arXiv research

Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.

Universal model for soft tissue mechanics under shock waves.

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constr…

This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{\infty}(M,N)$ of all smooth mappings from a finite dimensional Whitney manifold germ $M$ into a …

Continuum mechanics theory describes skin's complex anisotropic behavior.

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.

Continuum Dropout improves neural differential equations by preventing overfitting.

Affine deformations serve as basic examples in the continuum mechanics of deformable 3-dimensional bodies (referred as homogeneous deformations). They preserve parallelism and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by …

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…

New finite element method for complex forms in any dimension.

The paper shows objective derivatives are covariant derivatives on Riemannian metrics.

This paper is an attempt at understanding the quantum-like dynamics of financial markets in terms of non-differentiable price-time continuum having fractal properties. The main steps of this development are the statistical scaling, the non-differentiability hypothesis, and the equations of motion entailed by this hypot…

Survey explores cohomology's roles in applied math and sciences.

New framework reveals limits of flexible, periodic thin surfaces.

We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fréchet category. This study is motivated by the occurrence of such objects in the geometric Lag…

Completeness of the eigenfunctions of a quantum mechanical system is crucial for its probability interpretation. By using the method of contour integral we give properly normalized eigenfunctions for both discrete and continuum spectrum of the Morse potential, and explicitly prove the completeness relation. As an appli…

Smooth knots can be embedded into a specific Menger continuum.

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the…

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq…

We introduce the continuum self-similar tree (CSST) and characterize it topologically. We apply this to answer a question of Curien about the topology of the continuum random tree (CRT). We also give a topological characterization of other trees with branch points of finite or infinite valences.

We propose a continuum model for the description of buyer and seller dynamics in an Internet market. The relevant variables are the research effort of buyers and the sellers' reputation building process. We show that, if a commercial web-site gives consumers the possibility to rate credibly sellers they bargained with,…

Bayesian neural networks predict stress fields and uncertainty in materials.

The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Al…

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary sup…

Derives continuum model from discrete $\varepsilon$-graphs with connectivity functional.

An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally f…

Generalizes Alexandroff's $V^n$-continua to cohomological dimensions.

A machine learning model captures non-Newtonian fluid dynamics from molecular details.

Proves continuum limits of Lipschitz learning using Γ-convergence.

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

A mesh-free method solves continuum-marginal optimal transport problems.

We propose a stochastic process driven by memory effect with novel distributions including both exponential and leptokurtic heavy-tailed distributions. A class of distribution is analytically derived from the continuum limit of the discrete binary process with the renormalized auto-correlation and the closed form momen…

Optimal reinsurance contracts designed for a continuum of risk types.

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…

This work proves the continuum limit of t-SNE for data visualization.

A neural network speeds up bond-associated peridynamics simulations.

Gradient flows on graphons converge to curves on graphon space.

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, Continuum Mechanics and Differential Geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an inde…

Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, whi…

Many statistical learning problems can be posed as minimization of a sum of two convex functions, one typically a composition of non-smooth and linear functions. Examples include regression under structured sparsity assumptions. Popular algorithms for solving such problems, e.g., ADMM, often involve non-trivial optimiz…

Continuum transformers learn operators in context via gradient descent.

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and…

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…

A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to th…

We propose a stochastic process driven by the memory effect with novel distributions which include both exponential and leptokurtic heavy-tailed distributions. A class of the distributions is analytically derived from the continuum limit of the discrete binary process with the renormalized auto-correlation. The moment …

A crucial challenge in image-based modeling of biomedical data is to identify trends and features that separate normality and pathology. In many cases, the morphology of the imaged object exhibits continuous change as it deviates from normality, and thus a generative model can be trained to model this morphological con…