Related papers: Boundary Calculus, Rigidity Islands, and Deformati…
In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension $\geq 3$,…
In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points…
We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean $d$-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity.…
Large language models suffer from "hallucinations"-logical inconsistencies induced by semantic noise. We propose that current architectures operate in a "Metric Phase," where causal order is vulnerable to spontaneous symmetry breaking.…
We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity…
This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial…
The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the…
Continual learning systems operating in fixed-dimensional spaces face a fundamental geometric barrier: the flat manifold problem. When experience is represented as a linear trajectory in Euclidean space, the geodesic distance between…
Serrations are commonly employed to mitigate the turbulent boundary layer trailing-edge noise. However, significant discrepancies persist between model predictions and experimental observations. In this paper, we show that this results from…
In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle…
Bounds are obtained on the volume fraction in a two-dimensional body containing two elastically isotropic materials with known bulk and shear moduli. These bounds use information about the average stress and strain fields, energy,…
We present a numerical stability analysis of the immersed boundary(IB) method for a special case which is constructed so that Fourier analysis is applicable. We examine the stability of the immersed boundary method with the discrete Fourier…
Using the canonical Monte Carlo simulation technique, we study a Regge calculus model on triangulated spherical surfaces. The discrete model is statistical mechanically defined with the variables $X$, $g$ and $\rho$, which denote the…
This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each…
A general scheme for determining and studying integrable deformations of algebraic curves, based on the use of Lenard relations, is presented. We emphasize the use of several types of dynamical variables : branches, power sums and…
We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness…
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin…
Mathematical model of the turbulent flux in the three-layer boundary system is presented. Turbulence is described as a presence of the nonzero vorticity. Generalized advection-diffusion-reaction equation is derived for arbitrary number…
Models for fluid deformable surfaces provide valid theories to describe the dynamics of thin fluidic sheets of soft materials. To use such models in morphogenesis and development requires to incorporate active forces. We consider active…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…