(iii) For the case that both tangent frequencies ω and normal frequencies are inﬁnite dimensional, according to our knowledge, not only of elliptic type but also of hyperbolic type, there has not been any KAM theorem to deal with this situation. However, the ex- istence of almost-periodic solutions for the networks of **weakly** **coupled** pendulum equa- tions (1.1) needs to be proved. That is the problem we are most concerned with in this paper.

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Many CDA studies have already considered the **coupled** forecast initialization on timescales from seasonal to decadal (e.g. the Japan Agency for Marine-Earth Science and Tech- nology (JAMSTEC), Sugiura et al., 2008; Mochizuki et al., 2016; the National Oceanic and Atmospheric Administration Geophysical Fluid Dynamics Laboratory (NOAA/GFDL), Yang et al., 2013; Zhang et al., 2014). At the Met Of- fice, a **weakly** **coupled** data assimilation (WCDA) system has been developed (Lea et al., 2015) to improve the fore- cast skill from short range to seasonal timescales, though this system is not yet operational. It is based on using a **coupled** atmosphere–land–ocean–ice model to compute the background states for separate atmosphere and ocean analy- ses in a 6 h assimilation window. Generally speaking, WCDA at the Met Office performed reasonably well, providing re- sults very similar to the uncoupled DA. In that study, the au- thors identified two main problems in their implementation of WCDA: the ocean SST diurnal cycle issue and an erro- neous **coupled** river runoff. The latter issue led to degrada- tion in the salinity fields around some river basins. These re- sults are nevertheless encouraging considering that the CDA system is new and neither the atmospheric nor ocean data assimilation systems were adjusted as part of implementing CDA.

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Abstract. We investigate black hole evaporation in a **weakly** **coupled** model of two-dimensional dilaton gravity paying a particular attention to the validity of the semiclassical mean-field approximation. Our model is obtained by adding a reflecting boundary to the celebrated RST model describing N gravitating mass- less scalar fields to one-loop level. The boundary cuts off the region of strong coupling. Although our model is explicitly **weakly** **coupled**, we find that the mean field approximation inevitably fails at the end of black hole evaporation. We propose an alternative semiclassical method aiming at direct calculation of S-matrix elements and illustrate it in a simple shell model.

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Moreover, FGR and LS have a ratio independent of the temperature difference. In addi- tion, numerics show, that even in the **weakly** **coupled** case with small temperature difference, FGR and LS curves intersect rather than converge. This is due to the poor cut off in the Coulomb integral in LS. Thus, the FGR approach rather than LS formula should be used to compare with experiments or simulations if one searches for **coupled** mode effects in the data.

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Our main motivation is that there has been extensive lattice study of models close to the lower end of the conformal window because of their relevance for BSM model building in recent years [3]. Close to the lower end of the conformal window non-perturbative e ff ects are relevant because in the conformal case the fixed point coupling is large and in the chirally broken case the entire low energy dynamics is dictated by non-perturbative e ff ects similarly to QCD. In principle lattice simulations are an ideal tool to determine whether a given model is inside or outside the conformal window exactly because the lattice setup can capture all non-perturbative e ff ects. Nevertheless systematic e ff ects can be large sometimes leading to controversies for models close to the lower end of the conformal window. We study here the **weakly** **coupled** conformal case which can be tested on the lattice with predictable and controlled results in sufficient orders of perturbation theory. We are interested in how the lattice tool set is able to identify conformality, if there are any unexpected systematic effects and how ambiguous or unambiguous the lattice results are.

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We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some **coupled** implicit diﬀerential equations. In particular we show the persistence of such orbits connecting singularities in ﬁnite time provided a Melnikov like condition holds. Application is given to **coupled** nonlinear RLC system. MSC: Primary 34A09; secondary 34C23; 37G99

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scope of current models. Turning to recent high-level calculations of polyene excited states, we note that another triplet pair state, 1B u − , is predicted to lie just below the initial allowed 1B u transition. 55 The coupling between these states would likely be strong, with consequently fast relaxation into the triplet pair. Comparison with the more established polyacenes sheds some light onto the mechanism of singlet ﬁ ssion discussed here. In the most strongly **coupled** acenes (pentacene and TIPS-pentacene), singlet ﬁ ssion is described as adiabatic, with a rate independent of intermolecular coupling strength. For the majority of the more **weakly** **coupled** acenes, however, triplet formation appears to be highly dependent on intermolecular coupling. 15 By contrast, in astaxanthin similar sub-100 fs singlet ﬁ ssion kinetics are observed in strongly (I, V) and **weakly** (II, III) **coupled** systems with large (I) and small (V) energetic driving force. This suggests that singlet ﬁ ssion remains in the adiabatic regime in all aggregates and may re ﬂ ect a unique property of singlet ﬁssion in polyenes. This is perhaps not surprising given the widely accepted role of conical intersections in carotenoid photophysics, and it will be important to investigate how vibrational dynamics are implicated in singlet ﬁ ssion within the aggregates. A recent Figure 7. Ultrafast triplet formation. (a) Sub-30 fs TA measurements of aggregate II in the visible and NIR spectral regions show a direct transition from the initial singlet (solid) to triplets, already evident by 150 fs. No other states (such as 2A g ) are observed. Equivalent measurements on

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We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary diﬀerential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of **weakly** **coupled** reaction-di ﬀ usion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly **coupled** reaction-diﬀusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.

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A numerical test of the KKM formalism to treat a large number of channels statistically was done in Ref. [9] with 400 equidistant q-levels, 40 channels, with 20 equidistant coordinate points where HPQ is set to a Gaussian-distributed random interaction. The energy of the single-out P state was taken as E = 20 MeV, and we included 100 E points for the Lorentzian averaging between 18 and 22 MeV. The adopted value of I = 0.5 MeV and we considered s-wave scattering only, with Γ / D 1. Figures 3 and 4 show the average of the ﬂuctuating part of the T-matrix and the optical T-matrix. It is evident that for a large number of **weakly** **coupled** channels the KKM formalism yields a proper treatment of the S- and T-matrices. The advantage of the formalism is that one has only to consider the strongly **coupled** P-space states, with a simple average needed for the **weakly** **coupled** states which can be numerous.

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We present a constructive method for the study of observability or partial observability of weakly coupled linear distributed systems or, more generally, of compactly perturbed systems..[r]

This thesis examines the application of symmetric dynamical systems theory to two areas in applied mathematics: weakly coupled oscillators with symmetry, and bifurcations in flame front [r]

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The detailed dynamical structure factors for **weakly** **coupled** one dimensional plasma have been computed through involved space-time dependant correlation functions. Theoretical investigations have been performed for a wide range of wave-vectors: 2.0 to 10.0 cm -1 . Dynamical modes for plasma have been obtained as singularities of dynamical structure factor and yield dispersion relation. The dynamical modes have also been investigated for their temperature and density dependence and are observed to be altered significantly with change in number density of constituent particles.

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as a superposition of partial surface and volume fields, which are **coupled** on the lattice-vacuum interface. We discussed the structures of the volume and surface partial fields, and it was shown that the surface field excited is different from a whispering-gallery mode and has all the features of a surface mode. To define the structure of the eigenfield, the cylindrical lattice was substituted with a smooth cylindrical waveguide, partially loaded with a metadielectric. It was found that the properties of the metadielectric depend on the lattice and radiation parameters, and the conditions required to observe the elevation of the surface field above the lattice were discussed. Contour plots of the cavity eigenfield structures were demonstrated. By analyzing the dispersion of the **weakly** **coupled** partial fields, we illustrated that to observe coupling at a near cutoff frequency of the volume field, the surface field has to decay toward the center (i.e., it should have an imaginary transverse wave number), thus making its structure different from the whispering-gallery modes. Using the observed results, numerical studies of a Cherenkov oscillator based on a 2D PSL cavity and driven by an annular, electron beam were carried out. The dependence of the lattice parameters on the electron beam accelerating voltage was discussed and the required lattice parameters’ scaling (to maintain the operation of the Cherenkov oscillator) with variation of the electron beam voltage was shown. Using the 3D numerical code MAGIC , we demonstrated that single-mode steady-state operation of a high-power 200-GHz Cherenkov maser can be achieved.

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Figure 4.1: The performance of QCA, TUCC and the hybrid ansatz on a weakly coupled gapped system. In this figure the natural logarithm of the error log ( e ) is graphed as a function of [r]

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One aspect of earthquake sequencing that requires a close look is a model for the non-linear dynamics of earthquakes. In this paper, we investigate the synchronization behavior of **weakly** **coupled** “earthquake oscillations”. Such oscilla- tions in the earth’s crust and the epileptic brain show cer- tain commonalities in that the distributions of energies and recurrence times exhibit similar power-law behavior (Herz and Hopfield, 1995; Rundle et al., 2003; Osorio et al., 2010; Chialvo, 2010). A growing interest in understanding the be- havior of earthquakes and epileptic seizures with a view to exploring possible forecasting methods is one reason for the present study. In the case of epileptic seizures, the non-linear dynamics of pulse-**coupled** neuronal oscillations as an alter- native to the Kuramoto (1975) model are under close scrutiny (Rothkegel and Lehnertz, 2014). To our knowledge, neither a simple Kuramoto model nor a modification of it has been worked out for earthquake sequencing studies. Mirollo and Strogatz (1990), Kuramoto (1991) and Rothkegel and Lehn- ertz (2014) considered the synchronization of pulse-**coupled** oscillators in which single oscillators release energy rapidly when they reach a trigger threshold and become quiescent for some time until they reach the trigger threshold again. Examples falling into this category are earthquakes and spik- ing neuronal activities (Herz and Hopfield, 1995; Beggs and Plenz, 2003; Rundle et al., 2002, 2003; Scholz, 2010; Karsai et al., 2012; Rothkegel and Lehnertz, 2014). Herz and Hop- field (1995) studied the collective oscillations with pulse- **coupled** threshold elements on a fault system to capture the earthquake processes. There are two timescales: the first is given by the fault dynamics defining the duration of the earth- quake, and the second timescale is given by the recurrence time between “characteristic events”, the largest earthquakes on a fault. The known recurrence times on several fault sys- tems are 6 to 8 orders of magnitude longer than the dura- tion of single events. Rundle et al. (2002) examined the self- organization in “leaky” threshold systems such as networks of earthquake faults. In their paper, they argued that on the “macroscopic” scale of regional earthquake fault systems, self-organization leads to the appearance of phase dynam- ics and a state vector whose rotations would characterize the evolution of earthquake activity in the system. Scholz (2010) invoked the Kuramoto model to represent the fault interac- tions, although no numerical synchronization–simulation re- sults were presented. He postulated that the common oc-

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The detailed dynamical structure factors for **weakly** **coupled** one dimensional plasma have been computed through involved space-time dependant correlation functions. Theoretical investigations have been performed for a wide range of wave-vectors: 2.0 to 10.0 cm -1 . Dynamical modes for plasma have been obtained as singularities of dynamical structure factor and yield dispersion relation. The dynamical modes have also been investigated for their temperature and density dependence and are observed to be altered significantly with change in number density of constituent particles.

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In this paper, we present models with an N = 1 supersymmetric observable sector which, for both the weakly and strongly coupled heterotic string, has the following properties: Observable[r]

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The aim of present paper is to introduce the notion of t- conorm of H-type analogous to t-norm of H-type introduced by Hadzic [9] and using this notion we prove **coupled** fixed point theorems for **weakly** compatible mappings in intuitionistic fuzzy metric spaces.

Abstract—A systematic derivation of the **Coupled** Nonlinear Schrodinger Equations (CNLSE) governing nonlinear pulse propaga- tion in a **weakly** birefringent monomode optical fiber based on a multiple-scale perturbation solution of the semilinear vector wave equa- tion for the electric field in a (randomly) birefringent fiber medium is presented. The analysis of the nonlinear propagation characteristics of optical pulses based on a numerical solution of the CNLSE is deferred to the second part of this contribution.

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In this paper, we prove a common ﬁxed point theorem for **weakly** compatible mappings under φ -contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (Fixed Point Theory Appl. 2011:363716, 2011,

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