New Hesitant Fuzzy Operators

In this paper, four new operators $(O_1,O_2,O_3,O_4)$ are proposed, defined and considered to study the new properties and identities on hesitant fuzzy sets. Since these operators are useful for different operation on hesitant fuzzy sets, the various theorems are proved by using them. The study of the proposed new operators has opened a new area of research and applications.     

Lipschitz equivalence of self-similar sets

In 1997 David and Semmes asked whether there exists a bilipschitz map between the two compact self-similar subset M and $M^{\prime }$ of the real line defined by the relations $M=(M/5)\cup (M/5+2/5)\cup (M/5+4/5)$ and $M^{\prime }=(M^{\prime }/5)\cup (M^{\prime }/5+3/5)\cup (M^{\prime }/5+4/5)$. We answer this question positively. To cite this article: H. Rao et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the longest block in Lüroth expansion

In this paper, for a finite subset $A?\{2,3,?\}$, we introduce the notion of longest block function $L_n(x,A)$ for the Lüroth expansion of $x\in [0,1)$ with respect to A and consider the asymptotic behavior of $L_n(x,A)$ as n tends to ∞. We also obtain the Hausdorff dimensions of the level sets and exceptional set arising from the longest block function.     

The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is $O(n^{1/2})$, then there is a stacking order for which the visible perimeter is $\Omega (n^{2/3})$. We also show that this bound cannot be improved in the case of a sufficiently small $n^{1/2}\times n^{1/2}$ uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is $O(n^{3/4})$ with respect to any stacking order. This latter bound cannot be improved either.Finally, we address the case where no more than c disks can have a point in common.These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.     

Measurable sets with excluded distances

For a set of distances $D=\{d_1,\dots ,d_k\}$ a set A in the plane is called D-avoiding if no pair of points of A is at distance $d_i$ for some i. We show that the density of A is exponentially small in k provided the ratios $d_1/d_2,d_2/d_3,\dots ,d_{k-1}/d_k$ are all small enough. We also show that there exists a largest D-avoiding set, and give an algorithm to compute the maximum density of a D-avoiding set for any D.