Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of length to width of a flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been conjectured that the ribbonlength has an upper bound and a lower bound which are both linear in the crossing number of the knot. In the first part of the paper, we use grid diagrams to construct flat knotted ribbons and prove an explicit quadratic upper bound on the ribbonlength for all non-trivial knots. We then improve the quadratic upper bound to a linear upper bound for all non-trivial torus knots and twist knots. Our approach of using grid diagrams to study flat knotted ribbons is novel and can likely be used to obtain a linear upper bound for more general families of knots. In the second part of the paper, we obtain a sharper linear upper bound on the ribbonlength for nontrivial twist knots by constructing a flat knotted ribbon via folding the ribbon over itself multiple times to shorten the length.

No abstract uploaded!

The Lp dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Lp Brunn– Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for Lp dual curvature measure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the Lp dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.

Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat conductive gases in $\mathbb R^3$, provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density. Precisely, it is proved that for any strong solution to the Cauchy problem of the heat conductive compressible Navier--Stokes equations, the corresponding entropy keeps uniformly bounded and the $L^2$ regularities of the velocity and temperature can be propagated, up to the existing time of the solution, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{|x|^2})$. The main tools are some singularly weighted energy estimates and an elaborate De Giorgi type iteration technique. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

In this paper we analyze the Kahrobaei-Lam-Shpilrain (KLS) key exchange protocols that use extensions by endomorpisms of matrices over a Galois field proposed in \cite{Kahrobaei-Lam-Shpilrain:2014}. We show that both protocols are vulnerable to a simple linear algebra attack.