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The Friedmann Equations
"... Contents 1 Introduction 1 2 Cosmological field equations 2 2.1 Definition of curvature tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Christoffel symbols for the FRWmetric . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Components of the Ricci tensor . . . . . . . . . . ..."
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Einstein tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Energymomentum tensor for a perfect fluid . . . . . . . . . . . . . . . . . . . . . 9 2.6 The Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Cosmological
An Exploration of Symmetries in the Friedmann Equation
"... Abstract. The Friedmann Equation for a conformal scale factor a(t) is observed to be invariant under a Mobius transformation. Using that freedom, a synthetic scale factor z(t) is defined that obeys a modified Friedman equation invariant under the replacement z(t) → ±1 / z(t). If this is taken this ..."
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Abstract. The Friedmann Equation for a conformal scale factor a(t) is observed to be invariant under a Mobius transformation. Using that freedom, a synthetic scale factor z(t) is defined that obeys a modified Friedman equation invariant under the replacement z(t) → ±1 / z(t). If this is taken
Testing The Friedmann Equation: The Expansion of the Universe
, 2008
"... In conventional general relativity, the expansion rate H of a RobertsonWalker universe is related to the energy density by the Friedmann equation. Aside from the present day, the only epoch at which we can constrain the expansion history in a modelindependent way is during BigBang Nucleosynthesis ..."
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Cited by 4 (0 self)
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In conventional general relativity, the expansion rate H of a RobertsonWalker universe is related to the energy density by the Friedmann equation. Aside from the present day, the only epoch at which we can constrain the expansion history in a modelindependent way is during Big
The Technological Society
, 1964
"... A penetrating analysis of our technical civilization and of the effect of an increasingly standardized culture on the future of man ..."
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Cited by 259 (1 self)
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A penetrating analysis of our technical civilization and of the effect of an increasingly standardized culture on the future of man
Dark Energy as a Modification of the Friedmann Equation
, 2003
"... Dark energy could actually be the manifestation of a modification to the Friedmann equation arising from new physics (e.g., extra dimensions). Writing the correction as (1 − ΩM)H α /H α−2 0, we explore the phenomenology and detectability of such. We show that: (i) α must be ∼ < 1; (ii) such a cor ..."
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Cited by 17 (0 self)
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Dark energy could actually be the manifestation of a modification to the Friedmann equation arising from new physics (e.g., extra dimensions). Writing the correction as (1 − ΩM)H α /H α−2 0, we explore the phenomenology and detectability of such. We show that: (i) α must be ∼ < 1; (ii) such a
Geometric equations of state in FriedmannLema ître
, 2008
"... As a rule in General Relativity the spacetime metric fixes the Einstein tensor and through the Field Equations (FE) the energymomentum tensor. However one cannot write the FE explicitly until a class of observers has been considered. Every class of observers defines a decomposition of the energymo ..."
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of collineations. We examine this possibility for the FriedmannLemaîtreRobertsonWalker models admitting matter and Ricci collineations and determine the equations of state for the comoving observers. We find linear and nonlinear equations of state, which lead to solutions satisfying the energy conditions
Applications of Elliptic and Theta Functions to FriedmannRobertsonLemaîtreWalker
, 908
"... Elliptic functions are known to appear in many problems, applied and theoretical. However, a lesser known application is in the study of exact solutions to Einstein’s gravitational field equations in a FriedmannRobertson ..."
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Elliptic functions are known to appear in many problems, applied and theoretical. However, a lesser known application is in the study of exact solutions to Einstein’s gravitational field equations in a FriedmannRobertson
Probabilistic AuthorTopic Models for Information Discovery
 THE TENTH ACM SIGKDD INTERNATIONAL CONFERENCE ON KNOWLEDGE DISCOVERY AND DATA MINING
, 2004
"... We propose a new unsupervised learning technique for extracting information from large text collections. We model documents as if they were generated by a twostage stochastic process. Each author is represented by a probability distribution over topics, and each topic is represented as a probabilit ..."
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Cited by 171 (11 self)
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We propose a new unsupervised learning technique for extracting information from large text collections. We model documents as if they were generated by a twostage stochastic process. Each author is represented by a probability distribution over topics, and each topic is represented as a probability distribution over words for that topic. The words in a multiauthor paper are assumed to be the result of a mixture of each authors' topic mixture. The topicword and authortopic distributions are learned from data in an unsupervised manner using a Markov chain Monte Carlo algorithm. We apply the methodology to a large corpus of 160,000 abstracts and 85,000 authors from the wellknown CiteSeer digital library, and learn a model with 300 topics. We discuss in detail the interpretation of the results discovered by the system including specific topic and author models, ranking of authors by topic and topics by author, significant trends in the computer science literature between 1990 and 2002, parsing of abstracts by topics and authors and detection of unusual papers by specific authors. An online query interface to the model is also discussed that allows interactive exploration of authortopic models for corpora such as CiteSeer.
Reflections on places and placemaking in ChinaJohn Friedmann Reflections on Place and Placemaking in the Cities of China
"... This article is about the small spaces of the city we call ‘places’. Places are shaped by being lived in; they are spaces of encounter where the little histories of the city are played out. They are, of course, also shaped by the state through planning, supervision, ordinances, and so forth. The pat ..."
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This article is about the small spaces of the city we call ‘places’. Places are shaped by being lived in; they are spaces of encounter where the little histories of the city are played out. They are, of course, also shaped by the state through planning, supervision, ordinances, and so forth. The patterns and rhythms of life in the small spaces of the city are therefore not simply a straightforward projection of civil life. Places are also sites of resistance, contestation, and actions that are often thought to be illegal by the (local) state. After introducing the concept of place, the remainder of this article is a reflection on places and placemaking (but also placebreaking) in urban China. Because the patterns and rhythms of urban life have continuity, however, my approach to their study was historical. The story told here is roughly divided into four major periods: Imperial
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