Uniqueness for the signature of a path of bounded variation and the reduced path group by Hambly, B, Lyons, T

Partial regularity for BV minimizers by Gmeineder, F, Kristensen, J

“…We establish an ε -regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. …”

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws by Chen, G, Xiang, W, Zhang, Y

“…Finally, we extend the results to the case with non-compact support initial data of bounded variation. © 2013 Copyright Taylor and Francis Group, LLC.…”

The signature of a rough path: Uniqueness by Boedihardjo, H, Geng, X, Lyons, T, Yang, D

“…Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. …”

The initial–boundary value problem for general non-local scalar conservation laws in one space dimension by De Filippis, C, Goatin, P

“…We prove a global well-posedness result for a class of weak entropy solutions of bounded variation (BV) of scalar conservation laws with non-local flux on bounded domains, under suitable regularity assumptions on the flux function. …”

On the Morse-Sard property and level sets of Sobolev and BV functions by Bourgain, J, Korobkov, M, Kristensen, J

“…Using these results we prove that almost all level sets are finite disjoint unions of Lipschitz arcs whose tangent vectors are of bounded variation. In the case of W2,1 functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of C1 arcs whose tangent vectors are absolutely continuous along these arcs. © European Mathematical Society.…”

On the Morse–Sard property and level sets of Wn,1 Sobolev functions on ℝn by Bourgain, J, Korobkov, M, Kristensen, J

“…These results remain valid also within the larger space of functions of bounded variation BVn(Rn). For the proofs we establish and use some new results on Luzin-type approximation of Sobolev and BV-functions by Ck-functions, where the exceptional sets have small Hausdorff content.…”

On a discrete version of Tanaka's theorem for maximal functions by Bober, J, Carneiro, E, Hughes, K, Pierce, L

“…For the non-centered maximal operator M we prove that, given a function f: Z → R of bounded variation, =, where Var(f) represents the total variation of f. …”

L 1-Framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations by Chen, G, Karlsen, K

“…We apply our L 1-framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an L 1 error estimate for an upwind-central finite difference scheme. © 2004 American Mathematical Society.…”

Measure-theoretic analysis and nonlinear conservation laws by Chen, G, Torres, M, Ziemer, W

“…We focus particularly on entropy solutions without bounded variation and Cauchy fluxes on oriented surfaces which are used to formulate the balance law. …”

Stability of rarefaction waves and vacuum states for the multidimentional Euler equations by Chen, G, Chen, J

“…Our analysis applies to entropy solutions with arbitrarily large oscillation, and no bounded variation regularity is required. © World Scientific Publishing Company.…”

RIGHT INVERSES OF LEVY PROCESSES: THE EXCURSION MEASURE IN THE GENERAL CASE by Savov, M, Winkel, M

“…This description unifies known formulas in the case of a positive Gaussian coefficient and in the bounded variation case. While these known formulas relate to excursions away from a point starting negative continuously, and excursions started by a jump, the present description is in terms of excursions away from the supremum continued up to a return time. …”

Weighted signature kernels by Cass, T, Lyons, T, Xu, X

“…Suppose that γ and σ are two continuous bounded variation paths which take values in a finite-dimensional inner product space V. …”

The adaptive patched cubature filter and its implementation by Lee, W, Lyons, T

“…The algorithm can be performed by integrating along a number of carefully selected bounded variation paths. The iterated application of the KLV method has a tendency for the number of particles to increase. …”

Stability of conical shocks in the three-dimensional steady supersonic isothermal flows past lipschitz perturbed cones by Chen, G-QG, Kuang, J, Zhang, Y

“…We first formulate the shock stability problem as an initial boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and structural/asymptotic stability of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. …”

Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length by Chang, J, Lyons, T, Ni, H

“…Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the <i>n</i>-th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the <i>n</i>-th root of the norm of the <i>n</i>-th term in the normalised signature as <i>n</i> approaches infinity.…”