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  • M
    maththug 2024-09-18 12:22
    4.0
    Interesting:3,Helpfulness:4,Correctness:4
    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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  • M
    MayZhou 2024-09-14 12:33
    4.0
    Interesting:3,Helpfulness:4,Correctness:5
    Prompt: Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step.

    I asked Gemini to solve this graduate-level machine learning math problem "Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step". And Gemini provides correct and concise answers to this question. It breaks down into a few steps: 1. Initial Guess, which is to Let's start with an initial guess, say x 0=1. This is a reasonable guess since 1^2 is close to 2. 2. Newton's Method Formula 3. Iterations 4. Convergence and get the conclusion Newton's method converges quadratically, which means that the number of correct digits approximately doubles with each iteration. As we continue iterating, we'll get closer and closer to the exact value of \sqrt{2}.

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  • M
    MayZhou 2024-09-14 12:27
    4.0
    Interesting:4,Concise:4,Helpfulness:4,Correctness:5
    Prompt: Please Calculate the first order derivative of function f(x)=\frac{1}{x} at the point x=2.

    Gemini provides the correct answer to my high-school level math question about calculus "Calculate the first order derivative of function f(x)=\frac{1}{x} at the point x=2.". It uses the power rule to get the derivative function f'(x) and calculate its value at point x=2. And gets the correct answer -1/4. The answer is helpful, concise and easy to understand.

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  • M
    MayZhou 2024-09-14 12:22
    4.0
    Interesting:3,Concise:3,Helpfulness:4,Correctness:5
    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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  • A
    ai4science03 2024-09-13 08:44
    5.0
    Interesting:3,Helpfulness:5,Correctness:5
    Prompt: Let n be an even positive integer. Let p be a monic, real polynomial of degree 2n; that is to say, p(x)=x^{2n} + a_{2n-1}x^{2n-1} + ... + a_{1}x+ a_{0} for some real coefficients a_{0}, a_{1}, ..., a_{2n-1}. Suppose that p(1/k) = k^{2} for all integers k such as 1<=|k|<=n. Find all other real numbers x for which p(1/x)=x^2.

    Finally, OpenAI released o1 mdoel with stronger reasoning ability. And I looked through the detailed comparison of a math solving results on their website and the comparison between GPT4o vs OpenAI o1-preview on this Algebra problem. For the math question as the in the prompt, o1 uses a chain of thought when attempting to solve a problem, which is similar to how a human may think for a long time before responding to a difficult question. o1 response actually break down the question into a few steps: "Understanding the Given Information", "Defining a New Polynomial", "Properties of q(x)", "Constructing s(x)", "Matching Coefficients". Finding Additional Solutions and finally reaching Conclusion. The additional real numbers x satisfying p(1/x) = x^{2} are x=\frac{1}{n!} or -\frac{1}{n!}. Overall, the reasoning ability is quite complex compared to previous version, so it's helpful and the answers are correct.

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