Faster gossiping on butterfly networks

Gossiping has been considered intensively for butterflies and “simple” butterflies (which have no wrap-around connections). In the “telephone” communication model, for a butterfly of order k, the best previous gossiping algorithms require $2\frac{1}{2}k$ and $3k$ communication rounds, respectively. By new asymptotic methods we break through these bounds. We show that gossiping on a class of “column-based” networks, which also contains the cube-connected cycles, can be reduced to the simpler problem of “row-gossiping”. Row-gossiping in turn is reduced to “coherent row-broadcasting”. This latter problem is sufficiently simple to be solved by a sophisticated computer program for butterflies with up to $15\times 2^{15}$ nodes. Out of the produced solutions a pattern is distilled, which can be used to perform gossiping on butterflies and simple butterflies of order k in $2\frac{1}{4}k+\mathrm{o}(k)$ and $2\frac{1}{2}k+\mathrm{o}(k)$ rounds, respectively, for any k, considerably reducing the gap with the lower bounds. The new upper bounds also hold for gossiping in the weaker “telegraph” model.     

Equations of parametric surfaces with base points via syzygies

Let $S$ be a tensor product parametrized surface in $P^3$; that is, $S$ is given as the image of $\phi :P^1\times P^1\to P^3$. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of $S$ when certain base points are present. This work extends the algorithm provided by Cox [Cox, D.A., 2001. Equations of parametric curves and surfaces via syzygies. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemporary Mathematics vol. 286, pp. 1–20] for when $\phi$ has no base points, and it is analogous to some of the results of Busé et al. [Busé, L., Cox, D., D’Andrea, C., 2003. Implicitization of surfaces in $P^3$ in the presence of base points. J. Algebra Appl. 2 (2), 189–214] for the case of a triangular parametrization $\phi :P^2\to P^3$ with base points.     

A thermodynamic assessment of the Co–Sn system

A critical evaluation and thermodynamic modeling of the phase diagram and thermodynamic properties have been presented for the Co–Sn system. More attention has been paid to reproduce the thermodynamic properties of the liquid Co–Sn, including the appreciable temperature dependence in the mixing enthalpy and the sign-variable activity with composition observed by experiments. A four-sublattice model has been used to describe the ordering phenomenon between $\beta {\mathrm{Co}}_3{\mathrm{Sn}}_2$ and $\alpha {\mathrm{Co}}_3{\mathrm{Sn}}_2$.     

Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws

Explicit boundary dissipative conditions are given for the exponential stability in $L^2$-norm of one-dimensional linear hyperbolic systems of balance laws ${?}_t\xi +\Lambda {?}_x\xi ?M\xi =0$ over a finite interval, when the matrix $M$ is marginally diagonally stable. The result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised Saint–Venant–Exner equations.     

Corrigendum to “G2 B-spline interpolation to a closed mesh” [Comput Aided Des 43 (2011) 145–160]

A $G^2$ interpolation method to a closed mesh was proposed by Shi et al. in [1]. This note points out that their method has two theoretical defects, which may lead to the failure of $G^2$ continuity. We fix these defects and correct several formula mistakes.     

Suslin’s algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring $A$ if $(v_1\left(X\right),\dots ,v_n\left(X\right))\in {\left(A\left[X\right]\right)}^n$ is unimodular where $v_1$ is monic and $n\ge 3$, then there exist ${\gamma }_1,\dots ,{\gamma }_{\ell }\in E_{n-1}\left(A\left[X\right]\right)$ such that the ideal generated by $\mathrm{Res}(v_1,e_1.{\gamma }_1{}^t(v_2,\dots ,v_n)),\dots ,\mathrm{Res}(v_1,e_1.{\gamma }_{\ell }{}^t(v_2,\dots ,v_n))$ equals $A$. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where $A$ contains a set $E$ of cardinality greater than $deg{v}_1+1$ such that $y-y^{\prime }$ is invertible for each $y\ne y^{\prime }$ in $E$, we prove that the ${\gamma }_i$ can simply correspond to the elementary operations $L_1\to L_1+y_i{\sum }_{j=2}^{n-1}{u}_{j+1}{L}_j$, $1\le i\le \ell =deg{v}_1+1$, where $u_1{v}_1+\cdots +u_n{v}_n=1$. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in $K[X_1,\dots ,X_k]$ to ${}^t(1,0,\dots ,0)$ using elementary operations in the case where $K$ is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.