From the above proposition, it is very obvious that this research would have meaningful contributions in the realm of modern mathematical physics and theoretical physics and furthermore would have deep impact with long lasting implications and traces in the domain of fundamental researches. Nonetheless, at the frontier level of this fundamental research, it is an understatement to say that this research is already in the midst of achieving better understanding of its meaning, importance and benefits. Truly we think the current path is still haphazard, particularly to grasp the rigorous mathematical realization of Feynman integral theory & applications and the classification of physically significant **integrable** systems, and understandably more difficult to establish their relationship in the hypothetical domain of string theory or even M theory. There is as if a consensus amongst theoretical physicists and mathematical physicists that the real antidode to the above-mentioned problems (particularly the problem of unification between classical and quantum domains) is to rethink more meaningfully the concept of ‘space-time’, both from the geometrical as well as physical viewpoints (for example, probably by implementing appropriate extensions or modifications of Connes (1994) ‘non-commutative geometry’, or a more radical suggestion is to **completely** neglect the concept of ‘space-time’, Manin (1983)). Nevertheless, our point of view follows closely the suggestions of Atiyah (1994). This certainly rests largely upon the background of Witten’s conjectures and Kontsevich’s model. In precise and conjecturally, we believe that this approach would result in an alternative better concept of Feynman Path Integral: a well-defined mathematical entity with better calculational aspects. In our opinion, an acceptable existence theorem and a formal definition of a universal Feynman Path Integral would considerably dissolve this solid issue.

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Both the classical and quantum mechanics are characterized by two basic concepts: states and observables. The measurements are operations on the physical systems. In clas- sical mechanics, the state is a point in the manifold (symplectic) M (phase space) and observable is the function on M. In quantum mechanics, the state of a **system** probably cor- responds to a vector unit in a Hilbert space H while the observable quantity corresponds to the operator (self-adjoint and noncommutative) on H. Integrating special relativity to both the above-mentioned theories would direct classical mechanics towards general relativity, and quantum mechanics to quantum field theory. The path from classical mechanics to quantum mechanics is known as quantisation. Drinfeld and Witten studied the relationship between these theories through different perspectives. Following Drinfeld [4], the connec- tion between classical mechanics and quantum mechanics can be incorporated in terms of observables. In both cases, the observables form an associative algebra that is commutative in the classical sense and noncommutative in quantum perspective. Consequently, quanti- sation illustrates the transformation of commutative algebra to noncommutative algebra. States are described by Drinfeld [4] in terms of Hopf algebra. This algebraic approach of Drinfeld would bear the quantum group concept that eventually link to the **completely** **integrable** **system** in statistical mechanics, Yang-Baxter equation and deformations of Lie algebra. Witten’s approach is topological. Quantisation is expressed in terms of states, using the Feynman path integral method. Observables are represented by topological in- variants. Subsequently, Witten’s approach encompasses the string theory, topological quan- tum field theory, conformal field theory and Chern-Simons action function. The works by Jones are related to that of Drinfeld via Jones polynomial and associative representation of braid groups, their connection with the **integrable** **system** in statistical mechanics, and the combinatorical relationship between Yang-Baxter equation and Jones polynomial for loops. Similarly, the relationship between Jones’ works with that of Witten’s is exhibited through an interpretation of Jones polynomial in terms of topological quantum field theory. In ad- dition, Chern-Simons Lagrangian in the Feynman integral can be used to generate Jones polynomial, or otherwise can be used to retrace the formal functional integrals as explicit mathematical quantities. The afore-mentioned scenario has opened up application of meth- ods of field, string and **integrable** **system** theories to important progress, opening entirely new points of view in the context of Gromov-Witten invariants, Donaldson invariants and quantum-group invariants for knots and links. These would undoubtedly advance further our understanding of the bigger picture of string theory, i.e., the remarkable M-theory.

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u = u v u v , u 1 = u x t u 1 ( , ), 2 = u x t v 2 ( , ), 1 = v x t v 1 ( , ), 2 = v x t 2 ( , ) are complex-valued potential functions, ξ is a complex spectral parameter, i = − 1 . The relation between this 3rd-order complex spectral problem and the associated **completely** **integrable** **system** is considered. We derived the related evolu- tion equation hierarchy, one of which is often referred to on the literature as the coupled nonlinear Schrödinger equation:

In general, the mechanism of pattern formation in VC- SEL is very complicated and involves both a complex structure of a VCSEL cavity (including Bragg reflectors) [10, 11, 12], as well as peculiarities of light-matter in- teraction in active quantum well semiconductor layers [15, 16, 17, 18, 19, 20]. However, near threshold one can invoke the perturbation theory and obtain normal forms governing the evolution of the **system**. It was shown recently, that due to a slight spatial anisotropy of a VCSEL cavity only a few spatial modes come into play at threshold and the shape of these modes can be analyzed by the linear approximation of the normal form [10, 11, 12] (in contrast to spatially isotropic systems, where the whole degenerate family of modes have the same critical growth rate at threshold, and the selection process requires consideration of nonlinear competition even at threshold [21, 22]). The investigations of the problem using this point of view was started in [12] for VCSELs with circular aperture. In that work, transition from the ’flower-like’ modes dictated by circular bound- aries to ’stripe-like’ ones, which are required by spatial

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Manchester encoding is likewise called stage encoding. It can be utilized for a higher working recurrence. Manchester encoding is an extremely basic strategy and is likely the most ordinarily utilized. The signs can be transmitted serially. In Manchester encoding the normal power is dependably the same, regardless of what information is transmitted. Contrasted with all other encoding strategies, Manchester code takes after a calculation to encode the information. It generally delivers a progress at the focal point of the bit. It contains adequate data to recoup a clock. So if the information rate is twice, adequate clock data can be recuperated from the information stream with the goal that different timekeepers are not required. Subsequently, the electrical association utilizing Manchester code is effectively a galvanic partner isolator (it is the standard of detaching practical areas of electric frameworks to counteract current stream) utilizing a **system** isolator for straightforward balanced segregation change. In this manner, while transmitting the information, the quantity of wires is limited, which is utilized to decrease the clamor and transmission control. Rationale "1" speaks to the progress from HIGH to LOW. Rationale "0" speaks to the change from LOW to HIGH. To get a fast, give a synchronized information source as the principal clock beat for input information. While transmitting the information, it is an advanced encoding in which information transmission bits are spoken to by changes starting with one rationale then onto the next rationale. The length of each piece is set as default, and it expends the signs as self-timing. The heading of the progress chooses the condition of the bit. It is here and there important to have a change amidst a bit with the goal that the progress got toward the starting time frame is ignored.

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The identification of dromions [1-6] has triggered a renewed interest in the study of **integrable** models in (2+1) dimensions. Dromions arise essentially by virtue of coupling the field variable to a mean field / potential, thereby preventing wave collapse in (2+1) dimensions and they can, in general undergo inelastic collision unlike one dimensional solitons. The identification of a large number of arbitrary functions in the solutions of (2+1) dimensional **integrable** models has only added to the richness in the structure of them and hence construction of localized excitations in (2+1) dimensions continues to be a challenging and rewarding contemporary problem. In this equation we developed a very simple and straight forward procedure to generate a rather extended class of generic solutions of physical interest. In this 2LDW equation we utilize the local Laurent expansion of the general solution and truncate it at the constant level term (painleve truncation approach) and obtain solutions in terms of arbitrary functions. Through this procedure we generate various periodic and exponentially localized solutions. The novelty here is that the solution is generated through a simple procedure. But the solution obtained is rich in structure because of the arbitrary functions [9-14].

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Averaged exponential acceleration is achieved when the billiard parameters change along a non-trivial loop, for which the compression factor g of equation (29) is not identically one. A non-trivial loop bounds a non-empty interior when projected on a parameter plane in which one axis corresponds to the phase space volume of the chaotic zone and the other axis corresponds to the ﬂ ux between the chaotic and **integrable** components. An important con- clusion is that the exponential acceleration rate vanishes if the motion of the billiard boundary can be described by periodic oscillations of a single parameter (as is often done in numerical simulations of Fermi acceleration). Similarly, it vanishes if one of these two ingredients — the ﬂ ux between the ergodic components or the volume change of the ergodic components — is missing. When the exponential rate vanishes, we still expect to observe some slow accel- eration, typically quadratic in time (see e.g. [24, 25, 29, 30]).

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The introduction of frequency operators ~in the present terminology! provides a link to the most advanced papers of the ‘‘old quantum mechanics.’’ In 1925, Dirac @6# defined ‘‘q numbers,’’ which would fulfill the fundamental commu- tation relations of position and momentum. Basic algebraic consequences are derived, partly by exploiting the analogy to the classical Poisson brackets. One section of the paper is devoted to ‘‘multiply periodic systems’’ characterized by the existence of ‘‘uniformizing variables’’ ~which nowadays would be called action and angle operators! such that the Hamiltonian depends on the action operators only @cf. Eq. ~3!#. In the terminology of the present work multiply peri- odic systems are thus recognized as quantum **integrable** ones. Assuming the action-angle operators to fulfill the same com- mutation relations as momentum and position—which is un- tenable due to the defectiveness of a phase operator @18#— Dirac introduced two different types of frequencies. The first one is supposed to govern the time evolution of the phase operator itself, whereas the second one is associated with the time derivative of the exponentiated phase operator. It turns

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Currently, the Grad Supervisor simply marks 'graduated', 'left', or 'expelled' in a students status when the student leaves; instead, a **system** could be set that allows the supervisor to simply click a button and have the **system** automatically check which graduation requirements have been met, which are currently being worked on, and which are unmet. Other improvements to the **system** should relate to the maintenance of the **system**. There should be some method for the users of the **system** to correct mistakes made by themselves that do not require them to contact the webmaster to modify the mysql tables to fix the problem. Similarly, user account generation, password management, and course and advisor status should be modifiable from within the user interface for privileged users.

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In this thesis we develop a **system** to detect side effects in the drug reviews as a subtask of detecting implicit opinions in medical sources. We will also propose and compare the regular based and classification approaches for recognizing adverse side effects and discriminating them from disease symptoms.

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In [4–6] the correspondence, called the Bethe/gauge correspondence, between the two dimensional N = 2 supersymmetric gauge theories and quantum **integrable** systems was formulated in full generality. The novelty of this correspondence is the identification of the Planck constant of the quantum **integrable** **system** with a twisted mass parameter on the gauge theory side. The dictionary further identifies the Yang-Yang function of the quantum **integrable** **system** with the effective twisted superpotential W f eff of the gauge theory, which

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This paper is organized as follows. In section 2,by means of the gauge transformation of the Lax pair, we construct a Darboux transformation of Lax pair of the Eq.(5).To the best of our knowledge, Darboux transformation of Lax pair of this triangular **integrable** coupling has not been studied. In Section 3, as an application of Darboux transformation, an explicit solution of the Eq. (5) is deduced. Some conclusions and remarks are given in the final Section.

Riemann Integral: A bounded function f is said to be Riemann **integrable** (or simply **integrable**) over [a, b] if it's Upper and Lower integrals are equal and the common value of Upper and Lower integrals is called the Riemann integral f f on [a, b], is denoted by: (x)dx.The fact that f is **integrable** over [a, b], we express by writing f R[a, b] or R simply.

One particular aspect of quantum **integrable** field theories is the existence, by definition, of countably many independent conserved quantities. In 1 + 1 dimen- sions, nontrivial massive quantum field theories can be studied non-perturbatively thanks to the following results [1]: there is no particle production, the momenta are conserved individually and finally, any N -particle process can be decomposed as a sequence of two-particle processes. This last property is the so-called factorization property . It follows that the central ingredient for such theories is the two-body scattering matrix which has to satisfy the celebrated (quantum) Yang-Baxter equa- tion [2, 3].

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Abstract: We review some recent results concerning **integrable** quantum field theories in 1+1 space-time dimensions which contain unstable particles in their spectrum. Recalling first the main features of analytic scattering theories associated to **integrable** models, we subsequently propose a new bootstrap principle which allows for the construction of particle spectra involving unstable as well as stable particles. We describe the general Lie algebraic structure which underlies theories with unstable particles and formulate a decoupling rule, which predicts the renormalization group flow in dependence of the relative ordering of the resonance parameters. We extend these ideas to theories with an infinite spectrum of unstable particles. We provide new expressions for the scattering amplitudes in the soliton-antisoliton sector of the elliptic sine-Gordon model in terms of infinite products of q-deformed gamma functions. When relaxing the usual restriction on the coupling constants, the model contains additional bound states which admit an interpretation as breathers. For that situation we compute the complete S-matrix of all sectors. We carry out various reductions of the model, one of them leading to a new type of theory, namely an elliptic version of the minimal SO(n)-affine Toda field theory.

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In this paper, we have made an attempt to bring the rich hidden structure of **integrable** PDEs [4] to a new domain, namely, to differential equations on free associative algebras (see also [12, 13, 14]). We have formulated basic definitions and shown their correctness and efficiency. Like **integrable** PDEs, ODEs on associative algebras may have infinite hierarchies of symmetries and first in- tegrals and that asserts an algebraic definition of integrability. In the case of finite dimensional (matrix) representations of the algebra, the corresponding (matrix) systems of ODEs inherit these symmetries and first integrals and can be integrated in quadratures.

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Part (b) shows the analysis for a double defect **system** with one defect situated at x = 0 and the other at x = y. The double defect amplitudes are computed directly from (12) and (13) with the expression for the single defect (51) and (52). Since now both A and ˙ A appear explicitly in the formulae for the R’s and T ’s, it is clear that the expansion of the double defect can not be of type I, but it turns out to be of type II, i.e. of the form (42). Hence, we will now expect that besides the even also the odd multiples of ω will be filtered out, which is indeed visible in part (b) for various distances. Here we have only plotted a continuous spectrum for y = 0.5, whereas for reasons of clarity, we only drew the enveloping function which connects the maxima of the harmonics for the remaining distances. We observe that now not only odd multiples of the frequency emerge in addition to the ones in (a) as harmonics, but also that we obtain much higher harmonics and the cut-off is shifted further to the ultraviolet. Furthermore, we observe a regular pattern in the enveloping function, which appears to be independent of y. Similar patterns were observed before in the literature, as for instance in the context of atomic physics described by a Klein-Gordon formalism (see figure 2 in43).

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p-integrable slectors, Mtimeasres, Banach space, Radon-Nikodym property, Finite p-variation, L p spaces.. AMS MOS SUBJECT CLASSIFICATION 1970 Codes...[r]

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There exists at least one natural reason to expect that the integrability of the dilatation operator of the β-deformed N = 4 SYM is spoiled at some number of loops if Im(β) 6 = 0. The construction of the supergravity dual of this type of deformation required [35] the use of S-duality transformations. Even though the resulting string coupling is small (due to the use of the two S-duality transformations) this seems to imply that non-interacting strings in the deformed background should have a knowledge of interactions in the undeformed theory. We, however, do not expect the dilatation operator of the N = 4 SYM theory be **integrable** in such finite N regime. The fact that the dilatation operator in the 3-spin (TrΦ J 1

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Even though it has been commented upon the occurrence of unstable particles in **integrable** quantum field theories in 1+1 dimensions as early as the late seventies [19], only recently such type of theories have been investigated in more detail. For instance, scattering matrices for such type of theories have been constructed [20, 21, 22], their ultraviolet behaviour has been analyzed by means of the thermodynamic Bethe ansatz [23, 24, 25, 26, 27] and also form factors have been constructed which were used to compute correlation functions needed in various quantities [28, 29, 30, 31].

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