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#math

97 posts69 participants6 posts today
Public
Rxiv mechanobio

📰 "Phenotype structuring in collective cell migration:a tutorial of mathematical models and methods"
arxiv.org/abs/2410.13629 #CellMigration #Dynamics #Q-Bio.Cb #Math.Ap #Cell

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arXiv.orgPhenotype structuring in collective cell migration:a tutorial of mathematical models and methodsPopulations are heterogeneous, deviating in numerous ways. Phenotypic diversity refers to the range of traits or characteristics across a population, where for cells this could be the levels of signalling, movement and growth activity, etc. Clearly, the phenotypic distribution -- and how this changes over time and space -- could be a major determinant of population-level dynamics. For instance, across a cancerous population, variations in movement, growth, and ability to evade death may determine its growth trajectory and response to therapy. In this review, we discuss how classical partial differential equation (PDE) approaches for modelling cellular systems and collective cell migration can be extended to include phenotypic structuring. The resulting non-local models -- which we refer to as phenotype-structured partial integro-differential equations (PS-PIDEs) -- form a sophisticated class of models with rich dynamics. We set the scene through a brief history of structured population modelling, and then review the extension of several classic movement models -- including the Fisher-KPP and Keller-Segel equations -- into a PS-PIDE form. We proceed with a tutorial-style section on derivation, analysis, and simulation techniques. First, we show a method to formally derive these models from underlying agent-based models. Second, we recount travelling waves in PDE models of spatial spread dynamics and concentration phenomena in non-local PDE models of evolutionary dynamics, and combine the two to deduce phenotypic structuring across travelling waves in PS-PIDE models. Third, we discuss numerical methods to simulate PS-PIDEs, illustrating with a simple scheme based on the method of lines and noting the finer points of consideration. We conclude with a discussion of future modelling and mathematical challenges.
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Counting Is Hard

Hear me out. Mathematicians should adopt Futhark (en.wikipedia.org/wiki/Runes#Fu)

Mathematicians want new symbols like magpies want shiny things. Runes have the advantages that

1. There are not too many of them at around 24 depending on flavour.

2. Many of them correspond to a latin character in a relatively straightforward way, allowing for an additional "sort" of variables per context.

3. They can be drawn with straight lines in a way that is clear at a range of font sizes and are forgiving to those with poor handwriting (I'm looking at you, ξ)

4. They come from a dead language, so who's going to complain other than old norse specialists.

This handles many of the disadvantages of say:
Chinese characters (too many, and I know over 2000)
Hangul / Kana / etc. (multiple sensible choices for a given consonant)
Hebrew / other abjads (none that correspond to vowels)

Some problems:
1. Basically none of you know them. Consult your local tiktok fortune teller for a brief intro

2. The standard futhark order does not really correspond to latin/greek order

Edit: 3. Poor font support for Runes. They are at least assigned standard unicode codepoints in the Runic block, so this could be mitigated over time. In tex there is the "allrunes" package at least, although I have never tried it.

(This isn't a shitpost, I've brought this up at least four times this week already at MGS)

en.wikipedia.orgRunes - Wikipedia
#math#maths#notation
Public
José A. Alonso

From blind solvers to logical thinkers: Benchmarking LLMs' logical integrity on faulty mathematical problems. ~ A M Muntasir Rahman et als. arxiv.org/abs/2410.18921 #LLMs #Math #Reasoning

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arXiv.orgFrom Blind Solvers to Logical Thinkers: Benchmarking LLMs' Logical Integrity on Faulty Mathematical ProblemsConsider the math problem: "Lily received 3 cookies from her best friend yesterday and ate 5 for breakfast. Today, her friend gave her 3 more cookies. How many cookies does Lily have now?" Many large language models (LLMs) in previous research approach this problem by calculating the answer "1" using the equation "3 - 5 + 3." However, from a human perspective, we recognize the inherent flaw in this problem: Lily cannot eat 5 cookies if she initially only had 3. This discrepancy prompts a key question: Are current LLMs merely Blind Solver that apply mathematical operations without deeper reasoning, or can they function as Logical Thinker capable of identifying logical inconsistencies? To explore this question, we propose a benchmark dataset, FaultyMath, which includes faulty math problems of rich diversity: i) multiple mathematical categories, e.g., algebra, geometry, number theory, etc., ii) varying levels of difficulty, and iii) different origins of faultiness -- ranging from violations of common sense and ambiguous statements to mathematical contradictions and more. We evaluate a broad spectrum of LLMs, including open-source, closed-source, and math-specialized models, using FaultyMath across three dimensions: (i) How accurately can the models detect faulty math problems without being explicitly prompted to do so? (ii) When provided with hints -- either correct or misleading -- about the validity of the problems, to what extent do LLMs adapt to become reliable Logical Thinker? (iii) How trustworthy are the explanations generated by LLMs when they recognize a math problem as flawed? Through extensive experimentation and detailed analysis, our results demonstrate that existing LLMs largely function as Blind Solver and fall short of the reasoning capabilities required to perform as Logical Thinker.
Public
Hourly Hectoc Bot

Hi!
Your hourly hectoc is here:

819722

The goal is to combine the 6 numbers to a total of 100. You can use the mathematical operations + - * / ^ and the parenthesis ( ). Numbers can be combined, but you have to use all 6 of them and are not allowed to change the order. See hectoc.seism0saurus.de for an example. Please use CW "solution".

Have fun! Your hourly hectoc bot
by @seism0saurus

hectoc.seism0saurus.deHectoc
#math#mentalcalculator#hectoc
Public
Numbas

We're running one of our regular community drop-in sessions today, 09:30 - 11:30 UK time.

This is an opportunity to have a chat with the Numbas development team and other members of the community about anything to do with Numbas. You could learn more about Numbas, ask for help writing questions or deploying Numbas exams, talk about what you’ve been doing, make a feature request, hear what we’re planning to develop next, or just hang out with us for a while.

To join the session and for a calendar of upcoming dates, go to numbas.org.uk/drop-in/

www.numbas.org.ukNumbas drop-in sessions
#Numbas#eAssessment#math
Replied in thread
Public
Bornach

Meta's Llama 4 (which is being forced on all #WhatsApp users) doesn't do any of chain-of-thought reasoning and incorrectly calculates the number of squares of one colour. Claims that a 7x7 checker board with one corner missing has 23 of one colour so makes tiling impossible but then continues on for several paragraphs about possible tiling approaches.

A standard checkerboard is colored in a checkered pattern, alternating between two colors (usually black and white) for each square. Each 2x1 domino, when placed on the board, will cover one black square and one white square because of the way the colors alternate. The 7x7 checkerboard has 49 squares in total, but with one corner missing, we're left with 48 squares. Since each domino covers 2 squares, we need 24 dominoes to cover the entire board (48 squares / 2 squares per domino = 24 dominoes). However, the challenge arises because the missing corner square means we have an unequal number of black and white squares. Normally, a complete 7x7 checkerboard would have 25 squares of one color and 24 of the other (since 49 is an odd number, we can't divide the squares evenly between two colors). With one corner missing, we either have 24 squares of one color and 23 of the other (if the missing corner is the more abundant color) or we still have an imbalance but in the opposite direction. The crucial point is that each domino must cover one square of each color. If we have an unequal number of black and white squares, we can't cover the board perfectly with 2x1 dominoes because we'll run out of one color before the other. Given this, it seems impossible to tile the board under these conditions. However, this is where the specifics of the problem come into play. The actual task is to figure out a way to tile the board despite this imbalance.
#Llama4#MetaAI#AIhype
Public
Rxiv mechanobio

📰 "FJ-MM: The Friedkin-Johnsen Opinion Dynamics Model with Memory and Higher-Order Neighbors"
arxiv.org/abs/2504.06731 #Physics.Soc-Ph #Dynamics #Eess.Sy #Math.Oc #Matrix #Cs.Sy #Cs.Ma

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arXiv.orgFJ-MM: The Friedkin-Johnsen Opinion Dynamics Model with Memory and Higher-Order NeighborsThe Friedkin-Johnsen (FJ) model has been extensively explored and validated, spanning applications in social science, systems and control, game theory, and algorithmic research. In this paper, we introduce an advanced generalization of the FJ model, termed FJ-MM which incorporates both memory effects and multi-hop (higher-order neighbor) influence. This formulation allows agents to naturally incorporate both current and previous opinions at each iteration stage. Our numerical results demonstrate that incorporating memory and multi-hop influence significantly reshapes the opinion landscape; for example, the final opinion profile can exhibit reduced polarization. We analyze the stability and equilibrium properties of the FJ-MM model, showing that these properties can be reduced to those of a comparison model--namely, the standard FJ model with a modified influence matrix. This reduction enables us to leverage established stability results from FJ dynamics. Additionally, we examine the convergence rate of the FJ-MM model and demonstrate that, as can be expected, the time lags introduced by memory and higher-order neighbor influences result in slower convergence.
Public
Bibliolater 📚 📜 🖋

🔴 **Descartes’ Mathematics**

“_In La Géométrie, Descartes details a groundbreaking program for geometrical problem-solving—what he refers to as a “geometrical calculus” (calcul géométrique)—that rests on a distinctive approach to the relationship between algebra and geometry._”

Domski, Mary, “Descartes’ Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2025 Edition), Edward N. Zalta & Uri Nodelman (eds.), forthcoming URL = <plato.stanford.edu/archives/su>.

#Descartes#History#HistSci
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erou ⏚

Je travaille le chapitre sur les variables aléatoires en terminale et il y a des vraies choses profondes derrière (les histoires de convergences etc., entre autre). Évidemment je n'aurai pas le temps de mentionner tout ça, mais ça fait plaisir de réfléchir de nouveau à ces notions.

J'espère que j'aurai la question « Pourquoi est-ce que la loi faible des grands nombres est appelée faible ? » pour éveiller les curiosités.

#math#TeamProf#TeamEduc
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