On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

We investigate a periodic version of the Benjamin-Ono (BO) equation associated with a discrete Laplacian. We find some special solutions to this equation, and calculate the values of the first two integrals of motion _I1$I_1$ and _I2$I_2$ corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald q$q$-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to q=0$q=0$ in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions _In$I_n$ (n=1,2,…$n=1\text{,}2\text{,}\dots$), and successfully identify it with the eigenvalues of Macdonald operators in the limit q→0$q\to 0$, i.e. the limit where Macdonald polynomials tend to the Hall–Littlewood polynomials.     

Non-reducible descriptions for conditional Kolmogorov complexity

Assume that a program p$p$ on input a$a$ outputs b$b$. We are looking for a shorter program q$q$ having the same property (q(a)=b$q\left(a\right)=b$). In addition, we want q$q$ to be simple conditional to p$p$ (this means that the conditional Kolmogorov complexity K(q|p)$K\left(q\right|p)$ is negligible). In the present paper, we prove that sometimes there is no such program q$q$, even in the case when the complexity of p$p$ is much bigger than K(b|a)$K\left(b\right|a)$. We give three different constructions that use the game approach, probabilistic arguments and algebraic arguments, respectively.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

A palindromization map for the free group

We define a self-map Pal:_F2→_F2$Pal:F_2\to F_2$ of the free group on two generators a,b$a,b$, using automorphisms of _F2$F_2$ that form a group isomorphic to the braid group _B3$B_3$. The map Pal$Pal$ restricts to de Luca’s right iterated palindromic closure on the submonoid generated by a,b$a,b$. We show that Pal$Pal$ is continuous for the profinite topology on _F2$F_2$; it is the unique continuous extension of de Luca’s right iterated palindromic closure to _F2$F_2$. The values of Pal$Pal$ are palindromes and coincide with the elements g∈_F2$g\in F_2$ such that abg$abg$ and bag$bag$ are conjugate.     

Compatibility of unrooted phylogenetic trees is FPT

A collection of _T1,_T2,…,_Tk$T_1,T_2,\dots ,T_k$ of unrooted, leaf labelled (phylogenetic) trees, all with different leaf sets, is said to be compatible   if there exists a tree T$T$ such that each tree _Ti$T_i$ can be obtained from T$T$ by deleting leaves and contracting edges. Determining compatibility is NP-hard, and the fastest algorithm to date has worst case complexity of around Ω(n^k)$\mathrm{\Omega }(n^k)$ time, n$n$ being the number of leaves. Here, we present an O(nf(k))$\mathrm{O}(\mathit{nf}(k))$ algorithm, proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable   (FPT) with respect to the number k$k$ of trees.     

Approximation algorithms for orthogonal packing problems for hypercubes

Orthogonal packing problems are natural multidimensional generalizations of the classical bin packing problem and knapsack problem and occur in many different settings. The input consists of a set I={_r1,…,_rn}$I=\{r_1,\dots ,r_n\}$ of d$d$-dimensional rectangular items _ri=(_ai,1,…,_ai,d)$r_i=(a_{i,1},\dots ,a_{i,d})$ and a space Q$Q$. The task is to pack the items in an orthogonal and non-overlapping manner without using rotations into the given space. In the strip packing setting the space Q$Q$ is given by a strip of bounded basis and unlimited height. The objective is to pack all items into a strip of minimal height. In the knapsack packing setting the given space Q$Q$ is a single, usually unit sized bin and the items have associated profits _pi$p_i$. The goal is to maximize the profit of a selection of items that can be packed into the bin.     

Large independent sets in random regular graphs

We present algorithmic lower bounds on the size _sd$s_d$ of the largest independent sets of vertices in random d$d$-regular graphs, for each fixed d≥3$d\ge 3$. For instance, for d=3$d=3$ we prove that, for graphs on n$n$ vertices, _sd≥0.43475n$s_d\ge 0.43475n$ with probability approaching one as n$n$ tends to infinity.     

Adaptive timeliness of consensus in presence of crash and timing faults

The Δ$\mathrm{\Delta }$-timed uniform consensus is a stronger variant of the traditional consensus and it satisfies the following additional property: every correct process terminates its execution within a constant time Δ$\mathrm{\Delta }$ (Δ$\mathrm{\Delta }$-timeliness), and no two processes decide differently (uniformity). In this paper, we consider the Δ$\mathrm{\Delta }$-timed uniform consensus problem in presence of _fc$f_{\mathrm{c}}$ crash processes and _ft$f_{\mathrm{t}}$ timing-faulty processes, and propose a Δ$\mathrm{\Delta }$-timed uniform consensus algorithm. The proposed algorithm is adaptive in the following sense: it solves the Δ$\mathrm{\Delta }$-timed uniform consensus when at least _ft+1$f_{\mathrm{t}}+1$ correct processes exist in the system. If the system has less than _ft+1$f_{\mathrm{t}}+1$ correct processes, the algorithm cannot solve the Δ$\mathrm{\Delta }$-timed uniform consensus. However, as long as _ft+1$f_{\mathrm{t}}+1$ processes are non-crashed, the algorithm solves (non-timed) uniform consensus. We also investigate the maximum number of faulty processes that can be tolerated. We show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes requires that the system has at least _ft+1$f_{\mathrm{t}}+1$ correct processes. This impossibility result implies that the proposed algorithm attains the maximum resilience about the number of faulty processes. We also show that any Δ$\mathrm{\Delta }$-timed uniform consensus algorithm tolerating up to _ft$f_{\mathrm{t}}$ timing-faulty processes cannot solve the (non-timed) uniform consensus when the system has less than _ft+1$f_{\mathrm{t}}+1$ non-crashed processes. This impossibility result implies that our algorithm attains the maximum adaptiveness.     

Sampling methods for shortest vectors,closest vectors and successive minima

We study four problems from the geometry of numbers, the shortest vector problem  (Svp)$\left(\text{<small-caps>Svp</small-caps>}\right)$, the closest vector problem  (Cvp)$\left(\text{<small-caps>Cvp</small-caps>}\right)$, the successive minima problem  (Smp)$\left(\text{<small-caps>Smp</small-caps>}\right)$, and the shortest independent vectors problem   (Sivp$\text{<small-caps>Sivp</small-caps>}$). Extending and generalizing results of Ajtai, Kumar, and Sivakumar we present probabilistic single exponential time algorithms for all four problems for all _?p${?}_p$ norms. The results on Smp$\text{<small-caps>Smp</small-caps>}$ and Sivp$\text{<small-caps>Sivp</small-caps>}$ are new for all norms. The results on Svp$\text{<small-caps>Svp</small-caps>}$ and Cvp$\text{<small-caps>Cvp</small-caps>}$ generalize previous results of Ajtai et al. for the Euclidean _?2${?}_2$ norm to arbitrary _?p${?}_p$ norms. We achieve our results by introducing a new lattice problem, the generalized shortest vector problem   (GSvp$\text{<small-caps>GSvp</small-caps>}$). ¹ We describe a single exponential time algorithm for GSvp$\text{<small-caps>GSvp</small-caps>}$. We also describe polynomial time reductions from Svp,Cvp,Smp$\text{<small-caps>Svp</small-caps>},\text{<small-caps>Cvp</small-caps>},\text{<small-caps>Smp</small-caps>}$, and Sivp$\text{<small-caps>Sivp</small-caps>}$ to GSvp$\text{<small-caps>GSvp</small-caps>}$, establishing single exponential time algorithms for the four classical lattice problems. This approach leads to a unified algorithmic treatment of the lattice problems Svp,Cvp,Smp$\text{<small-caps>Svp</small-caps>},\text{<small-caps>Cvp</small-caps>},\text{<small-caps>Smp</small-caps>}$, and Sivp$\text{<small-caps>Sivp</small-caps>}$.     

Bounding the radii of balls meeting every connected component of semi-algebraic sets

We prove an explicit bound on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set S⊂R^k$S\subset {\mathbb{R}}^k$ defined by a weak sign condition involving s$s$ polynomials in Z[_X1,…,_Xk]$\mathbb{Z}[X_1,\dots ,X_k]$ having degrees at most d$d$, and whose coefficients have bitsizes at most τ$\tau$. Our bound is an explicit function of s,d,k$s,d,k$ and τ$\tau$, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of S$S$ (including the unbounded components). While asymptotic bounds of the form 2^τdO(k)$2^{\tau d^{O\left(k\right)}}$ on these quantities were known before, some applications require bounds which are explicit and which hold for all values of s,d,k$s,d,k$ and τ$\tau$. The bounds proved in this paper are of this nature.     

Minimum cost source location problem with local 3-vertex-connectivity requirements

Let G=(V,E)$G=(V,E)$ be a simple undirected graph with a set V$V$ of vertices and a set E$E$ of edges. Each vertex v∈V$v\in V$ has a demand d(v)∈_Z+$d\left(v\right)\in Z_{+}$ and a cost c(v)∈_R+$c\left(v\right)\in R_{+}$, where _Z+$Z_{+}$ and _R+$R_{+}$ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G$G$ requires finding a set S$S$ of vertices minimizing _∑v∈Sc(v)${\sum }_{v\in S}c\left(v\right)$ such that there are at least d(v)$d\left(v\right)$ pairwise vertex-disjoint paths from S$S$ to v$v$ for each vertex v∈V−S$v\in V-S$. It is known that if there exists a vertex v∈V$v\in V$ with d(v)≥4$d\left(v\right)\ge 4$, then the problem is NP-hard even in the case where every vertex has a uniform cost. In this paper, we show that the problem can be solved in O(|V|⁴log²|V|)$O\left({\left|V\right|}^4{log}^2\left|V\right|\right)$ time if d(v)≤3$d\left(v\right)\le 3$ holds for each vertex v∈V$v\in V$.     

Algorithms for near solutions to polynomial equations

Let F(x,y)$F(x,y)$ be a polynomial over a field K$K$ and m$m$ a nonnegative integer. We call a polynomial g$g$ over K$K$ an m$m$-near solution of F(x,y)$F(x,y)$ if there exists a c∈K$c\in K$ such that F(x,g)=cx^m$F(x,g)=c{x}^m$, and the number c$c$ is called an m$m$-value of F(x,y)$F(x,y)$ corresponding to g$g$. In particular, c$c$ can be 0. Hence, by viewing F(x,y)=0$F(x,y)=0$ as a polynomial equation over K[x]$K\left[x\right]$ with variable y$y$, every solution of the equation F(x,y)=0$F(x,y)=0$ in K[x]$K\left[x\right]$ is also an m$m$-near solution. We provide an algorithm that gives all m$m$-near solutions of a given polynomial F(x,y)$F(x,y)$ over K$K$, and this algorithm is polynomial time reducible to solving one variable equations over K$K$. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions.     

Choosing starting values for certain Newton–Raphson iterations

We aim at finding the best possible seed values when computing a^1/p$a^{1/p}$ using the Newton–Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function f(a)$f(a)$ in the interval [_amin,_amax]$[a_{\min },a_{\max }]$, by building the sequence _xn$x_n$ defined by the Newton–Raphson iteration, the natural choice consists in choosing _x0$x_0$ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between _x0$x_0$ and f(a)$f(a)$. And yet, if we perform n$n$ iterations, what matters is to minimize the maximum possible distance between _xn$x_n$ and f(a)$f(a)$. In several examples, the value of the best starting point varies rather significantly with the number of iterations.