Abstract In this paper, we study the validity of the sub-supersolution method for the equation \begin{equation*}\begin{cases}-\mbox{div}(K(x)\nabla u)=K(x)|x|^{\alpha-2}f(x,u) \,\mbox{in } {\mathbb{R}}^{N},\\u \gt 0 \,\mbox{in } {\mathbb{R}}^{N},\end{cases}\end{equation*} where $N \geq 3$ , $K(x)=exp(|x|^{\alpha}/4)$ , $\alpha\geq 2$ and $f$ is a continuous function, with hypotheses that will be given later. We apply the method to cases where $f$ is singular, where $f$ behaves like a logistic function, showing in both cases the existence and uniqueness of a positive solution.

Abstract We prove that a partially hyperbolic attracting set for a $C^2$ vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical/SRB measures, which are also Gibbs states along the central-unstable direction. This extends to continuous time systems a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set. In codimension two the same result holds, assuming only the trajectories on the trapping region admit a sequence of times with asymptotical sectional expansion, on a positive volume subset. We present several examples of application, including the existence of physical measures for asymptotically sectional hyperbolic attracting sets, and obtain physical measures in an alternative unified way for many known examples: Lorenz-like and Rovella attractors, and sectional-hyperbolic attracting sets (including the multidimensional Lorenz attractor).

Abstract Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space E . The relations between these lattices allow us to give multiple characterizations of when the underlying Banach space E is finite-dimensional and when it is reflexive. Furthermore, we show that lattice homomorphisms between free Banach lattices are always composition operators, and study how these operators behave on the scale of lattices of positively homogeneous functions.

Abstract In this paper, we will investigate the following inequality \begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*} which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.

Abstract Let f(z) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}^{2}f)$ be the symmetric square L-function attached to f(z). Suppose that $\lambda_{\mathrm{sym}^{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}^{2}f)$ . In this paper, we investigate the sum $\sum\limits_{n\leqslant x} \lambda^{j}_{\mathrm{sym}^{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{f }(n^{2})$ and get some similar results.

Abstract In this paper, the upper bounds of non-real eigenvalues of indefinite Sturm–Liouville (S-L) problems with boundary conditions depend on the eigenparameter are studied. The upper bounds of real parts, imaginary parts and absolute values of non-real eigenvalues are given under the condition that the coefficients are integrable.

Abstract Let a group Γ act on a paracompact, locally compact, Hausdorff space M by homeomorphisms and let 2 M denote the set of closed subsets of M. We endow 2 M with the Chabauty topology, which is compact and admits a natural Γ-action by homeomorphisms. We show that for every minimal Γ-invariant closed subset $\mathcal{Y}$ of 2 M consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact.

Abstract In this paper, we provide a characterization for a class of convex curves on the 3-sphere. More precisely, using a theorem that represents a locally convex curve on the 3-sphere as a pair of curves in $\mathbb S^2$ , one of which is locally convexand the other is an immersion, we are able of completely characterizing a class of convex curves on the 3-sphere.

Abstract For Γ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$ , we study the fibres of the Procesi bundle over the Γ-fixed points of the Hilbert scheme of n points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibres of the Procesi bundle, as an $(\mathfrak{S}_n \times \Gamma)$ -module, to the study of the fibres of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When Γ is of type A, our main result shows, as a corollary, that the fibre of the Procesi bundle over the monomial ideal associated with a partition λ is induced, as an $(\mathfrak{S}_n \times \Gamma)$ -module, from the fibre of the Procesi bundle over the monomial ideal associated with the core of λ. We give different proofs of this corollary in two edge cases using only representation theory and symmetric functions.