Judul: Determinan Dan Invers Matriks Circulant Dengan Vektor Utama Barisan Bilangan
Abstrak: Matriks Circulant adalah salah satu matriks segi dengan entri-entri pada setiap baris identik dengan entri-entri pada baris sebelumnya dengan bergeser atau berpindah posisi ke kanan. Vektor utama pada matriks circulant terletak pada baris pertama dan sering disebut dengan circulant vector. Vektor utama berupa barisan bilangan yang mempunyai bentuk rekursif akan memengaruhi pola nilai determinan dan invers matriks circulant. Pada karya ilmiah ini akan ditentukan determinan dan invers matriks circulant dengan vektor utama barisan bilangan.
Keyword: determinan, invers, matriks circulant, vektor utama barisan bilangan

Judul: Determinan dan Invers Matriks Circulant dengan entri Jacobsthal Numbers.
Abstrak: Matriks circulant adalah matriks yang entrinya bergeser satu kolom ke kanan secara berurutan, sehingga untuk mengetahui entri dari matriks circulant dapat dilihat dari satu kolom ataupun satu baris dari matriks tersebut. Salah satu sifat yang dimiliki matriks circulant, yaitu inversnya juga berupa circulant. Entrientri dari matriks circulant dapat diisi dengan berbagai entri, salah satunya adalah bilangan-bilangan yang membentuk suatu barisan bilangan. Salah satu barisan bilangan adalah Jacobsthal Number. Pada karya ilmiah ini ditentukan determinan matriks circulant dengan entri Jacobsthal Number menggunakan sifat determinan ketika matriks tersebut sudah dibentuk menjadi matriks segitiga atas dan mencari invers matriks tersebut cukup dengan mencari entri-entri pada baris terakhir dari matriks invers tersebut dengan memanfaatkan teorema invers dari matriks direct sum.
Keyword: bilangan Jacobsthal, determinan, inverse, matriks circulant

Judul: Utilization of hydrolized duckweed (Lemna minor) meal as a soybean replacement in Nile tilapia (Oreochromis niloticus) diet,
Abstrak: Tilapia is a fast growing fish widely cultured in many tropical and sub tropical countries. The recent production figures reported our global production estimates for 2015 is 5,576.8 metric tons. In developing countries the major challenges facing aquaculture industry is high cost of production associated with fish feeds. Feed represents 40-70% of operating costs for aquaculture operations, increasing expenses and potential declining supplies soybean meal have led scientists to search for less expensive but compatible alternative protein sources for fish feed; most of which are based on animal or plant protein sources. The objectives of study was to determine the incubation period for hydrolysis of duckweed and to evaluate the use of hydrolized duckweed as aprotein source for the growth performance of Nile tilapia Three experiments were conducted to determine the composition and digestibility of hydrolyzed duckweed meal by sheep rumen liquor as the replacement of soybean meal in the diet of Nile tilapia. The first trial examined the effectivity of sheep rumen liquor enzyme to decrease crude fiber contents of (DWM), the added volume of sheep rumen liquor was 100 ml/kg DWM and then it was incubated for 0, 12, 24 hr . The second trial was to evaluate the apparent digestibility of hydrolyzed DWM for Nile Tilapia, apparent digestibility were determined using chromic oxide indicator added to both reference diet and test diets. The feed ingredients used in the trial were hydrolyzed DWM (HDWM) and unhydrolyzed DWM (DWM). Ten fish with an average weight of 18±0.36g were used in the trial and reared in the aquarium of size 30×40×50cm for 15 days; feces were collected from three replicates using a fecal collection column attached to the aquarium. Third trial was to evaluate the growth performance of Tilapia fed test diet with hydrolyzed Duckweed meal (HDWM). The experimental design was set according to completely randomized design with four treatments and three replicates of HDWM inclusion levels that is 0%, 15%, 30% and 45%. Nile tilapia fish with initial body weight of 18±0.36g was kept in 12 units aquarium size of 30×40×50cm at stocking density of 10fish/aquarium, the fish were fed two times a day at 08:00am and 16:00pm to apparent satiation for a period of 60 days. The results indicated that duckweed meal (DWM) hydrolyzed with 100 mL/kg and incubated for 24 hrs showed the lowest crude fiber contents of 13.68% compared to other treatments. Total digestibility of hydrolyzed duckweed (HDWM) meal was 87.3% compared with unhydrolyzed duckweed (DWM) which was 66.8%. Total final weight (TFW), survival rate (SR), specific growth rate (SGR), feed conversional rate (FCR) and protein retention (PR) shows growth performance at 15% inclusion. The results of this study showed that 100mL/kg sheep rumen liquor at incubation period of 24 hr can be suitable for hydrolysis of duckweed to reduce crude fiber also hydrolyzed duckweed can substitute soybean meal at an incubation rate of 15% in the diet of Nile tilapia and had the same growth and FCR with control diet
Keyword: Digestibility, Duckweed meal, Growth performance, Sheep rumen