Hello all. I have recently been playing around with a “Terminating Sequence Game” that I have created. The rules are stated below. I have a few questions located at the bottom of my post that may spark a discussion in the comments. Thank you for reading!

INTRODUCTORY / BASICS

A sequence must be in the form a(b)c(d)e…x(y)z

Examples:

• 3(1)6

• 4(3)2(1)3

• 5(0)49

• 27(2)1(4)3(3)3

• The number inside the bracket we call the bracketed value. It must be any positive integer or 0.

• The numbers outside the brackets must be >0.

RULE 1 - EXPANSION

• Look at the leftmost instance of a(b)c in our sequence. (Example, 3(2)1(0)3 )

• Rewrite it as a(b-1)a(b-1)a…a(b-1)c (with a total a’s).

• Write out the rest of the sequence. In our case example, the rest is “(0)3”.

We are now left with : 3(1)3(1)3(1)1(0)3

SPECIAL CASE

If a(b)c where b=0, replace a(b)c with the sum of a and c.

Example :

1. 3(0)5(1)5

Turns into :

2. 8(1)5

RULE 2 - REPETITION

• Repeat “Rule 1” (including the special case when required) on the previous sequence each time.

• Eventually, a sequence will come down to a single value. Meaning that a sequence “terminates”.

EXAMPLE 1 : 2(2)3

2(2)3

2(1)2(1)3

2(0)2(0)2(1)3

4(0)2(1)3

6(1)3

6(0)6(0)6(0)6(0)6(0)6(0)3

12(0)6(0)6(0)6(0)6(0)3

18(0)6(0)6(0)6(0)3

24(0)6(0)6(0)3

30(0)6(0)3

36(0)3

39

EXAMPLE 2 : 1(3)2(1)2

1(3)2(1)2

1(2)2(1)2

1(1)2(1)2

1(0)2(1)2

3(1)2

3(0)3(0)3(0)2

6(0)3(0)2

9(0)2

11

EXAMPLE 3 : 2(3)2(1)1

2(3)2(1)1

2(2)2(2)2(1)1

2(1)2(1)2(2)2(1)1

2(0)2(0)2(1)2(2)2(1)1

4(0)2(1)2(2)2(1)1

6(1)2(2)2(1)1

6(0)6(0)6(0)6(0)6(0)6(0)2(2)2(1)1

…

38(2)2(1)1

…

Eventually terminates but takes a long time to do so.

EXAMPLE 4 : 3(2)3

3(2)3

3(1)3(1)3(1)3

3(0)3(0)3(0)3(1)3(1)3

6(0)3(0)3(1)3(1)3

9(0)3(1)3(1)3

12(1)3(1)3

12(0)12(0)…(0)12(0)12(1)3 (12 total 12’s)

…

147(1)3

147(0)147(0)…(0)147(0)3 (147 total 147’s)

…

21612

CONCLUDING RESULTS :

For a sequence a(1)c, a(1)c=a²+c

if we define a function SEQUENCE(n) as being n(n)n, I can also conclude that:

SEQUENCE(1)=2

SEQUENCE(2)=38

But I cannot figure out SEQUENCE(n) for n≥3 as the values simply get too large to handle. I am wondering, what are some lower/upper bounds for this? and more interestingly, how would one prove that every sequence of a finite length terminates in a finite amount of steps (if that is the case)?

What is t? t is completely unspecified and also how can u be raised to t? u cannot be multiplied with itself I thought so this whole paragraph doesn't make any sense to me.

I am trying to wrap my head around primitive recursive functions and on the book it is not entirely explained how addition, subtraction,multiplication, factorial and exponentiation can be defined solely using the constant zero, successor and projection operators along with the operator R. Can someone explain at least one of them to me?

in a given FOL statement that is NOT syntactically correct because it contains an unbound variable, is there a way to determine its truth within the domain??? is it always true or always false???? or does it remain ambiguous because of the unbound variable. (thanks in advance, i struggle with FOL a lot)

I'm developing FuturLang, a formal language for mathematical proofs. My initial goal was maintaining everything in truth-evaluable form for easy verification:

First Attempt (pure FuturLang):

∀r,x,y(Circle(r) ∧ x²+y² = r² → y = ±√(r²-x²))

∀θ(2cos²(θ) = 1 + cos(2θ))

∀r(Circle(r) →
  Area(r) = ∫_{-r}^r (√(r²-x²) - (-√(r²-x²)))dx ∧
  Area(r) = ∫_{-r}^r 2√(r²-x²)dx ∧
  Area(r) = ∫_{-r}^r 2r√(1-x²/r²)dx ∧
  (Let x = r*sin(θ) where θ ∈ [-π/2,π/2] →
    Area(r) = ∫_{-π/2}^{π/2} 2r²√(1-sin²(θ))cos(θ)dθ) ∧
  Area(r) = ∫_{-π/2}^{π/2} 2r²cos²(θ)dθ ∧
  Area(r) = r²∫_{-π/2}^{π/2} (1+cos(2θ))dθ ∧
  Area(r) = r²[θ + (1/2)sin(2θ)]_{-π/2}^{π/2} ∧
  Area(r) = πr²)

However, I realized we could also represent proofs programmatically while preserving their logical essence:

Define CircleEquation(x,y,r: Real) := {
  Input: Circle(r)
  Output: x² + y² = r²
}

Define IntegrationBySubstitution(f,g: Function, a,b: Real) := {
  Input: Integral(f(x))dx from a to b, x = g(t)
  Output: Integral(f(g(t))*g'(t))dt from g⁻¹(a) to g⁻¹(b)
}

Define DoubleAngleCosine(θ: Real) := {
  Input: angle(θ)
  Output: 2cos²(θ) = 1 + cos(2θ)
}

Define CircleAreaTheorem(r: Real) := {
  Input: Circle(r)
  Output: Area = πr²

  Proof:
  Apply(CircleEquation) →
  (Let y = ±√(r² - x²)) ∧
  (Area = ∫(√(r² - x²) - (-√(r² - x²)))dx from -r to r) →
  (Area = ∫(2√(r² - x²))dx from -r to r) →
  (Area = ∫(2r√(1 - x²/r²))dx from -r to r)

  Apply(IntegrationBySubstitution, x = r*sin(θ), θ ∈ [-π/2, π/2]) →
  (dx = r*cos(θ)dθ) →
  (Area = ∫(2r²√(1 - sin²(θ))*cos(θ))dθ from -π/2 to π/2) →
  (Area = ∫(2r²√(cos²(θ))*cos(θ))dθ from -π/2 to π/2) →
  (Area = ∫(2r²*cos²(θ))dθ from -π/2 to π/2)

  Apply(DoubleAngleCosine) →
  (Area = r²∫(1 + cos(2θ))dθ from -π/2 to π/2) →
  (Area = r²[θ + (1/2)sin(2θ)] from -π/2 to π/2) →
  (Area = πr²)
}

This dual representation approach could offer interesting possibilities:

1. The core format maintains pure logical statements for verification and AI training
2. The functional format provides practical tools for proof development and reuse
3. Proofs could be "compiled" between representations as needed
4. Complex proof structures could be built using proofs-as-functions while maintaining logical rigor

Looking for feedback on:

1. The syntax/structure of both formats
2. Potential use cases for each representation
3. How to maintain consistency between them
4. Additional features that would be useful

The goal is to create something that's both mathematically rigorous and practically useful for proof development. All input welcome, especially from those working with proof assistants or formal verification systems!

What do you think about this dual representation approach? Can you see potential applications or issues I should consider?​​​​​​​​​​​​​​​​

I am a philosophy student, and I had a logic course in which we were introduced to the basics of first-order logic, as well as basic notions of set theory to follow the explanations.

If mathematics feels like formal logic, I definitely don't know what I'm doing in philosophy. On the other hand, the texts that I have enjoyed the most are by Tarski (the one on "what are logical notions") and some by Frege, which are especially mathematical and most of my peers have hated them.

But aside from this last point... Does mathematics feel like formal logic? If you could send me a more or less simple text that I can understand, to see if I really like mathematics, I would be very grateful.

I'm still in time to change my studies.

In the recent years, several algorithms were proposed to leverage elliptic curves for lowering the degree of a finite field and thus allow to solve discrete logairthm modulo their largest suborder/subgroup instead of the original far larger finite field. https://arxiv.org/pdf/2206.10327 in part conduct a survey about those methods. Espescially since I don’t see why a large chararcteristics would be prone to fall in the trap being listed by the paper.

I do get the whole small characteristics alogrithms complexity makes those papers unsuitable for computing discrete logarithms in finite fields of large charateristics, but what does prevent applying the descent/degree shrinking part to large characteristics ? 

Hello everyone,

I am planning to take MTL733: Stochastic of Finance in the upcoming semester (Semester 6). However, I am aware that MTL106: Introduction to Probability Theory and Stochastic Processes is a prerequisite for MTL733, and I struggled to grasp the topics when I took MTL106 in my 4th semester. As a result, I feel my foundation is weak for the advanced topics in MTL733.

To bridge this gap, I want to use my one-month winter holidays to:

1. Revise and strengthen the key concepts of MTL106.
2. Get a head start on the essential topics in MTL733.

I am looking for guidance on resources and strategies to make the most out of this time.
Here's a summary of the syllabi for both courses for context:

MTL733: Stochastic Finance

• Stochastic Processes: Brownian motion, geometric Brownian motion, Lévy processes, jump-diffusion processes.
• Advanced Concepts: Conditional expectations, martingales, Ito integrals, Ito’s formula.
• Stochastic Differential Equations: Change of measure, Girsanov theorem, Martingale Representation Theorem, Feynman-Kac theorem.
• Applications in Finance: Option pricing, interest rate derivatives, and credit risk models with Levy processes.MTL733: Stochastic FinanceStochastic Processes: Brownian motion, geometric Brownian motion, Lévy processes, jump-diffusion processes. Advanced Concepts: Conditional expectations, martingales, Ito integrals, Ito’s formula. Stochastic Differential Equations: Change of measure, Girsanov theorem, Martingale Representation Theorem, Feynman-Kac theorem. Applications in Finance: Option pricing, interest rate derivatives, and credit risk models with Levy processes.

MTL106: Introduction to Probability Theory and Stochastic Processes

• Probability Theory: Axioms, probability space, conditional probability, independence, Bayes' rule.
• Random Variables: Common discrete and continuous distributions, moments, generating functions, distribution of functions of random variables.
• Multivariate Distributions: Two and higher dimensions, order statistics, covariance, correlation coefficient, conditional expectation.
• Convergence and Limit Theorems: Modes of convergence, laws of large numbers, central limit theorem.
• Stochastic Processes: Definitions, classifications, simple Markovian processes, Gaussian and stationary processes.
• Markov Chains: Discrete and continuous-time, classification of states, limiting distributions, birth-death processes, Poisson process, steady-state and transient distributions.
• Applications: Markovian queuing models (M/M/1, M/M/1/N, etc.).

My Goals:

1. Revise and understand key topics from MTL106 (e.g., probability, Markov chains, stochastic processes).
2. Build a foundation for the advanced mathematical tools in MTL733 (e.g., martingales, stochastic differential equations).

I’d appreciate suggestions for:

• Books or online resources for self-study.
• Video lectures or tutorials that explain these concepts clearly.
• Any structured study plans to effectively tackle these topics within a month.

Thank you in advance for your help! 🙏

From any point on a circle of radius R, move a distance r towards the centre, and draw a perpendicular to your path naming it h(r). h(R) must be 2R. I have taken the initial point on the very top. If I integrate h(r)dr, the horizontal rectangles on r distance from the point of the circle of dr thickness from r = 0 to r = R I should get the area of the semi circle. Consider this area function integrating h(r)dr from r=0 to r=r' Now using the fundamental theorem of calculus, if I differentiate both the sides with respect to dR, this area function at r=R will just give h(R) And the value of the area function at r=R is πR²/2, differentiating this wrt dR would give me πR. Which means, h(R)=πR Where is the mistake?

Hi, I'm a student who did a three-year degree in statistics in Bologna, I then did a year of Computer Science (but I realized that I didn't like the idea of ​​being a software engineer too much), and now I'll perhaps finish my master's degree in statistics. Let's say that my focus would be to produce mathematical/statistical models to better understand natural phenomena, or to make predictions, so it would be ideal for me to do a PhD in applied mathematics (in which I perhaps also do new subjects that are more interesting than a PhD in statistics which is very niche and maybe not too focused on these things). I wanted to ask you if I can do it in your opinion or do I have to integrate some mathematics credits to be able to do it? Maybe I'll enroll in the three-year course in Math to get some credits? I'm not sure how it works in these cases...

Hi Folks!

Please could someone help me understand the statement at the bottom i.e., "the right hand side is log-Laplace transform of a Bernoulli distribution with parameter $\frac{1}{N}\sum_{i=1}^{N}P(\sigma_{i})=R(\theta)$". For context, the author defines:

• $P(\sigma_{i})=\mathbb{E}_{P}[\sigma_{i}]$ i.e. the expectation of the random variable $\sigma_{i}$;
• There are N $(X_{i},Y_{i})$, $\mathcal{X}$ is infinite, $\mathcal{Y}$ is infinite

Please let me know if I am missing any context.

It is taken from here if interested: https://arxiv.org/pdf/0712.0248

https://preview.redd.it/jyubqisoys3e1.png?width=753&format=png&auto=webp&s=41d35b00b1db428147c25a5fc497deef58ca90ac

i am a community college student and have recently been accepted to a 4 year college where my major will be mathematics with an emphasis in science. i can choose from a number of sciences as my focus and i chose computer science as my science emphasis. i am thinking i really want to use my degree to get a job in the computer science field or data science field. i dont want to only limit myself to those things though i am open to many possible roles in the future but im not sure how you would even beginning to branch into other fields. like if youre in data science and have a math degree, did you take online courses to give you the knowledge to land a job in data science? i guess i’m really curious in hearing how you landed in the position you’re in today with your degree in mathematics.

my confusion might be really obvious to some but im feeling nervous and excited for the future and cant wait to see where my math degree will take me. any advice you could give me will be greatly appreciated!!

∀p(z)(Polynomial(p(z)) ∧ deg(p(z)) > 0 → (∃c∈ℂ(Root(p(z), c)) ∧ ∀k(1 ≤ k ≤ deg(p(z)) → ∃c∈ℂ(RootMultiplicity(p(z), c, k)) ∧ TotalRoots(p(z)) = deg(p(z)))))

(Assume ¬∃c∈ℂ(Root(p(z), c))) → (∀z(∃s(|z| > s → |p(z)| > 2|p₀|)) ∧ ∃t(|p(t)| = min(|p(z)|, |z| ≤ s))) ∧ (Define q(z) = p(z + t)) ∧ (q(0) = q₀ = |p(t)|) ∧ (q(z) = q₀ + qₘzᵐ + ∑_{k>m} qₖzᵏ) ∧ (∃r(Choose z = r(-q₀/qₘ)^(1/m))) ∧ (q(z) = q₀ - q₀rᵐ + ∑_{k>m} qₖzᵏ) ∧ (|q(z)| < |q₀| due to geometric decay of ∑_{k>m} qₖzᵏ) ∧ (Contradiction |q(0)| = min(|q(z)|)) → ¬(¬∃c∈ℂ(Root(p(z), c))) → ∃c∈ℂ(Root(p(z), c)).

(∃c∈ℂ(Root(p(z), c))) → (∀p(z)(p(z) = (z - c)q(z) ∧ deg(q(z)) = deg(p(z)) - 1)) → (∀n(Induction(n ≥ 1 ∧ deg(p(z)) = n → p(z) has exactly n roots counting multiplicities))) → ∀p(z)(deg(p(z)) = n → TotalRoots(p(z)) = n).

Hello, I'm a physics undergrad who's thinking about switching to maths, since I have problems doings physics when so little is specified about the systems. I would switch to an applied maths degree but unfortunately here in germany we just have general math degrees at a bachelors level. I'm pretty sure I want to do research in an applied area (i like biology), as I like the thought of my work having a positive influence on the life of ppl.

So my question is: If I would switch to maths, would for example proving things also improve my ability to extract mathematics from observed natural phenomena? I'm afraid of choosing the wrong degree, because I want to use the time my brain is most malleable

Thanks for answering in advance! I would also like to hear experiences of switching from general or pure maths to applied.

Edit: Thank you all for answering, really appreciate it :)

https://anilzen.github.io/post/hyperbolic-relativity/

Can I say that because special relativity is hyperbolic, the equations in Physics used to model special relativity follow the axiomatic system of hyperbolic geometry? Does that make sense?

I know it depends on the person, but let’s compare the average engineer to the average actuary. I’m just curious because I’d like to pursue something highly quantitative.

US based. Interested in career prospects and roles I should/could target upon completion of my degree.

Some background. I have about 2 and a half experience as a Data Analyst at a big 4 consulting firm. Another year as a data analyst for the federal reserve bank (contracting). I have an undergrad degree in business and an information systems masters.

I plan on focusing my applied mathematics studies on computational mathematics.

I’m curious about what Type of roles should I be targeting given this background and even what newer opportunities may lie ahead and potential gaps to account for.

Thanks guys! Also open to other Masters in AM career change stories!

I’m trying to make this shape out of 1” thick wood. I understand it’s several equilateral triangles of any size but if this is a three-dimensional hollow object, what’s the angle of the interior miters?

solving S (x+x) dx

we can make mathematics software which reflect human thinking processes. that is no surprise.

but now i recently was thinking about another view of mathematics, how a mathematics concept can be represented using algorithms [but that algorithm may have infinite loops and division by zero]

jerk as algorithm

linear algebra algorithm

WHAT ARE THESE

i proved using series formula that the numerical algorithm for jerk and integration gives the same result, if the numerical computation could be done perfectly somehow.

may be i am going to change my view about algorithms and mathematics their relationships very soon.

because this is going to be a new insight.

Hey everyone!
I just got accepted into a master's program in AI (Coursework), and also a bit nervous. I'm currently working as an app developer, but I want to prepare myself for the math side of things before I start.

Math has never been my strong suit (I’ve always been pretty average at it), and looking at the math for linear algebra reminds me of high school math, but I’m sure it’s more complex than that. I’m kind of nervous about what’s coming, and I really want to prepare so I’m not overwhelmed when my program starts.

I still remember when I tried to join a lab for AI in robotics. They told me I just needed "basic kinematics" to prepare—and then handed me problems on robotic hand kinematics! It was such a shock, and I don’t want to go through that again when I start my Master’s.

I know they’ll cover the foundations in the first semester, but I really want to be prepared ahead of time. Does anyone know of good websites or resources where I can practice linear algebra, statistics, and probability for machine learning? Ideally, something with key answers or explanations so I can learn effectively without feeling lost.

Does anyone have recommendations for sites, tools, or strategies that could help me prepare? Thanks in advance! 🙏

I couldnt add a photo to my last post (skill issue on my part)

Hey everyone, Im new to this group and as my title said my teacher has told me i am doing long divison the hard way and should break it down into smaller steps. The issue is i cannot comprehend in the slightest how to do it the way its taught (im autistic and they just are not breaking it down the way i need despite me asking) i was wondering if someone would be able to tell me if i did this equation correctly. (The equation is 1,375 ÷ 12.5) Ive went ahead and decided to solve it to the third decimal point (i think thats what its called, not too sure) I added the photo to show my steps.

Our theory of machines professor wants a small 2 page research about this theory and the sources have to be from mathematical books.

Do you guys know what is the name of the hat Fibonacci is wearing?? I cannot find any of these in the internet lol