Permutation polynomials over Galois fields of characteristic

In this paper, a class of permutation polynomial known as o-polynomial over Galois fields of characteristic 2 was studied. A necessary and sufficients condition for a monomial 𝑥2k to be an o-polynomial over F2t is given and two results obtained by Gupta and Sharma (2016) were deduced.

Non-isomorphic 2-groups with isomorphic modular group algebras

We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

Identically Self-Dual Matroids

<p>In this thesis we focus on identically self-dual matroids and their minors. We show that every sparse paving matroid is a minor of an identically self-dual sparse paving matroid. The same result is true if the property sparse paving is replaced with the property of representability and more specifically, F-representable where F is a field of characteristic 2, an algebraically closed field, or equal to GF(p) for a prime p = 3 (mod 4). We extend a result of Lindstrom [11] saying that no identically self-dual matroid is regular and simple. We assert that this also applies to all matroids which can be obtained by contracting an identically self-dual matroid. Finally, we present a characterisation of identically self-dual frame matroids and prove that the class of self-dual matroids is not axiomatisable.</p>

Identically Self-Dual Matroids

<p>In this thesis we focus on identically self-dual matroids and their minors. We show that every sparse paving matroid is a minor of an identically self-dual sparse paving matroid. The same result is true if the property sparse paving is replaced with the property of representability and more specifically, F-representable where F is a field of characteristic 2, an algebraically closed field, or equal to GF(p) for a prime p = 3 (mod 4). We extend a result of Lindstrom [11] saying that no identically self-dual matroid is regular and simple. We assert that this also applies to all matroids which can be obtained by contracting an identically self-dual matroid. Finally, we present a characterisation of identically self-dual frame matroids and prove that the class of self-dual matroids is not axiomatisable.</p>

Pengembangan Modul Berbasis Virtual Laboratory Terintegrasi Teknologi Tepat Guna pada Mata Kuliah Kimia Dasar

ABSTRACTThe aims of this research were to describe the validity and to describe the university students respons’ on appropriate technology integrated virtual laboratory based learning module on Acidic and Basic Solution Topic in Tadris IPA, Tarbiyah and Teacher Training Faculty, UIN KHAS Jember. Research method was R&D research by using ADDIE model which consisted of 4 steps: (1) Analyzing curricula and students’ university characteristic, (2) Designing content and lay out of learning module, (3) Developing appropriate technology integrated virtual laboratory based learning module, (4) Validating learning module by expert judging then implementating it to students’ university. The Evaluation steps was not already done by researcher because of time and fee limitation. Result of research showed that learning module got score 3.79 based on material eligibility, 3.88 based on design eigibility and 3,85 based on languange eligibility (for scale 1-4 ). Furthermore, the result of students’ university respon on learning module were 77.8% based on material eligibility, 86.7% based on design egibility, dan 91.1% based on languange eligibility.Keywords: virtual laboratory, appropriate technology, general chemistryABSTRAK Tujuan penelitian adalah untuk mendeskripsikan validitas dan mendeskripsikan hasil uji respon mahasiswa terhadap modul pembelajaran berbasis virtual laboratory terintegrasi Teknologi Tepat Guna pada materi Larutan Asam Basa pada program studi Tadris IPA, Tarbiyah and Teacher Training Faculty, UIN KHAS Jember. Penelitian ini menggunakan model penelitian dan pengembangan yang dimodifikasi dari model ADDIE yang terdiri dari 4 tahapan, yaitu: (1) Analisis Kurikulum dan Karakteristik Mahasiswa, (2) Perencanaan Konten dan Tata Letak Tampilan Modul, (3) Pembuatan Modul Teknologi Tepat Guna yang diintegrasikan dengan Laboratorium Maya (virtual laboratory) based learning module, (4) Validasi/ Uji Ahli Modul Pembelajaran yang dilanjutkan dengan Uji Coba pada Mahasiswa. Tahap kelima, Evaluasi, tidak dilakukan karena keterbatasan waktu dan biaya penelitian. Hasil validasi menunjukkan bahwa Modul Kimia Dasar berbasis Teknologi Tepat Guna terintegrasi Virtual Laboratory memperoleh skor 3.79 berdasarkan komponen kelayakan materi, 3.88 berdasarkan kelayakan desain modul dan 3.85 berdasarkan kelayakan komponen kebahasaan (skala 1-4 ). Selanjutnya, hasil uji respon menunjukkan bahwa modul memperoleh prosentase 77.8% untuk komponen kelayakan materi, 86.7% untuk komponen kelayakan desain modul, dan 91.1% untuk aspek kebahasaan.Kata kunci: virtual laboratory, teknologi tepat guna, kimia dasar

Classification of Simple Lie Superalgebras in Characteristic 2

All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.