In this study, dissipative singular q-Sturm-Liouville operators are studied in the Hilbert space $\mathscr{L}_{r,q}^{2}$(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, we prove the theorem on completeness of the system of eigenfunctions and associated functions (or root functions) of the dissipative q-Sturm-Liouville operator.

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.

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 Kⁿ⁻¹ ≠ 0. As main results, we classify such hyperspheres for dimension n ≤ 5.

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.

Let Q(Δ) be the set of all integrable holomorphic quadratic differentials on the unit disk Δ. The subset Q₀(Δ) of Q(Δ) is the set associated with T₀ classes in the universal Teichmüller space T(Δ). In this paper, it is shown that Q₀(Δ) is dense in Q(Δ). The infinitesimal version is also obtained.

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.

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.

Ruled real hypersurfaces in a nonflat complex space form $\tilde{M}_n(c) (n ≧ 2)$ 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 $\tilde{M}_n(c) (n ≧ 3)$ 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 $\tilde{M}_n(c)$.

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, μ₁(G) be the linear spectral gap of G, and λ₁(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 λ₁(G,X)/μ₁(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 X has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.

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.

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 and we apply the method to show that some of them are congruent.

Let f(t,z) = f₀(z) + tg(z) be a holomorphic function defined in a neighbourhood of the origin in C × Cⁿ. 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. Fernández de Bobadilla. In this paper, we show that Parusiński's result generalizes all the same to families of non-isolated singularities if the Lê numbers of the function f itself are defined and constant along the strata of an analytic stratification of C × (f₀⁻¹(0) $\cap$ g⁻¹(0)). Actually, it suffices to consider the strata that contain a critical point of f.

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.

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.