algebra

If the domains of functions is R^N then embedding theorem follows even if on bounded domains without extension operators.

There are several contexts in which it is of relevance that a certain property of a morphism is preserved (or stable) under pullback, i.e. also shared by the the morphism for any pullback diagram Geometers prefer to say “stable under base change”. Monomorphisms are always stable under pullback; that is, if is a monomorphism, then so is . Epimorphisms are not necessarily stable under pullback. In …

The Jordan-Hölder theorem says that every composition series of a given group, and every Jordan-Hölder sequence on a given object in an abelian category, has the same length, and the same simple factors, up to permutation. In particular says that the length of an object in an abelian category is well defined. More generally, a form of the theorem holds in any homological category. This is a proof…

Say that I have a representation $V_\ell$ of $\rm SO(3)$ , with angular momentum number $\ell$ . In Dirac notation, states in $V_\ell$ can be represented in the form $$ |v⟩ = \sum_{m=-\ell}^\ell c_m |\ell,m⟩, $$ where, if the representation is real-valued, we require $c_{-m} = (-1)^m c_m^*$ . I would like to randomly sample from this space , i.e., the vector $|v⟩$ and/or its coefficients $c_m$ , …

Given a -coalgebra-Galois extension of a -algebra , which is the appropriate generalization of a Hopf-Galois extension, where is faithfully flat over the base as a left -module, one constructs a coring, Ehresmann coring, out of these data. Its role is somewhat analogous to the gauge groupoid (see Atiyah Lie groupoid), and in Hopf-Galois case it is an intermediate stage in constructing another ana…

How did the letter x get its enduring role as a symbol of the unknown? A mathematician explains why it’s hard to say for sure.

_Zenodo_. 2026The directed Ihara (Bowen-Lanford) zeta function of the four-gon quiver Q, the four-vertex six-arrow strongly connected quiver with adjacency spectrum determined by det(I − uA) = 1 − u² − u³ − u⁴, is computed explicitly. The zeta function is ζ_Q(u) = 1/[(1 + u)(1 − u − u³)]. Its four poles occupy three distinct moduli {0.6823, 1, 1.2107}, which do not lie on a common circle. Three s…

There is a brief entry at bar construction together with a blog link There is some discussion of the bar-cobar adjointness as it relates to twisting cochains, at that entry. Here we will concentrate on the bar-cobar adjointness itself and start exploring the links with other parts of differential algebra. One of the earliest examples of a pair of adjoint functors studied in algebraic topology was…

In binary, the two's complement representation of a negative number is found by NOT b + 1 , with the leading bit being a sign bit. In general, prove that applying the operation NOT b + 1 twice to some n-bit binary number yields the original number b . I found this conceptually... Read more

homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Introduction to Basic Homotopy Theory Introduction to Abstract Homotopy Theory geometry of physics – homotopy types Definitions Paths and cylinders Homotopy groups B…

Fast Track Summary Core Distinction: Precalculus studies functions as fixed objects — their shapes, transformations and identities. Calculus studies how those same functions change and accumulate. Order of Study: Precalculus comes first. Every calculus topic assumes you can already manipulate the function you are differentiating or integrating. Which Is Harder: Calculus introduces harder ideas, b…

I am studying sheaves on topological spaces with values in $R\mathrm{-Mod}$. My definition of the sheafification of a presheaf is close to the étale spaces construction. In particular: Let ...

The Pedagogical Shift: Deconstructing Axler’s "Linear Algebra Done Right" In the landscape of undergraduate mathematics, the standard curriculum for linear algebra has historically been dominated by a matrix-centric approach. This traditional methodology emphasizes Gaussian elimination, determinant computation, and row-reduced echelon forms as the primary gateways to understanding vector spaces. …

dc.title: Intersection theory on Hurwitz spaces of low-degree covers dc.description.abstract: We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers H^1(H_{d,g}(μ)) vanishes for covers of degree d = 3 and deduce the same result for the classical Hurwitz space of simply-branched covers H^s_{3,g}. In degrees 4 and 5, we compute examples where H^1(H_{d,g}(μ)) …