arXiv subjects

Hiroki Mizuno

Publications and source records attributed to Hiroki Mizuno.

5 recordsLinked to original sources

arXiv2026

Underlying Stokes and de Rham structures for Arnold-type invariants

We introduce a framework on dual complexes for studying Arnold-type invariants of immersed curves and immersed surfaces via local finite-difference structures associated with Alexander numberings. For generic immersed plane curves and generic immersed surfaces, we define locally normalized maps $d^k ϕ$ on dual skeleta and show that suitable evaluations recover the Arnold-type invariants $St_{(1)}$ and $St_{(2)}$. In particular, we establish normalized discrete Stokes-type compatibilities between adjacent dual skeleta and derive corresponding Shumakovitch-type identities for curves and surfaces. The normalization coefficients are determined by finite-difference factorial structures together with multiplicities of local configurations. We further interpret the iterated-integral-type structures appearing in Shumakovitch-type identities through finite-difference structures and highest-degree local Stokes compatibilities on dual complexes. We also reinterpret the slice formula for $St_{(2)}$ and $St_{(1)}$ as a compatibility relation between slicing and local operations on the dual complex. These results provide a unified framework in which global Arnold-type invariants arise as distribution-type evaluations of local data on dual complexes. The framework further clarifies the distinction between untwisted local closures and globally twisted structures such as the original Arnold invariant $St$, and suggests the existence of higher-degree local operations associated with the same dual-complex structure.

math.GT↗

arXiv2026

An explicit slice formula for surface invariants via curve invariants

We give an explicit slice formula for a surface invariant of generic immersions in $\mathbb{R}^3$, expressed in terms of curve invariants arising from planar slices. Using a motion-picture viewpoint, we introduce differential measures that record local changes of the curve invariant $St_{(1)}$ and the surface invariant $St_{(2)}$ across singular slice transitions. Our main result shows that, for a quadruple-point event, if $j$ denotes the number of outward coorientations before the event, then the change of the surface invariant satisfies $dSt_{(2)} = 2j - 4$. This yields a computable and combinatorial description of the surface invariant via slice data. In particular, the formula makes explicit the relation between curve-level invariants and finite-order invariants of surface immersions in the sense of Nowik.

math.GT↗

arXiv2025

Complexity of the Existence of Constrained Secure Equilibria in Multi-Player Games

We consider a multi-player non-zero-sum turn-based game (abbreviated as multi-player game) on a finite directed graph. A secure equilibrium (SE) is a strategy profile in which no player has the incentive to deviate from the strategy because no player can increase her own payoff or lower the payoff of another player. SE is a promising refinement of Nash equilibrium in which a player does not care the payoff of another player. In this paper, we discuss the decidability and complexity of the problem of deciding whether a secure equilibrium with constraints (a payoff profile specifying which players must win) exists for a given multi-player game.

cs.GT↗

arXiv2025

Arnold Strangeness of surface immersions

It is known that for any smooth sphere eversion, the number of quadruple point jumps is always odd. In this paper, we define an integer-valued function that detects and classifies jumps involving quadruple points and triple-line tangencies. Our function provides a higher-dimensional analogue of the Arnold strangeness invariant for plane curves. It classifies quadruple point jumps into the five geometrically distinct cases based on coorientation data and reflects finer geometric features for generic immersions of closed surfaces into the 3-space.

math.GT↗

arXiv2012

In-plane deformation of a triangulated surface model with metric degrees of freedom

Using the canonical Monte Carlo simulation technique, we study a Regge calculus model on triangulated spherical surfaces. The discrete model is statistical mechanically defined with the variables $X$, $g$ and $ρ$, which denote the surface position in ${\bf R}^3$, the metric on a two-dimensional surface $M$ and the surface density of $M$, respectively. The metric $g$ is defined only by using the deficit angle of the triangles in {$M$}. This is in sharp contrast to the conventional Regge calculus model, where {$g$} depends only on the edge length of the triangles. We find that the discrete model in this paper undergoes a phase transition between the smooth spherical phase at $b to infty$ and the crumpled phase at $b to 0$, where $b$ is the bending rigidity. The transition is of first-order and identified with the one observed in the conventional model without the variables $g$ and $ρ$. This implies that the shape transformation transition is not influenced by the metric degrees of freedom. It is also found that the model undergoes a continuous transition of in-plane deformation. This continuous transition is reflected in almost discontinuous changes of the surface area of $M$ and that of $X(M)$, where the surface area of $M$ is conjugate to the density variable $ρ$.

cond-mat.stat-mech↗