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Algebra, matrices, and computers (2019)
Journal Article
Detinko, A., Flannery, D., & Hulpke, A. (2019). Algebra, matrices, and computers. Snapshots of modern mathematics from Oberwolfach, 5, 1-12. https://doi.org/10.14760/SNAP-2019-005-EN

What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix gr...

The strong approximation theorem and computing with linear groups (2019)
Journal Article
Detinko, A. S., Flannery, D. L., & Hulpke, A. (2019). The strong approximation theorem and computing with linear groups. Journal of Algebra, 529, 536-549. https://doi.org/10.1016/j.jalgebra.2019.04.011

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group for . More generally, we are able to com...

Practical computation with linear groups over infinite domains (2019)
Book Chapter
Detinko, A. S., & Flannery, D. L. (2019). Practical computation with linear groups over infinite domains. In C. Campbell, C. Parker, M. Quick, E. Robertson, & C. Roney-Dougal (Eds.), Groups St Andrews 2017 in Birmingham (261-270). Cambridge: Cambridge University Press. https://doi.org/10.1017/9781108692397.011

We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algorithms have...

Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field (2019)
Journal Article
Dommers, S., & Eichelsbacher, P. (in press). Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field. Stochastic processes and their applications, https://doi.org/10.1016/j.spa.2019.02.007

We study the inhomogeneous Curie-Weiss model with external field, where the inhomo-geneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their wei...

Linear groups and computation (2018)
Journal Article
Detinko, A. S., & Flannery, D. L. (2019). Linear groups and computation. Expositiones Mathematicae, 37(4), 454-484. https://doi.org/10.1016/j.exmath.2018.07.002

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for this class of groups are surveyed. We...

Primordial Gravitational Waves and Reheating in a New Class of Plateau-Like Inflationary Potentials (2018)
Journal Article
Chongchitnan, S. (2018). Primordial Gravitational Waves and Reheating in a New Class of Plateau-Like Inflationary Potentials. Universe, 4(7), Article 77. https://doi.org/10.3390/universe4070077

We study a new class of inflation model parametrized by the Hubble radius, such that aH∝exp(−αφ)n. These potentials are plateau-like, and reduce to the power-law potentials in the simplest case n=2. We investigate the range of model parameters that i...

Algorithms for experimenting with Zariski dense subgroups (2018)
Journal Article
Detinko, A. S., Flannery, D. L., & Hulpke, A. (2020). Algorithms for experimenting with Zariski dense subgroups. Experimental Mathematics, 29(3), 296-305. https://doi.org/10.1080/10586458.2018.1466217

We give a method to describe all congruence images of a finitely generated Zariski dense group . The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n; if n = 2 then we compute all congruence images only...

Large deviations for the annealed Ising model on inhomogeneous random graphs: spins and degrees (2018)
Journal Article
Dommers, S., Giardinà, C., Giberti, C., & Hofstad, R. V. D. (2018). Large deviations for the annealed Ising model on inhomogeneous random graphs: spins and degrees. Journal of statistical physics, 173(3-4), 1045-1081. https://doi.org/10.1007/s10955-018-2027-8

We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertice...

Congruences on direct products of transformation and matrix monoids (2018)
Journal Article
Araújo, J., Bentz, W., & Gomes, G. M. S. (2018). Congruences on direct products of transformation and matrix monoids. Semigroup Forum, 97(3), 384–416. https://doi.org/10.1007/s00233-018-9931-8

Malcev described the congruences of the monoid Tn of all full transformations on a finite set Xn={1,…,n}. Since then, congruences have been characterized in various other monoids of (partial) transformations on Xn, such as the symmetric inverse monoi...

High-dimensional limit theorems for random vectors in ℓpn-balls (2017)
Journal Article
Kabluchko, Z., Prochno, J., & Thäle, C. (2019). High-dimensional limit theorems for random vectors in ℓpn-balls. Communications in contemporary mathematics, 21(1), 1750092. https://doi.org/10.1142/S0219199717500924

In this paper, we prove a multivariate central limit theorem for ℓq-norms of high-dimensional random vectors that are chosen uniformly at random in an ℓnp-ball. As a consequence, we provide several applications on the intersections of ℓnp-balls in th...

On the testability of coarsening assumptions: a hypothesis test for subgroup independence (2017)
Journal Article
Plass, J., Cattaneo, M., Schollmeyer, G., & Augustin, T. (2017). On the testability of coarsening assumptions: a hypothesis test for subgroup independence. International Journal of Approximate Reasoning, 90, 292-306. https://doi.org/10.1016/j.ijar.2017.07.014

Since coarse(ned) data naturally induce set-valued estimators, analysts often assume coarsening at random (CAR) to force them to be single-valued. Focusing on a coarse categorical response variable and a precisely observed categorical covariate, we r...

Metastability in the reversible inclusion process (2017)
Journal Article
Bianchi, A., Dommers, S., & Giardinà, C. (2017). Metastability in the reversible inclusion process. Electronic journal of probability, 22, Article 70. https://doi.org/10.1214/17-EJP98

We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph S with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its...

The likelihood interpretation as the foundation of fuzzy set theory (2017)
Journal Article
Cattaneo, M. E. G. V. (2017). The likelihood interpretation as the foundation of fuzzy set theory. International Journal of Approximate Reasoning, 90, 333-340. https://doi.org/10.1016/j.ijar.2017.08.006

In order to use fuzzy sets in real-world applications, an interpretation for the values of membership functions is needed. The history of fuzzy set theory shows that the interpretation in terms of statistical likelihood is very natural, although the...

Zariski density and computing in arithmetic groups (2017)
Journal Article
Detinko, A., Flannery, D. L., & Hulpke, A. (2018). Zariski density and computing in arithmetic groups. Mathematics of Computation, 87(310), 967-986. https://doi.org/10.1090/mcom/3236

For n > 2, let Gamma _n denote either SL(n, {Z}) or Sp(n, {Z}). We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group H\leq \Gamma _n. This forms the main component of our methods for c...

Empirical interpretation of imprecise probabilities (2017)
Journal Article
Cattaneo, M. (2017). Empirical interpretation of imprecise probabilities. Proceedings of Machine Learning Research, 62, 61-72

This paper investigates the possibility of a frequentist interpretation of imprecise probabilities, by generalizing the approach of Bernoulli’s Ars Conjectandi. That is, by studying, in the case of games of chance, under which assumptions imprecise p...

Orbits of primitive k-homogenous groups on (N − k)-partitions with applications to semigroups (2017)
Journal Article
Araújo, J., Bentz, W., & Cameron, P. J. (2018). Orbits of primitive k-homogenous groups on (N − k)-partitions with applications to semigroups. Transactions of the American Mathematical Society, 371(1), 105-136. https://doi.org/10.1090/tran/7274

© 2018 American Mathematical Society. The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we exa...

The Birman exact sequence for 3--manifolds (2017)
Journal Article
Banks, J. (2017). The Birman exact sequence for 3--manifolds. Bulletin of the London Mathematical Society, 49(4), 604-629. https://doi.org/10.1112/blms.12051

We study the Birman exact sequence for compact 3–manifolds, obtaining a complete picture of the relationship between the mapping class group of the manifold and the mapping class group of the submanifold obtained by deleting an interior point. This c...

On the expectation of operator norms of random matrices (2017)
Book Chapter
Guédon, O., Hinrichs, A., Litvak, A. E., & Prochno, J. (2017). On the expectation of operator norms of random matrices. In Lecture Notes in Mathematics; Geometric Aspects of Functional Analysis (151-162). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-45282-1_10

We prove estimates for the expected value of operator norms of Gaussian random matrices with independent (but not necessarily identically distributed) and centered entries, acting as operators from ℓnp∗ to ℓ q m , 1 ≤ p∗ ≤ 2 ≤ q

On the geometry of projective tensor products (2017)
Journal Article
Giladi, O., Prochno, J., Schütt, C., Tomczak-Jaegermann, N., & Werner, E. (2017). On the geometry of projective tensor products. Journal of functional analysis, 273(2), 471-495. https://doi.org/10.1016/j.jfa.2017.03.019

© 2017 Elsevier Inc. In this work, we study the volume ratio of the projective tensor products ℓpn⊗πℓqn⊗πℓrnwith 1≤p≤q≤r≤∞. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a...