Periodic Harmonic Functions Research Articles

A Carleman inequality on product manifolds and applications to rigidity problems

Abstract In this article, we prove a Carleman inequality on a product manifold M × R M\times {\mathbb{R}} . As applications, we prove that (1) a periodic harmonic function on R 2 {{\mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.

선박용 디젤엔진 추진축에서 빙 충격 토크 기진에 의한 과도 비틀림 진동 응답

In recent years, there has been an increasing demand to apply the new IACS(International Association of Classification Societies) standards for ice and polar-classed ships. For ice-class vessel propulsion system, the ice impact torque design criterion is defined as a periodic harmonic function in relation to the number of the propeller blades. However, irregular or transient ice impact torque is assumed to occur likely in actual circumstances rather than these periodic loadings. In this paper, the reliability and torsional vibration characteristics of a comparatively large six-cylinder marine diesel engine for propulsion shafting system was examined and reviewed in accordance with current regulations. In this particular, the transient ice impact torque and excessive vibratory torque originating from diesel engine were interpreted and the resonant points identified through theoretical analysis. Several floating ice impacts were carried out to evaluate torque responses using the calculation method of classification rule requirement. The Newmark method was used for the transient response analysis of the whole system.

Stability analysis of a whirling rigid rotor supported by stationary grooved FDBs considering the five degrees of freedom of a general rotor-bearing system

This paper proposes a method to determine the stability of a whirling rotor supported by stationary grooved fluid dynamic bearings (FDBs), considering the five degrees of freedom of a general rotor-bearing system. Dynamic coefficients are calculated by using the finite element method and the perturbation method, and they are represented as periodic harmonic functions. Because of the periodic time-varying dynamic coefficients, the equations of motion of the rotor supported by FDBs can be represented as a parametrically excited system. The solution of the equations of motion can be assumed to be a Fourier series, allowing the equations of motion to be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Hill's infinite determinant is calculated by using these algebraic equations to determine the stability. Increasing rotational speed increases $$K_{xx}$$Kxx and $$K_{{\theta_{x} \theta_{x} }}$$K?x?x, which decreases the stability of the stationary grooved FDBs; increasing whirl radius increases the stability of the FDBs because the resulting increases in the averages and variations of $$C_{xx}$$Cxx and $$C_{{\theta_{x} \theta_{x} }}$$C?x?x increase the stability faster than the corresponding increases of $$K_{xx}$$Kxx and $$K_{{\theta_{x} \theta_{x} }}$$K?x?x decrease the stability. The proposed method was verified by investigating the convergence and divergence of the whirl radius after the equations of motion were solved using the fourth-order Runge---Kutta method.

Periodic harmonic functions on lattices and Chebyshev polynomials

We shall give an explicit expression of the dimension of the space of harmonic functions on the Cartesian product of path (resp. cycle) graphs in terms of Chebyshev polynomials of the second (resp. first) kind. As an application, we obtain several identities among the dimensions, some are new and some are known but obtained previously by other methods. Our motivation for this study is the “Lights Out” puzzle.

TKA wear testing input after kinematic and dynamic meta-analysis: Technique and proof of concept

Abstract Simulation under conditions that respect real kinematics and dynamics of applying a load to the components can be an important instrument to predict the behaviour of new prosthesis designs and materials during the time of wear. Various studies have shown that the gait of TKA patients is different from the gait of healthy subjects. Therefore the time behaviour which is proposed in the ISO standard does not really represent the loads in TKA patients. Most of the current testing laboratories simulate continuous walking according to ISO 14°243, but neglect everyday activities. It is impossible to simulate the real movement of specific patients. Hence, a loading mode (with respect to daily activities) was determined, based on experimental studies. The length and frequency of individual loading were also defined. The testing mode includes normal walk, sitting–standing, cycling, stair climbing and walking with extra load. The device is able to perform the ISO walk as well. The simulator has been designed to reproduce the configuration of the implanted prosthesis. The simulated motion is described by periodic harmonic function, which enables setting the proper position in required time. The duration of a compound load process reflecting daily activities lasted 240 min. The compound load consists of elementary sub cycles, each of them representing one motion phase. A relaxation phase is inserted between all active movements. Several differences were observed between known experimental results and the present standard ISO 14°243.

Resurgence flows in three-dimensional periodic porous media

Three-dimensional flows are addressed in porous media with resurgences. These media are characterized by a double structure, i.e., a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium. Spatially periodic harmonic functions with singularities at given points are constructed by means of the Berdichevskij functions; these functions extend to three dimensions the classical Weierstrass functions. The solution of the same medium with two vertices is obtained in two and three dimensions and thoroughly compared and discussed.

Periodic harmonic functions on lattices and points count in positive characteristic

Abstract This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.

A Mathematical Note: The Dipole Potential for the Swaying and Rolling Cylinder

In this study, the total velocity potential, for the swaying and rolling cylinder expressed as a sum of a series of linear multi-pole potentials and a dipole potential situated at the origin for incom- pressible, inviscid and irrotational flow. The non-dimensional expression of the dipole potential and its conjugated stream functions are re-obtained by using the mathematical expressions with complete alge- braic manipulations. The total velocity potential UT ðx; y; tÞ, for the swaying and rolling cylinder which is a periodic harmonic function of time t, may be expressed as a sum of a series of linear multi-pole potentials and a dipole potential situated at the origin (1),(2). The x-axis is horizontal which coincides with the free surface of the fluid and is perpendicular to the axis of the cylinder and the y-axis is vertical, positive downward and going through the mean position of the axis of the cylinder. De Jong (1) formulated the problem as follows: In a fluid of infinite depth a cylinder is considered which is oscil- lating one-dimensionally and harmonically with a frequency of x while the mean posi- tion of its axis is assumed to lie in the free surface of the undisturbed fluid. De Jong assumed the amplitude of the oscillation to be small with respect to the diam- eter of the cylinder and the length of the waves generated by the oscillation, so that in the linearized approximation, the values of all physical quantities could be referred to the mean position of the cylinders. Taking the cylinder very long with respect to the breadth or enclosing the cylinder at both ends between two infinitely long walls per- pendicular to the axis of the cylinder, De Jong neglected the velocity components par- allel to the axis of the cylinder and consequently the motion is two-dimensional. The determination of the motions of the fluid under influence of the harmonic oscillation of the cylinder could be reduced to the solution of a boundary-value prob- lem from the linear potential theory. So, the velocity potential UT ðx; y; tÞ is also a periodic harmonic function of time. Therefore, the potential expressed is given in the following form, using complex notation, UTðx; y; t Þ¼� i/ T ðx; yÞe ixt : ð1Þ

The hilbert transform of periodic distributions

Let f(t) be a periodic function with period 2τ defined on the real line R. A new definition of the Hilbert transform (Hf)(x) of this function is given by provided this limit exists; the integral in (1) is taken in the Cauchy principal-value sense. It is shown that the definition (1) is equivalent to the conventional definition, where the integral in (2) is taken in the Cauchy principal value sense too. We then use our results to extend the Hilbert transform to periodic distributions of period 2τ defined on the real line, i.e. to is the space of infinitely differentiable functions with period 2τ defined on the real line. An inversion formula over the space is proved. Amongst many other related results which are proved, we have proved the existence of a periodic harmonic function u(x y) (periodic in x with period 2τ) whose boundary values at a semicircular boundary are given.

One-Dimensional Heat Conduction in a Semi-infinite Solid With the Surface Temperature a Harmonic Function of Time: A Simple Approximate Solution for the Transient Behavior

This paper discusses a simple approximate solution for the transient behavior of the temperature distribution in a semi-infinite solid where the temperature at the surface origin is assumed to be a periodic harmonic function of time. The temperature distrbution in the medium is governed by the rate of heat conduction, the initial conditions, and the boundary conditions. The knowledge of the behavior of the transient part is necessary in some engineering problens such as the prediction of the transient behavior of recuperator or hardening furnaces, when modeling as a semi-infinite solid with a harmonic boundary condition holds. Therefore, and also to satisfy theoretical requirements, a detailed investigation of the thermal transient is presented.

Asymptotic behavior of periodic, periodic biharmonic and periodic harmonic functions

The behavior of periodic functions defined on domains containing the upper half space, { ( x 1 , x 2 , . . . , x n ) : x n > 0 } \left \{ {\left ( {{x^1},{x^2},...,{x^n}} \right ):{x^n} > 0} \right \} , is investigated as x n {x^n} approaches infinity. Bounds on some of the first order derivatives of these functions are obtained which are directly proportional to bounds on derivatives of arbitrary orders in certain directions. It is shown that a periodic biharmonic and a periodic harmonic function can be approximated, respectively, by a third degree and a first degree polynomial in the variable x n {x^n} and that, as x n {x^n} approaches infinity, the error in using this approximation vanishes faster than the reciprocal of x n {x^n} raised to any power.

Averaged description of a fluid containing gas bubbles

An averaged equation is derived that describes the irrotational flow of an ideal incompressible fluid due to the expansion and translational motion of bubbles at the sites of a periodic lattice. The calculation of the coefficients of the averaged equation reduces to the solution of a cell problem. An exact solution to the problem is constructed in the form of a series in periodic harmonic functions. An infinite system of equations is written down for the coefficients of the series, and the system is analyzed asymptotically at low volume concentrations c of the bubbles.

On Invariant Perforation in an Infinite Strip

Abstract The invariant perforation in an infinite strip can be classified into two groups. One is the finite group and the other is the infinite group. There are five cases in the finite group and nine cases in the infinite group. All the cases can be solved by the method of images. This method has, in fact, been used by the author to solve the stresses in an infinite strip containing either an unsymmetrically located single hole or a series of uniformly distributed equal holes. The solution is illustrated by working out in detail one of the cases in the infinite group, in which the strip contains two series of equal holes symmetrically staggered along the strip. The stress function is constructed by using a class of periodic harmonic functions derived from Weierstrass’ sigma function. Numerical examples also are given to show the effect of such a perforation on the stresses in the strip.

Stresses in a Perforated Strip

Abstract This paper presents an analytic solution of the classical problem dealing with the stresses in an infinite strip having an unsymmetrically located perforating hole. The solution is applicable to any stress system acting in the strip, which is symmetrical with respect to the line of symmetry of the strip. The required stress function is constructed by using four series of biharmonic functions and a bihamonic integral. The four series of biharmonic functions are formed from a class of periodic harmonic functions specially constructed for the purpose. The solution can be regarded as a complete solution of the problem in the sense that, unlike the previous solutions by Howland, Stevenson, and Knight for a symmetrically perforated strip, it is valid in the entire strip. Numerical examples are given for the fundamental cases of longitudinal tension and transverse bending.