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Counting dyadic equipartitions of the unit square

Jeffrey C. Lagarias, Joel H. Spencer, Jade P. Vinson
2002 Discrete Mathematics  
Let un count the number of ways of partitioning the unit square into 2 n dyadic rectangles, each of area 2 −n . One has u0 = 1, u1 = 2 and un = 2u 2 n−1 − u 4 n−2 .  ...  A dyadic interval is an interval of the form [j=2 k ; (j + 1)=2 k ], where j and k are integers, and a dyadic rectangle is a rectangle with sides parallel to the axes whose projections on the axes are  ...  Mario Szegedy and the referees for helpful comments and references, and to Ed Thurber for bringing Ref. [7] to our attention.  ... 
doi:10.1016/s0012-365x(02)00508-3 fatcat:jzk7wskrifcyhnkbpmhkwuhwvm

Page 5737 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
(I1-NY-C; New York, NY); Vinson, Jade P. (1-PRIN; Princeton, NJ) Counting dyadic equipartitions of the unit square. (English summary) Kleitman and combinatorics: a celebration (Cambridge, MA, 1999).  ...  Let u, count the number of ways of partitioning the unit square into 2” dyadic rectangles, each of area 2-". One has up = 1, u; =2 and u, = 2u?_, —us_,.  ... 

Diffuse elastic waves in a nearly axisymmetric body: Energy distribution, transport and dynamical localization

Richard L. Weaver, John Y. Yoritomo, J. Patrick Coleman
2017 The European Physical Journal Special Topics  
Aluminum cylinders with dimensions large compared to wavelength were excited by transient point sources at the center of one of the circular faces.  ...  Relaxation did not proceed all the way to uniformity and equipartition, behavior shown to be consistent with Enhanced Backscatter and Dynamical Anderson Localization.  ...  unit source at r on the x axis.  ... 
doi:10.1140/epjst/e2016-60186-7 fatcat:ianam55hw5br3npewlzycptoaq

Green's Dyadic, Spectral Function, Local Density of States, and Fluctuation Dissipation Theorem [article]

W.C. Chew, W.E.I Sha, Q.I. Dai
2015 arXiv   pre-print
The spectral functions are studied in conjunction with the dyadic Green's functions for various media.  ...  For the lossless media cases, the spectral functions can be directly related to the photon local density of states, and hence, to the electromagnetic energy density.  ...  Acknowledgements This work was supported in part by the USA NSF CCF Award 1218552, SRC Award 2012-IN-2347, at the Uni-versity of Illinois at Urbana-Champaign, by the Research Grants Council of Hong Kong  ... 
arXiv:1505.01586v1 fatcat:7b5ygujzynhopmbu62uoasqndu

GREEN'S DYADIC, SPECTRAL FUNCTION, LOCAL DENSITY OF STATES, AND FLUCTUATION DISSIPATION THEOREM

Weng Cho Chew, Wei E. I. Sha, Qi I. Dai
2019 Electromagnetic Waves  
The spectral functions are studied in conjunction with the dyadic Green's functions for various media.  ...  For the lossless media cases, the spectral functions can be directly related to the photon local density of states, and hence, to the electromagnetic energy density.  ...  In this manner, the LDOS can be directly related to this field correlation function as D(r, E)Θ(E)dE = D(r, E)Θ(E) dω = Tr[ 0 C(r, r, ω)]dω (41) In the above, C has the unit of field square per radian  ... 
doi:10.2528/pier19111801 fatcat:wyrp4bbarna4dktiav3iuu2ree

Limits of Latin squares [article]

Frederik Garbe, Robert Hancock, Jan Hladký, Maryam Sharifzadeh
2020 arXiv   pre-print
We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations.  ...  Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence.  ...  Graphons arise as limits of sequences of graphs, and could be informally understood as limits of their adjacency matrices rescaled into a square of unit area.  ... 
arXiv:2010.07854v2 fatcat:3sk3wbyqmnh7jea5vifygayyii

On the computability of graphons [article]

Nathanael L. Ackerman, Jeremy Avigad, Cameron E. Freer, Daniel M. Roy, Jason M. Rute
2018 arXiv   pre-print
We investigate the relative computability of exchangeable binary relational data when presented in terms of the distribution of an invariant measure on graphs, or as a graphon in either L^1 or the cut  ...  We show that 0' is necessary in general, but that in the case of random-free graphons, no oracle is necessary.  ...  Work on this publication was made possible through the support of United States Air Force Office of Scientific Research (AFOSR) Contract No.  ... 
arXiv:1801.10387v1 fatcat:5vg7wxb52rc7jjq476jqa2zt3a

Quantifying the negative impact of brain drain on the integration of European science

Omar A. Doria Arrieta, Fabio Pammolli, Alexander M. Petersen
2017 Science Advances  
The 2004/2007 European Union (EU) enlargement by 12 member states offers a unique opportunity to quantify the impact of EU efforts to expand and integrate the scientific competitiveness of the European  ...  These results challenge central tenets underlying ERA integration policies that unifying labor markets will increase the international competitiveness of the ERA, thereby calling attention to the need  ...  The set of units in our SCM analysis is divided into three subsets: (i) a balanced panel of 26 non-European countries (those with nonzero publication counts in each s for all t), (ii) the 10 countries  ... 
doi:10.1126/sciadv.1602232 pmid:28439544 pmcid:PMC5389781 fatcat:p7tv4friqvhbbltp5mnvzbhcbi

Field Theory for Magnetic Monopoles in (Square, Artificial) Spin Ice [article]

Cristiano Nisoli
2020 arXiv   pre-print
This suggests that the monopole plasma of square ice is a magnetic charge insulator.  ...  We show that the divergence of the correlation length at low temperature is exponential and inversely proportional to the mean square charge.  ...  Remembering thatγ αγα + ⊥γ α ⊥γ α = δ α,α one obtains F 2 = 1 2 (β H * ) · γγχ || + ⊥γ⊥γχ ⊥ · (β H ) (77) whereγ is the unit vector of γ and we have used dyadics inγ. The first term in Eq.  ... 
arXiv:2004.03735v2 fatcat:b47k32bp2faklpzzz6fr5ecraq

Master index to volumes 251-260

2003 Discrete Mathematics  
Vinson, Counting dyadic equipartitions of the unit square 257 (2002) 481-499 Laihonen, T., see I.  ...  Salvi, The observability of the Fibonacci and the Lucas cubes 255 353-359 Frankl, P. and N.  ... 
doi:10.1016/s0012-365x(02)00784-7 fatcat:2c4y6lmnybeuxcqr55ngld7r7e

Network Throughput Optimization via Error Correcting Codes [article]

Ratko V. Tomic
2013 arXiv   pre-print
the same number of ports provided and the same total throughput.  ...  The method is based on the exact equivalence discovered between the problem of maximizing network throughput (measured as bisection bandwidth) for a large class of practically interesting Cayley graphs  ...  equipartition H 0 +H 1 and in each case count the links between the two, then pick the one with the lowest count.  ... 
arXiv:1301.4177v1 fatcat:xmo55skta5gc3fedmt3bhmwzce

Diffuse wave density and directionality in anisotropic solids

Andrew N. Norris
2008 Journal of the Acoustical Society of America  
Several general results are derived for diffuse waves in anisotropic solids, including concise expressions for the modal density per unit volume and for the participation factor matrix G.  ...  The latter is a second order tensor which describes the orientational distribution of diffuse wave or reverberant energy, and reduces to the identity I under isotropy.  ...  The first integral in Eq. ͑12͒ is around the unit circle formed by the intersection of the plane n · x = 0 with the unit n sphere.  ... 
doi:10.1121/1.2836755 pmid:18345829 fatcat:j2zj6cuxdzgjbm4enpedtnfhj4

Page 1981 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
Counting dyadic equipartitions of the unit square. (English summary) Kleitman and combinatorics: a celebration (Cambridge, MA, 1999). Discrete Math. 257 (2002), no. 2-3, 481-499. (Edward A.  ...  Bender) 2003h:05018 05A16 (30D05, 37F99) — (with Janson, Svante; Randall, Dana) Random dyadic tilings of the unit square. (English summary) Random structures and algorithms (Poznan, 2001).  ... 

Noncontextual coloring of orthogonality hypergraphs [article]

Mohammad H. Shekarriz, Karl Svozil
2021 arXiv   pre-print
The latter ones have a quantum mechanical realization in terms of intertwined contexts or maximal observables.  ...  Structure reconstruction of these hypergraphs from their table of two-valued states is possible for a class of hypergraphs, namely perfectly separable hypergraphs.  ...  Thereby, unit vectors are interpreted as the orthogonal projection operators formed by the respective dyadic products.  ... 
arXiv:2105.08520v1 fatcat:ksqmzbcjzjgfta5r6h7t2dd46a

On the origin of cosmic magnetic fields

Russell M Kulsrud, Ellen G Zweibel
2008 Reports on progress in physics (Print)  
We review the literature concerning how the cosmic magnetic fields pervading nearly all galaxies actually got started. some observational evidence involves the chemical abundance of the light elements  ...  Seed fields, whose strength is of order 10^-20 gauss, easily sprung up in the era preceding galaxy formation. Several mechanisms are proposed to amplify these seed fields to microgauss strengths.  ...  * k ′ (t)v k (t ′ ) >= J(k)(I −kk)δ(k ′ − k)δ τ (t − t ′ ). (36) Here the angular brackets denote the ensemble average, of the velocities at different wave numbers and times, I is the unit dyadic, and  ... 
doi:10.1088/0034-4885/71/4/046901 fatcat:qltvczms7vblzdqdxtqaakvuke
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