Approximation in FEM, DG and IGA: A Theoretical Comparison

In this paper we compare approximation properties of degree $p$ spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, \smooth {p-1} splines provide better a priori error bounds for the approximation of functions in $H^{p+1}(0,1)$. Our result holds for all practically interesting cases when comparing \smooth {p-1} splines with \smooth {-1} (discontinuous) splines. When comparing \smooth {p-1} splines with \smooth 0 splines our proof covers almost all cases for $p\ge 3$, but we can not conclude anything for $p=2$. The results are generalized to the approximation of functions in $H^{q+1}(0,1)$ for $q<p$, to broken Sobolev spaces and to tensor product spaces.Comment: 21 pages, 4 figures. Fixed typos and improved the presentatio

Optimal spline spaces for L2L^2 nn-width problems with boundary conditions

In this paper we show that, with respect to the $L^2$ norm, three classes of functions in $H^r(0,1)$, defined by certain boundary conditions, admit optimal spline spaces of all degrees $\geq r-1$, and all these spline spaces have uniform knots.Comment: 17 pages, 4 figures. Fixed a typo. Article published in Constructive Approximatio

Telegram from Maria Joao Sande Lemos, Social Democratic Party of Portugal, to Geraldine Ferraro

Telegram from Maria Joao Sande Lemos, Social Democratic Party of Portugal, to Geraldine Ferraro. Telegram has handwritten notes.https://ir.lawnet.fordham.edu/vice_presidential_campaign_correspondence_1984_international/1363/thumbnail.jp

Sharp error estimates for spline approximation: explicit constants, nn-widths, and eigenfunction convergence

In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A

The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, \Lip^+-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of \Lip^+-unbounded initial data is worse than first-order.Comment: 10 pages, 5 figures, 2 tables. Fixed typos. Article published in Journal of Scientific Computin

Transverse Lattice QCD in 2+1 Dimensions

Following a suggestion due to Bardeen and Pearson, we formulate an effective light-front Hamiltonian for large-N gauge theory in (2+1)-dimensions. Two space-time dimensions are continuous and the remaining space dimension is discretised on a lattice. Eguchi-Kawai reduction to a (1+1)-dimensional theory takes place. We investigate the string tension and glueball spectrum, comparing with Euclidean Lattice Monte Carlo data.Comment: 4 pages LaTeX with 2 Postscript figures, uses boxedeps.tex and e spcrc2.sty. Poster session contribution to LATTICE96(poster). Minor changes in new versio

Pembagian Peran dalam Pengambilan Keputusan Pembelian Keluarga

Keluarga memiliki struktur sendiri dan kompleks, tiap anggota atau beberapa anggota dalam keluarga memainkan peran masing-masing dalam pengambilan keputusan pembelian. Karenanya bagaimana masing-masing anggota keluarga mengambil peran dalam pengambilan keputusan pembelian produk menjadi perhatian dalam pemasaran. Kajian ini bersifat deskriptif untuk memperoleh gambaran tentang peran anggota keluarga dalam pengambilan keputusan pembelian TV. Dapat disimpulkan bila peran initiator dominan dipegang oleh istri, peran influencer cenderung oleh anak. Sedangkan suami lebih memegang peran sebagai decider sekaligus buyer. Selanjutnya peran sebagai user sebagai penikmat TV baru adalah anak. Kata kunci: Pengambilan Keputusan Pembelian, Keluarga, Pera