Determining the doubly slice genera of prime knots with up to 12 crossings

For a knot [Formula: see text], the doubly slice genus [Formula: see text] is the minimal [Formula: see text] such that [Formula: see text] divides a closed, orientable, and unknotted surface of genus [Formula: see text] embedded in [Formula: see text]. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.

Braid group and leveling of a knot

Any knot [Formula: see text] in genus-[Formula: see text] [Formula: see text]-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the [Formula: see text]-length. We show that the [Formula: see text]-length equals the level number. We then find braid descriptions for [Formula: see text]-positions of all [Formula: see text]-bridge knots providing upper bounds for their level numbers and also show that the [Formula: see text]-pretzel knot has level number two.

Inflation of the universe by the non-minimal Y (R )F 2 models

The cosmological solutions of the non-minimal [Formula: see text] theory, which are compatible with FRW spacetime, are investigated. In order to avoid the isotropy violation of a vector field, it can be considered that the presence of a triplet of vector fields satisfy SO(3) symmetry or the average of randomly oriented electromagnetic fields over the sufficiently large volume. By considering the symmetry arguments, the inflation driven by the non-minimally coupled vector fields is studied. Then the cosmological solutions and corresponding models are obtained with the power-law, hyperbolic and hybrid scale functions.

The nilpotent variety of W(1;n)p is irreducible

In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic [Formula: see text] is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series [Formula: see text] and [Formula: see text]. In this paper, with the assumption that [Formula: see text], we confirm this conjecture for the minimal [Formula: see text]-envelope [Formula: see text] of the Zassenhaus algebra [Formula: see text] for all [Formula: see text].

Molecular trees with the sixth, seventh and eighth minimal irregularity values

The irregularity of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text] and [Formula: see text] is the edge set of [Formula: see text]. From the class of all [Formula: see text]-vertex (molecular) trees, graphs with the first five minimal [Formula: see text]-values have already been characterized in the literature. The main purpose of this paper is to determine the graphs with the sixth, seventh and eighth minimal [Formula: see text]-values among all the members of the aforementioned class for [Formula: see text], [Formula: see text] and [Formula: see text], respectively.

Effect of low anisotropy and supernova constraints on cosmological models

The 194 supernova Ia data and the effect of anisotropy are combined to reconstruct the dark energy equation of state parameter [Formula: see text] and the deceleration parameter [Formula: see text]. Using the supernovae type Ia data, we evaluate the anisotropy effects (although low) on dark energy parametrization [Formula: see text] and we compare the results with [Formula: see text]CDM model. Present supernova observations are analyzed using a standard [Formula: see text] method and the minimal [Formula: see text] values obtained for each model are compared. We confirm the difficulty of discriminating between these models using present SNIa data only. By means of the maximum likelihood method, we find that the best-fit dynamical [Formula: see text] and [Formula: see text] parameters [Formula: see text] are obtained from the SNIa dataset. In particular, we find the best-fit values of [Formula: see text]CDM model ([Formula: see text] = 0.013, [Formula: see text] = 197.559) for [Formula: see text] = 0.3 and ([Formula: see text] = 0.02, [Formula: see text] = 196.983) for [Formula: see text] = 0.27. Finally, we found that the presence of anisotropy is confirmed in mentioned models via SNIa dataset.

The canonical volume of minimal 3-folds of general type

Let [Formula: see text] be a nonsingular projective [Formula: see text]-fold of general type. Denote by [Formula: see text] the [Formula: see text]-canonical map of [Formula: see text] which is the rational map naturally associated to the complete linear system [Formula: see text]. Suppose that [Formula: see text] be a minimal [Formula: see text]-fold of [Formula: see text] and [Formula: see text] the pluricanonical section index. In this paper, we obtain the lower bounds of the canonical volume [Formula: see text] in term of [Formula: see text] for [Formula: see text]. In addition, we also classify the weighted baskets [Formula: see text] of [Formula: see text] satisfying [Formula: see text].

Solvable normal subgroups of 2-knot groups

If [Formula: see text] is an orientable, strongly minimal [Formula: see text]-complex and [Formula: see text] has one end, then it has no nontrivial locally finite normal subgroup. Hence, if [Formula: see text] is a 2-knot group, then (a) if [Formula: see text] is virtually solvable, then either [Formula: see text] has two ends or [Formula: see text], with presentation [Formula: see text], or [Formula: see text] is torsion-free and polycyclic of Hirsch length 4 (b) either [Formula: see text] has two ends, or [Formula: see text] has one end and the center [Formula: see text] is torsion-free, or [Formula: see text] has infinitely many ends and [Formula: see text] is finite, and (c) the Hirsch–Plotkin radical [Formula: see text] is nilpotent.

P4-equicoverable graphs which contain cycles with length at least 4

A graph is called [Formula: see text]-equicoverable if every minimal [Formula: see text]-covering of it is also its minimum [Formula: see text]-covering. In this paper, we consider how to characterize a graph to be [Formula: see text]-equicoverable where [Formula: see text] is [Formula: see text]. We obtained necessary and sufficient conditions for a graph contains a cycle with length greater than 3 but not contains any 3-cycles.

Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos

This paper introduces the notion of second minimal [Formula: see text]-periodic orbits of continuous maps on the interval according to whether [Formula: see text] is a successor of the minimal period of the map in the Sharkovski ordering. We pursue the classification of second minimal [Formula: see text]-orbits in terms of cyclic permutations and digraphs. It is proven that there are nine types of second minimal 7-orbits with accuracy up to inverses. The result is applied to the problem of the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. It is revealed that by fixing the maximum number of appearances of periodic windows, there is a universal pattern of distribution. In particular, the first appearance of all the orbits is always a minimal orbit, while the second appearance is a second minimal orbit. It is observed that the second appearance of the 7-orbit is a second minimal 7-orbit with a Type 1 digraph. The reason for the relevance of the Type 1 second minimal orbit is the fact that the topological structure of the unimodal map with a single maximum, is equivalent to the structure of the Type 1 piecewise monotonic endomorphism associated with the second minimal 7-orbit. Yet another important report of this paper is the revelation of universal pattern dynamics with respect to an increased number of appearances.