Norm form equations and linear divisibility sequences

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.

On the finiteness of solutions for polynomial-factorial Diophantine equations

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that {l!} is represented by {N_{A}(x)}, where {N_{A}} is a norm form constructed from the field norm of a field extension {K/\mathbf{Q}}. We also deal with the equation {N_{A}(x)=l!_{S}}, where {l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of {\mathbf{Z}} and for any polynomial {P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation {P(x)=l!_{S}} has only finitely many solutions {(x,l)}. For some special infinite subsets S of {\mathbf{Z}}, we can show the finiteness of solutions for the equation {P(x)=l!_{S}} unconditionally.

Computation of Euclidean minima in totally definite quaternion fields

We describe an algorithm that allows to compute the Euclidean minimum (for the norm form) of any order of a totally definite quaternion field over a number field [Formula: see text] of degree strictly greater than [Formula: see text]. Our approach is a generalization of the previous work dealing with number fields. The algorithm was practically implemented when [Formula: see text] has degree [Formula: see text].

Designing the Game to Play: Optimizing Payoff Structure in Security Games

We study Stackelberg Security Games where the defender, in addition to allocating defensive resources to protect targets from the attacker, can strategically manipulate the attacker’s payoff under budget constraints in weighted L^p-norm form regarding the amount of change. For the case of weighted L^1-norm constraint, we present (i) a mixed integer linear program-based algorithm with approximation guarantee; (ii) a branch-and-bound based algorithm with improved efficiency achieved by effective pruning; (iii) a polynomial time approximation scheme for a special but practical class of problems. In addition, we show that problems under budget constraints in L^0 and weighted L^\infty-norm form can be solved in polynomial time.

Perancangan Basis Data E-Library Program Studi Matematika FMIPA Universitas Udayana

Perancangan basis data merupakan proses membuat desain yang akan mendukung operasional dan tujuan suatu instansi. Pemanfaatan basis data pada e-library memungkinkan untuk menyimpan dan men-download e-book ataupun e-journal dengan mudah. Tujuan dari e-library itu sendiri terkait dengan efisiensi biaya penyimpanan fisik dan mempermudah civitas akademika FMIPA Udayana untuk mencari referensi-referensi yang akan digunakan dalam penelitian mereka. Metode yang digunakan dalam merancang basis data e-library adalah metode perancangan basis data relasional sampai dengan bentuk normal ke 3 (3-NF) dan skema basis data yang dihasilkan digambarkan dengan menggunakan SQLyog. Hasil perancangan sistem basis data e-library ini berupa tiga buah tabel yaitu tabel daftaranggota, tabel download, dan tabel katalog yang sudah memenuhi third norm form. Melalui rancangan basis data e-library ini pengguna dapat mengetahui laporan mengenai data mahasiswa yang menjadi anggota e-library, data download, dan data katalog berupa e-book ataupun e-journal yang tersedia, serta dapat mengetahui beberapa informasi lain yang bermanfaat bagi pengguna berkaitan dengan data pada tabel daftaranggota, tabel download, dan tabel katalog.