algebra

In Chapter 10 Section "Simplexes and Chains" Rudin introduces the concept of an affine oriented $k$-simplex $\sigma = [\mathbf p_0, \ldots, \mathbf p_k]$ with $\mathbf p_i \in \mathbb R^n$ ...

This problem is from the undergraduate olympiad "Elon Lima". Let $f_1(x) = x^2 + 4x + 2$ and let $f_{n+1}(x) = f(f_n(x))$ for $n \geq 1$. Let $s_n$ be the sum of coefficients of even power ...

Thomas J. Lada On homology of iterated loop spaces and the Dyer-Lashof operations?: On -algebras: Tom Lada, Jim Stasheff: Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993) 1087–1103 [doi:10.1007/BF00671791, arXiv:hep-th/9209099] Tom Lada, Martin Markl: Strongly homotopy Lie algebras, Communications in Algebra 23 6 (1995) [doi:10.1080/00927879508825335, arXiv:hep-th/940…

J. Peter May is a homotopy theorist at the University of Chicago, inventor of operads as a technique for studying infinite loop spaces and spectra. Peter May’s work makes extensive use of enriched- and model-category theory as power tools in algebraic topology/homotopy theory, notably in discussion of highly structured spectra in MMSS00‘s Model categories of diagram spectra (for exposition see In…

I have the following situation: $f \colon X \to Y$ is a quasi-finite morphism (i.e. $f$ has finite fibers) of connected and normal complex analytic spaces, and $Y$ is compact. Is it true that $f$ ...

_Zenodo_. 2026Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽…Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽³⁾_Πd as the unique structure satisfying the executive axiom syste…

Abstract This paper presents a complete proof of Goldbach's Conjecture by establishing an equivalence with the positivity of a density function D(n) over symmetric prime parametrizations, analyzed through Wilson's Theorem. We parametrize all possible prime pairs (p,q) with p+q=n as p=(n-m)/2 and q=(n+m)/2, where m is the symmetric distance parameter. Using Wilson's quotients k_p = ((p-1)!+1)/p an…

A couple months ago I wrote about how to compute the sine and cosine of a complex number using only real functions of real variables using the equations You can do something analogous for all the elementary functions, though some of the equations are quite a bit more complicated than the ones above. See the […] The post Building complex functions out of real parts first appeared on John D. Cook .

Let $G$ be a group and let $M$ be a $\mathbb{Q}[G]$-module. Assume that $H$ is another group acting on $M$ by $G$-equivariant maps, so for all $g \in G$ and $h \in H$ and $m \in M$ we have $$h \cdot ...

This paper proves a strengthened form of the strong Goldbach conjecture by showing that its negation is unprovable in ZFC, assuming this theory is sound. We reformulate the conjecture using an infinite set and show, based on properties of this set, that the assumption that there is a proof of its negation leads to a contradiction. Whereas the traditional approaches focus on the control over the d…

This paper proves that both the strong Goldbach conjecture and a strengthened form of it are independent of ZFC and Peano arithmetic (PA), with the latter result being a corollary of the former, assuming these theories are sound. For each of the conjectures, we define an infinite set with which we reformulate the conjecture and show that this set always remains the same, regardless of whether the…

I found the following two polynomial identities while experimenting with sums and differences of cubes. Identity 1 (sum of two cubes = perfect square): ...

I'm reading about the motivation behind the infinitesimal rotation generator for $\mathbb{R}^2$ while doing some self-study about Lie Algebras. I've noticed this jump in logic, and I'm curious as to ...

I am trying to find a way to compute the cohomology ring of a space of the form $X:=\mathrm{colim}_{I}X_i$. Here are the properties for this diagram of spaces: The index set $I$ is a finite poset. ...

Motivation: I am trying to find a variant of the Atiyah problem on configurations over a finite field. Let $\mathcal{P}_2$ denote the space of all polynomials of degree at most $2$ with coefficients ...

From Judson 2012: Coding theory is an application of algebra that has become increasingly important over the last several decades. When we transmit data, we are concerned about sending a message over a channel that could be affected by “noise.” We wish to be able to encode and decode the information in a manner that will allow the detection, and possibly the correction, of errors caused by noise.…

Set theory occupies the foundational stratum of modern mathematics, yet the question of whether its machinery reflects structural necessity or descriptive convenience has never been posed from outside the formal tradition itself. Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) is universally adopted as the axiomatic substrate for virtually all of contemporary mathematics, from analysis…

This page gives a detailed introduction to the Adams spectral sequence in its general spectral form (Adams-Novikov spectral sequence). For background on spectral sequences see Introduction to Spectral Sequences. For background on stable homotopy theory see Introduction to Stable homotopy theory. For background on complex oriented cohomology see Introduction to Cobordism and Complex Oriented Cohom…

In a series of works [1-6] an algebraic construction was developed that describes the dynamical change of the dimension of configuration space. In the present work these constructions are systematized in the form of a unified family of Khodakovsky superalgebras X_q, where the parameter q = 0,1,2,3,4,5 corresponds to the successive extension of the original algebra X (dimension n, rank m) through …

Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology cohomology with constant coefficients / with a local system of coefficients Special notions Variants differential cohomology Extra structure Operations Theorems representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, v…