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Geometry and Meaning

2006
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Computational Linguistics
*

-Plato

doi:10.1162/coli.2006.32.1.155
fatcat:dbgxycinrrg6ndkpgcbborhhke
*Geometry**and**Meaning*is an interesting book about a relationship between*geometry**and*logic defined on certain types of abstract spaces*and*how that intimate relationship might be exploited when ... foundations for taxonomy, analytical*geometry*, logic,*and*vector spaces, they had a more flexible*and*broader view of these subjects than is current. ...*means*of the vector concept, following the procedure of Grassmann's Ausdehnungslehre. ...##
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Riemannian geometry and matrix geometric means

2006
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Linear Algebra and its Applications
*

The geometric

doi:10.1016/j.laa.2005.08.025
fatcat:t4mkmvqxenatbeikdhjyyjkigi
*mean*of two positive definite matrices has been defined in several ways*and*studied by several authors, including Pusz*and*Woronowicz,*and*Ando. ... In some recent papers new understanding of the geometric*mean*of two positive definite matrices has been achieved by identifying the geometric*mean*of A*and*B as the midpoint of the geodesic (with respect ... The geometric*mean*has been linked to differential*geometry*in Corach-Porta-Recht [7]*and*Lawson-Lim [8] , for example. ...##
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Medians and means in Finsler geometry

2012
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LMS Journal of Computation and Mathematics
*

AbstractWe investigate existence

doi:10.1112/s1461157010000513
fatcat:23adbmzz6ncrrg2rzkgvgyw5pe
*and*uniqueness ofp-meansepand the mediane1of a probability measureμon a Finsler manifold, in relation with the convexity of the support ofμ. ... In this paper we will consider forward p-*means**and*we will call them p-*means*. ... Information*geometry*at its heart considers the differential*geometry*nature of probability distributions induced by a divergence function. ...##
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Neutrality and geometry of mean voting

2014
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Proceedings of the fifteenth ACM conference on Economics and computation - EC '14
*

Various connections are drawn between

doi:10.1145/2600057.2602898
dblp:conf/sigecom/LahaieS14
fatcat:bgbbwzod7jfwladyc5xga7lali
*mean*proximity rules*and*other prominent approaches to social choice. ... They embed all rankings into a Euclidean space, take the*mean*of the embeddings of the input votes,*and*return the ranking whose embedding is closest to the*mean*. ... He then makes a compelling case of using*geometry*in order to clarify*and*visualize voting rules. ...##
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Noncollapsing in mean-convex mean curvature flow

2012
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Geometry and Topology
*

Acknowledgement This research was partly supported by Discovery Projects grants DP0985802

doi:10.2140/gt.2012.16.1413
fatcat:324rv2asrnckxcxlckfm3op3hy
*and*DP120102462 of the Australian Research Council. ...*Geometry*& Topology, Volume 16 (2012) ... For convenience we denote by H x the*mean*curvature*and*x the outward unit normal at .x; t/,*and*we write d D jX.y; t/ X.x; t/j; w D X.y; t/ X.x; t/ d*and*@ x i D @X @x i : We compute the first*and*second ...##
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Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation
[chapter]

2012
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Matrix Information Geometry
*

Finally, we apply the medians

doi:10.1007/978-3-642-30232-9_8
fatcat:egac5nh2mncy7gwjdzsvqjqzpq
*and*the Riemannian*geometry*of Toeplitz covariance matrices to radar target detection. ... Stochastic*and*deterministic algorithms are proposed for computing Riemannian p-*means*. The rate of convergence*and*error estimates of these algorithms are also obtained. ... Particularly, the Karcher*mean*coincides with the Fréchet*mean*. The existence*and*uniqueness of p-*means*in Finsler*geometry*are recently proved by M. Arnaudon*and*F. Nielsen in [6] . ...##
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Sums over geometries and improvements on the mean field approximation

2007
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Physical Review D
*

In the case of the Efetov theory, the dominant

doi:10.1103/physrevd.76.105032
fatcat:y53fiotz3nflxc2bgiqqawk43i
*geometries*are locally tree-like,*and*the sum over*geometries*diverges in a way that is similar to quantum gravity's divergence when all topologies are included ... Expertise from the field of dynamically triangulated quantum gravity about sums over*geometries*may be able to remedy these defects*and*fulfill the Efetov theory's original promise. ... for its initial stages,*and*reading a late draft of this paper*and*making an important comment. ...##
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Medians and means in Riemannian geometry: existence, uniqueness and computation
[article]

2011
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arXiv
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pre-print

Finally, we apply the medians

arXiv:1111.3120v1
fatcat:66ay27wcz5c45fzkmrbzvodgje
*and*the Riemannian*geometry*of Toeplitz covariance matrices to radar target detection. ... Stochastic*and*deterministic algorithms are proposed for computing Riemannian p-*means*. The rate of convergence*and*error estimates of these algorithms are also obtained. ... Particularly, the Karcher*mean*coincides with the Fréchet*mean*. The existence*and*uniqueness of p-*means*in Finsler*geometry*are recently proved by M. Arnaudon*and*F. Nielsen in [6] . ...##
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Surfaces with constant mean curvature in Sol geometry

2011
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Differential geometry and its applications
*

The authors would like to thank the referees for all helpful comments

doi:10.1016/j.difgeo.2011.04.047
fatcat:qlojcyn5j5egxh6ioeca3wpb4a
*and*suggestions that have improved the quality of our initial manuscript. ... The*geometry*of Sol is often called solve-*geometry*(see [2] , [8] ). ... Let H*and*K ext be the*mean*curvature*and*the extrinsic Gaussian curvature of S, respectively. ...##
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Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

2011
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Journal of Geometry and Physics
*

We also give complete classifications of constant

doi:10.1016/j.geomphys.2011.04.008
fatcat:hikfzf5ezjgatlacamuu4etzqu
*mean*curvature proper biharmonic surfaces in 3-dimensional*geometries**and*in 3-dimensional Bianchi-Cartan-Vranceanu spaces,*and*a complete classifications ... We use this to show that a totally umbilical surface in Thurston's 3-dimensional*geometries*is proper biharmonic if*and*only if it is a part of S^2(1/√(2)) in S^3. ... Oniciuc for some invaluable discussions*and*email communications related to this work. ...##
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Rogers' mean value theorem for S-arithmetic Siegel transform and applications to the geometry of numbers
[article]

2021
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arXiv
*
pre-print

As applications, we obtain the random statements of Gauss circle problem for any convex sets in ℚ_S^d containing the origin

arXiv:1910.01824v5
fatcat:ksrskm7lizfadb4fe3ma3rnhgy
*and*of the effective Oppenheim conjecture for S-arithmetic quadratic forms. ... I would like to thank Seonhee Lim*and*Anish Ghosh for valuable advices*and*discussion. I am also grateful to Jayadev Athreya, Keivan Mallahi-Karai, Seungki Kim, Dmitry Kleinbock*and*Mishel Skenderi. ... For the computation of the*mean*value of S k (f ) over G/Γ, we need some notations. ...##
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Coalescing colony model: Mean-field, scaling, and geometry

2017
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Physical review. E
*

Assuming the primary colony to be always spherical of radius r(t)

doi:10.1103/physreve.96.062316
pmid:29347390
fatcat:ahirep3uwbbtlgvb26qjvte7sa
*and*the emission rate proportional to r(t)^θ where θ>0, we derive the*mean*-field equations governing the dynamics of the primary colony ... We analyze the coalescing model where a 'primary' colony grows*and*randomly emits secondary colonies that spread*and*eventually coalesce with it. ... Acknowledgments GC thanks the Complex Systems Institute in Paris (ISC-PIF) for hosting her during part of this work*and*for providing the OpenMole platform. ...##
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Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group
[article]

2000
*
arXiv
*
pre-print

More general quantum groups C(G^) U(g)

arXiv:hep-th/0006166v1
fatcat:yj7m7vahnjhkve6qasxz4n4wwu
*and*U_q(g) are also discussed. Finally, the generalisation from quantum groups to general quantum Riemannian*geometry*is outlined. ... The general*meaning*of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space. ... Acknowledgements This was a really great conference in Polanica*and*I'd like to thank all of the organisers*and*participants. ...##
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Equivariant mean field flow

2013
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Journal of Geometry and Physics
*

We consider a gradient flow associated to the

doi:10.1016/j.geomphys.2013.08.011
fatcat:5gpzimhes5e5fdqgi7ajkkoqxi
*mean*field equation on (M,g) a compact riemanniann surface without boundary. We prove that this flow exists for all time. ... Moreover, letting G be a group of isometry acting on (M,g), we obtain the convergence of the flow to a solution of the*mean*field equation under suitable hypothesis on the orbits of points of M under the ... Equation (1.1) is also related to conformal*geometry*. When (M, g) is the standard sphere*and*ρ = 8π, the problem to find a solution to equation (1.1) is called the Nirenberg Problem. ...##
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The topology and geometry of embedded surfaces of constant mean curvature

1988
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Journal of differential geometry
*

As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant

doi:10.4310/jdg/1214442008
fatcat:xii3amxmcnghllzrxpqkztosy4
*mean*curvature have a special*geometry**and*play an important role in global theorems. ... Hoffman*and*Meeks have developed a theory to deal with global problems concerning the*geometry*of properly embedded minimal surfaces M*and*, in particular, they show that most annular ends of M converge ... THE TOPOLOGY*AND**GEOMETRY*OF EMBEDDED SURFACES OF CONSTANT*MEAN*CURVATURE WILLIAM H. ...
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