Realistic GUT Yukawa couplings from a random clockwork model

We present realistic models of flavor in SU(5) and SO(10) grand unified theories (GUTs). The models are renormalizable and do not require any exotic representations in order to accommodate the necessary GUT breaking effects in the Yukawa couplings. They are based on a simple clockwork Lagrangian whose structure is enforced with just two (one) vectorlike U(1) symmetries in the case of SU(5) and SO(10) respectively. The inter-generational hierarchies arise spontaneously from products of matrices with order one random entries.

Linear k-power Preservers on Tensor Products of Matrices

Invariants and the study of the map preserving a certain invariant play vital roles in the study of the theoretical mathematics. The preserver problems are the researches on linear operators that preserve certain invariants between matrix sets. Based on the result of linear $k$-power preservers on general matrix spaces, in terms of the advantages of matrix tensor products which is not limited by the size of matrices as well as the immense actual background, the study of the structure of the linear $k$-power preservers on tensor products of matrices is essential, which is coped with in this paper. That is to characterize a linear map $f:M_{m_{1}\cdots m_{l}}\rightarrow M_{m_{1}\cdots m_{l}}$ satisfying $f(X_{1}\otimes \cdots \otimes X_{l})^{k}=f\left( (X_{1}\otimes \cdots \otimes X_{l})^{k}\right) $ for all $X_{1}\otimes \cdots \otimes X_{l}\in M_{m_{1}\cdots m_{l}}$.

Inequalities on weighted classical pythagorean means, Tracy-Singh products, and Khatri-Rao products for hermitian operators

We establish a number of operator inequalities between three kinds of means, namely, weighted arithmetic/harmonic/geometric means, and two kinds of operator products, namely, Tracy-Singh products and Khatri-Rao products. These results are valid under certain assumptions relying on (opposite) synchronization, comparability, and spectra of operators. Our results include tensor product of operators, and Tracy-Singh/Khatri-Rao products of matrices as special cases.

Zero-divisor graph of direct products of matrices over semirings

We prove that the zero-divisor graph of a direct product of matrices over finite zero-divisor free semirings uniquely determines the sizes of matrices and cardinalities of semirings in question. We also give an example that the semirings themselves are not necessarily uniquely determined.

Nubeam-dedup: a fast and RAM-efficient tool to de-duplicate sequencing reads without mapping

Summary We present Nubeam-dedup, a fast and RAM-efficient tool to de-duplicate sequencing reads without reference genome. Nubeam-dedup represents nucleotides by matrices, transforms reads into products of matrices, and based on which assigns a unique number to a read. Thus, duplicate reads can be efficiently removed by using a collisionless hash function. Compared with other state-of-the-art reference-free tools, Nubeam-dedup uses 50–70% of CPU time and 10–15% of RAM. Availability and implementation Source code in C++ and manual are available at https://github.com/daihang16/nubeamdedup and https://haplotype.org. Supplementary information Supplementary data are available at Bioinformatics online.