Entropy Scaling Law and the Quantum Marginal Problem

Abstract Heap’s Law https://dl.acm.org/citation.cfm?id=539986 Heaps, H S 1978 Information Retrieval: Computational and Theoretical Aspects (Academic Press). states that in a large enough text corpus, the number of types as a function of tokens grows as N = K{M^\beta } for some free parameters K, \beta . Much has been written http://iopscience.iop.org/article/10.1088/1367-2630/15/9/093033 Font-Clos, Francesc 2013 A scaling law beyond Zipf’s law and its relation to Heaps’ law (New Journal of Physics 15 093033)., http://iopscience.iop.org/article/10.1088/1367-2630/11/12/123015 Bernhardsson S, da Rocha L E C and Minnhagen P 2009 The meta book and size-dependent properties of written language (New Journal of Physics 11 123015)., http://iopscience.iop.org/article/10.1088/1742-5468/2011/07/P07013 Bernhardsson S, Ki Baek and Minnhagen 2011 A paradoxical property of the monkey book (Journal of Statistical Mechanics: Theory and Experiment, Volume 2011)., http://milicka.cz/kestazeni/type-token_relation.pdf Milička, Jiří 2009 Type-token & Hapax-token Relation: A Combinatorial Model (Glottotheory. International Journal of Theoretical Linguistics 2 (1), 99–110)., https://www.nature.com/articles/srep00943 Petersen, Alexander 2012 Languages cool as they expand: Allometric scaling and the decreasing need for new words (Scientific Reports volume 2, Article number: 943). about how this result and various generalizations can be derived from Zipf’s Law. http://dx.doi.org/10.1037/h0052442 Zipf, George 1949 Human behavior and the principle of least effort (Reading: Addison-Wesley). Here we derive from first principles a completely novel expression of the type-token curve and prove its superior accuracy on real text. This expression naturally generalizes to equally accurate estimates for counting hapaxes and higher n-legomena.

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Scaling law of radiative opacities for ICF elements

Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment ◽

10.1016/s0168-9002(01)00037-7 ◽

2001 ◽

Vol 464 (1-3) ◽

pp. 218-224 ◽

Cited By ~ 6

Author(s):

E Mı́nguez ◽

R Ruiz ◽

P Martel ◽

J.M Gil ◽

J.G Rubiano ◽

...

Keyword(s):

Scaling Law

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Jordan products of quantum channels and their compatibility

Nature Communications ◽

10.1038/s41467-021-22275-0 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Mark Girard ◽

Martin Plávala ◽

Jamie Sikora

Keyword(s):

Quantum State ◽

Sufficient Conditions ◽

The Other ◽

Independent Interest ◽

Quantum Channels ◽

Semidefinite Programs ◽

Jordan Product ◽

Marginal Problem ◽

Jordan Products ◽

Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

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Horizontal convection in a rectangular enclosure driven by a linear temperature profile

Applied Mathematics and Mechanics ◽

10.1007/s10483-021-2754-5 ◽

2021 ◽

Author(s):

Tianyong Yang ◽

Bofu Wang ◽

Jianzhao Wu ◽

Zhiming Lu ◽

Quan Zhou

Keyword(s):

Rayleigh Number ◽

Temperature Profile ◽

Scaling Law ◽

Thermal Plume ◽

Steady Regime ◽

Linear Temperature ◽

Square Enclosure ◽

Horizontal Convection ◽

Kinetic Boundary Layer ◽

The Mean

AbstractThe horizontal convection in a square enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically by using a finite difference method. The Prandtl number is fixed at 4.38, and the Rayleigh number Ra ranges from 107 to 1011. The convective flow is steady at a relatively low Rayleigh number, and no thermal plume is observed, whereas it transits to be unsteady when the Rayleigh number increases beyond the critical value. The scaling law for the Nusselt number Nu changes from Rossby’s scaling Nu ∼ Ra1/5 in a steady regime to Nu ∼ Ra1/4 in an unsteady regime, which agrees well with the theoretically predicted results. Accordingly, the Reynolds number Re scaling varies from Re ∼ Ra3/11 to Re ∼ Ra2/5. The investigation on the mean flows shows that the thermal and kinetic boundary layer thickness and the mean temperature in the bulk zone decrease with the increasing Ra. The intensity of fluctuating velocity increases with the increasing Ra.

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Fracture of Soft Elastic Foam

Journal of Applied Mechanics ◽

10.1115/1.4032050 ◽

2015 ◽

Vol 83 (3) ◽

Cited By ~ 6

Author(s):

Zhuo Ma ◽

Xiangchao Feng ◽

Wei Hong

Keyword(s):

Fracture Toughness ◽

Fracture Energy ◽

Cell Walls ◽

Scaling Law ◽

Phase Field Model ◽

Base Material ◽

Network Connectivity ◽

High Fracture Toughness ◽

High Fracture ◽

Order Of Magnitude

Consisting of stretchable and flexible cell walls or ligaments, soft elastic foams exhibit extremely high fracture toughness. Using the analogy between the cellular structure and the network structure of rubbery polymers, this paper proposes a scaling law for the fracture energy of soft elastic foam. To verify the scaling law, a phase-field model for the fracture processes in soft elastic structures is developed. The numerical simulations in two-dimensional foam structures of various unit-cell geometries have all achieved good agreement with the scaling law. In addition, the dependences of the macroscopic fracture energy on geometric parameters such as the network connectivity and spatial orientation have also been revealed by the numerical results. To further enhance the fracture toughness, a type of soft foam structures with nonstraight ligaments or folded cell walls has been proposed and its performance studied numerically. Simulations have shown that an effective fracture energy one order of magnitude higher than the base material can be reached by using the soft foam structure.

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