We give an explicit formula for Capelli-type identities of prehomogeneous vector spaces associated with sub-Hankel determinants. Moreover, we determine multivariate b-functions of these prehomogeneous vector spaces explicitly. As an application, we solve a conjecture on a b-function raised by Ishi and Kogiso (Seminar on Mathematical Sciences, 39. Keio University, 2016, pp. 83–93) affirmatively.

In this paper, we prove that the convergence speed of certain quadratic recurrence formulas in the Arimoto–Blahut algorithm is of order O(1/N). The Arimoto–Blahut algorithm is a well-known algorithm in information theory for computing the capacity of a discrete memoryless channel. There have been many studies on exponential convergence, whereas we found previously (Nakagawa et al. IEEE Trans. Inform. Theory 67(10) (2021), 6810–6831) that there exists a channel for which the convergence is of order O(1/N). In that reference, the convergence of order O(1/N) was analyzed by using quadratic recurrence formulas consisting of the first- and second-order terms of the Taylor expansion of the defining function of the Arimoto–Blahut algorithm. However, there we assumed an infinite number of inequalities and the proof was given under this assumption. In this present paper, we prove the convergence of order O(1/N) by assuming only a finite number of inequalities. The important contribution of this paper is the novelty of the proof. A key idea of the proof is to examine a continuous-time function (i.e., a function defined over the non-negative real numbers) obtained by interpolating the discrete-time function (i.e., a sequence defined over the non-negative integers) of the quadratic recurrence formulas with line segments. The correctness of the proof is demonstrated by several numerical examples.

The study of homogenization results has long been a central focus in the field of mathematical analysis, particularly for equations without lower-order terms. However, the importance of studying homogenization results for parabolic equations with lower-order terms cannot be understated. In this study, we aim to extend the analysis to homogenization for the general parabolic equation with random coefficients:

∂_𝑡𝑝^𝜖 ー ∇ · (𝐚(𝑥/𝜖, 𝑡/𝜖²) ∇ 𝑝^𝜖) ー 𝐛 (𝑥/𝜖, 𝑡/𝜖²) ∇ 𝑝^𝜖 ー 𝐝 (𝑥/𝜖, 𝑡/𝜖²) 𝑝^𝜖 = 0.

Moreover, we establish the Caccioppoli inequality and Meyers estimate for the generalized parabolic equation. By using the generalized Meyers estimate, we get the weak convergence of 𝑝^𝜖 in 𝐻¹.

We determine the topological complexity of a Khalimsky circle 𝕊¹_𝑛, a finite space of 2𝑛 points for all 𝑛 ≥ 2, which is weakly homotopy-equivalent to a circle 𝑆¹. This answers two conjectures on topological complexity for finite spaces raised by K. Tanaka (Algebr. Geom. Topol. 18(2)(2018), 779–796), namely, we obtain (1) tc(Sd^𝑘𝕊¹₂) = 1 for all 𝑘 ≥ 2, and (2) tc(𝕊¹_𝑛) < cat(𝕊¹_𝑛 ×𝕊¹_𝑛) for all 𝑛 ≥ 5.

The non-isothermal model of compressible nematic liquid crystals in a three-dimensional infinite layer is considered. We prove the existence of global strong solutions and their decay rates when the initial data is close to steady state. It turns out that the low-frequency part of the solution decays like a two-dimensional heat kernel. We can also see that the low-frequency part appears in the asymptotic reading part of the solution which is affected by nonlinear terms as t goes to infinity.

For finite connected graphs Γ and 𝐺, with Γ admitting a free involution 𝜏, we characterize the based homotopy classes 𝛼 ∈ [Γ, 𝐺] for which the Borsuk–Ulam property holds in the sense of Gonçalves, Guaschi, and Casteluber Laass, i.e., the homotopy classes 𝛼 such that each of the representatives 𝑓 ∈ 𝛼 satisfies 𝑓(𝑥) = 𝑓(𝜏 · 𝑥) for some 𝑥 ∈ Γ. This is attained through a graph-braid-group perspective aided by the use of discrete Morse theory.

We introduce a new class of von Neumann algebras by deforming the construction of the von Neumann algebras introduced by Bożejko and Speicher and discuss the factoriality of the same.

We present a new proof of the extended double shuffle relation for multiple zeta values which notably does not rely on the use of integrals. This proof is based on a formula recently obtained by Maesaka, Watanabe, and the present author.

We give some sufficient and necessary conditions for the existence of fold maps of orientable 6-dimensional manifolds into ℝ⁵ with a prescribed singular set. To get the results, we study the relative half Euler characteristic of linearly independent vector fields. As a corollary we obtain that there are fold maps of S⁶ into ℝ⁵ whose singular sets consist of aspherical homology 4-spheres.