Basic Properties of Periodic Functions

In this article we present definitions, basic properties and some examples of periodic functions according to [5].Li Bo - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaMen Yanhong - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Chuanzhang Chen. Mathematical Analysis. Higher Education Press, Beijing, 1978.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Peng Wang and Bo Li. Several differentiation formulas of special functions. Part V. Formalized Mathematics, 15(3):73-79, 2007, doi:10.2478/v10037-007-0009-4.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

Nilpotent Groups

This article describes the concept of the nilpotent group and some properties of the nilpotent groups.Li Dailu - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaMen Yanhong - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461-466, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Katarzyna Zawadzka. Solvable groups. Formalized Mathematics, 5(1):145-147, 1996

On Rough Subgroup of a Group

This article describes a rough subgroup with respect to a normal subgroup of a group, and some properties of the lower and the upper approximations in a group.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaWojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Some Properties of p-Groups and Commutative p-Groups

This article describes some properties of p-groups and some properties of commutative p-groups.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Marco Riccardi. The Sylow theorems. Formalized Mathematics, 15(3):159-165, 2007, doi:10.2478/v10037-007-0018-3.Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461-466, 1991.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Mapping and functional characterization of structural variation in 1060 pig genomes

BACKGROUND: Structural variations (SVs) have significant impacts on complex phenotypes by rearranging large amounts of DNA sequence.RESULTS: We present a comprehensive SV catalog based on the whole-genome sequence of 1060 pigs (Sus scrofa) representing 101 breeds, covering 9.6% of the pig genome. This catalog includes 42,487 deletions, 37,913 mobile element insertions, 3308 duplications, 1664 inversions, and 45,184 break ends. Estimates of breed ancestry and hybridization using genotyped SVs align well with those from single nucleotide polymorphisms. Geographically stratified deletions are observed, along with known duplications of the KIT gene, responsible for white coat color in European pigs. Additionally, we identify a recent SINE element insertion in MYO5A transcripts of European pigs, potentially influencing alternative splicing patterns and coat color alterations. Furthermore, a Yorkshire-specific copy number gain within ABCG2 is found, impacting chromatin interactions and gene expression across multiple tissues over a stretch of genomic region of ~200 kb. Preliminary investigations into SV's impact on gene expression and traits using the Pig Genotype-Tissue Expression (PigGTEx) data reveal SV associations with regulatory variants and gene-trait pairs. For instance, a 51-bp deletion is linked to the lead eQTL of the lipid metabolism regulating gene FADS3, whose expression in embryo may affect loin muscle area, as revealed by our transcriptome-wide association studies.CONCLUSIONS: This SV catalog serves as a valuable resource for studying diversity, evolutionary history, and functional shaping of the pig genome by processes like domestication, trait-based breeding, and adaptive evolution.</p

A compendium of genetic regulatory effects across pig tissues

The Farm Animal Genotype-Tissue Expression (FarmGTEx) project has been established to develop a public resource of genetic regulatory variants in livestock, which is essential for linking genetic polymorphisms to variation in phenotypes, helping fundamental biological discovery and exploitation in animal breeding and human biomedicine. Here we show results from the pilot phase of PigGTEx by processing 5,457 RNA-sequencing and 1,602 whole-genome sequencing samples passing quality control from pigs. We build a pig genotype imputation panel and associate millions of genetic variants with five types of transcriptomic phenotypes in 34 tissues. We evaluate tissue specificity of regulatory effects and elucidate molecular mechanisms of their action using multi-omics data. Leveraging this resource, we decipher regulatory mechanisms underlying 207 pig complex phenotypes and demonstrate the similarity of pigs to humans in gene expression and the genetic regulation behind complex phenotypes, supporting the importance of pigs as a human biomedical model.</p

Seismic analysis for the Lanzhou fireball

In this paper, we convert the Lanzhou fireball's trajectory using seismic data according to the analytical method presented in Pujol et al. (2005, 2006). Taking the same assumptions as Pujol et al., the position of the fireball burst at its terminal has been converted using a relative simple independent method. Both the trajectory and the position of burst are roughly coincident.The Meteoritics & Planetary Science archives are made available by the Meteoritical Society and the University of Arizona Libraries. Contact [email protected] for further information.Migrated from OJS platform February 202

Growing correlation length of moderate-sized earthquakes prior to two great earthquakes near Sumatra Island

Seismic correlation length for moderate earthquakes prior to two great earthquakes in the northern sea area of Sumatra Island (Mw9.1 in 2004 and Ms8. 6 in 2012) has been studied, using method of Single-Link-Cluster (SLC) analysis, and found to show a power-law growth about two years before their occurrences. No such growth was found for a magnitude 7 earthquake in the same area. This result suggests the occurrence of a physical process of critical-point characteristics in the source area before the great earthquakes

Nilpotent Groups

Summary. This article describes the concept of nilpotent group and some properties of the nilpotent groups