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Rozprawy doktorskie na temat „Equation”

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1

Thompson, Jeremy R. (Jeremy Ray). "Physical Motivation and Methods of Solution of Classical Partial Differential Equations." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277898/.

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We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.

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2

Howard, Tamani M. "Hyperbolic Monge-Ampère Equation." Thesis, University of North Texas, 2006. https://digital.library.unt.edu/ark:/67531/metadc5322/.

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In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equatio

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3

Vong, Seak Weng. "Two problems on the Navier-Stokes equations and the Boltzmann equation /." access full-text access abstract and table of contents, 2005. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?phd-ma-b19885805a.pdf.

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Thesis (Ph.D.)--City University of Hong Kong, 2005.<br>"Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy" Includes bibliographical references (leaves 72-77)

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4

Guan, Meijiao. "Global questions for evolution equations Landau-Lifshitz flow and Dirac equation." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/22491.

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This thesis concerns the stationary solutions and their stability for some evolution equations from physics. For these equations, the basic questions regarding the solutions concern existence, uniqueness, stability and singularity formation. In this thesis, we consider two different classes of equations: the Landau-Lifshitz equations, and nonlinear Dirac equations. There are two different definitions of stationary solutions. For the Landau-Lifshitz equation, the stationary solution is time-independent, while for the Dirac equation, the stationary solution, also called solitary wave solution or

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5

Jumarhon, Bartur. "The one dimensional heat equation and its associated Volterra integral equations." Thesis, University of Strathclyde, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342381.

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6

Banerjee, Paromita. "Numerical Methods for Stochastic Differential Equations and Postintervention in Structural Equation Models." Case Western Reserve University School of Graduate Studies / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=case1597879378514956.

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7

Wang, Jun. "Integral Equation Methods for the Heat Equation in Moving Geometry." Thesis, New York University, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10618746.

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<p> Many problems in physics and engineering require the solution of the heat equation in moving geometry. Integral representations are particularly appropriate in this setting since they satisfy the governing equation automatically and, in the homogeneous case, require the discretization of the space-time boundary alone. Unlike methods based on direct discretization of the partial differential equation, they are unconditonally stable. Moreover, while a naive implementation of this approach is impractical, several efforts have been made over the past few years to reduce the overall computation

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8

Grundström, John. "The Sustainability Equation." Thesis, Umeå universitet, Arkitekthögskolan vid Umeå universitet, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-133151.

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9

Gylys-Colwell, Frederick Douglas. "An inverse problem for the anisotropic time independent wave equation /." Thesis, Connect to this title online; UW restricted, 1993. http://hdl.handle.net/1773/5726.

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10

Shedlock, Andrew James. "A Numerical Method for solving the Periodic Burgers' Equation through a Stochastic Differential Equation." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103947.

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The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential eq

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11

Rogers, James W. Jr Sheng Qin. "Adaptive methods for the Helmholtz equation with discontinuous coefficients at an interface." Waco, Tex. : Baylor University, 2007. http://hdl.handle.net/2104/5122.

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12

Moyano, Garcia Iván. "Controllability of of some kinetic equations, of parabolic degenerated equations and of the Schrödinger equation via domain transformation." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX062/document.

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Ce mémoire présente les travaux réalisés au cours de ma thèse dans le but d'étudier la contrôlabilité de quelques équations aux dérivées partielles. La première partie de cette thèse est consacrée à l'étude de la contrôlabilité de quelques équations cinétiques en différents régimes. Dans un régime collisionnel, nous étudions la contrôlabilité de l'équation de Kolmogorov, un modèle de type Fokker-Planck cinétique, posée dans l'espace de phases $R^d times R^d$. Nous obtenons la contrôlabilité à zéro de cette équation grâce à l'utilisation d'une inégalité spectrale associée à l'opérateur Laplacie

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13

Sjölander, Filip. "Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297544.

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The aim of the report is to numerically construct solutions to two analytically solvable non-linear differential equations: the Korteweg–De Vries equation and the Boussinesq equation. To accomplish this, a range of numerical methods where implemented, including Galerkin methods. To asses the accuracy of the solutions, analytic solutions were derived for reference. Characteristic of both equations is that they support a certain type of wave-solutions called "soliton solutions", which admit an intuitive physical interpretation as solitary traveling waves. Theses solutions are the ones simulated.

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14

COMI, GIULIA. "Two Fractional Stochastic Problems: Semi-Linear Heat Equation and Singular Volterra Equation." Doctoral thesis, Università degli studi di Pavia, 2019. http://hdl.handle.net/11571/1292026.

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15

Voss, Alexander. "Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of state." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=976791641.

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16

Militaru, Mariana. "Sur les equations de navier-stokes deterministes et stochastiques et sur une equation elliptique." Clermont-Ferrand 2, 1997. http://www.theses.fr/1997CLF21922.

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Cette these, est constituee par trois problemes sur des conditions aux limites mixtes lies aux equations elliptiques et de navier-stokes. Dans la premiere partie, on montre l'existence d'une solution faible d'une equation de navier-stokes stochastique, lorsque la densite initiale s'annule. Apres avoir obtenu des estimations convenables sur des solutions approchees, on en deduit la convergence en loi dans un nouvel espace de probabilite. Par un passage a la limite elle obtient alors une solution verifiant une equation sous forme d'une esperance, donc une solution faible. Dans la seconde partie,

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17

Temimi, Helmi. "A Discontinuous Galerkin Method for Higher-Order Differential Equations Applied to the Wave Equation." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/26454.

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We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L2- norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first (m-1)-derivatives are strongly superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is superconvergent at the roots of (q+1-m)-degree Jacobi polynomial on each step. Furthermore, we use these results to construct asymptotically correct a posteriori error estimat

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18

Ubostad, Nikolai Høiland. "The Infinity Laplace Equation." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-20686.

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In this thesis, we prove that the Infinity-Laplace equation has a unique solution in the viscosity sense. We prove existence by approximating the equation by the p-Laplace equation, and uniqueness will be shown by use of the Theorem on Sums. We will also show that the viscosity solutions of the Infinity-Laplace equation enjoys comparison with cones, and vice versa.

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19

Shiono, Masaaki. "Investigations of Sayre's equation." Thesis, University of York, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329668.

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20

Capone, Lauren. "The Hat Lady Equation." ScholarWorks@UNO, 2014. http://scholarworks.uno.edu/td/1856.

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The Hat Lady Equation is a collection of poems by Lauren Capone. As influences she cites Elizabeth Bishop, John Berryman, among the exquisite minutiae of day-to-day living. The poems explore works of visual art by Alberto Giacometti, James Taylor Bonds, Chris Dennis, Blaine Capone (her brother), and creatures of the natural world including fish, the rhinoceros, a lettered olive shell. . . . Lauren shows a preoccupation with disassembling through the poems whether it's her identity, art, or happenings of everyday life.

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21

Kutahyalioglu, Aysen. "Oscillation Of Second Order Dynamic Equations On Time Scales." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605380/index.pdf.

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During the last decade, the use of time scales as a means of unifying and extending results about various types of dynamic equations has proven to be both prolific and fruitful. Many classical results from the theories of differential and difference equations have time scale analogues. In this thesis we derive new oscillation criteria for second order dynamic equations on time scales.

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22

Beech, Robert. "Extensions of the nonlinear Schrödinger equation using Mathematica." Thesis, View thesis, 2009. http://handle.uws.edu.au:8081/1959.7/46572.

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The aim of this thesis is to investigate the theory of the extensions of the Nonlinear Schrödinger Equation (NLSE), concentrating on the following main points: Developing further analytical techniques and properties under relativistic conditions. This thesis demonstrates numerical techniques that can be used to form numerical codes that can be applied to the very recent need for source and industrial application of laser-driven ion sources for ion implantation. The analytical and numerical evaluations of the nonlinear mechanisms are measured utilising various techniques that include computer p

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23

Knaub, Karl R. "On the asymptotic behavior of internal layer solutions of advection-diffusion-reaction equations /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/6772.

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24

Selcuk, Aysun. "Oscillation Of Second Order Matrix Equations On Time Scales." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605606/index.pdf.

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The theory of time scales is introduced by Stefan Hilger in his PhD thesis in 1998 in order to unify continuous and discrete analysis. In our thesis, by making use of the time scale calculus we study the oscillation of nonlinear matrix differential equations of second order. the first chapter is introductory in nature and contains some basic definitions and tools of the time scales calculus, while certain well-known results have been presented with regard to oscillation of the solutions of second order matrix equations and some new oscillation criteria for the same type equations have been est

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25

Sun, Weizhou. "LOCAL DISCONTINUOUS GALERKIN METHOD FOR KHOKHLOV-ZABOLOTSKAYA-KUZNETZOV EQUATION AND IMPROVED BOUSSINESQ EQUATION." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1480327264817905.

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26

Drake, Robert M. "Far field extrapolation technique using CHIEF enclosing sphere deduced pressures and velocities." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2003. http://library.nps.navy.mil/uhtbin/hyperion-image/03Dec%5FDrake.pdf.

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27

Grava, Tamara. "On the Cauchy Problem for the Whitham Equations." Doctoral thesis, SISSA, 1998. http://hdl.handle.net/20.500.11767/4352.

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28

Di, Cosmo Jonathan. "Nonlinear Schrödinger equation and Schrödinger-Poisson system in the semiclassical limit." Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209863.

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The nonlinear Schrödinger equation appears in different fields of physics, for example in the theory of Bose-Einstein condensates or in wave propagation models. From a mathematical point of view, the study of this equation is interesting and delicate, notably because it can have a very rich set of solutions with various behaviours.<p><p>In this thesis, we have been interested in standing waves, which satisfy an elliptic partial differential equation. When this equation is seen as a singularly perturbed problem, its solutions concentrate, in the sense that they converge uniformly to zero outsid

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29

Beech, Robert. "Extensions of the nonlinear Schrödinger equation using Mathematica." View thesis, 2009. http://handle.uws.edu.au:8081/1959.7/46572.

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Thesis (Ph.D.)--University of Western Sydney, 2009.<br>A thesis presented to the University of Western Sydney, College of Health and Science, School of Computing and Mathematics, in fulfilment of the requirements for the degree of Doctor of Philosophy (PhD). Includes bibliographies.

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30

Guzainuer, Maimaitiyiming. "Boundary Summation Equation Preconditioning for Ordinary Differential Equations with Constant Coefficients on Locally Refined Meshes." Thesis, Linköpings universitet, Matematiska institutionen, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102573.

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This thesis deals with the numerical solution of ordinary differential equations (ODEs) using finite difference (FD) methods. In particular, boundary summation equation (BSE) preconditioning for FD approximations for ODEs with constant coefficients on locally refined meshes is studied. Firstly, the BSE for FD approximations of ODEs with constant coefficients is derived on a locally refined mesh. Secondly, the obtained linear system of equations are solved by the iterative method GMRES. Then, the arithmetic complexity and convergence rate of the iterative solution of the BSE formulation are dis

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31

Kok, Tayfun. "Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equation." Thesis, University of York, 2017. http://etheses.whiterose.ac.uk/18070/.

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The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions

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32

Zhou, Dong. "High-order numerical methods for pressure Poisson equation reformulations of the incompressible Navier-Stokes equations." Diss., Temple University Libraries, 2014. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/295839.

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Mathematics<br>Ph.D.<br>Projection methods for the incompressible Navier-Stokes equations (NSE) are efficient, but introduce numerical boundary layers and have limited temporal accuracy due to their fractional step nature. The Pressure Poisson Equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity

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33

Klepel, Konrad Verfasser], and Dirk [Akademischer Betreuer] [Blömker. "Amplitude equations for the generalised Swift-Hohenberg equation with noise / Konrad Klepel. Betreuer: Dirk Blömker." Augsburg : Universität Augsburg, 2015. http://d-nb.info/107770562X/34.

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34

Ugail, Hassan. "3D facial data fitting using the biharmonic equation." ACTA Press, 2006. http://hdl.handle.net/10454/2684.

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This paper discusses how a boundary-based surface fitting approach can be utilised to smoothly reconstruct a given human face where the scan data corresponding to the face is provided. In particular, the paper discusses how a solution to the Biharmonic equation can be used to set up the corresponding boundary value problem. We show how a compact explicit solution method can be utilised for efficiently solving the chosen Biharmonic equation. Thus, given the raw scan data of a 3D face, we extract a series of profile curves from the data which can then be utilised as boundary conditions to solve

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35

Nordli, Anders Samuelsen. "On the Hunter-Saxton equation." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2012. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-19057.

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The Cauchy problem for a two-component Hunter-Saxton equation, begin{align*}(u_t+uu_x)_x&=frac{1}{2}u_x^2+frac{1}{2}rho^2,rho_t+(urho)_x) &= 0,end{align*}on $mathbb{R}times[0,infty)$ is studied. Conservative and dissipative weak solutions are defined and shown to exist globally. This is done by explicitly solving systems of ordinary differential equation in the Lagrangian coordinates, and using these solutions to construct semigroups of conservative and dissipative solutions.

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36

Kwek, Keng-Huat. "On Cahn-Hilliard type equation." Diss., Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/28819.

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37

Mugassabi, Souad. "Schrödinger equation with periodic potentials." Thesis, University of Bradford, 2010. http://hdl.handle.net/10454/4895.

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The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential

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38

Fedrizzi, Ennio. "Partial Differential Equation and Noise." Phd thesis, Université Paris-Diderot - Paris VII, 2012. http://tel.archives-ouvertes.fr/tel-00759355.

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Dans ce travail, nous présentons quelques exemples des effets du bruit sur la solution d'une équation aux dérivées partielles (EDP) dans trois contextes différents. Nous exam- inons d'abord deux équations aux dérivées partielles non linéaires dispersives, l'équation de Schrödinger non linéaire et l'équation de Korteweg - de Vries. Nous allons analyser les effets d'une condition initiale aléatoire sur certaines solutions spéciales, les solitons. Le deuxième cas considéré est une EDP linéaire, l'équation d'onde, avec conditions initiales aléatoires. Nous allons montrer qu'avec des conditions ini

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39

Moussa, Ridha. "On the generalized Ince equation." Thesis, The University of Wisconsin - Milwaukee, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3636427.

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<p> We investigate a Hill differential equation with trigonometric polynomial coefficients. we are interested in solutions which are even or odd and have period π or semi-period π. With one of the mentioned boundary conditions, our equation constitute a regular Sturm-Liouville eigenvalue problem. Using Fourier series representation each one of the four Sturm-Liouville operators is represented by an infinite banded matrix. In the particular cases of Ince and Lamé equations, the four infinite banded matrices become tridiagonal. We then investigate the problem of coexistence of peri

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40

Gatti, Antonio. "A gauge invariant flow equation." Thesis, University of Southampton, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.268629.

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41

Melia, F. "The cosmic equation of state." Springer Verlag, 2014. http://hdl.handle.net/10150/614766.

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The cosmic spacetime is often described in terms of the Friedmann-Robertson-Walker (FRW) metric, though the adoption of this elegant and convenient solution to Einstein's equations does not tell us much about the equation of state, $p=w\rho$, in terms of the total energy density $\rho$ and pressure $p$ of the cosmic fluid. $\Lambda$CDM and the $R_{\rm h}=ct$ Universe are both FRW cosmologies that partition $\rho$ into (at least) three components, matter $\rho_{\rm m}$, radiation $\rho_{\rm r}$, and a poorly understood dark energy $\rho_{\rm de}$, though the latter goes one step further

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42

Correia, Joaquim, Costa Fernando da, Sackmone Sirisack, and Khankham Vongsavang. "Burgers' Equation and Some Applications." Master's thesis, Edited by Thepsavanh Kitignavong, Faculty of Natural Sciences, National University of Laos, 2017. http://hdl.handle.net/10174/26615.

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In this thesis, I present Burgers' equation and some of its applications. I consider the inviscid and the viscid Burgers' equations and present different analytical methods for their study: the Method of Characteristics for the inviscid case, and the Cole-Hopf Transformation for theviscid one. Two applications of Burgers' equations are given: one in simple models of Traffic Flow (which have been introduced independently by Lighthill-Whitham and Richards) and another in Coagulation theory (in which we use Laplace Transform to obtain Burgers' equations from the original coagulation integro-diffe

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43

Al, Homsi Rania. "Equation Solving in Indian Mathematics." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-355870.

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44

Carroll, Andrew. "The stochastic nonlinear heat equation." Thesis, University of Hull, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.310216.

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45

Zhang, Henshui. "Local analysis of Loewner equation." Thesis, Orléans, 2018. http://www.theses.fr/2018ORLE2064.

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Dans cette thèse nous étudions le problème de la génération d’une courbe par l’équation de Loewner generalisée. Nous utilisons une transformation locale dans l’équation chordal de Loewner, analysons la solution de l’équation de Loewner, et obtenons trois résultats.En premier lieu, nous analysons la limite supérieure et la limite inférieure de l’ordre 1/2 à gauche de la fonction pilotant l’équation, nous prouvons ensuite un lemme basique qui assure que la courbe générée ne s’auto-intersecte pas localement. Ce lemme nous conduit à trois conclusions. Premièrement Lind a prouvé que lorsque la norm

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46

Daviau, Claude. "Equation de dirac non lineaire." Nantes, 1993. http://www.theses.fr/1993NANT2006.

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Partant de l'equation de dirac, le terme d'impulsion-energie de l'electron est simplifie. On obtient ainsi une equation d'ondes non lineaire, qui est etudiee en utilisant le formalisme de l'algebre de clifford d'espace-temps. La resolution est effectuee pour les ondes planes monochromatiques, qui sont sans energies negatives tant que la masse propre reste positive. Puis sont etudiees les invariances de l'equation: invariance relativiste, symetries p, t, c, invariances de jauge. La resolution de l'equation, pour l'atome d'hydrogene, est effectuee par separation des variables. La non linearite n

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Eti, Neslihan Pashaev Oktay. "Classical And Quantum Euler Equation/." [s.l.]: [s.n.], 2007. http://library.iyte.edu.tr/tezler/master/matematik/T000610.pdf.

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Valenciano, Alejandro A. "Imaging by wave-equation inversion /." May be available electronically:, 2008. http://proquest.umi.com/login?COPT=REJTPTU1MTUmSU5UPTAmVkVSPTI=&clientId=12498.

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Guidi, Chiara. "The Complex Monge-Ampère Equation." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9004/.

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In questa trattazione si studia la regolarità delle soluzioni viscose plurisubarmoniche dell’equazione di Monge-Ampère complessa. Si tratta di un’equazione alle derivate parziali del secondo ordine completamente non lineare il cui termine del secondo ordine è il determinante della matrice hessiana complessa di una funzione incognita a valori reali u. Il principale risultato della tesi è un nuovo controesempio di tipo Pogorelov per questa equazione. Si prova cioè l’esistenza di soluzioni viscose plurisubarmoniche e non classiche per un equazione di Monge-Ampère complessa.

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Murray, Patrick R. "The thermo-acoustic Fant equation." Thesis, Keele University, 2012. http://eprints.keele.ac.uk/3836/.

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A theoretical analysis is made of combustion instabilities of three combustor configurations. The equations governing aeroacoustics and combustion are derived, arriving at an acoustic analogy in terms of the pressure and total enthalpy. A solution for the acoustic analogy is determined in terms of a Green's function and initial instability results are presented for the pressure Green's function. These predictions are limited by assumptions made about the combustion zone. Finally a `reduced complexity' equation is derived accounting for a generalised combustion zone. The equation is nonlinear a

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