1.1. Definition. Let C be an abelian category. The derivative D(C) of C is defined as follows: The objects are exact sequences 0 −→ X0 iX −−→ X1 πX −−→ X0 −→ 0 of C, and the morphisms from such an object are morphisms of exact sequences whose two X parts coincide. The category D(C) is again abelian. An exact functor F : C1 −→ C2 gives rise to an induced (exact) functor D(F ) : D(C1) −→ D(C2). W...

We study the automorphism groups of the reduction X0(N)× F̄p of a modular curve X0(N) over primes p ∤ N .

The problem of constructing a measure on discrete set X taking values in positive cone bounded operators Hilbert space is considered. It assumed that projectionvalued function defined subset X0 the original initially given. aim study to find such scalar μ and continuation projector-valued from X, which results an operator-valued having density relative μ. In general, solved for |X| = 4 |X0| 2. ...

Repellent efficacy of the plant-based repellent, TT-4302 (5% geraniol), was compared with 16 other products in laboratory arm-in-cage trials against Aedes aegypti (L). Eight repellents (Badger, BioUD, Burt's bees, California Baby, Cutter Natural, EcoSMART, Herbal Armor, and SkinSmart) exhibited a mean repellency below 90% to Ae. aegypti at 0.5 h after application. Three repellents (Buzz Away Ex...

S d := {x = (x1, . . . , xd) ∈ R : 0 6 x1, . . . , xd 6 1, 0 6 x1 + · · ·+ xd 6 1} denote the standard simplex in R. We denote by ∂S the boundary of S. We will also use barycentric coordinates on the simplex which we denote by the boldface symbol x = (x0, x1, . . . , xd), x0 := 1−x1−· · ·−xd. We will use standard multiindex notation such as x := x0 0 x α1 1 · · ·xd d and α n := (α0 n , α1 n , ·...

In this note we discuss an analog of the classical Waring problem for C[x0,x1,...,x(n)]. Namely, we show that a general homogeneous polynomial p ∈ C[x0,x1,...,x(n)] of degree divisible by k≥2 can be represented as a sum of at most k(n) k-th powers of homogeneous polynomials in C[x0,x1,...,x(n)]. Noticeably, k(n) coincides with the number obtained by naive dimension count.

Consider a configuration of k points or landmarks in R, represented as a k × m matrix X0. For many purposes, it is not the configuration of X0 which is of importance but its shape, that is, its equivalence class under an appropriate group of transformations. One important type of shape is projective shape. Let X = [1k, X0] be a k × p matrix, where p = m + 1, containing the landmark positions in...