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We provide a quantum Laurent expansion formula for the quantum cluster algebras from unpunctured orbifolds with arbitrary coefficients and quantization. As an application, positivity for such a class of quantum cluster algebras is provided. Assume for technical reasons that the weights of the orbifold points are 1/2. [ABSTRACT FROM AUTHOR]

In this paper, we first develop a notion of dominated splitting for \mathbb {M}(2,\mathbb {C})-sequences and show it is a stable property under \|\cdot \|_\infty-perturbation. Then we show an energy parameter belongs to the spectrum of a Jacobi operator, possibly singular, if and only if the associated Jacobi cocycle does not admit dominated splitting. This generalizes the results obtained by the second author [J. Spectr. Theory 10 (2020), pp. 1471–1517] in the scenario of Schrödinger operators. Finally, we consider dynamically defined Jacobi operators whose base dynamics is only assumed to be topologically transitive. We show an energy parameter belongs to the spectrum of the operator defined by the base point with a dense orbit if and only if the dynamically defined Jacobi cocycle does not admit dominated splitting. This includes the original Johnson's theorem obtained by R. Johnson [J. Differential Equations 61 (1986), pp. 54–78] for Schrödinger operators and the main theorem obtained by C. Marx [Nonlinearity 27 (2014), pp. 3059–3072] for Jacobi operators as special cases. [ABSTRACT FROM AUTHOR]

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen–Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincaré–Birkhoff–Witt basis. Furthermore, we give counter-examples to the converse. [ABSTRACT FROM AUTHOR]

Building on results of Clemens and Kley, we find criteria for a continuous family of curves in a nodal K-trivial threefold Y0to deform to a scheme of finitely many smooth isolated curves in a general deformation Yt of Y0. As an application, we show the existence of smooth isolated curves of bounded genera and unbounded degrees in Calabi-Yau complete intersection threefolds. [ABSTRACT FROM AUTHOR]

The article presents a study on the topology of cobordism categories of manifolds with corners. These categories are relevant while studying open-closed topological and theories of conformal field. It describes the homotopy type of the category of strings with a background space and higher dimensional analogues.