algebra

I use the mismath package, which among other math operators, defines \card, but with a definition different from the one I want, which is simply: \DeclareMathOperator{\card}{card} To effect this ...

This paper studies the recursive complex-root relation obtained by repeatedly taking roots of the negative of the preceding value, with particular emphasis on the role of branch selection. We show that different interpretations of the recurrence define fundamentally different mathematical systems. For the principal branch, the dynamics is deterministic and exactly solvable: the modulus converges …

Special and general types Special notions Variants Extra structure Operations Theorems The universal coefficient theorem states how ordinary homology/ordinary cohomology determines homology/cohomology with arbitrary coefficients. For a chain complex (of abelian groups) and a field (the coefficient field), the homology group and the cohomology group are indeed related by dualization: . If the coef…

There are many ways to formulate the group axioms. A standard definition is this: A group is pair $(G,\circ)$ consisting of a set $G$ and a binary operation $\circ$ on $G$ satisfying the following ...

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A […] The post Proof of the rank-trace theorem first appeared on …

Victor Ginzburg (in some 1980s articles spelled Ginsburg) is a professor of mathematics at the University of Chicago. His thesis in Moscow was under Alexandre Kirillov. His main interests are representation theory, especially geometric representation theory, including more recently noncommutative algebraic geometry. Warning: there is another mathematician (global analysis, symplectic geometry), V…

Special and general types Special notions Variants Extra structure Operations Theorems 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…

During my second week at the Recurse Center, I've been trying to formalize Dummit and Foote's abstract algebra textbook (appropriately titled "Abstract Algebra") in Rocq. I had quite a hard time with the first proof exercise in the book because the stated proof goal is not true. This was both frustrating and exciting to figure out :) A function f from a set A to a set B (written "f: A -> B") is a…

Advanced mathematics is characterized by an apparent tension between semantic flexibility and formal rigor. Algebraic geometry provides an especially striking case. Its concepts are repeatedly reformulated through local criteria, universal properties, functors, sheaves, descent data, equivalences, and increasingly abstract categorical environments. Yet this proliferation of representations does n…

nLab join of simplicial sets Context Homotopy theory 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 Definitions Paths and cylinders Homotopy groups Basic facts Theorems Contents Idea The join …

denotes ℤ the spectrum of the commutative ring of integers. Its underlying topological space (also known as the “prime spectrum” or “Zariski spectrum” of ) has the prime ideals of as points and carries the Zariski topology. The closed points are the maximal ideals , for each prime number in ; the non-maximal prime ideal is a generic point as it has as closure the whole of . A subset of is closed …

- Ponder This This puzzle was suggested by Hugo Pfoertner - thanks Hugo! In a now-famous 2004 article, Ben Green and Terence Tao proved that arbitrarily long arithmetic progressions exist in the primes. This holds true analogously for other sets of numbers if their density is sufficiently high, for example, numbers that are the sum of two squares. There are also enough numbers of the form x² + y²…

Algebraic geometry relies on a highly structured categorical language whose semantic precision masks a deep interpretative flexibility. This paper develops a unified framework that integrates a philosophically grounded account—Contextual Structural Logic (CSL)—with a rigorous mathematical formalization—Interpretation Moduli Objects (IMO). CSL analyzes how algebraic-geometric language remains rigo…

Here I solved total four problem based on dimension, basis, arithmatic and geometric Eigen ector.

Mathematics assistants are useful when they reduce the mechanical burden of a problem without hiding the reasoning. They can transcribe an equation from an image, propose a substitution, expand an expression, or generate a first draft of a proof. The difficult part is not producing a plausible sequence of symbols. The difficult part is deciding whether every transformation preserves the original …

Nature, Published online: 28 August 2026; doi:10.1038/d41586-026-02688-x His awe-inspiring proof for the resolution of singularities — such as peaks, edges and self-crossings — revolutionized algebraic geometry.

Most of us learn the quadratic formula as a finished piece of algebra: x = (-b ± sqrt(b² - 4ac)) / (2a) For exact arithmetic, that formula is complete. For floating-point arithmetic, it is only the beginning. The problem is not that JavaScript implements the formula incorrectly. The problem is that the two algebraically equivalent branches can have very different numerical behavior. When b and sq…

Eric Friedlander is an American mathematician, who has worked extensively in algebraic K-theory, and its relationship with algebraic geometry. E. M. Friedlander, 1982, Étale homotopy of simplicial schemes , volume 104 of Annals of Mathematics Studies , Princeton University Press, Princeton, N.J. E. M. Friedlander, Fibrations in étale homotopy theory (numdam), Publ. Math. Inst. des Haut. Études Sc…