{ "0706.0207_19": "\n \n \n \n \n \n \n ⟨\n ψ\n |\n \n e\n \n -\n 𝒬\n \n \n |\n ψ\n ⟩\n \n =\n 1\n \n .\n \n (\n 19\n )\n \n \n \n \n \n", "0706.2428_9": "\n \n \n \n \n \n \n \n \n (\n 1.6\n )\n \n ⁢\n \n {\n \n θ\n j\n \n ,\n \n θ\n k\n \n }\n \n \n =\n 0\n \n ,\n \n \n {\n \n θ\n j\n \n ,\n \n \n 1\n 2\n \n ⁢\n \n log\n ⁡\n \n [\n \n \n μ\n k\n \n /\n \n μ\n n\n \n \n ]\n \n \n \n }\n \n =\n \n δ\n \n j\n ⁢\n k\n \n \n \n \n ,\n \n \n \n \n", "0706.2580_8": "", "0707.1945_9": "\n \n \n \n \n \n d\n ⁢\n \n s\n 2\n \n \n =\n \n \n \n \n U\n 2\n \n \n ρ\n 2\n \n \n ⁢\n \n \n (\n \n \n d\n ⁢\n t\n \n -\n \n P\n ⁢\n d\n ⁢\n ϕ\n \n \n )\n \n 2\n \n \n -\n \n \n \n \n sin\n 2\n \n ⁡\n θ\n \n \n ρ\n 2\n \n \n ⁢\n \n \n [\n \n \n \n (\n \n F\n +\n \n l\n 2\n \n \n )\n \n ⁢\n d\n ⁢\n ϕ\n \n -\n \n a\n ⁢\n d\n ⁢\n t\n \n \n ]\n \n 2\n \n \n -\n \n \n \n ρ\n 2\n \n \n U\n 2\n \n \n ⁢\n d\n ⁢\n \n r\n 2\n \n \n -\n \n \n ρ\n 2\n \n ⁢\n d\n ⁢\n \n θ\n 2\n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n \n", "0707.2579_10": "\n \n \n \n \n \n \n ⟨\n \n ⟨\n \n ℰ\n β\n \n (\n j\n )\n \n \n |\n 𝒪\n |\n \n 𝒟\n α\n \n (\n i\n )\n \n \n ⟩\n \n ⟩\n \n =\n \n 0\n   \n \n (\n \n \n λ\n α\n \n ≠\n \n λ\n β\n \n \n )\n \n \n \n .\n \n (\n 9\n )\n \n \n \n \n \n", "0707.3327_12": "\n \n \n \n \n G\n =\n \n {\n \n (\n x\n ,\n \n x\n \n n\n +\n 1\n \n \n )\n \n ∣\n \n \n \n u\n 1\n \n ⁢\n \n (\n x\n )\n \n \n <\n \n x\n \n n\n +\n 1\n \n \n <\n \n \n u\n 2\n \n ⁢\n \n (\n x\n )\n \n \n \n }\n \n \n \n \n \n", "0707.4247_10": "\n \n \n \n \n \n w\n =\n \n \n p\n \n \n 2\n ⁢\n X\n ⁢\n \n \n ∂\n ⁡\n p\n \n \n ∂\n ⁡\n X\n \n \n \n -\n p\n \n \n \n \n .\n \n (\n 11\n )\n \n \n \n \n \n", "0710.0193_2": "", "0711.1060_88": "", "0711.1095_65": "\n \n \n \n \n \n \n 1\n -\n \n p\n ⁢\n \n (\n ω\n )\n \n \n \n :=\n \n \n P\n ω\n b\n \n ⁢\n \n (\n \n \n τ\n ⁢\n \n (\n d\n )\n \n \n <\n \n \n τ\n +\n \n ⁢\n \n (\n b\n )\n \n \n \n )\n \n \n =\n \n \n ω\n b\n \n ⁢\n \n \n e\n \n V\n ⁢\n \n (\n b\n )\n \n \n \n \n \n ∑\n \n x\n =\n b\n \n \n d\n -\n 1\n \n \n \n e\n \n V\n ⁢\n \n (\n x\n )\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "0711.1359_228": "", "0711.2469_1": "\n \n \n \n \n \n f\n a\n \n =\n \n \n lim\n \n ϵ\n →\n 0\n \n \n ⁡\n \n \n q\n 2\n \n ⁢\n \n \n ∫\n \n -\n ∞\n \n \n τ\n -\n ϵ\n \n \n \n \n \n ∇\n a\n \n ⁡\n \n G\n \n r\n ⁢\n e\n ⁢\n t\n \n \n \n ⁢\n \n (\n x\n ,\n \n x\n ′\n \n )\n \n ⁢\n \n 𝑑\n \n τ\n ′\n \n \n ⁢\n \n (\n 2\n )\n \n \n \n \n \n \n \n \n \n", "0711.3396_22": "\n \n \n \n \n \n ℳ\n \n l\n ±\n \n α\n \n =\n \n \n \n \n 1\n 2\n \n ⁢\n \n [\n \n τ\n α\n \n ,\n \n τ\n 0\n \n ]\n \n ⁢\n \n ℳ\n \n l\n ±\n \n \n (\n -\n )\n \n \n \n +\n \n \n 1\n 2\n \n ⁢\n \n {\n \n τ\n α\n \n ,\n \n τ\n 0\n \n }\n \n ⁢\n \n ℳ\n \n l\n ±\n \n \n (\n +\n )\n \n \n \n +\n \n \n τ\n α\n \n ⁢\n \n \n ℳ\n \n l\n ±\n \n \n (\n 0\n )\n \n \n \n \n \n ,\n \n (\n 18\n )\n \n \n \n \n \n \n", "0711.4559_123": "\n \n \n \n \n \n \n c\n 12\n \n =\n \n \n \n [\n \n -\n \n \n 1\n 2\n \n ⁢\n \n \n (\n \n \n \n ∂\n 2\n \n ⁡\n F\n \n \n ∂\n ⁡\n \n κ\n 2\n \n \n \n )\n \n c\n \n \n \n ]\n \n \n 1\n /\n 2\n \n \n ⁢\n \n \n (\n \n \n ∂\n ⁡\n κ\n \n \n ∂\n ⁡\n \n ν\n 2\n \n \n \n )\n \n c\n \n \n \n ,\n \n \n \n \n", "0711.4734_35": "", "0712.3103_33": "\n \n \n \n \n \n ξ\n =\n \n \n u\n +\n V\n \n -\n \n V\n ∞\n \n \n \n .\n \n \n \n \n", "0801.0077_27": "\n \n \n \n \n \n C\n ∞\n \n =\n \n ⋂\n R\n \n \n Ω\n R\n \n \n (\n C\n )\n \n .\n =\n \n ⋂\n d\n \n \n Ω\n d\n K\n \n \n (\n C\n )\n \n .\n \n \n \n \n", "0801.0219_12": "", "0801.0438_56": "\n \n \n \n \n \n ρ\n ⁢\n \n (\n \n H\n ⁢\n \n (\n z\n ,\n S\n )\n \n \n )\n \n \n =\n \n \n ⟨\n \n H\n ⁢\n \n (\n z\n ,\n \n π\n ⁢\n \n (\n S\n )\n \n \n )\n \n ⁢\n ξ\n \n ,\n ξ\n ⟩\n \n ℋ\n \n \n \n \n \n", "0801.1366_16": "\n \n \n \n \n ϕ\n 1\n \n =\n \n \n \n \n i\n ⁢\n \n (\n \n \n p\n 2\n \n -\n p\n \n )\n \n \n \n p\n 3\n \n \n \n ⁢\n \n ϕ\n 1\n ′\n \n \n \n ,\n \n \n ϕ\n 2\n ′\n \n =\n \n \n \n \n i\n ⁢\n \n (\n \n \n p\n 2\n \n +\n p\n \n )\n \n \n \n p\n 3\n \n \n \n ⁢\n \n ϕ\n 2\n \n \n \n \n .\n \n (\n 33\n )\n \n \n", "0801.2895_35": "\n \n \n \n \n \n \n \n \n (\n \n \n x\n 2\n \n -\n \n a\n 2\n \n \n )\n \n ⁢\n \n (\n \n 1\n -\n \n d\n ⁢\n x\n \n \n )\n \n \n <\n \n \n 2\n ⁢\n \n (\n \n 1\n -\n \n a\n 2\n \n \n )\n \n ⁢\n x\n ⁢\n \n (\n \n x\n -\n d\n \n )\n \n \n ,\n i\n \n \n .\n e\n \n .\n ,\n \n \n \n \n", "0801.3702_26": "\n \n \n \n \n \n D\n c\n \n =\n \n \n \n σ\n 2\n \n ⁢\n \n P\n e\n \n ⁢\n \n (\n \n N\n u\n \n )\n \n ⁢\n \n [\n \n \n \n \n \n γ\n +\n β\n \n γ\n \n ⁢\n \n ln\n ⁡\n \n (\n \n 1\n +\n \n γ\n β\n \n \n )\n \n \n \n -\n \n 1\n γ\n \n \n +\n \n 1\n 2\n \n \n ]\n \n \n ,\n \n (\n 20\n )\n \n \n \n \n \n \n", "0802.0067_19": "\n \n \n \n \n V\n =\n \n C\n ⁢\n \n \n 4\n ⁢\n π\n \n 3\n \n ⁢\n \n R\n 1\n \n ⁢\n \n R\n 2\n \n ⁢\n \n R\n 3\n \n ⁢\n \n (\n 18\n )\n \n \n \n \n \n \n", "0802.0340_14": "\n \n \n \n \n \n \n \n v\n \n g\n ⁢\n j\n \n \n \n v\n \n D\n ⁢\n j\n \n \n \n ≈\n \n \n g\n \n Ω\n j\n \n \n ⁢\n \n \n \n Ω\n j\n \n ⁢\n L\n \n \n C\n \n s\n ⁢\n j\n \n 2\n \n \n \n ≈\n \n L\n H\n \n \n ,\n \n (\n 15\n )\n \n \n \n \n \n", "0802.0986_85": "", "0802.2408_76": "\n \n \n \n \n \n \n ∂\n t\n \n ⁢\n \n \n ∫\n 𝐑\n \n \n \n Ψ\n A\n \n ⁢\n \n (\n \n x\n -\n \n x\n ⁢\n \n (\n t\n )\n \n \n \n )\n \n ⁢\n \n ε\n 2\n \n \n \n \n ≤\n \n -\n \n c\n ⁢\n \n \n ∫\n 𝐑\n \n \n \n (\n \n \n ε\n 2\n \n +\n \n ε\n x\n 2\n \n \n )\n \n ⁢\n \n e\n \n -\n \n \n |\n \n x\n -\n \n x\n ⁢\n \n (\n t\n )\n \n \n \n |\n \n /\n A\n \n \n \n \n \n \n \n \n \n \n \n", "0802.2445_4": "\n \n \n \n \n \n \n 42\n 4097\n \n ,\n \n 54\n 4097\n \n ,\n \n 134\n 4097\n \n ,\n \n 190\n 4097\n \n \n .\n \n \n \n \n", "0802.3537_20": "\n \n \n \n Δ\n ⁢\n Q\n \n =\n \n -\n \n \n \n ∮\n 0\n \n 2\n ⁢\n π\n \n \n \n \n \n 𝑑\n \n k\n y\n \n \n ⁢\n \n \n \n \n ∂\n ⁡\n \n P\n x\n \n \n ⁢\n \n (\n \n k\n y\n \n )\n \n \n \n ∂\n ⁡\n \n k\n y\n \n \n \n \n \n \n \n \n .\n \n (\n 17\n )\n \n \n", "0803.2360_66": "\n \n \n \n \n \n v\n 0\n \n =\n \n 220\n ±\n \n \n \n 20\n ⁢\n k\n ⁢\n m\n \n /\n s\n \n ⁢\n \n (\n 64\n )\n \n \n \n \n \n \n \n", "0803.3766_96": "\n \n \n \n \n \n \n \n ∫\n Y\n \n γ\n \n =\n \n \n 1\n 2\n \n ⁢\n \n \n ∫\n \n Y\n ^\n \n \n \n γ\n ^\n \n \n \n =\n \n \n 1\n \n 2\n ⁢\n t\n \n \n ⁢\n \n \n ∫\n \n S\n ^\n \n \n \n γ\n ^\n \n \n \n \n .\n \n \n \n \n", "0804.0633_76": "", "0804.4241_16": "\n \n \n \n \n \n \n \n {\n α\n }\n \n \n -\n \n (\n \n N\n +\n 1\n \n )\n \n \n \n =\n \n K\n ⁢\n \n (\n α\n ,\n \n q\n N\n \n )\n \n \n \n .\n \n (\n 2.15\n )\n \n \n \n \n \n", "0805.0964_3": "\n \n \n \n \n \n ⟨\n \n ⟨\n \n n\n j\n \n ⟩\n \n ⟩\n \n =\n \n \n ∑\n \n i\n =\n 1\n \n 5\n \n \n \n R\n i\n \n ⁢\n \n \n ⟨\n \n n\n j\n \n ⟩\n \n \n q\n i\n \n \n ⁢\n \n (\n 21\n )\n \n \n \n \n \n \n \n", "0805.1971_37": "\n \n \n \n \n \n \n \n L\n α\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n sup\n ⁡\n \n {\n \n θ\n ∈\n \n \n [\n 0\n ,\n 1\n ]\n \n ⁢\n \n such\n \n ⁢\n that\n ⁢\n \n \n ∑\n \n i\n =\n x\n \n n\n \n \n \n (\n \n n\n i\n \n )\n \n ⁢\n \n θ\n i\n \n ⁢\n \n \n (\n \n 1\n -\n θ\n \n )\n \n \n n\n -\n i\n \n \n \n \n \n ≥\n α\n \n }\n \n \n \n .\n \n \n \n \n", "0805.3094_26": "\n \n \n \n \n \n \n y\n i\n \n =\n \n \n ∑\n j\n \n \n \n ℋ\n \n i\n ⁢\n j\n \n \n ⁢\n \n z\n j\n \n \n \n \n .\n \n (\n 28\n )\n \n \n \n \n \n", "0805.4403_1": "\n \n \n \n \n \n \n ∂\n ⁡\n u\n \n \n ∂\n ⁡\n t\n \n \n =\n \n \n \n Δ\n ⁢\n u\n \n +\n \n u\n p\n \n \n -\n \n V\n ⁢\n \n (\n x\n )\n \n ⁢\n u\n \n \n \n \n \n \n", "0806.3338_11": "\n \n \n \n \n \n \n Ψ\n ⁢\n \n (\n \n x\n 1\n \n ,\n \n t\n 2\n \n )\n \n \n =\n \n \n \n 2\n \n ⁢\n η\n ⁢\n \n e\n \n i\n ⁢\n \n (\n \n \n κ\n ⁢\n \n x\n 1\n \n \n +\n \n \n \n (\n \n \n κ\n 2\n \n -\n \n η\n 2\n \n \n )\n \n ⁢\n \n t\n 2\n \n \n /\n 2\n \n \n )\n \n \n \n \n \n cosh\n ⁡\n \n (\n \n η\n ⁢\n \n (\n \n \n x\n 1\n \n +\n \n κ\n ⁢\n \n t\n 2\n \n \n \n )\n \n \n )\n \n \n \n \n .\n \n \n \n \n", "0807.2523_16": "\n \n \n \n \n \n \n \n (\n \n \n \n D\n ~\n \n ρ\n \n ⁢\n \n T\n \n λ\n \n μ\n ⁢\n ν\n \n \n \n \n )\n \n ⁢\n \n ξ\n ρ\n \n \n =\n \n \n (\n \n \n \n h\n \n σ\n ⁢\n λ\n \n \n ⁢\n \n T\n \n ρ\n \n μ\n ⁢\n ν\n \n \n \n \n +\n \n \n h\n \n σ\n μ\n \n \n ⁢\n \n T\n \n \n λ\n ⁢\n ρ\n \n ν\n \n \n \n +\n \n \n h\n \n σ\n ν\n \n \n ⁢\n \n T\n \n μ\n \n \n \n λ\n \n ⁢\n ρ\n \n \n \n \n )\n \n ⁢\n \n \n D\n ~\n \n \n [\n ρ\n \n \n ⁢\n \n ξ\n \n σ\n ]\n \n \n \n \n .\n \n (\n 18\n )\n \n \n \n \n \n", "0807.2940_25": "", "0807.4832_17": "\n \n \n \n \n \n =\n \n \n 1\n \n \n ∥\n \n a\n n\n \n ∥\n \n 2\n \n \n ⁢\n \n \n ∫\n 0\n ∞\n \n \n \n e\n \n -\n t\n \n \n ⁢\n \n \n ∫\n \n {\n \n \n \n ∥\n x\n ∥\n \n \n a\n n\n \n \n =\n t\n \n }\n \n \n \n \n (\n \n \n ∏\n \n i\n =\n 1\n \n n\n \n \n \n |\n \n x\n i\n \n |\n \n \n \n a\n \n i\n ,\n n\n \n \n ⁢\n s\n \n \n \n )\n \n ⁢\n \n 𝑑\n \n ℋ\n \n n\n -\n 1\n \n \n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n 𝑑\n t\n \n \n \n \n \n \n \n \n \n \n", "0808.0696_47": "\n \n \n \n \n \n \n p\n ~\n \n ⁢\n \n (\n κ\n )\n \n \n =\n \n \n m\n k\n \n ⁢\n \n [\n \n \n Z\n ⁢\n \n (\n \n \n sn\n \n -\n 1\n \n \n ⁢\n \n (\n \n \n \n -\n 1\n \n +\n \n 1\n \n k\n 2\n \n \n +\n \n \n κ\n 2\n \n \n m\n 2\n \n \n \n \n ,\n k\n )\n \n \n ,\n k\n )\n \n \n +\n \n π\n \n 2\n ⁢\n 𝐊\n \n \n \n ]\n \n ⁢\n \n (\n 48\n )\n \n \n \n \n \n \n", "0808.1551_30": "", "0808.2029_3": "\n \n \n \n \n \n \n 𝐪\n \n e\n ⁢\n f\n ⁢\n f\n \n \n =\n \n \n \n \n k\n ^\n \n i\n \n ⁢\n \n [\n \n \n k\n i\n \n -\n \n δ\n ⁢\n k\n \n \n ]\n \n \n -\n \n \n \n k\n ^\n \n f\n \n ⁢\n \n [\n \n \n k\n f\n \n -\n \n δ\n ⁢\n k\n \n \n ]\n \n \n \n \n .\n \n (\n 4\n )\n \n \n \n \n \n", "0809.0190_63": "\n \n \n \n \n \n ∫\n 0\n ∞\n \n \n \n 𝑑\n E\n \n ⁢\n \n E\n s\n \n \n \n \n \n \n", "0809.0306_105": "\n \n \n \n \n \n \n x\n p\n \n ≥\n \n 𝟎\n p\n \n \n ,\n \n \n \n \n", "0809.2305_14": "\n \n \n \n \n \n \n (\n \n \n Z\n \n c\n ⁢\n r\n ⁢\n u\n ⁢\n s\n ⁢\n t\n \n \n \n Z\n \n l\n ⁢\n i\n ⁢\n t\n ⁢\n h\n \n \n \n )\n \n =\n \n \n 7\n 110\n \n ⁢\n \n (\n \n Q\n \n Q\n \n E\n ⁢\n a\n ⁢\n r\n ⁢\n t\n ⁢\n h\n \n \n \n )\n \n ⁢\n \n (\n \n \n Z\n \n c\n ⁢\n r\n ⁢\n u\n ⁢\n s\n ⁢\n t\n \n \n \n Z\n \n \n c\n ⁢\n r\n ⁢\n u\n ⁢\n s\n ⁢\n t\n \n ,\n \n E\n ⁢\n a\n ⁢\n r\n ⁢\n t\n ⁢\n h\n \n \n \n \n )\n \n \n \n .\n \n (\n 15\n )\n \n \n \n \n \n", "0809.2342_1": "\n \n \n \n \n \n \n χ\n ^\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n \n A\n 0\n \n +\n \n \n A\n 1\n \n ⁢\n x\n \n +\n \n \n A\n 2\n \n ⁢\n \n x\n 2\n \n \n +\n \n \n A\n 3\n \n ⁢\n \n x\n \n \n d\n ⁢\n ν\n \n -\n 1\n \n \n \n +\n \n ⋯\n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "0809.3326_9": "\n \n \n \n \n \n R\n ⁢\n \n (\n y\n )\n \n \n =\n \n \n \n C\n ⁢\n \n (\n \n α\n s\n \n )\n \n ⁢\n Σ\n ⁢\n \n (\n y\n ,\n \n α\n s\n \n )\n \n \n +\n \n D\n ⁢\n \n (\n y\n ,\n \n α\n s\n \n )\n \n \n \n ,\n \n (\n 10\n )\n \n \n \n \n \n \n", "0809.4405_110": "\n \n \n \n \n \n \n \n (\n 4.7\n )\n \n ⁢\n \n G\n n\n \n ⁢\n \n (\n i\n ,\n j\n )\n \n \n =\n \n \n \n \n (\n \n -\n 1\n \n )\n \n \n |\n \n i\n -\n j\n \n |\n \n \n ⁢\n \n Γ\n i\n \n -\n 1\n \n \n ⁢\n \n T\n i\n \n ⁢\n \n Γ\n \n i\n +\n 1\n \n \n -\n 1\n \n \n ⁢\n \n T\n \n i\n +\n 1\n \n \n ⁢\n ⋯\n ⁢\n \n Γ\n \n j\n -\n 1\n \n \n -\n 1\n \n \n ⁢\n \n T\n j\n \n \n ⋅\n \n \n (\n \n \n Γ\n j\n \n -\n \n \n T\n j\n \n ⁢\n \n \n G\n ^\n \n \n j\n +\n 1\n \n \n ⁢\n \n T\n j\n †\n \n \n \n )\n \n \n -\n 1\n \n \n \n \n ,\n \n \n \n \n", "0810.0094_8": "\n \n \n \n \n \n E\n \n \n s\n ⁢\n y\n ⁢\n n\n \n ,\n \n c\n ⁢\n u\n ⁢\n t\n \n \n \n ≈\n \n \n \n 3\n ×\n \n 10\n \n -\n 3\n \n \n \n ⁢\n \n B\n 100\n \n ⁢\n \n \n (\n \n \n E\n \n e\n ,\n \n m\n ⁢\n a\n ⁢\n x\n \n \n \n TeV\n \n )\n \n 2\n \n ⁢\n keV\n \n ,\n \n (\n 9\n )\n \n \n \n \n \n \n", "0810.0525_12": "\n \n \n \n \n \n \n ϕ\n 1\n \n ≃\n \n \n ±\n \n \n 2\n 3\n \n ⁢\n \n \n 7\n 3\n \n \n ⁢\n \n y\n \n -\n \n 1\n /\n 2\n \n \n \n \n \n +\n \n O\n ⁢\n \n (\n \n y\n \n -\n 1\n \n \n )\n \n \n \n \n .\n \n (\n 13\n )\n \n \n \n \n \n", "0810.3029_14": "\n \n \n \n \n \n \n p\n i\n \n =\n \n P\n ⁢\n \n (\n \n \n σ\n i\n \n =\n 1\n \n )\n \n \n =\n \n \n e\n \n U\n g\n \n \n \n \n e\n \n U\n g\n \n \n +\n \n e\n \n -\n \n U\n g\n \n \n \n \n \n \n ,\n \n (\n 4.3\n )\n \n \n \n \n \n", "0811.1540_1": "\n \n \n \n \n \n f\n \n e\n ⁢\n f\n ⁢\n g\n \n \n \n f\n \n \n c\n ⁢\n b\n ⁢\n a\n \n \n b\n \n \n \n +\n \n d\n \n \n \n f\n \n f\n ⁢\n e\n ⁢\n a\n \n \n \n f\n \n \n c\n ⁢\n b\n ⁢\n g\n \n \n b\n \n \n \n +\n \n d\n \n \n \n f\n \n \n *\n \n g\n ⁢\n a\n ⁢\n f\n \n \n \n \n f\n \n \n c\n ⁢\n e\n ⁢\n b\n \n \n b\n \n \n \n +\n \n d\n \n \n \n f\n \n \n *\n \n a\n ⁢\n g\n ⁢\n e\n \n \n \n \n f\n \n \n c\n ⁢\n f\n ⁢\n b\n \n \n b\n \n \n \n =\n \n d\n \n \n 0\n .\n \n (\n 2.2\n )\n \n \n \n \n \n", "0811.2856_25": "\n \n \n \n \n \n |\n \n φ\n i\n \n \n (\n t\n )\n \n )\n \n =\n \n ∑\n \n β\n =\n 1\n \n N\n \n |\n \n ψ\n β\n \n \n (\n t\n )\n \n )\n _̌\n β\n i\n (\n t\n )\n  \n ,\n ∑\n \n \n α\n \n v\n \n \n \n α\n ⁢\n i\n \n \n \n \n *\n \n (\n t\n )\n v\n \n \n \n α\n ⁢\n j\n \n \n \n \n *\n \n (\n t\n )\n =\n δ\n \n \n \n \n i\n ⁢\n j\n \n \n \n .\n (\n 18\n )\n \n \n \n \n", "0812.1081_19": "\n \n \n \n \n \n \n \n Γ\n 0\n \n \n α\n 3\n \n \n <\n \n \n 3\n \n ⁢\n \n \n \n [\n \n 17\n 24\n \n ]\n \n \n 3\n /\n 2\n \n \n \n ⁢\n \n 1\n \n \n κ\n \n ⁢\n \n \n (\n \n 1\n -\n \n n\n ⁢\n s\n \n \n )\n \n \n 3\n /\n 2\n \n \n \n \n \n \n .\n \n (\n 21\n )\n \n \n \n \n \n", "0812.2467_23": "\n \n \n \n \n \n \n ∂\n θ\n \n \n K\n =\n κ\n \n (\n r\n ,\n θ\n )\n \n .\n  \n \n \n \n ∂\n r\n \n ⁡\n K\n \n \n \n ∂\n θ\n \n ⁡\n K\n \n \n =\n λ\n \n (\n r\n ,\n θ\n )\n \n ,\n \n (\n 24\n )\n \n \n \n \n \n", "0901.0421_55": "\n \n \n \n \n \n \n Φ\n u\n \n ⁢\n \n (\n r\n )\n \n \n ≥\n \n μ\n -\n \n C\n ⁢\n \n r\n 0\n ϵ\n \n \n \n \n \n \n \n", "0901.1171_315": "\n \n \n \n \n \n ∥\n \n φ\n ⁢\n \n \n (\n t\n )\n \n \n -\n 1\n \n \n ⁢\n ψ\n ⁢\n \n (\n t\n )\n \n \n ∥\n \n ≤\n \n 1\n  \n \n (\n \n t\n ∈\n \n Ω\n 0\n \n \n )\n \n \n \n \n \n \n", "0901.3835_37": "\n \n \n \n \n \n \n \n S\n \n q\n \n r\n -\n 1\n \n \n \n \n S\n \n q\n r\n \n \n \n ≤\n \n r\n \n -\n 2\n \n \n \n .\n \n (\n 3.6\n )\n \n \n \n \n \n", "0902.0079_70": "\n \n \n \n \n \n γ\n ˙\n \n =\n \n \n \n γ\n ×\n ω\n \n ⁢\n \n (\n t\n )\n \n \n ,\n \n (\n 5.30\n )\n \n \n \n \n \n \n", "0902.0238_18": "\n \n \n \n \n \n \n \n \n ⟨\n A\n ⟩\n \n \n c\n ⁢\n l\n \n \n =\n \n \n \n p\n ↑\n \n ⁢\n \n A\n ↑\n \n \n +\n \n \n p\n ↓\n \n ⁢\n \n A\n ↓\n \n \n \n \n ,\n \n \n \n ⟨\n \n A\n 2\n \n ⟩\n \n \n c\n ⁢\n l\n \n \n =\n \n \n \n p\n ↑\n \n ⁢\n \n A\n ↑\n 2\n \n \n +\n \n \n p\n ↓\n \n ⁢\n \n A\n ↓\n 2\n \n \n \n \n \n .\n \n (\n 20\n )\n \n \n \n \n \n", "0902.0279_3": "", "0902.1306_1": "\n \n \n \n \n \n f\n ⁢\n \n (\n r\n ,\n \n u\n 0\n \n ,\n …\n ,\n \n u\n d\n \n )\n \n \n =\n \n a\n ⁢\n \n (\n λ\n ,\n d\n )\n \n ⁢\n \n \n Δ\n d\n \n \n ⁢\n \n \n r\n \n \n d\n 2\n \n -\n 1\n \n \n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n \n λ\n \n ⁢\n \n \n w\n d\n \n \n ⁢\n \n r\n d\n \n \n \n )\n \n \n \n \n \n \n \n", "0902.2965_20": "\n \n \n \n \n \n \n d\n ⁢\n \n ln\n ⁡\n \n p\n l\n \n \n \n =\n \n \n \n (\n \n \n \n μ\n riskless\n \n +\n \n l\n ⁢\n \n μ\n excess\n \n \n \n -\n \n \n \n l\n 2\n \n ⁢\n \n σ\n M\n 2\n \n \n 2\n \n \n )\n \n ⁢\n d\n ⁢\n t\n \n +\n \n l\n ⁢\n \n σ\n M\n \n ⁢\n d\n ⁢\n \n W\n t\n \n \n \n \n .\n \n (\n 17\n )\n \n \n \n \n \n", "0902.4371_163": "", "0903.0529_97": "\n \n \n \n \n \n \n \n (\n 3.27\n )\n \n ⁢\n \n h\n ˙\n \n \n ≤\n \n \n -\n \n h\n ⁢\n \n (\n \n 1\n -\n \n \n |\n \n a\n ˙\n \n |\n \n a\n \n \n )\n \n \n \n +\n \n \n |\n \n a\n ˙\n \n |\n \n ⁢\n \n ∥\n \n V\n δ\n \n ∥\n \n \n \n \n .\n \n \n \n \n", "0903.1079_1": "\n \n \n \n \n \n ⟨\n \n S\n z\n \n ⟩\n \n =\n \n \n S\n ⁢\n \n (\n \n 1\n +\n \n \n J\n ⁢\n ρ\n \n 2\n \n +\n \n O\n ⁢\n \n (\n \n J\n 2\n \n )\n \n \n \n )\n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "0903.4222_0": "\n \n \n \n \n A\n ⁢\n ∅\n \n \n →\n D\n \n \n ∅\n ⁢\n A\n \n \n ,\n \n \n \n ∅\n ⁢\n A\n \n \n →\n D\n \n \n A\n ⁢\n ∅\n \n \n ,\n \n \n \n A\n ⁢\n A\n \n \n →\n p\n \n \n ∅\n ⁢\n ∅\n \n \n ,\n \n \n \n A\n ⁢\n A\n ⁢\n ∅\n ⁢\n ∅\n \n \n →\n \n σ\n R\n \n \n \n A\n ⁢\n A\n ⁢\n A\n ⁢\n A\n \n \n ,\n \n \n ∅\n ⁢\n ∅\n ⁢\n A\n ⁢\n A\n \n \n →\n \n σ\n L\n \n \n \n A\n ⁢\n A\n ⁢\n A\n ⁢\n A\n \n \n \n \n \n \n ,\n \n", "0903.4494_114": "", "0903.4929_93": "", "0903.5062_57": "\n \n \n \n \n \n \n ρ\n =\n \n \n ±\n \n \n \n 2\n /\n b\n \n \n ⁢\n \n ]\n \n [\n o\n b\n j\n e\n c\n t\n O\n b\n j\n e\n c\n t\n \n \n ⁢\n \n \n (\n \n a\n +\n \n b\n ⁢\n \n σ\n 1\n \n \n \n )\n \n 2\n \n \n \n +\n 1\n \n \n ,\n \n ψ\n =\n \n \n \n ±\n \n \n arctan\n ⁡\n \n (\n \n a\n +\n \n b\n ⁢\n \n σ\n 1\n \n \n \n )\n \n \n /\n 2\n \n \n +\n ϕ\n \n ,\n \n \n ¿\n ¯\n \n ⁢\n 0\n \n \n \n \n .\n \n (\n 43\n )\n \n \n \n \n \n", "0904.0714_85": "\n \n \n \n \n \n S\n :=\n \n \n ∑\n \n c\n =\n 1\n \n q\n \n \n Δ\n ⁢\n \n \n (\n N\n ,\n q\n ,\n c\n )\n \n 2\n \n \n \n \n ,\n \n \n \n \n", "0904.1508_86": "", "0904.1891_81": "\n \n \n \n \n \n \n \n d\n CE\n \n ⁢\n \n \n (\n ω\n )\n \n j\n \n \n =\n \n \n \n \n j\n ⁢\n \n (\n \n j\n -\n 1\n \n )\n \n \n 2\n \n ⁢\n \n [\n ω\n ,\n ω\n ]\n \n \n ∧\n \n \n (\n ω\n )\n \n \n j\n -\n 2\n \n \n \n =\n \n \n -\n \n j\n ⁢\n \n (\n \n j\n -\n 1\n \n )\n \n ⁢\n d\n ⁢\n ω\n \n \n ∧\n \n \n (\n ω\n )\n \n \n j\n -\n 2\n \n \n \n =\n \n -\n \n j\n ⁢\n d\n ⁢\n \n (\n \n ω\n \n j\n -\n 1\n \n \n )\n \n \n \n \n .\n \n \n \n \n", "0904.2751_71": "\n \n \n \n \n \n \n \n \n sup\n \n v\n ∈\n \n ℬ\n \n q\n ×\n q\n \n \n δ\n ,\n ϵ\n \n \n \n \n ⁡\n \n ℋ\n ⁢\n \n (\n v\n )\n \n \n \n +\n \n c\n ⁢\n ℰ\n ⁢\n \n (\n v\n )\n \n \n \n ≤\n \n \n ℋ\n ⁢\n \n (\n \n v\n ¯\n \n )\n \n \n +\n \n α\n ⁢\n ℰ\n ⁢\n \n (\n \n v\n ¯\n \n )\n \n \n \n \n .\n \n \n \n \n", "0904.3400_92": "", "0904.4623_36": "", "0905.1579_75": "\n \n \n \n \n x\n =\n \n (\n \n r\n ⁢\n \n cos\n ⁡\n θ\n \n \n ,\n \n r\n ⁢\n \n sin\n ⁡\n θ\n \n \n )\n \n \n \n \n \n", "0905.2513_29": "\n \n \n \n \n \n S\n ⁢\n \n (\n ϵ\n )\n \n \n =\n \n \n \n v\n a\n \n ⁢\n \n (\n ϵ\n )\n \n ⁢\n \n L\n \n \n 2\n ⁢\n π\n ⁢\n v\n \n \n \n ⁢\n \n exp\n ⁡\n \n [\n \n -\n \n \n ϵ\n 2\n \n \n 2\n ⁢\n v\n \n \n \n ]\n \n \n \n ,\n \n (\n 30\n )\n \n \n \n \n \n \n", "0905.2895_18": "", "0905.3000_8": "", "0905.3484_67": "\n \n \n \n \n \n \n \n \n (\n 4.1\n )\n \n ⁢\n \n 𝚿\n \n (\n \n i\n ⁢\n j\n \n )\n \n \n \n =\n \n \n \n A\n \n i\n ⁢\n j\n ⁢\n \n (\n i\n )\n \n \n \n ⁢\n \n 𝚿\n \n (\n i\n )\n \n \n \n +\n \n \n A\n \n j\n ⁢\n i\n ⁢\n \n (\n j\n )\n \n \n \n ⁢\n \n 𝚿\n \n (\n j\n )\n \n \n \n +\n \n \n (\n \n 1\n -\n \n A\n \n i\n ⁢\n j\n ⁢\n \n (\n i\n )\n \n \n \n -\n \n A\n \n j\n ⁢\n i\n ⁢\n \n (\n j\n )\n \n \n \n \n )\n \n ⁢\n 𝚿\n \n \n \n ,\n \n i\n ≠\n j\n \n \n ,\n \n \n \n \n", "0906.0107_7": "\n \n \n \n \n \n \n c\n 1\n \n ⁢\n \n 2\n \n n\n +\n 1\n \n \n ⁢\n \n \n (\n \n n\n +\n 1\n \n )\n \n !\n \n \n ≤\n \n \n ∫\n \n -\n ∞\n \n \n +\n ∞\n \n \n \n …\n ⁢\n \n \n ∫\n \n -\n ∞\n \n \n +\n ∞\n \n \n \n \n d\n ⁢\n \n x\n 1\n \n ⁢\n …\n ⁢\n d\n ⁢\n \n x\n n\n \n \n \n \n \n (\n \n 1\n +\n \n x\n 1\n 2\n \n \n )\n \n ⁢\n \n (\n \n 1\n +\n \n \n (\n \n \n x\n 2\n \n -\n \n x\n 1\n \n \n )\n \n 2\n \n \n )\n \n ⁢\n …\n ⁢\n \n (\n \n 1\n +\n \n \n (\n \n \n x\n n\n \n -\n \n x\n \n n\n -\n 1\n \n \n \n )\n \n 2\n \n \n )\n \n ⁢\n \n (\n \n 1\n +\n \n x\n n\n 2\n \n \n )\n \n \n \n \n \n \n \n ≤\n \n \n \n \n \n", "0906.2919_26": "\n \n \n \n \n \n \n A\n 1\n \n =\n \n \n 8\n 3\n \n ⁢\n \n \n ∫\n \n -\n ∞\n \n \n +\n ∞\n \n \n \n \n \n (\n \n \n a\n ′′\n \n ⁢\n \n (\n s\n )\n \n \n )\n \n 2\n \n ⁢\n a\n ⁢\n \n (\n s\n )\n \n ⁢\n \n 𝑑\n \n s\n \n \n \n \n \n \n .\n \n (\n 29\n )\n \n \n \n \n \n", "0906.4157_18": "\n \n \n \n \n \n \n \n \n (\n 2.1\n )\n \n ⁢\n \n \n ∂\n ⁡\n g\n \n \n ∂\n ⁡\n t\n \n \n \n =\n \n \n -\n \n 2\n ⁢\n R\n ⁢\n c\n \n \n +\n \n \n 2\n 3\n \n ⁢\n R\n ⁢\n g\n \n \n \n ,\n \n \n g\n ⁢\n \n (\n 0\n )\n \n \n =\n \n g\n 0\n \n \n \n ,\n \n \n \n \n", "0906.4297_9": "\n \n \n \n \n \n \n f\n ⁢\n \n (\n t\n )\n \n \n =\n \n \n 1\n λ\n \n ⁢\n \n \n ∑\n n\n \n \n f\n ⁢\n \n (\n \n n\n λ\n \n )\n \n ⁢\n g\n ⁢\n \n (\n \n t\n -\n \n n\n λ\n \n \n )\n \n \n \n \n \n .\n \n (\n 1.2\n )\n \n \n \n \n \n", "0906.4939_27": "\n \n \n \n \n \n \n ∫\n Ω\n \n \n \n ⟨\n \n a\n ⁢\n \n (\n x\n ,\n \n D\n ⁢\n u\n \n )\n \n \n ,\n \n D\n ⁢\n φ\n \n ⟩\n \n ⁢\n \n 𝑑\n x\n \n \n \n =\n \n \n ∫\n Ω\n \n \n \n φ\n \n ⁢\n \n 𝑑\n μ\n \n \n \n \n \n \n \n", "0907.3182_7": "", "0907.3608_116": "\n \n \n \n \n \n \n R\n 1\n \n =\n \n R\n 2\n \n =\n \n \n R\n 3\n \n \n \n ,\n \n \n P\n \n A\n ⁢\n b\n \n \n =\n \n Q\n \n A\n ⁢\n b\n \n \n =\n \n 0\n \n \n ,\n \n \n P\n \n D\n ⁢\n c\n ⁢\n A\n ⁢\n b\n \n \n =\n 0\n \n \n \n \n \n", "0907.3902_37": "", "0907.5606_116": "\n \n \n \n \n \n H\n ∘\n \n ψ\n \n w\n \n k\n ,\n T\n \n \n \n \n \n w\n \n k\n ,\n T\n \n 2\n \n \n \n \n \n", "0908.0344_45": "\n \n \n \n \n \n 𝐓𝐞𝐫𝐦\n ≤\n \n \n N\n \n \n -\n 2\n \n +\n \n \n ⁢\n \n \n ∏\n \n i\n =\n 1\n \n 6\n \n \n \n ∥\n \n I\n ⁢\n \n ϕ\n i\n \n \n ∥\n \n \n X\n \n 1\n ,\n \n \n 1\n 2\n \n +\n \n \n δ\n \n \n \n \n \n ,\n \n \n \n \n", "0908.1079_6": "\n \n \n \n \n \n Φ\n =\n \n \n 1\n \n 2\n \n \n ⁢\n \n (\n \n \n \n \n v\n +\n h\n +\n \n i\n ⁢\n \n ϕ\n 0\n \n \n \n \n \n \n i\n ⁢\n \n 2\n \n ⁢\n \n ϕ\n +\n \n ⁢\n i\n ⁢\n \n 2\n \n ⁢\n \n ϕ\n -\n \n \n \n \n \n \n v\n +\n h\n \n -\n \n i\n ⁢\n \n ϕ\n 0\n \n \n \n \n \n \n )\n \n \n \n .\n \n (\n 7\n )\n \n \n \n \n \n", "0908.1300_7": "\n \n \n \n \n \n σ\n ⁢\n \n (\n \n u\n 0\n \n ,\n v\n )\n \n \n =\n \n \n σ\n ⁢\n \n (\n \n u\n 0\n \n ,\n \n v\n 0\n \n )\n \n \n +\n \n \n ∫\n \n v\n ⁢\n 0\n \n v\n \n \n \n \n \n \n r\n \n ,\n v\n v\n \n \n -\n \n \n r\n \n ⁢\n \n \n (\n \n Ψ\n \n ,\n v\n \n \n )\n \n 2\n \n \n \n +\n \n \n r\n \n ⁢\n \n \n (\n \n Φ\n \n ,\n v\n \n \n )\n \n 2\n \n \n \n \n \n 2\n \n ⁢\n \n r\n \n ,\n v\n \n \n \n \n ⁢\n \n 𝑑\n v\n \n ⁢\n \n (\n 8\n )\n \n \n \n \n \n \n \n \n", "0908.4520_8": "", "0909.1024_20": "\n \n \n \n \n \n ϕ\n ⁢\n ϕ\n ⁢\n Ω\n \n =\n \n \n \n c\n ϕ\n \n ⁢\n 𝟏\n ⁢\n Ω\n \n +\n \n \n (\n \n ϕ\n +\n \n \n ∑\n \n l\n =\n \n 2\n ,\n 4\n ,\n …\n \n \n \n \n ϕ\n \n \n μ\n 1\n \n ⁢\n …\n ⁢\n \n μ\n l\n \n \n \n \n \n )\n \n ⁢\n Ω\n \n \n \n \n \n \n", "0909.1052_0": "\n \n \n \n \n \n y\n \n i\n ⁢\n j\n \n \n =\n \n \n \n 2\n ⁢\n m\n ⁢\n i\n ⁢\n n\n ⁢\n \n (\n \n E\n i\n 2\n \n ,\n \n E\n j\n 2\n \n )\n \n ⁢\n \n (\n \n 1\n -\n \n cos\n ⁡\n \n θ\n \n i\n ⁢\n j\n \n \n \n \n )\n \n \n \n E\n \n v\n ⁢\n i\n ⁢\n s\n \n 2\n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "0909.4272_36": "\n \n \n \n \n \n \n F\n \n j\n ⁢\n k\n \n \n =\n \n \n 1\n 2\n \n ⁢\n \n ε\n \n i\n ⁢\n j\n ⁢\n k\n \n \n ⁢\n \n \n x\n i\n \n \n r\n 3\n \n \n ⁢\n \n \n T\n \n m\n →\n \n \n \n \n \n ,\n \n \n Φ\n 9\n \n =\n \n \n \n 1\n \n 2\n ⁢\n r\n \n \n ⁢\n \n T\n \n m\n →\n \n \n \n ,\n \n (\n 5.6\n )\n \n \n \n \n \n \n \n", "0909.4501_9": "\n \n \n \n \n \n w\n 1\n \n ,\n …\n ,\n \n w\n ℓ\n \n ,\n \n z\n \n k\n ,\n p\n ,\n q\n \n \n \n (\n n\n )\n \n ,\n k\n =\n 1\n ,\n 2\n ,\n p\n =\n 1\n ,\n .\n .\n ,\n \n b\n k\n \n ,\n q\n =\n 1\n ,\n …\n ,\n \n d\n \n i\n ,\n k\n \n \n -\n 1\n \n \n \n \n", "0909.4926_175": "\n \n \n \n \n \n \n η\n ⁢\n \n \n max\n \n 1\n ≤\n i\n ≤\n d\n \n \n ⁡\n \n ρ\n i\n \n \n ⁢\n \n (\n ℋ\n ,\n L\n )\n \n \n ≤\n \n \n 1\n 2\n \n ⁢\n \n \n max\n \n 1\n ≤\n i\n ≤\n d\n \n \n ⁡\n \n ρ\n i\n \n \n ⁢\n \n (\n ℋ\n ,\n L\n )\n \n \n ≤\n \n \n \n min\n \n 1\n ≤\n i\n ≤\n d\n \n \n ⁡\n \n ρ\n i\n \n \n ⁢\n \n (\n ℋ\n ,\n L\n )\n \n \n \n .\n \n \n \n \n", "0909.5442_3": "\n \n \n \n \n \n t\n rec\n \n =\n \n \n m\n ⋅\n \n \n \n d\n →\n \n ⋅\n \n p\n →\n \n \n \n \n |\n p\n |\n \n 2\n \n \n \n ,\n \n (\n 4\n )\n \n \n \n \n \n \n", "0910.4002_114": "\n \n \n \n \n \n \n \n \n (\n 3.27\n )\n \n ⁢\n \n \n max\n \n ∂\n ⁡\n \n B\n 1\n \n \n \n ⁡\n \n |\n \n u\n j\n \n |\n \n \n \n =\n 1\n \n ,\n \n \n so\n ⁢\n  0\n \n <\n \n -\n \n u\n j\n \n \n ≤\n \n \n |\n x\n |\n \n \n -\n \n α\n j\n \n \n \n \n \n .\n \n \n \n \n", "0911.0901_10": "\n \n \n \n \n \n \n \n A\n +\n \n :=\n \n \n \n ⋃\n \n i\n ∈\n \n I\n +\n \n \n \n \n \n A\n i\n \n \n \n ,\n \n \n A\n -\n \n :=\n \n \n \n ⋃\n \n i\n ∈\n \n I\n -\n \n \n \n \n \n A\n i\n \n \n \n \n .\n \n \n \n \n", "0911.3174_2": "\n \n \n \n \n \n ψ\n ⁢\n \n (\n t\n )\n \n \n =\n \n \n \n cos\n ⁡\n \n (\n \n t\n ⁢\n \n A\n \n \n )\n \n \n ⁢\n f\n \n +\n \n \n \n sin\n ⁡\n \n (\n \n t\n ⁢\n \n A\n \n \n )\n \n \n \n A\n \n \n ⁢\n g\n \n \n \n \n \n \n", "0911.4201_16": "\n \n \n \n \n \n |\n \n J\n \n x\n ⁢\n y\n \n \n |\n \n >\n \n \n min\n ⁡\n \n {\n \n \n ∑\n \n z\n :\n \n \n \n ⟨\n z\n ,\n x\n ⟩\n \n ∈\n E\n \n ,\n \n z\n ≠\n y\n \n \n \n \n \n |\n \n J\n \n x\n ⁢\n z\n \n \n |\n \n \n ,\n \n \n ∑\n \n z\n :\n \n \n \n ⟨\n z\n ,\n y\n ⟩\n \n ∈\n E\n \n ,\n \n z\n ≠\n x\n \n \n \n \n \n |\n \n J\n \n y\n ⁢\n z\n \n \n |\n \n \n }\n \n \n ,\n \n (\n 10\n )\n \n \n \n \n \n \n", "0911.4206_27": "\n \n \n \n \n \n H\n -\n \n =\n \n \n \n -\n \n \n d\n 2\n \n \n d\n ⁢\n \n x\n 2\n \n \n \n \n +\n \n \n V\n -\n \n ⁢\n \n (\n x\n )\n \n \n \n -\n \n \n E\n 0\n -\n \n ⁢\n \n (\n \n 6\n ⁢\n l\n ⁢\n m\n ⁢\n n\n ⁢\n q\n ⁢\n r\n ⁢\n s\n \n )\n \n \n \n \n \n \n \n", "0911.4478_51": "\n \n \n \n \n \n \n \n Ω\n k\n \n ⁢\n  c o k e r \n ⁢\n J\n \n ⟶\n \n \n Ω\n k\n \n ⁢\n \n Q\n 0\n \n ⁢\n \n S\n 0\n \n \n ⟶\n \n \n Ω\n k\n \n ⁢\n J\n \n \n .\n \n \n \n \n", "0912.0872_12": "", "0912.1142_64": "\n \n \n \n \n \n \n f\n N\n \n ⁢\n \n (\n ζ\n )\n \n \n >\n \n -\n \n \n B\n 3\n \n \n \n Im\n \n ⁢\n ζ\n \n \n \n \n \n \n \n", "0912.2096_4": "\n \n \n \n \n \n \n T\n \n μ\n ⁢\n ν\n \n \n =\n \n \n \n ρ\n 0\n \n ⁢\n h\n ⁢\n \n u\n μ\n \n ⁢\n \n u\n ν\n \n \n +\n \n P\n ⁢\n \n g\n \n μ\n ⁢\n ν\n \n \n \n \n \n ,\n \n \n \n \n", "1001.1677_0": "\n \n \n \n \n H\n =\n \n \n \n \n ∑\n i\n \n \n h\n i\n JC\n \n \n -\n \n μ\n ⁢\n N\n \n -\n \n J\n ⁢\n \n \n ∑\n \n ⟨\n \n i\n ⁢\n j\n \n ⟩\n \n \n \n \n a\n i\n †\n \n ⁢\n \n \n a\n j\n \n \n \n \n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "1001.5217_10": "\n \n \n \n \n \n \n [\n \n \n \n f\n NL\n \n ⁢\n \n \n ℱ\n ⁢\n \n (\n \n \n k\n →\n \n 1\n \n ,\n \n \n k\n →\n \n 2\n \n )\n \n \n \n D\n ⁢\n \n (\n z\n )\n \n \n \n \n +\n \n J\n ⁢\n \n (\n \n \n k\n →\n \n 1\n \n ,\n \n \n k\n →\n \n 2\n \n )\n \n \n +\n \n \n \n b\n 2\n \n ⁢\n \n (\n z\n )\n \n \n \n 2\n ⁢\n \n b\n 1\n \n ⁢\n \n (\n z\n )\n \n \n \n \n ]\n \n +\n \n c\n ⁢\n y\n ⁢\n c\n \n \n .\n \n \n \n \n", "1002.0030_112": "\n \n \n \n \n \n \n \n (\n 51\n )\n \n ⁢\n Prob\n ⁢\n \n (\n \n A\n \n u\n ,\n a\n \n \n )\n \n \n =\n \n \n Prob\n ⁢\n \n (\n \n {\n \n \n \n ∥\n w\n ∥\n \n ∞\n \n >\n \n \n u\n a\n \n +\n \n O\n ⁢\n \n (\n \n a\n ⁢\n \n S\n 2\n \n \n )\n \n \n \n \n }\n \n )\n \n \n +\n \n Ø\n ⁢\n \n (\n \n \n exp\n ⁡\n \n (\n \n \n \n α\n f\n \n ⁢\n S\n \n -\n \n \n S\n 2\n \n \n 2\n ⁢\n \n σ\n f\n 2\n \n \n \n \n )\n \n \n +\n \n exp\n ⁡\n \n (\n \n \n \n α\n h\n \n ⁢\n S\n \n -\n \n \n S\n 2\n \n \n 2\n ⁢\n \n σ\n h\n 2\n \n \n \n \n )\n \n \n \n )\n \n \n \n \n .\n \n \n \n \n", "1002.2990_4": "\n \n \n \n \n \n \n \n u\n 0\n \n ⁢\n \n (\n x\n )\n \n \n =\n 1\n \n ,\n \n \n \n \n u\n 1\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n 3\n \n ⁢\n \n (\n \n 1\n -\n \n 2\n ⁢\n x\n \n \n )\n \n \n \n ,\n \n \n \n u\n 2\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n \n 5\n \n ⁢\n \n (\n \n \n 1\n -\n \n 6\n ⁢\n x\n \n \n +\n \n 6\n ⁢\n \n x\n 2\n \n \n \n )\n \n \n ,\n \n …\n ⁢\n \n (\n 3.5\n )\n \n \n \n \n \n \n \n \n \n", "1003.0667_21": "\n \n \n \n \n \n N\n ⁢\n \n (\n γ\n ,\n t\n )\n \n \n ≈\n \n \n \n q\n 0\n \n ⁢\n t\n ⁢\n \n γ\n \n -\n p\n \n \n ⁢\n \n [\n \n 1\n -\n \n \n \n t\n 0\n \n t\n \n ⁢\n \n \n (\n \n 1\n -\n \n \n \n γ\n ⁢\n \n t\n 0\n \n \n \n τ\n \n s\n ⁢\n 0\n \n \n \n ⁢\n \n ln\n ⁡\n \n (\n \n t\n \n t\n 0\n \n \n )\n \n \n \n \n )\n \n \n p\n -\n 2\n \n \n \n \n ]\n \n \n ,\n \n (\n 22\n )\n \n \n \n \n \n \n", "1003.2681_14": "\n \n \n \n 𝒱\n \n (\n 0\n ,\n 0\n )\n \n \n =\n \n 𝒱\n \n (\n 1\n ,\n 0\n )\n \n \n =\n \n [\n \n \n \n \n (\n +\n +\n )\n \n \n \n \n \n \n (\n +\n -\n )\n \n \n \n \n ]\n \n \n ;\n \n \n 𝒱\n \n (\n 1\n ,\n 1\n )\n \n \n =\n \n [\n \n \n \n \n (\n +\n +\n +\n -\n )\n \n \n \n \n \n \n (\n +\n +\n -\n +\n )\n \n \n \n \n ]\n \n \n \n", "1003.2793_17": "\n \n \n \n \n \n \n \n Δ\n \n ξ\n ⁢\n η\n \n \n ⁢\n f\n \n =\n \n \n f\n ⁢\n \n (\n ⋅\n ,\n ξ\n )\n \n \n -\n \n f\n ⁢\n \n (\n ⋅\n ,\n η\n )\n \n \n \n \n .\n \n \n \n \n", "1003.2960_22": "\n \n \n \n \n \n |\n 𝒜\n |\n \n ≤\n \n \n \n 2\n n\n \n \n \n ∑\n \n i\n =\n 0\n \n k\n \n \n (\n \n n\n i\n \n )\n \n \n \n ⁢\n \n (\n \n n\n k\n \n )\n \n \n \n \n \n \n", "1004.1665_4": "\n \n \n \n \n \n \n ϕ\n ⁢\n \n (\n 𝐫\n ;\n ω\n )\n \n \n =\n \n \n \n ∑\n \n n\n =\n \n -\n ∞\n \n \n ∞\n \n \n \n ∫\n \n \n \n \n d\n 2\n \n ⁢\n \n 𝐪\n ∥\n \n \n \n \n (\n \n 2\n ⁢\n π\n \n )\n \n 2\n \n \n ⁢\n \n [\n \n \n e\n \n -\n \n \n q\n n\n \n ⁢\n z\n \n \n \n -\n \n \n g\n n\n \n ⁢\n \n (\n \n 𝐪\n ∥\n \n ;\n ω\n )\n \n ⁢\n \n e\n \n \n q\n n\n \n ⁢\n z\n \n \n \n \n ]\n \n ⁢\n \n e\n \n \n i\n ⁢\n \n q\n \n x\n ,\n n\n \n \n ⁢\n x\n \n +\n \n i\n ⁢\n \n q\n y\n \n ⁢\n y\n \n \n \n ⁢\n \n for\n \n ⁢\n z\n \n \n \n \n ∼\n <\n \n \n 0\n \n \n ,\n \n (\n 5\n )\n \n \n \n \n \n", "1004.1795_21": "", "1004.3189_3": "\n \n \n \n \n \n \n V\n \n x\n ,\n y\n \n \n ⁢\n \n (\n t\n )\n \n \n =\n \n \n \n S\n \n x\n ,\n y\n \n \n ⁢\n \n (\n t\n )\n \n \n +\n \n \n V\n \n x\n ,\n y\n \n S\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n (\n 7\n )\n \n \n \n \n \n \n \n", "1004.4630_13": "\n \n \n \n \n (\n \n -\n 1\n \n )\n \n \n ν\n i\n \n \n =\n \n \n \n ∏\n \n \n \n n\n i\n \n =\n 1\n \n ,\n \n \n n\n \n j\n ≠\n i\n \n \n =\n \n 0\n ,\n 1\n \n \n \n \n \n \n \n \n ∏\n \n m\n =\n 1\n \n 2\n \n \n \n \n ξ\n m\n \n ⁢\n \n (\n \n 𝚪\n \n \n 𝐧\n 𝟏\n \n ⁢\n \n 𝐧\n 𝟐\n \n ⁢\n \n 𝐧\n 𝟑\n \n \n \n )\n \n \n \n \n \n .\n \n (\n 𝟏𝟑\n )\n \n \n", "1005.0865_1": "\n \n \n \n \n \n \n S\n ˙\n \n =\n \n \n 1\n 2\n \n ⁢\n \n \n ∑\n \n i\n ,\n j\n \n \n \n \n [\n \n \n \n ω\n i\n j\n \n ⁢\n \n P\n j\n \n ⁢\n \n (\n t\n )\n \n \n -\n \n \n ω\n j\n i\n \n ⁢\n \n P\n i\n \n ⁢\n \n (\n t\n )\n \n \n \n ]\n \n ⁢\n \n ln\n ⁡\n \n [\n \n \n \n ω\n i\n j\n \n ⁢\n \n P\n j\n \n ⁢\n \n (\n t\n )\n \n \n \n \n ω\n j\n i\n \n ⁢\n \n P\n i\n \n ⁢\n \n (\n t\n )\n \n \n \n ]\n \n \n \n \n \n \n .\n \n (\n 2\n )\n \n \n \n \n \n", "1005.2206_43": "\n \n \n \n \n \n \n 2\n ⁢\n γ\n ⁢\n \n r\n 2\n \n \n <\n \n \n R\n -\n r\n \n 2\n \n \n .\n \n (\n 4\n )\n \n \n \n \n \n", "1005.4345_41": "\n \n \n \n \n \n \n (\n \n C\n \n -\n 1\n \n \n )\n \n =\n \n (\n \n \n \n 2.305\n \n \n 29.698\n \n \n \n -\n 1.33329.698\n \n \n \n 6825.27\n \n \n \n \n -\n 113.180\n \n -\n 1.333\n \n \n \n \n -\n 113.180\n \n \n \n 3.414\n \n \n \n )\n \n \n .\n \n (\n 42\n )\n \n \n \n \n \n", "1006.0749_19": "", "1006.0791_31": "\n \n \n \n \n \n \n A\n i\n \n (\n 3\n )\n \n \n =\n \n \n ∂\n j\n \n ⁡\n \n [\n \n \n \n Ψ\n †\n \n \n ⁢\n \n I\n \n j\n ⁢\n i\n \n \n \n \n ∗\n \n ⁢\n Ψ\n \n ]\n \n \n \n .\n \n (\n 30\n )\n \n \n \n \n \n", "1006.3305_190": "", "1007.1469_5": "\n \n \n \n \n \n \n A\n l\n \n =\n \n \n -\n 120.57\n \n +\n \n x\n ⁢\n \n \n 2\n ⁢\n B\n \n \n σ\n +\n 1\n \n \n \n \n \n ,\n \n \n A\n u\n \n =\n \n \n \n -\n 95.98\n \n -\n \n x\n ⁢\n \n \n 2\n ⁢\n B\n \n \n σ\n +\n 1\n \n \n \n \n ,\n \n (\n 6\n )\n \n \n \n \n \n \n \n", "1007.2304_11": "\n \n \n \n \n \n \n \n B\n ¯\n \n =\n \n B\n /\n \n x\n c\n \n \n \n ,\n \n \n \n B\n ¯\n \n ′\n \n =\n \n 1\n -\n \n \n B\n ′\n \n /\n \n \n x\n c\n \n \n \n \n \n \n .\n \n (\n 18\n )\n \n \n \n \n \n", "1007.3506_25": "\n \n \n \n \n \n \n D\n 𝐑\n \n μ\n ⁢\n ν\n \n \n ⁢\n \n (\n w\n )\n \n \n =\n \n \n \n \n \n K\n 0\n \n ⁢\n \n (\n \n \n R\n ℓ\n \n ⁢\n \n \n 2\n ⁢\n τ\n ⁢\n ω\n \n \n \n )\n \n \n \n 2\n ⁢\n π\n ⁢\n m\n ⁢\n τ\n ⁢\n \n ℓ\n 2\n \n \n \n ⁢\n \n δ\n \n μ\n ⁢\n ν\n \n \n \n ,\n \n (\n 25\n )\n \n \n \n \n \n \n", "1008.0019_85": "\n \n \n \n \n \n \n \n ∫\n 0\n π\n \n \n \n (\n \n \n \n |\n \n \n ∂\n θ\n \n ⁡\n ω\n \n |\n \n 2\n \n \n η\n 2\n \n \n )\n \n ⁢\n \n sin\n ⁡\n \n \n θ\n \n ⁢\n d\n ⁢\n θ\n \n \n \n \n =\n 4\n \n .\n \n (\n 91\n )\n \n \n \n \n \n", "1008.0132_20": "\n \n \n \n \n \n \n \n i\n ⁢\n \n ω\n ±\n \n ⁢\n \n (\n q\n )\n \n \n τ\n \n =\n \n \n \n \n 1\n 4\n \n ⁢\n \n (\n \n \n 2\n ⁢\n \n ⟨\n \n Ω\n 2\n \n ⟩\n \n \n -\n \n ⟨\n \n Ω\n x\n 2\n \n ⟩\n \n \n )\n \n \n +\n \n \n 1\n 2\n \n ⁢\n \n ⟨\n \n v\n x\n 2\n \n ⟩\n \n ⁢\n \n q\n 2\n \n \n \n ±\n \n \n \n \n \n ⟨\n \n Ω\n x\n 2\n \n ⟩\n \n 2\n \n 16\n \n +\n \n \n q\n 2\n \n ⁢\n \n \n ⟨\n \n \n Ω\n y\n \n ⁢\n \n v\n x\n \n \n ⟩\n \n 2\n \n \n \n \n \n \n .\n \n (\n 21\n )\n \n \n \n \n \n", "1008.1169_58": "", "1008.1478_99": "\n \n \n \n \n \n f\n \n H\n ⁢\n x\n ⁢\n c\n \n AA\n \n =\n \n \n \n ω\n S\n \n -\n \n \n ϵ\n \n a\n ,\n i\n \n \n \n \n ,\n \n (\n 87\n )\n \n \n \n \n \n \n", "1008.2404_28": "\n \n \n \n \n \n \n \n V\n \n t\n ,\n \n ϵ\n ′\n \n ,\n \n D\n F\n \n \n \n +\n \n |\n \n t\n ⁢\n \n (\n \n ϵ\n -\n \n ϵ\n ′\n \n \n )\n \n ⁢\n \n a\n *\n \n ⁢\n H\n \n |\n \n \n ⊂\n \n V\n \n t\n ,\n ϵ\n ,\n \n D\n F\n \n \n \n \n .\n \n \n \n \n", "1008.2997_15": "", "1008.3428_82": "", "1011.1588_26": "\n \n \n \n \n \n \n 10\n 8\n \n ⁢\n \n (\n \n \n n\n 0\n \n -\n 1\n \n )\n \n \n =\n \n 8015.514\n +\n \n 2368616\n \n 128.7459\n -\n \n 1\n /\n \n λ\n 2\n \n \n \n \n +\n \n \n 19085.73\n \n 50.01974\n -\n \n 1\n /\n \n λ\n 2\n \n \n \n \n ⁢\n \n (\n 27\n )\n \n \n \n \n \n \n \n", "1011.2841_54": "\n \n \n \n \n \n ∏\n i\n \n \n (\n \n \n ξ\n i\n \n x\n -\n \n z\n i\n \n -\n 1\n \n \n ⁢\n \n e\n \n ε\n ⁢\n \n (\n \n ξ\n i\n \n )\n \n ⁢\n t\n \n \n \n )\n \n \n \n \n \n", "1011.2927_32": "", "1011.2966_6": "", "1011.3204_2": "\n \n \n \n \n \n p\n ±\n \n =\n \n \n 1\n \n 2\n \n \n ⁢\n \n (\n \n E\n ±\n \n p\n z\n \n \n )\n \n ⁢\n \n (\n 3.2\n )\n \n \n \n \n \n \n", "1011.3429_6": "\n \n \n \n \n \n \n \n Σ\n *\n \n ∩\n \n Σ\n 0\n \n \n =\n 0\n \n .\n \n (\n 7\n )\n \n \n \n \n \n", "1011.4638_9": "\n \n \n \n \n \n \n |\n \n ℳ\n \n \n g\n ⁢\n g\n \n →\n \n g\n ⁢\n g\n \n \n \n |\n \n 2\n \n =\n \n \n \n 9\n 2\n \n ⁢\n \n g\n 4\n \n ⁢\n \n \n s\n 2\n \n \n t\n 2\n \n \n ⁢\n \n [\n \n \n a\n 7\n \n +\n \n \n a\n 8\n \n ⁢\n \n \n t\n 2\n \n \n s\n 2\n \n \n \n +\n \n \n a\n 9\n \n ⁢\n \n \n t\n 3\n \n \n s\n 3\n \n \n \n \n ]\n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "1011.5630_19": "\n \n \n \n \n \n \n \n \n h\n ~\n \n P\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n x\n ⁢\n \n g\n p\n \n ⁢\n \n [\n \n \n \n h\n ~\n \n R\n \n ⁢\n \n (\n x\n )\n \n \n ]\n \n \n +\n \n x\n ⁢\n \n \n ∑\n \n q\n ≥\n 2\n \n \n \n \n Π\n q\n \n ⁢\n \n η\n q\n \n ⁢\n \n p\n q\n \n ⁢\n \n {\n \n 1\n -\n \n \n [\n \n \n \n h\n ~\n \n R\n \n ⁢\n \n (\n x\n )\n \n \n ]\n \n q\n \n \n }\n \n \n \n \n \n \n .\n \n (\n 17\n )\n \n \n \n \n \n", "1012.0504_115": "\n \n \n \n \n \n \n d\n \n d\n ⁢\n t\n \n \n ⁢\n \n (\n \n \n t\n 2\n \n ⁢\n \n exp\n ⁡\n \n [\n \n 2\n ⁢\n \n \n ∫\n t\n 1\n \n \n \n \n \n α\n ⁢\n \n (\n s\n )\n \n \n s\n \n \n ⁢\n \n 𝑑\n s\n \n \n \n \n ]\n \n \n \n )\n \n \n =\n \n 2\n ⁢\n t\n ⁢\n \n (\n \n 1\n -\n \n α\n ⁢\n \n (\n t\n )\n \n \n \n )\n \n ⁢\n \n exp\n ⁡\n \n [\n \n 2\n ⁢\n \n \n ∫\n t\n 1\n \n \n \n \n \n α\n ⁢\n \n (\n s\n )\n \n \n s\n \n \n ⁢\n \n 𝑑\n s\n \n \n \n \n ]\n \n \n \n \n \n \n \n", "1012.1246_44": "\n \n \n \n \n \n \n cos\n ⁡\n \n i\n ′′\n \n \n =\n \n C\n \n v\n H\n \n \n \n .\n \n (\n 34\n )\n \n \n \n \n \n", "1012.1358_2": "", "1012.2736_379": "\n \n \n \n \n \n \n \n u\n 2\n \n -\n \n u\n 1\n \n \n =\n \n -\n \n L\n ⁢\n \n (\n \n u\n 1\n \n )\n \n ⁢\n \n \n ∫\n 0\n 1\n \n \n \n (\n \n 1\n -\n t\n \n )\n \n ⁢\n \n D\n \n u\n ⁢\n u\n \n 2\n \n ⁢\n P\n ⁢\n \n (\n g\n ,\n \n \n u\n 1\n \n +\n \n t\n ⁢\n \n (\n \n \n u\n 2\n \n -\n \n u\n 1\n \n \n )\n \n \n \n )\n \n ⁢\n \n (\n \n \n u\n 2\n \n -\n \n u\n 1\n \n \n ,\n \n \n u\n 2\n \n -\n \n u\n 1\n \n \n )\n \n ⁢\n \n 𝑑\n t\n \n \n \n \n \n \n .\n \n (\n 5.49\n )\n \n \n \n \n \n", "1012.3765_80": "\n \n \n \n \n \n \n \n ∂\n 2\n \n ⁡\n F\n \n \n \n ∂\n ⁡\n Ω\n \n ⁢\n \n (\n j\n ,\n \n t\n ′\n \n )\n \n ⁢\n \n ∂\n ⁡\n \n Ω\n ∗\n \n \n ⁢\n \n (\n i\n ,\n t\n )\n \n \n \n =\n \n ⟨\n \n ψ\n ⁢\n \n (\n i\n ,\n t\n )\n \n ⁢\n \n ψ\n †\n \n ⁢\n \n (\n j\n ,\n \n t\n ′\n \n )\n \n \n ⟩\n \n \n \n \n \n", "1012.4008_28": "\n \n \n \n \n \n ℐ\n ⁢\n \n (\n μ\n ,\n ν\n ,\n α\n )\n \n \n ≡\n \n \n \n ∫\n μ\n ν\n \n \n \n \n y\n 2\n \n ⁢\n d\n ⁢\n y\n \n \n \n 1\n +\n \n \n y\n 2\n \n ⁢\n α\n \n \n \n \n \n ,\n \n (\n 31\n )\n \n \n \n \n \n \n", "1012.4353_6": "\n \n \n \n \n \n \n A\n μ\n \n →\n \n A\n μ\n \n =\n \n \n \n A\n ~\n \n μ\n \n +\n \n \n c\n \n M\n P\n 2\n \n \n ⁢\n \n \n ∇\n ν\n \n ⁡\n \n F\n \n ν\n ⁢\n μ\n \n \n \n \n \n \n ,\n \n (\n 5\n )\n \n \n \n \n \n", "1012.4850_13": "\n \n \n \n \n \n \n \n (\n 1.1.13\n )\n \n ⁢\n U\n ⁢\n \n (\n x\n ,\n y\n )\n \n \n =\n \n \n α\n p\n \n ⁢\n \n (\n \n \n |\n y\n |\n \n -\n \n \n (\n \n \n p\n *\n \n -\n 1\n \n )\n \n ⁢\n \n |\n x\n |\n \n \n \n )\n \n ⁢\n \n \n (\n \n \n |\n x\n |\n \n +\n \n |\n y\n |\n \n \n )\n \n \n p\n -\n 1\n \n \n \n \n ,\n \n \n \n \n", "1012.4868_6": "\n \n \n \n \n L\n \n g\n ⁢\n p\n ⁢\n E\n \n \n =\n \n \n ϵ\n \n g\n ⁢\n p\n ⁢\n a\n ⁢\n b\n \n \n ⁢\n \n ϵ\n \n E\n ⁢\n A\n ⁢\n B\n \n \n ⁢\n \n q\n \n a\n ⁢\n A\n \n T\n \n ⁢\n \n C\n \n -\n 1\n \n \n ⁢\n \n \n \n 1\n +\n \n γ\n 5\n \n \n 2\n \n \n ⁢\n \n q\n \n b\n ⁢\n B\n \n \n \n \n ,\n \n \n m\n ⁢\n a\n ⁢\n t\n ⁢\n h\n ⁢\n b\n ⁢\n b\n ⁢\n \n R\n \n \n f\n ˙\n \n ⁢\n \n q\n ˙\n \n ⁢\n E\n \n \n \n =\n \n \n ϵ\n \n \n f\n ˙\n \n ⁢\n \n q\n ˙\n \n ⁢\n \n c\n ˙\n \n ⁢\n \n d\n ˙\n \n \n \n ⁢\n \n ϵ\n \n E\n ⁢\n A\n ⁢\n B\n \n \n ⁢\n \n q\n \n \n c\n ˙\n \n ⁢\n A\n \n T\n \n ⁢\n \n C\n \n -\n 1\n \n \n ⁢\n \n \n \n 1\n -\n \n γ\n 5\n \n \n 2\n \n \n ⁢\n \n q\n \n \n d\n ˙\n \n ⁢\n B\n \n \n \n \n \n .\n \n", "1101.0439_74": "\n \n \n \n \n \n \n ψ\n n\n \n ⁢\n \n (\n \n r\n →\n \n )\n \n \n =\n \n \n \n C\n 1\n \n ⁢\n \n \n ∑\n \n j\n =\n 1\n \n M\n \n \n \n \n e\n \n -\n \n i\n ⁢\n \n (\n \n j\n -\n 1\n \n )\n \n ⁢\n \n \n 2\n ⁢\n π\n ⁢\n n\n \n M\n \n \n \n \n ⋅\n g\n \n ⁢\n \n (\n \n |\n \n \n r\n →\n \n -\n \n \n R\n →\n \n j\n \n \n |\n \n ,\n κ\n )\n \n \n \n \n ,\n \n (\n 55\n )\n \n \n \n \n \n \n", "1101.1496_50": "\n \n \n \n \n \n \n Ω\n ¯\n \n ⁢\n \n (\n \n X\n ^\n \n ,\n \n Y\n ^\n \n )\n \n ⁢\n Z\n \n =\n \n \n τ\n ⁢\n \n (\n \n X\n ^\n \n ,\n \n [\n \n Y\n ^\n \n ,\n \n Z\n ^\n \n ]\n \n )\n \n \n +\n \n τ\n ⁢\n \n (\n \n Y\n ^\n \n ,\n \n [\n \n Z\n ^\n \n ,\n \n X\n ^\n \n ]\n \n )\n \n \n +\n \n τ\n ⁢\n \n (\n \n Z\n ^\n \n ,\n \n [\n \n X\n ^\n \n ,\n \n Y\n ^\n \n ]\n \n )\n \n \n \n \n \n \n \n", "1101.1575_143": "", "1101.2109_35": "\n \n \n \n \n \n δ\n ⁢\n \n r\n 2\n \n ⁢\n \n (\n t\n )\n \n \n ∼\n \n \n \n \n (\n \n t\n τ\n \n )\n \n x\n \n \n ,\n \n (\n 34\n )\n \n \n \n \n \n \n", "1101.2409_103": "\n \n \n \n \n \n \n \n ∑\n \n k\n =\n 0\n \n 15\n \n \n \n f\n k\n \n ⁢\n \n \n log\n 2\n \n ⁡\n \n f\n k\n \n \n \n \n ≠\n \n \n ∑\n \n k\n =\n 0\n \n 15\n \n \n \n f\n k\n ′\n \n ⁢\n \n \n log\n 2\n \n ⁡\n \n f\n k\n ′\n \n \n \n \n \n .\n \n (\n 94\n )\n \n \n \n \n \n", "1101.3440_138": "\n \n \n \n \n \n \n Θ\n ⁢\n \n (\n a\n ,\n b\n ,\n k\n )\n \n \n =\n \n \n \n (\n \n -\n 1\n \n )\n \n \n \n a\n +\n b\n +\n k\n \n 2\n \n \n ⁢\n \n \n \n \n [\n \n \n \n a\n +\n b\n \n -\n k\n \n 2\n \n ]\n \n !\n \n ⁢\n \n \n [\n \n \n \n a\n -\n b\n \n +\n k\n \n 2\n \n ]\n \n !\n \n ⁢\n \n [\n \n \n \n -\n a\n \n +\n b\n +\n k\n \n 2\n \n ]\n \n ⁢\n \n \n [\n \n \n \n a\n +\n b\n +\n k\n \n 2\n \n +\n 1\n \n ]\n \n !\n \n \n \n \n \n [\n a\n ]\n \n !\n \n ⁢\n \n \n [\n b\n ]\n \n !\n \n ⁢\n \n \n [\n k\n ]\n \n !\n \n \n \n \n \n .\n \n \n \n \n", "1101.3832_7": "\n \n \n \n \n \n \n K\n c\n \n ⁢\n \n [\n f\n ]\n \n ⁢\n \n (\n z\n )\n \n \n =\n \n z\n ⁢\n \n \n (\n \n \n f\n ⁢\n \n (\n z\n )\n \n \n z\n \n )\n \n c\n \n \n \n \n \n \n", "1101.4381_39": "\n \n \n \n \n \n \n \n \n (\n 17\n )\n \n ⁢\n \n β\n δ\n \n ⁢\n \n (\n u\n )\n \n \n =\n \n \n 1\n δ\n \n ⁢\n β\n ⁢\n \n (\n \n u\n δ\n \n )\n \n \n \n ,\n \n \n β\n ⁢\n \n (\n u\n )\n \n \n =\n \n u\n \n 1\n +\n \n 4\n ⁢\n \n u\n 4\n \n \n \n \n \n \n .\n \n \n \n \n", "1101.4823_16": "\n \n \n \n \n \n \n \n f\n r\n \n ⁢\n \n (\n m\n )\n \n \n :=\n \n E\n ⁢\n \n (\n m\n ,\n …\n ,\n m\n )\n \n \n =\n \n \n m\n \n r\n -\n 1\n \n \n ⁢\n \n \n ∏\n \n p\n ∣\n m\n \n \n \n \n \n (\n \n p\n -\n 1\n \n )\n \n ⁢\n \n h\n r\n \n ⁢\n \n (\n p\n )\n \n \n \n p\n \n r\n -\n 1\n \n \n \n \n \n \n .\n \n (\n 14\n )\n \n \n \n \n \n", "1102.3978_55": "\n \n \n \n \n \n \n ζ\n n\n \n \n (\n \n m\n -\n 1\n \n )\n \n ⁢\n \n (\n \n n\n 2\n \n )\n \n ⁢\n s\n \n \n \n \n ∏\n \n i\n =\n 0\n \n \n n\n -\n 1\n \n \n \n (\n \n 1\n -\n \n \n ζ\n n\n \n s\n ⁢\n i\n \n \n ⁢\n t\n \n \n )\n \n \n \n =\n \n \n ζ\n n\n \n \n (\n \n m\n -\n 1\n \n )\n \n ⁢\n \n (\n \n n\n 2\n \n )\n \n ⁢\n s\n \n \n \n \n (\n \n \n ∏\n \n i\n =\n 0\n \n \n \n n\n g\n \n -\n 1\n \n \n \n (\n \n 1\n -\n \n \n ζ\n n\n \n s\n ⁢\n i\n \n \n ⁢\n t\n \n \n )\n \n \n )\n \n g\n \n \n =\n \n \n ζ\n n\n \n \n (\n \n m\n -\n 1\n \n )\n \n ⁢\n \n (\n \n n\n 2\n \n )\n \n ⁢\n s\n \n \n \n \n (\n \n 1\n -\n \n t\n \n n\n g\n \n \n \n )\n \n g\n \n \n =\n \n \n \n \n \n", "1102.4410_73": "\n \n \n \n \n \n T\n +\n f\n \n =\n \n \n \n 8\n ⁢\n π\n ⁢\n G\n ⁢\n \n Q\n 2\n \n ⁢\n u\n ⁢\n \n (\n \n u\n +\n 4\n \n )\n \n \n \n \n r\n ∞\n 4\n \n ⁢\n \n \n (\n \n u\n +\n 1\n \n )\n \n 2\n \n \n \n ,\n \n (\n 52\n )\n \n \n \n \n \n \n", "1102.5772_3": "", "1108.1000_6": "\n \n \n \n \n \n \n \n \n Φ\n 2\n \n ⁢\n \n (\n \n \n \n λ\n ~\n \n +\n \n ⁢\n \n (\n v\n )\n \n \n )\n \n \n =\n \n c\n ⁢\n \n (\n v\n )\n \n ⁢\n \n \n λ\n ~\n \n +\n \n ⁢\n \n (\n v\n )\n \n \n \n ,\n \n v\n ∈\n \n 𝒪\n ⁢\n \n p\n \n ⁢\n \n Σ\n \n \n \n \n ,\n \n \n \n \n", "1108.2346_15": "\n \n \n f\n ⁢\n \n (\n ℰ\n )\n \n \n =\n \n \n {\n \n \n \n \n \n \n ρ\n 1\n \n \n \n (\n \n 2\n ⁢\n π\n ⁢\n \n σ\n 2\n \n \n )\n \n \n 3\n /\n 2\n \n \n \n ⁢\n \n (\n \n \n e\n \n ℰ\n /\n \n σ\n 2\n \n \n \n -\n 1\n \n )\n \n \n \n \n \n ℰ\n >\n \n 0\n ,\n 0\n \n \n \n \n \n \n ℰ\n ≤\n 0\n \n ,\n \n \n \n \n \n \n ⁢\n \n (\n 17\n )\n \n \n \n", "1108.3511_1": "\n \n \n \n \n \n \n n\n μ\n \n =\n \n (\n \n -\n α\n \n ,\n 0\n ,\n 0\n ,\n 0\n )\n \n \n .\n \n (\n 2\n )\n \n \n \n \n \n", "1108.4329_7": "\n \n \n \n \n \n b\n *\n \n ≈\n \n \n 1\n 2\n \n -\n \n \n \n [\n \n \n 3\n 2\n \n ⁢\n \n \n \n m\n k\n ′\n \n ⁢\n \n \n (\n \n 1\n /\n 2\n \n )\n \n 3\n \n \n \n ⌊\n \n k\n /\n 2\n \n ⌋\n \n \n ⁢\n \n P\n ~\n \n \n ]\n \n \n 1\n /\n 2\n \n \n ⁢\n \n (\n 8\n )\n \n \n \n \n \n \n \n", "1108.4536_131": "\n \n \n \n \n \n 𝐅\n \n 𝐑\n i\n \n \n es\n ,\n \n N\n ⁢\n E\n ⁢\n Q\n \n \n \n ≠\n \n \n 𝐅\n \n LS\n ,\n \n 𝐑\n i\n \n \n GCE\n \n ⁢\n \n (\n 130\n )\n \n \n \n \n \n \n", "1108.4939_119": "\n \n \n \n \n \n \n \n ∫\n 0\n T\n \n \n \n ∫\n Ω\n \n \n \n \n ρ\n ⁢\n div\n ⁢\n 𝐮\n \n ¯\n \n ⁢\n \n 𝑑\n x\n \n ⁢\n \n 𝑑\n t\n \n \n \n \n ≤\n \n \n \n ∫\n Ω\n \n \n \n ρ\n \n 0\n ,\n δ\n \n \n ⁢\n \n log\n ⁡\n \n \n ρ\n \n 0\n ,\n δ\n \n \n ⁢\n d\n ⁢\n x\n \n \n \n \n -\n \n \n ∫\n Ω\n \n \n \n \n ρ\n ⁢\n \n log\n ⁡\n ρ\n \n \n ¯\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n 𝑑\n x\n \n \n \n \n \n .\n \n (\n 5.8\n )\n \n \n \n \n \n", "1108.5515_14": "\n \n \n \n \n \n \n g\n i\n \n ⁢\n \n (\n \n x\n i\n \n )\n \n \n =\n \n \n 1\n -\n \n exp\n ⁡\n \n [\n \n \n k\n i\n \n ⁢\n \n x\n i\n \n ⁢\n \n (\n \n \n f\n i\n \n -\n 1\n \n )\n \n \n ]\n \n \n \n ,\n \n (\n 15\n )\n \n \n \n \n \n \n", "1108.5621_169": "\n \n \n \n \n \n \n \n [\n \n z\n n\n \n ]\n \n ⁢\n \n [\n \n ρ\n ⁢\n \n \n (\n z\n )\n \n j\n \n ⁢\n \n H\n 0\n \n ⁢\n \n (\n z\n )\n \n \n ]\n \n \n =\n \n \n [\n \n z\n \n n\n +\n j\n \n \n ]\n \n ⁢\n \n [\n \n \n z\n j\n \n ⁢\n ρ\n ⁢\n \n \n (\n z\n )\n \n j\n \n ⁢\n \n H\n 0\n \n ⁢\n \n (\n z\n )\n \n \n ]\n \n \n \n .\n \n (\n 5.7\n )\n \n \n \n \n \n", "1111.7104_46": "\n \n \n \n \n \n \n \n 2\n ⁢\n \n (\n \n \n 2\n \n \n 3\n 2\n \n ⁢\n k\n \n \n -\n 1\n \n )\n \n \n -\n \n k\n ⁢\n \n ln\n ⁡\n 2\n \n \n \n >\n \n k\n ⁢\n \n ln\n ⁡\n 2\n \n ⁢\n \n (\n \n \n 3\n ⋅\n \n 2\n \n \n 3\n 2\n \n ⁢\n k\n \n \n \n -\n 1\n \n )\n \n \n >\n 0\n \n .\n \n (\n 47\n )\n \n \n \n \n \n", "1112.1121_186": "", "1112.2773_229": "", "1112.5396_26": "\n \n \n \n \n \n 4\n -\n \n \n max\n i\n \n ⁡\n \n \n b\n \n i\n ,\n \n m\n ⁢\n a\n ⁢\n x\n \n \n \n \n B\n i\n \n \n \n \n 4\n \n \n \n \n", "1201.1387_25": "", "1201.2580_13": "", "1201.3875_0": "\n \n \n \n \n |\n G\n :\n Z\n \n (\n G\n )\n \n |\n ≥\n \n (\n \n ∏\n \n i\n =\n 0\n \n \n n\n -\n 1\n \n \n |\n D\n \n (\n \n x\n \n p\n i\n \n \n )\n \n :\n C\n \n (\n \n x\n \n p\n i\n \n \n )\n \n |\n )\n \n |\n C\n \n (\n x\n )\n \n :\n Z\n \n (\n G\n )\n \n |\n ≥\n \n ∏\n \n i\n =\n 0\n \n \n n\n -\n 1\n \n \n |\n Z\n \n (\n G\n )\n \n |\n \n p\n n\n \n .\n \n \n \n \n", "1201.4373_14": "\n \n \n \n \n \n \n ω\n =\n \n 𝒪\n ⁢\n \n (\n \n r\n \n -\n \n (\n \n \n d\n /\n 2\n \n +\n 3\n \n )\n \n \n \n )\n \n \n \n ,\n \n \n \n ψ\n i\n \n =\n \n 𝒪\n ⁢\n \n (\n \n r\n \n -\n \n (\n \n \n d\n /\n 2\n \n +\n 2\n \n )\n \n \n \n )\n \n \n \n ,\n \n \n \n ϕ\n \n i\n ⁢\n j\n \n \n =\n \n 𝒪\n ⁢\n \n (\n \n r\n \n -\n \n (\n \n \n d\n /\n 2\n \n +\n 2\n \n )\n \n \n \n )\n \n \n \n ,\n \n \n ϕ\n =\n \n 𝒪\n ⁢\n \n (\n \n r\n \n -\n \n (\n \n \n d\n /\n 2\n \n +\n 2\n \n )\n \n \n \n )\n \n \n \n ,\n \n \n \n ψ\n i\n ′\n \n =\n \n 𝒪\n ⁢\n \n (\n \n r\n \n -\n \n (\n \n \n d\n /\n 2\n \n +\n 1\n \n )\n \n \n \n )\n \n \n \n ,\n \n \n ω\n ′\n \n =\n \n 𝒪\n ⁢\n \n (\n \n r\n \n -\n \n (\n \n \n d\n /\n 2\n \n +\n 1\n \n )\n \n \n \n )\n \n \n \n \n \n \n \n \n .\n \n \n \n \n", "1201.4468_15": "\n \n \n \n \n \n \n |\n \n ℒ\n i\n ∗\n \n |\n \n =\n \n \n ∑\n \n a\n =\n 1\n \n i\n \n \n \n ∑\n \n \n 1\n \n \n =\n \n \n ∑\n \n a\n =\n 1\n \n i\n \n \n φ\n ⁢\n \n (\n a\n )\n \n \n \n \n ,\n \n \n \n \n", "1201.5339_37": "\n \n \n \n \n \n ℋ\n on\n \n =\n \n \n \n u\n arm\n \n ⁢\n \n (\n \n \n \n \n g\n on\n \n \n \n 00\n \n \n \n g\n on\n \n \n \n \n )\n \n \n ,\n \n (\n 32\n )\n \n \n \n \n \n \n", "1201.5559_9": "\n \n \n \n \n \n L\n =\n \n \n (\n \n \n S\n 1\n \n ⊕\n \n S\n 2\n \n ⊕\n ⋯\n ⊕\n \n S\n k\n \n \n )\n \n ⁢\n \n +\n ˙\n \n ⁢\n I\n \n \n .\n \n \n \n \n", "1201.5738_12": "\n \n \n \n \n \n ω\n \n \n =\n \n \n lim\n \n η\n →\n \n ∓\n ∞\n \n \n \n ⁡\n \n ω\n ⁢\n \n (\n η\n )\n \n ⁢\n \n (\n 13\n )\n \n \n \n \n \n \n \n", "1201.6247_60": "\n \n \n \n \n \n \n G\n \n Λ\n 1\n \n (\n n\n )\n \n \n \n ⁢\n \n (\n E\n )\n \n ⁢\n ψ\n \n =\n \n \n ψ\n ⁢\n \n G\n \n Λ\n 2\n \n (\n n\n )\n \n \n \n ⁢\n \n (\n E\n )\n \n \n +\n \n \n G\n \n Λ\n 1\n \n (\n n\n )\n \n \n \n ⁢\n \n (\n E\n )\n \n ⁢\n \n (\n \n \n \n (\n \n ∇\n ⁡\n ψ\n \n )\n \n ⋅\n ∇\n \n +\n \n ∇\n ⋅\n \n (\n \n ∇\n ⁡\n ψ\n \n )\n \n \n \n )\n \n ⁢\n \n G\n \n Λ\n 2\n \n (\n n\n )\n \n \n \n ⁢\n \n (\n E\n )\n \n \n \n \n \n \n \n", "1202.3614_113": "\n \n \n \n \n \n f\n ⁢\n \n (\n t\n )\n \n \n :=\n \n \n vol\n \n 2\n ⁢\n k\n \n \n ⁢\n \n (\n \n P\n ⁢\n \n ϕ\n t\n \n ⁢\n \n (\n B\n )\n \n \n )\n \n \n =\n \n \n vol\n \n 2\n ⁢\n k\n \n \n ⁢\n \n (\n \n P\n ⁢\n \n ψ\n t\n \n ⁢\n \n (\n B\n )\n \n \n )\n \n \n \n \n \n \n", "1203.0041_58": "\n \n \n \n \n \n \n \n \n F\n 1\n \n ⁢\n \n M\n \n \n -\n \n  2\n ⁢\n u\n ⁢\n \n (\n \n 1\n -\n u\n \n )\n \n ⁢\n \n \n \n d\n ⁢\n M\n \n \n d\n ⁢\n u\n \n \n \n \n \n =\n \n M\n ⁢\n \n T\n 1\n \n \n \n .\n \n (\n 6.2\n )\n \n \n \n \n \n", "1203.0090_4": "\n \n \n \n \n \n \n \n T\n \n U\n \n 3\n ,\n 5\n \n \n \n ⁢\n \n (\n x\n ,\n y\n )\n \n \n =\n \n \n x\n 3\n \n +\n \n 2\n ⁢\n \n x\n 2\n \n \n +\n \n 3\n ⁢\n x\n \n +\n \n 3\n ⁢\n y\n \n +\n \n y\n 2\n \n \n \n .\n \n \n \n \n", "1203.0708_160": "\n \n \n \n \n \n (\n \n \n \n α\n 1\n \n -\n \n \n α\n 1\n \n ⁢\n \n γ\n 2\n \n \n \n \n \n \n A\n 1\n \n -\n \n \n A\n 1\n \n ⁢\n \n γ\n 2\n \n \n \n +\n \n α\n 2\n \n \n \n ,\n \n \n α\n 2\n \n \n 1\n -\n \n γ\n 2\n \n \n \n )\n \n ,\n \n \n \n \n", "1203.4390_13": "\n \n \n \n \n \n \n \n \n |\n \n b\n 1\n \n >\n \n ⁢\n \n |\n \n 1\n ,\n 0\n \n >\n \n \n +\n \n \n |\n \n b\n 2\n \n >\n \n ⁢\n \n |\n \n 0\n ,\n 1\n \n >\n \n \n \n =\n \n (\n \n \n \n \n |\n \n b\n 1\n \n >\n \n \n \n \n \n \n |\n \n b\n 2\n \n >\n \n \n \n \n )\n \n \n .\n \n \n \n \n", "1203.4780_213": "", "1203.4780_66": "\n \n \n \n \n \n \n A\n ⁢\n \n (\n z\n )\n \n \n =\n \n 1\n +\n \n z\n ⁢\n A\n ⁢\n \n \n (\n z\n )\n \n \n k\n +\n 1\n \n \n \n \n \n .\n \n \n \n \n", "1203.5991_72": "\n \n \n \n \n *\n *\n \n \n {\n \n S\n θ\n \n }\n \n \n θ\n >\n 0\n \n \n :\n \n 𝒜\n ℓ\n 0\n \n \n (\n Ω\n )\n \n ⟶\n \n ∩\n \n s\n ≥\n 0\n \n \n \n 𝒜\n ℓ\n s\n \n \n (\n Ω\n )\n \n ,\n *\n *\n \n \n \n \n", "1203.6047_111": "\n \n \n \n \n \n T\n g\n \n m\n ,\n l\n \n \n \n (\n f\n )\n \n =\n exp\n \n (\n l\n -\n m\n )\n \n f\n \n (\n \n \n g\n \n -\n 1\n \n \n \n ⋅\n )\n \n \n \n \n \n", "1203.6848_21": "", "1204.0665_77": "\n \n \n \n \n \n \n \n \n ∥\n \n \n \n ∇\n ⁡\n f\n \n ⁢\n \n (\n x\n )\n \n \n -\n \n \n ∇\n ⁡\n f\n \n ⁢\n \n (\n y\n )\n \n \n \n ∥\n \n *\n \n ≤\n \n \n L\n ⁢\n \n ∥\n \n x\n -\n y\n \n ∥\n \n \n ,\n \n \n for\n \n ⁢\n all\n ⁢\n x\n \n \n \n ,\n \n y\n ∈\n Q\n \n \n ,\n \n \n \n \n", "1204.4283_149": "\n \n \n \n \n \n g\n q\n \n \n (\n λ\n )\n \n :=\n det\n \n \n (\n I\n +\n B\n R\n \n (\n λ\n ,\n \n A\n 0\n \n )\n \n )\n \n \n \n ⌈\n q\n ⌉\n \n \n \n ,\n B\n =\n A\n -\n \n A\n 0\n \n ∈\n \n 𝒮\n q\n \n ,\n \n \n \n \n", "1204.4747_20": "", "1205.1249_69": "\n \n \n \n \n \n \n \n ||\n \n X\n ~\n \n ||\n \n \n B\n ⁢\n M\n ⁢\n O\n ⁢\n \n (\n \n P\n ~\n \n )\n \n \n \n ≤\n \n \n (\n \n 1\n +\n \n \n \n 2\n \n 2\n \n ⁢\n \n \n ||\n \n M\n ~\n \n ||\n \n \n B\n ⁢\n M\n ⁢\n O\n ⁢\n \n (\n \n P\n ~\n \n )\n \n \n \n \n \n )\n \n ⁢\n \n \n ||\n X\n ||\n \n \n B\n ⁢\n M\n ⁢\n O\n ⁢\n \n (\n P\n )\n \n \n \n \n \n .\n \n (\n 17\n )\n \n \n \n \n \n", "1205.5361_91": "\n \n \n \n \n \n \n |\n \n \n ∫\n \n \n \n ℒ\n ~\n \n \n f\n ,\n ϕ\n \n n\n \n ⁢\n \n φ\n \n ⁢\n \n 𝑑\n ξ\n \n \n \n -\n \n ∫\n \n \n φ\n \n ⁢\n \n 𝑑\n \n μ\n \n f\n ,\n ϕ\n \n \n \n \n \n \n |\n \n ≤\n \n C\n ⁢\n \n τ\n n\n \n ⁢\n \n \n ∥\n φ\n ∥\n \n α\n \n \n \n .\n \n \n \n \n", "1205.5919_10": "\n \n \n \n \n \n \n \n (\n \n 1\n +\n \n t\n \n -\n 2\n \n \n +\n ⋯\n +\n \n \n (\n \n t\n \n -\n 2\n \n \n )\n \n \n n\n -\n 1\n \n \n \n )\n \n ⁢\n \n V\n ~\n \n ⁢\n \n (\n t\n )\n \n \n +\n \n \n t\n \n -\n \n 2\n ⁢\n n\n \n \n \n ⁢\n V\n ⁢\n \n (\n \n L\n 0\n \n ,\n t\n )\n \n \n \n ,\n \n \n \n \n", "1205.6845_11": "\n \n \n \n \n \n \n ∥\n \n \n x\n *\n \n -\n x\n \n ∥\n \n 2\n \n ≤\n \n \n \n \n C\n 0\n ′\n \n ⁢\n ϵ\n \n +\n \n \n C\n 1\n ′\n \n ⁢\n \n k\n \n -\n \n 1\n /\n 2\n \n \n \n ⁢\n \n (\n \n \n ω\n ⁢\n \n \n ∥\n \n x\n -\n \n x\n k\n \n \n ∥\n \n 1\n \n \n +\n \n \n (\n \n 1\n -\n ω\n \n )\n \n ⁢\n \n \n ∥\n \n x\n \n \n \n T\n ~\n \n c\n \n ∩\n \n T\n 0\n c\n \n \n \n ∥\n \n 1\n \n \n \n )\n \n \n \n ,\n \n (\n 11\n )\n \n \n \n \n \n \n", "1206.2620_135": "\n \n \n \n \n \n \n \n ψ\n f\n \n ⁢\n \n (\n \n X\n \n k\n ⁢\n Δ\n \n \n ,\n t\n )\n \n \n =\n \n \n f\n ′\n \n ⁢\n \n (\n \n X\n \n k\n ⁢\n Δ\n \n \n )\n \n ⁢\n \n (\n \n \n σ\n ⁢\n \n (\n \n X\n \n k\n ⁢\n Δ\n \n \n )\n \n ⁢\n \n \n ∫\n \n k\n ⁢\n Δ\n \n t\n \n \n 𝑑\n \n W\n s\n \n \n \n \n +\n \n ξ\n ⁢\n \n (\n \n X\n \n k\n ⁢\n Δ\n \n \n )\n \n ⁢\n \n \n ∫\n \n k\n ⁢\n Δ\n \n t\n \n \n z\n ⁢\n \n μ\n ~\n \n ⁢\n \n (\n \n d\n ⁢\n s\n \n ,\n \n d\n ⁢\n z\n \n )\n \n \n \n \n \n )\n \n \n \n .\n \n \n \n \n", "1207.1843_13": "\n \n \n \n \n \n α\n ⁢\n \n (\n t\n )\n \n \n =\n \n \n \n α\n 0\n \n +\n \n δ\n ⁢\n α\n ⁢\n \n (\n t\n )\n \n \n \n ,\n \n (\n 14\n )\n \n \n \n \n \n \n", "1207.3232_98": "\n \n \n \n \n \n (\n 4.49\n )\n \n \n A\n t\n \n =\n \n c\n 1\n \n \n \n ln\n ⁡\n \n (\n \n 1\n +\n t\n \n )\n \n \n \n 1\n +\n t\n \n \n +\n \n ℓ\n t\n \n \n k\n 1\n \n ln\n \n (\n 1\n +\n t\n )\n \n )\n \n \n 4\n \n \n \n \n \n", "1207.4608_20": "\n \n \n \n \n \n \n \n (\n 15\n )\n \n ⁢\n U\n ⁢\n \n (\n x\n ,\n τ\n )\n \n \n =\n \n \n ∑\n \n \n k\n \n j\n +\n 1\n \n \n =\n 0\n \n ∞\n \n \n \n b\n \n k\n \n j\n +\n 1\n \n \n \n ⁢\n \n sin\n ⁡\n \n (\n \n \n \n \n k\n \n j\n +\n 1\n \n \n ⁢\n π\n \n L\n \n ⁢\n x\n \n )\n \n \n ⁢\n \n e\n \n -\n \n \n \n (\n \n \n \n k\n \n j\n +\n 1\n \n \n ⁢\n π\n \n L\n \n )\n \n 2\n \n ⁢\n \n (\n \n τ\n -\n \n τ\n \n n\n -\n j\n \n \n \n )\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "1207.4734_9": "\n \n \n \n \n u\n =\n \n u\n ⁢\n \n (\n z\n )\n \n \n :=\n \n z\n ⋅\n \n N\n \n 1\n /\n 2\n \n \n \n \n \n \n \n", "1207.4980_37": "", "1207.5307_54": "\n \n \n \n \n \n \n \n rank\n ⁢\n v\n \n =\n r\n \n ,\n \n \n \n χ\n ⁢\n \n (\n v\n )\n \n \n =\n χ\n \n ,\n \n \n \n \n c\n 1\n \n ⁢\n \n (\n v\n )\n \n \n ⋅\n f\n \n =\n 1\n \n \n \n .\n \n \n \n \n", "1209.2276_4": "\n \n \n \n \n \n \n \n i\n ^\n \n =\n \n \n i\n \n ⁢\n \n \n σ\n ^\n \n 1\n \n \n \n ,\n \n \n \n j\n ^\n \n =\n \n \n i\n \n ⁢\n \n \n σ\n ^\n \n 2\n \n \n \n ,\n \n \n k\n ^\n \n =\n \n \n i\n \n ⁢\n \n \n σ\n ^\n \n 3\n \n \n \n \n \n .\n \n (\n 5\n )\n \n \n \n \n \n", "1209.2749_21": "\n \n \n \n \n \n \n (\n 4.1\n )\n \n ⁢\n 0\n \n →\n M\n →\n E\n →\n N\n →\n 0\n \n \n \n \n", "1209.3718_20": "\n \n \n \n \n \n △\n ⁢\n \n (\n \n P\n -\n \n p\n 1\n \n -\n \n p\n 2\n \n \n )\n \n \n =\n \n 0\n ,\n \n (\n 19\n )\n \n \n \n \n \n \n", "1209.5419_34": "", "1210.1000_343": "\n \n \n \n \n \n \n \n (\n 6.40\n )\n \n ⁢\n \n \n sup\n \n ω\n ∈\n Ω\n \n \n ⁡\n \n ∥\n \n S\n l\n \n ∥\n \n \n \n ≤\n \n \n L\n l\n \n ⁢\n \n e\n \n C\n ⁢\n L\n \n \n \n \n .\n \n \n \n \n", "1210.1896_31": "\n \n \n \n \n \n \n S\n 2\n \n =\n \n \n p\n 2\n \n ⁢\n \n [\n \n \n \n D\n 1\n \n ⁢\n \n (\n p\n )\n \n ⁢\n \n \n ∂\n ⁡\n \n f\n 0\n I\n \n \n \n ∂\n ⁡\n p\n \n \n \n -\n \n \n \n (\n \n \n d\n ⁢\n p\n \n \n d\n ⁢\n t\n \n \n )\n \n 1\n \n ⁢\n \n f\n 0\n I\n \n ⁢\n \n (\n p\n )\n \n \n \n ]\n \n \n \n .\n \n (\n 30\n )\n \n \n \n \n \n", "1210.2699_9": "\n \n \n \n \n \n \n \n sup\n n\n \n ⁡\n \n \n ∑\n \n k\n ≠\n n\n \n \n \n \n (\n \n \n ε\n n\n \n +\n \n ε\n k\n \n \n )\n \n ⁢\n \n |\n \n \n p\n \n n\n ,\n k\n \n \n -\n \n 1\n /\n \n p\n \n n\n ,\n k\n \n \n \n \n |\n \n \n \n \n <\n ∞\n \n .\n \n \n \n \n", "1210.5625_33": "\n \n \n \n \n \n \n ∑\n \n n\n =\n 0\n \n N\n \n \n \n \n (\n \n λ\n ⁢\n \n μ\n 1\n \n \n )\n \n n\n \n ⁢\n \n \n ∥\n \n ρ\n -\n \n ρ\n ′\n \n \n ∥\n \n 1\n \n \n \n ≤\n \n \n \n ∑\n \n n\n =\n 0\n \n N\n \n \n \n λ\n n\n \n ⁢\n \n \n ∥\n \n ρ\n -\n \n ρ\n ′\n \n \n ∥\n \n 1\n \n \n \n ,\n \n (\n 33\n )\n \n \n \n \n \n \n", "1210.7485_10": "\n \n \n \n \n \n \n \n E\n ⁢\n \n [\n \n \n h\n i\n ′\n \n ⁢\n \n (\n X\n ,\n Y\n )\n \n \n ]\n \n \n =\n \n E\n ⁢\n \n [\n \n \n h\n i\n \n ⁢\n \n (\n U\n ,\n V\n )\n \n \n ]\n \n \n =\n \n α\n i\n \n \n ,\n \n i\n =\n \n 1\n ,\n …\n ,\n k\n \n \n \n .\n \n \n \n \n", "1211.0342_121": "\n \n \n \n \n \n 1\n \n θ\n ∗\n \n \n =\n \n \n \n 1\n \n θ\n h\n \n \n +\n \n 1\n \n θ\n 𝐈𝐡\n \n \n \n ,\n \n (\n 96\n )\n \n \n \n \n \n \n", "1211.0648_1": "\n \n \n \n \n \n \n A\n x\n \n (\n n\n )\n \n \n :=\n \n \n ∏\n \n j\n =\n n\n \n 1\n \n \n A\n ⁢\n \n (\n \n \n θ\n j\n \n ⁢\n x\n \n )\n \n \n \n \n .\n \n \n \n \n", "1211.3227_117": "\n \n \n \n \n \n \n \n min\n x\n \n \n ⁡\n \n {\n \n \n f\n ⁢\n \n (\n x\n )\n \n \n +\n \n \n 1\n \n 2\n ⁢\n \n t\n i\n \n \n \n ⁢\n \n \n ∥\n \n x\n -\n \n x\n i\n \n \n ∥\n \n 2\n \n \n \n }\n \n \n .\n \n (\n 4.19\n )\n \n \n \n \n \n", "1211.4408_42": "\n \n \n \n \n \n \n \n \n d\n ⁢\n \n \n ∥\n v\n ∥\n \n 2\n \n \n \n d\n ⁢\n t\n \n \n +\n \n ν\n ⁢\n \n [\n \n \n \n ∥\n \n ∇\n ⁡\n v\n \n ∥\n \n 2\n \n +\n \n \n ∥\n \n \n ∂\n z\n \n ⁡\n v\n \n ∥\n \n 2\n \n \n ]\n \n \n \n ≤\n \n c\n ⁢\n \n \n ∥\n \n G\n f\n \n ∥\n \n 2\n \n \n \n ,\n \n \n \n \n", "1211.5073_10": "\n \n \n \n \n \n p\n ⁢\n 53\n \n +\n \n M\n ⁢\n d\n ⁢\n m\n ⁢\n 2\n ⁢\n \n ⟶\n \n W\n 5\n \n \n ⁢\n M\n ⁢\n d\n ⁢\n m\n ⁢\n 2\n ⁢\n \n (\n 11\n )\n \n \n \n \n \n \n", "1211.5598_27": "\n \n \n \n \n \n \n g\n ¯\n \n =\n \n \n g\n 0\n \n ±\n \n γ\n ⁢\n \n (\n \n \n \n z\n 1\n \n +\n \n \n V\n 0\n \n ⁢\n T\n \n \n ±\n \n \n 7\n 12\n \n ⁢\n \n g\n 0\n \n ⁢\n \n T\n 2\n \n \n \n )\n \n \n \n \n .\n \n (\n 26\n )\n \n \n \n \n \n", "1211.6266_76": "\n \n \n \n \n \n \n \n ∫\n H\n \n \n \n ∫\n H\n \n \n g\n ⁢\n \n (\n \n x\n +\n y\n \n )\n \n ⁢\n \n μ\n 2\n \n ⁢\n \n (\n \n d\n ⁢\n y\n \n )\n \n ⁢\n \n μ\n 1\n \n ⁢\n \n (\n \n d\n ⁢\n x\n \n )\n \n \n \n \n =\n \n E\n ⁢\n g\n ⁢\n \n (\n \n L\n ⁢\n \n (\n t\n )\n \n \n )\n \n \n <\n ∞\n \n .\n \n \n \n \n", "1211.6705_114": "\n \n \n \n \n \n \n \n Hess\n \n \n e\n \n 2\n ⁢\n s\n ⁢\n \n ω\n 0\n \n \n \n ⁢\n \n h\n 0\n \n \n \n ⁢\n \n (\n \n 𝒱\n n\n \n )\n \n ⁢\n \n (\n \n ω\n 0\n \n )\n \n \n ≤\n \n -\n \n \n c\n 0\n \n ⁢\n \n \n ||\n \n d\n ⁢\n \n ω\n 0\n \n \n ||\n \n \n L\n 2\n \n 2\n \n \n \n \n .\n \n \n \n \n", "1212.2053_236": "\n \n \n \n \n \n n\n ≤\n \n \n lim inf\n m\n \n ⁡\n \n \n λ\n m\n \n ⁢\n \n \n lim sup\n m\n \n ⁡\n \n \n D\n U\n \n ⁢\n \n (\n \n [\n \n \n \n m\n /\n \n c\n ε\n \n \n ⁢\n n\n \n \n ]\n \n )\n \n ⁢\n \n n\n \n \n \n \n \n ≤\n \n λ\n ⁢\n χ\n ⁢\n \n (\n \n 2\n +\n ε\n \n )\n \n ⁢\n \n n\n \n \n \n ,\n \n \n \n \n", "1212.2384_34": "\n \n \n \n \n \n δ\n ⁢\n \n g\n eff\n \n \n =\n \n \n \n \n ∂\n r\n \n ⁡\n Ψ\n \n +\n \n 4\n ⁢\n π\n ⁢\n G\n ⁢\n \n ρ\n 0\n \n ⁢\n \n ξ\n r\n \n \n \n -\n \n \n \n 2\n ⁢\n \n ξ\n r\n \n \n r\n \n ⁢\n \n g\n eff\n \n \n -\n \n \n ω\n 2\n \n ⁢\n \n ξ\n r\n \n ⁢\n \n (\n 30\n )\n \n \n \n \n \n \n \n", "1212.4026_59": "\n \n \n \n \n \n \n \n U\n \n Δ\n ⁢\n x\n \n \n |\n \n \n 𝒯\n i\n \n \n =\n \n \n \n ∑\n \n ℓ\n =\n 1\n \n k\n \n \n \n U\n ℓ\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n φ\n ℓ\n \n ⁢\n \n (\n ξ\n )\n \n \n \n ,\n \n (\n 53\n )\n \n \n \n \n \n \n", "1212.5036_77": "\n \n \n \n \n X\n =\n \n \n (\n \n \n \n A\n \n \n B\n \n \n C\n \n \n \n D\n ⁢\n B\n \n \n \n F\n \n \n D\n \n \n \n H\n ⁢\n m\n ⁢\n a\n ⁢\n t\n ⁢\n h\n ⁢\n b\n ⁢\n b\n ⁢\n C\n \n \n \n D\n \n \n K\n \n \n \n L\n ⁢\n D\n \n \n \n H\n \n \n L\n \n \n P\n \n \n \n )\n \n ⁢\n \n (\n 77\n )\n \n \n \n \n \n \n", "1212.6463_1": "\n \n \n \n \n χ\n =\n \n \n \n χ\n CW\n \n +\n \n χ\n T\n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "1301.1580_28": "\n \n \n \n \n \n \n \n \n |\n \n ∇\n ⁡\n \n C\n j\n \n \n |\n \n 2\n \n =\n \n \n (\n \n 1\n -\n \n C\n j\n 2\n \n \n )\n \n ⁢\n \n (\n \n \n C\n j\n 2\n \n -\n \n a\n j\n \n \n )\n \n \n \n ,\n \n \n \n Δ\n ⁢\n \n C\n j\n \n \n =\n \n \n C\n j\n \n ⁢\n \n (\n \n \n 2\n ⁢\n \n a\n j\n \n \n -\n 1\n -\n \n C\n j\n 2\n \n \n )\n \n \n \n ,\n \n j\n =\n \n 1\n ,\n 2\n \n \n \n \n .\n \n \n \n \n", "1301.1916_1": "\n \n \n \n \n \n \n a\n ¯\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n \n -\n \n \n ∫\n \n -\n ∞\n \n \n +\n ∞\n \n \n \n \n \n \n 𝑑\n x\n \n ⁢\n \n Ψ\n *\n \n ⁢\n \n (\n x\n ,\n t\n )\n \n ⁢\n \n 𝑑\n V\n \n ⁢\n \n (\n x\n )\n \n \n /\n \n 𝑑\n x\n \n \n ⁢\n Ψ\n ⁢\n \n (\n x\n ,\n t\n )\n \n \n \n \n ,\n \n (\n 5\n )\n \n \n \n \n \n \n", "1301.2147_20": "\n \n \n \n \n \n A\n =\n \n S\n ⁢\n \n [\n \n \n \n \n A\n 1\n \n \n \n \n I\n ⁢\n 0\n \n \n \n \n A\n 1\n \n \n \n \n ]\n \n ⁢\n \n \n S\n \n -\n 1\n \n \n \n ⁢\n a\n ⁢\n n\n ⁢\n \n d\n \n ⁢\n B\n \n =\n \n L\n ⁢\n \n [\n \n \n \n \n B\n 1\n \n \n \n \n I\n ⁢\n 0\n \n \n \n \n B\n 1\n \n \n \n \n ]\n \n ⁢\n \n L\n \n -\n 1\n \n \n \n \n .\n \n \n \n \n", "1402.2963_169": "\n \n \n \n \n P\n ⁢\n r\n ⁢\n \n (\n \n −\n ⁢\n e\n ⁢\n 2\n ⁢\n \n (\n \n \n n\n ⁢\n i\n \n +\n \n 1\n ⁢\n h\n ⁢\n i\n \n +\n 1\n \n )\n \n ⁢\n −\n ⁢\n ⋯\n ⁢\n −\n ⁢\n \n (\n \n n\n ⁢\n i\n ⁢\n −\n ⁢\n 1\n ⁢\n h\n ⁢\n i\n ⁢\n −\n ⁢\n 1\n \n )\n \n ⁢\n −\n ⁢\n e\n ⁢\n 1\n \n )\n \n \n \n \n \n", "1402.4267_77": "\n \n \n \n \n \n =\n \n \n lim\n \n α\n →\n \n +\n 0\n \n \n \n ⁡\n \n ℑ\n ⁡\n \n (\n \n \n ∫\n \n -\n ∞\n \n ∞\n \n \n \n \n ϕ\n ⁢\n \n (\n u\n )\n \n \n \n α\n -\n \n i\n ⁢\n u\n \n \n \n ⁢\n \n 𝑑\n u\n \n \n \n )\n \n \n \n \n \n \n \n", "1402.5709_72": "", "1403.3729_334": "\n \n \n \n \n \n \n \n I\n ⁢\n \n (\n \n λ\n 1\n *\n \n )\n \n \n <\n \n +\n ∞\n \n \n ,\n \n \n ∫\n \n \n log\n ⁡\n \n (\n \n 1\n +\n \n y\n 2\n \n \n )\n \n \n ⁢\n \n 𝑑\n \n λ\n 1\n *\n \n \n ⁢\n \n (\n y\n )\n \n \n \n <\n ∞\n \n \n .\n \n \n \n \n", "1403.4230_1": "\n \n \n \n \n \n h\n ⁢\n \n \n (\n z\n )\n \n \n -\n \n 7\n /\n 3\n \n \n \n ⁢\n \n (\n \n \n L\n \n X\n ,\n  500\n ,\n \n \n [\n \n 0.1\n -\n 2.4\n \n ]\n \n ⁢\n keV\n \n ,\n \n r\n ⁢\n e\n ⁢\n s\n ⁢\n t\n \n \n \n \n 10\n 44\n \n \n )\n \n \n =\n \n c\n ⁢\n \n \n (\n \n \n ℳ\n \n X\n ,\n  500\n \n \n \n \n 3\n ×\n \n 10\n 14\n \n \n ⁢\n \n M\n ⊙\n \n \n \n )\n \n α\n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n", "1403.4747_5": "", "1403.6894_97": "\n \n \n \n \n \n 0\n →\n \n \n 𝒟\n min\n \n ⁢\n \n (\n A\n )\n \n \n →\n \n \n H\n 𝒯\n 1\n \n ⁢\n \n (\n ℳ\n ;\n E\n )\n \n \n \n →\n \n γ\n A\n \n \n \n \n H\n \n 1\n -\n \n (\n \n \n x\n ⁢\n \n ∂\n x\n \n \n +\n \n 1\n /\n 2\n \n \n )\n \n \n \n ⁢\n \n (\n \n ∂\n ⁡\n ℳ\n \n ,\n 𝒯\n )\n \n \n →\n 0\n \n .\n \n \n \n \n", "1403.7479_110": "\n \n \n \n \n \n \n g\n n\n \n ⋅\n x\n \n ⁢\n \n ⟶\n \n n\n →\n \n +\n ∞\n \n \n \n ⁢\n p\n \n \n \n \n", "1404.0242_104": "", "1404.0344_56": "", "1404.0500_22": "\n \n \n \n \n \n \n K\n +\n \n θ\n ⁢\n \n (\n 0\n )\n \n \n \n ⊂\n \n \n \n ⋃\n \n i\n =\n 1\n \n n\n \n \n \n B\n q\n \n ⁢\n \n (\n \n a\n i\n \n ,\n \n δ\n \n a\n i\n \n \n )\n \n \n \n +\n \n θ\n ⁢\n \n (\n 0\n )\n \n \n \n =\n \n \n ⋃\n \n i\n =\n 1\n \n n\n \n \n \n B\n q\n \n ⁢\n \n (\n \n a\n i\n \n ,\n \n δ\n \n a\n i\n \n \n )\n \n \n \n ⊂\n \n \n ⋃\n \n i\n =\n 1\n \n n\n \n \n \n B\n q\n \n ⁢\n \n [\n \n a\n i\n \n ,\n \n δ\n \n a\n 1\n \n \n ]\n \n \n \n \n .\n \n \n \n \n", "1404.1275_7": "\n \n \n \n \n \n \n \n \n (\n 1.5\n )\n \n ⁢\n \n K\n \n -\n 1\n \n \n \n ≤\n q\n ≤\n K\n \n ,\n a\n .\n e\n .\n \n in\n ⁢\n Ω\n \n \n .\n \n \n \n \n", "1404.1356_48": "", "1404.1665_18": "\n \n \n \n \n \n \n 𝒞\n ′\n \n ⁢\n \n (\n n\n )\n \n \n :=\n \n {\n \n (\n \n f\n 0\n \n ,\n \n f\n 2\n \n )\n \n ∣\n \n \n \n f\n 0\n \n ≤\n \n f\n 2\n \n ≤\n \n 2\n ⁢\n \n f\n 0\n \n \n \n ,\n \n \n f\n 2\n \n ≥\n \n n\n -\n 1\n \n \n \n }\n \n \n \n \n \n", "1404.1700_8": "\n \n \n \n \n \n \n (\n 7\n )\n \n ⁢\n \n ρ\n \n ph\n -\n pairs\n \n \n \n =\n \n \n ∑\n \n \n \n \n n\n ′\n \n ,\n \n m\n ′\n \n ,\n n\n ,\n m\n \n =\n R\n \n ,\n L\n \n \n \n \n (\n \n ⟨\n \n \n |\n \n \n n\n ′\n \n ⁢\n \n m\n ′\n \n \n ⟩\n \n ⁢\n \n ⟨\n \n n\n ⁢\n m\n \n |\n \n \n ⟩\n \n )\n \n ⁢\n \n |\n \n \n n\n ′\n \n ⁢\n \n m\n ′\n \n \n ⟩\n \n ⁢\n \n ⟨\n \n n\n ⁢\n m\n \n |\n \n \n \n \n \n \n \n", "1404.2916_28": "\n \n \n L\n ▷\n \n (\n \n x\n ⋅\n y\n \n )\n \n \n =\n \n \n (\n \n \n L\n \n (\n 1\n )\n \n \n ▷\n x\n \n )\n \n ⋅\n \n (\n \n \n L\n \n (\n 2\n )\n \n \n ▷\n y\n \n )\n \n \n \n", "1404.3275_62": "\n \n \n \n \n \n \n τ\n coll\n \n ≈\n \n \n r\n 0\n 2\n \n D\n \n \n .\n \n (\n 52\n )\n \n \n \n \n \n", "1404.4561_334": "", "1405.1743_35": "\n \n \n \n \n \n \n \n -\n \n \n g\n \n k\n +\n 1\n \n ′\n \n ⁢\n \n (\n u\n )\n \n \n \n +\n \n 2\n ⁢\n \n f\n k\n \n ⁢\n \n (\n u\n )\n \n \n \n =\n \n \n Ψ\n \n \n k\n +\n 1\n \n ,\n 0\n \n \n ⁢\n \n (\n u\n )\n \n \n \n .\n \n (\n 3.10\n )\n \n \n \n \n \n", "1405.1934_562": "\n \n \n \n \n \n \n \n |\n \n \n g\n ^\n \n ⁢\n \n (\n k\n )\n \n \n |\n \n 2\n \n ≤\n \n \n (\n \n \n |\n \n \n g\n ^\n \n ⁢\n \n (\n ξ\n )\n \n \n |\n \n +\n \n |\n \n \n \n g\n ^\n \n ⁢\n \n (\n k\n )\n \n \n -\n \n \n g\n ^\n \n ⁢\n \n (\n ξ\n )\n \n \n \n |\n \n \n )\n \n 2\n \n ≤\n \n 2\n ⁢\n \n (\n \n \n \n |\n \n \n g\n ^\n \n ⁢\n \n (\n ξ\n )\n \n \n |\n \n 2\n \n +\n \n \n |\n \n \n \n g\n ^\n \n ⁢\n \n (\n k\n )\n \n \n -\n \n \n g\n ^\n \n ⁢\n \n (\n ξ\n )\n \n \n \n |\n \n 2\n \n \n )\n \n \n \n .\n \n \n \n \n", "1405.4056_11": "\n \n \n \n \n \n H\n \n μ\n ⁢\n ν\n \n \n =\n \n \n δ\n \n μ\n ⁢\n ν\n \n \n +\n \n \n (\n \n 1\n -\n \n δ\n \n μ\n ⁢\n ν\n \n \n \n )\n \n ⁢\n \n \n H\n ¯\n \n \n μ\n ⁢\n ν\n \n \n ⁢\n \n (\n 12\n )\n \n \n \n \n \n \n \n", "1406.5882_17": "\n \n \n \n \n \n \n tan\n ⁡\n θ\n \n =\n \n y\n \n y\n ′\n \n \n \n .\n \n (\n 15\n )\n \n \n \n \n \n", "1406.7278_72": "", "1407.0472_35": "\n \n \n \n \n \n \n \n \n C\n ˇ\n \n \n x\n ¯\n \n \n \n ⊗\n R\n \n X\n \n \n ⟶\n ∼\n \n \n \n \n C\n ˇ\n \n \n x\n ¯\n \n \n \n ⊗\n R\n \n \n \n Hom\n R\n \n ⁡\n \n (\n \n L\n ⋅\n R\n \n ,\n \n F\n R\n ⋅\n \n )\n \n \n \n \n .\n \n \n \n \n", "1407.2847_120": "\n \n \n \n \n \n \n 𝐤\n ˙\n \n =\n \n -\n 𝐄\n \n \n .\n \n (\n 120\n )\n \n \n \n \n \n", "1407.3667_47": "\n \n \n \n \n \n \n \n (\n \n \n \n ∂\n 2\n \n +\n \n \n 1\n 4\n \n ⁢\n R\n \n \n -\n \n m\n 2\n \n \n )\n \n ⁢\n G\n ⁢\n \n (\n x\n ,\n \n x\n ′\n \n )\n \n \n =\n \n -\n \n δ\n ⁢\n \n (\n x\n ,\n \n x\n ′\n \n )\n \n \n \n \n .\n \n (\n 45\n )\n \n \n \n \n \n", "1407.6528_38": "\n \n \n \n \n \n Γ\n ret\n \n \n (\n t\n )\n \n =\n \n {\n \n {\n \n \n \n \n 0\n  \n \n \n \n \n \n \n for\n  \n t\n \n <\n \n 0\n \n \n ,\n \n Γ\n ⁢\n \n (\n t\n )\n \n \n \n \n \n \n \n for\n  \n t\n \n >\n 0\n \n \n \n \n \n \n \n \n \n \n", "1408.3328_38": "\n \n \n \n \n \n \n \n |\n \n \n ℬ\n 0\n \n ⁢\n \n (\n u\n ,\n v\n )\n \n \n |\n \n ≤\n \n C\n ⁢\n \n min\n ⁡\n \n {\n 1\n ,\n \n \n \n λ\n ⁢\n p\n ⁢\n ε\n \n \n ⁢\n p\n \n }\n \n \n ⁢\n \n \n ∥\n u\n ∥\n \n \n 0\n ,\n I\n \n \n ⁢\n \n \n ∥\n v\n ∥\n \n \n 0\n ,\n \n I\n ε\n \n \n \n \n \n  \n \n \n \n ∀\n u\n \n ∈\n \n S\n 1\n \n \n ,\n \n v\n ∈\n \n S\n ε\n \n \n \n \n .\n \n \n \n \n", "1409.7648_39": "\n \n \n \n \n \n \n sup\n ⁡\n \n {\n \n area\n ⁢\n \n (\n \n Φ\n ⁢\n \n (\n x\n )\n \n \n )\n \n \n :\n \n x\n ∈\n \n ∂\n ⁡\n \n I\n 5\n \n \n \n }\n \n \n =\n \n 4\n ⁢\n π\n \n \n .\n \n \n \n \n", "1409.7957_91": "\n \n \n \n \n \n \n S\n y\n n\n \n ⁢\n A\n ⁢\n \n (\n y\n )\n \n \n =\n \n \n \n Q\n n\n \n ⁢\n A\n ⁢\n \n (\n y\n )\n \n \n +\n \n \n R\n n\n \n ⁢\n \n \n B\n ⁢\n \n (\n y\n )\n \n \n y\n \n \n \n \n \n \n \n", "1409.8629_25": "\n \n \n \n \n \n \n \n (\n \n \n \n \n (\n \n k\n +\n 1\n \n )\n \n ⁢\n ρ\n \n -\n 1\n \n \n ρ\n -\n 1\n \n \n )\n \n U\n \n =\n \n \n \n ∏\n \n t\n =\n 1\n \n \n ρ\n -\n 1\n \n \n \n U\n \n \n k\n ⁢\n ρ\n \n +\n t\n \n \n \n \n \n ∏\n \n t\n =\n 1\n \n \n ρ\n -\n 1\n \n \n \n U\n t\n \n \n \n \n .\n \n \n \n \n", "1410.0796_19": "\n \n \n \n \n \n \n u\n ⁢\n \n (\n \n  \n \n x \n \n  \n \n ,\n t\n )\n \n \n ≃\n \n \n u\n h\n \n ⁢\n \n (\n \n  \n \n x \n \n  \n \n ,\n t\n )\n \n \n =\n \n \n ⨁\n \n k\n =\n 1\n \n K\n \n \n \n u\n h\n k\n \n ⁢\n \n (\n \n  \n \n x \n \n  \n \n ,\n t\n )\n \n \n \n ∈\n \n V\n h\n \n =\n \n \n ⨁\n \n k\n =\n 1\n \n K\n \n \n \n P\n N\n 2\n \n ⁢\n \n (\n \n D\n k\n \n )\n \n \n \n \n ,\n \n (\n 2.5\n )\n \n \n \n \n \n", "1410.2041_23": "\n \n \n \n \n \n U\n =\n \n -\n \n 2\n ⁢\n \n log\n ⁡\n \n ψ\n 0\n \n \n \n \n \n .\n \n (\n 24\n )\n \n \n \n \n \n", "1410.6174_126": "\n \n \n \n \n \n \n 2\n ⁢\n π\n ⁢\n \n i\n \n ⁢\n \n Res\n \n x\n =\n \n \n i\n ⁢\n r\n \n /\n k\n \n \n \n ⁢\n \n (\n \n \n \n i\n \n ⁢\n \n \n e\n \n \n π\n ⁢\n \n i\n \n ⁢\n k\n ⁢\n τ\n ⁢\n \n x\n 2\n \n \n /\n 2\n \n \n \n ⁢\n \n e\n \n -\n \n π\n ⁢\n k\n ⁢\n u\n ⁢\n x\n \n \n \n \n \n 1\n -\n \n e\n \n \n π\n ⁢\n x\n \n -\n \n \n π\n ⁢\n i\n ⁢\n r\n \n /\n k\n \n \n \n \n \n )\n \n \n =\n \n \n 2\n \n ⁢\n \n e\n \n \n -\n \n π\n ⁢\n i\n ⁢\n r\n ⁢\n u\n \n \n -\n \n \n \n π\n ⁢\n i\n ⁢\n \n r\n 2\n \n ⁢\n τ\n \n /\n 2\n \n ⁢\n k\n \n \n \n \n \n .\n \n (\n 191\n )\n \n \n \n \n \n", "1410.6587_28": "\n \n \n \n \n \n \n y\n ⁢\n \n (\n \n \n a\n ⁢\n \n θ\n y\n \n \n +\n \n θ\n x\n \n \n )\n \n \n =\n 0\n =\n \n x\n ⁢\n \n (\n \n \n b\n ⁢\n \n θ\n x\n \n \n +\n \n θ\n y\n \n \n )\n \n \n \n ,\n \n \n \n \n", "1410.7006_36": "\n \n \n \n \n \n \n \n (\n 8\n )\n \n ⁢\n \n \n ∥\n \n \n 𝐆\n E\n \n ⁢\n V\n ⁢\n u\n \n ∥\n \n 2\n \n \n ≥\n \n \n \n 1\n 2\n \n ⁢\n \n \n ∥\n \n \n \n X\n ⟂\n \n ⁢\n u\n \n -\n \n V\n ⁢\n \n (\n λ\n )\n \n ⁢\n V\n ⁢\n u\n \n \n ∥\n \n 2\n \n \n -\n \n \n ∥\n \n V\n ⁢\n \n 𝐆\n E\n \n ⁢\n u\n \n ∥\n \n 2\n \n \n \n .\n \n \n \n \n", "1410.7496_8": "\n \n \n \n \n \n \n ∥\n \n ω\n ~\n \n ∥\n \n ≤\n \n \n \n ∑\n \n i\n =\n 2\n \n N\n \n \n \n (\n \n \n υ\n i\n \n +\n \n υ\n 1\n \n \n )\n \n 2\n \n \n \n \n .\n \n (\n 8\n )\n \n \n \n \n \n", "1411.1547_15": "\n \n \n \n \n \n \n χ\n 0\n \n ⁢\n \n (\n \n \n q\n =\n 0\n \n ,\n \n \n ω\n m\n \n =\n 0\n \n \n )\n \n \n =\n \n T\n ⁢\n \n \n ∑\n n\n \n \n \n ∑\n \n 𝐩𝐩\n ′\n \n \n \n \n Φ\n \n 𝐩𝐩\n ′\n \n \n ⁢\n \n (\n \n ε\n n\n \n )\n \n ⁢\n \n (\n 16\n )\n \n \n \n \n \n \n \n \n \n", "1411.1786_728": "\n \n \n \n \n \n \n \n (\n f\n ,\n P\n )\n \n ↦\n \n \n \n ∂\n β\n \n ⁡\n \n [\n \n \n J\n \n x\n ¯\n \n \n ⁢\n \n T\n \n Q\n ^\n \n \n ⁢\n \n (\n f\n ,\n P\n )\n \n \n ]\n \n \n ⁡\n \n (\n \n x\n ¯\n \n )\n \n \n \n  \n \n \n \n for\n \n ⁢\n \n every\n \n ⁢\n β\n \n ∈\n ℳ\n \n \n .\n \n \n \n \n", "1411.1909_122": "\n \n \n \n \n \n ν\n ~\n \n =\n \n \n \n \n λ\n ~\n \n ⁢\n \n χ\n \n Ω\n ~\n \n \n \n +\n \n \n μ\n ~\n \n ⁢\n \n χ\n \n \n ℳ\n ~\n \n ∖\n \n Ω\n ~\n \n \n \n \n \n ,\n \n (\n 6.1\n )\n \n \n \n \n \n \n", "1411.2716_110": "\n \n \n \n \n \n \n I\n ⁢\n \n d\n E\n \n \n =\n \n ∑\n \n \n s\n i\n \n ⊗\n \n s\n i\n \n *\n \n h\n ~\n \n \n \n \n \n \n ,\n \n \n \n \n", "1411.3694_15": "\n \n \n \n \n \n γ\n ⁢\n \n (\n t\n )\n \n \n =\n \n (\n \n \n b\n \n ⁢\n \n cos\n ⁡\n \n (\n \n t\n a\n \n )\n \n \n \n ,\n \n \n \n ∫\n 0\n \n t\n /\n a\n \n \n \n \n \n \n \n \n a\n 2\n \n -\n \n \n b\n 2\n \n ⁢\n \n sin\n ⁡\n \n (\n \n s\n a\n \n )\n \n \n \n \n \n \n ⁢\n \n 𝑑\n s\n \n \n \n )\n \n \n \n \n \n", "1412.6680_42": "\n \n \n \n \n \n \n f\n ˙\n \n ⁢\n \n (\n a\n )\n \n \n =\n \n \n \n \n \n C\n 1\n \n ⁢\n \n a\n 2\n \n \n +\n \n \n C\n 2\n \n ⁢\n a\n \n +\n \n C\n 3\n \n \n \n \n σ\n n\n 2\n \n ⁢\n ξ\n ⁢\n \n a\n 0\n 2\n \n ⁢\n \n \n (\n \n 1\n +\n a\n \n )\n \n 2\n \n \n \n ,\n \n (\n 51\n )\n \n \n \n \n \n \n", "1412.8240_159": "\n \n \n \n \n \n \n \n (\n 42\n )\n \n ⁢\n \n K\n *\n \n ⁢\n \n (\n \n \n C\n *\n \n ⁢\n \n (\n \n \n C\n 0\n \n ⁢\n \n (\n X\n ,\n D\n )\n \n \n ,\n α\n ;\n J\n )\n \n \n )\n \n \n ≅\n \n \n K\n *\n \n ⁢\n \n (\n X\n ,\n φ\n ,\n \n \n {\n \n α\n x\n \n }\n \n \n x\n ∈\n Δ\n \n \n ;\n Y\n )\n \n \n \n .\n \n \n \n \n", "1501.00036_3": "\n \n \n \n \n \n S\n h\n \n =\n \n \n 𝒜\n ⁢\n \n (\n \n \n \n z\n h\n \n ⁢\n \n e\n \n -\n \n i\n ⁢\n \n k\n 0\n \n ⁢\n h\n \n \n \n \n -\n 1\n \n )\n \n ⁢\n \n [\n \n \n e\n \n i\n ⁢\n \n \n \n k\n 0\n \n ⁢\n \n r\n \n o\n ⁢\n u\n ⁢\n t\n ⁢\n e\n ⁢\n r\n \n 2\n \n \n \n 2\n ⁢\n \n (\n \n X\n -\n h\n \n )\n \n \n \n \n \n -\n \n e\n \n i\n ⁢\n \n \n \n k\n 0\n \n ⁢\n \n r\n \n i\n ⁢\n n\n ⁢\n n\n ⁢\n e\n ⁢\n r\n \n 2\n \n \n \n 2\n ⁢\n \n (\n \n X\n -\n h\n \n )\n \n \n \n \n \n \n ]\n \n \n ,\n \n (\n 4\n )\n \n \n \n \n \n \n", "1501.00423_93": "\n \n \n \n \n \n \n \n ∀\n x\n \n ∈\n Ω\n \n ,\n \n \n -\n \n \n \n ∥\n l\n ∥\n \n ∞\n \n λ\n \n \n ≤\n \n \n u\n λ\n \n ⁢\n \n (\n x\n )\n \n \n ≤\n \n \n \n ∥\n l\n ∥\n \n ∞\n \n λ\n \n \n \n .\n \n (\n 35\n )\n \n \n \n \n \n", "1501.00551_1": "\n \n \n \n \n \n \n K\n 2\n \n ⁢\n \n (\n f\n ,\n Λ\n )\n \n \n :=\n \n Ker\n ⁡\n \n (\n \n \n f\n *\n \n :\n \n \n \n H\n 2\n \n ⁢\n \n (\n X\n ;\n Λ\n )\n \n \n →\n \n \n H\n 2\n \n ⁢\n \n (\n P\n ;\n Λ\n )\n \n \n \n \n )\n \n \n \n \n \n \n", "1501.03233_87": "", "1501.06443_22": "\n \n \n \n \n \n Ψ\n ⁢\n \n (\n 13\n ,\n 13\n ,\n 1\n )\n \n \n =\n \n 0.1\n ⁢\n …\n \n >\n \n 1\n /\n 15\n \n =\n \n 0.066\n ⁢\n …\n \n \n \n \n \n", "1501.06630_101": "\n \n \n \n \n \n \n f\n ⁢\n \n (\n ξ\n ;\n \n ψ\n *\n \n ,\n 0\n )\n \n \n ≤\n \n \n ∑\n \n l\n =\n 1\n \n s\n \n \n \n c\n \n l\n ,\n d\n \n \n ⁢\n f\n ⁢\n \n (\n ξ\n ;\n \n ψ\n \n l\n ,\n d\n \n \n ,\n \n π\n \n l\n ,\n d\n \n \n )\n \n ⁢\n \n ∀\n ξ\n \n \n \n \n ,\n \n \n \n \n", "1502.01902_28": "\n \n \n \n \n \n \n \n \n κ\n m\n \n ⁢\n \n log\n ⁡\n \n s\n 0\n \n \n ⁢\n \n (\n τ\n )\n \n \n +\n \n \n s\n 0\n \n ⁢\n \n (\n τ\n )\n \n \n \n =\n \n A\n -\n \n κ\n ⁢\n τ\n \n \n \n ,\n \n \n c\n 0\n \n =\n \n \n \n s\n 0\n \n \n \n κ\n m\n \n +\n \n s\n 0\n \n \n \n ,\n \n (\n 3.11\n )\n \n \n \n \n \n \n \n", "1502.03128_14": "\n \n \n \n \n \n C\n =\n \n {\n \n σ\n ⁢\n \n (\n \n \n n\n -\n k\n \n +\n 1\n \n )\n \n \n ,\n …\n ,\n \n σ\n ⁢\n \n (\n \n n\n +\n 1\n \n )\n \n \n }\n \n \n ,\n \n \n \n \n", "1502.03303_207": "\n \n \n \n \n lim\n \n T\n ↑\n ∞\n \n \n ⁡\n \n \n lim\n \n θ\n ↑\n ∞\n \n \n ⁡\n \n (\n \n \n \n ξ\n ⋅\n \n A\n \n T\n ,\n θ\n \n \n (\n \n \n p\n 0\n \n +\n p\n \n )\n \n \n \n ⁢\n ξ\n \n -\n \n \n ξ\n ⋅\n \n A\n \n T\n ,\n θ\n \n \n (\n \n p\n 0\n \n )\n \n \n \n ⁢\n ξ\n \n \n )\n \n \n \n =\n \n \n lim\n \n T\n ↑\n ∞\n \n \n ⁡\n \n (\n \n \n \n ξ\n ⋅\n \n A\n T\n \n (\n \n \n p\n 0\n \n +\n p\n \n )\n \n \n \n ⁢\n ξ\n \n -\n \n \n ξ\n ⋅\n \n A\n T\n \n (\n \n p\n 0\n \n )\n \n \n \n ⁢\n ξ\n \n \n )\n \n \n =\n \n \n \n ξ\n ⋅\n \n A\n hom\n \n (\n \n \n p\n 0\n \n +\n p\n \n )\n \n \n \n ⁢\n ξ\n \n -\n \n \n ξ\n ⋅\n \n A\n hom\n \n (\n \n p\n 0\n \n )\n \n \n \n ⁢\n ξ\n \n \n \n .\n \n (\n 5.20\n )\n \n \n", "1502.03557_27": "\n \n \n \n \n \n \n \n (\n 5\n )\n \n ⁢\n \n \n lim\n \n n\n →\n \n +\n ∞\n \n \n \n ⁡\n \n \n \n t\n λ\n \n ⁢\n \n (\n \n n\n ⁢\n x\n \n )\n \n \n n\n \n \n \n =\n \n \n lim\n \n n\n →\n \n +\n ∞\n \n \n \n ⁡\n \n \n \n σ\n λ\n \n ⁢\n \n (\n \n n\n ⁢\n x\n \n )\n \n \n n\n \n \n =\n \n \n μ\n λ\n \n ⁢\n \n (\n x\n )\n \n \n \n .\n \n \n \n \n", "1502.04398_11": "\n \n \n \n \n \n \n \n (\n 2\n )\n \n ⁢\n u\n ⁢\n \n (\n t\n )\n \n \n =\n \n \n \n e\n \n t\n ⁢\n Δ\n \n \n ⁢\n g\n \n +\n \n \n ∫\n 0\n t\n \n \n \n e\n \n \n (\n \n t\n -\n s\n \n )\n \n ⁢\n Δ\n \n \n ⁢\n m\n ⁢\n \n (\n s\n )\n \n ⁢\n \n u\n x\n 2\n \n ⁢\n \n (\n s\n )\n \n ⁢\n \n 𝑑\n s\n \n \n \n \n \n .\n \n \n \n \n", "1502.05136_39": "", "1502.05826_1": "\n \n \n \n \n \n W\n ch\n \n =\n \n \n (\n \n 1\n -\n \n c\n 2\n \n \n )\n \n 2\n \n \n \n \n \n", "1502.06172_0": "\n \n \n \n \n \n C\n \n i\n ⁢\n j\n \n \n =\n \n \n f\n ⁢\n \n (\n \n A\n \n i\n ⁢\n j\n \n \n )\n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "1502.06417_11": "", "1502.07713_21": "\n \n \n \n \n \n \n 1\n 4\n \n ⁢\n τ\n ⁢\n \n (\n G\n )\n \n \n \n ≤\n ∃\n \n \n \n κ\n ⁢\n \n (\n 𝒢\n )\n \n \n \n \n κ\n f\n \n ⁢\n \n (\n 𝒢\n )\n \n \n \n \n \n \n \n", "1503.00283_23": "\n \n \n \n \n \n \n \n \n \n \n (\n 2.19\n )\n \n ⁢\n \n \n ∂\n x\n m\n \n ⁡\n θ\n \n \n |\n \n \n Γ\n 1\n \n \n =\n 0\n \n ,\n \n \n \n \n \n and\n \n ⁢\n \n \n ∂\n y\n m\n \n ⁡\n θ\n \n \n |\n \n \n Γ\n 2\n \n \n =\n 0\n \n \n ,\n \n \n \n \n", "1503.05419_23": "", "1503.07319_11": "\n \n \n \n \n \n \n V\n g\n 2\n \n =\n \n \n V\n ⁢\n \n \n ∂\n ⁡\n \n V\n g\n \n \n \n ∂\n ⁡\n θ\n \n \n ⁢\n \n sin\n ⁡\n θ\n \n \n +\n \n V\n ⁢\n \n V\n g\n \n ⁢\n \n cos\n ⁡\n θ\n \n \n \n \n .\n \n (\n 12\n )\n \n \n \n \n \n", "1503.07348_190": "", "1504.03591_46": "\n \n \n \n \n \n \n \n ℒ\n ⁢\n W\n ⁢\n \n (\n z\n )\n \n \n =\n \n \n z\n k\n \n ⁢\n \n \n \n ∫\n 0\n ∞\n \n \n \n \n \n e\n \n -\n \n z\n ⁢\n x\n \n \n \n \n ⁢\n f\n ⁢\n \n (\n x\n )\n \n ⁢\n \n 𝑑\n x\n \n \n \n \n \n ,\n \n \n ℜ\n ⁡\n \n \n e\n \n ⁢\n z\n \n \n >\n σ\n \n \n .\n \n \n \n \n", "1504.05560_4": "\n \n \n \n \n \n Δ\n ⁢\n \n X\n T\n \n \n =\n \n \n \n 2\n ⁢\n \n R\n c\n \n \n -\n \n S\n ⁢\n \n cos\n ⁡\n \n [\n \n 2\n ⁢\n \n (\n \n \n \n R\n c\n \n ⁢\n K\n \n +\n \n π\n 4\n \n \n )\n \n \n ]\n \n \n \n \n +\n \n A\n ⁢\n \n cos\n ⁡\n \n (\n \n w\n ⁢\n τ\n \n )\n \n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n \n", "1504.05674_2": "\n \n \n \n \n \n \n α\n ⁢\n \n J\n α\n *\n \n ⁢\n \n (\n i\n )\n \n \n =\n \n \n min\n \n a\n ∈\n \n A\n ⁢\n \n (\n i\n )\n \n \n \n \n ⁡\n \n {\n \n \n c\n ⁢\n \n (\n i\n ,\n a\n )\n \n \n +\n \n \n ∑\n \n j\n ∈\n S\n \n \n \n \n J\n α\n *\n \n ⁢\n \n (\n j\n )\n \n ⁢\n q\n ⁢\n \n (\n \n j\n |\n \n i\n ,\n a\n \n \n )\n \n \n \n \n }\n \n \n \n .\n \n (\n 3\n )\n \n \n \n \n \n", "1504.06004_182": "\n \n \n \n \n \n u\n ∈\n \n \n D\n *\n \n ⁢\n F\n ⁢\n \n (\n \n x\n ¯\n \n ,\n \n y\n ¯\n \n )\n \n ⁢\n \n (\n v\n )\n \n ⁢\n \n and\n \n ⁢\n v\n \n ∈\n \n N\n ⁢\n \n (\n \n y\n ¯\n \n ;\n Θ\n )\n \n \n \n .\n \n \n \n \n", "1504.06101_204": "\n \n \n \n \n \n \n F\n ⁢\n \n (\n ξ\n )\n \n \n ≥\n \n \n F\n ⁢\n \n (\n \n \n \n θ\n \n ⁢\n ξ\n \n +\n \n \n (\n \n 1\n -\n θ\n \n )\n \n ⁢\n \n ξ\n ′\n \n \n \n )\n \n \n +\n \n \n (\n \n 1\n -\n θ\n \n )\n \n ⁢\n \n ⟨\n ζ\n ,\n \n ξ\n -\n \n ξ\n ′\n \n \n ⟩\n \n \n +\n \n \n \n μ\n 36\n \n \n ⁢\n \n \n (\n \n 1\n -\n θ\n \n )\n \n 2\n \n ⁢\n \n \n |\n \n \n ξ\n ′\n \n -\n ξ\n \n |\n \n 2\n \n \n \n \n ,\n \n \n \n \n", "1504.07554_19": "\n \n \n \n \n \n \n v\n ⁢\n \n (\n t\n ,\n τ\n )\n \n \n =\n \n \n α\n ⁢\n \n a\n 0\n \n ⁢\n \n sin\n ⁡\n \n (\n \n β\n ⁢\n \n ω\n 0\n \n ⁢\n t\n \n )\n \n \n ⁢\n \n cosh\n ⁡\n \n (\n \n β\n ⁢\n \n ω\n 0\n \n ⁢\n τ\n \n )\n \n \n \n -\n \n \n a\n 0\n \n ⁢\n \n sin\n ⁡\n \n (\n \n \n ω\n 0\n \n ⁢\n t\n \n )\n \n \n ⁢\n \n sinh\n ⁡\n \n (\n \n \n ω\n 0\n \n ⁢\n τ\n \n )\n \n \n \n \n \n .\n \n (\n 22\n )\n \n \n \n \n \n", "1505.01766_31": "\n \n \n \n \n \n \n \n s\n α\n \n ⁢\n E\n ⁢\n \n (\n F\n ;\n v\n )\n \n ⁢\n \n s\n β\n *\n \n \n =\n \n \n s\n α\n \n ⁢\n E\n ⁢\n \n (\n \n F\n ∪\n \n {\n \n α\n ⁢\n v\n \n ,\n \n β\n ⁢\n v\n \n }\n \n \n ;\n v\n )\n \n ⁢\n \n s\n β\n *\n \n \n \n .\n \n \n \n \n", "1505.02647_40": "\n \n \n \n \n \n \n \n \n Z\n ⁢\n \n (\n K\n )\n \n \n =\n \n \n 2\n \n K\n -\n 1\n \n \n ⁢\n \n \n e\n \n \n \n \n π\n 2\n \n \n ⁢\n K\n \n \n \n 12\n \n ⁢\n δ\n \n \n \n \n ⁢\n \n (\n \n \n e\n \n -\n \n \n φ\n ⁢\n \n (\n K\n )\n \n \n δ\n \n \n \n +\n \n e\n \n -\n \n \n χ\n ⁢\n \n (\n K\n )\n \n \n δ\n \n \n \n \n )\n \n \n \n \n \n ,\n \n (\n 36\n )\n \n \n \n \n \n", "1505.03807_27": "\n \n \n \n \n \n \n Tr\n A\n \n ⁢\n \n [\n \n \n f\n ′\n \n ⁢\n \n (\n \n ρ\n \n A\n ⁢\n B\n \n ′\n \n )\n \n \n ,\n \n ρ\n \n A\n ⁢\n B\n \n \n ]\n \n \n =\n \n \n 0\n \n ,\n \n (\n 27\n )\n \n \n \n \n \n \n", "1505.06785_83": "\n \n \n \n \n \n \n ℒ\n ⁢\n \n (\n \n \n Ext\n ⋅\n \n ⁢\n \n (\n F\n )\n \n \n )\n \n ⁢\n \n [\n v\n ,\n \n v\n ¯\n \n ]\n \n \n =\n \n 2\n ⁢\n \n \n ∫\n \n M\n 0\n \n \n \n \n \n |\n \n η\n v\n \n |\n \n 2\n \n \n |\n \n q\n 0\n \n |\n \n \n \n \n \n ,\n \n \n \n \n", "1505.07224_129": "", "1506.00023_61": "\n \n \n \n \n \n δ\n 1\n \n :=\n inf\n \n {\n \n \n (\n \n ℒ\n 1\n \n P\n ,\n P\n )\n \n \n L\n 2\n \n \n ;\n P\n ∈\n \n H\n 4\n \n ,\n ∥\n P\n \n ∥\n \n L\n 2\n \n \n =\n 1\n ,\n \n \n (\n P\n ,\n ϕ\n )\n \n \n L\n 2\n \n \n =\n \n \n (\n P\n ,\n \n ϕ\n ′\n \n )\n \n \n L\n 2\n \n \n =\n 0\n }\n \n .\n \n (\n 4.10\n )\n \n \n \n \n \n", "1506.00732_55": "\n \n \n \n \n \n ⊆\n \n \n R\n ⁢\n \n (\n M\n )\n \n \n +\n \n [\n \n [\n \n R\n ⁢\n \n (\n M\n )\n \n \n ,\n \n L\n ⁢\n D\n ⁢\n e\n ⁢\n \n r\n l\n \n ⁢\n \n (\n M\n )\n \n \n ]\n \n ,\n \n R\n ⁢\n \n (\n M\n )\n \n \n ]\n \n +\n \n [\n \n R\n ⁢\n \n (\n M\n )\n \n \n ,\n \n [\n \n R\n ⁢\n \n (\n M\n )\n \n \n ,\n \n L\n ⁢\n D\n ⁢\n e\n ⁢\n \n r\n l\n \n ⁢\n R\n ⁢\n \n (\n M\n )\n \n \n ]\n \n ]\n \n \n ⊆\n \n \n \n \n \n", "1506.01454_210": "\n \n \n \n \n \n \n (\n \n U\n ⊗\n 𝟏\n \n )\n \n ⁢\n v\n \n =\n \n w\n ⁢\n \n (\n \n U\n ⊗\n 𝟏\n \n )\n \n \n \n \n \n \n", "1506.03207_9": "\n \n \n \n \n \n \n Δ\n ⁢\n g\n \n =\n \n \n 4\n ⁢\n m\n ⁢\n \n g\n 0\n \n ⁢\n \n v\n 0\n \n \n /\n c\n \n \n .\n \n (\n 9\n )\n \n \n \n \n \n", "1506.04914_15": "\n \n \n \n \n \n \n f\n fluc\n \n =\n \n \n n\n f\n \n \n N\n s\n \n \n =\n \n α\n -\n \n \n \n β\n ⁢\n \n (\n α\n )\n \n \n \n N\n s\n \n \n \n \n \n ,\n \n (\n 16\n )\n \n \n \n \n \n", "1506.06260_9": "", "1506.07767_32": "\n \n \n \n \n \n \n \n ∫\n B\n \n \n \n g\n u\n p\n \n ⁢\n \n d\n μ\n \n \n \n ≤\n \n \n lim inf\n \n j\n →\n ∞\n \n \n ⁡\n \n \n ∫\n B\n \n \n \n g\n \n u\n j\n \n p\n \n ⁢\n \n d\n μ\n \n \n \n \n \n .\n \n \n \n \n", "1507.00050_6": "\n \n \n \n \n \n \n \n ∇\n T\n \n ⋅\n A\n ⋅\n \n ∇\n ⁡\n θ\n \n \n ⁢\n \n (\n x\n ,\n y\n ,\n z\n )\n \n \n =\n \n 0\n ,\n \n (\n 5\n )\n \n \n \n \n \n \n", "1507.00077_60": "\n \n \n \n \n \n (\n \n \n 0\n ~\n \n ⁢\n \n |\n \n det\n ⁡\n \n S\n ~\n \n \n |\n \n ⁢\n 0\n \n )\n \n =\n \n det\n ⁡\n \n \n Θ\n \n l\n ⁢\n m\n \n \n ⁢\n \n (\n \n \n 0\n ~\n \n ⁢\n \n |\n \n det\n ⁡\n \n (\n \n 1\n +\n \n \n 1\n \n 2\n ⁢\n π\n \n \n ⁢\n \n U\n T\n \n ⁢\n \n (\n k\n ,\n q\n )\n \n \n \n )\n \n \n |\n \n ⁢\n 0\n \n )\n \n ⁢\n \n (\n 40\n )\n \n \n \n \n \n \n \n", "1507.01742_1": "\n \n \n \n \n \n 𝐡\n ⁢\n \n (\n 𝐤\n )\n \n \n =\n \n \n (\n \n \n v\n F\n \n ⁢\n \n k\n x\n \n \n ,\n \n \n v\n F\n \n ⁢\n \n k\n y\n \n \n ,\n 0\n )\n \n ⁢\n \n (\n 2\n )\n \n \n \n \n \n \n", "1507.02365_3": "\n \n \n \n \n \n \n β\n n\n \n =\n \n \n (\n sgn\n )\n \n ⁢\n \n ψ\n n\n \n \n \n .\n \n \n \n \n", "1507.05296_0": "", "1507.05297_42": "\n \n \n \n \n \n \n |\n \n {\n i\n :\n \n x\n ∈\n \n B\n ⁢\n \n (\n \n x\n i\n \n ,\n \n r\n i\n \n )\n \n \n \n }\n \n |\n \n ≤\n \n \n c\n 4\n \n ⁢\n forall\n ⁢\n x\n \n ∈\n \n B\n ⁢\n \n (\n w\n ,\n R\n )\n \n \n \n .\n \n (\n 2.15\n )\n \n \n \n \n \n", "1507.05994_1": "\n \n \n \n \n \n 𝒚\n ℓ\n \n =\n \n \n \n \n \n ρ\n ⁢\n K\n \n \n ⁢\n \n 𝑯\n ℓ\n \n (\n N\n )\n \n \n ⁢\n \n 𝒛\n ℓ\n \n \n +\n \n 𝒏\n ℓ\n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "1508.00830_44": "\n \n \n \n \n \n \n \n (\n 21\n )\n \n ⁢\n H\n ⁢\n \n (\n \n P\n A\n \n )\n \n \n ≤\n \n \n L\n M\n \n ⁢\n H\n ⁢\n \n (\n P\n )\n \n ⁢\n \n \n ∏\n \n i\n =\n 1\n \n L\n \n \n h\n ⁢\n \n \n (\n \n 𝒙\n i\n \n )\n \n M\n \n \n \n \n \n .\n \n \n \n \n", "1508.03448_46": "\n \n \n \n \n \n \n Θ\n ⁢\n \n (\n \n 𝐮\n \n (\n j\n )\n \n \n )\n \n \n -\n \n Θ\n ⁢\n \n (\n \n \n 𝐮\n ^\n \n \n (\n j\n )\n \n \n )\n \n \n \n =\n \n \n \n ∑\n \n i\n =\n 1\n \n 8\n \n \n \n T\n ~\n \n i\n \n \n ,\n \n (\n 35\n )\n \n \n \n \n \n \n", "1508.03932_47": "\n \n \n \n \n \n \n \n \n σ\n k\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n 1\n \n k\n !\n \n \n ⁢\n \n \n ∫\n 0\n x\n \n \n \n y\n k\n \n ⁢\n \n \n e\n \n -\n y\n \n \n \n ⁢\n \n 𝑑\n y\n \n \n \n \n \n ,\n \n \n \n ω\n k\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n e\n \n -\n x\n \n \n ⁢\n \n \n ∑\n \n s\n =\n 0\n \n k\n \n \n \n x\n s\n \n \n s\n !\n \n \n \n \n \n \n .\n \n \n \n \n", "1508.04279_40": "\n \n \n \n \n \n \n Δ\n \n p\n /\n 2\n \n \n ⁢\n \n (\n \n \n \n (\n \n J\n ⁢\n \n A\n 0\n \n \n )\n \n *\n \n ⁢\n \n (\n \n J\n ⁢\n \n A\n 0\n \n \n )\n \n \n )\n \n \n =\n \n \n Δ\n \n p\n /\n 2\n \n \n ⁢\n \n (\n \n \n A\n 0\n *\n \n ⁢\n \n A\n 0\n \n \n )\n \n \n \n \n \n \n", "1508.04374_20": "\n \n \n \n \n \n \n \n \n (\n \n 1\n -\n \n \n (\n \n \n Λ\n j\n \n /\n \n Λ\n 0\n \n \n )\n \n \n 1\n /\n m\n \n \n \n )\n \n \n (\n \n 1\n -\n \n \n q\n ⁢\n \n Λ\n 0\n \n \n /\n \n Λ\n j\n \n \n \n )\n \n \n \n ⁢\n \n \n \n Q\n m\n \n m\n \n \n ⁢\n \n \n ϕ\n 0\n \n \n \n C\n \n 0\n ,\n j\n \n \n ⁢\n \n (\n m\n )\n \n \n \n \n ,\n \n \n \n \n", "1508.05068_45": "\n \n \n \n \n \n \n \n \n \n \n \n f\n 1\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n \n 52.0141\n +\n \n 10.3198\n ⁢\n x\n \n \n -\n \n 38.8957\n ⁢\n \n x\n 2\n \n \n \n +\n \n 4.70555\n ⁢\n \n x\n 3\n \n \n +\n \n 4.89777\n ⁢\n \n x\n 4\n \n \n +\n \n 4.6887\n ⁢\n \n x\n 5\n \n \n +\n \n 0.476241\n ⁢\n \n x\n 6\n \n \n +\n \n 7.49464\n ⁢\n \n x\n 7\n \n \n \n \n ,\n \n \n \n \n ⁢\n \n (\n 40\n )\n \n \n \n \n \n", "1508.06219_0": "\n \n \n \n \n \n R\n ⁢\n a\n \n =\n \n \n \n g\n ⁢\n α\n ⁢\n Δ\n ⁢\n T\n ⁢\n \n H\n 3\n \n \n \n ν\n ⁢\n κ\n \n \n ,\n \n (\n 1.0\n )\n \n \n \n \n \n \n", "1509.00590_70": "\n \n \n \n \n \n \n \n 1\n 12\n \n ⁢\n \n \n 6\n ⁢\n \n U\n 0\n \n \n \n \n <\n σ\n <\n \n \n 12\n ⁢\n \n U\n 0\n \n \n \n \n .\n \n (\n 54\n )\n \n \n \n \n \n", "1509.01678_61": "\n \n \n \n \n \n \n |\n \n \n \n z\n \n k\n +\n 1\n \n \n (\n j\n )\n \n \n ⁢\n \n (\n t\n ,\n a\n )\n \n \n -\n \n \n z\n k\n \n (\n j\n )\n \n \n ⁢\n \n (\n t\n ,\n a\n )\n \n \n \n |\n \n ≤\n \n \n \n K\n 1\n \n (\n j\n )\n \n \n ⁢\n \n |\n a\n |\n \n ⁢\n \n L\n 1\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n \n α\n ⁢\n t\n \n /\n 2\n \n \n )\n \n \n \n /\n \n (\n \n \n 2\n k\n \n ⁢\n α\n \n )\n \n \n \n .\n \n (\n 4.57\n )\n \n \n \n \n \n", "1509.02926_22": "\n \n \n \n \n \n \n [\n \n \n D\n ^\n \n a\n \n ,\n \n Γ\n ^\n \n ]\n \n ⁢\n \n ε\n ^\n \n \n =\n \n 0\n ,\n \n (\n 2.23\n )\n \n \n \n \n \n \n", "1509.03845_73": "\n \n \n \n \n \n \n t\n 3\n \n ⁢\n \n \n ∥\n \n \n \n ∂\n t\n \n ⁡\n u\n \n ⁢\n \n (\n t\n )\n \n \n ∥\n \n \n L\n 2\n \n 2\n \n \n ≤\n \n \n \n Q\n ⁢\n \n (\n \n \n ∥\n \n u\n 0\n \n ∥\n \n \n L\n 2\n \n \n )\n \n \n +\n \n Q\n ⁢\n \n (\n \n \n ∥\n g\n ∥\n \n \n L\n 2\n \n \n )\n \n \n \n ,\n \n ∈͡\n ⁢\n \n (\n 0\n ,\n 1\n ]\n \n ⁢\n \n (\n 4.17\n )\n \n \n \n \n \n \n \n", "1509.03948_49": "\n \n \n \n \n \n \n ω\n ⁢\n D\n \n +\n \n D\n ⁢\n ω\n \n \n =\n 0\n \n \n \n \n", "1509.05970_9": "", "1509.08463_17": "\n \n \n \n \n \n τ\n  s i m \n  T T + l o w P \n \n ≈\n \n \n 0.078\n -\n \n 0.0015\n ⁢\n \n (\n \n \n Δ\n ⁢\n h\n \n 0.6731\n \n )\n \n \n \n +\n \n …\n ⁢\n \n (\n 12\n )\n \n \n \n \n \n \n \n", "1509.08778_9": "\n \n \n \n \n \n E\n ⁢\n \n (\n \n X\n \n DIS\n -\n GW\n -\n AGG\n \n \n )\n \n \n =\n \n E\n ⁢\n \n (\n \n X\n \n DIS\n -\n SN\n -\n AGG\n \n \n )\n \n \n =\n \n \n I\n 1\n \n +\n \n 1\n ⁢\n \n (\n 10\n )\n \n \n \n \n \n \n \n", "1510.01776_45": "\n \n \n \n \n \n R\n i\n \n =\n \n \n \n R\n 1\n \n \n 1\n +\n \n \n ∑\n \n j\n =\n 2\n \n i\n \n \n \n n\n j\n \n \n n\n 1\n \n \n \n \n \n ⁢\n \n (\n 36\n )\n \n \n \n \n \n \n", "1510.02399_98": "\n \n \n \n \n \n \n E\n ⁢\n \n \n 𝐮\n ~\n \n t\n \n ⁢\n \n 𝐮\n \n t\n -\n k\n \n ′\n \n \n =\n \n \n 𝐆\n k\n \n ⁢\n \n 𝚪\n u\n \n \n \n .\n \n \n \n \n", "1510.03588_214": "\n \n \n \n \n \n \n x\n ⁢\n v\n ⁢\n \n (\n t\n ,\n x\n )\n \n \n =\n \n \n e\n \n \n K\n ′\n \n ⁢\n \n (\n \n s\n +\n \n )\n \n ⁢\n t\n \n \n ⁢\n u\n ⁢\n \n (\n t\n ,\n \n e\n \n \n K\n ′\n \n ⁢\n \n (\n \n s\n +\n \n )\n \n ⁢\n t\n \n \n )\n \n \n ∼\n \n \n C\n \n t\n \n \n ⁢\n \n e\n \n \n (\n \n \n \n \n K\n ′\n \n ⁢\n \n (\n s\n )\n \n ⁢\n \n (\n \n 1\n -\n s\n \n )\n \n \n +\n \n K\n ⁢\n \n (\n s\n )\n \n \n \n -\n 1\n \n )\n \n ⁢\n t\n \n \n \n \n ,\n \n \n \n \n", "1510.04385_6": "\n \n \n \n \n \n ℰ\n =\n \n ⟨\n H\n ⟩\n \n =\n \n \n ∂\n ⁡\n Ω\n \n \n ∂\n ⁡\n β\n \n \n \n ,\n \n (\n 10\n )\n \n \n \n \n \n", "1510.04681_56": "\n \n \n \n \n \n \n ν\n ⁢\n \n (\n \n \n \n lim sup\n \n n\n →\n ∞\n \n \n ⁡\n \n \n \n M\n n\n \n ⁢\n \n (\n x\n )\n \n \n \n u\n n\n \n \n \n =\n 0\n \n )\n \n \n =\n 1\n \n .\n \n \n \n \n", "1511.00570_2": "\n \n \n \n \n \n \n λ\n free\n \n ∼\n \n \n 1\n \n \n α\n 2\n \n ⁢\n T\n \n \n ⁢\n \n \n \n (\n \n \n m\n p\n \n T\n \n )\n \n \n 1\n /\n 2\n \n \n \n \n \n .\n \n (\n 3\n )\n \n \n \n \n \n", "1511.01215_192": "\n \n \n \n \n \n \n \n (\n 8.17\n )\n \n ⁢\n o\n ⁢\n s\n ⁢\n \n c\n Ω\n \n ⁢\n f\n \n :=\n \n \n sup\n \n \n x\n ,\n y\n \n ∈\n Ω\n \n \n ⁡\n \n |\n \n \n f\n ⁢\n \n (\n x\n )\n \n \n -\n \n f\n ⁢\n \n (\n y\n )\n \n \n \n |\n \n \n =\n \n \n sup\n \n 0\n ≤\n ϕ\n ≤\n \n 2\n ⁢\n π\n \n \n \n ⁡\n \n o\n ⁢\n s\n ⁢\n \n c\n Ω\n \n ⁢\n \n ℜ\n ⁡\n \n (\n \n \n e\n \n i\n ⁢\n ϕ\n \n \n ⁢\n f\n \n )\n \n \n \n \n \n .\n \n \n \n \n", "1511.02261_79": "\n \n \n \n \n \n \n L\n \n h\n ,\n t\n \n \n =\n \n \n t\n \n -\n \n a\n h\n \n \n \n ⁢\n \n L\n t\n \n ⁢\n \n (\n \n \n T\n h\n \n -\n 1\n \n \n ⁢\n \n (\n x\n ,\n y\n )\n \n \n )\n \n \n \n ,\n \n \n \n \n", "1511.02662_10": "\n \n \n \n \n \n \n \n \n \n π\n ~\n \n \n S\n ,\n γ\n \n \n ⁢\n \n (\n f\n )\n \n ⁢\n \n ξ\n \n t\n ¯\n \n \n \n =\n \n f\n ⁢\n \n (\n \n t\n ⁢\n x\n \n )\n \n ⁢\n \n ξ\n \n t\n ¯\n \n \n \n \n ,\n \n \n \n \n π\n ~\n \n \n S\n ,\n γ\n \n \n ⁢\n \n (\n \n u\n s\n \n )\n \n ⁢\n \n ξ\n \n t\n ¯\n \n \n \n =\n \n γ\n ⁢\n \n (\n s\n )\n \n ⁢\n \n ξ\n \n \n s\n ⁢\n t\n \n ¯\n \n \n \n \n \n ,\n \n \n \n \n", "1511.04069_14": "\n \n \n \n \n \n μ\n ⁢\n \n (\n \n \n l\n s\n \n ⁢\n f\n \n )\n \n \n =\n \n μ\n ⁢\n \n (\n f\n )\n \n \n \n \n \n \n", "1511.05057_26": "\n \n \n \n \n \n \n \n (\n \n \n \n n\n +\n m\n \n -\n 1\n \n m\n \n )\n \n \n m\n \n n\n -\n 1\n \n \n \n =\n \n \n \n (\n \n 1\n +\n \n \n n\n -\n 1\n \n m\n \n \n )\n \n ⁢\n ⋯\n ⁢\n \n (\n \n 1\n +\n \n 1\n m\n \n \n )\n \n \n \n \n (\n \n n\n -\n 1\n \n )\n \n !\n \n \n \n .\n \n \n \n \n", "1511.05554_72": "\n \n \n \n \n \n \n \n Ω\n ^\n \n ε\n \n =\n \n {\n \n 𝐦\n ∈\n \n \n Ω\n ^\n \n \n \n :\n \n \n \n m\n k\n \n ≥\n ε\n \n ,\n \n \n k\n =\n \n 1\n ,\n …\n ,\n J\n \n \n ,\n \n \n \n \n and\n \n ⁢\n J\n \n -\n \n m\n 1\n \n -\n ⋯\n -\n \n m\n J\n \n \n ≥\n ε\n \n \n \n }\n \n \n .\n \n \n \n \n", "1511.07339_4": "\n \n \n \n \n \n \n \n U\n α\n \n ⁢\n \n (\n 𝐫\n )\n \n \n =\n \n \n \n \n e\n \n i\n ⁢\n π\n ⁢\n α\n \n \n ⁢\n \n sin\n ⁡\n \n (\n \n π\n ⁢\n α\n \n )\n \n \n \n π\n \n ⁢\n \n \n ∑\n \n n\n =\n \n -\n ∞\n \n \n ∞\n \n \n \n \n U\n n\n \n ⁢\n \n (\n 𝐫\n )\n \n \n \n α\n -\n n\n \n \n \n \n \n .\n \n (\n 4\n )\n \n \n \n \n \n", "1511.07378_73": "\n \n \n \n \n \n \n \n Z\n φ\n R\n \n ⁢\n \n (\n ⋅\n )\n \n \n =\n \n \n Z\n ψ\n R\n \n ⁢\n \n (\n ⋅\n )\n \n ⁢\n \n if\n \n ⁢\n \n and\n \n ⁢\n \n only\n \n ⁢\n if\n ⁢\n φ\n \n =\n ψ\n \n ,\n \n \n \n \n", "1511.08582_9": "\n \n \n \n \n \n Φ\n ⁢\n \n (\n \n p\n →\n \n )\n \n ⁢\n \n exp\n ⁡\n \n (\n \n i\n ⁢\n \n p\n →\n \n ⁢\n \n r\n →\n \n \n )\n \n \n \n ,\n \n \n Φ\n +\n \n ⁢\n \n (\n \n p\n →\n \n )\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n i\n ⁢\n \n p\n →\n \n ⁢\n \n r\n →\n \n \n \n )\n \n \n \n \n \n \n \n", "1512.01498_59": "\n \n \n \n \n \n \n \n \n 2\n ⁢\n \n a\n 2\n \n \n -\n \n 2\n ⁢\n Δ\n \n \n +\n \n r\n ⁢\n \n Δ\n ′\n \n \n \n ≥\n 0\n \n .\n \n (\n 61\n )\n \n \n \n \n \n", "1512.02126_170": "\n \n \n \n \n \n \n \n (\n 170\n )\n \n ⁢\n \n \n (\n \n ∫\n \n \n \n |\n \n \n ∇\n \n (\n \n \n 2\n ⁢\n m\n \n -\n β\n \n )\n \n \n ⁡\n u\n \n |\n \n \n 2\n β\n ♯\n \n \n ⁢\n \n 𝑑\n x\n \n \n \n )\n \n \n 2\n /\n \n 2\n β\n ♯\n \n \n \n \n ≤\n \n C\n ⁢\n \n ∫\n \n \n \n |\n \n \n ∇\n \n 2\n ⁢\n m\n \n \n ⁡\n u\n \n |\n \n 2\n \n ⁢\n \n 𝑑\n x\n \n \n \n \n \n .\n \n \n \n \n", "1512.03239_65": "\n \n \n \n \n \n v\n ⁢\n \n (\n u\n )\n \n \n =\n \n \n \n [\n \n \n ∫\n \n B\n ⁢\n \n (\n u\n )\n \n ⁢\n \n e\n \n -\n \n \n ∫\n \n A\n ⁢\n \n (\n u\n )\n \n ⁢\n \n 𝑑\n u\n \n \n \n \n \n \n ⁢\n \n 𝑑\n u\n \n \n \n +\n \n v\n 0\n \n \n ]\n \n ⁢\n \n e\n \n \n ∫\n \n A\n ⁢\n \n (\n u\n )\n \n ⁢\n \n 𝑑\n u\n \n \n \n \n \n \n ,\n \n (\n 65\n )\n \n \n \n \n \n \n", "1512.04168_80": "", "1512.07934_7": "", "1512.09199_10": "\n \n \n \n \n \n *\n ρ\n \n ω\n =\n \n R\n ρ\n \n *\n \n R\n ρ\n \n ω\n .\n \n \n \n \n", "1601.02859_117": "", "1601.03342_80": "", "1601.03712_74": "", "1601.03853_192": "\n \n \n \n \n \n \n M\n ⁢\n \n (\n x\n )\n \n \n :=\n \n \n \n \n λ\n \n -\n 1\n \n \n ⁢\n \n 𝒆\n 1\n \n \n ⊗\n \n 𝒆\n 1\n \n \n +\n \n λ\n ⁢\n \n \n ∑\n \n \n i\n ,\n j\n \n =\n 1\n \n n\n \n \n \n \n ν\n i\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n ν\n j\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n 𝒆\n \n i\n +\n 1\n \n \n \n ⊙\n \n 𝒆\n \n j\n +\n 1\n \n \n \n \n \n \n \n .\n \n \n \n \n", "1601.04445_135": "", "1601.04737_178": "\n \n \n \n \n \n \n ∥\n \n 𝐠\n ⁢\n \n (\n \n 𝐱\n \n (\n k\n )\n \n \n )\n \n \n ∥\n \n ≤\n \n \n ∥\n \n \n ∇\n ⁡\n F\n \n ⁢\n \n (\n \n 𝐱\n \n (\n k\n )\n \n \n )\n \n \n ∥\n \n +\n \n ϵ\n 2\n \n \n \n ,\n \n \n \n \n", "1601.05950_130": "\n \n \n \n \n \n u\n 0\n \n ∈\n \n \n C\n ⁢\n \n (\n \n [\n 0\n ,\n T\n ]\n \n ,\n \n \n H\n \n 2\n +\n \n 2\n ⁢\n \n α\n ′\n \n \n \n \n ⁢\n \n (\n I\n )\n \n \n )\n \n \n ∩\n \n \n C\n 1\n \n ⁢\n \n (\n \n [\n 0\n ,\n T\n ]\n \n ,\n \n \n H\n \n 2\n ⁢\n \n α\n ′\n \n \n \n ⁢\n \n (\n I\n )\n \n \n )\n \n \n \n \n \n \n \n", "1602.02973_138": "\n \n \n \n \n \n f\n ⁢\n \n (\n x\n )\n \n \n ∈\n \n \n \n \n x\n ~\n \n 0\n \n +\n \n \n A\n ~\n \n ⁢\n 𝐛\n \n +\n \n 𝐫\n ~\n \n \n ,\n \n (\n 4.18\n )\n \n \n \n \n \n \n", "1602.03660_9": "\n \n \n \n \n \n Σ\n ⁢\n \n (\n θ\n )\n \n \n =\n \n \n \n \n \n (\n \n m\n ⁢\n \n α\n e\n \n ⁢\n I\n ⁢\n \n n\n m\n \n \n )\n \n 3\n \n ⁢\n \n \n (\n \n π\n ⁢\n a\n \n )\n \n 2\n \n \n \n 4\n ⁢\n ℏ\n ⁢\n \n p\n 5\n \n \n \n ⁢\n \n J\n z\n \n ⁢\n Q\n ⁢\n \n sin\n ⁡\n θ\n \n ⁢\n \n (\n 10\n )\n \n \n \n \n \n \n", "1602.04270_6": "\n \n \n H\n ⁢\n \n (\n i\n ,\n s\n ,\n q\n ,\n t\n )\n \n \n ≐\n \n f\n ⁢\n \n (\n \n i\n \n s\n ⁢\n t\n \n \n ,\n q\n )\n \n \n =\n \n (\n \n \n \n 01\n -\n \n g\n ⁢\n \n (\n j\n ,\n t\n )\n \n ⁢\n 1\n \n \n \n 11\n -\n \n g\n ⁢\n \n (\n j\n ,\n t\n )\n \n ⁢\n 0\n \n \n \n \n )\n \n \n", "1602.06383_64": "\n \n \n \n \n \n \n \n n\n \n ⁢\n \n (\n \n \n \n ϑ\n ^\n \n n\n \n (\n n\n )\n \n \n -\n \n ϑ\n n\n \n \n )\n \n \n =\n \n \n \n n\n \n ⁢\n \n \n ∑\n \n i\n =\n 1\n \n n\n \n \n \n α\n \n i\n ,\n n\n \n \n ⁢\n \n ℓ\n n\n \n ⁢\n \n (\n \n X\n i\n \n (\n n\n )\n \n \n ,\n \n ϑ\n n\n \n )\n \n \n \n \n +\n \n \n o\n p\n \n ⁢\n \n (\n 1\n )\n \n ⁢\n as\n ⁢\n n\n \n \n →\n ∞\n \n .\n \n \n \n \n", "1602.06470_85": "\n \n \n \n \n \n η\n =\n \n -\n \n \n i\n 2\n \n ⁢\n \n (\n \n \n \n ∑\n \n j\n =\n 1\n \n n\n \n \n (\n \n \n \n \n z\n j\n \n ¯\n \n ⁢\n d\n ⁢\n \n z\n j\n \n \n -\n \n \n z\n j\n \n ⁢\n d\n ⁢\n \n \n z\n j\n \n ¯\n \n \n \n )\n \n \n -\n \n (\n \n \n \n \n z\n \n n\n +\n 1\n \n \n ¯\n \n ⁢\n d\n ⁢\n \n z\n \n n\n +\n 1\n \n \n \n -\n \n \n z\n \n n\n +\n 1\n \n \n ⁢\n d\n ⁢\n \n \n z\n \n n\n +\n 1\n \n \n ¯\n \n \n \n )\n \n \n )\n \n \n \n \n .\n \n \n \n \n", "1602.06524_181": "\n \n \n \n \n \n \n \n (\n 4.16\n )\n \n ⁢\n 𝝍\n ⁢\n \n (\n t\n )\n \n \n =\n \n (\n \n ψ\n ⁢\n \n (\n t\n )\n \n \n ,\n \n \n ψ\n ˙\n \n ⁢\n \n (\n t\n )\n \n \n )\n \n :=\n \n \n \n \n ∂\n t\n \n ⁡\n 𝝋\n \n ⁢\n \n (\n t\n )\n \n \n -\n \n D\n ⁢\n E\n ⁢\n \n (\n \n 𝝋\n ⁢\n \n (\n t\n )\n \n \n )\n \n \n \n =\n \n (\n \n \n \n \n ∂\n t\n \n ⁡\n φ\n \n ⁢\n \n (\n t\n )\n \n \n -\n \n \n φ\n ˙\n \n ⁢\n \n (\n t\n )\n \n \n \n ,\n \n \n \n \n ∂\n t\n \n ⁡\n \n φ\n ˙\n \n \n ⁢\n \n (\n t\n )\n \n \n -\n \n (\n \n \n \n \n ∂\n r\n 2\n \n ⁡\n φ\n \n ⁢\n \n (\n t\n )\n \n \n +\n \n \n 1\n r\n \n ⁢\n \n \n ∂\n r\n \n ⁡\n φ\n \n ⁢\n \n (\n t\n )\n \n \n +\n \n \n 1\n \n r\n 2\n \n \n ⁢\n f\n ⁢\n \n (\n \n φ\n ⁢\n \n (\n t\n )\n \n \n )\n \n \n \n )\n \n \n )\n \n \n .\n \n \n \n \n", "1602.08404_52": "\n \n \n \n \n \n \n S\n ⁢\n \n R\n a\n \n \n ≡\n \n \n \n \n 2\n *\n \n m\n π\n \n \n ⁢\n \n m\n K\n \n \n \n \n s\n \n t\n ⁢\n h\n \n \n \n \n ⁢\n \n P\n \n ⁢\n \n \n ∫\n \n s\n \n t\n ⁢\n h\n \n \n ∞\n \n \n \n \n \n Im\n \n ⁢\n \n T\n -\n \n ⁢\n \n (\n \n s\n ′\n \n )\n \n \n \n \n s\n ′\n \n ⁢\n \n (\n \n \n s\n ′\n \n -\n \n s\n \n t\n ⁢\n h\n \n \n \n )\n \n \n \n ⁢\n \n 𝑑\n \n s\n ′\n \n \n \n \n \n \n .\n \n \n \n \n", "1603.00022_9": "\n \n \n \n \n \n \n \n \n 𝒥\n \n \n μ\n 1\n \n ⁢\n …\n ⁢\n \n μ\n s\n \n \n \n \n ∼\n \n \n \n ϕ\n a\n \n ⁢\n \n \n ∂\n \n μ\n 1\n \n \n ⁡\n …\n \n ⁢\n \n \n ∂\n \n μ\n s\n \n \n ⁡\n \n ϕ\n a\n \n \n \n +\n …\n \n \n ,\n \n a\n =\n \n 1\n ,\n …\n ,\n \n N\n \n \n \n \n .\n \n (\n 2.1\n )\n \n \n \n \n \n", "1603.00269_25": "", "1603.01246_542": "\n \n \n \n \n \n r\n ≤\n \n P\n ⁢\n \n (\n \n \n ⟨\n z\n ⟩\n \n n\n \n )\n \n \n ≤\n \n P\n ⁢\n \n (\n \n \n ⟨\n z\n ⟩\n \n \n n\n -\n 1\n \n \n ,\n x\n )\n \n \n ≤\n \n r\n +\n \n A\n ⁢\n \n [\n \n \n H\n ⁢\n \n (\n \n \n ⟨\n \n f\n ⁢\n \n (\n x\n )\n \n \n ⟩\n \n \n n\n -\n 1\n \n \n ,\n \n g\n ⁢\n \n (\n x\n )\n \n \n )\n \n \n -\n \n H\n ⁢\n \n (\n \n \n ⟨\n \n f\n ⁢\n \n (\n x\n )\n \n \n ⟩\n \n n\n \n )\n \n \n \n ]\n \n \n \n \n .\n \n \n \n \n", "1603.01908_121": "\n \n \n \n \n \n \n U\n ~\n \n ⁢\n \n (\n \n t\n ~\n \n ,\n \n y\n ~\n \n )\n \n \n :=\n \n U\n ⁢\n \n (\n \n \n t\n ~\n \n \n N\n i\n \n \n ,\n \n \n y\n ~\n \n \n \n N\n i\n \n ⁢\n \n N\n \n i\n -\n 1\n \n \n \n \n )\n \n \n \n \n \n \n", "1603.03689_6": "\n \n \n \n \n \n S\n M\n \n =\n \n \n ∫\n \n \n d\n 4\n \n ⁢\n x\n ⁢\n \n (\n \n \n \n Π\n a\n \n ⁢\n \n \n ∂\n t\n \n ⁡\n \n ϕ\n a\n \n \n \n -\n \n N\n ⁢\n \n ℋ\n M\n \n \n -\n \n \n N\n i\n \n ⁢\n \n ℋ\n M\n i\n \n \n \n )\n \n \n \n ,\n \n (\n 3.8\n )\n \n \n \n \n \n \n", "1603.04486_15": "\n \n \n \n \n \n \n \n J\n b\n \n ⁢\n \n (\n x\n ,\n \n η\n 1\n \n ,\n \n η\n 2\n \n )\n \n \n =\n \n \n κ\n 3\n \n ⁢\n \n \n ∫\n \n η\n 1\n \n \n η\n 2\n \n \n \n W\n ⁢\n \n (\n \n x\n -\n y\n \n )\n \n ⁢\n \n 𝑑\n y\n \n \n \n \n \n .\n \n \n \n \n", "1603.06128_84": "\n \n \n \n \n \n \n F\n ⁢\n \n (\n \n V\n ⊗\n \n A\n i\n \n \n )\n \n \n :=\n \n \n RHom\n \n 𝒫\n λ\n \n a\n ⁢\n \n f\n i\n \n \n \n \n ⁢\n \n (\n \n h\n λ\n V\n \n ,\n F\n )\n \n \n \n ,\n \n \n \n \n", "1603.07239_52": "\n \n \n \n \n  V \n =\n \n -\n \n \n ∫\n \n \n {\n \n 0\n ≤\n \n φ\n ⁢\n \n (\n ⋅\n ,\n \n t\n h\n \n )\n \n \n <\n \n \n (\n \n \n \n ∂\n t\n \n ⁡\n φ\n \n -\n γ\n \n )\n \n ⁢\n h\n \n \n }\n \n ∩\n \n B\n δ\n \n \n \n \n \n P\n h\n \n ⁢\n \n (\n y\n )\n \n ⁢\n \n 𝑑\n y\n \n \n \n \n =\n \n \n ∫\n \n \n {\n 0\n ≤\n φ\n \n (\n \n σ\n h\n \n 1\n /\n s\n \n \n ⋅\n ,\n \n t\n h\n \n )\n \n <\n \n (\n \n ∂\n t\n \n φ\n -\n γ\n )\n \n h\n }\n \n ∩\n \n B\n \n δ\n /\n \n σ\n h\n \n 1\n /\n s\n \n \n \n \n \n \n \n P\n ⁢\n \n (\n y\n )\n \n ⁢\n \n 𝑑\n y\n \n \n \n \n \n \n \n", "1603.09350_5": "\n \n \n \n \n \n ℒ\n \n E\n ⁢\n W\n \n \n =\n -\n \n \n e\n ⁢\n \n s\n θ\n \n \n \n 2\n ⁢\n \n c\n w\n \n ⁢\n \n s\n w\n \n \n \n \n Z\n μ\n \n \n \n u\n ¯\n \n †\n \n \n \n σ\n ¯\n \n μ\n \n \n U\n ¯\n \n -\n \n \n g\n ⁢\n \n s\n θ\n \n \n 2\n \n \n W\n μ\n -\n \n \n (\n \n \n u\n ¯\n \n †\n \n \n \n σ\n ¯\n \n μ\n \n \n D\n ¯\n \n +\n \n \n X\n ¯\n \n †\n \n \n \n σ\n ¯\n \n μ\n \n \n u\n ¯\n \n )\n \n +\n h\n .\n c\n .\n ,\n \n (\n 15\n )\n \n \n \n \n \n", "1603.09604_17": "\n \n \n \n \n \n \n \n H\n ^\n \n \n 2\n ⁢\n B\n \n \n =\n \n ∫\n \n \n d\n 2\n \n ⁢\n \n r\n \n ⁢\n g\n ⁢\n \n \n ψ\n ^\n \n ↑\n †\n \n ⁢\n \n (\n 𝐫\n )\n \n ⁢\n \n \n ψ\n ^\n \n ↓\n †\n \n ⁢\n \n (\n 𝐫\n )\n \n ⁢\n \n \n ψ\n ^\n \n ↓\n \n ⁢\n \n (\n 𝐫\n )\n \n ⁢\n \n \n ψ\n ^\n \n ↑\n \n ⁢\n \n (\n 𝐫\n )\n \n \n \n \n .\n \n (\n 14\n )\n \n \n \n \n \n", "1603.09626_84": "\n \n \n \n \n \n \n d\n ⁢\n \n (\n ⋆\n )\n \n \n =\n \n \n ∑\n \n k\n =\n 0\n \n ∞\n \n \n \n 1\n \n k\n !\n \n \n ⁢\n \n \n (\n \n ℏ\n i\n \n )\n \n k\n \n ⁢\n \n d\n k\n \n ⁢\n \n (\n ⋆\n )\n \n \n \n \n ,\n \n \n \n \n", "1604.00453_14": "\n \n \n \n \n \n \n H\n 2\n \n ≃\n \n \n 1\n 3\n \n ⁢\n \n ρ\n ϕ\n \n \n \n .\n \n (\n 13\n )\n \n \n \n \n \n", "1604.06380_29": "\n \n \n \n \n \n \n ζ\n =\n \n \n -\n \n \n ∫\n 0\n ∞\n \n \n \n \n \n u\n \n -\n \n \n 1\n /\n 2\n \n ⁢\n p\n \n \n \n ⁢\n \n ℒ\n ′\n \n ⁢\n \n (\n u\n )\n \n \n \n ℒ\n ⁢\n \n (\n u\n )\n \n \n \n ⁢\n \n 𝑑\n u\n \n \n \n \n ,\n and\n \n \n  \n \n \n 𝒱\n 1\n \n =\n \n \n \n C\n *\n \n ⁢\n \n C\n ℓ\n \n ⁢\n \n ξ\n 2\n \n ⁢\n \n σ\n 2\n \n ⁢\n \n (\n x\n )\n \n \n \n \n p\n *\n \n ⁢\n \n (\n 0\n )\n \n ⁢\n \n ξ\n 1\n 2\n \n ⁢\n \n λ\n \n \n 1\n +\n \n 2\n ⁢\n ρ\n ⁢\n p\n \n \n \n \n 2\n ⁢\n p\n \n -\n 1\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "1604.07451_42": "", "1605.01719_21": "\n \n \n \n \n \n \n E\n ⁢\n \n [\n \n u\n 0\n \n ]\n \n \n <\n 0\n \n .\n \n (\n 3.16\n )\n \n \n \n \n \n", "1605.04630_66": "\n \n \n \n \n \n Φ\n \n U\n ,\n W\n \n \n =\n \n \n \n a\n \n U\n ,\n W\n \n \n ⁢\n \n Id\n W\n \n \n +\n \n \n b\n \n U\n ,\n W\n \n \n ⁢\n \n x\n W\n \n \n \n \n \n \n \n", "1605.08839_93": "\n \n \n \n \n \n \n ∥\n \n \n 𝐀\n γ\n \n -\n \n 𝐁\n γ\n \n \n ∥\n \n ≤\n \n γ\n ⁢\n \n r\n \n γ\n -\n 1\n \n \n ⁢\n \n ∥\n \n 𝐀\n -\n 𝐁\n \n ∥\n \n \n \n .\n \n \n \n \n", "1605.09074_86": "\n \n \n \n \n \n \n \n Φ\n i\n \n ⁢\n \n (\n 𝐩\n )\n \n \n =\n \n \n d\n \n d\n ⁢\n y\n \n \n ⁢\n Φ\n ⁢\n \n (\n \n \n 𝐄\n 𝐩\n \n ⁢\n \n [\n \n f\n ⁢\n \n (\n 𝐗\n )\n \n \n ]\n \n \n )\n \n ⁢\n \n E\n 𝐩\n \n ⁢\n \n [\n \n f\n ⁢\n \n (\n 𝐗\n )\n \n ⁢\n \n S\n i\n \n ⁢\n \n (\n 𝐗\n ;\n 𝐩\n )\n \n \n ]\n \n \n \n .\n \n \n \n \n", "1606.02357_34": "\n \n \n \n \n \n \n \n (\n 4.24\n )\n \n ⁢\n \n |\n \n \n \n {\n \n \n R\n \n i\n +\n ℓ\n \n \n ⁢\n x\n \n }\n \n \n ℓ\n =\n 1\n \n \n \n (\n \n m\n -\n \n m\n ′\n \n \n )\n \n ⁢\n \n w\n k\n \n \n \n ∩\n \n (\n \n \n [\n \n -\n \n ∥\n \n \n q\n k\n \n ⁢\n α\n \n ∥\n \n \n ,\n 0\n )\n \n ∪\n \n [\n \n z\n -\n \n ∥\n \n \n q\n k\n \n ⁢\n α\n \n ∥\n \n \n ,\n z\n )\n \n \n )\n \n \n |\n \n \n =\n 1\n \n .\n \n \n \n \n", "1607.02305_12": "\n \n \n \n \n (\n \n \n N\n M\n \n \n )\n \n ⊗\n \n \n \n (\n \n \n N\n M\n \n \n )\n \n ¯\n \n \n \n =\n \n \n 1\n \n ⊕\n \n ⋯\n \n \n \n .\n \n (\n 12\n )\n \n \n", "1607.03851_285": "", "1607.04902_4": "\n \n \n \n \n \n \n D\n ⁢\n i\n ⁢\n a\n ⁢\n g\n ⁢\n \n (\n M\n )\n \n \n =\n \n {\n \n ϕ\n ⁢\n \n (\n \n c\n \n a\n ¯\n \n \n )\n \n \n :\n \n \n \n \n \n ϕ\n ⁢\n \n (\n \n x\n ¯\n \n )\n \n ⁢\n \n is\n \n ⁢\n \n a\n \n ⁢\n quantifier\n \n -\n \n \n free\n \n ⁢\n \n ℒ\n \n \n -\n formula\n \n ,\n \n ∪\n \n a\n ¯\n \n \n \n ⊆\n \n d\n ⁢\n o\n ⁢\n m\n ⁢\n \n (\n M\n )\n \n \n \n ,\n \n \n and\n ⁢\n M\n \n ⊧\n \n ϕ\n ⁢\n \n (\n \n a\n ¯\n \n )\n \n \n \n \n }\n \n \n .\n \n \n \n \n", "1607.05493_52": "\n \n \n \n \n {\n 0.960\n ,\n 1.056\n ,\n 0.921\n ,\n 1.112\n ,\n 0.842\n ,\n 1.224\n ,\n 0.683\n ,\n 1.448\n ,\n 0.367\n ,\n 1.252\n ,\n 0.644\n ,\n 1.504\n ,\n 0.287\n ,\n 0.981\n ,\n 1.027\n }\n \n \n \n \n", "1607.07925_20": "\n \n \n \n \n \n \n θ\n x\n \n (\n i\n )\n \n \n =\n \n \n \n {\n \n \n \n w\n i\n \n 1\n /\n 2\n \n \n ⁢\n \n z\n \n i\n ⁢\n i\n \n \n ⁢\n x\n \n +\n \n \n γ\n i\n T\n \n ⁢\n \n X\n ~\n \n ⁢\n B\n ⁢\n \n (\n \n β\n ^\n \n )\n \n \n +\n \n \n d\n i\n \n \n 2\n ⁢\n ϕ\n \n \n \n }\n \n ⁢\n \n e\n i\n \n ⁢\n \n (\n x\n )\n \n \n +\n \n \n \n z\n \n i\n ⁢\n i\n \n \n \n 2\n ⁢\n ϕ\n \n \n ⁢\n \n h\n i\n \n ⁢\n \n (\n x\n )\n \n \n \n \n .\n \n (\n 8\n )\n \n \n \n \n \n", "1607.08721_49": "\n \n \n \n \n \n \n sup\n \n x\n ∈\n \n dmn\n ⁢\n \n (\n Φ\n )\n \n \n \n \n ⁡\n \n {\n \n 𝐌\n ⁢\n \n (\n \n \n Φ\n i\n \n ⁢\n \n (\n x\n )\n \n \n )\n \n \n }\n \n \n ≤\n \n \n \n sup\n \n x\n ∈\n \n dmn\n ⁢\n \n (\n Φ\n )\n \n \n \n \n ⁡\n \n {\n \n 𝐌\n ⁢\n \n (\n \n Φ\n ⁢\n \n (\n x\n )\n \n \n )\n \n \n }\n \n \n +\n \n δ\n i\n \n \n \n \n \n \n", "1608.01519_15": "\n \n \n \n \n \n Δ\n ⁢\n F\n ⁢\n \n (\n n\n )\n \n \n =\n \n \n \n -\n \n n\n ⁢\n \n |\n \n Δ\n ⁢\n μ\n \n |\n \n \n \n +\n \n \n \n (\n \n 36\n ⁢\n π\n \n )\n \n \n 1\n /\n 3\n \n \n ⁢\n \n \n (\n \n n\n ⁢\n v\n \n )\n \n \n 2\n /\n 3\n \n \n ⁢\n γ\n \n \n ,\n \n (\n 15\n )\n \n \n \n \n \n \n", "1608.07334_40": "\n \n \n \n \n \n \n \n (\n 10\n )\n \n ⁢\n E\n ⁢\n \n [\n f\n ]\n \n \n =\n \n \n ∫\n M\n \n \n \n \n \n |\n \n d\n ⁢\n f\n ⁢\n \n (\n x\n )\n \n \n |\n \n 2\n \n \n ⁢\n \n 𝑑\n V\n \n ⁢\n o\n ⁢\n \n l\n g\n \n \n \n \n .\n \n \n \n \n", "1609.00244_122": "\n \n \n \n \n \n \n \n g\n x\n \n ⁢\n \n (\n z\n )\n \n \n =\n \n e\n \n \n x\n 2\n \n ⁢\n \n (\n \n z\n +\n \n 1\n z\n \n \n )\n \n \n \n =\n \n \n ∑\n \n k\n =\n \n -\n ∞\n \n \n \n +\n ∞\n \n \n \n \n I\n k\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n z\n k\n \n \n \n \n ;\n \n \n \n \n", "1609.00397_40": "\n \n \n \n \n \n \n \n \n π\n ^\n \n g\n \n (\n D\n )\n \n \n ,\n \n \n π\n ^\n \n f\n \n (\n D\n )\n \n \n ,\n \n H\n \n (\n D\n )\n \n \n ,\n \n Q\n \n (\n D\n )\n \n \n \n ∼\n \n t\n \n -\n \n (\n \n \n 2\n ⁢\n l\n \n +\n 1\n \n )\n \n \n \n \n .\n \n \n \n \n", "1609.00957_254": "", "1609.01320_83": "\n \n \n \n \n \n \n ∫\n \n (\n 0\n ,\n t\n ]\n \n \n \n \n |\n \n ⟨\n \n \n v\n ∗\n \n ⁢\n \n (\n s\n )\n \n \n ,\n φ\n ⟩\n \n |\n \n ⁢\n \n 𝑑\n A\n \n ⁢\n \n (\n s\n )\n \n \n \n ≤\n \n \n ∑\n i\n \n \n \n ∫\n \n (\n 0\n ,\n t\n ]\n \n \n \n \n |\n \n ⟨\n \n \n v\n i\n ∗\n \n ⁢\n \n (\n s\n )\n \n \n ,\n φ\n ⟩\n \n |\n \n ⁢\n \n 𝑑\n A\n \n ⁢\n \n (\n s\n )\n \n \n \n \n \n \n \n \n", "1609.03694_30": "\n \n \n \n \n p\n >\n \n \n max\n ⁡\n \n (\n \n ℓ\n ⁢\n \n (\n \n \n m\n \n +\n \n n\n \n \n )\n \n \n ,\n \n \n 2\n ⁢\n n\n \n -\n 5\n \n )\n \n \n ⁢\n \n (\n 4.2\n )\n \n \n \n \n \n \n", "1609.04246_20": "\n \n \n \n \n Ω\n =\n \n \n Ω\n \n B\n ⁢\n a\n ⁢\n r\n ⁢\n y\n ⁢\n o\n ⁢\n n\n ⁢\n i\n ⁢\n c\n \n \n +\n \n Ω\n \n D\n ⁢\n a\n ⁢\n r\n ⁢\n k\n ⁢\n e\n ⁢\n n\n ⁢\n e\n ⁢\n r\n ⁢\n g\n ⁢\n y\n \n \n \n ≈\n \n 0.77\n ⁢\n \n (\n 9\n )\n \n \n \n \n \n \n", "1609.04596_4": "\n \n \n \n \n \n \n 1\n θ\n \n ⁢\n \n \n |\n \n D\n ⁢\n ψ\n \n |\n \n θ\n \n \n ≤\n \n -\n 1\n \n \n \n \n \n", "1609.05722_13": "\n \n \n \n \n \n Γ\n i\n \n =\n \n diag\n ⁢\n \n (\n \n \n ρ\n ′′\n \n ⁢\n \n (\n \n \n (\n \n \n K\n i\n \n ⁢\n w\n \n )\n \n 1\n \n )\n \n \n ,\n ⋯\n ,\n \n \n ρ\n ′′\n \n ⁢\n \n (\n \n \n (\n \n \n K\n i\n \n ⁢\n w\n \n )\n \n N\n \n )\n \n \n )\n \n \n \n \n \n \n", "1609.06394_30": "", "1609.08118_73": "\n \n \n \n \n \n J\n ⁢\n \n (\n 1\n )\n \n ⁢\n \n (\n \n x\n +\n \n \n τ\n +\n \n ⁢\n \n θ\n 0\n \n \n \n ,\n \n θ\n 0\n \n )\n \n \n =\n \n exp\n ⁡\n \n (\n \n -\n \n \n ∫\n \n -\n \n τ\n +\n \n \n \n \n τ\n -\n \n ⁢\n \n (\n x\n ,\n \n θ\n 0\n \n )\n \n \n \n \n σ\n ⁢\n \n (\n \n x\n -\n \n s\n ⁢\n \n θ\n 0\n \n \n \n )\n \n ⁢\n \n 𝑑\n s\n \n \n \n \n )\n \n \n \n \n \n \n", "1609.09151_19": "\n \n \n \n \n g\n =\n \n \n k\n 1\n \n ⁢\n \n exp\n ⁡\n H\n \n ⁢\n \n (\n g\n )\n \n ⁢\n \n n\n 1\n \n ⁢\n \n (\n 2.3\n )\n \n \n \n \n \n \n", "1609.09726_1": "", "1610.01168_10": "\n \n \n \n \n \n \n Δ\n g\n \n ⁢\n u\n \n =\n \n f\n ,\n \n (\n 5\n )\n \n \n \n \n \n \n", "1610.02593_27": "\n \n \n \n \n \n \n \n N\n n\n \n ⁢\n \n (\n p\n )\n \n \n =\n \n \n ∑\n \n e\n ⊂\n \n R\n ⁢\n \n (\n n\n )\n \n \n \n \n \n 𝟏\n \n \n A\n n\n \n ⁢\n \n (\n p\n ,\n e\n )\n \n \n \n \n \n .\n \n \n \n \n", "1610.02882_29": "", "1610.03465_248": "\n \n \n \n \n \n E\n :=\n \n \n \n l\n ϵ\n \n \n l\n \n \n ⁢\n \n \n ∑\n \n n\n =\n 1\n \n \n l\n /\n 2\n \n \n \n \n \n (\n \n l\n n\n \n )\n \n \n 1\n /\n 4\n \n \n ⁢\n \n |\n \n \n ∑\n k\n \n \n h\n ⁢\n \n (\n \n \n 4\n ⁢\n k\n \n K\n \n )\n \n ⁢\n \n \n sin\n ⁡\n \n (\n \n \n \n (\n \n \n 4\n ⁢\n k\n \n -\n 1\n \n )\n \n ⁢\n \n arcsin\n ⁡\n \n (\n \n \n n\n /\n l\n \n \n )\n \n \n \n -\n \n π\n /\n 4\n \n \n )\n \n \n \n \n \n 4\n ⁢\n k\n \n -\n 1\n \n \n \n \n \n |\n \n \n \n \n \n .\n \n \n \n \n", "1610.03981_10": "\n \n \n \n \n \n \n Q\n 1\n \n =\n \n \n Q\n a\n \n +\n \n Q\n n\n \n \n \n .\n \n (\n 11\n )\n \n \n \n \n \n", "1610.04060_128": "\n \n \n \n \n \n \n ⟨\n \n \n \n \n p\n →\n \n 1\n \n \n ⁢\n \n δ\n 3\n \n ⁢\n \n (\n r\n )\n \n ⁢\n \n \n p\n →\n \n 2\n \n \n ⟩\n \n =\n \n -\n \n \n \n 1\n 2\n \n \n ⁢\n \n ⟨\n \n \n \n ∇\n 2\n \n ⁡\n \n δ\n 3\n \n \n ⁢\n \n (\n r\n )\n \n \n ⟩\n \n \n \n \n .\n \n (\n 106\n )\n \n \n \n \n \n", "1610.06109_14": "", "1611.00920_13": "\n \n \n \n \n \n \n \n F\n 2\n \n ⁢\n \n (\n u\n )\n \n \n =\n \n inf\n ⁡\n \n {\n \n \n F\n 1\n \n ⁢\n \n (\n w\n )\n \n \n |\n \n \n π\n ⁢\n \n (\n w\n )\n \n \n =\n u\n \n }\n \n \n \n .\n \n \n \n \n", "1611.03033_21": "", "1611.07192_21": "\n \n \n \n \n \n \n [\n \n 16\n ⁢\n π\n ⁢\n \n (\n \n -\n g\n \n )\n \n ⁢\n \n (\n \n \n T\n \n μ\n ⁢\n ν\n \n \n +\n \n t\n \n L\n ⁢\n L\n \n \n μ\n ⁢\n ν\n \n \n \n )\n \n \n ]\n \n \n ,\n ν\n \n \n =\n \n 0\n ,\n \n (\n 22\n )\n \n \n \n \n \n \n", "1611.10327_4": "\n \n \n \n \n \n W\n ⁢\n \n (\n ζ\n ,\n Φ\n )\n \n \n =\n \n \n \n \n m\n \n p\n ⁢\n ℓ\n \n 2\n \n ⁢\n \n W\n 2\n \n ⁢\n \n (\n z\n )\n \n \n +\n \n \n m\n \n p\n ⁢\n ℓ\n \n \n ⁢\n \n W\n 1\n \n ⁢\n \n (\n z\n )\n \n \n +\n \n \n W\n 0\n \n ⁢\n \n (\n z\n ,\n Φ\n )\n \n \n \n ,\n \n (\n 22\n )\n \n \n \n \n \n \n", "1612.01921_0": "\n \n \n \n \n \n \n H\n 2\n \n ⁢\n \n (\n a\n )\n \n \n =\n \n \n \n \n 8\n ⁢\n π\n ⁢\n G\n \n 3\n \n ⁢\n \n ∑\n \n \n ρ\n i\n \n ⁢\n \n (\n a\n )\n \n \n \n \n ,\n \n (\n 2.1\n )\n \n \n \n \n \n \n", "1612.04255_116": "\n \n \n \n \n \n δ\n ⁢\n \n U\n eff\n \n ℓ\n =\n 1\n \n \n \n =\n \n \n 1\n \n c\n 2\n \n \n ⁢\n \n ℬ\n \n a\n ⁢\n b\n \n \n ⁢\n \n \n \n S\n ^\n \n b\n \n \n ⁢\n \n x\n a\n \n ⁢\n \n (\n 116\n )\n \n \n \n \n \n \n", "1612.05541_159": "\n \n \n \n \n \n \n β\n i\n \n =\n \n \n β\n i\n \n +\n \n \n \n (\n \n Γ\n \n i\n ⁢\n i\n \n \n (\n U\n )\n \n \n )\n \n \n -\n 1\n \n \n ⁢\n \n \n ∑\n \n \n k\n =\n 1\n \n ,\n \n k\n ≠\n i\n \n \n \n \n \n Γ\n \n i\n ⁢\n k\n \n \n (\n U\n )\n \n \n ⁢\n \n β\n k\n \n (\n U\n )\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "1612.07342_86": "\n \n \n \n \n \n \n 𝒞\n \n K\n ⁢\n K\n \n \n ≡\n \n A\n ⁢\n \n λ\n 3\n \n ⁢\n \n [\n \n \n P\n \n K\n ⁢\n K\n \n \n (\n \n c\n ⁢\n t\n \n )\n \n \n +\n \n P\n ⁢\n \n A\n \n K\n ⁢\n K\n \n \n (\n \n c\n ⁢\n t\n \n )\n \n \n \n \n ]\n \n \n \n ,\n \n \n \n d\n \n K\n ⁢\n K\n \n \n ⁢\n \n e\n \n i\n ⁢\n \n θ\n \n K\n ⁢\n K\n \n \n \n \n \n ≡\n \n \n \n R\n b\n \n ⁢\n \n [\n \n \n \n P\n \n K\n ⁢\n K\n \n \n (\n \n u\n ⁢\n t\n \n )\n \n \n +\n \n P\n ⁢\n \n A\n \n K\n ⁢\n K\n \n \n (\n \n u\n ⁢\n t\n \n )\n \n \n \n \n \n \n P\n \n K\n ⁢\n K\n \n \n (\n \n c\n ⁢\n t\n \n )\n \n \n +\n \n P\n ⁢\n \n A\n \n K\n ⁢\n K\n \n \n (\n \n c\n ⁢\n t\n \n )\n \n \n \n \n \n ]\n \n \n ,\n \n (\n 83\n )\n \n \n \n \n \n \n \n", "1612.07391_1": "\n \n \n \n \n \n \n A\n ⁢\n \n (\n x\n ,\n t\n )\n \n \n =\n \n \n ∑\n \n n\n ≥\n 0\n \n \n \n \n A\n n\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n \n t\n n\n \n \n n\n !\n \n \n \n \n =\n \n \n 1\n -\n x\n \n \n \n e\n \n t\n ⁢\n \n (\n \n x\n -\n 1\n \n )\n \n \n \n -\n x\n \n \n \n .\n \n (\n 1\n )\n \n \n \n \n \n", "1612.07623_270": "\n \n \n \n \n \n \n 𝖽\n ⁢\n \n (\n \n γ\n 0\n \n ,\n \n γ\n 1\n \n )\n \n \n ≤\n \n \n 𝖽\n ⁢\n \n (\n \n γ\n 0\n \n ,\n p\n )\n \n \n +\n \n 𝖽\n ⁢\n \n (\n q\n ,\n \n γ\n 1\n \n )\n \n \n \n \n .\n \n \n \n \n", "1701.01109_10": "\n \n \n \n \n \n \n \n η\n ⁢\n c\n \n \n ν\n ⁢\n D\n \n \n =\n θ\n =\n \n Ω\n ⁢\n Δ\n ⁢\n t\n \n \n .\n \n (\n 10\n )\n \n \n \n \n \n", "1701.01254_112": "\n \n \n \n \n \n \n \n \n \n r\n 2\n \n \n ⁢\n \n \n \n D\n ′\n \n D\n \n \n ⁢\n \n u\n 0\n \n \n +\n \n r\n ⁢\n \n \n ∂\n ⁡\n \n u\n 0\n \n \n \n ∂\n ⁡\n r\n \n \n \n \n =\n 0\n \n .\n \n \n \n \n", "1701.03950_104": "\n \n \n \n \n m\n =\n \n \n \n m\n ^\n \n \n ,\n \n (\n 106\n )\n \n \n \n \n \n \n", "1701.04784_124": "\n \n \n \n \n \n \n (\n 7.4\n )\n \n ⁢\n \n q\n 1\n \n ⁢\n \n (\n \n r\n ⁢\n \n ω\n \n 1\n /\n 2\n \n \n ⁢\n \n ω\n k\n \n \n )\n \n \n =\n \n r\n ⁢\n \n ω\n \n 1\n /\n 2\n \n \n ⁢\n \n ω\n k\n \n ⁢\n \n p\n 1\n \n ⁢\n \n (\n \n -\n \n r\n T\n \n \n )\n \n \n \n \n \n \n", "1702.01409_26": "\n \n \n \n \n \n \n \n 1\n M\n \n \n ⁢\n \n \n ∑\n \n t\n =\n 1\n \n M\n \n \n \n C\n g\n \n ⁢\n \n (\n \n \n ℬ\n t\n \n |\n ψ\n \n )\n \n \n \n \n =\n \n \n 1\n -\n \n \n \n 1\n M\n \n \n ⁢\n \n \n ∑\n \n t\n =\n 1\n \n M\n \n \n \n p\n max\n \n ⁢\n \n (\n \n \n ℬ\n t\n \n |\n ψ\n \n )\n \n \n \n \n \n ,\n \n (\n 27\n )\n \n \n \n \n \n \n", "1702.01630_180": "\n \n \n \n \n \n \n ∥\n \n \n \n (\n \n -\n \n Δ\n p\n \n \n )\n \n s\n \n ⁢\n u\n \n ∥\n \n \n \n (\n \n \n W\n 0\n \n s\n ,\n p\n \n \n ⁢\n \n (\n Ω\n )\n \n \n )\n \n *\n \n \n =\n \n \n \n ∥\n u\n ∥\n \n \n \n W\n 0\n \n s\n ,\n p\n \n \n ⁢\n \n (\n Ω\n )\n \n \n \n p\n -\n 1\n \n \n ,\n \n (\n 6.1\n )\n \n \n \n \n \n \n", "1702.04658_27": "\n \n \n \n \n \n \n T\n ⁢\n \n (\n G\n )\n \n ⁢\n \n (\n v\n )\n \n \n =\n \n 12\n ⁢\n \n (\n \n G\n ⁢\n \n (\n v\n )\n \n ⁢\n −\n ⁢\n κ\n ⁢\n G\n ⁢\n \n (\n \n κ\n ⁢\n v\n \n )\n \n \n )\n \n \n =\n \n 12\n ⁢\n \n (\n \n \n G\n ⁢\n \n (\n v\n )\n \n \n +\n \n κ\n ⁢\n 2\n ⁢\n G\n ⁢\n \n (\n v\n )\n \n \n \n )\n \n \n =\n \n G\n ⁢\n \n (\n v\n )\n \n \n \n .\n \n \n \n \n", "1702.05858_3": "", "1702.07258_26": "", "1702.08124_169": "\n \n \n \n \n \n \n \n (\n \n 1\n -\n \n \n \n 5\n ⁢\n \n λ\n \n r\n +\n 1\n \n \n \n +\n μ\n \n \n \n 5\n ⁢\n \n λ\n \n r\n +\n 1\n \n \n \n +\n \n 3\n ⁢\n μ\n \n \n \n \n )\n \n ⁢\n \n H\n \n (\n t\n )\n \n \n \n ⪯\n \n \n \n ∇\n 2\n \n ⁡\n F\n \n ⁢\n \n (\n \n x\n \n (\n t\n )\n \n \n )\n \n \n ⪯\n \n \n (\n \n 1\n +\n \n 1\n 2\n \n \n )\n \n ⁢\n \n H\n \n (\n t\n )\n \n \n \n \n .\n \n \n \n \n", "1703.00292_33": "\n \n \n \n \n \n \n \n (\n 24\n )\n \n ⁢\n \n \n lim\n \n t\n →\n \n -\n ∞\n \n \n \n ⁡\n \n \n ∫\n X\n \n \n \n \n |\n \n w\n t\n \n |\n \n \n ω\n ,\n h\n \n 2\n \n ⁢\n \n e\n \n -\n ψ\n \n \n ⁢\n \n 𝑑\n \n V\n \n X\n ,\n ω\n \n \n \n \n \n \n \n =\n 0\n \n .\n \n \n \n \n", "1703.01068_26": "", "1703.03463_26": "\n \n \n \n \n \n \n I\n ⁢\n \n (\n \n \n \n L\n ⊙\n \n /\n p\n \n ⁢\n \n c\n 2\n \n \n )\n \n \n =\n \n exp\n ⁡\n \n [\n \n -\n \n \n (\n \n \n I\n ⁢\n \n (\n \n \n \n mags\n ⊙\n \n /\n a\n \n ⁢\n r\n ⁢\n c\n ⁢\n s\n ⁢\n e\n ⁢\n \n c\n 2\n \n \n )\n \n \n -\n \n M\n ⊙\n \n -\n 21.572\n \n )\n \n 2.5\n \n \n ]\n \n \n \n .\n \n (\n 27\n )\n \n \n \n \n \n", "1703.03571_5": "\n \n \n \n \n \n 0\n ≤\n \n e\n \n 0\n ,\n 1\n \n \n ≤\n 3\n \n .\n \n \n \n \n", "1703.04419_6": "", "1703.05006_0": "\n \n \n \n \n <\n <\n ∫\n I\n Φ\n \n (\n u\n )\n \n m\n \n (\n d\n u\n )\n \n ,\n φ\n >\n >\n =\n ∫\n I\n <\n <\n Φ\n u\n ,\n φ\n >\n >\n m\n \n (\n d\n u\n )\n \n .\n \n \n \n \n", "1703.05254_63": "\n \n \n \n \n \n \n \n ∫\n X\n \n \n \n min\n ⁡\n \n (\n \n \n h\n j\n \n -\n \n \n h\n ^\n \n j\n \n \n ,\n 1\n )\n \n \n ⁢\n MA\n ⁢\n \n (\n \n \n h\n ^\n \n j\n \n )\n \n \n \n =\n 0\n \n .\n \n \n \n \n", "1703.06262_68": "", "1703.06403_6": "\n \n \n \n \n \n \n Vol\n ⁢\n \n (\n C\n )\n \n \n ≤\n \n \n Vol\n ⁢\n \n (\n K\n )\n \n \n \n n\n +\n 1\n \n \n \n .\n \n \n \n \n", "1703.08147_16": "\n \n \n \n \n \n \n \n \n T\n ⁢\n r\n ⁢\n B\n \n /\n A\n \n ⁢\n \n (\n \n θ\n ⁢\n x\n ⁢\n j\n \n )\n \n \n =\n \n ∑\n k\n \n =\n \n 0\n ⁢\n r\n ⁢\n −\n ⁢\n 1\n ⁢\n a\n ⁢\n k\n ⁢\n p\n ⁢\n k\n \n \n ,\n \n \n \n \n T\n ⁢\n r\n ⁢\n B\n \n /\n A\n \n ⁢\n \n (\n \n θ\n ⁢\n y\n ⁢\n j\n \n )\n \n \n =\n \n ∑\n k\n \n =\n \n 0\n ⁢\n r\n ⁢\n −\n ⁢\n 1\n ⁢\n b\n ⁢\n k\n ⁢\n p\n ⁢\n k\n \n \n .\n \n (\n 10\n )\n \n \n \n \n \n", "1703.09088_145": "", "1703.10063_5": "", "1704.02918_78": "", "1704.06806_75": "\n \n \n \n \n \n χ\n τ\n \n :\n \n \n ℋ\n \n ⟶\n \n \n ℋ\n τ\n \n \n \n :\n \n \n x\n ⁢\n \n (\n t\n )\n \n \n ⟼\n \n \n \n x\n ⁢\n \n (\n t\n )\n \n \n |\n \n \n [\n \n -\n τ\n \n ,\n τ\n ]\n \n \n \n \n \n \n \n", "1704.07623_98": "\n \n \n \n \n \n k\n ⁢\n x\n \n ≪\n \n 1\n ,\n \n (\n 96\n )\n \n \n \n \n \n \n", "1704.08555_51": "\n \n \n \n \n π\n \n (\n u\n ,\n v\n )\n \n =\n l\n i\n m\n k\n →\n ∞\n ℱ\n −\n 1\n \n (\n ψ\n \n (\n 2\n −\n 𝓀\n ⋅\n )\n \n 𝓊\n ∧\n )\n \n ℱ\n −\n 1\n \n (\n ψ\n \n (\n 2\n −\n 𝓀\n ⋅\n )\n \n 𝓋\n ∧\n )\n \n \n (\n 2.43\n )\n \n \n \n \n \n", "1705.02208_73": "", "1705.03300_63": "\n \n \n \n \n \n \n \n (\n \n \n τ\n ∘\n \n F\n i\n \n \n ⁢\n \n (\n e\n )\n \n \n )\n \n ⁢\n \n (\n a\n )\n \n \n =\n \n τ\n ⁢\n \n (\n \n ℰ\n ⁢\n \n (\n \n \n Φ\n i\n \n ⁢\n \n (\n \n π\n ⁢\n \n (\n a\n )\n \n \n )\n \n \n )\n \n \n )\n \n \n =\n \n \n τ\n ′\n \n ⁢\n \n (\n \n \n Φ\n i\n \n ⁢\n \n (\n \n π\n ⁢\n \n (\n a\n )\n \n \n )\n \n \n )\n \n \n ≤\n \n \n τ\n ′\n \n ⁢\n \n (\n \n π\n ⁢\n \n (\n a\n )\n \n \n )\n \n \n =\n \n τ\n ⁢\n \n (\n a\n )\n \n \n \n ;\n \n \n \n \n", "1705.03538_11": "\n \n \n \n \n \n \n \n λ\n -\n 1\n \n 2\n \n ≥\n \n \n 𝒫\n ⁢\n \n (\n λ\n )\n \n \n -\n \n n\n k\n \n \n \n .\n \n (\n 4\n )\n \n \n \n \n \n", "1705.07430_11": "\n \n \n \n \n \n \n d\n ⁢\n g\n ⁢\n \n (\n μ\n )\n \n \n \n d\n ⁢\n \n ln\n ⁡\n μ\n \n \n \n =\n \n -\n \n \n β\n 0\n \n ⁢\n \n \n \n g\n 3\n \n ⁢\n \n (\n μ\n )\n \n \n \n 16\n ⁢\n \n π\n 2\n \n \n \n ⁢\n \n (\n 11\n )\n \n \n \n \n \n \n \n", "1707.01722_226": "\n \n \n \n \n \n \n ∥\n E\n ∥\n \n p\n p\n \n ⪯\n \n \n e\n \n \n -\n \n α\n ⁢\n p\n ⁢\n r\n \n \n 2\n \n \n ⁢\n \n \n ∑\n j\n \n \n μ\n ⁢\n \n \n (\n \n \n B\n g\n \n ⁢\n \n (\n \n a\n \n k\n j\n \n \n ,\n δ\n )\n \n \n )\n \n p\n \n ⁢\n \n (\n 5.7\n )\n \n \n \n \n \n \n \n \n", "1707.02664_31": "\n \n \n \n \n \n \n \n \n ∥\n \n \n Ω\n ⁢\n \n (\n x\n )\n \n \n -\n \n Ω\n ⁢\n \n (\n y\n )\n \n \n \n ∥\n \n ∞\n \n ≤\n \n \n 8\n ⁢\n a\n ⁢\n \n ρ\n 2\n \n ⁢\n F\n ⁢\n \n \n ∥\n \n y\n -\n x\n \n ∥\n \n ∞\n \n \n ,\n \n ∀\n x\n \n \n \n ,\n \n y\n ∈\n \n C\n ⁢\n \n (\n \n [\n \n -\n a\n \n ,\n a\n ]\n \n )\n \n \n \n \n ,\n \n \n \n \n", "1707.03851_15": "\n \n \n \n \n \n \n \n K\n \n e\n ⁢\n f\n ⁢\n f\n \n \n \n ≪\n \n \n Δ\n ⁢\n ϕ\n \n \n M\n P\n \n \n ≪\n \n 1\n \n \n .\n \n (\n 3.6\n )\n \n \n \n \n \n", "1707.03932_1": "\n \n \n \n \n \n \n φ\n \n ⁢\n \n (\n \n \n ⨁\n \n g\n ∈\n \n \n α\n \n -\n 1\n \n \n ⁢\n \n (\n h\n )\n \n \n \n \n \n A\n g\n \n \n )\n \n \n =\n \n B\n h\n \n \n \n \n \n", "1707.05283_73": "\n \n \n \n \n \n \n \n Θ\n 0\n \n =\n \n \n c\n \n -\n \n 1\n \n 1\n +\n α\n \n \n \n \n ⁢\n \n Γ\n 0\n \n m\n \n 1\n +\n α\n \n \n \n ⁢\n \n U\n 0\n \n \n 1\n +\n n\n \n \n 1\n +\n α\n \n \n \n \n \n ,\n \n \n Σ\n 0\n \n =\n \n \n c\n \n α\n \n 1\n +\n α\n \n \n \n ⁢\n \n Γ\n 0\n \n m\n \n 1\n +\n α\n \n \n \n ⁢\n \n U\n 0\n \n -\n \n \n α\n -\n n\n \n \n 1\n +\n α\n \n \n \n \n \n \n \n .\n \n \n \n \n", "1707.07776_72": "\n \n \n \n \n \n \n u\n k\n 2\n \n ≤\n \n \n \n u\n \n k\n -\n 1\n \n \n ⁢\n \n u\n \n k\n +\n 1\n \n \n \n   \n \n (\n \n k\n =\n \n 1\n ,\n 2\n ,\n …\n \n \n )\n \n \n \n .\n \n (\n 4.4\n )\n \n \n \n \n \n", "1707.08655_119": "\n \n \n \n \n \n \n \n \n ξ\n k\n \n ⁢\n \n ψ\n k\n \n \n ≤\n \n \n γ\n k\n 2\n \n ⁢\n \n \n ∥\n \n ∇\n ⁡\n \n ϕ\n k\n \n \n ∥\n \n 2\n \n \n \n     \n \n \n \n  f o r \n \n ⁢\n k\n \n =\n \n 1\n ,\n 2\n ,\n …\n \n \n \n .\n \n \n \n \n", "1708.02705_178": "\n \n \n \n \n \n \n \n (\n \n \n P\n \n r\n -\n k\n \n \n ⁢\n \n h\n \n n\n ,\n ϑ\n \n \n \n )\n \n ⁢\n \n (\n \n d\n \n 1\n :\n k\n \n \n )\n \n \n =\n \n \n (\n \n \n ∏\n \n j\n =\n 1\n \n k\n \n \n \n L\n \n b\n n\n \n \n ⁢\n \n (\n \n x\n -\n \n x\n j\n \n \n )\n \n \n \n )\n \n ⁢\n \n \n ∫\n \n \n [\n \n -\n 1\n \n ,\n 1\n ]\n \n \n m\n ⁢\n \n (\n \n r\n -\n k\n \n )\n \n \n \n \n \n \n φ\n \n [\n \n r\n -\n k\n \n ]\n \n \n ⁢\n \n (\n \n v\n \n 1\n :\n k\n \n \n ,\n \n x\n -\n \n \n b\n n\n \n ⁢\n \n x\n \n \n k\n +\n 1\n \n :\n r\n \n \n \n \n )\n \n ⁢\n \n (\n \n \n ∏\n \n j\n =\n \n k\n +\n 1\n \n \n r\n \n \n L\n ⁢\n \n (\n \n x\n j\n \n )\n \n \n \n )\n \n ⁢\n \n 𝑑\n \n x\n \n \n k\n +\n 1\n \n :\n r\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "1708.02974_5": "\n \n \n \n \n ∀\n a\n ,\n b\n ∈\n K\n \n [\n G\n ]\n \n ,\n \n (\n a\n ≤\n K\n ⁢\n \n [\n G\n ]\n \n b\n ⇔\n ∃\n c\n ∈\n K\n \n [\n G\n ]\n \n ,\n a\n +\n c\n =\n b\n )\n \n .\n \n \n \n \n", "1708.04079_9": "", "1708.08226_50": "", "1708.09271_12": "\n \n \n \n \n \n \n \n z\n 1\n \n n\n +\n 2\n \n \n ⁢\n \n P\n \n n\n ,\n 4\n \n \n ⁢\n \n (\n \n z\n 1\n \n )\n \n \n +\n \n \n f\n \n n\n ,\n 3\n \n ′\n \n ⁢\n \n (\n \n z\n 1\n \n )\n \n ⁢\n \n \n ∂\n ⁡\n G\n \n \n ∂\n ⁡\n ϵ\n \n \n ⁢\n \n (\n 0\n ,\n 0\n ,\n w\n )\n \n \n \n =\n 0\n \n \n \n \n", "1709.07478_14": "\n \n \n \n \n \n \n ∑\n k\n \n \n \n \n |\n \n E\n k\n \n |\n \n 2\n \n \n 16\n ⁢\n π\n ⁢\n \n W\n 0\n \n \n \n \n ≈\n \n \n \n (\n \n m\n M\n \n )\n \n \n (\n \n 1\n /\n 3\n \n )\n \n \n ⁢\n \n (\n 15\n )\n \n \n \n \n \n \n", "1709.10291_10": "\n \n \n \n \n \n \n \n J\n c\n \n ⁢\n \n (\n H\n )\n \n \n =\n \n \n J\n c0\n \n ⁢\n \n \n 1\n -\n \n \n |\n \n H\n /\n \n H\n \n c\n ⁢\n 2\n \n \n \n |\n \n α\n \n \n \n 1\n +\n \n \n |\n \n H\n /\n \n H\n 0\n \n \n |\n \n α\n \n \n \n ⁢\n \n \n (\n \n 1\n -\n \n |\n \n H\n /\n \n H\n irr\n \n \n |\n \n \n )\n \n n\n \n \n \n .\n \n (\n 11\n )\n \n \n \n \n \n", "1710.01414_114": "\n \n \n \n \n \n \n \n W\n 0\n \n 1\n ,\n p\n \n \n ⁢\n \n (\n D\n )\n \n \n ⊂\n \n \n L\n p\n \n ⁢\n \n (\n D\n )\n \n \n ⊂\n \n \n L\n 2\n \n ⁢\n \n (\n D\n )\n \n \n ⊂\n \n \n W\n 0\n \n 1\n ,\n p\n \n \n ⁢\n \n \n (\n D\n )\n \n *\n \n \n ⊂\n \n \n L\n q\n \n ⁢\n \n (\n D\n )\n \n \n \n ,\n \n \n \n \n", "1710.03335_192": "", "1710.07676_16": "\n \n \n \n \n \n (\n 12\n )\n \n ⁢\n \n F\n n\n \n ⁢\n \n (\n z\n )\n \n \n :=\n \n \n \n \n 1\n \n i\n ⁢\n π\n \n \n \n ⁢\n \n \n \n ∫\n Λ\n \n \n \n \n \n \n \n u\n n\n \n ⁢\n \n (\n ζ\n )\n \n \n \n ζ\n -\n z\n \n \n \n ⁢\n \n 𝑑\n ζ\n \n \n \n \n +\n \n \n \n 1\n \n i\n ⁢\n π\n \n \n \n ⁢\n \n \n \n ∫\n Λ\n \n \n \n \n {\n \n \n \n \n d\n ⁢\n ζ\n \n \n ζ\n -\n z\n \n \n \n -\n \n \n \n d\n ⁢\n \n ζ\n ¯\n \n \n \n \n ζ\n ¯\n \n -\n \n z\n ¯\n \n \n \n \n \n }\n \n ⁢\n \n \n \n ∑\n \n j\n =\n 1\n \n ∞\n \n \n \n \n u\n \n n\n -\n j\n \n \n ⁢\n \n (\n ζ\n )\n \n ⁢\n \n \n (\n \n \n \n \n ζ\n ¯\n \n -\n \n z\n ¯\n \n \n \n ζ\n -\n z\n \n \n \n )\n \n j\n \n \n \n \n \n \n \n \n ,\n \n z\n ∈\n \n Λ\n ∪\n L\n \n \n \n ,\n \n", "1710.07683_8": "\n \n \n \n \n \n \n ℰ\n P\n \n ⁢\n f\n ⁢\n \n (\n t\n ,\n x\n )\n \n \n =\n \n ∫\n \n \n e\n \n i\n ⁢\n \n (\n \n \n t\n ⁢\n P\n ⁢\n \n (\n \n |\n ξ\n |\n \n )\n \n \n +\n \n x\n ⁢\n ξ\n \n \n )\n \n \n \n ⁢\n f\n ⁢\n \n (\n ξ\n )\n \n ⁢\n \n 𝑑\n ξ\n \n \n \n \n \n \n \n", "1710.08403_66": "\n \n \n \n \n 0\n ≤\n \n log\n ⁡\n u\n \n ≤\n \n log\n ⁡\n \n (\n \n 1\n +\n \n 1\n \n p\n -\n 2\n \n \n \n )\n \n \n ≤\n \n 1\n \n p\n -\n 2\n \n \n \n \n \n \n", "1710.08811_71": "\n \n \n \n \n \n |\n \n \n \n w\n ε\n \n ⁢\n \n (\n y\n )\n \n \n -\n \n \n φ\n ε\n \n ⁢\n \n (\n y\n )\n \n \n \n |\n \n ≤\n \n \n C\n ⁢\n \n γ\n ε\n \n -\n 1\n \n \n ⁢\n \n (\n \n 1\n +\n \n O\n ⁢\n \n (\n \n r\n ε\n \n -\n 1\n \n \n )\n \n \n \n )\n \n ⁢\n \n μ\n ε\n \n ⁢\n \n ln\n ⁡\n \n \n \n μ\n ε\n \n +\n \n r\n ε\n \n \n \n \n μ\n ε\n \n +\n \n |\n y\n |\n \n \n \n \n \n +\n \n O\n ⁢\n \n (\n \n γ\n ε\n \n -\n 2\n \n \n )\n \n \n \n \n \n \n \n", "1711.02859_234": "\n \n \n \n \n \n \n \n \n d\n ⁢\n \n \n Q\n ~\n \n t\n \n \n \n d\n ⁢\n Q\n \n \n ⁢\n \n (\n w\n )\n \n \n =\n \n \n M\n t\n \n ⁢\n \n (\n w\n )\n \n \n \n .\n \n \n \n \n", "1711.03578_4": "\n \n \n \n \n \n \n μ\n k\n \n ⁢\n \n (\n ω\n )\n \n \n =\n \n \n \n n\n \n k\n +\n 1\n \n \n -\n \n n\n k\n \n \n \n g\n ⁢\n \n (\n \n n\n k\n \n )\n \n \n \n ≤\n \n \n n\n \n k\n +\n 1\n \n \n \n g\n ⁢\n \n (\n \n n\n k\n \n )\n \n \n \n <\n \n \n 2\n ⁢\n \n n\n \n k\n +\n 1\n \n \n \n \n g\n ⁢\n \n (\n \n \n n\n \n k\n +\n 1\n \n \n -\n 1\n \n )\n \n \n \n =\n \n \n \n 2\n ⁢\n \n (\n \n \n n\n \n k\n +\n 1\n \n \n -\n 1\n \n )\n \n \n \n g\n ⁢\n \n (\n \n \n n\n \n k\n +\n 1\n \n \n -\n 1\n \n )\n \n \n \n +\n \n 2\n \n g\n ⁢\n \n (\n \n \n n\n \n k\n +\n 1\n \n \n -\n 1\n \n )\n \n \n \n \n <\n \n δ\n +\n \n 2\n \n g\n ⁢\n \n (\n 0\n )\n \n \n \n \n \n \n \n \n", "1711.05673_64": "", "1711.08070_48": "\n \n \n \n \n \n g\n ⁢\n \n (\n x\n )\n \n \n =\n \n β\n ⁢\n \n \n ∫\n 0\n ∞\n \n \n \n \n \n f\n ⁢\n \n (\n x\n )\n \n \n -\n \n f\n ⁢\n \n (\n \n x\n -\n ξ\n \n )\n \n \n \n \n ξ\n \n 1\n +\n β\n \n \n \n ⁢\n \n 𝑑\n ξ\n \n ⁢\n \n (\n 5.1\n )\n \n \n \n \n \n \n \n \n", "1711.09788_101": "", "1711.09816_3": "\n \n \n \n \n \n \n \n Δ\n ⁢\n \n S\n m\n \n ⁢\n \n (\n \n T\n \n r\n ⁢\n 1\n \n \n )\n \n \n /\n Δ\n \n ⁢\n \n S\n m\n \n m\n ⁢\n a\n ⁢\n x\n \n \n \n =\n \n \n \n Δ\n ⁢\n \n S\n M\n \n ⁢\n \n (\n \n T\n \n r\n ⁢\n 2\n \n \n )\n \n \n /\n Δ\n \n ⁢\n \n S\n m\n \n m\n ⁢\n a\n ⁢\n x\n \n \n \n =\n \n h\n ⁢\n \n (\n \n 0\n <\n h\n <\n 1\n \n )\n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n", "1712.00624_51": "\n \n \n \n \n \n F\n m\n \n 1\n →\n \n 2\n ,\n \n S\n ⁢\n E\n ⁢\n P\n \n \n \n \n =\n \n \n \n 1\n \n 2\n ⁢\n \n (\n \n d\n +\n 1\n \n )\n \n \n \n +\n \n \n \n (\n \n 2\n +\n d\n \n )\n \n \n 2\n ⁢\n \n (\n \n d\n +\n 1\n \n )\n \n \n \n ⁢\n \n 1\n \n \n P\n ~\n \n m\n \n \n ⁢\n \n \n (\n \n \n ∑\n \n k\n =\n 0\n \n \n d\n -\n 1\n \n \n \n \n \n |\n \n α\n k\n \n |\n \n 2\n \n ⁢\n \n \n c\n ~\n \n \n k\n +\n m\n \n \n \n \n )\n \n 2\n \n \n \n ,\n \n (\n 48\n )\n \n \n \n \n \n \n", "1712.04383_22": "\n \n \n \n \n ε\n =\n \n f\n +\n \n T\n ⁢\n s\n ⁢\n \n (\n 23\n )\n \n \n \n \n \n \n \n", "1712.07902_24": "\n \n \n \n \n \n M\n ⁢\n \n (\n \n [\n \n N\n \n 3\n /\n 4\n \n \n ]\n \n )\n \n \n ≥\n \n \n c\n 2\n \n ⁢\n \n |\n \n \n u\n ⁢\n \n (\n p\n )\n \n \n -\n \n u\n ⁢\n \n (\n \n p\n ′\n \n )\n \n \n \n |\n \n ⁢\n \n N\n \n 3\n /\n 4\n \n \n \n ≥\n \n \n c\n 3\n \n ⁢\n \n N\n \n 1\n /\n 4\n \n \n \n ≥\n \n C\n 1\n \n \n \n \n \n", "1712.08150_96": "", "1712.08991_54": "\n \n \n \n \n \n -\n \n \n 𝟏\n \n {\n \n \n i\n 1\n \n =\n \n i\n 2\n \n ≠\n 0\n \n }\n \n \n ⁢\n \n 𝟏\n \n {\n \n \n j\n 1\n \n =\n \n j\n 2\n \n \n }\n \n \n ⁢\n \n ζ\n \n j\n 3\n \n \n (\n \n i\n 3\n \n )\n \n \n ⁢\n \n ζ\n \n j\n 4\n \n \n (\n \n i\n 4\n \n )\n \n \n ⁢\n \n ζ\n \n j\n 5\n \n \n (\n \n i\n 5\n \n )\n \n \n \n \n -\n \n \n \n 𝟏\n \n {\n \n \n i\n 1\n \n =\n \n i\n 3\n \n ≠\n 0\n \n }\n \n \n ⁢\n \n 𝟏\n \n {\n \n \n j\n 1\n \n =\n \n j\n 3\n \n \n }\n \n \n ⁢\n \n ζ\n \n j\n 2\n \n \n (\n \n i\n 2\n \n )\n \n \n ⁢\n \n ζ\n \n j\n 4\n \n \n (\n \n i\n 4\n \n )\n \n \n ⁢\n \n ζ\n \n j\n 5\n \n \n (\n \n i\n 5\n \n )\n \n \n \n -\n \n \n \n \n \n", "1712.09023_49": "\n \n \n \n \n \n \n \n ∥\n \n \n M\n q\n ♯\n \n ⁢\n \n (\n \n \n M\n b\n \n ⁢\n f\n \n )\n \n \n ∥\n \n \n \n L\n p\n \n ⁢\n \n (\n ω\n )\n \n \n \n ≤\n \n C\n ⁢\n \n \n ∥\n b\n ∥\n \n \n B\n ⁢\n M\n ⁢\n O\n \n \n ⁢\n \n [\n \n \n \n ∥\n \n \n M\n s\n \n ⁢\n \n (\n \n \n M\n ~\n \n ⁢\n f\n \n )\n \n \n ∥\n \n \n \n L\n p\n \n ⁢\n \n (\n ω\n )\n \n \n \n +\n \n \n ∥\n \n \n M\n \n L\n ⁢\n \n (\n \n l\n ⁢\n o\n ⁢\n g\n ⁢\n L\n \n )\n \n \n \n ⁢\n f\n \n ∥\n \n \n \n L\n p\n \n ⁢\n \n (\n ω\n )\n \n \n \n \n ]\n \n \n \n .\n \n \n \n \n", "1712.09235_24": "\n \n \n \n \n \n \n K\n j\n α\n \n =\n \n \n K\n j\n 1\n \n +\n \n K\n j\n 2\n \n +\n \n K\n j\n 3\n \n +\n \n K\n j\n 4\n \n \n \n ,\n \n \n \n \n", "1712.09461_106": "", "1712.09746_205": "\n \n \n \n \n \n +\n \n \n 𝟏\n \n {\n \n \n \n i\n 7\n \n =\n \n i\n 2\n \n ≠\n 0\n \n ,\n \n \n \n j\n 7\n \n =\n \n j\n 2\n \n \n ,\n \n \n i\n 1\n \n =\n \n i\n 6\n \n ≠\n 0\n \n \n ,\n \n \n j\n 1\n \n =\n \n j\n 6\n \n \n \n }\n \n \n ⁢\n \n \n ∏\n \n l\n =\n \n 3\n ,\n 4\n ,\n 5\n \n \n \n \n ζ\n \n j\n l\n \n \n (\n \n i\n l\n \n )\n \n \n \n \n \n +\n \n \n \n 𝟏\n \n {\n \n \n \n i\n 7\n \n =\n \n i\n 2\n \n ≠\n 0\n \n ,\n \n \n \n j\n 7\n \n =\n \n j\n 2\n \n \n ,\n \n \n i\n 3\n \n =\n \n i\n 4\n \n ≠\n 0\n \n \n ,\n \n \n j\n 3\n \n =\n \n j\n 4\n \n \n \n }\n \n \n ⁢\n \n \n ∏\n \n l\n =\n \n 1\n ,\n 5\n ,\n 6\n \n \n \n \n ζ\n \n j\n l\n \n \n (\n \n i\n l\n \n )\n \n \n \n \n +\n \n \n \n \n \n", "1712.10326_93": "\n \n \n \n \n \n ∑\n \n n\n =\n 1\n \n ∞\n \n \n \n 1\n \n \n n\n \n 3\n -\n \n 2\n ⁢\n p\n \n \n \n ⁢\n \n \n log\n \n 2\n ⁢\n \n [\n p\n ]\n \n \n \n ⁡\n \n (\n \n n\n +\n 1\n \n )\n \n \n \n \n ⁢\n \n \n ∫\n \n \n \n I\n ¯\n \n N\n \n ×\n \n I\n N\n \n \n \n \n \n \n |\n \n \n S\n \n n\n ,\n n\n \n \n ⁢\n a\n ⁢\n \n (\n x\n ,\n y\n )\n \n \n |\n \n p\n \n ⁢\n \n 𝑑\n μ\n \n ⁢\n \n (\n x\n ,\n y\n )\n \n \n \n \n \n \n \n \n", "1801.00120_18": "\n \n \n \n \n \n \n A\n ~\n \n \n n\n ,\n 1\n \n \n =\n 2\n \n B\n 1\n \n H\n \n (\n 2\n H\n -\n 1\n )\n \n \n n\n \n 2\n ⁢\n H\n \n \n \n ∫\n 0\n 1\n \n \n ∫\n 0\n 1\n \n \n t\n n\n \n \n s\n n\n \n \n (\n \n B\n s\n \n -\n \n B\n 1\n \n )\n \n |\n t\n -\n s\n \n |\n \n \n 2\n ⁢\n H\n \n -\n 2\n \n \n d\n s\n d\n t\n =\n :\n \n B\n 1\n \n \n A\n \n n\n ,\n 3\n \n \n ,\n \n \n \n \n", "1801.00461_38": "", "1801.00983_47": "\n \n \n \n \n \n C\n ⁢\n \n \n ∥\n \n \n χ\n ′\n \n ⁢\n v\n \n ∥\n \n \n \n L\n 1\n \n ⁢\n \n (\n \n -\n 2\n \n ,\n 2\n )\n \n \n \n \n .\n \n \n \n \n", "1801.01636_43": "\n \n \n \n \n \n \n \n G\n \n z\n ⁢\n z\n \n \n ⁢\n \n (\n ℓ\n ,\n t\n )\n \n \n =\n \n \n \n ∓\n 1\n \n \n 1\n ∓\n \n Re\n ⁢\n \n \n Pf\n ⁢\n \n [\n \n Φ\n NS\n \n z\n ⁢\n z\n \n \n ]\n \n \n \n \n \n \n ⁢\n Re\n ⁢\n \n (\n \n \n d\n ⁢\n \n \n Pf\n ⁢\n \n [\n \n Φ\n NS\n \n z\n ⁢\n z\n \n \n ]\n \n \n \n \n \n d\n ⁢\n t\n \n \n )\n \n \n \n ,\n \n \n \n \n", "1801.05114_15": "", "1801.05519_50": "\n \n \n \n \n \n t\n =\n \n \n \n e\n \n \n i\n ⁢\n E\n ⁢\n \n (\n \n d\n -\n w\n \n )\n \n \n /\n \n v\n 0\n \n \n \n ⁢\n \n v\n 0\n \n ⁢\n q\n \n \n \n \n v\n 0\n \n ⁢\n q\n ⁢\n \n cos\n ⁡\n \n (\n \n q\n ⁢\n w\n \n )\n \n \n \n -\n \n i\n ⁢\n \n (\n \n \n A\n z\n \n +\n E\n \n )\n \n ⁢\n \n sin\n ⁡\n \n (\n \n q\n ⁢\n w\n \n )\n \n \n \n \n \n \n .\n \n (\n 51\n )\n \n \n \n \n \n", "1801.06058_20": "\n \n \n \n \n \n \n u\n ^\n \n =\n \n \n [\n \n \n k\n ⁢\n \n u\n r\n \n \n -\n \n \n (\n \n k\n +\n 2\n \n )\n \n ⁢\n \n x\n ^\n \n \n -\n \n w\n ^\n \n \n ]\n \n /\n 3\n \n \n .\n \n (\n 25\n )\n \n \n \n \n \n", "1801.06338_3": "\n \n \n \n \n \n \n \n \n ∥\n \n \n T\n ρ\n \n ⁢\n f\n \n ∥\n \n 2\n \n ≤\n \n \n ∥\n f\n ∥\n \n \n 4\n /\n 3\n \n \n \n ,\n \n ρ\n =\n \n \n (\n \n \n 2\n ⁢\n k\n ⁢\n \n (\n \n n\n -\n k\n \n )\n \n \n \n n\n ⁢\n \n (\n \n n\n -\n 1\n \n )\n \n \n \n )\n \n \n C\n H\n \n \n \n \n .\n \n (\n 3\n )\n \n \n \n \n \n", "1801.06881_10": "", "1801.08783_37": "\n \n \n \n \n \n \n dist\n ⁡\n \n (\n \n \n f\n n\n \n ⁢\n \n (\n \n v\n 1\n \n )\n \n \n ,\n \n \n f\n n\n \n ⁢\n \n (\n \n w\n 1\n \n )\n \n \n )\n \n \n <\n \n \n \n 2\n ⁢\n ε\n \n /\n 3\n \n ⁢\n f\n ⁢\n o\n ⁢\n \n r\n \n ⁢\n all\n ⁢\n n\n \n ≥\n 0\n \n .\n \n (\n 10\n )\n \n \n \n \n \n", "1801.09276_40": "\n \n \n \n \n \n \n \n W\n 0\n \n ⁢\n \n (\n \n \n ψ\n ρ\n \n ⁢\n \n h\n g\n \n \n )\n \n \n -\n \n \n (\n \n 1\n -\n ε\n \n )\n \n ⁢\n \n W\n 0\n \n ⁢\n \n (\n \n z\n g\n \n )\n \n \n \n ≤\n \n 0\n ,\n \n (\n 3.7\n )\n \n \n \n \n \n \n", "1801.10554_33": "", "1802.00040_0": "\n \n \n \n \n \n (\n ω\n ,\n φ\n )\n \n =\n ∫\n M\n ω\n ∧\n ∗\n φ\n ¯\n ,\n \n (\n 1.1\n )\n \n \n \n \n \n", "1802.00643_1593": "\n \n \n \n \n \n ∫\n t\n T\n \n \n \n ψ\n k\n \n ⁢\n \n (\n \n t\n k\n \n )\n \n ⁢\n …\n ⁢\n \n \n ∫\n t\n \n t\n \n l\n +\n 2\n \n \n \n \n \n ψ\n \n l\n +\n 1\n \n \n ⁢\n \n (\n \n t\n \n l\n +\n 1\n \n \n )\n \n ⁢\n \n \n ∫\n t\n \n t\n \n l\n +\n 1\n \n \n \n \n \n ψ\n l\n \n ⁢\n \n (\n \n t\n \n l\n -\n 1\n \n \n )\n \n ⁢\n \n ψ\n \n l\n -\n 1\n \n \n ⁢\n \n (\n \n t\n \n l\n -\n 1\n \n \n )\n \n ⁢\n \n \n ∫\n t\n \n t\n \n l\n -\n 1\n \n \n \n \n \n ψ\n \n l\n -\n 2\n \n \n ⁢\n \n (\n \n t\n \n l\n -\n 2\n \n \n )\n \n ⁢\n …\n \n \n \n \n \n \n \n \n \n \n \n", "1802.03582_2": "", "1802.05537_64": "\n \n \n \n \n n\n s\n \n ⁢\n \n (\n q\n )\n \n \n -\n 1\n \n ≡\n \n \n \n (\n \n \n \n q\n \n \n 𝒫\n \n ℛ\n ,\n 0\n \n \n (\n S\n )\n \n \n ⁢\n \n (\n \n t\n q\n \n ,\n q\n )\n \n \n \n \n ⁢\n \n \n \n d\n ⁢\n \n 𝒫\n \n ℛ\n ,\n 0\n \n \n (\n S\n )\n \n \n ⁢\n \n (\n \n t\n q\n \n ,\n q\n )\n \n \n \n d\n ⁢\n q\n \n \n \n \n )\n \n \n q\n =\n \n ℋ\n ⁢\n \n (\n \n t\n q\n \n )\n \n \n \n \n ⁢\n \n (\n 65\n )\n \n \n \n", "1802.09011_7": "\n \n \n \n \n \n \n A\n μ\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n ∫\n \n \n d\n ³\n \n ⁢\n x\n ⁢\n \n (\n \n \n \n \n A\n μ\n \n ~\n \n ⁢\n \n (\n k\n )\n \n ⁢\n \n e\n \n i\n ⁢\n k\n ⁢\n x\n \n \n \n +\n \n \n \n \n (\n \n A\n μ\n \n )\n \n *\n \n ~\n \n ⁢\n \n (\n k\n )\n \n ⁢\n \n e\n \n -\n \n i\n ⁢\n k\n ⁢\n x\n \n \n \n \n \n )\n \n ⁢\n \n (\n 12\n )\n \n \n \n \n \n \n \n", "1802.09156_8": "", "1803.00434_4": "\n \n \n \n \n \n ρ\n f\n \n :\n \n \n G\n E\n \n →\n \n Aut\n ⁢\n \n (\n \n T\n f\n \n )\n \n \n \n \n \n \n \n", "1803.03199_222": "\n \n \n \n \n \n \n E\n ⁢\n \n [\n \n F\n ⁢\n \n (\n \n X\n \n t\n 1\n \n \n ,\n …\n ,\n \n X\n \n t\n n\n \n \n )\n \n \n ]\n \n \n =\n \n \n lim\n \n k\n →\n ∞\n \n \n ⁡\n \n \n E\n \n 1\n /\n k\n \n \n ⁢\n \n [\n \n F\n ⁢\n \n (\n \n X\n \n t\n 1\n \n \n ,\n …\n ,\n \n X\n \n t\n n\n \n \n )\n \n \n ]\n \n \n \n \n .\n \n \n \n \n", "1803.04583_76": "", "1803.05743_203": "\n \n \n \n \n \n \n Det\n ⁢\n \n (\n \n \n K\n 1\n \n ⁢\n \n (\n \n Λ\n ⁢\n \n (\n 𝒢\n )\n \n \n )\n \n \n )\n \n \n ⊆\n \n \n Hom\n ∗\n \n ⁢\n \n (\n \n \n R\n p\n \n ⁢\n \n (\n 𝒢\n )\n \n \n ,\n \n \n Λ\n c\n \n ⁢\n \n \n (\n \n Γ\n K\n \n )\n \n ×\n \n \n )\n \n \n \n .\n \n \n \n \n", "1803.05899_80": "", "1803.06543_11": "\n \n \n \n \n \n \n \n 𝐀\n t\n \n ⁢\n \n (\n x\n )\n \n \n :=\n \n \n (\n \n \n \n 𝐚\n t\n \n i\n ⁢\n j\n \n \n ⁢\n \n (\n x\n )\n \n \n -\n \n \n σ\n t\n \n i\n ⁢\n k\n \n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n σ\n t\n \n j\n ⁢\n k\n \n \n ⁢\n \n (\n x\n )\n \n \n \n )\n \n \n \n \n i\n ,\n j\n \n =\n 1\n \n ,\n \n …\n ,\n d\n \n \n \n \n .\n \n \n \n \n", "1803.07236_72": "\n \n \n \n \n \n \n lim\n \n ϵ\n →\n 0\n \n \n ⁡\n \n \n g\n t\n \n \n g\n -\n \n g\n x\n \n \n \n \n =\n \n \n \n \n \n e\n \n -\n x\n \n \n ⁢\n \n d\n \n n\n -\n 1\n \n \n ⁢\n \n (\n \n \n \n e\n x\n \n ⁢\n \n D\n \n \n \n N\n -\n n\n \n +\n 1\n \n ,\n t\n \n 2\n \n \n -\n \n 2\n ⁢\n \n D\n \n \n \n N\n -\n n\n \n +\n 2\n \n ,\n t\n \n 2\n \n \n \n )\n \n ⁢\n \n h\n n\n \n \n -\n \n \n d\n n\n \n ⁢\n \n (\n \n \n \n e\n x\n \n ⁢\n \n D\n \n \n N\n -\n n\n \n ,\n t\n \n 2\n \n \n -\n \n 2\n ⁢\n \n D\n \n \n \n N\n -\n n\n \n +\n 1\n \n ,\n t\n \n 2\n \n \n \n )\n \n ⁢\n \n h\n \n n\n -\n 1\n \n \n \n \n \n \n -\n \n \n h\n n\n \n ⁢\n \n d\n \n n\n -\n 1\n \n \n ⁢\n \n D\n \n \n N\n -\n n\n \n +\n 1\n \n 2\n \n \n \n +\n \n \n h\n \n n\n -\n 1\n \n \n ⁢\n \n d\n n\n \n ⁢\n \n e\n x\n \n ⁢\n \n D\n \n N\n -\n n\n \n 2\n \n \n \n \n ,\n \n (\n 4.10\n )\n \n \n \n \n \n \n", "1803.07631_26": "", "1803.10950_62": "\n \n \n \n \n \n \n \n (\n 43\n )\n \n ⁢\n \n \n lim sup\n \n n\n →\n ∞\n \n \n ⁡\n \n F\n ⁢\n \n (\n \n μ\n n\n \n )\n \n \n \n \n ≤\n \n F\n ⁢\n \n (\n μ\n )\n \n \n \n .\n \n \n \n \n", "1803.11065_19": "\n \n \n \n \n \n \n |\n ϕ\n ⟩\n \n =\n \n \n α\n ⁢\n \n |\n 00\n ⟩\n \n \n +\n \n β\n ⁢\n \n |\n 01\n ⟩\n \n \n +\n \n β\n ⁢\n \n |\n 10\n ⟩\n \n \n +\n \n δ\n ⁢\n \n |\n 11\n ⟩\n \n \n \n \n ,\n \n \n \n \n", "1803.11407_17": "\n \n \n \n \n \n \n A\n \n t\n ,\n \n t\n ′\n \n \n \n =\n \n \n 1\n \n dim\n ⁡\n \n (\n \n \n c\n \n t\n \n )\n \n \n \n ⁢\n \n \n ∑\n \n d\n =\n 1\n \n \n dim\n ⁡\n \n (\n \n \n c\n \n t\n \n )\n \n \n \n \n α\n \n \n t\n ′\n \n ,\n t\n \n d\n \n \n \n \n .\n \n \n \n \n", "1804.00328_109": "\n \n \n \n \n \n \n ⟨\n \n \n 𝖳\n \n q\n →\n p\n \n \n ⁢\n \n f\n \n J\n →\n x\n \n \n \n ,\n \n g\n \n J\n →\n y\n \n \n ⟩\n \n ≥\n \n \n \n μ\n q\n \n ⁢\n \n (\n \n f\n \n J\n →\n x\n \n \n )\n \n \n -\n \n δ\n 1\n \n \n \n ,\n \n \n \n \n", "1804.00851_86": "\n \n \n \n \n \n \n C\n N\n \n (\n α\n )\n \n \n ⁢\n \n (\n x\n ,\n y\n )\n \n \n =\n \n \n ∑\n \n k\n =\n 0\n \n N\n \n \n \n \n P\n ~\n \n k\n \n (\n α\n )\n \n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n \n P\n ~\n \n k\n \n (\n α\n )\n \n \n ⁢\n \n (\n y\n )\n \n \n \n \n \n \n \n", "1804.02268_4": "\n \n \n \n \n \n u\n \n i\n ,\n a\n \n \n -\n 1\n \n \n ⁢\n \n (\n w\n )\n \n \n -\n \n \n u\n \n i\n ,\n b\n \n \n -\n 1\n \n \n ⁢\n \n (\n w\n )\n \n \n \n =\n \n \n p\n \n i\n ,\n a\n \n \n -\n \n p\n \n i\n ,\n b\n \n \n \n =\n \n \n u\n \n i\n ,\n a\n \n \n -\n 1\n \n \n ⁢\n \n (\n \n v\n \n i\n ,\n b\n \n \n )\n \n \n \n .\n \n (\n 5\n )\n \n \n", "1804.06367_33": "", "1804.06975_166": "\n \n \n \n \n \n \n \n (\n 19\n )\n \n ⁢\n \n D\n \n Z\n ⁢\n \n (\n E\n )\n \n \n \n ⁢\n \n (\n \n N\n ⁢\n \n (\n Y\n )\n \n \n )\n \n \n =\n \n \n \n \n d\n \n d\n ⁢\n t\n \n \n ⁢\n \n (\n \n N\n ⁢\n \n (\n \n Y\n +\n \n t\n ⁢\n \n E\n Y\n \n \n \n )\n \n \n )\n \n \n |\n \n \n t\n =\n 0\n \n \n =\n \n (\n \n Y\n #\n \n ,\n \n E\n Y\n \n )\n \n =\n \n N\n ⁢\n \n (\n Y\n )\n \n ⁢\n \n tr\n ⁡\n \n (\n E\n )\n \n \n \n \n .\n \n \n \n \n", "1804.09034_82": "\n \n \n \n \n \n \n \n dim\n \n μ\n ,\n log\n \n 0\n \n ⁢\n \n (\n E\n )\n \n \n =\n \n \n dim\n log\n \n ⁢\n \n (\n E\n )\n \n \n =\n \n dim\n ⁢\n \n (\n E\n )\n \n \n \n ,\n \n \n \n \n", "1804.09349_24": "\n \n \n \n \n \n \n t\n \n ≥\n \n \n τ\n ν\n ϵ\n \n  \n ⟹\n \n \n  \n \n \n \n 1\n t\n \n ⁢\n \n log\n ⁡\n \n ∥\n \n \n ℰ\n t\n \n ⁢\n \n (\n \n A\n ϵ\n \n )\n \n \n ∥\n \n \n \n ≤\n \n \n (\n \n 1\n -\n ν\n \n )\n \n ⁢\n μ\n ⁢\n \n (\n \n \n A\n ^\n \n ∞\n ϵ\n \n )\n \n \n <\n \n  0\n ⁢\n \n (\n 18\n )\n \n \n \n \n \n \n \n", "1804.10417_58": "\n \n \n \n \n Q\n =\n \n -\n \n \n 1\n \n 2\n ⁢\n \n a\n 2\n \n \n \n ⁢\n \n \n \n Δ\n \n  \n x \n \n \n ⁢\n \n ρ\n \n \n \n ρ\n \n \n ⁢\n \n (\n 37\n )\n \n \n \n \n \n \n \n", "1805.01857_106": "\n \n \n \n \n \n \n \n (\n 8.5\n )\n \n ⁢\n \n dim\n ⁡\n \n T\n ⁢\n o\n ⁢\n p\n ⁢\n \n (\n ϕ\n )\n \n \n \n \n =\n \n \n \n q\n \n \n (\n \n e\n -\n 1\n \n )\n \n ⁢\n \n (\n \n m\n -\n 1\n \n )\n \n \n \n ⁢\n \n (\n \n \n q\n m\n \n +\n \n \n (\n \n -\n 1\n \n )\n \n \n m\n +\n 1\n \n \n \n )\n \n \n /\n \n (\n \n q\n +\n 1\n \n )\n \n \n \n .\n \n \n \n \n", "1805.03512_26": "\n \n \n \n \n \n \n \n \n \n \n ∥\n u\n |\n \n Ω\n \n ∥\n \n X\n \n ≤\n \n C\n ⁢\n \n ∥\n u\n ∥\n \n \n \n ,\n \n \n ∀\n u\n \n ∈\n \n \n 𝒟\n 0\n \n 1\n ,\n p\n \n \n ⁢\n \n (\n \n A\n \n R\n 1\n \n \n R\n 2\n \n \n ;\n L\n )\n \n \n \n \n .\n \n \n \n \n", "1805.04143_105": "\n \n \n \n \n \n \n \n sup\n \n t\n ∈\n \n (\n \n -\n 1\n \n ,\n 0\n )\n \n \n \n ⁡\n \n \n ∫\n \n B\n 1\n \n \n \n \n \n \n |\n \n D\n ⁢\n \n d\n \n i\n ⁢\n j\n \n \n ⁢\n \n (\n t\n ,\n x\n )\n \n \n |\n \n 2\n \n \n ⁢\n \n 𝑑\n x\n \n \n \n \n ≤\n \n 4\n ⁢\n \n \n sup\n \n t\n ∈\n \n (\n \n -\n 1\n \n ,\n 0\n )\n \n \n \n ⁡\n \n \n ∫\n \n B\n 1\n \n \n \n \n \n \n |\n \n u\n ⁢\n \n (\n t\n ,\n x\n )\n \n \n |\n \n 2\n \n \n ⁢\n \n 𝑑\n x\n \n \n \n \n \n <\n ∞\n \n ,\n \n \n \n ∀\n ı\n \n ,\n j\n \n =\n 1\n \n ,\n \n 2\n ,\n …\n ,\n d\n \n .\n \n (\n 5.4\n )\n \n \n \n \n \n", "1805.05493_147": "", "1805.06056_41": "\n \n \n \n \n \n \n A\n \n (\n c\n )\n \n \n =\n \n \n \n \n 4\n ⁢\n π\n \n \n p\n ⁢\n M\n \n \n ⁢\n \n δ\n \n (\n c\n )\n \n \n \n +\n ⋯\n \n =\n \n \n \n \n 4\n ⁢\n π\n ⁢\n \n a\n P\n \n (\n c\n )\n \n \n \n M\n \n ⁢\n \n p\n 2\n \n \n +\n \n O\n ⁢\n \n (\n \n p\n 4\n \n )\n \n \n \n \n ,\n \n (\n 43\n )\n \n \n \n \n \n", "1805.06528_58": "\n \n \n \n \n \n u\n t\n \n =\n \n \n \n \n L\n 1\n *\n \n ⁢\n u\n \n +\n \n \n a\n 11\n *\n \n ⁢\n \n (\n x\n )\n \n ⁢\n u\n ⁢\n \n (\n \n 1\n -\n u\n \n )\n \n \n \n ,\n \n (\n 2.13\n )\n \n \n \n \n \n \n", "1805.06567_9": "\n \n \n \n \n \n \n \n p\n ¯\n \n \n i\n ⁢\n d\n \n \n =\n \n \n \n ∑\n k\n \n \n n\n ⁢\n \n (\n k\n )\n \n ⁢\n \n \n (\n \n 1\n -\n \n \n F\n s\n \n ⁢\n \n (\n \n D\n t\n \n )\n \n \n \n )\n \n k\n \n \n \n \n \n ∑\n k\n \n \n n\n ⁢\n \n (\n k\n )\n \n \n \n \n \n .\n \n (\n 10\n )\n \n \n \n \n \n", "1805.06682_33": "\n \n \n \n \n \n λ\n \n M\n ⁢\n A\n \n \n ⁢\n \n (\n t\n ,\n \n ϑ\n \n M\n ⁢\n A\n \n \n )\n \n \n =\n \n \n λ\n 0\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n exp\n ⁡\n \n [\n \n \n ϑ\n 0\n \n M\n ⁢\n A\n \n \n +\n \n \n ϑ\n 1\n \n M\n ⁢\n A\n \n \n ⁢\n i\n ⁢\n \n (\n t\n )\n \n \n \n ]\n \n \n \n \n ,\n \n \n \n λ\n \n M\n ⁢\n B\n \n \n ⁢\n \n (\n t\n ,\n \n ϑ\n \n M\n ⁢\n B\n \n \n )\n \n \n =\n \n \n λ\n 0\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n exp\n ⁡\n \n [\n \n \n ϑ\n 0\n \n M\n ⁢\n B\n \n \n +\n \n \n ϑ\n 1\n \n M\n ⁢\n B\n \n \n ⁢\n i\n ⁢\n \n (\n t\n )\n \n \n \n ]\n \n \n \n \n \n ,\n \n", "1805.07087_87": "\n \n \n \n \n \n \n ε\n I\n \n =\n \n \n ε\n II\n \n ⁢\n \n (\n \n 1\n -\n r\n \n )\n \n \n =\n \n -\n \n \n \n σ\n -\n tot\n \n \n σ\n +\n tot\n \n \n ⁢\n \n (\n \n 1\n -\n r\n \n )\n \n \n \n \n ,\n \n (\n 80\n )\n \n \n \n \n \n", "1805.07630_20": "\n \n \n \n \n \n φ\n ⁢\n \n (\n \n x\n 1\n \n )\n \n \n =\n \n \n x\n 1\n \n ⁢\n  a n d \n ⁢\n φ\n ⁢\n \n (\n \n x\n i\n \n )\n \n \n =\n \n x\n \n i\n -\n 1\n \n \n \n \n \n \n", "1805.10206_89": "\n \n \n \n \n \n \n \n J\n ^\n \n ⁢\n \n [\n \n \n f\n ^\n \n 0\n \n ]\n \n \n =\n \n \n \n (\n \n \n 𝖤\n ⁢\n G\n ⁢\n \n (\n Y\n )\n \n \n -\n \n 𝖤\n ⁢\n G\n ⁢\n \n (\n Z\n )\n \n \n \n )\n \n 2\n \n \n \n 2\n \n ⁢\n \n δ\n 2\n \n \n \n \n .\n \n \n \n \n", "1805.10232_26": "\n \n \n \n \n \n R\n ⁢\n M\n ⁢\n S\n ⁢\n E\n ⁢\n \n (\n \n 𝐀\n ^\n \n )\n \n \n =\n \n \n 1\n N\n \n ⁢\n \n \n ∑\n \n k\n =\n 1\n \n N\n \n \n \n \n \n 1\n P\n \n ⁢\n \n \n ∑\n \n p\n =\n 1\n \n P\n \n \n \n (\n \n \n a\n \n p\n ⁢\n k\n \n \n -\n \n \n a\n ^\n \n \n p\n ⁢\n k\n \n \n \n )\n \n 2\n \n \n \n \n ⁢\n \n (\n 26\n )\n \n \n \n \n \n \n \n \n", "1805.12318_20": "", "1806.01077_0": "\n \n \n \n \n \n \n \n 𝑥\n ˙\n \n \n =\n \n P\n ⁢\n \n (\n x\n ,\n y\n )\n \n \n \n ,\n \n \n \n 𝑦\n ˙\n \n \n =\n \n \n Q\n ⁢\n \n (\n x\n ,\n y\n )\n \n \n ,\n \n (\n 1.1\n )\n \n \n \n \n \n \n \n", "1806.02513_27": "\n \n \n \n \n \n \n π\n 1\n \n ≃\n \n \n π\n 1\n \n ⊗\n ν\n \n \n .\n \n \n \n \n", "1806.03808_34": "\n \n \n \n \n \n \n α\n \n 0\n ,\n \n q\n -\n p\n \n \n \n =\n ⋯\n =\n \n α\n \n p\n ,\n q\n \n \n =\n 0\n \n .\n \n \n \n \n", "1806.05045_30": "\n \n \n \n \n \n λ\n R\n \n =\n \n \n 1\n -\n \n 2\n ⁢\n \n \n f\n R\n \n R\n \n \n \n +\n \n 3\n ⁢\n \n (\n \n \n n\n \n -\n 1\n \n )\n \n ⁢\n \n (\n 31\n )\n \n \n \n \n \n \n \n", "1806.06896_27": "", "1806.07756_16": "\n \n \n \n \n \n \n ℒ\n ⁢\n \n (\n \n f\n r\n \n ,\n z\n ;\n X\n )\n \n \n =\n \n ℒ\n ⁢\n \n (\n \n \n f\n ¯\n \n r\n \n ,\n z\n ;\n X\n )\n \n \n =\n 0\n \n ,\n \n \n \n \n", "1806.09772_98": "", "1807.00507_8": "\n \n \n \n \n \n \n ∑\n \n i\n =\n 1\n \n n\n \n \n \n x\n i\n \n ×\n \n d\n i\n \n \n \n =\n \n L\n ⁢\n \n (\n 8\n )\n \n \n \n \n \n \n", "1807.00711_104": "\n \n \n \n \n \n \n \n ζ\n \n n\n -\n 1\n \n ⋆\n \n ⁢\n \n (\n \n \n {\n 1\n }\n \n \n ℓ\n 1\n \n \n )\n \n ⁢\n \n ζ\n \n n\n -\n 1\n \n \n ⁢\n \n (\n \n \n {\n 1\n }\n \n \n ℓ\n 2\n \n \n )\n \n \n =\n \n \n ∑\n \n ∘\n =\n `\n `\n ,\n \n \n or\n \n \n \n ′\n \n ′\n \n `\n `\n +\n \n \n \n ′\n \n \n \n ′\n \n \n \n \n \n ζ\n \n n\n -\n 1\n \n \n ⁢\n \n (\n \n \n {\n 1\n }\n \n \n ℓ\n 2\n \n \n )\n \n ⁢\n \n ζ\n \n n\n -\n 1\n \n \n ⁢\n \n (\n \n \n \n \n \n 1\n ∘\n 1\n \n ⁢\n ⋯\n \n ∘\n 1\n \n ⏟\n \n \n ℓ\n 1\n \n \n )\n \n \n \n \n .\n \n \n \n \n", "1807.02246_18": "", "1807.02438_44": "", "1903.08066_15": "\n \n \n \n \n \n \n s\n 3\n \n ⁢\n \n (\n \n \n q\n 3\n \n -\n \n z\n 3\n \n \n )\n \n \n =\n \n \n \n \n \n s\n 1\n \n ⁢\n \n (\n \n \n q\n 1\n \n -\n \n z\n 1\n \n \n )\n \n \n ⋅\n \n s\n 2\n \n \n ⁢\n \n (\n \n \n q\n 2\n \n -\n \n z\n 2\n \n \n )\n \n \n ,\n \n (\n 11\n )\n \n \n \n \n \n \n", "1903.09509_18": "", "1903.11585_24": "", "1903.11969_6": "\n \n \n \n \n \n \n \n D\n M\n \n α\n ,\n β\n \n \n \n n\n \n ⁢\n y\n \n +\n \n \n p\n \n n\n -\n 1\n \n \n ⁢\n \n \n (\n t\n )\n \n \n n\n -\n 1\n \n \n ⁢\n \n D\n M\n \n α\n ,\n β\n \n \n ⁢\n y\n \n +\n …\n +\n \n \n p\n 2\n \n ⁢\n \n \n (\n t\n )\n \n 2\n \n ⁢\n \n D\n M\n \n α\n ,\n β\n \n \n ⁢\n y\n \n +\n \n \n p\n 1\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n D\n M\n \n α\n ,\n β\n \n \n ⁢\n y\n \n +\n \n \n p\n 0\n \n ⁢\n \n (\n t\n )\n \n ⁢\n y\n \n \n =\n \n 0\n ⁢\n \n (\n 1\n )\n \n \n \n \n \n \n", "1904.00638_14": "\n \n \n \n \n Irr\n \n \n (\n U\n )\n \n Σ\n \n :=\n \n {\n \n Inf\n \n U\n /\n \n X\n \n 𝒩\n Σ\n \n \n \n U\n \n \n (\n χ\n )\n \n ∣\n χ\n ∈\n Irr\n \n \n (\n U\n /\n \n X\n \n 𝒩\n Σ\n \n \n )\n \n Σ\n \n }\n \n .\n \n \n \n \n", "1904.06035_16": "\n \n \n \n \n \n X\n =\n \n \n R\n \n ⊕\n a\n \n \n ⊕\n \n I\n \n ⊕\n b\n \n \n ⊕\n \n J\n \n ⊕\n c\n \n \n ⊕\n \n (\n \n \n ⨁\n i\n \n \n M\n \n p\n i\n \n \n ⊕\n \n d\n i\n \n \n \n \n )\n \n ⊕\n \n (\n \n \n ⨁\n j\n \n \n N\n \n q\n j\n \n \n ⊕\n \n e\n j\n \n \n \n \n )\n \n \n \n .\n \n \n \n \n", "1904.06248_52": "\n \n \n \n \n \n \n c\n \n ϵ\n .\n 𝒲\n \n \n ⁢\n \n (\n \n k\n x\n \n ,\n \n v\n ⁢\n e\n ⁢\n c\n ⁢\n ξ\n \n )\n \n \n =\n \n ϵ\n ⁢\n \n (\n 0\n )\n \n ⁢\n \n r\n c\n \n ⁢\n \n \n \n 2\n \n \n 3\n /\n 2\n \n -\n \n 3\n ⁢\n β\n \n \n \n ⁢\n \n π\n \n \n \n Γ\n ⁢\n \n (\n \n 3\n ⁢\n β\n \n )\n \n \n \n ⁢\n \n ∫\n \n \n \n (\n \n 1\n +\n \n \n θ\n 2\n \n \n r\n c\n 2\n \n \n \n )\n \n \n \n 1\n /\n 2\n \n -\n \n 3\n ⁢\n β\n \n \n \n ⁢\n \n ℱ\n \n \n 3\n ⁢\n β\n \n -\n \n 1\n /\n 2\n \n \n \n ⁢\n \n (\n \n ω\n ⁢\n \n |\n \n k\n x\n \n |\n \n ⁢\n \n \n \n θ\n 2\n \n +\n \n r\n c\n 2\n \n \n \n \n )\n \n ⁢\n 𝒲\n ⁢\n \n (\n \n v\n ⁢\n e\n ⁢\n c\n ⁢\n θ\n \n )\n \n ⁢\n \n e\n \n -\n \n \n \n i\n ⁢\n ω\n ⁢\n v\n ⁢\n e\n ⁢\n c\n ⁢\n ξ\n \n ⋅\n v\n \n ⁢\n e\n ⁢\n c\n ⁢\n θ\n \n \n \n ⁢\n \n d\n v\n \n ⁢\n e\n ⁢\n c\n ⁢\n θ\n ⁢\n \n (\n 18\n )\n \n \n \n \n \n \n \n \n", "1904.07525_133": "\n \n \n \n \n \n \n x\n 1\n \n ≥\n \n π\n -\n \n \n \n 4\n ⁢\n π\n ⁢\n σ\n \n \n 3\n ⁢\n \n (\n \n \n σ\n ⁢\n A\n \n -\n \n A\n 2\n \n \n )\n \n \n \n ⋅\n \n 1\n R\n \n \n \n \n ,\n \n \n \n \n", "1904.08224_7": "\n \n \n \n \n \n r\n H\n \n =\n \n \n \n M\n +\n \n \n (\n \n M\n +\n \n β\n ⁢\n \n m\n \n p\n ⁢\n l\n \n \n \n \n )\n \n 2\n \n \n \n M\n ⁢\n \n m\n \n p\n ⁢\n l\n \n 2\n \n \n \n ⁢\n \n (\n 7\n )\n \n \n \n \n \n \n", "1904.10099_50": "", "1904.11645_190": "", "1904.13359_9": "\n \n \n \n \n \n \n \n p\n i\n \n ⁢\n \n (\n 𝐱\n ,\n ω\n )\n \n \n =\n \n \n a\n i\n \n ⁢\n \n (\n ω\n )\n \n ⁢\n \n exp\n ⁡\n \n (\n \n \n i\n ⁢\n \n 𝐤\n 𝐢\n \n \n ⋅\n 𝐱\n \n )\n \n \n \n =\n \n \n a\n i\n \n ⁢\n \n (\n ω\n )\n \n ⁢\n \n exp\n ⁡\n \n [\n \n i\n ⁢\n \n (\n \n \n \n k\n x\n \n i\n ⁢\n \n [\n 0\n ]\n \n \n \n ⁢\n x\n \n -\n \n \n k\n y\n \n i\n ⁢\n \n [\n 0\n ]\n \n \n \n ⁢\n y\n \n \n )\n \n \n ]\n \n \n \n \n ,\n \n (\n 11\n )\n \n \n \n \n \n", "1905.01728_1": "\n \n \n \n \n \n \n \n \n (\n 2\n )\n \n ⁢\n \n V\n k\n \n ⁢\n \n (\n K\n )\n \n \n =\n \n \n (\n \n n\n k\n \n )\n \n ⁢\n \n \n κ\n n\n \n \n \n κ\n k\n \n ⁢\n \n κ\n \n n\n -\n k\n \n \n \n \n ⁢\n \n \n ∫\n \n \n G\n n\n \n ,\n m\n \n \n \n \n \n Vol\n k\n \n ⁡\n \n (\n \n \n p\n ν\n \n ⁢\n \n (\n K\n )\n \n \n )\n \n \n ⁢\n \n 𝑑\n ω\n \n ⁢\n \n (\n ν\n )\n \n \n \n \n \n ,\n \n 1\n ≤\n k\n ≤\n n\n \n \n .\n \n \n \n \n", "1905.02290_37": "", "1905.04296_0": "\n \n \n \n \n \n \n (\n \n \n R\n ˙\n \n R\n \n )\n \n 2\n \n =\n \n \n \n \n \n 8\n ⁢\n π\n ⁢\n \n G\n 0\n \n \n 3\n \n ⁢\n ρ\n \n -\n \n \n \n k\n ⁢\n \n c\n 2\n \n \n \n R\n 2\n \n \n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "1905.06506_41": "\n \n \n \n \n \n \n B\n \n 2\n ,\n ψ\n \n \n =\n \n -\n \n 2\n ⁢\n H\n ⁢\n \n (\n 2\n ,\n p\n )\n \n \n \n \n ,\n \n \n \n \n", "1905.08182_4": "\n \n \n \n \n \n \n \n \n f\n ⁢\n \n (\n x\n )\n \n \n -\n \n f\n ⁢\n \n (\n y\n )\n \n \n \n ≥\n \n \n d\n ⁢\n \n f\n y\n \n ⁢\n \n (\n \n η\n ⁢\n \n (\n x\n ,\n y\n )\n \n \n )\n \n \n ,\n \n ∀\n x\n \n \n \n ,\n \n y\n ∈\n S\n \n \n .\n \n \n \n \n", "1905.08334_57": "\n \n \n \n \n \n \n cos\n ⁡\n B\n \n ≤\n \n -\n \n \n 2\n \n 2\n \n \n \n .\n \n (\n 6\n )\n \n \n \n \n \n", "1905.08521_76": "", "1905.10961_102": "\n \n \n \n \n \n 𝐲\n -\n \n 𝐮\n ⁢\n \n (\n \n k\n +\n 1\n \n )\n \n \n \n =\n \n \n (\n \n 1\n -\n η\n \n )\n \n ⁢\n \n (\n \n 𝐲\n -\n \n 𝐮\n ⁢\n \n (\n k\n )\n \n \n \n )\n \n ⁢\n \n (\n 108\n )\n \n \n \n \n \n \n", "1905.11454_58": "\n \n \n \n \n \n \n C\n ⁢\n \n (\n M\n ,\n g\n )\n \n \n ≡\n \n \n \n 6\n ⁢\n \n P\n 1\n 2\n \n ⁢\n \n (\n \n ν\n ⁢\n \n (\n g\n )\n \n \n )\n \n ⁢\n \n P\n 2\n 2\n \n ⁢\n \n (\n \n ν\n ⁢\n \n (\n g\n )\n \n \n )\n \n \n -\n \n 24\n ⁢\n \n P\n 2\n 3\n \n ⁢\n \n (\n \n ν\n ⁢\n \n (\n g\n )\n \n \n )\n \n \n \n \n 30\n ⁢\n \n P\n 3\n \n ⁢\n \n (\n \n ν\n ⁢\n \n (\n g\n )\n \n \n )\n \n \n \n \n .\n \n \n \n \n", "1905.13286_81": "", "1906.00034_36": "\n \n \n \n \n \n tr\n ⁢\n \n B\n \n j\n +\n 1\n \n \n \n ≤\n \n \n \n \n tr\n ⁢\n \n B\n s\n \n \n +\n \n \n ∑\n \n k\n =\n s\n \n j\n \n \n \n 1\n \n \n 𝐲\n k\n T\n \n ⁢\n \n 𝐬\n k\n \n \n \n ⁢\n \n \n ∥\n \n 𝐲\n k\n \n ∥\n \n 2\n \n \n \n \n -\n \n \n ∑\n \n k\n =\n s\n \n j\n \n \n \n 1\n \n \n 𝐬\n k\n T\n \n ⁢\n \n \n B\n ~\n \n k\n \n ⁢\n \n 𝐬\n k\n \n \n \n ⁢\n \n \n ∥\n \n \n \n B\n ~\n \n k\n \n ⁢\n \n 𝐬\n k\n \n \n ∥\n \n 2\n \n ⁢\n \n (\n \n 1\n -\n ϕ\n -\n \n \n ψ\n k\n \n ⁢\n ϕ\n \n \n )\n \n \n \n \n ,\n \n (\n 30\n )\n \n \n \n \n \n \n", "1906.02929_84": "\n \n \n \n \n lim inf\n \n n\n →\n ∞\n \n \n -\n \n \n \n 1\n n\n \n \n ⁢\n log\n ⁢\n \n \n sup\n \n \n P\n \n X\n ⁢\n Y\n \n \n ∈\n 𝒮\n \n \n ⁡\n \n \n \n max\n \n d\n ∈\n \n \n 𝒟\n ~\n \n n\n \n \n \n ⁡\n \n ε\n \n X\n ⁢\n \n Y\n \n (\n d\n )\n \n \n \n \n (\n n\n )\n \n \n \n ⁢\n \n (\n \n \n f\n ~\n \n n\n \n (\n 1\n )\n \n \n ,\n \n \n f\n ~\n \n n\n \n (\n 2\n )\n \n \n ,\n \n \n φ\n ~\n \n n\n \n )\n \n \n \n \n \n ≥\n \n \n sup\n \n δ\n >\n 0\n \n \n ⁡\n \n \n min\n \n i\n ∈\n \n {\n 1\n ,\n 2\n ,\n 3\n }\n \n \n \n ⁢\n \n \n sup\n \n 0\n ≤\n ρ\n ≤\n 1\n \n \n ⁡\n \n \n inf\n \n \n P\n \n X\n ⁢\n Y\n \n \n ∈\n \n 𝒮\n δ\n \n \n \n ⁡\n \n \n F\n i\n \n ⁢\n \n (\n ρ\n ,\n \n R\n i\n \n ,\n \n P\n \n X\n ⁢\n Y\n \n \n ,\n 0\n )\n \n \n \n \n \n \n \n ,\n \n", "1906.03119_23": "\n \n \n \n \n \n \n w\n 0\n \n ⁢\n \n (\n u\n )\n \n \n :=\n \n -\n \n \n \n c\n 2\n \n ⁢\n \n (\n u\n )\n \n \n \n η\n 2\n \n \n \n =\n \n -\n \n \n 1\n \n η\n 2\n \n \n ⁢\n \n (\n \n \n ρ\n ⁢\n η\n ⁢\n u\n \n -\n λ\n \n )\n \n \n \n \n \n \n \n", "1906.03759_38": "\n \n \n \n \n \n \n \n (\n 4.11\n )\n \n ⁢\n \n \n Δ\n ⁢\n \n (\n \n \n (\n \n \n (\n \n U\n ∖\n \n {\n 1\n }\n \n \n )\n \n ∪\n \n D\n ¯\n \n \n )\n \n ←\n \n )\n \n \n \n Δ\n ⁢\n \n (\n \n \n (\n \n \n (\n \n \n U\n ′\n \n ∖\n \n {\n 1\n }\n \n \n )\n \n ∪\n \n \n D\n ′\n \n ¯\n \n \n )\n \n ←\n \n )\n \n \n \n ⁢\n \n \n Δ\n ⁢\n \n (\n \n \n α\n ⁢\n U\n \n ∪\n \n \n β\n ⁢\n D\n \n ¯\n \n \n )\n \n \n \n Δ\n ⁢\n \n (\n \n \n α\n ⁢\n \n U\n ′\n \n \n ∪\n \n \n β\n ⁢\n \n D\n ′\n \n \n ¯\n \n \n )\n \n \n \n \n =\n \n \n \n Δ\n ⁢\n \n (\n \n U\n ∪\n \n D\n ¯\n \n \n )\n \n \n \n Δ\n ⁢\n \n (\n \n \n U\n ′\n \n ∪\n \n \n D\n ′\n \n ¯\n \n \n )\n \n \n \n ⁢\n \n \n Δ\n ⁢\n \n (\n \n \n (\n \n \n (\n \n \n α\n ⁢\n U\n \n ∖\n \n {\n 1\n }\n \n \n )\n \n ∪\n \n \n β\n ⁢\n D\n \n ¯\n \n \n )\n \n ←\n \n )\n \n \n \n Δ\n ⁢\n \n (\n \n \n (\n \n \n (\n \n \n α\n ⁢\n \n U\n ′\n \n \n ∖\n \n {\n 1\n }\n \n \n )\n \n ∪\n \n \n β\n ⁢\n \n D\n ′\n \n \n ¯\n \n \n )\n \n ←\n \n )\n \n \n \n \n \n ,\n \n \n \n \n", "1906.05564_86": "", "1906.06006_0": "\n \n \n \n \n \n \n Δ\n ⁢\n \n (\n C\n )\n \n \n =\n \n {\n σ\n ∣\n \n σ\n ⊂\n \n c\n ⁢\n \n for\n \n ⁢\n some\n ⁢\n c\n \n ∈\n C\n \n }\n \n \n .\n \n \n \n \n", "1906.06860_163": "\n \n \n \n \n \n (\n p\n ↔\n q\n ↔\n r\n )\n \n :=\n \n p\n g\n \n \n r\n \n g\n -\n 1\n \n \n \n q\n \n g\n -\n 2\n \n \n +\n \n q\n g\n \n \n p\n \n g\n -\n 1\n \n \n \n r\n \n g\n -\n 2\n \n \n +\n \n q\n g\n \n \n r\n \n g\n -\n 1\n \n \n \n p\n \n g\n -\n 2\n \n \n +\n \n r\n g\n \n \n p\n \n g\n -\n 1\n \n \n \n q\n \n g\n -\n 2\n \n \n +\n \n r\n g\n \n \n q\n \n g\n -\n 1\n \n \n \n p\n \n g\n -\n 2\n \n \n ,\n \n \n \n \n", "1906.07041_1": "\n \n \n \n \n \n \n \n \n max\n \n D\n ∈\n \n Φ\n ⁢\n \n (\n C\n )\n \n \n \n \n \n ⁡\n tr\n \n ⁢\n \n (\n \n U\n ⋅\n D\n ⋅\n Π\n \n )\n \n \n ,\n \n \n \n \n", "1906.09597_171": "", "1906.11494_1": "\n \n \n \n \n \n \n λ\n \n ν\n ,\n en\n \n \n ⁢\n \n (\n E\n ,\n  x \n )\n \n \n =\n \n \n \n \n \n c\n \n ⁢\n \n λ\n \n ν\n ,\n tot\n \n \n ⁢\n \n (\n E\n ,\n  x \n )\n \n \n \n \n ∑\n s\n \n \n \n χ\n \n ν\n ,\n ab\n ,\n s\n \n \n ⁢\n \n (\n E\n ,\n  x \n )\n \n \n \n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "1907.00308_15": "\n \n \n \n \n \n E\n k\n \n =\n \n \n \n 1\n 2\n \n ⁢\n \n (\n \n \n P\n k\n 2\n \n +\n \n \n ω\n k\n 2\n \n ⁢\n \n Q\n k\n 2\n \n \n \n )\n \n \n ,\n \n (\n 17\n )\n \n \n \n \n \n \n", "1907.02433_1": "\n \n \n \n \n \n \n ℒ\n ⁢\n \n [\n X\n ]\n \n \n =\n \n \n -\n \n i\n ⁢\n \n [\n H\n ,\n X\n ]\n \n \n \n +\n \n 𝒟\n ⁢\n \n [\n X\n ]\n \n \n \n \n .\n \n \n \n \n", "1907.03465_2": "\n \n \n \n \n \n L\n ⁢\n o\n ⁢\n s\n ⁢\n s\n \n =\n \n \n \n λ\n 1\n \n ⁢\n \n L\n \n c\n ⁢\n l\n ⁢\n s\n \n \n \n +\n \n \n λ\n 2\n \n ⁢\n \n L\n \n r\n ⁢\n e\n ⁢\n g\n \n \n \n +\n \n \n λ\n 3\n \n ⁢\n \n L\n \n t\n ⁢\n r\n ⁢\n i\n \n \n \n +\n \n \n λ\n 4\n \n ⁢\n \n \n L\n \n p\n ⁢\n u\n ⁢\n l\n ⁢\n l\n \n \n \n ⁢\n a\n ⁢\n d\n ⁢\n d\n ⁢\n @\n ⁢\n c\n ⁢\n e\n ⁢\n n\n ⁢\n t\n ⁢\n e\n ⁢\n r\n ⁢\n i\n ⁢\n n\n ⁢\n g\n ⁢\n \n (\n 3\n )\n \n \n \n \n \n \n \n", "1907.06220_3": "", "1907.06469_19": "", "1907.09159_28": "\n \n \n \n \n \n \n \n ℰ\n int\n \n ⁢\n \n [\n \n {\n \n x\n ⁢\n \n (\n s\n )\n \n \n }\n \n ]\n \n \n ≈\n \n \n J\n \n ⁢\n ζ\n ⁢\n \n (\n k\n )\n \n ⁢\n \n \n N\n \n k\n +\n 1\n \n \n \n ⁢\n \n ∫\n \n \n \n \n [\n \n \n ρ\n N\n \n ⁢\n \n (\n x\n )\n \n \n ]\n \n \n k\n +\n 1\n \n \n \n ⁢\n \n 𝑑\n \n x\n \n \n \n \n \n \n .\n \n (\n 29\n )\n \n \n \n \n \n", "1907.11928_203": "\n \n \n \n \n \n M\n :=\n \n \n max\n \n r\n ,\n \n p\n 1\n \n ,\n \n …\n ⁢\n \n p\n r\n \n \n \n \n ⁢\n \n \n ∏\n \n α\n =\n 1\n \n \n r\n -\n 1\n \n \n \n \n max\n \n \n \n ξ\n 1\n \n ,\n \n …\n ⁢\n \n ξ\n p\n \n \n \n ∈\n \n B\n R\n \n \n \n ⁡\n \n |\n \n \n P\n \n α\n ,\n \n p\n \n r\n -\n α\n \n \n \n \n ⁢\n \n (\n \n ξ\n 1\n \n ,\n …\n ,\n \n ξ\n p\n \n )\n \n \n |\n \n \n \n \n \n ,\n \n \n \n \n", "1907.13569_49": "\n \n \n \n \n \n \n I\n ⁢\n \n (\n P\n ,\n L\n )\n \n \n :=\n \n |\n \n {\n \n \n (\n p\n ,\n ℓ\n )\n \n ∈\n \n P\n ×\n L\n \n \n :\n \n p\n ∈\n ℓ\n \n }\n \n |\n \n ≤\n \n \n \n |\n P\n |\n \n ⁢\n \n \n |\n L\n |\n \n \n 1\n /\n 2\n \n \n \n +\n \n 2\n ⁢\n \n |\n L\n |\n \n \n \n \n .\n \n \n \n \n", "1907.13571_43": "\n \n \n \n \n \n \n \n 𝒟\n *\n \n ⋅\n \n F\n ~\n \n \n :=\n \n \n ∑\n \n j\n =\n 1\n \n d\n \n \n \n 𝒟\n \n e\n j\n \n *\n \n ⁢\n \n \n F\n ~\n \n j\n \n \n \n \n .\n \n \n \n \n", "1908.00115_137": "", "1908.01895_91": "\n \n \n \n \n P\n =\n \n \n [\n \n \n \n \n I\n ⁢\n ρ\n \n \n \n \n ]\n \n ⁢\n \n \n (\n \n I\n +\n \n \n ρ\n *\n \n ⁢\n ρ\n \n \n )\n \n \n -\n 1\n \n \n ⁢\n \n [\n \n \n \n I\n \n \n \n ρ\n *\n \n \n \n \n ]\n \n \n \n \n \n \n", "1908.03553_164": "\n \n \n \n \n \n \n (\n \n \n S\n I\n \n ⁢\n \n (\n x\n )\n \n \n )\n \n j\n i\n \n =\n \n -\n \n \n \n g\n \n Y\n ⁢\n M\n \n 2\n \n \n 4\n ⁢\n \n π\n 2\n \n \n \n ⁢\n \n [\n \n \n \n ∑\n \n K\n =\n 1\n \n \n W\n 1\n \n \n \n \n \n t\n K\n \n ⁢\n \n \n (\n \n 𝒴\n 1\n \n )\n \n K\n I\n \n ⁢\n \n \n χ\n ¯\n \n K\n i\n \n ⁢\n \n χ\n \n K\n ⁢\n j\n \n \n \n \n |\n \n x\n -\n \n x\n K\n \n \n |\n \n \n \n +\n \n \n ∑\n \n K\n =\n \n \n W\n 1\n \n +\n 1\n \n \n W\n \n \n \n \n t\n K\n \n ⁢\n \n \n (\n \n 𝒴\n 2\n \n )\n \n K\n I\n \n ⁢\n \n \n χ\n ¯\n \n K\n i\n \n ⁢\n \n χ\n \n K\n ⁢\n j\n \n \n \n \n |\n \n x\n -\n \n x\n K\n \n \n |\n \n \n \n \n ]\n \n ⁢\n \n (\n 6.27\n )\n \n \n \n \n \n \n \n", "1908.04435_50": "\n \n \n \n \n \n σ\n ⁢\n \n (\n \n \n 𝒮\n B\n \n ⁢\n \n (\n G\n )\n \n \n )\n \n \n =\n \n \n \n σ\n ⁢\n \n (\n G\n )\n \n \n ∪\n \n σ\n ⁢\n \n \n (\n \n S\n 1\n \n )\n \n \n p\n -\n 1\n \n \n \n \n ,\n \n (\n 12\n )\n \n \n \n \n \n \n", "1908.05741_39": "\n \n \n \n \n \n \n Q\n ~\n \n \n α\n ⁢\n μ\n ⁢\n ν\n \n \n =\n \n \n \n Ω\n 2\n \n ⁢\n \n (\n \n \n Q\n \n α\n ⁢\n μ\n ⁢\n ν\n \n \n +\n \n 2\n ⁢\n \n g\n \n μ\n ⁢\n ν\n \n \n ⁢\n \n \n \n ∂\n α\n \n ⁡\n ln\n \n ⁡\n Ω\n \n \n \n )\n \n \n ,\n \n (\n 39\n )\n \n \n \n \n \n \n", "1908.07895_232": "", "1908.08415_4": "\n \n \n \n \n \n Δ\n ⁢\n \n (\n t\n )\n \n \n ≡\n \n \n \n \n C\n τ\n \n ⁢\n \n (\n t\n )\n \n \n -\n \n C\n ∞\n \n \n ,\n \n (\n 4\n )\n \n \n \n \n \n \n", "1908.10681_611": "\n \n \n \n \n *\n *\n \n g\n \n i\n ⁢\n j\n \n \n \n t\n i\n \n \n t\n j\n \n =\n 1\n .\n *\n *\n \n \n \n \n", "1909.02409_7": "", "1909.08334_5": "\n \n \n \n \n \n \n \n P\n \n λ\n ,\n ν\n \n \n ⁢\n \n (\n Z\n ,\n U\n )\n \n \n =\n \n \n \n (\n \n \n det\n ⁡\n \n (\n \n I\n -\n \n Z\n ⁢\n \n Z\n ∗\n \n \n \n )\n \n \n \n \n ∣\n \n det\n ⁡\n \n (\n \n I\n -\n \n Z\n ⁢\n \n U\n ∗\n \n \n \n )\n \n \n ∣\n \n 2\n \n \n )\n \n \n \n \n \n i\n ⁢\n λ\n \n +\n n\n \n -\n ν\n \n 2\n \n \n ⁢\n \n \n (\n \n det\n ⁡\n \n (\n \n I\n -\n \n Z\n ⁢\n \n U\n ∗\n \n \n \n )\n \n \n )\n \n \n -\n ν\n \n \n \n \n .\n \n \n \n \n", "1909.08785_4": "\n \n \n \n \n \n C\n \n α\n ⁢\n β\n \n \n =\n \n \n ∑\n \n a\n ⁢\n b\n \n \n \n \n g\n \n a\n ⁢\n b\n \n \n ⁢\n \n (\n \n \n \n U\n \n a\n ⁢\n α\n \n \n ⁢\n \n U\n \n b\n ⁢\n β\n \n \n \n -\n \n \n U\n \n b\n ⁢\n α\n \n \n ⁢\n \n U\n \n a\n ⁢\n β\n \n \n \n \n )\n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n \n", "1909.10052_76": "", "1909.13369_40": "\n \n \n \n \n \n \n \n \n lim\n \n n\n →\n ∞\n \n \n ⁡\n \n \n μ\n ⁢\n \n (\n \n \n \n T\n \n -\n n\n \n \n ⁢\n \n (\n A\n )\n \n \n ∩\n B\n \n )\n \n \n \n μ\n ⁢\n \n (\n A\n )\n \n \n \n \n =\n \n \n μ\n ⁢\n \n (\n B\n )\n \n \n ,\n \n ∀\n A\n \n \n \n ,\n \n B\n ∈\n ℬ\n \n \n .\n \n \n \n \n", "1910.00624_81": "\n \n \n \n \n \n \n \n I\n 1\n \n ⁢\n \n (\n 0\n )\n \n \n =\n \n \n S\n 0\n \n ⁢\n \n M\n 1\n \n ⁢\n \n (\n 0\n )\n \n ⁢\n \n S\n 0\n \n \n =\n \n \n \n S\n 0\n \n ⁢\n Y\n ⁢\n \n S\n 0\n \n \n +\n \n \n i\n \n ⁢\n \n \n (\n \n Z\n ⁢\n \n S\n 0\n \n \n )\n \n *\n \n ⁢\n \n (\n \n Z\n ⁢\n \n S\n 0\n \n \n )\n \n \n \n \n ,\n \n \n \n \n", "1910.02524_18": "\n \n \n \n \n \n \n \n \n Q\n 1\n \n ¯\n \n ≅\n \n 2\n 4\n \n \n :\n \n \n \n S\n ⁢\n y\n ⁢\n \n m\n 5\n \n \n ,\n \n and\n ⁢\n \n \n Q\n 2\n \n ¯\n \n \n \n ≅\n \n \n (\n \n \n \n S\n ⁢\n \n L\n 2\n \n ⁢\n \n (\n 3\n )\n \n \n ∘\n S\n \n ⁢\n \n L\n 2\n \n ⁢\n \n (\n 3\n )\n \n \n )\n \n ⁢\n \n .2\n 2\n \n \n \n \n .\n \n \n \n \n", "1910.02758_11": "\n \n \n \n \n \n f\n ⁢\n \n (\n a\n ,\n b\n )\n \n \n =\n \n \n \n 0.1010101\n ⁢\n ⋯\n \n ×\n a\n \n +\n \n \n \n 0.01010\n ⁢\n ⋯\n \n ×\n b\n \n ⁢\n \n (\n 2\n )\n \n \n \n \n \n \n \n", "1910.03391_59": "\n \n \n \n \n lim\n \n n\n →\n \n +\n ∞\n \n \n \n ⁡\n \n \n ⟨\n \n \n T\n \n ♯\n A\n \n \n ⁢\n T\n ⁢\n \n x\n n\n \n \n ,\n \n \n S\n \n ♯\n A\n \n \n ⁢\n S\n ⁢\n \n x\n n\n \n \n ⟩\n \n A\n \n \n =\n \n \n \n \n ∥\n \n \n T\n \n ♯\n A\n \n \n ⁢\n T\n \n ∥\n \n A\n \n \n ⁢\n \n \n ∥\n \n \n S\n \n ♯\n A\n \n \n ⁢\n S\n \n ∥\n \n A\n \n \n \n ,\n \n", "1910.03691_233": "\n \n \n \n \n \n \n q\n ⁢\n \n (\n z\n )\n \n \n =\n \n \n ∫\n C\n \n \n \n e\n \n s\n ⁢\n z\n \n \n ⁢\n Y\n ⁢\n \n (\n s\n )\n \n ⁢\n \n 𝑑\n s\n \n \n \n \n .\n \n \n \n \n", "1910.05976_44": "\n \n \n \n P\n ⁢\n r\n ⁢\n \n (\n \n \n Z\n =\n z\n \n ,\n \n Y\n =\n y\n \n \n )\n \n \n =\n \n \n P\n \n z\n +\n y\n \n \n ⁢\n \n \n \n (\n \n n\n z\n \n )\n \n \n (\n \n \n n\n +\n 1\n \n \n z\n +\n y\n \n \n )\n \n \n \n \n \n .\n \n (\n 26\n )\n \n \n", "1910.07018_55": "\n \n \n \n \n \n \n \n ∫\n Θ\n \n \n h\n ⁢\n \n (\n θ\n )\n \n ⁢\n \n 𝑑\n ν\n \n \n \n ≥\n \n \n \n ∫\n Θ\n \n \n h\n ⁢\n \n (\n θ\n )\n \n ⁢\n \n 𝑑\n \n μ\n ∞\n \n \n \n \n -\n \n \n \n 2\n ⁢\n K\n \n ⋅\n \n d\n W\n \n \n ⁢\n \n (\n ν\n ,\n \n μ\n ∞\n \n )\n \n \n \n \n .\n \n \n \n \n", "1910.07626_6": "", "1910.09141_7": "\n \n \n \n \n \n \n \n ℒ\n 𝐘\n \n ⁢\n \n (\n 𝐗\n )\n \n \n =\n \n \n \n ℒ\n \n 𝐘\n R\n \n \n ⁢\n \n (\n \n 𝐗\n R\n \n )\n \n \n +\n \n \n ℒ\n \n 𝐘\n I\n \n \n ⁢\n \n (\n \n 𝐗\n I\n \n )\n \n \n \n \n .\n \n (\n 7\n )\n \n \n \n \n \n", "1910.09831_109": "\n \n \n \n \n \n \n \n \n ∥\n \n \n 𝒞\n ¯\n \n n\n \n ∥\n \n 2\n \n ≤\n \n \n \n c\n 4\n \n ⁢\n \n n\n \n \n  \n and\n \n \n  \n \n \n \n \n ∥\n \n ℰ\n ⁢\n \n (\n n\n ,\n m\n )\n \n \n ∥\n \n 2\n \n ≤\n \n \n \n c\n 4\n \n ⁢\n h\n ⁢\n \n (\n \n n\n +\n m\n \n )\n \n \n ,\n n\n \n \n ,\n \n m\n ≥\n 1\n \n \n \n .\n \n (\n 4.3\n )\n \n \n \n \n \n", "1910.10298_12": "\n \n \n \n \n \n \n \n (\n 3.12\n )\n \n ⁢\n \n min\n ⁡\n \n \n ∥\n \n \n 𝝉\n ~\n \n -\n 𝝉\n \n ∥\n \n F\n 2\n \n \n \n +\n \n λ\n ⁢\n R\n ⁢\n \n (\n W\n ,\n b\n )\n \n \n \n ,\n \n \n \n \n", "1910.10436_281": "\n \n \n \n \n \n \n \n ker\n ⁡\n \n \n d\n x\n \n ⁢\n \n F\n w\n \n \n \n |\n \n S\n \n =\n \n ker\n ⁡\n \n (\n \n R\n x\n *\n \n ,\n \n \n d\n x\n \n ⁢\n \n F\n w\n \n \n )\n \n \n \n \n \n \n", "1910.11098_133": "\n \n \n \n \n \n \n \n \n θ\n 2\n 2\n \n ⁢\n \n θ\n 1\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n -\n \n i\n ⁢\n \n \n t\n -\n 1\n \n \n t\n +\n 1\n \n \n \n \n \n ,\n \n \n \n \n θ\n 1\n \n ⁢\n \n θ\n 2\n 2\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n i\n ⁢\n \n \n t\n +\n 1\n \n \n t\n -\n 1\n \n \n \n \n ,\n \n \n \n \n θ\n 1\n \n ⁢\n \n θ\n 2\n 2\n \n ⁢\n \n θ\n 1\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n i\n ⁢\n \n \n t\n -\n 1\n \n \n t\n +\n 1\n \n \n \n \n ,\n \n \n \n \n θ\n 2\n 2\n \n ⁢\n \n θ\n 1\n \n ⁢\n \n θ\n 2\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n -\n \n 1\n t\n \n \n \n ,\n \n \n \n θ\n 1\n \n ⁢\n \n θ\n 2\n 2\n \n ⁢\n \n θ\n 1\n \n ⁢\n \n θ\n 2\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n 1\n t\n \n \n \n \n \n \n .\n \n \n \n \n", "1910.13863_57": "", "1910.14412_35": "\n \n \n \n \n \n \n \n \n \n [\n \n \n (\n \n 64.125\n +\n \n 32.0625\n ⁢\n j\n \n \n ,\n \n 48.1875\n -\n \n 15.75\n ⁢\n j\n \n \n )\n \n T\n \n ,\n \n \n (\n \n 48.1875\n -\n \n 15.75\n ⁢\n j\n \n \n ,\n \n 16.1250\n -\n \n 31.3125\n ⁢\n j\n \n \n )\n \n T\n \n ,\n \n \n (\n \n 16.1250\n -\n \n 31.3125\n ⁢\n j\n \n \n ,\n \n \n -\n 8.4375\n \n -\n \n 22.5\n ⁢\n j\n \n \n )\n \n T\n \n ]\n \n .\n \n \n \n \n ⁢\n \n (\n 45\n )\n \n \n \n \n \n", "1911.00471_91": "\n \n \n \n \n \n \n l\n ⁢\n \n (\n \n \n ρ\n |\n \n H\n \n )\n \n \n ≥\n \n 1\n ⁢\n f\n ⁢\n o\n ⁢\n \n r\n \n ⁢\n \n every\n \n ⁢\n \n (\n \n non\n -\n trivial\n \n )\n \n ⁢\n \n simple\n \n ⁢\n \n normal\n \n ⁢\n \n subgroup\n \n ⁢\n \n H\n \n ⁢\n \n of\n \n ⁢\n \n G\n \n \n \n .\n \n (\n \n B\n ⁢\n .7\n \n )\n \n \n \n \n \n", "1911.01656_52": "\n \n \n \n \n \n \n ρ\n ⁢\n \n (\n \n t\n |\n \n x\n 0\n \n \n )\n \n \n =\n \n -\n \n \n \n ∂\n ⁡\n S\n \n ⁢\n \n (\n \n t\n |\n \n x\n 0\n \n \n )\n \n \n \n ∂\n ⁡\n t\n \n \n \n =\n \n -\n \n \n 2\n π\n \n ⁢\n \n \n ∫\n 0\n ∞\n \n \n \n \n \n d\n ⁢\n q\n \n q\n \n \n ⁢\n \n sin\n ⁡\n \n (\n \n q\n ⁢\n \n x\n 0\n \n \n )\n \n \n ⁢\n \n Υ\n ′\n \n ⁢\n \n (\n t\n ;\n \n q\n 2\n \n )\n \n \n \n \n \n \n ,\n \n (\n 49\n )\n \n \n \n \n \n", "1911.01842_9": "\n \n \n \n \n \n \n \n (\n 4\n )\n \n ⁢\n A\n ⁢\n \n ρ\n p\n \n \n -\n \n B\n ⁢\n \n σ\n \n 2\n ⁢\n p\n \n \n \n \n =\n C\n \n \n \n \n", "1911.05657_40": "\n \n \n \n \n \n \n ∑\n \n n\n =\n 0\n \n ∞\n \n \n \n ∫\n 0\n 1\n \n \n ϕ\n ⁢\n \n (\n \n ∥\n \n d\n ⁢\n \n f\n n\n \n ⁢\n \n (\n t\n )\n \n \n ∥\n \n )\n \n ⁢\n \n 𝑑\n t\n \n \n \n \n <\n \n +\n ∞\n \n \n \n \n \n", "1911.05862_9": "\n \n \n \n \n \n U\n 1\n \n ,\n \n U\n 2\n \n ,\n …\n ,\n \n U\n k\n \n \n \n \n \n", "1911.09760_0": "\n \n \n \n \n \n -\n \n ln\n ⁡\n \n [\n \n P\n ⁢\n \n (\n \n M\n NS\n \n ,\n \n a\n *\n \n )\n \n \n ]\n \n \n \n =\n \n \n \n ∑\n i\n \n \n \n \n \n [\n \n \n ν\n \n prec\n ,\n i\n \n \n -\n \n ν\n \n HBO\n ,\n i\n \n \n \n ]\n \n 2\n \n +\n \n \n [\n \n \n ν\n \n vert\n ,\n i\n \n \n -\n \n ν\n \n \n u\n ⁢\n kHz\n \n ,\n i\n \n \n \n ]\n \n 2\n \n \n \n \n \n (\n \n \n σ\n \n HBO\n ,\n i\n \n \n ⁢\n \n cos\n ⁡\n \n Φ\n i\n \n \n \n )\n \n 2\n \n +\n \n \n (\n \n \n σ\n \n \n u\n ⁢\n kHz\n \n ,\n i\n \n \n ⁢\n \n sin\n ⁡\n \n Φ\n i\n \n \n \n )\n \n 2\n \n \n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "1911.11328_96": "\n \n \n \n \n \n \n d\n 2\n \n ≤\n \n C\n ⁢\n \n (\n n\n )\n \n ⁢\n \n exp\n ⁡\n \n (\n \n \n osc\n X\n \n ⁡\n \n f\n η\n \n \n )\n \n \n ⁢\n \n 4\n \n 2\n ⁢\n n\n \n \n \n \n .\n \n \n \n \n", "1912.01078_192": "\n \n \n \n \n d\n =\n \n \n \n a\n ⁢\n \n (\n \n s\n 0\n \n ,\n 0\n )\n \n \n -\n \n \n T\n m\n \n ⁢\n \n s\n 0\n \n \n \n +\n \n \n ∫\n \n s\n 0\n \n \n +\n ∞\n \n \n \n \n (\n \n \n \n \n ∂\n s\n \n ⁡\n a\n \n ⁢\n \n (\n s\n ,\n 0\n )\n \n \n -\n \n T\n m\n \n \n )\n \n ⁢\n \n 𝑑\n s\n \n ⁢\n τ\n \n \n \n =\n \n \n \n θ\n ~\n \n ⁢\n \n (\n \n s\n 0\n \n ,\n 0\n )\n \n \n +\n \n \n ∫\n \n s\n 0\n \n \n +\n ∞\n \n \n \n \n \n ∂\n s\n \n ⁡\n \n θ\n ~\n \n \n ⁢\n \n (\n s\n ,\n 0\n )\n \n ⁢\n \n 𝑑\n s\n \n \n \n \n \n \n \n \n", "1912.01318_6": "\n \n \n \n \n \n \n ∥\n \n \n \n |\n ∇\n |\n \n s\n \n ⁢\n \n P\n \n \n ≤\n N\n \n \n ⁢\n f\n \n ∥\n \n \n L\n x\n 2\n \n \n ≤\n \n \n N\n s\n \n ⁢\n \n \n ∥\n f\n ∥\n \n \n L\n x\n 2\n \n \n \n \n \n \n \n", "1912.03405_27": "\n \n \n \n \n \n \n (\n 3.9\n )\n \n ⁢\n \n T\n ε\n \n ⁢\n \n [\n \n u\n ε\n \n ]\n \n ⁢\n \n (\n \n x\n i\n \n )\n \n \n :=\n \n f\n ⁢\n \n (\n \n x\n i\n \n )\n \n \n \n \n \n \n", "1912.03499_32": "\n \n \n \n \n \n \n \n J\n |\n \n \n e\n →\n 0\n \n \n =\n \n |\n \n \n -\n \n \n 9\n ⁢\n M\n ⁢\n \n a\n 2\n \n ⁢\n \n (\n \n \n \n 5\n ⁢\n \n a\n 4\n \n \n -\n \n 15\n ⁢\n \n a\n 2\n \n ⁢\n \n p\n 2\n \n \n \n +\n \n 4\n ⁢\n \n p\n 4\n \n \n \n )\n \n ⁢\n e\n \n \n 4\n ⁢\n \n p\n 5\n \n ⁢\n \n \n (\n \n \n p\n 2\n \n -\n \n 3\n ⁢\n \n a\n 2\n \n \n \n )\n \n \n 3\n /\n 2\n \n \n ⁢\n \n (\n \n \n p\n 2\n \n +\n \n a\n 2\n \n \n )\n \n \n \n \n +\n \n O\n ⁢\n \n (\n \n e\n 2\n \n )\n \n \n \n |\n \n \n .\n \n (\n 69\n )\n \n \n \n \n \n", "1912.03895_29": "\n \n \n \n \n \n \n \n |\n \n \n \n (\n \n t\n +\n \n i\n ⁢\n s\n \n \n )\n \n 2\n \n -\n \n 4\n ⁢\n r\n ⁢\n \n (\n \n 1\n -\n r\n \n )\n \n \n \n |\n \n \n \n |\n \n \n \n c\n r\n \n ⁢\n \n (\n υ\n )\n \n \n -\n \n (\n \n t\n +\n \n i\n ⁢\n s\n \n \n )\n \n \n |\n \n \n =\n \n \n \n |\n \n \n \n c\n r\n \n ⁢\n \n (\n υ\n )\n \n \n +\n t\n +\n \n i\n ⁢\n s\n \n \n |\n \n \n |\n \n \n \n c\n r\n \n ⁢\n \n (\n v\n )\n \n \n -\n \n (\n \n t\n +\n \n i\n ⁢\n s\n \n \n )\n \n \n |\n \n \n \n ≤\n \n \n \n \n |\n \n \n \n c\n r\n \n ⁢\n \n (\n v\n )\n \n \n +\n t\n \n |\n \n +\n \n |\n s\n |\n \n \n \n \n |\n \n \n \n c\n r\n \n ⁢\n \n (\n υ\n )\n \n \n -\n t\n \n |\n \n ∨\n \n |\n s\n |\n \n \n \n \n \n \n \n \n", "1912.09082_24": "\n \n \n \n \n \n H\n ⁢\n \n (\n z\n ,\n t\n )\n \n \n =\n \n (\n \n \n \n h\n 1\n \n ⁢\n \n (\n z\n )\n \n \n +\n \n \n X\n 1\n \n ⁢\n \n (\n z\n )\n \n ⁢\n ρ\n ⁢\n \n (\n t\n )\n \n \n \n ,\n \n \n \n h\n 2\n \n ⁢\n \n (\n z\n )\n \n \n +\n \n \n X\n 2\n \n ⁢\n \n (\n z\n )\n \n ⁢\n ρ\n ⁢\n \n (\n t\n )\n \n \n \n ,\n t\n )\n \n \n \n \n \n", "1912.09688_18": "", "1912.10684_247": "\n \n \n \n \n \n \n \n \n ℒ\n Z\n \n ⁢\n V\n \n =\n \n \n \n \n ∇\n ~\n \n Z\n \n ⁢\n V\n \n -\n V\n \n \n ,\n \n \n \n ℒ\n \n Z\n ¯\n \n \n ⁢\n V\n \n =\n \n \n \n ∇\n ~\n \n \n Z\n ¯\n \n \n ⁢\n V\n \n \n \n ,\n \n \n \n \n", "1912.11703_17": "\n \n \n \n \n \n λ\n =\n \n \n O\n p\n \n ⁢\n \n (\n \n n\n \n -\n \n r\n /\n \n (\n \n 1\n +\n \n 2\n ⁢\n r\n \n \n )\n \n \n \n \n )\n \n ⁢\n and\n ⁢\n \n λ\n \n -\n 1\n \n \n \n =\n \n \n O\n p\n \n ⁢\n \n (\n \n n\n \n r\n /\n \n (\n \n \n 2\n ⁢\n r\n \n +\n 1\n \n )\n \n \n \n )\n \n \n \n .\n \n \n \n \n", "2001.00545_14": "\n \n \n \n \n \n \n Ω\n 𝐪\n \n =\n \n \n \n B\n ∥\n \n ⁢\n \n q\n ∥\n 2\n \n \n +\n \n \n B\n z\n \n ⁢\n \n q\n z\n 2\n \n \n \n ≡\n \n \n \n q\n ∥\n 2\n \n \n 2\n ⁢\n \n M\n ∥\n \n \n \n +\n \n \n \n q\n z\n 2\n \n \n 2\n ⁢\n \n M\n z\n \n \n \n \n \n \n ,\n \n (\n 15\n )\n \n \n \n \n \n", "2001.07033_110": "\n \n \n \n \n \n \n 1\n h\n \n ≥\n \n \n 1\n -\n b\n \n \n θ\n b\n \n \n \n .\n \n \n \n \n", "2001.07664_56": "\n \n \n \n \n \n \n E\n ⁢\n e\n ⁢\n −\n ⁢\n t\n ⁢\n ⁢\n ⁢\n J\n ⁢\n α\n ⁢\n ⁢\n ⁢\n \n [\n \n S\n ⁢\n α\n ⁢\n ⁢\n ⁢\n \n (\n x\n )\n \n \n ]\n \n \n =\n \n η\n ⁢\n α\n ⁢\n \n (\n t\n )\n \n ⁢\n \n ∫\n \n κ\n ⁢\n −\n ⁢\n x\n ⁢\n ∞\n ⁢\n e\n ⁢\n −\n ⁢\n t\n ⁢\n ⁢\n ⁢\n z\n ⁢\n U\n ⁢\n α\n ⁢\n \n (\n \n d\n ⁢\n z\n \n )\n \n \n \n \n \n ,\n \n \n ∀\n t\n \n ≥\n \n 0\n ⁢\n \n (\n 46\n )\n \n \n \n \n \n \n \n", "2001.09197_166": "\n \n \n \n \n \n ∥\n \n w\n *\n \n ∥\n \n =\n \n \n \n \n 1\n \n D\n φ\n \n \n ⁢\n \n (\n \n \n 4\n ⁢\n \n D\n φ\n 2\n \n \n +\n \n v\n 2\n \n \n )\n \n ⁢\n \n (\n \n \n cos\n ⁡\n α\n \n -\n \n 2\n ⁢\n \n D\n φ\n \n \n \n )\n \n \n \n ⁢\n \n (\n 160\n )\n \n \n \n \n \n \n", "2001.10233_1": "", "2001.10970_45": "", "2002.00112_217": "", "2002.00462_190": "", "2002.01084_58": "\n \n \n \n \n \n \n \n \n (\n 4.8\n )\n \n ⁢\n \n \n X\n ^\n \n T\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n ∫\n 0\n ∞\n \n \n \n e\n \n -\n \n z\n ⁢\n \n u\n ′\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n \n Y\n ^\n \n T\n \n \n \n \n ⁢\n ν\n ⁢\n \n (\n \n d\n ⁢\n z\n \n )\n \n \n \n \n ,\n \n x\n >\n 0\n \n \n .\n \n \n \n \n", "2002.01866_17": "\n \n \n \n \n \n β\n ⁢\n \n U\n \n e\n ⁢\n l\n ⁢\n a\n ⁢\n s\n \n \n \n =\n \n \n β\n ⁢\n \n \n κ\n s\n \n 2\n \n ⁢\n \n \n \n (\n \n A\n -\n \n A\n 0\n \n \n )\n \n 2\n \n \n A\n 0\n \n \n \n ,\n \n (\n 18\n )\n \n \n \n \n \n \n", "2002.03376_68": "\n \n \n \n \n \n \n C\n =\n \n \n θ\n \n ρ\n 2\n \n \n   \n and\n \n \n   \n \n D\n =\n \n \n \n \n θ\n 2\n \n +\n \n \n 2\n ⁢\n \n ρ\n 2\n \n \n η\n \n \n \n \n ρ\n 2\n \n \n \n \n ,\n \n \n \n \n", "2002.06438_98": "\n \n \n \n \n \n φ\n =\n \n \n y\n \n \n 1\n 2\n \n -\n ν\n \n \n ⁢\n \n K\n \n |\n ν\n |\n \n \n ⁢\n \n (\n y\n )\n \n \n \n ,\n \n ξ\n =\n \n -\n \n \n y\n \n \n 1\n 2\n \n -\n ν\n \n \n ⁢\n \n K\n \n |\n \n ν\n -\n 1\n \n |\n \n \n ⁢\n \n (\n y\n )\n \n ⁢\n \n (\n 5.22\n )\n \n \n \n \n \n \n \n \n", "2002.07140_1": "\n \n \n \n \n \n s\n ⁢\n p\n ⁢\n e\n ⁢\n \n c\n ε\n \n ⁢\n \n (\n G\n )\n \n \n =\n \n {\n \n ε\n 1\n \n m\n 1\n \n \n ,\n \n ε\n 2\n \n m\n 2\n \n \n ,\n …\n ,\n \n ε\n k\n \n m\n k\n \n \n }\n \n \n \n \n \n", "2002.08941_84": "\n \n \n \n \n \n |\n \n E\n ⁢\n \n (\n x\n )\n \n \n |\n \n ≤\n \n \n k\n 0\n \n ⁢\n \n \n |\n x\n |\n \n \n \n -\n 1\n \n -\n τ\n \n \n \n \n \n \n \n", "2002.10630_0": "\n \n \n \n \n \n \n [\n \n \n \n (\n \n ω\n -\n \n m\n ⁢\n Ω\n \n \n )\n \n 2\n \n -\n \n κ\n 2\n \n \n ]\n \n ⁢\n \n [\n \n \n \n (\n \n ω\n -\n \n m\n ⁢\n Ω\n \n \n )\n \n 2\n \n -\n \n n\n ⁢\n \n Ω\n ⊥\n 2\n \n \n \n ]\n \n \n =\n \n \n \n k\n 2\n \n ⁢\n \n c\n s\n 2\n \n ⁢\n \n \n (\n \n ω\n -\n \n m\n ⁢\n Ω\n \n \n )\n \n 2\n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "2002.11454_73": "\n \n \n \n \n \n \n \n (\n 31\n )\n \n ⁢\n \n curl\n ⁡\n \n (\n w\n )\n \n \n \n :=\n \n \n (\n \n \n ∂\n 2\n \n ⁡\n w\n \n ,\n \n -\n \n \n ∂\n 1\n \n ⁡\n w\n \n \n )\n \n   \n and\n \n \n   \n \n \n rot\n ⁡\n \n (\n v\n )\n \n \n :=\n \n \n -\n \n \n ∂\n 2\n \n ⁡\n \n v\n 1\n \n \n \n +\n \n \n ∂\n 1\n \n ⁡\n \n v\n 2\n \n \n \n \n \n \n \n \n", "2003.04061_141": "\n \n \n \n \n \n \n \n μ\n Ω\n \n ⁢\n \n (\n E\n )\n \n \n ≤\n \n \n \n q\n E\n Ω\n \n ⁢\n \n (\n \n v\n 0\n \n )\n \n \n \n \n ∥\n \n v\n 0\n \n ∥\n \n \n \n L\n 2\n \n ⁢\n \n (\n Ω\n )\n \n \n 2\n \n \n =\n \n \n \n \n \n ∥\n \n ∇\n ⁡\n \n v\n 0\n \n \n ∥\n \n \n \n L\n 2\n \n ⁢\n \n (\n Ω\n )\n \n \n 2\n \n +\n \n E\n ⁢\n \n \n ∥\n \n v\n 0\n \n ∥\n \n \n \n L\n 2\n \n ⁢\n \n (\n \n ∂\n ⁡\n Ω\n \n )\n \n \n 2\n \n \n \n \n \n ∥\n \n v\n 0\n \n ∥\n \n \n \n L\n 2\n \n ⁢\n \n (\n Ω\n )\n \n \n 2\n \n \n -\n \n E\n 2\n \n \n \n ,\n \n (\n 22\n )\n \n \n \n \n \n", "2003.04252_4": "\n \n \n \n \n \n \n ∂\n ⁡\n \n L\n lv\n \n \n \n ∂\n ⁡\n t\n \n \n =\n \n \n \n ∇\n ⋅\n \n [\n \n \n \n h\n 3\n \n \n 3\n ⁢\n η\n \n \n ⁢\n \n ∇\n ⁡\n \n \n δ\n ⁢\n Ω\n \n \n δ\n ⁢\n \n L\n lv\n \n \n \n \n \n ]\n \n \n -\n \n \n k\n lv\n \n ⁢\n \n \n δ\n ⁢\n Ω\n \n \n δ\n ⁢\n \n L\n lv\n \n \n \n \n \n +\n \n \n k\n sl\n \n ⁢\n \n \n Δ\n ⁢\n ρ\n \n \n ρ\n l\n \n \n ⁢\n \n \n δ\n ⁢\n Ω\n \n \n δ\n ⁢\n \n L\n sl\n \n \n \n ⁢\n \n (\n \n 4\n ⁢\n b\n \n )\n \n \n \n \n \n \n \n", "2003.08281_235": "\n \n \n \n \n \n (\n C\n .2\n )\n \n q\n \n (\n ξ\n )\n \n ≤\n 0\n \n (\n resp\n .\n ≥\n 0\n )\n \n .\n \n \n \n \n", "2003.09471_33": "", "2003.10492_32": "\n \n \n \n \n \n T\n  s t e p \n \n =\n \n \n min\n \n j\n ∈\n \n {\n 1\n ,\n ⋯\n ,\n R\n }\n \n \n \n ⁢\n \n t\n j\n  n e x t \n \n ⁢\n \n (\n 20\n )\n \n \n \n \n \n \n", "2003.12838_27": "\n \n \n \n inf\n \n Ψ\n \n (\n i\n )\n \n \n \n ⁡\n \n {\n \n \n \n E\n 0\n \n (\n i\n )\n \n \n ⁢\n \n Ψ\n \n (\n i\n )\n \n \n \n +\n \n \n \n 1\n \n |\n \n ℱ\n 0\n \n |\n \n \n \n ⁢\n \n \n \n ∑\n \n f\n ∈\n \n ℱ\n 0\n \n \n \n \n \n \n E\n f\n \n (\n i\n )\n \n \n ⁢\n \n (\n \n 1\n -\n \n Ψ\n \n (\n i\n )\n \n \n \n )\n \n \n \n \n \n }\n \n \n ≥\n \n \n \n (\n \n 1\n -\n \n C\n ⁢\n \n η\n n\n \n \n \n )\n \n 2\n \n ,\n \n (\n 3.5\n )\n \n \n \n", "2003.14374_166": "\n \n \n \n \n \n \n \n X\n ±\n \n ⁢\n \n (\n z\n )\n \n \n =\n \n \n 1\n \n 2\n ⁢\n π\n ⁢\n i\n \n \n ⁢\n \n \n ∫\n Σ\n \n \n H\n ⁢\n \n (\n w\n )\n \n ⁢\n \n \n d\n ⁢\n w\n \n \n w\n -\n \n z\n ±\n \n \n \n \n \n \n \n .\n \n \n \n \n", "2004.00301_19": "", "2004.00416_120": "", "2004.00549_90": "\n \n \n \n \n \n \n \n \n \n (\n \n -\n Δ\n \n )\n \n s\n \n ⁢\n u\n ⁢\n \n (\n x\n )\n \n \n +\n \n a\n ⁢\n \n (\n x\n ,\n u\n )\n \n \n \n =\n 0\n \n .\n \n \n \n \n", "2004.00607_1": "\n \n \n \n Im\n ⁢\n \n 𝒜\n \n \n γ\n ⁢\n p\n \n →\n \n V\n ⁢\n p\n \n \n \n ⁢\n \n (\n W\n )\n \n \n =\n \n \n \n ∫\n 0\n 1\n \n \n \n \n d\n β\n \n ⁢\n \n \n ∫\n \n \n d\n 2\n \n ⁢\n \n \n r\n ⟂\n \n \n ⁢\n \n Ψ\n V\n †\n \n ⁢\n \n (\n β\n ,\n \n r\n ⟂\n \n )\n \n ⁢\n \n Ψ\n γ\n \n ⁢\n \n (\n β\n ,\n \n r\n ⟂\n \n )\n \n ⁢\n \n σ\n \n q\n ⁢\n \n q\n ¯\n \n \n \n ⁢\n \n (\n x\n ,\n \n r\n ⟂\n \n )\n \n \n \n \n \n \n \n ,\n \n x\n =\n \n \n \n \n \n M\n V\n 2\n \n \n W\n 2\n \n \n \n \n ,\n \n (\n 2\n )\n \n \n \n \n", "2004.04029_10": "\n \n \n \n \n \n 𝒮\n \n \n k\n ⁢\n j\n \n ,\n \n r\n ⁢\n i\n \n \n \n =\n \n -\n \n \n 𝒮\n \n \n k\n ⁢\n j\n \n ,\n \n i\n ⁢\n r\n \n \n \n ⁢\n \n (\n 11\n )\n \n \n \n \n \n \n \n", "2004.04763_142": "", "2004.06298_13": "\n \n \n \n \n \n \n 𝖡\n ⁢\n \n (\n \n 𝒢\n ×\n \n {\n μ\n }\n \n \n ,\n \n ℛ\n κ\n \n 0\n ,\n 1\n \n \n )\n \n \n ≤\n \n 1\n -\n \n \n ρ\n ⁢\n \n ⌊\n \n \n d\n \n 2\n ⁢\n p\n \n \n -\n 1\n \n ⌋\n \n ⁢\n V\n \n /\n 3\n \n \n \n .\n \n \n \n \n", "2004.06944_77": "", "2004.07661_6": "\n \n \n \n \n \n \n S\n \n M\n ,\n \n H\n ⁢\n L\n \n \n \n =\n \n -\n \n \n 1\n 2\n \n ⁢\n \n ∫\n \n \n 𝑑\n t\n \n ⁢\n \n d\n 3\n \n ⁢\n \n \n x\n →\n \n \n ⁢\n \n [\n \n \n \n F\n \n 0\n ⁢\n i\n \n \n ⁢\n \n F\n \n 0\n ⁢\n i\n \n \n \n +\n \n \n 1\n \n 2\n ⁢\n \n Λ\n HL\n \n 2\n ⁢\n \n (\n \n z\n -\n 1\n \n )\n \n \n \n \n \n ⁢\n \n F\n \n i\n ⁢\n j\n \n \n ⁢\n \n \n (\n \n -\n Δ\n \n )\n \n \n z\n -\n 1\n \n \n ⁢\n \n F\n \n i\n ⁢\n j\n \n \n \n \n ]\n \n \n \n \n \n \n .\n \n (\n 6\n )\n \n \n \n \n \n", "2004.07698_3": "\n \n \n \n \n \n \n s\n |\n \n U\n \n :\n \n \n \n R\n m\n l\n \n ×\n \n R\n k\n n\n \n \n →\n \n \n R\n k\n n\n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n \n", "2004.10007_280": "\n \n \n \n \n \n \n \n ∏\n \n j\n =\n 1\n \n N\n \n \n (\n \n X\n -\n \n β\n j\n \n \n )\n \n \n =\n \n \n \n ∑\n \n k\n =\n 0\n \n \n N\n -\n 1\n \n \n \n \n a\n k\n \n ⁢\n \n X\n k\n \n \n \n +\n \n X\n N\n \n \n \n .\n \n (\n \n A\n ⁢\n .1\n \n )\n \n \n \n \n \n", "2005.00735_3": "\n \n \n \n \n \n \n H\n ⁢\n \n (\n t\n )\n \n \n =\n \n i\n ⁢\n \n ∑\n \n \n ϵ\n \n m\n ,\n n\n \n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n γ\n m\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n γ\n n\n \n ⁢\n \n (\n t\n )\n \n \n \n \n \n .\n \n (\n \n S\n ⁢\n 4\n \n )\n \n \n \n \n \n", "2005.01154_10": "\n \n \n \n \n \n \n \n 𝒟\n i\n \n ⁢\n \n (\n \n u\n ∧\n v\n \n )\n \n \n =\n \n \n \n ∑\n \n j\n =\n 0\n \n i\n \n \n \n 𝒟\n j\n \n ⁢\n u\n \n \n ∧\n \n \n D\n \n i\n -\n j\n \n \n ⁢\n v\n \n \n \n ,\n \n \n \n \n", "2005.01255_23": "\n \n \n \n \n \n \n p\n \n 1\n -\n p\n \n \n =\n \n 1\n \n 2\n 6\n \n \n \n .\n \n (\n 23\n )\n \n \n \n \n \n", "2005.01354_36": "\n \n \n \n \n \n b\n >\n \n \n \n -\n \n (\n \n \n a\n 2\n \n -\n a\n -\n 2\n \n )\n \n \n +\n \n \n \n \n (\n \n \n a\n 2\n \n -\n a\n -\n 1\n \n )\n \n 2\n \n +\n \n 4\n ⁢\n a\n ⁢\n \n (\n \n a\n +\n 1\n \n )\n \n ⁢\n \n (\n \n \n \n 2\n ⁢\n a\n \n -\n 1\n \n +\n \n 3\n ⁢\n e\n \n \n )\n \n \n \n \n \n \n 2\n ⁢\n a\n ⁢\n \n (\n \n a\n +\n 1\n \n )\n \n \n \n :=\n \n \n g\n 1\n \n ⁢\n \n (\n a\n )\n \n \n \n .\n \n \n \n \n", "2005.02232_141": "", "2005.06209_1": "\n \n \n \n \n \n ℒ\n \n s\n ⁢\n s\n \n \n =\n \n ℱ\n ⁢\n \n (\n \n ℐ\n ~\n \n ,\n ℐ\n )\n \n \n =\n \n ℱ\n ⁢\n \n (\n \n π\n ⁢\n \n (\n \n ℐ\n ⁢\n \n †\n \n \n ,\n \n K\n ⁢\n \n †\n \n \n ,\n \n R\n |\n \n t\n ,\n K\n ,\n d\n \n \n )\n \n \n ,\n ℐ\n )\n \n ⁢\n \n (\n 2\n )\n \n \n \n \n \n \n", "2005.07417_105": "", "2005.08280_281": "", "2005.09465_157": "", "2005.13557_5": "\n \n \n \n \n \n \n lcm\n ⁡\n \n (\n 𝐱\n ,\n 𝐲\n )\n \n \n \n gcd\n ⁡\n \n (\n 𝐱\n ,\n 𝐲\n )\n \n \n \n =\n \n u\n ⁢\n v\n \n \n \n \n \n", "2005.13915_16": "", "2006.02495_21": "\n \n \n \n \n \n \n \n y\n r\n \n ⁢\n \n (\n 0\n )\n \n \n =\n \n \n \n C\n r\n \n ⁢\n \n \n x\n ~\n \n r\n \n ⁢\n \n (\n 0\n )\n \n \n +\n \n \n [\n \n \n \n D\n \n \n \n \n \n 1\n \n β\n ⁢\n \n \n 2\n ⁢\n α\n \n \n \n \n \n ⁢\n C\n ⁢\n \n X\n 0\n \n \n \n \n \n ]\n \n ⁢\n \n u\n ~\n \n ⁢\n \n (\n 0\n )\n \n \n \n =\n \n \n D\n ⁢\n u\n ⁢\n \n (\n 0\n )\n \n \n +\n \n C\n ⁢\n \n X\n 0\n \n ⁢\n \n z\n 0\n \n \n \n =\n \n y\n ⁢\n \n (\n 0\n )\n \n \n \n .\n \n \n \n \n", "2006.06194_29": "\n \n \n \n \n \n \n \n \n K\n ~\n \n c\n \n +\n \n \n Q\n 2\n \n ⁢\n \n K\n ~\n \n \n \n >\n \n \n K\n ~\n \n c\n \n -\n 1\n \n \n \n .\n \n (\n 18\n )\n \n \n \n \n \n", "2006.11405_5": "\n \n \n \n \n \n \n l\n \n d\n ⁢\n a\n \n \n ⁢\n \n (\n m\n ,\n n\n )\n \n \n =\n \n \n 1\n \n 4\n ⁢\n \n d\n s\n 2\n \n \n \n ⁢\n \n \n ∥\n \n \n C\n m\n s\n \n -\n \n C\n n\n s\n \n \n ∥\n \n F\n 2\n \n \n \n \n \n \n", "2006.12377_41": "\n \n \n \n \n \n \n {\n v\n :\n \n \n \n P\n n\n \n ⁢\n \n (\n v\n )\n \n \n =\n \n 0\n ⁢\n o\n ⁢\n r\n ⁢\n \n Q\n n\n \n ⁢\n \n (\n v\n )\n \n \n =\n 0\n \n }\n \n ⊂\n \n (\n \n -\n \n 2\n ⁢\n \n b\n \n 1\n 2\n \n \n \n \n ,\n \n 2\n ⁢\n \n b\n \n 1\n 2\n \n \n \n )\n \n ⊂\n \n (\n \n -\n \n (\n \n b\n +\n 1\n \n )\n \n \n ,\n \n (\n \n b\n +\n 1\n \n )\n \n )\n \n \n ,\n \n (\n 5.40\n )\n \n \n \n \n \n", "2006.12551_3": "", "2006.15499_100": "\n \n \n \n \n Δ\n ′\n =\n \n ⟨\n z\n 1\n ,\n z\n 2\n ,\n z\n 3\n :\n z\n 1\n p\n =\n z\n 22\n ⁢\n p\n =\n z\n 32\n ⁢\n q\n =\n z\n 1\n z\n 2\n z\n 3\n =\n 1\n ⟩\n \n \n \n \n \n", "2007.02644_222": "", "2007.04096_101": "\n \n \n \n \n \n \n δ\n ~\n \n =\n \n \n lim inf\n \n n\n →\n \n +\n ∞\n \n \n \n ⁡\n \n \n |\n \n ω\n ∩\n \n [\n \n -\n \n \n \n 2\n ⁢\n n\n \n +\n 1\n \n \n \n ,\n \n \n \n 2\n ⁢\n n\n \n +\n 1\n \n \n ]\n \n \n |\n \n \n n\n \n \n \n >\n 0\n \n .\n \n \n \n \n", "2007.04382_25": "\n \n \n \n \n \n \n \n \n ∥\n \n (\n 1\n ,\n 0\n )\n \n ∥\n \n X\n \n =\n 2\n =\n \n \n ∥\n \n (\n 0\n ,\n 1\n )\n \n ∥\n \n Y\n \n \n ,\n \n \n \n ∥\n \n (\n 0\n ,\n 1\n )\n \n ∥\n \n X\n \n =\n \n 1\n /\n 2\n \n =\n \n \n ∥\n \n (\n 1\n ,\n 0\n )\n \n ∥\n \n Y\n \n \n \n ,\n \n \n \n \n", "2007.07734_19": "\n \n \n \n \n \n \n \n lim\n \n \n p\n \n 1\n |\n 0\n \n \n →\n \n 0\n +\n \n \n \n ⁡\n \n \n Q\n σ\n \n M\n ,\n s\n \n \n ⁢\n \n (\n \n p\n \n 1\n |\n 0\n \n \n )\n \n \n \n =\n \n \n lim\n \n \n p\n \n 1\n |\n 0\n \n \n →\n \n s\n -\n \n \n \n ⁡\n \n \n Q\n σ\n \n M\n ,\n s\n \n \n ⁢\n \n (\n \n p\n \n 1\n |\n 0\n \n \n )\n \n \n \n =\n 0\n \n .\n \n (\n 22\n )\n \n \n \n \n \n", "2007.15342_70": "\n \n \n \n \n \n \n ⟨\n Ω\n ⟩\n \n \n n\n ≥\n 1\n \n \n =\n \n \n \n 1\n N\n \n ⁢\n \n [\n \n \n \n N\n 1\n \n ⁢\n \n γ\n 1\n \n \n +\n \n \n N\n 2\n \n ⁢\n \n γ\n 2\n \n \n +\n \n \n ∑\n \n \n n\n i\n \n ≥\n 3\n \n \n \n Ω\n i\n \n \n \n ]\n \n \n ,\n \n (\n 23\n )\n \n \n \n \n \n \n", "2008.00073_130": "\n \n \n \n \n \n v\n =\n \n \n y\n i\n \n ⁢\n \n ∂\n \n x\n i\n \n \n \n \n .\n \n (\n 131\n )\n \n \n \n \n \n", "2008.00945_106": "\n \n \n \n \n \n \n \n \n ℳ\n n\n \n ⁢\n δ\n ⁢\n \n 𝒩\n n\n d\n \n \n +\n \n \n ℳ\n s\n \n ⁢\n δ\n ⁢\n \n 𝒩\n s\n d\n \n \n \n =\n 0\n \n .\n \n (\n 107\n )\n \n \n \n \n \n", "2008.04328_11": "\n \n \n \n \n ρ\n ^\n \n 𝗉𝗋𝗈𝖻𝖾\n 𝗌𝖼\n \n ∝\n \n \n 𝖳𝗋\n λ\n \n ⁢\n \n [\n \n \n T\n ^\n \n ⁢\n \n \n ρ\n ^\n \n 𝗂𝗇\n \n ⁢\n \n \n T\n ^\n \n †\n \n \n ]\n \n \n \n ,\n \n", "2008.10374_151": "\n \n \n \n \n \n \n \n \n ∥\n \n \n \n |\n ξ\n |\n \n γ\n \n ⁢\n \n u\n \n h\n ⁢\n i\n ⁢\n g\n ⁢\n h\n \n \n ⁢\n \n (\n t\n ,\n s\n ,\n ⋅\n )\n \n \n ∥\n \n \n L\n 2\n \n \n ≤\n \n C\n ⁢\n \n \n (\n \n 1\n +\n t\n \n )\n \n \n -\n \n μ\n 2\n \n \n \n ⁢\n \n \n (\n \n 1\n +\n s\n \n )\n \n \n μ\n 2\n \n \n ⁢\n \n \n ∥\n \n u\n 1\n \n ∥\n \n \n L\n 2\n \n \n \n ≤\n \n C\n ⁢\n \n \n (\n \n 1\n +\n t\n \n )\n \n \n 1\n -\n μ\n -\n \n n\n \n 2\n ⁢\n σ\n \n \n -\n \n γ\n σ\n \n \n \n ⁢\n \n \n (\n \n 1\n +\n s\n \n )\n \n \n \n μ\n -\n 1\n \n +\n \n n\n \n 2\n ⁢\n σ\n \n \n +\n \n γ\n σ\n \n \n \n ⁢\n \n \n ∥\n \n u\n 1\n \n ∥\n \n \n L\n 2\n \n \n \n \n ,\n \n q\n ≥\n 2\n \n \n .\n \n \n \n \n", "2008.12453_53": "", "2008.13408_85": "\n \n \n \n \n \n N\n =\n \n \n (\n \n n\n \n r\n 1\n \n \n )\n \n ⁢\n \n (\n \n \n n\n -\n \n r\n 1\n \n \n \n r\n 2\n \n \n )\n \n ⁢\n ⋯\n ⁢\n \n (\n \n \n n\n -\n \n r\n 1\n \n -\n ⋯\n -\n \n r\n \n k\n -\n 1\n \n \n \n \n r\n k\n \n \n )\n \n ⁢\n \n \n (\n \n q\n -\n 1\n \n )\n \n r\n \n \n =\n \n \n \n n\n !\n \n \n \n \n (\n \n n\n -\n r\n \n )\n \n !\n \n ⁢\n \n \n r\n 1\n \n !\n \n ⁢\n ⋯\n ⁢\n \n \n r\n k\n \n !\n \n \n \n ⁢\n \n \n (\n \n q\n -\n 1\n \n )\n \n r\n \n \n \n .\n \n \n \n \n", "2009.00778_86": "", "2009.07330_5": "\n \n \n \n \n \n \n \n \n minimize\n 𝑝\n \n \n \n \n \n \n 𝒥\n \n ⁢\n \n (\n p\n )\n \n \n ,\n \n \n subject\n \n ⁢\n to\n \n \n \n \n \n \n \n \n 𝒞\n \n ⁢\n \n (\n p\n )\n \n \n =\n 0\n \n ,\n \n \n \n \n ⁢\n \n (\n 5\n )\n \n \n \n \n \n", "2009.07673_16": "\n \n \n \n \n \n \n K\n ⁢\n u\n \n =\n \n f\n ⁢\n on\n ⁢\n Ω\n \n \n .\n \n \n \n \n", "2009.10108_252": "\n \n \n \n \n \n ψ\n i\n \n =\n \n \n ∑\n \n j\n =\n 1\n \n J\n \n \n \n α\n \n i\n ⁢\n j\n \n \n ⁢\n \n x\n \n -\n \n 1\n 2\n \n \n \n ⁢\n \n φ\n j\n \n \n \n \n \n \n \n", "2009.13045_14": "", "2009.14200_7": "\n \n \n \n \n \n |\n \n Ψ\n ν\n \n ⟩\n \n =\n \n \n \n ∏\n 𝐤\n \n \n \n (\n \n \n ∏\n \n j\n =\n 1\n \n \n \n (\n \n ν\n +\n 4\n \n )\n \n /\n 2\n \n \n \n \n c\n \n 𝐤\n ,\n +\n ,\n \n η\n j\n \n ,\n \n s\n j\n \n \n †\n \n ⁢\n \n c\n \n 𝐤\n ,\n -\n ,\n \n η\n j\n \n ,\n \n s\n j\n \n \n †\n \n \n \n )\n \n ⁢\n \n |\n 0\n ⟩\n \n \n \n ,\n \n (\n 7\n )\n \n \n \n \n \n \n", "2010.00264_104": "\n \n \n \n \n \n Δ\n ⁢\n w\n \n <\n \n \n -\n \n \n 2\n ⁢\n π\n ⁢\n \n N\n ~\n \n \n \n Vol\n \n g\n 0\n \n \n \n \n -\n \n \n 1\n 2\n \n ⁢\n λ\n ⁢\n \n e\n \n -\n \n v\n 0\n δ\n \n \n \n ⁢\n F\n ⁢\n \n (\n \n \n 2\n ⁢\n w\n \n +\n \n u\n 0\n δ\n \n \n )\n \n ⁢\n \n (\n 5.9\n )\n \n \n \n \n \n \n \n", "2010.04557_60": "\n \n \n \n \n \n c\n =\n \n \n (\n \n 1\n +\n η\n \n )\n \n ⁢\n \n (\n \n 1\n +\n \n \n ∥\n K\n ∥\n \n 1\n \n \n )\n \n ⁢\n \n \n ∥\n \n K\n ′\n \n ∥\n \n 2\n \n ⁢\n \n \n \n a\n ⁢\n T\n \n 2\n \n \n \n \n ,\n \n \n \n \n", "2010.05319_61": "\n \n \n \n \n \n \n J\n ⁢\n \n (\n p\n ,\n \n |\n x\n |\n \n )\n \n \n =\n \n \n \n J\n +\n \n ⁢\n \n (\n p\n ,\n \n |\n x\n |\n \n )\n \n \n -\n \n \n J\n -\n \n ⁢\n \n (\n p\n ,\n \n |\n x\n |\n \n )\n \n \n \n \n ,\n \n \n \n \n", "2010.08374_70": "\n \n \n \n \n \n \n \n (\n 5.2\n )\n \n ⁢\n \n w\n r\n \n ⁢\n \n \n (\n G\n ;\n ℰ\n )\n \n p\n \n \n ≤\n \n C\n ⁢\n \n (\n p\n ,\n d\n ,\n r\n ,\n L\n ,\n \n ε\n 0\n \n )\n \n \n <\n ∞\n \n .\n \n \n \n \n", "2010.10411_24": "\n \n \n \n \n \n A\n =\n \n \n \n ∑\n \n P\n =\n 1\n \n N\n \n \n \n ∑\n \n Q\n =\n 1\n \n N\n \n \n J\n \n P\n ⁢\n Q\n \n \n \n \n +\n \n \n A\n 2\n \n ⁢\n \n [\n \n equation\n ⁢\n \n (\n \n M\n ⁢\n .11\n \n )\n \n \n ]\n \n \n +\n \n \n A\n 3\n \n ⁢\n \n [\n \n equation\n ⁢\n \n (\n \n M\n ⁢\n .12\n \n )\n \n \n ]\n \n \n \n \n .\n \n (\n \n M\n ⁢\n .13\n \n )\n \n \n \n \n \n", "2010.11770_134": "\n \n \n \n \n \n \n \n (\n \n C\n ⁢\n .6\n \n )\n \n ⁢\n α\n ⁢\n \n (\n \n \n x\n /\n 4\n \n +\n \n \n ⌈\n \n \n \n (\n \n x\n /\n 4\n \n )\n \n /\n α\n \n ⁢\n \n (\n x\n )\n \n \n ⌉\n \n ⁢\n α\n ⁢\n \n (\n x\n )\n \n \n \n )\n \n \n >\n \n \n 2\n \n \n \n (\n \n x\n /\n 4\n \n )\n \n /\n α\n \n ⁢\n \n (\n x\n )\n \n \n \n ⁢\n α\n ⁢\n \n (\n \n x\n /\n 4\n \n )\n \n \n \n .\n \n \n \n \n", "2010.13767_134": "\n \n \n \n \n \n \n \n Ψ\n ~\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n e\n \n -\n \n \n \n i\n ⁢\n q\n \n ℏ\n \n ⁢\n χ\n ⁢\n \n (\n x\n )\n \n \n \n \n ⁢\n Ψ\n ⁢\n \n (\n x\n )\n \n \n \n .\n \n (\n 128\n )\n \n \n \n \n \n", "2010.15089_11": "", "2010.15514_4": "\n \n \n \n \n \n \n \n y\n ⁢\n \n (\n x\n )\n \n \n =\n \n 1\n -\n \n x\n λ\n \n \n \n ,\n \n λ\n >\n 0\n \n \n .\n \n (\n 5\n )\n \n \n \n \n \n", "2011.03825_103": "\n \n \n \n \n F\n \n (\n ⋅\n )\n \n =\n ∑\n k\n =\n 1\n K\n <\n P\n N\n ⋅\n ,\n p\n k\n >\n W\n N\n u\n f\n k\n ∈\n W\n 2\n −\n /\n q\n 1\n ,\n q\n \n (\n \n Γ\n \n )\n \n ;\n G\n \n (\n ⋅\n )\n \n =\n P\n q\n \n (\n m\n \n (\n ∑\n k\n =\n 1\n K\n <\n P\n N\n ⋅\n ,\n q\n k\n >\n W\n N\n u\n u\n k\n )\n \n τ\n )\n \n ∈\n L\n σ\n q\n \n (\n Ω\n )\n \n \n (\n 5.4\n a\n )\n \n \n \n \n \n", "2011.10399_59": "\n \n \n \n \n \n \n \n \n Ω\n ~\n \n ⁢\n \n (\n μ\n )\n \n ⁢\n \n (\n \n [\n 0\n ,\n t\n ]\n \n )\n \n \n =\n \n \n ∫\n \n (\n 0\n ,\n ∞\n )\n \n \n \n \n \n e\n \n -\n \n x\n /\n t\n \n \n \n \n ⁢\n \n 𝑑\n μ\n \n ⁢\n \n (\n x\n )\n \n \n \n =\n \n exp\n ⁡\n \n (\n \n \n C\n μ\n \n ⁢\n \n (\n \n -\n \n t\n \n -\n 1\n \n \n \n )\n \n \n )\n \n \n \n ,\n \n t\n ∈\n \n (\n 0\n ,\n ∞\n )\n \n \n \n .\n \n \n \n \n", "2011.12116_150": "\n \n \n \n \n \n \n G\n ⁢\n \n (\n y\n )\n \n \n -\n \n U\n ⁢\n \n (\n y\n )\n \n \n \n =\n \n \n ∑\n \n k\n ≥\n 1\n \n \n \n \n ξ\n k\n \n ⁢\n \n φ\n k\n \n ⁢\n \n (\n y\n )\n \n \n \n \n \n \n \n", "2011.14865_177": "", "2012.00574_42": "\n \n \n \n \n \n \n Y\n ′\n \n :=\n \n \n L\n 1\n \n ×\n \n L\n 2\n \n ×\n \n {\n \n \n π\n 3\n \n ⁢\n \n (\n u\n )\n \n \n }\n \n ×\n ⋯\n ×\n \n {\n \n \n π\n k\n \n ⁢\n \n (\n u\n )\n \n \n }\n \n \n ⊂\n Y\n \n .\n \n \n \n \n", "2012.02065_236": "\n \n \n \n \n \n (\n 6.15\n )\n \n ±\n \n min\n ⁡\n \n {\n \n \n (\n \n \n β\n ⁢\n \n |\n δ\n |\n \n \n +\n \n \n C\n ′\n \n ⁢\n K\n \n \n )\n \n ⁢\n \n r\n \n -\n 2\n \n \n \n ,\n \n \n C\n ′\n \n ⁢\n K\n ⁢\n \n r\n \n -\n 2.1\n \n \n \n }\n \n \n \n \n \n \n", "2012.12159_229": "\n \n \n \n \n \n \n  L e n g t h \n ⁢\n \n (\n γ\n )\n \n \n =\n \n \n ∫\n Γ\n \n \n α\n η\n \n \n =\n \n \n ∫\n Γ\n \n \n \n α\n ¯\n \n θ\n \n \n =\n \n  L e n g t h \n ⁢\n \n (\n \n γ\n ¯\n \n )\n \n \n \n .\n \n \n \n \n", "2101.04987_255": "", "2101.06854_39": "\n \n \n \n \n \n \n 𝝈\n j\n z\n \n ≡\n \n \n 𝐈\n 1\n \n ⊗\n ⋯\n ⊗\n \n \n \n \n 𝐈\n \n j\n -\n 1\n \n \n ⊗\n \n 𝝈\n j\n z\n \n ⊗\n \n 𝐈\n \n j\n +\n 1\n \n \n \n ⏟\n \n \n \n vertex\n \n ⁢\n j\n \n \n ⊗\n ⋯\n ⊗\n \n 𝐈\n b\n \n \n \n .\n \n \n \n \n", "2101.09148_120": "\n \n \n \n \n \n \n \n μ\n ~\n \n =\n \n μ\n +\n \n α\n ⁢\n f\n ⁢\n \n (\n r\n )\n \n \n \n \n ,\n \n \n ν\n ~\n \n =\n \n ν\n +\n \n β\n ⁢\n h\n ⁢\n \n (\n r\n )\n \n \n \n \n \n ,\n \n \n \n \n", "2101.09278_462": "", "2101.10262_108": "", "2101.11558_2": "\n \n \n \n \n \n \n D\n ⁢\n \n (\n \n A\n ∘\n B\n \n )\n \n ⁢\n E\n \n =\n \n \n (\n \n D\n ⁢\n A\n ⁢\n E\n \n )\n \n ∘\n B\n \n =\n \n \n (\n \n D\n ⁢\n A\n \n )\n \n ∘\n \n (\n \n B\n ⁢\n E\n \n )\n \n \n =\n \n \n (\n \n A\n ⁢\n E\n \n )\n \n ∘\n \n (\n \n D\n ⁢\n B\n \n )\n \n \n =\n \n A\n ∘\n \n (\n \n D\n ⁢\n B\n ⁢\n E\n \n )\n \n \n \n .\n \n \n \n \n", "2102.02349_42": "\n \n \n \n \n \n \n \n μ\n k\n \n ⁢\n \n (\n C\n )\n \n \n ≤\n \n \n ∑\n \n j\n ∈\n \n [\n \n \n d\n 2\n \n -\n \n r\n k\n \n -\n 1\n \n ]\n \n \n \n \n (\n \n \n d\n -\n \n r\n k\n \n -\n 3\n \n \n \n r\n k\n \n -\n 1\n \n \n )\n \n \n =\n \n \n (\n \n \n d\n 2\n \n -\n \n r\n k\n \n -\n 1\n \n )\n \n ⁢\n \n (\n \n \n d\n -\n \n r\n k\n \n -\n 3\n \n \n \n r\n k\n \n -\n 1\n \n \n )\n \n \n <\n \n \n e\n k\n \n ⁢\n \n (\n C\n )\n \n \n \n .\n \n \n \n \n", "2102.04111_89": "\n \n \n \n \n \n \n b\n 1\n \n +\n \n b\n 2\n \n +\n \n b\n 3\n \n \n =\n \n \n ξ\n 1\n \n +\n \n ξ\n 2\n \n +\n \n ξ\n 3\n \n \n =\n \n \n 3\n ⁢\n \n a\n 1\n \n \n +\n \n a\n 2\n \n +\n \n \n 1\n /\n 2\n \n ⁢\n \n (\n 5.1\n )\n \n \n \n \n \n \n \n", "2102.05538_484": "", "2102.05834_7": "\n \n \n \n \n A\n =\n \n (\n \n A\n \n τ\n ⁢\n γ\n \n \n )\n \n =\n \n (\n \n \n ∑\n \n i\n =\n 1\n \n 3\n \n \n \n z\n τ\n i\n \n ⁢\n \n z\n γ\n i\n \n \n \n )\n \n \n \n \n \n", "2102.07586_109": "", "2102.07591_7": "", "2102.12757_55": "\n \n \n \n n\n m\n \n =\n \n \n \n ∑\n \n s\n =\n 1\n \n L\n \n \n \n n\n s\n m\n \n \n \n ,\n \n \n \n ρ\n m\n \n =\n \n \n \n ∑\n \n s\n =\n 1\n \n L\n \n \n \n ρ\n s\n m\n \n \n \n ,\n \n \n \n ρ\n s\n m\n \n =\n \n \n m\n s\n m\n \n ⁢\n \n n\n s\n m\n \n \n \n ,\n \n \n s\n =\n \n 1\n ,\n ⋯\n \n \n ,\n \n \n \n L\n ⁢\n \n u\n m\n \n \n =\n \n \n \n 1\n \n ρ\n m\n \n \n \n ⁢\n \n \n \n ∑\n \n s\n =\n 1\n \n L\n \n \n \n \n ρ\n s\n m\n \n ⁢\n \n u\n s\n m\n \n \n \n \n \n ,\n \n \n 3\n ⁢\n \n n\n m\n \n ⁢\n R\n ⁢\n \n T\n m\n \n \n =\n \n \n 3\n ⁢\n \n \n \n ∑\n \n s\n =\n 1\n \n L\n \n \n \n \n n\n s\n m\n \n ⁢\n R\n ⁢\n \n T\n s\n m\n \n \n \n \n +\n \n \n \n ∑\n \n s\n =\n 1\n \n L\n \n \n \n \n ρ\n s\n m\n \n ⁢\n \n \n |\n \n \n u\n s\n m\n \n -\n \n u\n m\n \n \n |\n \n 2\n \n \n \n \n \n \n \n \n \n \n", "2103.00557_50": "\n \n \n \n \n \n \n \n C\n ¯\n \n \n ⁢\n \n 1\n \n L\n ^\n \n \n ⁢\n \n \n ∑\n \n i\n =\n 1\n \n N\n \n \n \n ∑\n \n j\n =\n 1\n \n M\n \n \n \n Z\n \n i\n ⁢\n j\n \n \n ⁢\n \n f\n μ\n \n ⁢\n \n (\n \n W\n \n i\n ⁢\n j\n \n \n )\n \n ⁢\n \n →\n 𝑑\n \n ⁢\n N\n ⁢\n \n (\n 0\n ,\n \n \n μ\n T\n \n ⁢\n \n Γ\n \n O\n ⁢\n L\n ⁢\n S\n \n \n ⁢\n μ\n \n )\n \n \n \n \n \n ,\n \n \n \n \n", "2103.01191_114": "\n \n \n \n \n \n 𝒜\n ⁢\n \n (\n W\n )\n \n \n :=\n \n {\n \n α\n ∈\n 𝒜\n \n :\n \n \n Im\n ⁢\n \n (\n α\n )\n \n \n ⊂\n W\n \n }\n \n \n \n \n \n", "2103.01264_311": "\n \n \n \n \n \n \n \n (\n \n \n -\n \n e\n \n 2\n ⁢\n π\n ⁢\n n\n ⁢\n t\n \n \n \n +\n \n e\n \n -\n \n 2\n ⁢\n π\n ⁢\n n\n ⁢\n t\n \n \n \n \n )\n \n \n z\n 2\n n\n \n \n =\n \n \n (\n \n \n \n e\n \n π\n ⁢\n n\n ⁢\n t\n \n \n ⁢\n \n (\n \n 1\n +\n \n e\n \n -\n \n 2\n ⁢\n π\n ⁢\n n\n ⁢\n t\n \n \n \n \n )\n \n \n \n \n z\n 1\n \n ⁢\n \n \n (\n \n \n z\n 2\n \n \n z\n 1\n \n \n )\n \n n\n \n \n \n )\n \n ⁢\n \n \n (\n \n \n -\n \n e\n \n π\n ⁢\n n\n ⁢\n t\n \n \n \n +\n \n e\n \n -\n \n π\n ⁢\n n\n ⁢\n t\n \n \n \n \n )\n \n \n z\n 1\n \n n\n -\n 1\n \n \n \n \n =\n \n \n \n (\n \n \n -\n \n e\n \n π\n ⁢\n n\n ⁢\n t\n \n \n \n +\n \n e\n \n -\n \n π\n ⁢\n n\n ⁢\n t\n \n \n \n \n )\n \n \n z\n 1\n \n n\n -\n 1\n \n \n \n ⋅\n o\n \n \n ,\n \n \n \n \n", "2103.01740_3": "\n \n \n \n \n \n \n τ\n \n rt\n -\n \n \n =\n \n \n \n 2\n ⁢\n L\n \n c\n \n ⁢\n \n (\n \n 1\n -\n \n h\n 2\n \n \n )\n \n \n \n .\n \n \n \n \n", "2103.03338_6": "\n \n \n \n \n \n \n lim\n \n n\n →\n \n +\n ∞\n \n \n \n ⁡\n \n \n \n y\n ⁢\n \n (\n \n \n t\n 0\n \n +\n \n \n (\n \n n\n +\n 1\n \n )\n \n ⁢\n T\n \n \n )\n \n \n -\n \n y\n ⁢\n \n (\n \n \n t\n 0\n \n +\n \n n\n ⁢\n T\n \n \n )\n \n \n \n T\n \n \n =\n \n \n v\n 0\n \n ⁢\n \n (\n 𝒢\n )\n \n \n \n \n \n \n", "2103.04484_138": "\n \n \n \n \n \n \n \n \n (\n \n -\n Δ\n \n )\n \n \n α\n 2\n \n \n ⁢\n \n ρ\n ~\n \n ⁢\n \n (\n t\n )\n \n \n ≥\n \n C\n ⁢\n \n (\n α\n ,\n d\n ,\n p\n )\n \n ⁢\n \n \n \n ρ\n ~\n \n \n 1\n +\n \n \n p\n ⁢\n α\n \n d\n \n \n \n \n \n ∥\n ρ\n ∥\n \n \n L\n p\n \n \n \n p\n ⁢\n α\n \n d\n \n \n \n \n \n .\n \n (\n 4.13\n )\n \n \n \n \n \n", "2103.04507_10": "\n \n \n \n \n \n \n W\n 𝒜\n \n =\n \n \n \n \n arg\n ⁡\n min\n \n \n W\n ,\n γ\n \n \n \n ⁢\n \n ℒ\n \n train\n \n \n ⁢\n \n (\n \n 𝒩\n ⁢\n \n (\n 𝒜\n ,\n W\n ,\n γ\n )\n \n \n )\n \n \n \n .\n \n \n \n \n", "2103.06314_256": "\n \n \n \n \n \n \n \n [\n \n K\n \n i\n ⁢\n j\n \n \n ]\n \n =\n \n 0\n ,\n or\n \n \n  \n \n \n \n \n [\n \n K\n \n a\n ⁢\n b\n \n \n ]\n \n ⁢\n \n [\n \n K\n \n a\n ⁢\n b\n \n \n ]\n \n \n -\n \n 3\n ⁢\n R\n \n \n =\n \n 3\n \n 2\n ⁢\n β\n \n \n \n \n .\n \n (\n 5.15\n )\n \n \n \n \n \n", "2103.08072_73": "\n \n \n \n \n \n \n γ\n tot\n \n -\n 1\n \n \n =\n \n τ\n ⁢\n \n R\n 0\n \n -\n 1\n \n \n ⁢\n \n \n Nu\n C\n \n \n Nu\n T\n \n \n \n ≃\n \n 0.3\n ⁢\n \n R\n 0\n \n -\n 1\n \n \n \n \n ,\n \n \n \n \n", "2103.08753_50": "", "2103.09893_123": "", "2103.09986_9": "\n \n \n \n \n \n \n (\n \n \n ∂\n ⁡\n T\n \n \n ∂\n ⁡\n V\n \n \n )\n \n S\n \n =\n \n -\n \n T\n ⁢\n \n \n \n (\n \n \n ∂\n ⁡\n P\n \n \n ∂\n ⁡\n T\n \n \n )\n \n V\n \n \n \n (\n \n \n ∂\n ⁡\n E\n \n \n ∂\n ⁡\n T\n \n \n )\n \n V\n \n \n \n \n \n \n \n \n", "2103.10913_10": "\n \n \n \n \n \n ∂\n \n ∂\n ⁡\n t\n \n \n p\n =\n \n \n D\n eff\n \n \n \n \n R\n \n -\n 1\n \n \n \n \n \n \n ∂\n 2\n \n \n ∂\n ⁡\n \n R\n 2\n \n \n \n \n \n (\n \n R\n \n p\n )\n \n  \n ,\n \n (\n S\n 2\n )\n \n \n \n \n \n", "2103.10952_31": "\n \n \n \n \n \n \n G\n \n \n \n 2\n ⁢\n n\n \n -\n 2\n \n ,\n s\n \n \n =\n \n \n ∑\n \n t\n ≠\n s\n \n \n \n \n ∏\n \n i\n ≠\n s\n \n \n \n a\n i\n \n (\n \n k\n i\n \n )\n \n \n \n \n =\n \n \n ∑\n \n t\n ≠\n s\n \n \n 1\n \n =\n \n n\n -\n 1\n \n \n ,\n \n 1\n ≤\n s\n ≤\n n\n \n .\n \n (\n \n S\n ⁢\n 27\n \n )\n \n \n \n \n \n", "2103.12726_4": "", "2103.13387_37": "\n \n \n \n \n \n \n \n (\n 2.4\n )\n \n ⁢\n \n \n ∫\n N\n \n \n v\n ⁢\n \n (\n s\n )\n \n ⁢\n \n d\n \n λ\n N\n \n \n ⁢\n \n (\n \n \n x\n \n -\n 1\n \n \n ⁢\n s\n ⁢\n x\n \n )\n \n \n \n \n =\n \n \n σ\n N\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n \n ∫\n N\n \n \n v\n ⁢\n \n (\n s\n )\n \n ⁢\n \n d\n \n λ\n N\n \n \n ⁢\n \n (\n s\n )\n \n \n \n \n \n ,\n \n \n \n \n", "2103.14296_54": "\n \n \n \n J\n ~\n \n ⁢\n \n (\n k\n ;\n x\n ,\n t\n )\n \n \n =\n \n \n {\n \n \n \n \n \n \n \n \n  G \n ⁢\n \n (\n k\n ;\n L\n ,\n 0\n )\n \n ⁢\n J\n ⁢\n \n (\n k\n ;\n x\n ,\n t\n )\n \n \n ,\n k\n \n ∈\n \n Σ\n +\n \n \n ,\n \n \n \n J\n ⁢\n \n (\n k\n ;\n x\n ,\n t\n )\n \n ⁢\n  G \n ⁢\n \n \n (\n k\n ;\n L\n ,\n 0\n )\n \n \n -\n 1\n \n \n \n ,\n k\n \n ∈\n \n Σ\n -\n \n \n \n .\n \n \n \n \n \n \n ⁢\n \n (\n 3.32\n )\n \n \n \n", "2103.14573_166": "", "2103.14665_299": "\n \n \n \n \n \n \n \n ∫\n \n \n \n ∏\n \n j\n =\n 1\n \n \n k\n -\n 1\n \n \n \n \n 𝒱\n j\n \n \n \n \n \n 𝟏\n \n {\n \n \n t\n k\n \n >\n 0\n \n }\n \n \n ⁢\n \n 𝑑\n \n Σ\n \n \n k\n -\n 1\n \n ,\n m\n \n k\n \n \n ⁢\n \n (\n \n t\n k\n \n )\n \n \n \n ≤\n \n \n ∑\n \n p\n =\n 1\n \n N\n \n \n \n ∫\n \n {\n \n \n \n Exactly\n \n ⁢\n \n p\n \n ⁢\n \n number\n \n ⁢\n \n of\n \n ⁢\n \n v\n j\n \n \n ∈\n \n \n 𝒱\n j\n \n \n \n 1\n -\n η\n \n \n 2\n ⁢\n \n (\n \n 1\n +\n η\n \n )\n \n \n \n ⁢\n δ\n \n \n \n \n }\n \n \n \n \n 𝟏\n \n {\n \n \n t\n k\n \n >\n 0\n \n }\n \n \n ⁢\n \n 𝑑\n \n Σ\n \n \n k\n -\n 1\n \n ,\n m\n \n k\n \n \n ⁢\n \n (\n \n t\n k\n \n )\n \n \n \n \n \n .\n \n (\n 2.134\n )\n \n \n \n \n \n", "2103.15300_323": "\n \n \n \n \n \n \n \n \n I\n \n \n d\n ⁢\n i\n ⁢\n s\n ⁢\n c\n \n ,\n \n ϕ\n ˙\n \n \n \n (\n \n \n G\n ˙\n \n ~\n \n ,\n \n \n η\n ˙\n \n ~\n \n )\n \n \n ⁢\n \n (\n \n \n f\n ˙\n \n ~\n \n )\n \n \n =\n \n \n 2\n \n ⁢\n ι\n ⁢\n \n (\n \n \n G\n ˙\n \n ~\n \n ,\n \n \n H\n ˙\n \n ~\n \n )\n \n ⁢\n \n S\n \n \n d\n ⁢\n i\n ⁢\n s\n ⁢\n c\n \n ,\n \n \n ϕ\n ˙\n \n \n H\n ˙\n \n \n \n \n \n H\n ˙\n \n ~\n \n \n ⁢\n \n (\n \n \n \n f\n ˙\n \n ~\n \n \n \n H\n ˙\n \n ~\n \n \n )\n \n \n \n ,\n \n \n \n f\n ˙\n \n ~\n \n ∈\n \n \n ℋ\n ¯\n \n ⁢\n \n (\n \n \n G\n ˙\n \n ~\n \n ,\n \n χ\n ~\n \n )\n \n \n \n \n .\n \n \n \n \n", "2103.15891_9": "\n \n \n ζ\n ≡\n \n \n \n α\n \n G\n ⁢\n B\n \n 2\n \n \n κ\n ⁢\n β\n ⁢\n \n M\n 4\n \n \n \n \n \n .\n \n", "2103.16301_214": "\n \n \n \n \n \n \n \n \n R\n \n ±\n ,\n δ\n \n \n ⁢\n \n (\n s\n )\n \n \n =\n \n \n \n Y\n \n ±\n ,\n δ\n \n \n ⁢\n \n (\n s\n )\n \n \n +\n \n \n \n Π\n \n ±\n ,\n δ\n \n \n ⁢\n \n (\n \n h\n ⋆\n \n )\n \n \n \n s\n -\n \n h\n ⋆\n \n \n \n \n \n :\n \n \n \n Ω\n c\n ∙\n \n ⁢\n \n (\n \n M\n δ\n \n )\n \n \n →\n \n \n 𝒟\n \n ′\n ⁣\n ∙\n \n \n ⁢\n \n (\n \n M\n δ\n \n )\n \n \n \n \n .\n \n \n \n \n", "2104.03098_41": "", "2104.03153_19": "", "2104.04372_91": "\n \n \n \n \n \n \n p\n ⁢\n \n (\n s\n )\n \n \n ≤\n \n \n C\n ⁢\n \n s\n m\n \n \n ,\n and\n \n \n  \n \n \n \n p\n ′\n \n ⁢\n \n (\n s\n )\n \n \n ≥\n \n \n \n s\n \n m\n -\n 1\n \n \n C\n \n ,\n \n (\n 2.6\n )\n \n \n \n \n \n \n \n", "2104.05589_32": "", "2104.05810_24": "\n \n \n \n \n \n \n J\n i\n \n =\n \n \n D\n i\n \n -\n \n \n γ\n i\n \n ⁢\n \n |\n \n D\n i\n \n |\n \n \n -\n \n \n \n r\n ⁢\n \n ϵ\n 0\n \n \n -\n \n \n ∑\n \n i\n =\n 1\n \n r\n \n \n \n γ\n i\n \n ⁢\n \n |\n \n D\n i\n \n |\n \n \n \n \n r\n \n \n \n .\n \n \n \n \n", "2104.05880_27": "\n \n \n \n \n \n E\n =\n \n \n E\n vor\n \n +\n \n E\n core\n \n +\n \n E\n boundary\n \n \n \n .\n \n \n \n \n", "2104.06159_0": "", "2104.09152_6": "\n \n \n \n \n \n ℒ\n \n U\n ⁢\n E\n \n \n =\n \n \n ℒ\n \n C\n ⁢\n E\n \n \n +\n \n λ\n ⁢\n \n ℒ\n \n K\n ⁢\n L\n \n \n \n \n \n \n \n \n", "2104.10491_47": "\n \n \n \n \n \n \n \n ∫\n \n -\n q\n \n q\n \n \n \n d\n \n λ\n \n \n ⁢\n \n ρ\n p\n \n ⁢\n \n (\n λ\n )\n \n \n \n =\n n\n \n .\n \n \n \n \n", "2104.10498_106": "\n \n \n \n \n \n \n \n \n 1\n ε\n \n ⁢\n f\n ⁢\n \n (\n \n ε\n ⁢\n \n w\n ε\n \n ⁢\n \n (\n x\n )\n \n \n )\n \n \n →\n \n \n α\n ⁢\n w\n ⁢\n \n (\n x\n )\n \n \n   \n \n \n for\n \n ⁢\n a\n \n \n \n .\n e\n .\n \n x\n ∈\n \n Ω\n \n \n \n .\n \n \n \n \n", "2104.11196_67": "", "2104.13125_161": "\n \n \n \n \n \n \n b\n n\n \n =\n \n \n \n \n 2\n ⁢\n n\n \n +\n 1\n \n \n 4\n ⁢\n π\n ⁢\n \n D\n 1\n \n ⁢\n \n (\n \n n\n +\n 1\n \n )\n \n \n \n ⁢\n \n \n \n (\n \n \n \n D\n 1\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n β\n ⁢\n \n V\n 1\n \n \n \n )\n \n \n \n -\n \n \n D\n 2\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n β\n ⁢\n \n V\n 2\n \n \n \n )\n \n \n \n \n )\n \n ⁢\n \n \n (\n \n 1\n -\n δ\n \n )\n \n \n \n 2\n ⁢\n n\n \n +\n 1\n \n \n \n \n \n \n \n D\n 1\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n β\n ⁢\n \n V\n 1\n \n \n \n )\n \n \n \n +\n \n \n n\n \n n\n +\n 1\n \n \n ⁢\n \n D\n 2\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n β\n ⁢\n \n V\n 2\n \n \n \n )\n \n \n \n \n -\n \n \n (\n \n \n \n D\n 1\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n β\n ⁢\n \n V\n 1\n \n \n \n )\n \n \n \n -\n \n \n D\n 2\n \n ⁢\n \n exp\n ⁡\n \n (\n \n -\n \n β\n ⁢\n \n V\n 2\n \n \n \n )\n \n \n \n \n )\n \n ⁢\n \n \n (\n \n 1\n -\n δ\n \n )\n \n \n \n 2\n ⁢\n n\n \n +\n 1\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "2104.14249_129": "\n \n \n \n \n \n \n \n ∏\n \n i\n =\n 1\n \n \n \n j\n \n 𝐚\n ,\n l\n \n \n -\n l\n \n \n \n r\n \n a\n \n l\n +\n i\n \n \n \n \n ≤\n \n 1\n \n n\n \n 2\n /\n \n \n dim\n S\n \n ⁡\n \n (\n Φ\n )\n \n \n \n \n \n \n .\n \n (\n 3.17\n )\n \n \n \n \n \n", "2105.01536_11": "\n \n \n \n \n \n d\n ⁢\n \n (\n x\n )\n \n \n =\n \n \n ∑\n \n j\n =\n 1\n \n \n n\n R\n \n \n \n \n α\n j\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n (\n \n \n g\n ⁢\n \n (\n \n x\n +\n \n v\n j\n \n \n )\n \n \n -\n \n g\n ⁢\n \n (\n x\n )\n \n \n \n )\n \n \n \n \n \n \n \n", "2105.01936_43": "\n \n \n \n \n \n \n \n ∏\n \n i\n =\n 1\n \n n\n \n \n \n ∏\n \n j\n =\n 1\n \n N\n \n \n \n \n (\n \n \n x\n i\n \n ⁢\n \n z\n j\n \n \n ;\n q\n )\n \n ∞\n \n \n \n (\n \n \n t\n \n -\n 1\n \n \n ⁢\n \n x\n i\n \n ⁢\n \n z\n j\n \n \n ;\n q\n )\n \n ∞\n \n \n \n \n =\n \n \n ∑\n \n \n l\n ⁢\n \n (\n μ\n )\n \n \n ≤\n n\n \n \n \n \n t\n \n -\n \n |\n μ\n |\n \n \n \n ⁢\n \n P\n μ\n \n ⁢\n \n (\n x\n )\n \n ⁢\n \n Q\n μ\n \n ⁢\n \n (\n z\n )\n \n \n \n \n ,\n \n \n \n \n", "2105.02267_213": "\n \n \n \n \n \n \n p\n b\n \n =\n \n \n \n \n i\n \n j\n 1\n \n \n +\n ⋯\n +\n \n i\n \n j\n m\n \n \n +\n \n i\n j\n \n \n -\n 2\n \n \n (\n \n \n i\n a\n \n -\n 1\n \n )\n \n \n \n .\n \n \n \n \n", "2105.02487_158": "", "2105.02528_180": "\n \n \n \n \n \n \n ∇\n \n B\n ⊗\n H\n \n \n φ\n B\n h\n \n \n =\n \n ∇\n \n B\n ⊗\n H\n \n \n φ\n B\n \n \n \n .\n \n \n \n \n", "2105.04926_233": "\n \n \n \n \n \n \n ⋀\n \n \n α\n 1\n \n <\n \n α\n 2\n \n \n \n \n ε\n α\n \n \n =\n \n \n ε\n 1 2\n \n ∧\n ⋯\n ∧\n \n ε\n \n \n 1\n \n ⁢\n n\n \n \n ∧\n \n ε\n 2 3\n \n ∧\n ⋯\n ∧\n \n ε\n \n \n 2\n \n ⁢\n n\n \n \n ∧\n ⋯\n ∧\n \n ε\n \n n\n -\n \n \n 1\n \n ⁢\n n\n \n \n \n \n \n \n \n \n", "2105.07204_201": "\n \n \n \n \n \n \n ∥\n \n \n π\n I\n \n ⁢\n \n (\n \n \n g\n ⁢\n \n (\n 0\n ,\n \n x\n 1\n \n )\n \n \n -\n \n g\n ⁢\n \n (\n 0\n ,\n \n x\n 0\n \n )\n \n \n \n )\n \n \n ∥\n \n ≤\n \n \n (\n \n \n \n L\n g\n 2\n \n ⁢\n \n L\n f\n 1\n \n \n +\n \n L\n g\n 1\n \n \n )\n \n ⁢\n \n ∥\n \n \n x\n 1\n \n -\n \n x\n 0\n \n \n ∥\n \n \n \n .\n \n \n \n \n", "2105.08203_141": "\n \n \n \n \n \n \n \n lim\n \n n\n →\n ∞\n \n \n ⁡\n \n \n ∥\n \n \n θ\n n\n \n -\n θ\n \n ∥\n \n \n \n L\n T\n ∞\n \n ⁢\n \n H\n β\n \n \n \n \n =\n 0\n \n .\n \n \n \n \n", "2105.10474_33": "\n \n \n \n \n \n \n \n log\n 10\n \n ⁡\n \n (\n \n f\n scale\n \n )\n \n \n =\n \n \n f\n \n scale\n ,\n 0\n \n \n +\n \n \n f\n \n scale\n ,\n 1\n \n \n ⁢\n \n (\n \n a\n -\n 1\n \n )\n \n \n \n \n .\n \n \n \n \n", "2105.11009_190": "", "2105.11092_9": "", "2105.11425_43": "\n \n \n \n \n \n γ\n ∼\n \n 𝒩\n ⁢\n \n (\n 0\n ,\n \n Var\n ⁢\n \n [\n \n \n 𝖿\n ^\n \n p\n \n ]\n \n \n )\n \n \n \n .\n \n (\n \n B\n ⁢\n .3\n \n )\n \n \n \n \n \n", "2105.13129_70": "\n \n \n \n \n ∥\n 0\n ,\n x\n i\n −\n y\n ,\n y\n −\n x\n i\n ∥\n ≤\n r\n s\n \n (\n F\n )\n \n ,\n ∥\n 0\n ,\n x\n i\n −\n T\n x\n i\n ,\n T\n x\n i\n −\n x\n i\n ∥\n ≤\n r\n s\n \n (\n F\n )\n \n ,\n ∥\n 0\n ,\n T\n x\n i\n −\n y\n ,\n y\n −\n T\n x\n i\n ∥\n ≤\n r\n s\n \n (\n F\n )\n \n \n \n \n \n", "2105.14915_3": "\n \n \n \n \n \n P\n ⁢\n \n (\n t\n )\n \n \n =\n \n p\n ⁢\n l\n ⁢\n a\n ⁢\n n\n ⁢\n n\n ⁢\n i\n ⁢\n n\n ⁢\n g\n ⁢\n \n (\n \n B\n ⁢\n \n (\n t\n )\n \n \n ,\n \n I\n ⁢\n V\n ⁢\n \n (\n t\n )\n \n \n ,\n \n G\n ⁢\n \n (\n t\n )\n \n \n ,\n \n K\n ⁢\n W\n \n ,\n \n A\n ⁢\n I\n \n )\n \n \n \n \n \n \n", "2106.03373_2": "\n \n \n \n \n \n \n f\n ⁢\n \n (\n q\n ,\n d\n )\n \n \n =\n \n \n \n max\n \n i\n =\n 1\n \n m\n \n ⁢\n \n P\n i\n \n \n ⋅\n \n C\n ′\n \n \n \n .\n \n \n \n \n", "2106.04112_2": "", "2106.04293_5": "", "2106.04785_211": "", "2106.06205_22": "\n \n \n \n \n \n \n f\n ≤\n \n h\n /\n g\n \n \n ⇔\n \n \n f\n ⁢\n g\n \n ≤\n h\n \n ⇔\n \n g\n ≤\n \n f\n \\\n h\n \n \n \n .\n \n \n \n \n", "2106.09298_21": "\n \n \n \n \n \n \n C\n t\n 1\n \n ≡\n C\n =\n \n \n ∫\n 0\n τ\n \n \n \n 𝑑\n t\n \n ⁢\n \n ∥\n \n \n H\n \n L\n ⁢\n E\n ⁢\n O\n \n \n ⁢\n \n (\n t\n )\n \n \n ∥\n \n \n \n \n .\n \n \n \n \n", "2106.10074_68": "\n \n \n \n \n \n \n ∫\n \n \n d\n 4\n \n ⁢\n \n x\n \n ⁢\n \n \n [\n \n \n ψ\n \n 0\n ⁢\n μ\n \n \n ⁢\n \n ψ\n \n 0\n ⁢\n μ\n \n \n \n ]\n \n \n N\n f\n \n \n \n \n ∼\n \n \n T\n \n \n 3\n ⁢\n \n N\n f\n \n \n -\n 1\n \n \n ⁢\n \n \n ∫\n 0\n ∞\n \n \n \n 𝑑\n r\n \n ⁢\n  4\n ⁢\n π\n ⁢\n \n r\n \n 2\n -\n \n N\n f\n \n \n \n ⁢\n \n e\n \n -\n \n 2\n ⁢\n \n N\n f\n \n ⁢\n π\n ⁢\n T\n ⁢\n r\n \n \n \n \n \n \n \n ,\n \n \n \n \n", "2106.10957_17": "\n \n \n \n \n \n z\n =\n \n \n α\n 0\n 2\n \n \n \n ρ\n 0\n \n ⁢\n \n κ\n 0\n \n \n \n \n .\n \n \n \n \n", "2106.11356_161": "\n \n \n \n \n \n \n T\n X\n vir\n \n =\n \n \n [\n \n E\n 0\n \n ]\n \n -\n \n [\n \n E\n 1\n \n ]\n \n \n ∈\n \n \n K\n 0\n \n ⁢\n \n (\n X\n )\n \n \n \n .\n \n \n \n \n", "2106.11637_17": "\n \n \n \n \n \n \n \n ϵ\n ⁢\n \n (\n k\n )\n \n \n \n J\n ~\n \n \n =\n \n \n \n -\n \n \n \n cos\n ⁡\n θ\n \n 4\n \n ⁢\n \n (\n \n N\n -\n 2\n \n )\n \n \n \n +\n \n \n sin\n ⁡\n θ\n \n ⁢\n \n cos\n ⁡\n \n 2\n ⁢\n k\n \n \n \n \n -\n \n \n μ\n \n 2\n ⁢\n \n J\n ~\n \n \n \n \n \n \n .\n \n \n \n \n", "2106.13307_75": "\n \n \n \n \n B\n :=\n B\n 1\n +\n B\n 2\n =\n p\n 0\n \n (\n t\n ,\n x\n −\n y\n )\n \n \n (\n a\n β\n \n (\n θ\n ,\n α\n ,\n y\n )\n \n +\n O\n \n (\n |\n x\n −\n y\n |\n −\n 1\n )\n \n a\n s\n |\n x\n |\n →\n ∞\n ,\n \n \n \n \n \n", "2106.13661_5": "\n \n \n \n \n \n \n \n R\n i\n A\n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n )\n \n \n =\n \n \n \n Z\n ⁢\n \n f\n i\n \n p\n /\n A\n \n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n )\n \n \n +\n \n N\n ⁢\n \n f\n i\n \n n\n /\n A\n \n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n )\n \n \n \n \n \n Z\n ⁢\n \n f\n i\n p\n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n )\n \n \n +\n \n N\n ⁢\n \n f\n i\n n\n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n )\n \n \n \n \n \n ,\n \n \n \n \n", "2106.14142_49": "\n \n \n \n \n \n \n \n ∑\n \n d\n ≤\n x\n \n \n \n |\n \n g\n ⁢\n \n (\n d\n )\n \n \n |\n \n \n =\n \n \n ∑\n \n l\n ≤\n \n x\n \n \n \n \n \n μ\n 2\n \n ⁢\n \n (\n l\n )\n \n \n \n ≍\n \n x\n \n \n ,\n \n \n \n \n", "2106.15062_25": "\n \n \n \n \n \n \n γ\n 5\n \n =\n \n i\n ⁢\n \n γ\n 0\n \n ⁢\n \n γ\n 1\n \n ⁢\n \n γ\n 2\n \n ⁢\n \n γ\n 3\n \n \n =\n \n \n γ\n 4\n \n ⁢\n \n γ\n 1\n \n ⁢\n \n γ\n 2\n \n ⁢\n \n γ\n 3\n \n \n =\n \n -\n \n \n γ\n 1\n \n ⁢\n \n γ\n 2\n \n ⁢\n \n γ\n 3\n \n ⁢\n \n γ\n 4\n \n \n \n \n ,\n \n \n \n \n", "2106.15173_145": "\n \n \n \n \n \n \n \n ∏\n \n 1\n ≤\n \n 2\n r\n \n ≤\n n\n \n \n \n \n (\n \n 1\n -\n \n δ\n \n 2\n r\n \n \n \n )\n \n ⁢\n \n (\n \n 1\n -\n \n δ\n \n 2\n \n r\n -\n 1\n \n \n \n \n )\n \n \n \n ≥\n \n c\n 4\n \n \n ,\n \n \n \n \n", "2107.01459_40": "\n \n \n \n \n ∑\n \n \n ℓ\n =\n \n \n \n ℓ\n 1\n \n +\n \n ℓ\n 2\n \n \n -\n \n ℓ\n 3\n \n \n \n \n \n \n \n ℓ\n 1\n 2\n \n +\n \n ℓ\n 2\n 2\n \n \n -\n \n ℓ\n 3\n 2\n \n -\n \n ℓ\n 2\n \n \n =\n 0\n \n \n \n \n \n \n z\n \n (\n \n m\n 1\n \n ,\n \n ℓ\n 1\n \n )\n \n \n ⁢\n \n z\n \n (\n \n m\n 2\n \n ,\n \n ℓ\n 2\n \n )\n \n \n ⁢\n \n \n z\n ¯\n \n \n (\n \n m\n 3\n \n ,\n \n ℓ\n 3\n \n )\n \n \n ⁢\n \n \n y\n ¯\n \n \n (\n m\n ,\n ℓ\n )\n \n \n \n \n =\n \n \n \n ∑\n \n \n ℓ\n =\n \n \n \n ℓ\n 1\n \n +\n \n ℓ\n 2\n \n \n -\n \n ℓ\n 3\n \n \n \n \n \n \n \n ℓ\n 1\n 2\n \n +\n \n ℓ\n 2\n 2\n \n \n -\n \n ℓ\n 3\n 2\n \n -\n \n ℓ\n 2\n \n \n =\n 0\n \n \n \n \n \n \n z\n \n (\n m\n ,\n \n ℓ\n 1\n \n )\n \n \n ⁢\n \n z\n \n (\n \n m\n 2\n \n ,\n \n ℓ\n 2\n \n )\n \n \n ⁢\n \n \n z\n ¯\n \n \n (\n \n m\n 2\n \n ,\n \n ℓ\n 3\n \n )\n \n \n ⁢\n \n \n y\n ¯\n \n \n (\n m\n ,\n ℓ\n )\n \n \n \n \n \n", "2107.02837_145": "", "2107.04121_9": "\n \n \n \n \n \n \n \n m\n c\n h\n \n ⁢\n \n (\n \n \n v\n ¯\n \n h\n \n ,\n \n \n u\n ¯\n \n h\n \n )\n \n \n =\n \n \n 𝗏\n a\n k\n \n ⁢\n \n (\n \n \n ∫\n T\n \n \n \n δ\n \n i\n ⁢\n a\n \n \n ⁢\n \n ϕ\n k\n \n ⁢\n \n (\n \n ξ\n ¯\n \n )\n \n ⁢\n \n δ\n \n i\n ⁢\n b\n \n \n ⁢\n \n ϕ\n m\n \n ⁢\n \n (\n \n ξ\n ¯\n \n )\n \n ⁢\n \n |\n \n \n J\n c\n \n ⁢\n \n (\n \n ξ\n ¯\n \n )\n \n \n |\n \n \n \n )\n \n ⁢\n \n 𝗎\n b\n m\n \n \n =\n \n \n 𝗏\n a\n k\n \n ⁢\n \n 𝖬\n \n a\n ⁢\n k\n ⁢\n b\n ⁢\n m\n \n \n ⁢\n \n \n 𝗎\n b\n m\n \n \n \n \n .\n \n \n \n \n", "2107.06572_5": "\n \n \n \n \n \n \n \n ∑\n k\n \n \n \n P\n i\n \n ⁢\n \n (\n \n x\n k\n \n )\n \n ⁢\n D\n ⁢\n \n (\n \n \n \n P\n f\n \n ⁢\n \n (\n \n x\n k\n \n )\n \n \n =\n \n 1\n ∥\n \n P\n i\n \n \n \n )\n \n \n \n =\n \n H\n ⁢\n \n (\n \n \n P\n i\n \n ⁢\n \n (\n X\n )\n \n \n )\n \n \n \n .\n \n (\n 6\n )\n \n \n \n \n \n", "2107.08257_56": "\n \n \n \n \n \n \n \n sup\n \n R\n ≥\n 1\n \n \n ⁡\n \n \n 1\n \n R\n 2\n \n \n ⁢\n \n \n (\n \n \n ∫\n \n \n B\n \n 2\n ⁢\n R\n \n +\n \n ∖\n \n B\n R\n +\n \n \n \n \n \n \n z\n \n 1\n -\n \n 2\n ⁢\n s\n \n \n \n \n ⁢\n \n 𝑑\n \n x\n \n \n ⁢\n \n 𝑑\n z\n \n \n \n )\n \n \n \n (\n \n γ\n -\n 1\n \n )\n \n /\n γ\n \n \n \n \n <\n \n +\n ∞\n \n \n .\n \n \n \n \n", "2107.08489_79": "", "2107.08568_416": "\n \n \n \n \n \n \n \n P\n ⁢\n \n u\n n\n \n \n +\n \n \n b\n i\n \n ⁢\n \n D\n \n v\n i\n \n \n ⁢\n \n u\n n\n \n \n +\n \n c\n ⁢\n \n u\n n\n \n \n +\n \n λ\n ⁢\n \n u\n n\n \n \n \n =\n \n f\n n\n \n \n .\n \n (\n 7.13\n )\n \n \n \n \n \n", "2107.09643_2": "\n \n \n \n \n \n \n m\n ⁢\n \n \n \n d\n 2\n \n ⁢\n \n r\n →\n \n \n \n d\n ⁢\n \n t\n 2\n \n \n \n \n =\n \n \n \n -\n \n \n d\n ⁢\n \n r\n →\n \n \n \n d\n ⁢\n t\n \n \n \n -\n \n \n [\n \n f\n +\n \n a\n ⁢\n \n sin\n ⁡\n \n (\n \n ω\n ⁢\n t\n \n )\n \n \n \n \n ]\n \n ⁢\n \n x\n ^\n \n \n \n +\n \n \n ξ\n →\n \n ⁢\n \n (\n t\n )\n \n \n \n \n .\n \n (\n 3\n )\n \n \n \n \n \n", "2107.10200_65": "\n \n \n \n \n \n 2\n ⁢\n i\n ⁢\n \n ℓ\n 1\n \n \n =\n \n \n \n 𝒱\n ⋅\n ℋ\n \n ⁢\n ℓ\n \n ,\n \n (\n )\n \n \n \n \n \n \n", "2107.12687_4": "\n \n \n \n \n \n \n \n K\n \n -\n 1\n \n \n ⁢\n \n |\n b\n |\n \n \n ≤\n \n \n f\n 2\n \n ⁢\n \n (\n b\n )\n \n \n ≤\n \n K\n ⁢\n \n (\n \n 1\n +\n \n |\n b\n |\n \n \n )\n \n \n \n ;\n \n \n \n \n", "2108.03876_107": "\n \n \n \n \n \n \n \n \n U\n ~\n \n +\n \n ⁢\n \n (\n ϕ\n )\n \n \n =\n \n -\n \n \n 2\n ⁢\n t\n ⁢\n z\n \n \n \n z\n 2\n \n +\n \n 2\n ⁢\n \n π\n 2\n \n ⁢\n \n ϕ\n 2\n \n \n \n \n \n \n .\n \n (\n 111\n )\n \n \n \n \n \n", "2108.04229_3": "\n \n \n \n \n \n Attention\n ⁢\n \n (\n Q\n ,\n K\n ,\n V\n )\n \n \n =\n \n softmax\n ⁢\n \n (\n \n \n Q\n ⁢\n \n K\n T\n \n \n \n \n d\n k\n \n \n \n )\n \n ⁢\n V\n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n", "2108.05551_1947": "", "2108.06496_41": "\n \n \n \n \n \n \n φ\n 2\n \n =\n \n \n [\n \n \n \n \n l\n -\n a\n \n b\n \n ⁢\n \n C\n 1\n \n ⁢\n \n ln\n ⁡\n r\n \n \n +\n \n \n \n C\n 1\n \n +\n \n \n (\n \n l\n -\n a\n \n )\n \n ⁢\n \n C\n 2\n \n \n \n b\n \n \n ]\n \n ⁢\n \n r\n l\n \n \n \n .\n \n \n \n \n", "2108.08913_5": "\n \n \n \n \n \n \n ϕ\n ¨\n \n +\n \n 3\n ⁢\n H\n ⁢\n \n ϕ\n ˙\n \n \n \n =\n \n \n -\n \n V\n \n ,\n ϕ\n \n \n \n -\n \n \n A\n \n ,\n ϕ\n \n \n ⁢\n \n \n ρ\n ^\n \n ν\n \n ⁢\n \n (\n 6\n )\n \n \n \n \n \n \n \n", "2108.09607_27": "", "2108.11552_16": "", "2108.12239_15": "\n \n \n \n \n \n \n \n ⋂\n \n i\n =\n 1\n \n k\n \n \n \n η\n ′\n \n ⁢\n \n (\n \n E\n i\n \n \n S\n \n F\n i\n \n \n \n \n )\n \n \n \n ⊆\n \n \n ⋂\n \n i\n =\n 1\n \n k\n \n \n \n η\n ′\n \n ⁢\n \n (\n \n E\n i\n \n \n S\n i\n \n 𝒵\n \n \n \n \n )\n \n \n \n \n .\n \n \n \n \n", "2108.13000_0": "", "2109.02483_150": "", "2109.03834_33": "\n \n \n \n \n Υ\n ≃\n \n \n v\n k\n \n \n c\n s\n \n \n ≃\n \n 14\n ⁢\n \n \n (\n \n z\n /\n \n z\n s\n \n \n )\n \n \n -\n \n (\n \n \n \n f\n q\n \n -\n 1\n \n \n +\n 3\n \n -\n α\n \n )\n \n \n \n ⁢\n \n β\n \n -\n 2\n \n \n ⁢\n \n \n \n (\n \n \n T\n h\n \n \n \n 2\n ×\n \n 10\n 6\n \n \n ⁢\n K\n \n \n )\n \n \n -\n \n 1\n /\n 2\n \n \n \n \n ⁢\n \n (\n 33\n )\n \n \n \n \n \n \n", "2109.06167_63": "\n \n \n \n \n \n \n \n a\n ¨\n \n a\n \n =\n \n \n \n \n [\n \n \n 4\n ⁢\n \n a\n 3\n \n ⁢\n \n \n (\n \n 32\n -\n \n 3\n ⁢\n \n b\n 3\n \n \n \n )\n \n 2\n \n \n +\n \n 9\n ⁢\n \n (\n \n 16\n -\n \n 3\n ⁢\n \n b\n 3\n \n \n \n )\n \n ⁢\n \n b\n 3\n \n \n \n ]\n \n \n 3\n ⁢\n \n \n [\n \n \n 9\n ⁢\n \n b\n 3\n \n \n -\n \n 8\n ⁢\n \n a\n 3\n \n ⁢\n \n (\n \n \n 3\n ⁢\n \n b\n 3\n \n \n -\n 32\n \n )\n \n \n \n ]\n \n 2\n \n \n \n ⁢\n \n κ\n 2\n \n ⁢\n \n ϕ\n 2\n \n \n -\n \n \n κ\n 6\n \n ⁢\n \n (\n \n 1\n +\n \n 3\n ⁢\n w\n \n \n )\n \n ⁢\n \n ρ\n \n \n \n \n .\n \n (\n 118\n )\n \n \n \n \n \n", "2109.06685_111": "", "2109.09197_15": "", "2109.14161_39": "\n \n \n \n \n \n \n \n ∫\n M\n \n \n \n ∏\n \n i\n =\n 1\n \n n\n \n \n \n t\n ⁢\n \n x\n i\n \n ⁢\n \n (\n \n 1\n +\n \n y\n ⁢\n \n e\n \n -\n \n t\n ⁢\n \n x\n i\n \n \n \n \n \n \n )\n \n \n \n 1\n -\n \n e\n \n -\n \n t\n ⁢\n \n x\n i\n \n \n \n \n \n \n \n \n =\n \n \n t\n n\n \n ⁢\n \n \n ∫\n M\n \n \n \n ∏\n \n i\n =\n 1\n \n n\n \n \n \n \n x\n i\n \n ⁢\n \n (\n \n 1\n +\n \n y\n ⁢\n \n e\n \n -\n \n x\n i\n \n \n \n \n \n )\n \n \n \n 1\n -\n \n e\n \n -\n \n x\n i\n \n \n \n \n \n \n \n \n =\n \n \n t\n n\n \n ⁢\n \n χ\n y\n \n ⁢\n \n (\n M\n )\n \n \n \n .\n \n \n \n \n", "2110.02166_10": "\n \n \n \n \n \n \n a\n μ\n \n =\n \n \n μ\n A\n *\n \n \n μ\n \n agg\n ,\n B\n \n *\n \n \n \n .\n \n (\n 12\n )\n \n \n \n \n \n", "2110.02272_177": "\n \n \n \n \n \n |\n \n \n \n ℰ\n 2\n \n ⁢\n \n (\n \n T\n c\n \n )\n \n \n \n \n ϕ\n -\n \n ⁢\n \n (\n \n T\n c\n \n )\n \n \n \n |\n \n ≈\n \n \n |\n \n \n \n r\n 1\n \n r\n \n ⁢\n \n (\n \n \n \n (\n \n \n 2\n ⁢\n Υ\n \n -\n \n 4\n 3\n \n \n )\n \n \n Υ\n ⁢\n \n \n (\n \n Υ\n +\n 2\n \n )\n \n 3\n \n \n \n +\n \n \n 3\n ⁢\n \n (\n \n Υ\n -\n 2\n \n )\n \n \n \n \n \n (\n \n Υ\n +\n 2\n \n )\n \n 3\n \n ⁢\n F\n \n \n \n )\n \n ⁢\n \n j\n 1\n 3\n \n \n |\n \n ⁢\n \n (\n 279\n )\n \n \n \n \n \n \n", "2110.04768_1": "\n \n \n \n \n \n \n \n y\n ^\n \n k\n \n =\n \n \n \n α\n k\n A\n \n ⁢\n \n s\n k\n A\n \n \n +\n \n \n α\n k\n B\n \n ⁢\n \n s\n k\n B\n \n \n \n \n .\n \n \n \n \n", "2110.07974_16": "\n \n \n \n \n \n \n \n (\n 1.14\n )\n \n ⁢\n 𝒩\n \n ∈\n \n UC\n ⁡\n \n [\n \n ω\n 1\n \n ]\n \n \n \n ,\n \n \n \n \n", "2110.08523_27": "\n \n \n \n \n \n \n \n U\n L\n \n ⁢\n \n (\n \n e\n \n ı\n ⁢\n θ\n \n \n )\n \n \n =\n \n [\n \n \n \n \n \n \n e\n \n -\n \n ı\n ⁢\n L\n ⁢\n θ\n \n \n \n ⁢\n \n e\n \n ı\n ⁢\n L\n ⁢\n θ\n \n \n \n \n \n \n \n ]\n \n \n .\n \n \n \n \n", "2110.09517_91": "\n \n \n \n \n \n \n \n \n \n ∥\n \n \n u\n a\n \n -\n \n u\n j\n a\n \n \n ∥\n \n \n \n L\n T\n ∞\n \n ⁢\n \n (\n \n B\n \n 2\n ,\n 1\n \n 2\n \n )\n \n \n \n +\n \n \n ∥\n \n \n τ\n a\n \n -\n \n τ\n j\n a\n \n \n ∥\n \n \n \n L\n T\n ∞\n \n ⁢\n \n (\n \n B\n \n 2\n ,\n 1\n \n 1\n \n )\n \n \n \n \n ≤\n \n ϵ\n 3\n \n \n ,\n \n \n f\n ⁢\n o\n ⁢\n r\n ⁢\n a\n ⁢\n n\n ⁢\n y\n ⁢\n a\n \n ≥\n 0\n \n \n .\n \n \n \n \n", "2111.05256_153": "\n \n \n \n \n \n \n S\n 26\n \n =\n \n \n \n h\n ⁢\n \n (\n \n -\n p\n \n )\n \n \n 2\n \n ⁢\n \n [\n \n \n 2\n -\n \n (\n \n 2\n p\n \n )\n \n -\n \n 2\n ⁢\n \n (\n \n 3\n p\n \n )\n \n \n \n +\n \n (\n \n 6\n p\n \n )\n \n \n ]\n \n \n \n ,\n \n \n \n \n", "2111.06592_73": "\n \n \n \n \n \n \n P\n ⁢\n \n Y\n ′\n \n ⁢\n R\n ⁢\n Λ\n ⁢\n \n R\n \n -\n 1\n \n \n \n +\n \n X\n ⁢\n \n W\n x\n \n \n \n =\n \n \n Y\n ′\n \n ,\n \n (\n 67\n )\n \n \n \n \n \n \n", "2111.08905_298": "\n \n \n \n \n \n \n σ\n ⁢\n \n (\n z\n )\n \n \n =\n \n \n ∑\n \n \n γ\n 3\n \n ∈\n \n S\n 3\n \n \n \n \n \n ν\n 3\n \n ⁢\n \n (\n \n γ\n 3\n \n )\n \n ⁢\n \n \n \n e\n z\n \n ⁢\n \n (\n \n γ\n 3\n \n )\n \n \n \n deg\n ⁡\n \n γ\n 3\n \n \n \n \n \n \n .\n \n \n \n \n", "2111.13206_18": "\n \n \n \n \n \n \n \n lim\n \n u\n →\n ∞\n \n \n ⁡\n \n P\n ⁢\n \n (\n \n \n \n u\n \n -\n 1\n \n \n ⁢\n \n X\n ε\n \n \n >\n \n 1\n +\n \n \n 1\n 2\n \n ⁢\n \n ε\n 2\n \n ⁢\n \n 1\n \n R\n ⁢\n \n (\n 0\n )\n \n \n \n ⁢\n \n \n sup\n \n 0\n ≤\n t\n ≤\n ε\n \n \n ⁡\n \n \n R\n ′′\n \n ⁢\n \n (\n t\n )\n \n ⁢\n \n |\n \n X\n 0\n \n >\n \n ⁢\n u\n \n \n \n \n \n )\n \n \n \n =\n \n 0\n \n \n .\n \n \n \n \n", "2111.13221_89": "\n \n \n \n \n \n \n ℛ\n \n i\n ⁢\n i\n \n \n =\n \n 6\n ⁢\n \n (\n \n 2\n -\n \n s\n 2\n \n \n )\n \n \n \n ,\n \n \n ℛ\n \n α\n ⁢\n α\n \n \n =\n \n \n \n \n 2\n +\n \n \n 4\n \n ⁢\n \n s\n 4\n \n \n \n \n s\n 2\n \n \n \n ,\n \n (\n 100\n )\n \n \n \n \n \n \n \n", "2111.15092_4": "\n \n \n \n \n \n \n \n \n I\n 0\n N\n \n ⁢\n \n (\n 𝟎\n )\n \n \n =\n 1\n \n ,\n \n \n and\n ⁢\n \n I\n 0\n N\n \n ⁢\n \n (\n 𝐱\n )\n \n \n =\n \n 0\n ⁢\n f\n ⁢\n o\n ⁢\n \n r\n \n ⁢\n all\n ⁢\n 𝐱\n \n ≠\n 𝟎\n \n \n ;\n \n (\n 3\n )\n \n \n \n \n \n", "2111.15310_10": "\n \n \n \n \n \n Ψ\n a\n \n =\n \n \n (\n \n \n Γ\n 12\n \n ⊗\n \n Γ\n 34\n \n \n )\n \n ⁢\n \n \n G\n 0\n \n (\n 2\n ,\n 2\n )\n \n \n \n ⁢\n \n Φ\n a\n \n ⁢\n \n (\n 10\n )\n \n \n \n \n \n \n", "2112.00419_4": "\n \n \n \n \n \n E\n x\n \n :=\n \n \n \n H\n 0\n \n ⁢\n \n (\n \n \n π\n \n -\n 1\n \n \n ⁢\n \n (\n x\n )\n \n \n ,\n \n \n ℒ\n |\n \n \n \n π\n \n -\n 1\n \n \n ⁢\n \n (\n x\n )\n \n \n \n )\n \n \n ,\n \n (\n 1.5\n )\n \n \n \n \n \n \n", "2112.04654_10": "\n \n \n \n \n \n \n Σ\n ⁢\n \n (\n y\n )\n \n \n =\n \n \n ⋃\n \n z\n ∈\n \n \n Max\n ⁡\n \n σ\n α\n \n \n ⁢\n \n (\n \n x\n |\n y\n \n )\n \n \n \n \n \n Σ\n ⁢\n \n (\n z\n )\n \n \n \n \n .\n \n (\n 2.1\n )\n \n \n \n \n \n", "2112.05338_40": "\n \n \n \n \n \n \n (\n \n α\n ∧\n β\n \n )\n \n ∨\n \n (\n \n α\n ∧\n γ\n \n )\n \n \n =\n \n α\n ∧\n \n [\n \n α\n ∨\n \n (\n \n β\n ∧\n γ\n \n )\n \n \n ]\n \n ∧\n \n [\n \n β\n ∨\n \n (\n \n γ\n ∧\n α\n \n )\n \n \n ]\n \n ∧\n \n [\n \n γ\n ∨\n \n (\n \n α\n ∧\n β\n \n )\n \n \n ]\n \n \n \n \n \n \n", "2112.08146_2": "\n \n \n \n \n \n \n V\n α\n \n 0\n ⁢\n ν\n \n \n ⁢\n \n (\n \n r\n \n a\n ⁢\n b\n \n \n )\n \n \n =\n \n \n \n 1\n \n g\n A\n 2\n \n \n ⁢\n \n ∫\n \n \n \n \n d\n 3\n \n ⁢\n 𝐪\n \n \n \n (\n \n 2\n ⁢\n π\n \n )\n \n 3\n \n \n ⁢\n \n e\n \n \n i\n ⁢\n 𝐪\n \n ⋅\n \n 𝐫\n \n a\n ⁢\n b\n \n \n \n \n ⁢\n \n V\n α\n \n 0\n ⁢\n ν\n \n \n ⁢\n \n (\n \n 𝐪\n 2\n \n )\n \n \n \n \n ,\n \n (\n 3\n )\n \n \n \n \n \n \n", "2112.12420_28": "\n \n \n \n \n \n \n ⟨\n \n ϵ\n \n i\n ⁢\n j\n \n *\n \n ⟩\n \n 2\n \n =\n \n \n 1\n \n V\n 2\n \n \n ⁢\n \n \n ∫\n \n Ω\n 2\n \n \n \n \n (\n \n \n ε\n \n i\n ⁢\n j\n \n *\n \n +\n \n \n κ\n \n i\n ⁢\n j\n ⁢\n k\n \n *\n \n ⁢\n \n x\n k\n \n \n \n )\n \n ⁢\n \n 𝑑\n v\n \n \n \n \n =\n \n \n ε\n \n i\n ⁢\n j\n \n *\n \n ⁢\n \n [\n \n 5\n ⁢\n p\n ⁢\n t\n \n ]\n \n ⁢\n \n \n ⟨\n \n \n ϵ\n \n i\n ⁢\n j\n \n *\n \n ⁢\n \n x\n k\n \n \n ⟩\n \n 2\n \n \n =\n \n \n 1\n \n V\n 2\n \n \n ⁢\n \n \n ∫\n \n Ω\n 2\n \n \n \n \n (\n \n \n \n ε\n \n i\n ⁢\n j\n \n *\n \n ⁢\n \n x\n k\n \n \n +\n \n \n κ\n \n i\n ⁢\n j\n ⁢\n l\n \n *\n \n ⁢\n \n x\n l\n \n ⁢\n \n x\n k\n \n \n \n )\n \n ⁢\n \n 𝑑\n v\n \n \n \n \n =\n \n \n κ\n \n i\n ⁢\n j\n ⁢\n l\n \n *\n \n ⁢\n \n E\n \n l\n ⁢\n k\n \n \n (\n 2\n )\n \n \n ⁢\n \n (\n 27\n )\n \n \n \n \n \n \n", "2112.13767_38": "\n \n \n \n \n \n \n a\n \n r\n ⁢\n o\n ⁢\n t\n \n \n =\n \n b\n \n r\n ⁢\n o\n ⁢\n t\n \n \n =\n \n R\n ⁢\n \n (\n \n 1\n +\n \n \n ϵ\n M\n \n 3\n \n \n )\n \n \n \n ;\n \n \n c\n \n r\n ⁢\n o\n ⁢\n t\n \n \n =\n \n R\n ⁢\n \n (\n \n 1\n -\n \n \n 2\n ⁢\n \n ϵ\n M\n \n \n 3\n \n \n )\n \n \n \n .\n \n (\n 0\n )\n \n \n \n \n \n", "2112.14107_35": "", "2112.15197_6": "\n \n \n \n \n \n \n W\n \n 2\n ⁢\n s\n \n \n ⁢\n \n (\n E\n ;\n φ\n ,\n T\n )\n \n \n =\n \n \n \n ∫\n 0\n E\n \n \n \n ω\n \n 2\n ⁢\n s\n \n \n ⁢\n \n (\n \n E\n ′\n \n ;\n φ\n ,\n T\n )\n \n ⁢\n \n 𝑑\n \n E\n ′\n \n \n \n \n ,\n \n (\n 7\n )\n \n \n \n \n \n \n", "2201.00104_43": "\n \n \n \n \n \n \n \n \n 1\n 2\n \n ⁢\n \n ⌈\n \n L\n \n x\n 1\n \n \n ⌉\n \n \n ⋅\n \n (\n \n \n ⌈\n \n L\n \n x\n 1\n \n \n ⌉\n \n -\n 1\n \n )\n \n \n ≤\n \n \n L\n 2\n \n \n x\n 1\n 2\n \n \n \n .\n \n \n \n \n", "2201.02100_2": "", "2201.04592_27": "", "2201.04745_74": "", "2201.10773_0": "\n \n \n \n \n λ\n →\n \n \n λ\n ⁢\n F\n ⁢\n \n (\n t\n )\n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "2201.12529_88": "\n \n \n \n \n \n -\n \n \n ω\n 2\n \n ⁢\n \n u\n x\n \n ⁢\n \n (\n z\n )\n \n \n \n =\n \n \n \n \n G\n ⁢\n \n (\n ω\n )\n \n \n \n ρ\n l\n \n \n ⁢\n \n \n \n \n d\n \n ⁡\n \n u\n x\n \n \n \n 2\n \n \n ⁢\n \n (\n z\n )\n \n \n \n \n d\n \n ⁡\n \n z\n 2\n \n \n \n \n ,\n \n (\n 92\n )\n \n \n \n \n \n \n", "2201.12908_5": "\n \n \n \n \n \n \n λ\n 1\n \n ≥\n \n λ\n 2\n \n ≥\n …\n ≥\n \n \n λ\n k\n \n \n \n .\n \n \n \n \n", "2201.13120_66": "\n \n \n \n \n \n \n G\n 00\n \n ⁢\n \n g\n 00\n \n \n =\n \n \n \n 8\n ⁢\n π\n \n \n c\n 4\n \n \n ⁢\n \n T\n 00\n \n ⁢\n \n g\n 00\n \n \n \n \n \n \n", "2201.13199_22": "", "2201.13339_87": "\n \n \n \n \n \n \n A\n \n s\n ,\n δ\n \n \n \n n\n +\n 1\n \n \n n\n +\n 2\n \n \n \n ≤\n \n \n \n ||\n \n e\n F\n \n ||\n \n \n p\n 0\n \n \n ⁢\n C\n ⁢\n \n (\n \n q\n 0\n \n ,\n β\n )\n \n ⁢\n v\n ⁢\n o\n ⁢\n \n l\n \n \n 1\n \n q\n 0\n \n \n ⁢\n \n (\n \n 1\n -\n \n 1\n β\n \n \n )\n \n \n \n ⁢\n \n (\n \n Ω\n s\n \n )\n \n \n \n .\n \n \n \n \n", "2202.00823_63": "\n \n \n \n \n \n \n lim\n \n \n l\n Γ\n \n →\n \n 0\n +\n \n \n \n ⁡\n \n \n d\n ⁢\n \n Γ\n \n i\n ⁢\n j\n \n \n \n \n d\n ⁢\n \n l\n Γ\n \n \n \n \n =\n \n \n 0\n ⁢\n \n ∀\n ı\n \n \n ,\n \n j\n ⁢\n \n (\n 54\n )\n \n \n \n \n \n \n \n", "2208.10054_76": "\n \n \n \n \n \n \n 𝐭\n \n m\n ⁢\n i\n ⁢\n x\n \n \n f\n 2\n \n \n =\n \n 𝒪\n ⁢\n \n (\n \n \n N\n 3\n \n ⁢\n \n (\n \n \n ln\n ⁡\n 4\n \n +\n \n β\n ⁢\n \n (\n \n c\n -\n \n ℋ\n *\n \n \n )\n \n \n +\n \n π\n 2\n \n +\n \n N\n ⁢\n \n ln\n ⁡\n 2\n \n \n \n )\n \n \n )\n \n \n \n .\n \n \n \n \n", "2208.11479_20": "\n \n \n \n \n \n \n I\n 2\n \n λ\n \n ∼\n \n \n 1\n 2\n \n ⁢\n \n (\n 21\n )\n \n \n \n \n \n \n", "2208.13190_33": "", "2208.13964_0": "\n \n \n \n \n γ\n \n (\n G\n -\n v\n )\n \n =\n γ\n \n (\n G\n )\n \n -\n 1\n ⇔\n γ\n \n (\n G\n -\n v\n )\n \n ≤\n γ\n \n (\n G\n )\n \n -\n 1\n .\n \n (\n 1.1\n )\n \n \n \n \n \n", "2209.00512_193": "\n \n \n \n \n \n \n \n h\n top\n \n \n log\n 3\n \n ⁡\n 2\n \n \n ⁢\n \n (\n π\n ,\n Ω\n ,\n σ\n )\n \n \n =\n \n log\n ⁡\n \n (\n \n 1\n +\n \n 2\n \n \n log\n 3\n \n ⁡\n 2\n \n \n \n )\n \n \n \n .\n \n \n \n \n", "2209.03573_18": "\n \n \n \n \n \n \n \n (\n \n g\n *\n g\n \n )\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n [\n \n x\n =\n 0\n \n ]\n \n \n .\n \n \n \n \n", "2210.05685_8": "\n \n \n \n \n \n \n ρ\n χ\n \n ⁢\n \n (\n r\n )\n \n \n =\n \n \n \n ρ\n s\n \n \n \n r\n /\n \n r\n s\n \n \n ⁢\n \n \n (\n \n 1\n +\n \n r\n /\n \n r\n s\n \n \n \n )\n \n 2\n \n \n \n ⁢\n \n (\n 9\n )\n \n \n \n \n \n \n", "2210.06873_2": "\n \n \n \n \n \n h\n ⁢\n \n (\n \n x\n q\n \n )\n \n \n <\n \n θ\n ,\n \n (\n 3\n )\n \n \n \n \n \n \n", "2210.08609_1": "\n \n \n \n \n \n \n |\n \n ⟨\n \n cos\n ⁡\n \n (\n \n 2\n ⁢\n \n ϕ\n T\n \n \n )\n \n \n ⟩\n \n |\n \n \n x\n ,\n \n Q\n 2\n \n ,\n \n k\n T\n \n \n \n ∝\n \n \n \n \n q\n T\n 2\n \n \n 2\n ⁢\n \n M\n p\n 2\n \n \n \n ⁢\n \n \n \n h\n 1\n \n ⟂\n ,\n g\n \n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n ,\n \n k\n T\n \n )\n \n \n \n \n f\n 1\n g\n \n ⁢\n \n (\n x\n ,\n \n Q\n 2\n \n ,\n \n k\n T\n \n )\n \n \n \n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "2210.08939_79": "\n \n \n \n \n 0\n ≤\n \n \n 2\n ⁢\n b\n \n -\n 1\n \n ≤\n b\n \n \n \n \n", "2210.09971_0": "\n \n \n \n \n \n \n \n d\n H\n \n ⁢\n \n (\n A\n ,\n B\n )\n \n \n =\n \n max\n ⁡\n \n {\n \n \n sup\n \n a\n ∈\n A\n \n \n ⁡\n \n \n inf\n \n b\n ∈\n B\n \n \n ⁡\n \n d\n ⁢\n \n (\n a\n ,\n b\n )\n \n \n \n \n ,\n \n \n sup\n \n b\n ∈\n B\n \n \n ⁡\n \n \n inf\n \n a\n ∈\n A\n \n \n ⁡\n \n d\n ⁢\n \n (\n a\n ,\n b\n )\n \n \n \n \n }\n \n \n \n .\n \n \n \n \n", "2210.10747_21": "", "2210.12684_10": "\n \n \n \n \n \n \n \n Ω\n ^\n \n =\n \n Ω\n -\n \n \n ϑ\n 2\n \n ⁢\n \n P\n β\n \n \n \n \n ,\n \n \n β\n ^\n \n =\n \n β\n +\n \n \n ϑ\n 2\n \n ⁢\n \n P\n Ω\n \n \n \n \n \n .\n \n (\n 11\n )\n \n \n \n \n \n", "2211.11605_89": "", "2211.12218_4": "\n \n \n \n \n \n \n \n ∥\n f\n ∥\n \n \n L\n ω\n \n p\n ⁢\n \n (\n ⋅\n )\n \n \n \n \n :=\n \n \n ∥\n \n f\n ⁢\n ω\n \n ∥\n \n \n L\n \n p\n ⁢\n \n (\n ⋅\n )\n \n \n \n \n \n .\n \n \n \n \n", "2211.12836_142": "\n \n \n \n \n \n \n d\n ⁢\n \n p\n ⋆\n \n ⁢\n \n μ\n H\n \n ⁢\n \n (\n \n u\n 1\n \n ,\n …\n ,\n \n u\n n\n \n )\n \n \n =\n \n \n 1\n \n \n (\n \n 2\n ⁢\n π\n \n )\n \n k\n \n \n ⁢\n \n \n ∏\n \n l\n <\n m\n \n \n \n \n |\n \n \n e\n \n i\n ⁢\n \n u\n l\n \n \n \n -\n \n e\n \n i\n ⁢\n \n u\n m\n \n \n \n \n |\n \n 2\n \n \n \n =\n \n \n 1\n \n \n (\n \n 2\n ⁢\n π\n \n )\n \n k\n \n \n ⁢\n \n \n |\n \n V\n ⁢\n \n (\n \n u\n →\n \n )\n \n \n |\n \n 2\n \n \n \n .\n \n \n \n \n", "2211.13834_19": "\n \n \n \n \n \n \n \n \n ℛ\n κ\n \n |\n \n \n \n 𝒩\n =\n \n 2\n *\n \n \n ,\n \n \n m\n f\n \n =\n 0\n \n \n \n ∝\n \n ∓\n \n \n 1\n \n \n T\n -\n \n T\n \n c\n ⁢\n r\n ⁢\n i\n ⁢\n t\n \n \n \n \n \n \n \n \n .\n \n (\n 1.20\n )\n \n \n \n \n \n", "2301.04890_99": "", "2301.06023_104": "\n \n \n \n \n \n \n w\n ⁢\n \n (\n 0\n ,\n t\n )\n \n \n =\n \n \n ℓ\n 01\n \n ⁢\n \n (\n t\n )\n \n \n \n ,\n \n \n \n w\n ⁢\n \n (\n τ\n ,\n 0\n )\n \n \n ∈\n \n R\n 0\n \n \n ,\n \n \n w\n ⁢\n \n (\n τ\n ,\n 1\n )\n \n \n ∈\n \n \n \n R\n 1\n \n \n ⁢\n \n \n lim\n \n τ\n →\n ∞\n \n \n ⁡\n \n w\n ⁢\n \n (\n τ\n ,\n t\n )\n \n \n \n \n =\n \n γ\n ⁢\n \n (\n t\n )\n \n \n \n \n \n \n \n \n", "2301.06524_105": "\n \n \n \n \n \n \n (\n 26\n )\n \n -\n h\n \n ≤\n \n \n \n E\n \n \n (\n \n z\n \n α\n ,\n h\n \n \n )\n \n ,\n δ\n \n \n ⁢\n \n (\n u\n ,\n \n ψ\n α\n \n ,\n \n x\n α\n \n ,\n \n t\n α\n \n )\n \n \n -\n \n \n E\n \n \n (\n \n z\n \n α\n ,\n h\n \n \n )\n \n ,\n δ\n \n \n ⁢\n \n (\n v\n ,\n \n φ\n α\n \n ,\n \n y\n α\n \n ,\n \n r\n α\n \n )\n \n \n \n \n \n \n \n", "2301.12080_28": "\n \n \n \n \n \n \n ∥\n \n A\n ⁢\n R\n ⁢\n \n (\n μ\n ,\n A\n )\n \n \n ∥\n \n ≤\n \n K\n μ\n \n \n .\n \n \n \n \n", "2308.13344_109": "\n \n \n \n \n \n \n \n \n B\n ~\n \n ε\n \n ⁢\n \n (\n z\n )\n \n ⁢\n f\n ⁢\n \n (\n t\n )\n \n ⁢\n \n (\n \n x\n Σ\n \n )\n \n \n =\n \n \n ∫\n \n -\n 1\n \n 1\n \n \n \n ∫\n Σ\n \n \n \n G\n z\n \n ⁢\n \n (\n \n \n \n x\n Σ\n \n +\n \n ε\n ⁢\n t\n ⁢\n ν\n ⁢\n \n (\n \n x\n Σ\n \n )\n \n \n \n -\n \n y\n Σ\n \n -\n \n ε\n ⁢\n s\n ⁢\n ν\n ⁢\n \n (\n \n y\n Σ\n \n )\n \n \n \n )\n \n ⁢\n f\n ⁢\n \n (\n s\n )\n \n ⁢\n \n (\n \n y\n Σ\n \n )\n \n ⁢\n \n 𝑑\n σ\n \n ⁢\n \n (\n \n y\n Σ\n \n )\n \n ⁢\n \n 𝑑\n s\n \n \n \n \n \n .\n \n (\n 3.18\n )\n \n \n \n \n \n", "2308.14989_106": "\n \n \n \n \n \n \n R\n ¯\n \n ≡\n \n (\n \n \n R\n ¯\n \n 1\n \n ,\n \n \n R\n ^\n \n \n -\n 1\n \n \n )\n \n \n .\n \n \n \n \n", "2308.16340_27": "\n \n \n \n \n \n \n \n \n ∑\n 1\n 3\n \n \n pdist\n ⁢\n \n (\n \n a\n i\n \n ,\n D\n )\n \n \n \n -\n \n \n pper\n D\n \n ⁢\n \n (\n \n \n a\n 1\n \n ⁢\n \n a\n 2\n \n ⁢\n \n a\n 3\n \n \n )\n \n \n \n ≤\n \n \n 3\n ⁢\n d\n ⁢\n i\n ⁢\n a\n ⁢\n m\n ⁢\n \n (\n D\n )\n \n \n -\n \n per\n ⁢\n \n (\n \n \n a\n 1\n \n ⁢\n \n a\n 2\n \n ⁢\n \n a\n 3\n \n \n )\n \n \n \n ≤\n \n diam\n ⁢\n \n (\n D\n )\n \n \n \n .\n \n \n \n \n", "2309.01596_19": "\n \n \n \n \n \n 𝐁\n i\n \n =\n \n \n -\n \n \n (\n \n \n ∑\n j\n \n \n \n 𝐫\n \n i\n ⁢\n j\n \n \n ⊗\n \n ∇\n ⁡\n \n \n W\n \n i\n ⁢\n j\n \n \n ⁢\n \n w\n j\n \n \n \n \n \n )\n \n \n -\n 1\n \n \n \n ,\n \n (\n 20\n )\n \n \n \n \n \n \n", "2309.02015_104": "\n \n \n \n \n \n \n Tr\n ⁡\n Q\n \n =\n \n Tr\n ⁡\n \n (\n \n tr\n ⁡\n Q\n \n )\n \n \n \n .\n \n \n \n \n", "2309.03597_30": "\n \n \n \n \n \n \n \n \n supp\n ⁡\n ϕ\n \n ⁢\n \n (\n t\n ,\n ⋅\n )\n \n \n ,\n \n \n supp\n ⁡\n a\n \n ⁢\n \n (\n t\n ,\n ⋅\n )\n \n \n \n ⊂\n K\n \n .\n \n \n \n \n", "2309.06054_1": "", "2309.06446_83": "\n \n \n \n \n \n |\n \n ∂\n ⁡\n \n Ω\n 0\n \n \n |\n \n \n |\n \n Ω\n 0\n \n |\n \n \n \n =\n \n \n \n \n \n d\n ⁢\n \n λ\n 0\n \n \n \n d\n ⁢\n α\n \n \n \n |\n \n \n α\n =\n 0\n \n \n =\n \n \n lim\n \n α\n →\n 0\n \n \n ⁡\n \n \n \n \n \n λ\n 0\n \n ⁢\n \n (\n α\n )\n \n \n -\n \n \n λ\n 0\n \n ⁢\n \n (\n 0\n )\n \n \n \n α\n \n \n \n =\n \n \n lim\n \n α\n →\n 0\n \n \n ⁡\n \n \n \n \n \n ||\n \n ∇\n ⁡\n \n ψ\n 0\n \n \n ||\n \n \n \n L\n 2\n \n ⁢\n \n (\n \n Ω\n 0\n \n )\n \n \n 2\n \n +\n \n α\n ⁢\n \n \n ||\n \n ψ\n 0\n \n ||\n \n \n \n L\n 2\n \n ⁢\n \n (\n \n ∂\n ⁡\n \n Ω\n 0\n \n \n )\n \n \n 2\n \n \n \n α\n \n \n \n \n .\n \n", "2309.09317_7": "\n \n \n \n \n \n \n 𝐣\n ⁢\n \n (\n t\n )\n \n \n =\n \n \n d\n ⁢\n 𝐚\n ⁢\n \n (\n t\n )\n \n \n \n d\n ⁢\n t\n \n \n =\n \n \n \n d\n 2\n \n ⁢\n 𝐯\n ⁢\n \n (\n t\n )\n \n \n \n d\n ⁢\n \n t\n 2\n \n \n \n =\n \n \n \n d\n 3\n \n ⁢\n 𝐱\n ⁢\n \n (\n t\n )\n \n \n \n d\n ⁢\n \n t\n 3\n \n \n \n \n .\n \n (\n 8\n )\n \n \n \n \n \n", "2309.10343_72": "", "2309.10545_18": "\n \n \n \n \n \n \n \n ∑\n \n i\n =\n 1\n \n N\n \n \n \n f\n i\n \n ⁢\n \n (\n \n x\n 1\n \n ,\n ⋯\n ,\n \n x\n k\n \n )\n \n ⁢\n \n \n ∂\n \n x\n i\n \n \n \n \n \n ⟼\n \n \n ∑\n \n i\n =\n 1\n \n k\n \n \n \n f\n i\n \n ⁢\n \n (\n \n x\n 1\n \n ,\n ⋯\n ,\n \n x\n k\n \n )\n \n ⁢\n \n \n ∂\n \n x\n i\n \n \n \n \n \n \n ,\n \n \n \n \n", "2309.11114_29": "\n \n \n \n \n \n \n ρ\n pseudo\n \n ⁢\n \n (\n ω\n )\n \n \n =\n \n \n \n ∑\n \n i\n =\n 1\n \n 2\n \n \n \n A\n i\n \n ⁢\n \n exp\n ⁡\n \n [\n \n \n \n (\n \n ω\n -\n \n M\n i\n \n \n )\n \n 2\n \n \n Γ\n i\n \n \n ]\n \n \n \n \n ,\n \n (\n 30\n )\n \n \n \n \n \n \n", "2309.12116_40": "\n \n \n \n \n \n \n \n (\n 5.7\n )\n \n ⁢\n \n ⟨\n \n ⟨\n \n \n ⁣\n ∗\n x\n \n ⋅\n b\n \n ,\n y\n ⟩\n \n ⟩\n \n ⁢\n \n (\n h\n )\n \n \n ∈\n \n \n \n K\n ⁢\n r\n ⁢\n ⁢\n ⁢\n \n (\n h\n )\n \n \n ⋅\n B\n \n ⁢\n h\n \n =\n \n K\n ⁢\n h\n \n \n .\n \n \n \n \n", "2309.12256_1": "\n \n \n \n \n \n \n \n \n ℋ\n CG\n \n ⁢\n \n {\n \n S\n i\n \n }\n \n \n =\n \n \n 1\n 8\n \n ⁢\n \n \n ∑\n \n i\n ≠\n j\n \n \n \n \n \n e\n 2\n \n \n κ\n ⁢\n \n |\n \n \n \n r\n i\n \n →\n \n -\n \n \n r\n j\n \n →\n \n \n |\n \n \n \n ⁢\n \n (\n \n \n \n S\n i\n \n +\n 1\n \n -\n \n 2\n ⁢\n K\n \n \n )\n \n ⁢\n \n (\n \n \n \n S\n j\n \n +\n 1\n \n -\n \n 2\n ⁢\n K\n \n \n )\n \n \n \n \n \n ,\n \n \n S\n i\n \n =\n \n ±\n 1\n \n \n \n .\n \n (\n 2\n )\n \n \n \n \n \n", "2309.13454_14": "", "2309.13669_0": "\n \n \n \n \n I\n \n G\n r\n \n =\n \n {\n \n {\n \n \n \n \n \n \n \n N\n \n A\n ,\n u\n ,\n r\n \n \n \n N\n \n A\n ,\n r\n \n \n \n \n ,\n \n \n \n \n \n \n if\n ⁢\n \n N\n \n A\n ,\n r\n \n \n \n >\n \n 0\n ⁢\n \n [\n \n 5\n ⁢\n p\n ⁢\n t\n \n ]\n \n ⁢\n 0\n \n \n ,\n \n \n \n otherwise\n \n \n \n \n \n (\n 1\n )\n \n \n \n \n \n \n", "2309.15026_4": "\n \n \n \n \n \n \n 𝖣𝖳\n ⁢\n \n (\n f\n )\n \n \n :=\n \n \n \n min\n \n 𝒯\n :\n \n 𝒯\n ⁢\n  i s a D T c o m p u t i n g \n ⁢\n f\n \n \n \n ⁡\n  d e p t h \n \n ⁢\n \n (\n 𝒯\n )\n \n \n \n .\n \n \n \n \n", "2310.00778_27": "", "2310.00778_37": "\n \n \n \n \n \n \n \n \n \n \n d\n 2\n \n \n d\n ⁢\n \n t\n 2\n \n \n \n |\n \n \n t\n =\n 0\n \n \n ⁢\n \n \n ∑\n \n i\n =\n 1\n \n 2\n \n \n ℰ\n ⁢\n \n (\n U\n ,\n \n h\n i\n t\n \n )\n \n \n \n \n ≥\n 0\n \n .\n \n \n \n \n", "2310.02237_12": "", "2310.04214_37": "\n \n \n \n \n \n \n ∂\n α\n \n ⁡\n \n (\n \n f\n ⁢\n g\n \n )\n \n \n =\n \n \n \n \n α\n !\n \n \n \n \n α\n 1\n \n !\n \n ⁢\n \n \n α\n 2\n \n !\n \n \n \n ⁢\n \n (\n \n \n ∂\n \n α\n 1\n \n \n ⁡\n f\n \n )\n \n ⁢\n \n (\n \n \n ∂\n \n α\n 2\n \n \n ⁡\n g\n \n )\n \n \n ,\n \n (\n 3.5\n )\n \n \n \n \n \n \n", "2310.06688_36": "", "2310.07486_14": "\n \n \n \n \n \n \n Q\n 2\n \n ⁢\n \n (\n \n τ\n ≫\n \n \n η\n e\n \n ,\n \n η\n o\n \n \n \n )\n \n \n =\n \n \n \n Q\n \n 2\n ,\n e\n \n \n +\n \n \n [\n \n \n Q\n \n 2\n ,\n o\n \n \n -\n \n Q\n \n 2\n ,\n e\n \n \n \n ]\n \n ⁢\n \n \n \n ζ\n ⁢\n ϵ\n \n \n 1\n +\n \n ζ\n ⁢\n ϵ\n \n \n \n \n \n \n ,\n \n (\n 14\n )\n \n \n \n \n \n \n", "2310.07993_3": "\n \n \n \n \n \n ⟨\n \n ψ\n ⁢\n \n [\n \n C\n p\n \n ]\n \n ⁢\n …\n \n ⟩\n \n ,\n \n (\n 48\n )\n \n \n \n \n \n", "2310.09974_50": "", "2310.09980_72": "\n \n \n \n \n \n \n α\n =\n \n \n α\n 1\n \n +\n \n α\n 2\n \n +\n ⋯\n +\n \n α\n j\n \n \n \n ,\n \n \n α\n i\n \n ∈\n \n 𝒪\n K\n +\n \n \n \n ,\n \n \n \n \n", "2310.10873_5": "\n \n \n \n \n \n \n 𝒮\n i\n \n -\n \n 𝒮\n j\n \n \n =\n \n \n {\n \n 𝐚\n 1\n \n ,\n …\n ,\n \n 𝐚\n r\n \n }\n \n ⁢\n \n (\n 6\n )\n \n \n \n \n \n \n", "2310.13601_86": "", "2310.14073_26": "\n \n \n \n \n \n \n \n \n \n e\n ˙\n \n \n (\n t\n )\n \n =\n \n A\n K\n \n χ\n \n (\n t\n )\n \n +\n K\n y\n \n (\n t\n )\n \n +\n \n A\n K\n \n P\n \n (\n t\n )\n \n +\n ϕ\n \n (\n y\n ,\n u\n )\n \n +\n +\n \n [\n \n A\n K\n \n Ω\n \n (\n t\n )\n \n +\n G\n \n (\n y\n ,\n u\n )\n \n ]\n \n θ\n -\n A\n x\n \n (\n t\n )\n \n -\n ϕ\n \n (\n y\n ,\n u\n )\n \n -\n -\n G\n \n (\n y\n ,\n u\n )\n \n θ\n =\n \n A\n K\n \n \n (\n χ\n \n (\n t\n )\n \n +\n P\n \n (\n t\n )\n \n +\n Ω\n \n (\n t\n )\n \n \n θ\n \n -\n x\n \n (\n t\n )\n \n )\n \n +\n +\n K\n δ\n \n (\n t\n )\n \n =\n \n A\n K\n \n e\n \n (\n t\n )\n \n +\n K\n δ\n \n (\n t\n )\n \n ,\n e\n \n (\n \n t\n 0\n \n )\n \n =\n \n χ\n 0\n \n -\n \n x\n 0\n \n .\n \n \n \n \n ⁢\n \n (\n 17\n )\n \n \n \n \n \n", "2310.14132_202": "\n \n \n \n \n \n \n \n (\n 7.2\n )\n \n ⁢\n \n P\n \n -\n 1\n \n \n \n =\n \n [\n \n \n \n A\n \n \n \n B\n ⁢\n \n B\n *\n \n \n \n \n \n D\n ′\n \n \n \n \n ]\n \n \n .\n \n \n \n \n", "2310.18441_73": "\n \n \n \n \n \n \n S\n \n \n [\n \n \n \n μ\n 1\n \n ⁢\n \n μ\n 2\n \n ⁢\n …\n ⁢\n \n \n μ\n \n n\n +\n 1\n \n \n \n \n |\n ν\n \n ]\n \n ⁢\n ρ\n \n \n =\n 0\n \n ,\n \n \n S\n \n σ\n ⁢\n \n [\n \n \n \n μ\n 1\n \n ⁢\n \n μ\n 2\n \n ⁢\n …\n ⁢\n \n \n μ\n n\n \n \n \n |\n \n ν\n ⁢\n ρ\n \n \n ]\n \n \n \n =\n \n 0\n ⁢\n \n (\n 66\n )\n \n \n \n \n \n \n \n", "2311.00325_7": "\n \n \n \n \n \n \n 𝐆\n t\n \n =\n \n [\n \n 𝐠\n \n t\n ,\n 0\n \n \n ,\n …\n ,\n \n 𝐠\n \n t\n ,\n \n K\n -\n 1\n \n \n \n ]\n \n \n .\n \n (\n 4\n )\n \n \n \n \n \n", "2311.00351_75": "\n \n \n \n \n \n \n G\n 20\n \n =\n \n \n \n P\n 2\n \n \n m\n 0\n 2\n \n \n ⁢\n \n (\n \n 3\n -\n \n c\n v\n \n \n )\n \n \n \n ,\n \n \n G\n 30\n \n =\n \n \n \n \n 3\n ⁢\n T\n ⁢\n \n P\n 2\n \n \n \n m\n 0\n 2\n \n \n ⁢\n \n (\n \n \n e\n P\n \n -\n \n c\n v\n \n \n )\n \n \n ,\n \n (\n 69\n )\n \n \n \n \n \n \n \n", "2311.01215_39": "", "2311.01568_13": "", "2311.02388_2": "\n \n \n \n \n \n \n \n x\n +\n n\n \n -\n \n 2\n ⁢\n x\n \n \n +\n \n \n ∑\n \n i\n =\n 1\n \n n\n \n \n t\n i\n \n \n \n =\n \n 2\n -\n \n 2\n ⁢\n j\n \n \n ⟹\n x\n =\n \n \n (\n \n n\n -\n 2\n \n )\n \n +\n \n 2\n ⁢\n j\n \n +\n \n \n ∑\n \n i\n =\n 1\n \n n\n \n \n t\n i\n \n \n \n \n \n \n \n", "2311.03643_2": "", "2311.07173_51": "", "2311.09962_0": "\n \n \n \n \n \n \n ℒ\n \n c\n ⁢\n o\n ⁢\n n\n ⁢\n t\n ⁢\n r\n ⁢\n a\n ⁢\n s\n ⁢\n t\n ⁢\n i\n ⁢\n v\n ⁢\n e\n \n \n =\n \n -\n \n \n ∑\n \n i\n =\n 1\n \n N\n \n \n log\n ⁡\n \n \n exp\n ⁡\n \n (\n \n \n s\n ⁢\n i\n ⁢\n m\n ⁢\n \n (\n \n 𝐳\n i\n \n ,\n \n \n 𝐳\n ~\n \n i\n \n )\n \n \n /\n τ\n \n )\n \n \n \n \n ∑\n \n j\n =\n 1\n \n N\n \n \n \n (\n \n 1\n -\n \n δ\n \n j\n ,\n k\n \n \n \n )\n \n ⁢\n \n [\n \n \n exp\n ⁡\n \n (\n \n \n s\n ⁢\n i\n ⁢\n m\n ⁢\n \n (\n \n 𝐳\n j\n \n ,\n \n 𝐳\n k\n \n )\n \n \n /\n τ\n \n )\n \n \n +\n \n exp\n ⁡\n \n (\n \n \n s\n ⁢\n i\n ⁢\n m\n ⁢\n \n (\n \n 𝐳\n j\n \n ,\n \n \n 𝐳\n ~\n \n k\n \n )\n \n \n /\n τ\n \n )\n \n \n \n ]\n \n \n \n \n \n \n \n \n .\n \n (\n 1\n )\n \n \n \n \n \n", "2311.11286_10": "\n \n \n \n \n D\n =\n \n \n [\n \n \n max\n \n x\n ,\n y\n \n \n ⁡\n \n {\n \n \n \n r\n ^\n \n \n C\n \n f\n 1\n \n \n \n ⁢\n \n (\n x\n ,\n y\n )\n \n \n }\n \n \n ,\n …\n ,\n \n \n max\n \n x\n ,\n y\n \n \n ⁡\n \n {\n \n \n \n r\n ^\n \n \n C\n \n f\n λ\n \n \n \n ⁢\n \n (\n x\n ,\n y\n )\n \n \n }\n \n \n ]\n \n ⁢\n \n (\n 11\n )\n \n \n \n \n \n \n", "2311.11568_158": "\n \n \n \n \n \n \n φ\n \n n\n ,\n j\n \n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n e\n \n i\n ⁢\n 2\n ⁢\n π\n ⁢\n n\n ⁢\n x\n \n \n +\n \n \n \n A\n m\n ∗\n \n ⁢\n \n (\n \n E\n \n n\n ,\n j\n ,\n m\n \n \n )\n \n \n +\n \n \n \n \n \n \n", "2311.14975_6": "", "2311.15461_8": "\n \n \n \n \n \n \n 3\n \n 2\n ⁢\n \n K\n 2\n \n \n \n =\n \n \n w\n \n 1\n -\n n\n \n \n +\n \n \n w\n ¯\n \n \n 1\n -\n n\n \n \n +\n \n b\n ⁢\n \n \n ln\n \n ⁡\n \n \n |\n w\n |\n \n 2\n \n \n \n +\n Const\n \n \n .\n \n \n \n \n", "2311.18235_13": "\n \n \n \n \n \n \n T\n \n S\n 0\n 2\n \n \n =\n \n \n ∑\n α\n \n \n \n \n S\n α\n \n ⁢\n T\n \n ⊗\n \n S\n α\n \n \n \n \n .\n \n \n \n \n", "2312.00800_123": "\n \n \n \n \n \n \n \n 1\n \n c\n d\n \n \n ⁢\n \n (\n \n -\n Δ\n \n )\n \n ⁢\n \n φ\n μ\n \n \n =\n \n μ\n \n \n .\n \n \n \n \n", "2312.01048_126": "\n \n \n \n \n \n \n (\n \n \n δ\n \n i\n m\n \n \n \n δ\n \n \n i\n m\n \n -\n 1\n \n \n \n )\n \n ⁢\n \n (\n \n \n δ\n \n \n i\n m\n \n -\n 1\n \n \n \n δ\n \n \n i\n m\n \n -\n 2\n \n \n \n )\n \n ⁢\n ⋯\n ⁢\n \n (\n \n \n δ\n \n \n i\n \n m\n -\n 1\n \n \n +\n 1\n \n \n \n δ\n \n i\n \n m\n -\n 1\n \n \n \n \n )\n \n \n ,\n \n \n \n \n", "2312.01591_64": "", "2312.02354_3": "\n \n \n \n \n \n \n \n f\n i\n \n ⁢\n \n (\n p\n ,\n x\n )\n \n \n =\n \n \n \n \n f\n i\n eq\n \n ⁢\n \n (\n p\n )\n \n \n +\n \n δ\n ⁢\n \n f\n i\n \n ⁢\n \n (\n p\n ,\n x\n )\n \n \n \n   \n with\n \n \n   \n \n \n \n f\n i\n eq\n \n ⁢\n \n (\n p\n )\n \n \n =\n \n \n 1\n \n \n e\n \n β\n ⁢\n \n p\n μ\n \n ⁢\n \n u\n μ\n \n \n \n ∓\n 1\n \n \n ,\n \n (\n 4\n )\n \n \n \n \n \n \n \n", "2312.02455_40": "\n \n \n \n \n \n \n \n \n h\n 1\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n lim\n \n n\n →\n ∞\n \n \n ⁡\n \n \n ∫\n \n D\n ∖\n V\n \n \n \n \n G\n \n D\n ∩\n U\n \n \n ⁢\n \n (\n x\n ,\n z\n )\n \n ⁢\n \n μ\n n\n \n ⁢\n \n (\n \n d\n ⁢\n z\n \n )\n \n \n \n \n \n ,\n \n x\n ∈\n \n D\n ∩\n V\n \n \n \n .\n \n (\n 2.8\n )\n \n \n \n \n \n", "2312.03337_57": "\n \n \n \n \n \n \n ∥\n \n \n u\n \n k\n n\n \n \n δ\n n\n \n \n -\n \n u\n †\n \n \n ∥\n \n ≤\n \n \n \n ∏\n \n j\n =\n k\n \n \n \n k\n n\n \n -\n 1\n \n \n \n \n (\n \n 1\n -\n \n λ\n j\n \n \n )\n \n ⁢\n \n ∥\n \n \n u\n k\n \n δ\n n\n \n \n -\n \n u\n †\n \n \n ∥\n \n \n \n +\n \n 4\n ⁢\n ρ\n ⁢\n \n \n 1\n +\n \n \n L\n 2\n \n \n κ\n 2\n \n \n \n \n ⁢\n \n (\n \n 1\n -\n \n \n ∏\n \n j\n =\n \n \n k\n n\n \n -\n 1\n -\n k\n \n \n \n \n k\n n\n \n -\n 1\n \n \n \n (\n \n 1\n -\n \n λ\n j\n \n \n )\n \n \n \n )\n \n \n \n \n .\n \n (\n 2.21\n )\n \n \n \n \n \n", "2312.06636_241": "\n \n \n \n \n \n (\n 8.10\n )\n \n ⁢\n \n \n \n \n \n χ\n ⁢\n \n (\n \n h\n 1\n \n ,\n \n h\n 2\n \n )\n \n \n \n ∼\n \n \n O\n \n d\n ,\n ϵ\n \n \n ⁢\n \n (\n 1\n )\n \n \n \n \n \n χ\n ⁢\n \n (\n \n h\n 2\n \n ,\n \n h\n 1\n \n )\n \n \n mod\n \n \n Ξ\n p\n 2\n \n ⁢\n \n (\n \n \n L\n \n -\n 1\n \n \n ⁢\n \n (\n \n Λ\n 4\n \n )\n \n \n )\n \n \n \n \n \n \n \n \n \n \n \n", "2312.07032_27": "", "2312.11787_0": "\n \n \n \n \n \n \n \n \n (\n 0.1\n )\n \n ⁢\n \n \n ∥\n \n \n \n (\n \n -\n Δ\n \n )\n \n \n s\n /\n 2\n \n \n ⁢\n U\n \n ∥\n \n 2\n 2\n \n \n -\n \n \n 𝒮\n \n s\n ,\n n\n \n \n ⁢\n \n \n ∥\n U\n ∥\n \n \n \n 2\n ⁢\n n\n \n \n n\n -\n \n 2\n ⁢\n s\n \n \n \n 2\n \n \n \n ≥\n \n \n C\n \n n\n ,\n s\n \n \n ⁢\n \n d\n 2\n \n ⁢\n \n (\n U\n ,\n \n ℳ\n s\n \n )\n \n \n \n ,\n \n \n \n \n", "2312.12281_83": "\n \n \n \n \n \n \n \n λ\n j\n \n ⁢\n \n (\n \n sup\n ⁡\n \n {\n \n \n g\n i\n \n ⁢\n \n (\n ω\n )\n \n \n ∣\n \n i\n ∈\n I\n \n }\n \n \n )\n \n \n =\n \n sup\n ⁡\n \n {\n \n \n λ\n j\n \n ⁢\n \n (\n \n g\n i\n \n )\n \n ⁢\n \n (\n ω\n )\n \n \n ∣\n \n i\n ∈\n I\n \n }\n \n \n \n .\n \n \n \n \n", "2312.12901_7": "\n \n \n \n \n \n λ\n int1\n ′\n \n =\n \n -\n \n ln\n ⁡\n \n (\n \n 1\n -\n \n p\n int1\n ′\n \n \n )\n \n \n \n =\n \n -\n \n ln\n ⁡\n \n (\n \n 1\n -\n 0.0521\n \n )\n \n \n \n =\n \n 0.0535\n ⁢\n \n (\n 3.6\n )\n \n \n \n \n \n \n", "2312.14258_11": "\n \n \n \n \n \n \n \n \n r\n ˙\n \n 2\n \n +\n \n \n \n (\n \n α\n +\n 1\n \n )\n \n ⁢\n \n (\n \n \n r\n 4\n \n -\n \n r\n h\n 4\n \n \n )\n \n ⁢\n \n L\n 2\n \n \n \n \n r\n 4\n \n ⁢\n \n (\n \n \n r\n 2\n \n -\n \n 2\n ⁢\n α\n ⁢\n \n r\n h\n 2\n \n \n \n )\n \n \n \n \n =\n \n E\n 2\n \n \n .\n \n (\n 13\n )\n \n \n \n \n \n", "2312.14449_36": "", "2312.17102_25": "\n \n \n \n \n \n \n \n G\n ±\n \n ≡\n \n \n {\n \n \n \n [\n \n G\n 0\n ±\n \n ]\n \n \n -\n 1\n \n \n -\n \n Σ\n ±\n \n \n }\n \n \n -\n 1\n \n \n \n ,\n \n \n \n Σ\n ±\n \n ≡\n \n \n Δ\n ∓\n \n ⁢\n \n G\n 0\n ∓\n \n ⁢\n \n Δ\n ±\n \n \n \n ,\n \n \n Ξ\n ±\n \n ≡\n \n -\n \n \n G\n ∓\n \n ⁢\n \n Δ\n ±\n \n ⁢\n \n G\n 0\n ±\n \n \n \n =\n \n -\n \n \n G\n 0\n ∓\n \n ⁢\n \n Δ\n ±\n \n ⁢\n \n G\n ±\n \n \n \n \n \n \n ,\n \n (\n \n A\n ⁢\n 7\n \n )\n \n \n \n \n \n", "2401.00270_25": "\n \n \n \n \n \n \n \n \n e\n γ\n \n π\n \n \n =\n \n 0.7529495060464205959354997575876048108386\n ⁢\n …\n \n \n ,\n \n \n \n \n", "2401.00648_4": "", "2401.01714_125": "", "2401.03247_99": "\n \n \n \n \n 𝒯\n ≈\n \n \n ∑\n n\n \n \n \n 𝐮\n n\n \n ⁢\n \n σ\n n\n \n ⁢\n \n 𝐯\n n\n †\n \n ⁢\n \n (\n 90\n )\n \n \n \n \n \n \n \n", "2401.03409_78": "", "2401.05601_1": "", "2401.05795_32": "", "2401.06322_2": "\n \n \n \n \n \n \n \n ρ\n L\n \n ⁢\n \n (\n z\n )\n \n \n =\n \n \n \n (\n \n \n ∏\n \n j\n =\n 0\n \n \n N\n -\n 1\n \n \n \n θ\n ⁢\n \n (\n \n \n ϕ\n j\n \n ⁢\n \n (\n z\n )\n \n \n )\n \n \n \n )\n \n \n 1\n /\n N\n \n \n ⋅\n \n \n ∏\n \n j\n =\n 0\n \n \n N\n -\n 2\n \n \n \n \n |\n \n \n ϕ\n ′\n \n ⁢\n \n (\n \n \n ϕ\n j\n \n ⁢\n \n (\n z\n )\n \n \n )\n \n \n |\n \n \n \n (\n \n N\n -\n 1\n -\n j\n \n )\n \n /\n N\n \n \n \n \n \n .\n \n \n \n \n", "2401.06621_5": "\n \n \n \n \n \n \n G\n ⁢\n \n (\n x\n ,\n y\n )\n \n \n ≤\n \n \n C\n \n \n |\n \n x\n -\n y\n \n |\n \n \n d\n -\n 2\n \n \n \n ⁢\n \n (\n \n 1\n ∧\n \n \n d\n x\n \n \n |\n \n x\n -\n y\n \n |\n \n \n \n )\n \n ⁢\n \n (\n \n 1\n ∧\n \n \n d\n y\n \n \n |\n \n x\n -\n y\n \n |\n \n \n \n )\n \n \n \n ,\n \n \n \n \n", "2401.06895_25": "\n \n \n \n \n \n d\n ⁢\n σ\n ⁢\n \n (\n \n \n a\n ⁢\n b\n \n →\n \n c\n ⁢\n κ\n \n \n )\n \n \n \n d\n ⁢\n t\n \n \n \n ∼\n \n \n 1\n \n t\n \n ′\n ⁣\n 2\n \n \n \n \n ∼\n \n \n \n A\n ′\n \n \n \n [\n \n 1\n +\n \n \n (\n \n \n m\n c\n 2\n \n +\n \n c\n T\n 2\n \n \n )\n \n /\n \n M\n \n T\n ⁢\n 0\n \n 2\n \n \n \n ]\n \n \n n\n /\n 2\n \n \n \n \n \n ,\n \n (\n 21\n )\n \n \n", "2401.08853_63": "\n \n \n \n \n \n \n \n M\n \n j\n ,\n \n e\n ⁢\n x\n \n \n 0\n \n =\n \n 2\n ⁢\n \n \n 1\n /\n \n β\n j\n \n \n \n \n \n ,\n \n j\n =\n \n \n a\n ⁢\n i\n \n ,\n \n a\n ⁢\n o\n \n ,\n v\n ,\n \n c\n ⁢\n i\n \n ,\n \n c\n ⁢\n o\n \n \n \n \n .\n \n \n \n \n", "2401.09287_42": "\n \n \n \n \n \n \n 𝒞\n 2\n \n ≻\n \n 𝒞\n 1\n \n ≻\n \n 𝒞\n 5\n \n ≻\n \n 𝒞\n 6\n \n ≻\n \n 𝒞\n 3\n \n ≻\n \n 𝒞\n 4\n \n \n .\n \n \n \n \n", "2401.09702_75": "\n \n \n \n \n lim sup\n \n n\n →\n \n +\n ∞\n \n \n \n ⁡\n \n \n \n 1\n n\n \n \n ⁢\n \n log\n ⁡\n \n ∥\n \n \n \n ∂\n x\n \n ⁡\n \n Φ\n \n 0\n ,\n n\n \n \n δ\n x\n \n \n \n ⁢\n \n (\n ω\n ,\n y\n )\n \n \n ∥\n \n \n \n \n ≤\n \n C\n ′\n \n \n .\n \n", "2401.10711_1": "", "2401.12166_9": "\n \n \n \n \n \n χ\n \n H\n -\n \n H\n ⁢\n e\n \n \n 2\n \n =\n \n ∑\n \n i\n =\n 1\n \n 2\n \n \n C\n P\n \n |\n \n \n \n P\n i\n *\n \n -\n \n P\n i\n \n \n \n P\n i\n \n \n |\n +\n \n C\n T\n \n |\n \n \n \n T\n i\n *\n \n -\n \n T\n i\n \n \n \n T\n i\n \n \n |\n  \n ,\n \n (\n 10\n )\n \n \n \n \n \n", "2401.14028_32": "\n \n \n \n \n \n \n \n Ω\n  a o n \n \n ⁢\n \n (\n 𝐩\n )\n \n \n =\n \n \n ∑\n \n s\n ≥\n 2\n \n \n \n (\n \n \n \n ω\n \n s\n ⁢\n 1\n \n \n ⁢\n 1\n ⁢\n \n {\n \n 𝐩\n =\n \n (\n s\n )\n \n \n }\n \n \n +\n \n \n ω\n \n s\n ⁢\n 0\n \n \n ⁢\n 1\n ⁢\n \n {\n \n \n \n \n ∥\n 𝐩\n ∥\n \n 0\n \n ≥\n 2\n \n ;\n \n \n \n ∥\n 𝐩\n ∥\n \n 1\n \n =\n s\n \n \n }\n \n \n \n )\n \n \n \n .\n \n \n \n \n", "2401.15137_8": "\n \n \n \n \n \n \n \n \n |\n \n B\n 1\n \n |\n \n 2\n \n =\n \n w\n \n \n 45\n ∘\n \n ,\n \n 45\n ∘\n \n \n \n \n ,\n \n \n \n |\n \n B\n 2\n \n |\n \n 2\n \n =\n \n w\n \n \n 45\n ∘\n \n ,\n \n 135\n ∘\n \n \n \n t\n ⁢\n o\n ⁢\n t\n \n \n =\n \n 2\n ⁢\n \n w\n \n \n 45\n ∘\n \n ,\n \n 135\n ∘\n \n \n \n \n =\n \n 2\n ⁢\n \n w\n \n \n 135\n ∘\n \n ,\n \n 45\n ∘\n \n \n \n \n \n \n ,\n \n \n and\n  \n \n \n |\n \n B\n 3\n \n |\n \n 2\n \n \n =\n \n w\n \n \n 135\n ∘\n \n ,\n \n 135\n ∘\n \n \n \n \n .\n \n (\n 9\n )\n \n \n \n \n \n", "2401.16930_41": "\n \n \n \n \n \n \n \n g\n i\n \n ⁢\n \n (\n N\n ,\n v\n ,\n P\n )\n \n \n =\n \n \n g\n j\n \n ⁢\n \n (\n N\n ,\n v\n ,\n P\n )\n \n \n \n .\n \n \n \n \n", "2401.17473_9": "", "2402.00062_160": "\n \n \n \n \n \n \n (\n \n \n P\n k\n \n (\n 0\n )\n \n \n ⁢\n \n (\n σ\n )\n \n ⁢\n f\n \n )\n \n ⁢\n \n (\n x\n )\n \n \n =\n \n 0\n ,\n \n (\n 5.10\n )\n \n \n \n \n \n \n", "2402.00062_190": "\n \n \n \n \n \n \n \n f\n \n \n (\n \n i\n ⁢\n n\n \n )\n \n ,\n σ\n \n \n ⁢\n \n (\n x\n )\n \n \n =\n \n \n \n f\n \n -\n ,\n σ\n \n \n ⁢\n \n (\n x\n )\n \n \n +\n \n \n ∑\n \n n\n =\n 1\n \n ∞\n \n \n \n L\n \n n\n ,\n σ\n \n \n ⁢\n \n (\n x\n )\n \n \n \n \n \n ,\n \n \n \n \n", "2402.00505_15": "\n \n \n \n \n \n \n ln\n ⁡\n ℒ\n \n ⁢\n \n (\n \n \n DM\n \n o\n ⁢\n b\n ⁢\n s\n \n \n |\n Θ\n \n )\n \n \n ∝\n \n \n -\n \n \n 1\n 2\n \n ⁢\n \n \n ∑\n i\n \n N\n frb\n \n \n \n [\n \n \n \n (\n \n \n \n DM\n \n model\n ,\n i\n \n \n ⁢\n \n (\n Θ\n )\n \n \n -\n \n DM\n \n obs\n ,\n i\n \n \n \n )\n \n 2\n \n \n σ\n i\n 2\n \n \n ]\n \n \n \n \n ,\n \n (\n 15\n )\n \n \n \n \n \n \n", "2402.01455_119": "\n \n \n \n \n \n O\n ⁢\n \n (\n \n \n \n (\n \n ℓ\n ⁢\n X\n ⁢\n T\n \n )\n \n ϵ\n \n ⋅\n \n (\n \n \n \n T\n \n 3\n 2\n \n \n ⁢\n X\n \n +\n \n \n ℓ\n \n 1\n 4\n \n \n ⁢\n T\n ⁢\n X\n \n +\n \n T\n 2\n \n +\n \n \n ℓ\n \n 1\n 4\n \n \n ⁢\n \n T\n \n 3\n 2\n \n \n \n \n )\n \n \n )\n \n \n ,\n \n \n \n \n", "2402.02118_15": "\n \n \n \n \n ∑\n σ\n \n (\n i\n v\n ↑\n ↓\n d\n σ\n †\n f\n L\n ⁢\n σ\n ¯\n +\n i\n v\n ↑\n ↓\n f\n R\n ⁢\n σ\n †\n d\n σ\n +\n H\n .\n c\n .\n )\n \n +\n ∑\n σ\n \n (\n t\n s\n c\n f\n L\n ⁢\n σ\n †\n f\n R\n ⁢\n σ\n +\n H\n .\n c\n .\n )\n \n .\n \n (\n 16\n )\n \n \n \n \n \n", "astro-ph_0407175_21": "\n \n \n \n \n \n b\n =\n \n \n \n b\n 0\n \n \n ⁢\n \n (\n \n 1\n -\n \n \n \n 31\n 12\n \n \n ⁢\n \n \n c\n 2\n \n \n a\n 0\n 2\n \n \n \n \n )\n \n \n \n ,\n \n \n \n _\n ¯\n \n ⁢\n 0\n \n =\n \n \n \n \n 2\n 3\n \n \n ⁢\n \n a\n 0\n \n \n ,\n \n (\n 22\n )\n \n \n \n \n \n \n \n", "astro-ph_0410682_8": "\n \n \n \n \n \n μ\n e\n \n =\n \n \n 0.248\n ⁢\n \n \n m\n s\n 2\n \n μ\n \n \n -\n \n 0.007\n ⁢\n \n \n m\n s\n 4\n \n \n μ\n 3\n \n \n \n -\n \n 0.034\n ⁢\n \n \n m\n s\n 6\n \n \n μ\n 5\n \n \n ⁢\n \n (\n 9\n )\n \n \n \n \n \n \n \n", "astro-ph_0412408_2": "\n \n \n \n \n U\n =\n \n \n (\n \n \n \n \n cos\n ⁡\n θ\n \n \n \n \n \n sin\n ⁡\n θ\n \n -\n \n sin\n ⁡\n θ\n \n \n \n \n \n cos\n ⁡\n θ\n \n \n \n \n )\n \n ,\n \n (\n 2\n )\n \n \n \n \n \n \n", "astro-ph_0510492_2": "\n \n \n \n \n \n \n \n ρ\n q\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n \n \n 1\n 2\n \n ⁢\n \n \n ϕ\n ˙\n \n 2\n \n \n +\n V\n \n \n ;\n \n \n \n \n p\n q\n \n ⁢\n \n (\n t\n )\n \n \n =\n \n \n \n 1\n 2\n \n ⁢\n \n \n ϕ\n ˙\n \n 2\n \n \n -\n V\n \n \n ;\n \n \n w\n q\n \n =\n \n \n \n 1\n -\n \n (\n \n \n 2\n ⁢\n V\n \n /\n \n \n ϕ\n ˙\n \n 2\n \n \n )\n \n \n \n 1\n +\n \n (\n \n \n 2\n ⁢\n V\n \n /\n \n \n ϕ\n ˙\n \n 2\n \n \n )\n \n \n \n ⁢\n \n (\n 3\n )\n \n \n \n \n \n \n \n \n", "astro-ph_0608064_14": "\n \n \n \n \n \n \n \n ℒ\n i\n \n ⁢\n \n (\n \n \n v\n ⁢\n e\n ⁢\n c\n ⁢\n d\n \n |\n \n π\n i\n \n \n )\n \n \n =\n \n log\n ⁡\n \n {\n \n \n [\n \n \n ∏\n \n j\n ≠\n i\n \n \n \n ∫\n \n d\n \n π\n j\n \n \n \n \n ]\n \n ⁢\n \n exp\n ⁡\n \n [\n \n ℒ\n ⁢\n \n (\n \n \n 𝓋\n ⁢\n ℯ\n ⁢\n 𝒸\n ⁢\n 𝒹\n \n |\n \n 𝓋\n ⁢\n ℯ\n ⁢\n 𝒸\n ⁢\n π\n \n \n )\n \n \n ]\n \n \n \n }\n \n \n \n .\n \n (\n 16\n )\n \n \n \n \n \n", "astro-ph_0609234_12": "\n \n \n \n \n d\n d\n ⁢\n z\n \n (\n ρ\n v\n n\n 2\n )\n \n )\n =\n −\n F\n .\n \n \n \n \n", "astro-ph_0610474_11": "\n \n \n \n \n \n p\n =\n \n \n (\n \n γ\n -\n 1\n \n )\n \n ⁢\n \n (\n \n e\n -\n ρ\n \n )\n \n \n \n .\n \n (\n 12\n )\n \n \n \n \n \n", "astro-ph_0612211_4": "\n \n \n \n \n β\n =\n \n \n \n \n \n \n μ\n app\n 2\n \n ⁢\n \n d\n 2\n \n ⁢\n \n \n (\n \n 1\n -\n \n β\n ⁢\n \n cos\n ⁡\n θ\n \n \n \n )\n \n 2\n \n \n \n c\n 2\n \n \n +\n \n \n (\n \n β\n ⁢\n \n cos\n ⁡\n θ\n \n \n )\n \n 2\n \n \n \n ,\n \n (\n 5\n )\n \n \n \n \n \n \n", "astro-ph_0702117_9": "\n \n \n \n \n \n \n I\n ν\n \n ⁢\n \n (\n r\n ,\n ψ\n ,\n ϕ\n )\n \n \n =\n \n \n ρ\n ⁢\n \n (\n r\n ,\n ψ\n ,\n ϕ\n )\n \n ⁢\n Ψ\n ⁢\n \n (\n ψ\n ,\n ϕ\n )\n \n ⁢\n Φ\n ⁢\n \n (\n ϕ\n )\n \n \n ,\n \n (\n \n A\n ⁢\n 3\n \n )\n \n \n \n \n \n \n", "astro-ph_9809315_5": "\n \n \n \n \n \n \n a\n c\n \n =\n \n \n \n (\n \n 2.278\n ±\n 0.008\n \n )\n \n +\n \n \n (\n \n 3.824\n ±\n 0.33\n \n )\n \n ⁢\n e\n \n \n -\n \n \n (\n \n 1.71\n ±\n 0.10\n \n )\n \n ⁢\n \n e\n 2\n \n \n \n \n .\n \n (\n 3\n )\n \n \n \n \n \n", "astro-ph_9811156_51": "\n \n \n \n \n \n \n δ\n c\n \n =\n \n \n δ\n \n c\n ⁢\n 0\n \n \n -\n \n \n 1\n 2\n \n ⁢\n h\n \n \n \n .\n \n (\n 52\n )\n \n \n \n \n \n", "astro-ph_9811200_0": "\n \n \n \n \n Σ\n =\n \n \n \n \n N\n e\n \n ⁢\n k\n ⁢\n T\n ⁢\n \n [\n \n \n σ\n e\n id\n \n +\n \n σ\n ee\n x\n \n +\n \n σ\n ee\n c\n \n \n ]\n \n \n +\n \n \n N\n i\n \n ⁢\n k\n ⁢\n T\n ⁢\n \n [\n \n \n σ\n i\n id\n \n +\n \n σ\n ii\n c\n \n +\n \n σ\n ii\n cq\n \n +\n \n σ\n ie\n c\n \n \n ]\n \n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "astro-ph_9909300_5": "\n \n \n \n \n \n \n \n Y\n ^\n \n i\n S\n \n =\n \n \n N\n ^\n \n i\n \n \n ,\n \n ı\n =\n \n 1\n ,\n 2\n ,\n …\n ,\n \n L\n \n ,\n \n (\n 7\n )\n \n \n \n \n \n \n \n", "cond-mat_0002045_14": "\n \n \n \n \n \n η\n =\n \n \n \n ⟨\n \n \n f\n \n j\n ⁢\n γ\n \n \n ⁢\n \n (\n A\n ,\n J\n )\n \n ⁢\n \n \n |\n \n H\n j\n \n |\n \n γ\n \n ⁢\n \n \n H\n j\n \n \n |\n \n H\n j\n \n |\n \n \n \n ⟩\n \n \n A\n ⁢\n J\n \n \n \n \n ⟨\n \n \n f\n \n j\n ⁢\n γ\n \n \n ⁢\n \n (\n A\n ,\n J\n )\n \n ⁢\n \n \n |\n \n H\n j\n \n |\n \n γ\n \n \n ⟩\n \n \n A\n ⁢\n J\n \n \n \n \n .\n \n (\n 11\n )\n \n \n \n \n \n", "cond-mat_0005200_2": "\n \n \n \n \n \n Γ\n ⁢\n \n (\n θ\n )\n \n \n ≃\n \n \n \n \n τ\n ⁢\n \n \n (\n θ\n )\n \n 2\n \n \n ℏ\n \n ⁢\n \n \n [\n \n π\n \n \n E\n r\n \n ⁢\n \n k\n B\n \n ⁢\n T\n \n \n ]\n \n \n 1\n /\n 2\n \n \n ⁢\n e\n ⁢\n x\n ⁢\n p\n ⁢\n \n (\n \n -\n \n \n \n E\n f\n \n /\n \n k\n B\n \n \n ⁢\n T\n \n \n )\n \n \n ,\n \n (\n 1\n )\n \n \n \n \n \n \n", "cond-mat_0005458_5": "\n \n \n \n \n \n g\n ⁢\n \n (\n x\n )\n \n \n ∝\n \n \n {\n \n \n \n \n \n c\n ⁢\n o\n ⁢\n n\n ⁢\n s\n ⁢\n t\n \n ,\n \n \n \n \n \n x\n ≫\n \n 1\n ⁢\n \n x\n \n -\n \n 1\n /\n 2\n \n \n \n \n \n ,\n \n \n \n \n \n x\n ≪\n 1\n \n .\n \n \n \n \n \n \n ⁢\n \n (\n 6\n )\n \n \n \n \n \n \n", "cond-mat_0011248_39": "\n \n \n \n \n \n \n ∏\n \n i\n =\n 1\n \n \n 2\n ⁢\n N\n \n \n \n cos\n ⁡\n \n (\n \n \n \n Δ\n i\n \n ⁢\n ξ\n \n 2\n \n )\n \n \n \n =\n \n \n cos\n \n 2\n ⁢\n N\n \n \n ⁡\n \n (\n \n \n \n Δ\n 0\n \n ⁢\n ξ\n \n 2\n \n )\n \n \n =\n \n \n 2\n \n -\n \n 2\n ⁢\n N\n \n \n \n ⁢\n \n \n ∑\n \n k\n =\n 0\n \n \n 2\n ⁢\n N\n \n \n \n \n C\n \n 2\n ⁢\n N\n \n k\n \n ⁢\n \n cos\n ⁡\n \n [\n \n \n Δ\n 0\n \n ⁢\n \n (\n \n N\n -\n k\n \n )\n \n ⁢\n ξ\n \n ]\n \n \n ⁢\n \n (\n 28\n )\n \n \n \n \n \n \n \n \n", "cond-mat_0012334_19": "\n \n \n \n \n \n U\n 3\n \n =\n \n e\n \n i\n ⁢\n \n π\n 4\n \n ⁢\n \n σ\n z\n 1\n \n ⁢\n \n σ\n z\n 3\n \n \n \n \n \n \n \n", "cond-mat_0103298_6": "\n \n \n \n \n \n Ξ\n =\n \n \n ⟨\n \n \n ∑\n \n i\n =\n 1\n \n N\n \n \n e\n \n -\n \n β\n ⁢\n \n [\n \n \n E\n c\n m\n \n -\n \n \n E\n c\n m\n \n ⁢\n \n (\n i\n )\n \n \n \n ]\n \n \n \n \n \n ⟩\n \n \n {\n N\n ,\n \n x\n i\n N\n \n }\n \n \n =\n \n Tr\n ⁢\n \n ∫\n \n \n 𝑑\n \n 𝐏\n \n \n ⁢\n 𝚺\n \n \n \n \n ,\n \n (\n 12\n )\n \n \n \n \n \n", "cond-mat_0104166_19": "\n \n \n \n \n \n \n p\n ⁢\n \n (\n x\n ,\n t\n )\n \n \n =\n \n \n F\n \n -\n 1\n \n \n ⁢\n \n L\n \n -\n 1\n \n \n ⁢\n \n p\n ^\n \n ⁢\n \n (\n k\n ,\n s\n )\n \n \n ≡\n \n \n F\n \n -\n 1\n \n \n ⁢\n \n L\n \n -\n 1\n \n \n ⁢\n \n [\n \n \n \n 1\n -\n \n ψ\n ^\n \n \n s\n \n ⁢\n \n 1\n \n 1\n -\n \n λ\n ⁢\n \n (\n k\n )\n \n ⁢\n \n ψ\n ^\n \n ⁢\n \n (\n s\n )\n \n \n \n \n \n ]\n \n \n \n ,\n \n (\n 20\n )\n \n \n \n \n \n", "cond-mat_0106091_16": "\n \n \n \n \n \n \n 𝐩\n ⁢\n \n (\n 𝐫\n ,\n z\n )\n \n \n =\n \n -\n \n e\n ⁢\n \n \n ∑\n j\n \n \n \n (\n \n 𝐫\n +\n \n z\n ⁢\n \n 𝐳\n ^\n \n \n \n )\n \n ⁢\n \n Ψ\n j\n †\n \n ⁢\n \n (\n 𝐫\n ,\n z\n )\n \n ⁢\n \n Ψ\n j\n \n ⁢\n \n (\n 𝐫\n ,\n z\n )\n \n \n \n \n \n \n .\n \n (\n 13\n )\n \n \n \n \n \n", "cond-mat_0111372_6": "\n \n \n \n \n H\n =\n \n ∑\n Q\n \n \n H\n Q\n \n +\n N\n \n (\n \n \n \n \n n\n A\n \n ⁢\n \n h\n A\n \n \n +\n \n \n n\n B\n \n ⁢\n \n h\n B\n \n \n \n 2\n \n )\n \n   \n ,\n \n (\n 8\n )\n \n \n \n \n \n", "cond-mat_0204236_57": "\n \n \n \n \n \n \n \n λ\n c\n \n ⁢\n \n (\n 0\n )\n \n \n =\n \n \n Ω\n s\n \n ⁢\n \n \n (\n \n \n σ\n c\n \n D\n ⁢\n C\n \n \n ⁢\n \n (\n \n T\n c\n +\n \n )\n \n \n )\n \n \n -\n \n \n (\n \n 2\n +\n z\n \n )\n \n /\n 4\n \n \n \n \n \n ,\n \n \n Ω\n s\n \n =\n \n \n \n γ\n \n 0\n ,\n 0\n \n \n ⁢\n \n λ\n \n \n a\n ⁢\n b\n \n ,\n 0\n \n \n ⁢\n \n (\n 0\n )\n \n ⁢\n \n \n (\n \n \n 4\n ⁢\n \n e\n 2\n \n ⁢\n \n σ\n 0\n \n \n \n h\n ⁢\n \n γ\n \n \n T\n c\n \n ,\n 0\n \n 2\n \n \n \n )\n \n \n \n (\n \n 2\n +\n z\n \n )\n \n /\n 4\n \n \n \n ,\n \n (\n 57\n )\n \n \n \n \n \n \n \n", "cond-mat_0209625_26": "\n \n \n \n \n \n \n w\n c\n \n =\n \n \n \n w\n -\n \n \n \n 2\n ⁢\n \n w\n -\n \n \n +\n 1\n \n \n \n \n .\n \n (\n 27\n )\n \n \n \n \n \n", "cond-mat_0211626_5": "\n \n \n \n \n \n \n e\n \n -\n \n b\n ⁢\n \n x\n j\n \n \n \n \n -\n 1\n \n =\n \n \n \n m\n \n a\n ⁢\n b\n \n \n ⁢\n \n β\n 2\n \n ⁢\n \n sech\n 2\n \n ⁢\n \n (\n \n \n κ\n ⁢\n j\n \n ±\n \n β\n ⁢\n t\n \n \n )\n \n \n ;\n \n (\n 6\n )\n \n \n \n \n \n \n", "cond-mat_0212469_19": "\n \n \n \n \n \n \n \n P\n ~\n \n ⁢\n \n (\n k\n )\n \n \n =\n \n \n \n A\n 4\n \n \n \n (\n \n 1\n -\n A\n \n )\n \n 2\n \n \n ⁢\n \n e\n \n -\n \n \n k\n +\n 2\n \n \n k\n 0\n \n \n \n \n ⁢\n \n \n ∑\n \n s\n =\n 0\n \n k\n \n \n \n (\n \n s\n +\n 1\n \n )\n \n ⁢\n \n (\n \n \n k\n +\n 1\n \n -\n s\n \n )\n \n \n \n \n \n .\n \n (\n 22\n )\n \n \n \n \n \n", "cond-mat_0307446_273": "\n \n \n \n \n \n \n \n ℋ\n r\n \n ⁢\n \n (\n \n a\n j\n \n )\n \n \n =\n \n -\n \n \n \n κ\n r\n \n 2\n \n ⁢\n \n \n ∑\n i\n \n \n cos\n ⁡\n \n (\n \n \n ϵ\n \n i\n ⁢\n j\n \n \n ⁢\n \n \n ∂\n i\n \n ⁡\n \n a\n j\n \n \n \n )\n \n \n \n \n \n \n .\n \n (\n 236\n )\n \n \n \n \n \n", "cond-mat_0310065_16": "\n \n \n \n \n \n \n K\n \n \n ′\n \n \n ⁢\n \n (\n n\n ,\n θ\n ,\n \n θ\n \n \n ′\n \n \n )\n \n \n =\n \n π\n ⁢\n \n \n \n \n cos\n ⁡\n \n (\n θ\n )\n \n \n ⁢\n \n sin\n ⁡\n \n (\n \n 2\n ⁢\n π\n ⁢\n \n (\n \n \n r\n ⁢\n \n cos\n ⁡\n \n (\n \n θ\n \n \n ′\n \n \n )\n \n \n \n -\n \n \n cos\n ⁡\n \n (\n θ\n )\n \n \n /\n 2\n \n \n )\n \n \n )\n \n \n \n +\n \n \n sin\n ⁡\n \n (\n θ\n )\n \n \n ⁢\n \n sinh\n ⁡\n \n (\n \n 2\n ⁢\n π\n ⁢\n \n (\n \n \n n\n +\n \n r\n ⁢\n \n sin\n ⁡\n \n (\n \n θ\n \n \n ′\n \n \n )\n \n \n \n \n -\n \n \n sin\n ⁡\n \n (\n θ\n )\n \n \n /\n 2\n \n \n )\n \n \n )\n \n \n \n \n \n \n cosh\n ⁡\n \n (\n \n 2\n ⁢\n π\n ⁢\n \n (\n \n \n n\n +\n \n r\n ⁢\n \n sin\n ⁡\n \n (\n \n θ\n \n \n ′\n \n \n )\n \n \n \n \n -\n \n \n sin\n ⁡\n \n (\n θ\n )\n \n \n /\n 2\n \n \n )\n \n \n )\n \n \n -\n \n cos\n ⁡\n \n (\n \n 2\n ⁢\n π\n ⁢\n \n (\n \n \n r\n ⁢\n \n cos\n ⁡\n \n (\n \n θ\n \n \n ′\n \n \n )\n \n \n \n -\n \n \n cos\n ⁡\n \n (\n θ\n )\n \n \n /\n 2\n \n \n )\n \n \n )\n \n \n \n \n ⁢\n \n (\n 16\n )\n \n \n \n \n \n \n", "cond-mat_0311017_483": "\n \n \n \n \n \n \n (\n \n n\n \n ,\n \n \n \n e\n \n \n τ\n \n ⁢\n \n \n L\n \n k\n \n \n \n \n ⁢\n \n \n f\n \n *\n \n ⁢\n \n (\n \n \n n\n k\n \n ⁢\n \n (\n t\n )\n \n \n )\n \n \n )\n \n =\n \n \n 1\n 2\n \n ⁢\n \n (\n \n n\n \n ,\n \n \n \n e\n \n \n τ\n \n ⁢\n \n \n L\n \n k\n \n \n \n \n ⁢\n \n n\n \n \n )\n \n ⁢\n \n n\n k\n \n ⁢\n \n (\n t\n )\n \n \n \n .\n \n (\n 319\n )\n \n \n \n \n \n", "cond-mat_0401393_46": "\n \n \n \n \n H\n R\n ⁢\n D\n ⁢\n N\n =\n ∫\n d\n 𝐱\n Σ\n \n [\n 1\n +\n \n (\n ∇\n l\n )\n \n 2\n −\n 1\n ]\n \n +\n s\n e\n c\n α\n W\n \n (\n s\n e\n c\n α\n \n [\n l\n −\n ψ\n \n (\n 𝐱\n ]\n \n )\n \n \n (\n 49\n )\n \n \n \n \n \n \n", "cond-mat_0407180_2": "\n \n \n \n \n \n ∂\n ⁡\n P\n \n ⁢\n \n (\n V\n ,\n t\n )\n \n \n \n ∂\n ⁡\n t\n \n \n \n =\n \n \n \n \n ∑\n \n n\n =\n 1\n \n ∞\n \n \n \n \n \n \n \n (\n \n -\n 1\n \n )\n \n n\n \n \n n\n !\n \n \n \n ⁢\n \n \n (\n \n \n d\n \n d\n ⁢\n V\n \n \n \n )\n \n n\n \n ⁢\n \n {\n \n \n a\n n\n \n ⁢\n \n (\n V\n )\n \n ⁢\n P\n ⁢\n \n (\n V\n ,\n t\n )\n \n \n }\n \n \n \n ,\n \n (\n 3\n )\n \n \n \n", "cond-mat_0412751_27": "\n \n \n \n \n \n \n Z\n ⁢\n \n (\n \n η\n ^\n \n ,\n η\n )\n \n \n =\n \n ∫\n \n D\n ⁢\n \n ψ\n ^\n \n ⁢\n \n (\n x\n ,\n t\n )\n \n ⁢\n D\n ⁢\n ψ\n ⁢\n \n (\n x\n ,\n t\n )\n \n ⁢\n \n e\n \n -\n \n \n 1\n ϵ\n \n ⁢\n \n (\n \n S\n +\n \n \n ∫\n \n \n 𝑑\n x\n \n ⁢\n \n 𝑑\n t\n \n ⁢\n \n [\n \n \n \n η\n ^\n \n ⁢\n \n (\n x\n ,\n t\n )\n \n ⁢\n ψ\n ⁢\n \n (\n x\n ,\n t\n )\n \n \n +\n \n \n ψ\n ^\n \n ⁢\n \n (\n x\n ,\n t\n )\n \n ⁢\n η\n ⁢\n \n (\n x\n ,\n t\n )\n \n \n \n ]\n \n \n \n \n \n )\n \n \n \n \n \n \n \n .\n \n (\n 25\n )\n \n \n \n \n \n", "cond-mat_0502112_65": "\n \n \n \n \n \n \n \n \n x\n 2\n \n +\n \n y\n 2\n \n \n \n ⁢\n \n \n \n \n \n sin\n 2\n \n ⁡\n x\n \n +\n \n \n sinh\n 2\n \n ⁡\n y\n \n \n \n \n ⁢\n \n sin\n ⁡\n \n (\n \n \n -\n \n 2\n ⁢\n k\n ⁢\n x\n ⁢\n \n (\n \n y\n +\n \n η\n 0\n \n \n )\n \n \n \n +\n α\n \n )\n \n \n \n ,\n \n (\n 66\n )\n \n \n \n \n \n", "cond-mat_0508240_3": "\n \n \n \n \n 𝐇\n =\n \n ∫\n \n \n d\n d\n \n ⁢\n \n x\n \n ⁢\n \n {\n \n \n Ψ\n ⁢\n H\n ⁢\n Ψ\n \n +\n \n \n U\n 2\n \n ⁢\n \n \n (\n \n Ψ\n ⁢\n Ψ\n \n )\n \n 2\n \n \n \n }\n \n ⁢\n \n (\n 4\n )\n \n \n \n \n \n \n \n", "cond-mat_0510071_4": "\n \n \n \n \n L\n =\n \n \n \n m\n 2\n \n ⁢\n \n u\n 2\n \n \n +\n \n \n \n α\n E\n \n 2\n \n ⁢\n \n E\n \n ′\n ⁣\n 2\n \n \n \n +\n \n \n \n α\n B\n \n 2\n \n ⁢\n \n c\n 2\n \n ⁢\n \n B\n \n ′\n ⁣\n 2\n \n \n ⁢\n \n (\n 5\n )\n \n \n \n \n \n \n \n" }