Related papers: Knot types of generalized Kirchhoff rods
Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We…
Hopf solitons in the Skyrme-Faddeev model are string-like topological solitons classified by the integer-valued Hopf charge. In this paper we introduce an approximate description of Hopf solitons in terms of elastic rods. The general form…
We generalize Birkhoff's Theorem in the following fashion. We find necessary and sufficient conditions for any spherically symmetric space-time to be static in terms of the eigenvalues of the stress-energy tensor. In particular, we…
We study the dynamics of critical spin fluctuations and hot electrons at the metallic antiferromagnetic quantum critical points with $Z_2$ and $O(2)$ spin symmetries, building upon earlier works on the $O(3)$ symmetric theory. The…
We prove that, for certain extensions of valued fields which admit a sensible theory of ramification groups, there exist canonical towers that correspond to the break-points of their Herbrand function. In particular, each of the…
Using ideas from algebraic topology and statistical mechanics, we generalize Kirchhoff's network and matrix-tree theorems to finite CW complexes of arbitrary dimension. As an application, we give a formula expressing Reidemeister torsion as…
We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schr\"oder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under…
We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as…
Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two…
In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a…
The exact form of the kinetic energy functional has remained elusive in orbital-free models of density functional theory (DFT). This has been the main stumbling block for the development of a general-purpose framework on this basis. Here,…
Free energy functionals of Ginzburg-Landau type lie at the heart of a broad class of continuum dynamical models, such as the Cahn-Hilliard and Swift-Hohenberg equations. Despite the wide use of such models, the assumptions embodied in the…
The average-density approximation is used to construct a nonlocal kinetic energy functional for an inhomogeneous two-dimensional Fermi gas. This functional is then used to formulate a Thomas-Fermi von Weizs\"acker-like theory for the…
On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg-Landau and a perimeter term. Our scaling allows low-energy states to be…
A new class of rational parametrization has been developed and it was used to generate a new family of rational functions B-splines $\displaystyle{{\left({}^{\alpha}{\mathbf B}_{i}^{k} \right)}_{i=0}^{k}}$ which depends on an index $\alpha…
We compute the Chekanov-Eliashberg contact homology of what we call the Legendrian closure of a positive braid. We also construct an augmentation for each such link diagram. Then we apply the monodromy techniques established in an earlier…
In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the…
A Kirchhoff graph is a vector graph with orthogonal cycles and vertex cuts. An algorithm has been developed that constructs all the Kirchhoff graphs up to a fixed edge multiplicity. This algorithm is used to explore the structure of prime…
A general class of solutions is obtained which describe a spherically symmetric wormhole system. The presence of arbitrary functions allows one to describe infinitely many wormhole systems of this type. The source of the stress-energy…
This article aims to extend classical homological results about the rational normal curves to analogues in weighted projective spaces. Results include determinantality and nonstandard versions of quadratic generation and the Koszul…