In this short note, we construct an explicit embedding of the rescaling of the p-sum (Koplus _p K^{circ }) of the centrally symmetric convex domain K and it’s polar (K^{circ }) to the product (K times K^{circ }). The rescaling constant is sharp in some cases. Additionally, we comment about the strong Viterbo conjecture for (Koplus _p K^{circ }).
We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets corresponding to an eigenvalue of multiplicity 2. We establish geometric properties, such as number of nodal domains, presence of crossings, and boundary intersections, of nodal sets for a large class of boundary deformations and study how these properties change along each eigenvalue branch for small perturbations. We show that internal crossings of the nodal set break under generic deformations and obtain estimates on the location and regularity of the nodal sets on the perturbed rectangle.
In this paper, we introduce the notion of circular orderability for quandles. We show that the set of all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right (respectively left) orderings of a quandle embeds in its space of right (respectively left) circular orderings. Examples of quandles that are not left circularly orderable and examples of quandles that are neither left nor right circularly orderable are given.
We correct the faulty formulas given in a previous article and we compute the defect group for the Iwasawa (lambda ) invariants attached to the S-ramified T-decomposed abelian pro-(ell )-extensions over the ({{mathbb {Z}}_ell })-cyclotomic extension of a number field. As a consequence, we extend the results of Itoh, Mizusawa and Ozaki on tamely ramified Iwasawa modules for the cyclotomic ({{mathbb {Z}}_ell })-extension of abelian fields.
In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue (sigma _1) of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.
This article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3, 2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4, 1) from [13], where the thinness of 7 other cases was already proven. The O(4, 1) type groups were predicted to be all thin and our result leaves just one case open.
We associate to a filtered (varphi )-module (mathcal {D}) a sub-(mathbb Z_p[[x]])-module of convergent series on the open unit disk in which the p-adic L-functions of the Galois representation associated to (mathcal {D}) live (if they exist). This generalizes the already known case where (mathcal {D}) is of dimension 2, for example associated to an elliptic curve or a modular form.
In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat 3-manifolds ((M_i, g_i)) with nonnegative scalar curvature and ADM mass (m(g_i)) tending to zero, by subtracting some open subsets (Z_i), whose boundary area satisfies (textrm{Area}(partial Z_i) le C m(g_i)^{frac{1}{2}- varepsilon }), for any base point (p_i in M_i{setminus } Z_i), ((M_i{setminus } Z_i, g_i, p_i)) converges to the Euclidean space (({mathbb {R}}^3, g_E, 0)) in the (C^0) modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then ((M_i, g_i, p_i)) converges to (({mathbb {R}}^3, g_E, 0)) in the pointed Gromov–Hausdorff topology.
We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution in Euclidean space with two boundary components, each isometric to ({mathbb {S}}^{n-1}). For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound (B_n(L)) (that depends only on the dimension (n ge 3) and the meridian length (L>0)) which is reached by a degenerated metric (g^*) that we compute explicitly. We also give a sharp upper bound (B_n) which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.
We give a description of the growth rates of (L^2)-normalized Laplace eigenfunctions on the unit disk with Dirichlet and Neumann boundary conditions. In particular, we show that the growth rates of both Dirichlet and Neumann eigenfunctions are bounded away from zero. Our approach starts with P. Sarnak growth exponents and uses several key asymptotic formulas for Bessel functions or their zeros.
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