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Shingo Takeuchi

Publications and source records attributed to Shingo Takeuchi.

At least 19 recordsLinked to original sources
arXiv2026

Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal $p$-Laplacian problem

We classify the arithmetic nature of the Dirichlet eigenvalues of a homogeneous nonlocal one-dimensional $p$-Laplacian problem with power-type nonlinearity $|u|^{q-2}u$. For rational $p,q>1$, either all eigenvalues are algebraic or all are transcendental. Algebraicity holds exactly when the interval length is an algebraic multiple of a distinguished scale determined by a beta value. This scale is transcendental. When $p=q$, the criterion reduces to algebraicity of the interval length divided by $π$,independently of the rational exponent.

math.CA↗
arXiv2026

The Remnant of a Rotating Regular Black Hole from the Generalized Entropy in the Final Stage of Evaporation

We express the generalized entropy (GE) of the Hawking radiation in the final stage of an evaporating rotating regular black hole (BH) by writing the mass of the BH as $m_{\rm ext} + α$, where $m_{\rm ext}$ represents the mass at the extremal limit and $α$ is a parameter. Generally, entropy is non-negative. Based on this fact, we assume that, in the GE considered in this study, the signs of the contributions from the area term and the correction term remain unchanged throughout the entire evaporation process of the BH. Therefore, we regard $α$ at which the correction term vanishes as its lower bound and determine it. As a result, we find that such a value of $α$ is finite. Denoting this value by $α_2$, this result indicates that the mass of the BH cannot become smaller than $m_{\rm ext} + α_2$, which can be interpreted as the emergence of a remnant at the final stage of BH evaporation. The BH considered in this study is a rotating regular BH. The regularization is motivated by the possibility that the fine structure of the central region becomes relevant in the final stage of evaporation. A rotating BH is considered from the viewpoint of generality.

hep-th↗
arXiv2026

Anomaly Equation of the Large U(1) Chiral Symmetry

In this study, we first heuristically construct the charges corresponding to the chiral transformation associated with the large U(1) gauge symmetry. We refer to these as the large chiral charges, and to the chiral transformation they generate as the large chiral transformations. Then, showing that these large chiral charges can be obtained based on Noether's theorem, we obtain the anomaly equation associated with these large chiral transformations. Subsequently, considering the one-loop diagrams of the fermionic field coupled to multiple classical gauge fields (which constitute the effective action of the model in this study with regard to the gauge field), we perform an axialization. Then, defining the BRS transformations for the large U(1) gauge symmetry (we refer to these as the large BRS transformations), we perform these transformations to these axialized one-loop diagrams, and demonstrate that the anomaly equations mentioned above can be derived by evaluating these diagrams. We further confirm that these anomaly equations can be derived using the Fujikawa method. Finally, we discuss the breaking of unitarity and the low-energy effective model associated with the large chiral anomaly in this study, and comment on potential future developments arising from them.

hep-th↗
arXiv2025

Remarks on Redheffer's inequality

Redheffer's inequality and its extensions are applied to study the behavior and estimates of the first eigenvalue of $p$-Laplacian with respect to $p$. Furthermore, a Redheffer-type inequality for the generalized trigonometric function is extended to a broader class.

math.GM↗
arXiv2025

Properties of generalized Jacobi elliptic functions with three parameters

Jacobi elliptic functions and complete elliptic integrals are generalized using three parameters. These generalized functions and integrals are closely related to ordinary differential equations involving $p$-Laplacian. In this paper, Wallis-type integral formulae are constructed for the generalized Jacobi elliptic functions. Moreover, for the generalized complete elliptic integrals, a Legendre-type relation is derived, which is equivalent to Elliott's identity for Gaussian hypergeometric series, along with its implications. In addition, nontrivial inequalities on binomial expansions of generalized Jacobi elliptic functions are given.

math.CA↗
arXiv2025

Canonical Quantization of the U(1) Gauge Field in the Rindler Coordinates

This paper describes the canonical quantization of the U(1) gauge field across all four regions in the Rindler coordinates in the Lorentz-covariant gauge. Concretely, in the four regions (future, past, left and right Rindler-wedges) in the Rindler coordinates, the gauge-fixed Lagrangian in the Lorentz-covariant gauge is obtained, which is composed of the U(1) gauge field, the B-field and ghost fields. Since the U(1) gauge and B-fields are decoupled from the ghost fields by the property of the U(1) gauge theory, the U(1) gauge field and the B-field are examined in this study. Then, by solving the equations of motion obtained from the gauge-fixed Lagrangian, the solutions of each mode of the U(1) gauge field and the B-field can be obtained. Following this, with the Klein-Gordon inner-product defined in the Rindler coordinates, the normalization constants of each of those mode-solutions are determined. Subsequently, formulating the canonical commutation relations of the U(1) gauge field and its canonical conjugate momentum, the equal-time commutation relations of the coefficient of each mode-solution in each direction of the U(1) gauge field in each region of the Rindler coordinates are obtained. From these, it can be seen that those coefficients have physical meaning as creation/annihilation operators. The polarization vectors in each region of the Rindler coordinates are also given in this study.

hep-th↗
arXiv2024

Canonical Quantization of the U(1) Gauge Field in the right Rindler-wedge in the Rindler Coordinates

In this study, the canonical quantization of the U(1) gauge field in the Lorentz-covariant gauge in the right Rindler-wedge (RRW) of the four-dimensional Rindler coordinates is performed. Specifically, we obtain the gauge-fixed Lagrangian by the Lorentz-covariant gauge in the RRW of the Rindler coordinates, which is composed of the U(1) gauge field and B-field. Then, we obtain the mode-solutions of the U(1) gauge field and B-field by solving the equations of motion obtained from that gauge-fixed Lagrangian. Subsequently, defining the Klein-Gordon inner-product in the RRW of the Rindler coordinates, we determine the normalization constants of all directions of the mode-solutions of the U(1) gauge field and B-field. Then, for the U(1) gauge field given by those normalized mode-expanded solutions, we obtain the commutation relations of the creation and annihilation operators defined in the RRW of the Rindler coordinates by formulating the canonical commutation relations. In addition, we provide a polarization vector for the annihilation operators obtained in this way. Using these result, we show that the Minkowski ground state can be expressed as the outer-product of the left and right Rindler-wedges state on which those creation and annihilation operators act. Then, tracing out the left Rindler states of that Minkowski ground state, we obtain the density matrix of the U(1) gauge field in the RRW. From this, we show that the U(1) gauge field in a constant accelerated system will feel the Unruh temperature as well.

hep-th↗
arXiv2023

Bose-Einstein condensation in the Rindler space

Based on the Unruh effect, we calculate the critical acceleration for the Bose-Einstein condensation in a free complex scalar field at finite density in the Rindler space. Our model corresponds to an ideal gas performing constantly accelerating motion in a Minkowski spacetime at zero temperature, where the gas is composed of the complex scalar particles, and is supposed to be in a thermal bath at some temperature by the Unruh effect. The critical Unruh temperature we obtain agrees with the usual result in the 4D Euclid space at finite temperature.

hep-th↗
arXiv2023

A 4D asymptotically flat rotating black hole solution including supertraslation correction

In this study, beginning with the 4D asymptotically flat Kerr black hole solution in the Boyer-Lindquist (BL) coordinate system, then by using the technique of frame-dragging and some coordinate transformation to incorporate the linear-order supertranslation correction, we finally obtain a 4D asymptotically flat black hole solution with an arbitrary $a$ in the linear-order supertranslated BL coordinate system. We can confirm that this satisfies the Einstein equation. At the end, we calculate the angular momentum of the spacetime we obtained in this study including the linear-order correction. Since the supertranslation is a type of the general coordinate transformation, which is usually defined by linear-order, the corrections we incorporate in this study is sufficient, considering the fact that $a$ is arbitrary. This work is interesting as a generalization of the supertranslated spacetime to the rotating system.

hep-th↗
arXiv2022

Josephson junction formed in the wormhole space time from the analysis for the critical temperature of Bose-Einstein condensate

In this study, considering some gas in the Morris-Thorne traversable wormhole space time, we analyze the critical temperature of the Bose-Einstein condensate in the vicinity of its throat. As a result, we obtain the result that it is zero. Then, from this result, we point out that an analogous state to the Josephson junction is always formed at any temperatures in the vicinity of its throat. This would be interesting as a gravitational phenomenology.

hep-th↗
arXiv2022

Applications of a duality between generalized trigonometric and hyperbolic functions II

Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of Wilker-type inequalities, Huygens-type inequalities, and (relaxed) Cusa-Huygens-type inequalities for the generalized functions. In addition, multiple- and double-angle formulas not previously obtained are also given.

math.CA↗
arXiv2021

Hawking Flux from a Black Hole with Nonlinear Supertranslation Hair

We study the Hawking flux from a black hole with soft hair by the anomaly cancellation method proposed by Robinson and Wilczek. Unlike the earlier studies considering the black hole with linear supertranslation hair, our study takes into account the supertranslation hair to the quadratic order, which then yields the angular dependent horizon. As a result, highly nontrivial kinetic-mixings appear among the spherical Kaluza-Klein modes of the (1+1)d near-horizon reduced theory, which obscures the traditional derivation of the Hawking flux. However, after a series of field re-definitions, we can disentangle the mode-mixings into canonical normal modes, but the reduced metrics for these normal modes are mode-dependent. Despite of this, the resultant Hawking flux turns out to be mode-independent and remains the same as the Schwarzschild's one. Thus, one cannot tell the black holes with nonlinear supertranslation hairs from the Schwarzschild's one by examining the Hawking flux, so that the nonlinear soft hairs can be thought as the microstates.

hep-th↗
arXiv2020

Applications of a duality between generalized trigonometric and hyperbolic functions

It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties for the other. In this way, Mitrinović-Adamović-type inequalities, multiple-angle formulas, and double-angle formulas for both can be produced.

math.CA↗
arXiv2020

Lyapunov-type inequalities for a Sturm-Liouville problem of the one-dimensional $p$-Laplacian

This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian \begin{align*} \begin{cases} \left(\vert u'\vert^{p-2}u'\right)'+\left(λ+r(x)\right)\vert u\vert ^{p-2}u=0,\,\, x\in (0,π_{p}),\\ u(0)=u(π_{p})=0, \end{cases} \end{align*} where $1<p<\infty$, $π_{p}$ is the generalized $π$ given by $π_{p}=2π/\left(p\sin(π/p)\right)$, $r\in C[0,π_{p}]$ and $λ<p-1$. Sharp Lyapunov-type inequalities, which are necessary conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.

math.CA↗
arXiv2020

Coarse-graining of the Einstein-Hilbert Action rewritten by the Fisher information metric

In this study, considering the Fisher information metric (Fisher metric) given by a specific form, which is the form of weights in statistical physics, we rewrite the Einstein-Hilbert (EH) action. Then, determining the transformation rules of the Fisher metric, etc under the coarse-graining, we perform the coarse-graining toward that rewritten EH action. We finally show an existence of a trivial fixed-point. Here, the existence of a trivial fixed-point is not trivial for us because we consider the metric given by the Fisher metric, which is not the normal metric and has to satisfy some constraint in the formalism of the the Fisher metric. We use the path-integral in our analysis. At this time we have to accept that a fundamental constraint in the formalism of the Fisher metric is broken at the quantum level. However we consider we can accept this with the thought that some constraints and causal relations held at the classical level usually get broken at the quantum level. We finds some problems of the Fisher metric. The space-time we consider is two-dimensional.

hep-th↗
arXiv2020

Remarkable properties of the $\text{sinc}_{p, q}$ functions and related integrals

Different types of sinc integrals are investigated when the standard sine function is replaced by the generalised $\sin_{p,q}$ in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising but interesting generalisation of the identity between the Dirichlet integral and that of the integrand $\text{sinc}^2$ is discovered. Moreover, an asymptotically sharpened form of Ball's integral inequality is obtained in terms of two parameters.

math.CA↗