In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual 𝑡-motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some explicit sufficient conditions for the linear independence of Carlitz polylogarithms at algebraic points in both ∞-adic and 𝑣-adic settings.

We investigate the compact submanifolds in Riemannian space forms of nonnegative sectional curvature that satisfy a lower bound on the Ricci curvature, that bound depending solely on the length of the mean curvature vector of the immersion. While generalizing the results, we give a positive answer to a conjecture by H. Xu and J. Gu in (2013, Geom. Funct. Anal. 23). Our main accomplishment is the elimination of the need for the mean curvature vector field to be parallel.

In Part I, the present authors introduced the notion of a quasi-Galois point, for investigating the automorphism groups of plane curves. In this second part, the number of quasi-Galois points for smooth plane curves is described. In particular, sextic or quartic curves with many quasi-Galois points are characterized.

In this article, we will investigate a generalization of the Dirichlet form associated with a one-dimensional diffusion process. In this generalization, the scale function, which determines the expression of the Dirichlet form, is only required to be non-decreasing. While this generalized form is almost a Dirichlet form, it does not satisfy regularity in general. Consequently, it cannot be directly associated with a process in probability theory. To tackle this issue, we adopt Fukushima's regular representation method, which enables to find a family of strong Markov processes that are homeomorphic to each other and related to the generalized form in a certain sense. Additionally, this correspondence reveals the connection between this generalized form and a quasidiffusion. Moreover, we interpret the probabilistic implications behind the regular representation through two intuitive transformations. These transformations offer us the opportunity to obtain another symmetric non-strong Markov process with continuous sample paths. The Dirichlet form of this non-strong Markov process is precisely the non-regular generalized form we previously analyzed. Furthermore, the strong Markov process obtained from the regular representation is its Ray–Knight compactification.

The Bloch–Torrey operator −ℎ² Δ + 𝑒^𝑖𝛼 𝑥₁ on a bounded smooth planar domain, subject to Dirichlet boundary conditions, is analyzed. Assuming 𝛼 ∈ [0, 3𝜋/5) and a non-degeneracy assumption on the left-hand side of the domain, asymptotics of eigenvalues in the limit ℎ → 0 are derived. The strategy is a backward complex scaling and the reduction to a tensorized operator involving a real Airy operator and a complex harmonic oscillator.

Höhn and Mason classified the groups acting symplectically on an irreducible holomorphic symplectic (IHS) manifold of K3^[2]-type, finding that ℤ₃⁴ : 𝒜₆ is the one with the largest order. In this paper we study IHS manifolds of K3^[2]-type with a symplectic action of ℤ₃⁴ : 𝒜₆ which also admit a non-symplectic automorphism. We characterize such IHS manifolds and prove their existence. We also prove that the order of a finite group acting on an IHS manifold of K3^[2]-type is bounded by 174960, this bound is sharp and there is a unique IHS manifold of K3^[2]-type acted by a group of this order, which is the Fano variety of lines of the Fermat cubic fourfold.

Let 𝐷 be a strongly self-absorbing 𝐶^*-algebra. In previous work, we showed that locally trivial bundles with fibers 𝒦 ⊗ 𝐷 over a finite CW-complex 𝑋 are classified by the first group 𝐸¹_𝐷(𝑋) in a generalized cohomology theory 𝐸^*_𝐷(𝑋). In this paper, we establish a natural isomorphism 𝐸¹_{𝐷 ⊗ 𝒪_∞}(𝑋) ≅ 𝐻¹(𝑋;ℤ/2) ×_{_𝑡𝑤} 𝐸¹_𝐷(𝑋) for stably-finite 𝐷. In particular, 𝐸¹_{𝒪_∞}(𝑋) ≅ 𝐻¹(𝑋;ℤ/2) ×_{_𝑡𝑤} 𝐸¹_𝒵(𝑋), where 𝒵 is the Jiang–Su algebra. The multiplication operation on the last two factors is twisted in a manner similar to Brauer theory for bundles with fibers consisting of graded compact operators. The proof of the isomorphism described above made it necessary to extend our previous results on generalized Dixmier–Douady theory to graded 𝐶^*-algebras. More precisely, for complex Clifford algebras ℂℓ_𝑛, we show that the classifying spaces of the groups of graded automorphisms of ℂℓ_𝑛 ⊗ 𝒦 ⊗ 𝐷 possess compatible infinite loop space structures. These structures give rise to a cohomology theory \hat{𝐸}^*_𝐷(𝑋). We establish isomorphisms \hat{𝐸}¹_𝐷(𝑋) ≅ 𝐻¹(𝑋;ℤ/2) ×_{_𝑡𝑤} 𝐸¹_𝐷(𝑋) and \hat{𝐸}¹_𝐷(𝑋) ≅ 𝐸¹_{𝐷 ⊗ 𝒪_∞}(𝑋) for stably finite 𝐷. Together, these isomorphisms represent a crucial step in the integral computation of 𝐸¹_{𝐷 ⊗ 𝒪_∞}(𝑋).

For each field 𝔽 and positive integers 𝑚, 𝑛, 𝑑 with (𝑚, 𝑛) ≠ (1, 1), Farb and Wolfson defined the certain affine variety Poly^𝑑,𝑚_𝑛(𝔽) as generalizations of spaces first studied by Arnol'd, Vassiliev, Segal and others. As a natural generalization, for each fan Σ and 𝑟-tuple 𝐷 = (𝑑₁, …, 𝑑_𝑟) of positive integers, the authors also defined and considered a more general space Poly^𝐷,Σ_𝑛(𝔽), where 𝑟 is the number of one dimensional cones in Σ. This space can also be regarded as a generalization of the space Hol^*_𝐷(𝑆², 𝑋_Σ) of based rational curves from the Riemann sphere 𝑆² to the toric variety 𝑋_Σ of degree 𝐷, where 𝑋_Σ denotes the toric variety (over ℂ) corresponding to the fan Σ.

In this paper, we define a space Q^𝐷,Σ_𝑛(𝔽) (𝔽 = ℝ or ℂ) which is its real analogue and can be viewed as a generalization of spaces considered by Arnol'd, Vassiliev and others in the context of real singularity theory. We prove that homotopy stability holds for this space and compute the stability dimension explicitly.

We investigate the relation between essential divisors and F-blowups, in particular, address the problem whether all essential divisors appear on the 𝑒-th F-blowup for large enough 𝑒. Focusing on the case of normal affine toric varieties, we establish a simple sufficient condition for a divisor over the given toric variety to appear on the normalized limit F-blowup as a prime divisor. As a corollary, we show that if a normal toric variety has a crepant resolution, then the above problem has a positive answer, provided that we use the notion of essential divisors in the sense of Bouvier and Gonzalez-Sprinberg. We also provide an example of toric threefold singularities for which a non-essential divisor appears on an F-blowup.

Given a totally real number field 𝐹 and its adèle ring 𝔸_𝐹, let 𝜋 vary in the set of irreducible cuspidal automorphic representations of PGL₂(𝔸_𝐹) corresponding to primitive Hilbert modular forms of a fixed weight. We determine the symmetry type of the one-level density of low-lying zeros of the symmetric power 𝐿-functions 𝐿(𝑠, Sym^𝑟(𝜋)) weighted by special values of the symmetric square 𝐿-functions 𝐿((𝑧+1)/2, Sym²(𝜋)) at 𝑧 ∈ [0, 1] in the level aspect. If 0 < 𝑧 ≤ 1, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power 𝐿-functions for 𝐹 = ℚ with harmonic weight. Hence our result is regarded as a 𝑧-interpolation of Ricotta and Royer's result. If 𝑧 = 0, the density of low-lying zeros weighted by central values is of a different type only when 𝑟 = 2.

It is shown that if the universal enveloping algebra of a simple ℤ^𝑛-graded Lie algebra is Noetherian, then the Lie algebra is finite-dimensional.

We present a new notion, the upper Aikawa codimension, and establish its equivalence with the upper Assouad codimension in a metric space with a doubling measure. To achieve this result, we first prove a variant of a local fractional Hardy inequality.

Honda, Izawa and Suwa define Čech–Dolbeault representation of hyperfunctions and an embedding of distributions to the space of hyperfunctions. With this embedding, we can regard 𝐶^∞ functions as hyperfunctions in the framework of Čech–Dolbeault cohomology.

This article aims to characterize a Čech–Dolbeault representative which corresponds to the image of the embedding of a 𝐶^∞ function, and also to construct the inverse map of the embedding of 𝐶^∞ functions.