Judul: Hubungan linear antara bilangan fibonacci, bilangan lucas dan golden section di himpunan bilangan real tak negatif
Abstrak: Leonardo of Pisa (Fibonacci), pada tahun 1202 menemukan barisan bilangan yang diperoleh dari bertambahnya jumlah pasangan kelinci tiap bulan. Kelinci tersebut diasumsikan hidup selamanya. Bilangan pada barisan tcrscbut empat abad kemudian oleh Kepler dinyatakan dalam rumus rekursif berikut: F(0) = 0, F(I) = I, F(n+2)=F(n) + F(n+I), dengan n = 0, I, 2, ..... Oleh Eduard Lucas, bilangan pada barisan tersebut diberi nama Bilangan Fibonacci. Kemudian Lucas menemukan barisan bilangan yang mirip dengan bilangan Fibonacci, yaitu : L(0) = 2, l(I) = I, l(n+2)=l(n) + l(n+I), dengan n = 0, I, 2, .... lebih dari ernpat puluh persamaan yang menyatakan hubungan linear antara bilangan Fibonacci, bilangan Lucas dan Golden Section. Sebagian besar dapat dibuktikan dengan prinsip induksi matematika dan aljabar di himpunan bilangan real.
Keyword: Fibonacci numbers, number line, mathematical induction

Judul: Nilai Eigen, Determinan dan Norm Matriks Circulant dengan Entri Bilangan k-Fibonacci
Abstrak: Matriks circulant adalah matriks yang entrinya bergeser satu kolom ke kanan secara berurutan pada baris berikutnya. Entri–entri matriks circulant dapat diisi dengan berbagai barisan bilangan, salah satunya adalah bilangan k–Fibonacci. Berdasarkan definisinya, pada karya ilmiah ini, nilai eigen, determinan dan norm matriks circulant dengan entri bilangan k–Fibonacci diformulasikan secara eksplisit. Kemudian, contoh kalkulasi berdasarkan rumusan tersebut diberikan sebagai ilustrasi., A circulant matrix is a matrix whose entries are shifted one column to the right in succession in the next row. Circulant matrix entries can be filled in various sequence of numbers, one of which is k–Fibonacci number. In this scientific paper, based on the definition, the eigenvalues, determinants and norms of the circulant matrix with the entries of k–Fibonacci numbers are formulated explicitly. Then, examples of the calculation of the obtained formulas are given as illustrations.
Keyword: circulant matrix, determinants, eigen values, matrix norms, k-Fibonacci number

Judul: Heterologous Expression Of Endo-Β-Xylanase From Bacillus Pumilus Strain Bcrs-01 In Escherichia Coli.
Abstrak: Xylanases are glycoside hydrolase family 11 which depolymerise xylan in the plant cell wall, the most excessive form of hemicellulose. Endo-β-xylanases (EC 3.2.1.8) are most important to breakdown β-1,4-glicoside bond of xylan to produce disaccaride xyloolygosaccaride (XOs) and xylose (monomer). Bovine rumen fluids have potential to obtain endo-β-xylanase produced of bacteria. The objectives of this study were to obtain xylanase producing bacteria and to express xynA gene encoding endo-β-xylanase from bacterium in Escherichia coli. Twenty nine isolate bacteria have been succesfully isolated from bovine rumen fluid in agar medium containing birchwood xylan. Isolate BCRS-01 bacterium was previously screened for the high production of xylanase. Based on 16S rRNA gene sequence analysis, BCRS-01 showed high similarity (99%) to Bacillus pumilus (accession number NR043241). The crude xylanase had optimal activity at 55oC, pH 7.0. Furthermore, the crude xylanase exhibited broad pH stability from 4.0 to 10.0 and remained more than 60% and 40% of its activity after incubation at pH 4.0 until 10.0, respectively for 60 min. It also showed a stability at temperature 25-50oC since it retained 80% and 50% of its activity when incubated at 50oC or 30 and 60 min, respectively. The xylanase activity was strongly inhibited by Ca2+ and Cu2+ at concentration 10 mM, and as little as 1 mM for Mn2+. Therefore, this xylanase has potentially used as additive feeds in animal feed industry. The DNA sequence xynA gene encoded endo-β-xylanase was amplified and cloned into the expression vector pET-15by under the control of T7 promoter. The recombinant protein had succesfully expressed in Escherichia coli BL21 (DE3) pLysS. A protein recombinant of approximately, 26.7 kDa was observed on the SDS-PAGE analysis. Crude extract of recombinant endo-β-xylanase demonstrated activity of enzyme at 56 U mL-1 using birchwood xylan as a substrate at 55oC for 10 min.
Keyword: Bacillus pumilus, bovine rumen fluid, endo-β-xylanase, heterologous expression.