Sin Kx Research Articles

Optimal Sine and Sawtooth Inequalities

We determine the optimal inequality of the form $$\sum _{k=1}^m a_k\sin kx\le 1$$ , in the sense that $$\sum _{k=1}^m a_k$$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle.

Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients

The sum of a sine series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ . He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a sine series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.

Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients

It is known that the sum of the sine series $$g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}} $$bk sin kx whose coefficients constitute a convex sequence b is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function $$\sigma \left( {{\mathop{\rm b}\nolimits} ,\,x} \right) = {1 \over {m\left( x \right)}}\sum\limits_{k = 1}^{m\left( x \right) - 1} {{k^2}\left( {{b_k} - {b_{k + 1}}} \right),\,\,\,\,\,\,\,\,\,\,\,m\left( x \right) = \left[ {{{\rm{\pi }} \over x}} \right].} $$ He showed that the difference g(b, x) − (bm(x)/2)cot(x/2) in a neighborhood of zero admits a two-sided estimate in terms of the function a(b,x) with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.

AdS Q-soliton and inhomogeneously mass-deformed ABJM model

We study dual geometries to a deformed ABJM model with spatially dependent source functions at finite temperature. These source functions are proportional to the mass function m(x) = m0 sin kx and its derivative m′(x). As dual geometries, we find hairy black branes and AdS solitons corresponding to deconfinement phase and confining phase of the dual field theory, respectively. It turns out that the hairy AdS solitons have lower free energy than the black branes when the Hawking temperature is smaller than the confining scale. Therefore the dual system undergoes the first order phase transition. Even though our study is limited to the so-called Q-lattice ansatz, the solution space contains a set of solutions dual to a supersymmetric mass deformation. As a physical quantity to probe the confining phase, we investigate the holographic entanglement entropy and discuss its behavior in terms of modulation effect.

A Reliable Wave Convention for Oppositely Traveling Waves

Care must be taken representing waves traveling in opposite directions. Erroneous results can occur when combining waves with the popular convention where a wave traveling to the right is ψR(x,t) = A sin (kx–ωt) and a left-traveling wave is ψL(x,t) = A sin (kx + ωt). An easily overlooked pitfall is pointed out and a way to avoid it with a secure convention is presented. As an exercise in critical thinking, students can be presented with the pitfall and asked to find a way out.

Topological bounds for Fourier coefficients and applications to torsion

Let Ω⊂R2 be a bounded convex domain in the plane and consider−Δu=1inΩu=0on∂Ω. If u assumes its maximum in x0∈Ω, then the eccentricity of level sets close to the maximum is determined by the Hessian D2u(x0). We prove that D2u(x0) is negative definite and give a quantitative bound on the spectral gapλmax(D2u(x0))≤−c1exp⁡(−c2diam(Ω)inrad(Ω))for universalc1,c2>0. This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if f:T→R is continuous and has n sign changes, then∑k=0n/2|〈f,sin⁡kx〉|+|〈f,cos⁡kx〉|≳n|f‖L1(T)n+1‖f‖L∞(T)n. This statement immediately implies estimates on higher derivatives of harmonic functions u in the unit ball: if u is very flat in the origin, then the boundary function u(cos⁡t,sin⁡t):T→R has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with f:T→R cannot decay faster than ∼exp⁡(−(#sign changes)2t/4).

Addition to V. P. Zastavnyi’s paper “Estimates for sums of moduli of blocks in trigonometric Fourier series”

Results on the integrability and integrability with power weight for sums of moduli of blocks from the series Ʃ1/k sin kx are extended to the class of series Ʃb k sin kx with coefficients b k of a more general form.

Hybrid guiding-centre/full-orbit simulations in non-axisymmetric magnetic geometry exploiting general criterion for guiding-centre accuracy

To identify under what conditions guiding-centre or full-orbit tracing should be used, an estimation of the spatial variation of the magnetic field is proposed, not only taking into account gradient and curvature terms but also parallel currents and the local shearing of field-lines. The criterion is derived for general three-dimensional magnetic equilibria including stellarator plasmas. Details are provided on how to implement it in cylindrical coordinates and in flux coordinates that rely on the geometric toroidal angle. A means of switching between guiding-centre and full-orbit equations at first order in Larmor radius with minimal discrepancy is shown. Techniques are applied to a MAST (mega amp spherical tokamak) helical core equilibrium in which the inner kinked flux-surfaces are tightly compressed against the outer axisymmetric mantle and where the parallel current peaks at the nearly rational surface. This is put in relation with the simpler situation B(x, y, z) = B0[sin(kx)ey + cos(kx)ez], for which full orbits and lowest order drifts are obtained analytically. In the kinked equilibrium, the full orbits of NBI fast ions are solved numerically and shown to follow helical drift surfaces. This result partially explains the off-axis redistribution of neutral beam injection fast particles in the presence of MAST long-lived modes (LLM).

Bases of trigonometric polynomials consisting of shifts of Dirichlet kernels

The system of shifts of Dirichlet kernel on \(\frac{{2k\pi }} {{2n + 1}} \), k = 0, ± 1, …, ± n, and the system of such shifts of conjugate Dirichlet kernels with \(\frac{1} {2} \) are orthogonal bases in the space of trigonometric polynomials of degree n. The system of shifts of the kernels \(\Sigma _{k = m}^n \) cos kx and \(\Sigma _{k = m}^n \) sin kx on \(\frac{{2k\pi }} {{n - m + 1}} \), k = 0, 1, …, n−m, is an orthogonal basis in the space of trigonometric polynomials with the components from m ⩾ 1 to n. There is no orthogonal basis of shifts of any function in this space for 0 < m < n.

Некоторые специальные двумерные ряды по системе {sin x sin kx} и их аппроксимативные свойства

Некоторые специальные двумерные ряды по системе {sin x sin kx} и их аппроксимативные свойства

Дискретные преобразования со свойством прилипания на основе системы {sin x sin kx} и системы полиномов Чебышёва второго рода

Дискретные преобразования со свойством прилипания на основе системы {sin x sin kx} и системы полиномов Чебышёва второго рода

О свойствах модулей блоков членов ряда Σ 1/k sin kx

О свойствах модулей блоков членов ряда Σ 1/k sin kx

Remarks on the uniform and L 1-convergence of trigonometric series

Chaundry and Jolliffe [1] proved that if ak is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series Σk=1∞ak sin kx is limk→∞kak = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L1-convergence of Fourier series.

Generalization of an elementary inequality in Fourier analysis

The inequality \( \mathop {\sup }\limits_{n \geqslant 1} \left| {\sum\limits_{k = 1}^n {\frac{{\sin kx}} {k}} } \right| \leqslant 3\sqrt \pi , \) plays an important role in Fourier analysis and approximation theory. It has recently been generalized by Telyakovskii and Leindler. This paper further generalizes and improves their results by introducing a new class of sequences called γ-piecewise bounded variation sequence (γ-PBVS).

Perturbed periodic solution for boussinesq equation

We consider the solution of the good Boussinesq equation Utt-Uxx+Uxxxx=(U2)xx,-∞<x<∞, t≥0, with periodic initial value U(x,0)=ɛ(μ+ϕ(x)),Ut(x,0)=ɛψ(x),-∞<x<∞, where μ ≠ 0, ϕ(x) and ψ(x) are 27π-periodic functions with 0-average value in [0, 2π], and ɛ is small. A two parameter Backhand transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ɛ, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form ϕ(x)=μ + a sin kx, ψ(x)= an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T > 0, the difference between the true solution u(x,t;ɛ) and the N-th partial sum of the asymptotic series is bounded by φN+1 multiplied by a constant depending on T and N, for all −∞ <x < ∞, 0 ⩽ |ɛ|t and 0⩽|ɛ| ⩽ ɛ0.

A Two-Parameter Trigonometric Series

00 B. Given a trigonometric series Ek=l (Ak cos kx + Bk sin kx), find a function f (x) such that the given trigonometric series is its Fourier series (then we can find the sum of the given trigonometric series). Upon learning the theory of Fourier series, we know questions of type A are straightforward but questions of type B are not, for a given trigonometric series may not even be a Fourier series, as

Response of Large Membrane Optical Surfaces to Acoustic Environment

Acoustic launch vibration is often the critical source of vibration for space telescopes. The large surfaces of membrane optics act as receivers of acoustic energy and can experience significant stresses on launch and significant displacements during laboratory testing. Using the finite element software MSC.NASTRAN, several membrane reflector geometries were analyzed under applied axial tension to the membrane surfaces. The pressure field was applied with a sin (kx)/(kx) spatial correlation. Stress and displacements were found at selected points on the membrane surface. Recommendations regarding membrane launch and laboratory testing states are presented in this paper.

Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem u tt - u xx + u = ε ( u t - 1 3 u t 3 ) , - ∞ < x < ∞ , t ⩾ 0 , u ( x , 0 ) = sin kx , u t ( x , 0 ) = 0 , when t is restricted to the interval [ 0 , T / ε ] , where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u ( t , x ; ε ) and the Nth partial sum of the asymptotic series is bounded by ε N + 1 multiplied by a constant depending on T but not on x and t.

Spin-Triplet Superconductivity Induced by Charge Fluctuations in Extended Hubbard Model

The pairing symmetry in the electron mechanism for superconductivity is explored when charge fluctuations coexist with spin fluctuations. The extended Hubbard model is adopted to obtain, with the fluctuation exchange approximation, a phase diagram against the on-site Coulomb repulsion U and the off-site repulsion V for the square lattice with second-neighbor hopping t'. We have found that (i) for large U(>9) a triplet superconductivity with a sin(kx+ky) symmetry can appear just below the charge density wave phase. The pairing is degenerate with sin(kx-ky), so a chiral sin(kx+ky)+isin(kx-ky) that breaks the time reversal symmetry should result, which is a candidate for the gap function on the gamma band of Sr2RuO4 and is consistent with a recent measurement of the specific heat. (ii) By systematically deforming the Fermi surface with varied t', we have identified the region where the triplet pairing is most favored to be the region where the Fermi surface traverses the van Hove singularity with the charge susceptibility strongly enhanced.

Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission

A recently proposed concept suggests that a matched periodic modulation of both the refractive index and the gain/loss of the media breaks the coupling symmetry of the two co-propagating modes and allows only a unidirectional coupling from the i-th mode to j-the mode but not the opposite. This concept has been used to design a ring resonator coupled through a complex grating composed of both real (index) and imaginary (loss/gain) parts according to Euler relation: n = n0 exp(-jkx) = n0 (cos(kx) - j sin(kx)). Such asymmetrical coupling allows light to be coupled into the ring without letting it out. We present a detailed theoretical analysis of the ring resonator in the linear regime, and we investigate its linear temporal dynamics. Three possible states of the complex grating leads to the possibility of developing a dynamic optical memory cell where, for example, a data modulated train of optical pulses can be stored. This data can be accessed without destroying it, and can also be erased thus permitting the storage of a new bit. Finally, the ring can be used for pulse retiming.