Let LWPC denote the identity (xy · x) · xz = x((yx · x)z), and RWPC the mirror identity. Phillips proved that a loop satisfies LWPC and RWPC if and only if it is a WIP PACC loop. Here, it is proved that a loop Q fulfils LWPC if and only if it is a left conjugacy closed (LCC) loop that fulfils the identity (xy · x)x = x(yx · x). Similarly, RWPC is equivalent to RCC and x(x · yx) = (x · xy)x. If a loop satisfies LWPC or RWPC, then it is power associative (PA). The smallest nonassociative LWPC-loop was found to be unique and of order 6 while there are exactly 6 nonassociative LWPC-loops of order 8 up to isomorphism. Methods of construction of nonassociative LWPC-loops were developed.

A transitive subgroup G ≤ SN is called a genus zero group if there exist non identity elements x1 , . . . , xr∈G satisfying G =<x1, . . . , xr>, x1·...·xr=1 and ind x1+...+ind xr = 2N − 2. The Hurwitz space Hinr(G) is the space of genus zero coverings of the Riemann sphere P1 with r branch points and the monodromy group G.In this paper, we assume that G is a finite group with PSp(4, q) ≤ G ≤ Aut(PSp(4, q)) and G acts on the projective points of 3-dimensional projective geometry PG(3, q), q is a prime power. We show that G possesses no genus zero group if q > 5. Furthermore, we study the connectedness of the Hurwitz space Hinr(G) for a given group G and q ≤ 5.

The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length.

For a continuous and positive function w (λ), λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform D (w, µ) (T ) := ∫0∞w (λ) (λ + T ) −1 dµ (λ) , where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if A ≥ m 1 > 0, B ≥ m 2 > 0, then ||D (w, µ) (B) − D (w, µ) (A) − D (D (w, µ)) (A) (B − A)|| ≤|B − A|2×[D(w,µ)(m2)−D(w,µ)(m1)−(m2- m1)D’(w,µ)(m1)]/(m2−m1)2 if m1≠m2, ≤ D’’(w, µ)(m)/2 if m1=m2=m, where D (D (w, µ)) is the Fréchet derivative of D (w, µ) as a function of operator and D’’(w, µ) is the second derivative of D (w, µ) as a real function. We also prove the norm integral inequalities for power r ∈ (0, 1] and A, B ≥ m > 0, ||∫01((1−t)A+tB)r−1dt−((A+B)/2)r−1|| ≤ (1−r) (2−r) mr−3||B−A||2/24 and ||((Ar−1+Br−1 )/2) − ∫01((1−t) A+tB)r−1dt|| ≤ (1−r) (2−r) mr−3||B − A||2/12.

Extensions and crossed modules of Lie type superalgebras are introduced and studied. We construct homology and cohomology theories of Lie-type superalgebras. The notion of left super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.

Motivated by the classifications of extreme and exposed 2-homogeneous polynomials on the plane with the hexagonal norm ||(x, y)|| = max{|y|, |x| + |y|/2} (see [15, 16]), we classify all smooth 2-homogeneous polynomials on R2 with the hexagonal norm.

We present a general survey on formal power series over topological algebras, along with some perspectives which are not easily found in the literature.

In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have been able to show some results about arbitrary bases in complete metric spaces and also about regular open subsets in Hausdorff regular topological spaces.

What has category theory to offer to Banach spacers? In this second part survey-like paper we will focus on very much needed advanced categorical and homological elements, such as Kan extensions, derived category and derived functor or Abelian hearts of Banach spaces.

We investigate in this paper the isolated points of the approximate point spectrum of a closed linear relation acting on a complex Banach space by using the concepts of quasinilpotent part and the analytic core of a linear relation.