# Mathematics Faculty Research and Publications

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Item Open Access A RIGID STAMP INDENTATION INTO A SEMIPLANE WITH A CURVATURE-DEPENDENT SURFACE TENSION ON THE BOUNDARY(2016-03-17) Walton, Jay R.; Zemlyanova, Anna Y.; azemIt 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.Item Open Access Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots(2015-06-19) Arone, G.; Turchin, Victor; turchinItem Open Access A symmetry result for strictly convex domainsRamm, Alexander G.; rammItem Open Access Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies(2015-09-08) Ramm, Alexander G.; rammItem Open Access Existence and uniqueness of the global solution to the Navier-Stokes equations(Elsevier, 2015-11-01) Ramm, Alexander G.; rammA proof is given of the global existence and uniqueness of a weak solution to Navier–Stokes equations in unbounded exterior domains.Item Open Access Inverse scattering on the half-line revisited(Elsevier, 2015-10-01) Ramm, Alexander G.; rammThe 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.Item Open Access Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies(American Institute of Physics (AIP), 2015-09-08) Ramm, Alexander G.; rammA 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]Item Open Access Representation of vector fields(Science Publishing Corporation, 2015-06-01) Ramm, Alexander G.; rammA 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.Item Open Access EM Wave Scattering by Many Small Impedance Particles and Applications to Materials Science(Bentham Open, 2015-04-25) Ramm, Allexander G.; rammItem Open Access A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Particles(World Academic Publishing) Ramm, Alexander G.; Tran, Nhan Thanh; rammScalar 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.Item Unknown Creating Media with Prescribed Permeability Using the Asymptotic Solution to EM Wave Scattering Problem(IEEE Xplore Digital Library, 2014-06-14) Ramm, Alexander G.; Andriychuk, Mykhaylo; rammAsymptotic solution to scattering problem of electromagnetic (EM) waves by many small impedance particles, embedded in a homogeneous medium, is applied for creating media with prescribed permeability. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the resulting medium. The proposed theory allows one to create a medium with a desirable spatially inhomogeneous permeability. The main new physical result is the explicit analytic formula for the permeability μ(x) of the limiting medium. The computational results confirm a possibility to create the media with various distributions of μ(x).Item Unknown Electromagnetic wave scattering by small perfectly conducting particles and applications(American Institute of Physics, 2014-07-25) Ramm, Alexander G.; rammA rigorous theory of electromagnetic (EM) wave scattering by one and many perfectly conducting small bodies of an arbitrary shape is developed. Equation for the effective field is derived in amedium in which many small particles are distributed. A method is given to change the refraction coefficient of a given medium in a desired direction by embedding into this mediummany small particles.Item Unknown Electromagnetic Wave Scattering by Small Impedance Particles of an Arbitrary Shape and Applications(Multidisciplinary Digital Publishing Institute, 2014-02-07) Ramm, Alexander G.; rammThe proposal deals with electromagnetic (EM) wave scattering by one and many small impedance particles of an arbitrary shape. Analytic formula is derived for EM wave scattering by one small impedance particle of an arbitrary shape and an integral equation for the effective field in the medium where many such particles are embedded. These results are applied for creating a medium with a desired refraction coefficient. The proposed theory has no analogs in the literature. (Mathematical Subject Classification: 35J05, 35J25, 65N12, 78A25, 78A48.)Item Open Access Recovery of the Potential from I-Function(Elsevier, 2014-10-01) Ramm, Alexander G.; rammThe inverse scattering problem on half-axis, in other words, in the spherically symmetric case, consists of finding the unknown potential from a suitable class from the scattering data S. The other problem of practical interest is to find this potential from the spectral data dp(λ). In the literature there are recovery procedures for finding the potential from the spectral or from the scattering data. Define the I-function: I(k) := f'(0,k)/f(k) , where f(x,k) is the Jost solution. Constructive ways to find I(k) from dp(λ) and vice versa, and I(k) from S and vice versa are given. The theory of Riemann problem is used as an essential tool. If I(k) is found, then our methods allow one to construct the scattering data and from these recover the potential by the known procedure. Alternatively, one can construct from I(k) the spectral data and from these find the potential by the known procedure. MSC: 34A55Item Open Access On deformations of pasting diagrams, II(2014-06-01) Shrestha, Tej Bahadur; Yetter, David; dyetterWe continue the development of the infinitesimal deformation theory of pasting diagrams of k-linear categories begun in TAC, Vol 22, #2. In that article the standard result that all obstructions are cocycles was established only for the elementary, composition-free parts of pasting diagrams. In the present work we give a proof for pasting diagrams in general. As tools we use the method developed by Shrestha of simultaneously representing formulas for obstructions, along with the corresponding cocycle and cobounding conditions by suitably labeled polygons, giving a rigorous exposition of the previously heuristic method; and deformations of pasting diagrams in which some cells are required to be deformed trivially.Item Open Access On deformations of pasting diagrams(2009-06-01) Yetter, David; dyetterWe adapt the work of Power to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in the 2-category of k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack, proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov.Item Open Access Relative reproducing kernel Hilbert spaces(2014-07-17) Alpay, Daniel; Jorgensen, Palle; Volok, Dan; danvolokWe introduce a reproducing kernel structure for Hilbert spaces of functions where differences of point evaluations are bounded. The associated reproducing kernels are characterized in terms of conditionally negative functions.Item Open Access The Lind Lehmer constant for Z(p)(n)(2014-02-26) De Silva, Dilum P.; Pinner, Christopher G.; cpinnerWe determine the Lind Lehmer constant for groups of the form Z(np).Item Open Access Waring's number for large subgroups of double-struck Z_p(2014-11-25) Cochrane, Todd E.; Hart, Derrick; Pinner, Christopher G.; Spencer, Craig; cochrane; cpinner; cvsLet p be a prime, Z_p be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero k-th powers in Z_p. The goal of this paper is to determine, for a given positive integer s, a value t_s such that if |A| ≫ t_s then every element of Z_p is a sum of s k-th powers. We obtain t_4 = p^{\frac{22}{39} + \in}, t_5 = p^{\frac{15}{29} + \in} and for s s ≥ 6, t_s = p^{\frac{9s+45}{29s+33} + \in}. For s ≥ 24 further improvements are made, such as t_32 = p^{\frac{5}{16} + \in} and t_128 = p^{\frac{1}{4}}.Item Open Access Inverse scattering problem with underdetermined data(2014-07-17) Ramm, Alexander G.; ramm