Some Reflections on the Chronological Problems of the Mahābhārata

大叙事詩マハーバーラタのおほよその成立年代を推定しようとする場合は、同叙事詩の叙述の枠組およびそれぞれの枠組を具へた傳本の形成の過程にも注目せねばならない。諸傳本の成立については次の諸段階が想定される。一. ジャナメージャヤ王のサルパサトラ祭場におけるヴァイシャムパーヤナによるヴィヤーサ叙事詩朗誦を枠とする傳本Ｖの成立。二. ナイミシャ林におけるウグラシュラヴァスとシャウナカとの對話を枠とする傳本Ｕの成立。この枠は傳本Ｖにアースティーカ物語が附加された時に設けられたものである。三. 現行ハリヴァンシャの一部を成すバヴィシヤトの編者による傳本Ｕの枠組の踏襲。四. パルヴァサングラハパルヴァンが傳本Ｕに附加されたことによるナイミシャ林對話の「二重導入」の成立。バヴィシヤトでは、婆羅門出身であつたとおぼしきシュンガ王朝開祖プシャミトラ王のアシュヴァメーダ祭擧行が暗示され、アシュヴァメーダ祭からのクシャトリヤ階層の疎外といふ未曾有の事態のもたらした危機感が全篇の主題となつてゐる。この危機感や興奮のいまだ醒めやらぬシュンガ朝中期後期がバヴィシヤトの成立時であつたと思はれる。とすればバヴィシヤトに先行するはずのマハーバーラタＵ傳本がシュンガ王朝期より後に成つたとは考へられない。Ｕ傳本は遅くともシュンガ朝初期中期には成立してゐたと見るべきであり、Ｕ傳本よりさらに古いＶ傳本はすでにマウリヤ朝時代には形成されてゐたと考へるのが妥當である。ただし、このＶ傳本の成立が前マウリヤ朝期まで遡るかいなかは定かではない。一方パルヴァサングラハパルヴァンなど現行マハーバーラタの初三章はシュンガ王朝期より後に順次追加されていつたはずである。すなはち「二重導入」は後シュンガ朝期になつてはじめて成立したと想はれる。本稿では、もつぱら語りの枠組に留意して構想された大叙事詩成立年代論を提示した。本来は、ほかのさまざまなマハーバーラタ成立年代論をも吟味檢討すべきであつたが、その作業は別の機會に俟たねばならない

TorusE: Knowledge Graph Embedding on a Lie Group

Knowledge graphs are useful for many artificial intelligence (AI) tasks. However, knowledge graphs often have missing facts. To populate the graphs, knowledge graph embedding models have been developed. Knowledge graph embedding models map entities and relations in a knowledge graph to a vector space and predict unknown triples by scoring candidate triples. TransE is the first translation-based method and it is well known because of its simplicity and efficiency for knowledge graph completion. It employs the principle that the differences between entity embeddings represent their relations. The principle seems very simple, but it can effectively capture the rules of a knowledge graph. However, TransE has a problem with its regularization. TransE forces entity embeddings to be on a sphere in the embedding vector space. This regularization warps the embeddings and makes it difficult for them to fulfill the abovementioned principle. The regularization also affects adversely the accuracies of the link predictions. On the other hand, regularization is important because entity embeddings diverge by negative sampling without it. This paper proposes a novel embedding model, TorusE, to solve the regularization problem. The principle of TransE can be defined on any Lie group. A torus, which is one of the compact Lie groups, can be chosen for the embedding space to avoid regularization. To the best of our knowledge, TorusE is the first model that embeds objects on other than a real or complex vector space, and this paper is the first to formally discuss the problem of regularization of TransE. Our approach outperforms other state-of-the-art approaches such as TransE, DistMult and ComplEx on a standard link prediction task. We show that TorusE is scalable to large-size knowledge graphs and is faster than the original TransE.Comment: accepted for AAAI-1

Double-winding Wilson loop in SU(N)SU(N) Yang-Mills theory: A criterion for testing the confinement models

We examine how the average of double-winding Wilson loops depends on the number of color $N$ in the $SU(N)$ Yang-Mills theory. In the case where the two loops $C_1$ and $C_2$ are identical, we derive the exact operator relation which relates the double-winding Wilson loop operator in the fundamental representation to that in the higher dimensional representations depending on $N$. By taking the average of the relation, we find that the difference-of-areas law for the area law falloff recently claimed for $N=2$ is excluded for $N \geq 3$, provided that the string tension obeys the Casimir scaling for the higher representations. In the case where the two loops are distinct, we argue that the area law follows a novel law $(N - 3)A_1/(N-1)+A_2$ with $A_1$ and $A_2 (A_1<A_2)$ being the minimal areas spanned respectively by the loops $C_1$ and $C_2$, which is neither sum-of-areas ($A_1+A_2$) nor difference-of-areas ($A_2 - A_1$) law when ($N\geq3$). Indeed, this behavior can be confirmed in the two-dimensional $SU(N)$ Yang-Mills theory exactly.Comment: 6 pages, 2 figures, presented at the 35th International Symposium on Lattice Field Theory (Lattice 2017), 18-24 June 2017, Granada, Spai