Sin Kx Research Articles

Некоторые свойства р ядов по синусам с моно тонными коэффициентами

We consider the sine series $$\mathop \sum \limits_{k = 1}^\infty a_k \sin kx$$ with monotone coefficients tending to zero and denote byg(x) its sum. We establish estimates of the integral ∝¦g¦dx over a given subinterval of (0,π]. These estimates are uniform with respect to the coefficientsak and the endpoints of the subinterval. In the particular case wheng is not integrable over the period, we get an asymptotic estimate of the growth order of the integral over [∈, π] as∈↓+0. It is of the same form as in the case of series with convex coefficients.

Modified newton–cotes formulae for numerical quadrature of oscillatory integrals with two independent variable frequencies

It is shown how a function can be approximated in a unique way by an interpolation function fn (x) which is a linear combination of cos kx, sin kx, cos k′x, sin k′xand a polynomial of {n– 4)th degree, such that fn coincides with f(x) in (n + 1) equidistant points, and whereby kand k′are two arbitrary real, purely imaginary or complex conjugate parameters. Several equivalent expressions of the interpolation function fn are given, and also the error term is derived in closed form. With this type of interpolation a set of modified Newton–Cotes quadrature rules is established and the total truncation error associated with these rules is discussed. Subsequently, the five-point quadrature rule is treated in full detail and several formulae are derived which facilitate numerical computation. Finally, the proposed formulae are implemented for certain illustrative numerical examples. In particular, a technique is established for attributing values to the parameters k and k′ such that the total truncation error is minimized.

On a correction of Numerov-like eigenvalue approximations for Sturm-Liouville problems

For computing approximations for the eigenvalues of Sturm-Liouville problems, Numerov's method and other finite-difference methods are often used, giving rise to an algebraic eigenvalue problem of which the smallest eigenvalues approximate the fundamental eigenvalue and the first few harmonics of the original problem. Recently the authors have derived a modified Numerov method with step-dependent coefficients, which integrates exactly the functions 1, x, x 2, x 3, sin kx and cos kx. The parameter k is calculated by minimizing the error term associated to the above modified multistep method equation. We shall show that this modified Numerov method delivers much more accurate eigenvalues than the classical Numerov method which is based on a same grid pattern. Several numerical experiments will be presented.

Dynamic periodical distortions of stripe domains in garnet films (abstract)

One of the problems arising in the development of VBL memory is to make stripe domains (SD) resistant to bending distortions. The paper considers a theoretical and experimental study of the mechanism of bending sinusoidal distortions of isolated SD caused by the sequence of bias fields (Hz) spatially homogeneous pulses compressing SD. The SD distortion dynamics was studied by the method of high speed photography with exposure of 10 ns. The SD critical width has been discovered. When reached, it first causes small distortions of the domain wall (DW). Further effects of the Hz pulses increases them, and after 10–12 field pulses the irreversible statically stable sinusoidal distortions of SD are finally formed. After that each subsequent Hz pulse just inverts the spatial structure of the distortions (i.e., transfers it from one bistable sin kx-type state to another: −sin kx). The performed theoretical analysis shows that the above effect is connected with the DB structure transformation observed when some critical SD width w* is reached. It has been experimentally discovered that the formed bending distortions of SD remain stable during the effect of the Hz pulse where SD width w≳w*, and tend to straighten when width w<w* is reached. The sequence of the above changes in the DW structure leads to the transition from one bistable state to another through an intermediate state with a small SD width. Experimentally found has been a close-to-linear dependence of spatial amplitude of distortions on the value of Hz. It has also been discovered that the distortions period and amplitude linearly increase together with the duration of Hz. The relation between the time of reaching w* and time of bending formation has been established.<UFMMM>

On a class of modified Newton—Cotes quadrature formulae based upon mixed-type interpolation

We present quadrature rules which integrate exactly not only polynomials up to a certain degree, but also the functions sin kx and cos kx (where k is a free parameter). The formulae we obtain are modified Newton—Cotes formulae. They are derived by replacing the integrand by an interpolation function of the form a cos kx + b sin kx + Σ n-2 j = 0 c j x j based on equally spaced nodes. The total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified Simpson's 3 8 rule. Numerical examples show the efficiency of the modified rules and the importance of the error term.

On a new type of mixed interpolation

We approximate every function f by a function f n ( x) of the form a cos kx + b sin kx + Σ n−2 i=0 c i x i so that f( jh) = f n ( jh) for the n + 1 equidistant points jh, j = 0,…, n. That interpolation function f n ( x) is proved to be unique and can be written as the sum of the nth-degree interpolation polynomial based on the same points and two correction terms. The error term is also discussed. The results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0. This new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations.

Periodic magnetic modulation of a two-dimensional electron gas: Transport properties

Electrical transport of the two-dimensional electron gas (2DEG) is considered in the presence of a perpendicular magnetic field( B) which is modulated weakly and periodically along one direction by − B 0 sin( Kx). The magnetoresistivity exhibits novel oscillations which: 1) are 90° shifted with respect to similar oscillations discovered for a 1D electrical periodic potential, and 2) their amplitude is an order of magnitude larger than for a corresponding electrical potential modulation with strength equal to h ̵ ω c0 = h ̵ eB 0/m ∗c .

Nonlinear oscillations in warm plasmas with initial velocity perturbations

The Vlasov equation is solved by a phase-space boundary integration method in order to investigate the nonlinear frequency of an electron plasma mode. Unlike the familiar discrete particle simulation codes in which a limited number of particles may be considered, the present model represents a more realistic plasma case in which a very large number (practically unlimited) of plasma particles evolving in their self-consistent collective field is investigated. The unperturbed collisionless one-dimensional plasma system consists of warm electrons having a water-bag distribution to simulate equilibrium, and of static ions. The initial perturbation is introduced by changing the boundaries of the electron plasma such that υu = υ0 (1 + α.sin kx) and υl = − υ0(1 − α sin kx), υu and υl representing the upper and the lower boundaries, respectively. This corresponds to a standing wave perturbation. The results obtained for different wavelength perturbations λ ≡ 2π/k, and for several perturbation amplitudes α, are presented and discussed.

Necessary and sufficient conditions for $L\sp{1}$ convergence of trigonometric series

It is shown that for the class of cosine series satisfying a(n)log n = o(l) and Aa(n) > 0 that integrability and L' convergence occur together. Relaxing the monotonicity to bounded variation we show that our previous result cannot be extended. It is well known that the condition a(n)log n = o(l) is both necessary and sufficient for L' convergence for some classes of Fourier cosine series. Here we show, for the class of cosine series satisfying a(n)log n = o(l) and Aa(n) > 0, that integrability and L' convergence occur together. Relaxing the monotonicity to bounded variation we show that our previous result [1] cannot be extended. Finally we show that a cosine series with Aan > 0 is integrable if the norm of the derivative of the partial sums of its conjugate series are bounded. In what follows f(x) = limnoo Sn(x) where n Sn(X) = 2a(0) + , [a(k)cos kx + b(k)sin kx]. k= 1 We denote an(x) = l/(n + l)EnX 0 Sk(x), and Sn(x) is the derivative of the conjugate of Sn (x). THEOREM 1. Let a(n)log n = o(l), b(n)log n = o(l), Aa(n) > 0, and Ab(n) > 0. Then I ISnII = o(n). PROOF. 1Sn|| E [ka(k)cos kx + kb(k)sin kx] k = 1 =| E1 {[kAa(k) a(k + l)1[Dk(x)+ [kAb(k) b(k + 1) ] Dk (x)} + na(n) [Dn (x) I + nb (n) Dn (x) Received by the editors August 6, 1975. AMS (MOS) subject classifications (1970). Primary 42A20, 42A32.

On 𝐿¹ convergence of certain cosine sums

Rees and Stanojevic introduced a new class of modified cosine sums {gn(x) = 2 '=O Aa(k) + 2k=1 2jk Aa(j)cos kx) and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series f with coefficients of bounded variation, a Rees-Stanojevic cosine sum gn can be associated such that gn converges to f pointwise, and a necessary and sufficient condition for L1 convergence of gn to f is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions. Theorem A gives a necessary and sufficient condition for a sine series with coefficients of bounded variation and converging to zero to be the Fourier series of its sum, or equivalently, for its sum to be integrable. Theorem B shows that if such a series is a Fourier series then its convergence is good, that is, convergence in the L1 metric. THEOREM A [1]. Let f(x) = 2 ??I b(n)sin nx where l\b(n) > 0 [?Ab(n) = b(n) b(n + 1)] and lim b(n) = 0. Then f E L1 [0, T] or, equivalently, b(n) sin nx is the Fourier series off if and only if n A Ib(n)I log n < oo. THEOREM B [1]. Let f(x) be as in Theorem A. If f E L'[O,] then k= I b(k) sin kx converges to f in the L1 metric. There is no known analogue of Theorem A for the cosine series. Theorems C and D only give sufficient conditions for the cosine series to be the Fourier series of its sum. In what follows we will denote by C the cosine series 00 2a(O) + 2 a(n)cos nx n=l where limn a(n) = 0 and n? Izla(n)l < 00. Partial sums of C will be denoted by Sn (x), and f (x) = limnOo Sn (x) THEOREM C [1]. If n '1 zIla(n) Ilog n < x, thenf E L1 [O, ] or, equivalently, C is the Fourier series off. THEOREM D [1]. If 2 ??1 I \2a(n)I(n + 1) < oo, then f e L1 [0, ] or, equivalently, C is the Fourier series of f. Received by the editors October 21, 1974 and, in revised form, January 9, 1975. AMS (MOS) subject classifications (1970). Primary 42A20, 42A32.

Methods of summation and best approximation

Let λ={λ k n } be a triangular method of summation,f e Lp (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f − Un (f, λ) ∥ Lp in terms of a best approximation En (f) Lp in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.

Variable Notch Antenna Patterns

Results of antenna pattern measurements of a notch antenna are presented for variable notch lengths and different modes of notch excitation. The notch is an open slot normal to the edge of a 12° metallic wedge. Antenna patterns of the Es component are measured in perpendicular planes with the intersection line coincident with the notch axis. The Eθ component in the φ = 180° plane is compared with analytical expressions for a notch cut in a semi‐infinite conducting sheet. The excitation modes, V(x), in the analytical notch expressions are Vo cos kx, Vo sin kx, Voeikx, and Voe−ikz. The experimental verification of the assumptions in the analytical formulations is very good.

Laminar Boundary Layer under a Wave

The laminar boundary-layer flow resulting from a wave potential flow of form U = αc sin(kx — ωt), with α a parameter and c the wave velocity, is considered. It is shown, after suitable transformation, that there is an exact solution of the unsteady boundary-layer equations which is of the form of a power series in the phase kx — ωt. The coefficients φn are functions of a similarity variable, and are the solutions of an infinite set of linear third-order differential equations with nonlinear forcing terms. The forcing term in the equation for φn is a function of φ0, φ1, …, φn−1 and their derivatives. Solutions for φ0, φ1, and φ2 have been computed and are presented. The theory is applied to the laminar boundary layer under a progressive shallow-water wave, where α = a/h, and compared to a linearized theory. It is concluded that if α ≪ 1, or for any α in a sufficiently small region near kx — ωt = 0, the linearized theory is valid. Otherwise, the linear theory does not provide an adequate description of the flow.

A Method for Determining the Ultimate Strength of a Column Swaged at Ends

Let ABC, Fig. 2, be a pin ended strut of length La deflected under its Euler load . Let DBE be a pin ended strut with a smaller moment of inertia, a shorter length Lb, but with the same ultimate column strength, deflected under its Euler load . It is assumed that the ends of the two struts lie on the straight line which is the line of action of the force P and that they are so deflected by this force that their elastic axes are tangent at B. On page 564, Vol. II , of Strength of Materials by Timoshenko, it is shown that at any point the deflection of a strut fixed at one end and with a sufficiently large axial load applied at the other end is given by the equation y=5(l —cos Kx). The origin for and x is at the fixed end and 6 is the maximum deflection which occurs a t the free end. For convenience the deflection at any point may be measured from the line of action of the force which passes through the free end and is parallel to the x axis: d — y = 5cosKx. It is evident that the fixed ended strut may be considered to be half of a symmetrical pin ended strut and that the value, d cos Kx, yields the ordinates to the elastic curve of a pin ended strut when the origin is at the intersection of a line drawn through the ends and the plane of symmetry. If the origin is a t the end of the strut, the value of the ordinate to the elastic curve at any point measured from the line of action of the force P through the ends becomes 5 sin Kx. AtB, Fig. 2, we have the deflection equal to