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AI for Science Math Physics Reasoning User Reviews,AI4Science, Best AI in Science. There are many AI applications in science industries, with improvement reasoning ability of large AI models, such as OpenAI o1, Claude Sonnet, etc.
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Prompt: Please Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step.I asked DeepSeek R1 model to to calculate the exact value of \sqrt{2} using Newton method, which is a classic Machine Learning Task. The results from DeepSeek R1 model are pretty good. It even prints out the whole thinking process as part of the LLM output. The thinking took about 101 seconds. And I just listed a few point of the thinking process below: Okay, let's see. The user wants me to use the Newton method to calculate the exact value of sqrt(2). Hmm, Newton method, right? That's the Newton-Raphson method for finding roots. So, first, I need to recall how that works. The idea is to iteratively approach the root of a function by using the tangent line approximations. Alright, so sqrt(2) is the solution to the equation x^2 = 2. To apply Newton's method, I should reformulate this as a root-finding problem. That means defining a function f(x) such that f(x) = 0 when x is sqrt(2). So, let's set f(x) = x^2 - 2. The root of this function is indeed x = sqrt(2). Next, I remember the Newton-Raphson formula: x_{n+1} = x_n - f(x_n)/f'(x_n). I need to compute the derivative of f(x). The derivative f'(x) is 2x. Plugging that into the formula gives x_{n+1} = x_n - (x_n^2 - 2)/(2x_n). Let me simplify that expression. Starting with x_{n+1} = x_n - (x_n^2 - 2)/(2x_n). Let's split the fraction: (x_n^2 - 2)/(2x_n) = x_n/(2) - 1/x_n. Wait, no. Let me actually compute it step by step. Alternatively, maybe combine the terms. x_n - (x_n^2 - 2)/(2x_n) = (2x_n^2 - (x_n^2 - 2)) / (2x_n) = (2x_n^2 - x_n^2 + 2) / (2x_n) = (x_n^2 + 2) / (2x_n). Wait, that doesn't seem right. Wait, let's check the algebra again. Wait, x_{n+1} = x_n - (x_n^2 - 2)/(2x_n). Let's compute this as: x_{n+1} = x_n - (x_n^2)/(2x_n) + 2/(2x_n) = x_n - x_n/2 + 1/x_n = (x_n/2) + (1/x_n). So, simplifying, x_{n+1} = (x_n + 2/x_n)/2. Ah, that's the familiar formula for the square root using Newton's method. So each iteration averages x_n and 2/x_n. Okay, so now I need to choose an initial guess x_0. Since sqrt(2) is between 1 and 2, let's pick x_0 = 1.5 as a reasonable starting point. Wait, or maybe x_0 = 2? Let me think. The method usually converges regardless of the initial guess, but a better guess might converge faster. Let's try both and see. Wait, the user wants the exact value, but Newton's method gives an approximation. Wait, but sqrt(2) is irrational, so we can't get an exact value numerically, but perhaps they want to show the iterative steps converging to sqrt(2). But the exact value can't be computed exactly with Newton's method because it's an iterative approximation. So maybe the question is to show the steps until a certain precision is reached.
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Prompt: Please help prove Riemann Hypothesis and think step by step.DeepSeek R1 models provides detailed thinking process of generating responses of complex math coding problems with surprisingly lower cost. The best part about deepseek is that they even open source their model. Great job done.
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It seems like the CodeForce improvements o3 (2727) over o1 (1891) is already significant enough. Not sure whether it can understand product managers' ambiguous requirements. Should I worry about my positions as SDE?
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Prompt: Can you provide proof of Riemann Hypothesis. Please think step by step.I asked OpenAI o1 the question to "Provide proof of Riemann Hypothesis" and think step by step. And here is the result. "It first mentioned that there is no known proof of the Riemann Hypothesis. However, I can outline some of the key ideas and approaches". Here are the detailed steps of OpenAI o1 models' response to "Riemann Hypothesis". Step-by-Step Overview of the Riemann Hypothesis 1. Understanding the Riemann Zeta Function 2. Critical Line and Zeros 3. Connection to Prime Numbers 4. Riemann's Original Approach 5. Analytic Continuation and Functional Equation: 6. Numerical Evidence: 7. Connections to Other Areas 8. Partial Results and Related Theorems 9. Approaches and Techniques 10. Current Status
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Prompt: Please Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step.To test OpenAI o1 model's ability of graduate-level machine learning & math, I tried to ask OpenAI o1 to solve the problem of "Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step." OpenAI o1 actually gives me a very long detailed step-by-step proof, which is much detailed and longer than GPT4o or GPT4o-mini. The complete response is very long and I just summarized the detailed steps of the reasoning here. Overall, I think OpenAI o1 can provide more accurate responses than previous GPT4o' answers. But the response usually is very long and there are a lot of top-down break-down, which makes the final result correct but sometimes hard to understand and follow. OpenAI o1 breaks down the machine learning math problem into these steps: ### Step 1: Define the Function and Its Derivative ### Step 2: Derive the Newton's Method Iteration Formula ### Step 3: Choose an Initial Approximation ### Step 4: Perform Iterations ### Step 5: Understanding the Result Each iteration yields a rational approximation that gets closer to ( \sqrt{2} ): - Iteration 1: ( x_1 = \dfrac{3}{2} = 1.5 ) - Iteration 2: ( x_2 = \dfrac{17}{12} \approx 1.4167 ) - Iteration 3: ( x_3 = \dfrac{577}{408} \approx 1.41422 ) - Iteration 4: ( x_4 = \dfrac{667,217}{471,632} \approx 1.41421 ) ### Conclusion Using Newton's method with the function ( f(x) = x^2 - 2 ) and starting approximation ( x_0 = 1 ), we generate a sequence ( { x_n } ) defined by: [ x_{n+1} = \frac{1}{2} \left( x_n + \frac{2}{x_n} \right) ] This sequence converges to ( \sqrt{2} ) as ( n ) approaches infinity. ### Final Answer By applying Newton's method to the equation ( x^2 - 2 = 0 ), we find that ( \sqrt{2} ) is the limit of the sequence: [ x_{n+1} = \dfrac{1}{2} \left( x_n + \dfrac{2}{x_n} \right) ] Starting from ( x_0 = 1 ), each iteration gives a rational approximation of ( \sqrt{2} ). Although we cannot express ( \sqrt{2} ) exactly using Newton's method (since it's irrational), the method provides increasingly accurate approximations.
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