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AbstractAbstract

[en] A computational classification of contact symmetries and higher-order local symmetries that do not commute with $t,x$, as well as local conserved densities that are not invariant under $t,x$ is carried out for a generalized version of the Krichever–Novikov (KN) equation. Several new results are obtained. First, the KN equation is explicitly shown to have a local conserved density that contains $t,x$. Second, apart from the dilational point symmetries known for special cases of the KN equation and its generalized version, no other local symmetries with low differential order are found to contain $t,x$. Third, the basic Hamiltonian structure of the KN equation is used to map the local conserved density containing $t,x$ into a nonlocal symmetry that contains $t,x$. Fourth, a recursion operator is applied to this nonlocal symmetry to produce a hierarchy of nonlocal symmetries that have explicit dependence on $t,x$. When the inverse of the Hamiltonian map is applied to this hierarchy, only trivial conserved densities are obtained. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/49/10/105201; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 49(10); [29 p.]

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Ipsen, J R, E-mail: jipsen@math.uni-bielefeld.de

AbstractAbstract

[en] We discuss the product of independent induced quaternion (β = 4) Ginibre matrices, and the eigenvalue correlations of this product matrix. The joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single Ginibre matrix, but with a more complicated weight function. We find the skew-orthogonal polynomials corresponding to the weight function of the product matrix, and use the method of skew-orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size. The radial behavior of the density of states is studied in the limit of large matrices, and the macroscopic density is discussed. The microscopic limit at the origin, at the edge(s) and in the bulk is also discussed for the radial behavior of the density of states. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/46/26/265201; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(26); [16 p.]

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Molinari, Luca Guido, E-mail: luca.molinari@mi.infn.it

AbstractAbstract

[en] This paper is about analytic properties of single transfer matrices originating from general block-tridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity. If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of the inverse of banded matrices. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/46/25/254004; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(25); [15 p.]

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AbstractAbstract

[en] In this paper, we use the finite-size Lyapunov exponent (FSLE) to characterize Lagrangian coherent structures in three-dimensional (3D) turbulent flows. Lagrangian coherent structures act as the organizers of transport in fluid flows and are crucial to understand their stirring and mixing properties. Generalized maxima (ridges) of the FSLE fields are used to locate these coherent structures. 3D FSLE fields are calculated in two phenomenologically distinct turbulent flows: a wall-bounded flow (channel flow) and a regional oceanic flow obtained by the numerical solution of the primitive equations where two-dimensional (2D) turbulence dominates. In the channel flow, autocorrelations of the FSLE field show that the structure is substantially different from the near wall to the mid-channel region and relates well to the more widely studied Eulerian coherent structure of the turbulent channel flow. The ridges of the FSLE field have complex shapes due to the 3D character of the turbulent fluctuations. In the oceanic flow, strong horizontal stirring is present and the flow regime is similar to that of 2D turbulence where the domain is populated by coherent eddies that interact strongly. This in turn results in the presence of high FSLE lines throughout the domain leading to strong non-local mixing. The ridges of the FSLE field are quasi-vertical surfaces, indicating that the horizontal dynamics dominates the flow. Indeed, due to rotation and stratification, vertical motions in the ocean are much less intense than horizontal ones. This suppression is absent in the channel flow, as the 3D character of the FSLE ridges shows. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/46/25/254022; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(25); [20 p.]

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Scimiterna, C; Levi, D, E-mail: levi@roma3.infn.it

AbstractAbstract

[en] We consider a class of nonlinear partial difference equations defined on three points of a plane lattice. We construct conditions for this class of partial difference equations to be linearizable through a point or a Cole–Hopf transformation. Using these conditions we are able to classify all multilinear linearizable equations belonging to this class. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/46/2/025205; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(2); [13 p.]

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Scullard, Christian R; Jacobsen, Jesper Lykke, E-mail: scullard1@llnl.gov, E-mail: jesper.jacobsen@ens.fr

AbstractAbstract

[en] Percolation thresholds have recently been studied by means of a graph polynomial P

_{B}(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P_{B}(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P_{B}(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P_{B}(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 8^{2}), kagome, and (3, 12^{2}) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p_{c}(4, 8^{2}) = 0.676 803 329…, p_{c}(kagome) = 0.524 404 998…, p_{c}(3, 12^{2}) = 0.740 420 798…, comparable to the best simulation results. We also show that the alternative definition of P_{B}(p) can be applied to study site percolation problems. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday. (paper)Primary Subject

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Available from http://dx.doi.org/10.1088/1751-8113/45/49/494004; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 45(49); [23 p.]

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Alfimov, G L, E-mail: galfimov@yahoo.com

AbstractAbstract

[en] The paper focuses on 2L-periodic solutions, U

_{L}(x), of the nonlocal equation Hu_{x}-u+u^{p}=0 (p > 2 is integer, H is the Hilbert transform). We give numerical evidence for the existence of a continuous family of these solutions parametrized by L for p = 3, 4, 5. The features of the solutions U_{L}(x) are discussed. In particular, we offer strong numerical arguments that U_{L}(x) can be continued analytically from the real axis to some strip in the complex plane, U_{L}(x) → U_{L}(z). The singularities which arise under analytical continuation of these solutions into the complex plane are considered. It is proved (theorem 1) that U_{L}(z) cannot be any power of a meromorphic function. The formula which describes the asymptotic behavior of U_{L}(z) in a neighborhood of the singularity is given. It agrees with the numerical results; computations also allow us to locate the closest to the real axis singularity of U_{L}(z) and estimate the width of the analyticity strip. (paper)Primary Subject

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Available from http://dx.doi.org/10.1088/1751-8113/45/39/395205; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 45(39); [13 p.]

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AbstractAbstract

[en] We consider Novikov’s Camassa–Holm type equation with cubic nonlinearity. In particular, we present a compact parametric representation of the smooth bright multisolution solutions on a constant background and investigate their structure. We find that the tau-functions associated with the solutions are closely related to those of a model equation for shallow-water waves (SWW) introduced by Hirota and Satsuma. This novel feature is established by applying the reciprocal transformation to the Novikov equation. We also show by specifying a complex phase parameter that the smooth soliton is converted to a novel singular soliton with single cusp and double peaks. We demonstrate that both the smooth and singular solitons converge to a peakon as the background field tends to zero, whereby we employ a method that has been developed for performing a similar limiting procedure for the multisoliton solutions of the Camassa–Holm equation. In the subsequent asymptotic analysis of the two- and N-soliton solutions, we confirm their solitonic behavior. Remarkably, the formulas for the phase shifts of the solitons as well as their peakon limits coincide formally with those of the Degasperis–Procesi equation. Last, we derive an infinite number of conservation laws of the Novikov equation by using a relation between solutions of the Novikov equation and those of the SWW equation. In appendix, we prove various bilinear identities associated with the tau-functions of the multisoliton solutions of the SWW equation. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/46/36/365203; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(36); [27 p.]

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Kamioka, Shuhei; Takagaki, Tomoaki, E-mail: kamioka.shuhei.3w@kyoto-u.ac.jp

AbstractAbstract

[en] Combinatorial expressions are presented of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules. For the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of non-intersecting paths, for which two results in combinatorics, Flajolet’s interpretation of continued fractions and Gessel–Viennot’s lemma on determinants, are applied. By ultradiscretizing the subtraction-free expression, the solution to the ultradiscrete Toda molecule is obtained. It is finally shown that the initial value problem of the ultradiscrete Toda molecule is exactly solved in terms of shortest paths on a specific graph. The behavior of the solution is also investigated in comparison with the box–ball system. (paper)

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Available from http://dx.doi.org/10.1088/1751-8113/46/35/355203; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(35); [19 p.]

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Agliari, E; Annibale, A; Barra, A; Coolen, A C C; Tantari, D, E-mail: agliari@fis.unipr.it, E-mail: alessia.annibale@kcl.ac.uk, E-mail: adriano.barra@roma1.infn.it, E-mail: ton.coolen@kcl.ac.uk, E-mail: tantari@mat.uniroma1.it

AbstractAbstract

[en] Associative network models featuring multi-tasking properties have been introduced recently and studied in the low-load regime, where the number P of simultaneously retrievable patterns scales with the number N of nodes as P ∼ log N. In addition to their relevance in artificial intelligence, these models are increasingly important in immunology, where stored patterns represent strategies to fight pathogens and nodes represent lymphocyte clones. They allow us to understand the crucial ability of the immune system to respond simultaneously to multiple distinct antigen invasions. Here we develop further the statistical mechanical analysis of such systems, by studying the medium-load regime, P ∼ N

^{δ}with δ ∈ (0, 1]. We derive three main results. First, we reveal the nontrivial architecture of these networks: they exhibit a high degree of modularity and clustering, which is linked to their retrieval abilities. Second, by solving the model we demonstrate for δ < 1 the existence of large regions in the phase diagram where the network can retrieve all stored patterns simultaneously. Finally, in the high-load regime δ = 1 we find that the system behaves as a spin-glass, suggesting that finite-connectivity frameworks are required to achieve effective retrieval. (paper)Primary Subject

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Available from http://dx.doi.org/10.1088/1751-8113/46/33/335101; Country of input: International Atomic Energy Agency (IAEA)

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Journal Article

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Journal of Physics. A, Mathematical and Theoretical (Online); ISSN 1751-8121; ; v. 46(33); [33 p.]

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