/r/math

This subreddit is for discussion of mathematics. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.

This subreddit is for discussion of mathematics. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.

**Please read the FAQ before posting.**

**Rule 1: Stay on-topic**

All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.

In particular, any political discussion on /r/math should be directly related to mathematics - all threads and comments should be about concrete events and how they affect mathematics. Please avoid derailing such discussions into *general political discussion*, and report any comments that do so.

**Rule 2: Questions should spark discussion**

Questions on /r/math should spark discussion. For example, if you think your question can be answered quickly, you should instead post it in the Quick Questions thread.

Requests for calculation or estimation of real-world problems and values are best suited for the Quick Questions thread, /r/askmath or /r/theydidthemath.

If you're asking for help learning/understanding something mathematical, post in the Quick Questions thread or /r/learnmath. This includes reference requests - also see our list of free online resources and recommended books.

**Rule 3: No homework problems**

Homework problems, practice problems, and similar questions should be directed to /r/learnmath, /r/homeworkhelp or /r/cheatatmathhomework. Do not ask or answer this type of question in /r/math. If you ask for help cheating, you will be banned.

**Rule 4: No career or education related questions**

If you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread.

**Rule 5: No low-effort image/video posts**

Image/Video posts should be on-topic and should promote discussion. Memes and similar content are not permitted.

If you upload an image or video, *you must explain why it is relevant* by posting a comment providing additional information that prompts discussion.

**Rule 6: Be excellent to each other**

Do not troll, insult, antagonize, or otherwise harass. This includes not only comments directed at users of /r/math, but at any person or group of people (e.g. racism, sexism, homophobia, hate speech, etc.).

Unnecessarily combative or unkind comments may result in an immediate ban.

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Filters: Hide Image Posts Show All Posts

**Recurring Threads and Resources**

*What Are You Working On?* - every Monday

*Discussing Living Proof* - every Tuesday

*Quick Questions* - every Wednesday

*Career and Education Questions* - every Thursday

*This Week I Learned* - every Friday

*A Compilation of Free, Online Math Resources*.

*Click here to chat with us on IRC!*

**Using LaTeX**

To view LaTeX on reddit, install *one* of the following:

MathJax userscript (userscripts need Greasemonkey, Tampermonkey or similar)

TeX all the things Chrome extension (configure inline math to use [*;* *;*] delimiters)

`[; e^{\pi i} + 1 = 0 ;]`

Post the equation above like this:

`[*;* e^{\pi i}+1=0 *;*]`

**Using Superscripts and Subscripts**

x*_sub_* makes x*sub*

x*`sup`* and x^(sup) both make x^{sup}

x*_sub_`sup`* makes x*sub*`sup`

x*`sup`_sub_* makes x`sup`

*sub*

**Useful Symbols**

Basic Math Symbols

≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °

Geometry Symbols

∠ ∟ ° ≅ ~ ‖ ⟂ ⫛

Algebra Symbols

≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗 ⊲ ⊳

Set Theory Symbols

∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟

Logic Symbols

¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣

Calculus and Analysis Symbols

∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ

Greek Letters

**Other Subreddits**

**Math**

- /r/learnmath
- /r/mathbooks
- /r/cheatatmathhomework
- /r/matheducation
- /r/casualmath
- /r/puremathematics
- /r/mathpics
- /r/mathriddles
- /r/mathmemes

**Tools**

**Related fields**

/r/math

6

I know it is a tool that is currently being applied to make computations in integral p-adic Hodge theory, but what do we ultimately hope to gain from this seemingly peculiar construction?

Lars Hesselholdt has a paper where he computes the zeta function of a smooth projective variety over finite fields using negative cyclic homology, can we hope a variant thereof would give a cohomological interpretation of the Riemann Zeta function?

Educate me.

2 Comments

2024/11/07

23:55 UTC

23:55 UTC

1

Summary: I have looked everywhere, and so far, can't find a satisfactory explanation of one aspect of Theorem 9.8-3 in Kreyszig "Functional Analysis with Applications": in particular, what is the motivation for using the **"positive part"** of the operator T_lambda instead of T_lambda itself, and what goes wrong if we had just used T_lamba itself instead?

here is a screenshot of the theorem in question:

where the "positive part" is defined as T^+ = 1/2 (B+T) where B = (T^2)^(1/2).

my question is: why is the "positive part" used?

If we consider just the operator T_lambda itself, then the projection onto the nullspace of T-lambdaI is just the projection onto the "eigenspace" (with the caveat that lambda could be a non-eigenvalue spectral value i.e. in the continuous spectrum of T) corresponding to the spectral value lambda; this to me is somewhat reminiscent of the finite-dimensional case; which involves a sum over projections onto the eigenspaces of T.

So why is the additional definition of the "positive part" of this operator needed? What goes wrong when we use the nullspace of T_lambda itself; why do we need to use the "positive part" operator T_lambda^+ ?

0 Comments

2024/11/07

23:40 UTC

23:40 UTC

1

I am a grad (ms) student in mathematics. I have recently got to know about TDA and found it interesting. I want to get a research paper published by the time I complete my MS. I want opinions on whether it would be a good idea to do research in the direction of TDA. I also am looking for suggestions on some research topics involving TDA and Machine learning that I can work on and complete in 2/2.5 semesters. Thank you.

0 Comments

2024/11/07

22:46 UTC

22:46 UTC

63

So at my university, there's this math library with a small bookshelf that said "free books." One of the books that caught my attention was this one in Russian (*Asymptotic methods for linear ordinary differential equations* by M.V. Fedoryuk), so I picked it up and put it in my backpack thinking it would be a cool book to just keep around cause why not (I can't even read Russian💀). I noticed that it had a few papers in it but didn't think much of it until I got home and pulled out the papers. To my surprise, it includes the following:

- A handwritten letter from January 1995. It said "FEDORYUK. According to Sergai Slavyanov, Fedoryuk died when he fell off a railway platform and landed on the back of his head. Might have been pushed? Mightc have had hangover?"
- A printed out email with a bunch of people from Harvard, MIT, Berkeley, Duke, etc. (i can provide the names but i dont wanna dox people from like 30 years ago) just saying that they saw a sticker on a truck that said "JESUS IS COMING: LOOK BUSY" that weekend. A few of the recipients have passed away but a couple are still alive and well.
- I have no idea how this ended up at my university (UMN)

- A few summaries of the book from Springer.
- A handwritten summary that has the table of contents.
- What I find most interesting: Inside the book, someone wrote "To professor F.W.J Olver with compliments from the author." Unfortunately, I couldn't find anything from Fedoryuk himself :(
- The book was valued at $130 at the time and I feel like a thief when I found out because it was free

2 Comments

2024/11/08

02:57 UTC

02:57 UTC

20

I've been reading Reid's Undergraduate Commutative Algebra, and it's been a really enjoyable read. I appreciate the effort he puts into motivating the development of ring and module theory. The use of first person is unusual, but this is one of the few cases where I don't mind seeing so much of the author's personality. And Atiyah and MacDonald's exceedingly terse text feels somewhat more penetrable after reading this book.

That said, the book teases and hints at a lot of advanced math, which is cool but frustrating. For instance, in the opening chapters he draws a couple of pictures of things like Spec k[X,Y] and Spec Z[X]. He tells the reader that it's okay if you don't understand these pictures at this point.

I feel like a novitiate and I'm being shown a Zen riddle by a master, who tells me that I will understand it someday when I'm enlightened enough. What does it take to really understand these pictures? Do algebraic geometers each have their own way of visualizing prime spectra? (Specifically, are these pictures just trying to depict the Zariski topology, or is it deeper than that?)

Isn't there a really famous rendition of Spec Z[X] in Mumford's Red Book?

20 Comments

2024/11/07

23:11 UTC

23:11 UTC

0

It makes so much more sense to me. Even as a small child I have thought about this. I know that converting would have some push back, but we went metric. Again, I don't think we should have.

53 Comments

2024/11/07

21:39 UTC

21:39 UTC

0

When thinking about people making good and bad predictions, I got to wondering how one could design a system of rating people's predictions and the corresponding observation about whether they were right.

To make things simple, let's say that a person has a probability distribution p over a random variable X, which is a prediction of a future event. So for example X could be a roll of a die with support {1,2,3,4,5,6} and they believe the die is biased with probability distribution 1/10, 1/10, 1/10, 1/10, 1/10, 1/2.

We roll the die once and the outcome is observed.

How do we rate the credibility of the person based on their probability distribution and whatever the observed result is? I.e. how can we assign them a score for their prediction, with the following obviously desirable properties.

They earn a positive score when their prediction seems to agree with the outcome, however that is measured.

If they assign to event X=i the probability 50%, then they neither earn nor lose points if X=i occurs.

The score should somehow reflect the value of the person as a source of information and prediction. The two points above are not the only important considerations -- another would be informativeness of the prediction. For example, if someone predicts that the sun will come up tomorrow, that's not very informative, so you shouldn't earn many points when the sun rises. But I'll think about how to incorporate those kinds of considerations later.

Is this the kind of thing people have already studied?

3 Comments

2024/11/07

21:23 UTC

21:23 UTC

52

i’m in calc 2 and i know all these cool methods of integration - integration by parts, partial fractions, and so on. We also have the power rule, and other rules to actually make antiderivative easier.

But when newton and liebniz did integration - did THEY have those tools? If not, how was area computed then? I watched a video saying newton used the power rule when finding an approximation for pi

24 Comments

2024/11/07

20:13 UTC

20:13 UTC

12

I’m curious how people remain engaged with math if they’re not in academia. For context, I have very solid math skills — I took very advanced coursework at a top 5 university and published some papers in reputable journals as an undergrad — but I graduated a number of years ago and now work in industry without much math application. Lately, I’ve been missing the feeling from doing math coursework and research, but don’t have a good idea for how to start back up again.

If you don’t have a connection with an advisor, how do you find interesting problems for research? Some textbooks have open problems, but I figure that they’re either too hard to approach or too easy that they’ve already been solved since the book’s publication. I’m aware of some specific books that contain open problems, but I don’t have a good criterion for discerning which problems are good to tackle.

Would appreciate any advice.

4 Comments

2024/11/07

18:35 UTC

18:35 UTC

102

Sometimes I fantasize about creating a whole new field in mathematics, with some cool name (algebraic probability ?) that would attract fellow mathematicians to actually consider it as interesting and worthy, I am wondering if this is normal or I am just spending a lot of time thinking about mathematics.

34 Comments

2024/11/07

19:54 UTC

19:54 UTC

8

I know that alphaproof is not available to the public. Are there any open source projects for lean or coq (for example ) to integrate automated provers such as z3 so that theorem proving can be at least semi-automated?

10 Comments

2024/11/07

19:52 UTC

19:52 UTC

52

My understanding of the Seifert-van Kampen theorem is that for two spaces U and V, pi_1(U \cup V) can be written as a free product of pi_1(U) and pi_1(V), modulo pi_1(U \cap V). Intuitively, the free product is the naive way of combining the fundamental groups of two spaces, but it leads to overcounting the loops, so we then rein in our guess by quotienting out the stuff we double counted.

This feels remarkably similar to the inclusion-exclusion theorem, that |A \cup B| = |A| + |B| - |A \cap B|. Or the similar theorem for vector spaces, that dim(U + V) = dim(U) + dim(V) - dim(U \cap V). Is my intuition that these are related correct? Is there some broader way of generalizing these notions?

8 Comments

2024/11/07

18:34 UTC

18:34 UTC

1

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#DataScience #BiomedicalScience #BiomedicalResearch #GraduateProgram #StJude

0 Comments

2024/11/07

15:46 UTC

15:46 UTC

4

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

2 Comments

2024/11/07

17:00 UTC

17:00 UTC

14

It's not a big secret that good books on the history of relatively modern mathematics are few and far between. Sure, there are some memoirs, autobiographies, overviews of some particular fields, collections of anecdotes, and a few books on the history of mathematics in general, but little of what professional historians would call a serious history text — something that would concern the institutions, politics, economics, and other extra-mathematical contexts involved in the development of modern mathematics as a historically-grounded enterprise.

This probably shouldn't come as too big of a surprise given the comparatively small number of academic mathematicians, the seemingly parochial, obscure, esoteric nature of the field in the eyes of historians, and the fact that few of the working professionals would have enough of historical “knack” to write a reliable history.

Yet still, there are many questions that could be easily asked and less easily answered regarding the every day matters of institutional mathematics.

How would they justify themselves to the government in the matters of, say, funding? How would they justify themselves to the universities? How did they attract students to the programs? What were the typical career paths of math students in, say, mid-20th century? What was the demographics of math departments: age, class, gender? What was expected to know from a freshman, a bachelor, phd candidate? When, why, where the pure math programs were created? How do external factors come into play — is it an accident, for example, that planned-economy era soviet mathematicians were dominant in optimisation and probability? And so on, and so on, and so on.

If you have any readings that could shed some light on those matters, any resources, even if indirect (personal diaries, biographies, statistics, old reports, etc), I'd be immensely thankful if you share them here.

Anything in major european languages is fine, though english language materials are preferable.

4 Comments

2024/11/07

11:47 UTC

11:47 UTC

58

Some background: I’m a master’s student in mathematics, and during my bachelor’s degree, I took a course on stochastic calculus. I took it because I enjoyed both measure theory and measure-theoretic probability theory and was interested in seeing how they are used, for example, in mathematical finance. However, I found the course more difficult than any other that I had taken up to that point, so much so that I decided to drop it halfway through. Concretely, I had a hard time keeping track of all the very technical definitions and developing an intuition for the presented concepts.

Fast forward to my master’s studies, and I chose to take a course on numerical methods for mathematical finance, with only measure-theoretic probability theory as a prerequisite. Halfway through the course, we started discussing the basics of Brownian motion, stochastic calculus and SDEs, and once again, I found myself struggling in the same way I did during the stochastic calculus course.

All of this has got me wondering if maybe stochastic calculus “isn’t for me.” Has anyone else had similar experiences?

27 Comments

2024/11/07

08:06 UTC

08:06 UTC

1

For a long time, I've been intrigued by these. I've also seen many charts of both, including one showing all 230 3D space groups! However, I really don't know much about these groups. Is there a general formula for the analogous numbers of such groups of arbitrary dimensions? If not, what's the largest dimension for which they're known? Is there a systematic algorithm for finding these groups, or is it mainly a matter of trial and error? And what about quasi-lattices and quasicrystals? Is there a natural way to fit these in?

0 Comments

2024/11/07

06:23 UTC

06:23 UTC

0

I'm in my final year of undergraduate mathematics, though it is only my second year as a mathematics major (long story short, I switched majors a few times). What sparked my interest was how beautiful I found the subject of calculus to be, but, while currently suffering through a course in abstract linear algebra, I am beginning to realize two things especially: (1) how naive I was of the abstract logical structures underlying mathematics, and (2) how mathematically immature I am. The former is a wonderful surprise; the latter, not so much. When I discuss the homework problems with my classmates, I'll discuss my prolonged and awkward-but-successful proofs, and most of the time they'll laugh and show me their 3-line proof, making it seem trivial. It is good that I am surrounded by people more experienced in higher mathematics, and it has only made me hungrier for improvement. I'm often pessimistic due to my frequent inability to focus, but I want to change my mindset and gain confidence.

My most conspicuous shortcoming: problem-solving, the heart of mathematics itself. I'm poor at it, and I want to be better. So, how would you suggest I go about improving this skill? I've taken courses in calculus (my first love in mathematics) and differential equations, real analysis, abstract algebra, and linear algebra. I feel like my knowledge of mathematics is sparse, so I'd like to develop a routine familiarity with the major reappearing concepts, motifs, and whatnot. Above all, I want to practice, practice, practice. Where can I find problems to solve? Of course, there are textbook exercises, but what are some other resources? They don't have to be immediately solvable (like arithmetic or integration); I'm more looking for challenging proof-problems and the like.

3 Comments

2024/11/07

06:15 UTC

06:15 UTC

1

Hi,few folks from my institute want to start an active discussion group on Algebraic Geometry and Algebraic Number Theory.

The aim will be first 3 chapters of Hartshorne and first four chapters of Janusz.

This will be academically demanding and may lead to future collaborations.There will be weekly meetings, discussions of problems, presentations of theory.

We will also motivate each other and simply encourage each other too.

If interested,please fill out this Google form asking for some basic information (hope this is allowed)

4 People with the best background and motivation will be asked to join by email.

Any suggestions are welcome.

0 Comments

2024/11/07

05:19 UTC

05:19 UTC

3

I'm working on a problem where I'm packing circles of unequal radii sequentially into a rectangle

because I'm doing this sequentially, it would help if I can somehow the capture of how much effective area is remaining after I place a certain number of circles into the rectangle

for instance, for 4 circles I could:

- pack all 4 in 1 corner
- pack 2 in 1 corner, pack 2 in another
- pack all 4 right in the centre of the rectangle

for all 3 methods, the area that remains afterward is the same, but methods 1 and 2 are clearly superior because the area that remains still allows for more circles of larger areas to be placed, whereas method 3 prevents that (i.e. the effective area that remains is reduced)

so, is there a way I can capture this notion? I thought of "what's the largest circle that can be placed in the remaining area", but that would be cumbersome to compute (especially repeatedly after each circle placement)

7 Comments

2024/11/07

16:14 UTC

16:14 UTC

19

Hey there, I have real analysis as a course in the upcoming study block in my uni. I want to prepare for it in advance. What is a good video lecture seseries and/or online resource for real analysis (specifically for Understanding Analysis, by Abbott since that's the textbook the course uses)?

8 Comments

2024/11/07

11:52 UTC

11:52 UTC

17

Say I have a polynomial f(x) with real coefficients and degree d.

Also, I have the points set 0 = x_1 < ... < x_n = 1 with uniformly spread points, i.e. delta x = 1/(n-1).

I am looking for a lower bound of the cardinality of {f(x_1), ..., f(x_n)} in terms of n and d.

Clearly, ceil(n/d) works, but is it possible to do better? Indeed, this bound does not assume anything about the structure of the points, but I am specifically interested in the case of uniformly spread points.

11 Comments

2024/11/07

11:07 UTC

11:07 UTC

7

I'm a sophomore, and I will be taking a PDE class next semester which covers things such as fourier series, perturbation theory, and of course PDE. Thing is that I did well in diff eq, but I for sure was confused towards the end, and knowing that the subject is hard and the professor at my school is rather notorious, I'm trying to learn some of the material before the actual class. I've seen some of the textbooks such as evans and strauss, but I was wondering if you guys had any (perhaps lesser known) textbooks that make learning the subject easy. Bonus points if it explains every step thoroughly and if there's a lot of practice problems with solutions. Appreciate the help

5 Comments

2024/11/07

02:23 UTC

02:23 UTC

99

To all the professors out there, which elementary skills do you see students most commonly lack?

i.E poor trigonometry foundation for Calculus I

23 Comments

2024/11/07

00:00 UTC

00:00 UTC

1

Is it possible to determine whether a text was written by humans or by LLM by using Zipf‘s law where the frequency of any word is inversely proportional to its rank in the frequency table?

2 Comments

2024/11/06

23:34 UTC

23:34 UTC

9

Hello,

I work at a K-12 special education school. Kids are low to low-moderate and have some form of ADHD/ADD, are on the spectrum, or have dyslexia/dyscalculia to some degree. Our math program is in need of restructuring and is a subject students struggle with the most, so I've been tasked with compiling a list of math software apps and looking into a good few options. The highest level we go to is Algebra 1 and Geometry in high school. Some problem areas:

- Elementary - developing number sense/numeracy. Our lowest kids have difficulty adding/subtracting without counting
- Middle school - fractions are the biggest thing that confuses students
- High School - lack of basic arithmetic knowledge/numeracy. Some kids in geometry can't do basic operations with fractions.

Either way, across the board, numeracy and basic arithmetic is something that is a struggle for some of our students.

The tricky part is not every student starts in elementary. Because we're an NPS, districts send their kids to us from all different grade levels and knowledge (but they all have some form of learning disability). We may get kids enrolling as early as 1st or late as 12th grade. So some of those basic numeracy skills need to be honed for our high schoolers who are really behind.

Here is a working list of apps I've been looking into. I'd love any additional feedback on some apps that you've worked with or are familiar with--any other suggestions not on the list are welcome!

Of course, critical feedback is more than welcome. There are so many of these out there it can be hard to choose, but I wanted to cast a wide net because we are different from your traditional public school.

**Banzai**- seems more applied (financial literacy) with real-life focused problems**Beestar**- parents can monitor performance online. also gives motivational recognitions every week to encourage students**GeoGebra**- was looking into an option for geometry and this was the top result**Illustrative Mathematics**- has built-in assessments and hands-activities, review-focused**Gimkit/Prodigy**- very gamified, can create own questions and students have to answer them to proceed in the game or get 'energy' etc.**iReady**- Good as a diagnostic tool. Helpful in knowing a student's grade level and gaps in knowledge**IXL**- great for review and practice. Not as visual as other apps**Khan Academy**- particularly looking into Khan Academy Kids for K-8 but seems like a strong resource for geometry**Nessy Numbers (Woodin)**- seems good for building numeracy/number sense**Splashlearn**- very visual and engaging**Zearn**- K-5, seems good for early math review/practice

5 Comments

2024/11/06

23:25 UTC

23:25 UTC

64

I’m about to wrap up my first semester in an elite applied math PhD program and wanted to ask whether this experience is common. There are 10 students in my year, and although we’ve grown close, it seems almost all of us are having second thoughts — lots of talk of mastering out/switching to another program.

Most of us were originally enrolled 3 courses, but pretty much all of us have since dropped to 2 due to being overwhelmed. Even taking just 2 courses is comparably difficult to my undergrad where I was able to manage 5 without too much difficulty.

Between coursework and 10-20 hours per week of TA duties, I haven’t found time to start getting involved in research, and worry I am slipping behind on this front. In speaking to my cohort, it seems most of us are having similar concerns.

For those who finished their programs, did you also have these worries? How did you deal with them? What kind of attrition rate did you see year to year?

13 Comments

2024/11/06

22:31 UTC

22:31 UTC

11

I've been thinking of this question for a few months now, and all the talk about the elections have only made me think more about this, so here it goes.

Let's say there's a two-party system in a country, where P1 and P2 are the parties. Let's say I define a quantity called "vote density" for each of these parties, p1 and p2 - p1(x,y)dx dy would be the number of votes that the party P1 receives in an infinitesimal rectangular area of side lengths dx, dy, at the point (x,y), divided by the total population in that infinitesimal area- similar to a 2d probability density. The vote density p2(x,y) is defined in an analogous fashion. In general, there could be less than perfect voter turnout, so the sum of these voter densities, p1+p2 would be less than or equal to 1 for all (x,y).

Now let us say that the elections in this country work by splitting the country into smaller constituencies/counties. Whichever party gets higher votes in a particular county is declared the "winner" of that county, and whoever wins in the majority of the counties is declared the winner and forms the government. Further let us assume the vote density functions p1 and p2 are known, and that the county borders are given.

Let us define the total vote share of a party to be the integral of its vote density function. Can it be possible for party P1 to have a lesser vote share than party P2, but still win the elections - maybe in a way where P1 loses heavily in a select few counties and wins marginally in a larger number of counties, by virtue of which, the total vote share of P1 is lesser, but it is still declared the winner. What are the conditions for this to happen?

Related to my above question, let's say the total vote share of P1 is less than that of P2. Can I always come up with a bordering of the counties such that P1 still manages to win? Or is there any "critical vote share" for P1, which if it fails to reach, no design of country borders can help it win? If yes, what is this critical vote share? If no, how do we construct such a county bordering?

Now let's say we only know the total vote shares of both the parties, and not the vote density functions themselves. I want to design my counties in such a way that leading in the vote share correlates to winning the elections - I wish to avoid a system where a party like P1, which has lower vote share, gets a "rogue" win due to a poor design of borders. Can I come up with an optimal county design which minimizes the chances of these "rogue" wins? If yes, under what conditions, and how do we design it? Does the chance of these rogue wins also depend on the voter turnout? That is, does a higher value of p1(x,y)+p2(x,y) result in lesser chances of rogue wins?

Do these questions also have answers for multi party systems?

Do any of these questions come under any particular field as such? Are there any resources to read more about these ideas? Also, I guess I haven't been too rigorous in formulating my problem, but I've tried to keep it as intuitive as possible.

5 Comments

2024/11/06

21:16 UTC

21:16 UTC