242 research outputs found
Transversality Conditions for Infinite Horizon Variational Problems on Time Scales
We consider problems of the calculus of variations on unbounded time scales.
We prove the validity of the Euler-Lagrange equation on time scales for
infinite horizon problems, and a new transversality condition.Comment: Submitted 6-October-2009; Accepted 19-March-2010 in revised form; for
publication in "Optimization Letters"
A General Backwards Calculus of Variations via Duality
We prove Euler-Lagrange and natural boundary necessary optimality conditions
for problems of the calculus of variations which are given by a composition of
nabla integrals on an arbitrary time scale. As an application, we get
optimality conditions for the product and the quotient of nabla variational
functionals.Comment: Submitted to Optimization Letters 03-June-2010; revised 01-July-2010;
accepted for publication 08-July-201
Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales
We obtain Euler-Lagrange and transversality optimality conditions for
higher-order infinite horizon variational problems on a time scale. The new
necessary optimality conditions improve the classical results both in the
continuous and discrete settings: our results seem new and interesting even in
the particular cases when the time scale is the set of real numbers or the set
of integers.Comment: This is a preprint of a paper whose final and definite form will
appear in Journal of Optimization Theory and Applications (JOTA). Paper
submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for
publication 15-April-201
Euler-Lagrange equations for composition functionals in calculus of variations on time scales
In this paper we consider the problem of the calculus of variations for a
functional which is the composition of a certain scalar function with the
delta integral of a vector valued field , i.e., of the form
. Euler-Lagrange
equations, natural boundary conditions for such problems as well as a necessary
optimality condition for isoperimetric problems, on a general time scale, are
given. A number of corollaries are obtained, and several examples illustrating
the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems
(DCDS-B); revised 10-March-2010; accepted 04-July-201
Visibility graphs of fractional Wu-Baleanu time series
[EN] We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.J.A. Conejero is supported Ministerio de Economia y Competitividad Grant Project MTM2016-75963-P. Carlos Lizama is supported by CONICYT, under Fondecyt Grant number 1180041. Cristobal Rodero-Gomez is funded by European Commission H2020 research and Innovation programme under the Marie Sklodowska-Curie grant agreement No. 764738.Conejero, JA.; Lizama, C.; Mira-Iglesias, A.; Rodero-Gómez, C. (2019). Visibility graphs of fractional Wu-Baleanu time series. The Journal of Difference Equations and Applications. 25(9-10):1321-1331. https://doi.org/10.1080/10236198.2019.1619714S13211331259-10Anand, K., & Bianconi, G. (2009). Entropy measures for networks: Toward an information theory of complex topologies. Physical Review E, 80(4). doi:10.1103/physreve.80.045102Barabási, A.-L., & Albert, R. (1999). Emergence of Scaling in Random Networks. Science, 286(5439), 509-512. doi:10.1126/science.286.5439.509Brzeziński, D. W. (2017). Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition. Applied Mathematics and Nonlinear Sciences, 2(1), 237-248. doi:10.21042/amns.2017.1.00020Brzeziński, D. W. (2018). Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus. Applied Mathematics and Nonlinear Sciences, 3(2), 487-502. doi:10.2478/amns.2018.2.00038DONNER, R. V., SMALL, M., DONGES, J. F., MARWAN, N., ZOU, Y., XIANG, R., & KURTHS, J. (2011). RECURRENCE-BASED TIME SERIES ANALYSIS BY MEANS OF COMPLEX NETWORK METHODS. International Journal of Bifurcation and Chaos, 21(04), 1019-1046. doi:10.1142/s0218127411029021Edelman, M. (2015). On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(7), 073103. doi:10.1063/1.4922834Edelman, M. (2018). On stability of fixed points and chaos in fractional systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(2), 023112. doi:10.1063/1.5016437Gao, Z.-K., Small, M., & Kurths, J. (2016). Complex network analysis of time series. EPL (Europhysics Letters), 116(5), 50001. doi:10.1209/0295-5075/116/50001Iacovacci, J., & Lacasa, L. (2016). Sequential visibility-graph motifs. Physical Review E, 93(4). doi:10.1103/physreve.93.042309Indahl, U. G., Naes, T., & Liland, K. H. (2018). A similarity index for comparing coupled matrices. Journal of Chemometrics, 32(10), e3049. doi:10.1002/cem.3049Kantz, H., & Schreiber, T. (2003). Nonlinear Time Series Analysis. doi:10.1017/cbo9780511755798Lacasa, L., & Iacovacci, J. (2017). Visibility graphs of random scalar fields and spatial data. Physical Review E, 96(1). doi:10.1103/physreve.96.012318Lacasa, L., Luque, B., Ballesteros, F., Luque, J., & Nuño, J. C. (2008). From time series to complex networks: The visibility graph. Proceedings of the National Academy of Sciences, 105(13), 4972-4975. doi:10.1073/pnas.0709247105Lizama, C. (2015). lp-maximal regularity for fractional difference equations on UMD spaces. Mathematische Nachrichten, 288(17-18), 2079-2092. doi:10.1002/mana.201400326Lizama, C. (2017). The Poisson distribution, abstract fractional difference equations, and stability. Proceedings of the American Mathematical Society, 145(9), 3809-3827. doi:10.1090/proc/12895Luque, B., Lacasa, L., Ballesteros, F., & Luque, J. (2009). Horizontal visibility graphs: Exact results for random time series. Physical Review E, 80(4). doi:10.1103/physreve.80.046103Luque, B., Lacasa, L., Ballesteros, F. J., & Robledo, A. (2011). Feigenbaum Graphs: A Complex Network Perspective of Chaos. PLoS ONE, 6(9), e22411. doi:10.1371/journal.pone.0022411Luque, B., Lacasa, L., & Robledo, A. (2012). Feigenbaum graphs at the onset of chaos. Physics Letters A, 376(47-48), 3625-3629. doi:10.1016/j.physleta.2012.10.050May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459-467. doi:10.1038/261459a0Núñez, Á. M., Luque, B., Lacasa, L., Gómez, J. P., & Robledo, A. (2013). Horizontal visibility graphs generated by type-I intermittency. Physical Review E, 87(5). doi:10.1103/physreve.87.052801Ravetti, M. G., Carpi, L. C., Gonçalves, B. A., Frery, A. C., & Rosso, O. A. (2014). Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph. PLoS ONE, 9(9), e108004. doi:10.1371/journal.pone.0108004Robledo, A. (2013). Generalized Statistical Mechanics at the Onset of Chaos. Entropy, 15(12), 5178-5222. doi:10.3390/e15125178Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379-423. doi:10.1002/j.1538-7305.1948.tb01338.xSong, C., Havlin, S., & Makse, H. A. (2006). Origins of fractality in the growth of complex networks. Nature Physics, 2(4), 275-281. doi:10.1038/nphys266West, J., Lacasa, L., Severini, S., & Teschendorff, A. (2012). Approximate entropy of network parameters. Physical Review E, 85(4). doi:10.1103/physreve.85.046111Wu, G.-C., & Baleanu, D. (2013). Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), 283-287. doi:10.1007/s11071-013-1065-7Wu, G.-C., & Baleanu, D. (2014). Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics, 80(4), 1697-1703. doi:10.1007/s11071-014-1250-3Zhang, J., & Small, M. (2006). Complex Network from Pseudoperiodic Time Series: Topology versus Dynamics. Physical Review Letters, 96(23). doi:10.1103/physrevlett.96.23870
Universality in Systems with Power-Law Memory and Fractional Dynamics
There are a few different ways to extend regular nonlinear dynamical systems
by introducing power-law memory or considering fractional
differential/difference equations instead of integer ones. This extension
allows the introduction of families of nonlinear dynamical systems converging
to regular systems in the case of an integer power-law memory or an integer
order of derivatives/differences. The examples considered in this review
include the logistic family of maps (converging in the case of the first order
difference to the regular logistic map), the universal family of maps, and the
standard family of maps (the latter two converging, in the case of the second
difference, to the regular universal and standard maps). Correspondingly, the
phenomenon of transition to chaos through a period doubling cascade of
bifurcations in regular nonlinear systems, known as "universality", can be
extended to fractional maps, which are maps with power-/asymptotically
power-law memory. The new features of universality, including cascades of
bifurcations on single trajectories, which appear in fractional (with memory)
nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201
The relationship between the phosphate and structural carbonate fractionation of fallow deer bioapatite in tooth enamel
The species‐specific relationship between phosphate (δ18OP values) and structural carbonate (δ18OC values) oxygen isotope ratios has been established for several modern and fossil animal species but until now it has not been investigated in European fallow deer (Dama dama dama). This study describes the relationship between phosphate and structural carbonate bioapatite in tooth enamel of extant fallow deer, which will help us further understand the species' unique environmental and cultural history.
Methods
The oxygen isotope composition of phosphate (δ18OP value) and structural carbonate (δ18OC value) of hydroxylapatite was determined in 51 modern fallow deer tooth enamel samples from across Europe and West Asia. The δ18OC values were measured on a GV IsoPrime dual‐inlet mass spectrometer and the δ18OP values on a temperature‐controlled elemental analyser (TC/EA) coupled to a DeltaPlus XL isotope ratio mass spectrometer via a ConFlo III interface.
Results
This study establishes a direct and linear relationship between the δ18OC and δ18OP values from fallow deer tooth enamel (δ18OC = +9.244(±0.216) + 0.958 * δ18OP (±0.013)). Despite the successful regression, the variation in δ18O values from samples collected in the same geographical area is greater than expected, although the results cluster in broad climatic groupings when Koppen‐Geiger classifications are taken into account for the individuals' locations.
Conclusions
This is the first comprehensive study of the relationship between ionic forms of oxygen (phosphate oxygen and structural carbonate) in fallow deer dental enamel. The new equation will allow direct comparison with other herbivore data. Variable δ18O values within populations of fallow deer broadly reflect the ecological zones they are found in which may explain this pattern of results in other euryphagic species
The Hahn Quantum Variational Calculus
We introduce the Hahn quantum variational calculus. Necessary and sufficient
optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange
problems, are studied. We also show the validity of Leitmann's direct method
for the Hahn quantum variational calculus, and give explicit solutions to some
concrete problems. To illustrate the results, we provide several examples and
discuss a quantum version of the well known Ramsey model of economics.Comment: Submitted: 3/March/2010; 4th revision: 9/June/2010; accepted:
18/June/2010; for publication in Journal of Optimization Theory and
Application
Hardness and approximation for the geodetic set problem in some graph classes
In this paper, we study the computational complexity of finding the
\emph{geodetic number} of graphs. A set of vertices of a graph is a
\emph{geodetic set} if any vertex of lies in some shortest path between
some pair of vertices from . The \textsc{Minimum Geodetic Set (MGS)} problem
is to find a geodetic set with minimum cardinality. In this paper, we prove
that solving the \textsc{MGS} problem is NP-hard on planar graphs with a
maximum degree six and line graphs. We also show that unless , there is
no polynomial time algorithm to solve the \textsc{MGS} problem with
sublogarithmic approximation factor (in terms of the number of vertices) even
on graphs with diameter . On the positive side, we give an
-approximation algorithm for the \textsc{MGS}
problem on general graphs of order . We also give a -approximation
algorithm for the \textsc{MGS} problem on the family of solid grid graphs which
is a subclass of planar graphs
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