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Geometry and Meaning
2006
Computational Linguistics
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-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
2006
Linear Algebra and its Applications
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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
2012
LMS Journal of Computation and Mathematics
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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
2014
Proceedings of the fifteenth ACM conference on Economics and computation - EC '14
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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
2012
Geometry and Topology
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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]
2012
Matrix Information Geometry
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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
2007
Physical Review D
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2007 accepted arXiv:0704.3129v2 fatcat:vbl6wdamcbc7peb4vg6rnooffaOther Versions
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]
2011
arXiv
pre-print
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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
2011
Differential geometry and its applications
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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
2011
Journal of Geometry and Physics
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2011 pre-print arXiv:1101.3282v1 fatcat:ehzwf4xxdbf7lmgkaak4pb4oei
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]
2021
arXiv
pre-print
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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
2017
Physical review. E
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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]
2000
arXiv
pre-print
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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
2013
Journal of Geometry and Physics
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2012 pre-print arXiv:1212.1854v1 fatcat:phjdlstta5gtln2difg2gtvq6u
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
1988
Journal of differential geometry
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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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