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While deep learning methods have achieved state-of-the-art performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach,…

The convex feasibility problem asks to find a point in the intersection of a collection of nonempty closed convex sets. This problem is of basic importance in mathematics and the physical sciences, and projection (or splitting) methods…

We study variational obstacle avoidance problems on complete Riemannian manifolds and apply the results to the construction of piecewise smooth curves interpolating a set of knot points in systems with impulse effects. We derive the…

This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…

We introduce the problem of knot-based inverse perceptual art. Given multiple target images and their corresponding viewing configurations, the objective is to find a 3D knot-based tubular structure whose appearance resembles the target…

Averaging certain class of quasiperiodic monotone operators can be simplified to the periodic homogenization setting by mapping the original quasiperiodic structure onto a periodic structure in a higher dimensional space using cut-and…

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of…

Automatically determining knot number and positions is a fundamental and challenging problem in B-spline approximation. In this paper, the knot placement is abstracted as a mapping from initial knots to the optimal knots. We innovatively…

In recent years, considerable progress has been made for the task of rigid object pose estimation from a single RGB-image, but achieving robustness to partial occlusions remains a challenging problem. Pose refinement via rendering has shown…

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using…

A new exact projective penalty method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing…

In this paper we consider projective and injective resolutions of Koszul complexes and give several applications to the study of Koszul homology modules.

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous…

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a…

We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It…

This article focuses on numerical efficiency of projection algorithms for solving linear optimization problems. The theoretical foundation for this approach is provided by the basic result that bounded finite dimensional linear optimization…

In this paper, we propose a new method that combines the inexact Newton method with a procedure to obtain a feasible inexact projection for solving constrained smooth and nonsmooth equations. The local convergence theorems are established…

We propose a new learning-based approach to solve ill-posed inverse problems in imaging. We address the case where ground truth training samples are rare and the problem is severely ill-posed - both because of the underlying physics and…

Approximate solutions of Urysohn integral equations using projection methods involve integrals which need to be evaluated using a numerical quadrature formula. It gives rise to the discrete versions of the projection methods. For $r \geq…

We consider the problem of selective prediction (also known as reject option) in deep neural networks, and introduce SelectiveNet, a deep neural architecture with an integrated reject option. Existing rejection mechanisms are based mostly…