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In this paper, we study the interior C^{2} estimates for a type of Hessian quotient equation. If k

Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of Christianson and Metcalfe [Indiana Univ. Math. J. 63 (2014), pp. 969–992] on inflection-transmission type trapping on warped product manifolds to the case of multi -warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping. [ABSTRACT FROM AUTHOR]

A unified geometric approach for the stability analysis of traveling pulse solutions for reaction diffusion equations with skew-gradient structure has been established in a previous paper (see Paul Cornwell [Indiana Univ. Math. J. 68 (2019), pp. 1801-1832]), but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equations. [ABSTRACT FROM AUTHOR]

In [Indiana Univ. Math. J. 60 (2011), pp. 847-857] the first author introduced second order necessary conditions for a commuting square to admit sequential deformations in the moduli space of non-isomorphic commuting squares. In this paper we investigate these conditions for commuting squares CG constructed from finite groups G. We are especially interested in the case G = Zn, since deformations of CZn correspond to deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. We show that for G = Zn the second order conditions follow automatically from the first order conditions, but this is not necessarily true for other finite abelian groups G. Our result gives a complete description of the second order deformations of the Fourier matrix Fn in the moduli space of non-equivalent complex Hadamard matrices. [ABSTRACT FROM AUTHOR]

Let \mathbf {G} be the General Linear or Special Linear group with entries from the finite quotients of the ring of integers of a non-archimedean local field and \mathbf {U} be the subgroup of \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta) of \mathbf {G} obtained from a non-degenerate character \theta of \mathbf {U} is multiplicity free for all \ell \geq 2. This is analogous to the multiplicity one theorem regarding Gelfand-Graev representation for the finite Chevalley groups. We prove that for many cases the regular representations of \mathbf {G} are characterized by the property that these are the constituents of the induced representation \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta) for some non-degenerate character \theta of \mathbf {U}. We use this to prove that the restriction of a regular representation of General Linear groups to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases. [ABSTRACT FROM AUTHOR]

We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in \mathbb {R}^d. Our bound is explicit and captures the correct order of growth in d, improving upon the existing results of Giorgi and Smits [Indiana Univ. Math. J. 59 (2010), pp. 987–1011] and Biswas and Lőrinczi [J. Differential Equations 267 (2019), pp. 267–306]. Along the way, we make progress towards a torsion analogue of Chen and Song's [J. Funct. Anal. 226 (2005), pp. 90–113] two-sided eigenvalue estimates for subordinate Brownian motion. [ABSTRACT FROM AUTHOR]