Nearly Sasakian manifolds revisited

We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.

Analisis Pemahaman Konsep Matematika Mahasiswa dalam Menyelesaikan Soal Teori Grup

Tujuan penelitian ini untuk menganalisis pemahaman konsep mahasiswa, menganalisis kesalahan dan penyebab mahasiswa melakukan kesalahan dalam menyelesaikan soal Teori Grup. Penelitian ini menggunakan metode deskriptif kualitatif. Subjek dalam penelitian ini adalah mahasiswa semester IV kelas D Program Studi Pendidikan Matematika Universitas Singaperbangsa Karawang. Instrumen dalam penelitian ini berupa uraian dan wawancara. Metode yang digunakan dalam penelitian ini adalah metode deskriptif kualitatif. Dari hasil analisis data penelitian, diperoleh persentase pemahaman konsep mahasiswa dalam menyelesaikan soal Teori Grup pada indikator menyatakan ulang sebuah konsep 73,46%, indikator mengklasifikasikan objek menurut sifat tertentu 37,69%, indikator memberikan contoh dan non contoh dari konsep 90,38%, indikator menyajikan konsep dalam berbagai bentuk representasi matematis 99,04%, indikator mengembangkan syarat perlu atau syarat cukup suatu konsep 14,1%, indikator menggunakan, memanfaatkan, dan memilih prosedur atau operasi tertentu 67,18% dan indikator mengaplikasikan konsep atau algoritma pemecahan masalah 41,15%. Kesalahan yang dilakukan mahasiswa meliputi kesalahan konsep, kesalahan data, kesalahan menggunakan logika untuk menarik kesimpulan, kesalahan menggunakan definisi atau teorema, penyelesaian tidak diperiksa kembali dan kesalahan strategi. Penyebab mahasiswa melakukan kesalahan dalam menyelesaikan soal tersebut yaitu: mahasiswa kurang teliti dalam membaca soal, kurang mampu memahami soal abstrak, kurang paham dengan konsep pembuktian, kurang paham bagaimana cara mengawali sebuah pembuktian, kesulitan dalam memahami dan mengingat definisi pada konsep pembuktian, kesulitan memanfaatkan definisi dalam menyusun sebuah pembuktian dan kurang percaya diri dalam menjawab sebuah soal. Kata kunci: pemahaman konsep, kesalahan siswa, teori grup ABSTRACT The purpose of this study is to analyze the students’ concept and students’ mistake while solving the Theory Group task. This experiment used the qualitative descriptive methode. The Subjects of this experiment were the students in 4th semester on D Class of Mathematic Education in Singaperbangsa Karawang University. data were collected by observation and interview method. The result shows that the percentage of students’ concept understanding while solving the Theory Group is 73,46%, the result of object clarification according to another character is 37,69%, giving sample and non sample is 90,38%, presenting concept by the right form mathematic representation is 99,04%, developing the need-qualification and enough-qualification concept is 14,1%, using the benefit and chosing the certain procedure is 67,18% and applying concept and algorithm to solve problem is 41,15%. The students’ mistakes while solving the Teory Group task are: conceptual mistake, calculation mistake, using the wrong logic to draw the conclusion, using the wrong theorem, not checking the task. Some indicators such as: the students don’t understand enough about the abstract question, conceptual proof, how to start the proof, they get difficulties to remember the definition in conceptual proofing and use them to construct the proof, the students are not paying attention while reading the question, and they are having less confidence to answer the questions are the causes of the students’ mistake while solving Theory Group. Keyword: conceptual Understanding, students’ mistake, theory group.

On the consistency of the spacings test for multivariate uniformity, including on manifolds

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ ℝd that is based on the maximal spacing generated by independent and identically distributed points X1, . . ., Xn in K, i.e. the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the d > 1 case are only availabe under uniformity, a key element of the proof is a suitable coupling. The proof is general enough to cover the case of testing for uniformity on compact Riemannian manifolds with spacings defined by geodesic balls.

A Short Conceptual Proof of Narayana's Path-Counting Formula

We deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.

Drinfeld Realization of Quantum Twisted Affine Algebras via Braid Group

The Drinfeld realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig’s braid group action.

A Note on Statistical Averages for Oscillating Tableaux

Oscillating tableaux are certain walks in Young's lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits. We show that the average weight of all oscillating tableaux of shape $\lambda$ and length $|\lambda|+2n$ (where $|\lambda|$ denotes the size of $\lambda$ and $n \in \mathbb{N}$) has a surprisingly simple formula: it is a quadratic polynomial in $|\lambda|$ and $n$. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm of Propp and Roby may lead to a more conceptual proof of this result and reveal a hidden symmetry in the set of perfect matchings.

Prime Decomposition of Three-Dimensional Manifolds into Boundary Connected Sum

In 2003 Matveev suggested a new version of the Diamond Lemma suitable for topological applications. We apply this result to different situations and get a new conceptual proof of theorem on decomposition of three-dimensional manifolds into boundary connected sum of prime components.