arXiv subjects

Takumi Noda

Publications and source records attributed to Takumi Noda.

5 recordsLinked to original sources

arXiv2026

Polylogarithmic Analogues of Euler's Constant

We introduce a family of constants \[ C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac1k\right) - \log n \right), \] which may be regarded as polylogarithmic analogues of Euler's constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants $C_m$ to special values of certain Dirichlet series involving the Riemann zeta function.

math.NT↗

arXiv2026

On Dirichlet Series Involving $ζ(s)$ and Extensions of the Euler-Mascheroni Constant

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of the Euler--Mascheroni constant and express certain special values explicitly in terms of the Bendersky constants. These results provide a unified framework for evaluating Dirichlet series involving the Riemann zeta-function at integer arguments, together with the associated number-theoretic constants.

math.NT↗

arXiv2023

Textile-based conformable and breathable ultrasound imaging probe

Daily monitoring of internal tissues with conformable and breathable ultrasound (US) imaging probes is promising for early detection of diseases. In recent years, textile substrates are widely used for wearable devices since they satisfy both conformability and breathability. However, it is not currently possible to use textile substrates for US probes due to the reflection or attenuation of US waves at the air gaps in the textiles. In this paper, we fabricated a conformable and breathable US imaging probe by sandwiching the US elements between two woven polyester textiles on which copper electrodes were formed through electroless plating. The air gaps between the fibers at the electrode parts were filled with copper, allowing for high penetration of US waves. On the other hand, the non-electrode parts retain air gaps, leading to high breathability. The fabricated textile-based probe showed low flexural rigidity ($0.066 \times 10^{-4} N \cdot m^2/m$) and high air permeability ($11.7 cm^3 / cm^2 \cdot s$). Human neck imaging demonstrated the ability of the probe to monitor the pulsation of the common carotid artery and change in the internal jugular vein diameter, which lead to the early detection of health issues such as arteriosclerosis and dehydration.

physics.med-ph↗

arXiv2023

Blind Signal Separation for Fast Ultrasound Computed Tomography

Breast cancer is the most prevalent cancer with a high mortality rate in women over the age of 40. Many studies have shown that the detection of cancer at earlier stages significantly reduces patients' mortality and morbidity rages. Ultrasound computer tomography (USCT) is considered as a promising screening tool for diagnosing early-stage breast cancer as it is cost-effective and produces 3D images without radiation exposure. However, USCT is not a popular choice mainly due to its prolonged imaging time. USCT is time-consuming because it needs to transmit a number of ultrasound waves and record them one by one to acquire a high-quality image. We propose FastUSCT, a method to acquire a high-quality image faster than traditional methods for USCT. FastUSCT consists of three steps. First, it transmits multiple ultrasound waves at the same time to reduce the imaging time. Second, it separates the overlapping waves recorded by the receiving elements into each wave with UNet. Finally, it reconstructs an ultrasound image with a synthetic aperture method using the separated waves. We evaluated FastUSCT on simulation on breast digital phantoms. We trained the UNet on simulation using natural images and transferred the model for the breast digital phantoms. The empirical result shows that FastUSCT significantly improves the quality of the image under the same imaging time to the conventional USCT method, especially when the imaging time is limited.

eess.IV↗

arXiv2023

Asymptotic expansions for a class of generalized holomorphic Eisenstein series, Ramanujan's formula for $ζ(2k+1)$, Weierstrass' elliptic and allied functions

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new variants of the celebrated formulae of Euler and of Ramanujan for specific values of the Riemann zeta-function (Theorem~4 and Corollaries~4.1--4.5), and to various modular type relations for the classical Eisenstein series of any even integer weight (Corollary~4.6) as well as for Weierstraß' elliptic and allied functions (Corollaries~4.7--4.9). Crucial r{ô}les in the proofs are played by certain Mellin-Barnes type integrals, which are manipulated with several properties of confluent hypergeometric functions.

math.NT↗