「世紀難題-考拉茲猜想」 考拉茲猜想中循環的探討
自1930年代以來,考拉茲猜想(Collatz conjecture)一直是個未解之謎,其敘述如下:選定一個自然數,如果是偶數,則用2來除;如果是奇數,則乘以3再加1,經過有限次迭代,最後一定得到1。也就是說會得到1,4,2,1,4,2,…的數列,稱之為1-2-4循環。即使此猜想敘述簡單,卻是個橫跨世紀的難題,至近幾年才有一些證明方法出現。 其中一種證明考拉茲猜想的想法為證明所有不符合考拉茲猜想的狀況為假,而其中一種狀況為除了1-2-4循環還有其他組循環,即有些正整數在經過數次考拉茲猜想的計算後,會進入一組非1-2-4循環的循環。 因此,在此篇報告中我們透過討論每一個奇數在經由乘3再加1的計算後,所得到的偶數的2的冪次,再經由反證法證明除了1-2-4循環不會有其他組循環。
Microfossil association of the Štíty locality
My thesis focuses on studying Cretaceous microfossil specimens from the excavation of former brickworks in Štíty, especially foraminifera. In the theoretical part, I have covered the structure of the Bohemian Cretaceous Basin area, especially Bystřice Lithofacial Development. I have also processed previous paleontological researches from the locality. Emphasis was placed on field research and subsequently on laboratory research of the site. I have examined the present state of the location and gathered samples of silt clay containing a wide variety of fossils. I have acquired the microfossils, determined them, and ordered them systematically. The most important part of the thesis is the systematic and palaeoecological processing of the collection of microfossils from the locality. The thesis continues the research of the last year of SOČ, where I have gathered a collection of fossil macrofauna, flora, and ichnofauna. My collection is supplemented mainly by benthic and planktonic foraminifers. I have confirmed that the specimens found are typical representatives of marine fauna belonging to the Upper Cretaceous Coniacian. The paleoecological characteristics of the locality correspond to a nutrient-rich shallow-water environment, occasionally disturbed by storm waves.
Locus of the Points on Circumference of the n-th Circle that Formed by Moving the Center of any Radius Circles on the Outermost Circumference of Preceding set of Circles
This project aimed to study the motion which occurred from the end point on the circumference of the outermost circle by moving the center on the circumference of a preceding circle and the center of an innermost circle at origin. According to the study, when angular velocity was changed, it caused the different of loci. Based on the above information, finding the locus of the point on circumference of n-th circle that formed by moving the center of any radius circles on circumference of preceding set of circles was studied to get general equation. A set of circle and locus were created with GSP program. First, set the same radius circles on the X-axis with the first circle at origin, then found the relationship that occurred from the characteristics of locus. The result showed that if the ratios of angular velocity are 1:1:1, 2:2:2, 3:3:3, ..., …, n:n:n or 1:2:3, 2:4:6, 3:6:9, …,nw1:nw2:nw3, the characteristics of locus will be the same, while the others will be different. Finally, the equation of locus was found as follow: (x,y) = { ..........see in abstract...........} when .........see in abstract........... Where ri is the radius of i-th circle, zeta i is an angle between the radius of i-th circle and X-axis, wi is the angular velocity, t is elapsed time and alpha i is a starting angle between the radius of i-th circle and X-axis.