Study of metric anomalies in uniform elastic solids without stress.
problem Understanding metric anomalies in uniform elastic solids.
method Introducing a quasi-plastic deformation framework and deriving a general form of metric anomalies.
result Derivation of a general form of metric anomalies yielding zero stress in uniform solids.
The present paper is devoted to the joint motion of two immiscible incompressible liquids in porous media. The liquids have different densities and initially separated by a surface of strong discontinuity (free boundary). We discuss the results of numerical simulations for exact free boundary problems on the microscopi…
Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.
problem Geometric formulation of solid and fluid mechanics.
method Port-Hamiltonian framework, Dirac structures, Hamiltonian reduction theory.
result Systematic derivation of port-Hamiltonian models for solid and fluid mechanics.
The paper connects fluid mechanics, elasticity, and geometry to study wrinkled solutions.
problem Existence of wrinkled solutions in nonlinear partial differential equations.
method Develops connections between fluids, elasticity, and isometric embeddings, mapping mechanical equations into geometric frameworks.
result Geometric theory provides a method for addressing admissibility criteria in nonlinear conservation laws.
We discuss some differential geometry pertaining to continuum mechanics and the route recently taken by D.N. Arnold, R.S. Falk, and R. Winther in deriving new improved finite element schemes in linear elasticity from constructions in projective geometry.
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.
Deep learning framework improves accuracy in solid mechanics.
problem Improving accuracy in solid mechanics simulations.
method Physics Informed Neural Networks (PINN) with multi-network model.
result PINN framework leads to more accurate predictions and improved robustness.
Lie groupoids and algebroids help analyze material uniformity and homogeneity.
problem Analyzing uniformity and homogeneity of material bodies.
method Associated Lie groupoids and algebroids to elastic materials, using them to characterize uniformity and homogeneity.
result Characterized uniformity and homogeneity of materials using Lie groupoids and algebroids.
New methods for estimating and inferring nonparametric structural functions and elasticities.
problem Estimating and inferring nonparametric structural functions and their derivatives.
method Data-driven sieve dimension choice and uniform confidence bands construction.
result Optimal estimation and inference procedures with minimax rates of convergence.
Geometrically characterizes two strain measures in solid mechanics.
problem Characterizing isotropic strain measures in nonlinear elasticity.
method Purely geometric methods based on geodesic distance on the general linear group.
result Identifies the Hencky strain tensor as the natural nonlinear extension of the linear strain tensor.
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions …
The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…
In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show t…
New geometric decompositions reveal volumes of hyperbolic tiling links.
problem Understanding volumes of hyperbolic tiling links.
method Explicit geometric decompositions and bipyramid constructions.
result Volumes of hyperbolic tiling links are dense in a specific interval.
Study on material inhomogeneity and strain compatibility in thin elastic shells.
problem Understanding material inhomogeneity and strain compatibility in thin elastic shells.
method Developed a relationship between inhomogeneity and incompatibility measures using both 3D and 2D theories, derived intrinsic dislocation density tensors, and formulated governing equations for residual stress fields.
result Explicit forms of intrinsic dislocation density tensors characterizing inhomogeneity of dislocated Cosserat shells and simplified governing equations for residual stress fields.
Multiple kernel learning (MKL), structured sparsity, and multi-task learning have recently received considerable attention. In this paper, we show how different MKL algorithms can be understood as applications of either regularization on the kernel weights or block-norm-based regularization, which is more common in str…
The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case Functionally Graded Media, which allows us to obtain some homogeneity conditions.
New phenomena in knot thickness revealed by Legendrian large cables.
problem Understanding non-uniformly thick knots and their Legendrian large cables.
method Definition and analysis of Legendrian large cables, and construction of knots.
result Existence of knots that are not uniformly thick and have virtually overtwisted contact structures.
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.
We theoretically investigate the convergence rate and support consistency (i.e., correctly identifying the subset of non-zero coefficients in the large sample limit) of multiple kernel learning (MKL). We focus on MKL with block-l1 regularization (inducing sparse kernel combination), block-l2 regularization (inducing un…
Deformational structures, in many aspects generalizing standard elasticity theory, are investigated in abstract form. Within free deformational structures we define algebra of deformations, classify them by its special properties, define motions and conformal motions together with deformational decomposition of manifol…
Study on materials with disclinations, limiting their size.
problem Limiting the size of disclinations in materials with symmetries.
method Defining material-uniform hyperelastic bodies with disclinations, rigorously analyzing their properties.
result The size of disclinations is limited by the symmetries of the constitutive relation.
Our goal is to identify the type and number of static equilibrium points of solids arising from fine, equidistant n-discretrizations of smooth, convex surfaces. We assume uniform gravity and a frictionless, horizontal, planar support. We show that as n approaches infinity these numbers fluctuate around specific val…
Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.
problem Gradient obstacle problems on compact Riemannian manifolds.
method Uniform semiconcavity estimates and fine convergence results for solutions and free boundaries.
result The elastic and λ-elastic sets of solutions converge to the cut locus and λ-cut locus of the manifold. Gaussian process regression speeds up nudged elastic band calculations for transitions.
problem Reducing computational effort for calculating minimum energy paths in thermalized systems.
method Approximate energy surface generation and refinement using Gaussian process regression.
result The number of energy and force evaluations can be reduced by an order of magnitude.
Universal model for soft tissue mechanics under shock waves.
problem Modeling shock wave mechanics in soft biological tissues.
method Continuum mixture theory with phase-field mechanics.
result Universal thermodynamically consistent formulation for soft porous tissues.
We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals consid…
Improved performance in shape identification tasks using elastic metrics in t-SNE and UMAP.
problem Improper metrics in dimensionality reduction techniques lead to poor performance in machine learning applications.
method Incorporating elastic metrics into t-SNE and UMAP for functional data.
result Improved F1 scores on shape identification tasks for three benchmark datasets.
Elastic-InfoGAN learns object identity in class-imbalanced data.
problem Learning disentangled representations in class-imbalanced data.
method Invariance to identity-preserving transformations to learn object identity.
result Effectiveness in disentangling object identity in imbalanced datasets.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
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.
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.
New self-shrinkers of Platonic solids found.
problem Finding new embedded self-shrinkers of specific genus.
method Variational methods, numerical discovery by D. Chopp.
result Constructed self-shrinkers resembling doublings of Platonic solids.
New decompositions reveal volumes of tiling links in hyperbolic and spherical settings.
problem Understanding volumes of tiling links in different geometries.
method Generalizing angle structures and using truncated bipyramids to compute volumes.
result Volumes of hyperbolic tiling links are dense in a specific interval.
Proves NP and co-NP status for knot core recognition in solid torus.
problem Determining if a knot is the core of a solid torus.
method Alternate proof and corollary of Hopf link recognition problem.
result Proves NP and co-NP status for solid torus core recognition problem.
Rational knots and links in solid torus characterized by continued fractions.
problem Characterizing rational knots and links in solid torus.
method Using rational tangles and continued fractions, and generalizing to skein module invariants.
result Rational links in solid torus fully characterized by rational tangles and continued fractions.
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.
We find a Weierstrass-like formula for 2D elastic maps.
problem Understanding elastic maps between planar domains.
method Develop a Weierstrass representation for critical points of certain energy functionals.
result Elastic maps admit a Weierstrass representation in terms of holomorphic functions.
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 theorem for 4D links simplifies characterisation problem.
problem Long-standing open problem in link characterisation.
method Reidemeister Theorem for solid ribbon torus links.
result Complete characterisation of a related class of links.
We consider the solid angle that a planar compact subset subtends at a point in a level set of height h and study two extremal problems for the solid angle. One of the variables is a point in such a plane, that is, we study the properties of the solid angle maximizer. The other is the pair of a planar compact subset an…
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.
Study solid angles to find Seifert hypersurfaces.
problem Understanding solid angles and their relation to Seifert hypersurfaces.
method Analyze the solid angle function and its critical levels.
result Non-critical level sets of the solid angle function are Seifert hypersurfaces.
The elastic trefoil is the twice covered circle, a key finding in knot elasticity.
problem Characterizing the elastic behavior of knotted loops of springy wire.
method Minimizing bending energy and ropelength to penalize self-intersection.
result The elastic trefoil is the twice covered circle, not the round circle.
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.