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Symmetry breaking: A tool to unveil the topology of chaotic scattering with three degrees of freedom  [PDF]
C. Jung,W. P. Karel Zapfe,O. Merlo,T. H. Seligman
Physics , 2010, DOI: 10.1063/1.3537863
Abstract: We shall use symmetry breaking as a tool to attack the problem of identifying the topology of chaotic scatteruing with more then two degrees of freedom. specifically we discuss the structure of the homoclinic/heteroclinic tangle and the connection between the chaotic invariant set, the scattering functions and the singularities in the cross section for a class of scattering systems with one open and two closed degrees of freedom.
Spherical membranes in Matrix theory  [PDF]
Daniel Kabat,Washington Taylor
Mathematics , 1997,
Abstract: We consider membranes of spherical topology in uncompactified Matrix theory. In general for large membranes Matrix theory reproduces the classical membrane dynamics up to 1/N corrections; for certain simple membrane configurations, the equations of motion agree exactly at finite N. We derive a general formula for the one-loop Matrix potential between two finite-sized objects at large separations. Applied to a graviton interacting with a round spherical membrane, we show that the Matrix potential agrees with the naive supergravity potential for large N, but differs at subleading orders in N. The result is quite general: we prove a pair of theorems showing that for large N, after removing the effects of gravitational radiation, the one-loop potential between classical Matrix configurations agrees with the long-distance potential expected from supergravity. As a spherical membrane shrinks, it eventually becomes a black hole. This provides a natural framework to study Schwarzschild black holes in Matrix theory.
Degrees of freedom in lasso problems  [PDF]
Ryan J. Tibshirani,Jonathan Taylor
Statistics , 2011, DOI: 10.1214/12-AOS1003
Abstract: We derive the degrees of freedom of the lasso fit, placing no assumptions on the predictor matrix $X$. Like the well-known result of Zou, Hastie and Tibshirani [Ann. Statist. 35 (2007) 2173-2192], which gives the degrees of freedom of the lasso fit when $X$ has full column rank, we express our result in terms of the active set of a lasso solution. We extend this result to cover the degrees of freedom of the generalized lasso fit for an arbitrary predictor matrix $X$ (and an arbitrary penalty matrix $D$). Though our focus is degrees of freedom, we establish some intermediate results on the lasso and generalized lasso that may be interesting on their own.
Degrees of Freedom and Model Search  [PDF]
Ryan J. Tibshirani
Statistics , 2014,
Abstract: Degrees of freedom is a fundamental concept in statistical modeling, as it provides a quantitative description of the amount of fitting performed by a given procedure. But, despite this fundamental role in statistics, its behavior not completely well-understood, even in some fairly basic settings. For example, it may seem intuitively obvious that the best subset selection fit with subset size k has degrees of freedom larger than k, but this has not been formally verified, nor has is been precisely studied. In large part, the current paper is motivated by this particular problem, and we derive an exact expression for the degrees of freedom of best subset selection in a restricted setting (orthogonal predictor variables). Along the way, we develop a concept that we name "search degrees of freedom"; intuitively, for adaptive regression procedures that perform variable selection, this is a part of the (total) degrees of freedom that we attribute entirely to the model selection mechanism. Finally, we establish a modest extension of Stein's formula to cover discontinuous functions, and discuss its potential role in degrees of freedom and search degrees of freedom calculations.
Degrees of Freedom in Multiuser MIMO  [PDF]
Syed A. Jafar,Maralle J. Fakhereddin
Mathematics , 2005,
Abstract: We explore the available degrees of freedom for various multiuser MIMO communication scenarios such as the multiple access, broadcast, interference, relay, X and Z channels. For the two user MIMO interference channel, we find a general inner bound and a genie-aided outer bound that give us the exact number of degrees of freedom in many cases. We also study a share-and-transmit scheme for transmitter cooperation. For the share-and-transmit scheme, we show how the gains of transmitter cooperation are entirely offset by the cost of enabling that cooperation so that the available degrees of freedom are not increased.
Degrees of Freedom in Massive Gravity  [PDF]
D. Comelli,M. Crisostomi,F. Nesti,L. Pilo
Physics , 2012, DOI: 10.1103/PhysRevD.86.101502
Abstract: We study in a systematic way a generic nonderivative (massive) deformation of general relativity using the Hamiltonian formalism. The number of propagating degrees of freedom is analyzed in a nonperturbative and background independent way. We show that the condition of having only five propagating degrees of freedom can be cast in a set of differential equations for the deforming potential. Though the conditions are rather restrictive, many solutions can be found.
Biological and synthetic membranes: What can be learned from a coarse-grained description?  [PDF]
Marcus Mueller,Kirill Katsov,Michael Schick
Physics , 2006, DOI: 10.1016/j.physrep.2006.08.003
Abstract: We discuss the role coarse-grained models play in the investigation of the structure and thermodynamics of bilayer membranes, and we place them in the context of alternative approaches. Because they reduce the degrees of freedom and employ simple and soft effective potentials, coarse-grained models can provide rather direct insight into collective phenomena in membranes on large time and length scales. We present a summary of recent progress in this rapidly evolving field, and pay special attention to model development and computational techniques. Applications of coarse-grained models to changes of the membrane topology are illustrated with studies of membrane fusion utilizing simulations and self-consistent field theory.
Hadronic degrees of freedom  [PDF]
Marco Traini
Physics , 2001,
Abstract: I report on the research activities performed under the (italian) MURST-PRIN project "Fisica Teorica del Nucleo e dei sistemi a pi\'u corpi" covering part of the topics on hadronic degrees of freedom. The most recent achievements in the field are summarized focusing on the specific role of the nuclear physics community.
Are there Thermodynamical Degrees of Freedom of Gravitation?  [PDF]
H. -H. v. Borzeszkowski,T. Chrobok
Physics , 2003,
Abstract: In discussing fundamentals of general-relativistic irreversible continuum thermodynamics, this theory is shown to be characterized by the feature that no thermodynamical degrees of freedom are ascribed to gravitation. However, accepting that black hole thermodynamics seems to oppose this harmlessness of gravitation one is called on consider other approaches. Therefore, in brief some gravitational and thermodynamical alternatives are reviewed.
On the degrees of freedom of lattice electrodynamics  [PDF]
Bo He,F. L. Teixeira
Physics , 2004, DOI: 10.1016/j.physleta.2005.01.001
Abstract: Using Euler's formula for a network of polygons for 2D case (or polyhedra for 3D case), we show that the number of dynamic\textit{\}degrees of freedom of the electric field equals the number of dynamic degrees of freedom of the magnetic field for electrodynamics formulated on a lattice. Instrumental to this identity is the use (at least implicitly) of a dual lattice and of a (spatial) geometric discretization scheme based on discrete differential forms. As a by-product, this analysis also unveils a physical interpretation for Euler's formula and a geometric interpretation for the Hodge decomposition.
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