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 Similar Comment: http://www.deepnlp.org/store/pub/pub-openai-o1

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.

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.

I tested ChatGPT Strawberry or o1 model this high school calculus math problem, which is to "Calculate the first order derivative of function f(x)=\frac{1}{x} at the point x=2.". ChatGPT o1 gives me the precision answer in detailed step of what's the derivative of function of f(x) and what's the value at point x=2. The calculation is very detailed and helpful. By the way, I am using the huggingface model apps of OpenAI o1 API, which doesn't requires the subscription.

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'.

This humanoid female from Midjourney looks beautiful. The eyes and expressions look surreal. I would very much like to change prompts with different eye colors next time.

I used Midjourney to generate a picture of "Lamborghini Poison" in the setting of the sunset color skies. The overall results looks really cool. I would tried different colors and backgrounds of this AI generated Lamborghini Poison in more settings.

The picture of robotics in the dark backgrounds generated by Midjourney look really cool. And I like the red-color eyes of female robots in the first picture.

The picture of the female soldier wearing red headband look great. The facial expressions look amazing.

The female characters generated by mj look great. The color look correct just like the scene and tune in the Game of Throne TV series. The eyes of the characters also look sharp.

The pictures of imaginary Monkey Viking Museum looks surreal. The first picture is the cover of a brochure of the museum. And the second picture has the statues of the monkey viking soldiers. The interiors decoration of the museums look great.

The color of turquoise water in the generated sand beach image look clean and bright. And the details of the hammock between two trunks have so many details. Really Nice Picture.

I think the pictures generated seems okay. But the color tune in the second pic "iron man" seems a little bit weird, especially the color has some "red" tune in the left and the remaining seems like a mix of black and purple and the color is a little bit nasty....

The dark robots are amazing, especially some details in the red eyes of the robots. Well done!

The picture of Elon Musk as joker looks really cool and evil. The color is bright and the facial expression looks just like a joker. Nice Pic.

I like this picture of "Donald Trump dressed up as super hero" a lot. The picture has many details. The hair-style (wind-blowing hairstyle, light blonde color) looks so much like Donald Trump.

The details of lady gaga's picture is good. And the color tune is a little bit dark. It will be better if it has bright tune.

Well, the picture of professor fox at the animals academy looks good. The details in the hands are correct. The only draw back is the color is too dark and have blurs to some extent.

I think the flying house of details makes me really uncomfortable. Maybe due to the color and messy backgrounds? Not recommending this type of picture generated by midjourney. I looks like some scene coming out of Tim Burton's movies.

The basketball generated by midjourney looks extremely real. And the tones of warm color are also correct and aligned with the prompts. Great picture.

The color looks great and I really like the pink tune in the backgrounds. So I will give it a 4 star ratings.

Generate Cartoons in the wild backgrounds. It seems not very aligned. Too good enough. Prompt: pikachu, game screenshot —sref (link to a breath of the wild screenshot here)

Mansions in Alps designed for different styles. The quality of picture is good. But it's not very interesting designs and seems a little bit too plain and normal without and surprises. Prompts to Share: 1. a giant mansion in the swiss alps designed in a bauhaus style. It is morning the sun is rising. --stylize 1000 2. a giant mansion in the swiss alps designed in a post modern style made entirely of high quality different types of wood. --stylize 800 3.a photo of a giant mansion in the swiss alps designed in a functionalist style. It is evening the sun is setting. --stylize 1000 4.a giant mansion in the swiss alps designed in an ultra minimalist cyberpunk style with neon green and purple lights . It is getting dark and the sun is setting. --stylize 1000 5.a giant mansion in the swiss alps designed in the ultra minimalist style. It is getting dark and the sun is setting. --stylize 1000 6. a giant mansion in the swiss alps designed in the constructivism style. --stylize 700 7.a giant mansion in the swiss alps designed in steampunk style. --stylize 600 8.a giant mansion in the swiss alps designed in the deconstructive style. --stylize 700 9.a giant mansion in the swiss alps designed in the bauhaus style. It is morning and spring --stylize 1000

Amazing photos generated by Midjourney. I want to generate some dogs in city life the same as the ones in Paw Patrol dogs but in a realistic manners. I searched the web and find the prompts "A real life dalmation dog wearing a red firefighter suit with a red firefighter helmet. Behind him is smoke from a burning building and firemen trying to quench the flames. Photorealistic". The overall result are amazing except some details are not correct. But overall it's still stunning!!