├── .gitignore ├── LICENSE ├── README.md └── resources ├── Category Theory └── category-theory.pdf ├── Chaos ├── Manifest.toml ├── Project.toml ├── README.md ├── chaos.pdf └── cobweb.jl ├── Differential Geometry ├── README.md ├── curvature.pdf ├── frenet-serret.ggb ├── manifolds.pdf └── parallel-transport.pdf ├── Functional Data Analysis ├── README.md └── fda.pdf ├── Information Geometry ├── README.md └── information-geometry.pdf ├── Johnson-Lindenstrauss Lemma ├── README.md └── johnsonlindenstrauss.pdf ├── Morse Theory ├── Eleuther.pdf └── README.md ├── Multilinear Algebra ├── README.md ├── multilinear-alegbra.pdf └── tensors (english version).pdf ├── ODEs and State-Space Models ├── AntsaklisMichel_Primer_up to Chapter 6.pdf ├── README.md ├── diffeq.pdf └── ssms.pdf ├── Papers ├── Geometric Algebra Transformers │ ├── GATr.pdf │ └── README.md ├── Harmonic Networks │ ├── Harmonic_Neural_Networks.pdf │ └── README.md ├── Kolmogorov Arnold Networks │ └── README.md └── Neural Tangent Kernels │ ├── README.md │ └── ntkoverview.pdf ├── Random Matrix Theory ├── README.md ├── RMT-1.pdf ├── RMT-2.pdf ├── RMT-3.pdf ├── RMT-chap1.jl ├── RMT-chap2.jl └── RMT-chap4.jl ├── Representation Theory └── README.md ├── Signature Transform ├── README.md └── signatures.pdf └── Topology ├── README.md ├── compactness.pdf └── topology.pdf /.gitignore: -------------------------------------------------------------------------------- 1 | .vscode/* 2 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Creative Commons Legal Code 2 | 3 | CC0 1.0 Universal 4 | 5 | CREATIVE COMMONS CORPORATION IS NOT A LAW FIRM AND DOES NOT PROVIDE 6 | LEGAL SERVICES. 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Affirmer understands and acknowledges that Creative Commons is not a 120 | party to this document and has no duty or obligation with respect to 121 | this CC0 or use of the Work. 122 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # EAI-Math-Reading-Group 2 | 3 | Machine Learning already relies on a lot of math, and with more and more researchers starting to bring mathematical tools from pure mathematics and physics into the field, many practitioners are starting to feel like they need to level up. Similarly, many people wanting to get started into alignment find themselves wishing they knew more advanced math. 4 | 5 | If the previous paragraph sounds just like you, or you just like to learn math in general, consider joining this reading group. We usually meet on Sundays at 1PM Pacific on the EAI Discord server. 6 | 7 | On discord, you can reach out to genetyx8 or join the [thread](https://discord.com/channels/729741769192767510/1089231591785639946). 8 | 9 | Recordings of the meetings can be found on the [EleutherAI Youtube channel](https://www.youtube.com/playlist?list=PLvtrkEledFjodK_UKua2h4exNMcWNGJrs) 10 | 11 | See the [resources](https://github.com/irregular-rhomboid/EAI-Math-Reading-Group/tree/main/resources) folder for the slides and links to the documents we read as part of the reading group. 12 | 13 | ## Organisation 14 | Since there are many interesting topics, and sticking for months to a single one is difficult, our current plan is to have sessions on long-running topics like Geometric Deep Learning interspersed with some shorter ones. 15 | 16 | While we'll generally try to stay close to ML, some topics need a basic introduction, so we'll have some of those as well. 17 | 18 | ## Contributing 19 | - If there are particular topics you'd like us to cover, feel free to let us know or create an issue on this repo, and we'll add them to the list. 20 | - If you are knowledgeable in some topic, and/or are motivated to present one or more sessions on it, we'd love to have you. Just open an issue on this repo on the topic, or reply to the existing issue for that topic. 21 | 22 | ## Guidelines for contributors 23 | In order to prepare sessions on a topic, it's important to have some source material available online that attendees can read. You can find some [here](https://github.com/rossant/awesome-math), but there's plenty more out there. 24 | 25 | A topic will be considered ready to be covered once it has reading material, people willing to present and an idea of how many sessions it should take, at which point it will be scheduled. 26 | 27 | ## List of suggested topics 28 | 29 | - ~~Representation Theory~~ 30 | - ~~Geometric Deep Learning~~ 31 | - ~~Random Matrix Theory~~ 32 | - Partial Differential Equations (Theory + ML applications) 33 | - Scientific Machine Learning 34 | - Undergraduate Physics (Electromagnetism, Statistical Mechanics, Quantum Mechanics) 35 | - Advanced Linear Algebra 36 | - ~~Dynamical Systems~~ 37 | - ~~Topology~~ 38 | - Measure Theory 39 | - Game Theory 40 | - ~~Information Geometry~~ 41 | - ~~Functional Data Analysis~~ 42 | - ~~Topological Data Analysis~~ 43 | -------------------------------------------------------------------------------- /resources/Category Theory/category-theory.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Category Theory/category-theory.pdf -------------------------------------------------------------------------------- /resources/Chaos/Manifest.toml: -------------------------------------------------------------------------------- 1 | # This file is machine-generated - editing it directly is not advised 2 | 3 | julia_version = "1.10.2" 4 | manifest_format = "2.0" 5 | project_hash = "f576800cc22d86375927d6f2df313404bb26378d" 6 | 7 | [[deps.ADTypes]] 8 | git-tree-sha1 = "016833eb52ba2d6bea9fcb50ca295980e728ee24" 9 | uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" 10 | version = "0.2.7" 11 | 12 | [[deps.AbstractFFTs]] 13 | deps = ["LinearAlgebra"] 14 | git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef" 15 | uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" 16 | version = "1.5.0" 17 | weakdeps = ["ChainRulesCore", "Test"] 18 | 19 | [deps.AbstractFFTs.extensions] 20 | AbstractFFTsChainRulesCoreExt = "ChainRulesCore" 21 | AbstractFFTsTestExt = "Test" 22 | 23 | [[deps.AbstractLattices]] 24 | git-tree-sha1 = "222ee9e50b98f51b5d78feb93dd928880df35f06" 25 | uuid = "398f06c4-4d28-53ec-89ca-5b2656b7603d" 26 | version = "0.3.0" 27 | 28 | [[deps.AbstractTrees]] 29 | git-tree-sha1 = "2d9c9a55f9c93e8887ad391fbae72f8ef55e1177" 30 | uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c" 31 | version = "0.4.5" 32 | 33 | [[deps.Accessors]] 34 | deps = ["CompositionsBase", "ConstructionBase", "Dates", "InverseFunctions", "LinearAlgebra", "MacroTools", "Markdown", "Test"] 35 | git-tree-sha1 = "c0d491ef0b135fd7d63cbc6404286bc633329425" 36 | uuid = "7d9f7c33-5ae7-4f3b-8dc6-eff91059b697" 37 | version = "0.1.36" 38 | 39 | [deps.Accessors.extensions] 40 | AccessorsAxisKeysExt = "AxisKeys" 41 | AccessorsIntervalSetsExt = "IntervalSets" 42 | AccessorsStaticArraysExt = "StaticArrays" 43 | AccessorsStructArraysExt = "StructArrays" 44 | AccessorsUnitfulExt = "Unitful" 45 | 46 | [deps.Accessors.weakdeps] 47 | AxisKeys = "94b1ba4f-4ee9-5380-92f1-94cde586c3c5" 48 | IntervalSets = "8197267c-284f-5f27-9208-e0e47529a953" 49 | Requires = "ae029012-a4dd-5104-9daa-d747884805df" 50 | StaticArrays = "90137ffa-7385-5640-81b9-e52037218182" 51 | StructArrays = "09ab397b-f2b6-538f-b94a-2f83cf4a842a" 52 | Unitful = "1986cc42-f94f-5a68-af5c-568840ba703d" 53 | 54 | [[deps.Adapt]] 55 | deps = ["LinearAlgebra", "Requires"] 56 | git-tree-sha1 = "6a55b747d1812e699320963ffde36f1ebdda4099" 57 | uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" 58 | version = "4.0.4" 59 | weakdeps = ["StaticArrays"] 60 | 61 | [deps.Adapt.extensions] 62 | AdaptStaticArraysExt = "StaticArrays" 63 | 64 | [[deps.Animations]] 65 | deps = ["Colors"] 66 | git-tree-sha1 = "e81c509d2c8e49592413bfb0bb3b08150056c79d" 67 | uuid = "27a7e980-b3e6-11e9-2bcd-0b925532e340" 68 | version = "0.4.1" 69 | 70 | [[deps.ArgTools]] 71 | uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" 72 | version = "1.1.1" 73 | 74 | [[deps.ArnoldiMethod]] 75 | deps = ["LinearAlgebra", "Random", "StaticArrays"] 76 | git-tree-sha1 = "62e51b39331de8911e4a7ff6f5aaf38a5f4cc0ae" 77 | uuid = "ec485272-7323-5ecc-a04f-4719b315124d" 78 | version = "0.2.0" 79 | 80 | [[deps.ArrayInterface]] 81 | deps = ["Adapt", "LinearAlgebra", "SparseArrays", "SuiteSparse"] 82 | git-tree-sha1 = "44691067188f6bd1b2289552a23e4b7572f4528d" 83 | uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9" 84 | version = "7.9.0" 85 | 86 | [deps.ArrayInterface.extensions] 87 | ArrayInterfaceBandedMatricesExt = "BandedMatrices" 88 | ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices" 89 | ArrayInterfaceCUDAExt = "CUDA" 90 | ArrayInterfaceChainRulesExt = "ChainRules" 91 | ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore" 92 | ArrayInterfaceReverseDiffExt = "ReverseDiff" 93 | ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore" 94 | ArrayInterfaceTrackerExt = "Tracker" 95 | 96 | [deps.ArrayInterface.weakdeps] 97 | BandedMatrices = "aae01518-5342-5314-be14-df237901396f" 98 | BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0" 99 | CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba" 100 | ChainRules = "082447d4-558c-5d27-93f4-14fc19e9eca2" 101 | GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527" 102 | ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" 103 | StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c" 104 | Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c" 105 | 106 | [[deps.ArrayLayouts]] 107 | deps = ["FillArrays", "LinearAlgebra"] 108 | git-tree-sha1 = "6404a564c24a994814106c374bec893195e19bac" 109 | uuid = "4c555306-a7a7-4459-81d9-ec55ddd5c99a" 110 | version = "1.8.0" 111 | weakdeps = ["SparseArrays"] 112 | 113 | [deps.ArrayLayouts.extensions] 114 | ArrayLayoutsSparseArraysExt = "SparseArrays" 115 | 116 | [[deps.Artifacts]] 117 | uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" 118 | 119 | [[deps.Attractors]] 120 | deps = ["BlackBoxOptim", "Clustering", "ComplexityMeasures", "Distances", "Distributions", "DynamicalSystemsBase", "LinearAlgebra", "Mmap", "Neighborhood", "Optim", "ProgressMeter", "Random", "Reexport", "StateSpaceSets"] 121 | git-tree-sha1 = "895833094aa539c856ba686381909d7c7e535b3e" 122 | uuid = "f3fd9213-ca85-4dba-9dfd-7fc91308fec7" 123 | version = "1.15.0" 124 | weakdeps = ["Makie"] 125 | 126 | [deps.Attractors.extensions] 127 | AttractorsVisualizations = "Makie" 128 | 129 | [[deps.Automa]] 130 | deps = ["PrecompileTools", "TranscodingStreams"] 131 | git-tree-sha1 = "588e0d680ad1d7201d4c6a804dcb1cd9cba79fbb" 132 | uuid = "67c07d97-cdcb-5c2c-af73-a7f9c32a568b" 133 | version = "1.0.3" 134 | 135 | [[deps.AxisAlgorithms]] 136 | deps = ["LinearAlgebra", "Random", "SparseArrays", "WoodburyMatrices"] 137 | git-tree-sha1 = "66771c8d21c8ff5e3a93379480a2307ac36863f7" 138 | uuid = "13072b0f-2c55-5437-9ae7-d433b7a33950" 139 | version = "1.0.1" 140 | 141 | [[deps.AxisArrays]] 142 | deps = ["Dates", "IntervalSets", "IterTools", "RangeArrays"] 143 | git-tree-sha1 = "16351be62963a67ac4083f748fdb3cca58bfd52f" 144 | uuid = "39de3d68-74b9-583c-8d2d-e117c070f3a9" 145 | version = "0.4.7" 146 | 147 | [[deps.Base64]] 148 | uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" 149 | 150 | [[deps.BitTwiddlingConvenienceFunctions]] 151 | deps = ["Static"] 152 | git-tree-sha1 = "0c5f81f47bbbcf4aea7b2959135713459170798b" 153 | uuid = "62783981-4cbd-42fc-bca8-16325de8dc4b" 154 | version = "0.1.5" 155 | 156 | [[deps.BlackBoxOptim]] 157 | deps = ["CPUTime", "Compat", "Distributed", "Distributions", "JSON", "LinearAlgebra", "Printf", "Random", "Requires", "SpatialIndexing", "StatsBase"] 158 | git-tree-sha1 = "9c203a2515b5eeab8f2987614d2b1db83ef03542" 159 | uuid = "a134a8b2-14d6-55f6-9291-3336d3ab0209" 160 | version = "0.6.3" 161 | 162 | [deps.BlackBoxOptim.extensions] 163 | BlackBoxOptimRealtimePlotServerExt = ["HTTP", "Sockets"] 164 | 165 | [deps.BlackBoxOptim.weakdeps] 166 | HTTP = "cd3eb016-35fb-5094-929b-558a96fad6f3" 167 | Sockets = "6462fe0b-24de-5631-8697-dd941f90decc" 168 | 169 | [[deps.Bzip2_jll]] 170 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 171 | git-tree-sha1 = "9e2a6b69137e6969bab0152632dcb3bc108c8bdd" 172 | uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0" 173 | version = "1.0.8+1" 174 | 175 | [[deps.CEnum]] 176 | git-tree-sha1 = "389ad5c84de1ae7cf0e28e381131c98ea87d54fc" 177 | uuid = "fa961155-64e5-5f13-b03f-caf6b980ea82" 178 | version = "0.5.0" 179 | 180 | [[deps.CPUSummary]] 181 | deps = ["CpuId", "IfElse", "PrecompileTools", "Static"] 182 | git-tree-sha1 = "601f7e7b3d36f18790e2caf83a882d88e9b71ff1" 183 | uuid = "2a0fbf3d-bb9c-48f3-b0a9-814d99fd7ab9" 184 | version = "0.2.4" 185 | 186 | [[deps.CPUTime]] 187 | git-tree-sha1 = "2dcc50ea6a0a1ef6440d6eecd0fe3813e5671f45" 188 | uuid = "a9c8d775-2e2e-55fc-8582-045d282d599e" 189 | version = "1.0.0" 190 | 191 | [[deps.CRC32c]] 192 | uuid = "8bf52ea8-c179-5cab-976a-9e18b702a9bc" 193 | 194 | [[deps.CRlibm]] 195 | deps = ["CRlibm_jll"] 196 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"RingLists", "SimplePartitions", "SimplePolynomials", "SimpleRandom", "SparseArrays", "Statistics"] 1869 | git-tree-sha1 = "f65caa24a622f985cc341de81d3f9744435d0d0f" 1870 | uuid = "55797a34-41de-5266-9ec1-32ac4eb504d3" 1871 | version = "0.8.6" 1872 | 1873 | [[deps.SimpleNonlinearSolve]] 1874 | deps = ["ADTypes", "ArrayInterface", "ConcreteStructs", "DiffEqBase", "FastClosures", "FiniteDiff", "ForwardDiff", "LinearAlgebra", "MaybeInplace", "PrecompileTools", "Reexport", "SciMLBase", "StaticArraysCore"] 1875 | git-tree-sha1 = "873a1bf90744acfa615e45cd5dddfd0ee89a094f" 1876 | uuid = "727e6d20-b764-4bd8-a329-72de5adea6c7" 1877 | version = "1.5.0" 1878 | 1879 | [deps.SimpleNonlinearSolve.extensions] 1880 | SimpleNonlinearSolveChainRulesCoreExt = "ChainRulesCore" 1881 | SimpleNonlinearSolvePolyesterForwardDiffExt = "PolyesterForwardDiff" 1882 | SimpleNonlinearSolveStaticArraysExt = "StaticArrays" 1883 | 1884 | [deps.SimpleNonlinearSolve.weakdeps] 1885 | ChainRulesCore = 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git-tree-sha1 = "5d7e3f4e11935503d3ecaf7186eac40602e7d231" 1910 | uuid = "699a6c99-e7fa-54fc-8d76-47d257e15c1d" 1911 | version = "0.9.4" 1912 | 1913 | [[deps.SimpleUnPack]] 1914 | git-tree-sha1 = "58e6353e72cde29b90a69527e56df1b5c3d8c437" 1915 | uuid = "ce78b400-467f-4804-87d8-8f486da07d0a" 1916 | version = "1.1.0" 1917 | 1918 | [[deps.Sixel]] 1919 | deps = ["Dates", "FileIO", "ImageCore", "IndirectArrays", "OffsetArrays", "REPL", "libsixel_jll"] 1920 | git-tree-sha1 = "2da10356e31327c7096832eb9cd86307a50b1eb6" 1921 | uuid = "45858cf5-a6b0-47a3-bbea-62219f50df47" 1922 | version = "0.1.3" 1923 | 1924 | [[deps.Sockets]] 1925 | uuid = "6462fe0b-24de-5631-8697-dd941f90decc" 1926 | 1927 | [[deps.SortingAlgorithms]] 1928 | deps = ["DataStructures"] 1929 | git-tree-sha1 = "66e0a8e672a0bdfca2c3f5937efb8538b9ddc085" 1930 | uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c" 1931 | version = "1.2.1" 1932 | 1933 | [[deps.SparseArrays]] 1934 | deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"] 1935 | uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" 1936 | version = "1.10.0" 1937 | 1938 | [[deps.SparseDiffTools]] 1939 | deps = ["ADTypes", "Adapt", "ArrayInterface", "Compat", "DataStructures", "FiniteDiff", "ForwardDiff", "Graphs", "LinearAlgebra", "PackageExtensionCompat", "Random", "Reexport", "SciMLOperators", "Setfield", "SparseArrays", "StaticArrayInterface", "StaticArrays", "Tricks", "UnPack", "VertexSafeGraphs"] 1940 | git-tree-sha1 = "a616ac46c38da60ac05cecf52064d44732edd05e" 1941 | uuid = "47a9eef4-7e08-11e9-0b38-333d64bd3804" 1942 | version = "2.17.0" 1943 | 1944 | [deps.SparseDiffTools.extensions] 1945 | SparseDiffToolsEnzymeExt = "Enzyme" 1946 | SparseDiffToolsPolyesterExt = "Polyester" 1947 | SparseDiffToolsPolyesterForwardDiffExt = "PolyesterForwardDiff" 1948 | SparseDiffToolsSymbolicsExt = "Symbolics" 1949 | SparseDiffToolsZygoteExt = "Zygote" 1950 | 1951 | [deps.SparseDiffTools.weakdeps] 1952 | Enzyme = "7da242da-08ed-463a-9acd-ee780be4f1d9" 1953 | Polyester = "f517fe37-dbe3-4b94-8317-1923a5111588" 1954 | PolyesterForwardDiff = "98d1487c-24ca-40b6-b7ab-df2af84e126b" 1955 | Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7" 1956 | Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" 1957 | 1958 | [[deps.Sparspak]] 1959 | deps = ["Libdl", "LinearAlgebra", "Logging", "OffsetArrays", "Printf", "SparseArrays", "Test"] 1960 | git-tree-sha1 = "342cf4b449c299d8d1ceaf00b7a49f4fbc7940e7" 1961 | uuid = "e56a9233-b9d6-4f03-8d0f-1825330902ac" 1962 | version = "0.3.9" 1963 | 1964 | [[deps.SpatialIndexing]] 1965 | git-tree-sha1 = "84efe17c77e1f2156a7a0d8a7c163c1e1c7bdaed" 1966 | uuid = "d4ead438-fe20-5cc5-a293-4fd39a41b74c" 1967 | version = "0.1.6" 1968 | 1969 | [[deps.SpecialFunctions]] 1970 | deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"] 1971 | git-tree-sha1 = "e2cfc4012a19088254b3950b85c3c1d8882d864d" 1972 | uuid = "276daf66-3868-5448-9aa4-cd146d93841b" 1973 | version = "2.3.1" 1974 | weakdeps = ["ChainRulesCore"] 1975 | 1976 | [deps.SpecialFunctions.extensions] 1977 | SpecialFunctionsChainRulesCoreExt = "ChainRulesCore" 1978 | 1979 | [[deps.StableHashTraits]] 1980 | deps = ["Compat", "PikaParser", "SHA", "Tables", "TupleTools"] 1981 | git-tree-sha1 = "10dc702932fe05a0e09b8e5955f00794ea1e8b12" 1982 | uuid = "c5dd0088-6c3f-4803-b00e-f31a60c170fa" 1983 | version = "1.1.8" 1984 | 1985 | [[deps.StackViews]] 1986 | deps = ["OffsetArrays"] 1987 | git-tree-sha1 = "46e589465204cd0c08b4bd97385e4fa79a0c770c" 1988 | uuid = "cae243ae-269e-4f55-b966-ac2d0dc13c15" 1989 | version = "0.1.1" 1990 | 1991 | [[deps.StateSpaceSets]] 1992 | deps = ["Distances", "LinearAlgebra", "Neighborhood", "Random", "StaticArraysCore", "Statistics"] 1993 | git-tree-sha1 = "fcfc8b9ce0f43bcb39e6bd8664855dc501a63886" 1994 | uuid = "40b095a5-5852-4c12-98c7-d43bf788e795" 1995 | version = "1.4.5" 1996 | 1997 | [[deps.Static]] 1998 | deps = ["IfElse"] 1999 | git-tree-sha1 = "d2fdac9ff3906e27f7a618d47b676941baa6c80c" 2000 | uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3" 2001 | version = "0.8.10" 2002 | 2003 | [[deps.StaticArrayInterface]] 2004 | deps = ["ArrayInterface", "Compat", "IfElse", "LinearAlgebra", "PrecompileTools", "Requires", "SparseArrays", "Static", "SuiteSparse"] 2005 | git-tree-sha1 = "5d66818a39bb04bf328e92bc933ec5b4ee88e436" 2006 | uuid = "0d7ed370-da01-4f52-bd93-41d350b8b718" 2007 | version = "1.5.0" 2008 | weakdeps = ["OffsetArrays", "StaticArrays"] 2009 | 2010 | [deps.StaticArrayInterface.extensions] 2011 | StaticArrayInterfaceOffsetArraysExt = "OffsetArrays" 2012 | StaticArrayInterfaceStaticArraysExt = "StaticArrays" 2013 | 2014 | [[deps.StaticArrays]] 2015 | deps = ["LinearAlgebra", "PrecompileTools", "Random", "StaticArraysCore"] 2016 | git-tree-sha1 = "bf074c045d3d5ffd956fa0a461da38a44685d6b2" 2017 | uuid = "90137ffa-7385-5640-81b9-e52037218182" 2018 | version = "1.9.3" 2019 | weakdeps = ["ChainRulesCore", "Statistics"] 2020 | 2021 | [deps.StaticArrays.extensions] 2022 | StaticArraysChainRulesCoreExt = "ChainRulesCore" 2023 | StaticArraysStatisticsExt = "Statistics" 2024 | 2025 | [[deps.StaticArraysCore]] 2026 | git-tree-sha1 = "36b3d696ce6366023a0ea192b4cd442268995a0d" 2027 | uuid = "1e83bf80-4336-4d27-bf5d-d5a4f845583c" 2028 | version = "1.4.2" 2029 | 2030 | [[deps.Statistics]] 2031 | deps = ["LinearAlgebra", "SparseArrays"] 2032 | uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" 2033 | version = "1.10.0" 2034 | 2035 | [[deps.StatsAPI]] 2036 | deps = ["LinearAlgebra"] 2037 | git-tree-sha1 = "1ff449ad350c9c4cbc756624d6f8a8c3ef56d3ed" 2038 | uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0" 2039 | version = "1.7.0" 2040 | 2041 | [[deps.StatsBase]] 2042 | deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"] 2043 | git-tree-sha1 = "1d77abd07f617c4868c33d4f5b9e1dbb2643c9cf" 2044 | uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" 2045 | version = 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git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" 2320 | uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" 2321 | version = "0.9.10+4" 2322 | 2323 | [[deps.Xorg_libpthread_stubs_jll]] 2324 | deps = ["Artifacts", "JLLWrappers", "Libdl"] 2325 | git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9" 2326 | uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" 2327 | version = "0.1.1+0" 2328 | 2329 | [[deps.Xorg_libxcb_jll]] 2330 | deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] 2331 | git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d" 2332 | uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" 2333 | version = "1.15.0+0" 2334 | 2335 | [[deps.Xorg_xtrans_jll]] 2336 | deps = ["Artifacts", "JLLWrappers", "Libdl"] 2337 | git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77" 2338 | uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" 2339 | version = "1.5.0+0" 2340 | 2341 | [[deps.Zlib_jll]] 2342 | deps = ["Libdl"] 2343 | uuid = "83775a58-1f1d-513f-b197-d71354ab007a" 2344 | version = "1.2.13+1" 2345 | 2346 | [[deps.isoband_jll]] 2347 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 2348 | git-tree-sha1 = "51b5eeb3f98367157a7a12a1fb0aa5328946c03c" 2349 | uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4" 2350 | version = "0.2.3+0" 2351 | 2352 | [[deps.libaom_jll]] 2353 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 2354 | git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" 2355 | uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" 2356 | version = "3.4.0+0" 2357 | 2358 | [[deps.libass_jll]] 2359 | deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] 2360 | git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" 2361 | uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" 2362 | version = "0.15.1+0" 2363 | 2364 | [[deps.libblastrampoline_jll]] 2365 | deps = ["Artifacts", "Libdl"] 2366 | uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" 2367 | version = "5.8.0+1" 2368 | 2369 | [[deps.libfdk_aac_jll]] 2370 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 2371 | git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" 2372 | uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" 2373 | version = "2.0.2+0" 2374 | 2375 | [[deps.libpng_jll]] 2376 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"] 2377 | git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4" 2378 | uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" 2379 | version = "1.6.43+1" 2380 | 2381 | [[deps.libsixel_jll]] 2382 | deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"] 2383 | git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c" 2384 | uuid = "075b6546-f08a-558a-be8f-8157d0f608a5" 2385 | version = "1.10.3+0" 2386 | 2387 | [[deps.libvorbis_jll]] 2388 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] 2389 | git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" 2390 | uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" 2391 | version = "1.3.7+1" 2392 | 2393 | [[deps.nghttp2_jll]] 2394 | deps = ["Artifacts", "Libdl"] 2395 | uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" 2396 | version = "1.52.0+1" 2397 | 2398 | [[deps.p7zip_jll]] 2399 | deps = ["Artifacts", "Libdl"] 2400 | uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" 2401 | version = "17.4.0+2" 2402 | 2403 | [[deps.x264_jll]] 2404 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 2405 | git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" 2406 | uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" 2407 | version = "2021.5.5+0" 2408 | 2409 | [[deps.x265_jll]] 2410 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 2411 | git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" 2412 | uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" 2413 | version = "3.5.0+0" 2414 | -------------------------------------------------------------------------------- /resources/Chaos/Project.toml: -------------------------------------------------------------------------------- 1 | [deps] 2 | ChaosTools = "608a59af-f2a3-5ad4-90b4-758bdf3122a7" 3 | DynamicalSystems = "61744808-ddfa-5f27-97ff-6e42cc95d634" 4 | GLMakie = "e9467ef8-e4e7-5192-8a1a-b1aee30e663a" 5 | Makie = "ee78f7c6-11fb-53f2-987a-cfe4a2b5a57a" 6 | OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" 7 | 8 | [compat] 9 | Makie = "0.19" 10 | -------------------------------------------------------------------------------- /resources/Chaos/README.md: -------------------------------------------------------------------------------- 1 | # Dynamical Systems and Chaos Theory 2 | 3 | Main resources: 4 | 5 | - [The Chaos Hypertextbook](https://hypertextbook.com/chaos) 6 | - [Stephen Strogatz's lecture (and book)](https://www.youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V) 7 | 8 | Additional resources (Thanks to Carter Hinsley): 9 | 10 | - [Shadowing lemma](http://www.scholarpedia.org/article/Shadowing_lemma_for_flows) 11 | - [Numerical orbits of chaotic processes represent true orbits](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-19/issue-2/Numerical-orbits-of-chaotic-processes-represent-true-orbits/bams/1183554729.full) 12 | - [Shadowing Theorem](https://mathworld.wolfram.com/ShadowingTheorem.html) 13 | -------------------------------------------------------------------------------- /resources/Chaos/chaos.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Chaos/chaos.pdf -------------------------------------------------------------------------------- /resources/Chaos/cobweb.jl: -------------------------------------------------------------------------------- 1 | using DynamicalSystems 2 | using OrdinaryDiffEq 3 | using GLMakie 4 | 5 | 6 | 7 | # the second range is a convenience for intermittency example of logistic 8 | rrange = 1:0.001:4.0 9 | # rrange = (rc = 1 + sqrt(8); [rc, rc - 1e-5, rc - 1e-3]) 10 | 11 | lo = Systems.logistic(0.4; r = rrange[1]) 12 | fig1 = interactive_cobweb(lo, rrange, 5) 13 | 14 | fig2 = interactive_orbitdiagram(lo, 1, 0.0, 4.0; u0 = [0.5]) 15 | 16 | 17 | 18 | lorenz = Systems.lorenz() 19 | diffeq = (alg = Vern9(), abstol = 1e-9, reltol = 1e-9) 20 | 21 | ds = CoupledODEs(lorenz, diffeq) 22 | 23 | 24 | u0s = [ 25 | [0.0,1.0,1.0], 26 | [0.0,1.0,1.01], 27 | ] 28 | 29 | fig3, dsob = interactive_trajectory_timeseries( 30 | lorenz, 31 | [1,2,3], 32 | u0s; 33 | statespace_axis = true, Δt = 0.01, 34 | tail = 1000, 35 | figure = (size = (1100, 650),) 36 | ) 37 | 38 | fig3 39 | 40 | u0 = [10 .*rand(3) for i in 1:10] 41 | 42 | trs = [trajectory(ds, 50000, u0)[1][:, SVector(1,2,3)] for u0 ∈ u0s] 43 | j = 2 # the dimension of the plane 44 | 45 | interactive_poincaresos_scan(trs, j; 46 | linekw = (transparency = true,), 47 | scatterkw = (markersize=3,) 48 | ) 49 | 50 | tent = Systems.tentmap(0.2,1.0) 51 | 52 | function tent_rule(u, p, n) 53 | x = u[1] 54 | μ = p[1] 55 | y = x < 0.5 ? μ*x : μ*(1-x) 56 | return SVector(y) 57 | end 58 | 59 | tent = DeterministicIteratedMap(tent_rule, 0.25, [2.0]) 60 | 61 | fig4 = interactive_cobweb(tent, 0.0:0.01:4.0, 5) 62 | -------------------------------------------------------------------------------- /resources/Differential Geometry/README.md: -------------------------------------------------------------------------------- 1 | 2 | # Differential Geometry 3 | 4 | ## Manifolds 5 | 6 | - http://mysite.science.uottawa.ca/rossmann/Differential%20Geometry%20book_files/Diffgeo.pdf 7 | - https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf 8 | - http://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf 9 | - https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS 10 | 11 | ## Geometric Deep Learning 12 | 13 | - https://arxiv.org/abs/1611.08097 14 | - https://arxiv.org/abs/2104.13478 15 | - https://geometricdeeplearning.com/ 16 | 17 | -------------------------------------------------------------------------------- /resources/Differential Geometry/curvature.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Differential Geometry/curvature.pdf -------------------------------------------------------------------------------- /resources/Differential Geometry/frenet-serret.ggb: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Differential Geometry/frenet-serret.ggb -------------------------------------------------------------------------------- /resources/Differential Geometry/manifolds.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Differential Geometry/manifolds.pdf -------------------------------------------------------------------------------- /resources/Differential Geometry/parallel-transport.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Differential Geometry/parallel-transport.pdf -------------------------------------------------------------------------------- /resources/Functional Data Analysis/README.md: -------------------------------------------------------------------------------- 1 | # Functional Data Analysis 2 | 3 | Main Resource: 4 | https://arxiv.org/abs/2312.05523 5 | 6 | https://github.com/GAA-UAM/scikit-fda -------------------------------------------------------------------------------- /resources/Functional Data Analysis/fda.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Functional Data Analysis/fda.pdf -------------------------------------------------------------------------------- /resources/Information Geometry/README.md: -------------------------------------------------------------------------------- 1 | # Information Geometry 2 | 3 | ## Main Resources: 4 | - https://www.mdpi.com/1099-4300/22/10/1100 5 | - https://math.ucr.edu/home/baez/information/ 6 | 7 | ## Additional Resources: 8 | See the resources for Differential Geometry -------------------------------------------------------------------------------- /resources/Information Geometry/information-geometry.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Information Geometry/information-geometry.pdf -------------------------------------------------------------------------------- /resources/Johnson-Lindenstrauss Lemma/README.md: -------------------------------------------------------------------------------- 1 | # Johnson-Lindenstrauss Lemma 2 | 3 | Main resources: 4 | - https://arxiv.org/abs/2103.00564 5 | 6 | Additional resources: 7 | - https://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf 8 | - https://cgi.di.uoa.gr/~optas/papers/jl.pdf 9 | - https://users.ics.aalto.fi/ella/publications/randproj_kdd.pdf 10 | - https://www.cs.cornell.edu/jeh/book.pdf 11 | - https://www.cs.princeton.edu/~chazelle/pubs/FJLT-sicomp09.pdf 12 | - https://web.archive.org/web/20220802090638id_/http://proceedings.mlr.press/v134/li21c/li21c.pdf 13 | - https://arxiv.org/abs/1012.1577 14 | https://arxiv.org/abs/2006.04768 15 | - https://colab.research.google.com/github/TevenLeScao/blog/blob/master/_notebooks/2020-06-18-JL-Lemma-%2B-Linformer.ipynb 16 | - https://github.com/dell/jlt 17 | -------------------------------------------------------------------------------- /resources/Johnson-Lindenstrauss Lemma/johnsonlindenstrauss.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Johnson-Lindenstrauss Lemma/johnsonlindenstrauss.pdf -------------------------------------------------------------------------------- /resources/Morse Theory/Eleuther.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Morse Theory/Eleuther.pdf -------------------------------------------------------------------------------- /resources/Morse Theory/README.md: -------------------------------------------------------------------------------- 1 | # Morse Theory 2 | 3 | Resources: 4 | * https://www.maths.ed.ac.uk/~v1ranick/papers/milnmors.pdf 5 | * https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/morse-theory.pdf 6 | * http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf -------------------------------------------------------------------------------- /resources/Multilinear Algebra/README.md: -------------------------------------------------------------------------------- 1 | # Multilinear Algebra 2 | 3 | Main resources: 4 | * https://metaphor.ethz.ch/x/2021/fs/401-0164-00L/Script/multilinear_algebra.pdf 5 | 6 | Additional Resources: 7 | * https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG 8 | * https://transformer-circuits.pub/2021/framework/index.html 9 | * https://haskellformaths.blogspot.com/2011/02/tensor-products-of-vector-spaces-part-1.html -------------------------------------------------------------------------------- /resources/Multilinear Algebra/multilinear-alegbra.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Multilinear Algebra/multilinear-alegbra.pdf -------------------------------------------------------------------------------- /resources/Multilinear Algebra/tensors (english version).pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Multilinear Algebra/tensors (english version).pdf -------------------------------------------------------------------------------- /resources/ODEs and State-Space Models/AntsaklisMichel_Primer_up to Chapter 6.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/ODEs and State-Space Models/AntsaklisMichel_Primer_up to Chapter 6.pdf -------------------------------------------------------------------------------- /resources/ODEs and State-Space Models/README.md: -------------------------------------------------------------------------------- 1 | # ODEs and State-Space Models 2 | 3 | Main resources: 4 | - [https://www.jirka.org/diffyqs/diffyqs.pdf]() 5 | - [https://users.math.msu.edu/users/gnagy/teaching/ode.pdf]() 6 | - [https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf]() 7 | - [https://arxiv.org/pdf/1806.07366.pdf]() 8 | - [https://arxiv.org/abs/2111.00396]() 9 | - [https://arxiv.org/abs/2312.00752]() -------------------------------------------------------------------------------- /resources/ODEs and State-Space Models/diffeq.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/ODEs and State-Space Models/diffeq.pdf -------------------------------------------------------------------------------- /resources/ODEs and State-Space Models/ssms.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/ODEs and State-Space Models/ssms.pdf -------------------------------------------------------------------------------- /resources/Papers/Geometric Algebra Transformers/GATr.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Papers/Geometric Algebra Transformers/GATr.pdf -------------------------------------------------------------------------------- /resources/Papers/Geometric Algebra Transformers/README.md: -------------------------------------------------------------------------------- 1 | # Geometric Algebra Transformers 2 | 3 | Main resources: 4 | - https://arxiv.org/abs/2305.18415 5 | - https://github.com/Qualcomm-AI-research/geometric-algebra-transformer 6 | 7 | Additional resources 8 | - https://www.youtube.com/watch?v=60z_hpEAtD8 9 | - https://www.youtube.com/watch?v=0i3ocLhbxJ4 10 | - https://galgebra.readthedocs.io/en/latest/galgebra_guide.html 11 | - https://marctenbosch.com/quaternions/ 12 | - https://www.youtube.com/watch?v=60z_hpEAtD8 13 | - https://www.youtube.com/watch?v=0i3ocLhbxJ4 14 | - https://projectivegeometricalgebra.org/projgeomalg.pdf 15 | - https://en.wikipedia.org/wiki/Plane-based_geometric_algebra 16 | - https://en.wikipedia.org/wiki/Conformal_geometric_algebra 17 | - https://math.stackexchange.com/questions/4590369/is-projective-geometric-algebra-a-strict-subset-of-conformal-geometric-algebra -------------------------------------------------------------------------------- /resources/Papers/Harmonic Networks/Harmonic_Neural_Networks.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Papers/Harmonic Networks/Harmonic_Neural_Networks.pdf -------------------------------------------------------------------------------- /resources/Papers/Harmonic Networks/README.md: -------------------------------------------------------------------------------- 1 | # Harmonic Networks 2 | 3 | Main Resources: 4 | * https://arxiv.org/abs/1612.04642 5 | 6 | Additional Resources: 7 | * https://proceedings.neurips.cc/paper/2021/file/f18224a1adfb7b3dbff668c9b655a35a-Paper.pdf -------------------------------------------------------------------------------- /resources/Papers/Kolmogorov Arnold Networks/README.md: -------------------------------------------------------------------------------- 1 | # Kolmogorov Arnold Networks 2 | 3 | Main resources: 4 | - https://arxiv.org/abs/2404.19756 5 | 6 | Additional resources: 7 | - https://discord.com/channels/729741769192767510/1089231591785639946/1250099202638479410 8 | -------------------------------------------------------------------------------- /resources/Papers/Neural Tangent Kernels/README.md: -------------------------------------------------------------------------------- 1 | # Neural Tangent Kernels 2 | 3 | Main Reading 4 | - https://arxiv.org/abs/1806.07572 5 | 6 | Additional resources 7 | - https://link.springer.com/chapter/10.1007/978-1-4612-0745-0_2 8 | - https://cdn.discordapp.com/attachments/1089231591785639946/1103399853272350720/neal1996.pdf 9 | - https://www.cs.toronto.edu/~duvenaud/cookbook/ 10 | -------------------------------------------------------------------------------- /resources/Papers/Neural Tangent Kernels/ntkoverview.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Papers/Neural Tangent Kernels/ntkoverview.pdf -------------------------------------------------------------------------------- /resources/Random Matrix Theory/README.md: -------------------------------------------------------------------------------- 1 | # Random Matrix Theory 2 | 3 | Main resources: 4 | * https://cdn.discordapp.com/attachments/1089231591785639946/1113876605396340906/Potters_Bouchaud_-_A_First_Course_in_Random_Matrix_Theory-Cambridge_University_Press_2020.pdf 5 | 6 | Additional Resources: 7 | * https://www.youtube.com/watch?v=Y4mnlIvVJEs 8 | * https://arxiv.org/abs/1712.07903.pdf 9 | * https://arxiv.org/abs/2102.06740 10 | * https://arxiv.org/abs/1901.08276 11 | * https://arxiv.org/abs/1810.01075 12 | 13 | Check out the Julia notebooks. They have some nice visualizations. You'll need to install Julia and the Pluto package to run them. 14 | 15 | -------------------------------------------------------------------------------- /resources/Random Matrix Theory/RMT-1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Random Matrix Theory/RMT-1.pdf -------------------------------------------------------------------------------- /resources/Random Matrix Theory/RMT-2.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Random Matrix Theory/RMT-2.pdf -------------------------------------------------------------------------------- /resources/Random Matrix Theory/RMT-3.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Random Matrix Theory/RMT-3.pdf -------------------------------------------------------------------------------- /resources/Random Matrix Theory/RMT-chap1.jl: -------------------------------------------------------------------------------- 1 | ### A Pluto.jl notebook ### 2 | # v0.19.27 3 | 4 | using Markdown 5 | using InteractiveUtils 6 | 7 | # This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). 8 | macro bind(def, element) 9 | quote 10 | local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end 11 | local el = $(esc(element)) 12 | global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) 13 | el 14 | end 15 | end 16 | 17 | # ╔═╡ 8770247a-fa57-11ed-3bcd-f548a0b9a3d4 18 | begin 19 | using LinearAlgebra 20 | using Plots 21 | using StatsPlots 22 | using Distributions 23 | using PlutoUI 24 | using LaTeXStrings 25 | using Random 26 | end 27 | 28 | # ╔═╡ b5cea2b2-30c0-4641-bca6-82489344ae5e 29 | """ 30 | random_orthogonal_matrix(n) 31 | 32 | Generate a random orthogonal matrix of size `n`. 33 | 34 | The generated matrices are uniformly distributed over the set of real orthogonal matrices. 35 | """ 36 | function random_orthogonal_matrix(n::Integer) 37 | A = randn(n,n) 38 | return qr(A).Q 39 | end 40 | 41 | # ╔═╡ e303c065-b61f-43a3-97eb-23004bd52f5f 42 | """ 43 | random_symmetric_spectral(λ) 44 | 45 | Generate a random symmetric matrix with eigenvalues `λ`. 46 | """ 47 | function random_symmetric_spectral(λ) 48 | Q = random_orthogonal_matrix(length(λ)) 49 | return Symmetric(Q * Diagonal(λ) * Q') 50 | end 51 | 52 | # ╔═╡ bb1e77cc-da9e-4be5-9f96-af7b2761e7a1 53 | function random_symmetric_spectral(d::UnivariateDistribution, n) 54 | λ = rand(d, n) 55 | return random_symmetric_spectral(λ) 56 | end 57 | 58 | # ╔═╡ a1594bc3-aba9-423d-a2e6-208b8f9b4783 59 | md""" 60 | ## Exercise 1.1.1 Instability of eigenvalues of non-symmetric matrices 61 | """ 62 | 63 | # ╔═╡ f470f591-d4cc-4550-b152-e3c3d9723e90 64 | function M₀(n) 65 | A = zeros(n,n) 66 | for i in 1:n-1 67 | A[i,i+1] = 2.0 68 | end 69 | return A 70 | end 71 | 72 | # ╔═╡ 443ee9df-dfb9-4ba3-99c6-110295ef85e6 73 | function M₁(n) 74 | A = M₀(n) 75 | A[n,1] = 0.5^(n-1) 76 | return A 77 | end 78 | 79 | # ╔═╡ d84f32be-6a8f-45ed-9a34-0686922cbc4c 80 | md""" 81 | To show that ``M_0^N = 0``, one can note that it is essentially the adjacency matrix of a directed path of length ``N-1``, hence the equality holds as there are no walks of length larger than ``N-1`` in this graph. 82 | 83 | A more direct approach is to compute the powers explicitly and find that ``M_0^k`` has a ``k``-th upper diagonal filled with ``2^k`` with zeroes everywhere else, and that there are only ``N-1`` upper diagonals. 84 | """ 85 | 86 | # ╔═╡ e2e93f77-5202-4a0d-a049-d40df0472634 87 | all(x -> x ≈0.0, eigvals(M₀(100))) 88 | 89 | # ╔═╡ 5b670729-1fd0-4e6f-9fd4-563c61f6a225 90 | @bind N Slider(1:100, default=50, show_value=true) 91 | 92 | # ╔═╡ 0e18c236-8017-48cd-a658-dedcea335746 93 | begin 94 | A_1 = M₀(N) 95 | O = random_orthogonal_matrix(N) 96 | λ_0 = eigvals(O * A_1 * O') 97 | scatter([0+0im], label=L"\sigma (M_0)") 98 | scatter!(λ_0, label = L"\sigma (O M_0 O')") 99 | end 100 | 101 | # ╔═╡ d1251e6e-ed9d-40fc-9869-af09b140c932 102 | begin 103 | A_2 = M₁(N) 104 | λ_1 = eigvals(O * A_2 * O') 105 | scatter(λ_0, label=L"\sigma(O M_0 O')") 106 | scatter!(λ_1, label=L"\sigma (O M_1 O')") 107 | end 108 | 109 | # ╔═╡ 53172f2f-0e35-4d67-b8f1-8c9de407ce50 110 | md""" 111 | ## Exercise 1.2.1 Gershgorin and Perron-Frobenius 112 | """ 113 | 114 | # ╔═╡ cce96a7e-a401-4cdd-abb4-65d20daeedff 115 | md""" 116 | By the Perron-Frobenius theorem, the dominant eigenvalue ``\lambda_\max`` of a positive matric ``A`` is bounded above by ``\max_i \sum_j A_{ij}``. 117 | 118 | By the Gershgorin theorem, we already know that the spectral radius is smaller than ``\max_i |A_{ii}| + R_i = \max_i \sum_j |A_{ij}|``, which yields the desired bound under the hypotheses of the Perron-Frobenius theorem. 119 | """ 120 | 121 | # ╔═╡ fd7c00fa-5fa1-46b0-8749-e82bd55e39c3 122 | md""" 123 | ## Exercise 1.2.2 Sherman-Morrison 124 | By multiplying the right hand side with ``(A + uv')`` we obtain 125 | 126 | ```math 127 | \begin{align*} 128 | &\left( A^{-1} - \frac{A^{-1}uv' A^{-1}}{1 + v' A^{-1} u}\right) (A + uv') \\ 129 | &= I + A^{-1}uv' - \frac{A^{-1}uv'}{1 + v' A^{-1} u} + \frac{A^{-1}u(v' A^{-1}u)v'}{1 + v' A^{-1} u} \\ 130 | &= I + A^{-1}uv' - \frac{(1+v'A^{-1}u)A^{-1}uv'}{1+ v'A^{-1}u} \\ 131 | &= I 132 | 133 | \end{align*} 134 | ``` 135 | """ 136 | 137 | # ╔═╡ 4f2d645b-78d6-439e-8964-9919ffb3c217 138 | md""" 139 | ## Exercise 1.2.3 Combining Schur and Sherman-Morrison 140 | 141 | Let $n=1$ in Schur's complement formula. Use the Sherman-Morrisson formula to show in that case that 142 | ```math 143 | Q_{22} = (M_{22})^{-1} + \frac{(M_{22})^{-1} M_{21}M_{12}(M_{22})^{-1}}{M_{11} + M_{12}(M_{22})^{-1} M_{21}} 144 | ``` 145 | 146 | By definition, we have by writing the product of $M$ and $M^{-1}$ block by block that 147 | ```math 148 | \begin{align} 149 | M_{11}Q_{11} + M_{12}Q_{21} &= 1, \\ 150 | M_{21}Q_{12} + M_{22}Q_{22} &= I, \\ 151 | M_{11}Q_{12} + M_{12}Q_{22} &= 0, \\ 152 | M_{21}Q_{11} + M_{22}Q_{21} &= 0, 153 | \end{align} 154 | ``` 155 | from which we obtain after some manipulations 156 | ```math 157 | Q_{12} = -M_{12}Q_{22} / M_{11}. 158 | ``` 159 | We can inject this in the second equation to obtain 160 | ```math 161 | \left(M_{22} - \frac{M_{21}M_{12}}{M_{11}} \right)Q_{22} = I, 162 | ``` 163 | i.e. $Q_{22} = \left(M_{22} - \frac{M_{21}M_{12}}{M_{11}} \right)^{-1}$, so by the Sherman-Morisson formula 164 | ```math 165 | \begin{align} 166 | Q_{22} &= (M_{22})^{-1} + \frac{(M_{22})^{-1} \frac{M_{21} M_{12}}{M_{11} (M_{22})^{-1}} }{1 + \frac{1}{M_{11}} M_{12}(M_{22})^{-1} M_{21}}, \\ 167 | &= (M_{22})^{-1} + \frac{(M_{22})^{-1} M_{21} M_{12} (M_{22})^{-1} }{M_{11} + M_{12}(M_{22})^{-1} M_{21}}, 168 | \end{align} 169 | ``` 170 | """ 171 | 172 | # ╔═╡ e57dcd2f-aac1-4e30-9331-d6b8b44371d6 173 | md""" 174 | ## Exercise 1.2.4 Random Matrices 175 | 1. Let $M$ be a random real symmetric orthogonal matrix. Show that all its eigenvalues are equal to $\pm 1$. 176 | 177 | This is trivial, since as an orthogonal matrix, all the eigenvalues of $M$ have modulus equal to one, and as a symmetric matrix, its eigenvalues are real, so they must be either $1$ or $-1$. 178 | 179 | 2. Let $X$ be a Wigner matrix, i.e. a $N\times N$ real symmetric matrix whose diagonal and upper triangular entries are i.i.d. $\mathcal{N}(0,\sigma^2/N)$. Such a matrix can be generated as $X = \sigma(H'+H)/\sqrt(2N)$, where $H$ is a non-symmetric matrix with entries i.i.d. $\mathcal{N}(0,1)$ 180 | 181 | 182 | 183 | 3. The matrix $P_+$ is defined as $P_+ = \frac{1}{2}(M + I)$ Convince yourself that $P_+$ is the projector onto the eigenspace of $M$ with eigenvalue $1$. Explain the effect of the matrix $P_+$ on eigenvectors of $M$. 184 | 185 | This can easily be seen by taking the eigenvalue decomposition of $M$: 186 | ```math 187 | M = O \begin{bmatrix} I_+ & \\ & -I_- \end{bmatrix} O', 188 | ``` 189 | where $O$ is an orthonormal basis of eigenvectors, and $I_+$ and $I_-$ are identity matrices of appropriate size. Then 190 | ```math 191 | \frac{1}{2}(M+I) = \frac{1}{2} O \begin{bmatrix} 2I_+ & \\ & 0 \end{bmatrix} O' = O \begin{bmatrix} I_+ & \\ & 0 \end{bmatrix} O' 192 | ``` 193 | which is the orthogonal projector onto the eigenspace associated with the eigenvalue $1$. 194 | """ 195 | 196 | # ╔═╡ a8f612b1-fdcb-463e-add4-972b7f178b08 197 | function random_wigner(n, σ=1.0) 198 | H = randn(n,n) 199 | return Symmetric((H+H') * σ / sqrt(2*n)) 200 | end 201 | 202 | # ╔═╡ 2bed026f-0e69-4d5f-b624-c4e2f9e5adf5 203 | function random_symmetric_orthogonal(n) 204 | W = random_wigner(n) 205 | eig = eigen(W) 206 | λ = Diagonal(sign.(eig.values)) 207 | return Symmetric(eig.vectors * λ * eig.vectors') 208 | end 209 | 210 | # ╔═╡ 3774764c-b6fe-4749-9b9a-fd245519ecea 211 | function orthogonal_projector(M) 212 | return (M+I)/2 213 | end 214 | 215 | # ╔═╡ 08d8bceb-c4d0-4464-bab1-5004361f74a0 216 | M = random_symmetric_orthogonal(10) 217 | 218 | # ╔═╡ ae2442b3-a01d-4356-b195-7891b1af8766 219 | eigvals(M) 220 | 221 | # ╔═╡ ff292549-4418-4b88-bf1e-60066973790c 222 | M * M' 223 | 224 | # ╔═╡ 2b6961f7-1cab-4f17-8454-2490c11065dc 225 | random_wigner(10) 226 | 227 | # ╔═╡ 582a9ed3-b6f7-496b-a121-67dd8733f6e6 228 | function perturb_M!(E, M, σ) 229 | n = size(M,1) 230 | X = random_wigner(n, σ) 231 | E .= M .+ X 232 | end 233 | 234 | # ╔═╡ a99f0b12-b658-460e-946d-8f4e070b8130 235 | md""" 236 | (a) Plot a histogram of the eigenvalues of $E$, for a single sample first, and then for many samples (say 100). 237 | """ 238 | 239 | # ╔═╡ dd01eea5-75f4-4e0f-be4c-b261b76824c1 240 | begin 241 | local n = 1000 242 | local M = random_symmetric_orthogonal(n) 243 | E = zeros(n,n) 244 | λs = Array{Float64,2}(undef, n, 3) 245 | σs = [0.1, 1.0, 10.0] 246 | for i in 1:3 247 | perturb_M!(E,M,σs[i]) 248 | λs[:,i] .= eigvals(E) 249 | end 250 | end 251 | 252 | # ╔═╡ 3ab0493e-fc39-4b88-9428-7914c441b9c7 253 | begin 254 | local p1 = histogram(λs[:,1]) 255 | local p2 = histogram(λs[:,2]) 256 | local p3 = histogram(λs[:,3]) 257 | plot(p1, p2, p3) 258 | end 259 | 260 | # ╔═╡ 6956cec8-12ef-43b9-b437-d22ff24f9b7f 261 | begin 262 | local n = 1000 263 | local M = random_symmetric_orthogonal(n) 264 | #E_rep = zeros(n,n) 265 | λs_rep = Array{Float64,3}(undef, n, 100, 3) 266 | #σs = [0.1, 1.0, 10.0] 267 | for i in 1:3, j in 1:100 268 | perturb_M!(E,M,σs[i]) 269 | λs_rep[:,j,i] .= eigvals(E) 270 | end 271 | end 272 | 273 | # ╔═╡ 7edcbbea-88be-4362-90ec-405d421de796 274 | begin 275 | local p1 = histogram(vec(λs_rep[:,:,1])) 276 | local p2 = histogram(vec(λs_rep[:,:,2])) 277 | local p3 = histogram(vec(λs_rep[:,:,3])) 278 | plot(p1, p2, p3) 279 | end 280 | 281 | # ╔═╡ c7a0c42a-74f4-4052-bc5b-df47db1090f1 282 | md""" 283 | (c) For every normalized eigenvector $v_i$ of $E$, compute the norm of the vector $P_+ v_i$. For a single sample, do a scatter plot of $\|P_+ v_i\|^2$ vs $\lambda_i$. Turn your scatter plot into an approximate conditional expectation value (using a histogram) including data from many samples. 284 | """ 285 | 286 | # ╔═╡ 810c971a-81c4-447e-8097-4f48ee78018c 287 | begin 288 | local n = 1000 289 | local M = random_symmetric_orthogonal(n) 290 | P = orthogonal_projector(M) 291 | local E = zeros(n,n) 292 | V = Array{Float64,3}(undef, n,n, 3) 293 | Λ = Array{Float64,2}(undef, n,3) 294 | for i in 1:3 295 | perturb_M!(E,M,σs[i]) 296 | local eig = eigen(E) 297 | V[:,:,i] .= P * eig.vectors 298 | Λ[:,i] .= eig.values 299 | end 300 | end 301 | 302 | # ╔═╡ 1b384db3-1050-4774-882d-b842e6de387f 303 | norm.(eachcol(V[:,:,1])) 304 | 305 | # ╔═╡ 8d589c74-5fe3-43d3-a4f9-cc7a5c8af335 306 | begin 307 | local p1 = scatter(Λ[:,1], norm.(eachcol(V[:,:,1]))) 308 | local p2 = scatter(Λ[:,2], norm.(eachcol(V[:,:,2]))) 309 | local p3 = scatter(Λ[:,3], norm.(eachcol(V[:,:,3]))) 310 | plot(p1, p2, p3) 311 | end 312 | 313 | # ╔═╡ 24e0d796-7c48-4ba6-879a-d95f082417e5 314 | md""" 315 | (e) Show numerically that the eigenvalues of $E$ are not i.i.d. 316 | - For each sample $E$, rank its eigenvalues $\lambda_1 < \dots < \lambda_N$. Consider the eigenvalue spacing $s_k = \lambda_k - \lambda_{k-1}$ for eigenvalues in the bulk ($0.2N < k < 0.3N$ and $0.7N < k < 0.8N$). 317 | - Make a histogram of $\{s_k\}$ including data from 100 samples. 318 | - Make 100 pseudo-IID samples: mix eigenvalues for 100 different samples and randomly choose $N$ for the $100N$ possibilities, do not choose the same eigenvalue twice for a given pseudo-IID sample. 319 | - For each pseudo-IID sample, compute $s_k$ in the bulk and make a histogram of the values using data from all 100 pseudo-IID samples. 320 | - (Bonus) Try to fit an exponential distribution to these two histograms. The IID case should be well fitted by the exponential, but not the original data. 321 | """ 322 | 323 | # ╔═╡ 6b01ed6e-3dc1-4a0d-94ba-c4ec7637dc40 324 | begin 325 | local n = 1000 326 | bulk_idx = [round(Int,0.2*n):round(Int,0.8*n); round(Int,0.7*n):round(Int,0.8*n)] 327 | s = [λs_rep[k,i,j]-λs_rep[k-1,i,j] for k in bulk_idx, i in axes(λs_rep, 2), j in axes(λs_rep,3)] 328 | end 329 | 330 | # ╔═╡ f7a3e318-20f6-4f21-ac29-bec6d850aad9 331 | begin 332 | local p1 = scatterhist(vec(s[:,:,1]), normalize=true) 333 | local p2 = scatterhist(vec(s[:,:,2]), normalize=true) 334 | local p3 = scatterhist(vec(s[:,:,3]), normalize=true) 335 | plot(p1, p2, p3) 336 | end 337 | 338 | # ╔═╡ 3dbc7984-35a9-4a2b-bc7e-f91e78be18b8 339 | function pseudo_iid_sample(λ, n) 340 | p = randperm(length(λ)) 341 | return λ[p[1:n]] 342 | end 343 | 344 | # ╔═╡ ad15b839-ec36-427c-965c-dd92ddee4160 345 | begin 346 | λ_iid = similar(λs_rep) 347 | for j in axes(λ_iid,3), i in axes(λ_iid, 2) 348 | λ_iid[:,i,j] .= pseudo_iid_sample(view(λs_rep, :,:,j), size(λ_iid,1)) 349 | sort!(view(λ_iid, :,i,j)) 350 | end 351 | end 352 | 353 | # ╔═╡ 9c011b4f-31bc-450f-be2d-4dc54a72acb9 354 | s_iid = [λ_iid[k,i,j]-λ_iid[k-1,i,j] for k in bulk_idx, i in axes(λ_iid, 2), j in axes(λ_iid,3)] 355 | 356 | # ╔═╡ f3ff0674-e706-4834-803e-a5dc7e49cbc6 357 | begin 358 | #=local p1 = scatterhist(vec(s[:,:,1]), normalize=true, 359 | label = "Original Data", 360 | markersize=3) 361 | scatterhist!(p1, vec(s_iid[:,:,1]), normalize=true, 362 | label = "Pseudo IID Resample", 363 | markersize=3) 364 | γ_rep = 1/mean(s[:,:,1]) 365 | γ_iid = 1/mean(s_iid[:,:,1]) 366 | plot!(p1, 0.0:0.005:0.13, t -> γ_rep*exp(-γ_rep*t), label="Exponential fit") 367 | plot!(p1, 0.0:0.005:0.35, t -> γ_iid*exp(-γ_iid*t), label="Exponential fit") 368 | local p2 = scatterhist(vec(s[:,:,1]), normalize=true, 369 | label = "Original Data", 370 | markersize=3) 371 | scatterhist!(p2, vec(s_iid[:,:,2]), normalize=true, 372 | label = "Pseudo IID Resample", 373 | markersize=3) 374 | γ_rep = 1/mean(s[:,:,2]) 375 | γ_iid = 1/mean(s_iid[:,:,2]) 376 | plot!(p2, 0.0:0.005:0.13, t -> γ_rep*exp(-γ_rep*t), label="Exponential fit") 377 | plot!(p2, 0.0:0.005:0.35, t -> γ_iid*exp(-γ_iid*t), label="Exponential fit") 378 | =# 379 | local p3 = scatterhist(vec(s[:,:,3]), normalize=true, 380 | label = "Original Data", 381 | markersize=3) 382 | scatterhist!(p3, vec(s_iid[:,:,3]), normalize=true, 383 | label = "Pseudo IID Resample", 384 | markersize=3) 385 | γ_rep = 1/mean(s[:,:,3]) 386 | γ_iid = 1/mean(s_iid[:,:,3]) 387 | plot!(p3, 0.0:0.005:0.13, t -> γ_rep*exp(-γ_rep*t), label="Exponential fit") 388 | plot!(p3, 0.0:0.005:0.35, t -> γ_iid*exp(-γ_iid*t), label="Exponential fit") 389 | plot(p3) 390 | end 391 | 392 | # ╔═╡ 7b19a04b-11dd-4341-b4b1-8cfaf6152ed3 393 | wigner_density(x, σ) = abs(x) > 2σ ? 0.0 : sqrt(4σ^2 - x^2)/(2π*σ^2) 394 | 395 | # ╔═╡ 669bc90c-685f-4e5f-b91d-f1746086de9d 396 | begin 397 | local x = -2.0:0.01:2.0 398 | plot(x, wigner_density.(x,1), label="") 399 | end 400 | 401 | # ╔═╡ 00000000-0000-0000-0000-000000000001 402 | PLUTO_PROJECT_TOML_CONTENTS = """ 403 | [deps] 404 | Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" 405 | LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" 406 | LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" 407 | Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" 408 | PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" 409 | Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" 410 | StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd" 411 | 412 | [compat] 413 | Distributions = "~0.25.94" 414 | LaTeXStrings = "~1.3.0" 415 | Plots = "~1.38.12" 416 | PlutoUI = "~0.7.51" 417 | StatsPlots = "~0.15.5" 418 | """ 419 | 420 | # ╔═╡ 00000000-0000-0000-0000-000000000002 421 | PLUTO_MANIFEST_TOML_CONTENTS = """ 422 | # This file is machine-generated - editing it directly is not advised 423 | 424 | julia_version = "1.9.3" 425 | manifest_format = "2.0" 426 | project_hash = 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"91a5bcdd-55d7-5caf-9e0b-520d859cae80" 1159 | version = "1.38.12" 1160 | 1161 | [deps.Plots.extensions] 1162 | FileIOExt = "FileIO" 1163 | GeometryBasicsExt = "GeometryBasics" 1164 | IJuliaExt = "IJulia" 1165 | ImageInTerminalExt = "ImageInTerminal" 1166 | UnitfulExt = "Unitful" 1167 | 1168 | [deps.Plots.weakdeps] 1169 | FileIO = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549" 1170 | GeometryBasics = "5c1252a2-5f33-56bf-86c9-59e7332b4326" 1171 | IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a" 1172 | ImageInTerminal = "d8c32880-2388-543b-8c61-d9f865259254" 1173 | Unitful = "1986cc42-f94f-5a68-af5c-568840ba703d" 1174 | 1175 | [[deps.PlutoUI]] 1176 | deps = ["AbstractPlutoDingetjes", "Base64", "ColorTypes", "Dates", "FixedPointNumbers", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "MIMEs", "Markdown", "Random", "Reexport", "URIs", "UUIDs"] 1177 | git-tree-sha1 = "b478a748be27bd2f2c73a7690da219d0844db305" 1178 | uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8" 1179 | version = "0.7.51" 1180 | 1181 | [[deps.PrecompileTools]] 1182 | deps = ["Preferences"] 1183 | git-tree-sha1 = "259e206946c293698122f63e2b513a7c99a244e8" 1184 | uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a" 1185 | version = "1.1.1" 1186 | 1187 | [[deps.Preferences]] 1188 | deps = ["TOML"] 1189 | git-tree-sha1 = "7eb1686b4f04b82f96ed7a4ea5890a4f0c7a09f1" 1190 | uuid = "21216c6a-2e73-6563-6e65-726566657250" 1191 | version = "1.4.0" 1192 | 1193 | [[deps.Printf]] 1194 | deps = ["Unicode"] 1195 | uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7" 1196 | 1197 | [[deps.Qt5Base_jll]] 1198 | deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"] 1199 | git-tree-sha1 = "0c03844e2231e12fda4d0086fd7cbe4098ee8dc5" 1200 | uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1" 1201 | version = "5.15.3+2" 1202 | 1203 | [[deps.QuadGK]] 1204 | deps = ["DataStructures", "LinearAlgebra"] 1205 | git-tree-sha1 = "6ec7ac8412e83d57e313393220879ede1740f9ee" 1206 | uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc" 1207 | version = "2.8.2" 1208 | 1209 | [[deps.REPL]] 1210 | deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"] 1211 | uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb" 1212 | 1213 | [[deps.Random]] 1214 | deps = ["SHA", "Serialization"] 1215 | uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" 1216 | 1217 | [[deps.Ratios]] 1218 | deps = ["Requires"] 1219 | git-tree-sha1 = "6d7bb727e76147ba18eed998700998e17b8e4911" 1220 | uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439" 1221 | version = "0.4.4" 1222 | weakdeps = ["FixedPointNumbers"] 1223 | 1224 | [deps.Ratios.extensions] 1225 | RatiosFixedPointNumbersExt = "FixedPointNumbers" 1226 | 1227 | [[deps.RecipesBase]] 1228 | deps = ["PrecompileTools"] 1229 | git-tree-sha1 = "5c3d09cc4f31f5fc6af001c250bf1278733100ff" 1230 | uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01" 1231 | version = "1.3.4" 1232 | 1233 | [[deps.RecipesPipeline]] 1234 | deps = ["Dates", "NaNMath", "PlotUtils", "PrecompileTools", "RecipesBase"] 1235 | git-tree-sha1 = "45cf9fd0ca5839d06ef333c8201714e888486342" 1236 | uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c" 1237 | version = "0.6.12" 1238 | 1239 | [[deps.Reexport]] 1240 | git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b" 1241 | uuid = "189a3867-3050-52da-a836-e630ba90ab69" 1242 | version = "1.2.2" 1243 | 1244 | [[deps.RelocatableFolders]] 1245 | deps = ["SHA", "Scratch"] 1246 | git-tree-sha1 = "90bc7a7c96410424509e4263e277e43250c05691" 1247 | uuid = "05181044-ff0b-4ac5-8273-598c1e38db00" 1248 | version = "1.0.0" 1249 | 1250 | [[deps.Requires]] 1251 | deps = ["UUIDs"] 1252 | git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7" 1253 | uuid = "ae029012-a4dd-5104-9daa-d747884805df" 1254 | version = "1.3.0" 1255 | 1256 | [[deps.Rmath]] 1257 | deps = ["Random", "Rmath_jll"] 1258 | git-tree-sha1 = "f65dcb5fa46aee0cf9ed6274ccbd597adc49aa7b" 1259 | uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa" 1260 | version = "0.7.1" 1261 | 1262 | [[deps.Rmath_jll]] 1263 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1264 | git-tree-sha1 = "6ed52fdd3382cf21947b15e8870ac0ddbff736da" 1265 | uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f" 1266 | version = "0.4.0+0" 1267 | 1268 | [[deps.SHA]] 1269 | uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" 1270 | version = "0.7.0" 1271 | 1272 | [[deps.Scratch]] 1273 | deps = ["Dates"] 1274 | git-tree-sha1 = "30449ee12237627992a99d5e30ae63e4d78cd24a" 1275 | uuid = "6c6a2e73-6563-6170-7368-637461726353" 1276 | version = "1.2.0" 1277 | 1278 | [[deps.SentinelArrays]] 1279 | deps = ["Dates", "Random"] 1280 | git-tree-sha1 = "77d3c4726515dca71f6d80fbb5e251088defe305" 1281 | uuid = "91c51154-3ec4-41a3-a24f-3f23e20d615c" 1282 | version = "1.3.18" 1283 | 1284 | [[deps.Serialization]] 1285 | uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" 1286 | 1287 | [[deps.SharedArrays]] 1288 | deps = ["Distributed", "Mmap", "Random", "Serialization"] 1289 | uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383" 1290 | 1291 | [[deps.Showoff]] 1292 | deps = ["Dates", "Grisu"] 1293 | git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de" 1294 | uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f" 1295 | version = "1.0.3" 1296 | 1297 | [[deps.SimpleBufferStream]] 1298 | git-tree-sha1 = "874e8867b33a00e784c8a7e4b60afe9e037b74e1" 1299 | uuid = "777ac1f9-54b0-4bf8-805c-2214025038e7" 1300 | version = "1.1.0" 1301 | 1302 | [[deps.Sockets]] 1303 | uuid = "6462fe0b-24de-5631-8697-dd941f90decc" 1304 | 1305 | [[deps.SortingAlgorithms]] 1306 | deps = ["DataStructures"] 1307 | git-tree-sha1 = "a4ada03f999bd01b3a25dcaa30b2d929fe537e00" 1308 | uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c" 1309 | version = "1.1.0" 1310 | 1311 | [[deps.SparseArrays]] 1312 | deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"] 1313 | uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" 1314 | 1315 | [[deps.SpecialFunctions]] 1316 | deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"] 1317 | git-tree-sha1 = "ef28127915f4229c971eb43f3fc075dd3fe91880" 1318 | uuid = "276daf66-3868-5448-9aa4-cd146d93841b" 1319 | version = "2.2.0" 1320 | weakdeps = ["ChainRulesCore"] 1321 | 1322 | [deps.SpecialFunctions.extensions] 1323 | SpecialFunctionsChainRulesCoreExt = "ChainRulesCore" 1324 | 1325 | [[deps.StaticArrays]] 1326 | deps = ["LinearAlgebra", "Random", "StaticArraysCore", "Statistics"] 1327 | git-tree-sha1 = "8982b3607a212b070a5e46eea83eb62b4744ae12" 1328 | uuid = "90137ffa-7385-5640-81b9-e52037218182" 1329 | version = "1.5.25" 1330 | 1331 | [[deps.StaticArraysCore]] 1332 | git-tree-sha1 = "6b7ba252635a5eff6a0b0664a41ee140a1c9e72a" 1333 | uuid = "1e83bf80-4336-4d27-bf5d-d5a4f845583c" 1334 | version = "1.4.0" 1335 | 1336 | [[deps.Statistics]] 1337 | deps = ["LinearAlgebra", "SparseArrays"] 1338 | uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" 1339 | version = "1.9.0" 1340 | 1341 | [[deps.StatsAPI]] 1342 | deps = ["LinearAlgebra"] 1343 | git-tree-sha1 = "45a7769a04a3cf80da1c1c7c60caf932e6f4c9f7" 1344 | uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0" 1345 | version = "1.6.0" 1346 | 1347 | [[deps.StatsBase]] 1348 | deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"] 1349 | git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916" 1350 | uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" 1351 | version = "0.33.21" 1352 | 1353 | [[deps.StatsFuns]] 1354 | deps = ["HypergeometricFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"] 1355 | git-tree-sha1 = "f625d686d5a88bcd2b15cd81f18f98186fdc0c9a" 1356 | uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c" 1357 | version = "1.3.0" 1358 | 1359 | [deps.StatsFuns.extensions] 1360 | StatsFunsChainRulesCoreExt = "ChainRulesCore" 1361 | StatsFunsInverseFunctionsExt = "InverseFunctions" 1362 | 1363 | [deps.StatsFuns.weakdeps] 1364 | ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" 1365 | InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112" 1366 | 1367 | [[deps.StatsPlots]] 1368 | deps = ["AbstractFFTs", "Clustering", "DataStructures", "Distributions", "Interpolations", "KernelDensity", "LinearAlgebra", "MultivariateStats", "NaNMath", "Observables", "Plots", "RecipesBase", "RecipesPipeline", "Reexport", "StatsBase", "TableOperations", "Tables", "Widgets"] 1369 | git-tree-sha1 = "14ef622cf28b05e38f8af1de57bc9142b03fbfe3" 1370 | uuid = "f3b207a7-027a-5e70-b257-86293d7955fd" 1371 | version = "0.15.5" 1372 | 1373 | [[deps.SuiteSparse]] 1374 | deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"] 1375 | uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9" 1376 | 1377 | [[deps.SuiteSparse_jll]] 1378 | deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"] 1379 | uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c" 1380 | version = "5.10.1+6" 1381 | 1382 | [[deps.TOML]] 1383 | deps = ["Dates"] 1384 | uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76" 1385 | version = "1.0.3" 1386 | 1387 | [[deps.TableOperations]] 1388 | deps = ["SentinelArrays", "Tables", "Test"] 1389 | git-tree-sha1 = "e383c87cf2a1dc41fa30c093b2a19877c83e1bc1" 1390 | uuid = "ab02a1b2-a7df-11e8-156e-fb1833f50b87" 1391 | version = "1.2.0" 1392 | 1393 | [[deps.TableTraits]] 1394 | deps = ["IteratorInterfaceExtensions"] 1395 | git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39" 1396 | uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c" 1397 | version = "1.0.1" 1398 | 1399 | [[deps.Tables]] 1400 | deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"] 1401 | git-tree-sha1 = "1544b926975372da01227b382066ab70e574a3ec" 1402 | uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c" 1403 | version = "1.10.1" 1404 | 1405 | [[deps.Tar]] 1406 | deps = ["ArgTools", "SHA"] 1407 | uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e" 1408 | version = "1.10.0" 1409 | 1410 | [[deps.TensorCore]] 1411 | deps = ["LinearAlgebra"] 1412 | git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6" 1413 | uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50" 1414 | version = "0.1.1" 1415 | 1416 | [[deps.Test]] 1417 | deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] 1418 | uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" 1419 | 1420 | [[deps.TranscodingStreams]] 1421 | deps = ["Random", "Test"] 1422 | git-tree-sha1 = "9a6ae7ed916312b41236fcef7e0af564ef934769" 1423 | uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa" 1424 | version = "0.9.13" 1425 | 1426 | [[deps.Tricks]] 1427 | git-tree-sha1 = "aadb748be58b492045b4f56166b5188aa63ce549" 1428 | uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775" 1429 | version = "0.1.7" 1430 | 1431 | [[deps.URIs]] 1432 | git-tree-sha1 = "074f993b0ca030848b897beff716d93aca60f06a" 1433 | uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4" 1434 | version = "1.4.2" 1435 | 1436 | [[deps.UUIDs]] 1437 | deps = ["Random", "SHA"] 1438 | uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" 1439 | 1440 | [[deps.Unicode]] 1441 | uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" 1442 | 1443 | [[deps.UnicodeFun]] 1444 | deps = ["REPL"] 1445 | git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf" 1446 | uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1" 1447 | version = "0.4.1" 1448 | 1449 | [[deps.Unzip]] 1450 | git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78" 1451 | uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d" 1452 | version = "0.2.0" 1453 | 1454 | [[deps.Wayland_jll]] 1455 | deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"] 1456 | git-tree-sha1 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c" 1457 | uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89" 1458 | version = "1.21.0+0" 1459 | 1460 | [[deps.Wayland_protocols_jll]] 1461 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1462 | git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da" 1463 | uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91" 1464 | version = "1.25.0+0" 1465 | 1466 | [[deps.Widgets]] 1467 | deps = ["Colors", "Dates", "Observables", "OrderedCollections"] 1468 | git-tree-sha1 = "fcdae142c1cfc7d89de2d11e08721d0f2f86c98a" 1469 | uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62" 1470 | version = "0.6.6" 1471 | 1472 | [[deps.WoodburyMatrices]] 1473 | deps = ["LinearAlgebra", "SparseArrays"] 1474 | git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3" 1475 | uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6" 1476 | version = "0.5.5" 1477 | 1478 | [[deps.XML2_jll]] 1479 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"] 1480 | git-tree-sha1 = "93c41695bc1c08c46c5899f4fe06d6ead504bb73" 1481 | uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a" 1482 | version = "2.10.3+0" 1483 | 1484 | [[deps.XSLT_jll]] 1485 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"] 1486 | git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a" 1487 | uuid = "aed1982a-8fda-507f-9586-7b0439959a61" 1488 | version = "1.1.34+0" 1489 | 1490 | [[deps.Xorg_libX11_jll]] 1491 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"] 1492 | git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527" 1493 | uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc" 1494 | version = "1.6.9+4" 1495 | 1496 | [[deps.Xorg_libXau_jll]] 1497 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1498 | git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e" 1499 | uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec" 1500 | version = "1.0.9+4" 1501 | 1502 | [[deps.Xorg_libXcursor_jll]] 1503 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"] 1504 | git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd" 1505 | uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724" 1506 | version = "1.2.0+4" 1507 | 1508 | [[deps.Xorg_libXdmcp_jll]] 1509 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1510 | git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4" 1511 | uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05" 1512 | version = "1.1.3+4" 1513 | 1514 | [[deps.Xorg_libXext_jll]] 1515 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] 1516 | git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3" 1517 | uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" 1518 | version = "1.3.4+4" 1519 | 1520 | [[deps.Xorg_libXfixes_jll]] 1521 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] 1522 | git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4" 1523 | uuid = "d091e8ba-531a-589c-9de9-94069b037ed8" 1524 | version = "5.0.3+4" 1525 | 1526 | [[deps.Xorg_libXi_jll]] 1527 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"] 1528 | git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246" 1529 | uuid = "a51aa0fd-4e3c-5386-b890-e753decda492" 1530 | version = "1.7.10+4" 1531 | 1532 | [[deps.Xorg_libXinerama_jll]] 1533 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"] 1534 | git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123" 1535 | uuid = "d1454406-59df-5ea1-beac-c340f2130bc3" 1536 | version = "1.1.4+4" 1537 | 1538 | [[deps.Xorg_libXrandr_jll]] 1539 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"] 1540 | git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631" 1541 | uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484" 1542 | version = "1.5.2+4" 1543 | 1544 | [[deps.Xorg_libXrender_jll]] 1545 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] 1546 | git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" 1547 | uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" 1548 | version = "0.9.10+4" 1549 | 1550 | [[deps.Xorg_libpthread_stubs_jll]] 1551 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1552 | git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" 1553 | uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" 1554 | version = "0.1.0+3" 1555 | 1556 | [[deps.Xorg_libxcb_jll]] 1557 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] 1558 | git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6" 1559 | uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" 1560 | version = "1.13.0+3" 1561 | 1562 | [[deps.Xorg_libxkbfile_jll]] 1563 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] 1564 | git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2" 1565 | uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a" 1566 | version = "1.1.0+4" 1567 | 1568 | [[deps.Xorg_xcb_util_image_jll]] 1569 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] 1570 | git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97" 1571 | uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b" 1572 | version = "0.4.0+1" 1573 | 1574 | [[deps.Xorg_xcb_util_jll]] 1575 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"] 1576 | git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1" 1577 | uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5" 1578 | version = "0.4.0+1" 1579 | 1580 | [[deps.Xorg_xcb_util_keysyms_jll]] 1581 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] 1582 | git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00" 1583 | uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7" 1584 | version = "0.4.0+1" 1585 | 1586 | [[deps.Xorg_xcb_util_renderutil_jll]] 1587 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] 1588 | git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" 1589 | uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" 1590 | version = "0.3.9+1" 1591 | 1592 | [[deps.Xorg_xcb_util_wm_jll]] 1593 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] 1594 | git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" 1595 | uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" 1596 | version = "0.4.1+1" 1597 | 1598 | [[deps.Xorg_xkbcomp_jll]] 1599 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] 1600 | git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" 1601 | uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" 1602 | version = "1.4.2+4" 1603 | 1604 | [[deps.Xorg_xkeyboard_config_jll]] 1605 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] 1606 | git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" 1607 | uuid = "33bec58e-1273-512f-9401-5d533626f822" 1608 | version = "2.27.0+4" 1609 | 1610 | [[deps.Xorg_xtrans_jll]] 1611 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1612 | git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" 1613 | uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" 1614 | version = "1.4.0+3" 1615 | 1616 | [[deps.Zlib_jll]] 1617 | deps = ["Libdl"] 1618 | uuid = "83775a58-1f1d-513f-b197-d71354ab007a" 1619 | version = "1.2.13+0" 1620 | 1621 | [[deps.Zstd_jll]] 1622 | deps = ["Artifacts", "JLLWrappers", "Libdl"] 1623 | git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c" 1624 | uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" 1625 | version = "1.5.5+0" 1626 | 1627 | [[deps.fzf_jll]] 1628 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1629 | git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56" 1630 | uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09" 1631 | version = "0.29.0+0" 1632 | 1633 | [[deps.libaom_jll]] 1634 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1635 | git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" 1636 | uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" 1637 | version = "3.4.0+0" 1638 | 1639 | [[deps.libass_jll]] 1640 | deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] 1641 | git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" 1642 | uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" 1643 | version = "0.15.1+0" 1644 | 1645 | [[deps.libblastrampoline_jll]] 1646 | deps = ["Artifacts", "Libdl"] 1647 | uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" 1648 | version = "5.8.0+0" 1649 | 1650 | [[deps.libfdk_aac_jll]] 1651 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1652 | git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" 1653 | uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" 1654 | version = "2.0.2+0" 1655 | 1656 | [[deps.libpng_jll]] 1657 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] 1658 | git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" 1659 | uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" 1660 | version = "1.6.38+0" 1661 | 1662 | [[deps.libvorbis_jll]] 1663 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] 1664 | git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" 1665 | uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" 1666 | version = "1.3.7+1" 1667 | 1668 | [[deps.nghttp2_jll]] 1669 | deps = ["Artifacts", "Libdl"] 1670 | uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" 1671 | version = "1.48.0+0" 1672 | 1673 | [[deps.p7zip_jll]] 1674 | deps = ["Artifacts", "Libdl"] 1675 | uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" 1676 | version = "17.4.0+0" 1677 | 1678 | [[deps.x264_jll]] 1679 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1680 | git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" 1681 | uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" 1682 | version = "2021.5.5+0" 1683 | 1684 | [[deps.x265_jll]] 1685 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] 1686 | git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" 1687 | uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" 1688 | version = "3.5.0+0" 1689 | 1690 | [[deps.xkbcommon_jll]] 1691 | deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] 1692 | git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" 1693 | uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" 1694 | version = "1.4.1+0" 1695 | """ 1696 | 1697 | # ╔═╡ Cell order: 1698 | # ╠═8770247a-fa57-11ed-3bcd-f548a0b9a3d4 1699 | # ╠═b5cea2b2-30c0-4641-bca6-82489344ae5e 1700 | # ╠═e303c065-b61f-43a3-97eb-23004bd52f5f 1701 | # ╠═bb1e77cc-da9e-4be5-9f96-af7b2761e7a1 1702 | # ╟─a1594bc3-aba9-423d-a2e6-208b8f9b4783 1703 | # ╠═f470f591-d4cc-4550-b152-e3c3d9723e90 1704 | # ╠═443ee9df-dfb9-4ba3-99c6-110295ef85e6 1705 | # ╟─d84f32be-6a8f-45ed-9a34-0686922cbc4c 1706 | # ╠═e2e93f77-5202-4a0d-a049-d40df0472634 1707 | # ╠═5b670729-1fd0-4e6f-9fd4-563c61f6a225 1708 | # ╠═0e18c236-8017-48cd-a658-dedcea335746 1709 | # ╠═d1251e6e-ed9d-40fc-9869-af09b140c932 1710 | # ╟─53172f2f-0e35-4d67-b8f1-8c9de407ce50 1711 | # ╟─cce96a7e-a401-4cdd-abb4-65d20daeedff 1712 | # ╟─fd7c00fa-5fa1-46b0-8749-e82bd55e39c3 1713 | # ╟─4f2d645b-78d6-439e-8964-9919ffb3c217 1714 | # ╟─e57dcd2f-aac1-4e30-9331-d6b8b44371d6 1715 | # ╠═a8f612b1-fdcb-463e-add4-972b7f178b08 1716 | # ╠═2bed026f-0e69-4d5f-b624-c4e2f9e5adf5 1717 | # ╠═3774764c-b6fe-4749-9b9a-fd245519ecea 1718 | # ╠═08d8bceb-c4d0-4464-bab1-5004361f74a0 1719 | # ╠═ae2442b3-a01d-4356-b195-7891b1af8766 1720 | # ╠═ff292549-4418-4b88-bf1e-60066973790c 1721 | # ╠═2b6961f7-1cab-4f17-8454-2490c11065dc 1722 | # ╠═582a9ed3-b6f7-496b-a121-67dd8733f6e6 1723 | # ╟─a99f0b12-b658-460e-946d-8f4e070b8130 1724 | # ╠═dd01eea5-75f4-4e0f-be4c-b261b76824c1 1725 | # ╠═3ab0493e-fc39-4b88-9428-7914c441b9c7 1726 | # ╠═6956cec8-12ef-43b9-b437-d22ff24f9b7f 1727 | # ╠═7edcbbea-88be-4362-90ec-405d421de796 1728 | # ╟─c7a0c42a-74f4-4052-bc5b-df47db1090f1 1729 | # ╠═810c971a-81c4-447e-8097-4f48ee78018c 1730 | # ╠═1b384db3-1050-4774-882d-b842e6de387f 1731 | # ╠═8d589c74-5fe3-43d3-a4f9-cc7a5c8af335 1732 | # ╟─24e0d796-7c48-4ba6-879a-d95f082417e5 1733 | # ╠═6b01ed6e-3dc1-4a0d-94ba-c4ec7637dc40 1734 | # ╠═f7a3e318-20f6-4f21-ac29-bec6d850aad9 1735 | # ╠═3dbc7984-35a9-4a2b-bc7e-f91e78be18b8 1736 | # ╠═ad15b839-ec36-427c-965c-dd92ddee4160 1737 | # ╠═9c011b4f-31bc-450f-be2d-4dc54a72acb9 1738 | # ╠═f3ff0674-e706-4834-803e-a5dc7e49cbc6 1739 | # ╠═7b19a04b-11dd-4341-b4b1-8cfaf6152ed3 1740 | # ╠═669bc90c-685f-4e5f-b91d-f1746086de9d 1741 | # ╟─00000000-0000-0000-0000-000000000001 1742 | # ╟─00000000-0000-0000-0000-000000000002 1743 | -------------------------------------------------------------------------------- /resources/Random Matrix Theory/RMT-chap2.jl: -------------------------------------------------------------------------------- 1 | ### A Pluto.jl notebook ### 2 | # v0.19.27 3 | 4 | using Markdown 5 | using InteractiveUtils 6 | 7 | # This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). 8 | macro bind(def, element) 9 | quote 10 | local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end 11 | local el = $(esc(element)) 12 | global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) 13 | el 14 | end 15 | end 16 | 17 | # ╔═╡ 937c36f6-5541-11ee-3863-d33f0332ab86 18 | using Pkg; Pkg.activate() 19 | 20 | # ╔═╡ f86162f7-d852-449f-b637-0f53f2cbef37 21 | begin 22 | using LinearAlgebra 23 | using Plots 24 | #using PlotlyJS # You'll need to install it to run the notebook but after that Plots will handle things 25 | using StatsPlots 26 | using Distributions 27 | using Statistics 28 | using PlutoUI 29 | using LaTeXStrings 30 | using Random 31 | end 32 | 33 | # ╔═╡ 7d15e3fa-cd70-449c-9110-bdf6a9595b7c 34 | md""" 35 | # Random Matrix Theory: Wigner Matrices and the semi-circle law 36 | 37 | This notebook contains the solutions to some of the exercises in the second chapter of the book, but mostly some nice interactive visualizations of the Stieltjes transform. 38 | """ 39 | 40 | # ╔═╡ 80ee1731-4da1-47ad-8a4e-b3f239c03177 41 | md"Comment out the following cell if you want Pluto to install the packages for you." 42 | 43 | # ╔═╡ 90763c30-5c1e-4118-bf94-e206423169da 44 | gr() 45 | 46 | # ╔═╡ c528e9d5-2acb-4149-8033-6e16e27c8e2d 47 | md""" 48 | ## Exercise 2.3.1 - Stieltjes Transform for shifted and scaled matrices 49 | Let $A$ be a random matrix drawn from a well-behaved ensemble with Stieltjes transform $g(z)$. What are the Siteltjes transforms of the random matrices $\alpha A$ and $A + \beta I$, where $\alpha$ and $\beta$ are non-zero real numbers. 50 | """ 51 | 52 | # ╔═╡ 25da0e70-d4c1-4d6b-9081-011fb4de9f01 53 | md""" 54 | By definition of the Stieltjes transform, 55 | ```math 56 | g^A(z) = \frac{1}{N} \text{tr}((zI - A)^{-1} ), 57 | ``` 58 | and $(zI - \alpha A) = \alpha (\frac{z}{\alpha} I - A)$ 59 | hence 60 | ```math 61 | g^{\alpha A}(z) = \frac{1}{N} \frac{1}{\alpha} \text{tr} ((\frac{z}{\alpha}I - A)^{-1}) = \frac{1}{\alpha} g^A(\frac{z}{\alpha}) 62 | ``` 63 | Similarly, 64 | ```math 65 | g^{A + \beta I}(z) = \frac{1}{N} \text{tr} (((z-\beta)I - A)^{-1}) = g(z-\beta) 66 | ``` 67 | """ 68 | 69 | # ╔═╡ a6e30f43-809d-4996-925c-22a5d9251201 70 | stieltjes_transform(λs,z) = mean(1/(z-λ) for λ in λs) 71 | 72 | # ╔═╡ 3ed04034-42b1-4b9c-bbb5-8d06a9dc3996 73 | stieltjes_transform(A::AbstractMatrix,z) = stieltjes_transform(eigvals(A), z) 74 | 75 | # ╔═╡ 1738150e-a074-4c0a-82b0-2759d39858d5 76 | function random_wigner(n, σ=1.0) 77 | H = randn(n,n) 78 | return Symmetric((H+H') * σ / sqrt(2*n)) 79 | end 80 | 81 | # ╔═╡ ae5ba9db-5277-4600-8cf4-8cf40ddb2688 82 | begin 83 | xmin = -4.0 84 | xmax = 4.0 85 | ymin = -4.0 86 | ymax = 4.0 87 | step = 0.01 88 | xs = xmin:step:xmax 89 | ys = ymin:step:ymax 90 | end 91 | 92 | # ╔═╡ cb27a1e8-c440-4f48-adba-cd873fa23d3b 93 | md""" 94 | You can move the slider below to change the size of the matrix used for the plot. 95 | """ 96 | 97 | # ╔═╡ a28aa970-8441-4b68-9d95-d1c804f7d4c4 98 | @bind n Slider(1:100; default=5, show_value=true) 99 | 100 | # ╔═╡ 3d552c8d-88af-4e60-b82a-f55492d0957c 101 | begin 102 | plotlyjs() # switch to plotlyjs backend for this plot 103 | local A = random_wigner(n) 104 | local λs = eigvals(A) 105 | local z = (x,y) -> min(4, abs(stieltjes_transform(λs, x+y*im))) 106 | p = Plots.surface(xs,ys, z, 107 | title="Stieltjes Transform of a Wigner Matrix (N=$n)", 108 | size=(700,600), 109 | xlabel="Re(z)", 110 | ylabel="Im(z)", 111 | zlabel="|g(z)|" 112 | ) 113 | local η_1 = fill(-1/n, length(xs)) 114 | local η_2 = fill(-1/sqrt(n), length(xs)) 115 | #scatter!(p, xs, η_1, z.(xs,η_1), label=L"x-i/N") 116 | #scatter!(p, xs, η_2, z.(xs,η_2), label=L"x-i/\sqrt{N}") 117 | p 118 | end 119 | 120 | # ╔═╡ c23aec92-41a6-4071-924b-1215d05185f2 121 | g_(z::Complex,σ=1) = (z - z*sqrt(1 -4(σ^2)/z^2))/(2σ^2) 122 | 123 | # ╔═╡ 488e41f9-8ceb-41da-a709-075cbb20e162 124 | g_(x,y) = g_(x+y*im) 125 | 126 | # ╔═╡ 770b9719-ad78-460a-8098-9b9ef494c6e4 127 | begin 128 | plotlyjs() 129 | local z = (x,y) -> min(4, abs(g_(x,y))) 130 | local p = Plots.surface(xs,ys, z, 131 | title="Stieltjes Transform of a Wigner Matrix (Analytical limit)", 132 | size=(700,600), 133 | xlabel="Re(z)", 134 | ylabel="Im(z)", 135 | zlabel="|g(z)|" 136 | ) 137 | p 138 | end 139 | 140 | # ╔═╡ aac7f646-d0d7-43d1-a4d2-12669d39ddd8 141 | md""" 142 | Note that it is different than in the video! Not sure if it was a typo or just complex functions being finicky. 143 | """ 144 | 145 | # ╔═╡ 9dc436c2-f7f0-42a3-96b7-adb4b4bad369 146 | md""" 147 | Here's the plot of the imaginary part. We can clearly see how it approaches the semi-circle density as we approach from the real axis from below. 148 | """ 149 | 150 | # ╔═╡ 9c0ebc3a-225c-454d-ac18-33810e6f33b5 151 | begin 152 | plotlyjs() 153 | local z = (x,y) -> min(4, imag(g_(x,y))) 154 | local p = Plots.surface(xs,ys, z, 155 | title="Stieltjes Transform of a Wigner Matrix (Analytical limit)", 156 | size=(700,600), 157 | xlabel="Re(z)", 158 | ylabel="Im(z)", 159 | zlabel="Im g(z)" 160 | ) 161 | p 162 | end 163 | 164 | # ╔═╡ d0dbbe01-af4e-44b8-8c64-ecde5b5e4242 165 | md""" 166 | ## Exercise 2.3.2 Finite ``N`` approximation and small imaginary part 167 | ``\text{Im} g_N(x-i\eta)/\pi`` is a good approximation of ``\rho(x)`` for small positive ``\eta`` where ``g_N(z)`` is the sample Stieltjes transform. 168 | 169 | Numerically generate a Wigner matrix of size ``N`` and ``\sigma^2=1``. 170 | 171 | a) For ``\eta \in \{ 1/N, 1/\sqrt{N}, 1 \}``, plot $\text{Im} g_N(x-i\eta)/\pi$ and the theoretical $\rho(x)$ on the same plot. 172 | """ 173 | 174 | # ╔═╡ 2a24ce45-563e-4a42-81ee-e3d873860ffd 175 | ρ(x, σ=1) = abs(x) > 2σ ? 0.0 : sqrt(4σ^2 - x^2)/(2π*σ^2) 176 | 177 | # ╔═╡ 31051193-340d-4893-9850-04aebdf2d0cc 178 | ρ_approx(λ) = (x,y) -> imag(stieltjes_transform(λ, x-y*im)) / pi 179 | 180 | # ╔═╡ f1c34391-de42-4d7d-bbcf-006b00a377e3 181 | begin 182 | gr() 183 | local n = 400 184 | local A = random_wigner(n) 185 | local λs = eigvals(A) 186 | local g = ρ_approx(λs) 187 | local p = plot(xs, ρ, label="Analytical density") 188 | plot!(p, xs, g.(xs, 1.0), label=L"\eta=1") 189 | plot!(p, xs, g.(xs, 1/sqrt(n)), label=L"\eta=1/\sqrt{N}") 190 | #plot!(p, xs, g.(xs, 1/n), label=L"\eta=1/N") 191 | p 192 | end 193 | 194 | # ╔═╡ 5b19482a-883c-4183-99f3-18229b10fc1e 195 | md""" 196 | b) Compute the error as a function of ``\eta`` where the error is 197 | ```math 198 | (\rho(x)-\text{Im}g_N(x-i\eta)/\pi)^2 199 | ``` 200 | integrated over ``[-3,3]`` (use equidistant point with $0.01$ spacing to approximate the integral). Plot this error for $\eta$ between $1/N$ and $1$. You should see that $1/\sqrt{N}$ is close to the minimum of this function. 201 | """ 202 | 203 | # ╔═╡ afb6fd9d-3e91-4c58-a96f-a6dbaa58e532 204 | xs_err = -3:0.01:3 205 | 206 | # ╔═╡ f39bf84c-b21a-475a-ae34-4379fe07f4c4 207 | err_fun(g,x,η) = (ρ(x)-g(x,η))^2 208 | 209 | # ╔═╡ 890230ae-216f-4d2d-b895-b7fbb770dcba 210 | err(g,η) = sum(err_fun(g,x,η) for x in xs_err) 211 | 212 | # ╔═╡ c3588633-07fc-4cf1-a08d-f806c7817701 213 | begin 214 | local n = 400 215 | local A = random_wigner(n) 216 | local λs = eigvals(A) 217 | local g = ρ_approx(λs) 218 | ηs = (1/n):0.01:1 219 | plot(ηs, err.(g, ηs), label="") 220 | vline!([1/sqrt(n)], label=L"1/\sqrt{N}") 221 | end 222 | 223 | # ╔═╡ 342379ba-be1f-4362-9cf3-b6b8edeaaa31 224 | md""" 225 | ## Ginibre Ensemble 226 | Let $X \in \mathbb{R}^{N\times N}$ be a random matrix with $X_{ij} \sim N(0,\sigma^2/N)$, then the eigenvalue density of $X$ follows the Girko circular law. Eigenvalues are uniformly distributed in a circle centered at $0$ and of radius $\sigma^2$ 227 | """ 228 | 229 | # ╔═╡ cf025540-4644-4a2d-8868-bfcc63614d27 230 | begin 231 | local n = 1000 232 | local A = randn(n,n) / sqrt(n) 233 | local λs = eigvals(A) 234 | scatter(λs, label="", title="Girko circular law") 235 | local t = 0.0:0.01:2π 236 | plot!(cos.(t), sin.(t), label="", linestyle=:dash) 237 | end 238 | 239 | # ╔═╡ ea1a543c-7896-46c0-8f47-1cb35bb3f69f 240 | begin 241 | local n = 1000 242 | local A = randn(n,n) / sqrt(2n) 243 | local B = randn(n,n) / sqrt(2n) 244 | local λs = eigvals(A+im*B) 245 | scatter(λs, label="", title="Girko circular law (Complex Gaussian)") 246 | local t = 0.0:0.01:2π 247 | plot!(cos.(t), sin.(t), label="", linestyle=:dash) 248 | end 249 | 250 | # ╔═╡ 1338402d-0905-46a2-95a8-42c3f50d5918 251 | # ╠═╡ disabled = true 252 | #=╠═╡ 253 | md""" 254 | ## Exercise 2.3.3 From the moments to the density 255 | """ 256 | ╠═╡ =# 257 | 258 | # ╔═╡ 70ed779e-417f-48b4-899b-4df3f59db796 259 | # ╠═╡ disabled = true 260 | #=╠═╡ 261 | md""" 262 | ## Exercise 3.1.1 Quaternionic matrices of size one 263 | """ 264 | ╠═╡ =# 265 | 266 | # ╔═╡ ce009503-175b-4c74-94e6-e8036f9ff6b3 267 | # ╠═╡ disabled = true 268 | #=╠═╡ 269 | md""" 270 | ## Exercise 3.1.2 Three quarter-circle laws 271 | """ 272 | ╠═╡ =# 273 | 274 | # ╔═╡ 401a26a1-8363-45c6-9a87-58ee1bf4ff55 275 | # ╠═╡ disabled = true 276 | #=╠═╡ 277 | md""" 278 | ## Exercise 3.2.1 Non-crossing pair partitions of eight elements 279 | """ 280 | ╠═╡ =# 281 | 282 | # ╔═╡ Cell order: 283 | # ╟─7d15e3fa-cd70-449c-9110-bdf6a9595b7c 284 | # ╟─80ee1731-4da1-47ad-8a4e-b3f239c03177 285 | # ╠═937c36f6-5541-11ee-3863-d33f0332ab86 286 | # ╠═f86162f7-d852-449f-b637-0f53f2cbef37 287 | # ╠═90763c30-5c1e-4118-bf94-e206423169da 288 | # ╟─c528e9d5-2acb-4149-8033-6e16e27c8e2d 289 | # ╟─25da0e70-d4c1-4d6b-9081-011fb4de9f01 290 | # ╠═a6e30f43-809d-4996-925c-22a5d9251201 291 | # ╠═3ed04034-42b1-4b9c-bbb5-8d06a9dc3996 292 | # ╠═1738150e-a074-4c0a-82b0-2759d39858d5 293 | # ╠═ae5ba9db-5277-4600-8cf4-8cf40ddb2688 294 | # ╟─cb27a1e8-c440-4f48-adba-cd873fa23d3b 295 | # ╠═a28aa970-8441-4b68-9d95-d1c804f7d4c4 296 | # ╠═3d552c8d-88af-4e60-b82a-f55492d0957c 297 | # ╠═c23aec92-41a6-4071-924b-1215d05185f2 298 | # ╠═488e41f9-8ceb-41da-a709-075cbb20e162 299 | # ╠═770b9719-ad78-460a-8098-9b9ef494c6e4 300 | # ╟─aac7f646-d0d7-43d1-a4d2-12669d39ddd8 301 | # ╟─9dc436c2-f7f0-42a3-96b7-adb4b4bad369 302 | # ╠═9c0ebc3a-225c-454d-ac18-33810e6f33b5 303 | # ╟─d0dbbe01-af4e-44b8-8c64-ecde5b5e4242 304 | # ╠═2a24ce45-563e-4a42-81ee-e3d873860ffd 305 | # ╠═31051193-340d-4893-9850-04aebdf2d0cc 306 | # ╠═f1c34391-de42-4d7d-bbcf-006b00a377e3 307 | # ╟─5b19482a-883c-4183-99f3-18229b10fc1e 308 | # ╠═afb6fd9d-3e91-4c58-a96f-a6dbaa58e532 309 | # ╠═f39bf84c-b21a-475a-ae34-4379fe07f4c4 310 | # ╠═890230ae-216f-4d2d-b895-b7fbb770dcba 311 | # ╠═c3588633-07fc-4cf1-a08d-f806c7817701 312 | # ╟─342379ba-be1f-4362-9cf3-b6b8edeaaa31 313 | # ╠═cf025540-4644-4a2d-8868-bfcc63614d27 314 | # ╠═ea1a543c-7896-46c0-8f47-1cb35bb3f69f 315 | # ╟─1338402d-0905-46a2-95a8-42c3f50d5918 316 | # ╟─70ed779e-417f-48b4-899b-4df3f59db796 317 | # ╟─ce009503-175b-4c74-94e6-e8036f9ff6b3 318 | # ╟─401a26a1-8363-45c6-9a87-58ee1bf4ff55 319 | -------------------------------------------------------------------------------- /resources/Random Matrix Theory/RMT-chap4.jl: -------------------------------------------------------------------------------- 1 | ### A Pluto.jl notebook ### 2 | # v0.19.29 3 | 4 | using Markdown 5 | using InteractiveUtils 6 | 7 | # This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). 8 | macro bind(def, element) 9 | quote 10 | local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end 11 | local el = $(esc(element)) 12 | global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) 13 | el 14 | end 15 | end 16 | 17 | # ╔═╡ b51af2cc-653e-11ee-19cc-bb1f00f82e59 18 | using Pkg; Pkg.activate() 19 | 20 | # ╔═╡ 6d86c249-016e-4ac0-ae42-5ee5afa305d9 21 | begin 22 | using LinearAlgebra 23 | using Plots 24 | #using PlotlyJS # You'll need to install it to run the notebook but after that Plots will handle things 25 | using StatsPlots 26 | using Distributions 27 | using Statistics 28 | using PlutoUI 29 | using LaTeXStrings 30 | using Random 31 | end 32 | 33 | # ╔═╡ 86d7fbae-2b06-4506-b635-0f3ce5cc4c4e 34 | md"# Random Matrix Theory: Wishart Matrices and the Marcenko-Pastur distribution" 35 | 36 | # ╔═╡ 47183ef7-a6a1-4626-8449-12950733c6cc 37 | plotlyjs() 38 | 39 | # ╔═╡ 84849894-e7a0-4698-a8e7-9ab08db88838 40 | begin 41 | xmin = -2.0 42 | xmax = 4.0 43 | ymin = -2.0 44 | ymax = 2.0 45 | step = 0.01 46 | xs = xmin:step:xmax 47 | ys = ymin:step:ymax 48 | end 49 | 50 | # ╔═╡ de3d0390-844e-4053-996c-c83349b82503 51 | λ₋(q) = (1-sqrt(q))^2 52 | 53 | # ╔═╡ c2b60494-6de3-4e31-8dbb-911457d818e6 54 | λ₊(q) = (1+sqrt(q))^2 55 | 56 | # ╔═╡ e8aad398-2dff-422f-b754-2e35dea55019 57 | g(z,q) = (z - (1-q) - sqrt((z-(1-sqrt(q))^2))*sqrt((z-(1+sqrt(q))^2))) / (2*q*z) 58 | 59 | # ╔═╡ 67013fbf-8b8b-4c8c-a395-9152aa760054 60 | @bind q Slider(0.0:0.01:2.0, default=1.0, show_value=true) 61 | 62 | # ╔═╡ 62e2f4ab-9d93-49b3-a0b8-8c28156beb86 63 | g_(x,y) = g(x+y*im, q) 64 | 65 | # ╔═╡ 90ff0901-8b0e-4306-b9c5-5d5bb7b2872a 66 | begin 67 | #plotlyjs() 68 | local z = (x,y) -> min(4, abs(g_(x,y))) 69 | local p = Plots.surface(xs,ys, z, 70 | title="Stieltjes Transform of a White Wishart Matrix", 71 | size=(700,600), 72 | xlabel="Re(z)", 73 | ylabel="Im(z)", 74 | zlabel="|g(z)|" 75 | ) 76 | p 77 | end 78 | 79 | # ╔═╡ 2ce15f2f-6e14-4a80-be68-acc92c7ea43f 80 | begin 81 | #plotlyjs() 82 | local z = (x,y) -> min(10, abs(g_(x,y))) 83 | local p = Plots.contour(xs,ys, z, 84 | title="Stieltjes Transform of a White Wishart Matrix", 85 | size=(700,600), 86 | xlabel="Re(z)", 87 | ylabel="Im(z)", 88 | #zlabel="|g(z)|" 89 | ) 90 | p 91 | end 92 | 93 | # ╔═╡ ee10d94e-9290-4d10-a794-b3e57d5c7e12 94 | begin 95 | #plotlyjs() 96 | local z = (x,y) -> clamp(imag(g_(x,y)/π), -4,4) 97 | local p = Plots.surface(xs,ys, z, 98 | title="Stieltjes Transform of a White Wishart Matrix", 99 | size=(700,600), 100 | xlabel="Re(z)", 101 | ylabel="Im(z)", 102 | zlabel="Im g(z)" 103 | ) 104 | p 105 | end 106 | 107 | # ╔═╡ 0a5c0410-0451-4697-bc97-51594d2a63ba 108 | relu(x) = max(x,0.0) 109 | 110 | # ╔═╡ 19d530ab-6943-4e42-b522-e7dd72827d6c 111 | ρ(x,q) = (sqrt ∘ relu)((λ₊(q)- x) * (x-λ₋(q))) / (2π*q*x) 112 | 113 | # ╔═╡ 17449982-0813-49b5-8c3f-254b20562adc 114 | begin 115 | local ts = 0.0:0.01:6.0 116 | plot(ts, ρ.(ts,q), 117 | label="", 118 | title = "Marcenko-Pastur Density", 119 | xlabel="λ", 120 | ylabel="ρ(λ)", 121 | ) 122 | end 123 | 124 | # ╔═╡ Cell order: 125 | # ╟─86d7fbae-2b06-4506-b635-0f3ce5cc4c4e 126 | # ╠═b51af2cc-653e-11ee-19cc-bb1f00f82e59 127 | # ╠═6d86c249-016e-4ac0-ae42-5ee5afa305d9 128 | # ╠═47183ef7-a6a1-4626-8449-12950733c6cc 129 | # ╠═84849894-e7a0-4698-a8e7-9ab08db88838 130 | # ╠═de3d0390-844e-4053-996c-c83349b82503 131 | # ╠═c2b60494-6de3-4e31-8dbb-911457d818e6 132 | # ╠═e8aad398-2dff-422f-b754-2e35dea55019 133 | # ╠═67013fbf-8b8b-4c8c-a395-9152aa760054 134 | # ╠═62e2f4ab-9d93-49b3-a0b8-8c28156beb86 135 | # ╠═90ff0901-8b0e-4306-b9c5-5d5bb7b2872a 136 | # ╠═2ce15f2f-6e14-4a80-be68-acc92c7ea43f 137 | # ╠═ee10d94e-9290-4d10-a794-b3e57d5c7e12 138 | # ╠═0a5c0410-0451-4697-bc97-51594d2a63ba 139 | # ╠═19d530ab-6943-4e42-b522-e7dd72827d6c 140 | # ╠═17449982-0813-49b5-8c3f-254b20562adc 141 | -------------------------------------------------------------------------------- /resources/Representation Theory/README.md: -------------------------------------------------------------------------------- 1 | # Representation Theory 2 | 3 | Main resources: 4 | - https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf 5 | - https://mat.uab.cat/~pitsch/ReadingSeminar/Fulton-Harris.pdf 6 | - https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf 7 | 8 | Additional resources: 9 | - https://www.youtube.com/watch?v=Q9OsEZV5YX8&list=PL8yHsr3EFj51AM_VmB0VygPIzcUCH_jUG 10 | * https://colab.research.google.com/github/dashstander/sn-grok/blob/main/Sn_Circuits.ipynb -------------------------------------------------------------------------------- /resources/Signature Transform/README.md: -------------------------------------------------------------------------------- 1 | 2 | # Signature Transform 3 | 4 | Main resources: 5 | 6 | - [A Primer on the Signature Method in Machine Learning](https://arxiv.org/abs/1603.03788) 7 | - [Theoretical Foundations of Deep Selective State-Space Models](https://arxiv.org/abs/2402.19047) 8 | 9 | Additional resources: 10 | 11 | - [Deep Signature Transforms](https://papers.nips.cc/paper_files/paper/2019/file/d2cdf047a6674cef251d56544a3cf029-Paper.pdf) 12 | - [Path Development Network with Finite-dimensional Lie Group](https://openreview.net/forum?id=YoWBLu74TL) 13 | -------------------------------------------------------------------------------- /resources/Signature Transform/signatures.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Signature Transform/signatures.pdf -------------------------------------------------------------------------------- /resources/Topology/README.md: -------------------------------------------------------------------------------- 1 | # Topology 2 | 3 | Main Resources: 4 | * https://math.bme.hu/~kalex/Teaching/Spring10/Topology/TopNotes_Spring10.pdf 5 | * https://web.math.ku.dk/~moller/e03/3gt/notes/gtnotes.pdf 6 | 7 | Additional Resources: 8 | * https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/ -------------------------------------------------------------------------------- /resources/Topology/compactness.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Topology/compactness.pdf -------------------------------------------------------------------------------- /resources/Topology/topology.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/irregular-rhomboid/EAI-Math-Reading-Group/2a3e44ca8403fee1d3348f1dff478c761e930c2a/resources/Topology/topology.pdf --------------------------------------------------------------------------------