Generalized models as a universal approach to the analysis of nonlinear dynamical systems

We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of generalized models by considering examples from three different disciplines of science: a socio-economic model of dynastic cycles in china, a model for a coupled laser system and a general ecological food web.Comment: 15 pages, 2 figures, (Fig. 2 in color

Time-dependent quantum many-body theory of identical bosons in a double well: Early time ballistic interferences of fragmented and number entangled states

A time-dependent multiconfigurational self-consistent field theory is presented to describe the many-body dynamics of a gas of identical bosonic atoms confined to an external trapping potential at zero temperature from first principles. A set of generalized evolution equations are developed, through the time-dependent variational principle, which account for the complete and self-consistent coupling between the expansion coefficients of each configuration and the underlying one-body wave functions within a restricted two state Fock space basis that includes the full effects of the condensate's mean field as well as atomic correlation. The resulting dynamical equations are a classical Hamiltonian system and, by construction, form a well-defined initial value problem. They are implemented in an efficient numerical algorithm. An example is presented, highlighting the generality of the theory, in which the ballistic expansion of a fragmented condensate ground state is compared to that of a macroscopic quantum superposition state, taken here to be a highly entangled number state, upon releasing the external trapping potential. Strikingly different many-body matter-wave dynamics emerge in each case, accentuating the role of both atomic correlation and mean-field effects in the two condensate states.Comment: 16 pages, 5 figure

Everything, and then some

On its intended interpretation, logical, mathematical and metaphysical discourse sometimes seems to involve absolutely unrestricted quantification. Yet our standard semantic theories do not allow for interpretations of a language as expressing absolute generality. A prominent strategy for defending absolute generality, influentially proposed by Timothy Williamson in his paper ‘Everything’ (2003), avails itself of a hierarchy of quantifiers of ever increasing orders to develop non-standard semantic theories that do provide for such interpretations. However, as emphasized by Øystein Linnebo and Agustín Rayo (2012), there is pressure on this view to extend the quantificational hierarchy beyond the finite level, and, relatedly, to allow for a cumulative conception of the hierarchy. In his recent book, Modal Logic as Metaphysics (2013), Williamson yields to that pressure. I show that the emerging cumulative higher-orderist theory has implications of a strongly generality-relativist flavour, and consequently undermines much of the spirit of generality absolutism that Williamson set out to defend

Division Algebras and Extended N=2,4,8 SuperKdVs

The first example of an N=8 supersymmetric extension of the KdV equation is here explicitly constructed. It involves 8 bosonic and 8 fermionic fields. It corresponds to the unique N=8 solution based on a generalized hamiltonian dynamics with (generalized) Poisson brackets given by the Non-associative N=8 Superconformal Algebra. The complete list of inequivalent classes of parametric-dependent N=3 and N=4 superKdVs obtained from the ``Non-associative N=8 SCA" is also furnished. Furthermore, a fundamental domain characterizing the class of inequivalent N=4 superKdVs based on the "minimal N=4 SCA" is given.Comment: 14 pages, LaTe

LT revisited : explanation-based learning and the logic of Principia mathematica

This paper describes an explanation-based learning (EBL) system based on a version of Newell, Shaw and Simon's LOGIC-THEORIST (LT). Results of applying this system to propositional calculus problems from Principia Mathematica are compared with results of applying several other versions of the same performance element to these problems. The primary goal of this study is to characterize and analyze differences between not learning, rote learning (LT's original learning method), and EBL. Another aim is to provide base-line characterizations of the performance of a simple problem solver in the context of the Principa problems, in the hope that these problems can be used as a benchmark for testing improved learning methods, just as problems like chess and the eight puzzle have been used as benchmarks in research on search methods

Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a polygonal path composed of line segments with two adjacent directions from a given set $S$ of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply $NP$-hardness for any $S$ with $|S|\ge 4$ even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where $S$ contains precisely the horizontal and vertical directions, can be solved in polynomial time [GD'09]. Our results imply that this is incorrect unless $P=NP$. Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer and \v{S}tefankovi\v{c}. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati and Roselli.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017