Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.
We study a class of elastic energy functionals for maps between planar domains (among them the so-called squared distance functional) whose critical points (elastic maps) allow a far more complete theory than one would expect from general elasticity theory. For some of these functionals elastic maps even admit a "Weier…
Study of red blood cells using elastic surface theory.
problem Understanding the shape of red blood cells.
method Used Helfrich-Canham functional to model red blood cells as elastic surfaces.
result Cassinian ovals, except for the round sphere, do not solve the shape equation.
Sparse elasticity reconstruction from local displacements reduces error.
problem Reconstructing elasticity from limited data.
method Sparse elasticity reconstruction theory, local clustering, alternating optimization.
result Higher spatial resolution elasticity distribution estimation.
New discrete curves defined in space forms with geometric properties.
problem Defining discrete elastic and constrained elastic curves in space forms.
method Extending discrete Euclidean curvature to space forms and using Bäcklund transformations.
result Discrete elastic and constrained elastic curves are elements of a curve hierarchy.
We derive a dimensionally-reduced limit theory for an n-dimensional nonlinear elastic body that is slender along k dimensions. The starting point is to view an elastic body as an n-dimensional Riemannian manifold together with a not necessarily isometric W1,2-immersion in n-dimensional Euclidean space. The…
Stable discretizations for elastic flow on Riemannian manifolds.
problem Discretizing elastic flow on curved spaces.
method Conformally flat Riemannian manifolds discretization.
result Robust and quadratic convergence of the method.
Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Mo…
Improves parameter selection for denoising with elastic net.
problem Challenges in selecting optimal parameters for denoising.
method Combines statistical learning theory and regularisation theory to approximate optimal elastic net parameters.
result Explicit error bounds on accuracy of approximated parameter and regularisation solution.
The study examines elastic curves with self-intersections and their properties.
problem Understanding self-intersecting elastic curves in Euclidean space.
method Review of classical elastica theory and discussion of open problems.
result Proof of Li--Yau type inequality for self-intersecting curves.
Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.
problem Rigidity of hyperbolic shells and their \(Γ\)-limit behavior.
method Nonlinear rigidity estimates for \(H^1\) deformations and hyperbolic shells with clamped lateral boundary.
result Derives the optimal exponent \(h^{-4/3}\) for hyperbolic shells.
We consider an infinite 3-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which give…
New method weaves paper strips for designing curved surfaces with elasticity.
problem Designing general curved surfaces with geometrical elasticity.
method Shape optimization of paper strips using nonlinear elasticity theory.
result Demonstrated creation of catenoid and helicoid surfaces with 54 paper strips.
Geometrically reformulates elasticity theory using exterior calculus.
problem Formulating nonlinear elasticity theory geometrically.
method Using exterior calculus and bundle-valued differential forms.
result Equivalence to standard tensor calculus formulations.
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate tha…
Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie for the selection of groups of correlated variables. To investigate on the statistical properties of this scheme and in particular on its consistency properties, we set up …
This work extends elasticity theory to curved spaces, solving stress potentials.
problem Addressing elasticity in curved spaces with boundary.
method Using double forms and bilaplacian operator regularity, solving biharmonic equations.
result Stress potentials can be used in non-Euclidean geometries.
Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension red…
Unified theory solves strain compatibility and elasticity of origami metamaterials.
problem Understanding and controlling the morphing paths of origami metamaterials.
method Unified theory for a wide array of origami tessellations, solving strain compatibility and elasticity.
result Origami metamaterials exhibit equal but opposite in-plane and out-of-plane Poisson's ratios and bending energy depends on strain gradient.
Theory models nonlinear soft tissue elasticity and remodeling using extended Finsler geometry.
problem Understanding and predicting the behavior of nonlinear soft tissues, especially in biologic contexts.
method Formulated a continuum mechanical theory incorporating extended Finsler geometry to describe the complex behaviors of fibrous soft solids.
result The model quantifies residual strains from growth, remodeling, and degradation, and predicts equilibrium configurations.
Unified theory for curved shell deformations with elastic and inelastic components.
problem Coupled nonlinear elastic and inelastic deformations of curved thin shells.
method Multiplicative decomposition of surface deformation gradient, detailed kinematics analysis, surface balance laws, constitutive relations derived from thermodynamics.
result Unified constitutive relations for growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity of shells.
We develop a theory of axisymmetric surfaces minimizing a combination of surface tension and nematic elastic energies which may be suitable for describing simple film and bubble shapes. As a function of the elastic constant and the applied tension on the bubbles, we find the analogues of the unduloid, sphere, and nodoi…
Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.
problem Understanding complex wrinkling patterns in thin elastic hyperbolic surfaces.
method Non-Euclidean plate theory and investigation of branch points.
result Branch points are natural defects in hyperbolic sheets, influencing their morphology robustly.
Corrects errors in previous work on spectral asymptotics in elasticity.
problem Two-term asymptotics of elastic eigenvalues on Riemannian manifolds.
method Strongly continuous semigroups and pseudodifferential operators.
result Theorem 1.1 in \cite{Liu-21} is rigorously proven.
Unified treatment of elastic metrics for curves in any dimension.
problem Defining metrics on spaces of Euclidean curves for statistical analysis.
method Developing a unified approach to elastic metrics, extending results on existence of solutions and algorithms for computing distances and geodesics.
result Unified treatment of elastic metrics for all parameter choices, extending previous work.
Researchers prove a generalized Piola identity for Riemannian manifolds.
problem Generalizing the Piola identity to Riemannian settings.
method Two approaches: Euclidean derivation and variational interpretation via Null-Lagrangians.
result Proof of generalized Piola identity for Riemannian manifolds.
We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a 3-dimensional theory, with small thickness, as well as a 2-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two des…
Financial derivatives have often been criticized as casino-style betting instruments. It turns out that many naive ways of making them are indeed equivalent to gambling. Fortunately, this inadvertent effect can be understood and prevented. We present a theory of product design which achieves that.
We suggest an alternative mathematical model for the massless neutrino. Consider an elastic continuum in 3-dimensional Euclidean space and assume that points of this continuum can experience no displacements, only rotations. This framework is a special case of the so-called Cosserat theory of elasticity. Rotations of p…
The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.
problem Comparing and analyzing shapes of curves.
method Construction and theoretical properties of quotient elastic metrics, special case of square root velocity metric, numerical approaches for estimation.
result Simplified expression for the square root velocity metric distance.
Analyzes properties of stiffness tensors for elastic wave imaging.
problem Characterizing stiffness tensor fields for elastic wave imaging.
method Finsler-geometric methods applied to anisotropic stiffness tensor fields.
result Conditions for Finsler-geometric methods to be applicable.
Elastic Cash adjusts money supply to stabilize interest rates.
problem Stabilizing interest rates in a decentralized system.
method Modifies supply to keep interest rate fixed by public market.
result Improves elasticity of US Dollar and new cryptocurrencies.
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…
Proposes using elastic demand to improve forecasting accuracy.
problem Mismatch between planned supply and actual demand due to demand variance.
method Reallocate historical elastic demand to reduce forecasting variance.
result Improves forecasting and supply planning effectiveness.
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
Revisits elastic string model to explain interest rate correlations.
problem Describing the forward interest rate curve using an elastic string model.
method Reinterprets Baaquie and Bouchaud's (2004) model to highlight market forces.
result Model accurately reproduces FRC correlation structure with minimal parameters.
Approximate 3D elastic curves with exact constraints
problem Designing and approximating 3D elastic curves
method Numerically stable method for recovering 11 parameters
result Fast and stable approximation of arbitrary curves
New algorithm selects genes for cancer classification using adaptive elastic net and conditional mutual information.
problem Selecting informative genes for microarray cancer classification.
method Adaptive Elastic Net with Conditional Mutual Information (AEN-CMI).
result AEN-CMI achieves the best classification performance with fewer genes.
The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.
problem Rigidity and continuity properties of elastic bodies in non-Euclidean settings.
method Geometric rigidity estimates, asymptotic rigidity of elastic membranes, simplified geometric proof of continuous dependence.
result Established geometric rigidity estimate and proved asymptotic rigidity of elastic membranes.
Study preserves planar and graphical properties of curves under elastic flow.
problem Maintaining planar and graphical properties of non-compact curves under elastic flow.
method Extended recent work on adapted elastic energy to derive thresholds for planar and graphical embeddedness.
result Derived new Li--Yau type inequality for complete planar curves.
Theory of packing diabolic domains in liquid crystals.
problem Understanding the packing of diabolic domains in liquid crystals.
method Lorentz transformations and geometric analysis.
result Diabolic domains can lower the elastic energy of the system.
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli m…
Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pau…
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Study on migrating elastic flows of curves across half-planes.
problem Migrating elastic flows of curves from upper to lower half-planes.
method Analytical and numerical construction of migrating elastic flows.
result Construction of various migrating elastic flows.
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.
Corrects errors in previous work on linear elasticity calculations.
problem Incorrect formulas and conclusions in previous research on linear elasticity.
method Identifying and correcting errors in previous calculations and conclusions.
result Proves the correctness of the main theorem in a previous paper.