Authors:Jacopo Gandini Pages: 299 - 340 Abstract: We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety. PubDate: 2018-03-01 DOI: 10.1007/s10114-018-7162-2 Issue No:Vol. 34, No. 3 (2018)

Authors:Bernhard Krötz; Henrik Schlichtkrull Pages: 341 - 370 Abstract: We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces Z = G/H attached to a real reductive Lie group G. A special emphasis is made to the case where Z is real spherical. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-6557-9 Issue No:Vol. 34, No. 3 (2018)

Authors:Nicolas Perrin Pages: 371 - 416 Abstract: These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-7163-6 Issue No:Vol. 34, No. 3 (2018)

Authors:Guido Pezzini Pages: 417 - 438 Abstract: These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-7214-z Issue No:Vol. 34, No. 3 (2018)

Authors:Stéphanie Cupit-Foutou; Dmitry A. Timashev Pages: 439 - 453 Abstract: Let G be a complex semisimple algebraic group and X be a complex symmetric homogeneous G-variety. Assume that both G, X as well as the G-action on X are defined over real numbers. Then G(ℝ) acts on X(ℝ) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of Borel and Ji. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-7184-1 Issue No:Vol. 34, No. 3 (2018)

Authors:Kiumars Kaveh; Christopher Manon Pages: 454 - 465 Abstract: This is a survey of some recent results on spherical tropical geometry. PubDate: 2018-03-01 DOI: 10.1007/s10114-018-7238-z Issue No:Vol. 34, No. 3 (2018)

Authors:Shin-Young Kim; Kyeong-Dong Park Pages: 466 - 487 Abstract: Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional F4-homogeneous manifold associated to a short simple root. PubDate: 2018-03-01 DOI: 10.1007/s10114-018-7165-z Issue No:Vol. 34, No. 3 (2018)

Authors:Bernhard Krötz; Eitan Sayag; Henrik Schlichtkrull Pages: 488 - 531 Abstract: We provide L p -versus L∞-bounds for eigenfunctions on a real spherical space Z of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on Z. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-7164-5 Issue No:Vol. 34, No. 3 (2018)

Authors:Duo Li Pages: 532 - 541 Abstract: We study two-dimensional irreducible projective smooth algebraic semigroups. Minimal surface semigroups with Kodaira dimension at most one are partially classified. We also calculate the local dimension of the moduli scheme parameterizing all algebraic semigroup laws on a fixed minimal surface. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-7182-3 Issue No:Vol. 34, No. 3 (2018)

Authors:Boris Pasquier Pages: 542 - 562 Abstract: In a previous work, we described the Minimal Model Program in the family of ℚ-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (X, Δ) when X is a projective horospherical variety. PubDate: 2018-03-01 DOI: 10.1007/s10114-017-6558-8 Issue No:Vol. 34, No. 3 (2018)

Authors:Kay Paulus; Guido Pezzini; Bart Van Steirteghem Pages: 563 - 596 Abstract: Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties for G = SL(2) × ℂ × and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop’s classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward’s result that every reflective Delzant polytope is the moment polytope of such a manifold. PubDate: 2018-03-01 DOI: 10.1007/s10114-018-7244-1 Issue No:Vol. 34, No. 3 (2018)

Authors:Hui Qiao Pages: 171 - 193 Abstract: This paper presents a theorem of existence of \((N - N_{0} + 1)2^{N_{0}}\) noncontractible periodic orbits of any given rotational vector with zero components for the planar forced N-pendulum equation, where N 0 denotes the number of zero components of given rotational vector. We make special arrangements for masses and length instead of the nondegenerate assumption. Moreover, for rotational vectors with identical components, similar results is also obtained. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-7037-y Issue No:Vol. 34, No. 2 (2018)

Authors:Tai Xiang Sun Pages: 194 - 208 Abstract: Let G be a graph and f: G → G be a continuous map. Denote by h(f), P(f),AP(f),R(f) and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent: (1) h(f) > 0. (2) There exists an x ∈ G such that ω(x, f) ∩ P(f) ≠ Ø and ω(x, f) is an infinite set. (3) There exists an x ∈ G such that ω(x, f) contains two minimal sets. (4) There exist x, y ∈ G such that ω(x, f) − ω(y, f) is an uncountable set and ω(y, f) ∩ ω(x, f) ≠ Ø. (5) There exist an x ∈ G and a closed subset A ⊂ ω(x, f) with f(A) ⊂ A such that ω(x, f) − A is an uncountable set. (6) R(f) − AP(f) ≠ Ø. (7) f P ( f ) is not pointwise equicontinuous. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-7236-6 Issue No:Vol. 34, No. 2 (2018)

Authors:Ke Feng; Ya Long Shi; Yi Yan Xu Pages: 209 - 220 Abstract: We study the Dirichlet problem of the n-dimensional complex Monge—Ampère equation det(u ij̅ ) = F/ z 2α , where 0 < α < n. This equation comes from La Nave—Tian’s continuity approach to the Analytic Minimal Model Program. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-7148-5 Issue No:Vol. 34, No. 2 (2018)

Authors:Akhilesh Prasad; Kanailal Mahato Pages: 221 - 232 Abstract: This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-7151-x Issue No:Vol. 34, No. 2 (2018)

Authors:Aurelien Fouetio; Jean Louis Woukeng Pages: 233 - 254 Abstract: For a family of linear hyperbolic damped stochastic wave equations with rapidly oscillating coefficients, we establish the homogenization result by using the sigma-convergence method. This is achieved under an abstract assumption covering special cases like the periodicity, the almost periodicity and some others. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-6436-4 Issue No:Vol. 34, No. 2 (2018)

Authors:Ya Ming Lu Pages: 255 - 264 Abstract: In this paper, we will prove for 0.9993 < γ < 1 that there are infinitely primes p of the form [n 1/γ ] with p + 2 having at most four prime factors. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-7030-5 Issue No:Vol. 34, No. 2 (2018)

Authors:Bao Jian Qiu; Ji Hui Wang; Yan Liu Pages: 265 - 274 Abstract: Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′∑(G). Flandrin et al. proposed the following conjecture that χ′∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C 5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < \(\tfrac{{37}} {{12}}\) and Δ(G) ≥ 7. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-6491-x Issue No:Vol. 34, No. 2 (2018)

Authors:Jian Jun Chen; Xiao Feng Wang; Jin Xia; Guang Fu Cao Pages: 288 - 296 Abstract: The Sarason’s Toeplitz product problem asks when the Toeplitz product operator T u T v̅ , with analytic symbols u and v, is bounded on Hilbert space of analytic functions. In this paper, we deal with this problem on the Fock–Sobolev space and have a complete solution that u = e q , v = Ce −q , where q is a linear complex polynomial and C is a nonzero constant. PubDate: 2018-02-01 DOI: 10.1007/s10114-017-5780-8 Issue No:Vol. 34, No. 2 (2018)