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The goal of this paper is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artés et al. (Mem Am Math Soc 134:639, 1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincaré disc modulo limit cycles in this family, and Artés et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification of planar quadratic systems of codimension two in the structural stability. Combining the sets of codimension-one quadratic vector fields one to each other, we obtain ten new sets. Here we consider set AA obtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of set AA and prove their realization. We got 34 new topologically distinct phase portraits in the Poincaré disc modulo limit cycles. Moreover, in this paper we correct a mistake made by the authors in the book of Artés et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) and we reduce to 203 the number of topologically distinct phase portrait of codimension one modulo limit cycles. [ABSTRACT FROM AUTHOR]

The stabilization properties of dissipative Timoshenko systems have been attracted the attention and efforts of researchers over the years. In the past 20 years, the studies in this scenario distinguished primarily by the nature of the coupling and the type or strength of damping. Particularly, under the premise that the Timoshenko beam model is a two-by-two system of hyperbolic equations, a large number of papers have been devoted to the study of the so-called partially damped Timoshenko systems by assuming damping effects acting only on the angle rotation or vertical displacement (Almeida Júnior et al. in Math Methods Appl Sci 36:1965–1976, 2013; in Z Angew Math Phys 65:1233–1249, 2014; Alves et al. in SIAM J Math Anal 51(6):4520–4543, 2019; Ammar-Khodja et al. in J Differ Equ 194:82–115, 2003; Muñoz Rivera and Racke in Discrete Contin Dyn Syst Ser B 9:1625–1639, 2003; J Math Anal Appl 341:1068–1083, 2008; Santos et al. in J Differ Equ 253(9):2715–2733, 2012). In these cases, the desired exponential decay property of the energy solutions is achieved when the non-physical equal wave speed assumption plays the role to stabilization according since the pioneering Soufyane's paper (C R Acad Sci Paris 328(8):731–734, 1999). Recent results due to Almeida Júnior et al. (Z Angew Math Phys 68(145):1–31, 2017; Z Angew Math Mech 98(8):1320–1333, 2018; IMA J Appl Math 84(4):763–796, 2019; Acta Mech 231:3565–3581, 2020) show that the second vibration mode or simply second spectrum of frequency and it's damaging consequences appears as a lost element in analysis of stabilization and now it's more clear that the damping importance into stabilization scenario of Timoshenko type systems. This paper considers a one-dimensional viscoelastic Timoshenko type system in the light of the second spectrum of frequency where the equal wave speed assumption is not needed for getting the exponential decay property. Precisely, we consider the so-called truncated version for the Timoshenko system according studies due to Elishakoff (Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010; ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11, 2015; Int J Solids Struct 109:143–151, 2017; J Sound Vib 435:409–430, 2017; Int J Eng Sci 116:58–73, 2017; Acta Mech 229:1649–1686, 2018; Z Angew Math Mech 98(8):1334–1368, 2018; Math Mech Solids, 2019) and we added a viscoelastic damping acting on shear force. We firstly prove the global well-posedness of the system by Faedo–Galerkin approximation. By assuming minimal conditions on the relaxation function, we establish an optimal explicit and energy decay rate for which exponential and polynomial rates are special cases. This result is new and substantially improves earlier results in the literature where the equal wave speeds plays the role for getting the stability properties. It is likely to open more research areas to Timoshenko system and probably others. [ABSTRACT FROM AUTHOR]

It is well known that, for fixed Diophantine frequencies and generic small smooth or analytic quasiperiodic potentials, both continuous and discrete Schrödinger operators have Cantor spectrum. Although there have been several examples of Schrödinger operators with Cantor spectrum since Moser's pioneering work (Bellissard, in: Luck, Moussa, Waldschmidt (eds) Number theory and physics (Les Houches, 1989), Springer, Berlin, 1990; Bellissard et al. in Phys Rev Lett 49:701–704, 1982; Damanik et al. in Ann Henri Poincaré 15:1123–1144, 2014, J Spectr Theory 7:1101–1118, 2017; Moser in Comment Math Helv 56:198–224, 1981); however, so far there is no concrete quasiperiodic example in the continuous case, and there is no concrete quasiperiodic example in the discrete case besides the cosine-like potentials (Avila and Jitomirskaya in Ann Math 170:303–342, 2009; Puig in Commun Math Phys 244:297–309, 2004; Sinai in J Stat Phys 46:861–909; 1987; Wang and Zhang in Int Math Res Not 2017:2300–2336, 2017). In this paper, we present a strategy for explicitly constructing quasiperiodic Schrödinger operators with Cantor spectrum. [ABSTRACT FROM AUTHOR]

Let A be an abelian category, and X and Y subcategories. We derive in this paper an additive functor T (- , -) using proper X -resolutions (respectively, proper Y -coresolutions). Under certain conditions, we establish balance results for such relative homology (respectively, cohomology) over A . Our main theorem simultaneously recovers theorems of Sather-Wagstaff et al. (J Math Kyoto Univ 48(3):571–596, 2008), Enochs et al. (Relative homological algebra. De Gruyter, Berlin 2000) and Di et al. (Bull Korean Math Soc 52(1):137–147, 2015) as special cases. [ABSTRACT FROM AUTHOR]