A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann type according to their causal characters, and as a corollary, we prove that if a zero mean curvature surface of Riemann type has exactly two causal characters, then the lightlike part of the surface is a part of a straight line.

The aim of this paper is to give a shorter proof to the main theorem of Yamasaki-Nomura in 2016, stating that a homogeneous cone of rank r is an irreducible symmetric cone if and only if the basic relative invariants of the cone and of the dual cone both have the degrees 1, 2,... ,r, up to permutations. Our proof is based on the multiplier matrix of homogeneous cones studied in detail by the present author.

In this paper, we obtain several interesting results on submanifolds of conformal Kenmotsu manifolds. In addition to this we consider submanifolds of a conformal Kenmotsu manifold of which the Ricci tensor is parallel, Lie ξ-parallel or recurrent. We also present an illustration example of a three-dimensional conformal Kenmotsu manifold that is not Kenmotsu.

While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group Γ and an arbitrary representation R of Γ in GL (n, C), their existence, as far as we know, has been established in the literature only when restrictions are imposed on Γ or R.In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group Γ and any n-dimensional complex representation R of Γ. To this end, we realize these automorphic forms as global sections of a special rank n vector bundle built using solutions to the Riemann-Hilbert problem over various non-compact Riemann surfaces and Kodaira's vanishing theorem.

Let ∑^∞_n=0 a_n zⁿ ∈ Q[[z]] be a G-function, and, for any n ≥ 0, let δ_n ≥ 1 denote the least integer such that δ_na₀, δ_na₁,..., δ_na_n are all algebraic integers. By the definition of a G-function, there exists some constant c ≥ 1 such that δ_n ≤ cⁿ⁺¹ for all n ≥ 0. In practice, it is observed that δ_n always divides D^s_bn C ⁿ⁺¹ where D_n = lcm{1, 2,..., n}, b, C are positive integers and s ≥ 0 is an integer. We prove that this observation holds for any G-function provided the following conjecture is assumed: let K be a number field, and L ∈ K[z,d/dz] be a G-operator; then the generic radius of solvability R_v (L) is equal to one, for all finite places v of K except a finite number. The proof makes use of very precise estimates in the theory of p-adic differential equations, in particular the Christol-Dwork theorem. Our result becomes unconditional when L is a geometric differential operator, a special type of G-operators for which the conjecture is known to be true. The famous Bombieri-Dwork conjecture asserts that any G-operator is of geometric type, hence it implies the above conjecture.

Downarowicz and Maass have defined the topological rank for all Cantor minimal homeomorphisms. On the other hand, Gambaudo and Martens have expressed all Cantor minimal continuous surjections as the inverse limits of certain graph coverings. Using the aforementioned results, we previously extended the notion of topological rank to all Cantor minimal continuous surjections. In this paper, we show that taking natural extensions of Cantor minimal continuous surjections does not increase their topological ranks. Further,we apply the result to some minimal symbolic cases.

We use the Björling formula in Lorentz-Minkowski space to construct explicit parametrizations of maximal surfaces containing a circle and a helix. For Frenet curves, the orthogonal vector field along the core curve is a linear combination of the principal normal and binormal vectors where the coefficients are hyperbolic trigonometric functions. In the particular case that these coefficients are constant, we obtain all rotational maximal surfaces.

We investigate the Weierstrass representation of these surfaces.

We give the monodromy representations of local systems of twisted homology groups associated with Lauricella's system F_D(a,b,c) of hypergeometric differential equations under mild conditions on parameters. Our representation is effective even in some cases when the system F_D(a,b,c) is reducible. We show that invariant subspaces under our monodromy representations are given by the kernel or image of a natural map from a finite twisted homology group to locally finite one.

Poisson wavelets are a powerful tool in the analysis of spherical signals. In order to have a deeper characterization of them, we compute their uncertainty product, a quantity introduced for the first time by Narcowich and Ward [Nonstationary spherical wavelets for scattered data. Approximation Theory VIII, Vol. 2 (College Station, TX, 1995) (Ser. Approx. Decompos., 6). World Scientific Publishing, River Edge, NJ, 1995, pp. 301-308] and used to measure the trade-off between the space and frequency localization of a function. Surprisingly, the uncertainty product of Poisson wavelets tends to the minimal value in some limiting cases. This shows that in the case of spherical functions, it is not only the Gauss kernel that has this property.

The Abel-Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration-differentiation process. In this article, we use the Abel-Tauber process to establish an interesting asymptotic expansion for the Riesz sums of arithmetic functions with best possible error estimate. The novelty of our paper is that we incorporate the Selberg type divisor problem in this process by viewing the contour integral as part of the residual function. The novelty also lies in the uniformity of the error term in the additional parameter which varies according to the cases. Generalization of the famous Selberg Divisor problem to arithmetic progression has been made by Rieger [Zum Teilerproblem von Atle Selberg. Math. Nachr. 30 (1965), 181-192], Marcier [Sums of the form Σ g(n)/f(n). Canad. Math. Bull. 24 (1981), 299-307], Nakaya [On the generalized division problem in arithmetic progressions. Sci. Rep. Kanazawa Univ. 37 (1992), 23-47] and around the same time Nowak [Sums of reciprocals of general divisor functions and the Selberg division problem, Abh. Math. Sem. Univ. Hamburg 61 (1991), 163-173] studied the related subject of reciprocals of an arithmetic function and obtained an asymptotic formula with the Vinogradov-Korobov error estimate with the main term as a finite sum of logarithmic terms. We shall also elucidate the situation surrounding these researches and illustrate our results by rich examples.

For invertible bounded positive operators a, b the operator a : b = (a⁻¹ + b⁻¹)⁻¹ is known as the parallel sum. In this article a theory of parallel sums of unbounded positive self-adjoint operators is developed.

In this paper, asymptotic behavior of the uncertainty product for a family of zonal spherical wavelets is computed. The family contains the most popular wavelets, such as Gauss-Weierstrass, Abel-Poisson and Poisson wavelets and Mexican needlets. Boundedness of the uncertainty constant is in general not given, but it is a property of some of the wavelets from this class.