The purpose of the paper is to consider an equivalence problem of second order partial differential equations for one unknown function of two independent variables under scale transformations. For this equivalence problem, explicit forms of invariant functions are given. In particular, if all of these invariant functions vanish, then PDEs are equivalent to the flat equation.

A Banach space operator T ∈ B(¥cal{X}) is hereditarily polaroid, T ∈ (¥cal{HP}), if the isolated points of the spectrum of every part T_p of the operator are poles of the resolvent of T_p; T is hereditarly normaloid, T ∈ (¥cal{HN}), if every part T_p of T is normaloid. Let (¥cal{HNP}) denote the class of operators T ∈ B(¥cal{X}) such that T ∈ (¥cal{HP}) ∩ (¥cal{HN}). (¥cal{HNP}) operators such that the Berberian-Quigley extension T° of T is also in (¥cal{HNP}) satisfy Bishop's property (β). Given Hilbert space operators A, B^* ∈ B(¥cal{H}), let d_AB ∈ B(B(¥cal{H})) stands for either of the elementary operators δ_AB(X) = AX - XB and Δ_AB(X) = AXB - X. If A, B^* ∈ (¥cal{HP}) satisfy property (β), and the eigenspaces corresponding to distinct eigenvalues of A (resp., B^*) are orthogonal, then f(d_AB) satisfies Weyl's theorem and f(d_AB)^* satisfies a-Weyl's theorem for every function f which is analytic on a neighbourhood of σ(d_AB). Finally, we characterize perturbations of d_AB by quasinilpotent and algebraic operators A, B ∈ B(¥cal{H}).

If the log canonical divisor on a projective variety with only Kawamata log terminal singularities is numerically equivalent to some semi-ample Q-divisor, then it is semi-ample.

In this paper, we define a perspective projection of a given immersed n-dimensional hypersurface as a C^∞ map via a C^∞ immersion from the given n-manifold to Sⁿ⁺¹, and characterize when and only when such a perspective projection is non-singular.

In order to obtain such characterizations, we consider an immersion from an n-dimensional manifold to Sⁿ⁺¹. We first obtain equivalent conditions for a given point P of Sⁿ⁺¹ to be outside the union of tangent great hyperspheres of a given immersed n-dimensional manifold r(N) in Sⁿ⁺¹ (Theorem 2.4). It turns out that if such a point P exists then the given manifold N must be diffeomorphic to Sⁿ and in the case that n ≥ 2 the given immersion r: N → Sⁿ⁺¹ must be an embedding. Then, we obtain characterizations of a perspective projection of a given immersed n-dimensional manifold to be non-singular.

Next, we obtain one more equivalent condition in terms of hedgehogs when the given N isSⁿ and the given immersion is an embedding (Theorem 3.3). We also explain why we consider these equivalent conditions for an embedding r: Sⁿ → Sⁿ⁺¹ instead of an embedding ¥widetilde{r}: Sⁿ → ¥mathbb{R}ⁿ⁺¹ in terms of hedgehogs.

Let ¥cal{A} be an irreducible Coxeter arrangement and k be a multiplicity of ¥cal{A}. We study the derivation module D(¥cal{A}, k). Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action of the Coxeter group. In this paper, we will explicitly construct a basis for D(¥cal{A}, k) assuming k is constant on each orbit. Consequently we will determine the exponents of (¥cal{A}, k) under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.

In the framework of a pre-inner product C^*-module over a unital C^*-algebra, we show several reverse Cauchy-Schwarz type inequalities of additive and multiplicative types, by using some ideas in N. Elezović et al. [Math. Inequal. Appl., 8 (2005), no.2, 223-231]. We apply our results to give Klamkin-Mclenaghan, Shisha-Mond and Cassels type inequalities. We also present a Grüss type inequality.

We study the Grassmann geometry of surfaces in the special real linear group SL(2,R).

In this paper, we characterize 42 ovoids with a certain property in a projective space PG(3,2) described in Yucas [6]. As a corollary, we construct the Steiner 4-wise balanced design S(4,{5,6},17) with 252 blocks which is an extension of the point-plane design A of an affine space AG(4,2). The construction leads to not only the uniqueness of such an extension, but also a (usual) extension of the 2-repeated design 2.A.