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

Basic Properties of Even and Odd Functions

In this article we present definitions, basic properties and some examples of even and odd functions [6].Li Bo - Qingdao University of Science and Technology, ChinaMen Yanhong - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 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.Pacharapokin Chanapat, Kanchun, and Hiroshi Yamazaki. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(2):139-141, 2004.Chuanzhang Chen. Mathematical Analysis. Higher Education Press, Beijing, 1978.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Xiquan Liang and Bing Xie. Inverse trigonometric functions arctan and arccot. Formalized Mathematics, 16(2):147-158, 2008, doi:10.2478/v10037-008-0021-3.Takashi Mitsuishi and Yuguang Yang. Properties of the trigonometric function. Formalized Mathematics, 8(1):103-106, 1999.Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.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.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

Practical Lessons on Optimizing Sponsored Products in eCommerce

In this paper, we study multiple problems from sponsored product optimization in ad system, including position-based de-biasing, click-conversion multi-task learning, and calibration on predicted click-through-rate (pCTR). We propose a practical machine learning framework that provides the solutions to such problems without structural change to existing machine learning models, thus can be combined with most machine learning models including shallow models (e.g. gradient boosting decision trees, support vector machines). In this paper, we first propose data and feature engineering techniques to handle the aforementioned problems in ad system; after that, we evaluate the benefit of our practical framework on real-world data sets from our traffic logs from online shopping site. We show that our proposed practical framework with data and feature engineering can also handle the perennial problems in ad systems and bring increments to multiple evaluation metrics

1,1,3-Trimethyl-3-(4-nitro­phen­yl)indane

In the title compound, C18H19NO2, the five-membered ring of the indane fragment adopts an envelope conformation, with the unsubstituted C atom, acting as the flap atom, deviating by 0.412 (3) Å from the plane through the remaining four atoms. The dihedral angle between the nitro­phenyl ring and the indane benzene ring is 72.5 (1)°. The distances from the two O atoms to the plane of the adjacent benzene ring are 0.113 (4) and 0.064 (4) Å

Dental-Derived Mesenchymal Stem Cells: State of the Art

Mesenchymal stem cells (MSCs) could be identified in mammalian teeth. Currently, dental-derived MSCs (DMSCs) has become a collective term for all the MSCs isolated from dental pulp, periodontal ligament, dental follicle, apical papilla, and even gingiva. These DMSCs possess similar multipotent potential as bone marrow-derived MSCs, including differentiation into cells that have the characteristics of odontoblasts, cementoblasts, osteoblasts, chondrocytes, myocytes, epithelial cells, neural cells, hepatocytes, and adipocytes. Besides, DMSCs also have powerful immunomodulatory functions, which enable them to orchestrate the surrounding immune microenvironment. These properties enable DMSCs to have a promising approach in injury repair, tissue regeneration, and treatment of various diseases. This review outlines the most recent advances in DMSCs’ functions and applications and enlightens how these advances are paving the path for DMSC-based therapies

Several Integrability Formulas of Special Functions. Part II

In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function [3].Bo Li - Qingdao University of Science and Technology, ChinaYanping Zhuang - Qingdao University of Science and Technology, ChinaYanhong Men - Qingdao University of Science and Technology, ChinaXiquan Liang - Qingdao University of Science and Technology, Chin

Effect of Lycii fructus polysaccharides on ovulation failure in rats

Purpose: To investigate the effect of Lycii Fructus polysaccharides (LFPS) on ovulation failure.Methods: A rat model of ovulation failure was established by intragastric administration of hydroxyurea (300 mg/kg). Rats with ovulation failure then received LFPS via oral administration at doses of 100, 200, or 400 mg/kg. The body, uterus and ovary of each rat were weighed using electronic scales. The hypothalamic-pituitary-ovarian (HPO) axis hormones, including estradiol (E2) level, follicle-stimulating hormone (FSH) activity, and luteinizing hormone (LH) activity in the serum of each rat were determined by enzyme-linked immunosorbent assay (ELISA). The levels of pro-apoptotic proteins (Fas, FasL, FADD, c-caspase-8, c-caspase-10, c-caspase-3, c-caspase-6, and c-caspase-7) in the ovarian tissue of each rat were detected by western blot.Results: Hydroxyurea reduced significantly (p < 0.01) uterus and ovary indices (uterus or ovary weight/body weight) (0.119 and 0.026 %), E2 level (3.42 pmol/L), and FSH and LH activities (2.28 and 2.76 U/L), compared with those in the normal group (0.169 and 0.039 %; 6.72 pmol/L; 2.76 and 3.75 U/L). Hydroxyurea increased significantly (p < 0.01) the levels of the above-mentioned pro-apoptotic proteins relative to those in the normal group. LFPS (100, 200, or 400 mg/kg) reversed significantly (p < 0.05 or 0.01) the effect of hydroxyurea on all of the above indices.Conclusion: LFPS exhibits a protective effect on hydroxyurea-induced ovulation failure by regulating the HPO axis hormones and death receptor-mediated apoptotic pathway.Keywords: Lycii Fructus polysaccharides, Ovulation failure, Hypothalamic-pituitary-ovarian axis, Death receptor-mediated apoptotic pathwa