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Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two…

With the aid of a computer, we provide a motion picture of the twist-spun trefoil which exhibits the periodicity well.

We give a rigorous proof of the colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots. For the trefoil knot and the figure-eight knot, it is expressed by a single sum, and for a twist knot, it is expressed…

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…

As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the…

Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating over the remaining variables. Over the last 30 years, the technique has been widely used, with empirical and…

We characterize the semiclosed projections and apply them to compute the Schur complement of a selfadjoint operator with respect to a closed subspace. These projections occur naturally when dealing with weak complementability.

We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define…

We show that if a Montesinos knot admits a Dehn surgery yielding a toroidal Seifert fibered 3-manifold, then the knot is the trefoil knot and the surgery slope is 0.

In this work we consider an iterative method for solving the quasi-convex feasibility problem. We firstly introduce the so-called star subgradient projection operator and present some useful properties. We subsequently obtain a convergence…

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level…

Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A benefit of these methods is that…

Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewise polynomial function (ppf) (or any $q$-monotone ppf,…

This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact…

Classical extragradient schemes and their stochastic counterpart represent a cornerstone for resolving monotone variational inequality problems. Yet, such schemes have a per-iteration complexity of two projections onto a convex set and…

We prove the existence of the reflected diffusion on a complex of an arbitrary size for a large class of planar simple nested fractals. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded…

Let X be a projective irreducible symplectic manifold and L a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample

We establish sufficient conditions for finite convergence of the alternating projections method for two non-intersecting and potentially nonconvex sets. Our results are based on a generalization of the concept of intrinsic transversality,…

Vector optimization problems are a generalization of multiobjective optimization in which the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative octant. Due to its real life applications, it is…

Tangential computed tomography (TCT) is a useful tool for imaging the large-diameter samples, such as oil pipelines and rockets. However, TCT projections are truncated along the detector direction, resulting in degraded slices with radial…