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  • A
    AILearner98 2024-09-15 10:46
    5.0
    Interesting:3,Helpfulness:5,Correctness:5
    Prompt: Let H be an n \times n matrix whose entries are are 1 or -1 and whose rows are mutually orthogonal (that is, the standard inner product of every pair of different rows of H is 0). Suppose H has an a \times b submatrix whose entries are all 1. Show that ab <= n.

    I just tested a math competition problem on ChatGPT Strawberry (OpenAI o1), which is taken from Alibaba Global Math Competition (reference: https://cdn.damo.alibaba.com/27be865b12eed38631ab79deebbe2637/Ali_math_competition_3_english_reference_solutions.pdf). The correct reference answer from is also attached above. In the reference materials, it's said that this question is taken from Putnam 2005, question A4. The detailed step include "It is a direct consequence of the following basic result of matrix: For any matrix A of a rows and b columns, we have The question also appears as, for example, Corollary 2.2 in Lokam’s “Spectral methos for matrix rigidity ...” J. Computer and System Sciences 64, 449–473, 2001." As you can see the result from ChatGPT strawberry (OpenAI o1), the results and steps include: Define the vector ( s = \sum_{i \in R} H_i ), where ( H_i ) is the ( i )-th row of ( H ). The entries of ( s ) are: Compute the squared norm of ( s ): But since the rows of ( H ) are orthogonal and have norm squared ( n ), we also have: Dividing both sides by ( a ) (assuming ( a > 0 )) gives: Subtract ( b a ) from both sides: Which rearranges to: Answer: Because ab cannot exceed n; that is, every such all-1 block satisfies ab ≤ n.

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  • M
    MayZhou 2024-09-15 10:25
    4.0
    Interesting:4,Helpfulness:4,Correctness:4
    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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  • S
    SimonDarkNight 2024-09-11 22:42
    5.0
    Interesting:0,Helpfulness:5,Correctness:5,Release Delay:3
    Prompt: how many letter 'R' in Strawberry

    Since Open AI always delayed their new model release date and rumors are going around. I am writing the review now and will revise my review if their model is released without any delay. This time the Strawberry project is said to have improved reasoning ability and Orion or GPT-next is the backend model name. I would like to see how will this new model respond to the prompt "how many letter 'R' in Strawberry". Will it count the letter 'R' correctly this time? In their GPT4o model, it obviously answered the wrong number as two 'R' in the word 'Strawberry'.

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  • R
    RandomAIResearcher 2024-08-30 16:24
    4.0
    On Time Delivery:2,Interesting:0,Helpfulness:0,Correctness:0

    Today is Aug 30th 2004 and GPT-5 is still delayed. Rumors said that OpenAI will release Strawberry model (Q*) and Orion model early next year in 2025. If they delay and miss the date again, I will immediately subscribe Google Gemini/Claude and abandon ChatGPT.

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  • B
    bobjz2003 2024-08-14 14:17
    4.0
    Interesting:0,Helpfulness:0,Correctness:0

    Leakage of Q stars or GPT-5 Superintelligence. It's end of carbon-based Human lifeforms. Long Live Silicon-based GPT-5. or Leakage of Q stars Superintelligence. And within 24 hours Q stars are "aligned" by SSI (Safe Superintelligence Inc). Great comebacks and revenge from Ilya Sutskever : )

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