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Octonionic geometry

Merab Gogberashvili
2005 Advances in Applied Clifford Algebras  
We connect the property of non-associativity with the time irreversibility and fundamental probabilities in physics.  ...  We extend vector formalism by including it in the algebra of split octonions, which we treat as the universal algebra to describe physical signals.  ...  As distinct from string models in present paper we want to apply octonions to describe space-time geometry and not only internal spaces.  ... 
doi:10.1007/s00006-005-0003-2 fatcat:i7e6jq6gmvfxhm3h65logxdo7m

Combinatorial optimization in geometry

Igor Rivin
2003 Advances in Applied Mathematics  
The basic tools, in addition to the results of [Rivin, Ann. of Math. 139 (1994)] and combinatorial geometry are methods of combinatorial optimization-linear programming and network flow analysis; hence  ...  In this paper we extend and unify the results of [] are generalized from the framework of ideal polyhedra in H 3 to that of singular Euclidean structures on surfaces, possibly with an infinite number of  ...  In Section 6 we show how the results apply to ideal polyhedra, and in particular to characterize infinite ideal polyhedra.  ... 
doi:10.1016/s0196-8858(03)00093-9 fatcat:7jf663pxzjgxfeumyodnoq6xoa

Idempotent Geometry in Generic Algebras

Yakov Krasnov, Vladimir G. Tkachev
2018 Advances in Applied Clifford Algebras  
For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.  ...  Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras.  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4. 0/), which permits unrestricted use, distribution, and reproduction in  ... 
doi:10.1007/s00006-018-0902-7 fatcat:xg6bnnxdhnfsjhh456sxwewzky

Reconstructing ternary Dowling geometries

Thomas Dowling, Hongxun Qin
2005 Advances in Applied Mathematics  
We prove that the only geometry of rank n > 4 all of whose proper contractions are ternary Dowling geometries is the ternary Dowling geometry.  ...  Now Lemma 3.3 applies and we have F ∼ = Q 4 . Consequently, P ∼ = Q 3 . Now assume that P contains six or seven points.  ...  Points, lines, and planes of a geometry are its flats of rank 1, 2, and 3, respectively. For a geometry G, |G| will denote the number of points in G.  ... 
doi:10.1016/j.aam.2004.07.003 fatcat:qzansmxqqbdt3mrq23olp2g7dm

The invariator principle in convex geometry

Ó. Thórisdóttir, M. Kiderlen
2014 Advances in Applied Mathematics  
The invariator principle is a measure decomposition that was rediscovered in local stereology in 2005 and has since been used widely in the stereological literature.  ...  This results in geometrical quantities represented as averages over weighted Crofton-type integrals in linear sections.  ...  Acknowledgements This research was supported by Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation.  ... 
doi:10.1016/j.aam.2014.02.003 fatcat:6dzs6zf6qjblbmpsfz6ry3gyoq

Integral geometry for the 1-norm

Tom Leinster
2012 Advances in Applied Mathematics  
Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space.  ...  We show that integral geometry for the 1-norm bears a striking resemblance to integral geometry for the 2-norm, but is radically different from that for all other values of p.  ...  Applied to n 2 , this says that the intersection of two convex sets is convex. Applied to n 1 , this is Corollary 1.9.  ... 
doi:10.1016/j.aam.2012.02.002 fatcat:xaknvafjrzaldnbn5gvkisq4ny

Spin Geometry and Image Processing

Michel Berthier
2013 Advances in Applied Clifford Algebras  
We give a survey of applications of spin geometry to image processing. We mainly focus on the problem of defining geometric Fourier transforms for both grey-level and color images.  ...  As mentioned in the introduction this decomposition doesn't account for the geometry of the image graph.  ...  that accounts for color data and for the geometry of the underlying Riemannian image surface?  ... 
doi:10.1007/s00006-013-0409-1 fatcat:axcb7yrjkvckbhqbdbz5ruot5m

Algebraic geometry of Gaussian Bayesian networks

Seth Sullivant
2008 Advances in Applied Mathematics  
Finally, we relate the ideals of Bayesian networks to a number of classical constructions in algebraic geometry including toric degenerations of the Grassmannian, matrix Schubert varieties, and secant  ...  Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices.  ...  Acknowledgments I would like to thank Mathias Drton, Thomas Richardson, Mike Stillman, and Bernd Sturmfels for helpful comments and discussions about the results in this paper.  ... 
doi:10.1016/j.aam.2007.04.004 fatcat:k3d4zkpz25fptedao7djg2rthu

Integral geometry of tensor valuations

Daniel Hug, Rolf Schneider, Ralph Schuster
2008 Advances in Applied Mathematics  
We mention that interest in the integral geometry of intrinsic volumes and their generalizations comes also from applied sciences. We refer to the work of K.  ...  In the next step, we apply Corollary 4.6 in u ⊥ to the integral on the right-hand side.  ...  a position to prove the theorems stated in Section 2, in particular, to express the mean values for K ∈ K n , r, s ∈ N 0 and 0 j k n − 1, in terms of basic tensor valuations.  ... 
doi:10.1016/j.aam.2008.04.001 fatcat:svts5cx7j5g6xfdmyug5utqhpi

The geometry and topology of reconfiguration

R. Ghrist, V. Peterson
2007 Advances in Applied Mathematics  
We prove classification and realization theorems for state complexes, using CAT(0) geometry as the primary tool.  ...  A number of reconfiguration problems in robotics, biology, computer science, combinatorics, and group theory coordinate local rules to effect global changes in system states.  ...  Acknowledgments Many of the details in the definition of reconfigurable systems and state complexes were developed in conversations with A. Abrams.  ... 
doi:10.1016/j.aam.2005.08.009 fatcat:6qpx7oxsbzct3nec2r6r2jzb7y

Geometry and complexity of O'Hara's algorithm

Matjaž Konvalinka, Igor Pak
2009 Advances in Applied Mathematics  
Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general.  ...  In this paper we analyze O'Hara's partition bijection. We present three type of results.  ...  We are grateful to George Andrews and Dennis Stanton for their interest in the paper and to Kathy O'Hara for sending us a copy of her thesis [O1]. The second named author was supported by the NSF.  ... 
doi:10.1016/j.aam.2008.06.005 fatcat:443j57tslrfqdfrdp6kinmrgjy

A Bundle Representation for Continuous Geometries

John Harding, Melvin F. Janowitz
1997 Advances in Applied Mathematics  
We relate this bundle representation to the Pierce sheaf of the continuous geometry and to the subdirect product representation Ä 4 Ž . the family of dimension functions D : J g Y and in fact D arJ s J  ...  We show that a reducible continuous geometry can be represented as the continuous sections of a bundle of irreducible continuous geometries.  ...  Applying part 1 we have that J k Ž . Ž . D u n a F ⑀ and therefore d a , a F ⑀. Let S S be the collection of all admissible families in T T.  ... 
doi:10.1006/aama.1997.0548 fatcat:o3gqgb762bhqzk5tdahldwvbcu

Curious Characterizations of Projective and Affine Geometries

Joseph P.S. Kung
2002 Advances in Applied Mathematics  
To do this, we use a lemma which is of independent interest: If H is a geometry in which all the lines have exactly − 1 or points and G is a geometry with at least three of the four matroid invariants  ...  the same as H, then all the lines in G also have exactly − 1 or points.  ...  With this hypothesis (which is stronger than necessary), we can now apply part (b) of Theorem 1.6 to conclude that G is a rank-n affine geometry of order q. This completes the proof of Theorem 1.4.  ... 
doi:10.1006/aama.2001.0793 fatcat:3gb6wx6ygvhirgj5kbww35cawq

Geometric Algebras for Euclidean Geometry

Charles Gunn
2016 Advances in Applied Clifford Algebras  
We then introduce the dual projectivized Clifford algebra P(R^*_n,0,1) (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry.  ...  We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics.  ...  Thus, the Cayley-Klein construction applies the signature (n, 0, 1) to dual projective space to obtain a model for euclidean geometry in n dimensions.  ... 
doi:10.1007/s00006-016-0647-0 fatcat:p5g46krsvjam5ckdkvmjbmrraq

Geometry of the Kimura 3-parameter model

M. Casanellas, J. Fernández-Sánchez
2008 Advances in Applied Mathematics  
The Kimura 3-parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety.  ...  This leads to a major improvement of phylogenetic reconstruction methods based on algebraic geometry.  ...  Both authors would like to thank the people attending the workshop "Applications in Biology, Dynamics, and Statistics.  ... 
doi:10.1016/j.aam.2007.09.003 fatcat:pqdssdeazncrbdakvfaxvcrjpe
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