algebra-over-a-field

The notion of strong dinatural transformation is a notion of natural transformation between pairs of functors that is stronger than that of dinatural transformations. Unlike dinatural transformations, strong dinatural transformations can always be composed. They have close connections to parametricity in computer science. Let be functors. A strong dinatural transformation consists of, for each , …

I am having troubles on the spectral sequence of a double complex, as treated in Weibel's AITHA. In Example 1.2.4 it is defined a double complex in an abelian category $\sf A$: a double complex ...

Let $f(z)$ be an entire function of exponential type $1$, and let $s,t>0$, $s+t=1$. Can we always find entire functions $g,h$ of exponential types $s,t$ such that $f=gh$ ?

I'm trying to solve the following problem from a commutative algebra book (Álgebra comutativa em quatro movimentos by Borges and Tengan). The question has 30 concrete examples of rings (mostly ...

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras A Lie 2-algebra is to a Lie 2-group as a Lie algebra is to a Lie group. Thus, it is a vertical categorification of a Lie algebra. A (“semistrict”) Lie 2-alge…

Let $$ P_n(X)=\prod_{\substack{\mu_1,\dots,\mu_n\in\{\pm1\}\\ \mu_1\cdots \mu_n=1}} \left( X-\sum_{k=1}^n \mu_k e^{k\pi i/n} \right) $$ be a polynomial of degree $d=2^{n-1}$. Let $c_n''$ be the ...

I'm learning linear algebra and am confused about how subspaces are usually defined and thought about. Suppose my host vector space is $R=\mathbb R^3$. Consider the plane $z=0$ passing through the ...

Suppose $R$ is a reduced Noetherian ring. We know that $f \in R$ is a zero divisor if and only if $V(f)$ contains an irreducible component of $\mathrm{Spec} R$. I would like to know if one could use ...

Throughout the question, let $X$ denote a set together with a binary operation $\cdot $ on it. The below is the standard definition of a group structure on $X$ that I first saw: (G1) $x \cdot(y\cdot ...

I’ve been testing Claude’s ability to generate Lean 4 code to prove theorems. I’ve written about a couple experiments that verified calculations. I did not write about my failed attempt to get Claude to formalize a proof of the pqr theorem for seminorms. This time I asked Claude to formally prove the theorem from the […] The post Formalizing a ring theorem with Lean 4 and Claude first appeared on…

Suppose you have a Lie group. The generators of this group, along with their commutation relations, form a Lie algebra. The maximal commuting subset of this is called a Cartan Sub-Algebra (CSA). Suppose you have a CSA with 20 generators. Given some vector space V, each of the 20 generators can be associated with 20 matrices. If you change V, you change the 20 matrices If a vector A of the vector …

This is a basic question. What do we actually mean when we say "we can describe the curve parametrically" or "we parametrize the curve with a parameter $t$ or $\theta$."? What is ...

Nearly everyone who as seen partial fraction decomposition was introduced to it as a way to compute integrals. If P(x) and Q(x) are polynomials, then you can break their ratio P(x)/Q(x) into a sum of terms that can each be integrated in closed form. As with most topics in a calculus class, partial fractions go by in […] The post Partial fraction decomposition first appeared on John D. Cook .

I am happy to announce the third SAIR challenge, which is focused on obtaining numerical data for the infamous inverse Galois problem. This is a collaborative project with the L-functions and modular forms database (LMFDB), and is organized by John Jones, Jen Paulhus, David Roe, Andrew Sutherland, and myself. The challenge is somewhat similar to […]

Clearly the statement above is evident to anyone with a grounding in mathematics, but out of curiosity, I tried to think of the underlying definition that allows this, and found I really just 'knew ...

Researchers around the world are racing to develop new quantum-based systems for sensing, communication, computing and control that have the promise of outperforming traditional systems. Creating stable, measurable, distinguishable quantum states—which would be the heart of any such system—is a daunting task. Quantum states possess unique properties that can be exploited to develop novel informat…

On simplicial de Rham cohomology?: Johan Louis Dupont: Simplicial de Rham Cohomology and characteristic classes of flat bundles, Topology 15 3 (1976) 233–245 [doi:10.1016/0040-9383(76)90038-0] Johan Louis Dupont: A dual simplicial de Rham complex, in: Algebraic Topology, Rational Homotopy, Lecture Notes in Mathematics 1318 (1988) [doi:10.1007/BFb0077796] On regulators and characteristic classes o…

Raoul Bott (1923–2005) was one of the great 20th century topologists and geometers. Among his famous works, one should mention the Bott periodicity theorem (of importance in K-theory), studies in Morse theory (including the study of Bott–Morse functions), the Borel–Weil–Bott theorem in geometric representation theory, the study of fixed point (localization) formulas (the Atiyah–Bott fixed point t…

On the simplicial de Rham complex and equivariant de Rham cohomology:

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras superalgebra and (synthetic ) supergeometry A super Lie algebra which is a polyvector extension of the super Poincaré Lie algebra (supersymmetry) in for supe…