1.
Integral inequalities for Lipschitzian mappings between two Banach spaces and applications - Dragomir, Sever S.
In this paper we obtain some inequalities of Ostrowski and Hermite-Hadamard type for Lipschitzian mappings between two Banach spaces. Applications for functions of norms in Banach spaces and functions defined by power series in Banach algebras are provided as well.

2.
A new form of the generalized complete elliptic integrals - Takeuchi, Shingo
Generalized trigonometric functions are applied to Legendre's form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized π in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for π by Brent and Salamin. Moreover, an elementary alternative proof of Ramanujan's cubic transformation is also given.

3.
Topological triviality of linear deformations with constant Lê numbers - Eyral, Christophe; Ruas, Maria Aparecida Soares
Let f(t,z) = f_{0}(z) + tg(z) be a holomorphic function defined in a neighbourhood of the origin in C × C^{n}. It is well known that if the one-parameter deformation family {f_{t}} defined by the function f is a μ-constant family of isolated singularities, then {f_{t}} is topologically trivial—a result of A. Parusiński. It is also known that Parusiński's result does not extend to families of non-isolated singularities in the sense that the constancy of the Lê numbers of f_{t} at 0, as t varies, does not imply the topological triviality of the family f_{t} in general—a result of J....

4.
Congruence of minimal surfaces - Kassabov, Ognian
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for minimal surfaces. Here we explain not only how this method works but also how we can find the correspondence between the minimal surfaces, if they are congruent. We show that two families of minimal surfaces which are proved to be conjugate actually coincide and coincide with their associated surfaces. We also consider another family of minimal polynomial surfaces of degree 6...

5.
The structure Jacobi operator of three-dimensional real hypersurfaces in non-flat complex space forms - Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
In this paper new results concerning three dimensional real hypersurfaces in non-flat complex space forms in terms of their stucture Jacobi operator are presented. More precisely, the conditions of 1) the structure Jacobi operator being of Codazzi type with respect to the generalized Tanaka-Webster connection and commuting with the shape operator and 2) η-invariance of the structure Jacobi operator and commutativity of it with the shape operator are studied. Furthermore, results concerning Hopf hypersurfaces and ruled hypersurfaces of dimension greater than three satisfying the previous conditions are also included.

6.
Fixed point property for a CAT(0) space which admits a proper cocompact group action - Toyoda, Tetsu
We prove that if a geodesically complete CAT(0) space X admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of X is less than 1. Let G be a finite connected graph, μ_{1}(G) be the linear spectral gap of G, and λ_{1}(G,X) be the nonlinear spectral gap of G with respect to such a CAT(0) space X. Then, the result implies that the ratio λ_{1}(G,X)/μ_{1}(G) is bounded from below by a positive constant which is independent of the graph G. It follows that any isometric action of a random group of the graph model on such...

7.
Ruled real hypersurfaces having the same sectional curvature as that of an ambient nonflat complex space form - Maeda, Sadahiro; Tanabe, Hiromasa; Kim, Young Ho
Ruled real hypersurfaces in a nonflat complex space form $\widetilde{M}_n$ (c) (n ≥ 2) are obtained by having a one-codimensional foliation whose leaves are totally geodesic complex hypersurfaces of the ambient space. Motivated by a fact that the sectional curvature K of every ruled real hypersurface M in $\widetilde{M}_n$ (c) (n ≥ 3) satisfies |c/4| ≤ |K(X, Y)| ≤ |c| for an arbitrary pair of orthonormal vectors X and Y that are tangent to the leaf at each point x of M, we study ruled real hypersurfaces having the sectional curvature K with |c/4| ≤ |K| ≤ |c| in $\widetilde{M}_n$...

8.
On the degree of Fano schemes of linear subspaces on hypersurfaces - Hiep, Dang Tuan
In this paper we propose and prove an explicit formula for computing the degree of Fano schemes of linear subspaces on general hypersurfaces. The method used here is based on the localization theorem and Bott's residue formula in equivariant intersection theory.

9.
Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids - Nakamoto, Kazunori; Torii, Takeshi
In this paper we study the topology of the moduli spaces of representations of degree 2 for free monoids. We calculate the virtual Hodge polynomials of the character varieties for several types of 2-dimensional representations. Furthermore, we count the number of isomorphism classes for each type of 2-dimensional representations over any finite field F_{q}, and show that the number coincides with the virtual Hodge polynomial evaluated at q.

10.
A note on holomorphic quadratic differentials on the unit disk - Yao, Guowu
Let Q(Δ) be the set of all integrable holomorphic quadratic differentials on the unit disk Δ. The subset Q_{0}(Δ) of Q(Δ) is the set associated with T_{0} classes in the universal Teichmüller space T(Δ). In this paper, it is shown that Q_{0}(Δ) is dense in Q(Δ). The infinitesimal version is also obtained.

11.
The Jones polynomial of rational links - Qazaqzeh, Khaled; Yasein, Moh'd; Abu-Qamar, Majdoleen
We use the Tutte polynomial to give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of its slope.

12.
Locally homogeneous affine hyperspheres with constant sectional curvature - Li, Cece
In this paper, we study the n-dimensional locally homogeneous affine hyperspheres with constant sectional curvature, vanishing Pick invariant and the difference tensor K satisfying K^{n−1} ≠ 0. As main results, we classify such hyperspheres for dimension n ≤ 5.

13.
Lower bound of admissible functions on the Grassmannian G_{
m,nm
} (C) - Abdesselem, Adnène Ben; Adouani, Ines
We prove the existence of an "extremal" function lower bounding all admissible functions (ie plurisubharmonic functions modulo a metric) with supremum equal to zero on the complex Grassmann manifold G_{m,nm}(C). The functions considered are invariant under a suitable automorphisms group. This gives a conceptually simple method to compute Tian's invariant in the case of a non toric manifold.

14.
Spectral problems of non-self-adjoint q-Sturm-Liouville operators in limit-point case - Allahverdiev, Bilender P.
In this study, dissipative singular q-Sturm-Liouville operators are studied in the Hilbert space $\mathcal{L}_{r,q}^{2}$ (R_{q,+}), that the extensions of a minimal symmetric operator in limit-point case. We construct a self-adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax-Phillips. Then, we create a functional model of the maximal dissipative operator via the incoming spectral representation and define its characteristic function in terms of the Weyl-Titchmarsh function (or scattering function of the dilation) of a self-adjoint q-Sturm-Liouville operator. Finally,...

15.
The existence of quasiconformal homeomorphism between planes with countable marked points - Fujino, Hiroki
We consider quasiconformal deformations of C $\backslash$ Z. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to C $\backslash$ Z. In particular, we characterize the closed subsets of R whose compliments are quasiconformally equivalent to C $\backslash$ Z.

16.
Nilpotent admissible indigenous bundles via Cartier operators in characteristic three - Hoshi, Yuichiro
In the present paper, we study the p-adic Teichmüller theory in the case where p = 3. In particular, we discuss nilpotent admissible/ordinary indigenous bundles over a projective smooth curve in characteristic three. The main result of the present paper is a characterization of the supersingular divisors of nilpotent admissible/ordinary indigenous bundles in characteristic three by means of various Cartier operators. By means of this characterization, we prove that, for every nilpotent ordinary indigenous bundle over a projective smooth curve in characteristic three, there exists a connected finite étale covering of the curve on which the indigenous bundle is not...

18.
Two normality criteria and counterexamples to the converse of Bloch's principle - Charak, Kuldeep Singh; Singh, Virender
In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman [Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1 (2001), 289-299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.

19.
Some inverse fractional legendre transforms of gamma function form - Ansari, Alireza
In this paper, using the asymptotic expansion of the ratio of gamma functions, we obtain new inversion formulas for the fractional Legendre transforms. These formulas are given in terms of some integral representations of the inverse Mellin transforms. Also, the closed forms of solutions of the fractional Laplace and Helmholtz equations are obtained by these inversion formulas.

20.
Equinormalizability and topologically triviality of deformations of isolated curve singularities over smooth base spaces - Lê, Công-Trình
We give a δ-constant criterion for equinormalizability of deformations of isolated (not necessarily reduced) curve singularities over smooth base spaces of dimension ≥ 1. For one-parametric families of isolated curve singularities, we show that their topologically triviality is equivalent to the admission of weak simultaneous resolutions.