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Yuan Xu

Publications and source records attributed to Yuan Xu.

At least 199 records · Page 11Linked to original sources
arXiv2008

Discrete Fourier analysis on fundamental domain of $A_d$ lattice and on simplex in $d$-variables

A discrete Fourier analysis on the fundamental domain $Ω_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $Ω_2$ is the regular hexagon and $Ω_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.

math.CA↗
arXiv2008

Cubature formula and interpolation on the cubic domain

Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$.

math.NA↗
arXiv2008

Spin transfer torques in nonlocal lateral spin valve

We report a theoretical study on the spin and electron transport in the nonlocal lateral spin valve with non-collinear magnetic configuration. The nonlocal magnetoresistance, defined as the voltage difference on the detection lead over the injected current, is derived analytically. The spin transfer torques on the detection lead are calculated. It is found that spin transfer torques are symmetrical for parallel and antiparallel magnetic configurations, which is different from that in conventional sandwiched spin valve.

cond-mat.mes-hall↗
arXiv2008

Spin-transfer torques in anti-ferromagnetic metals from first principles

In spite of the absence of a macroscopic magnetic moment, an anti-ferromagnet is spin-polarized on an atomic scale. The electric current passing through a conducting anti-ferromagnet is polarized as well, leading to spin-transfer torques when the order parameter is textured, such as in anti-ferromagnetic non-collinear spin valves and domain walls. We report a first principles study on the electronic transport properties of anti-ferromagnetic systems. The current-induced spin torques acting on the magnetic moments are comparable with those in conventional ferromagnetic materials, leading to measurable angular resistances and current-induced magnetization dynamics. In contrast to ferromagnets, spin torques in anti-ferromagnets are very nonlocal. The torques acting far away from the center of an anti-ferromagnetic domain wall should facilitate current-induced domain wall motion.

cond-mat.mtrl-sci↗
arXiv2008

New cubature formulae and hyperinterpolation in three variables

A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube.

math.NA↗
arXiv2008

Cesàro means of Jacobi expansions on the parabolic biangle

We study Cesàro $(C,δ)$ means for two-variable Jacobi polynomials on the parabolic biangle $B=\{(x_1,x_2)\in{\mathbb R}^2:0\leq x_1^2\leq x_2\leq 1\}$. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Cesàro operator can be interpreted as a convolution operator. We then show that the Cesàro $(C,δ)$ means of the orthogonal expansion on the biangle are uniformly bounded if $δ>α+β+1$, $α-\frac 12\geqβ\geq 0$. Furthermore, for $δ\geqα+2β+\frac 32$ the means define positive linear operators.

math.CA↗
arXiv2008

Discrete Fourier analysis on a dodecahedron and a tetrahedron

A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^3$.

math.NA↗
arXiv2008

On a two variable class of Bernstein-Szego measures

The one variable Bernstein-Szego theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.

math.CA↗
arXiv2008

Fourier series and approximation on hexagonal and triangular domains

Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of the objective of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that allow an extensive theory for Fourier expansions and approximation.

math.CA↗
arXiv2008

First principles study on the spin transfer torques

An efficient first principles method was developed to calculate spin transfer torques in layered system with noncollinear magnetization. The complete scattering wave function is determined by matching the wave function in the scattering region with the Bloch states in the leads. The spin transfer torques are obtained with aid of the scattering wave function. We applied our method to the ferromagnetic spin valve and found that the material (Co, Ni and Ni80Fe20) dependence of the spin transfer torques could be well understood by the Fermi surface. Ni has much longer spin injection penetration length than Co. Interfacial disorder is also considered. It is found that the spin transfer torques could be enhanced by the interfacial disorder in some system.

cond-mat.mes-hall↗
arXiv2007

Krall--type Orthogonal Polynomials in several variables

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.

math.CA↗
arXiv2007

Orthogonal polynomials and partial differential equations on the unit ball

Orthogonal polynomials of degree $n$ with respect to the weight function $W_μ(x) = (1-\|x\|^2)^μ$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ Δ- \la x, \nabla \ra^2 - (2 μ+d) \la x, \nabla \ra \right ] P = -n(n+2 μ+d) P $$ for $μ> -1$. The singular case of $μ= -1,-2, ...$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $ν= -2,-3,...$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed.

math.CA↗
arXiv2007

Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.

math.NA↗
arXiv2007

Spin Dependence of Interfacial Reflection Phase Shift at Cu/Co Interface

The spin dependent reflection at the interface is the key element to understand the spin transport. By completely solving the scattering problem based on first principles method, we obtained the spin resolved reflectivity spectra. The comparison of our theoretical results with experiment is good in a large energy scale from Fermi level to energy above vacuum level. It is found that interfacial distortion is crucial for understanding the spin dependence of the phase gain at the Cu$|$Co interface. Near the Fermi level, image state plays an important role to the phase accumulation in the copper film.

cond-mat.mtrl-sci↗
arXiv2007

Boundedness of projection operators and Cesàro means in weighted $L^p$ space on the unit sphere

For the weight function $\prod_{i=1}^{d+1}|x_i|^{2\k_i}$ on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Cesàro $(C,δ)$ means in the weighted $L^p$ space for $δ$ below the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex.

math.CA↗
arXiv2007

Cesàro means of orthogonal expansions in several variables

Cesàro $(C,δ)$ means are studied for orthogonal expansions with respect to the weight function $\prod_{i=1}^{d}|x_i|^{2\k_i}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the $(C,\d)$ kernel with $\d > -1$ and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Cesàro means and the projection operator on these domains.

math.CA↗
arXiv2007

Reconstruction from Radon projections and orthogonal expansion on a ball

The relation between Radon transform and orthogonal expansions of a function on the unit ball in $\RR^d$ is exploited. A compact formula for the partial sums of the expansion is given in terms of the Radon transform, which leads to algorithms for image reconstruction from Radon data. The relation between orthogonal expansion and the singular value decomposition of the Radon transform is also exploited.

math-ph↗