A cobordism realizing crossing change on sl2\mathfrak{sl}_2 tangle homology and a categorified Vassiliev skein relation

In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Natan homology, and Naot's universal Khovanov homology as well as Khovanov homology with arbitrary coefficients. We also verify that the invariant satisfies categorified analogues of Vassiliev skein relation and the FI relation.Comment: 35 pages, 5 figures. Changed title, Refinement of some part

Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model

Recent studies on the Lorentzian version of the type IIB matrix model show that (3+1)D expanding universe emerges dynamically from (9+1)D space-time predicted by superstring theory. Here we study a bosonic matrix model obtained by omitting the fermionic matrices. With the adopted simplification and the usage of a large-scale parallel computer, we are able to perform Monte Carlo calculations with matrix size up to $N=512$, which is twenty times larger than that used previously for the studies of the original model. When the matrix size is larger than some critical value $N_{\rm c}\simeq 110$, we find that (3+1)D expanding universe emerges dynamically with a clear large-$N$ scaling property. Furthermore, at sufficiently late times, we observe a power-law behavior $t^{1/2}$ of the spatial extent with respect to time $t$, which is reminiscent of the expanding behavior of the Friedmann-Robertson-Walker universe in the radiation dominated era. We discuss possible implications of this result on the original model including fermionic matrices.Comment: 22 pages, 21 figures, 4 tables, (v2) typos correcte

Bifurcation analysis based on a material model with stress-rate dependency and non-associated flow rule for fracture prediction in metal forming

Recent increasing application of advanced high-strength metals causes grow-ing demand for accurate fracture prediction in metal forming simulation. However, since the construction of objective and reliable fracture prediction method is generally difficult, essential progress in fundamental theory that supports evolution of fracture rediction framework is required. In this study, a fracture prediction framework based on the bifurcation theory is pre- sented. The main achievement is a novel material model based on stress-rate dependency related with non-associate flow rule. This model is based on non-associated flow rule with independent arbitrary higher-order yield function and plastic potential function for any anisotropic materials. And this formulation is combined with the stress-rate depen- dency plastic constitutive equation, which is known as Ito-Goya model, to construct a generalized plastic constitutive model in which non-normality and non-associativity are reasonably considered. Then, by adopting the three-dimensional bifurcation theory, which is known as the 3D localized bifurcation theory, more accurate prediction of the initiation of shear band is realized, leading to general and reliable construction of forming limit dia- gram. Then, by using virtual material data, numerical simulation is carried out to exhibit fracture limit diagram for demonstrating the generality and reliability of the proposed methodology. In particular, the effect of stress-rate dependency on the bifurcation analy- sis is investigated, and the order of the yield function is used to investigate the influence on the forming limit prediction