Mathematics Faculty Research and Publicationshttps://hdl.handle.net/2097/78732024-06-23T17:44:22Z2024-06-23T17:44:22Z1131A RIGID STAMP INDENTATION INTO A SEMIPLANE WITH A CURVATURE-DEPENDENT SURFACE TENSION ON THE BOUNDARYWalton, Jay R.Zemlyanova, Anna Y.https://hdl.handle.net/2097/339842021-04-07T20:33:23Z2016-03-17T00:00:00Zdc.title: A RIGID STAMP INDENTATION INTO A SEMIPLANE WITH A CURVATURE-DEPENDENT SURFACE TENSION ON THE BOUNDARY
dc.contributor.author: Walton, Jay R.; Zemlyanova, Anna Y.
dc.description.abstract: It has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience, fracture propagation, and contact mechanics. This paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature-dependent surface tension acting on the boundary of the semiplane. Cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered. Using the method of integral transforms, each problem is reduced to a system of singular integro-differential equations, which is further reduced to one or two weakly singular integral equations. It has been shown that the introduction of the curvature-dependent surface tension eliminates the classical singularities of the order 1/2 of the stresses and strains at the end-points of the contact interval. The numerical solution of the problem is obtained by approximation of unknown functions with Taylor polynomials.
dc.description: Citation: Walton, J. R., & Zemlyanova, A. Y. (2016). A RIGID STAMP INDENTATION INTO A SEMIPLANE WITH A CURVATURE-DEPENDENT SURFACE TENSION ON THE BOUNDARY. Siam Journal on Applied Mathematics, 76(2), 618-640. doi:10.1137/15m1044096
2016-03-17T00:00:00ZGraph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knotsArone, G.Turchin, Victorhttps://hdl.handle.net/2097/323422021-04-07T20:33:57Z2015-06-19T00:00:00Zdc.title: Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots
dc.contributor.author: Arone, G.; Turchin, Victor
dc.description: Citation: Arone, G., & Turchin, V. (2015). Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots. Annales de l'Institut Fourier, 65(1), 1-62. Retrieved from http://www.scopus.com/inward/record.url?eid=2-s2.0-84937572773&partnerID=40&md5=4e6f2452f6ac30f193989d2479d66d44; We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on the different ways the calculations can be done. In particular we describe three different graph-complexes computing these rational homotopy groups. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.
2015-06-19T00:00:00ZA symmetry result for strictly convex domainsRamm, Alexander G.https://hdl.handle.net/2097/323412021-04-07T20:43:32Zdc.title: A symmetry result for strictly convex domains
dc.contributor.author: Ramm, Alexander G.
dc.description: Citation: Ramm, A. G. (2015). A symmetry result for strictly convex domains. Analysis, 35(1), 29-32. doi:10.1515/anly-2014-1257; Let D?R2 be a strictly convex domain with C2-smooth boundary. Assume that D eix yn dxdy=0 for all sufficiently large n. In this paper, we will prove that D is a disc. © 2015 by De Gruyter.
Scattering of electromagnetic waves by many small perfectly conducting or impedance bodiesRamm, Alexander G.https://hdl.handle.net/2097/323402021-04-07T20:44:41Z2015-09-08T00:00:00Zdc.title: Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies
dc.contributor.author: Ramm, Alexander G.
dc.description: Citation: Ramm, A. G. (2015). Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies. Journal of Mathematical Physics, 56(9), 21. doi:10.1063/1.4929965; A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an arbitrary shape, an explicit analytical formula is derived for the scattering amplitude. The formula holds as a -> 0, where a is a characteristic size of the small particle and the wavelength is arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to a(2-k), where k epsilon [0,1) is a parameter which can be chosen by an experimenter as he/she wants. The boundary impedance of a small particle is assumed to be of the form zeta = ha(-k), where h = const, Reh >= 0. The scattering amplitude for a small perfectly conducting particle is proportional to a(3), and it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions a << d << lambda, where d is the minimal distance between neighboring particles and lambda is the wavelength. The distribution law for the small impedance particles is N(Delta) similar to 1/a(2-k) integral N-Delta(x) dx as a -> 0. Here, N(x) >= 0 is an arbitrary continuous function that can be chosen by the experimenter and N(.) is the number of particles in an arbitrary sub-domain Delta. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as a -> 0 and a differential equation is derived for the limiting field. On this basis, a recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material. (C) 2015 AIP Publishing LLC.
2015-09-08T00:00:00ZExistence and uniqueness of the global solution to the Navier-Stokes equationsRamm, Alexander G.https://hdl.handle.net/2097/204842021-04-07T20:46:18Z2015-11-01T00:00:00Zdc.title: Existence and uniqueness of the global solution to the Navier-Stokes equations
dc.contributor.author: Ramm, Alexander G.
dc.description.abstract: A proof is given of the global existence and uniqueness of a weak solution to Navier–Stokes equations in unbounded exterior domains.
2015-11-01T00:00:00ZInverse scattering on the half-line revisitedRamm, Alexander G.https://hdl.handle.net/2097/204832021-04-07T20:48:13Z2015-10-01T00:00:00Zdc.title: Inverse scattering on the half-line revisited
dc.contributor.author: Ramm, Alexander G.
dc.description.abstract: The inverse scattering problem on the half-line has been studied in the literature in detail. V. Marchenko presented the solution to this problem. In this paper, the invertibility of the steps of the inver-sion procedure is discussed and a new set of necessary and suﬃcient conditions on the scattering data is given for the scattering data to be generated by a potential q ∈ L1,1. Our proof is new and in con-trast with Marchenko’s proof does not use equations on the negative half-line.
2015-10-01T00:00:00ZScattering of electromagnetic waves by many small perfectly conducting or impedance bodiesRamm, Alexander G.https://hdl.handle.net/2097/204822021-04-07T20:50:13Z2015-09-08T00:00:00Zdc.title: Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies
dc.contributor.author: Ramm, Alexander G.
dc.description.abstract: A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an arbitrary shape, an explicit analytical formula is derived for the scattering amplitude. The formula holds as a → 0, where a is a characteristic size of the small particle and the wavelength is arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to a2−κ, where κ ∈ [0,1) is a parameter which can be chosen by an experimenter
as he/she wants. The boundary impedance of a small particle is assumed to be of the form ζ = ha−κ, where h = const, Reh ≥ 0. The scattering amplitude for a small perfectly conducting particle is proportional to a3, and it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions a ≪ d ≪ λ, where d is the minimal distance between neighboring particles and λ is the wavelength. The distribution law for the small
impedance particles is N(∆) ∼ 1/a2−κ∆ N(x)dx as a → 0. Here, N(x) ≥ 0 is an
arbitrary continuous function that can be chosen by the experimenter and N(∆)
is the number of particles in an arbitrary sub-domain ∆. It is proved that the EM field in the medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as a → 0 and a differential equation is derived for the limiting field. On this basis, a recipe is given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4929965]
2015-09-08T00:00:00ZRepresentation of vector fieldsRamm, Alexander G.https://hdl.handle.net/2097/201282021-04-07T20:52:13Z2015-06-01T00:00:00Zdc.title: Representation of vector fields
dc.contributor.author: Ramm, Alexander G.
dc.description.abstract: A simple proof is given for the explicit formula which allows one to recover a C2 – smooth vector field A=A(x) in R3, decaying at infinity, from the knowledge of its ∇×A and ∇⋅A. The representation of A as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded C2 - smooth domain.
2015-06-01T00:00:00ZEM Wave Scattering by Many Small Impedance Particles and Applications to Materials ScienceRamm, Allexander G.https://hdl.handle.net/2097/201272021-04-07T20:57:17Z2015-04-25T00:00:00Zdc.title: EM Wave Scattering by Many Small Impedance Particles and Applications to Materials Science
dc.contributor.author: Ramm, Allexander G.
2015-04-25T00:00:00ZA Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of ParticlesRamm, Alexander G.Tran, Nhan Thanhhttps://hdl.handle.net/2097/201262021-04-07T20:59:55Zdc.title: A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Particles
dc.contributor.author: Ramm, Alexander G.; Tran, Nhan Thanh
dc.description.abstract: Scalar wave scattering by many small particles of arbitrary shapes with impedance boundary condition is studied. The problem is solved asymptotically and numerically under the assumptions a d where k = 2 = is the wave number, is the wave length, a is the characteristic size of the particles, and d is the smallest distance between neighboring particles. A fast algorithm for solving this wave scattering problem by billions of particles is presented. The algorithm comprises the derivation of the (ORI) linear system and makes use of Conjugate Orthogonal Conjugate Gradient method and Fast Fourier Transform. Numerical solutions of the scalar wave scattering problem with 1, 4, 7, and 10 billions of small impedance particles are achieved for the first time. In these numerical examples, the problem of creating a material with negative refraction coefficient is also described and a recipe for creating materials with a desired refraction coefficient is tested.