Cos Kx Research Articles

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).

On a cosine polynomial of Turán and its sine counterpart

In 1953, Turan published the inequality $$\begin{aligned} \sum _{k=1}^n \frac{1\cdot 3 \cdots (2k-1)}{2\cdot 4 \cdots 2k}\cos (kx)>-1 \quad {(n\in \mathbf {N}; \, 0<x<\pi )}. \end{aligned}$$ We prove a refinement of this result and offer inequalities for the corresponding sine polynomial. Moreover, we present an application to the theory of absolutely monotonic functions.

Oscillating cosmological model with a varying Λ $\varLambda $ term in Barber’s second self-creation theory of gravitation

We investigate the spatially homogeneous and isotropic FRW cosmological model with a time varying \(\varLambda \) term in Barber’s second self-creation theory. The field equations of this theory are solved by using a time periodic varying deceleration parameter \(q=m\cos kx -1\) with \(m\) and \(k\) being positive constants. From the derivation of the expressions for the statefinder parameters \(r\) and \(s\), it is obtained that the values of \(r\) and \(s\) of the present model can reduce to statefinder parameters of standard \(\varLambda \)CDM model \(r=1\) and \(s=0\) only when \(m=\frac{3}{2}\).

Effect of interlayer single-particle hoppings on the superconducting transition temperature

It is well known that the superconducting transition temperature of high-Tc cuprates depends on the number of CuO2 planes in the unit cell. The multilayer structure implies the possibility of interlayer hopping. Under the assumption that the interlayer hopping can be specified by the parameter t⊥(k) = t⊥(cos(kx) − cos(ky))2, the quasiparticle excitation spectrum for the bilayer cuprate in the superconducting state has been determined in the framework of the t − t′ − t″ − t⊥ − J* model using the generalized mean-field approximation. It turns out that the interlayer hoppings does not create any additional mechanism of the Cooper paring and does not lead to an increase in Tc. The splitting of the upper Hubbard quasiparticle band attributed to the interlayer hoppings is manifested as two peaks in the doping dependence of the superconducting transition temperature at temperatures below the maximum Tc value for a single-layer cuprate. It has been found that antiferromagnetic interlayer correlations suppress the interlayer splitting. This probably leads to the common doping dependence of Tc for both single-layer and bilayer cuprates.

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

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series \(\sum\nolimits_{k = 1}^\infty {a_k \sin kx}\) and \(\frac{{a_0 }}{2} + \sum\nolimits_{k = 1}^\infty {a_k \cos kx}\) whose coefficients satisfy the condition \(a_k = a_{n_m }\) for \(n_{m - 1} < k \leqslant n_m\), where the sequence \(\left\{ {n_m } \right\}\) can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If \(ka_k \to 0\) as \(k \to \infty\), then the series \(\sum\nolimits_{k = 1}^\infty {ak} sinkx\) is uniformly convergent. If \(k\left| {a_k } \right| \leqslant C\) for all \(k\), then the sequence of partial sums of this series is uniformly bounded. If the series \(\frac{{a_0 }}{2} + \sum\nolimits_{k = 1}^\infty {a_k \cos kx}\) is convergent for \(x = 0\) and \(ka_k \to 0\) as \(k \to \infty\), then this series is uniformly convergent. If the sequence of partial sums of the series \(\frac{{a_0 }}{2} + \sum\nolimits_{k = 1}^\infty {a_k \cos kx}\) for \(x = 0\) is bounded and \(k\left| {a_k } \right| \leqslant C\) for all \(k\), then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence \(\left\{ {n_m } \right\}\) is lacunary. In the general case, they are not necessary.

Trace Formula for Sturm--Liouville Operators with Singular Potentials

Suppose that u(x) is a function of bounded variation on the closed interval [0,π], continuous at the endpoints of this interval. Then the Sturm—Liouville operator Sy=−y″+q(x) with Dirichlet boundary conditions and potential q(x)=u′(x) is well defined. (The above relation is understood in the sense of distributions.) In the paper, we prove the trace formula $$\sum\limits_{k = 1}^\infty {\left( {\lambda _k^2 - k^2 + b_{2k} } \right)} = - \frac{1}{8}\sum {h_j^2 } , b_k = \frac{1}{{\pi }}\int_0^\pi cos kx du (x),$$ where the λk are the eigenvalues of S and hj are the jumps of the function u(x). Moreover, in the case of local continuity of q(x) at the points 0 and π the series \(\sum\nolimits_{k = 1}^\infty {\left( {\lambda _{\,k} - k^2 } \right)}\) is summed by the mean-value method, and its sum is equal to $$ - \frac{{\left( {q\left( 0 \right) + q\left( \pi \right)} \right)}}{4} - \frac{1}{8}\sum {h_j^2 } .$$

PROXIMITY OF THE METAL-INSULATOR/MAGNETIC TRANSITION AND ITS IMPACT ON THE ONE-ELECTRON SPECTRAL FUNCTION: A DOPING-DEPENDENT ARPES STUDY

The doping dependence of the low-temperature spectral function is precisely determined from angle resolved photoemision (ARPES) measurements. It is found that, as the doping decreases, the maximum of the superconducting gap increases, but the slope of the gap near the nodes decreases. Though consistent with d-wave symmetry, the gap with underdoping cannot be fit by the simple cos(kx)-cos(ky) form. We suggest that this arises due to the increasing importance of long range interactions as one approaches the insulator. It is also shown that the shape of the spectral function at the (π, 0) point below T c can be explained by the interaction of the electrons with a collective mode whose energy matches that of the magnetic resonance as obtained by inelastic neutron scattering data, and points to the intimate relation of magnetic correlatins to high Tc superconductivity.

CHANGES IN SUPERCONDUCTING GAP ANISOTROPY WITH DOPING AND IMPLICATIONS FOR THE PENETRATION DEPTH

The momentum dependence of the superconducting gap at low temperatures is precisely determined from angle resolved photoemission (ARPES) measurements on overdoped and underdoped samples of Bi2212. As the doping decreases, the maximum gap increases, but the slope of the gap near the nodes decreases. Though consistent with d-wave symmetry, the gap with underdoping cannot be fit by the simple cos (kx)- cos (ky) form. We suggest that this arises due to the increasing importance of long range interactions as one approaches the insulator. Furthermore, a comparison of our ARPES results with available penetration depth data clearly indicates that the renormalization of the linear T suppression of the superfluid density at low temperatures due to quasiparticle excitations around the d-wave nodes is large and doping dependent.

Many-dimensional periodicity and turbulence of Rayleigh–Taylor and Richtmyer–Meshkov instabilities

The work begins with a presentation of results concerning periodic Rayleigh-Taylor (RT)and Richtmyer–Meshkov (RM) dynamics. The periodic RT flows are considered in 2D and 3D cases. The periodic RM flows are considered in a 2D case. Nonstationary and steadystate statements of RT and RM problems are given. Nonstationary statements of RT and RM problems are given in the2D case. Steady-state statements of RT and RM problems are considered in the 2D and 3D cases (RT) and in the 2D case (RM). Questions about turbulence are given in the second part of the work. In the nonstationary statement, the beginning of a transition from an equilibrium close to hydrostatics and behavior of a dynamic system when it is close to the steady-state solution are considered. To study the smooth evolution of the dynamic system in the RT case (at last during an initial of order of unity t ≃ 1 stage) we have to cut of f an inverse turbulent cascade incoming from the small scales (inverse means that the typical k decreases in time during the cascade). The cascade starts from some small scale k ≫ 1 (here g = 1) and appears in the scales k ≃ 1 after time t ≃ 1 after start. We have to consider exponentially smoothed initial data to cut it off. This means that amplitudes of initial spectrum of perturbations have to be exponentially decreased in the direction of high k. The steadystates have been studied in two ways. In both of them the spectral decomposition of a velocity field and an elevation of the contact surface has been used. In one of them, corresponding to the nonstationary situation, the ordinary differential system for time dependent spectral components has been integrated. The other solutions of the algebraic equations for constant components have been studied. These results have been compared with a computer simulation that has been done by large particles and artificial compressibility methods. High-order nonlinear terms corresponding to the interaction of a wide spectrum of harmonics are considered in both ways. The spatial structure of the many dimensional periodic solutions is studied. It is known that there are two important features in the flows under consideration: the bubble envelope and the jet “phase.” Spatial structures of the 2D and 3D flows are compared in the report. It is shown that the apices of the bubbles are isolated points and the bubble envelopes in both flows are qualitatively similar; the difference has a quantitative character. On the contrary, the jet “phases” differ qualitatively in the 2D and 3D cases. In the 2D case in the plane picture, the jets are separated from each other. In this picture they are isolated fingers. In the 3D case, near the bubble envelope after some time of development following after cos kx cos qy type initial perturbation, they are not the isolated fingers. They form a structure of the intersecting walls. We see, thus, that the 2D and 3D cases are topologically different. Most powerful ejections of dense fluid are in the points where these wall jets intersect each other.

Large-scale and periodic modes of rectangular cell flow

Linear stability of the rectangular cell flow: Ψ=cos kx cos y (0&amp;lt;k&amp;lt;1), is studied, both numerically and analytically. Owing to its spatial periodicity, the disturbances are characterized by the Floquet exponents (α,β). Based on numerical results, it is found that two types of the critical modes with vanishingly small exponents exist. One type (large-scale mode) has an almost uniform spatial structure. The other type (periodic mode) has a structure with the same periodicity as the main flow. The large-scale mode gives the critical Reynolds number in a more isotropic case (i.e., k≳0.6), while the periodic mode does so in the less isotropic case (i.e., k&amp;lt;0.6). Asymptotic expansions from (α,β)=(0,0) agree with the numerical results. Using the periodic mode, a possible explanation is given for the merging process of a pair of counter-rotating vortices observed in the experiments of a linear array of vortices by Tabeling et al. [J. Fluid Mech. 213, 511 (1990)].

Faraday waves: rolls versus squares

Weakly nonlinear capillary—gravity waves of frequency ω and wavenumber k that are induced on the surface of a liquid in a square cylinder that is subjected to the vertical displacement a0 cos 2ωt are studied on the assumptions that: 0 &lt; δ &lt; ka0, where δ is the linear damping ratio; the dominant modes are cos kx and cos ky, where x and y are Cartesian coordinates in a horizontal plane. The formulation extends those of Simonelli &amp; Gollub (1989), Feng &amp; Sethna (1989), Nagata (1989, 1991) and Umeki (1991) by incorporating capillarity, cubic forcing and cubic damping. The results are also applicable to a laterally unbounded fluid, but the basic symmetry then is hypothetical rather than imposed by the boundaries. Canonical evolution equations for the modal amplitudes are determined from an average Lagrangian. The fixed points of the evolution equations comprise: (i) the null solution; (ii) an orthogonal pair of rolls described by either cos kx or cos ky; (iii) an orthogonal pair of squares described by either cos kx + cos ky or cos kx – cos ky; (iv) coupled-mode solutions for which both modes are active and neither in phase nor in antiphase. The solutions for squares are isomorphic to those for rolls through a linear transformation of the coefficients in the Hamiltonian. The fixed points for rolls and squares lie on separate loci in an energy–frequency plane that intersect the null solution at a pair of pitchfork bifurcations, one of which is definitely supercritical and the other of which may be either subcritical or supercritical. The parametric domain of the various solutions includes subdomains in which squares/rolls are stable/unstable and conversely. In the limiting case of deep-water capillary waves in the threshold domain 0 &lt; ka0 — δ &lt; 8δ3/9 all of the rolls and coupled-mode solutions are unstable, while squares are stable except for fixed points between the subcritical bifurcation (if it exists) and the corresponding turning point.

On the differential geometry approach to geophysical flows

Abstract It is shown that a variety of geophysically relevant flows admit a geometric interpretation and may be treated as geodesic flows on certain infinite-dimensional manifolds. As an illustration of this approach we present an example of the quasi-geostrophic flows. The curvature tensor determining the separation of geodesics and, hence, stability is calculated. It is shown that on the ƒ-plane the curvature in the sections containing a stationary periodic flow with a stream function given by cos( kx + ly ) and a perturbation of the same functional form is negative, like in the case of conventional 2-d hydrodynamics. On the contrary, for any stationary shear flow on the β-plane the curvature is always positive for large enough values of β.

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.

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.

Quasiparticles and photoemission spectra in correlated fermion systems

We present some selected numerical results for the spectral function of two models currently discussed for HTSC. For the t-J model we find a quasiparticle band with dispersion ϵk→ ≈ J2(cos kx + cos ky)2 well separated from a broad continuum of states (7t). For the Emery model photoemission and inverse photoemission spectra for the insulating (undoped) and for doped cases are discussed. In the doped case new states with strong O-character appear in the charge transfer gap, which appear to be responsible for conductivity in hole-doped HTSC.

Arbitrarily slow decay of correlations in quasiperiodic systems

For diffusive motion in random media it is widely believed that the velocity autocorrelation function c(t) exhibits power law decay as time t ..-->.. infinity. We demonstrate that the decay of c(t) in quasiperiodic media can be arbitrarily slow within the class of integrable functions. For example, in d = 1 with a potential V(x) = cos x + cos kx, there is a dense set of irrational k's such that the decay of c(k, t) is slower than 1/t/sup (1+epsilon/)/ for any epsilon > 0. The irrationals producing such a slow decay of c(k, t) are very well approximated by rationals.

The instability of rhombic cell flows

Instability of two-dimensional periodic flows with rhombic cell structure represented by the stream function ψ=cos kx+cos y is investigated. Stability characteristics are obtained for the Reynolds number R=1, 2, 3 and 4 and the ratio of the diagonals of the cell k=1, ½ and ¼. Variation of the critical Reynolds number Rc with k is obtained, and the square cell flow (k=1) is found to be most stable (Rc=√2). It is found that Rc → 1 as k → 0, which leads to a finite gap between this limiting Rc and Rc=√2 for K=0 (ψ=cos y).