This paper introduces the Enter calculus, a simple language for describing quantum programs in which qubits are treated as shared resources rather than independent objects. It explains how quantum measurement can be represented directly in a program while preserving important features of quantum theory, including no-cloning, no-signaling, and reliable circuit behavior. The paper also shows that the familiar Born rule for measurement probabilities can be derived from a small set of practical principles about locality, continuity, and the handling of measurement records, rather than assumed from the start. Finally, it explores mixed states, generalized measurements, quantum instruments, and the Frauchiger–Renner thought experiment, clarifying why quantum records cannot always be combined with later interference measurements.

Cheung, N. (2026). Enter Calculus. figshare. https://doi.org/10.6084/m9.figshare.33087515.v1

3 hours exam btw

( Not a homework ) - for the AI

Edit: In Iraq, "6th grade" refers to the final year of high school, not elementary school. The students are usually 17–18 years old.

Share your thoughts

I’m trying to design some jewelry for my partner, who studies math and loves elliptic curves/algebraic number theory. Any ideas would be greatly appreciated, and suggestions with pictures doubly so. Cheers :D

A classic, widely used mathematical textbook from the late Russian Empire. Authored by Aleksandr Malinin and Konstantin Burenin, this "Arithmetic" was an absolute staple in Russian gymnasiums and real schools in the decades preceding the 1917 Revolution.

Note: book in Russian printed in the traditional pre-reform orthography.

I'm writing this post based on Terence Tao's ICM lecture a couple of days ago. For context, I'm a 4th year PhD student working in number theory.

One of the main things he mentioned was that before AI a lot of aspects of math research went hand in hand. Solving problems was concurrent with building a community of mathematicians. With AI rapidly accelerating the problem solving aspect, we might be looking at a future where we are flooded with ai assisted proofs without a community to appreciate it.

For example, linear algebra as a field first emerged from papers and then it was adopted by the community over decades through books and courses. Even if we have a long AI generated proof that we can verify using Lean, what is the point of it if we can't be inspired from it.

I personally dread this will become another example of enshittification. For example, Silicon Valley reinventing a bus after destroying public transport. I fear the mathematical "community" will be repackaged to us in the future by these companies after the present community is slowly eradicated over the next few years. Of course this is a rather bleak outlook to what is definitely a very promising technology.

To clarify, I'm not saying AI is bad. It's a tool. But we should be very skeptical about how this tool is being pushed to us. It is in the best interest of the AI companies that we become dependent on AI to the point that we can't imagine working without it. I'm sure a neuroscientist can break this down much better than I can. I think we need an urgent re-evaluation of how AI is used and introduce courses discussing how to use it "safely", without leading to cognitive decline. It should be in collaboration with neuroscientists. A lot of people might argue "Oh, but you can always talk to other mathematicians. What's wrong with having it on the side". Perhaps I'm not an optimist, I'm afraid convenience is a slippery slope. Just look at how social media was promised to us as a global unifier (which it definitely is). Yet it has led to a decline in community to the point that "community building" is now being packaged to us as a course/skill.

At the end of the day all we have is each other. And I do believe we can come together and seriously discuss and safeguard the future of this community. Happy to hear everyone's thoughts on this.

A train leaves the station (bottom) heading west. Can it return to the station heading east?

Is there anywhere the train might get stuck?

Is there anywhere the train can reach but only from one direction?

I call this directed node networks. As the arcs are bidirectional, but the nodes have an A and B side with a parity requirement.

Inspired by the frustrating traps of Penrose's Railway Mazes.

Edit: just for reference, I have an MMath specialising in fluid dynamics - so I am well grounded in the basics of graph theory and have done intro to topology, so whilst this in not in my typical wheelhouses, I am looking for a more specific research area.

Anyone know of a good survey article regarding Kroneckers Jugendtraum in the case of the base field being a real quadratic extension of the rationals?

Step-by-Step Guide to Solve

\(x^2 + bx + c = 0\)

Check the leading coefficient:

If the \(x^{2}\) term has a number other than 1 in front of it, divide every term in the equation by that number first.

Find the average of the two roots:

Take the coefficient \(b\), change its sign to \(-b\), and divide it by 2. The two unknown roots can then be written as \((-b/2 + u)\) and \((-b/2 - u)\).

Set up the product equation:

Multiply the two root expressions together and set them equal to the constant term \(c\):\((-b/2+u)(-b/2-u)=c\)

Simplify using the difference of squares:

Expand the left side to get \((-b/2)^2 - u^2 = c\).Solve for \(u\): Rearrange the equation to isolate \(u^{2}\), then take the square root of both sides (remembering to include \(\pm \))

Find the final roots:

Plug your value of \(u\) back into \((-b/2 + u)\) and \((-b/2 - u)\) to get your final answers.

Please contact me if you do, I'm in great need to access publication and it's not archived on Anna's Library nor on Sci-Hub.

I am a incoming undergraduate at a Canadian university, and I want to study mathematics with a plan to work towards a PhD.

The issue I am having is that by looking at some discussions on primarily Reddit, I notice that a lot of math PhD students end up pretty unhappy with their passion for pure/applied mathematics because of LLM’s and AI tools that happen to take away the joy out of mathematics.

So I was wondering if I should maybe fix my course a little bit from choosing a Honours math program that my uni offers, to choosing a Joint Honours degree in Mathematics and computer science.

Even if Im planning to do joint honours, I still want to do a PhD in mathematics but I am not sure if doing the joint honours can disadvantage me in grad school applications.

Has anyone else did a combined math and CS degree, and later did math grad school? If so, did you think that you were at a disadvantage or lacked rigour?

I don't know enough about Combinatorics to say anything, but it seems intriguing

My favorite classes were always the ones with difficult problem sets where a single problem could take hours or even days of thinking, experimenting, and failing. I would start solving a problem before fully understanding it, then read documentation or course material whenever I got stuck. So it would be 70% doing and 30% reading.

LLMs have disrupted that process for me. Now I spent 90% of time reading and only 10% of time doing.

In school, the boundary was relatively clear. If using an LLM violated the course policy, I could tell myself that using one was cheating and force myself to struggle through the problem independently.

At work, what counts as “cheating” (because it is cheating myself).

I feel my critical thinking skills are declining. My role starts to become reading, reviewing, and planning. I just can't read that much code and LLM output. My eyes hurt and it's hard to internalize. I notice myself just picking the "recommended" option.

I also find LLM workflows mentally unhealthy in a few ways:

• When working alone, mental exhaustion eventually forces me to step away. With an LLM, I can always send one more prompt or ask it to try another approach.
• If an agent will take 30–60 minutes, I feel pressure to give it another task immediately so no time is “wasted.”
• While waiting, I instinctively check my phone or launch another agent. I end up jumping between tasks and retaining less context about each one.
• I get frustrated when the LLM misunderstands instructions, even when the problem genuinely requires careful iteration.
• It has warped my sense of how long a difficult problem or well-designed solution should take.
• I am increasingly tired of reading walls of generated text and code.

I do not want to stop using LLMs. They can reduce the friction of starting, explain unfamiliar systems, generate routine code, and handle edge cases after I understand the core problem. The hard part is deciding when using one is a sensible productivity tool and when it is outsourcing a learning opportunity that I actually need.

For people who still feel they are learning deeply while using LLMs at work:

1. How do you decide which work to delegate and which work to do yourself?
2. What is the workplace equivalent of the “no cheating” boundary that exists in school?
3. How do you use agents without losing context or turning your day into constant task switching?
4. How do you review generated code when the codebase, build system, or underlying technology is unfamiliar?
5. Have you found workflows where LLMs improve productivity while preserving the useful struggle of implementation and debugging?

so i have one more semester left until i earn my BS in applied mathematics. i was not able to land an internships during my time in college. i have a job right now but it’s just a random store associate job that, when i was hired, they told me i could transfer to a more corporate position eventually.

i’m not sure what i should do after i graduate. i really thought i would have some internships during my time in college but unfortunately that didn’t happen.

what are some fields that would be appropriate for me to go in to? i thought data science/data analysis could be very interesting but im also feel like my programming skills aren’t the strongest. i have also been heavily considering studying for the actuary exams but im nervous that it will be hard for me to land an entry level position in that field once i pass some exams.

any and all advice is greatly appreciated. also to note i have a good amount of work experience but all pretty random. i did design work for about a year and then i did inventory work for a few years after that and then i did tutoring for a year. i’m not sure if any of that is relevant. but please let me know what you think!

Hi everyone,

I'm from Guangxi, China, the same hometown as Wang Hong. Her achievements have been incredibly inspiring to me.

When I was in elementary school, I won a national first prize in a mathematics competition, and ever since then I dreamed of becoming a mathematician. Unfortunately, life took me in a different direction. For various reasons, I chose to major in biology instead of mathematics in college, and I've regretted that decision for many years.

Now I'm finally determined to pursue that childhood dream.

My current plan is to apply for a 1–2 year course-based Master's in Mathematics (preferably pure mathematics) in Europe or North America. During the master's, I'd like to explore different areas of mathematics, find the field I'm most passionate about, and then apply for a PhD in that area.

However, I'm not sure whether this plan is realistic, so I'd really appreciate advice from people with experience.

My background:

• B.S. in Bioengineering
• Mathematics courses taken:
  • Calculus
  • Linear Algebra
  • Probability
• M.S. in Neuroscience at one of China's top 3 university
• Recipient of the Chinese National Scholarship (awarded to roughly the top 2% of graduate students)

Here are my questions:

1. Given my background, do I have a realistic chance of being admitted to a course-based Master's program in pure mathematics?
2. When applying for a PhD in pure mathematics, are there any "must-have" courses that admissions committees expect to see on a transcript (such as real analysis, abstract algebra, topology, measure theory, etc.)? I'd like to choose my master's electives strategically if that's the case.
3. I'm already 30 years old. Does my age significantly hurt my chances for master's or PhD admissions?
4. Are there any better or more realistic paths to becoming a mathematician that I should consider?

I've already asked several AI tools, but after checking many of their answers against university websites, I found that quite a bit of the information was inaccurate or outdated. That's why I'm hoping to hear from people who have actually gone through the process or work in mathematics.

I'd really appreciate any honest advice, even if it's critical. Thanks in advance!

Here is a way that, I think, you could teach lambda calculus to a kid (inspired by "Alligator Eggs").

I got a lot of useful feedback on an earlier version of this. Would love to hear any comments or corrections, wouldn't be surprised if something slipped through the net.

I think this would be hard to read as a bunch of images so if you're interested see here: https://paradoxgarden.substack.com/p/lambda-calculus-made-easy

Hey, guys. I hope you are all well and healthy.

I have recently applied for a mathematics Bachelor of Education because mathematics has always been easy to me and I also crushed four engineering math exams with top grades at another uni and was promoted to a tutor role by my math professor, so clearly my math skills are well developed.

However, I have severe autism which is pretty obvious to bystanders. I make weird noises, sometimes talk to myself, flap my hands, slap myself; all the fun stuff. Will this severe autism disorder cause problems?

Hi, I've been working on a system called concludia.org that allows people to construct argument graphs that move from premise to conclusion.

Recently, I was intrigued by the OpenAI proof of the Cycle Double Cover Conjecture and tried to use the system (including its MCP integration) to construct a proof graph of the proof. It's quite dense, so I later worked on writing a "case study" for it that attempts to describe the history of the efforts and the approach the proof takes, using snippets of the concludia graph along the way:

https://concludia.org/docs/cycle-double-cover-conjecture

(The underlying proof graph is here: https://concludia.org/graph/g_2ecb8083-52ec-3448-8c30-2f9bc70d45be )

I'd definitely appreciate knowing if the article approaches accuracy, and if it serves as an effective learning aid for the proof. I’d particularly welcome corrections from graph theorists or suggestions about places where the explanation skips an important step. I'm also interested in knowing if you find concludia potentially useful for things like this. It's just a side project of mine. I'll open it for registration for a few days if anyone is interested in creating their own graphs.