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Geometry and Meaning

C. J. "Keith" van Rijsbergen
2006 Computational Linguistics  
-Plato 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.  ... 
doi:10.1162/coli.2006.32.1.155 fatcat:dbgxycinrrg6ndkpgcbborhhke

Riemannian geometry and matrix geometric means

Rajendra Bhatia, John Holbrook
2006 Linear Algebra and its Applications  
The geometric 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.  ... 
doi:10.1016/j.laa.2005.08.025 fatcat:t4mkmvqxenatbeikdhjyyjkigi

Medians and means in Finsler geometry

Marc Arnaudon, Frank Nielsen
2012 LMS Journal of Computation and Mathematics  
AbstractWe investigate existence 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.  ... 
doi:10.1112/s1461157010000513 fatcat:23adbmzz6ncrrg2rzkgvgyw5pe

Neutrality and geometry of mean voting

Sebastien Lahaie, Nisarg Shah
2014 Proceedings of the fifteenth ACM conference on Economics and computation - EC '14  
Various connections are drawn between 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.  ... 
doi:10.1145/2600057.2602898 dblp:conf/sigecom/LahaieS14 fatcat:bgbbwzod7jfwladyc5xga7lali

Noncollapsing in mean-convex mean curvature flow

Ben Andrews
2012 Geometry and Topology  
Acknowledgement This research was partly supported by Discovery Projects grants DP0985802 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  ... 
doi:10.2140/gt.2012.16.1413 fatcat:324rv2asrnckxcxlckfm3op3hy

Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation [chapter]

Marc Arnaudon, Frédéric Barbaresco, Le Yang
2012 Matrix Information Geometry  
Finally, we apply the medians 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] .  ... 
doi:10.1007/978-3-642-30232-9_8 fatcat:egac5nh2mncy7gwjdzsvqjqzpq

Sums over geometries and improvements on the mean field approximation

Vincent E. Sacksteder
2007 Physical Review D  
In the case of the Efetov theory, the dominant 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.  ... 
doi:10.1103/physrevd.76.105032 fatcat:y53fiotz3nflxc2bgiqqawk43i

Medians and means in Riemannian geometry: existence, uniqueness and computation [article]

Marc Arnaudon
2011 arXiv   pre-print
Finally, we apply the medians 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] .  ... 
arXiv:1111.3120v1 fatcat:66ay27wcz5c45fzkmrbzvodgje

Surfaces with constant mean curvature in Sol geometry

Rafael López, Marian Ioan Munteanu
2011 Differential geometry and its applications  
The authors would like to thank the referees for all helpful comments 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.  ... 
doi:10.1016/j.difgeo.2011.04.047 fatcat:qlojcyn5j5egxh6ioeca3wpb4a

Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

Ye-Lin Ou, Ze-Ping Wang
2011 Journal of Geometry and Physics  
We also give complete classifications of constant 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.  ... 
doi:10.1016/j.geomphys.2011.04.008 fatcat:hikfzf5ezjgatlacamuu4etzqu

Rogers' mean value theorem for S-arithmetic Siegel transform and applications to the geometry of numbers [article]

Jiyoung Han
2021 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 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.  ... 
arXiv:1910.01824v5 fatcat:ksrskm7lizfadb4fe3ma3rnhgy

Coalescing colony model: Mean-field, scaling, and geometry

Giulia Carra, Kirone Mallick, Marc Barthelemy
2017 Physical review. E  
Assuming the primary colony to be always spherical of radius r(t) 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.  ... 
doi:10.1103/physreve.96.062316 pmid:29347390 fatcat:ahirep3uwbbtlgvb26qjvte7sa

Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group [article]

S. Majid
2000 arXiv   pre-print
More general quantum groups C(G^) U(g) 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.  ... 
arXiv:hep-th/0006166v1 fatcat:yj7m7vahnjhkve6qasxz4n4wwu

Equivariant mean field flow

Jean-baptiste Castéras
2013 Journal of Geometry and Physics  
We consider a gradient flow associated to the 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.  ... 
doi:10.1016/j.geomphys.2013.08.011 fatcat:5gpzimhes5e5fdqgi7ajkkoqxi

The topology and geometry of embedded surfaces of constant mean curvature

William H. Meeks, III
1988 Journal of differential geometry  
As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant 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.  ... 
doi:10.4310/jdg/1214442008 fatcat:xii3amxmcnghllzrxpqkztosy4
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