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Register a new userLattice equations arising from discrete Painleve systems (I): $(A_2+A_1)^{(1)}$ and $(A_1+A_1)^{(1)}$ cases
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We introduce the concept of $omega$-lattice, constructed from $tau$ functions of Painleve systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painleve systems corresponding affine Weyl group symmetries are of $(A_2+A_1)^{(1)}$- and $(A_1+A_1)^{(1)}$-types, respectively.
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We find a two-parameter family of coupled Painleve systems in dimension four with affine Weyl group symmetry of type $A_4^{(2)}$. For a degenerate system of $A_4^{(2)}$ system, we also find a one-parameter family of coupled Painleve systems in dimension four with affine Weyl group symmetry of type $A_1^{(1)}$. We show that for each system, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new. Moreover, we find a one-parameter family of partial differential systems in three variables with $W(A_1^{(1)})$-symmetry. We show the relation between its polynomial Hamiltonian system and an autonomous version of the system of type $A_1^{(1)}$.
We show that the a_1-rho-pi Lagrangian is a decisive element for obtaining a good phenomenological description of the three-pion decays of the tau lepton. We choose it in a two-component form with a flexible mixing parameter sin(theta). In addition to the dominant a_1-> pho pi intermediate states, the a_1->pi sigma ones are included. When fitting the three-pion mass spectra, three data sets are explored: (1) ALEPH 2005 pi-pi-pi+ data, (2) ALEPH 2005 pi-pi0pi0 data, and (3) previous two sets combined and supplemented with the ARGUS 1993, OPAL 1997, and CLEO 2000 data. The corresponding confidence levels are (1) 28.3%, (2) 100%, and (3) 7.7%. After the inclusion of the a_1(1640) resonance, the agreement of the model with data greatly improves and the confidence level reaches 100% for each of the three data sets. From the fit to all five experiments [data set (3)] the following parameters of the a_1(1260) are obtained m_{a_1}=(1233+/-18) MeV, Gamma_{a_1}=(431+/-20) MeV. The optimal value of the Lagrangian mixing parameter sin(theta)= 0.459+/-0.004 agrees with the value obtained recently from the e+e- annihilation into four pions.
In this paper we have examined hyperbolicity and relative hyperbolicity of $Gamma_n := mathsf{Out}(G_n)$ , where $G_n = A_1*...*A_n$, is a finite free product and each $A_i$ is a finite group. We have used the $mathsf{Out}(G_n)$ action on the Guirardel-Levitt deformation space, to find a virtual generating set and prove quasi isometric embedding of a large class of subgroups. We have used ideas from works of Mosher-Handel and Alibegovic to prove non-distortion. We have used these subgroups to prove that $Gamma_n$ is thick for higher complexities. Thickness was developed by Behrstock-Druc{t}u-Mosher and thickness implies that the groups are non-relatively hyperbolic.
It is found that in presence of electroweak interactions the gauge covariant diagonalization of the axial-vector -- pseudoscalar mixing in the effective meson Lagrangian leads to a deviation from the vector meson and the axial-vector meson dominance of the entire hadronic electroweak current. The essential features of such a modification of the theory are investigated in the framework of the extended Nambu-Jona-Lasinio model with explicit breaking of chiral $U(2) times U(2)$ symmetry. The Schwinger-DeWitt method is used as a major tool in our study of the real part of the relevant effective action. Some straightforward applications are considered.
For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twisted G character variety of C, when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.
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