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Abstract factorization theorems with applications to idempotent factorizations
Full text linkLet ⪯ be a preorder on a monoid H with identity 1H and s be an integer ≥ 2. The ⪯-height of an element x ∈ H is the supremum of the integers k ≥ 1 for which there is a (strictly) ⪯-decreasing sequence x1, ..., xk of ⪯-non-units of H with x1 = x, where u ∈ H is a ⪯-unit if u ⪯ 1H ⪯ u and a ⪯-non-unit otherwise. We say H is ⪯-artinian if there is no infinite ⪯-decreasing sequence of elements of H, and strongly ⪯-artinian if the ⪯-height of each element is finite.
We establish that, if H is ⪯-artinian, then each ⪯-non-unit x ∈ H factors through the ⪯-irreducibles of degree s, where a ⪯-irreducible of degree s is a ⪯-non-unit a ∈ H that cannot be written as a product of s or fewer ⪯-non-units each of which is (strictly) smaller than a with respect to ⪯. In addition, we show that, if H is strongly ⪯-artinian, then x factors through the ⪯-quarks of H, where a ⪯-quark is a ⪯-minimal ⪯-non-unit. In the process, we obtain upper bounds for the length of a shortest factorization of x into ⪯-irreducibles of degree s (resp., ⪯-quarks) in terms of its ⪯-height.
Next, we specialize these results to the case in which (i) H is the multiplicative submonoid of a ring R formed by the zero divisors of R (and the identity 1R) and (ii) a ⪯ b if and only if the right annihilator of 1R − b is contained in the right annihilator of 1R − a. If H is ⪯-artinian (resp., strongly ⪯-artinian), then every zero divisor of R factors as a product of ⪯-irreducibles of degree s (resp., ⪯-quarks); and we prove that, for a variety of right Rickart rings, either the ⪯-quarks or the ⪯-irreducibles of degree 2 or 3 are coprimitive idempotents (an idempotent e ∈ R is coprimitive if 1R − e is primitive). In the latter case, we also derive sharp upper bounds for the length of a shortest idempotent factorization of a zero divisor x ∈ R in terms of the ⪯-height of x and the uniform dimension of RR. In particular, we can thus recover and improve on classical theorems of J. A. Erdos (1967), R.J.H. Dawlings (1981), and J. Fountain (1991) on idempotent factorizations in the endomorphism ring of a free module of finite rank over a skew field or a commutative DVD (e.g., we find that every singular n-by-n matrix over a commutative DVD, with n ≥ 2, is a product of 2n − 2 or fewer idempotent matrices of rank n − 1)
On the notions of upper and lower density
No full textLet be the power set of N. We say that a function is an upper density if, for all X, Y † N and h, k N+, the following hold: (f1); (f2) if X † Y; (f3); (f4), where k · X: = {kx: x X}; and (f5). We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ -1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)-(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that for every X † N. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory
On the finiteness of certain factorization invariants
Full text linkLet H be a monoid and πH be the unique extension of the identity map on H to a monoid homomorphism F(H)→H, where we denote by F(X) the free monoid on a set X.Given A⊆H,anA-word z (i.e., an element of F(A)) is minimal if πH(z)=πH(z′) for every permutation z′ of a proper subword of z. The minimal A-elasticity of H is then the supremum of all rational numbers m/n with m,n∈N+ such that there exist minimal A-words a and b of length m and n, resp., with πH(a)=πH(b). Among other things, we show that if H is commutative and A is finite, then the minimal A-elasticity of H is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where H is cancellative, commutative, and f initely generated modulo units, and A is the set A(H)ofatomsofH. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, f initely generated monoid with trivial group of units whose minimal A (H)-elasticity is infinite
On the commutation of generalized means on probability spaces
Full text linkLet f and g be real-valued continuous injections defined on a non-empty real interval I, and let (X,L,λ) and (Y,M,μ) be probability spaces in each of which there is at least one measurable set whose measure is strictly between 0 and 1. We say that (f,g) is a (λ,μ)-switch if, for every L⊗M-measurable function h:X×Y→R for which h[X×Y] is contained in a compact subset of I, it holds f−1(∫Xf(g−1(∫Yg∘hdμ))dλ)=g−1(∫Yg(f−1(∫Xf∘hdλ))dμ), where f−1 is the inverse of the corestriction of f to f[I], and similarly for g−1. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that (f,g) is a (λ,μ)-switch if and only if f=ag+b for some a,b∈R, a≠0
Factorization under local finiteness conditions
Full text linkIt has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called <=-factorizations, for the <=-non-units of a (multiplicatively written) monoid H endowed with a preorder <=, where an element is an element of is a <=-unit if u <= 1(H) <= and a <=-non-unit otherwise. The "building blocks" of these factorizations are the <=-irreducibles of H (i.e., the <=-non-units is an element of that cannot be written as a product of two <=-non-units each of which is strictly <=-smaller than a); and it is interesting to look for sufficient conditions for the <=-factorizations of a <=-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal <=-factorizations (i.e., a refinement of <=-factorizations used to counter the "blow-up phenomena" that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the "local behavior" of the pair (H, <=). Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where <= is the divisibility preorder on H (and hence in the setup of the classical theory)
On the density of sumsets, II
Full text linkArithmetic quasi-densities are a large family of real-valued set functions
partially defined on the power set of , including the asymptotic
density, the Banach density, the analytic density, etc.
Let be a non-empty set covering residue
classes modulo as (e.g., the primes or the perfect powers).
We show that, for each , there is a set such that, for every arithmetic quasi-density , both and
the sumset are in the domain of and, in addition, . The proof relies on the properties of a little known density first
considered by Buck in 1946
Il Maugeri Stress Index questionnaire per la valutazione dello stress lavoro correlato
No full textViene presentata la validazione di un questionario costruito per la valutazione del rischio stress correlato e delle componenti che lo possono determinare.
Per la costruzione del questionario si sono seguite le seguenti fasi: 1) analisi critica degli strumenti presenti in letteratura; 2) selezione di item rilevanti e formulazione di un questionario preliminare; 3) analisi della comprensibilità degli item mediante focus group; 4) e successiva somministrazione ad un campione di 329 soggetti operanti in strutture pubbliche e private dell’industria e del terziario ed ad un campione di 29 soggetti che lamentavano stress e vessazioni in ambito lavorativo.
Il Maugeri Strees Index raggiunge un valore di affidabilità (alfa di Cronbach) pari a 0.93. L’analisi fattoriale ha permesso di individuare una struttura a 5 fattori: Benessere, Adattamento, Supporto, Irritabilità, Evitamento. Si riscontra differenza statisticamente significativa, sia nel punteggio totale di MSI sia nelle sottoscale, tra soggetti che non riferivano stress e quelli che erano connotati da presenza di stress lavorativo percepito. In conclusione, il questionario Maugeri Stress Index presenta buone caratteristiche di attendibilità e di validità di costrutto; l’analisi fattoriale ha confermato una struttura pluridimensionale, caratterizzata da cinque sottoscale
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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