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Register a new userComplex Dynamics of the Difference Equation $z_{n+1}=frac{alpha}{z_{n}}+ frac{beta}{z_{n-1}}$
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The dynamics of the second order rational difference equation in the title with complex parameters and arbitrary complex initial conditions is investigated. Two associated difference equations are also studied. The solutions in the complex plane of such equations exhibit many rich and complicated asymptotic behavior. The analysis of the local stability of these three difference equations and periodicity have been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which does not arise when we consider the same equation with positive real parameters and initial conditions. Many interesting observations led us to pose several open problems and conjectures of paramount importance regarding chaotic and higher order periodic solutions and global asymptotic convergence of such difference equations. It is our hope that these observations of these complex difference equations would certainly be new add-ons to the present art of research in rational difference equations in understanding the behavior in the complex domain.
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We discuss an effective way for analyzing the system on the magnetized twisted orbifolds in operator formalism, especially in the complicated cases $T^{2}/Z_{3}$, $T^{2}/Z_{4}$ and $T^{2}/Z_{6}$. We can obtain the exact and analytical results which can be applicable for any larger values of the quantized magnetic flux M, and show that the (non-diagonalized) kinetic terms are generated via our formalism and the number of the surviving physical states are calculable in a rigorous manner by simply following usual procedures in linear algebra in any case. Our approach is very powerful when we try to examine properties of the physical states on (complicated) magnetized orbifolds $T^{2}/Z_{3}$, $T^{2}/Z_{4}$, $T^{2}/Z_{6}$ (and would be in other cases on higher-dimensional torus) and could be an essential tool for actual realistic model construction based on these geometries.
We study the ground-state properties of a class of $mathbb{Z}_n$ lattice gauge theories in 1 + 1 dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutator for the $mathrm{U}(1)$ group, preserve the unitary character of the minimal coupling, and have therefore the property of formally approximating lattice quantum electrodynamics in one spatial dimension in the large-$n$ limit. The numerical study of such approximated theories is important to determine their effectiveness in reproducing the main features and phenomenology of the target theory, in view of implementations of cold-atom quantum simulators of QED. In this paper we study the cases $n = 2 div 8$ by means of a DMRG code that exactly implements Gauss law. We perform a careful scaling analysis, and show that, in absence of a background field, all $mathbb{Z}_n$ models exhibit a phase transition which falls in the Ising universality class, with spontaneous symmetry breaking of the $CP$ symmetry. We then perform the large-$n$ limit and find that the asymptotic values of the critical parameters approach the ones obtained for the known phase transition the zero-charge sector of the massive Schwinger model, which occurs at negative mass.
We give another proof for [ sum_{n=1}^{infty}frac{1}{n^2}=frac{pi^2}{6} ] that basically follows from the theory of difference equations.
The aim of this paper is to establish global Calder{o}n--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -operatorname{div}(| abla u|^{p-2} abla u) = operatorname{div} (|F|^{p-2}F)~text{in}~Omegatimes (0,T)subset mathbb{R}^{n+1}, $$ proving that $$Fin L^qRightarrow abla uin L^q,$$ for any $q>max{p,frac{n(2-p)}{2}}$ and $p>1$. Acerbi and Mingione cite{Acerbi07} proved this estimate in the case $p>frac{2n}{n+2}$. In this article we settle the case $1<pleq frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm.
Considered are ${cal N}=2, SU(N_c)$ or $U(N_c)$ SQCD with $N_F<2N_c-1$ equal mass quark flavors. ${cal N}=2$ supersymmetry is softly broken down to ${cal N}=1$ by the mass term $mu_{rm x}{rm Tr},(X^2)$ of colored adjoint scalar partners of gluons, $mu_{rm x}llLambda_2$ ($Lambda_2$ is the scale factor of the $SU(N_c)$ gauge coupling). There is a large number of different types of vacua in these theories with both unbroken and spontaneously broken global flavor symmetry, $U(N_F)rightarrow U({rm n}_1)times U({rm n}_2)$. We consider in this paper the large subset of these vacua with the unbroken non-trivial $Z_{2N_c-N_Fgeq 2}$ discrete symmetry, at different hierarchies between the Lagrangian parameters $mgtrlessLambda_2,,, mu_{rm x}gtrless m$. The forms of low energy Lagrangians, charges of light particles and mass spectra are described in the main text for all these vacua. The calculations of power corrections to the leading terms of the low energy quark and dyon condensates are presented in two important Appendices. The results agree with also presented in these Appendices independent calculations of these condensates using roots of the Seiberg-Witten spectral curve. This agreement confirms in a non-trivial way a self-consistency of the whole approach. Our results differ essentially from corresponding results in e.g. recent related papers arXiv:1304.0822, arXiv:1403.6086 and arXiv:1704.06201 of M.Shifman and A.Yung (and in a number of their previous numerous papers on this subject), and we explain in the text the reasons for these differences. (See also the extended critique of a number of results of these authors in section 8 of arXiv:1308.5863).
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