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A temporal-theoretic formalism for understanding game theory is described where a strict ordering relation on a set of time points $T$ defines a game on $T$. Using this formalism, a proof of Zermelo's Theorem, which states that every finite 2-player zero-sum game is determined, is given and an exhaustive analysis of the game of Nim is presented. Furthermore, a combinatorial analysis of games on a set of arbitrary time points is given; in particular, it is proved that the number of distinct games on a set $T$ with cardinality $n$ is the number of partial orders on a set of $n$ elements. By generalizing this theorem from temporal modal frames to S5 modal frames, it is proved that the number of isomorphism classes of S5 modal frames $\mathcal{F} = \ < W, R \ >$ with $|W|=n$ is equal to the partition function $p(n)$. As a corollary of the fact that the partition function is asymptotic to the Hardy-Ramanujan number $$\frac{1}{4\sqrt{3}n}e^{\pi \sqrt{2n/3}}$$ the number of isomorphism classes of S5 modal frames $\mathcal{F} = \ < W, R \ >$ with $|W|=n$ is asymptotically the Hardy-Ramanujan number. Lastly, we use these results to prove that an arbitrary modal frame is an S5 modal frame with probability zero.

We summarize a new realist interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. Our interpretation---which we claim is ultimately compatible with Lorentz invariance---reformulates wave-function collapse in terms of an underlying interpolating dynamics, makes it possible to derive the Born rule from deeper principles, and resolves several open questions regarding ontological stability and dynamics.

The algorithm to compute theory prime implicates, a generalization of prime implicates, in propositional logic has been suggested in \cite{Marquis}. In this paper we have extended that algorithm to compute theory prime implicates of a knowledge base $X$ with respect to another knowledge base $\Box Y$ using \cite{Bienvenu}, where $Y$ is a propositional knowledge base and $X\models Y$, in modal system $\mathcal{T}$ and we have also proved its correctness. We have also proved that it is an equivalence preserving knowledge compilation and the size of theory prime implicates of $X$ with respect to $\Box Y$ is less than the size of the prime implicates of $X\cup\Box Y$.

proof theory of modal logic

The coherence of a shallow-water acoustic field transverse to the propagation path is represented using adiabatic normal modes, and path-integral theory is applied to fluctuations of mode amplitudes to obtain the coherence function. The problem is formulated in the horizontal plane where ray paths are nearly straight having no interaction with lateral boundaries, and requires the spectrum of environmental fluctuations that are provided by a shallow-water Garrett-Munk internal-wave model. The coherence function of the acoustic field is shown to depend on phase-structure functions of the normal modes. Environmental fluctuations with wavevectors transverse to the acoustic path were found to govern the structure functions. The impact of internal waves on coherence is proportional to the internal-wave energy, and is strongest for the lowest internal-wave modes in conjunction with those acoustic modes having maxima in the vicinity of high buoyancy. The transverse phase-structure function is found to behave like a sinc function, which provides the transition between limiting solutions for small and large hydrophone separations. Results show the coherence predicted by the theory decays too slowly with hydrophone separation to describe the observed ocean.

In a series of recent work, we have introduced a general framework for quantitative reasoning in specification theories. The contribution of this paper is to show how this framework can be applied to yield a robust specification theory for timed specifications.

A multivariable synthesis procedure based on eigenvalue/eigenvector assignment is reviewed and is employed to develop a systematic design procedure to meet the lateral handling qualities design objectives of a fighter aircraft over a wide range of flight conditions. The closed loop modal characterization developed provides significant insight into the design process and plays a pivotal role in the synthesis of robust feedback systems. The simplicity of the synthesis algorithm yields an efficient computer aided interactive design tool for flight control system synthesis.

NAS 1.60:1234 Digitized from IA1177307-12-0062 .

A unified approach is introduced using system realization theory to derive and correlate modal parameter identification methods for flexible structures. Several different time-domain and frequency-domain methods are analyzed and treated. A basic mathematical foundation is presented which provides insight into the field of modal parameter identification for comparison and evaluation. The relation among various existing methods is established and discussed. This report serves as a starting point to stimulate additional research towards the unification of the many possible approaches for modal parameter identification.

A unified approach is introduced using system realization theory to derive and correlate modal parameter identification methods for flexible structures. Several different time-domain and frequency-domain methods are analyzed and treated. A basic mathematical foundation is presented which provides insight into the field of modal parameter identification for comparison and evaluation. The relation among various existing methods is established and discussed. This report serves as a starting point to stimulate additional research towards the unification of the many possible approaches for modal parameter identification.

A unified electrodynamic approach to the guided wave excitation by external sources in the waveguiding structures with bianisotropic media is developed. Effect of electric, magnetic, and magneto- electric losses in such media manifests itself in the special form of eigenmode orthogonality referred to as the quasi-orthogonality relation. It reflects the existence of the cross-power flow and for any pair of modes which are rigidly linked to each other by this relation. The quasi-orthogonality relation remains true in the limiting case of lossless waveguides yielding the customary relations of orthogonality and normalization for propagating (active) modes and also their generalization for nonpropagating (reactive) modes. It is shown that the eigenmode set for a waveguiding structure is complete only outside the region of exciting sources. Inside this region the modal expansions of fields are incomplete and must be supplemented with the orthogonal complementary fields which extend the proper Hilbert space spanned by waveguide eigenfunctions. Among exciting sources there are the external bulk sources (currents, fields, and medium perturbations) and the external surface currents. Besides, the orthogonal complementary fields generate the effective surface currents on boundaries of the bulk exciting sources. The problem of waveguide excitation by external sources is solved by means of determining both the mode amplitudes for the modal field expansions and the orthogonal complementary fields inside the source region. The equations of mode excitation are derived on the basis of three approaches applying the direct use of Maxwell's equations, the electrodynamic analogy with the mathematical method of variation of constants, and the conjugate reciprocity theorem.

A unified electrodynamic approach to the guided wave excitation by external sources in the waveguiding structures with bianisotropic media is developed. Effect of electric, magnetic, and magneto- electric losses in such media manifests itself in the special form of eigenmode orthogonality referred to as the quasi-orthogonality relation. It reflects the existence of the cross-power flow and for any pair of modes which are rigidly linked to each other by this relation. The quasi-orthogonality relation remains true in the limiting case of lossless waveguides yielding the customary relations of orthogonality and normalization for propagating (active) modes and also their generalization for nonpropagating (reactive) modes. It is shown that the eigenmode set for a waveguiding structure is complete only outside the region of exciting sources. Inside this region the modal expansions of fields are incomplete and must be supplemented with the orthogonal complementary fields which extend the proper Hilbert space spanned by waveguide eigenfunctions. Among exciting sources there are the external bulk sources (currents, fields, and medium perturbations) and the external surface currents. Besides, the orthogonal complementary fields generate the effective surface currents on boundaries of the bulk exciting sources. The problem of waveguide excitation by external sources is solved by means of determining both the mode amplitudes for the modal field expansions and the orthogonal complementary fields inside the source region. The equations of mode excitation are derived on the basis of three approaches applying the direct use of Maxwell's equations, the electrodynamic analogy with the mathematical method of variation of constants, and the conjugate reciprocity theorem.

Modal density estimates for sandwich panels - theory and experiment

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation's characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ALBA to the setting of monotone modal logic, and we use a certain duality-induced encoding of finite lattices as monotone neighbourhood frames to translate lattice terms into formulas in monotone modal logic.

The linear and renormalized nonlinear kinetic theory of drift instability of plasma shear flow across the magnetic field, which has the Kelvin's method of shearing modes or so-called non-modal approach as its foundation, is developed. The developed theory proves that the time-dependent effect of the finite ion Larmor radius is the key effect, which is responsible for the suppression of drift turbulence in an inhomogeneous electric field. This effect leads to the non-modal decrease of the frequency and growth rate of the unstable drift perturbations with time. We find that turbulent scattering of the ion gyrophase is the dominant effect, which determines extremely rapid suppression of drift turbulence in shear flow.

Using the restricted three body problem, the equations of motion (EOM) and Hamiltonian are computed for the moon's orbit in physical variables. A periodic orbit is found in the vicinity of the moon's orbit, and classical Floquet theory is applied to the periodic orbit to give stability information and the complete solution to the equations of variation. Floquet theory also supplies a transformation from physical variables to modal variables. This transformation to modal variables is made canonical by constraining the initial transformation matrix to be symplectic. Actual lunar data is used to calculate the modes for the real moon's orbit. Once satisfied that the moon's real-world modes are in (or near) the linear regime of the periodic orbit, the modal EOM are found by doing a perturbation expansion on the new modal Hamiltonian. The modal results from the real lunar orbit are compared with the modal EOM/ expansion results. The modal expansion proves to be an accurate solution to the moon's orbit given enough expansion terms.

We present a discrete model theory similar in structure to ordinary quantum mechanics, but based on a finite field instead of complex amplitudes. The interpretation of this theory involves only the "modal" concepts of possibility and necessity rather than quantitative probability measures. Despite its simplicity, our model theory includes entangled states and has versions of both Bell's theorem and the no cloning theorem.

The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a lambda-calculus. Modal (boxed) types are intuitively interpreted as `closed syntax of the calculus'. This lambda-calculus is called modal type theory --- this is the basic case of a more general contextual modal type theory, or CMTT. CMTT has never been given a denotational semantics in which modal types are given denotation as closed syntax. We show how this can indeed be done, with a twist. We also use the denotation to prove some properties of the system.

Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The proofs of these analogue theorems are similar to, but much simpler than, the originals. We also show that the structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.

This episode focuses on Modal Mixture or using multiple modes over one chord using upward and downward modulations. This episode should be watched after https://youtu.be/cPYpgHDyQEw This incorporates modal theory with knowledge of the circle of 5ths. This is great for improvising on jazz tunes and writing contemporary classical on popular music. Links To Follow: If you are interested in purchasing The Beato Book please with me at [email protected] Patreon - http://patreon.com/user?u=3400819 YouTube - https://www.youtube.com/c/RickBeato Facebook - https://www.facebook.com/rickbeatopro... Instagram - https://www.instagram.com/rickbeato1/ Twitter - @rickbeato www.nuryl.com www.rickbeato.com Source: https://www.youtube.com/watch?v=y-Wnt-dmNnA Uploader: Rick Beato

This episode focuses on Modal Mixture or using multiple modes over one chord using upward and downward modulations. This episode should be watched after https://youtu.be/cPYpgHDyQEw This incorporates modal theory with knowledge of the circle of 5ths. This is great for improvising on jazz tunes and writing contemporary classical on popular music. Links To Follow: If you are interested in purchasing The Beato Book please with me at [email protected] Patreon - http://patreon.com/user?u=3400819 YouTube - https://www.youtube.com/c/RickBeato Facebook - https://www.facebook.com/rickbeatopro... Instagram - https://www.instagram.com/rickbeato1/ Twitter - @rickbeato www.nuryl.com www.rickbeato.com Source: https://www.youtube.com/watch?v=y-Wnt-dmNnA Uploader: Rick Beato

This episode focuses on Modal Mixture or using multiple modes over one chord using upward and downward modulations. This episode should be watched after https://youtu.be/cPYpgHDyQEw This incorporates modal theory with knowledge of the circle of 5ths. This is great for improvising on jazz tunes and writing contemporary classical on popular music. Links To Follow: If you are interested in purchasing The Beato Book please with me at [email protected] Patreon - http://patreon.com/user?u=3400819 YouTube - https://www.youtube.com/c/RickBeato Facebook - https://www.facebook.com/rickbeatopro... Instagram - https://www.instagram.com/rickbeato1/ Twitter - @rickbeato www.nuryl.com www.rickbeato.com Source: https://www.youtube.com/watch?v=y-Wnt-dmNnA Uploader: Rick Beato

This episode focuses on Modal Mixture or using multiple modes over one chord using upward and downward modulations. This episode should be watched after https://youtu.be/cPYpgHDyQEw This incorporates modal theory with knowledge of the circle of 5ths. This is great for improvising on jazz tunes and writing contemporary classical on popular music. Links To Follow: If you are interested in purchasing The Beato Book please with me at [email protected] Patreon - http://patreon.com/user?u=3400819 YouTube - https://www.youtube.com/c/RickBeato Facebook - https://www.facebook.com/rickbeatopro... Instagram - https://www.instagram.com/rickbeato1/ Twitter - @rickbeato www.nuryl.com www.rickbeato.com Source: https://www.youtube.com/watch?v=y-Wnt-dmNnA Uploader: Rick Beato

In recent work, Benjamin Schumacher and Michael D. Westmoreland investigate a version of quantum mechanics which they call modal quantum theory. This theory is obtained by instantiating the mathematical framework of Hilbert spaces with a finite field instead of the field of complex numbers. This instantiation collapses much the structure of actual quantum mechanics but retains several of its distinguishing characteristics including the notions of superposition, interference, and entanglement. Furthermore, modal quantum theory excludes local hidden variable models, has a no-cloning theorem, and can express natural counterparts of quantum information protocols such as superdense coding and teleportation. We show that the problem of UNIQUE-SAT --- which decides whether a given Boolean formula is unsatisfiable or has exactly one satisfying assignment --- is deterministically solvable in any modal quantum theory in constant time. The solution exploits the lack of orthogonality in modal quantum theories and is not directly applicable to actual quantum theory.

This article is from Pakistan Journal of Medical Sciences , volume 30 . Abstract Objective: To evaluate assessment system of the 'Research Methodology Course' using utility criteria (i.e. validity, reliability, acceptability, educational impact, and cost-effectiveness). This study demonstrates comprehensive evaluation of assessment system and suggests a framework for similar courses.Methods: Qualitative and quantitative methods used for evaluation of the course assessment components (50 MCQ, 3 Short Answer Questions (SAQ) and research project) using the utility criteria. Results of multiple evaluation methods for all the assessment components were collected and interpreted together to arrive at holistic judgments, rather than judgments based on individual methods or individual assessment.Results: Face validity, evaluated using a self-administered questionnaire (response rate-88.7%) disclosed that the students perceived that there was an imbalance in the contents covered by the assessment. This was confirmed by the assessment blueprint. Construct validity was affected by the low correlation between MCQ and SAQ scores (r=0.326). There was a higher correlation between the project and MCQ (r=0.466)/SAQ (r=0.463) scores. Construct validity was also affected by the presence of recall type of MCQs (70%; 35/50), item construction flaws and non-functioning distractors. High discriminating indices (>0.35) were found in MCQs with moderate difficulty indices (0.3-0.7). Reliability of the MCQs was 0.75 which could be improved up to 0.8 by increasing the number of MCQs to at least 70. A positive educational impact was found in the form of the research project assessment driving students to present/publish their work in conferences/peer reviewed journals. Cost per student to complete the course was US$164.50.Conclusions: The multi-modal evaluation of an assessment system is feasible and provides thorough and diagnostic information. Utility of the assessment system could be further improved by modifying the psychometrically inappropriate assessment items.

The non-modal approach for a linearized system differs from a normal mode analysis by following the temporal evolution of some perturbed equilibria, and therefore includes transient effects. We employ a non-modal approach for studying the stability of a bi-Maxwellian magnetized plasma using the Landau fluid model, which we briefly describe. We show that bi-Maxwellian stable equilibria can support transient growth of some physical quantities, and we study how these transients behave when an equilibrium approaches its marginally stable condition. This is relevant to anisotropic plasma, that are often observed in the solar wind with a temperature anisotropy close to values that can trigger a kinetic instability. The results obtained with a non-modal approach are relevant to a re-examination of the concept of linear marginal stability. Moreover, we discuss the topic of the dissipation of turbulent fluctuations, suggesting that the non-modal approach should be included in future studies.

We introduce a spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1+1 dimensional phi^4 theory, identifying both one-particle and multi-particle contributions. We discuss the computational limits of existing diagonalization algorithms and suggest new quasi-sparse eigenvector methods to handle very large Fock spaces and higher dimensional field theories.

In a recent article, Dieks has proposed a way to implement the modal interpretation of (nonrelativistic) quantum theory in relativistic quantum field theory. We show that his proposal fails to yield a well-defined prescription for which observables in a local spacetime region possess definite values. On the other hand, we demonstrate that there is a well-defined and unique way of extending the modal interpretation to the local algebras of relativistic quantum field theory. This extension, however, faces a potentially serious difficulty in connection with ergodic states of a field.

Plates can have more than one buckled solution for a fixed set of boundary conditions. The theory for the identification and the computation of multiple solutions in buckled plates is examined. The theory predicts modal interaction (which is also called change in buckle pattern or secondary buckling) in experiments on certain plates with multiple theoretical solutions. A set of coordinate functions is defined for Galerkin's method so that the von Karman plate equations are reduced to a coupled set of cubic equations in generalized coordinates that are uncoupled in the linear terms. An iterative procedure for solving modal interaction problems is suggested based on this cubic form.

We present an adaptation of input/output conformance (ioco) testing principles to families of similar implementation variants as appearing in product line engineering. Our proposed product line testing theory relies on Modal Interface Automata (MIA) as behavioral specification formalism. MIA enrich I/O-labeled transition systems with may/must modalities to distinguish mandatory from optional behavior, thus providing a semantic notion of intrinsic behavioral variability. In particular, MIA constitute a restricted, yet fully expressive subclass of I/O-labeled modal transition systems, guaranteeing desirable refinement and compositionality properties. The resulting modal-ioco relation defined on MIA is preserved under MIA refinement, which serves as variant derivation mechanism in our product line testing theory. As a result, modal-ioco is proven correct in the sense that it coincides with traditional ioco to hold for every derivable implementation variant. Based on this result, a family-based product line conformance testing framework can be established.

We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.

We review recent developments in non-perturbative field theory using modal field methods. We discuss Monte Carlo results as well as a new diagonalization technique known as the quasi-sparse eigenvector method.

The general aim of this paper is to extend the Modal-Hamiltonian interpretation of quantum mechanics to the case of relativistic quantum mechanics with gauge U(1) elds. In this case we propose that the actual- valued observables are the Casimir operators of the Poincar\'e group and of the group U(1) of the internal symmetry of the theory. Moreover, we also show that the magnitudes that acquire actual values in the relativistic and in the non-relativistic cases are correctly related through the adequate limit.

The general aim of this paper is to extend the Modal-Hamiltonian interpretation of quantum mechanics to the case of relativistic quantum mechanics with gauge U(1) elds. In this case we propose that the actual- valued observables are the Casimir operators of the Poincar\'e group and of the group U(1) of the internal symmetry of the theory. Moreover, we also show that the magnitudes that acquire actual values in the relativistic and in the non-relativistic cases are correctly related through the adequate limit.

Possibility theory focuses on quantifying the degree to which a statement or proposition is possible. It deals with the measurement of possibility and necessity in a similar manner to how probability theory deals with likelihood. While the concept of fuzzy logic has proven valuable in addressing uncertainties and imprecision, its main drawback arises when dealing with situations that involve not only uncertainty but also indeterminacy, as well as the coexistence of truth, falsity, and indeterminacy within a single statement.  In this paper, we suggest using neutrosophic logic as a mathematical framework for reasoning with ambiguity and vagueness. We propose utilizing Kripke structures for neutrosophic propositions as conceptual abstract models, providing an alternative method to describe possibility theory in a neutrosophic environment. An illustrative scenario in the context of medical diagnosis is presented in order to demonstrate the efficacy and flexibility of our method. This novel approach not only enriches our knowledge of uncertainty, but it also provides pathways for more comprehensive and nuanced analysis in other domains such as in artificial intelligence, decision support systems, knowledge representation, cognitive computing etc., thus highlighting the potential benefit of merging neutrosophic logic with possibility and modal structures.

The implementation of a general three-dimensional hydroelastic capability in the NASTRAN computer program is described. A method for analyzing the combined mode shapes of arbitrary fluid and structure finite element models was developed. The fluid was modeled with three-dimensional solid elements with options for tetrahedron, wedge, and hexahedron shapes. The elements were then connected to fluid grid points which defined the pressure in the fluid at the specified location. Matrices were formulated to provide efficient solutions for large-order problems. The structure matrices were processed separately and could be reduced using matrix condensation procedures or through a modal formulation using the normal modes of the empty structure as solution coordinates. The fluid/structure interface and the free surface were defined by the user with special NASTRAN boundary elements. A special purpose mesh generator program was provided to generate the actual NASTRAN data cards for the fluid, the structure, and the boundary elements for typical tank-type models.

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments.

The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, unlike the situation with other statistical problems as regression or classification, for some of the cluster methodologies it is quite difficult to specify a population goal to which the data-based clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of the usual nonparametric approach to clustering, which understands clusters as regions of high density, by presenting an explicit formulation for the ideal population clustering.

Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hubbard model with 25/64 filling.

Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hubbard model with 25/64 filling.

The average polarization properties of conal single and double profiles directly reflect the polarization-modal structure of the emission beams which produce them. These average properties require that the circulating subbeam systems which produce conal beams entail paired PPM and SPM emission elements which are offset from each other in both magnetic azimuth and magnetic colatitude. A clear delineation of the modal polarization topology of the conal beam promises to address fundamental questions about the nature and origin of this modal emission--and the modal parity at the outer beam edges is a fact of considerable significance. The different angular dependences of the modal ``beamlets'' suggests that the polarization modes are generated via propagation effects. This argument may prove much stronger if the modal emission is fundamentally only partially polarized. Several theories now promise quantitative comparison with the observations.

An investigation was made of a relationship between modal and linear optimal control theories to determine whether the modal feedback gain matrix would be of help in finding the weighting matrices of the optimal steady-state criterion. Modal and optimal control theories are reviewed, and the concepts of steady-state optimal control, and single-input and multiple-input modal control are developed.

An investigation was made of a relationship between modal and linear optimal control theories to determine whether the modal feedback gain matrix would be of help in finding the weighting matrices of the optimal steady-state criterion. Modal and optimal control theories are reviewed, and the concepts of steady-state optimal control, and single-input and multiple-input modal control are developed.