https://taboege.de Blog taboege.de Wed, 6 May 2026 12:19:47 +0000 https://cyber.harvard.edu/rss/rss.html Tobias Boege /blog/2025/09/Maximum-entropy-principle--Copy-lemma-and-information-inequalities <p>I present the Maximum entropy principle in information theory as a way to tweak probability distributions into more special positions while preserving some of their marginal entropies.</p> Tobias copy-lemma, information-inequalities, maximum-entropy, conditional-independence /blog/2025/09/Maximum-entropy-principle--Copy-lemma-and-information-inequalities Fri, 12 Sep 2025 00:00:00 +0000 /blog/2025/02/Counting-combinatorial-objects-using-SAT-solvers <p>This is a short article on how to compute the numbers of combinatorial objects using a model counter. At the end I provide challenges in the form of CNF files for open counting problems.</p> Tobias antimatroids, gaussoids, semigraphoids, matroids, forknesses, sat-solvers, oeis /blog/2025/02/Counting-combinatorial-objects-using-SAT-solvers Fri, 28 Feb 2025 00:00:00 +0000 /blog/2025/01/Computing-the-characteristic-set-of-a-quasiaffine-variety <p>Given a system of polynomial equations and inequations with integer coefficients, one can compute the set of characteristics of fields over which the system has a solution. This is proved constructively in books on model theory, but the given procedure is usually rather complicated to implement. This article shows how to do it in practice using Gröbner bases as a blackbox and presents a toy application computing the characteristic sets of some small matroids.</p> Tobias algebraic-geometry, groebner-basis, model-theory, matroid /blog/2025/01/Computing-the-characteristic-set-of-a-quasiaffine-variety Wed, 8 Jan 2025 00:00:00 +0000 /blog/2024/02/Keeping-mathematical-software-working-and-available <p>This article is about the use of containers in the context of mathematical research data. Container image build files document the often complex setup of custom mathematical software and make it repeatable. The resulting image can be shared to avoid any setup at all. The image also acts as an archive recording the versions of the software and all its dependencies which made the computation work.</p> Tobias mathematical-software, containers, research-data, reproducibility /blog/2024/02/Keeping-mathematical-software-working-and-available Sat, 3 Feb 2024 00:00:00 +0000 /blog/2022/05/On-prose-in-math-papers <p>This is an apology for writing prose in mathematics papers (and a clarification of what I mean by “prose”).</p> Tobias writing-mathematics /blog/2022/05/On-prose-in-math-papers Sat, 28 May 2022 00:00:00 +0000 /blog/2022/05/First-order-objects-in-algebraic-geometry <p>Some properties of varieties over the complex numbers do not change when the variety (via its defining equations which must have integral coefficients) is instead considered over sufficiently large finite fields of sufficiently large characteristic. This phenomenon can be explained via symbolic algorithms in computer algebra, particularly Gröbner bases.</p> Tobias algebraic-geometry, model-theory, lefschetz-principle, finite-field /blog/2022/05/First-order-objects-in-algebraic-geometry Sat, 21 May 2022 00:00:00 +0000 /blog/2022/05/Document-preparation-at-home-vs--the-arXiv <p>I describe the two more subtle problems I used to run into when submitting papers to the arXiv and their solutions.</p> Tobias arxiv, latex, publishing /blog/2022/05/Document-preparation-at-home-vs--the-arXiv Tue, 17 May 2022 00:00:00 +0000 /blog/2022/05/App--Muletracks--Downloader-for-Nugs-net <p>I present a Nugs.net downloader I wrote and which is now usable.</p> Tobias perl, mojo, music, muletracks /blog/2022/05/App--Muletracks--Downloader-for-Nugs-net Sun, 15 May 2022 00:00:00 +0000 /blog/2022/02/Locally-finite-cover-on-a-primary-basic-semialgebraic-set <p>Any open cover of a primary basic semialgebraic set can be refined to a locally finite one. Here is an explicit proof of this fact when the cover consists of neighborhoods indexed by points from the set itself and which, after an initial adjustment of the sizes of the open sets, constructs a locally finite subcover, i.e., just removes redundant open sets. This detail matters when the points indexing the original cover have special significance elsewhere.</p> Tobias semialgebraic-geometry, topology, axiom-of-choice, open-cover /blog/2022/02/Locally-finite-cover-on-a-primary-basic-semialgebraic-set Fri, 11 Feb 2022 00:00:00 +0000 /blog/2022/02/On-stable-equivalence-of-semialgebraic-sets <p>I give a counterexample to a claim in the book “Realization spaces of polytopes”. This counterexample shows that a fundamental definition in this book is (slightly, technically) wrong. I explain what causes the failure and how it might be fixed in a forthcoming note.</p> Tobias semialgebraic-geometry, topology, stable-equivalence, universality /blog/2022/02/On-stable-equivalence-of-semialgebraic-sets Wed, 9 Feb 2022 00:00:00 +0000 /blog/2021/07/On-experimental-mathematics <p>Experimental mathematics uses computers to inspire research in terms of finding examples and counterexamples, suggesting conjectures or informing proof strategies. I present a strange phenomenon in this practice at two examples: it happens that mathematical software confirms conjectured properties of very large objects very quickly. Much, much more quickly than one might expect a general problem of that type to be solved. In these cases, the software apparently knows a “proof” of the conjecture and the human can either look it up in the source code or at least be more confident in the conjecture.</p> Tobias mathematical-software, experimental-math, heuristics, open-source /blog/2021/07/On-experimental-mathematics Sat, 17 Jul 2021 00:00:00 +0000 /blog/2020/03/A-rational-realization-of-Simecek-s--85 <p>In his article “Gaussian Representation of Independence Models over Four Random Variables” Petr Šimeček gives realizations of all conditional independence models arising from (not necessarily regular) Gaussian distributions in dimension 4. Only one of these models, M85, is given with irrational correlations. Here I give a rational representation of this model. The question whether all Gaussian CI models are realizable by rational correlation matrices is still open and M85 was the only concrete undecided example I am aware of.</p> Tobias ci, rationality, gaussian /blog/2020/03/A-rational-realization-of-Simecek-s--85 Fri, 20 Mar 2020 00:00:00 +0000 /blog/2020/02/tappp-hpp-v0-2-0-released <p>tappp.hpp is a header-only TAP producer for C++17.</p> Tobias c++, tappp, testing /blog/2020/02/tappp-hpp-v0-2-0-released Thu, 27 Feb 2020 00:00:00 +0000 /blog/2019/09/The-computation-of-symmetry-orbits-of-LUBF-gaussoids-in-dimension-8 <p>The symmetry reduction of <code>LUBF</code>-gaussoids in dimension 8 was a tough nut. I learned that precomputing invariants of the symmetry classes first allows more intelligent distribution of tasks in a parallel symmetry reduction algorithm. In this case, I achieved a 160-fold speedup.</p> Tobias perl, gaussoids, postgresql, mojo, sagemath, symmetry /blog/2019/09/The-computation-of-symmetry-orbits-of-LUBF-gaussoids-in-dimension-8 Wed, 4 Sep 2019 00:00:00 +0000 /blog/2019/01/bah--Another-static-site-generator <p>Yet another static site generator – for a blog and homepage.</p> Tobias perl, vim, web /blog/2019/01/bah--Another-static-site-generator Wed, 30 Jan 2019 00:00:00 +0000