Infinite Diameter Research Articles

Quotients of the curve complex

We consider three kinds of quotients of the curve complex, which are obtained by coning off uniformly quasiconvex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the mapping class group on those quotients are strongly weakly properly discontinuously (WPD), which implies that the actions are non-elementary and those quotients are of infinite diameter.

The Artin monoid Cayley graph

In this paper, we investigate properties of the Artin monoid Cayley graph. This is the Cayley graph of an Artin group A_{\Gamma} with respect to the (infinite) generating set given by the associated Artin monoid A^+_{\Gamma} . In a previous paper, the first three authors introduced a monoid Deligne complex and showed that this complex is contractible for all Artin groups. In this paper, we show that the Artin monoid Cayley graph is quasi-isometric to a modification of the Deligne complex for A_{\Gamma} obtained by coning off translates of the monoid Deligne complex. We then address the question of when the monoid Cayley graph has infinite diameter. We conjecture that this holds for all Artin groups of infinite type. We give a set of criteria that imply infinite diameter, and using existing solutions to the word problem for large type Artin groups and 3-free Artin groups, we prove that the conjecture holds for any Artin group containing a 3-generator subgroup of one of these two types.

Laminations and 2-filling rays on infinite type surfaces

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays have strong filling properties while failing to correspond to points on the boundary of the loop graph. As such they may be thought of as “fake boundary points.” We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.

Detonation performance of urea‐hydrogen peroxide (UHP)**

AbstractCarbamide Peroxide, an adduct of Urea and Hydrogen Peroxide, is commonly used in the cosmetic and pharmaceutical industries as a solid source of hydrogen peroxide. However, it exhibits explosive properties and can be easily manufactured from readily available household chemicals, making it a potential emerging threat. We carried out a detailed performance assessment, combining experiments, thermochemical calculations and numerical simulations and highlighted a good level of agreement between experimental data from lab, field and underwater firings. A maximum detonation velocity of 3.65 km/s was recorded for unconfined 25 kg UHP charges at 0.85 g/cm3(200 mm charge diameter). We determined in these conditions an infinite diameter detonation velocity of 3.94 km/s. These results are also consistent with previous results obtained at small scale under heavy confinement. Airblast measurements highlighted an average 40 % TNT equivalence for impulse and 55 % for peak overpressure at short distance, which are in good agreement with the 57 % (Power Index) calculated from Explo5, while 50 % for bubble energy (explosive power) and 20 % for shock pressure (brisance) were obtained from underwater experiments. The use of different experimental approaches has proven useful to characterise the performances parameters of a non‐ideal explosive for risk assessment purposes.

Explosive growth for a constrained Hastings–Levitov aggregation model

We consider a constrained version of the HL(0) Hastings-Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $\sqrt{t\log n}$, uniformly in $t\in [0,T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained HL(0).

Graphs of curves for surfaces with finite-invariance index \(1\)

In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with fi­ni­te-invariance index \(1\) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups.

Metric properties of incomparability graphs with an emphasis on paths

We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the incomparability graph of a poset is connected and has infinite diameter, then it contains an infinite induced path. Furthermore, if the diameter of the set of vertices of degree at least $3$ is infinite, then the graph contains as an induced subgraph either a comb or a kite.

Topological Veech dichotomy and tessellations of the hyperbolic plane

For every half-translation surface with marked points (M, Σ), we construct an associated tessellation Π(M, Σ) of the Poincaré upper half plane whose tiles have finitely many sides and area at most π. The tessellation Π(M, Σ) is equivariant with respect to the action of PSL(2, ℝ), and invariant with respect to (half-)translation covering. In the case (M, Σ) is the torus ℂ/ℤ2 with a one marked point, Π(ℂ/ℤ2, {0}) coincides with the iso-Delaunay tessellation introduced by Veech [25] (see also [1, 2]) as both tessellations give the Farey tessellation. As application, we obtain a bound on the volume of the corresponding Teichmüller curve in the case (M, Σ) is a Veech surface (lattice surface). Under the assumption that (M, Σ) satisfies the topological Veech dichotomy, there is a natural graph $${\cal G}$$ underlying Π(M, Σ) on which the Veech group Γ acts by automorphisms. We show that $${\cal G}$$ has infinite diameter and is Gromov hyperbolic. Furthermore, the quotient $$\overline {\cal G}:= {\cal G}/\Gamma $$ is a finite graph if and only if (M, Σ) is actually a Veech surface, in which case we provide an algorithm to determine the graph $$\overline {\cal G} $$ explicitly. This algorithm also allows one to get a generating family and a “coarse” fundamental domain of the Veech group Γ.

Infinite diameter confidence sets in Hedges’ publication bias model

Meta-analysis, the statistical analysis of results from separate studies, is a fundamental building block of science. But the assumptions of classical meta-analysis models are not satisfied whenever publication bias is present, which causes inconsistent parameter estimates. Hedges’ selection function model takes publication bias into account, but estimating and inferring with this model is tough for some datasets. Using a generalized Gleser–Hwang theorem, we show there is no confidence set of guaranteed finite diameter for the parameters of Hedges’ selection model. This result provides a partial explanation for why inference with Hedges’ selection model is fraught with difficulties.

Аппроксимации ациклических графов

In this paper, approximations of acyclic graphs are studied. It is proved that any theory of an acyclic graph (tree) of finite diameter is pseudofinite with respect to acyclic graphs (trees), that is, any such theory is approximated by theories of finite structures (acyclic graphs, trees). It is also proved that an acyclic graph of infinite diameter with infinite number of rays is pseudofinite.

Calculated Critical and Subcritical Data for Revision of ANSI/ANS 8.12

Calculated critical and subcritical data for systems of mixed oxides of plutonium and uranium are presented for review and subsequent incorporation into a revision of ANSI/ANS 8.12. The system specifications were prepared in parallel with the ISO standard ISO 11311:2011, involving experience from international mixed oxide fuel fabrication facilities. Data are presented for two mass fractions of plutonium to uranium plus plutonium (at 12.5% and 35%), damp mixed oxide density up to a theoretical density of 11.03 g/cm3 (for the 12.5% mass fraction, ignoring the variation due to plutonium isotopic content) and up to a density of 3.5 g/cm3 (for the 35% mass fraction), three plutonium isotopic mass fractions (100% 239Pu, 95% 239Pu with 5% 240Pu, and a reactor grade composition with 20% 240Pu and the balance 239Pu, 241Pu, and 242Pu), systems at two moderation conditions (water mass fractions at 3% and at optimal conditions) and systems at two water reflector conditions (nominal 2.5 cm and full 30.0 cm). Parameters of interest presented are values of volume (L), infinite cylinder diameter (cm), infinite slab thickness (cm), mass (kg), cylinder linear density (g/cm), and slab areal density (g/cm2). Values are presented at calculated multiplication factors of 1.000, 0.980 and 0.950. Calculations were made by various subject matter experts with various computational programs and cross section libraries. Validation methods were reviewed and discussed herein though data propagated to revision of ANSI/ANS 8.12 will reflect a conservative subcritical margin selected based on expert judgment.

The compression body graph has infinite diameter

We show that the compression body graph has infinite diameter.

Parabolic subgroups acting on the additional length graph

Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical type. We show that periodic elements of $A$ and the elements preserving some parabolic subgroup of $A$ act elliptically on the additional length graph $\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $\langle P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic subgroup $P$ of $A$. The length of $g$ is uniformly bounded with respect to the Garside generators, independently of $A$. This allows us to show that, in contrast with the Artin generators case, the sequence $\{\omega(A_n,\mathcal{S})\}_{n\in \mathbb{N}}$ of exponential growth rates of braid groups with respect to the Garside generating set, goes to infinity.

Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text]. The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text]. Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text]. On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

Cylinder curves in finite holonomy flat metrics

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order [Formula: see text], the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when [Formula: see text] is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case [Formula: see text] and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if [Formula: see text] and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

The group \({\text {Diff}}({\mathcal {M}})\) of diffeomorphisms of a closed manifold \({\mathcal {M}}\) is naturally equipped with various right-invariant Sobolev norms \(W^{s,p}\). Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when \(sp\le \dim {\mathcal {M}}\) and \(s<1\)). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when \((s-1)p\ge \dim {\mathcal {M}}\), and that for spheres the diameter is finite when \((s-1)p<1\). In particular, this gives a full characterization of the diameter of \({\text {Diff}}(S^1)\). In addition, we show that for \({\text {Diff}}_c({\mathbb {R}}^n)\), if the diameter is not zero, it is infinite.

Tetraamminecopper Perchlorate (TACP): Explosive Properties

AbstractTetraamminecopper perchlorate (TACP) is one of the most interesting explosive copper perchlorate‐ammonia complexes. Despite the fact that this complex has been known for more than 100 years there is very little information in the literature about its detonation parameters. Impact sensitivity of TACP is slightly higher than that for PETN while friction sensitivity is between those for PETN and RDX. Detonation velocity for infinite diameter is 3230 m s−1 (density 0.9 g cm−3). Critical diameter of TACP is low – less than 2 mm. Based on the heat of combustion measurement the previously published erroneous enthalpy of formation was corrected to −496 kJ mol−1. Detonation heat of TACP is 4322 kJ kg−1 and Gurney velocity is 1536 m s−1. Experimentally measured detonation parameters of TACP were also compared with theoretically calculated values by the EXPLO5 code.

Geometry of the graphs of nonseparating curves: covers and boundaries

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenstädt identifying the Gromov boundary with the space of ending laminations on full genus subsurfaces. We introduce several tools based around the analysis of bicorn curves and laminations which may be of independent interest for studying the geometry of nonseparating curve graphs of infinite type surfaces and their boundaries.

Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.

On groups with unbounded Cayley graphs

We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.