mathematics

Let ξ(s) = ½s(s−1)π^(−s/2)Γ(s/2)ζ(s) be the completed Riemann zeta function and let {γ_ρ} denote the positive imaginary parts of its nontrivial zeros. We first prove a universal product formula: for all s∈ℂ (under the Riemann Hypothesis), ξ(s)/ξ(½) = ∏_{γ_ρ>0} (1 + (s−½)²/γ_ρ²), expressing the ratio ξ(s)/ξ(½) as a product over the Riemann zeros with coupling a(s) = |s−½|. The formula has two regi…

In my procrastination, I've been trying to understand some basic topology which started with the definition of topological spaces, which I pretty much understand with the definition via open sets on ...

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…

The homology of (iterated) based loop spaces (ordinary homology or generalized homology) carries special structure, reflecting the ∞-group-structure of based loop spaces. In particular, under mild technical conditions (see Milnor-Moore 65, p. 262, Halperin 92) the Pontrjagin ring-structure induced by concatenation of loops enhances the homology coalgebra induced by the diagonal maps to that of a …

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…

It seems clear at this point that AI will change mathematics in many ways and we should be well-prepared for it. However, reading recent trending posts about Math and AI on social media, blogs and ...

_Zenodo_. 2026Essay V of the Gradient Fractals suite executes the Topological layer of the ten-layer derivational chain. The four preceding essays established the Gradient Fractal Field’s ontological necessity (GF-I), algebraic-computational spine (GF-II), geometric character D = 93/40 (GF-III), and informational constitution dS/dτ = log₂(3) (GF-IV). GF Essay V now asks: what is the topological c…

Alexander Karabegov sends me new puzzles from time to time. This time, however, it is not a puzzle but a math joke. Joke. If a woman gives birth to a child at the age of 30, then 60 years earlier, her child was twice as old as she was. Whatever that means. Share:

This was asked before, but no answer was given. I would like to give a name to a manifold that can be covered by a single coordinate chart. Something other than "coordinate chart". In the ...

A connective spectrum is a connective object in the stable -category of spectra, hence a spectrum whose homotopy groups in all negative degrees are trivial: . These are equivalently: Connective spectra form a sub-(∞,1)-category of spectra There are objects in Spectra, though, that do not come from “naively” delooping a topological space infinitely many times. These are the non-connective spectra.…

A battle between “slimes” and “zoglins” could be the best way to calculate pi—at least for fans of this megahit game

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum. What does this mean/imply?

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 $f$ ...

In Riemann integration, one defines both lower and upper sums $ L(f,P), U(f,P), $ and declares a bounded function $f:[a,b]\to\mathbb{R} $ to be integrable if $ \sup_P L(f,P)=\inf_P U(f,P). $ On the ...

An especially strong motivating case for the usage of spatial probability models comes from the mining industry. During exploration for mineral resources, prospectors will take geologic samples by drilling holes and examining the resulting material for presence or concentration of valuable ores. These data typically show strong spatial correlation, but constructing a fully-detailed geophysical mo…

We formalize a structural perspective on the Riemann zeta function, highlighting a natural balance between the "inside" (integer counting) and "outside" (density/reflection) perspectives of the number system. Using divisor pairings, information measures, and the functional equation, we show that s = 1/2 is a unique symmetry axis where information is balanced. While this framework does not prove t…

This work presents a philosophical and mathematical research highlight on the ontological structure of space understood as a compact, simply connected, boundaryless three-dimensional manifold: M ≃ S³. Instead of treating space as a local product of physical emergence, the treatise argues that the manifold should be regarded as a persistent global structure, while cosmological evolution describes …