In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.
The representation category of a conformal net is a unitary modular tensor category. We investigate the reconstruction program: whether all unitary modular tensor categories are representation categories of conformal nets. We give positive evidence: the fruitful theory of multi-interval Jones-Wassermann subfactors on conformal nets is also true for modular tensor categories. We construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are symmetrically self-dual. It generalizes and categorifies the self-duality of finite abelian groups. We call this duality the modular self-duality, because the modularity of the modular tensor category appears in a crucial way. For each unitary modular tensor category, we obtain a sequence of unitary fusion categories. The cyclic group case gives examples of Tambara-Yamagami categories.
In this article, we classify all standard invariants that can arise from a composed inclusion of an A 3 with an A 4 subfactor. More precisely, if N P is an A 3 subfactor and P M is an A 4 subfactor, then only four standard invariants can arise from the composed inclusion N M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A 4 subfactors.
The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the HausdorffYoung inequality, Young's inequality, the HirschmanBeckner uncertainty principle, the DonohoStark uncertainty principle. We characterize the minimizers of the uncertainty principles and then we prove Hardy's uncertainty principle by using minimizers. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's <i></i>-lattices, modular tensor categories, etc.
A time-dependent HermiteGalerkin spectral method (THGSM) is investigated in this paper for the nonlinear convectiondiffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theoretical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Kortewegde VriesBurgers (KdVB) equation and its special cases, including the heat equation and the Burgers equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.
We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of (N−12)×(N−12) inside triangles with vertices (0,0),(0,λβ−−√) and (λ/β−−√,0) with respect to β>0, for fixed λ≥0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N+σ)×(N+τ), for σ,τ>−1. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, σ=τ=−12, is a critical case.
We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries (mn−−−√log(mn))−1 for m,n≥2. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with [0,π]. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.
We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the “radius” approaches infinity. In particular, the result implies that among all p-ellipses (or Lamé curves), the p-circle encloses the most first-quadrant lattice points as the radius approaches infinity, for 1<p<∞.
The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0<p<1 by building on our results here.
The case p=1 remains open, and is closely related to minimizing energy levels of harmonic oscillators: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity?
We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order <i>n</i> 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order <i>n</i> 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.