Amenable dynamical systems over locally compact groups

We establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$ . In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$ .

THE DIAMETER AND RADIUS OF RADIALLY MAXIMAL GRAPHS

A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. Harary and Thomassen [‘Anticritical graphs’, Math. Proc. Cambridge Philos. Soc.79(1) (1976), 11–18] proved that the radius r and diameter d of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton et al. [‘Changing and unchanging of the radius of a graph’, Linear Algebra Appl.217 (1995), 67–82] rediscovered this result with a different proof and conjectured that the converse is true, that is, if r and d are positive integers satisfying $r\le d\le 2r-2,$ then there exists a radially maximal graph with radius r and diameter $d.$ We prove this conjecture and a little more.

The M Waves Emitted by an Expanding Phase Boundary Create a “Lacuna”

The M waves introduced by Burridge and Willis (1969, “The Self-Similar Problem of the Expanding Crack in an Anisotropic Solid,” Math. Proc. Cambridge Philos. Soc., 66(2), pp. 443–468) are emitted by the surface of a self-similarly expanding elliptical crack, and they give Rayleigh waves at the corresponding crack speed. In the analysis for the self-similarly expanding spherical inclusion with phase change (dynamic Eshelby problem), the M waves are related to the waves obtained on the basis of the dynamic Green’s function containing the contribution from the latest wavelets emitted by the expanding boundary of phase discontinuity, and they satisfy the Hadamard jump conditions for compatibility and linear momentum across the moving phase boundary of discontinuity. In the interior of the expanding inclusion, they create a “lacuna” with zero particle velocity by canceling the effect of the P and S. It is shown that the “lacuna” and Eshelby properties are also valid for a Newtonian fluid undergoing phase change in a self-similarly expanding ellipsoidal region of a fluid with different viscosity.

Weakly divergent partial quotients

We study the real numbers with partial quotients diverging to infinity in a subsequence. We show that if the subsequence has positive density then such sets have Hausdorff dimension equal to 1/2. This generalizes one of the results obtained in [C. Y. Cao, B. W. Wang and J. Wu, The growth speed of digits in infinite iterated function systems, Studia. Math. 217(2) (2013) 139–158; I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc. 37 (1941) 199–228].

Circuit presentation and lattice stick number with exactly four z-sticks

The lattice stick number [Formula: see text] of a link [Formula: see text] is defined to be the minimal number of straight line segments required to construct a stick presentation of [Formula: see text] in the cubic lattice. Hong, No and Oh [Upper bound on lattice stick number of knots, Math. Proc. Cambridge Philos. Soc. 155 (2013) 173–179] found a general upper bound [Formula: see text]. A rational link can be represented by a lattice presentation with exactly 4 [Formula: see text]-sticks. An [Formula: see text]-circuit is the disjoint union of [Formula: see text] arcs in the lattice plane [Formula: see text]. An [Formula: see text]-circuit presentation is an embedding obtained from the [Formula: see text]-circuit by connecting each [Formula: see text] pair of vertices with one line segment above the circuit. By using a two-circuit presentation, we can easily find the lattice presentation with exactly four [Formula: see text]-sticks. In this paper, we show that an upper bound for the lattice stick number of rational [Formula: see text]-links realized with exactly four [Formula: see text]-sticks is [Formula: see text]. Furthermore, it is [Formula: see text] if [Formula: see text] is a two-component link.

On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings

We give presentations of the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. As the volume conjecture states, the leading terms of the expansions present the hyperbolic volume and the Chern–Simons invariant of the complements of the knots. As coefficients of the expansions, we obtain a series of new invariants of the knots. This paper is a continuation of the previous papers [T. Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the [Formula: see text] knot, Quantum Topol. 7 (2016) 669–735; T. Ohtsuki and Y. Yokota, On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings, to appear in Math. Proc. Cambridge Philos. Soc.], where the asymptotic expansions of the Kashaev invariant are calculated for hyperbolic knots with five and six crossings. A technical difficulty of this paper is to use 4-variable saddle point method, whose concrete calculations are far more complicated than the previous papers.

Characterizing local rings via perfect and coperfect modules

Let [Formula: see text] be a Noetherian ring and let [Formula: see text] be a semidualizing [Formula: see text]-module. In this paper, by using the classes [Formula: see text] and [Formula: see text], we extend the notions of perfect and coperfect modules introduced by Rees [The grade of an ideal or module, Proc. Cambridge Philos. Soc. 53 (1957) 28–42] and Jenda [The dual of the grade of a module, Arch. Math. 51 (1988) 297–302]. First, we study the basic properties of these modules and relations between them. Next, we characterize local rings in terms of the existence of special perfect (respectively, coperfect) modules.

The homological content of the Jones representations at q = −1

We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at [Formula: see text], are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of [Formula: see text], Math. Proc. Cambridge Philos. Soc. 141(3) (2006) 477–488] by extending their original argument for the sphere with four marked points to our more general case.

Cross-connections of completely simple semigroups

A completely simple semigroup [Formula: see text] is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that [Formula: see text] is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup [Formula: see text] (cf. D. Rees, On semigroups, Proc. Cambridge Philos. Soc. 36 (1940) 387–400). In the study of structure theory of regular semigroups, Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left (right) ideals using cross-connections. A normal category [Formula: see text] is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories [Formula: see text] and [Formula: see text] is a local isomorphism [Formula: see text] where [Formula: see text] is the normal dual of the category [Formula: see text]. In this paper, we identify the normal categories associated with a completely simple semigroup [Formula: see text] and show that the semigroup of normal cones [Formula: see text] is isomorphic to a semi-direct product [Formula: see text]. We characterize the cross-connections in this case and show that each sandwich matrix [Formula: see text] correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.

A dimensional result in continued fractions

Given x ∈ (0, 1), let [a1(x), a2(x), a3(x),…] be the continued fraction expansion of x and [Formula: see text] be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {an(x), n ≥ 1} and proved that for any β > 0, the set [Formula: see text] is of Hausdorff dimension [Formula: see text]. In this paper, we consider, for any β > 0, the set [Formula: see text] and show that the Hausdorff dimension of F(β) is [Formula: see text].