從有限三角和公式研究偶次調和級數之遞迴公式及其相關等式之推廣與應用
本研究中,我們將提出一些新穎結果,著重討論其在三角中的應用;同時,找出其遞迴關係式,得出三角展開式與其所對應之多項式分解式,進而討論出多種的規律性及所涵蓋的內容及推廣性質,我得到很多高中數學公式無法推導出在【4】和【8】中的漂亮公式及創新的結果,且這些等式都是由我們不太瞭解的無理數所構成的。
主要是討論我們在【7】中所得到的收穫與經驗;複數是三角、幾何、代數互動的橋樑,我是以不同的角度及嶄新的方法來綜合探討在【6】中相關的應用。提出關於正整數平方的倒數和公式更為精簡且基本的證明,將 sin−2 x 表示成級數形式的部分分式,進而應用在(a,b) = 1的機率問題上;並研究相關的等式,直接透過三角與代數來研究關於 2p 次方的倒數之求和問題,得出級數 之和的有用遞迴公式,並與最重要的常數扯上關係。
For one thing, we present diverse methods to evaluate finite trigonometric summation and related sums. Trigonometric summations over the angles equally divided on the upper half plane are investigated systematically. Several related trigonometric identities are also exhibited.
What is more, we use methods of calculus, and make several surprising and unexpected transformations. A useful recursive formula for obtaining the infinite sums of even order harmonic series, infinite sums of a few even order harmonic series, which are calculated using the recursive formulas, are tabulated for easy references. Furthermore, is there any interesting results and applications?
Finally, the purpose of this paper is to develop a new proof of and related identities, but their derivations are more complicated. The following studies are completed under the instruction of the professor.
分子篩與過氧化氫感測器
目前市面上缺乏簡單而精準的過氧化氫檢測法,我們參考Fenton Reaction 中,鐵離子對過氧化氫分解的催化模式,利用鐵、鈷、鎳、錳、鋅分子篩,測試他們催化過氧化氫分解反應的效率,從成本、毒性、活化能的多方考量下,選擇以分子篩來固定金屬催化離子,作為我們後續實驗的研究主軸。實驗的初步,我們選擇過氧化氫作為自由基,並著重於過氧化氫的分解反應。利用濃差電池的原理,設計出一套濃度檢測系統,由分子篩作為電極。鑒於粉末狀的分子篩容易流失,我們製備出陶土鐵分子篩,以陶土固定分子篩,並以此作為電極,搭配白金絲,透過能士特方程式,測出過氧化氫的濃度,且藉由電路調控放大倍率,可以直接控制檢測範圍。從實驗結果得知,鐵分子篩在處理過氧化氫的時候,不會有鐵離子溶出的現象,且其催化性在酸性液中可以維持,能不斷的使用,長時間來看,分子篩相當有經濟與環境保護上的價值。We attempted to provide a system for quickly determining the concentration of free radicals. The existing methods or techniques are inefficient or need expensive equipment, therefore, an inexpensive system is being sought for. As a preliminary study, we focused on the decomposition of hydrogen peroxide. Taking the Fenton reaction as reference, we designed a measuring system. This system includes a catalyst containing Fe or Pt ions for catalyzing hydrogen peroxide decomposition reaction. The Fe- and Pt-zeolite were prepared to hold Fe and Pt ions to avoid losing. Because the electrically induced potential would decrease with the decrease in the concentration of hydrogen peroxide, we could measure the concentration of hydrogen peroxide by monitoring the electrical potential. We determined the initial concentration of hydrogen peroxide in water from the initial electrical potential measured through the equation obtained from the calibration line. The practicability of this system has been assured after a series of experiments. We will further develop the technique for measuring other free radicals. We anticipate that this technique will be further developed for measuring other free radicals. Although there are several problems and limitations to be solved and conquered, one thing is for sure: this system is an environment-friendly and cost-effective facility for determining the concentration of free radicals in an aqueous solution.
牛魔王的故鄉-台東利吉惡地之探討
本研究針對利吉惡地進行探討,研究此區之泥岩含水量、有機質含量、pH值、比重、可溶性陽離子含量、滲水特性和該區之植物種類,並探討坡度、水量對沖蝕率、山脊密度和溝痕形成之影響。 研究結果發現: 一、表層泥岩之含水率較高,深層泥岩最低。中層泥岩之有機質含量較高,表層與深層泥岩較低。各層泥岩pH值約8.1;無植被採樣點之 pH較高,有植被採樣點偏中性。有植被採樣點,其 Ca2+含量較高。 二、此區共發現十九種植物,其中銀合歡、相思樹屬優勢種。 三、坡度增加時,沖蝕率亦增加;水量增加時,沖蝕率、溝痕寬度也隨之增加,兩者呈高度正相關。第一區坡度較緩,山脊密度較大,第二區坡度較陡,山脊密度小,表面較平坦,溝痕較淺。Our research discusses with the contents of water in mudstone, the organic content, pH, the specific weight, the contents of dissoluble cation, dankness and the category of plants in Ligiligi Badland. Dissecting it`s slope, the abrasion of water, the density of mountain ridge and the formation of scuff mark. Outcome of our research: 1.Solum of mudstone is the dampest. Intermediate of mudstone has more organic content than others. Every bed of mudstone`s PH is 8.1. Having plants area is indifferent and having more Ca2+.2.We found nineteen categories of plants. For example Leucaena glauca and Taiwan acacia. 3.The more augmentation of gradient,the more increase of the abrasion of water and the breadth of the density and the abrasion of water too.
佛手瓜卷鬚之向觸性及其參與蛋白質之探討
本研究利用佛手瓜的卷鬚探討向觸性的原理。本研究大致分為兩部份,一方面我們在卷鬚中發現了含量極為豐富的構造,此一螺旋狀構造分布於維管束中,且用雙縮?詴劑檢測後發現其含有蛋白質,且不具有運輸水分的功能;並發現此一構造的分布疏密,會影響到螺旋內側外側以及切割後片段泡溫水的彎曲方向。此外,在進行卷鬚蛋白質電泳的過程中,我們發現使用含尿素的緩衝液萃取蛋白質的效果最佳,1克的卷鬚乾重約可萃取到5毫克的蛋白質,且蛋白質總量會隨著卷鬚的成熟而遞減。利用軟體比對及質譜分析八個蛋白質點,得知此八點的蛋白質為:malate dehydrogenase, oxygen-evolving enhancer protein 1, oxyen-evolving enhancer protein 2, calreticulin, peroxidase, stromal 70 kDa heat shock-related protein, and AP2/ERF and B3 domain-containing transcription repressor。由此可知,向觸性為植物經過一連串訊號傳遞後,對外界刺激的順應。
丹尼爾寶寶的酗酒日記--酒精對斑馬魚胚胎頭骨與腦下垂體基因的影響
長期以來,臨床研究發現酒精會影響人類胚胎的正常發育,但是其分子機 制尚不清楚。在本研究中利用RNA 定位雜交的方式觀察酒精對於胚胎發育過程 中shh、sox9a、sox9b、col2a1、hand2 的影響,發現這些基因的表現均會受到酒 精的抑制。這項結果顯示在胚胎發育過程中,酒精透過對上述基因的影響,造成 神經脊細胞減少,細胞遷移異常,以及干擾軟骨細胞分化的現象,進而造成頭骨 發育的嚴重缺陷。此外,實驗中亦發現生長激素在腦下垂體的表現亦受到酒精抑 制。這項研究的結果成功地從基因的層次深入了解胎兒酒精中毒症候群造成頭骨 畸形及生長遲緩的病理機制。 It was known that prenatal alcohol exposure may cause serious birth defects and developmental disabilities. The molecular mechanism of this fetal alcohol syndrome still remains unclear. As revealed by whole mount RNA in situ hybridization, it was shown that expression of a number of craniofacial cartilage-related genes, including shh, sox9a, sox9b, col2a1 and hand2, were all inhibited in zebrafish embryo by alcohol exposure. It suggests that alcohol exposure may result in reducing neural crest cell production, interfering neural crest migration, preventing chondrogenesis and eventually cause craniofacial defects. In addition, the transcriptional profile of pituitary hormones were investigated by RNA in situ hybridization. It appears that only growth hormone, but not prolactin and thyroid stimulating hormone, was inhibited by alcohol exposure. The reduction of growth hormone transcription was also confirmed by real time PCR. It also appears that the expression of upstream transcription factor pit1 and downstream target gene igf1 remains unchanged. It suggests that the reduction of gh transcription is mediated by a PIT1-independent pathway. The transcriptional profile of alcohol-exposed embryo was investigated by gene microarray analysis. It appears that the expression profiles of a number of development, cellular signaling, cell growth and apoptosis related genes have be affected by 1.5% alcohol treatment. It was noted that a number of retinal-specific genes were all repressed significantly. It consists with histochemical observation that alcohol exposure results in loss lamination and disturbed differentiation. This study help us understanding the molecular mechanism of fetal alcohol syndrome.
AtbZIPs 轉錄因子及其下游基因啟動子的特定序列之研究
Arabidopsis thaliana bZIPs(AtbZIPs)是一群影響層面相當廣泛的轉錄因子,一半以上的 AtbZIPs 基因表現受到光的調控,且近九成的分子機制尚未明瞭,因此探討 AtbZIPs 在植物光調控機制中所扮演的角色將是個有趣的課題。AtbZIP16 與 AtbZIP17 皆被推測會參與光的調控機制,然而迄今少有文獻針對這二個轉錄因子進行更多的研究。因此,我們想藉由細菌單雜合系統(Bacterial one hybrid system)的方法,找出能與 AtbZIP16 與 AtbZIP17轉錄因子結合的 DNA 序列,以瞭解此二轉錄因子調節下游基因表現的分子機制,並探討其在光訊息傳導途徑中所扮演的角色。針對 AtbZIP16 與 AtbZIP17,本實驗分別找到了 7 與10 種可能的結合序列。首先,經由資料庫比對分析,我們發現其序列上帶有的 motifs 功能,主要參與在光調控、環境逆境反應機制、組織發育、賀爾蒙調節、病原菌防禦、鈣離子訊息傳遞等方面,其中又以光調控佔最大的比例。再者,藉由將 motifs 的功能繪製成文氏圖,並與 HY5 (AtbZIP56)做比較,結果顯示,這三個轉錄因子雖同屬於 AtbZIP family,據推測皆受到光的調控,可能參與某些相似的生理調節過程,但都各自具備不同的功能,影響植物體的發育。如此的差異,表示他們有實質上的不同,值得我們更深入的研究。整體而言,本實驗結果除了說明 AtbZIPs 的功能確實廣泛之外,也顯示AtbZIP16 與AtbZIP17 是執行光訊號傳導很重要的調控因子。Arabidopsis thaliana bZIPs (AtbZIPs) is a group of transcription factors affecting a wide range of responses in Arabidopsis. The expression of more than half of the AtbZIPs is regulated by light, and the molecular mechanism for roughly 90% of these AtbZIPs remains unknown. Therefore, the roles AtbZIPs play in Arabidopsis light signal transduction is an interesting topic to pursue. AtbZIP16 and AtbZIP17 have been suggested to participate in the regulation mechanism mediated by light. However, only limited studies for these two transcription factors have been previously performed. For this reason, we intended to determine the DNA-binding sequences for AtbZIP16 and AtbZIP17 via the bacterial one hybrid system to reveal their target binding sites in the promoter region of their downstream genes and to speculate their possible biological function especially in light signal transduction pathway. We have identified 7 and 10 possible recognition sequences for AtbZIP16 and AtbZIP17, respectively. Using motif-finding programs analyses, we found the motifs identified are mainly involved in light and stress signaling, tissue development, hormone regulation, pathogen defense and Ca2+ signaling. Among these regulation pathways, sequences involved in light regulation owns the highest proportion. Furthermore, a Venn diagram was generated to compare functions of genes regulated by AtbZIP16, AtbZIP17 and HY5. Results revealed that, although these three transcription factors all belong to the AtbZIP family and are predicted to be regulated by some similar physiological regulation process (e.g. light), they still possess distict biological functions in plant development. Further studies are thus required to put these transcription factors into their shared and unique biological context. Taken together, the results of this experiment not only indicated light is a key regulation factor for AtbZIP16 and AtbZIP17, but also showed the function of AtbZIPs could be diverse.