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model of a spin glass physical review letters 35 26 1792 1796 bibcode 1975phrvl 35 1792s doi 10 1103 physrevlett 35 1792 issn 0031 9007 hopfield j j 1982 neural networks and physical systems with emergent collective computational abilities proceedings of the national academy of sciences 79 8 2554 2558 bibcode 1982pnas 79 2554h doi 10 1073 pnas 79 8 2554 pmc 346238 pmid 6953413 hopfield j j 1984 neurons with graded response have collective computational properties like those of two state neurons proceedings of the national academy of sciences 81 10 3088 3092 bibcode 1984pnas 81 3088h doi 10 1073 pnas 81 10 3088 pmc 345226 pmid 6587342 engel a broeck c van den 2001 statistical mechanics of learning cambridge uk new york ny cambridge university press isbn 978 0 521 77307 2 seung h s sompolinsky h tishby n 1992 04 01 statistical mechanics of learning from examples physical review a 45 8 6056 6091 bibcode 1992phrva 45 6056s doi 10 1103 physreva 45 6056 pmid 9907706 yi ping ma ivan sudakov 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gliozzi f meineri m others 2013 line defects in the 3d ising model jhep 1307 7 055 arxiv 1304 4110 bibcode 2013jhep 07 055b doi 10 1007 jhep07 2013 055 s2cid 119226610 cosme catarina lopes j m viana parente penedones joao 2015 conformal symmetry of the critical 3d ising model inside a sphere journal of high energy physics 2015 8 22 arxiv 1503 02011 bibcode 2015jhep 08 022c doi 10 1007 jhep08 2015 022 s2cid 53710971 zhu wei han chao huffman emilie hofmann johannes s he yin chen 2023 uncovering conformal symmetry in the 3d ising transition state operator correspondence from a quantum fuzzy sphere regularization physical review x 13 2 021009 arxiv 2210 13482 bibcode 2023phrvx 13b1009z doi 10 1103 physrevx 13 021009 s2cid 253107625 delamotte bertrand tissier matthieu wschebor nicolás 2016 scale invariance implies conformal invariance for the three dimensional ising model physical review e 93 12144 012144 arxiv 1501 01776 bibcode 2016phrve 93a2144d doi 10 1103 physreve 93 012144 pmid 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05 27 at the wayback machine and vol ii archived 2008 05 27 at the wayback machine kleinert h and schulte frohlinde v 2001 critical properties of φ 4 theories world scientific singapore paperback isbn 981 02 4658 7 also available online archived 2011 07 16 at the wayback machine lenz w 1920 beiträge zum verständnis der magnetischen eigenschaften in festen körpern physikalische zeitschrift 21 613 615 barry m mccoy and tai tsun wu 1973 the two dimensional ising model harvard university press cambridge massachusetts isbn 0 674 91440 6 montroll elliott w potts renfrey b ward john c 1963 correlations and spontaneous magnetization of the two dimensional ising model journal of mathematical physics 4 2 308 322 bibcode 1963jmp 4 308m doi 10 1063 1 1703955 issn 0022 2488 mr 0148406 retrieved 2009 10 25 citation cs1 maint deprecated archival service link onsager lars 1944 crystal statistics i a two dimensional model with an order disorder transition physical review series ii 65 3 4 117 149 bibcode 1944phrv 65 117o doi 10 1103 physrev 65 117 mr 0010315 onsager lars 1949 discussion supplemento al nuovo cimento 6 261 john palmer 2007 planar ising correlations birkhäuser boston isbn 978 0 8176 4248 8 istrail sorin 2000 statistical mechanics three dimensionality and np completeness i universality of intractability for the partition function of the ising model across non planar surfaces extended abstract pdf proceedings of the thirty second annual acm symposium on theory of computing acm pp 87 96 doi 10 1145 335305 335316 isbn 978 1 58113 184 0 mr 2114521 s2cid 7944336 yang c n 1952 the spontaneous magnetization of a two dimensional ising model physical review series ii 85 5 808 816 bibcode 1952phrv 85 808y doi 10 1103 physrev 85 808 mr 0051740 glasser m l 1970 exact partition function for the two dimensional ising model american journal of physics 38 8 1033 1036 bibcode 1970amjph 38 1033g doi 10 1119 1 1976530 jellito r j 1979 the ising model on a closed cayley tree physica 99a 1 268 280 bibcode 1979phya 99 268j doi 10 1016 0378 4371 79 90134 1 krizan j e barth p f glasser m l 1983 exact phase transitions for the ising model on the closed cayley tree physica 119a north holland publishing co 230 242 doi 10 1016 0378 4371 83 90157 7 glasser m l goldberg m 1983 the ising model on a closed cayley tree physica 117a 2 670 672 bibcode 1983phya 117 670g doi 10 1016 0378 4371 83 90138 3 süzen mehmet 2014 effective ergodicity in single spin flip dynamics physical review e 90 3 032141 arxiv 1405 4497 bibcode 2014phrve 90c2141s doi 10 1103 physreve 90 032141 pmid 25314429 external links edit wikimedia commons has media related to ising model ising model at the net advance of physics barry arthur cipra the ising model is np complete siam news vol 33 no 6 online edition pdf science world article on the ising model a dynamical 2d ising java applet by ucsc archived 2020 02 21 at the wayback machine a dynamical 2d ising java applet a larger more complicated 2d ising java applet archived 2020 11 25 at the wayback machine i sing well tempered the ising model a simple model for critical behavior in a system of spins by dirk brockman is an interactive simulation that allows users to export the working code to a presentation slide ising model simulation by enrique zeleny the wolfram demonstrations project phase transitions on lattices three dimensional proof for ising model impossible sandia researcher claims interactive monte carlo simulation of the ising xy and heisenberg models with 3d graphics requires webgl compatible browser ising model code image denoising example with ising model david tong s lecture notes provide a good introduction the cartoon picture of magnets that has transformed science quanta magazine article about ising model simulation of the 2 dimensional ising model in julia https github com cossio squareisingmodel jl v t e stochastic processes discrete time bernoulli process branching process chinese restaurant process galton watson 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differences local sub super random dynamical system regenerative process renewal process stochastic chains with memory of variable length white noise fields and other dirichlet process gaussian random field gibbs measure hopfield model ising model potts model boolean network markov random field percolation pitman yor process point process cox determinantal poisson random field random graph time series models autoregressive conditional heteroskedasticity arch model autoregressive integrated moving average arima model autoregressive ar model autoregressive moving average arma model generalized autoregressive conditional heteroskedasticity garch model moving average ma model financial models binomial options pricing model black derman toy black karasinski black scholes chan karolyi longstaff sanders ckls chen constant elasticity of variance cev cox ingersoll ross cir garman kohlhagen heath jarrow morton hjm heston ho lee hull white korn kreer lenssen libor market rendleman bartter sabr volatility vašíček wilkie actuarial models bühlmann cramér lundberg risk process sparre anderson queueing models bulk fluid generalized queueing network m g 1 m m 1 m m c properties càdlàg paths continuous continuous paths ergodic exchangeable feller continuous gauss markov markov mixing piecewise deterministic predictable progressively measurable self similar stationary time reversible limit theorems central limit theorem donsker s theorem doob s martingale convergence theorems ergodic theorem fisher tippett gnedenko theorem large deviation principle law of large numbers weak strong law of the iterated logarithm maximal ergodic theorem sanov s theorem zero one laws blumenthal borel cantelli engelbert schmidt hewitt savage kolmogorov lévy inequalities burkholder davis gundy doob s martingale doob s upcrossing kunita watanabe marcinkiewicz zygmund tools cameron martin theorem convergence of random variables doléans dade exponential doob decomposition theorem doob meyer decomposition 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