Many factors governed the penetration efficacy of prehistoric projectile weaponry. Archaeologists broadly focus their efforts on understanding the effect of stone weapon tips because these specimens are often the only part of the weapon system that survives in the archaeological record. The tip cross‐sectional area (TCSA) and perimeter (TCSP) of stone weapon tips have been shown to correlate with target penetration depth. Here, using results from both static and dynamic penetration testing, we compare TCSA and TCSP against other tip geometry metrics: lateral surface area (LSA) and volume (V). Our analyses broadly show that using a single‐point geometry metric evaluated at multiple locations along the length of the point, or using multiple geometry metrics evaluated at a single location, better predicts required energy than using a single‐point geometry metric evaluated at a single location. Our results also show that in the case where a single geometry metric evaluated at multiple locations is used LSA provided the most robust prediction models. Finally, our results show that for the case where all geometry metrics evaluated at a single location are used the location that provides the most robust prediction model is dependent on how far the point penetrated the target.
COARSE AND FINE GEOMETRY OF THE THURSTON METRIC
We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.
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- NSF-PAR ID:
- 10169325
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 8
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Let $M$ be a compact 3-manifold and $\Gamma =\pi _1(M)$. Work by Thurston and Culler–Shalen established the ${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$ character variety $X(\Gamma )$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(\Gamma )$, as well as distinguished points on the canonical component, when $\Gamma $ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.more » « less
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Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol. , to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint , 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint , 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.more » « less
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Let M be a compact 3-manifold and = π1(M). The work by Thurston and Culler– Shalen established the SL2(C) character variety X() as fundamental tool in the study of the geometry and topology of M. This is particularly the case when M is the exterior of a hyperbolic knot K in S3. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of X(), as well as distinguished points on the canonical component, when is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2020.3
