We investigate the use of artificial neural networks (NNs) as an alternative tool to current analytical methods for recognizing knots in a given polymer conformation. The motivation is twofold. First, it is of interest to examine whether NNs are effective at learning the global and sequential properties that uniquely define a knot. Second, knot classification is an important and unsolved problem in mathematical and physical sciences, and NNs may provide insights into this problem. Motivated by these points, we generate millions of polymer conformations for five knot types: 0, 3_1, 4_1, 5_1, and 5_2, and we design various NN models for classification. Our best model achieves a five-class classification accuracy of above 99% on a polymer of 100 monomers. We find that the sequential modeling ability of recurrent NNs is crucial for this result, as it outperforms feed-forward NNs and successfully generalizes to differently sized conformations as well. We present our methods and suggest that deep learning may be used in specific applications of knot detection where some error is permissible. Hopefully, with further development, NNs can offer an alternative computational method for knot identification and facilitate knot research in mathematical and physical sciences.

Let$f=g$_{t}+$h$_{t}be the optimal decomposition for calculating the exact value of the$K$-functional$K(t, f$; $$\bar X$$ ) of an element$f$with respect to a couple $$\bar X$$ =($X$_{0},$X$_{1}) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spaces$X$_{0}and$X$_{1}is either$L$^{∞}or$L$^{1}. Many examples are given of couples of lattices $$\bar X$$ for which |$g$_{t}| increases monotonically a.e. with respect to$t$. It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak$K$-divisibility constant γ( $$\bar X$$ ) for the couple. But it is also shown that certain couples $$\bar X$$ do not have this property. These also provide examples of couples of lattices for which γ( $$\bar X$$ ).

We use a positive parameter to develop a dierentiable perturbed reconstruction model to solve the L1–L2 magnetic resonance image (MRI) reconstruction problem. We use Bregman iterative formulation to solve the dierentiable perturbed L1–L2 model, and lagged diusivity xed-point iteration to solve the minimization problem in the Bregman iteration. Two Fourier transforms and an inverse Fourier transform are used to accelerate L1–L2 MRI reconstruction. Real MR images are used to test the method in numerical experiments. The results demonstrate that the proposed method is very ecient for L1–L2MRI reconstruction.

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n.1, 1).We construct an arithmetic theta lift from harmonic Maass forms of weight 2 . n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla– Rapoport divisors, equipped with the Green function given by the regularized theta lift of f . Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f ) and the theta function of a positive definite hermitian lattice of rank n − 1. When specialized to the case n = 2, this result can be viewed as a variant of the Gross–Zagier formula for Shimura curves associated to unitary groups of signature (1, 1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

We consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect. The dimensionality was first reduced via a high-dimensional factor model implemented by the principal component analysis. Using the extracted factors, we develop a novel forecasting method called the sufficient forecasting, which provides a set of sufficient predictive indices, inferred from high-dimensional predictors, to deliver additional predictive power. The projected principal component analysis will be employed to enhance the accuracy of inferred factors when a semi-parametric factor model is assumed. Our method is also applicable to cross-sectional sufficient regression using extracted factors. The connection between the sufficient forecasting and the deep learning architecture is explicitly stated. The sufficient forecasting correctly estimates projection indices of the underlying factors even