We introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets, we study the behavior of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. To this end we associate each atom with a natural notion or arity. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of $k+1$-ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any k-ary atoms, whence the $k+1$-ary atoms are not definable in terms of any atoms of smaller arity.

Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$-dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $, which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$. The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

The ongoing developments of psychiatric classification systems have largely improved reliability of diagnosis, including that of schizophrenia. However, with an unknown pathophysiology and lacking biomarkers, its validity still remains low, requiring further advancements. Research has helped establish multiple sclerosis (MS) as the central nervous system (CNS) disorder with an established pathophysiology, defined biomarkers and therefore good validity and significantly improved treatment options. Before proposing next steps in research that aim to improve the diagnostic process of schizophrenia, it is imperative to recognize its clinical heterogeneity. Indeed, individuals with schizophrenia show high interindividual variability in terms of symptomatic manifestation, response to treatment, course of illness and functional outcomes. There is also a multiplicity of risk factors that contribute to the development of schizophrenia. Moreover, accumulating evidence indicates that several dimensions of psychopathology and risk factors cross current diagnostic categorizations. Schizophrenia shares a number of similarities with MS, which is a demyelinating disease of the CNS. These similarities appear in the context of age of onset, geographical distribution, involvement of immune-inflammatory processes, neurocognitive impairment and various trajectories of illness course. This article provides a critical appraisal of diagnostic process in schizophrenia, taking into consideration advancements that have been made in the diagnosis and management of MS. Based on the comparison between the two disorders, key directions for studies that aim to improve diagnostic process in schizophrenia are formulated. All of them converge on the necessity to deconstruct the psychosis spectrum and adopt dimensional approaches with deep phenotyping to refine current diagnostic boundaries.

For a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$, then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$. The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $.

Traditional categorical approaches to classifying personality disorders are limited in important ways, leading to a shift in the field to dimensional approaches to conceptualizing personality pathology. Different areas of psychology – personality, developmental, and psychopathology – can be leveraged to understand personality pathology by examining its structure, development, and underlying mechanisms. However, an integrative model that encompasses these distinct lines of inquiry has not yet been proposed. In order to address this gap, we review the latest evidence for dimensional classification of personality disorders based on structural models of maladaptive personality traits, provide an overview of developmental theories of pathological personality, and summarize the Research Domain Criteria (RDoC) initiative, which seeks to understand underlying mechanisms of psychopathology. We conclude by proposing an integrative model of personality pathology development that aims to elucidate the developmental pathways of personality pathology and its underlying mechanisms.

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgroup of positive dimension, and a pseudofinite group of dimension 2 contains a soluble subgroup of dimension 2.

Taxometric procedures have been used extensively to investigate whether individual differences in personality and psychopathology are latently dimensional or categorical (‘taxonic’). We report the first meta-analysis of taxometric research, examining 317 findings drawn from 183 articles that employed an index of the comparative fit of observed data to dimensional and taxonic data simulations. Findings supporting dimensional models outnumbered those supporting taxonic models five to one. There were systematic differences among 17 construct domains in support for the two models, but psychopathology was no more likely to generate taxonic findings than normal variation (i.e. individual differences in personality, response styles, gender, and sexuality). No content domain showed aggregate support for the taxonic model. Six variables – alcohol use disorder, intermittent explosive disorder, problem gambling, autism, suicide risk, and pedophilia – emerged as the most plausible taxon candidates based on a preponderance of independently replicated findings. We also compared the 317 meta-analyzed findings to 185 additional taxometric findings from 96 articles that did not employ the comparative fit index. Studies that used the index were 4.88 times more likely to generate dimensional findings than those that did not after controlling for construct domain, implying that many taxonic findings obtained before the popularization of simulation-based techniques are spurious. The meta-analytic findings support the conclusion that the great majority of psychological differences between people are latently continuous, and that psychopathology is no exception.

This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.

In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.