In this paper, we aim at developing scalable neural network-type learning systems. Motivated by the idea of constructive neural networks in approximation theory, we focus on constructing rather than training feed-forward neural networks (FNNs) for learning, and propose a novel FNNs learning system called the constructive FNN (CFN). Theoretically, we prove that the proposed method not only overcomes the classical saturation problem for constructive FNN approximation, but also reaches the optimal learning rate when the regression function is smooth, while the state-of-the-art learning rates established for traditional FNNs are only near optimal (up to a logarithmic factor). A series of numerical simulations are provided to show the efficiency and feasibility of CFN.

Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We prove that the symmetry-extension method saturates certain anomalies, but also prove that neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fully trivialized, with the background 1-form field $A$, Pontryagin square $\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates an obstruction to constructing a fully 1-form center and 0-form chiral symmetry (full discrete axial symmetry) preserving 4d TQFT with confinement, a no-go scenario via symmetry-extension for specific higher anomalies. We comment on the implications and constraints on deconfined quantum critical points (dQCP), quantum spin liquids (QSL) or quantum fermionic liquids in condensed matter, and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.

We present an algorithm for shape reconstruction from incomplete 3D scans by fusing together two acquisition modes: 2D photographs and 3D scans. The two modes exhibit complementary characteristics: scans have depth information, but are often sparse and incomplete; photographs, on the other hand, are dense and have high resolution, but lack important depth information. In this work we fuse the two modes, taking advantage of their complementary information, to enhance 3D shape reconstruction from an incomplete scan with a 2D photograph. We compute geometrical and topological shape properties in 2D photographs and use them to reconstruct a shape from an incomplete 3D scan in a principled manner. Our key observation is that shape properties such as boundaries, smooth patches and local connectivity, can be inferred with high confidence from 2D photographs. Thus, we register the 3D scan with the 2D photograph and use scanned points as 3D depth cues for lifting 2D shape structures into 3D. Our contribution is an algorithm which significantly regularizes and enhances the problem of 3D reconstruction from partial scans by lifting 2D shape structures into 3D. We evaluate our algorithm on various shapes which are loosely scanned and photographed from different views, and compare them with state-of-the-art reconstruction methods.

Let$D$⊂$C$be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. For$R$>0, let$ω$_{$D$}($R$) denote the harmonic measure at 0 of the set {$z$:|$z$|≽$R$}⋔∂$D$. Then it is shown that$there exist$β>0$and C$>0$such that for each such D$,$ω$_{$D$}($R$)≤$Ce$^{−$βR$},$for every R$>0. Thus a natural question is: What is the supremum of all β′s , call it β_{0}, for which the above inequality holds for every such$D$? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β_{0}is found. Upper bounds for β_{0}can be obtained by constructing examples of domains$D$. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.

We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time H\"older continuity of the weak solution is also discussed as a byproduct.