Information Geometry Under Monotone Embedding. Part II: Geometry.

The results of this analysis are displayed in Table II . Under these conditions, the EW3DLC method is only slightly (30%) faster. ... In the Appendix we give explicit formulas for the ELC-term contributions to the force and the diagonal part of the stress tensor. II. ...

doi:10.1063/1.1491954 fatcat:ibfg4vorezdgdlsoksozh7jpvy

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(ii) For every vertex k of K, the image of {k} ⊲ under f is a p-coCartesian morphism of X. (iii) There exists a vertex k of K such that the image of {k} ⊲ under f is a p-coCartesian morphism of X. ... More informally, a functor f : N(J 0 ) → C ⊗ belongs to E 0 (C) if there exists an object C = f ( 1 * ) ∈ C, such that for all n ≥ 0, the object f ( n * ) ∈ C ⊗ n * corresponds to (C, C, . . . , C) under ... Assertion (i) follows immediately from the stability of C under finite coproducts. We now prove (ii). Suppose given a pair of maps f : X → N and g : Y → N , respectively. ...

arXiv:math/0702299v5 fatcat:6tdcjvtqtjctvlqxahu7kvrtcu

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comprised by the three-level quantum systems endowed with the Bures metric, we numerically approximate the integrals over the manifold of several functions of the curvature and of its (anti-)self-dual parts ... II. ... 2.0692 1.5519 3.62111 1.41 1.09736 self-dual part 18.6228 7.75951 26.3823 5.15961 3.29208 TABLE II. ...

arXiv:math-ph/0102032v1 fatcat:t7mwdwrpnvhwlazcm3klic7pxi

The standard model of information geometry, expressed as Fisher-Rao metric and Amari-Chensov tensor, reflects an embedding of probability density by log-transform. ... The present paper studies parametrized statistical models and the induced geometry using arbitrary embedding functions, comparing single-function approaches (Eguchi's U-embedding and Naudts' deformed-log ... Acknowledgements The project is supported in part by DARPA/ARO Grant W911NF-16-1-0383 ("Information Geometry: Geometrization of Science of Information", PI: Zhang). ...

doi:10.1007/s41884-018-0004-6 fatcat:ibuabvrptncp7jwvwvorlcbxqa

Step (iv) Our task is now to compare the information found in steps (ii) and (iii). Suppose that k is odd. ... In Part II we continue where we left off. We begin, in Section 2, by restating Theorem B, or rather, stating a corollary of it which is our jumping-off point for this part. ...

doi:10.2140/gt.2008.12.1461 fatcat:soluwp4lanfhllgbeulhovelwe

Szczepanski Multiple Versions

Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. ... (ii) The function c(a)/a is nonincreasing. The first property follows directly from the (Monotonicity) axiom. ... One part of his pioneering work has grown into what is now called Gromov-Witten theory, see e.g. [73] for an introduction. ...

arXiv:math/0506191v1 fatcat:5plrjwxrtreo5lsrvv2yo6nv6y

In particular, we provide a codimension 2 isometric embedding which naturally gives rise to the previously introduced global coordinates. ... under {H, C, D, N } thought of as elements of so(2, 2) via the above embedding of Lie algebras. ... Thus N can either arise from a null translation in the translational part of the isometry algebra of the embedding space or from a null rotation. ...

doi:10.1007/jhep07(2010)069 fatcat:zkoki6t6xjgi5nciv6agbuscaq

DOAJ Multiple Versions

Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. ... Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. ... II offers a possible explanation and a mechanism for the routing of information in the brain. ...

arXiv:2001.03241v2 fatcat:n3kqsgmpxffr5klzoihs525mrm

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Building on the analysis of critical curves and caustics presented in the first part of this work, we proceed to explore the geometry of images formed by the lens. ... In the case of our lens model, we describe the occurrence of zero-shear points and specify the conditions under which they become umbilic points. ... From this value the shear decreases outward monotonically. ...

arXiv:2109.02495v2 fatcat:5co5xyhmnnfnno7k7z3cxoa3ia

Open Access Multiple Versions

We investigate the statistical geometry of inherent structures ͑mechanically stable arrangements of particles generated by a steepest-descent mapping of equilibrium configurations to local potential minima͒ ... For a wide range of densities, including some higher than the triple point density, inherent structures are found to display remarkably heterogeneous geometry, with an apparently bicontinuous structure ... II. . . . SASTRY, DEBENEDETTI, AND STILLINGER STATISTICAL GEOMETRY OF . . . . II. . . . ...

doi:10.1103/physreve.56.5533 fatcat:ho46m6aiezhovedwloj66ihlei

In this second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) we discuss results that refer to the following three topics: bodies of constant Minkowski ... This paper was written under a grant from the DFG-NRF agreement. It was partially written ... In Part II we survey three topics from Minkowski Geometry that are very geometric in nature and show interesting relations to further disciplines, such as Classical Convexity, Abstract Convexity, Computational ...

doi:10.1016/s0723-0869(04)80009-4 fatcat:enhlkgtkobhwzgltxeemz7aici

Szczepanski

Information from the external world goes through various transformations. ... For illustrative purposes an example is presented for sensory fusion: the geometry of the 3D world is learned using the outputs of two cameras. ... Henceforth we shall assume that function 9 is monotonic with respect to the inclusion X Fig. 5. Learnt neighbouring relations in 3 dimensions, II. ...

doi:10.1016/0925-2312(95)00116-6 fatcat:eqoa4rwn6nc7liq7zoincx5lju

Having previously developed a differential geometry framework for analyzing and conceptualizing Multidisciplinary Design Optimization (MDO) problems and methods, we now apply that framework to consider ... In Part II, we now provide that example through our analysis of the Quasi-Separable Decomposition (QSD) architecture. ... Introduction In Part I of this paper, we developed a differential geometry (DG) framework for analyzing and developing Multidisciplinary Design Optimization (MDO) solution methods: we outlined the underlying ...

doi:10.1007/s00158-014-1170-3 fatcat:mnmuwiyahjgc7lrhqulukdchbi

Monotonicity thus reduces to the elementary estimate pK ′2 < π 2 /4. ... (What other pseudo-Riemannian geometries-not just symmetric subspaces-are naturally embedded in Λ via our constructions is one of the general issues raised in [C-L] which we intend to address elsewhere ... The simplest example of such exceptional behavior was already given in the last part of Example 1. ...

doi:10.1080/10586458.2007.10129010 fatcat:npqhxjfaprfnpgpt4ieq3aag5y

Szczepanski

Rao of the geometric meaning of the Fisher information. Most of the research is focused on parametric statistical models. ... In a second part we discuss two specific topics, differential equations and the metric connection. The position of this line of research with respect to other approaches is briefly discussed. ... Plan Part I Exponential manifold Part II Vector bundles Part III Deformed exponential Part I Exponential manifold Sets of densities Definition P 1 is the set of real random variables f such that f dµ ...

doi:10.1007/978-3-642-40020-9_3 fatcat:pvxyqtnezjd7bk42v4ny4s5xaq

Electrostatics in periodic slab geometries. II

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Derived Algebraic Geometry II: Noncommutative Algebra [article]

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Bures geometry of the three-level quantum systems. II [article]

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Rho–tau embedding and gauge freedom in information geometry

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Lagrangian matching invariants for fibred four-manifolds: II

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Quantitative symplectic geometry [article]

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Geometry of Schrödinger space-times II: particle and field probes of the causal structure

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Network Geometry [article]

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Gravitational Lensing By a Massive Object in a Dark Matter Halo. II. Shear, Phase, and Image Geometry [article]

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Statistical geometry of particle packings. II. "Weak spots" in liquids

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The geometry of minkowski spaces — A survey. Part II

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Approximate geometry representations and sensory fusion

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Differential geometry tools for multidisciplinary design optimization, part II: application to QSD

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Schwarz Reflection Geometry II: Local and Global Behavior of the Exponential Map

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Nonparametric Information Geometry [chapter]

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