Floor, ceiling, slopes, and K-theory

Some explicit counter-examples to Weibel's conjecture

We give two related but different methods for constructing rings $R$ of Krull dimension 1 and $K_{-d}(R) \neq 0$. The first works for $d = 2$, and the second works for any $d \geq 2$.

More than five-twelfths of the zeros of $ζ$ are on the critical line

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(μ\star Λ_1^{\star k_1} \star Λ_2^{\star k_2} \star \cdots...

Categories for the Working Mathematician

Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

Sur quelques points d'algèbre homologique, I

The rich are loopy: a spectral framework for loop structure in the rich club

The rich club coefficient $\phi(k)$ quantifies whether high-degree nodes interconnect more densely than expected, but it is blind to the structure of those interconnections and offers no principled way to select the degree threshold $k$. We introduce the *cycle core* $H_k$, the maximal subgraph of vertices with degree at least $k$ that forms cycles among themselves. The nested sequence $H_1 \supseteq H_2 \supseteq \cdots$ reveals how the cycle core evolves as the degree threshold increases, capturing structure that $\phi(k)$ cannot see. We develop two spectral tools for this sequence. The first is structural: the *$k$-blocking operator* $\mathbf{M}_k$ generalizes the non-backtracking matrix by blocking $k - 1$ directions at each node. We prove that its spectrum is equivalent to that of a modified non-backtracking matrix weighted with a specific function of the nodes' degrees. Thus, $\mathbf{M}_k$ couples cycle structure with the degree sequence of the graph. We show the spectral radius of $\mathbf{M}_k$ peaks at an intrinsic threshold $k^*$, yielding the *optimal rich club* $H_{k^*}$ without null models or parameter tuning. The second tool is dynamical: the $k$-blocking walk defines a sub-stochastic Markov chain whose leading eigenvalue $\lambda_1(k)$ measures how long a process can persist on the cycle core before being expelled to the periphery. We evaluate the framework on fifteen networks spanning different domains, wherein $H_{k^*}$ closely recovers the published rich clubs.

Jacobian Conjecture -- from Wolfram MathWorld

The Jacobian conjecture asserts that a polynomial map F:C^n->C^n having a nonzero constant Jacobian determinant is an automorphism. In the plane, first stated by Keller (1939), it says that a ring map F of C[x,y] (the polynomial ring in two variables over the complex numbers) to itself that fixes C and sends x, y to f, g, respectively, is an automorphism iff the Jacobian f_xg_y-f_yg_x is a nonzero constant. The condition is easily shown to be necessary. There have been at least five...

Homotopical patch theory | ACM SIGPLAN Notices

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a ...

Computational category theory

Computational category theory by D. E. Rydeheard, 1988, Prentice Hall edition, in English

the aligned distinct ring · sol mark construction

one ray, repeated ten times. one ring, solved from its center and its air. a construction of whole angles, simple fractions, and φ.

Trees, maps, and theorems

Trees, maps, and theorems

Notes from the Lab - March 2025 · folk-js folkjs · Discussion #36

This month, we restructured the repository in preparation for publishing. We extracted and optimized geometry utilities into a separate package, implemented Bounding Volume Hierarchies for efficien...

General Theory of Natural Equivalences

Samuel Eilenberg, Saunders MacLane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society, Vol. 58, No. 2 (Sep., 1945), pp. 231-294