Shankar C Venkataramani

Interests

Research

PDEs and calculus of variations, geometry, probability, statistical physics and applications to data science.

Teaching

Analysis and its applications, PDEs, Dynamical systems, Differential geometry, Mathematical Physics, Probability

Courses

Scholarly Contributions

Chapters

• Sethuraman, S., & Venkataramani, S. C. (2019). On the Growth of a Superlinear Preferential Attachment Scheme.. In Probability and Analysis in Interacting Physical Systems(pp 243-265). Cham: Springer. doi:https://doi.org/10.1007/978-3-030-15338-0_9
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  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, among other results, we establish law of large numbers type limits in probability, as n\to \infty$, for the counts Z_k(n) of the number of nodes with degree k\geq 1 at time n\geq 1. These limits rigorously verify a physical picture of Krapivisky, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.
• Sethuraman, S., Sethuraman, S., & Venkataramani, S. C. (2016). On the Growth of a Superlinear Preferential Attachment Scheme. In International Conference in Honor of the 75th Birthday of SRS Varadhan. Springer New York LLC. doi:10.1007/978-3-030-15338-0_9
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  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 \(k\ge 1\) at time \(n\ge 1\). These limits rigorously verify and extend a physical picture of Krapivisky et al. (Phys Rev Lett 85:4629–4632, 2000 [16]) on how the condensation arises with respect to the degree distribution.
• Ott, E., Sommerer, J. C., Antonsen, T. M., & Venkataramani, S. C. (1995). Blowout bifurcations: Symmetry breaking of spatially symmetric chaotic states. In Lévy Flights and Related Topics in Physics. Springer, Berlin, Heidelberg. doi:10.1007/3-540-59222-9_34
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  We consider a dynamical system that possesses a smooth manifold M on which the dynamics is chaotic. This general situation occurs in spatially symmetric extended systems that exhibit chaotic behavior of a spatially symmetric pattern. (Motion on M corresponds to spatial symmetry of the pattern.) Loss of stability transverse to M has been called a blowout bifurcation and corresponds to spatial symmetry breaking. It is shown that a blowout bifurcation is typically associated with either one of two novel types of dynamical behavior: (a) “on-off-intermittency” (in which short bursts of motion far from M occur between epoches where the orbit is exceedingly close to M), or (b) “riddling” of the basin of attraction of an attractor on (in which all points in the basin of the attractor on M have arbitrarily nearby points in the basin of another attractor not on M).

Journals/Publications

• Venkataramani, S. C. (2022). An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects. Discrete and Continuous Dynamical Systems B, 15(9), 2719-2746. doi:10.3934/dcdss.2022101
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  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. [Journal_ref: ]
• Doehrman, T., Sethuraman, S., & Venkataramani, S. C. (2021). Remarks on the range and multiple range of a random walk up to the time of exit. Rocky Mountain Journal of Mathematics, 51(5), 1603 -- 1614.
• Venkataramani, S. C. (2021). Computing with non-orientable defects: nematics, smectics and natural patterns. Physica D. doi:10.1016/j.physd.2020.132828
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  Defects are a ubiquitous feature of ordered media. They have certainuniversal features, independent of the underlying physical system, reflectingtheir topological origins. While the topological properties of defects arerobust, they appear as `unphysical' singularities, with non-integrable energydensities in coarse-grained macroscopic models. We develop a principledapproach for enriching coarse-grained theories with enough of the`micro-physics' to obtain thermodynamically consistent, well-set models, thatallow for the investigations of dynamics and interactions of defects inextended systems. We also develop associated numerical methods that areapplicable to computing energy driven behaviors of defects across theamorphous-soft-crystalline materials spectrum. Our methods can handle orderparameters that have a head-tail symmetry, i.e. director fields, in systemswith a continuous translation symmetry, as in nematic liquid crystals, and insystems where the translation symmetry is broken, as in smectics and convectionpatterns. We illustrate our methods with explicit computations.[Journal_ref: ]
• Venkataramani, S. C., & Jaramillo, G. (2021). A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures. Multiscale Modeling & Simulation, 19(2), 886-920. doi:10.1137/19m1306907
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  The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valu...
• Venkataramani, S. C., & Newell, A. C. (2021). Disk galaxies and their dark halos as self-organized patterns. Physics Letters B, 813, 136060. doi:10.1016/j.physletb.2020.136060
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  Abstract Galaxies are built by complex physical processes with significant inherent stochasticity. It is therefore surprising that the inferred dark matter distributions in galaxies are 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. The model applies to rotation supported systems and, for them, we compute galactic rotation curves, obtain a radial acceleration relation with two branches, and deduce the Freeman limit for central surface brightness.
• Venkataramani, S. C., & Newell, A. C. (2021). Pattern dark matter and galaxy scaling relations. The European Physical Journal Special Topics, 230(9), 2139-2165.
• Venkataramani, S. C., & Shearman, T. L. (2021). Distributed Branch Points and the Shape of Elastic Surfaces with Constant Negative Curvature. Journal of Nonlinear Science, 31(1). doi:10.1007/s00332-020-09657-2
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  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 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.
• Yamamoto, K. K., Venkataramani, S. C., Struckmeyer, E. J., Shearman, T. L., & Gemmer, J. A. (2021). Nature's forms are frilly, flexible, and functional.. The European physical journal. E, Soft matter, 44(7), 95. doi:10.1140/epje/s10189-021-00099-6
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  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.
• Acharya, A., & Venkataramani, S. C. (2020). Mechanics of moving defects in growing sheets: 3-d, small deformation theory.. Materials Theory. doi:10.1186/s41313-020-00018-w
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  Growth processes in many living organisms create thin, soft materials with anintrinsically hyperbolic geometry. These objects support novel types ofmesoscopic defects - discontinuity lines for the second derivative and branchpoints - terminating defects for these line discontinuities. These higher-orderdefects move "easily", and thus confer a great degree of flexibility to thinhyperbolic elastic sheets. We develop a general, higher-order, continuummechanical framework from which we can derive the dynamics of higher orderdefects in a thermodynamically consistent manner. We illustrate our frameworkby obtaining the explicit equations for the dynamics of branch points in anelastic body.[Journal_ref: ]
• Newell, A. C., & Venkataramani, S. C. (2019). Pattern Universes. Comptes Rendus Mecanique, 347(4), 318-331.
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  In this essay we explore analogies between macroscopic patterns, which resultfrom a sequence of phase transitions/instabilities starting from a homogeneousstate, and similar phenomena in cosmology, where a sequence of phasetransitions in the early universe is believed to have separated the fundamentalforces from each other, and also shaped the structure and distribution ofmatter in the universe. We discuss three distinct aspects of this analogy: (i)Defects and topological charges in macroscopic patterns are analogous to spinsand charges of quarks and leptons; (ii) Generic (3+1) stripe patterns carry an(energy) density that accounts for phenomena that are currently attributed todark matter; (iii) Space-time patterns of interacting nonlinear waves displaybehaviors reminiscent of quantum phenomena including inflation, entanglementand dark energy.[Journal_ref: ]
• Venkataramani, S. C. (2019). Buckling sheets open a door to understanding self-organization in soft matter. Proceedings of the National Academy of Sciences of the United States of America, 116(5), 1477-1479. doi:10.1073/pnas.1820937116
• Yamamoto, K. K., & Venkataramani, S. C. (2019). On the unusual swimming gaits of sea-slugs.. Bulletin of the American Physical Society, 2019.
• Ram'irez, J. M., Moghimi, S., Restrepo, J. M., & Venkataramani, S. C. (2018). Modelling the mass exchange dynamics of oceanic surface and subsurface oil. Ocean Modeling, 129(1), 1-12. doi:https://doi.org/10.1016/j.ocemod.2018.06.004
• Ramirez, J., Moghimi, S., Restrepo, J. M., & Venkataramani, S. C. (2018). Mass Exchange Dynamics of Surface and Subsurface Oil in Shallow-Water Transport. Ocean Modeling, 129, 1-12. doi:https://doi.org/10.1016/j.ocemod.2018.06.004
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  We formulate a model for the mass exchange between oil at and below the seasurface. This is a particularly important aspect of modeling oil spills.Surface and subsurface oil have different chemical and transportcharacteristics and lumping them together would compromise the accuracy of theresulting model. Without observational or computational constraints, it is thusnot possible to quantitatively predict oil spills based upon partial fieldobservations of surface and/or sub-surface oil. The primary challenge incapturing the mass exchange is that the principal mechanisms are on themicroscale. This is a serious barrier to developing practical models for oilspills that are capable of addressing questions regarding the fate of oil atthe large spatio-temporal scales, as demanded by environmental questions. Weuse upscaling to propose an environmental-scale model which incorporates themass exchange between surface and subsurface oil due to oil droplet dynamics,buoyancy effects, and sea surface and subsurface mechanics. While the massexchange mechanism detailed here is generally applicable to oil transportmodels, 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 largespatio-temporal scales. It accomplishes this goal by specializing toshallow-water environments, in which depth averaging is a perfectly goodapproximation for the flow, while at the same time retaining mass conservationof oil over the whole oceanic domain.[Journal_ref: ]
• Restrepo, J. M., Venkataramani, S. C., Restrepo, J. M., & Dawson, C. (2018). Nearshore Sticky Waters. Bulletin of the American Physical Society, 2018.
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  Abstract Wind- and current-driven flotsam, oil spills, pollutants, and nutrients, approaching the nearshore will frequently appear to slow down/park just beyond the break zone, where waves break. Moreover, the portion of these tracers that beach will do so only after a long time. Explaining why these tracers park and at what rate they reach the shore has important implications on a variety of different nearshore environmental issues, including the determination of what subscale processes are essential in computer models for the simulation of pollutant transport in the nearshore. Using a simple model we provide an explanation for the underlying mechanism responsible for the parking of tracers, not subject to inertial effects, the role played by the bottom topography, and the non-uniform dispersion which leads, in some circumstances, to the eventual landing of all or a portion of the tracers. We refer to the parking phenomenon in this environment as nearshore sticky waters.
• Venkataramani, S. C. (2018). Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling. Nonlinearity, 31(9), 4162. doi:10.1088/1361-6544/aac9a6
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  We analyze the effect of adding a weak, localized, inhomogeneity to a twodimensional array of oscillators with nonlocal coupling. We propose and alsojustify a model for the phase dynamics in this system. Our model is ageneralization of a viscous eikonal equation that is known to describe thephase modulation of traveling waves in reaction-diffusion systems. We show theexistence of a branch of target pattern solutions that bifurcates from thespatially homogeneous state when $\varepsilon$, the strength of theinhomogeneity, is nonzero and we also show that these target patterns have anasymptotic wavenumber that is small beyond all orders in $\varepsilon$. The strategy of our proof is to pose a good ansatz for an approximate form ofthe solution and use the implicit function theorem to prove the existence of asolution in its vicinity. The analysis presents two challenges. First, thelinearization about the homogeneous state is a convolution operator ofdiffusive type and hence not invertible on the usual Sobolev spaces. Second, aregular perturbation expansion in $\varepsilon$ does not provide a good ansatzfor applying the implicit function theorem since the nonlinearities play amajor role in determining the relevant approximation, which also needs to be"correct" to all orders in $\varepsilon$. We overcome these two points byproving Fredholm properties for the linearization in appropriate Kondratievspaces and using a refined ansatz for the approximate solution, which obtainedusing matched asymptotics.[Journal_ref: Nonlinearity, vol. 31, no. 9, pg. 4162 (2018)]
• Newell, A. C., & Venkataramani, S. C. (2016). Elastic sheets, phase surfaces and pattern universes. Studies in Applied Mathematics. doi:10.1111/sapm.12184
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  We connect the theories of the deformation of elastic surfaces and phasesurfaces arising in the description of almost periodic patterns. In particular,we show striking parallels between expansions for the energy of elasticsurfaces in powers of the thickness $h$ and the free energy averaged over aperiod of an almost periodic pattern expanded in powers of $\epsilon$, theinverse aspect ratio of the pattern field. In both cases, the resulting energycan be expressed in terms of the first and second fundamental forms of thesurfaces involved, the elastic surface in the former case and the phase surfacein the latter. We discuss various results that are obtained by exploiting thisanalogy and also address some of the outstanding questions. One common resultof particular interest concerns the condensation of the Gaussian curvature ontoisolated point defects in two dimensions and onto loop filaments in threedimensions. We also lay out an ambitious and somewhat speculative program to build amulti-scale model of the universe inspired by patterns, in which the short(spatial and temporal) scale structure on the Planck scales is given by anearly periodic microstructure, and macroscopic/slowly varying/averagedbehaviors on scales much larger than the Planck scale leads to a hierarchy ofstructures and features including analogues of quarks, leptons, dark matter,dark energy and inflationary cosmology.[Journal_ref: ]
• Newell, A. C., & Venkataramani, S. C. (2017). Elastic Sheets, Phase Surfaces, and Pattern Universes. STUDIES IN APPLIED MATHEMATICS, 139(2), 322-368.
• Rosenthal, W. S., Venkataramani, S. C., Mariano, A., & Restrepo, J. M. (2016). Displacement Data Assimilation. J. Comp. Phys..
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  We show that modifying a Bayesian data assimilation scheme by incorporatingkinematically-consistent displacement corrections produces a scheme that isdemonstrably better at estimating partially observed state vectors in a settingwhere feature information important. While the displacement transformation isnot tied to any particular assimilation scheme, here we implement it within anensemble Kalman Filter and demonstrate its effectiveness in trackingstochastically perturbed vortices.[Journal_ref: ]
• Rosenthal, W. S., Venkataramani, S., Mariano, A. J., & Restrepo, J. M. (2017). Displacement data assimilation. JOURNAL OF COMPUTATIONAL PHYSICS, 330, 594-614.
• Venkataramani, S. C., Venkataramani, R. C., & Restrepo, J. M. (2016). Dimension reduction for systems with slow relaxation. J. Stat Phys..
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  We develop reduced, stochastic models for high dimensional, dissipativedynamical systems that relax very slowly to equilibrium and can encode longterm memory. We present a variety of empirical and first principles approachesfor model reduction, and build a mathematical framework for analyzing thereduced models. We introduce the notions of universal and asymptotic filters tocharacterize `optimal' model reductions for sloppy linear models. We illustrateour methods by applying them to the practically important problem of modelingevaporation in oil spills.[Journal_ref: ]
• Venkataramani, S. C., Venkataramani, R. C., & Restrepo, J. M. (2017). Dimension Reduction for Systems with Slow Relaxation. JOURNAL OF STATISTICAL PHYSICS, 167(3-4), 892-933.
• Choi, J., Sethuraman, S., & Venkataramani, S. C. (2016). A scaling limit for the degree distribution in sublinear preferential attachment schemes. RANDOM STRUCTURES & ALGORITHMS, 48(4), 703-731.
• Gemmer, J., Sharon, E., Shearman, T., & Venkataramani, S. C. (2016). Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets. EPL, 114(2).
• Restrepo, J. M., & Venkataramani, S. (2016). Stochastic longshore current dynamics. ADVANCES IN WATER RESOURCES, 98, 186-197.
• Venkataramani, S. C. (2016). Stochastic Longshore Current Dynamics. Advances in Water Resources, 98, 186-197. doi:10.1016/j.advwatres.2016.11.002
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  We develop a stochastic parametrization, based on a `simple' deterministicmodel for the dynamics of steady longshore currents, that produces ensemblesthat are statistically consistent with field observations of these currents.Unlike deterministic models, stochastic parameterization incorporatesrandomness and hence can only match the observations in a statistical sense.Unlike statistical emulators, in which the model is tuned to the statisticalstructure of the observation, stochastic parametrization are not directly tunedto match the statistics of the observations. Rather, stochasticparameterization combines deterministic, i.e physics based models withstochastic models for the "missing physics" to create hybrid models, that arestochastic, but yet can be used for making predictions, especially in thecontext of data assimilation. We introduce a novel measure of the utility ofstochastic models of complex processes, that we call {\em consistency ofsensitivity}. We show, in the context of data assimilation, the stochasticparametrization of longshore currents achieves good results in capturing thestatistics of observation {\em that were not used} in tuning the model.[Journal_ref: Advances in Water Resources, Vol. 98, December 2016, pp. 186-197]
• Restrepo, J. M., Ramirez, J. M., & Venkataramani, S. C. (2015). An Oil Fate Model for Shallow-Waters. J. Mar. Sci. Eng., 3(4), 1504-1543. doi:doi:10.3390/jmse3041504
• Gemmer, J. A., Venkataramani, S. C., Durfee, C. G., & Moloney, J. V. (2014). Optical beam shaping and diffraction free waves: A variational approach. PHYSICA D-NONLINEAR PHENOMENA, 283, 15-28.
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  We investigate the problem of shaping radially symmetric annular beams into desired intensity patterns along the optical axis. Within the Fresnel approximation, we show that this problem can be expressed in a variational form equivalent to the one arising in phase retrieval. Using the uncertainty principle we prove various rigorous lower bounds on the functional; these lower bounds estimate the L-2 error for the beam shaping problem in terms of the design parameters. We also use the method of stationary phase to construct a natural ansatz for a minimizer in the short wavelength limit. We illustrate the implications of our results by applying the method of stationary phase coupled with the Gerchberg-Saxton algorithm to beam shaping problems arising in the remote delivery of beams and pulses. (C) 2014 Elsevier B.V. All rights reserved.
• Kent, S., & Venkataramani, S. C. (2014). Sharp interfaces in two-dimensional free boundary problems: Interface calculation via matched conformal maps. PHYSICAL REVIEW E, 90(1).
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  We use conformal maps to study a free boundary problem for a two-fluid electromechanical system, where the interface between the fluids is determined by the combined effects of electrostatic forces, gravity, and surface tension. The free boundary in our system develops sharp corners or singularities in certain parameter regimes, and this is an impediment to using existing "single-scale" numerical conformal mapping methods. The difficulty is due to the phenomenon of crowding, i.e., the tendency of nodes in the preimage plane to concentrate near the sharp regions of the boundary, leaving the smooth regions of the boundary poorly resolved. A natural idea is to exploit the scale separation between the sharp regions and smooth regions to solve for each region separately and then stitch the solutions together. However, this is not straightforward as conformal maps are rigid "global" objects, and it is not obvious how one would patch two conformal maps together to obtain a new conformal map. We develop a "multiscale" (i.e., adaptive) conformal mapping method that allows us to carry out this program of stitching conformalmaps on different scales together. We successfully apply our method to the electromechanical model problem.
• Kent, S., & Venkataramani, S. C. (2014). Sharp interfaces in two-dimensional free boundary problems: interface calculation via matched conformal maps. Physical review. E, Statistical, nonlinear, and soft matter physics, 90(1), 012407.
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  We use conformal maps to study a free boundary problem for a two-fluid electromechanical system, where the interface between the fluids is determined by the combined effects of electrostatic forces, gravity, and surface tension. The free boundary in our system develops sharp corners or singularities in certain parameter regimes, and this is an impediment to using existing "single-scale" numerical conformal mapping methods. The difficulty is due to the phenomenon of crowding, i.e., the tendency of nodes in the preimage plane to concentrate near the sharp regions of the boundary, leaving the smooth regions of the boundary poorly resolved. A natural idea is to exploit the scale separation between the sharp regions and smooth regions to solve for each region separately and then stitch the solutions together. However, this is not straightforward as conformal maps are rigid "global" objects, and it is not obvious how one would patch two conformal maps together to obtain a new conformal map. We develop a "multiscale" (i.e., adaptive) conformal mapping method that allows us to carry out this program of stitching conformal maps on different scales together. We successfully apply our method to the electromechanical model problem.
• Restrepo, J. M., Flaschka, H., Venkataramani, S. C., Restrepo, J. M., Flaschka, H., & Comeau, D. (2014). How can you tell whether Earth is warming Up. Bulletin of the American Physical Society, 2014.
• Restrepo, J. M., Venkataramani, S. C., & Dawson, C. (2014). Nearshore sticky waters. OCEAN MODELLING, 80, 49-58.
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  Wind- and current-driven flotsam, oil spills, pollutants, and nutrients, approaching the nearshore will frequently appear to slow down/park just beyond the break zone, where waves break. Moreover, the portion of these tracers that beach will do so only after a longtime. Explaining why these tracers park and at what rate they reach the shore has important implications on a variety of different nearshore environmental issues, including the determination of what subscale processes are essential in computer models for the simulation of pollutant transport in the nearshore. Using a simple model we provide an explanation for the underlying mechanism responsible for the parking of tracers, not subject to inertial effects, the role played by the bottom topography, and the non-uniform dispersion which leads, in some circumstances, to the eventual landing of all or a portion of the tracers. We refer to the parking phenomenon in this environment as nearshore sticky waters. (C) 2014 Elsevier Ltd. All rights reserved.
• Restrepo, J. M., Venkataramani, S., Comeau, D., & Flaschka, H. (2014). Defining a trend for time series using the intrinsic time-scale decomposition. NEW JOURNAL OF PHYSICS, 16.
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  We propose criteria that define a trend for time series with inherent multi-scale features. We call this trend the tendency of a time series. The tendency is defined empirically by a set of criteria and captures the large-scale temporal variability of the original signal as well as the most frequent events in its histogram. Among other properties, the tendency has a variance no larger than that of the original signal; the histogram of the difference between the original signal and the tendency is as symmetric as possible; and with reduced complexity, the tendency captures essential features of the signal. To find the tendency we first use the intrinsic time-scale decomposition (ITD) of the signal, introduced in 2007 by Frei and Osorio, to produce a set of candidate tendencies. We then apply the criteria to each of the candidates to single out the one that best agrees with them. While the criteria for the tendency are independent of the signal decomposition scheme, it is found that the ITD is a simple and stable methodology, well suited for multi-scale signals. The ITD is a relatively new decomposition and little is known about its outcomes. In this study we take the first steps towards a probabilistic model of the ITD analysis of random time series. This analysis yields details concerning the universality and scaling properties of the components of the decomposition.
• Özgökmen, T. M., Beron-Vera, F. J., Bogucki, D., Chen, S. S., Dawson, C., Dewar, W., Griffa, A., Haus, B. K., Haza, A. C., Huntley, H., Iskandarani, M., Jacobs, G., Jagers, B., Kirwan, A. D., Laxague, N., Lipphardt, B., MacMahan, J., Mariano, A. J., Olascoaga, J., , Novelli, G., et al. (2014). Research Overview of the Consortium for Advanced Research on Transport of Hydrocarbon in the Environment (CARTHE). International Oil Spill Conference Proceedings, 2014(1), 544-560.
• Gemmer, J. A., & Venkataramani, S. C. (2013). Shape selection in non-Euclidean plates. PHYSICA D-NONLINEAR PHENOMENA, 240(19), 1536-1552.
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  We investigate isometric immersions of disks with constant negative curvature into R-3, and the minimizers for the bending energy, i.e. the L-2 norm of the principal curvatures over the class of W-2.2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H-2 into R-3. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L-infinity norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W-2.2, and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disk. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates. (C) 2011 Elsevier B.V. All rights reserved.
• Gemmer, J., & Venkataramani, S. C. (2013). Shape transitions in hyperbolic non-Euclidean plates. Soft Matter, 9(34), 8151-8161.
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  Abstract: A non-Euclidean plate is a thin elastic object whose intrinsic geometry is not flat and hence has residual stresses arising from being embedded in three dimensional space. Recently, there has been interest in using localized swelling to induce residual stresses that shape flat objects into desired three dimensional structures. A fundamental question is whether we can use the mathematical theory of non-Euclidean plates to deduce the three dimensional configuration of the swelling sheet given the exact knowledge of the imposed geometry. We present and summarize the results of recent mathematical studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both annular and disc geometries. We show in the Föppl-von Kármán approximation to the elastic energy there are only two types of global minimizers-flat and saddle shaped deformations-with localized regions of stretching near the boundary of the domain. We also show that there exist n-wave local minimizers that closely resemble experimental observations and have additional regions of stretching near lines of inflection. Furthermore, in the Kirchhoff approximation to the elastic energy, we show that there exists exact isometric immersions with periodic profiles. The number of waves in these configurations is set by the condition that the bending energy remains finite and grows approximately exponentially with the radius of the annulus. For large radii, these shape are energetically favorable over saddle shapes and could explain why wavy shapes are selected in crochet models of the hyperbolic plane. The predicted morphologies however differ from what is observed in experiments on hydrogel disks highlighting the need for further theoretical studies. © 2013 The Royal Society of Chemistry.
• Graf, T., Moloney, J., & Venkataramani, S. (2013). Asymptotic analysis of weakly nonlinear Bessel-Gauß beams. Physica D: Nonlinear Phenomena, 243(1), 32-44.
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  Abstract: In this paper we investigate the propagation of conical waves in nonlinear media. In particular, we are interested in the effects resulting from applying a Gaussian apodization to an ideal nondiffracting wave. First, we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear Schrödinger type. From these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution. Moreover, we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results. © 2012 Elsevier B.V. All rights reserved.
• Klein, Y., Venkataramani, S., & Sharon, E. (2013). Experimental Study of Shape Transitions and Energy Scaling in Thin Non-Euclidean Plates. PHYSICAL REVIEW LETTERS, 106(11).
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  We present the first quantitative measurements of shape and energy variation in non-Euclidean plates. Using environmentally responsive gel, we construct non-Euclidean disks of constant imposed Gaussian curvature, K(tar). We vary the disks' thickness t(0) and measure the dependence of configurations, surface curvature, and energy content on t(0). For K(tar) < 0, configurations are of a single wavy mode and undergo a set of bifurcations that leads to their refinement with decreasing thickness. This leads to sharp increase in the amount of surface bending as t(0) -> 0, and to a slow decay of both bending and stretching energies. Both vary like t(0)(2), compared with t(0)(3) of the bending energy in disks with K(tar) > 0.
• Gemmer, J. A., & Venkataramani, S. C. (2012). Defects and boundary layers in non-Euclidean plates. Nonlinearity, 25(12), 3553-3581.
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  Abstract: We investigate the behaviour of non-Euclidean plates with constant negative Gaussian curvature using the Föppl-von Kármán reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers - deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet. © 2012 IOP Publishing Ltd & London Mathematical Society.
• Graf, T., Christodoulides, D. N., Mills, M. S., Moloney, J. V., Venkataramani, S. C., & Wright, E. M. (2012). Propagation of Gaussian-apodized paraxial beams through first-order optical systems via complex coordinate transforms and ray transfer matrices. Journal of the Optical Society of America A: Optics and Image Science, and Vision, 29(9), 1860-1869.
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  PMID: 23201942;Abstract: We investigate the linear propagation of Gaussian-apodized solutions to the paraxial wave equation in free-space and first-order optical systems. In particular, we present complex coordinate transformations that yield a very general and efficient method to apply a Gaussian apodization (possibly with initial phase curvature) to a solution of the paraxial wave equation. Moreover, we show how this method can be extended from free space to describe propagation behavior through nonimaging first-order optical systems by combining our coordinate transform approach with ray transfer matrix methods. Our framework includes several classes of interesting beams that are important in applications as special cases. Among these are, for example, the Bessel-Gauss and the Airy-Gauss beams, which are of strong interest to researchers and practitioners in various fields. © 2012 Optical Society of America.
• Gemmer, J. A., & Venkataramani, S. C. (2011). Shape selection in non-Euclidean plates. Physica D: Nonlinear Phenomena, 240(19), 1536-1552.
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  Abstract: We investigate isometric immersions of disks with constant negative curvature into R3, and the minimizers for the bending energy, i.e. the L2 norm of the principal curvatures over the class of W2 ,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H2 into R3. In elucidating the connection between these immersions and the non-existence/ singularity results of Hilbert and Amsler, we obtain a lower bound for the L∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W2,2, and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disk. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates. © 2011 Elsevier B.V. All rights reserved.
• Graf, T., Moloney, J., & Venkataramani, S. (2011). Asymptotic analysis of weakly nonlinear Bessel-Gauss beams. PHYSICA D-NONLINEAR PHENOMENA, 243(1), 32-44.
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  In this paper we investigate the propagation of conical waves in nonlinear media. In particular, we are interested in the effects resulting from applying a Gaussian apodization to an ideal nondiffracting wave. First, we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear Schrodinger type. From these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution. Moreover, we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results. (c) 2012 Elsevier B.V. All rights reserved.
• Klein, Y., Venkataramani, S., & Sharon, E. (2011). Experimental study of shape transitions and energy scaling in thin non-Euclidean plates. Physical review letters, 106(11), 118303.
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  We present the first quantitative measurements of shape and energy variation in non-Euclidean plates. Using environmentally responsive gel, we construct non-Euclidean disks of constant imposed Gaussian curvature, K(tar). We vary the disks' thickness t(0) and measure the dependence of configurations, surface curvature, and energy content on t(0). For K(tar)0.
• Klein, Y., Venkataramani, S., & Sharon, E. (2011). Experimental study of shape transitions and energy scaling in thin non-euclidean plates. Physical Review Letters, 106(11).
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  Abstract: We present the first quantitative measurements of shape and energy variation in non-Euclidean plates. Using environmentally responsive gel, we construct non-Euclidean disks of constant imposed Gaussian curvature, K tar. We vary the disks' thickness t0 and measure the dependence of configurations, surface curvature, and energy content on t 0. For Ktar0. © 2011 American Physical Society.
• Venkataramani, S. C., & Kent, S. (2011). The 2D Selective Withdrawal Transition: Analogies with MEM Systems. Bulletin of the American Physical Society.
• Venkataramani, S. C., & Gemmer, J. (2010). Shape Selection in Non-Euclidean Plates. Bulletin of the American Physical Society, 2010.
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  Abstract We investigate isometric immersions of disks with constant negative curvature into R 3 , and the minimizers for the bending energy, i.e. the L 2 norm of the principal curvatures over the class of W 2 , 2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H 2 into R 3 . In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L ∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W 2 , 2 , and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disk. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates.
• Ercolani, N. M., Venkataramani, S. C., & Ercolani, N. M. (2009). A Variational Theory for Point Defects in Patterns. Journal of Nonlinear Science, 19(3), 267-300. doi:10.1007/s00332-008-9035-9
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  We derive a rigorous scaling law for minimizers in a natural version of the regularized Cross-Newell model for pattern formation far from threshold. These energy-minimizing solutions support defects having the same character as what is seen in experimental studies of the corresponding physical systems and in numerical simulations of the microscopic equations that describe these systems.
• Berkenbuseh, M. K., Claus, I., Dunn, C., Kadanoff, L. P., Nicewicz, M., & Venkataramani, S. C. (2004). Discrete charges on a two dimensional conductor. Journal of Statistical Physics, 116(5-6), 1301-1358.
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  Abstract: We investigate the electrostatic equilibria of N discrete charges of size 1/N on a two dimensional conductor (domain). We study the distribution of the charges on symmetric domains including the ellipse, the hypotrochoid and various regular polygons, with an emphasis on understanding the distributions of the charges, as the shape of the underlying conductor becomes singular. We find that there are two regimes of behavior, a symmetric regime for smooth conductors, and a symmetry broken regime for "singular" domains. For smooth conductors, the locations of the charges can be determined, to within O(√logN/N2) by an integral equation due to Pommerenke [Math. Ann., 179: 212-218, (1969)]. We present a derivation of a related (but different) integral equation, which has the same solutions. We also solve the equation to obtain (asymptotic) solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps. Conductors with sharp cusps and singularities show qualitatively different behavior, where the symmetry of the problem is broken, and the distribution of the discrete charges does not respect the symmetry of the underlying domain. We investigate the symmetry breaking both theoretically, and numerically, and find good agreement between our theory and the numerics. We also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime, although this distribution is very different from the one given by the integral equation. © 2004 Springer Science+Business Media, Inc.
• Venkataramani, S. C. (2004). Lower bounds for the energy in a crumpled elastic sheet - A minimal ridge. Nonlinearity, 17(1), 301-312.
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  Abstract: We study the linearized Föppl-von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle 2α. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness, σ, and the bending angle. We show that the minimum energy scales as σ5/3σ7/3. This scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions, where the energy scales as σ.
• DiDonna, B. A., Witten, T. A., Venkataramani, S. C., & Kramer, E. M. (2002). Singularities, structures, and scaling in deformed m-dimensional elastic manifolds. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 65(1), 016603/1-016603/25.
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  Abstract: The crumpling of a thin sheet can be understood as the condensation of elastic energy into a network of ridges that meet in vertices. Elastic energy condensation should occur in response to compressive strain in elastic objects of any dimension greater than 1. We study elastic energy condensation numerically in two-dimensional elastic sheets embedded in spatial dimensions three or four and three-dimensional elastic sheets embedded in spatial dimensions four and higher. We represent a sheet as a lattice of nodes with an appropriate energy functional to impart stretching and bending rigidity. Minimum energy configurations are found for several different sets of boundary conditions. We observe two distinct behaviors of local energy density falloff away from singular points, which we identify as cone scaling or ridge scaling. Using this analysis, we demonstrate that there are marked differences in the forms of energy condensation depending on the embedding dimension. © 2001 The American Physical Society.
• Gopinathan, A., Witten, T. A., & Venkataramani, S. C. (2002). Trapping of vibrational energy in crumpled sheets. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 65(3), 036613/1-036613/11.
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  Abstract: We investigate the propagation of transverse elastic waves in crumpled media. We. set up the wave equation for transverse waves on a generic curved, strained surface via a Langrangian formalism and use this to study the scaling behavior of the dispersion curves near the ridges and on the flat facets. This analysis suggests that ridges act as barriers to wave propagation and that modes in a certain frequency regime could be trapped in the facets. A simulation study of the wave propagation qualitatively supported our analysis and showed interesting effects of the ridges on wave propagation. © 2002 The American Physical Society.
• Kim, J. W., Vaishnav, J. Y., Ott, E., Venkataramani, S. C., & Losert, W. (2001). Front propagation of spatiotemporal chaos. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 64(1 II), 016215/1-016215/5.
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  Abstract: A continuum map model for the study of the dynamics of a front separating a region of spatiotemporal chaos from a stable steady region is presented. The model is applicable to periodically forced systems.
• Venkataramani, S. C., & Ott, E. (2001). Pattern selection in extended periodically forced systems: A continuum coupled map approach. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 63(4 II), 462021-4620227.
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  Abstract: The vibrated granular layers were investigated by continuum coupled maps (CCM). A CCM model was presented with the consideration of dimensional analysis and the symmetries underlying the systems. The analysis of pattern selection in CCM models was presented by a truncated model. A procedure for the analysis of these models was also suggested.
• Venkataramani, S. C., Witten, T. A., Kramer, E. M., & Geroch, R. P. (2000). Limitations on the smooth confinement of an unstretchable manifold. Journal of Mathematical Physics, 41(7), 5107-5128.
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  Abstract: We prove that an m-dimensional unit ball Dm in the Euclidean space Rm cannot be isometrically embedded into a higher-dimensional Euclidean ball Bdr⊂.ℝd of radius r
• Brenner, M. P., Constantin, P., Kadanoff, L. P., Schenkel, A., & Venkataramani, S. C. (1999). Diffusion, attraction and collapse. Nonlinearity, 12(4), 1071-1098.
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  Abstract: We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2 < d < 10 there are two stable blowup modalities (self-similar and Burgers-like) and one stable steady state. On unbounded domains, there exists a one-parameter family of unstable steady solutions and a countable number of unstable blowup behaviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections between one unstable blowup and two different stable blowups. There is a topological and stability correspondence between the various asymptotic behaviours and this suggests the possibility of constructing a global phase portrait for the system that treats the global in time solutions and the blowing up solutions on an equal footing.
• Povinelli, M. L., Coppersmith, S. N., Kadanoff, L. P., Nagel, S. R., & Venkataramani, S. C. (1999). Noise stabilization of self-organized memories. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 59(5 B), 4970-4982.
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  PMID: 11969451;Abstract: We investigate a nonlinear dynamical system which "remembers" preselected values of a system parameter. The deterministic version of the system can encode many parameter values during a transient period, but in the limit of long times, almost all of them are forgotten. Here we show that a certain type of stochastic noise can stabilize multiple memories, enabling many parameter values to be encoded permanently. We present analytic results that provide insight both into the memory formation and into the noise-induced memory stabilization. The relevance of our results to experiments on the charge-density wave material NbSe3 is discussed. ©1999 The American Physical Society.
• Venkataramani, S. (1998). Cones, creases and crumpled sheets. Physics World, 11(7), 19-20.
• Venkataramani, S. C., & Ott, E. (1998). Spatiotemporal bifurcation phenomena with temporal period doubling: Patterns in vibrated sand. Physical Review Letters, 80(16), 3495-3498.
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  Abstract: This paper examines the consequences of the interaction between temporal period doubling and spatial pattern formation. We propose a simple discrete time, spatially continuous system, where the discrete time dynamics incorporates period doubling and the spatial operator imposes patterning at a preferred length scale. We find that this model displays a variety of bifurcations between different spatiotemporal states, and these bifurcations are generic in that they do not depend on the details of the model. The results from our simple model bear remarkable similarities with recent experiments on a vertically vibrated granular layer.
• Venkataramani, S. C., Antonsen Jr., T. M., & Ott, E. (1998). Anomalous diffusion in bounded temporally irregular flows. Physica D: Nonlinear Phenomena, 112(3-4), 412-440.
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  Abstract: We investigate the longitudinal dispersion of passive tracers in a transversely bounded shear flow. The flow is assumed to be temporally irregular, but to possess smooth large scale spatial structure without significant small spatial scale motions. We show that the existence of a region where the longitudinal velocity goes to zero linearly with distance from the wall causes the tracers to "stick" to the walls. This results in a superdiffusive dispersion of the tracer. We analyze the growth of the variance and show that σ2(t) ∼ tv with v = 3/2. We verify this result by numerical simulation and compare the results from this model with experimental observations. Copyright © 1998 Elsevier Science B.V. All rights reserved.
• Coppersmith, S. N., Jones, T. C., Kadanoff, L. P., Levine, A., McCarten, J. P., Nagel, S. R., Venkataramani, S. C., & Xinlei, W. u. (1997). Self-organized short-term memories. Physical Review Letters, 78(21), 3983-3986.
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  Abstract: We report short-term memory formation in a nonlinear dynamical system with many degrees of freedom. The system r"emembers" a sequence of impulses for a transient period, but it coarsens and eventually "forgets" nearly all of them. The memory duration increases as the number of degrees of freedom in the system increases. We demonstrate the existence of these transient memories in a laboratory experiment.
• Venkataramani, S. C., Antonsen Jr., T. M., & Ott, E. (1997). Lévy flights in fluid flows with no kolmogorov-arnold-moser surfaces. Physical Review Letters, 78(20), 3864-3867.
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  Abstract: We investigate the Lévy flights observed experimentally in the transport of tracers in a temporally irregular flow that has no Kolmogorov-Arnold-Moser surfaces, and show that the Lévy flights are due to the sticking of the tracers near the walls. The tracer is found to spread superdiffusively with an exponent ν = 3/2, in reasonable agreement with the experiments.
• Venkataramani, S. C., Levine, A. J., Wu, X., Venkataramani, S. C., Nagel, S. R., Mccarten, J. P., Levine, A. J., Kadanoff, L. P., Jones, T. C., & Coppersmith, S. N. (1997). SELF-ORGANIZED SHORT-TERM MEMORIES. Physical Review Letters, 78(21), 3983-3986. doi:10.1103/physrevlett.78.3983
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  We report short-term memory formation in a nonlinear dynamical system with many degrees of freedom. The system ``remembers'' a sequence of impulses for a transient period, but it coarsens and eventually ``forgets'' nearly all of them. The memory duration increases as the number of degrees of freedom in the system increases. We demonstrate the existence of these transient memories in a laboratory experiment.
• Lai, Y., Grebogi, C., Yorke, J. A., & Venkataramani, S. C. (1996). Riddling bifurcation in chaotic dynamical systems. Physical Review Letters, 77(1), 55-58.
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  Abstract: When a chaotic attractor lies in an invariant subspace, as in systems with symmetry, riddling can occur. Riddling refers to the situation where the basin of a chaotic attractor is riddled with holes that belong to the basin of another attractor. We establish properties of the riddling bifurcation that occurs when an unstable periodic orbit embedded in the chaotic attractor, usually of low period, becomes transversely unstable. An immediate physical consequence of the riddling bifurcation is that an extraordinarily low fraction of the trajectories in the invariant subspace diverge when there is a symmetry breaking.
• Venkataramani, S. C., Antonsen Jr., T. M., Ott, E., & Sommerer, J. C. (1996). On-off intermittency: Power spectrum and fractal properties of time series. Physica D: Nonlinear Phenomena, 96(1-4), 66-99.
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  Abstract: Some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic. This situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators. The invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow-out value. This could result in an extreme form of temporally intermittent bursting called on-off intermittency. We propose a model that incorporates the universal features of systems that display on-off intermittency. We study this model both with and without additive noise and we derive scaling results for the power spectral density of the on-off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold. We then present numerical simulations realizing these results. Copyright © 1996 Elsevier Science B.V. All rights reserved.
• Venkataramani, S. C., Hunt, B. R., & Ott, E. (1996). Bubbling transition. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 54(2), 1346-1360.
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  Abstract: Recently, physically important examples of dynamical systems that have a chaotic attractor embedded in an invariant submanifold have been pointed out, and the unusual dynamical consequences of this situation have been studied. As a parameter 6 of the system is increased, a periodic orbit embedded in the attractor on the invariant manifold can become unstable for perturbations transverse to the invariant manifold. This bifurcation is called the bubbling transition, and it can lead to the occurrence of a recently discovered, new kind of basin of attraction, called a riddled basin. In this paper we study the effects of noise and asymmetry on the bubbling transition. We find that, in the presence of noise or asymmetry, the attractor is replaced either by a chaotic transient or an intermittently bursting time evolution, and we derive scaling relations, valid near the bubbling transition, for the characteristic time (i.e., the average chaotic transient lifetime or the average interburst time interval) as a function of the strength of the asymmetry and the variance of the additive noise. We also present numerical evidence for the predicted scalings.
• Venkataramani, S. C., Hunt, B. R., Ott, E., Gauthier, D. J., & Bienfang, J. C. (1996). Transitions to bubbling of chaotic systems. Physical Review Letters, 77(27), 5361-5364.
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  Abstract: Certain dynamical systems exhibit a phenomenon called bubbling, whereby small perturbations induce intermittent bursting. In this Letter we show that, as a parameter is varied through a critical value, the transition to bubbling can be "hard" (the bursts appear abruptly with large amplitude) or "soft" (the maximum burst amplitude increases continuously from zero), and that the presence or absence of symmetry in the unperturbed system has a fundamental effect on these transitions. These results are confirmed by numerical and physical experiments. © 1996 The American Physical Society.
• Jacobson, T., & Venkataramani, S. (1995). Topology of event horizons and topological censorship. Classical and Quantum Gravity, 12(4), 1055-1061.
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  Abstract: We prove that, under certain conditions, the topology of the event horizon of a four-dimensional asymptotically flat black-hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole. More precisely, let M be a four-dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication to the future of a cut K of is globally hyperbolic. Suppose further that a Cauchy surface for is a topological 3-manifold with compact boundary in M, and is a compact submanifold of with spherical boundary in (and possibly other boundary components in ). Then we prove that the homology group must be finite. This implies that either consists of a disjoint union of 2-spheres, or is non-orientable and contains a projective plane. Furthermore, , and will be a cross section of the horizon as long as no generator of becomes a generator of . In this case, if is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.
• Venkataramani, S. C., Antonsen Jr., T. M., Ott, E., & Sommerer, J. C. (1995). Characterization of on-off intermittent time series. Physics Letters A, 207(3-4), 173-179.
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  Abstract: We outline results from a model that incorporates the universal features of systems that display on-off intermittency, and present numerical experiments realizing these results. The particular results investigated are the scaling of the power spectral density and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold. We also report the effect of additive noise on these results. © 1995.

Proceedings Publications

• Venkataramani, S. C., & Ott, E. (2001). Continuum Coupled Maps: a Model for Patterns in Vibrated Sand. In Proceedings of the 5th Experimental Chaos Conference, 143-153.

Presentations

• Venkataramani, S. C. (2021, July). Hyperbolic Monge-Ampere equations, Lorentz surfaces and thin sheet elasticity.. SIAM Materials Science Conference (SIAM MS21). Bilbao, Spain/Online: Society for Industrial and Applied Mathematics.
• Venkataramani, S. C. (2021, June). On branch points and C^{1,1} pseudospherical immersions. Online seminar in Geometric Analysis. Zoom: U. Pittsburgh, U. Freiburg, U. Salzburg and TU Chemnitz.
• Venkataramani, S. C. (2021, March). On leaves, flowers, and sea slugs. APS March meeting. Nashville/Zoom: American Physical Society.
• Venkataramani, S. C. (2021, May). Higher-Order Field theories, Constraints and the Ostrogradsky instability.. SIAM Dynamical Systems DS21. Online: Society for Industrial and Applied Mathematics.
• Venkataramani, S. C. (2020, April). Geometry and Topology in the "real world".. Mathematics Colloquium. Dallas TX/Zoom: Southern Methodist University.
• Venkataramani, S. C. (2020, February). On leaves, flowers and sea-slugs. Aspen Winter conference on Low dimensional solids in hard and soft condensed matter. Aspen, CO.: Aspen Center for Physics.
• Venkataramani, S. C. (2020, October). Integrability on Sea-slugs. Utah-Arizona Math Physics seminar. Tucson AZ/Zoom: UA/U. Utah.
• Venkataramani, S. C. (2019, December). Phase Extraction and Defect Dynamics in Convection Patterns. SIAM Analysis of PDE conference. La Quinta, CA: Society for Industrial and Applied Mathematics.
• Venkataramani, S. C. (2019, February). Hyperbolic Monge-Ampere equations and applications to thin sheet elasticity. EDDy Seminar. Aachen, Germany: RWTH, Aachen.
• Venkataramani, S. C. (2019, February). Hyperbolic Monge-Ampere equations and applications to thin sheet elasticity. Hausdorff Forum. Bonn, Germany: University of Bonn.
• Venkataramani, S. C. (2019, February). Leaves, flowers, and sea-slugs: From discrete geometry to discrete mechanics. DGD Seminar. Berlin, Germany: TU Berlin.
• Venkataramani, S. C. (2019, JUNE). The geometry of defects in patterns. Condensed matter seminar. Philadelphia, PA: University of Pennsylvania.
• Venkataramani, S. C. (2019, January). Exotic continua: Leaves, flowers, sea-slugs, and hyperbolic origami.. Nonlinear physics seminar. Jerusalem, Israel: Hebrew University.
• Venkataramani, S. C. (2019, January). Hyperbolic Monge-Ampere equations and applications to thin sheet elasticity. Analysis-Probabiliy Oberseminar. Leipzig, Germany: Max Planck Institute.
• Venkataramani, S. C. (2019, January). On the mechanics of leaves, flowers and sea slugs.. Clore seminar. Rehovot, Israel: Weizmann Institute.
• Venkataramani, S. C. (2019, July). On leaves, flowers and sea-slugs: geometry and mechanics.. Baglunch seminar. Chicago, IL: University of Chicago.
• Venkataramani, S. C. (2019, March). On the unusual swimming gaits of sea-slugs. APS March meeting. Boston, MA: American Physical Society.
• Venkataramani, S. C. (2019, May). Topologically protected defects. SIAM Dynamical Systems DS19. Snowbird, UT: Society for Industrial and Applied Mathematics.
• Venkataramani, S. C. (2018, February). Singular limits: Analysis in the "real world".. RTG/Case studies.. Tucson, AZ: UA math department and program in applied math..
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  RTG seminar talk for the grad students in both programs.
• Venkataramani, S. C. (2018, July). On the Geometry of Leaves, Flowers and Sea Slugs. SIAM Materials Science Conference (SIAM MS18). Portland, OR: Society for Industrial and Applied Mathematics.
• Venkataramani, S. C. (2018, July). Phase Fields, Defects, and Patterns in Finite Domains. SIAM Annual meeting (SIAM AN18). Portland, OR: Society for Industrial and Applied Mathematics.
• Venkataramani, S. C. (2018, March). Manipulating geometries for fun and profit.. Folded forms lecture series.. Tucson, AZ: Tucson Botanical gardens and UA College of Science.
• Venkataramani, S. C. (2018, November). On discrete leaves, flowers and sea slugs.. OCIAM (Oxford Center for Industrial and Applied Math) seminar. Oxford, UK: OCIAM, Oxford University.
• Venkataramani, S. C. (2018, October). Flowers, Sea Slugs, and Robots: a journey through nature's geometry.. Everything is Math lecture series.. Tucson, AZ: UA math department.
• Venkataramani, S. C. (2018, September). Existence and regularity for solutions to Hyperbolic Monge-Ampere equations with applications to Non- Euclidean elasticity.. Center for Nonlinear Analysis (CNA) seminar. Pittsburgh, PA: CNA, Carnegie Mellon University.
• Venkataramani, S. C. (2018, September). On the Geometry of Leaves, Flowers and Sea Slugs. Mechanics and Computation seminar. Pittsburgh, PA: MSE/CEE, Carnegie Mellon University.
• Jaramillo, G., & Venkataramani, S. C. (2017). Ribbons of Infinite Length. APS March meeting.
• Venkataramani, S. C. (2017, September). Non-euclidean elasticity, discrete differential geometry and a generalized Sine-Gordon equation. AMS Fall sectional (Central). Denton, TX: American Mathematical Society.
• Venkataramani, S. C. (2016, March). Limit problems for thin sheets: A mathematical perspective. KITP Program: Geometry, elasticity, fluctuations, and order in 2D soft matter. Kavli Institute for Theoretical Physics, UC Santa Barbara: KITP.
• Venkataramani, S. C. (2016, May). Geometric defects in thin elastic structures. SIAM Conference on Mathematical Aspects of Materials science. Philadelphia, PA: SIAM.
• Venkataramani, S. C. (2016, November). Structures and universality: Rough solutions in geometric analysis and the calculus of variations. AMS Southeastern sectional meeting. Raleigh, NC: AMS.
• Venkataramani, S. C., & Kent, S. (2015, December). Multi-Scale Conformal Maps for Singular Interfaces in Free Boundary Problems. SIAM conference on the Analysis of PDE. Scottsdale, AZ.
• Venkataramani, S. C., & Kent, S. (2015, July). Numerical conformal mappings and free boundary problems,. Math department seminar, Indian Institute of Technology Madras, Chennai, India.. Chennai.
• Venkataramani, S. C., Dawson, C., Rosenthal, S., & Restrepo, J. (2015, December). Dimension Reduction, Stochastic Parametrization and Data Assimilation for Transport in the Ocean. SIAM conference on the Analysis of PDE. Scottsdale, AZ.
• Venkataramani, S. C., Gemmer, J., & Sharon, E. (2015, May). Geometry and mechanics of non-Euclidean thin sheets. Pan-American Congress of Applied Mechanics XV. Urbana-Champaign, IL..
• Venkataramani, S. C., Venkataramani, R. C., & Restrepo, J. (2015, May). Low dimensional models for the transport and aging of Oil in the environment. MRSEC seminar, James Franck Inst., University of Chicago. Chicago.
• Venkataramani, S. C., & Restrepo, J. M. (2014, May 1). Towards efficient parameterization of oil transport from first principles. Consortium for advanced research on the transport of hydrocarbons in the environment (CARTHE) annual meeting. Miami: Gulf of Mexico Research initiative (GoMRI).
• Venkataramani, S. C., Gemmer, J. A., Moloney, J. V., & Durfee, C. G. (2014, March). Phase shaping for linear and weakly nonlinear conical beams. MURI semi-annual review. U. Arizona: AFOSR.
• Venkataramani, S. C., Gemmer, J. A., Graf, T., Durfee, C. G., & Moloney, J. V. (2015, February). The mathematics of conical beams: Progress and questions. MURI semi-annual review. U. Arizona: AFOSR.

Poster Presentations

• Venkataramani, S., & Restrepo, J. M. (2015, February 2015). Research Overview of the Consortium for Advanced Research on Transport of Hydrocarbon in the Environment (CARTHE). International Oil Spills and Environmental science Conference. Houston, TX.

Reviews

• Gemmer, J., & Venkataramani, S. C. (2011. Shape transitions in hyperbolic non-Euclidean plates(pp 8151-8161).
  More info

  A non-Euclidean plate is a thin elastic object whose intrinsic geometry is not flat and hence has residual stresses arising from being embedded in three dimensional space. Recently, there has been interest in using localized swelling to induce residual stresses that shape flat objects into desired three dimensional structures. A fundamental question is whether we can use the mathematical theory of non-Euclidean plates to deduce the three dimensional configuration of the swelling sheet given the exact knowledge of the imposed geometry. We present and summarize the results of recent mathematical studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both annular and disc geometries. We show in the Foppl-von Karman approximation to the elastic energy there are only two types of global minimizers - flat and saddle shaped deformations - with localized regions of stretching near the boundary of the domain. We also show that there exist n-wave local minimizers that closely resemble experimental observations and have additional regions of stretching near lines of inflection. Furthermore, in the Kirchhoff approximation to the elastic energy, we show that there exists exact isometric immersions with periodic profiles. The number of waves in these configurations is set by the condition that the bending energy remains finite and grows approximately exponentially with the radius of the annulus. For large radii, these shape are energetically favorable over saddle shapes and could explain why wavy shapes are selected in crochet models of the hyperbolic plane. The predicted morphologies however differ from what is observed in experiments on hydrogel disks highlighting the need for further theoretical studies.