On factorization invariants and Hilbert functions

Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of the natural numbers), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, $\omega$-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain.

The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, Garc\'ia-S\'anchez and Leamer construct a monoid S_\Gamma^s whose elements correspond to arithmetic sequences in a numerical monoid \Gamma of step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and investigate certain factorization invariants. Furthermore, when S_\Gamma^s is an arithmetical Leamer monoid, we give an exact description of its atoms and use this to provide explicit formulae for its Delta set and catenary degree.

Studying certain combinatorial properties of non-unique factorizations have been a subject of recent literature. Little is known about two combinatorial invariants, namely the catenary degree and the tame degree, even in the case of numerical monoids. In this paper we compute these invariants for a certain class of numerical monoids generated by generalized arithmetic sequences. We also show that the difference between the tame degree and the catenary degree can be arbitrary large even if the number of minimal generators is fixed.

The objective of this study was to develop the Chinese version of the biopsychosocial impact scale (BPIm-S) to assess functional limitation and psychosocial distress in orofacial pain (OFP) patients in mainland China, and investigate the factor structure, reliability and validity, measurement invariance, as well as scores differences across genders, age and educational status among OFP patients. The BPIm-S was developed and evaluated in four stages: (1) concept selection and item generation; (2) a pilot study assessing face and content validity; (3) the factors structure, reliability, convergent validity, and measurement invariance; and (4) concurrent validity and clinical responsiveness. Exploratory (EFA) and confirmatory factor analyses (CFA) were performed on data gathered from 406 OFP patients to assess construct validity. Composite Reliability (CR) and the Average Variance Extracted (AVE) were used to assess internal convergent validity. CR, internal consistency, and split-half reliability were also performed to determine the reliability. Multigroup CFA (MGCFA) was used to assess measurement invariance across genders, age and educational status. Mann-Whitney test compared scores across different genders, age and educational status. Participants completed the BPIm-S, visual analog scale (VAS), brief pain inventory facial (BPI-F), General Anxiety Disorder-7 (GAD-7) and Patient Health Questionnaire-9 (PHQ-9), and spearman's correlation coefficient was used to evaluate the concurrent validity and item-total correlations. A total of 12 patients with OFP completed the BPIm-S twice to test clinical responsiveness. To conduct the CFA and measurement invariance analysis, Mplus 8.4 was used. IBM SPSS Statistics 21 software and SPSSAU, a web-based data science algorithm platform tool, were used for all additional studies. For the preliminary version, 17 items were chosen. A total of four items were removed following the pilot research. The remaining 13 items of the BPIm-S comprised an overall summary scale. Excellent reliability (Item-to-total correlations ranged from 0.763 to 0.912) and strong internal consistency (Cronbach's α = 0.970, functional limitation, 0.962, and psychosocial distress, 0.977) were discovered. CFA also validated the structural validity of the 13-item scale. EFA was performed and a two-factor structure was investigated. In addition, MGCFA corroborated the measurement invariance of the BPIm-S across gender, age, and educational status. Patients over the age of 30, those with a medium level of education, and those with a low level of education showed substantially greater levels of functional limitation and psychological distress (Wilcoxon test, p < 0.001). Both concurrent validity and clinical responsiveness were assessed to be of good quality. The BPIm-S demonstrated good psychometric qualities and is a reliable tool that can now be used by clinicians to evaluate functional limitation and psychosocial distress among OFP patient.

ABSTRACTWe compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.

The prevalence of psychotic experiences (PEs) is higher in low-and-middle-income-countries (LAMIC) than in high-income countries (HIC). Here, we examine whether this effect is explicable by measurement bias. A community sample from 13 countries (N = 7141) was used to examine the measurement invariance (MI) of a frequently used self-report measure of PEs, the Community Assessment of Psychic Experiences (CAPE), in LAMIC (n = 2472) and HIC (n = 4669). The CAPE measures positive (e.g. hallucinations), negative (e.g. avolition) and depressive symptoms. MI analyses were conducted with multiple-group confirmatory factor analyses. MI analyses showed similarities in the structure and understanding of the CAPE factors between LAMIC and HIC. Partial scalar invariance was found, allowing for latent score comparisons. Residual invariance was not found, indicating that sum score comparisons are biased. A comparison of latent scores before and after MI adjustment showed both overestimation (e.g. avolition, d = 0.03 into d = -0.42) and underestimation (e.g. magical thinking, d = -0.03 into d = 0.33) of PE in LAMIC relative to HIC. After adjusting the CAPE for MI, participants from LAMIC reported significantly higher levels on most CAPE factors but a significantly lower level of avolition. Previous studies using sum scores to compare differences across countries are likely to be biased. The direction of the bias involves both over- and underestimation of PEs in LAMIC compared to HIC. Nevertheless, the study confirms the basic finding that PEs are more frequent in LAMIC than in HIC.

The Posttraumatic Growth Inventory (PTGI) assesses positive changes after a traumatic or serious life crisis. However, there are differing views regarding its factor structure and little understanding if it captures the positive changes experienced among individuals diagnosed with a chronic disease. Using confirmatory factor analysis (CFA), the proposed five-factor structure and measurement invariance of the PTGI was examined using two chronic illness samples: arthritis and inflammatory bowel disease (IBD). Individuals diagnosed with arthritis (n = 301) or IBD (n = 544) recruited from the community and the Internet completed the PTGI. Using a pooled sample, CFA tested five hypothesized models of the underlying factors structure of the PTGI. A stepwise procedure for testing the measurement invariance across the two groups evaluated the factor structure, factorial invariance, and latent mean invariance. Using the pooled sample, the CFA supported the hypothesized five-factor model, revealing the PTGI is multidimensional. Multigroup CFA supported invariance of the PTGI across the two groups although there were significant differences in latent means. The findings support that the PTGI assesses five related dimensions of PTG and that different chronic disease groups experience different types of positive changes. PTG may therefore be relevant as a meaningful treatment goal for people with chronic diseases as it is for people affected by other traumatic events.

We examine the factorial structure of the Security Scale Questionnaire (SSQ), exploring measurement invariance across mother-father-child attachment relationships, child sex, and country. We used the new 21-item SSQ version that integrates both safe haven and secure base behaviors in a two factors structure. Participants were 457 children (224 girls and 233 boys), ranging from 9 to 14 years old (M = 10.84, SD = 1.02) from Portuguese and USA samples. We confirmed the SSQ’s two-factor structure, although four items were unrelated to the latent structure and excluded from the final model. Results showed that SSQ can be used to study both mother/child and father/child attachment relationships. Multi-group analyses suggested measurement invariance between boys and girls and between Portuguese and USA samples. Our findings suggest that the SSQ can be considered a valid and cost-effective tool to measure perceived attachment security in middle childhood for both mother/child and father/child relationships.

Factorization theory: From commutative to noncommutative settings

The catenary degree of an element [Formula: see text] of a cancellative commutative monoid [Formula: see text] is a nonnegative integer measuring the distance between the irreducible factorizations of [Formula: see text]. The catenary degree of the monoid [Formula: see text], defined as the supremum over all catenary degrees occurring in [Formula: see text], has been studied as an invariant of nonunique factorization. In this paper, we investigate the set [Formula: see text] of catenary degrees achieved by elements of [Formula: see text], focusing on the case where [Formula: see text] is finitely generated (where [Formula: see text] is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of [Formula: see text] that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for [Formula: see text].

Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

Abstract: We develop and validate a short self-report measure of test anxiety, the Trait Test Anxiety Inventory – Short (TTAI-S) following the Kane (2013) validation framework. Data were collected from three independent samples of young adults in the US ( N = 629; Mage = 22.25 years). Evidence was gathered to support three aspects of the validity argument (i.e., scoring, extrapolation, and generalization). Good internal consistency and confirmed structure of a single factor supported scoring inferences. Scalar measurement invariance between different samples (Internet vs. undergraduate students) and demographic subgroups (i.e., gender, race/ethnicity, and parental educational attainment) provided evidence for generalization inferences. Significant associations between the TTAI-S score and theoretically relevant (state test anxiety, performance expectation, and self-confidence in math) and weaker associations with less relevant constructs (enjoyment, motivation, and values in learning math) substantiated extrapolation inferences. Having established measurement invariance, we examined demographic differences and found that students historically underserved or underrepresented in STEM disciplines reported greater test anxiety than their counterparts. These findings support the validity of the TTAI-S, a concise measure that is easy to administer and easy to score. The TTAI-S may be used to further investigate trait test anxiety for a diverse population, particularly factors that may contribute to or mitigate group differences.

Empathy is a key sociocognitive skill that reflects the ability to perceive and experience others’ emotional states. In the present research, we aimed to validate the Adolescent Measure of Empathy and Sympathy (AMES) in a sample of 534 14- to 19-year-old Portuguese adolescents (207 boys, M = 16.22 years, SD = .92). Construct validity, reliability (internal consistency), measurement invariance, and convergent and known-groups validity were assessed. The original structure of three-related factors was confirmed (comparative fit index [CFI] = .99, Tucker–Lewis index [TLI] = .99, root mean square error of approximation [RMSEA] = .04, standardized root mean square residual [SRMR] = .05), with all factors showing acceptable internal consistency. Measurement invariance considering the sex and the school year of participants was supported. The AMES also revealed good convergent validity, having significantly positive associations with all subscales of the Basic Empathy Scale. Finally, girls showed higher cognitive and affective empathy and sympathy than boys, and 12th graders revealed higher cognitive and affective empathy and sympathy than 10th and 11th graders, thus attesting AMES known-groups validity. Overall, the Portuguese version of the AMES appears to be an adequate measure of empathy and sympathy for adolescents.

There is a growing recognition of the importance of testing for measurement equivalence when comparing latent constructs—such as political trust—across countries and over time. Indeed, equivalence of measurements across countries and time points is a precondition for making meaningful and valid comparisons of means, scores, and relationships between constructs. Until recently, most efforts in this area of comparative research have focused on establishing measurement equivalence across groups, such as countries or cultural regions. In contrast, scarce attention has been paid to examining the measurement equivalence of constructs across levels of data. Rather, most empirical studies that use cross-national survey data assume that the factorial structure of a given construct is isomorphic, in other words, similar in measurement and meaning at the individual and country level. This assumption is not always justified, as the dimensions found at the individual level of data do not always generalize to the country level. In such cases, the results and substantive conclusions based on the assumption of measurement isomorphism may be misleading. In this article, we emphasize the importance of examining measurement isomorphism when working with cross-national survey data and describe a testing procedure using multilevel confirmatory factor analysis. We illustrate the procedure by examining cross-level equivalence of the political trust scale included in the European Social Survey (2008–2009). The results of the analysis indicate that the structure of political trust factor can be considered configurally isomorphic across individual and country levels. In addition, the measurement scale across the two levels is found to be partially isomorphic. These findings could be regarded as an encouraging result for the applied researchers who use the aggregated individual scores of political trust. At the same time, we demonstrate that measurement isomorphism cannot be simply assumed and hence should be examined prior to analysis to ensure valid and meaningful results.

We generalize the geometric sequence {ap, ap−1b, ap−2b2,…, bp} to allow the p copies of a (resp. b) to all be different. We call the sequence {a1a2a3… ap, b1a2a3…ap, b1b2a3…ap,…, b1b2b3…bp} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.