• Volume 127, Issue 4

September 2017,   pages  551-751

• Some inequalities for the Bell numbers

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faà di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.

• Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Let $F(X, Y) = \sum^{s}_{i=0}a_{i}X^{r_i}Y^{r−r_i} \in \mathbb{Z}[X, Y]$ be a form of degree $r = r_{s} \geq 3$, irreducible over $\mathbb{Q}$ and having at most $s + 1$ non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality $$\mid F(X, Y) \mid \leq h$$ is $\ll s^{2}h^{2/r}(1+log h^{1/r})$. They conjectured that $s^{2}$ may be replaced by $s$. Let $$\Psi = \mathop{\max}\limits_{0\leq i\leq s} max \left(\sum^{i-1}_{w=0} \frac{1}{r_{i}-r_{w}}, \sum^{s}_{w=i+1}\frac{1}{r_{w}-r_{i}}\right)$$. Then we show that $s^2$ may be replaced by ${max(s\log^{3} s, se^{\Psi})}$. We also show that if $\mid{a_0}\mid = \mid{a_s}\mid$ and $\mid{a_i} \leq \mid{a_0}\mid$ for $1 \leq i \leq s − 1$, then $s^2$ may be replaced by $s\log^{3/2} s$. In particular, this is true if $a_{i}\in {−1, 1}$.

• Twisting formula of epsilon factors

For characters of a non-Archimedean local field we have explicit formula for epsilon factors. But in general, we do not have any generalized twisting formula of epsilon factors. In this paper, we give a generalized twisting formula of epsilon factorsvia local Jacobi sums.

• On bigraded regularities of Rees algebra

For any homogeneous ideal $I$ in $K[x_{1}, . . . , x_{n}]$ of analytic spread $\ell$, we show that for the Rees algebra $R(I)$, $\rm{reg^{syz}_ {(0,1)}}\sl(R(I)) = \rm{reg^{T}_{(0,1)}}\sl(R(I))$. We compute a formula for the (0, 1)-regularity of $R(I)$, which is a bigraded analog of Theorem1.1 of Aramova and Herzog $(\it{Am. J. Math.} \bf{122(4)} (2000) 689–719)$ and Theorem 2.2of Römer $(\it{Ill. J. Math.} \bf{45(4)} (2001) 1361–1376)$ to $R(I)$. We show that if the defect sequence, $e_{k} := {\rm reg}(I^k)− k \rho(I)$, is weakly increasing for $k\geq\rm{reg^{syz}_{(0,1)}}\sl(R(I))$, then $\rm{reg}\sl(I^{j}) = j\rho(I) + e$ for $j\geq \rm{reg^{syz}_{(0,1)}}\sl(R(I)) + \ell$, where $\ell = {\rm min}\{\mu(J)\mid J \subseteq I$ a graded minimal reduction of $I\}$. This is an improvement of Corollary 5.9(i) of [16].

• $M$-curves and symmetric products

Let $(X , \sigma)$ be a geometrically irreducible smooth projective $M$-curve of genus $g$ defined over the field of real numbers.We prove that the $n$-th symmetric product of $(X , \sigma)$ is an $M$-variety for $n$ = 2 ,3 and $n \geq 2g − 1$.

• Homological algebra in $n$-abelian categories

In this paper, we study the homological theory in $n$-abelian categories. First, we prove some useful properties of $n$-abelian categories, such as $(n+2)×(n+2)$-lemma, 5-lemma and $n$-Horseshoes lemma. Secondly, we introduce the notions of right(left) $n$-derived functors of left(right) $n$-exact functors, $n$-(co)resolutions, and $n$-homologicaldimensions of $n$-abelian categories. For an $n$-exact sequence, we show that the long$n$-exact sequence theorem holds as a generalization of the classical long exact sequence theorem. As a generalization of $\sf{Ext^\ast}$(−,−), we study the $n$-derived functor $\sf{nExt^\ast}$(−,−) of hom-functor Hom(−,−). We give an isomorphism between the abelian group of equivalent classes of $m$-fold $n$-extensions $\sf{nE}^{m}(A, B)$ of $A$, $B$ and $\sf{nExt}^{m}_\mathcal{A}(A, B)$ using $n$-Baer sum for $m$, $n \geq 1$.

• Strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in multi-Banach space

Let $C$ be a bounded closed convex subset of a uniformly convex multi-Banach space $X$ and let $\Im_{j}$ = ${T_{j} (t) : t \in G}$ be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from $C$ into itself. In this paper, we prove the strong mean ergodic convergence theorem for the almost-orbit of $\Im$. Our results extend and unify many previously known results especially (Dong et al. On the strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in Banach space, preprint).

• Spectral properties and stability of perturbed Cartesian product

Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras, and let $T : \mathcal{B → A}$ be an algebra homomorphism with $\|T\| \leq 1$. Then $T$ induces a Banach algebra product $^\times{T}$ perturbing the coordinatewise product on the Cartesian product space $\mathcal{A \times B}$. We show that the spectral properties like spectral extension property, unique uniform norm property, regularity, weak regularity as well as Ditkin’s condition are stable with respectto this product.

• Involutions and trivolutions on second dual of algebras related to locally compact groups and topological semigroups

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $\ast$-semigroup $S$ when it is infinite non-discrete cancellative, $M_{a}(S)^{\ast\ast}$ does not admit an involution, and $M_{a}(S)^{\ast\ast}$ has atrivolution with range $M_{a}(S)$ if and only if $S$ is discrete. We also show that when $G$ isan amenable group, the second dual of the Fourier algebra of $G$ admits an involutionextending one of the natural involutions of $A(G)$ if and only if $G$ is finite. However,$A(G)^{\ast\ast}$ always admits trivolution.

• Hypersurfaces in nearly Kaehler manifold $\mathbb{S}^3\times \mathbb{S}^3$

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold $\mathbb{S}^3\times \mathbb{S}^3$ with a conformal vector field. There are three almost contact metric structures on a hypersurface of $\mathbb{S}^3\times \mathbb{S}^3$, and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in $\mathbb{S}^3\times \mathbb{S}^3$.

• Minimal surfaces in symmetric spaces with parallel second fundamental form

In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is $S^2$, we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We alsoconsider isometric minimal two-spheres immersed in complex two-dimensional Kählersymmetric spaces with parallel second fundamental form, and prove that the immersionis totally geodesic with constant Kähler angle if it is neither holomorphic nor antiholomorphicwith Kähler angle $\alpha\neq 0$ (resp. $\alpha\neq \pi$) everywhere on $S^2$.

• Hamiltonian cycles in polyhedral maps

We present a necessary and sufficient condition for existence of a contractible, non-separating and non-contractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces.We also present algorithms to construct such cycles whenever it exists where one of them is linear time and another is exponential time algorithm.

• # Proceedings – Mathematical Sciences

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November 2017