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We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean…
We leverage path differentiability and a recent result on nonsmooth implicit differentiation calculus to give sufficient conditions ensuring that the solution to a monotone inclusion problem will be path differentiable, with formulas for…
Several deterministic and stochastic multi-variable global optimization algorithms (Conjugate Gradient, Nelder-Mead, Quasi-Newton, and Global) are investigated in conjunction with energy minimization principle to resolve the pressure and…
Here we study the nonlinear hyperbolic equations of the type of equations from theory of flows on networks, for which we prove the solvability theorem under the appropriate conditions and also investigate the behaviour of the solution.
The DC Power Flow approximation has been widely used for decades in both industry and academia due to its computational speed and simplicity, but suffers from inaccuracy, in part due to the assumption of a lossless network. Here we present…
Many optimization problems in power transmission networks can be formulated as polynomial problems with complex variables. A polynomial optimization problem with complex variables consists in optimizing a real-valued polynomial whose…
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
This paper proposes a method to embed the AC power flow problem with voltage magnitude constraints in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the…
We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link…
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to…
We employ a coarse-graining approach to analyze nonlinear cascades in Boussinesq flows using high-resolution simulation data. We derive budgets which resolve the evolution of energy and potential enstrophy simultaneously in space and in…
An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting…
We consider the problem of deriving an explicit approximate solution of the nonlinear power equations that describe a balanced power distribution network. We give sufficient conditions for the existence of a practical solution to the power…
The flow equation approach investigated by Wegner et al. is applied to an unbounded Hamiltonian system with a generalization. We show that a well-known quantized complex energy eigenvalues which is related to decay widths can be given with…
Automating the solutions of multiple network information theory problems, stretching from fundamental concerns such as determining all information inequalities and the limitations of linear codes, to applied ones such as designing coded…
With respect to earlier investigations, the theory of multi-component, concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and extended, including subsystems with nonzero density on the boundary and subsystems with…
We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded…
Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related…
The Lie point symmetries and corresponding invariant solutions are obtained for a Gaussian, irrotational, compressible fluid flow. A supersymmetric extension of this model is then formulated through the use of a superspace and superfield…
We give a short survey on computational techniques which can be used to solve the representation conversion problem for polyhedra up to symmetries. We in particular discuss decomposition methods, which reduce the problem to a number of…