We show the existence of constant mean curvature-one surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior.

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We study the model of the spreading of infection for an infinite system of independent Brownian motions. In this model we give a sufficient condition for recurrence (i.e., all the neighbors of the origin will be visited by infected particles infinitely often a.s.).

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Let ƒ(τ) be a Hauptmodul for Γ₀(2) that has zero at the cusp infinity. We construct the 2-adic modular form ƒ from ƒ(τ) by using Atkin’s method, which is an eigenfunction of the Hecke operator U . We expand ƒ as a formal power series in ƒ(τ). We can compute explicitly the 2-adic ordinal of these coefficients of the ƒ -expansion of ƒ. We show that ƒ coincides with the 2-adic modular form constructed from the modular invariant j(τ) by using Atkin’s method.

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Let D be a smoothly bounded pseudoconvex domain of finite type in C².Let D_M be a one-dimensional submanifold of D which meets the boundary of D transversally. Then any function in a Hardy space H^p(D_M) extends to the function in the corresponding Hardy space H^p(D).

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For the simply connected compact exceptional Lie group E₈, we determine the structure of subgroup (E₈)^σ,σ′ of E₈ that is the intersection (E₈)^σ ∩ (E₈)^σ′ . Then the space E₈/(E₈)^σ,σ′ is the exceptional Z₂ × Z_2- symmetric space of type EVIII-VIII-VIII, and we give two involutions σ, σ′ for the space E₈/(E₈)^σ,σ′ concretely.

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The principal aim of the present paper is to provide a new lower bound for linear forms in two p-adic elliptic logarithms. We refine a result due to Rémond and Urfels. Our improvement lies in the dependence on the height of algebraic coefficients of the linear forms. Our bound is the best possible one concerning the height of the coefficients, where all the relevant constants are explicit. We use the argument that relies on the interpolation method of Laurent, on the variable change introduced by Chudnovsky, and on Faà di Bruno’s formula adapted to matrices whose elements are p-adic elliptic logarithmic functions. The bounds would be useful to determine the set of S-integer points on elliptic curves defined over a number field.

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Surfaces with parallel mean curvature vector play important roles in the theory of harmonic maps, differential geometry as well as in physics. Surfaces with parallel mean curvature vector in Riemannian space forms were classified in the early 1970s by Chen and Yau. Recently, space-like surfaces with parallel mean curvature vector in arbitrary indefinite space forms were completely classified by Chen in two papers in 2009. In this paper, we completely classify Lorentz surfaces with parallel mean curvature vector in a pseudo-Euclidean space E^m_s with arbitrary dimension m and arbitrary index s. Our main result states that there are 23 families of Lorentz surfaces with parallel mean curvature vector in a pseudo-Euclidean m-space E^m_s . Conversely, every Lorentz surface with parallel mean curvature vector in E^m_s is obtained from the 23 families.

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Assume the system of differential equations E₄(a, b, c, c′; X, Y) satisfied by Appell’s hypergeometric function F₄(a, b, c, c′; X, Y) has a finite irreducible monodromy group M₄(a, b, c, c′). The monodromy matrix Γ_3∗ derived from a loop Γ₃ surrounding once the irreducible component C ={(X, Y)|(X - Y)² -2(X + Y) +1 =0} of the singular locus of E₄ is a complex reflection. The minimal normal subgroup N_C of M₄ containing Γ_3∗ is, by definition, a finite complex reflection group of rank four. Let P(G) be the projective monodromy group of the Gauss hypergeometric differential equation ₂E₁(a, b, c). It is known that N_C is reducible if ε :=c +c′-a -b -1 ∉Z or if ε ∈Z and P(G) is a dihedral group. We prove that, if ε ∈Z,then N_C is the (irreducible) Coxeter group W(D₄), W(F₄)and W(H₄) according as P(G) is the tetrahedral, octahedral and icosahedral group, respectively.

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In 1975 Horikawa introduced a method of resolving singularities of double covers over a smooth surface, called the canonical resolution. Ashikaga gave a similar method for certain triple covers in 1992, and Tan constructed the canonical resolution for any triple covers in 2002. These methods are useful for the global or local study of branched covers of surfaces. In this paper, we consider similar resolution for 4-fold covers over a smooth surface, which is based on Lagrange’s method to solve quartic equations. By using this method, we compute the Chern numbers c²₁ and c₂ of certain 4-fold covers over a smooth projective surface.

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he Macdonald polynomials with prescribed symmetry are obtained from the non-symmetric Macdonald polynomials via the operations of t-symmetrization, t-antisymmetrization and normalization. Motivated by corresponding results in Jack polynomial theory we proceed to derive an expansion formula and a related normalization. Eigenoperator methods are used to relate the symmetric and antisymmetric Macdonald polynomials, and we discuss how these methods can be extended to special classes of the prescribed symmetry polynomials in terms of their symmetric counterpart. We compute the explicit form of the normalization with respect to the constant term inner product. Surpassing our original motivation, this is used to provide a derivation of a special case of a conjectured q-constant term identity.

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