181 research outputs found
Solvable and/or integrable and/or linearizable N-body problems in ordinary (three-dimensional) space. I
Several N-body problems in ordinary (3-dimensional) space are introduced
which are characterized by Newtonian equations of motion (``acceleration equal
force;'' in most cases, the forces are velocity-dependent) and are amenable to
exact treatment (``solvable'' and/or ``integrable'' and/or ``linearizable'').
These equations of motion are always rotation-invariant, and sometimes
translation-invariant as well. In many cases they are Hamiltonian, but the
discussion of this aspect is postponed to a subsequent paper. We consider
``few-body problems'' (with, say, \textit{N}=1,2,3,4,6,8,12,16,...) as well as
``many-body problems'' (N an arbitrary positive integer). The main focus of
this paper is on various techniques to uncover such N-body problems. We do not
discuss the detailed behavior of the solutions of all these problems, but we do
identify several models whose motions are completely periodic or multiply
periodic, and we exhibit in rather explicit form the solutions in some cases
Discovering Causal Relations and Equations from Data
Physics is a field of science that has traditionally used the scientific
method to answer questions about why natural phenomena occur and to make
testable models that explain the phenomena. Discovering equations, laws and
principles that are invariant, robust and causal explanations of the world has
been fundamental in physical sciences throughout the centuries. Discoveries
emerge from observing the world and, when possible, performing interventional
studies in the system under study. With the advent of big data and the use of
data-driven methods, causal and equation discovery fields have grown and made
progress in computer science, physics, statistics, philosophy, and many applied
fields. All these domains are intertwined and can be used to discover causal
relations, physical laws, and equations from observational data. This paper
reviews the concepts, methods, and relevant works on causal and equation
discovery in the broad field of Physics and outlines the most important
challenges and promising future lines of research. We also provide a taxonomy
for observational causal and equation discovery, point out connections, and
showcase a complete set of case studies in Earth and climate sciences, fluid
dynamics and mechanics, and the neurosciences. This review demonstrates that
discovering fundamental laws and causal relations by observing natural
phenomena is being revolutionised with the efficient exploitation of
observational data, modern machine learning algorithms and the interaction with
domain knowledge. Exciting times are ahead with many challenges and
opportunities to improve our understanding of complex systems.Comment: 137 page
States and sequences of paired subspace ideals and their relationship to patterned brain function
It is found here that the state of a network of coupled ordinary differential equations is partially localizable through a pair of contractive ideal subspaces, chosen from dual complete lattices related to the synchrony and synchronization of cells within the network. The first lattice is comprised of polydiagonal subspaces, corresponding to synchronous activity patterns that arise from functional equivalences of cell receptive fields. This lattice is dual to a transdiagonal subspace lattice ordering subspaces transverse to these network-compatible synchronies.
Combinatorial consideration of contracting polydiagonal and transdiagonal subspace pairs yields a rich array of dynamical possibilities for structured networks. After proving that contraction commutes with the lattice ordering, it is shown that subpopulations of cells are left at fixed potentials when pairs of contracting subspaces span the cells' local coordinates - a phenomenon named glyph formation here. Treatment of mappings between paired states then leads to a theory of network-compatible sequence generation.
The theory's utility is illustrated with examples ranging from the construction of a minimal circuit for encoding a simple phoneme to a model of the primary visual cortex including high-dimensional environmental inputs, laminar speficicity, spiking discontinuities, and time delays. In this model, glyph formation and dissolution provide one account for an unexplained anomaly in electroencephalographic recordings under periodic flicker, where stimulus frequencies differing by as little as 1 Hz generate responses varying by an order of magnitude in alpha-band spectral power.
Further links between coupled-cell systems and neural dynamics are drawn through a review of synchronization in the brain and its relationship to aggregate observables, focusing again on electroencephalography. Given previous theoretical work relating the geometry of visual hallucinations to symmetries in visual cortex, periodic perturbation of the visual system along a putative symmetry axis is hypothesized to lead to a greater concentration of harmonic spectral energy than asymmetric perturbations; preliminary experimental evidence affirms this hypothesis.
To conclude, connections drawn between dynamics, sensation, and behavior are distilled to seven hypotheses, and the potential medical uses of the theory are illustrated with a lattice depiction of ketamine xylazine anaesthesia and a reinterpretation of hemifield neglect
Change in Hamiltonian General Relativity from the Lack of a Time-like Killing Vector Field
In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. Attention to the gauge generator of Rosenfeld, Anderson, Bergmann, Castellani \emph{et al.}, a specially \emph{tuned sum} of first-class constraints, facilitates seeing that a solitary first-class constraint in fact generates not a gauge transformation, but a bad physical change in electromagnetism (changing the electric field) or General Relativity. The change spoils the Lagrangian constraints, Gauss's law or the Gauss-Codazzi relations describing embedding of space into space-time, in terms of the physically relevant velocities rather than auxiliary canonical momenta. While Maudlin and Healey have defended change in GR much as G. E. Moore resisted skepticism, there remains a need to exhibit the technical flaws in the no-change argument.
Insistence on Hamiltonian-Lagrangian equivalence, a theme emphasized by Mukunda, Castellani, Sugano, Pons, Salisbury, Shepley and Sundermeyer among others, holds the key. Taking objective change to be ineliminable time dependence, one recalls that there is change in vacuum GR just in case there is no time-like vector field xi^{alpha} satisfying Killing's equation pounds_{xi}g_{mu\nu}=0, because then there exists no coordinate system such that everything is independent of time. Throwing away the spatial dependence of GR for convenience, one finds explicitly that the time evolution from Hamilton's equations is real change just when there is no time-like Killing vector. The inclusion of a massive scalar field is simple. No obstruction is expected in including spatial dependence and coupling more general matter fields. Hence change is real and local even in the Hamiltonian formalism.
The considerations here resolve the Earman-Maudlin standoff over change in Hamiltonian General Relativity: the Hamiltonian formalism is helpful, and, suitably reformed, it does not have absurd consequences for change.
Hence the classical problem of time is resolved, apart from the issue of observables, for which the solution is outlined. The Lagrangian-equivalent Hamiltonian analysis of change in General Relativity is compared to Belot and Earman's treatment. The more serious quantum problem of time, however, is not automatically resolved due to issues of quantum constraint imposition
Generalization Through the Lens of Learning Dynamics
A machine learning (ML) system must learn not only to match the output of a
target function on a training set, but also to generalize to novel situations
in order to yield accurate predictions at deployment. In most practical
applications, the user cannot exhaustively enumerate every possible input to
the model; strong generalization performance is therefore crucial to the
development of ML systems which are performant and reliable enough to be
deployed in the real world. While generalization is well-understood
theoretically in a number of hypothesis classes, the impressive generalization
performance of deep neural networks has stymied theoreticians. In deep
reinforcement learning (RL), our understanding of generalization is further
complicated by the conflict between generalization and stability in widely-used
RL algorithms. This thesis will provide insight into generalization by studying
the learning dynamics of deep neural networks in both supervised and
reinforcement learning tasks.Comment: PhD Thesi
Discovering causal relations and equations from data
Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws, and principles that are invariant, robust, and causal has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventions on the system under study. With the advent of big data and data-driven methods, the fields of causal and equation discovery have developed and accelerated progress in computer science, physics, statistics, philosophy, and many applied fields. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for data-driven causal and equation discovery, point out connections, and showcase comprehensive case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is revolutionised with the efficient exploitation of observational data and simulations, modern machine learning algorithms and the combination with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems
Mathemathical methods of theoretical physics
Course material for mathematical methods of theoretical physics intended for
an undergraduate audience.Comment: 287 pages, revised, some further (relative to previous edition)
proofs and sections (on differential operators in orthogonal curvilinear
coordinates) adde
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