Relevant bibliographies by topics / Doubling property

Academic literature on the topic 'Doubling property'

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Published: 9 March 2023

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Journal articles on the topic "Doubling property"

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Su, Bo. "Doubling property of elliptic equations." Communications on Pure & Applied Analysis 7, no. 1 (2008): 143–47. http://dx.doi.org/10.3934/cpaa.2008.7.143.

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Wang, Liqiu, and Mingtian Xu. "Property of period-doubling bifurcations." Chaos, Solitons & Fractals 24, no. 2 (April 2005): 527–32. http://dx.doi.org/10.1016/j.chaos.2004.09.045.

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Shkredov, I. D. "On sets with small doubling property." Mathematical Notes 84, no. 5-6 (December 2008): 859–78. http://dx.doi.org/10.1134/s000143460811028x.

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Aimar, Hugo, Marilina Carena, and Bibiana Iaffei. "Gradual doubling property of Hutchinson orbits." Czechoslovak Mathematical Journal 65, no. 1 (March 2015): 191–205. http://dx.doi.org/10.1007/s10587-015-0168-3.

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Le Donne, Enrico. "Doubling Property for BiLipschitz Homogeneous Geodesic Surfaces." Journal of Geometric Analysis 21, no. 4 (August 4, 2010): 783–806. http://dx.doi.org/10.1007/s12220-010-9167-7.

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Jin, Renling. "Freiman's inverse problem with small doubling property." Advances in Mathematics 216, no. 2 (December 2007): 711–52. http://dx.doi.org/10.1016/j.aim.2007.06.002.

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Zhu, Jiuyi. "Doubling Property and Vanishing Order of Steklov Eigenfunctions." Communications in Partial Differential Equations 40, no. 8 (April 16, 2015): 1498–520. http://dx.doi.org/10.1080/03605302.2015.1025980.

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Baudoin, Fabrice, and Nicola Garofalo. "Perelman’s Entropy and Doubling Property on Riemannian Manifolds." Journal of Geometric Analysis 21, no. 4 (September 14, 2010): 1119–31. http://dx.doi.org/10.1007/s12220-010-9180-x.

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Heer, Loreno. "Some Invariant Properties of Quasi-Möbius Maps." Analysis and Geometry in Metric Spaces 5, no. 1 (September 2, 2017): 69–77. http://dx.doi.org/10.1515/agms-2017-0004.

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Abstract We investigate properties which remain invariant under the action of quasi-Möbius maps of quasimetric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)Möbius maps. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)Möbius maps as well.
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Caffarelli, Luis A., and Michael G. Crandall. "Relations between geometric convexity, doubling measures and property $\Gamma $." Proceedings of the American Mathematical Society 142, no. 7 (March 21, 2014): 2395–406. http://dx.doi.org/10.1090/s0002-9939-2014-11940-x.

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Dissertations / Theses on the topic "Doubling property"

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Alcaraz, Barrera Rafael. "Topological and symbolic dynamics of the doubling map with a hole." Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/topological-and-symbolic-dynamics-of-the-doubling-map-with-a-hole(b6f17b43-5285-4e35-883a-baf4708993bc).html.

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This work motivates the study of open dynamical systems corresponding to the doubling map. In particular, the dynamical properties of the attractor of the doubling map when a symmetric, centred open interval is removed are studied. Using the arithmetical properties of the binary expansion of the points on the boundary of the removed interval, we study properties such as topological transitivity, the specification property and intrinsic ergodicity. The properties of the function that associates to each hole $(a,b)$ the topological entropy of the attractor of the considered dynamical system are also shown. For these purposes, a subshift corresponding to an element of the lexicographic world is associated to each attractor and the mentioned properties are studied symbolically.
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SALOGNI, FRANCESCA. "Harmonic Bergman spaces, Hardy-type spaces and harmonic analysis of a symmetric diffusion semigroup on R^n." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2013. http://hdl.handle.net/10281/41814.

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This thesis is divided into two parts, which deal with quite diverse subjects. The first part is, in turn, divided into two chapters. The first focuses on the development of new function spaces in $R^n$, called generalized Bergman spaces, and on their application to the Hardy space $H^1(R^n)$. The second is devoted to the theory of Bergman spaces on noncompact Riemannian manifolds which possess the doubling property and to its relationships with spaces of Hardy type. The latter are tailored to produce endpoint estimates for interesting operators, mainly related to the Laplace-Beltrami operator. The second part is devoted to the study of some interesting properties of the operator $A f = -1/2 \Delta f- x \cdot \nabla f \forall f \in C_c^\infty (R^n)$, which is essentially self-adjoint with respect to the measure $d \gamma_{-1}(x) = \pi^{n/2} \e^{|x|^2} d \lambda (x) \forall x \in R^n$, where $\lambda$ denotes the Lebesgue measure, and of the semigroup that $A$ generates.
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Feng, Zhaosheng. "Some results on the 1D linear wave equation with van der Pol type nonlinear boundary conditionsand the Korteweg-de Vries-Burgers equation." Diss., Texas A&M University, 2004. http://hdl.handle.net/1969.1/1078.

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Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt − wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F (un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result.
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Books on the topic "Doubling property"

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Machery, Edouard. Doubling Down on the Nomological Notion of Human Nature. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198823650.003.0002.

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This chapter defends the nomological notion of human nature according to which human nature is the set of properties that humans tend to possess as a result of the evolution of their species. In particular, I show why it is appropriate to single out the traits that are typical of human beings (the ‘universality proposal’) and the traits that have evolved (the ‘evolution proposal’). According to the former, traits that belong to human nature must be typical of human beings; according to the latter, they must have evolved. This proposal has been extensively criticized, and the goal of this chapter is to address these criticisms and to improve the nomological notion of human nature. In particular, the chapter aims to distinguish those traits that can be proper targets of explanations appealing to evolutionary information from those that can’t.
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Book chapters on the topic "Doubling property"

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Smid, Michiel. "The Weak Gap Property in Metric Spaces of Bounded Doubling Dimension." In Lecture Notes in Computer Science, 275–89. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-03456-5_19.

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Senapati, Apurbalal, Soumen Maji, and Arunendu Mondal. "Limitations and Implications of Doubling Time Approach in COVID-19 Infection Spreading Study." In Data Preprocessing, Active Learning, and Cost Perceptive Approaches for Resolving Data Imbalance, 137–48. IGI Global, 2021. http://dx.doi.org/10.4018/978-1-7998-7371-6.ch007.

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To control the spread of COVID-19, around the world, many countries imposed lockdowns. Numerous studies were reported on COVID-19 in different disciplines with various aspects. The doubling time is a mathematical technique to estimate the current rate of spread of the disease. Researchers used the doubling technique to address the COVID-19 pandemic situation. The larger doubling period represents a low spreading rate, whereas the smaller doubling period represents a high spreading rate. In other words, high infection implies the low doubling period and low infection implies the high doubling period. So, there is an inverse relationship between doubling time and the infection rate. But the real-life data does not follow such a rule properly in various domains. The data shows that after a certain time when the infection is high, the doubling period is also high, which misleads our general concept of doubling time. This chapter addressed this issue by investigating the real-time COVID-19 data. To overcome this limitation, a gradient smoothing technique has been proposed.
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Chatzopoulou, Katerina. "Renewal and stability." In Negation and Nonveridicality in the History of Greek, 173–208. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198712404.003.0006.

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This chapter discusses the Greek negator transformations in relation to Jespersen’s Cycle. The developments of NEG1 and NEG2 in Greek do not properly qualify as instances of Jespersen’s Cycle in the traditional understanding of the phenomenon, as it did not manifest a doubling stage. A new approach for Jespersen’s Cycle is proposed, which accommodates not only Greek, but also various other atypical languages that deviate in one way or another from the traditional morphosyntactic description of the phenomenon. It is proposed that Jespersen’s Cycle is a diachronic phenomenon whose regularities are to be found in the semantics. An overview is also provided of the diachronically stable functions of NEG2, which are the COMP-related functions of NEG2 μη‎. It is argued that NEG2 μη‎ did not eventually renew, because of the inertial pressures of its several nonnegative functions, which, being nonnegative, were not affected by Jespersen’s Cycle phenomena.
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Adams, David. "Italian Diction." In A Handbook of Diction for Singers, 31–130. 3rd ed. Oxford University PressNew York, 2022. http://dx.doi.org/10.1093/oso/9780197639504.003.0002.

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Abstract This chapter presents the sounds of Italian: vowels, consonants, and glides, along with the relevant phonetic symbols. It explains diacritical marks, vowel length, syllabification, word stress, and apocopation. There is extensive discussion of diphthongs and triphthongs, both within words and across words boundaries (phrasal diphthongs). Double consonants and phrasal doubling across word boundaries are presented in detail. There are examples of words with translations and phonetic transcriptions, as well as musical examples. Problems specific to singing are discussed, such as proper value of vowels in diphthongs and triphthongs and how to execute double consonants and consonant clusters. Resources for Italian and sample vocal texts with phonetic transcriptions and translations are provided. There are exercises interspersed throughout the chapter. Following the main chapter is an appendix, primarily meant as a reference, that discusses the two sounds each for the letters e and o in Italian, the open sound and the close sound.
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Mukhala, Elijah. "Food and Agriculture Organization and Agricultural Droughts." In Monitoring and Predicting Agricultural Drought. Oxford University Press, 2005. http://dx.doi.org/10.1093/oso/9780195162349.003.0044.

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The Food and Agriculture Organization (FAO) of the United Nations was founded in 1945 with a mandate to raise levels of nutrition and standards of living, to improve agricultural productivity, and to improve the condition of rural populations in the world. Today, FAO is the largest specialized agency in the United Nations system and is the lead agency for agriculture and rural development. FAO is composed of eight departments: Agriculture, Economic and Social, Fisheries, Forestry, Sustainable Development, Technical Cooperation, General Affairs, and Information and Administration and Finance. As an intergovernmental organization, FAO has 183 member countries plus one member organization, the European Union. Since its inception, FAO has worked to alleviate poverty and hunger by promoting agricultural development, improved nutrition, and the pursuit of food security—defined as the access of all people at all times to the food they need for an active and healthy life. Food production in the world has increased at an unprecedented rate since FAO was founded, outpacing the doubling of the world’s population over the same period. Since the early 1960s, the proportion of hungry people in the developing world has been reduced from more than 50% to less than 20%. Despite these progressive developments, more than 790 million people in the developing world— more than the total population of North America and Western Europe combined—still go hungry (FAO, 2004). FAO strives to reduce food insecurity in the world, especially in developing countries. In 1996, the World Food Summit convened by FAO in Rome adopted a plan of action aimed to reduce the number of the world’s hungry people in half by 2015. While the proper foundation of this goal lies, among others, in the increase of food production and ensuring access to food, there is also a need to monitor the current food supply and demand situation, so that timely interventions can be planned whenever the possibility of drought, famine, starvation, or malnutrition exists. With an imminent food crisis, actions need to be taken as early as possible because it takes time to mobilize resources, and logistic operations are often hampered by adverse natural or societal conditions, including war and civil strife.
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Maynard Smith, John, and Eors Szathmary. "The origin of sex and the nature of species." In The Major Transitions in Evolution. Oxford University Press, 1997. http://dx.doi.org/10.1093/oso/9780198502944.003.0013.

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By sex in eukaryotes, we understand a more-or-less regular succession of meiosis and syngamy. A natural consequence of this is the alternation of haploid and diploid phases in the life cycle. Eukaryotic sex significantly differs from prokaryotic sex in two crucial respects: the cellular mechanisms are quite different, and the transfer of genetic material in prokaryotes is less frequent and more localized (Maynard Smith et al., 1991). However, there seems to be significant continuity in the molecular mechanisms: sex in either case requires recombination enzymes, many of which are active in repair of damaged DNA as well. Thus, it seems plausible that recombinational repair was a preadaptation for sexual recombination. We mention in passing that there is a theory that selection for the recombinational repair of doublestrand DNA damage is responsible for the current maintenance of eukaryotic sex (Bernstein et al., 1981, 1988), but there are severe theoretical as well as factual problems with this theory; we will mention some factual difficulties later. Although an alternation of haploid and diploid phases follows from sex, a clue to the origin problem may lie in the idea that this alternation existed before the evolution of sexual recombination proper. The first hint that this may have been so comes from the classic paper by Cleveland (1947), where he proposed that the haploid-diploid cycle may have started with a spontaneous diploidization by endomitosis: that is, without syngamy. His suggestions were based on original observations on primitive flagellates (hypermastigotes and polymastigotes). Among them, Barbulanympha has a regular endomitosis-meiosis cycle. Margulis & Sagan (1986) called renewed attention to Cleveland’s ideas. In particular, they argued that the alternation of ploidy phases could have a primarily ecological explanation: if the environment alternates in some important factors, this may drive a haploid-diploid cycle, provided the phases are adaptations to different environments. For example, diploids have a smaller relative surface area than haploids, which may confer higher metabolic efficiency. We shall come back to such ideas soon. We focus first on the possible cellular mechanisms connecting the two phases. It is important that some protists have a one-step rather than a two-step meiosis: after syngamy, the two homologous chromosomes become disjunct without premeiotic doubling.
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Conference papers on the topic "Doubling property"

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Yao, Jianquan, Yu Li, and Dapeng Zhang. "Effect of the polarizing property in a frequency-doubled YAG laser." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1987. http://dx.doi.org/10.1364/oam.1987.wr9.

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For type II phase-matched frequency doubling in a biaxial crystal, we proved that the amplitude ratio of two vertical fundamental wavefields in the crystal is an important factor affecting the frequency doubling efficiency (FDE). The plot of FDE as a function of the ratio has been drawn by numerically solving the coupling wave equations. The optimum orientation of the frequency-doubling crystal to the laser rod has been determined in which the FDE is the highest. When KT/OPO4(KTP) crystal is used as the frequency doubler in a YAG laser, the optimum orientation is the position in which the ratio is 0.976.
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Honda, Norihiro, Katsunori Ishii, Akinori Kimura, Makoto Sakai, and Kunio Awazu. "Determination of optical property changes by laser treatments using inverse adding-doubling method." In SPIE BiOS: Biomedical Optics, edited by Steven L. Jacques, E. Duco Jansen, and William P. Roach. SPIE, 2009. http://dx.doi.org/10.1117/12.810029.

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Cheng, L. T. "Organic Nonlinear Materials for Frequency Conversion: Property Trade-off and Recent Applications." In Compact Blue-Green Lasers. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/cbgl.1992.thd2.

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Frequency doubling of short wavelength diode lasers has significant technological applications in imaging and optical recording. One of the major hurdles towards achieving efficient doubling remains to develop highly nonlinear materials with appropriate linear optical properties such as birefringence and transparency. One approach is to engineer materials by doping or grafting nonlinear organic molecules into transparent polymeric matrixes. Second-order nonlinearity is obtained by the acentric alignment of these molecules along their dipole moments under an external electric field. With this approach, several molecular properties are important, including large molecular dipole moment and hyperpolarizability inner product (μβμ) as well as very low absorption at the first and second harmonic wavelengths of diode lasers (near 800 and 400 nm). An established guideline in identifying organic molecules with large hyperpolarizabilities is by the presence of low-lying strong charge-transfer (CT) electronic transitions. However, such an approach is in conflict with the requirement of transparency.
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He, Qi-Wei, Shi-Jian Zhu, Jing-Jun Lou, and Lin He. "Application of Air Spring in Controlling Line Spectra." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84276.

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Application of air spring in controlling line spectra in the radiated noise of marine vessels was studied. Starting with the route to chaos and the scaling property of the power spectrum in the cascade of period-doubling bifurcations, the method of chaotic vibration isolation was advanced. The performance indices for the presented method were also given. The method was experimentally verified.
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Stone, Bryan D., and G. W. Forbes. "Novel techniques for ray tracing in inhomogeneous media." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1989. http://dx.doi.org/10.1364/oam.1989.mt1.

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Tracing rays through inhomogeneous media generally involves numerically integrating a differential equation. Accuracy doubling,1 which takes advantage of the extremal property of a ray (namely, Fermat's principle) in a novel way, provides a framework for deriving new higher-order integration schemes from existing integration schemes. These accuracy doubled integration schemes will, in general, be of twice the order as the scheme from which they are derived. This technique has been applied to a commonly used fourth-order Runge-Kutta scheme introduced by Sharma et al.2 In the context of inhomogeneous ray tracing, however, the form of the differential equation makes it possible to derive a more efficient fourth-order Runge-Kutta scheme on which to apply the techniques of accuracy doubling. The resulting scheme is eighth order, but it requires fewer computations to achieve a given accuracy than the eighth-order scheme derived from Sharma's method. It so happens that, to derive these accuracy doubled schemes, the underlying Runge-Kutta scheme must be modified to provide differential ray information about the base ray being traced. A method of determining this differential ray information is presented, and this procedure has uses outside the context of accuracy doubling.
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Yu, Xiang, Shijian Zhu, and Jingjun Lou. "Bifurcation Analysis of Nonlinear Vibration Isolation System With Hard Stiffness by Cell Mapping." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48497.

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Period-doubling bifurcation is one of the major routes to chaos, but the methods widely used have some shortages in analyzing the bifurcation of the nonlinear vibration isolation system with hard stiffness. The location and property of bifurcation solution can be obtained conveniently by using numerical methods. Therefore, numerical research is important. In this paper, global bifurcation diagram is achieved by using Poincare´ mapping method. Subsequently, cell mapping as a useful numerical method is applied to analyze the static and dynamic bifurcation of nonlinear vibration isolation system with hard stiffness.
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Kao, Y. H., and H. M. Chen. "Instability and Routes to Chaos in Laser Diodes Due to External Optical Feedback." In Nonlinear Optics. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nlo.1992.we7.

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It is well known that the semiconductor lasers, are susceptible to optical feedback from discrete interfaces or from distributed scattering such as in optical fiber medium. Feedback will affect the spectral property and the stability of the laser output. A number of dynamical behaviors, period doubling route, quasiperiodic route, and bistability, have been found[1–4]. In this letter we report the studies of routes to chaos of single mode laser diode output subjected to a variety of external cavity length. The studies of such effects are important not only for understanding of laser dynamical properties but also for applications of lasers in various systems.
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Jiang, L., and H. L. Tsai. "Ultrafast Photon-Electron Interactions in Dielectrics by a Single Laser Pulse." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-59288.

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This study develops a quantum mechanical model to investigate energy absorption in ultrafast laser of dielectrics. The model investigates the optical property variations, electron temperature, and density changes at femtosecond scales. The ionizations and electron heating are two major factors considered for pulse absorption occurring within the pulse duration. The flux-doubling model is employed to calculate the free electron generation mainly through impact ionization and photoionization. The quantum mechanical treatments are used to account for the specific heat and the relaxation time for free electrons. The time and space dependent optical properties of the dense plasma generated by the ultrafast laser pulse are calculated. The predictions of ablation threshold and ablation depth of fused silica and barium aluminum borosilicate (BBS) are in good agreements with published experimental data. The model greatly improves the accuracy in predicting the ablation depth and can predict the crater shape.
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Arneodo, A., F. Argoul, and P. Richetti. "Symbolic dynamics in the Belousov-Zhabotinskii reaction: from Rössler’s intuition to experimental evidence for Shil’nikov homoclinic chaos." In Nonlinear Dynamics in Optical Systems. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/nldos.1990.is2.

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The Belousov-Zhabotinskii reaction has revealed most of the well-known scenarios to chaos including period-doubling, intermittency, quasiperiodicity, frequency locking, fractal torus …. However, although the data have been shown to display unambiguous features of deterministic chaos, the understanding of the nature and the origin of the observed behavior has been incomplete. In 1976, Rössler suggested an intuitive interpretation to explain chemical chaos. His feeling was that nonperiodic wandering trajectories might arise in chemical systems from a pleated slow manifold (Fig. 1a), if the flow on the lower surface of the pleat had the property of returning trajectories to a small neighborhood of an unstable focus lying on the upper surface. In this communication, we intend to revisit the terminology introduced by Rössler of “spiral-type”, “screw-type” and “funnel-type” strange attractors in terms of chaotic orbits that occur in nearly homoclinic conditions. According to a theorem by Shil’nikov, there exist uncountably many nonperiodic trajectories in systems which display a homoclinic orbit biasymptotic to a saddle-focus O, providing the following condition is fulfilled: ρ/λ < 1, where the eigenvalue of O are (−λ, ρ ± iω). This subset of chaotic trajectories is actually in one to one correspondance with a shift automorphism with an infinite number of symbols. Since homoclinic orbits are structurally unstable objects which lie on codimension-one hypersurfaces in the constraint space, one can reasonably hope to cross these hypersurfaces when following a one-parameter path. The bifurcation structure encountered near homoclinicity involves infinite sequences of saddle-node and period-doubling bifurcations. The aim of this paper is to provide numerical and experimental evidences for Shil’nikov homoclinic chaos in nonequilibrium chemical systems.
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Arafat, Haider N., Ali H. Nayfeh, and Char-Ming Chin. "Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4028.

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Abstract The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its “exural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to the two integro-partial-differential equations of Crespo da Silva and Glynn. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. The modulation equations exhibit the symmetry property found by Feng and Leal by analytically manipulating the interaction coefficients in the modulation equations obtained by Nayfeh and Pai by applying the method of multiple scales to the governing integro-partial-differential equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.
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