Every braid admits a short sigma-definite representative

A result by Dehornoy (1992) says that every nontrivial braid admits a sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a sigma-definite word representative that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form

Exploring the tree of numerical semigroups

In this paper we describe an algorithm visiting all numerical semigroups up to a given genus using a well suited representation. The interest of this algorithm is that it fits particularly well the architecture of modern computers allowing very large optimizations: we obtain the number of numerical semigroups of genus g 67 and we confirm the Wilf conjecture for g 60.Comment: 14 page

The rotating normal form of braids is regular

Defined on Birman-Ko-Lee monoids, the rotating normal form has strong connections with the Dehornoy's braid ordering. It can be seen as a process for selecting between all the representative words of a Birman-Ko-Lee braid a particular one, called rotating word. In this paper we construct, for all n 2, a finite-state automaton which recognizes rotating words on n strands, proving that the rotating normal form is regular. As a consequence we obtain the regularity of a $\sigma$-definite normal form defined on the whole braid group

A divisibility result on combinatorics of generalized braids

For every finite Coxeter group $\Gamma$, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of $\Gamma$, the so-called Garside-normal form.The study of the associated adjacency matrix $Adj(\Gamma)$ allows to count the number of Garside-normal form of a given length.In this paper we prove that the characteristic polynomial of $Adj(B_n)$ divides the one of $Adj(B_{n+1})$. The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type $A$. We observe that this does not hold for type $D$. The other Coxeter types ($I$, $E$, $F$ and $H$) are also studied.Comment: 28 page

Geometrical analysis of thread milling – Part 1: Evaluation of tool angles

Thread milling is a method which is increasingly used for machining thread. For this operation, a helical interpolation is required. Furthermore, the thread mill is a tool whose geometry is rather complex. Its envelope profile is linked to the thread profile and a single tooth of the thread mill is composed of three continuous cutting edges. The present study proposes a geometrical model and an analytical formulation to define the rake face and the cutting edge. Further, the calculations of cutting planes and cutting angles are explained. The analysis shows specific aspects of thread mills, in particular the fact that the flute angle may lead to a negative rake angle. This study is a contribution to cutting geometry aspect and constitutes a step for cutting force model in thread milling

Investigation of tool geometry effect and penetration strategies on cutting forces during thread milling

The application of thread milling is increasing in industry because of its inherent advantages over other thread cutting techniques. The objective of this study is to investigate the effect of milling cutter tool geometry on cutting forces during thread milling. The proposed method can compare the performance of milling cutters in spite of the different number of tooth. The best thread milling cutter among the studied tools was determined from the cutting forces point of view. Furthermore, this study also pinpoints the best penetration strategy that provides minimum cutting forces. Lower cutting force variations will lead to fewer vibrations of the tool which in turn will produce accurate part.Postdoc de V Sharma financé par la région Bourgogn

Analysis and modeling of green wood milling: Chip production by slabber

During the primary transformation of wood, logs are faced with slabber heads. Chips produced are raw materials for pulp paper and particleboard industries. Efficiency of these industries is partly due to particle size distribution. Command of this distribution is no easy matter because of great dependence on cutting conditions and variability in material. This study aimed a better understanding and predictionof chip fragmentation. It starts with a detailed description of cutting kinematic and interaction between knife and log. This leads to the numerical development of a generic slabber head. Chip fragmentation phenomena were studied through experiments in dynamic conditions. These experiments were carried out thanks to a pendulum (Vc = 400 m/min). It was instrumented with piezoelectric force sensors and high speed camera. Obtained results agreed very well with previous quasi-static experiments

The effect of an organic pentasulfide EP additive in turning and milling operations

Coopération avec Ecole Centrale LyonBecause a cutting fluid could be equally used for different cutting operations, this study proposed to investigate the behavior of a well-known extreme-pressure additive (pentasulfide) in both turning and milling operations of a steel workpiece. The experimental approach is based on the coupling of mechanical tests (turning, milling, and tribological tests) with physico-chemical characterizations (Auger Electron Spectroscopy and X-Ray Photoelectron Spectroscopy) of the friction surfaces (chip and tool). In the case of milling, it was shown that the presence of a pentasulfide additive has a beneficial effect on the specific cutting energy (kc) and flank wear (Vb). These results are correlated with the presence of iron sulfides (FeS and FeS2) on the flank face of the cutter mill and on the chip face in contact with the mill. No such additive effects are found in case of turning. A lubrication model is proposed for the case of milling based on an indirect lubrication of the tool/workpiece and tool/chip contacts due to the transfer of iron and its reaction with sulfur compounds to produce iron sulfides. Because milling is a discontinuous cutting process, this lubrication mechanism is much more efficient than that observed in turning. Indeed, the tool faces are re-fed iron sulfides each time they leave the workpiece