arXiv subjects

Hayato Saigo

Publications and source records attributed to Hayato Saigo.

At least 19 recordsLinked to original sources

arXiv2026

Computational Implementation of a Model of Category-Theoretic Metaphor Comprehension

In this study, we developed a computational implementation for a model of metaphor comprehension based on the theory of indeterminate natural transformation (TINT) proposed by Fuyama et al. We simplified the algorithms implementing the model to be closer to the original theory and verified it through data fitting and simulations. The outputs of the algorithms are evaluated with three measures: data-fitting with experimental data, the systematicity of the metaphor comprehension result, and the novelty of the comprehension (i.e. the correspondence of the associative structure of the source and target of the metaphor). The improved algorithm outperformed the existing ones in all the three measures.

cs.CL↗

arXiv2024

Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations

We extend a certain type of identities on sums of $I$-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by $I$-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of $I$-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that $I$-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on $\mathbb{Z}^n$ whose initial condition is given by a linear code.

math-ph↗

arXiv2023

Reformalizing the notion of autonomy as closure through category theory as an arrow-first mathematics

Life continuously changes its own components and states at each moment through interaction with the external world, while maintaining its own individuality in a cyclical manner. Such a property, known as "autonomy," has been formulated using the mathematical concept of "closure." We introduce a branch of mathematics called "category theory" as an "arrow-first" mathematics, which sees everything as an "arrow," and use it to provide a more comprehensive and concise formalization of the notion of autonomy. More specifically, the concept of "monoid," a category that has only one object, is used to formalize in a simpler and more fundamental way the structure that has been formalized as "operational closure." By doing so, we show that category theory is a framework or "tool of thinking" that frees us from the habits of thinking to which we are prone and allows us to discuss things formally from a more dynamic perspective, and that it should also contribute to our understanding of living systems.

math.CT↗

arXiv2022

The limit theorem with respect to the matrices on non-backtracking paths of a graph

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

math.CO↗

arXiv2022

Conserved relativistic Ertel's current generating the vortical and thermodynamic aspects of spacetime

Motivated by Aoki et al.'s recent research on conserved charges and entropy current, we investigated the conservation of relativistic Ertel's current, which has received little attention outside the field of geophysical fluid dynamics. Ertel's charge is an important indicator of the correlation between vortex vectors and entropy gradient fields in Earth's meridional heat transport. We first show that in the generalized Hamiltonian structure of baroclinic fluids, the duality between the total energy and the Casimir as a function of Ertel's charge plays an important role in the nonrelativistic case. Then, by extending the result to relativistic cases, we show that this finding has far-reaching implications not only for spacetime issues in cosmology but also for the foundation of quantum field theory. An especially important finding is that, as an unreported dual form of the Einstein field equation, we identify a special equation satisfied not only by the vortex tensor field generated by the conserved charge but also by the Weyl tensor in interpreting the physical nature of the metric tensor $g^{μν}$, which appears in the cosmological term $Λg^{μν}$.

gr-qc↗

arXiv2022

On the Equivalence of Causal Models: A Category-Theoretic Approach

We develop a category-theoretic criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. Following Jacobs et al. (2019), we define a causal model as a probabilistic interpretation of a causal string diagram, i.e., a functor from the ``syntactic'' category $\textsf{Syn}_G$ of graph $G$ to the category $\textsf{Stoch}$ of finite sets and stochastic matrices. The equivalence of causal models is then defined in terms of a natural transformation or isomorphism between two such functors, which we call a $Φ$-abstraction and $Φ$-equivalence, respectively. It is shown that when one model is a $Φ$-abstraction of another, the intervention calculus of the former can be consistently translated into that of the latter. We also identify the condition under which a model accommodates a $Φ$-abstraction, when transformations are deterministic.

cs.LG↗

arXiv2021

Quantum Fields as Category Algebras

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 discussed.

math-ph↗

arXiv2021

Order recognition by Schubert polynomials generated by optical near-field statistics via nanometre-scale photochromism

We have previously observed an irregular spatial distribution of photon transmission through a photochromic crystal photoisomerized by a local optical near-field excitation, manifesting complex branching processes via the interplay of deformation of the material and near-field photon transfer therein. Furthermore, by combining such naturally constructed complex photon transmission with a simple photon detection protocol, Schubert polynomials, the foundation of versatile permutation operations in mathematics, have been generated. In this study, we demonstrate an order recognition algorithm inspired by Schubert calculus using optical near-field statistics via nanometre-scale photochromism. More specifically, by utilizing Schubert polynomials generated via optical near-field patterns, we show that the order of slot machines with initially unknown reward probability is successfully recognized. We emphasize that, unlike conventional algorithms in the literature, the proposed principle does not estimate the reward probabilities. Instead, it exploits the inversion relations contained in the Schubert polynomials. To quantitatively evaluate the impact of the Schubert polynomials generated from an optical near-field pattern, order recognition performances are compared with uniformly distributed and spatially strongly skewed probability distributions, where the optical near-field pattern outperforms the others. We found that the number of singularities contained in Schubert polynomials and that of the given problem or considered environment exhibits a clear correspondence, indicating that superior order recognition performances may be attained if the singularity of the given problem is presupposed. This study paves a new way toward nanophotonic intelligent devices and systems by the interplay of complex natural processes and mathematical insights gained by Schubert calculus.

physics.optics↗

arXiv2021

Category Algebras and States on Categories

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functionals defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that states on categories as linear functionals defined on category algebras turn out to be generalized of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain representations of category algebras of $^{\dagger}$-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.

math.CT↗

arXiv2021

Categorical Nonstandard Analysis

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor $\mathcal{U}$ on a topos of sets $S$ together with a natural transformation $\upsilon$, instead of the terms as "standard", "internal" or "external". Moreover, we propose a general notion of a space called $\mathcal{U}$-space, and the category $\mathcal{U}space$ whose objects are $\mathcal{U}$-spaces and morphisms are functions called $\mathcal{U}$-spatial morphisms. The category $\mathcal{U}Space$, which is shown to be cartesian closed, will give a unified viewpoint toward topological and coarse geometric structure. It will also useful to study symmetries/asymmetries of the systems with infinite degrees of freedom such as quantum fields.

math.CT↗

arXiv2020

Entangled and correlated photon mixed strategy for social decision making

Collective decision making is important for maximizing total benefits while preserving equality among individuals in the competitive multi-armed bandit (CMAB) problem, wherein multiple players try to gain higher rewards from multiple slot machines. The CMAB problem represents an essential aspect of applications such as resource management in social infrastructure. In a previous study, we theoretically and experimentally demonstrated that entangled photons can physically resolve the difficulty of the CMAB problem. This decision-making strategy completely avoids decision conflicts while ensuring equality. However, decision conflicts can sometimes be beneficial if they yield greater rewards than non-conflicting decisions, indicating that greedy actions may provide positive effects depending on the given environment. In this study, we demonstrate a mixed strategy of entangled- and correlated-photon-based decision-making so that total rewards can be enhanced when compared to the entangled-photon-only decision strategy. We show that an optimal mixture of entangled- and correlated-photon-based strategies exists depending on the dynamics of the reward environment as well as the difficulty of the given problem. This study paves the way for utilizing both quantum and classical aspects of photons in a mixed manner for decision making and provides yet another example of the supremacy of mixed strategies known in game theory, especially in evolutionary game theory.

quant-ph↗

arXiv2020

Entangled N-photon states for fair and optimal social decision making

Situations involving competition for resources among entities can be modeled by the competitive multi-armed bandit (CMAB) problem, which relates to social issues such as maximizing the total outcome and achieving the fairest resource repartition among individuals. In these respects, the intrinsic randomness and global properties of quantum states provide ideal tools for obtaining optimal solutions to this problem. Based on the previous study of the CMAB problem in the two-arm, two-player case, this paper presents the theoretical principles necessary to find polarization-entangled N-photon states that can optimize the total resource output while ensuring equality among players. These principles were applied to two-, three-, four-, and five-player cases by using numerical simulations to reproduce realistic configurations and find the best strategies to overcome potential misalignment between the polarization measurement systems of the players. Although a general formula for the N-player case is not presented here, general derivation rules and a verification algorithm are proposed. This report demonstrates the potential usability of quantum states in collective decision making with limited, probabilistic resources, which could serve as a first step toward quantum-based resource allocation systems.

quant-ph↗

arXiv2020

Rational Choice Hypothesis as X-point of Utility Function and Norm Function

Towards the realization of a sustainable, fair and inclusive society, we proposed a novel decision-making model that incorporates social norms in a rational choice model from the standpoints of deontology and utilitarianism. We proposed a hypothesis that interprets choice of action as the X-point for individual utility function that increases with actions and social norm function that decreases with actions. This hypothesis is based on humans psychologically balancing the value of utility and norms in selecting actions. Using the hypothesis and approximation, we were able to isolate and infer utility function and norm function from real-world measurement data of actions on environmental conditions and elucidate the interaction between the both functions that led from current status to target actions. As examples of collective data that aggregate decision-making of individuals, we looked at the changes in power usage before and after the Great East Japan Earthquake and the correlation between national GDP and CO2 emission in different countries. The first example showed that the perceived benefits of power (i.e., utility of power usage) was stronger than the power usage restrictions imposed by norms after the earthquake, contrary to our expectation. The second example showed that a reduction of CO2 emission in each country was not related to utility derived from GDP but to norms related to CO2 emission. Going forward, we will apply this new X-point model to actual social practices involving normative problems, and design the approaches for the diagnosis, prognosis and intervention of social systems by IT systems.

econ.GN↗

arXiv2020

Sustainability and Fairness Simulations Based on Decision-Making Model of Utility Function and Norm Function

We introduced a decision-making model based on value functions that included individualistic utility function and socio-constructivistic norm function and proposed a norm-fostering process that recursively updates norm function through mutual recognition between the self and others. As an example, we looked at the resource-sharing problem typical of economic activities and assumed the distribution of individual actions to define the (1) norm function fostered through mutual comparison of value/action ratio based on the equity theory (progressive tax-like), (2) norm function proportional to resource utilization (proportional tax-like) and (3) fixed norm function independent of resource utilization (fixed tax-like). By carrying out numerical simulation, we showed that the progressive tax-like norm function (i) does not increase disparity for the distribution of the actions, unlike the other norm functions, and (ii) has high resource productivity and low Gini coefficient. Therefore the progressive tax-like norm function has the highest sustainability and fairness.

econ.GN↗

arXiv2020

Quantum Walk and Dressed Photon

A physical model called dressed photons, a composite system of photons and excitation of electrons in the nano-particles, is effectively used in the realm of near-field optics. Many interesting behaviors of dressed photons are known, especially the rapid energy transfer and the accumulation to singular points, e.g., points with strong dissipation. We propose a new modelling of dressed photons based on quantum walks, especially Grover walks on semi-infinite graphs which we call jellyfish graphs, and show a universal accumulation phenomena around the point with strong dissipation.

math-ph↗

arXiv2019

Entangled-photon decision maker

The competitive multi-armed bandit (CMAB) problem is related to social issues such as maximizing total social benefits while preserving equality among individuals by overcoming conflicts between individual decisions, which could seriously decrease social benefits. The study described herein provides experimental evidence that entangled photons physically resolve the CMAB in the 2-arms 2-players case, maximizing the social rewards while ensuring equality. Moreover, we demonstrated that deception, or outperforming the other player by receiving a greater reward, cannot be accomplished in a polarization-entangled-photon-based system, while deception is achievable in systems based on classical polarization-correlated photons with fixed polarizations. Besides, random polarization-correlated photons have been studied numerically and shown to ensure equality between players and deception prevention as well, although the CMAB maximum performance is reduced as compared with entangled photon experiments. Autonomous alignment schemes for polarization bases were also experimentally demonstrated based only on decision conflict information observed by an individual without communications between players. This study paves a way for collective decision making in uncertain dynamically changing environments based on entangled quantum states, a crucial step toward utilizing quantum systems for intelligent functionalities.

physics.optics↗

arXiv2019

Generation of Schubert polynomial series by nanophotonics

Generation of irregular time series based on physical processes is indispensable in computing and artificial intelligence. In this report, we propose and experimentally demonstrate the generation of Schubert polynomials, which is the foundation of versatile permutations in mathematics, via optical near-field processes introduced in a photochromic crystal of diarylethene, which optical near-field excitation on the surface of a photochromic single crystal yields a chain of local photoisomerization, forming a complex pattern on the opposite side of the crystal. The incoming photon travels through the nanostructured photochromic crystal, and the exit position of the photon exhibits a versatile pattern. We experimentally generated Schubert matrices, corresponding to Schubert polynomials, via optical near-field density mapping. The versatility and correlations of the generated patterns could be reconfigured in either a soft or hard manner by adjusting the photon detection sensitivity. This is the first study of Schubert polynomial generation by physical processes or nanophotonics, paving the way toward future nano-scale intelligence devices and systems.

physics.optics↗

arXiv2018

Category theory as a foundation for soft robotics

Soft robotics is an emerging field of research where the robot body is composed of compliant and soft materials. It allows the body to bend, twist, and deform to move or to adapt its shape to the environment for grasping, all of which are difficult for traditional hard robots with rigid bodies. However, the theoretical basis and design principles for soft robotics are not well-founded despite their recognized importance. For example, the control of soft robots is outsourced to morphological attributes and natural processes; thus, the coupled relations between a robot and its environment are particularly crucial. In this paper, we propose a mathematical foundation for soft robotics based on category theory, which is a branch of abstract math where any notions can be described by objects and arrows. It allows for a rigorous description of the inherent characteristics of soft robots and their relation to the environment as well as the differences compared to conventional hard robots. We present a notion called the category of mobility that well describes the subject matter. The theory was applied to a model system and analysis to highlight the adaptation behavior observed in universal grippers, which are a typical example of soft robotics. This paper paves the way to developing a theoretical background and design principles for soft robotics.

cs.RO↗