shows that a classical limit may be taken, again within the same approximation. (shrink)

Recent accounts of actual causation are stated in terms of extended causal models. These extended causal models contain two elements representing two seemingly distinct modalities. The first element are structural equations which represent the or mechanisms of the model, just as ordinary causal models do. The second element are ranking functions which represent normality or typicality. The aim of this paper is to show that these two modalities can be unified. I do so by formulating two constraints under which (...) extended causal models with their two modalities can be subsumed under so called which contain just one modality. These two constraints will be formally precise versions of Lewissystem of weights or priorities” governing overall similarity between possible worlds. (shrink)

Barut showed us how it is possible to get a Poincaré invariant n-body equation with a single time. Starting from the Barut equation for n-free particles, we show how to generalize it when they interact through Dirac oscillators with different frequencies. We then particularize the problem to n=2 and consider the particle-antiparticle system whose frequencies are respectively ω and −ω. We indicate how the resulting equation can be solved by perturbation theory, though the spectrum and its comparison with that of (...) the mesons will be given in another publication. (shrink)

A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence relative to some class of algebras. We characterize logics admitting an equational completeness theorem that are either locally tabular or have some tautology. In particular, it is shown that a protoalgebraic logic admits an equational completeness theorem precisely when it has two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem is shown to (...) be decidable both for logics presented by a finite set of finite matrices and for locally tabular logics presented by a finite Hilbert calculus, it becomes undecidable for arbitrary logics presented by finite Hilbert calculi. (shrink)

The potential role of locus equations in three existing models of human classification behavior is examined. Locus equations can play a useful role in single-prototype and boundary-based models for human consonant recognition by reducing model complexity.

The remainder equations were introduced by S. O. Hansson. In this paper, we will give an exhaustive characterization of the set of solutions for remainder equations. Moreover, solutions to some unsolved problems proposed by Hansson are reported.

Hall [(2007), Philosophical Studies, 132, 109–136] offers a critique of structural equations accounts of actual causation, and then offers a new theory of his own. In this paper, I respond to Hall’s critique, and present some counterexamples to his new theory. These counterexamples are then diagnosed.

This paper provides characterisations of the equational theory of the PER model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words, we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model, a version of the Kreisel-Lacombe-Shoenfield theorem allows us to (...) transfer the result from the domain model to the PER model. (shrink)

Hybrid identities are sentences in a special second order language with identity. The model classes of sets of hybrid identities are called hybrid solid varieties. We give a Birkhoff-type-characterization of hybrid solid varieties and develop a hybrid equational logic.

It is well known that the model categories of universal Horn theories are locally presentable, hence essentially algebraic . In the special case of quasivarieties a direct translation of the implicational syntax into the essentially equational one is known . Here we present a similar translation for the general case, showing at the same time that many relationally presented Horn classes are in fact quasivarieties.

The increasing digitalization in the field of psychological and educational testing opens up new opportunities to innovate assessments in many respects (e.g., new item formats, flexible test assembly, efficient data handling). In particular, computerized adaptive testing provides the opportunity to make tests more individualized and more efficient. The newly developed continuous calibration strategy (CCS) from Fink, Born, Spoden, and Frey (2018) makes it possible to construct computerized adaptive tests in application areas where separate calibration studies are not feasible. Due to (...) the goal of reporting on a common metric across test cycles, the equating is crucial for the CCS. The quality of the equating depends on the common items selected and the scale transformation method applied. Given the novelty of the CCS, the aim of the study was to evaluate different equating setups in the CCS and to derive practical recommendations. The impact of different equating setups on the precision of item parameter estimates and on the quality of the equating was examined in a Monte Carlo simulation, based on a fully crossed design with the factors common item difficulty distribution (bimodal, normal, uniform), scale transformation method (mean/mean, mean/sigma, Haebara, Stocking-Lord), and sample size per test cycle (50, 100, 300). The quality of the equating was operationalized by three criteria (proportion of feasible equatings, proportion of drifted items, and error of transformation constants). The precision of the item parameter estimates increased with increasing sample size per test cycle, but no substantial difference was found with respect to the common item difficulty distribution and the scale transformation method. With regard to the feasibility of the equatings, no differences were found for the different scale transformation methods. However, when using the moment methods (mean/mean, mean/sigma), quite extrem levels of error for the transformation constants A and B occurred. Among the characteristic curve method the performance of the Stocking-Lord method was slightly better than for the Haebara method. Thus, while no clear recommendation can be made with regard to the common item difficulty distribution, the characteristic curve methods turned out to be the most favorable scale transformation methods within the CCS. (shrink)

Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL + is (...) also finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product. (shrink)

In the article, in which religious attitudes and behaviours in rural life and agricultural activities in Antalya have been studied, the religious mentality of the farmers engaged in agricultural activities has been analysed in the context of agricultural perceptions, agricultural production problems and religious attitudes scales. Human being has cultivated the soil to meet his most basic physiological needs, and the stages in these cultivation processes constitute the development stages of human history. In this way, it could be said that (...) the history of farming is the history of humanity. Farming, which has undergone a structural transformation with mechanization, is experiencing its most difficult period in the production-health equation today. The ever-increasing population in the world has led to the need to diversify products as well as to produce more. This brought the improvement of seeds, analysis of the soil, spraying in the production process all from the traditional form to a modern form, which brought the problem of trust towards the product and the producer to the fore. In our study, which focuses on the question of what is the effect of religious attitudes on the solution of agricultural production problems, the data have been obtained through face-to-face interviews with the farmers of Antalya and three scales have been developed as agricultural perception, agricultural production problems and religious attitudes. While the agricultural perceptions scale is dealt with three stages: farmers’ perceptions of soil, profession and technology, religious attitude scale is dealt with in four stages: fate, beliefs, traditional practices, and fiqh rules. The scale of agricultural production problems is discussed as one-dimensional. Data collected through each of these three scales has been analysed with the help of standard deviation and mean, t-test results and structural equation modelling (SEM) techniques. Within the framework of the structural equation model, four hypotheses have been developed and these hypotheses have been confirmed. According to structural equation modelling data, fate perception elaborates 17% of variance in farming perception and 12% of variance in religious attitude perception. While farming perception verbalises 31 % of variance in agricultural production problems, religious attitude perception explains 6 % of variance in agricultural production problems. (shrink)

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, (...) the paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains. (shrink)

Critiques of Neoclassical Economics extend, unsurprisingly, to its mathematical structure. The discussion has largely focused on General Equilibrium Theory (GET), a formalism developed by Leon Walras over a century ago. Internally consistent, but highly unrealistic, GET lacks predictive power, and has been a historical failure. As an alternative, this article proposes a methodology largely developed by Grabner et al. (2019), in which Agent-Based Models (ABMs) are linked with existing Equation-Based Models (EBMs) as a means of developing a more powerful formalism. (...) The approach is illustrated by application to the Arrow- Debreu (AD) model of Neoclassical theory, and the Kuznets Curve of Developmental Economics. Broader implications for the social and natural sciences are briefly considered. (shrink)

Although neural encoding by bats and owls presents seductive analogies, the major contribution of locus equations and orderly output constraints discussed by Sussman et al. is the demonstration that important acoustic information for speech perception can be captured by elegant and neurally-plausible learning processes.

In its original form Dirac's equations have been expressed by use of the γ-matrices γμ, μ=0, 1, 2, 3. They are elements of the matrix algebra M 4 (ℂ). As emphasized by Hestenes several times, the γ-matrices are merely a (faithful) matrix representation of an orthonormal basis of the orthogonal spaceℝ 1,3, generating the real Clifford algebra Cl 1,3 . This orthonormal basis is also denoted by γμ, μ=0, 1, 2, 3. The use of the matrix algebra M 4 (...) (ℂ) to represent Cl 1,3 has some unsatisfactory aspects. The γ-matrices contain imaginary numbers as entries whereas Cl 1,3 is real. Moreover, as a matrix algebra Cl 1,3 is M 2 (ℍ) but only a part of M 4 (ℂ). For that reason we investigate in this paper several forms of Dirac's equations in terms of M 2 (ℍ) instead of M 4 (ℂ). In Section1 we survey Dirac's equations describing the interaction of matter with electromagnetic, electroweak, and strong fields. Section2 deals with electromagnetic/weak interactions employing M 2 (ℍ). Finally, in Section3 we deal with Dirac's equations for strong interactions between quarks. In contrast to su(2) ⊕ u(1), the Lie algebra su(3) is not isomorphic to any subalgebra of Cl 1,3 . Therefore we do not give a description of strong interactions by use of M 2 (ℍ). Instead of such an approach we describe these interactions using the space of quadruples of bivector fields in Cl 1,3 . The thus obtained description has remarkable formal resemblance to the original Dirac equations using wave functions with values in the linear spaceℂ 4. (shrink)

We consider algebras on binary relations with two main operators: relational composition and dynamic negation. Relational composition has its standard interpretation, while dynamic negation is an operator familiar to students of Dynamic Predicate Logic (DPL) (Groenendijk and Stokhof, 1991): given a relation R its dynamic negation R is a test that contains precisely those pairs (s,s) for which s is not in the domain of R. These two operators comprise precisely the propositional part of DPL.This paper contains a finite equational (...) axiomatization for these dynamic relation algebras. The completenessresult uses techniques from modal logic. We also lookat the variety generated by the class of dynamic relation algebras and note that there exist nonrepresentable algebras in this variety, ones which cannot be construedas spaces of relations. These results are also proved for an extension to a signature containing atomic tests and union. (shrink)

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

Applying the resolution–scale relativity principle to develop a mechanics of non-differentiable dynamical paths, we find that, in one dimension, stationary motion corresponds to an Itô process driven by the solutions of a Riccati equation. We verify that the corresponding Fokker–Planck equation is solved for a probability density corresponding to the squared modulus of the solution of the Schrödinger equation for the same problem. Inspired by the treatment of the one-dimensional case, we identify a generalization to time dependent problems in any (...) number of dimensions. The Itô process is then driven by a function which is identified as establishing the link between non-differentiable dynamics and standard quantum mechanics. This is the basis for the scale relativistic interpretation of standard quantum mechanics and, in the case of applications to chaotic systems, it leads us to identify quantum-like states as characterizing the entire system rather than the motion of its individual constituents. (shrink)

We consider equational theories based on axioms for recursively defining functions, with rules for equality and substitution, but no form of induction—we denote such equational theories as PETS for pure equational theories with substitution. An example is Cook’s system PV without its rule for induction. We show that the Bounded Arithmetic theory $\mathrm {S}^{1}_2$ proves the consistency of PETS. Our approach employs model-theoretic constructions for PETS based on approximate values resembling notions from domain theory in Bounded Arithmetic, which may be (...) of independent interest. (shrink)

Equational logic for total functions is a remarkable fragment of first-order logic. Rich enough to lend itself to many uses, it is also quite austere. The only predicate symbol is one for a notion of equality, and there are no logical connectives. Proof theory for equational logic therefore is different from proof theory for other logics and, in some respects, more transparent. The question therefore arises to what extent a logic with a similar proof theory can be constructed when expressive (...) power is increased.The increase mainly studied here allows one both to consider arbitrary partial functions and to express the condition that a function be total. A further increase taken into account is equivalent to a change to universal Horn sentences for partial and for total functions.Two ways of increasing expressive power will be considered. In both cases, the notion of equality is modified and nonlogical function symbols are interpreted as ranging over partial functions, instead of ranging only over total functions. In one case, the only further change is the addition of symbols that denote logical functions, such as the binary projection functionAethat maps each pair ‹a0,a1› of elements of a setAinto the elementa0. An addition of this kind results in a language, and also in a system of logic based on this language, which we callequationalIn the other case, instead of adding a symbol forAe, one admits those special universal Horn sentences in which the conditions expressed by the antecedent are, in a sense, pure conditions of existence. Languages and systems of logic that result from a change of this kind will be callednear-equational. According to whether the number of existence conditions that one may express in antecedents is finite or arbitrary, the resulting language and logic shall befiniteorinfinitary, respectively. Each of our finite near-equational languages turns out to be equivalent to one of our equational languages, and vice versa. (shrink)

In classifications of vocabulary knowledge, vocabulary size and depth have often been separately conceptualized (Schmitt, 2014). Although size and depth are known to be substantially correlated, it is not clear whether they are a single construct or two separate components of vocabulary knowledge (Yanagisawa & Webb, 2020). This issue has not been addressed extensively in the literature and can be better examined using structural equation modeling (SEM), with measurement error modeled separately from the construct of interest. The current study reports (...) on conventional and Bayesian SEM approaches (e.g., Muthén & Asparouhov, 2012) to examine the factor structure of the size and depth of second language vocabulary knowledge of Japanese adult learners of English. A total of 255 participants took five vocabulary tests. One test was designed to measure vocabulary size in terms of the number of words known, while the remaining four were designed to measure vocabulary depth in terms of word association, polysemy, and collocation. All tests used a multiple-choice format. The size test was divided into three subtests according to word frequency. Results from conventional and Bayesian SEM show that a correlated two-factor model of size and depth with three and four indicators, respectively, fit better than a single-factor model of size and depth. In the two-factor model, vocabulary size and depth were strongly correlated (r =.945 for conventional SEM and.943 for Bayesian SEM with cross loadings), but they were distinct. The implications of these findings are discussed. (shrink)

We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.

We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We survey related results.

This study shows how precise simple analytical solutions for the generalized Dirac equation with repulsive vector and attractive energy-dependent Lorentz scalar potentials, position-dependent mass potential, and a tensor interaction term can be obtained within the framework of both similarity transformation and the asymptotic iteration methods. These methods yield a significant improvement over existing approaches and provide more plausible and applicable ways in explaining the pseudospin symmetry’s breaking mechanism in nuclei.

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established.

This paper concerns modal logics of provability — Gödel-Löb systemGL and Solovay logicS — the smallest and the greatest representation of arithmetical theories in propositional logic respectively. We prove that the decision problem for admissibility of rules (with or without parameters) inGL andS is decidable. Then we get a positive solution to Friedman''s problem forGL andS. We also show that A. V. Kuznetsov''s problem of the existence of finite basis for admissible rules forGL andS has a negative solution. Afterwards we (...) give an algorithm deciding the solvability of logical equations inGL andS and constructing some solutions. (shrink)

This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that (...) this view is claimed in Fisher’s “fundamental theorem of natural selection”. These claims are refuted, and various incorrect statements about the meaning and interpretation of the fundamental theorem of natural selection, in this sequence and in other papers by other authors, are discussed. Next, much of the work in this sequence rests on the first Price equation. In the deterministic (infinite population) case this equation is no more than the standard classical equation relating to changes in gene frequencies. In the stochastic case the equation gives the change in gene frequencies as the sum of two terms (the second of which vanishes in the deterministic case). These two terms are of essentially equal importance in the situation considered in the focal article, yet one of Grafen’s results ignores the second term in the stochastic analysis. This is associated with a wavering between deterministic and stochastic analyses and the use of the Price fitness concept and the classical fitness concept. These comments cast doubts on Grafen’s optimization theory. (shrink)

Spinor equations, previously found valid and interesting in dealing with plane waves of light, are applied to spherical waves. It is found that the spinors pertaining to light do not form outgoing spherical waves, as the vectors do, but they can form standing spherical waves, which the vectors usually cannot. The spinors disclose details (“hidden variables”) which are hidden from the accepted theories of the subatomic scale.

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.

Structural equation modelling (SEM) is outlined and compared with two non-linear alternatives, artificial neural networks and ''fast and frugal'' models. One particular non-linear decision-making situation is discussed, that exemplified by a lexicographic semi-order. We illustrate the use of SEM on a dataset derived from 539 volunteers' responses to questions about food-related risks. Our conclusion is that SEM is a useful member of the armoury of techniques available to the student of human judgement: it subsumes several multivariate statistical techniques and permits (...) their flexible combination, and it provides robust goodness-of-fit statistics and is available in (generally) easy-to-use computer packages. Although the number of tasks for which SEM provides a persuasive psychological model is small, it is very useful in identifying the important variables and their inter-relations that contribute to task performance, and thus can constitute a valuable intermediate staging point between raw data and a fully fledged psychological theory. (shrink)

The Maxwell equations are shown to be the one-photon spin-one quantum equations. All Maxwell equations (without sources) are derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. The wavefunction is a linear combination of the electric and magnetic fields. The procedure is not unique, there are ambiguities of adding a scalar field. A quaternionic representation of the Maxwell equations (with sources) is constructed, a covariant reformulation of (...) which is presented. Whittaker potentials are analysed. Conservation laws are derived using a method of “pseudo-Lagrangians.”. (shrink)

The appearance of the time derivative of the acceleration in the equation of motion (EOM) of an electric charge is studied. It is shown that when an electric charge is accelerated, a stress force exists in the curved electric field of the accelerated charge, and in the case of a constant linear acceleration, this force is proportional to the acceleration. This stress force acts as a reaction force which is responsible for the creation of the radiation (instead of the “radiation (...) reaction force” that actually does not exist at low velocities). Thus the initial acceleration should be supplied as an initial condition for the solution of the EOM of an electric charge. (shrink)

It is shown in this article that the two sides of an equation in the medieval Arabic algebra are aggregations of the algebraic “numbers” (powers) with no operations present. Unlike an expression such as our 3x + 4, the Arabic polynomial “three things and four dirhams” is merely a collection of seven objects of two different types. Ideally, the two sides of an equation were polynomials so the Arabic algebraists preferred to work out all operations of the enunciation to a (...) problem before stating an equation. Some difficult problems which involve square roots and divisions cannot be handled nicely by this basic method, so we do find square roots of polynomials and expressions of the form “A divided by B” in some equations. But rather than initiate a reconsideration of the notion of equation, these developments were used only for particularly complex problems. Also, the algebraic notation practiced in the Maghreb in the later middle ages was developed with the “aggregations” interpretation in mind, so it had no noticeable impact on the concept of polynomial. Arabic algebraists continued to solve problems by working operations before setting up an equation to the end of the medieval period. (shrink)

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ∇ S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable χ. Using (...) these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out. (shrink)