Compositions into Powers of bb: Asymptotic Enumeration and Parameters

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated

Adaptive Computation of the Swap-Insert Correction Distance

The Swap-Insert Correction distance from a string $S$ of length $n$ to another string $L$ of length $m\geq n$ on the alphabet $[1..d]$ is the minimum number of insertions, and swaps of pairs of adjacent symbols, converting $S$ into $L$. Contrarily to other correction distances, computing it is NP-Hard in the size $d$ of the alphabet. We describe an algorithm computing this distance in time within $O(d^2 nm g^{d-1})$, where there are $n_\alpha$ occurrences of $\alpha$ in $S$, $m_\alpha$ occurrences of $\alpha$ in $L$, and where $g=\max_{\alpha\in[1..d]} \min\{n_\alpha,m_\alpha-n_\alpha\}$ measures the difficulty of the instance. The difficulty $g$ is bounded by above by various terms, such as the length of the shortest string $S$, and by the maximum number of occurrences of a single character in $S$. Those results illustrate how, in many cases, the correction distance between two strings can be easier to compute than in the worst case scenario.Comment: 16 pages, no figures, long version of the extended abstract accepted to SPIRE 201

Finite-State Abstractions for Probabilistic Computation Tree Logic

Probabilistic Computation Tree Logic (PCTL) is the established temporal logic for probabilistic verification of discrete-time Markov chains. Probabilistic model checking is a technique that verifies or refutes whether a property specified in this logic holds in a Markov chain. But Markov chains are often infinite or too large for this technique to apply. A standard solution to this problem is to convert the Markov chain to an abstract model and to model check that abstract model. The problem this thesis therefore studies is whether or when such finite abstractions of Markov chains for model checking PCTL exist. This thesis makes the following contributions. We identify a sizeable fragment of PCTL for which 3-valued Markov chains can serve as finite abstractions; this fragment is maximal for those abstractions and subsumes many practically relevant specifications including, e.g., reachability. We also develop game-theoretic foundations for the semantics of PCTL over Markov chains by capturing the standard PCTL semantics via a two-player games. These games, finally, inspire a notion of p-automata, which accept entire Markov chains. We show that p-automata subsume PCTL and Markov chains; that their languages of Markov chains have pleasant closure properties; and that the complexity of deciding acceptance matches that of probabilistic model checking for p-automata representing PCTL formulae. In addition, we offer a simulation between p-automata that under-approximates language containment. These results then allow us to show that p-automata comprise a solution to the problem studied in this thesis

Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution)

Experimental study of coated carbide tools behaviour: application for Ti-5-5-5-3 turning

The goal of this paper is to study the relation between the input data (conditions and geometry of cut) and answers (wear of tool, forces and cutting temperatures) when machining the Ti-5-5-5-3 alloy treated. This study has shown that the cutting process is different and that the slip forces are preponderates. Compared with other materials, the specific cutting pressure is higher and does not vary according to the cutting speed but depend on feed rate. Moreover, both edge preparation and feed rate have an influence on cutting force direction. Besides, cutting temperatures are high and almost similar to those provided by high speed machining with low cutting speed. Finally, we have shown that failure modes are different from those obtained when machining other titanium alloys. Built-up edge is the most deteriorating phenomenon and no flank wear was met in our study context

Experimental characterization of behavior laws for titanium alloys: application to Ti5553

The aim of this paper is to study the machinability of a new titanium alloy: Ti-5AL-5Mo-5V-3CR used for the production of new landing gear. First, the physical and mechanical properties of this material will be presented. Second, we show the relationship between material properties and machinability. Third, the Ti5553 will be compared to Ti64. Unless Ti64 is α+β alloy group and Ti5553 is a metastable, we have chosen to compare these two materials. Ti64 is the most popular of titanium alloys and many works were been made on its machining. After, we have cited the Ti5553 properties and detailed the behavior laws. They are used in different ways: with or without thermal softening effect or without dynamic terms. The goal of the paper is to define the best cutting force model. So, different models are compared for two materials (steel and titanium alloy). To define the model, two methods exist that we have compared. The first is based on machining test; however the second is based on Hopkinson bar test. These methods allow us to obtain different ranges of strain rate, strain and temperature. This comparison will show the importance of a good range of strain rate, strain and temperature for behavior law, especially in titanium machining