A fourth-order finite difference method based on uniform mesh for singular two-point boundary-value problems

AbstractWe present a new fourth-order finite difference method based on uniform mesh for the (weakly) singular two-point boundary value problem: (xαy′)′ = f(x, y), 0 < x ⩽ 1, y(0) = A, y(1) = B, 0 < α < 1. Our method provides O(h4)-convergent approximations for all α ∈ (0, 1); for α = 0 it reduces to the well-known fourth-order method of Numerov for y″ = ƒ(x, y)

Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

AbstractWe discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: [p(x)y′]′ = f(x, y), α0y(a) - α1y′(a) = A, β0y(b) + β1y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method

An extended trapezoidal formula for the diffusion equation

AbstractA finite-difference scheme for the diffusion equation that has enjoyed great popularity is the Crank-Nicolson scheme [1] based on the classical trapezoidal formula for integration in time. For problems with discontinuities in the boundary conditions and the initial conditions, the Crank-Nicolson scheme can give unwanted oscillations in the computed solution. We present an alternative scheme based on the extended trapezoidal formula for integration in time. The resulting Extended Trapezoidal Formula Finite Difference Scheme (ETF-FDS) is third order in time, unconditionally stable and, unlike the Crank-Nicolson scheme, ETF-FDS can cope with discontinuities in the boundary conditions and the initial conditions as demonstrated by the numerical examples considered

A fourth-order spline method for singular two-point boundary-value problems

AbstractThis paper describes two methods for the solution of (weakly) singular two-point boundary-value problems: Consider the uniform mesh xi = ih, h = 1/N, i = 0(1)N. Define the linear functionals Li(y) = y(xi) and Mi(y) = (x−α(xαy′)′\xv;x=xi. In both these methods a piecewise ‘spline’ solution is obtained in the form s(x) = si(x), x\wE; [xi−1, xi], i = 1(1)N, where in each subinterval si(x) is in the linear span of a certain set of (non-polynomial) basis functions in the representation of the solution y(x) of the two-point boundary value problem and satisfies the interpolation conditions: Li−1(s) = Li−1(y), Li(y), Mi−1(s) = Mi−1(y), Mi(s) = Mi(y). By construction s and x−α(xαs′)′ \wE; C[0,1]. Conditions of continuity are derived to ensure that xαs′ \wE; C[0, 1]. It follows that the unknown parameters yi and Mi(y), i = 1(1)N − 1, must satisfy conditions of the form: The first method consists in replacing Mi(y) by fnof(xi, yi) and solving (*) to obtain the values yi; this method is generalization of the idea of Bickley [2] for the case of (weakly) singular two-point boundary-value problems and provides order h2 uniformly convergent approximations over [0, 1]. As a modification of the above method, in the second method we generate the solution yi at the nodal points by adapting the fourth-order method of Chawla [3] and then use the conditions of continuity (*) to obtain the corresponding smoothed approximations for Mi(y) needed for the construction of the spline solution. We show that the resulting new spline method provides order h4 uniformly convergent approximations over [0, 1]. The second-order and the fourth-order methods are illustrated computationally

Anisotropic flow of charged hadrons, pions and (anti-)protons measured at high transverse momentum in Pb-Pb collisions at sNN=2.76\sqrt{s_{\rm NN}}=2.76 TeV

The elliptic, $v_2$, triangular, $v_3$, and quadrangular, $v_4$, azimuthal anisotropic flow coefficients are measured for unidentified charged particles, pions and (anti-)protons in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV with the ALICE detector at the Large Hadron Collider. Results obtained with the event plane and four-particle cumulant methods are reported for the pseudo-rapidity range $|\eta|<0.8$ at different collision centralities and as a function of transverse momentum, $p_{\rm T}$, out to $p_{\rm T}=20$ GeV/$c$. The observed non-zero elliptic and triangular flow depends only weakly on transverse momentum for $p_{\rm T}>8$ GeV/$c$. The small $p_{\rm T}$ dependence of the difference between elliptic flow results obtained from the event plane and four-particle cumulant methods suggests a common origin of flow fluctuations up to $p_{\rm T}=8$ GeV/$c$. The magnitude of the (anti-)proton elliptic and triangular flow is larger than that of pions out to at least $p_{\rm T}=8$ GeV/$c$ indicating that the particle type dependence persists out to high $p_{\rm T}$.Comment: 16 pages, 5 captioned figures, authors from page 11, published version, figures at http://aliceinfo.cern.ch/ArtSubmission/node/186

Centrality dependence of charged particle production at large transverse momentum in Pb-Pb collisions at sNN=2.76\sqrt{s_{\rm{NN}}} = 2.76 TeV

The inclusive transverse momentum ($p_{\rm T}$) distributions of primary charged particles are measured in the pseudo-rapidity range $|\eta|<0.8$ as a function of event centrality in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}=2.76$ TeV with ALICE at the LHC. The data are presented in the $p_{\rm T}$ range $0.15<p_{\rm T}<50$ GeV/$c$ for nine centrality intervals from 70-80% to 0-5%. The Pb-Pb spectra are presented in terms of the nuclear modification factor $R_{\rm{AA}}$ using a pp reference spectrum measured at the same collision energy. We observe that the suppression of high-$p_{\rm T}$ particles strongly depends on event centrality. In central collisions (0-5%) the yield is most suppressed with $R_{\rm{AA}}\approx0.13$ at $p_{\rm T}=6$-7 GeV/$c$. Above $p_{\rm T}=7$ GeV/$c$, there is a significant rise in the nuclear modification factor, which reaches $R_{\rm{AA}} \approx0.4$ for $p_{\rm T}>30$ GeV/$c$. In peripheral collisions (70-80%), the suppression is weaker with $R_{\rm{AA}} \approx 0.7$ almost independently of $p_{\rm T}$. The measured nuclear modification factors are compared to other measurements and model calculations.Comment: 17 pages, 4 captioned figures, 2 tables, authors from page 12, published version, figures at http://aliceinfo.cern.ch/ArtSubmission/node/284

Measurement of charm production at central rapidity in proton-proton collisions at s=2.76\sqrt{s} = 2.76 TeV

The $p_{\rm T}$-differential production cross sections of the prompt (B feed-down subtracted) charmed mesons D$^0$, D$^+$, and D$^{*+}$ in the rapidity range $|y|<0.5$, and for transverse momentum $1< p_{\rm T} <12$ GeV/$c$, were measured in proton-proton collisions at $\sqrt{s} = 2.76$ TeV with the ALICE detector at the Large Hadron Collider. The analysis exploited the hadronic decays D$^0 \rightarrow$K$\pi$, D$^+ \rightarrow$K$\pi\pi$, D$^{*+} \rightarrow$D$^0\pi$, and their charge conjugates, and was performed on a $L_{\rm int} = 1.1$ nb$^{-1}$ event sample collected in 2011 with a minimum-bias trigger. The total charm production cross section at $\sqrt{s} = 2.76$ TeV and at 7 TeV was evaluated by extrapolating to the full phase space the $p_{\rm T}$-differential production cross sections at $\sqrt{s} = 2.76$ TeV and our previous measurements at $\sqrt{s} = 7$ TeV. The results were compared to existing measurements and to perturbative-QCD calculations. The fraction of cdbar D mesons produced in a vector state was also determined.Comment: 20 pages, 5 captioned figures, 4 tables, authors from page 15, published version, figures at http://aliceinfo.cern.ch/ArtSubmission/node/307

Particle-yield modification in jet-like azimuthal di-hadron correlations in Pb-Pb collisions at sNN\sqrt{s_{\rm NN}} = 2.76 TeV

The yield of charged particles associated with high-$p_{\rm T}$ trigger particles ($8 < p_{\rm T} < 15$ GeV/$c$) is measured with the ALICE detector in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV relative to proton-proton collisions at the same energy. The conditional per-trigger yields are extracted from the narrow jet-like correlation peaks in azimuthal di-hadron correlations. In the 5% most central collisions, we observe that the yield of associated charged particles with transverse momenta $p_{\rm T}> 3$ GeV/$c$ on the away-side drops to about 60% of that observed in pp collisions, while on the near-side a moderate enhancement of 20-30% is found.Comment: 15 pages, 2 captioned figures, 1 table, authors from page 10, published version, figures at http://aliceinfo.cern.ch/ArtSubmission/node/350

Phase 1 study of the MDM2 inhibitor AMG 232 in patients with advanced P53 wild-type solid tumors or multiple myeloma

_Background_ This open-label, first-in-human, phase 1 study evaluated AMG 232, an oral selective MDM2 inhibitor in patients with TP53 wild-type (P53WT), advanced solid tumors or multiple myeloma (MM). _Methods_ In the dose escalation (n = 39), patients with P53WT refractory solid tumors enrolled to receive once-dailyAMG 232 (15, 30, 60, 120, 240, 480, and 960 mg) for seven days every 3 weeks (Q3W). In the dose expansion (n = 68), patients with MDM2-amplified (well-differentiated and dedifferentiated liposarcomas [WDLPS and DDLPS], glioblastoma multiforme [GBM], or other solid tumors [OST]), MDM2-overexpressing ER+ breast cancer (BC), or MM received AMG 232 at the maximum tolerated dose (MTD). Safety, pharmacokinetics, pharmacodynamics, and efficacy were assessed. _Results_ AMG 232 had acceptable safety up to up to 240 mg