arXiv subjects

Tatsuki Seto

Publications and source records attributed to Tatsuki Seto.

8 recordsLinked to original sources

arXiv2026

A Leibniz rule of distributional pairing and hyperforce sum rule

We reformulate and generalize the equilibrium hyperforce sum rule, a generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, by employing the Schwartz space and its dual. We show that the hyperforce sum rule for the Euclidean space and the equilibrium BBGKY hierarchy at arbitrary level are derived through the Leibniz rule of the derivative for the pairing of tempered distributions and Schwartz functions. We also apply the Leibniz rule to obtain the hyperforce sum rule for systems with periodic boundary conditions.

math-ph↗

arXiv2024

A Young-type integration on self-similar sets in intervals

We introduce a generalization of the Young integration on self-similar sets defined in a closed interval and give a sufficient condition of its integrability. We also prove integration by substitution, integration by parts and term-by-term integration and give examples of the properties.

math.FA↗

arXiv2022

A combinatorial integration on the Cantor dust

In this paper, we generalize the Cantor function to $2$-dimensional cubes and construct a cyclic $2$-cocycle on the Cantor dust. This cocycle is non-trivial on the pullback of the smooth functions on the $2$-dimensional torus with the generalized Cantor function while it vanishes on the Lipschitz functions on the Cantor dust. The cocycle is calculated through the integration of $2$-forms on the torus by using a combinatorial Fredholm module.

math.KT↗

arXiv2019

A combinatorial Fredholm module on self-similar sets built on $n$-cubes

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the Fredholm module.

math.KT↗

arXiv2017

A cyclic cocycle and relative index theorems on partitioned manifolds

In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We also prove two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of [M. Karami-A.H.S. Sadegh-M.E. Zadeh, arXiv:1411.6090, Theorem 3.3].

math.DG↗

arXiv2016

A note on Atiyah's $\Gamma$-index theorem in Heisenberg calculus

In this note, we prove an index theorem on Galois covering for Heisenberg elliptic differential operators, which is not elliptic, analogous to Atiyah's $\Gamma$-index theorem. This note also contains an example of Heisenberg differential operators with non-trivial $\Gamma$-index.

math.DG↗

arXiv2014

Toeplitz operators and the Roe-Higson type index theorem

Let $M$ be a complete Riemannian manifold and assume that $M$ is partitioned by a hypersurface $N$. In this paper we introduce a novel class of functions $C_{\mathrm{w}}(M)$ on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of $\phi$ that belongs to $C_{\mathrm{w}}(M)$ we construct an index class $\mathrm{Ind}(\phi , D)$ in $K_{1}$-group of the Roe algebra of $M$ by using the Kasparov product. It is supposed to be a counterpart of Roe's odd index class. We finally prove that Connes' pairing of $\mathrm{Ind}(\phi , D)$ and Roe's cyclic $1$-cocycle is equal to the Fredholm index of a Toeplitz operator on $N$. This is an extension of the Roe-Higson index theorem to even-dimensional partitioned manifold.

math.DG↗

arXiv2014

Toeplitz operators and the Roe-Higson type index theorem in Riemannian surfaces

We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator. In the proof of this paper, we use some properties of a circle and use Higson's argument. In the last section, there is a example of partitioned manifold, which is not a cylinder, with non-trivial pairing.

math.DG↗