QUES: Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincare founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i["] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.

What are usually analyzed by associating groups to them and studying the elements of the corresponding groups?
What is the answer?
ANS: Mathematical objects
QUES: The papacy further refined the practice in the Mass in the Late Middle Ages, holding that the clergy alone was allowed to partake of the wine in the Eucharist. This further distanced the secular laity from the clergy. The laity continued the practices of pilgrimages, veneration of relics, and belief in the power of the Devil. Mystics such as Meister Eckhart (d. 1327) and Thomas a Kempis (d. 1471) wrote works that taught the laity to focus on their inner spiritual life, which laid the groundwork for the Protestant Reformation. Besides mysticism, belief in witches and witchcraft became widespread, and by the late 15th century the Church had begun to lend credence to populist fears of witchcraft with its condemnation of witches in 1484 and the publication in 1486 of the Malleus Maleficarum, the most popular handbook for witch-hunters.

When did Thomas a Kempis die?
What is the answer?
ANS: 1471
QUES: As in Brazil, the mandolin has played an important role in the Music of Venezuela. It has enjoyed a privileged position as the main melodic instrument in several different regions of the country. Specifically, the eastern states of Sucre, Nueva Esparta, Anzoategui and Monagas have made the mandolin the main instrument in their versions of Joropo as well as Puntos, Jotas, Polos, Fulias, Merengues and Malaguenas. Also, in the west of the country the sound of the mandolin is intrinsically associated with the regional genres of the Venezuelan Andes: Bambucos, Pasillos, Pasodobles, and Waltzes. In the western city of Maracaibo the Mandolin has been played in Decimas, Danzas and Contradanzas Zulianas; in the capital, Caracas, the Merengue Rucaneao, Pasodobles and Waltzes have also been played with mandolin for almost a century. Today, Venezuelan mandolists include an important group of virtuoso players and ensembles such as Alberto Valderrama, Jesus Rengel, Ricardo Sandoval, Saul Vera, and Cristobal Soto.

Who are the not famous Venezuelen mandolinist?
What is the answer?
ANS:
unanswerable