mathematics

nLab numerable open cover Contents Context Topology topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Introduction Basic concepts - open subset, closed subset, neighbourhood - topological space, locale - base for the topology, neighbourhood base - finer/coarser topology - cl…

The notion of a normal subobject is the proper generalization of a normal subgroup to other algebraic categories. The notion was found relatively late. In the category of groups, there are two equivalent descriptions of a normal subgroup: as the kernel of a homomorphism of groups and as the equivalence class of the unit of some (necessarily unique) congruence. Given a category admitting finite li…

The question states "Given the relation of position w.r.t. time, find the time at which velocity is zero." The relation is: $$x = \sqrt{t} + 2$$ First, I tried differentiating with respect to $t$ , which led to this: $$\begin{align*} \frac{dx}{dt} &= \frac{d}{dt}\left(\sqrt{t}+2\right) \\[1ex] v &= \frac{1}{2\sqrt{t}} \end{align*}$$ Now obviously for $v=0$ , either the numerator needs to be $0$ ,…

We propose a three-layered framework for mathematical specialization and research strategy in an era of rapidly advancing artificial intelligence. First, we introduce the "Diffraction Profile," a finite-resource cognitive model that uses a squared-sinc kernel to mathematically describe how deep specialization in a core domain generates non-monotonic, structurally resonant "side lobes" of secondar…

Here's a problem I just came up with. I am trying to determine the minimum area of a quadrilateral inscribed in a square, while respecting the given areas of the four right-angled triangles as shown ...

Twenty five Fields medalists were initial signatories of the letter “A Severe Misalignment of AI in Mathematics” in which they opine that “The goals of the AI companies and the goals of the mathematical community are severely misaligned”. They are right. As they say, the goals of AI companies do not seem to be aligned with “the primary goal of conceptual understanding and insight”, and indeed, th…

A statement by 25 Fields Medalists in mathematics. Some of their case applies, I would venture, to philosophy. An excerpt, but do read the whole statement: [T]he push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI […]

Every finite bridgeless cubic simple graph (Mathlib SimpleGraph with quantified Fintype and DecidableRel instances, IsRegularOfDegree 3, no edge an IsBridge) has a family Fin 6 → G.Subgraph of perfect matchings (Subgraph.IsPerfectMatching) such that for every edge the set of indi Berge–Fulkerson conjecture launch-corpus · graph-theory · open-problem · cubic-graphs · perfect-matchings · edge-colou…

This entry is about the basic notion of “monic” in relation to morphisms in category theory. For the notion of “monic” in relation to polynomials in commutative algebra, see at monic polynomial. The notion of monomorphism is the generalization of the notion of injective map of sets from the category Set to arbitrary categories. The formally dual concept is that of epimorphism, which similarly gen…

Matrix calculus pops up on social media from time to time, primarily due to the increased popularity of machine learning. It lets you differentiate functions of vector or matrix variables even if you are not a fan of automatic differentiation. It involves all kinds of complex rules, depending on whether your vectors are row or column vectors, and it gets even worse for matrices. Let’s not speak o…

Researchers at Binghamton University have developed a mathematical strategy that can solve Wordle with a 99% success rate. Instead of simply guessing words packed with common letters or choosing the most likely answer, the method uses Shannon entropy, a measure from information theory, to identify guesses that reveal the most useful information. Each guess is designed to rapidly shrink the pool o…

Why should you care? Sorting is one of the most common problems in programming. You may need to sort: Student marks Product prices Names Search results Timestamps Database records Large datasets A simple sorting algorithm might work well for a small array, but performance becomes important when the input grows. Merge Sort can sort n elements in: O(n log n) More importantly, Merge Sort introduces …