We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π (SU(2)θ=π YM). Its higher 't Hooft anomalies of generalized global symmetries indicate that the 4d SU(2)θ=π YM, in order to realize all global symmetries locally, necessarily couples to a 5d higher symmetry-protected topological state (SPTs, as an invertible TQFT, or as a 5d 1-form-center-symmetry-protected interacting "topological superconductor" in condensed matter). We revisit the 4d SU(2)θ=π YM-5d SRE-higher-SPTs coupled systems in [arXiv:1812.11968] and find their "Fantastic Four Siblings" with four sets of new higher anomalies associated with the Kramers singlet/doublet and bosonic/fermionic properties of Wilson lines. Following Weyl's gauge principle, by dynamically gauging the 1-form center symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)θ=π YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. Apply the tool introduced in [arXiv:1612.09298], we derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, which physically characterize the 5d SETs. We discover new triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge TQFTs, hinting a new intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide lattice simplicial complex regularizations and "condensed matter" realizations.

This paper presents an algorithm for extending B-spline curves and surfaces. Based on the unclamping algorithm for B-spline curves, we propose a new algorithm for extending B-spline curves that extrapolates using the recurrence property of the de Boor algorithm. This algorithm provides a nice extension, with maximum continuity, to the original curve segment. Moreover, it can be applied to the extension of B-spline surfaces. Extension to both single and multiple target points/curves are considered in this paper.

We consider the numerical solution of the time-fractional diffusion-wave equation on two-dimensional and three-dimensional unbounded spatial domains. Introduce an artificial boundary and find the exact and approximate artificial boundary conditions for the given problem, which lead to a bounded computational domain. Using the exact or approximating boundary conditions on the artificial boundary, the original problem is reduced to an initial-boundary-value problem on the bounded computational domain which is respectively equivalent to or approximates the original problem. Finite difference methods are used to solve the reduced problems on the bounded computational domain and the stability of these finite difference methods are proved. The numerical results demonstrate that the method given in this paper is effective and feasible.

In this article, we consider the time-dependent Maxwell’s equations modeling wave propagation in metamaterials. One-order higher global superclose results in the L2 norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L ∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue.

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