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This study suggests a new general scheme of high convergence order for approximating the solutions of nonlinear systems. The proposed scheme is the extension of an earlier study of Parhi and Gupta. This method requires two vector-function, two Jacobian matrices, two inverse matrices, and one frozen inverse matrix per iteration. Convergence error, computational efficiency, and numerical experiments are performed to verify the applicability and validity of the proposed methods compared with existing methods. Finally, we discuss the strange fixed points and conjugacy functions on a particular case of our scheme. The basins of attractions also demonstrate the dynamical behavior of this special case in the neighborhood of required roots and also assert the theoretical outcomes.

The 3x+1 map T is defined on the 2-adic integers by T(x)=x/2 for even x and T(x)=(3x+1)/2 for odd x and the 3x+1 conjecture states that the T-orbit of any positive integer contains 1. We define and study properties of the unique nontrivial autoconjugacy Ω of T. This autoconjugacy sends x to the unique 2-adic integer whose parity vector is the one's complement of the parity vector of x. We prove that if Ω maps rational numbers to rational numbers then there are no divergent T-orbits of positive integers. The map Ω is then used to restate the 3x+1 conjecture in a parity neutral form. We derive a necessary and sufficient condition for a cycle to be self-conjugate and show that self-conjugate cycles contain only positive elements. It is then shown that the only self-conjugate cycle of integers is {1,2}. Finally, we prove that for any rational 2-adic integer x, lim̄κn(x)n+limκn(Ω(x))/n=1 where κn(x) is the number of ones in the first n digits of the parity vector of x, and we use this along with generalizations of known restrictions on lim̄κn(x)/n to prove most of the results in the paper.

Let $S=\{s_i\in\mathbb N\cup\{0\}:0\leq s_i, Comment: This paper has been withdrawn due to a flaw in Theorem 3.2. The correct version with some minor results will be replaced

We consider equations on the monoid of factorial languages on the binary alphabet. We use the notion of a canonical decomposition of a factorial language and previous results by Avgustinovich and the author to solve several simple equations on binary factorial languages including Xn=Yn, the commutation equation XY=YX and the conjugacy equation XZ=ZY. At the end of the paper, we discuss the difficulties hindering the reduction of equations on factorial languages to equations on words and the extension of the alphabet considered.

Let $N$ be a finitely generated nilpotent group. Algorithm is constructed such, that for every automorphism $\phi \in Aut(N)$ defines the Reidemeister number $R(\phi).$ It is proved that any free nilpotent group of rank $r = 2$ or $r = 3$ and class $c \geq 4r,$ or rank $r \geq 4$ and class $c \geq 2r,$ belongs to the class $R_{\infty}.$, Comment: 8 pages

To an absolute plane (E,L,=,@a) in the general sense of Karzel et al. [Einfuhrung in die Geometrie, UTB 184, Vandenhoeck, Gottingen, 1973, Section 16] there will be associated an ordered commutative group (W,+

International audience; Loopless triangulations of a polygon with $k$ vertices in $k+2n$ triangles (with interior points and possibly multiple edges) were enumerated by Mullin in 1965, using generating functions and calculations with the quadratic method. In this article we propose a simple bijective interpretation of Mullin's formula. The argument rests on the method of conjugacy classes of trees, a variation of the cycle lemma designed for planar maps. In the much easier case of loopless triangulations of the sphere ($k=3$), we recover and prove correct an unpublished construction of the second author.

International audience; A one-sided (resp. two-sided) shift of finite type of dimension one can be described as the set of infinite (resp. bi-infinite) sequences of consecutive edges in a finite-state automaton. While the conjugacy of shifts of finite type is decidable for one-sided shifts of finite type of dimension one, the result is unknown in the two-sided case. In this paper, we study the shifts of finite type defined by infinite ranked trees. Indeed, infinite ranked trees have a natural structure of symbolic dynamical systems. We prove a decomposition Theorem for these tree-shifts, i.e. we show that a conjugacy between two tree-shifts can be broken down into a finite sequence of elementary transformations called in-splittings and in-amalgamations. We prove that the conjugacy problem is decidable for tree-shifts of finite type. This result makes the class of tree-shifts closer to the class of one-sided shifts of sequences than to the class of two-sided ones. Our proof uses the notion of bottom-up tree automata.

The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it cycling problem} as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type. On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USS), using left normal forms of braids. But one can equally use right normal forms and compute right-USS's. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USS's. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type., 20 pages

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w ( G ) ⩽ τ , can act properly on R τ ∖ { 0 } such that R τ ∖ { 0 } contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ⩽τ. The metric cones Cone ( G / H ) with H ⊂ G a compact subgroup such that G / H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H ⊂ G does not exceed the weight of G.

All continuous endomorphisms f"~ of the shift dynamical system S on the 2-adic integers Z"2 are induced by some f:B"n->{0,1}, where n is a positive integer, B"n is the set of n-blocks over {0, 1}, and f"~(x)=y"0y"1y"2... where for all i@?N, y"i=f(x"ix"i"+"1...x"i"+"n"-"1). Define D:Z"2->Z"2 to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z"2->Z"2 by V(x)=-1-x. We prove that D, V@?D, S, and V@?S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z"2->Z"2 to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m@?Z^+, there exists some n@?N such that R^-^1(m) has binary representation of the form x"0x"1...x"2"^"n"-"1@? or x"0x"1x"2...x"2"^"n@?.