567 research outputs found

    Lotka--Volterra Type Equations and their Explicit Integration

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    In the present note we give an explicit integration of some two--dimensionalised Lotka--Volterra type equations associated with simple Lie algebras, other than the familiar AnA_n case, possessing a representation without branching. This allows us, in particular, to treat the first fundamental representations of ArA_r, BrB_r, CrC_r, and G2G_2 on the same footing.Comment: 3 pages LATEX fil

    Riccati-type equations, generalised WZNW equations, and multidimensional Toda systems

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    We associate to an arbitrary Z\mathbb Z-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer--Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.Comment: LaTeX2e, 18 page

    The Digital Silicon Photomultiplier

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    The Silicon Photomultipliers (SiPMs) are the new step in the development of the modern detection structures in the area of low photon flux detection with a unique capability of detection up to the single photons. The Silicon Photomultiplier intrinsically represents a digital signal source on the elementary cell level. The materials and technology of SiPMs are consistent with the modern electronics technology. We present the realization and implementation of a fully digital Silicon Photomultiplier Imager with an enclosed readout and processing on the basis of modern 3D technology

    Continuous approximation of binomial lattices

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    A systematic analysis of a continuous version of a binomial lattice, containing a real parameter γ\gamma and covering the Toda field equation as γ→∞\gamma\to\infty, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like approximate solution is also obtained which reproduces the Eguchi-Hanson instanton configuration for γ→∞\gamma\to\infty. Furthermore, the equation under consideration is extended to (n+1)(n+1)--dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these (n=4)(n=4) enables us to establish a connection with the Euclidean Yang-Mills equations, another appears in the context of Differential Geometry in relation to the socalled Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation.Comment: 30 pages, LaTeX, no figures. Submitted to Annals of Physic

    Multidimensional Toda type systems

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    On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed.Comment: 29 pages, LaTeX fil

    WW--geometry of the Toda systems associated with non-exceptional simple Lie algebras

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    The present paper describes the WW--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the B,CB,C and DD series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Pl\"ucker embedding of the AA-case to the flag manifolds associated with the fundamental representations of BnB_n, CnC_n and DnD_n, and a direct proof that the corresponding K\"ahler potentials satisfy the system of two--dimensional finite non-periodic (conformal) Toda equations. It is shown that the WW--geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of CPNCP^N with appropriate choices of NN. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Pl\"ucker embedding. These conditions are automatically fulfiled when Toda equations hold.Comment: 30 pages, no figur

    Topological gravity on plumbed V-cobordisms

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    An ensemble of cosmological models based on generalized BF-theory is constructed where the role of vacuum (zero-level) coupling constants is played by topologically invariant rational intersection forms (cosmological-constant matrices) of 4-dimensional plumbed V-cobordisms which are interpreted as Euclidean spacetime regions. For these regions describing topology changes, the rational and integer intersection matrices are calculated. A relation is found between the hierarchy of certain elements of these matrices and the hierarchy of coupling constants of the universal (low-energy) interactions. PACS numbers: 0420G, 0240, 0460Comment: 29 page

    The sl(2n|2n)^(1) Super-Toda Lattices and the Heavenly Equations as Continuum Limit

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    The n→∞n\to\infty continuum limit of super-Toda models associated with the affine sl(2n∣2n)(1)sl(2n|2n)^{(1)} (super)algebra series produces (2+1)(2+1)-dimensional integrable equations in the S1×R2{\bf S}^{1}\times {\bf R}^2 spacetimes. The equations of motion of the (super)Toda hierarchies depend not only on the chosen (super)algebras but also on the specific presentation of their Cartan matrices. Four distinct series of integrable hierarchies in relation with symmetric-versus-antisymmetric, null-versus-nonnull presentations of the corresponding Cartan matrices are investigated. In the continuum limit we derive four classes of integrable equations of heavenly type, generalizing the results previously obtained in the literature. The systems are manifestly N=1 supersymmetric and, for specific choices of the Cartan matrix preserving the complex structure, admit a hidden N=2 supersymmetry. The coset reduction of the (super)-heavenly equation to the I×R(2)=(S1/Z2)×R2{\bf I}\times{\bf R}^{(2)}=({\bf S}^{1}/{\bf Z}_2)\times {\bf R}^2 spacetime (with I{\bf I} a line segment) is illustrated. Finally, integrable N=2,4N=2,4 supersymmetrically extended models in (1+1)(1+1) dimensions are constructed through dimensional reduction of the previous systems.Comment: 12 page
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