19,511 research outputs found

    I\u27m the oak tree, rounded

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    A Degeneracy in DRW Modelling of AGN Light Curves

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    Individual light curves of active galactic nuclei (AGNs) are nowadays successfully modelled with the damped random walk (DRW) stochastic process, characterized by the power exponential covariance matrix of the signal, with the power β=1\beta=1. By Monte Carlo simulation means, we generate mock AGN light curves described by non-DRW stochastic processes (0.5β1.50.5\leq\beta\leq 1.5 and β1\beta\neq1) and show they can be successfully and well-modelled as a single DRW process, obtaining comparable goodness of fits. A good DRW fit, in fact, may not mean that DRW is the true underlying process leading to variability and it cannot be used as a proof for it. When comparing the input (non-DRW) and measured (DRW) process parameters, the recovered time scale (amplitude) increases (decreases) with the increasing input β\beta. In practice, this means that the recovered DRW parameters may lead to biased (or even non-existing) correlations of the variability and physical parameters of AGNs if the true AGN variability is caused by non-DRW stochastic processes. The proper way of identifying the processes leading to variability are model-independent structure functions and/or power spectral densities and then using such information on the covariance matrix of the signal in light curve modelling.Comment: 3 pages, 2 figures, accepted for publication in MNRAS, final versio

    Opportunity for Regulating the Collective Effect of Random Expansion with Manifestations of Finite Size Effects in a Moderate Number of Finite Systems

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    One reports computational study revealing a set of general requirements, fulfilling of which would allow employing changes in ambient conditions to regulate accomplishing the collective outcome of emerging active network patterns in an ensemble of a moderate number of finite discrete systems. The patterns within all these component systems emerge out of random expansion process governed by certain local rule. The systems modeled are of the same type but different in details, finite discrete spatial domains of the expansion within the systems are equivalent regular hexagonal arrays. The way in which elements of a component system function in the local information transmission allows dividing them into two classes. One class is represented by zero-dimensional entities coupled into pairs identified at the array sites being nearest neighbors. The pairs preserve their orientation in the space while experiencing conditional hopping to positions close by and transferring certain information portions. Messenger particles hopping to signal the pairs for the conditional jumping constitute the other class. Contribution from the hopping pairs results in finite size effects being specific feature of accomplishing the mean expected network pattern representing the collective outcome. It is shown how manifestations of the finite size effects allow using changes in parameters of the model ambient conditions of the ensemble evolution to regulate accomplishing the collective outcome representation.Comment: 22 pages, 10 eps figures, corrected URL address placing in text, minor editorial correction in sec.2, author e-mail change

    On condensation properties of Bethe roots associated with the XXZ chain

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    I prove that the Bethe roots describing either the ground state or a certain class of "particle-hole" excited states of the XXZ spin-1/21/2 chain in any sector with magnetisation m[0;1/2]\mathfrak{m} \in [0;1/2] exist and form, in the infinite volume limit, a dense distribution on a subinterval of R\mathbb{R}. The results holds for any value of the anisotropy Δ1\Delta \geq -1 . In fact, I establish an even stronger result, namely the existence of an all order asymptotic expansion of the counting function associated with such roots. As a corollary, these results allow one to prove the existence and form of the infinite volume limit of various observables attached to the model -the excitation energy, momentum, the zero temperature correlation functions, so as to name a few- that were argued earlier in the literature.Comment: 54 pages, 2 figures. Some details in proof adde

    fractals and the Big Bang

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    On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

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    This work extracts, by means of an exact analysis, the singular behaviour of the dynamical response functions -- the Fourier transforms of dynamical two-point functions -- in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes which describe the local behaviour of the response function near a threshold. The singular behaviour is derived starting from first principle considerations: the method of analysis \textit{does not rely, at any stage}, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behaviour and of the associated edge exponents which arise in the vicinity of the dispersion relation of one massive excitation (hole, particle or bound state). In addition, the present analysis shows that, due to the lack of strict convexity of the particles dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These origin from multi-particle/hole/bound state excitations maximising the energy at fixed momentum.Comment: 115 pages, 6 figure

    On the emptiness formation probability of the open XXZ spin-\tf{1}{2} chain

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    This paper is devoted to the study of the emptiness formation probability \tau\pa{m} of the open XXZ chain. We derive a closed form for \tau\pa{m} at \Delta=\tf{1}{2} when the boundary field vanishes. Moreover we obtain its leading asymptotics for an arbitrary boundary field at the free fermion point. Finally, we compute the first term of the asymptotics of \ln\pa{\tau\pa{m}} in the whole massless regime 1<Δ<1-1<\Delta<1.Comment: 18 page

    On lacunary Toeplitz determinants

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    By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants detN[camb[f]]\det_N\big[ c_{\ell_a-m_b}[f] \big] generated by holomorhpic symbols, where a=a\ell_a=a (resp. mb=bm_b=b) except for a finite subset of indices a=h1,,hna=h_1,\dots, h_n (resp. b=t1,,trb=t_1,\dots, t_r). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size n+rn+r.Comment: 11 page
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