The Frohman Kania–Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime $p \ge 5$. We give some estimates of this ideal. We also calculate this invariant for some 3-manifolds constructed by doing surgery on a knot in the complement of another knot.

In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function $f$ belongs to the class ${\bf S}$ and its Julia set is locally connected, then the residual Julia set of $f$ is empty if and only if its Fatou set $F(f)$ has a completely invariant component or consists of only two components. We also show that if $f$ is a meromorphic function which is not of the form $\alpha + (z\,-\,\alpha)^{-k}e^{g(z)}$, where $k$ is a natural number, $\alpha$ is a complex number and $g$ is an entire function, then $f$ has buried components provided that $f$ has no completely invariant components and its Julia set $J(f)$ is disconnected. Moreover, if $F(f)$ has an infinitely connected component, then the singleton buried components are dense in $J(f)$. This generalizes a result of Baker and Domínguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.

It is shown that, if $\Sigma$ is a closed orientable surface and $\varphi\!: \Sigma \to \Sigma$ a homeomorphism, then one can find an ordering of $\pi_1(\Sigma)$ which is invariant under left- and right-multiplication, as well as under $\varphi_* \colon \pi_1(\Sigma) \to \pi_1(\Sigma)$, provided all the eigenvalues of the map induced by $\varphi$ on the integral first homology groups of $\Sigma$ are real and positive. As an application, if $M^3$ is a closed orientable 3-manifold which fibres over the circle, then its fundamental group is bi-orderable if the associated homology monodromy has all eigenvalues real and positive. This holds, in particular, if the monodromy is in the Torelli subgroup of the mapping class group of $\Sigma$.

We consider Exel's new construction of a crossed product of a $C^*$-algebra $A$ by an endomorphism $\alpha$. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz–Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from $A$ into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.

Suppose that $P(D)$ is a linear differential operator with complex coefficients and $\{f_k\}_{k \in \mathbb{Z}}$ is a two-sided sequence of complex-valued functions on $\mathbb{R}$ such that $f_{k+1} \,{=}\, P(D)f_k$ with the following growth condition: there exist $a\,{\in}\,[0,1)$ and $N\,{\geq}\,0$ such that $|f_k(x)|\,{\leq}\,M_{|k|}\exp (N|x|^a)$, where the sequence $\{M_k\}_{k=0}^{\infty}$ satisfies that for any $\varepsilon\,{>}\,0$, the sequence $\{{M_k}/{(1+\varepsilon)^{k}}\}_{k=0}^{\infty}$ has a bounded subsequence. Then $f_0$ is an entire function of an exponential growth. This result is a generalization of those of Roe, Burkill and Howard.

We show examples that mutants may not be visible as such from minimal crossing diagrams, and of knots with the same polynomial invariants and same hyperbolic volume, which are not mutants.

An example illustrates (a) that the use of indefinite inner products can illuminate Probabilistic Wiener–Hopf Theory in symmetrizable cases, and (b) that half-winding probabilities should be thought of as branching measures for Ray processes. The paper's wide variety of techniques could make it instructive for research students.

We study the spectrum containment of almost periodic solutions to differential equations with piecewise constant argument (EPCA, for short). We find an important property, which is different from that given by Cartwright for Ordinary Differential Equations (ODE). Some known (periodic, quasi-periodic) results may be expanded. As a corollary, it is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. As a consequence, a new class of quasi-periodic functions is described. This new phenomenon is due to the piecewise constant argument and illustrates a crucial difference between ODE and EPCA.

In two-dimensional knot theory, the relationship between surface-links and surface braids has been studied. A surface braid is described by a monodromy representation on a punctured disk which maps into the classical braid group. In this paper, we study a surface braid via a linear representation of the classical braid group and give a remarkable example of surface braids.

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.

Global existence results in the past time direction of cosmological models with collisionless matter and a massless scalar field are presented. It is shown that the singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is approached. In the case without Vlasov matter, the singularity is velocity dominated and the generalized Kasner exponents converge at each spatial point as the singularity is approached.

In a recent paper we defined a new filtration of the mapping class group – the Lagrangian filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive invariant of finite-type (e.g. the Casson invariant) and any level of the Lagrangian filtration, there is a homology 3-sphere which has a Heegaard decomposition whose gluing diffeomorphism lies at that level, on which this invariant is non-zero. In a final section we examine the relationship between the Johnson and Lagrangian filtrations.

In this paper, a generalized predator-prey system with delays is considered. By using coincidence degree theory, we establish the existence result of multiple positive periodic solutions for the system.

Two commuting real matrices $A$ and $B$ have commuting exponentials $\exp A$ and $\exp B$, a fact observed for instance in linear algebra or differential equations courses. The converse implication is false. A clarification of this phenomenon is proposed that makes use of the theory of the exponential function $\exp\colon{\fam\euffam g}\to G$ of a real Lie group $G$ and its singularities. In Section 1, a catalog of low-dimensional examples illustrates various ways that, for two elements $X, Y\in{\fam\euffam g}$, the commuting of $\exp X$ and $\exp Y$ in $G$ may fail to entail the commuting of $X$ and $Y$ in ${\fam\euffam g}$. In Section 2, consequences of the relation $[\exp X,\exp Y]={\bf 1}$ are inspected, whereby certain regularity assumptions on $X$ and $Y$ are made. A regular element $Y$ of the Lie algebra ${\fam\euffam g}$ determines a Cartan subalgebra ${\fam\euffam h}={\fam\euffam g}^0(Y)$ of ${\fam\euffam g}$ and a certain subgroup ${\cal W}_Y$ of the (finite!) Weyl group of ${\fam\euffam g}$ with respect to the Cartan subalgebra ${\fam\euffam h}$. If, additionally, the exponential function is regular at $X$ and at $Y$, then the ordered pair $(X,Y)$ is said to be in general position. If $(X,Y)$ is in general position, then the relation $[\exp X,\exp Y]={\bf 1}$ in $G$ permits the definition of a certain element $w(X,Y)\in{\cal W}_Y$. Let ${\fam\euffam z}({\fam\euffam g})$ denote the center of ${\fam\euffam g}$. It is shown that, if $\exp X$ and $\exp Y$ commute in $G$ for $(X,Y)$ in general position, then $[X,Y]\in{\fam\euffam z}({\fam\euffam g})\cap[{\fam\euffam h},{\fam\euffam h}]$ iff $w(X,Y)={\bf 1}$. Write $H\defi\exp{\fam\euffam h}$, and let $Z(G)$ denote the center of $G$. If the identity component of $Z(G)\cap[H,H]$ is simply connected, and if $\exp X$ and $\exp Y$ commute for $(X,Y)$ in general position, then $[X,Y]=0$ iff $w(X,Y)={\bf 1}$. If $G$ is simply connected compact, then $[\exp X,\exp Y]={\bf 1}$ and $[X,Y]=0$ are equivalent for all pairs $(X,Y)$ in general position. In ${\mathop{\rm SO}\nolimits}(3)$ this is not the case; here $|{\cal W}_Y|=2$. In Section 3, examples show that the validity of the equation $\exp X\exp Y=\exp\!(X+Y)$ has no implications whatsoever in the direction of the commuting of $\exp X$ and $\exp Y$. Finally, in Section 4, it is shown that, for a simply connected Lie group $G$, the commuting of $X, Y\in{\fam\euffam g}$ and that of $\exp X,\exp Y\in G$ are equivalent properties for all$X$ and $Y$ if and only if the exponential function is injective. This class of Lie groups was characterized in terms of other properties by Dixmier and by Saito, independently, in 1957.

Conflict sets are loci of intersecting wave fronts emanating from $l$ different manifolds. We show that for generic manifolds their conflict sets are projections of smooth conic Lagrangian manifolds. Thus conflict sets locally admit the structure of wave fronts. Simple stable singularities for this problem in $\mathbb{R}^n$ occur when $0\leq n-l \leq 4$. Relations with centre sets, billiards, pedals and orthotomics are also discussed. Throughout canonical relations are used as an essential tool to carry out these geometrical constructions.

Fractal sums of pulses are defined on $\mathop{\mathbb R}^D$ as follows: $F(t)=\sum_{n=1}^\infty n^{-H/D} G\big(n^{1/D}\,(t-X_n)\big)$ where the $X_n$ are independent random variables, $H\in [0,1]$ and $G$ is the elementary “bump” or “pulse”. If the $X_n$ are uniform on a cube and $G$ sufficiently regular we prove the (almost sure) existence of $F$ and show that the box dimension of its graph is $2-H$.

We show that if $f$ is a transcendental meromorphic function with a finite number of poles and $f$ has a cycle of Baker domains of period $p$, then there exist $C > 1$ and $r_0>0$ such that $\bigg\{z:\frac1C r\lt |z|\lt Cr\bigg\}\cap \mbox{sing} (f^{-p})\ne\varnothing,{\for}r\ge r_0.$ We also give examples to show that this result fails for transcendental meromorphic functions with infinitely many poles.