In this paper, by considering a special case of the spacelike mean curvature flow investigated by Li and Salavessa [Math. Z. 269 (2011), 697-719], we obtain a condition for the existence of smooth solutions of the Dirichlet problem for the minimal surface equation in arbitrary codimension. We also show that our condition is sharper than Wang's [Comm. Pure Appl. Math. 57 (2004), 267-281] provided that the hyperbolic angle θ of the initial spacelike submanifold M₀ satisfies max_M0 cosh θ> √2.

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In this paper, we realize any homogeneous cone by assembling uniquely determined subcones. These subcones are realized in the cones of positive-definite real symmetric matrices of minimal possible sizes. The subcones are found through the oriented graphs drawn by using the data of the given homogeneous cones. We also exhibit several interesting examples of our realizations of homogeneous cones. These are of rank 5, of dimension 19, of dimension 11 of continuously many inequivalent homogeneous cones, and some of the low-dimensional homogeneous cones.

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Let X be a smooth complex projective variety of dimension n and let L₁,... , L_n−1 be nef and big line bundles on X. In this paper, we prove the non-negativity of the sectional genus g₁(X, L₁,..., L_n−1). We also study multi-quasi-polarized manifolds (X, L₁,..., L_n−1) with g₁(X, L₁,..., L_n−1) = 0.

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We study the minimal rank of an infinite upper triangular matrix over a field. We characterize all linear maps on this space that preserve the minimal rank. Moreover, we give a partial solution to some open problems concerning this notion.

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We prove that if we consider p : Y → X to be an unbranched Riemann domain between two complex spaces with isolated singularities, X is Stein and p is locally q-complete with corners, then Y is q-complete with corners.

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We discuss the problem of the existence of envelopes of holomorphy of the new cross-type objects, the A -crosses, which leads us to the far-reaching generalizations of the famous Hartogs theorem. We also take under consideration the issue of the existence of ‘nice' general descriptions of envelopes of holomorphy of the cross-like objects in terms of the relative extremal function, which seems to be very natural in the light of the extension theory of separately holomorphic functions on classical crosses and (N, k)-crosses.

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We show that any C¹-diffeomorphism with a basic set has a C¹-neighborhood satisfying the following properties. A generic element in the neighborhood has a unique measure with maximum total exponent which is of zero entropy and fully supported on the continuation of the basic set. To the contrary, we show that for r ≥ 2 any C^r-diffeomorphism with a basic set does not have a C^r-neighborhood satisfying the above properties.

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In this paper, the conjecture on the Zalcmanness of Cⁿ (n ≥ 2) and (C^∗)², whichis posed in our previous work, is proved in the case where the derivatives of limit holomorphic curves are bounded. Moreover, several criteria for normality of families of holomorphic mappings are given.

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Let ĉ_n be the nth Cauchy number of the second kind, defined by the generating function x/(1 + x) ln(1 + x) = Σ^∞_n=0 ĉ_nxⁿ/n!. We obtain explicit expressions for (ĉ_l + ĉ_m)ⁿ := Σⁿ_j=0 (ⁿ_j)ĉ_l+jĉ_m+n−j with arbitrary fixed integers l, m ≥ 0. The corresponding expression for Bernoulli numbers has been studied by Agoh and Dilcher, and that for Cauchy numbers of the first kind by the author.

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We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line (−∞, 0] × {0} before time n. Let X = (X₁, X₂) be the increment of the two-dimensional random walk. For an aperiodic random walk with moment conditions E [X₂] = 0 and E [|X₁|^δ] < ∞, E [|X₂|^2+δ] < ∞ for some δ ∈ (0, 1), we obtain an asymptotic estimate (as n →∞) of this probability by assuming the behavior of the characteristic function of X near zero.

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Suggested by Shepp [Ann. Math. Statist. 36(4) (1965), 1107-1112] we defined a sequence space Λ_p(f) determined by a single function f(≠ 0) ∈ L_p(R, dx), 1 ≤ p < +∞, and discussed the structure of it. The problems are the linearity and the visible sequential representation of Λ_p(f). In this paper we name Λ_p(f) a Shepp space and discuss the problems in the case of p = 2 by defining an inner approximation Λ⁰₂(f) and an outer approximation Λ^φ₂(f) of Λ₂(f), and we give a necessary and sufficient condition for Λ⁰₂(f) = Λ^φ₂(f) in terms of doubling dimension. In this case Λ₂(f) is a linear space and those approximationsare its visible sequential representations. We also give an example such that Λ₂(f) is a linear space but Λ⁰₂(f) ≠ Λ^φ₂(f).

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In this paper, we study the distribution behavior of integer n-tuples (a₁,.. .,a_n) satisfying a₁ ···a_n ≡ 1 (mod p) and |a_i − a_j| ≤ A (for all i, j), an asymptotic formula will be given for the corresponding counting function.

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We consider the generalized hypergeometric function _m+1F_m and the differential equation _m+1E_m that it satisfies. We use the twisted (co)homology groups associated with an Euler-type integral representation. We evaluate the intersection numbers of the twisted cocycles that are defined as the mth exterior productsof logarithmic 1-forms.We also provide the twisted cycles corresponding to the local solutions to _m+1E_m around the origin, and we evaluate their intersection numbers. The intersection numbers of the twisted (co)cycles lead to the twisted period relations between two fundamental systems of solutions to _m+1E_m.

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A stochastic oscillatory integral is a probabilistic counterpart to a Feynman path integral, and the stationary phase method for stochastic oscillatory integrals relates to the semi-classical limits in the Feynman path integral theory. A complete stationary phase method for stochastic oscillatory integrals has not been achieved yet. In this paper, a new class of amplitude functions of stochastic oscillatory integrals is introduced, and it is shown that the stationary phase method works for the class.

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