The generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws

In this paper, we study the nonlinear initial–boundary Riemann problem and the generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the domain $\{(t,x)|t⩾0,x⩾0\}$. Under the assumption that each positive eigenvalue is either linearly degenerate or genuinely nonlinear, we get the existence and uniqueness of the self-similar solution to the nonlinear initial–boundary Riemann problem and of the global piecewise $C^1$ solution containing only shocks and (or) contact discontinuities to the corresponding generalized nonlinear initial–boundary Riemann problem. It shows that the self-similar solution to the nonlinear initial–boundary Riemann problem possesses the global structural stability.     

The Camassa–Holm equation on the half-line

We study the initial-boundary-value problem for the Camassa–Holm equation on the half-line by associating to it a matrix Riemann–Hilbert problem in the complex k-plane; the jump matrix is determined in terms of the spectral functions corresponding to the initial and boundary values. We prove that if the boundary values $u(0,t)$ are ?0 for all t then the corresponding initial-boundary-value problem has a unique solution, which can be expressed in terms of the solution of the associated RH problem. In the case $u(0,t)<0$, the compatibility of the initial and boundary data is explicitly expressed in terms of an algebraic relation to be satisfied by the spectral functions. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

We consider a one-dimensional solidification of a pure substance which is initially in liquid state in a bounded interval $[0,l]$. Initially, the liquid is above the freezing temperature, and cooling is applied at $x=0$ while the other end $x=l$ is kept adiabatic. At the time $t=0$, the temperature of the liquid at $x=0$ comes down to the freezing point and solidification begins, where $x=s\left(t\right)$ is the position of the solid–liquid interface. As the liquid solidifies, it shrinks ($0<r<1$) or expands ($r<0$) and appears a region between $x=0$ and $x=rs\left(t\right)$, with $r<1$. Temperature distributions of the solid and liquid phases and the position of the two free boundaries ($x=rs\left(t\right)$ and $x=s\left(t\right)$) in the solidification process are studied. For three different cases, changing the condition on the free boundary $x=rs\left(t\right)$ (temperature boundary condition, heat flux boundary condition and convective boundary condition) an explicit solution is obtained. Moreover, the solution of each problem is given as a function of a parameter which is the unique solution of a transcendental equation and for two of the three cases a condition on the parameter must be verified by data of the problem in order to have an instantaneous phase-change process. In all the cases, the explicit solution is given by a representation of the similarity type.     

Cayley’s hyperdeterminant: A combinatorial approach via representation theory

Cayley’s hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a $2\times 2\times 2$ array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra $\mathfrak{s}{l}_2(\mathbb{C})$ to reduce the problem of finding the invariant polynomials for a $2\times 2\times 2$ array to a combinatorial problem on the enumeration of $2\times 2\times 2$ arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley’s hyperdeterminant generates all the invariants. In the last section we discuss the application of our methods to general multidimensional arrays.     

Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem

We consider a second-order hyperbolic equation on an open bounded domain $\Omega$in ${\mathbb{R}}^n$ for $n\ge 2$, with $C^2$-boundary $\Gamma =?\Omega =\overline{{\Gamma }_0\cup {\Gamma }_1}$, ${\Gamma }_0\cap {\Gamma }_1=0?$, subject to non-homogeneous Neumann boundary conditions on the entire boundary $\Gamma$. We then study the inverse problem of determining both the interior damping and potential coefficients of the equation in one shot by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit sub-portion ${\Gamma }_1$ of the boundary $\Gamma$, and over a computable time interval $T>0$. Under sharp conditions on the complementary part ${\Gamma }_0=\Gamma ?{\Gamma }_1$, $T>0$, and under weak regularity requirements on the data, we establish the two canonical results of the inverse problem: (i) global uniqueness and (ii) stability. The latter (ii) is the main result of the paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the $H^1\times L_2$-level for second-order hyperbolic equations (Lasiecka et al. (2000) [3]); (b) a correspondingly implied continuous observability inequality at the same energy level [3]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data (Lasiecka and Triggiani (1990, 1991, 1994) [20], [21], [29] and Tataru (1998) [24]). The proof of the linear uniqueness result (Section 4, step 5) also takes advantage of a convenient tactical route “post-Carleman estimates” suggested by Isakov (2006) in [12, Thm. 8.2.2, p. 231].     

Tighter approximation bounds for LPT scheduling in two special cases

$Q||C_{\max }$ denotes the problem of scheduling n jobs on m machines of different speeds such that the makespan is minimized. In the paper two special cases of $Q||C_{\max }$ are considered: case I, when $m?1$ machine speeds are equal, and there is only one faster machine; and case II, when machine speeds are all powers of 2 (2-divisible machines). Case I has been widely studied in the literature, while case II is significant in an approach to design so called monotone algorithms for the scheduling problem.We deal with the worst case approximation ratio of the classic list scheduling algorithm ‘Largest Processing Time (LPT)’. We provide an analysis of this ratio $\mathit{Lpt}/\mathit{Opt}$ for both special cases: For ‘one fast machine’, a tight bound of $(\sqrt{3}+1)/2\approx 1.3660$ is given. For 2-divisible machines, we show that in the worst case $1.3673<\mathit{Lpt}/\mathit{Opt}<1.4$. Besides, we prove another lower bound of $955/699>(\sqrt{3}+1)/2$ when LPT breaks ties arbitrarily.To our knowledge, the best previous lower and upper bounds were $(4/3,3/2?1/2m]$ in case I [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166], respectively $[4/3?1/3m,3/2]$ in case II [R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics 17 (1969) 416–429; A. Kovács, Fast monotone 3-approximation algorithm for scheduling related machines, in: Proc. 13th Europ. Symp. on Algs. (ESA), in: LNCS, vol. 3669, Springer, 2005, pp. 616–627]. Moreover, Gonzalez et al. conjectured the lower bound 4/3 to be tight in the ‘one fast machine’ case [T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM Journal on Computing 6 (1) (1977) 155–166].     

Local regularity criteria of a suitable weak solution to MHD equations

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are H¨older continuous near boundary provided that the scaled mixed L_(x,t)~(p,q) -norm of the velocity vector field with 3/p + 2/q ≤ 2,2 q ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ∈ L_(x,t)~(p,q) with 1≤3/p+2/q≤3/2, 3 p ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/p+2/q-1).     

The pentavalent three-geodesic-transitive graphs

A complete classification is given of pentavalent 3-geodesic-transitive graphs which are not 3-arc-transitive, which shows that a pentavalent 3-geodesic-transitive but not 3-arc-transitive graph is one of the following graphs: $\overline{(2\times 6)\text{-}\mathsf{grid}}$, $\mathrm{H}(5,2)$, the icosahedron, the incidence graph of the 2-$(11,5,2)$-design, the Wells graph and the Sylvester graph.