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"Matrix analytic methods are popular as modeling tools because they give one the ability to construct and analyze a wide class of queuing models in a unified and algorithmically tractable way. The authors present the basic mathematical ideas and algorithms of the matrix analytic theory in a readable, up-to-date, and comprehensive manner. In the current literature, a mixed bag of techniques is used-some probabilistic, some from linear algebra, and some from transform methods. Here, many new proofs that emphasize the unity of the matrix analytic approach are included. The authors begin by describing several examples of quasi-birth-and-death (QBD) processes. These examples give the reader an idea of the variety of models which are hidden by the general block notation as well as reinforce some of the terminology and notation used throughout the text. These same examples are used as illustrations later. The second part of the book deals with phase-type distributions and related-point processes, which provide a versatile set of tractable models for applied probability. Part three reviews birth-and-death processes, and points out that the arguments for these processes carry over to more general processes in a parallel manner and are based on Markov renewal theory. Part four covers material where algorithmic and probabilistic reasoning are most intimately connected. In three steps, the authors take you from one of the simplest iterative procedures to the fastest, relating the successive approximations to the dynamic behavior of the stochastic process itself. The final part goes beyond simple QBDs with a sequence of short chapters where the authors discuss various extensions to the analyzed processes. Their intention is to show that the fundamental ideas extend beyond simple homogeneous QBD."

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"Graf merupakan himpunan simpul dan busur dengan setiap busurnya menghubungkan dua simpul. Graf dapat direpresentasikan dalam sebuah matriks. Matriks representasi graf di antaranya yaitu matriks ketetanggaan, matriks jarak, matrik kehadiran, dan matriks Laplacian. Matriks ketetanggaan merepresentasikan ada tidaknya busur yang menghubungkan dua buah simpul. Matriks jarak merepresentasikan jarak lintasan terpendek antara dua simpul pada graf. Pada graf berdiameter dua, yaitu jarak terpanjang di antara dua simpul adalah dua. Graf berdiameter dua di antaranya yaitu graf bipartit, graf roda, dan graf kipas. Pada tesis ini akan dibahas hubungan antara matriks ketetanggaan dan matriks jarak dari suatu graf berdiameter dua, dan sifat-sifat matriks jarak pada graf berdiameter dua, serta polinomial karakteristik dari matriks jarak pada kelas graf khusus berdiameter dua yaitu graf bipartit lengkap 𝐾𝑛,𝑛.

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Graph is the set of vertices and edges where each edge connects two vertices. The graph can be represented by a matrix. There are several matrix representation of graph, such as adjacency matrix, distance matrix, incidence matrix, and Laplacian matrix. The adjacency matrix represents the presence or absence of an arc connecting two vertices. Distance matrix represent the shortest path between two vertices on a graph. The example of two-diameter graphs are bipartite graphs, wheel graphs, and fan graphs. In this thesis we discuss the relationship between the adjacency matrix and the distance matrix of a two-diameter graph, and the properties of the distance matrix in the twodiameter graph, and the characteristic polynomial of the distance matrix of special family of two-diameter graph that is complete bipartite graph 𝐾𝑛,𝑛."

"Misalkan T adalah gelanggang matriks segitiga formal dan T[z; θ, d] adalah gelanggang polinomial miring atas T. Dengan mempelajari homomorfisma khusus dan derivasi miring pada gelanggang matriks segitiga formal, dapat dibuktikanbahwa gelanggang polinomial miring T[z; θ, d] memiliki representasi matriks segitiga formal.

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Let T be a formal triangular matrix ring and T[z; θ, d] be a skew polynomial ring over T. By studying a particular ring homomorphism and skew derivation on formal triangular matrix ring, one can show that the skew polynomial ring T[z; θ, d] has formal triangular matrix representation."

"Training merupakan salah satu aspek yang diatur di dalam CPOB yang menjadi hal penting untuk dilakukan dan diberikan oleh industri farmasi kepada setiap personil untuk menyiapkan personil yang terkualifikasi. Adanya keperluan akan dokumentasi terkait daftar training untuk operator produksi yang selama ini belum terdokumentasikan dengan baik di PT. Darya-Varia Laboratoria Plant Gunung Putri, sehingga dilakukannya pengerjaan tugas khusus ini untuk menyusun dan membentuk matriks training plan operator produksi. Metode penyusunan dan pembentukan matriks training plan dilakukan dengan membentuk klasifikasi bagian produksi sesuai penerapannya di PT. Darya Varia Laboratoria Plant Gunung Putri kemudian dilakukan pemilahan dan pengklasifikasian SOP dan batch manufacturing record sebagai dasar pelatian dan langkah terakhir dilakukan pembuatan materi training beserta bahan evaluasi training yang dilengkapi dengan matriks kualifikasi operator sesuai hasil evaluasi training yang dilakukan. Hasil dari penyusunan dan pembentukan matriks training plan untuk bagian general pharmacy didapatkan sebanyak 83 training plan untuk bagian mixing, 73 training plan untuk bagian filling, dan 33 training plan untuk bagian tulle, sedangkan untuk bagian soft capsule didapatka 58 training plan untuk bagian gelatin, 62 training plan untuk bagian mixing dan compounding, 63 training plan untuk bagian enkapsulasi, 49 training plan untuk bagian inspeksi dan drying, 56 training plan untuk bagian sortir dan printing, 52 training plan untuk bagian polishing, dan 59 training plan untuk bagian striping. Bahan evaluasi training yang dihasilkan sebanyak 71 bagian sesuai dengan jumlah SOP bagian produksi dengan masing masing bagian terdiri dari 10 hingga 20 soal dan juga terbentuk kriteria kualifikasi operator berdasarkan nilai evaluasi training.

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Training is one of the aspects regulated in GMP which is an important thing to be carried out and given by the pharmaceutical industry to each personnel to prepare qualified personnel. There is a need for documentation related to the training list for production operators which has not been well documented at PT. Darya-Varia Laboratoria Plant Gunung Putri. The method to create a matrix training plan was carried out by form a classification of the production section according to its application at PT. Darya Varia Laboratoria Plant Gunung Putri was then sorted and classified into SOPs and batch manufacturing records as the basis for the training. The last step was the preparation of training materials along with training evaluation form supplemented by an operator qualification matrix according to the results of the training evaluation carry out. The results of the preparation and formation of the matrix training plan for the general pharmacy section obtained 83 training plans for the mixing section, 73 training plans for the filling, and 33 training plans for the tulle production, while for the soft capsule section, 58 training plans for the gelatin section, 62 training plans for mixing and compounding, 63 training plans for encapsulation, 49 training plans for inspection and drying, 56 training plans for sorting and printing, 52 training plans for polishing, and 59 training plans for striping. The resulting training evaluation form are 71 sections with consisting of 10 to 20 questions and operator qualification criteria based on the training evaluation value."

"Pesatnya perkembangan teknologi telah mengubah perdagangan konvensional menjadi sistem perdagangan modern. Agar e-commerce berhasil harus dikembangkan menggunakan sistem yang akurat. Salah satu metode pada pendekatan collaborative filtering yaitu latent variable models berdasarkan faktorisasi matriks. Jika setiap vector pu yang menyatakan ketertarikan pengguna ke-u terhadap variabel tersembunyi dan setiap vector qi yang menyatakan hubungan item i dengan variabel tersembunyi dapat ditentukan, maka tingkat ketertarikan antara semua pengguna u pada setiap item i dapat ditentukan. Untuk menghindari terjadinya overfitting proses update pu dan qi dilakukan menggunakan metode gradient descent dengan regularisasi. Penelitian ini menentukan parameter k (banyaknya variabel tersembunyi) dan parameter (nilai regularisasi) agar model optimal.

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The rapid development of technology has changed the conventional trade into a modern trading system. In order for successful e-commerce must be developed using an accurate system. One method in collaborative filtering approach that is latent variable models based on matrix factorization. If any vector pu that expressed interest u user to a hidden variable and each qi vector expressing the relation item i with hidden variables can be determined, then the level of interest among all users u on every item i can be determined. To avoid overfitting the update process on pu vector and qi vector performed using gradient descent method with regularization. This study determines the parameter k (the number of hidden variables) and parameter (value regularization) that makes the model becomes optimal."

"Markov processes are processes that have limited memory. In particular, their dependence on the past is only through the previous state. They are used to model the behavior of many systems including communications systems, transportation networks, image segmentation and analysis, biological systems and DNA sequence analysis, random atomic motion and diffusion in physics, social mobility, population studies, epidemiology, animal and insect migration, queueing systems, resource management, dams, financial engineering, actuarial science, and decision systems.Covering a wide range of areas of application of Markov processes, this second edition is revised to highlight the most important aspects as well as the most recent trends and applications of Markov processes. The author spent over 16 years in the industry before returning to academia, and he has applied many of the principles covered in this book in multiple research projects."