• Title, Summary, Keyword: arclength

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A UNIFORM ESTIMATE ON CONVOLUTION OPERATORS WITH THE ARCLENGTH MEASURE ON NONDEGENERATE SPACE CURVES

  • Choi, Youngwoo
    • Korean Journal of Mathematics
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    • v.6 no.2
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    • pp.291-298
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    • 1998
  • The $L^p-L^q$ mapping properties of convolution operators with measures supported on curves in $\mathbb{R}^3$ have been studied by many authors. Oberlin provided examples of nondegenerate compact space curves whose arclength measures enjoy $L^p$-improving properties. This was later extended by Pan who showed that such properties hold for all nondegenerate compact space curves. In this paper, we will prove that the operator norm of the convolution operator with the arclength measure supported on a nondegenerate compact space curve depends only on certain quantities of the underlying curve.

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CONVOLUTION OPERATORS WITH THE AFFINE ARCLENGTH MEASURE ON PLANE CURVES

  • Choi, Young-Woo
    • Journal of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.193-207
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    • 1999
  • Let ${\gamma}$ : Ilongrightarrow R2 be a sufficiently smooth curve and $\sigma$${\gamma}$ be the affine arclength measure supported on ${\gamma}$. In this paper, we study the Lp - improving properties of the convolution operators T$\sigma$${\gamma}$ associated with $\sigma$${\gamma}$ for various curves ${\gamma}$. Optimal results are obtained for all finite type plane curves and homogeneous curves (possibly blowing up at the origin). As an attempt to extend this result to infinitely flat curves we give and example of a family of flat curves whose affine arclength measure has same Lp-improvement property. All of these results will be based on uniform estimates of damping oscillatory integrals.

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Internal resonance and nonlinear response of an axially moving beam: two numerical techniques

  • Ghayesh, Mergen H.;Amabili, Marco
    • Coupled systems mechanics
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    • v.1 no.3
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    • pp.235-245
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    • 2012
  • The nonlinear resonant response of an axially moving beam is investigated in this paper via two different numerical techniques: the pseudo-arclength continuation technique and direct time integration. In particular, the response is examined for the system in the neighborhood of a three-to-one internal resonance between the first two modes as well as for the case where it is not. The equation of motion is reduced into a set of nonlinear ordinary differential equation via the Galerkin technique. This set is solved using the pseudo-arclength continuation technique and the results are confirmed through use of direct time integration. Vibration characteristics of the system are presented in the form of frequency-response curves, time histories, phase-plane diagrams, and fast Fourier transforms (FFTs).

LOGHARMONIC MAPPINGS WITH TYPICALLY REAL ANALYTIC COMPONENTS

  • AbdulHadi, Zayid;Alarifi, Najla M.;Ali, Rosihan M.
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1783-1789
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    • 2018
  • This paper treats the class of normalized logharmonic mappings $f(z)=zh(z){\overline{g(z)}}$ in the unit disk satisfying ${\varphi}(z)=zh(z)g(z)$ is analytically typically real. Every such mapping f admits an integral representation in terms of its second dilatation function and a function of positive real part with real coefficients. The radius of starlikeness and an upper estimate for arclength are obtained. Additionally, it is shown that f maps the unit disk into a domain symmetric with respect to the real axis when its second dilatation has real coefficients.

Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support

  • Ghayesh, Mergen H.;Amabili, Marco
    • Coupled systems mechanics
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    • v.2 no.2
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    • pp.159-174
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    • 2013
  • The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.

Dynamical behaviour of electrically actuated microcantilevers

  • Farokhi, Hamed;Ghayesh, Mergen H.
    • Coupled systems mechanics
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    • v.4 no.3
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    • pp.251-262
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    • 2015
  • The current paper aims at investigating the nonlinear dynamical behaviour of an electrically actuated microcantilever. The microcantilever is excited by a combination of AC and DC voltages. The nonlinear equation of motion of the microcantilever is obtained by means of force and moment balances. A high-dimensional Galerkin scheme is then applied to reduce the equation of motion to a discrete model. A numerical technique, based on the pseudo-arclength continuation method, is used to solve the discretized model. The electrostatic deflection of the microcantilever and static pull-in instabilities, due to the DC voltage, are analyzed by plotting the so-called DC voltage-deflection curves. At the simultaneous presence of the DC and AC voltages, the nonlinear dynamical behaviour of the microcantilever is analyzed by plotting frequency-response and force-response curves.

Research on Numerical Calculation of Normal Modes in Nonlinear Vibrating Systems (비선형 진동계 정규모드의 수치적 계산 연구)

  • Lee, Kyoung-Hyun;Han, Hyung-Suk;Park, Sungho;Jeon, Soohong
    • Transactions of the Korean Society for Noise and Vibration Engineering
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    • v.26 no.7
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    • pp.795-805
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    • 2016
  • Nonlinear normal modes(NNMs) is a branch of periodic solution of nonlinear dynamic systems. Determination of stable periodic solution is very important in many engineering applications since the stable periodic solution can be an attractor of such nonlinear systems. Periodic solutions of nonlinear system are usually calculated by perturbation methods and numerical methods. In this study, numerical method is used in order to calculate the NNMs. Iteration of the solution is presented by multiple shooting method and continuation of solution is presented by pseudo-arclength continuation method. The stability of the NNMs is analyzed using Floquet multipliers, and bifurcation points are calculated using indirect method. Proposed analyses are applied to two nonlinear numerical models. In the first numerical model nonlinear spring-mass system is analyzed. In the second numerical model Jeffcott rotor system which has unstable equilibria is analyzed. Numerical simulation results show that the multiple shooting method can be applied to self excited system as well as the typical nonlinear system with stable equilibria.