We present a broad conceptual introduction to some new ideas in nonperturbative quantum field theory (QFT) that have led to progress toward an understanding of quark confinement in gauge theories ...and, more broadly, toward a nonperturbative continuum definition of QFTs. We first present exact orbifold equivalences of supersymmetric and nonsupersymmetric QFTs in the large-
N
limit and exact equivalences of large-
N
theories in infinite volume to large-
N
theories in finite volume, or even at a single point. We discuss principles by which calculable QFTs are continuously connected to strong-coupling QFTs, allowing understanding of the physics of confinement or the absence thereof. We discuss the role of particular saddle solutions, termed bions, in weak-coupling calculable regimes. The properties of bions motivate an extension of semiclassical methods used to evaluate functional integrals to include families of complex saddles (Picard-Lefschetz theory). This analysis leads us to the resurgence program, which may provide a framework for combining divergent perturbation series with semiclassical instanton and bion/renormalon contributions. This program could provide a nonperturbative definition of the path integral.
Recovering low-rank matrices from incomplete observations is a fundamental problem with many applications, especially in recommender systems. In theory, under certain conditions, this problem can be ...solved by convex or nonconvex relaxation. However, most existing provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a novel fuzzy double trace norm minimization (DTNM) method for recommender systems. We first present a tractable DTNM model, in which we can integrate both the user social relationship and the user reputation information using a fuzzy weighting way and coupling fuzzy matrix factorization. In essence, our model is a Schatten-<inline-formula><tex-math notation="LaTeX">{1/2}</tex-math> </inline-formula> quasi-norm minimization problem. Moreover, we develop two efficient augmented Lagrangian algorithms to solve the proposed problems, and prove the convergence of our algorithms. Finally, we investigate the empirical recoverability properties of our model and its advantage over classical trace norm. Extensive experimental results on both synthetic and real-world data sets verified both the efficiency and effectiveness of our method compared with the state-of-the-art algorithms.
In 2013 a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in Gradišar et al. (Nat Chem Bio 9:362–366,
...2013
). It was since observed that a polyhedron
P
can be realized by interlocking pairs of polypeptide chains if its corresponding graph
G
(
P
) admits a strong trace. It was since also demonstrated that a similar strategy can also be expanded to self-assembly of designed DNA (Kočar, Nat commun 7:1–8,
2016
). In this direction, in the present paper we characterize graphs which admit closed walk which traverses every edge exactly once in each direction and for every vertex
v
, there is no subset
N
of its neighbors, with
1
≤
|
N
|
≤
d
, such that every time the walk enters
v
from
N
, it also exits to a vertex in
N
. This extends Thomassen’s characterization (Thomassen, J Combin Theory Ser B 50:198–207,
1990
) for the case
d
=
1
.
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