Neural Implementation of Shape-Invariant Touch Counter Based on Euler Calculus

Zeldovich’s stretch–twist–fold (STF) dynamo provided a breakthrough in conceptual understanding of fast dynamos, including the small-scale fluctuation dynamos. We study the evolution and saturation behaviour of two types of generalized Baker’s map dynamos, which have been used to model Zeldovich’s STF dynamo process. Using such maps allows one to analyse dynamos at much higher magnetic Reynolds numbers $\mathit{Re}_{M}$ as compared to direct numerical simulations. In the two-strip map dynamo there is constant constructive folding, while the four-strip map dynamo also allows the possibility of a destructive reversal of the field. Incorporating a diffusive step parametrized by $\mathit{Re}_{M}$ into the map, we find that the magnetic field $B(x)$ is amplified only above a critical $\mathit{Re}_{M}=R_{\mathit{crit}}\sim 4$ for both types of dynamos. The growing $B(x)$ approaches a shape-invariant eigenfunction independent of initial conditions, whose fine structure increases with increasing $\mathit{Re}_{M}$. Its power spectrum $M(k)$ displays sharp peaks reflecting the fractal nature of $B(x)$ above the diffusive scale. We explore the saturation of these dynamos in three ways: via a renormalized reduced effective $\mathit{Re}_{M}$ (case I) or due to a decrease in the efficiency of the field amplification by stretching, without changing the map (case IIa), or changing the map (case IIb), and a combination of both effects (case III). For case I, we show that $B(x)$ in the saturated state, for both types of maps, approaches the marginal eigenfunction, which is obtained for $\mathit{Re}_{M}=R_{\mathit{crit}}$ independent of the initial $\mathit{Re}_{M}=R_{M0}$. On the other hand, in case II, for the two-strip map, we show that $B(x)$ saturates, preserving the structure of the kinematic eigenfunction. Thus the energy is transferred to larger scales in case I but remains at the smallest resistive scales in case II, as can be seen from both $B(x)$ and $M(k)$. For the four-strip map, $B(x)$ oscillates with time, although with a structure similar to the kinematic eigenfunction. Interestingly, the saturated state in case III shows an intermediate behaviour, with $B(x)$ similar to the kinematic eigenfunction at an intermediate $\mathit{Re}_{M}=R_{\mathit{sat}}$, with $R_{M0}>R_{\mathit{sat}}>R_{\mathit{crit}}$. The $R_{\mathit{sat}}$ value is determined by the relative importance of the increased diffusion versus the reduced stretching. These saturation properties are akin to the range of possibilities that have been discussed in the context of fluctuation dynamos.

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Quantum cosmology: From hidden symmetries towards a new (supersymmetric) perspective

International Journal of Modern Physics D ◽

10.1142/s0218271816300093 ◽

2016 ◽

Vol 25 (03) ◽

pp. 1630009 ◽

Cited By ~ 4

Author(s):

S. Jalalzadeh ◽

T. Rostami ◽

P. V. Moniz

Keyword(s):

New York ◽

Boundary Conditions ◽

Quantum Cosmology ◽

Ad Hoc ◽

Hidden Symmetries ◽

Shape Invariance ◽

Hidden Symmetry ◽

Shape Invariant ◽

Model Dynamics ◽

Lecture Notes

We review pedagogically some of the basic essentials regarding recent results intertwining boundary conditions, the algebra of constraints and hidden symmetries in quantum cosmology. They were extensively published in Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014), S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439 and T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212], where complete discussions and full details can be found. More concretely, in Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014) and S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439] it has been shown that specific boundary conditions can be related to the algebra of Dirac observables. Moreover, a process afterwards associated to the algebra of existent hidden symmetries, from which the boundary conditions can be selected, was introduced. On the other hand, in Ref. [T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212] it was subsequently argued that some factor ordering choices can be extracted from the hidden symmetries structure of the minisuperspace model.In Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014), S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439 and T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212], we proceeded gradually towards less simple models, ranging from a FLRW model with a perfect fluid [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541] up to a conformal scalar field content [T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212]. We envisage that we could extend this framework towards a class of shape invariant potentials, which could include well known analytically solvable cosmological cases. Provided, we identify integrability in terms of the shape invariance conditions, we could eventually consider to import features of supersymmetric quantum mechanics towards quantum cosmology [P. V. Moniz, Quantum Cosmology-the Supersymmetric Perspective-Vol. 1: Fundamentals, Lecture Notes in Physics, Vol. 803 (Springer-Verlag, Berlin, 2010), P. V. Moniz, Quantum Cosmology-the Supersymmetric Perspective-Vol. 2: Advanced Topics, Lecture Notes in Physics, Vol. 804 (Springer, New York, 2010)], which we will also discuss in this review.Another point to emphasize is that by means of a hidden symmetry and then an algebra of Dirac observables, boundary conditions are extracted (and not ad hoc formulated) within a framework intrinsic to each model dynamics. Therefore, meeting DeWitt’s conjecture [B. S. DeWitt, Phys. Rev. 160 (1967) 1113] that “the constraints are everything” and nothing else but the constraints should be needed.

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Generalized Robertson intelligent states and squeezing for supersymmetric and shape-invariant systems: an algebraic construction

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/40/19/011 ◽

2007 ◽

Vol 40 (19) ◽

pp. 5105-5126 ◽

Cited By ~ 10

Author(s):

A N F Aleixo ◽

A B Balantekin

Keyword(s):

Algebraic Construction ◽

Shape Invariant

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The representation of colored objects in macaque color patches

10.1101/205104 ◽

2017 ◽

Author(s):

Le Chang ◽

Pinglei Bao ◽

Doris Y. Tsao

Keyword(s):

Object Recognition ◽

Color Vision ◽

The Other ◽

Color Patch ◽

Shape Information ◽

Object Shape ◽

Shape Invariant ◽

High Concentrations ◽

The Brain

AbstractAn important question about color vision is: how does the brain represent the color of an object? The recent discovery of “color patches” in macaque inferotemporal (IT) cortex, the part of brain responsible for object recognition, makes this problem experimentally tractable. Here we record neurons in three color patches, middle color patch CLC (central lateral color patch), and two anterior color patches ALC (anterior lateral color patch) and AMC (anterior medial color patch), while presenting images of objects systematically varied in hue. We found that all three patches contain high concentrations of hue-selective cells, and the three patches use distinct computational strategies to represent colored objects: while all three patches multiplex hue and shape information, shape-invariant hue information is much stronger in anterior color patches ALC/AMC than CLC; furthermore, hue and object shape specifically for primate faces/bodies are over-represented in AMC but not in the other two patches.

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