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Shankar C. Venkataramani

Publications and source records attributed to Shankar C. Venkataramani.

At least 19 recordsLinked to original sources
arXiv2025

Hyperbolic Monge-Ampère Equation on a Cylinder: Well-Posedness and Stability

This paper develops a rigorous analytic framework for the hyperbolic Monge-Ampère equation on strip-like domains, which model wrinkled patterns in thin elastic sheets. Our work addresses the rigid side of the classical rigidity-flexibility dichotomy by defining this regime not by high smoothness, but by the more fundamental property of partial convexity. The hodograph transformation is the natural tool for this setting, as its validity is predicated on partial convexity. It converts the nonlinear Monge-Ampère equation into a linear damped wave equation, allowing us to formulate a well-posed Cauchy-Goursat problem. A key challenge is the corner singularity that arises where characteristic and non-characteristic boundary data meet. To resolve this, we develop a parametrix-corrector decomposition that captures the solution's inherent singular behavior. This method recasts the problem as a singular Volterra integral equation, for which we prove the existence and uniqueness of a new class of hodograph weak solutions. Finally, we derive energy estimates to establish the quantitative stability of these rigid solutions under perturbations of the underlying curvature function.

math.AP↗
arXiv2025

Discrete Differential Geometry for $C^{1,1}$ Hyperbolic Surfaces of Non-Constant Curvature

We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and numerically integrate a version of the classical Lelieuvre formulas that apply to immersions of $C^{1,1}$ hyperbolic surfaces of non-constant curvature. In contrast to the constant curvature case, these formulas do not provide an explicit method for constructing an immersion but rather describe an immersion via an implicit set of equations. We propose an iterative method for resolving these equations. Because we are interested in scenarios where the curvature is a function of the intrinsic material coordinates, in particular, on the geodesic distance from an origin or from an edge, we suggest a fast marching method for computing geodesic distance on manifolds. We apply our methods to generate surfaces of non-constant curvature and demonstrate how one can introduce branch points to account for the multi-generational buckling and subwrinkling observed in many applications.

math.DG↗
arXiv2022

An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects

We investigate stripe patterns formation far from threshold using a combination of topological, analytic, and numerical methods. We first give a definition of the mathematical structure of `multi-valued' phase functions that are needed for describing layered structures or stripe patterns containing defects. This definition yields insight into the appropriate `gauge symmetries' of patterns, and leads to the formulation of variational problems, in the class of special functions with bounded variation, to model patterns with defects. We then discuss approaches to discretize and numerically solve these variational problems. These energy minimizing solutions support defects having the same character as seen in experiments.

math-ph↗
arXiv2021

Nature's forms are frilly, flexible, and functional

A ubiquitous motif in nature is the self-similar hierarchical buckling of a thin lamina near its margins. This is seen in leaves, flowers, fungi, corals, and marine invertebrates. We investigate this morphology from the perspective of non-Euclidean plate theory. We identify a novel type of defect, a branch-point of the normal map, that allows for the generation of such complex wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of stretching. We argue that branch points are the natural defects in hyperbolic sheets, they carry a topological charge which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating elastic energy. We develop a theory for branch points and investigate their role in determining the mechanical response of hyperbolic sheets to weak external forces.

cond-mat.soft↗
arXiv2021

Pattern dark matter and galaxy scaling relations

We argue that a natural explanation for a variety of robust galaxy scaling relations comes from the perspective of pattern formation and self-organization as a result of symmetry breaking. We propose a simple Lagrangian model that combines a conventional model for normal matter in a galaxy with a conventional model for stripe pattern formation in systems that break continuous translation invariance. We show that the energy stored in the pattern field acts as an effective dark matter. Our theory reproduces the gross features of elliptic galaxies as well as disk galaxies (HSB and LSB) including their detailed rotation curves, the radial acceleration relation (RAR), and the Freeman law. We investigate the stability of disk galaxies in the context of our model and obtain scaling relations for the central dispersion for elliptical galaxies. A natural interpretation of our results is that (1) `dark matter' is potentially a collective, emergent phenomenon and not necessarily an as yet undiscovered particle, and (2) MOND is an effective theory for the description of a self-organized complex system rather than a fundamental description of nature that modifies Newton's second law.

astro-ph.GA↗
arXiv2020

Computing with non-orientable defects: nematics, smectics and natural patterns

Defects are a ubiquitous feature of ordered media. They have certain universal features, independent of the underlying physical system, reflecting their topological origins. While the topological properties of defects are robust, they appear as `unphysical' singularities, with non-integrable energy densities in coarse-grained macroscopic models. We develop a principled approach for enriching coarse-grained theories with enough of the `micro-physics' to obtain thermodynamically consistent, well-set models, that allow for the investigations of dynamics and interactions of defects in extended systems. We also develop associated numerical methods that are applicable to computing energy driven behaviors of defects across the amorphous-soft-crystalline materials spectrum. Our methods can handle order parameters that have a head-tail symmetry, i.e. director fields, in systems with a continuous translation symmetry, as in nematic liquid crystals, and in systems where the translation symmetry is broken, as in smectics and convection patterns. We illustrate our methods with explicit computations.

cond-mat.soft↗
arXiv2020

Distributed branch points and the shape of elastic surfaces with constant negative curvature

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating energy. We develop a discrete differential geometric (DDG) approach to study the deformations of hyperbolic objects with distributed branch points. We present evidence that the maximum curvature of surfaces with geodesic radius $R$ containing branch points grow sub-exponentially, $O(e^{c\sqrt{R}})$ in contrast to the exponential growth $O(e^{c' R})$ for surfaces without branch points. We argue that, to optimize norms of the curvature, i.e. the bending energy, distributed branch points are energetically preferred in sufficiently large pseudospherical surfaces. Further, they are distributed so that they lead to fractal-like recursive buckling patterns.

math.DG↗
arXiv2020

Disk galaxies and their dark halos as self-organized patterns

Galaxies are built by complex physical processes with significant inherent stochasticity. It is therefore surprising that the inferred dark matter distributions in galaxies are strongly correlated with the observed baryon distributions leading to various `Baryon-Halo conspiracies'. The fact that no dark matter candidate has been definitively identified invites a search for alternative explanations for such correlations and we present an approach motivated by the behaviors of self organized patterns. We propose a nonlocal relativistic Lagrangian theory for a `pattern field' which acts as an `effective dark matter', built on the idea that defects in this pattern field couple to the baryonic matter distribution. Our theory accounts for the gross structure of cold disk galaxies. We compute galactic rotation curves and derive various galaxy scaling relations including Renzo's rule, the radial acceleration relation, and the existence of the Freeman limit for central surface brightness.

astro-ph.GA↗
arXiv2020

Remarks on the range and multiple range of random walk up to the time of exit

We consider the scaling behavior of the range and $p$-multiple range, that is the number of points visited and the number of points visited exactly $p\geq 1$ times, of simple random walk on ${\mathbb Z}^d$, for dimensions $d\geq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $D\subset {\mathbb R}^d$, as $N\uparrow\infty$. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case $D$ is a cube in $d\geq 3$, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in $d\geq 2$, both weakly converge to proportional exit times of Brownian motion from $D$, and that the corresponding limit moments are `polyharmonic', solving a hierarchy of Poisson equations.

math.PR↗
arXiv2019

Continuum mechanics of moving defects in growing bodies

Growth processes in many living organisms create thin, soft materials with an intrinsically hyperbolic geometry. These objects support novel types of mesoscopic defects - discontinuity lines for the second derivative and branch points - terminating defects for these line discontinuities. These higher-order defects move "easily", and thus confer a great degree of flexibility to thin hyperbolic elastic sheets. We develop a general, higher-order, continuum mechanical framework from which we can derive the dynamics of higher order defects in a thermodynamically consistent manner. We illustrate our framework by obtaining the explicit equations for the dynamics of branch points in an elastic body.

cond-mat.soft↗
arXiv2018

Pattern Universes

In this essay we explore analogies between macroscopic patterns, which result from a sequence of phase transitions/instabilities starting from a homogeneous state, and similar phenomena in cosmology, where a sequence of phase transitions in the early universe is believed to have separated the fundamental forces from each other, and also shaped the structure and distribution of matter in the universe. We discuss three distinct aspects of this analogy: (i) Defects and topological charges in macroscopic patterns are analogous to spins and charges of quarks and leptons; (ii) Generic (3+1) stripe patterns carry an (energy) density that accounts for phenomena that are currently attributed to dark matter; (iii) Space-time patterns of interacting nonlinear waves display behaviors reminiscent of quantum phenomena including inflation, entanglement and dark energy.

nlin.PS↗
arXiv2017

Elastic sheets, phase surfaces and pattern universes

We connect the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns. In particular, we show parallels between asymptotic expansions for the energy of elastic surfaces in powers of the thickness $h$ and the free energy for almost periodic patterns expanded in powers of $ε$, the inverse aspect ratio of the pattern field. For sheets as well as patterns, the resulting energy can be expressed in terms of natural geometric invariants, the first and second fundamental forms of the elastic surface, respectively the phase surface. We discuss various results for these energies and also address some of the outstanding questions. We extend previous work on point (in 2D) and loop (in 3D) disclinations and connect their topological indices with the condensation of Gaussian curvature of the phase surface. Motivated by this connection with the charge and spin of pattern quarks and leptons, we lay out an ambitious program to build a multi-scale universe inspired by patterns in which the short (spatial and temporal) scales are given by a nearly periodic microstructure and whose macroscopic/slowly varying/averaged behaviors lead to a hierarchy of structures and features on much longer scales including analogues to quarks and leptons, dark matter, dark energy and inflationary cosmology. One of our new findings is an interpretation of dark matter as the energy density in a pattern field. The associated gravitational forces naturally result in galactic rotation curves that are consistent with observations, while simultaneously avoiding some of the small-scale difficulties of the standard $Λ$CDM paradigm in cosmology.

nlin.PS↗
arXiv2017

On the growth of a superlinear preferential attachment scheme

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as $n\uparrow\infty$, for the counts of the number of nodes with degree $k\geq 1$ at time $n\geq 1$. These limits rigorously verify and extend a physical picture of Krapivisky, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.

math.PR↗
arXiv2017

Mass Exchange Dynamics of Surface and Subsurface Oil in Shallow-Water Transport

We formulate a model for the mass exchange between oil at and below the sea surface. This is a particularly important aspect of modeling oil spills. Surface and subsurface oil have different chemical and transport characteristics and lumping them together would compromise the accuracy of the resulting model. Without observational or computational constraints, it is thus not possible to quantitatively predict oil spills based upon partial field observations of surface and/or sub-surface oil. The primary challenge in capturing the mass exchange is that the principal mechanisms are on the microscale. This is a serious barrier to developing practical models for oil spills that are capable of addressing questions regarding the fate of oil at the large spatio-temporal scales, as demanded by environmental questions. We use upscaling to propose an environmental-scale model which incorporates the mass exchange between surface and subsurface oil due to oil droplet dynamics, buoyancy effects, and sea surface and subsurface mechanics. While the mass exchange mechanism detailed here is generally applicable to oil transport models, it addresses the modeling needs of a particular to an oil spill model [1]. This transport model is designed to capture oil spills at very large spatio-temporal scales. It accomplishes this goal by specializing to shallow-water environments, in which depth averaging is a perfectly good approximation for the flow, while at the same time retaining mass conservation of oil over the whole oceanic domain.

physics.ao-ph↗
arXiv2017

Dimension reduction for systems with slow relaxation

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize `optimal' model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.

cond-mat.stat-mech↗
arXiv2016

Stochastic Longshore Current Dynamics

We develop a stochastic parametrization, based on a `simple' deterministic model for the dynamics of steady longshore currents, that produces ensembles that are statistically consistent with field observations of these currents. Unlike deterministic models, stochastic parameterization incorporates randomness and hence can only match the observations in a statistical sense. Unlike statistical emulators, in which the model is tuned to the statistical structure of the observation, stochastic parametrization are not directly tuned to match the statistics of the observations. Rather, stochastic parameterization combines deterministic, i.e physics based models with stochastic models for the "missing physics" to create hybrid models, that are stochastic, but yet can be used for making predictions, especially in the context of data assimilation. We introduce a novel measure of the utility of stochastic models of complex processes, that we call {\em consistency of sensitivity}. We show, in the context of data assimilation, the stochastic parametrization of longshore currents achieves good results in capturing the statistics of observation {\em that were not used} in tuning the model.

physics.data-an↗
arXiv2016

Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key role of branch-point (or "monkey-saddle") singularities in generating complex wrinkling patterns in isometric immersions, and show how they arise naturally from minimizing the elastic energy.

cond-mat.soft↗