Palmated Antlers of Moose May Serve as A Parabolic Reflector of Sounds

It has been postulated that the excellent sense of hearing in moose is mostly due to: (1) the large surface of the external ear, (2) better stereophony due to the large distance between ears, (3) independently movable, extremely adjustable pinna, and (4) the amplification of sounds reflected by the palms of the antlers. The last factor, possible reflection of sounds into pinna by the palm of the antlers, was tested in this study on a large antler trophy of Alaskan moose. The reception of a standard tone, broadcast from the frontally placed speaker, was recorded by a sound level meter located in an artificial moose ear. Three locations of the ear, as positioned relative to the speaker, e.g., frontward, sideward, and backward, were tested. The weakest reception was recorded in the backward position of the ear. If the sound pressure measured in the frontward position was set as 100%, the sound pressure in the backward position was 79%. The strongest reception was recorded when the artificial ear was positioned toward the center of the antler palm. In this position, the sound pressure was 119% relative to the frontward position. These findings strongly indicate that the palm of moose antlers may serve as an effective, parabolic reflector which increases the acoustic pressure of the incoming sound

Performance of a Broadcast Packet Switch

This paper reports the results of a simulation study undertaken to evaluate a high performance packet switching fabric supporting point-to-point and multipoint communications. This switching fabric contains several components each based on conventional binary routing networks. The most novel element is the Copy Network which performs the packet replication needed for multipoint connections. We present results characterizing the performance of the Copy Network, in particular quantifying its dependence on fanout and the location of active sources. We also evaluate several architectural alternatives for conventional binary routing networks. For example, we quantify the performance gains obtainable by using cut-through switching in the context of binary routing networks with small buffers. One surprising result is that networks constructed for nodes with more than two input and output ports can perform less well than those constructed form binary nodes. We quantify and explain this result, showing that it is a consequences of a subtle effect of the FIFO queueing discipline used in the nodes. We also show that substantially better performance can be obtained by relaxing the strict FIFo discipline

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more citations, to appear in Mathematische Zeitschrif

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Categorification of persistent homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational Geometr

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

A convenient category of locally preordered spaces

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.Comment: 26 pages, 0 figures, partially presented at GETCO 2005; changes: claim of Prop. 5.11 weakened to finite case and proof changed due to problems with proof of Lemma 3.26, now removed; Eg. 2.7, statement before Lem. 2.11, typos, and other minor problems corrected throughout; extensive rewording; proof of Lem. 3.31, now 3.30, adde

Persistent topology for natural data analysis - A survey

Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and classification of cells, lesions, music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brain connections, languages, handwritten and gestured letters are shown

Quantifying similarity of pore-geometry in nanoporous materials

In most applications of nanoporous materials the pore structure is as important as the chemical composition as a determinant of performance. For example, one can alter performance in applications like carbon capture or methane storage by orders of magnitude by only modifying the pore structure. For these applications it is therefore important to identify the optimal pore geometry and use this information to find similar materials. However, the mathematical language and tools to identify materials with similar pore structures, but different composition, has been lacking. We develop a pore recognition approach to quantify similarity of pore structures and classify them using topological data analysis. This allows us to identify materials with similar pore geometries, and to screen for materials that are similar to given top-performing structures. Using methane storage as a case study, we also show that materials can be divided into topologically distinct classes requiring different optimization strategies