No Arabic abstract
We proposed, in our previous paper, to characterize the Hirota-Miwa equation by means of the theory of triangulated category. We extend our argument in this paper to support the idea. In particular we show in detail how the singularity confinement, a phenomenon which was proposed to characterize integrable maps, can be associated with the projective resolution of the triangulated category.
In the present paper we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution structures. By utilizing category algebras and states on categories instead of simply considering categories, we can directly integrate relativity as a category theoretic structure and quantumness as a noncommutative probabilistic structure. Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT), are also be discussed.
We present a quantum description of black holes with a matter core given by coherent states of gravitons. The expected behaviour in the weak-field region outside the horizon is recovered, with arbitrarily good approximation, but the classical central singularity cannot be resolved because the coherent states do not contain modes of arbitrarily short wavelength. Ensuing quantum corrections both in the interior and exterior are also estimated by assuming the mean-field approximation continues to hold. These deviations from the classical black hole geometry could result in observable effects in the gravitational collapse of compact objects and both astrophysical and microscopic black holes.
We consider the resolvent of a second order differential operator with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents unusual powers of $lambda$ which depend on the singularity. The consequences for the pole structure of the $zeta$-function, and the small-$t$ asymptotic expansion of the heat-kernel, are also discussed.
The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $mathbb{R}^n to mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case ($n>1$) are described. Existence or nonexistence of blow-ups in different dimensions, foundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson $n$. Several concrete examples in two- and three-dimensional cases are analysed. Properties of $mathbb{R}^n_{underline{u}} to mathbb{R}^n_{underline{x}}$ mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.
In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field.