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Hiroyasu Miyazaki

Publications and source records attributed to Hiroyasu Miyazaki.

17 recordsLinked to original sources
arXiv2026

A Tractable Continuous-Time Model for Designing Interventions for Time-Inconsistent Agents

Designing effective goals and rewards for time-inconsistent agents is a central problem in many long-term tasks, such as learning, exercise, work, and project completion. An agent may initially plan to complete a task, but later abandon it because, under non-exponential discounting, the perceived trade-off between immediate effort and delayed reward changes over time. This paper develops a tractable continuous-time model for analyzing and designing interventions for such agents in deadline-constrained progress-based tasks. In the model, an agent repeatedly chooses a future progress trajectory that minimizes perceived cost and then follows its infinitesimal initial direction. Although this leads to a continuous-time dynamic behavior defined through a variational problem, we show that the resulting trajectory admits a concise analytical representation under generalized hyperbolic discounting, a broad class of discount functions that includes exponential and hyperbolic discounting as special cases. Using this representation, we characterize when the agent completes the task, abandons it immediately, or exhibits time-inconsistent abandonment after making partial progress. We then study two intervention design problems: optimal goal setting and optimal reward scheduling. For goal setting, we derive optimal goals both when exploitative rewards are allowed and when they are prohibited, and we identify conditions under which exploitative rewards are ineffective. For reward scheduling, we show that, for a fixed number of stages, equal-length periods and equal rewards are optimal, and that finer reward splitting monotonically improves final progress up to a discount-independent limit. These results provide a continuous-time framework for intervention design for time-inconsistent agents and clarify how optimal interventions differ from those in existing discrete-time models.

cs.GT↗
arXiv2025

Motivic homotopy theory with ramification filtrations

We construct a generalization of Morel--Voevodsky's motivic homotopy theory that captures non-$\mathbb{A}^1$-homotopy-invariant phenomena, such as wild ramification and irregular singularity. In the first part, we develop our motivic homotopy theory over quasi-compact and quasi-separated schemes, which satisfies the fundamental properties such as the projective bundle formula, the blow-up sequence, the Gysin sequence, and the Thom isomorphism when the base is normal. Moreover, we compare our theory with existing frameworks. In particular, we recover Morel--Voevodsky's motivic homotopy category and Binda--Park--{\O}stv{\ae}r's logarithmic motivic homotopy category as reflective localizations of our category over normal bases. Furthermore, we construct adjoint functors connecting Annala--Iwasa's category of motivic spectra with ours. In the second part, we equip several non-$\mathbb{A}^1$-homotopy invariant cohomology theories, such as Hodge cohomology, Hodge--Witt cohomology, rank $1$ integrable connections, and unramified cohomology, with canonical filtrations that encode arithmetic and geometric information such as irregular singularities and wild ramification, and prove that these cohomology theories with filtrations are representable in our motivic homotopy category. We also compute some of those filtrations explicitly, and show that they recover known constructions, including a ramification filtration on the Pontryagin dual of the abelian \'etale fundamental group, and an irregularity filtration on the sheaf of rank $1$ connections.

math.AG↗
arXiv2023

A motivic construction of the de Rham-Witt complex

The theory of reciprocity sheaves due to Kahn-Saito-Yamazaki is a powerful framework to study invariants of smooth varieties via invariants of pairs $(X,D)$ of a variety $X$ and a divisor $D$. We develop a generalization of this theory where $D$ can be a $\mathbb{Q}$-divisor. As an application, we provide a motivic construction of the de Rham-Witt complex, which is analogous to the motivic construction of the Milnor $K$-theory due to Suslin-Voevodsky.

math.AG↗
arXiv2023

Hodge cohomology with a ramification filtration, I

We consider a filtration on the cohomology of the structure sheaf indexed by (not necessarily reduced) divisors ``at infinity''. We show that the filtered pieces have transfers morphisms, fpqc descent, and are so called cube invariant. In the presence of resolution of singularities and weak factorisation they are invariant under blowup ``at infinity''. As such, they lead to a realisation functor from Kahn, Miyazaki, Saito and Yamazaki's category of motives with modulus over a characteristic zero base field.

math.AG↗
arXiv2023

Hodge cohomology with a ramification filtration, II

As a sequel of Part I, we consider a filtration of Hodge cohomology groups indexed by divisors "at infinity", and prove that they are represented in the category of motives with modulus. In particular, we obtain a realisation functor of the Hodge cohomology groups.

math.AG↗
arXiv2023

Modulus triples

We develop a theory of modulus triples, for future motivic applications.

math.AG↗
arXiv2021

Structural reduction of chemical reaction networks based on topology

We develop a model-independent reduction method of chemical reaction systems based on the stoichiometry, which determines their network topology. A subnetwork can be eliminated systematically to give a reduced system with fewer degrees of freedom. This subnetwork removal is accompanied by rewiring of the network, which is prescribed by the Schur complement of the stoichiometric matrix. Using homology and cohomology groups to characterize the topology of chemical reaction networks, we can track the changes of the network topology induced by the reduction through the changes in those groups. We prove that, when certain topological conditions are met, the steady-state chemical concentrations and reaction rates of the reduced system are ensured to be the same as those of the original system. This result holds regardless of the modeling of the reactions, namely chemical kinetics, since the conditions only involve topological information. This is advantageous because the details of reaction kinetics and parameter values are difficult to identify in many practical situations. The method allows us to reduce a reaction network while preserving its original steady-state properties, thereby complex reaction systems can be studied efficiently. We demonstrate the reduction method in hypothetical networks and the central carbon metabolism of Escherichia coli.

q-bio.MN↗
arXiv2021

Motives with modulus, III: The categories of motives

We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of proper modulus pairs, introduced in Part I of this work. To such a modulus pair we associate its motive in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$. In some cases the $\mathrm{Hom}$ group in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ between the motives of two modulus pairs can be described in terms of Bloch's higher Chow groups.

math.AG↗
arXiv2021

Modulus sheaves with transfers

We generalise Kahn, Miyazaki, Saito, Yamazaki's theory of modulus pairs to pairs $(X, D)$ consisting of a qcqs scheme $X$ equipped with an effective Cartier divisor $D$ representing a ramification bound. We develop theories of sheaves on such pairs for modulus versions of the Zariski, Nisnevich, étale, fppf, and qfh-topologies. We extend the Suslin-Voevodsky theory of correspondances to modulus pairs, under the assumption that the interior $U = X \setminus D$ is Noetherian. The resulting point of view highlights connections to (Raynaud-style) rigid geometry, and potentially provides a setting where wild ramification can be compared with irregular singularities. This framework leads to a homotopy theory of modulus pairs $\underline{M}H(X,D)$ and a theory of motives with modulus $\underline{M}DM^{eff}(X,D)$ over a general base $(X, D)$. For example, the case where $X$ is the spectrum of a rank one valuation ring (of mixed or equal characteristic) equipped with a choice $D$ of pseudo-uniformiser is allowed.

math.AG↗
arXiv2020

Topologies on schemes and modulus pairs

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs which were introduced respectively in [9] and [3]. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs. This is a revised version of arXiv:1809.05851 [math.AG].

math.AG↗
arXiv2020

Nisnevich topology with modulus

In Voevodsky's theory of motives, the Nisnevich topology on smooth schemes is used as an important building block. In this paper, we introduce a Grothendieck topology on proper modulus pairs, which will be used to construct a non-homotopy invariant generalization of motives. We also prove that the topology satisfies similar properties to the Nisnevich topology.

math.AG↗
arXiv2018

Mayer-Vietoris triangles for motives with modulus

We construct "MV squares" in the category $\mathbf{MCor}$ of modulus pairs which was introduced in arXiv:1511.07124 [math:AG]. They allow us to describe the category $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ of loc. cit. in a similar way as Voevodskys category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$, thus sharpening the results of the quoted paper.

math.AG↗
arXiv2017

Cube invariance of higher Chow groups with modulus

The higher Chow group with modulus was introduced by Binda-Saito as a common generalization of Bloch's higher Chow group and the additive higher Chow group. In this paper, we study invariance properties of the higher Chow group with modulus. First, we formulate and prove "cube invariance," which generalizes $\mathbb{A}^1$-homotopy invariance of Bloch's higher Chow group. Next, we introduce the nilpotent higher Chow group with modulus, as an analogue of the nilpotent algebraic $K$-group, and define a module structure on it over the big Witt ring of the base field. We deduce from the module structure that the higher Chow group with modulus with appropriate coefficients satisfies $\mathbb{A}^1$-homotopy invariance. We also prove that $\mathbb{A}^1$-homotopy invariance implies independence from the multiplicity of the modulus divisors.

math.AG↗
arXiv2016

Suslin's moving lemma with modulus

The moving lemma of Suslin states that a cycle on $X\times \mathbb{A} ^n$ meeting all faces properly can be moved so that it becomes equidimensional over $\mathbb{A}^n$. This leads to an isomorphism of motivic Borel-Moore homology and higher Chow groups. In this short paper we formulate and prove a variant of this. It leads to an isomorphism of Suslin homology with modulus and higher Chow groups with modulus, in an appropriate pro setting.

math.AG↗
arXiv2014

Special values of zeta functions of varieties over finite fields via higher Chow groups

We study special values of zeta functions of singular varieties over finite fields. We give a new formula of special values by constructing a morphism of homology theories, which we call regulator, from higher Chow group to weight homology. Our regulator is defined by using the notion of weight complex for varieties over a perfect field, which was introduced by Gillet and Soule. The main idea of the proof of our formula of special values is to use weight spectral sequence of homology theories, whose E1 terms are homology groups for smooth projective schemes. Also, to calculate special values, we prove that the weight complex for any variety over a perfect field is bounded. This boundedness result was known by Gillet and Soule in the case that the base field admits resolution of singularities, but not in general.

math.NT↗