Smooth topologically transitive dynamic systems

The paper proves the existence of smooth topologically transitive dynamic systems-flows and cascades for any connected n-dimensional region (n3), and cascades for two-dimensional regions diffeomorphic to the unit circle.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 751–759, December, 1968.The author wishes to thank D. V. Anosov for his advice and his interest in this work.     

Chaos in a topologically transitive system

The chaotic phenomena have been studied in a topologically transitive system and it has been shown that the erratic time dependence of orbits in such a topologically transitive system is more complicated than what described by the well-known technology "Li-Yorke chaos". The concept "sensitive dependency on initial conditions" has been generalized, and the chaotic phenomena has been discussed for transitive systems with the generalized sensitive dependency property.     

A note on infinite transitive graphs

In [4] Jung and Watkins proved that for a connected infinite graph X either κ_∞(X) = ∞ holds or X is a strip, if Aut(X) contains a transitive abelian subgroup G. Here we prove the same result under weaker assumptions.     

Knots and topologically transitive flows on 3-manifolds

Suppose that ? is a nonsingular (fixed point free) flow on a smooth three-dimensional manifold M. Suppose the orbit though a point p∈M is dense in M. Let D be an imbedded disk in M containing p which is transverse to the flow. Suppose that q∈D is a point in the forward orbit of p. Under certain assumptions on M, which include the case M=S³, we prove that if q is sufficiently close to p then the orbit segment from p to q together with a compact segment in D from p to q forms a nontrivial prime knot in M.     

A note on barely transitive groups

We study some properties for barely transitive group.     

Existence of topologically cylindrical shocks

In this Note the multidimensional stability of cylindrical shock profiles and the existence of a nearby perturbed structure is presented for the full Euler equations. This provides an example of a nonplanar structure for which the uniform Kreiss–Lopatinski–Majda stability condition can be explicitly verified. To cite this article: N. Costanzino, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A note on decompositions of transitive tournaments

For any positive integer n, we determine all connected digraphs G of size at most four, such that a transitive tournament of order n is G-decomposable. Among others, these results disprove a generalization of a theorem of Sali and Simonyi [Orientations of self-complementary graphs and the relation of Sperner and Shannon capacities, European J. Combin. 20 (1999), 93–99].     

Periodic prime knots and topologically transitive flows on 3-manifolds

Suppose that φ is a nonsingular (fixed point free) C¹ flow on a smooth closed 3-dimensional manifold M with H₂(M)=0. Suppose that φ has a dense orbit. We show that there exists an open dense set N⊆M such that any knotted periodic orbit which intersects N is a nontrivial prime knot.     

Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = × Z of the formU (, z)=(T,, z+f ()) ( , z Z), where is a complete separable metric space, T is a t.t. homeomorphism of onto itself, Z is a separable banach space, andf is a continuous map: z. For the special case W = S¹×R, T=+ ( is incommensurable with 2) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S¹R for which the corresponding homeomorphism (1) is t.t. for all incommensurable with 2.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.The author thanks D. V. Anosov for advice and interest in the work.     

A note on an embedding problem in transitive tournaments

Let TT_n be a transitive tournament on n vertices. It is known Görlich, Pil?niak, Wo?niak, (2006) [3] that for any acyclic oriented graph of order n and size not greater than , two graphs isomorphic to are arc-disjoint subgraphs of TT_n. In this paper, we consider the problem of embedding of acyclic oriented graphs into their complements in transitive tournaments. We show that any acyclic oriented graph of size at most is embeddable into all its complements in TT_n. Moreover, this bound is generally the best possible.     

A note on transitive permutation groups of prime degree

LetG be a nonsolvable transitive permutation group of prime degreep. LetP be a Sylow-p-subgroup ofG and letq be a generator of the subgroup ofN _G(P) fixing one point. Assume that |N _G(P)|=p(p?1) and that there exists an elementj inG such thatj ^?1qj=q^(p+1)/2. We shall prove that a group that satisfies the above condition must be the symmetric group onp points, andp is of the form 4n+1.     

A note on finite groups in which c-permutability is transitive

A subgroup H of G is c-permutable in G if there exists a permutable subgroup P of?G such that HP=G and H∩P≦H _pG , where H _pG is the largest permutable subgroup of?G contained in?H. A?group G is called CPT-group if c-permutability is transitive in?G. A?number of new characterizations of finite solvable CPT-groups are given.     

A note on transitive orientations with maximum sets of sources and sinks

Given a transitive orientation of a comparability graph G, a vertex of G is a source (sink) if it has indegree (outdegree) zero in , respectively. A source set of G is a subset of vertices formed by sources of some transitive orientation . A pair of subsets S,TV(G) is a source–sink pair of G when the vertices of S and T are sources and sinks, of some transitive orientation , respectively. We describe algorithms for finding a transitive orientation with a maximum source–sink pair in a comparability graph. The algorithms are applications of modular decomposition and are all of linear-time complexity.     

A note about the Catalan triangle

A new identity is obtained for the Catalan triangle introduced by Shapiro.     

A transitive closure algorithm

An algorithm is given for computing the transitive closure of a directed graph in a time no greater thana ₁ N ₁ n+a ₂ n ² for largen wherea ₁ anda ₂ are constants depending on the computer used to execute the algorithm,n is the number of nodes in the graph andN ₁ is the number of arcs (not counting those arcs which are part of a cycle and not counting those arcs which can be removed without changing the transitive closure). For graphs where each arc is selected at random with probabilityp, the average time to compute the transitive closure is no greater than min{a ₁ pn ³+a ₂ n ², 1/2a ₁ n ² p ^–2+a ₂ n ²} for largen. The algorithm will compute the transitive closure of an undirected graph in a time no greater thana ₂ n ² for largen. The method uses aboutn ²+n bits and 5n words of storage (where each word can holdn+2 values).     

A graph which is edge transitive but not arc transitive

A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.     

A note about the dominating circuit conjecture

The dominating circuit conjecture states that every cyclically 4-edge-connected cubic graph has a dominating circuit. We show that this is equivalent to the statement that any two edges of such a cyclically 4-edge-connected graph are contained in a dominating circuit.