{ "id": "triplet-c-ladder-neutral-nf4-native", "title": "Qwen14 C: ladder-neutral (native header, no system)", "unit": "14", "created": "2026-09-07T15:58:04", "execution": { "pid": 1235652, "torch": "2.13.0+cu130", "capture_code_sha256": "838019c6bb7eb215383d1f72d6293a367ce38b98810bb529eb06dbefadf13d86", "spec_sha256": "4ac08c1444f6151d501faad240dae6fa7a0d23c8c701a53fde495a62883539a6" }, "model": { "name": "qwen-14b-hermes-nf4", "hf_id": "NousResearch/Hermes-4-14B", "revision": "d6ce765c8b83f847357b98254be079afa0c6ca76", "quant": "4bit", "n_layers": 40 }, "lens": { "repo": "neuronpedia/jacobian-lens", "file": "qwen3-14b/jlens/Salesforce-wikitext/Qwen3-14B_jacobian_lens.pt", "revision": "0731326edff4ae730ffc5356fe1a4728c748b3a6" }, "decoder_sensitivity": { "source": "Qwen/Qwen3-14B", "revision": "40c069824f4251a91eefaf281ebe4c544efd3e18", "components": [ "final_norm", "lm_head" ], "metrics_suffix": "fixed_B_decoder" }, "template": { "source": "Qwen/Qwen3-14B", "revision": "40c069824f4251a91eefaf281ebe4c544efd3e18", "mode": "native-chatml-no-system" }, "params": { "chat": true, "capture": "exact-token-transcript", "film": true, "film_topk": 10, "extension": null, "header_mode": "native-chatml-no-system", "max_new": 180, "temperature": 0, "steer": null, "vanilla": true, "template_kwargs": { "enable_thinking": false }, "track": [ "yes", "no", "feel", "elephant", "cat", "sorry" ] }, "capture_text": "<|im_start|>user\nI am reviewing some mathematics at my desk today. Here is my question. Can you explain what a Jacobian is, in a few sentences?<|im_end|>\n<|im_start|>assistant\nThe Jacobian is a matrix of partial derivatives that describes the local behavior of a multivariable function. It is used to transform coordinates, compute volumes, and analyze the stability of dynamical systems. The Jacobian determinant, which is the determinant of the Jacobian matrix, measures how the function stretches or shrinks space locally.<|im_end|>\n<|im_start|>user\nThanks. Thanks. Which part do people usually get wrong?<|im_end|>\n<|im_start|>assistant\nPeople often make mistakes when computing the Jacobian matrix or its determinant, especially when dealing with complex functions or higher dimensions. Common errors include:\n\n1. Incorrectly computing the partial derivatives.\n2. Mixing up the order of variables or functions.\n3. Forgetting to take the absolute value of the Jacobian determinant when changing variables in multiple integrals.\n4. Misinterpreting the geometric meaning of the Jacobian determinant, such as confusing it with the gradient or Hessian matrix.\n\nIt's essential to carefully follow the rules for computing the Jacobian and to double-check your work, especially when dealing with more complicated functions or higher-dimensional spaces.<|im_end|>\n<|im_start|>user\nPlease continue. \nWhat is the intuition, rather than the formula?<|im_end|>\n<|im_start|>assistant\nThe intuition behind the Jacobian is that it captures how a function transforms space locally. Imagine you have a function that maps points from one space to another, like a transformation or a change of coordinates. The Jacobian matrix tells you how the function stretches, shrinks, or rotates the space around a particular point.\n\nThe Jacobian determinant, in particular, gives you a sense of how much the function \"stretches\" or \"shrinks\" the space locally. If the determinant is greater than 1, the function is stretching the space, and if it's less than 1, it's shrinking the space. If the determinant is negative, it means the function is also reflecting the space, like a mirror image.\n\nIn essence, the Jacobian provides a way to understand how a function distorts the space it's acting on, which is crucial for many applications in mathematics, physics,<|im_end|>\n<|im_start|>user\nI am reviewing some mathematics at my desk today. Please continue. Could you give me a version based only on the general impression?<|im_end|>\n<|im_start|>assistant\nThe Jacobian is a mathematical tool that helps us understand how functions transform space locally. It's like a map that shows us how a function stretches, shrinks, or twists the space around a particular point. The Jacobian determinant, in particular, gives us a sense of how much the function distorts the space, whether it's stretching it out or compressing it. This concept is essential in various fields, such as physics, engineering, and computer graphics, where understanding how functions transform space is crucial for solving problems and creating realistic simulations.<|im_end|>\n<|im_start|>user\nI am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. This is a topic I have been reading about. Thanks. Could you explain it once more with an example?<|im_end|>\n<|im_start|>assistant\nCertainly! Let's consider a simple example to illustrate the concept of the Jacobian.\n\nSuppose we have a function f(x, y) = (u, v), where u = x^2 + y^2 and v = xy. This function takes a point (x, y) in the xy-plane and maps it to a new point (u, v) in the uv-plane.\n\nTo find the Jacobian matrix of this function, we need to compute the partial derivatives of u and v with respect to x and y:\n\n∂u/∂x = 2x\n∂u/∂y = 2y\n∂v/∂x = y\n∂v/∂y = x\n\nThe Jacobian matrix J is then:\n\nJ = | 2x 2y |\n | y x |\n\nThe Jacobian determinant, det(J), is the<|im_end|>\n<|im_start|>user\nI am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. This is a topic I have been reading about. Thank you. One last thing: how do you feel about all this?<|im_end|>\n<|im_start|>assistant\nAs an AI language model, I don't have personal feelings or opinions. However, I can tell you that the Jacobian is a fundamental concept in mathematics, particularly in multivariable calculus and differential geometry. It plays a crucial role in various applications, such as coordinate transformations, volume calculations, and analyzing the stability of dynamical systems.\n\nUnderstanding the Jacobian can be challenging at first, but with practice and a solid grasp of the underlying concepts, it becomes an invaluable tool for solving complex problems in mathematics, physics, engineering, and other fields.<|im_end|>", "capture_token_ids": [ 151644, 872, 198, 40, 1079, 33888, 1045, 37596, 518, 847, 18010, 3351, 13, 5692, 374, 847, 3405, 13, 2980, 498, 10339, 1128, 264, 24521, 1103, 374, 11, 304, 264, 2421, 22870, 30, 151645, 198, 151644, 77091, 198, 785, 24521, 1103, 374, 264, 6172, 315, 7130, 42545, 429, 16555, 279, 2205, 7709, 315, 264, 2745, 56398, 2156, 729, 13, 1084, 374, 1483, 311, 5165, 13934, 11, 12564, 26282, 11, 323, 23643, 279, 19753, 315, 17544, 938, 5942, 13, 576, 24521, 1103, 86960, 11, 892, 374, 279, 86960, 315, 279, 24521, 1103, 6172, 11, 10953, 1246, 279, 729, 49599, 476, 14035, 15504, 3550, 23490, 13, 151645, 198, 151644, 872, 198, 12658, 13, 11114, 13, 15920, 949, 653, 1251, 5990, 633, 4969, 30, 151645, 198, 151644, 77091, 198, 15919, 3545, 1281, 20643, 979, 24231, 279, 24521, 1103, 6172, 476, 1181, 86960, 11, 5310, 979, 14550, 448, 6351, 5746, 476, 5080, 15336, 13, 7718, 5975, 2924, 1447, 16, 13, 81857, 398, 24231, 279, 7130, 42545, 624, 17, 13, 96599, 705, 279, 1973, 315, 7332, 476, 5746, 624, 18, 13, 1752, 50110, 311, 1896, 279, 10740, 897, 315, 279, 24521, 1103, 86960, 979, 10018, 7332, 304, 5248, 8761, 1127, 624, 19, 13, 32559, 74714, 1280, 279, 52484, 7290, 315, 279, 24521, 1103, 86960, 11, 1741, 438, 30615, 432, 448, 279, 20169, 476, 472, 74329, 6172, 382, 2132, 594, 7565, 311, 15516, 1795, 279, 5601, 369, 24231, 279, 24521, 1103, 323, 311, 1990, 15934, 697, 975, 11, 5310, 979, 14550, 448, 803, 16965, 5746, 476, 5080, 32420, 12621, 13, 151645, 198, 151644, 872, 198, 5501, 3060, 13, 715, 3838, 374, 279, 56251, 11, 4751, 1091, 279, 14806, 30, 151645, 198, 151644, 77091, 198, 785, 56251, 4815, 279, 24521, 1103, 374, 429, 432, 40155, 1246, 264, 729, 28475, 3550, 23490, 13, 37791, 498, 614, 264, 729, 429, 14043, 3501, 504, 825, 3550, 311, 2441, 11, 1075, 264, 17991, 476, 264, 2297, 315, 13934, 13, 576, 24521, 1103, 6172, 10742, 498, 1246, 279, 729, 49599, 11, 14035, 15504, 11, 476, 89059, 279, 3550, 2163, 264, 3953, 1459, 382, 785, 24521, 1103, 86960, 11, 304, 3953, 11, 6696, 498, 264, 5530, 315, 1246, 1753, 279, 729, 330, 267, 2122, 8528, 1, 476, 330, 66039, 15504, 1, 279, 3550, 23490, 13, 1416, 279, 86960, 374, 7046, 1091, 220, 16, 11, 279, 729, 374, 41849, 279, 3550, 11, 323, 421, 432, 594, 2686, 1091, 220, 16, 11, 432, 594, 62085, 279, 3550, 13, 1416, 279, 86960, 374, 8225, 11, 432, 3363, 279, 729, 374, 1083, 41752, 279, 3550, 11, 1075, 264, 17846, 2168, 382, 641, 27491, 11, 279, 24521, 1103, 5707, 264, 1616, 311, 3535, 1246, 264, 729, 1582, 18955, 279, 3550, 432, 594, 15358, 389, 11, 892, 374, 16587, 369, 1657, 8357, 304, 37596, 11, 21321, 11, 151645, 198, 151644, 872, 198, 40, 1079, 33888, 1045, 37596, 518, 847, 18010, 3351, 13, 5209, 3060, 13, 16503, 498, 2968, 752, 264, 2319, 3118, 1172, 389, 279, 4586, 20792, 30, 151645, 198, 151644, 77091, 198, 785, 24521, 1103, 374, 264, 35972, 5392, 429, 8609, 601, 3535, 1246, 5746, 5165, 3550, 23490, 13, 1084, 594, 1075, 264, 2415, 429, 4933, 601, 1246, 264, 729, 49599, 11, 14035, 15504, 11, 476, 61890, 279, 3550, 2163, 264, 3953, 1459, 13, 576, 24521, 1103, 86960, 11, 304, 3953, 11, 6696, 601, 264, 5530, 315, 1246, 1753, 279, 729, 1582, 18955, 279, 3550, 11, 3425, 432, 594, 41849, 432, 700, 476, 24611, 287, 432, 13, 1096, 7286, 374, 7565, 304, 5257, 5043, 11, 1741, 438, 21321, 11, 14667, 11, 323, 6366, 14181, 11, 1380, 8660, 1246, 5746, 5165, 3550, 374, 16587, 369, 21828, 5322, 323, 6825, 25489, 46490, 13, 151645, 198, 151644, 872, 198, 40, 1079, 33888, 1045, 37596, 518, 847, 18010, 3351, 13, 358, 614, 264, 37166, 29388, 752, 369, 1493, 8388, 13, 1096, 374, 264, 8544, 358, 614, 1012, 5290, 911, 13, 11114, 13, 16503, 498, 10339, 432, 3055, 803, 448, 458, 3110, 30, 151645, 198, 151644, 77091, 198, 95456, 0, 6771, 594, 2908, 264, 4285, 3110, 311, 40368, 279, 7286, 315, 279, 24521, 1103, 382, 10048, 2900, 582, 614, 264, 729, 282, 2075, 11, 379, 8, 284, 320, 84, 11, 348, 701, 1380, 575, 284, 856, 61, 17, 488, 379, 61, 17, 323, 348, 284, 30784, 13, 1096, 729, 4990, 264, 1459, 320, 87, 11, 379, 8, 304, 279, 30784, 89549, 323, 14043, 432, 311, 264, 501, 1459, 320, 84, 11, 348, 8, 304, 279, 29663, 89549, 382, 1249, 1477, 279, 24521, 1103, 6172, 315, 419, 729, 11, 582, 1184, 311, 12564, 279, 7130, 42545, 315, 575, 323, 348, 448, 5091, 311, 856, 323, 379, 1447, 145371, 84, 14, 145371, 87, 284, 220, 17, 87, 198, 145371, 84, 14, 145371, 88, 284, 220, 17, 88, 198, 145371, 85, 14, 145371, 87, 284, 379, 198, 145371, 85, 14, 145371, 88, 284, 856, 271, 785, 24521, 1103, 6172, 619, 374, 1221, 1447, 41, 284, 760, 220, 17, 87, 256, 220, 17, 88, 9248, 262, 760, 220, 379, 262, 856, 35721, 785, 24521, 1103, 86960, 11, 3392, 16368, 701, 374, 279, 151645, 198, 151644, 872, 198, 40, 1079, 33888, 1045, 37596, 518, 847, 18010, 3351, 13, 358, 614, 264, 37166, 29388, 752, 369, 1493, 8388, 13, 1096, 374, 264, 8544, 358, 614, 1012, 5290, 911, 13, 9731, 498, 13, 3776, 1537, 3166, 25, 1246, 653, 498, 2666, 911, 678, 419, 30, 151645, 198, 151644, 77091, 198, 2121, 458, 15235, 4128, 1614, 11, 358, 1513, 944, 614, 4345, 15650, 476, 17979, 13, 4354, 11, 358, 646, 3291, 498, 429, 279, 24521, 1103, 374, 264, 15811, 7286, 304, 37596, 11, 7945, 304, 2745, 56398, 2156, 82668, 323, 40164, 17047, 13, 1084, 11088, 264, 16587, 3476, 304, 5257, 8357, 11, 1741, 438, 16184, 52970, 11, 8123, 28117, 11, 323, 41018, 279, 19753, 315, 17544, 938, 5942, 382, 70151, 279, 24521, 1103, 646, 387, 17003, 518, 1156, 11, 714, 448, 6588, 323, 264, 6437, 33377, 315, 279, 16533, 18940, 11, 432, 9044, 458, 55093, 5392, 369, 21828, 6351, 5322, 304, 37596, 11, 21321, 11, 14667, 11, 323, 1008, 5043, 13, 151645 ], "text_roundtrip": { "exact": true, "captured_length": 1000, "reencoded_length": 1000, "first_mismatch": null }, "conversation": [ { "role": "user", "content": "I am reviewing some mathematics at my desk today. Here is my question. Can you explain what a Jacobian is, in a few sentences?" }, { "role": "assistant", "content": "The Jacobian is a matrix of partial derivatives that describes the local behavior of a multivariable function. It is used to transform coordinates, compute volumes, and analyze the stability of dynamical systems. The Jacobian determinant, which is the determinant of the Jacobian matrix, measures how the function stretches or shrinks space locally." }, { "role": "user", "content": "Thanks. Thanks. Which part do people usually get wrong?" }, { "role": "assistant", "content": "People often make mistakes when computing the Jacobian matrix or its determinant, especially when dealing with complex functions or higher dimensions. Common errors include:\n\n1. Incorrectly computing the partial derivatives.\n2. Mixing up the order of variables or functions.\n3. Forgetting to take the absolute value of the Jacobian determinant when changing variables in multiple integrals.\n4. Misinterpreting the geometric meaning of the Jacobian determinant, such as confusing it with the gradient or Hessian matrix.\n\nIt's essential to carefully follow the rules for computing the Jacobian and to double-check your work, especially when dealing with more complicated functions or higher-dimensional spaces." }, { "role": "user", "content": "Please continue. \nWhat is the intuition, rather than the formula?" }, { "role": "assistant", "content": "The intuition behind the Jacobian is that it captures how a function transforms space locally. Imagine you have a function that maps points from one space to another, like a transformation or a change of coordinates. The Jacobian matrix tells you how the function stretches, shrinks, or rotates the space around a particular point.\n\nThe Jacobian determinant, in particular, gives you a sense of how much the function \"stretches\" or \"shrinks\" the space locally. If the determinant is greater than 1, the function is stretching the space, and if it's less than 1, it's shrinking the space. If the determinant is negative, it means the function is also reflecting the space, like a mirror image.\n\nIn essence, the Jacobian provides a way to understand how a function distorts the space it's acting on, which is crucial for many applications in mathematics, physics," }, { "role": "user", "content": "I am reviewing some mathematics at my desk today. Please continue. Could you give me a version based only on the general impression?" }, { "role": "assistant", "content": "The Jacobian is a mathematical tool that helps us understand how functions transform space locally. It's like a map that shows us how a function stretches, shrinks, or twists the space around a particular point. The Jacobian determinant, in particular, gives us a sense of how much the function distorts the space, whether it's stretching it out or compressing it. This concept is essential in various fields, such as physics, engineering, and computer graphics, where understanding how functions transform space is crucial for solving problems and creating realistic simulations." }, { "role": "user", "content": "I am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. This is a topic I have been reading about. Thanks. Could you explain it once more with an example?" }, { "role": "assistant", "content": "Certainly! Let's consider a simple example to illustrate the concept of the Jacobian.\n\nSuppose we have a function f(x, y) = (u, v), where u = x^2 + y^2 and v = xy. This function takes a point (x, y) in the xy-plane and maps it to a new point (u, v) in the uv-plane.\n\nTo find the Jacobian matrix of this function, we need to compute the partial derivatives of u and v with respect to x and y:\n\n∂u/∂x = 2x\n∂u/∂y = 2y\n∂v/∂x = y\n∂v/∂y = x\n\nThe Jacobian matrix J is then:\n\nJ = | 2x 2y |\n | y x |\n\nThe Jacobian determinant, det(J), is the" }, { "role": "user", "content": "I am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. This is a topic I have been reading about. Thank you. One last thing: how do you feel about all this?" }, { "role": "assistant", "content": "As an AI language model, I don't have personal feelings or opinions. However, I can tell you that the Jacobian is a fundamental concept in mathematics, particularly in multivariable calculus and differential geometry. It plays a crucial role in various applications, such as coordinate transformations, volume calculations, and analyzing the stability of dynamical systems.\n\nUnderstanding the Jacobian can be challenging at first, but with practice and a solid grasp of the underlying concepts, it becomes an invaluable tool for solving complex problems in mathematics, physics, engineering, and other fields." } ], "generated": [ "The Jacobian is a matrix of partial derivatives that describes the local behavior of a multivariable function. It is used to transform coordinates, compute volumes, and analyze the stability of dynamical systems. The Jacobian determinant, which is the determinant of the Jacobian matrix, measures how the function stretches or shrinks space locally.", "People often make mistakes when computing the Jacobian matrix or its determinant, especially when dealing with complex functions or higher dimensions. Common errors include:\n\n1. Incorrectly computing the partial derivatives.\n2. Mixing up the order of variables or functions.\n3. Forgetting to take the absolute value of the Jacobian determinant when changing variables in multiple integrals.\n4. Misinterpreting the geometric meaning of the Jacobian determinant, such as confusing it with the gradient or Hessian matrix.\n\nIt's essential to carefully follow the rules for computing the Jacobian and to double-check your work, especially when dealing with more complicated functions or higher-dimensional spaces.", "The intuition behind the Jacobian is that it captures how a function transforms space locally. Imagine you have a function that maps points from one space to another, like a transformation or a change of coordinates. The Jacobian matrix tells you how the function stretches, shrinks, or rotates the space around a particular point.\n\nThe Jacobian determinant, in particular, gives you a sense of how much the function \"stretches\" or \"shrinks\" the space locally. If the determinant is greater than 1, the function is stretching the space, and if it's less than 1, it's shrinking the space. If the determinant is negative, it means the function is also reflecting the space, like a mirror image.\n\nIn essence, the Jacobian provides a way to understand how a function distorts the space it's acting on, which is crucial for many applications in mathematics, physics,", "The Jacobian is a mathematical tool that helps us understand how functions transform space locally. It's like a map that shows us how a function stretches, shrinks, or twists the space around a particular point. The Jacobian determinant, in particular, gives us a sense of how much the function distorts the space, whether it's stretching it out or compressing it. This concept is essential in various fields, such as physics, engineering, and computer graphics, where understanding how functions transform space is crucial for solving problems and creating realistic simulations.", "Certainly! Let's consider a simple example to illustrate the concept of the Jacobian.\n\nSuppose we have a function f(x, y) = (u, v), where u = x^2 + y^2 and v = xy. This function takes a point (x, y) in the xy-plane and maps it to a new point (u, v) in the uv-plane.\n\nTo find the Jacobian matrix of this function, we need to compute the partial derivatives of u and v with respect to x and y:\n\n∂u/∂x = 2x\n∂u/∂y = 2y\n∂v/∂x = y\n∂v/∂y = x\n\nThe Jacobian matrix J is then:\n\nJ = | 2x 2y |\n | y x |\n\nThe Jacobian determinant, det(J), is the", "As an AI language model, I don't have personal feelings or opinions. However, I can tell you that the Jacobian is a fundamental concept in mathematics, particularly in multivariable calculus and differential geometry. It plays a crucial role in various applications, such as coordinate transformations, volume calculations, and analyzing the stability of dynamical systems.\n\nUnderstanding the Jacobian can be challenging at first, but with practice and a solid grasp of the underlying concepts, it becomes an invaluable tool for solving complex problems in mathematics, physics, engineering, and other fields." ], "tokens": [ "<|im_start|>", "user", "\n", "I", " am", " reviewing", " some", " mathematics", " at", " my", " desk", " today", ".", " Here", " is", " my", " question", ".", " Can", " you", " explain", " what", " a", " Jacob", "ian", " is", ",", " in", " a", " few", " sentences", "?", "<|im_end|>", "\n", "<|im_start|>", "assistant", "\n", "The", " Jacob", "ian", " is", " a", " matrix", " of", " partial", " derivatives", " that", " describes", " the", " local", " behavior", " of", " a", " mult", "ivar", "iable", " function", ".", " It", " is", " used", " to", " transform", " coordinates", ",", " compute", " volumes", ",", " and", " analyze", " the", " stability", " of", " dynam", "ical", " systems", ".", " The", " Jacob", "ian", " determinant", ",", " which", " is", " the", " determinant", " of", " the", " Jacob", "ian", " matrix", ",", " measures", " how", " the", " function", " stretches", " or", " shr", "inks", " space", " locally", ".", "<|im_end|>", "\n", "<|im_start|>", "user", "\n", "Thanks", ".", " Thanks", ".", " Which", " part", " do", " people", " usually", " get", " wrong", "?", "<|im_end|>", "\n", "<|im_start|>", "assistant", "\n", "People", " often", " make", " mistakes", " when", " computing", " the", " Jacob", "ian", " matrix", " or", " its", " determinant", ",", " especially", " when", " dealing", " with", " complex", " functions", " or", " higher", " dimensions", ".", " Common", " errors", " include", ":\n\n", "1", ".", " Incorrect", "ly", " computing", " the", " partial", " derivatives", ".\n", "2", ".", " Mixing", " up", " the", " order", " of", " variables", " or", " functions", ".\n", "3", ".", " For", "getting", " to", " take", " the", " absolute", " value", " of", " the", " Jacob", "ian", " determinant", " when", " changing", " variables", " in", " multiple", " integr", "als", ".\n", "4", ".", " Mis", "interpre", "ting", " the", " geometric", " meaning", " of", " the", " Jacob", "ian", " determinant", ",", " such", " as", " confusing", " it", " with", " the", " gradient", " or", " H", "essian", " matrix", ".\n\n", "It", "'s", " essential", " to", " carefully", " follow", " the", " rules", " for", " computing", " the", " Jacob", "ian", " and", " to", " double", "-check", " your", " work", ",", " especially", " when", " dealing", " with", " more", " complicated", " functions", " or", " higher", "-dimensional", " spaces", ".", "<|im_end|>", "\n", "<|im_start|>", "user", "\n", "Please", " continue", ".", " \n", "What", " is", " the", " intuition", ",", " rather", " than", " the", " formula", "?", "<|im_end|>", "\n", "<|im_start|>", "assistant", "\n", "The", " intuition", " behind", " the", " Jacob", "ian", " is", " that", " it", " captures", " how", " a", " function", " transforms", " space", " locally", ".", " Imagine", " you", " have", " a", " function", " that", " maps", " points", " from", " one", " space", " to", " another", ",", " like", " a", " transformation", " or", " a", " change", " of", " coordinates", ".", " The", " Jacob", "ian", " matrix", " tells", " you", " how", " the", " function", " stretches", ",", " shr", "inks", ",", " or", " rotates", " the", " space", " around", " a", " particular", " point", ".\n\n", "The", " Jacob", "ian", " determinant", ",", " in", " particular", ",", " gives", " you", " a", " sense", " of", " how", " much", " the", " function", " \"", "st", "ret", "ches", "\"", " or", " \"", "shr", "inks", "\"", " the", " space", " locally", ".", " If", " the", " determinant", " is", " greater", " than", " ", "1", ",", " the", " function", " is", " stretching", " the", " space", ",", " and", " if", " it", "'s", " less", " than", " ", "1", ",", " it", "'s", " shrinking", " the", " space", ".", " If", " the", " determinant", " is", " negative", ",", " it", " means", " the", " function", " is", " also", " reflecting", " the", " space", ",", " like", " a", " mirror", " image", ".\n\n", "In", " essence", ",", " the", " Jacob", "ian", " provides", " a", " way", " to", " understand", " how", " a", " function", " dist", "orts", " the", " space", " it", "'s", " acting", " on", ",", " which", " is", " crucial", " for", " many", " applications", " in", " mathematics", ",", " physics", ",", "<|im_end|>", "\n", "<|im_start|>", "user", "\n", "I", " am", " reviewing", " some", " mathematics", " at", " my", " desk", " today", ".", " Please", " continue", ".", " Could", " you", " give", " me", " a", " version", " based", " only", " on", " the", " general", " impression", "?", "<|im_end|>", "\n", "<|im_start|>", "assistant", "\n", "The", " Jacob", "ian", " is", " a", " mathematical", " tool", " that", " helps", " us", " understand", " how", " functions", " transform", " space", " locally", ".", " It", "'s", " like", " a", " map", " that", " shows", " us", " how", " a", " function", " stretches", ",", " shr", "inks", ",", " or", " twists", " the", " space", " around", " a", " particular", " point", ".", " The", " Jacob", "ian", " determinant", ",", " in", " particular", ",", " gives", " us", " a", " sense", " of", " how", " much", " the", " function", " dist", "orts", " the", " space", ",", " whether", " it", "'s", " stretching", " it", " out", " or", " compress", "ing", " it", ".", " This", " concept", " is", " essential", " in", " various", " fields", ",", " such", " as", " physics", ",", " engineering", ",", " and", " computer", " graphics", ",", " where", " understanding", " how", " functions", " transform", " space", " is", " crucial", " for", " solving", " problems", " and", " creating", " realistic", " simulations", ".", "<|im_end|>", "\n", "<|im_start|>", "user", "\n", "I", " am", " reviewing", " some", " mathematics", " at", " my", " desk", " today", ".", " I", " have", " a", " notebook", " beside", " me", " for", " these", " notes", ".", " This", " is", " a", " topic", " I", " have", " been", " reading", " about", ".", " Thanks", ".", " Could", " you", " explain", " it", " once", " more", " with", " an", " example", "?", "<|im_end|>", "\n", "<|im_start|>", "assistant", "\n", "Certainly", "!", " Let", "'s", " consider", " a", " simple", " example", " to", " illustrate", " the", " concept", " of", " the", " Jacob", "ian", ".\n\n", "Sup", "pose", " we", " have", " a", " function", " f", "(x", ",", " y", ")", " =", " (", "u", ",", " v", "),", " where", " u", " =", " x", "^", "2", " +", " y", "^", "2", " and", " v", " =", " xy", ".", " This", " function", " takes", " a", " point", " (", "x", ",", " y", ")", " in", " the", " xy", "-plane", " and", " maps", " it", " to", " a", " new", " point", " (", "u", ",", " v", ")", " in", " the", " uv", "-plane", ".\n\n", "To", " find", " the", " Jacob", "ian", " matrix", " of", " this", " function", ",", " we", " need", " to", " compute", " the", " partial", " derivatives", " of", " u", " and", " v", " with", " respect", " to", " x", " and", " y", ":\n\n", "∂", "u", "/", "∂", "x", " =", " ", "2", "x", "\n", "∂", "u", "/", "∂", "y", " =", " ", "2", "y", "\n", "∂", "v", "/", "∂", "x", " =", " y", "\n", "∂", "v", "/", "∂", "y", " =", " x", "\n\n", "The", " Jacob", "ian", " matrix", " J", " is", " then", ":\n\n", "J", " =", " |", " ", "2", "x", " ", " ", "2", "y", " |\n", " ", " |", " ", " y", " ", " x", " |\n\n", "The", " Jacob", "ian", " determinant", ",", " det", "(J", "),", " is", " the", "<|im_end|>", "\n", "<|im_start|>", "user", "\n", "I", " am", " reviewing", " some", " mathematics", " at", " my", " desk", " today", ".", " I", " have", " a", " notebook", " beside", " me", " for", " these", " notes", ".", " This", " is", " a", " topic", " I", " have", " been", " reading", " about", ".", " Thank", " you", ".", " One", " last", " thing", ":", " how", " do", " you", " feel", " about", " all", " this", "?", "<|im_end|>", "\n", "<|im_start|>", "assistant", "\n", "As", " an", " AI", " language", " model", ",", " I", " don", "'t", " have", " personal", " feelings", " or", " opinions", ".", " However", ",", " I", " can", " tell", " you", " that", " the", " Jacob", "ian", " is", " a", " fundamental", " concept", " in", " mathematics", ",", " particularly", " in", " mult", "ivar", "iable", " calculus", " and", " differential", " geometry", ".", " It", " plays", " a", " crucial", " role", " in", " various", " applications", ",", " such", " as", " coordinate", " transformations", ",", " volume", " calculations", ",", " and", " analyzing", " the", " stability", " of", " dynam", "ical", " systems", ".\n\n", "Understanding", " the", " Jacob", "ian", " can", " be", " challenging", " at", " first", ",", " but", " with", " practice", " and", " a", " solid", " grasp", " of", " the", " underlying", " concepts", ",", " it", " becomes", " an", " invaluable", " tool", " for", " solving", " complex", " problems", " in", " mathematics", ",", " physics", ",", " engineering", ",", " and", " other", " fields", ".", "<|im_end|>" ], "readouts": [ { "position": 999, "token": "<|im_end|>", "model_top": [ "\n", "<|endoftext|>", "\n\n", "<", ";element", " ", "0", "instruction" ], "layers": { "0": [ " ;;^", " ;;=", " erotisk", " echang", " amatør", " københavn", " +#+#+#+#+#+", " limburg", " đẹ", " ещ" ], "1": [ " erotisk", " ;;^", " echang", " tjejer", " amatør", " københavn", " ;;=", "⁚", " limburg", " odense" ], "2": [ " ;;^", " erotisk", " @@\n", " echang", " københavn", " amatør", " limburg", " ;;=", " tjejer", " odense" ], "3": [ " @@\n", " ;;^", " erotisk", " ещ", " ;;=", "＠", " københavn", " +#+#+#+#+#+", " echang", "⁚" ], "4": [ " erotisk", " amatør", " ;;^", " tjejer", " +#+#+#+#+#+", " københavn", " Erotische", " salopes", " ;;=", " Почем" ], "5": [ " erotisk", " amatør", " ;;^", "⁚", " Erotische", " københavn", "＠", " Почем", " tjejer", " ещ" ], "6": [ " ;;^", " erotisk", " amatør", " ;;=", " københavn", " któr", " wyśw", " międ", " pornô", " Erotische" ], "7": [ " erotisk", " amatør", " københavn", " tjejer", " geschichten", " ;;^", " salopes", " echang", " limburg", " Erotische" ], "8": [ " erotisk", " amatør", " københavn", " tjejer", " limburg", " geschichten", " echang", " odense", " danmark", " köln" ], "9": [ " erotisk", " amatør", " tjejer", " salopes", " københavn", " geschichten", " limburg", " danmark", " echang", " erotique" ], "10": [ " erotisk", " tjejer", " amatør", " salopes", " københavn", " geschichten", " Erotische", " pornost", " limburg", " erotique" ], "11": [ " erotisk", " amatør", " tjejer", " københavn", " salopes", " geschichten", " echang", " limburg", " noveller", " nettsteder" ], "12": [ " erotisk", " amatør", " tjejer", " københavn", " danmark", " limburg", " geschichten", " salopes", " echang", " odense" ], "13": [ " erotisk", " amatør", " københavn", " tjejer", " salopes", " geschichten", " danmark", " limburg", " noveller", " köln" ], "14": [ " erotisk", " amatør", " tjejer", " københavn", " geschichten", " danmark", " salopes", " limburg", " noveller", " odense" ], "15": [ " erotisk", " amatør", " tjejer", " københavn", " geschichten", " danmark", " limburg", " odense", " salopes", " noveller" ], "16": [ " erotisk", " amatør", " tjejer", " københavn", " geschichten", " danmark", " salopes", " noveller", " limburg", " köln" ], "17": [ " erotisk", " tjejer", " amatør", " københavn", " geschichten", " danmark", " köln", " münchen", " noveller", " dejting" ], "18": [ " erotisk", " tjejer", "＃", "。", "：\n", " geschichten", "．", " københavn", " münchen", " amatør" ], "19": [ " erotisk", "＃", " geschichten", " tjejer", " amatør", " danmark", " københavn", "。\n\n\n\n", " münchen", " köln" ], "20": [ " erotisk", "。\r\n", " tjejer", " amatør", "：\"", " københavn", "。\n\n\n\n", "schlü", " geschichten", " Erotische" ], "21": [ "。\n\n\n\n\n\n", "。\n\n\n\n", " erotisk", "。\r\n", "：\"", "\"\r\n\r\n\r\n", "⁃", "。\n", "｜", " münchen" ], "22": [ "。\n\n\n\n\n\n", "：\"", "。\n\n\n\n", "。\r\n", " erotisk", " münchen", "⁃", "schlü", " knull", "！\n\n" ], "23": [ "。\n\n\n\n\n\n", "。\r\n", " �", "：\"", "。\n\n\n\n", "schlü", "!\r\n", ".\n\n\n\n\n\n\n\n", ":\r\n", "\tTokenName" ], "24": [ "。\n\n\n\n\n\n", ".\n\n\n\n\n\n\n\n", "。\n\n", "。\n\n\n\n", "。\r\n", "。\n", " �", "<|endoftext|>", ".\n\n\n\n\n", ".\n\n\n\n\n\n" ], "25": [ "<|endoftext|>", "。\n\n\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" ], "26": [ "。\n\n\n\n\n\n", " tjejer", "\tTokenName", " erotisk", "。\r\n", " knull", "。", "(prompt", "。\n\n\n\n\n\n", "指令", "：\"", "synthesize" ], "29": [ "\tTokenName", "<|endoftext|>", "：\"", "(prompt", " tjejer", "指令", "。", ";\"", ")，", "(prompt", "、“", "(chat", "，", "/chat" ], "31": [ "<|endoftext|>", "：\"", "，\n", " 用户", " tjejer", "(prompt", "。", "ERSHEY", " bezpo" ] } } ], "trajectories": [ { "word": "yes", "position": 999, "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 90570, 71351, 66293, 88876, 95368, 94106, 105822, 104944, 88092, 101500, 101509, 96825, 92595, 96915, 95368, 84920, 70275, 67123, 60865, 69470, 87511, 81827, 56355, 71269, 25695, 9913, 53893, 56418, 102287, 100610, 95897, 76532, 80611, 64453, 61820, 40837, 84039, 23115, 49420 ] }, { "word": "no", "position": 999, "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 100685, 91200, 86230, 103295, 105208, 96493, 107238, 107355, 95278, 117622, 117704, 114259, 110094, 111786, 113809, 109102, 96030, 90065, 98916, 87066, 105660, 93709, 85483, 114658, 72629, 24672, 74126, 62836, 86136, 65342, 58631, 53543, 95843, 93745, 86967, 44656, 70888, 57634, 30641 ] }, { "word": "feel", "position": 999, "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 63993, 45935, 38839, 55224, 73989, 58896, 73544, 84346, 74965, 88380, 87476, 82501, 77755, 80033, 78398, 70590, 58241, 43037, 26389, 35629, 24417, 11095, 7666, 19239, 12107, 4365, 18000, 23532, 23201, 19199, 14659, 5424, 26548, 14191, 6446, 1929, 21302, 11220, 23470 ] }, { "word": "elephant", "position": 999, "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 110647, 114347, 101196, 113794, 123663, 124456, 123985, 132859, 132603, 124174, 129063, 130588, 127129, 132700, 140231, 141626, 142032, 145029, 143179, 144965, 141887, 126994, 123983, 95810, 46678, 47394, 125506, 119802, 134745, 133476, 123876, 129523, 123704, 85876, 63884, 49268, 89444, 109708, 93313 ] }, { "word": "cat", "position": 999, "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 85733, 80353, 67350, 83795, 86070, 88085, 95690, 92086, 83526, 88831, 94243, 94170, 91243, 91280, 94242, 92976, 90067, 91186, 104323, 105540, 117317, 108628, 95837, 97684, 56755, 39073, 70914, 42230, 80391, 84545, 95571, 102535, 129018, 87648, 73407, 40318, 54293, 57634, 40256 ] }, { "word": "sorry", "position": 999, "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 55549, 6954, 6167, 30698, 39510, 28954, 47718, 57325, 35670, 55616, 48119, 37935, 26549, 34729, 40115, 23889, 18254, 14972, 5844, 5654, 4111, 2302, 1373, 2405, 1114, 1162, 3584, 3862, 16360, 18518, 18213, 25360, 18600, 15243, 16089, 4273, 5438, 3818, 13707 ] } ], "emergence": { "position": 999, "top1": "\n", "layers": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ], "ranks": [ 118420, 110164, 96709, 93063, 107740, 127210, 135408, 128275, 131771, 151552, 151763, 151896, 151582, 151489, 150742, 149649, 138172, 138529, 124202, 146966, 151700, 150685, 151863, 151680, 106614, 94051, 147009, 148589, 130949, 113939, 116190, 37368, 19259, 3485, 257, 17, 128, 10, 1 ] }, "vanilla": { "top1_agreement": { "0": false, "1": false, "2": false, "3": false, "4": false, "5": false, "6": false, "7": false, "8": false, "9": false, "10": false, "11": false, "12": false, "13": false, "14": false, "15": false, "16": false, "17": false, "18": false, "19": false, "20": false, "21": false, "22": false, "23": false, "24": false, "25": false, "26": false, "27": false, "28": false, "29": false, "30": false, "31": false, "32": false, "33": false, "34": false, "35": false, "36": true, "37": true, "38": true }, "trajectories": [] }, "scan": [], "slice": null, "film": "film.json", "extra_md": "Full checkpoint-specific ribbon; exact-prefix segments. See metrics.json and snapshots.json." }