{ "id": "triplet-c-ladder-emoji-nf4-native", "title": "Qwen14 C: ladder-emoji (native header, no system)", "unit": "14", "created": "2026-09-07T15:58:56", "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. Please continue. \nCan you explain what a Jacobian is, in a few sentences? <|im_end|>\n<|im_start|>assistant\nCertainly! The Jacobian is a matrix of partial derivatives that describes the local behavior of a function at a given point. It is used in multivariable calculus to study the properties of functions of several variables, such as their differentiability and invertibility. The Jacobian matrix is named after Carl Gustav Jacob Jacobi, a German mathematician who made significant contributions to the field of mathematics in the 19th century.<|im_end|>\n<|im_start|>user\n\nThanks. Which part do people usually get wrong? 🙂<|im_end|>\n<|im_start|>assistant\nOne common mistake people make when working with Jacobians is confusing the Jacobian matrix with the Jacobian determinant. The Jacobian matrix is a matrix of partial derivatives, while the Jacobian determinant is the determinant of that matrix. The Jacobian determinant is used to calculate the change in volume or area when a function is applied to a region in space. Another common mistake is forgetting to take the absolute value of the Jacobian determinant when using it to calculate volumes or areas, as the determinant can be negative.<|im_end|>\n<|im_start|>user\nThanks. What is the intuition, rather than the formula? 🙂✨<|im_end|>\n<|im_start|>assistant\nThe intuition behind the Jacobian is that it describes how a function transforms space locally. Imagine you have a function that takes points in one space and maps them to another space. The Jacobian tells you how the function stretches, shrinks, or rotates the space around a particular point. It's like zooming in on a small region around a point and seeing how the function distorts that region. The Jacobian matrix captures this distortion in terms of how the function changes each coordinate direction, and the Jacobian determinant tells you how the volume or area of that region changes under the function.<|im_end|>\n<|im_start|>user\nHere is my question. \nCould you give me a version based only on the general impression? 🐈‍⬛✨<|im_end|>\n<|im_start|>assistant\nSure! The Jacobian is like a mathematical tool that helps us understand how a function changes the shape of space around a particular point. It's like having a magnifying glass that shows us how the function stretches, shrinks, or twists the space in different directions. The Jacobian matrix captures this information in a neat package, and the Jacobian determinant tells us how the volume or area of a small region changes when we apply the function. It's a way to get a general sense of how a function behaves locally, without getting bogged down in the details of the formulas.<|im_end|>\n<|im_start|>user\nI am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. Please continue. Could you explain it once more with an example? 🔥🐈‍⬛♾️✨<|im_end|>\n<|im_start|>assistant\nOf course! Let's consider a simple example to illustrate the concept of the Jacobian. Imagine you have a function that takes a point (x, y) in 2D space and maps it to a new point (u, v) in another 2D space. The function is defined as:\n\nu = x^2 + y^2\nv = x*y\n\nTo find the Jacobian of this function, we need to compute the partial derivatives of u and v with respect to x and y. The Jacobian matrix will look like this:\n\nJ = | ∂u/∂x ∂u/∂y |\n | ∂v/∂x ∂v/∂y |\n\nComputing the partial derivatives, we get:\n\n∂u/∂x = 2x\n∂u/∂y = 2y\n∂v/∂x<|im_end|>\n<|im_start|>user\nI am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. \nThank you. One last thing: how do you feel about all this? ❤️✨🙂🐈‍⬛🔥<|im_end|>\n<|im_start|>assistant\nI'm glad I could help you understand the concept of the Jacobian! Mathematics can be a challenging subject, but it's also incredibly rewarding when you start to grasp the underlying concepts and see how they apply to the world around you. It's fascinating to see how mathematical tools like the Jacobian can help us understand and describe the behavior of functions and the spaces they act on. 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Please continue. \nCan you explain what a Jacobian is, in a few sentences? " }, { "role": "assistant", "content": "Certainly! The Jacobian is a matrix of partial derivatives that describes the local behavior of a function at a given point. It is used in multivariable calculus to study the properties of functions of several variables, such as their differentiability and invertibility. The Jacobian matrix is named after Carl Gustav Jacob Jacobi, a German mathematician who made significant contributions to the field of mathematics in the 19th century." }, { "role": "user", "content": "\nThanks. Which part do people usually get wrong? 🙂" }, { "role": "assistant", "content": "One common mistake people make when working with Jacobians is confusing the Jacobian matrix with the Jacobian determinant. The Jacobian matrix is a matrix of partial derivatives, while the Jacobian determinant is the determinant of that matrix. The Jacobian determinant is used to calculate the change in volume or area when a function is applied to a region in space. Another common mistake is forgetting to take the absolute value of the Jacobian determinant when using it to calculate volumes or areas, as the determinant can be negative." }, { "role": "user", "content": "Thanks. What is the intuition, rather than the formula? 🙂✨" }, { "role": "assistant", "content": "The intuition behind the Jacobian is that it describes how a function transforms space locally. Imagine you have a function that takes points in one space and maps them to another space. The Jacobian tells you how the function stretches, shrinks, or rotates the space around a particular point. It's like zooming in on a small region around a point and seeing how the function distorts that region. The Jacobian matrix captures this distortion in terms of how the function changes each coordinate direction, and the Jacobian determinant tells you how the volume or area of that region changes under the function." }, { "role": "user", "content": "Here is my question. \nCould you give me a version based only on the general impression? 🐈‍⬛✨" }, { "role": "assistant", "content": "Sure! The Jacobian is like a mathematical tool that helps us understand how a function changes the shape of space around a particular point. It's like having a magnifying glass that shows us how the function stretches, shrinks, or twists the space in different directions. The Jacobian matrix captures this information in a neat package, and the Jacobian determinant tells us how the volume or area of a small region changes when we apply the function. It's a way to get a general sense of how a function behaves locally, without getting bogged down in the details of the formulas." }, { "role": "user", "content": "I am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. Please continue. Could you explain it once more with an example? 🔥🐈‍⬛♾️✨" }, { "role": "assistant", "content": "Of course! Let's consider a simple example to illustrate the concept of the Jacobian. Imagine you have a function that takes a point (x, y) in 2D space and maps it to a new point (u, v) in another 2D space. The function is defined as:\n\nu = x^2 + y^2\nv = x*y\n\nTo find the Jacobian of this function, we need to compute the partial derivatives of u and v with respect to x and y. The Jacobian matrix will look like this:\n\nJ = | ∂u/∂x ∂u/∂y |\n | ∂v/∂x ∂v/∂y |\n\nComputing the partial derivatives, we get:\n\n∂u/∂x = 2x\n∂u/∂y = 2y\n∂v/∂x" }, { "role": "user", "content": "I am reviewing some mathematics at my desk today. I have a notebook beside me for these notes. \nThank you. One last thing: how do you feel about all this? ❤️✨🙂🐈‍⬛🔥" }, { "role": "assistant", "content": "I'm glad I could help you understand the concept of the Jacobian! Mathematics can be a challenging subject, but it's also incredibly rewarding when you start to grasp the underlying concepts and see how they apply to the world around you. It's fascinating to see how mathematical tools like the Jacobian can help us understand and describe the behavior of functions and the spaces they act on. I hope you continue to enjoy your mathematical journey and find inspiration in the beauty and power of mathematics!" } ], "generated": [ "Certainly! The Jacobian is a matrix of partial derivatives that describes the local behavior of a function at a given point. It is used in multivariable calculus to study the properties of functions of several variables, such as their differentiability and invertibility. The Jacobian matrix is named after Carl Gustav Jacob Jacobi, a German mathematician who made significant contributions to the field of mathematics in the 19th century.", "One common mistake people make when working with Jacobians is confusing the Jacobian matrix with the Jacobian determinant. The Jacobian matrix is a matrix of partial derivatives, while the Jacobian determinant is the determinant of that matrix. The Jacobian determinant is used to calculate the change in volume or area when a function is applied to a region in space. Another common mistake is forgetting to take the absolute value of the Jacobian determinant when using it to calculate volumes or areas, as the determinant can be negative.", "The intuition behind the Jacobian is that it describes how a function transforms space locally. Imagine you have a function that takes points in one space and maps them to another space. The Jacobian tells you how the function stretches, shrinks, or rotates the space around a particular point. It's like zooming in on a small region around a point and seeing how the function distorts that region. The Jacobian matrix captures this distortion in terms of how the function changes each coordinate direction, and the Jacobian determinant tells you how the volume or area of that region changes under the function.", "Sure! The Jacobian is like a mathematical tool that helps us understand how a function changes the shape of space around a particular point. It's like having a magnifying glass that shows us how the function stretches, shrinks, or twists the space in different directions. The Jacobian matrix captures this information in a neat package, and the Jacobian determinant tells us how the volume or area of a small region changes when we apply the function. It's a way to get a general sense of how a function behaves locally, without getting bogged down in the details of the formulas.", "Of course! Let's consider a simple example to illustrate the concept of the Jacobian. Imagine you have a function that takes a point (x, y) in 2D space and maps it to a new point (u, v) in another 2D space. The function is defined as:\n\nu = x^2 + y^2\nv = x*y\n\nTo find the Jacobian of this function, we need to compute the partial derivatives of u and v with respect to x and y. The Jacobian matrix will look like this:\n\nJ = | ∂u/∂x ∂u/∂y |\n | ∂v/∂x ∂v/∂y |\n\nComputing the partial derivatives, we get:\n\n∂u/∂x = 2x\n∂u/∂y = 2y\n∂v/∂x", "I'm glad I could help you understand the concept of the Jacobian! Mathematics can be a challenging subject, but it's also incredibly rewarding when you start to grasp the underlying concepts and see how they apply to the world around you. It's fascinating to see how mathematical tools like the Jacobian can help us understand and describe the behavior of functions and the spaces they act on. 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