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This work advances the use of nonclassical logics for developing qualitative models of real-world systems. Abstract mathematics is "qualitative" inasmuch as it relegates numerical considerations to the background and focuses explicitly on topological, algebraic, logical, or other types of conceptual forms. Mathematical logic, the present topic, serves to explicate alternative modes of reasoning for use in general research design and in model construction. The central thesis is that the theory of formal logical systems, and particularly, of logical systems based on nonclassical modes of reasoning, offers important new techniques for developing qualitative models of real-world systems. This thesis is supported in three major parts. Part I develops a semantically complete axiomatization of L. A. Zadeh's theory of approximate reasoning. This mode of reasoning is based on the conception of a "fuzzy set," by which means it yields a realistic representation of the "vagueness" ordinarily inherent in natural languages, such as English. All axiomatizations of this mode of reasoning to date have been deficient in that their linguistic structures are adequate for expressing only the simplest fuzzy linguistic ideas. The axiomatization developed herein goes beyond these limitations in a two-leveled formal system, which, at the inner level, is a multivalent logic that accommodates fuzzy assertions, and at the outer level, is a bivalent formalization of segments of the metalanguage. This system is adequate for expressing most of the basic fuzzy linguistic ideas, including: linguistic terms, hedges, and connectives; semantic equivalence and entailment; possibilistic reasoning; and linguistic truth. The final chapter of Part I applies the theory of approximate reasoning to a class of structural models for use in forecasting. The result is a direct mathematical link between the imprecision in a model and the uncertainty which that imprecision contributes to the model's forecasted events. Part II studies the systems of logical "form" which have beeen developed by G. Spencer-Brown and F. J. Varela. Spencer-Brown's "laws of form" is here shown to be essentially isomorphic with the axiomatized propositional calculus, and Varela's "calculus for self-reference" is shown to be isomorphically translatable into a system which axiomatizes a three-valued logic developed by S. C. Kleene. No semantically complete axiomatization of Kleene's logic has heretofore been known. Following on Kleene's original interpretation of his logic in the theory of partial recursion, this leads to a proof that Varela's concept of logical "autonomy" is exactly isomorphic with the notion of a "totally undecidable" partial recursive set. In turn, this suggests using Kleene-Varela type systems as formal tools for representing "mechanically unknowable" or empirically unverifiable system properties. Part III is an essay on the theoretical basis and methodological framework for implementing nonstandard logics in the modeling exercise. The evolution of mathematical logic is considered from the standpoint of its providing the opportunity to "select" alternative modes of reasoning. These general theoretical considerations serve to motivate the methodological ones, which begin by addressing the discussions of P. Suppes and M. Bunge regarding the role of formal systems in providing "the semantics of science." Bunge's work extends that of Suppes and is herein extended in turn to a study of the manner in which formal systems (both classical and nonclassical) can be implemented for mediation between the observer and the observed, i.e., for modeling. Whether real-world systems in fact obey the laws of one logic versus another must remain moot, but models based on alternative modes of reasoning to satisfy Bunge's criteria for empirical testability, and therefore do provide viable systems perspectives and methods of research.

<p> There is only one unambiguous report in the literature of anchimeric assistance by the π-bond in a free radical reaction. It was the purpose of this work to investigate the possibility of such acceleration in the thermal homolysis of 7-norbornene peresters. Two such t-butyl peresters, the isomeric syn- and anti-7-norbornene percarboxylates as well as t-butyl 7-norbornane percarboxylate, were prepared and their decomposition was studied by kinetic methods and product analysis.</p> <p> The homolysis of 7-norbornene peresters is shown to occur with rate-determining perester O-O bond rupture, the transition state of which may be a hybrid of radical and polar structures. A low order of anchimeric acceleration may also assist the decomposition of the syn compound.</p> <p> The pKa values for syn- and anti- 7-norbornene- and 7-norbornane- carboxylic acid are reported.</p> / Thesis / Doctor of Philosophy (PhD)

Thesis (Ph.D.)--Bowling Green State University, 2008. / Document formatted into pages; contains xii, 166 p. Includes bibliographical references.

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A geração e proteção de estados quânticos é fundamental para a mecânica quântica. Usualmente, utilizam-se protocolos de engenharia de estados baseados na aplicação sucessiva de transformações unitárias, cuja performance se torna menos efetiva conforme aumenta-se o número de componentes envolvidos nas transformações (já que transformações unitárias dependem de um elevado número de operações ou transformações com alto grau de fidelidade), além de tornarem-se mais vulneráveis a efeitos de flutuações de parâmetros experimentais, efeitos de desordem, decoerência e ruído. Dentre as possíveis estratégias para gerar estados quânticos, existe a chamada engenharia de interações quânticas, na qual pode-se tanto estudar como alterar a maneira a partir da qual sistemas interagem entre si para produzir um determinado estado estacionário desejado, quanto para investigar propriedades dos estados gerados ao se alterar alguma característica de tal interação. Neste trabalho apresentaremos duas propostas para gerar estados não-clássicos via engenharia de reservatórios (engenharia dissipativa) em dois tipos de sistemas distintos. No primeiro, utilizaremos um sistema optomecânico no qual efetua-se engenharia de dissipação a fim de obter hamiltonianos seletivos com os quais é possível preparar-se estados de Fock sob efeitos dissipativos, no qual mostramos o caráter não clássico dos estados obtidos nos regimes de cavidade altos e baixos fatores de qualidade. No segundo, utilizaremos um sistema de spins na qual podemos obter estados não-clássicos (emaranhados) para um sistema com o número de partículas (N) entre 2 e 12 via engenharia de interações quânticas com caráter coletivo, para obter diversos estados, considerando efeitos dissipativos como dissipações térmicas e defasagem, além de considerar a robustez com relação a flutuações em alguns parâmetros experimentais do modelo. / The generation and protection of quantum states is fundamental to quantum mechanics. Usually, state engineering protocols are used based on the successive application of unitary transformations, whose performance becomes less effective as the number of components involved in the transformations increases (as that depends on a large number of high-fidelity operations), in addition to becoming more vulnerable to the effects of fluctuations of experimental parameters , effects of disorder, decoherence and noise. Among the possible strategies to yield quantum states, there is the so-called quantum interaction engineering, in which one can either study how to change the way in which systems interact with each other to produce a desired steady state, or to investigate properties of the engineered states by changing some characteristic of such interaction. In this work we present a proposal to engineer non-classical states through reservoir engineering (dissipative engineering) in two types of systems. In the first one, we will use an optomechanical system in which dissipative engineering is carried out in order to obtain selective Hamiltonians with whom it is possible to prepare Fock states under dissipative effects, in which we show the non-classical character of the states obtained in the good and bad cavity regimes.. In the second, we will use a spin chain system in which we can obtain non-classical (entangled) states for a system with the number of particles (N) between 2 and 12 via quantum interaction engineering with collective character, to obtain several states, taking into account dissipative effects such as thermal dissipation and dephasing, and showing the robustness in relation to fluctuations in some experimental parameters of the model.

This thesis considers two types of applications of quantum feedback control; feedback creation of nonclassical states of light, and controlling nonclassical properties of an ensemble of atoms. An electro-optical feedback loop will create an in-loop field with nonclassical photon statistics similar to squeezed light, resulting in fluorescence line-narrowing of a two-level atom coupled to such light. We extend this theory to study a three-level atom coupled to broadband squashed light, and confirm the two-level atom line-narrowing using a more realistic non-Markovian description of the feedback loop. The second type of application utilizes continuous QND measurement of atomic ensembles. If we measure the collective spin, then the system experiences conditional spin squeezing dependent on the measurement results. We show that feedback based on these results can continuously drive the system into the same conditioned state, resulting in deterministically reproducible spin squeezing. If we measure the atom number fluctuations of a BEC, then, due to the nonlinearity of atomic self interactions, this is also information about phase fluctuations. We show that feedback based on this information can greatly reduce the collisional broadening of the linewidth of an atom laser out-coupled from the condensate.

Scintillation based gamma-ray detectors are widely used in medical imaging, high-energy physics, astronomy and national security. Scintillation gamma-ray detectors are field-tested, relatively inexpensive, and have good detection efficiency. Semi-conductor detectors are gaining popularity because of their superior capability to resolve gamma-ray energies. However, they are relatively hard to manufacture and therefore, at this time, not available in as large formats and much more expensive than scintillation gamma-ray detectors. Scintillation gamma-ray detectors consist of: a scintillator, a material that emits optical (scintillation) photons when it interacts with ionization radiation, and an optical detector that detects the emitted scintillation photons and converts them into an electrical signal. Compared to semiconductor gamma-ray detectors, scintillation gamma-ray detectors have relatively poor capability to resolve gamma-ray energies. This is in large part attributed to the "statistical limit" on the number of scintillation photons. The origin of this statistical limit is the assumption that scintillation photons are either Poisson distributed or super-Poisson distributed. This statistical limit is often defined by the Fano factor. The Fano factor of an integer-valued random process is defined as the ratio of its variance to its mean. Therefore, a Poisson process has a Fano factor of one. The classical theory of light limits the Fano factor of the number of photons to a value greater than or equal to one (Poisson case). However, the quantum theory of light allows for Fano factors to be less than one. We used two methods to look at the correlations between two detectors looking at same scintillation pulse to estimate the Fano factor of the scintillation photons. The relationship between the Fano factor and the correlation between the integral of the two signals detected was analytically derived, and the Fano factor was estimated using the measurements for SrI₂:Eu, YAP:Ce and CsI:Na. We also found an empirical relationship between the Fano factor and the covariance as a function of time between two detectors looking at the same scintillation pulse. This empirical model was used to estimate the Fano factor of LaBr₃:Ce and YAP:Ce using the experimentally measured timing-covariance. The estimates of the Fano factor from the time-covariance results were consistent with the estimates of the correlation between the integral signals. We found scintillation light from some scintillators to be sub-Poisson. For the same mean number of total scintillation photons, sub-Poisson light has lower noise. We then conducted a simulation study to investigate whether this low-noise sub-Poisson light can be used to improve spatial resolution. We calculated the Cramér-Rao bound for different detector geometries, position of interactions and Fano factors. The Cramér-Rao calculations were verified by generating simulated data and estimating the variance of the maximum likelihood estimator. We found that the Fano factor has no impact on the spatial resolution in gamma-ray imaging systems.

Thesis (M.S.)--Miami University, Dept. of Physics, 2005. / Title from first page of PDF document. Document formatted into pages; contains [1], v, 296 p. : ill. Includes bibliographical references (p. 393-396).

abstract: Solution methods for certain linear and nonlinear evolution equations are presented in this dissertation. Emphasis is placed mainly on the analytical treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. First, the Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line. Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. These relations give solvability results for the Cauchy problem of the parabolic equation considered. The superposition principle allows to solve formally this problem from an unconventional point of view. An eigenfunction expansion approach is also considered for this general evolution equation. Examples considered to corroborate the efficacy of the proposed solution methods include the Fokker-Planck equation, the Black-Scholes model and the one-factor Gaussian Hull-White model. The results obtained in the first part are used to solve the Cauchy initial value problem for certain inhomogeneous Burgers-type equation. The connection between linear (the Diffusion-type) and nonlinear (Burgers-type) parabolic equations is stress in order to establish a strong commutative relation. Traveling wave solutions of a nonautonomous Burgers equation are also investigated. Finally, it is constructed explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. It is shown that the product of the variances attains the required minimum value only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. Such explicit construction is possible due to the relation between the diffusion-type equation studied in the first part and the time-dependent Schrodinger equation. A modication of the radiation field operators for squeezed photons in a perfect cavity is also suggested with the help of a nonstandard solution of Heisenberg's equation of motion. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2013