61,510 research outputs found
The Price equation program: simple invariances unify population dynamics, thermodynamics, probability, information and inference
The fundamental equations of various disciplines often seem to share the same
basic structure. Natural selection increases information in the same way that
Bayesian updating increases information. Thermodynamics and the forms of common
probability distributions express maximum increase in entropy, which appears
mathematically as loss of information. Physical mechanics follows paths of
change that maximize Fisher information. The information expressions typically
have analogous interpretations as the Newtonian balance between force and
acceleration, representing a partition between direct causes of change and
opposing changes in the frame of reference. This web of vague analogies hints
at a deeper common mathematical structure. I suggest that the Price equation
expresses that underlying universal structure. The abstract Price equation
describes dynamics as the change between two sets. One component of dynamics
expresses the change in the frequency of things, holding constant the values
associated with things. The other component of dynamics expresses the change in
the values of things, holding constant the frequency of things. The separation
of frequency from value generalizes Shannon's separation of the frequency of
symbols from the meaning of symbols in information theory. The Price equation's
generalized separation of frequency and value reveals a few simple invariances
that define universal geometric aspects of change. For example, the
conservation of total frequency, although a trivial invariance by itself,
creates a powerful constraint on the geometry of change. That constraint plus a
few others seem to explain the common structural forms of the equations in
different disciplines. From that abstract perspective, interpretations such as
selection, information, entropy, force, acceleration, and physical work arise
from the same underlying geometry expressed by the Price equation.Comment: Version 3: added figure illustrating geometry; added table of symbols
and two tables summarizing mathematical relations; this version accepted for
publication in Entrop
Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory
The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table
First-principles calculations in crystalline structures are often performed
with a planewave basis set. To make the number of basis functions tractable two
approximations are usually introduced: core electrons are frozen and the
diverging Coulomb potential near the nucleus is replaced by a smoother
expression. The norm-conserving pseudopotential was the first successful method
to apply these approximations in a fully ab initio way. Later on, more
efficient and more exact approaches were developed based on the ultrasoft and
the projector augmented wave formalisms. These formalisms are however more
complex and developing new features in these frameworks is usually more
difficult than in the norm-conserving framework. Most of the existing tables of
norm- conserving pseudopotentials, generated long ago, do not include the
latest developments, are not systematically tested or are not designed
primarily for high accuracy. In this paper, we present our PseudoDojo framework
for developing and testing full tables of pseudopotentials, and demonstrate it
with a new table generated with the ONCVPSP approach. The PseudoDojo is an open
source project, building on the AbiPy package, for developing and
systematically testing pseudopotentials. At present it contains 7 different
batteries of tests executed with ABINIT, which are performed as a function of
the energy cutoff. The results of these tests are then used to provide hints
for the energy cutoff for actual production calculations. Our final set
contains 141 pseudopotentials split into a standard and a stringent accuracy
table. In total around 70.000 calculations were performed to test the
pseudopotentials. The process of developing the final table led to new insights
into the effects of both the core-valence partitioning and the non-linear core
corrections on the stability, convergence, and transferability of
norm-conserving pseudopotentials. ...Comment: abstract truncated, 17 pages, 25 figures, 8 table
Contrasting SYK-like Models
We contrast some aspects of various SYK-like models with large- melonic
behavior. First, we note that ungauged tensor models can exhibit symmetry
breaking, even though these are 0+1 dimensional theories. Related to this, we
show that when gauged, some of them admit no singlets, and are anomalous. The
uncolored Majorana tensor model with even is a simple case where gauge
singlets can exist in the spectrum. We outline a strategy for solving for the
singlet spectrum, taking advantage of the results in arXiv:1706.05364, and
reproduce the singlet states expected in . In the second part of the
paper, we contrast the random matrix aspects of some ungauged tensor models,
the original SYK model, and a model due to Gross and Rosenhaus. The latter,
even though disorder averaged, shows parallels with the Gurau-Witten model. In
particular, the two models fall into identical Andreev ensembles as a function
of . In an appendix, we contrast the (expected) spectra of AdS quantum
gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta
function.Comment: 45 pages, 17 figures; v2: minor improvements and rearrangements, refs
adde
Towards an Intelligent Tutor for Mathematical Proofs
Computer-supported learning is an increasingly important form of study since
it allows for independent learning and individualized instruction. In this
paper, we discuss a novel approach to developing an intelligent tutoring system
for teaching textbook-style mathematical proofs. We characterize the
particularities of the domain and discuss common ITS design models. Our
approach is motivated by phenomena found in a corpus of tutorial dialogs that
were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor
for textbook-style mathematical proofs can be built on top of an adapted
assertion-level proof assistant by reusing representations and proof search
strategies originally developed for automated and interactive theorem proving.
The resulting prototype was successfully evaluated on a corpus of tutorial
dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453
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