On the magnitude of spheres, surfaces and other homogeneous spaces

In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold. In the particular case of a homogeneous surface the form of the asymptotics can be given exactly up to vanishing terms and this involves just the area and Euler characteristic in the way conjectured for subsets of Euclidean space in previous work.Comment: 21 pages. Main change from v1: details added to proof of Theorem

Euler-Bessel and Euler-Fourier Transforms

We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible \zed-valued functions to continuous $\real$-valued functions over a vector space. Core contributions include: the definition of the topological Bessel transform; a relationship in terms of the logarithmic blowup of the topological Fourier transform; and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization (SVA) to mitigate sidelobe phenomena

The equation of state for two-dimensional hard-sphere gases: Hard-sphere gases as ideal gases with multi-core boundaries

The equation of state for a two-dimensional hard-sphere gas is difficult to calculate by usual methods. In this paper we develop an approach for calculating the equation of state of hard-sphere gases, both for two- and three-dimensional cases. By regarding a hard-sphere gas as an ideal gas confined in a container with a multi-core (excluded sphere) boundary, we treat the hard-sphere interaction in an interacting gas as the boundary effect on an ideal quantum gas; this enables us to treat an interacting gas as an ideal one. We calculate the equation of state for a three-dimensional hard-sphere gas with spin $j$, and compare it with the results obtained by other methods. By this approach the equation of state for a two-dimensional hard-sphere gas can be calculated directly.Comment: 9 pages, 1 figur

О собственном времени очага сильного землетрясения

The physics of earthquakes was contriubuted to by the concept of proper time of the source of a strong earthquake, which is different from universal (calendar) time. The earlier idea of proper time was implicit and has been considered only in relation to the physics of aftershocks. The present paper extends the applicability of the concept of proper time, proposes a possible way of its measuring, and provides an example to illustrate the procedure for sequential ordering of earthquakes by proper time. The object of this study is a global activity of strong (M≥7) earthquakes. We consider the sequence of earthquakes as a Poisson-type random process. Comparatively weak earthquakes are used as the "underground clock", the tick of which marks the proper time. The Poisson distribution is compared with the distributions for two sequences of strong earthquakes. One of the sequences is ordered by universal time, and another - by proper time. The studies indicate the distribution of events ordered by proper time is closer to the Poisson distribution than that of events ordered by universal time. We attribute this to the non-stationarity of the geological medium, which is an immanent property of the Earth's lithosphere.В физику землетрясений введено понятие о собственном времени очага сильного землетрясения, отличном от универсального (календарного) времени. Ранее использовалась идея о собственном времени, но неявно и только лишь в узкой области, относящейся к физике афтершоков. В данной работе расширена область применимости представления о собственном времени, указан возможный способ его измерения и приведен пример, иллюстрирующий процедуру упорядочивания последовательности землетрясений в собственном времени. В качестве объекта исследования выбрана глобальная активность сильных землетрясений (М≥7). Последовательность землетрясений мы рассматриваем как случайный процесс пуассоновского типа. В качестве «подземных часов», тиканье которых отмечает ход собственного времени, использованы сравнительно слабые землетрясения. Распределение Пуассона сопоставлено с распределениями для двух последовательностей сильных землетрясений, одна из которых упорядочена по универсальному времени, а другая - по собственному. Результат испытания показал, что распределение событий, упорядоченных по собственному времени, ближе к распределению Пуассона, чем распределение событий, упорядоченных по универсальному времени. Авторы объясняют это нестационарностью геологической среды, которая является имманентным свойством литосферы Земли

Beyond genus statistics: a unifying approach to the morphology of cosmic structure

The genus statistics of isodensity contours has become a well-established tool in cosmology. In this Letter we place the genus in the wider framework of a complete family of morphological descriptors. These are known as the Minkowski functionals, and we here apply them for the first time to isodensity contours of a continuous random field. By taking two equivalent approaches, one through differential geometry, the other through integral geometry, we derive two complementary formulae suitable for numerically calculating the Minkowski functionals. As an example we apply them to simulated Gaussian random fields and compare the outcome to the analytically known results, demonstrating that both are indeed well suited for numerical evaluation. The code used for calculating all Minkowski functionals is available from the authors.Comment: 8 pages plus 1 figure; uses aaspp4.sty and flushrt.sty. Matches version accepted for publication in Ap. J. Let

Emergence of Secondary Motifs in Tube-Like Polymers in a Solvent

We study the effects of two kinds of interactions in tube-like polymers and demonstrate that they result in the formation of secondary motifs. The first has an entropic origin and is a measure of the effective space available to the solvent. The second arises from solvophobic interactions of the solvent with the polymers and leads to an energy proportional to the contact surface between the tube and solvent particles. The solvent molecules are modeled as hard spheres and the two interactions are considered separately with the solvent density affecting their relative strength. In addition to analytical calculations, we present the results of numerical simulations in order to understand the role played by the finite length of short polymers and the discrete versus continuum descriptions of the system in determining the preferred conformation.Comment: 5 pages, 2 figures, 1 table. Accepted by Phys. Rev.

Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.Comment: 68 pages; minor change

Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure