Rational two-parameter families of spheres and rational offset surfaces

The present paper investigates two-parameter families of spheres in R³${\mathbb{R}}^3$ and their corresponding two-dimensional surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$. Considering a rational surface Φ$\Phi$ in R⁴${\mathbb{R}}^4$, the envelope surface Ψ$\Psi$ of the corresponding family of spheres in R³${\mathbb{R}}^3$ is typically non-rational. Using a classical sphere-geometric approach, we prove that the envelope surface Ψ$\Psi$ and its offset surfaces admit rational parameterizations if and only if Φ$\Phi$ is a rational sub-variety of a rational isotropic hyper-surface in R⁴${\mathbb{R}}^4$. The close relation between the envelope surfaces Ψ$\Psi$ and rational offset surfaces in R³${\mathbb{R}}^3$ is elaborated in detail. This connection leads to explicit rational parameterizations for all rational surfaces Φ$\Phi$ in R⁴${\mathbb{R}}^4$ whose corresponding two-parameter families of spheres possess envelope surfaces admitting rational parameterizations. Finally we discuss several classes of surfaces sharing this property.     

State-complexity hierarchies of uniform languages of alphabet-size length

We study the state complexity of certain simple languages. If A$A$ is an alphabet of k$k$ letters, then a k$k$-language   is a nonempty set of words of length k$k$, that is, a uniform language of length k$k$. We show that the minimal state complexity of a k$k$-language is k+2$k+2$, and the maximal, (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. We prove constructively that, for every i$i$ between the minimal and maximal bounds, there is a language of state complexity i$i$. We introduce a class of automata accepting sets of words that are permutations of A$A$; these languages define a complete hierarchy of complexities between k²−k+3$k^2-k+3$ and 2^k+1$2^k+1$. The languages of another class of automata, based on k$k$-ary trees, define a complete hierarchy of complexities between 2^k+1$2^k+1$ and (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. This provides new examples of uniform languages of maximal complexity.     

Computational topology for isotopic surface reconstruction

New computational topology techniques are presented for surface reconstruction of 2-manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope   (as defined herein) of the original surface. For any compact C²$C^2$-manifold M$M$ embedded in R³${\mathbf{R}}^3$, it is shown that its envelope is C^1,1$C^{1,1}$. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M$M$, whenever M$M$ is orientable. The emphasis of this paper is upon the formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to non-orientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.     

Optimal number of annuli for maximizing the lifetime of sensor networks

The most effective way to maximize the lifetime of a wireless sensor network (WSN) is to allocate initial energy to sensors such that they exhaust their energy at the same time. The lifetime of a WSN as well as an optimal initial energy allocation are determined by a network design. The main contribution of the paper is to show that the lifetime of a WSN can be maximized by an optimal network design. We represent the network lifetime as a function of the number m$m$ of annuli and show that m$m$ has significant impact on network lifetime. We prove that if the energy consumed by data transmission is proportional to d^α+c$d^{\alpha }+c$, where d$d$ is the distance of data transmission and α$\alpha$ and c$c$ are some constants, then for a circular area of interest with radius R$R$, the optimal number of annuli that maximizes the network lifetime is m=R((α−1)/c)^1/α$m=R{((\alpha -1)/c)}^{1/\alpha }$ for an arbitrary sensor density function.     

Paths and trails in edge-colored graphs

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s$s$ and t$t$ in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t$s-t$ path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s$s$ and t$t$ for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k$k$ pairwise vertex/edge disjoint properly edge-colored s−t$s-t$ paths/trails in a c$c$-edge-colored graph G^c$G^c$ is NP-complete even for k=2$k=2$ and c=Ω(n²)$c=\Omega \left(n^2\right)$, where n$n$ denotes the number of vertices in G^c$G^c$. Moreover, we prove that these problems remain NP-complete for c$c$-edge-colored graphs containing no properly edge-colored cycles and c=Ω(n)$c=\Omega \left(n\right)$. We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.     

Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor

We consider a two-edge connected, undirected graph G=(V,E)$G=(V,E)$, with n$n$ nodes and m$m$ non-negatively real weighted edges, and a single source shortest paths tree (SPT) T$T$ of G$G$ rooted at an arbitrary node r$r$. If an edge in T$T$ is temporarily removed, it makes sense to reconnect the nodes disconnected from the root by adding a single non-tree edge, called a swap edge  , instead of rebuilding a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge. In this paper we focus on the most prominent one, that is the minimization of the average distance between the root and the disconnected nodes. To this respect, we present an O(mlog²n)$O\left(m{log}^{2}n\right)$ time and O(m)$O\left(m\right)$ space algorithm to find a best swap edge for every edge of T$T$, thus improving for m=o(n²/log²n)$m=o(n^2/{log}^{2}n)$ the previously known O(n²)$O\left(n^2\right)$ time and space complexity algorithm.     

Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates

In this paper, two delayed SEIR epidemic models with continuous and impulsive vaccination and saturating incidence are investigated. The dynamical behaviors of the disease are analyzed. For continuous vaccination, we obtain a basic reproductive number _R1$R_1$ and prove that if _R1≤1$R_1\le 1$ then the disease-free equilibrium is globally attractive and if _R1>1$R_1>1$ then the disease is permanent by using the Lyapunov functional method. For impulsive vaccination, we obtain two thresholds R^∗$R^{\ast }$ and _R∗$R_{\ast }$ and prove that if R^∗<1$R^{\ast }<1$ then the disease-free periodic solution is globally attractive and if _R∗>1$R_{\ast }>1$ then the disease is permanent by using the comparison theorem of impulsive differential equation and the Lyapunov functional method. Lastly, we compared the effects of two vaccination strategies.     

Efficient pebbling for list traversal synopses with application to program rollback

We show how to support efficient back traversal in a unidirectional list, using small memory and with essentially no slowdown in forward steps. Using O(lgn)$O(lgn)$ memory for a list of size n$n$, the i$i$’th back-step from the farthest point reached so far takes O(lgi)$O(lgi)$ time in the worst case, while the overhead per forward step is at most ?$?$ for arbitrary small constant ?>0$?>0$. An arbitrary sequence of forward and back steps is allowed. A full trade-off between memory usage and time per back-step is presented: k$k$ vs. kn^1/k$k{n}^{1/k}$ and vice versa. Our algorithms are based on a novel pebbling technique which moves pebbles on a virtual binary, or n^1/k$n^{1/k}$-ary, tree that can only be traversed in a pre-order fashion.     

On the minimum of a positive polynomial over the standard simplex

We present a new positive lower bound for the minimum value taken by a polynomial P$P$ with integer coefficients in k$k$ variables over the standard simplex of R^k${\mathbb{R}}^k$, assuming that P$P$ is positive on the simplex. This bound depends only on the number of variables k$k$, the degree d$d$ and the bitsize τ$\tau$ of the coefficients of P$P$ and improves all the previous bounds for arbitrary polynomials which are positive over the simplex.