On the Stanley Depth of Squarefree Veronese Ideals

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted \sdepth(M), and conjectured that \depth(M) \le \sdepth(M) for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M = I / J$ with $J \subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I_{n,d}$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1\le d\le n < 5d+4$, then \sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d, and if $d\geq 1$ and $n\ge 5d+4$, then d+3\le \sdepth(I_{n,d}) \le \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d.Comment: 10 page

Algorithmic and Hardness Results for the Colorful Components Problems

In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want $G'$ to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is $NP$-hard (assuming $P \neq NP$). Then, we show that the second problem is $APX$-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is $NP$-hard. Finally, we show that the third problem does not admit polynomial time approximation within a factor of $|V|^{1/14 - \epsilon}$ for any $\epsilon > 0$, assuming $P \neq NP$ (or within a factor of $|V|^{1/2 - \epsilon}$, assuming $ZPP \neq NP$).Comment: 18 pages, 3 figure

A domain-theoretic investigation of posets of sub-sigma-algebras (extended abstract)

Given a measurable space (X, M) there is a (Galois) connection between sub-sigma-algebras of M and equivalence relations on X. On the other hand equivalence relations on X are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-sigma-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question

The second fundamental form of the real Kaehler submanifolds

Let $f\colon M^{2n}\to\R^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng conjectured in \cite{YZ} that if the codimension is $p\leq 11$ then, along any connected component of an open dense subset of $M^{2n}$, the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\R^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension three and then by Yan and Zheng for codimension four. In this paper we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the non-holomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate

PERANAN CHENG HO DALAM PERKEMBANGAN AGAMA ISLAM DI INDONESIA TAHUN 1405-1433

Skripsi ini berjudul “Peranan Cheng Ho dalam Perkembangan Agama Islam di Indonesia Tahun 1405-1433”. Permasalahan yang dibahas dalam skripsi ini adalah mengenai latar belakang kehidupan Cheng Ho, peran Cheng Ho dalam perkembangan agama Islam di Indonesia Tahun 1405-1433 dan dampak peran Cheng Ho dalam perkembangan agama Islam di Indonesia Tahun 1405-1433. Metode penelitian yang digunakan adalah metode historis yaitu dimulai dengan mengumpulkan berbagai sumber tulisan maupun lisan, kritik sumber secara internal dan eksternal, interpretasi dan historiografi. Dalam melakukan penelitian penulis banyak menggunakan teknik studi litelatur yaitu mengumpulkan berbagai sumber tulisan yang relevan dengan kajian skripsi, dan teknik wawancara untuk melengkapi sumber tulisan. Cheng Ho merupakan seorang laksamana yang berasal dari China, lahir pada tahun 1371 M dari sebuah keluarga Muslim Cheng Ho kecil belajar mengenai ajaran Islam dan juga dunia kelautan dari Ayahnya yang sudah melaksanakan ibadah haji ke Mekkah, bernama Ma Haji yang merupakan seorang pelaut dan hal tersebut menginspirasi Cheng Ho untuk melakukan pelayaran. Ketika masih anak-anak berusia belasan tahun Cheng Ho ditangkap tentara Ming dan bekerja di istana dengan mengabdi terhadap putra kaisar yang keempat, Zhu Di (Yong Le). Ketika naik tahta menjadi kaisar, Yong Le memperintahkan Cheng Ho untuk memimpin misi pelayaran akbar Dinasti Ming ke Samudera Barat dengan tujuan perdagangan dan persahabatan. Pelayaran muhibah Dinasti Ming yang dipimpin oleh Cheng Ho dengan misi perdagangan dan persahabatan dilakukan dengan berkunjung ke berbagai negara termasuk Indonesia dilakukan selama 7 kali, dari tahun 1405-1433. Ketika berada di Indonesia, Cheng Ho dipercaya tidak hanya melaksanakan misi Dinasti Ming, tetapi juga mempunyai misi pribadi yaitu menyebarkan agama Islam. Peran Cheng Ho dalam perkembangan agama Islam di Indonesia diantaranya adalah melakukan syiar Islam, memberikan fasilitas kepada komunitas Muslim China bermazhab hanafi, membangun masjid-masjid, membantu dalam proses Islamisasi yang kebanyakan masyarakat China perantauan dan mengamalkan perbuatan yang sesuai dengan ajaran agama Islam. Peran Cheng dalam kegiatan agama Islam di berbagai negara, termasuk Indonesia tersebut hanya sedikit yang tercatat dalam catatan-catatan Dinasti Ming. Ada beberapa hal kenapa peran Cheng Ho dalam agama Islam tidak tercatat, seperti misi Cheng Ho dalam penyebaran agama Islam bukan merupakan misi Dinasti Ming dan Islam bukan agama mayoritas masyarakat China maupun Kaisar Dinasti Ming dan pejabat di kalangan istana. Kapan Cheng Ho wafat masih diperdebatkan oleh para sejarawan, antara tahun 1433, 1434 atau 1435. Dampak peran Cheng Ho dalam perkembangan agama Islam di Nusantara, diantaranya muncul beberapa komunitas Muslim China, pembangunan masjid-masjid, komunitas Muslim China mazhab Hanafi yang ada di Indonesia lebih terorganisir keberadaannya setelah dibimbing serta diarahkan oleh Cheng Ho, sedangkan kehidupan bersama secara rukun dan damai hidup berdampingan menjadi warisan terbesar Cheng Ho di Asia Tenggara. The study is based on the author's concerns of theory the arrival of Islam to Indonesia contained in textbooks of history in schools. The entry of Islam into Indonesia mentioning mostly origin of Indian, Arabic and Persian, but the theory of China was never mentioned in textbooks. Therefor, the authors wanted to examine the role of one of China's Muslim leaders and never stepped foot on the archipelago, which Admiral Zheng He in the development of Islam in the archipelago in 1405-1433. The problems discussed are the background of the life of Zheng, Zheng role in the development of Islam in Indonesia in 1405-1433 and the impact of Cheng Ho's role in the development of Islam in Indonesia in 1405-1433. Zheng He was an admiral from China, was born in the year 1371 AD from a Muslim family and a sailor. Zheng He had served the emperor's fourth son, Zhu Di (Yong Le). When ascended the throne as emperor, Zheng Yong Le gave an order of mission to lead a grand voyage of the Ming Dynasty to the Western Ocean with the purpose of trade and friendship. The shipping is done by visiting various countries including Indonesia conducted over 7 times, from the years 1405 to 1433. When in Indonesia, Zheng He believed that a personal mission to spread Islam. Impact Zheng role in the development of Islam in the archipelago, of which emerged some Muslim communities of China, the construction of mosques, while living together in harmony and peaceful coexistence be the greatest legacy of Zheng He in Southeast Asia. Keyword: Cheng Ho, Islamisasi, Dakwah, Pelayaran Muhibah