An Optimal Estimate for the Anisotropic Logarithmic Potential

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.

A holistic approach to evaluating Parkinson’s disease, using the Delphi method: a linear evaluation index

ABSTRACT Background: Parkinson’s disease (PD) is a chronic disease that presents a multitude of symptoms, with symptoms of both motor and nonmotor nature. The Delphi method is widely used to create consensuses among experts in a field of knowledge. Objective: In order to reach a consensus on the values that should be assigned to the different motor and nonmotor manifestations of Parkinson’s disease, a linear evaluation index (LEI) was created. Subsequently, the metric properties of this index were studied. Methods: 120 consecutive patients with a Parkinson’s diagnosis were chosen in accordance with the UKPDSBB criteria. The Delphi method was used to reach a consensus among experts regarding the values of each of the manifestations included. Subsequently, the following attributes were analyzed: quality and acceptability of the data; reliability, in terms of internal consistency, reliability index, Cronbach’s alpha and standard error of measurement; and validity, in terms of convergent validity and validity for known groups. Results: Twenty-five experts participated. The importance factor did not differ between the first round and the second round (chi-square test). We analyzed the responses that assigned percentage values to the 10 dimensions of the LEI. Both in the first and in the second round, the values of the scattering coefficient Vr were always close to 0. The homogeneity index was 0.36; the corrected-item total correlation values ranged from 0.02 to 0.7; Cronbach’s α was 0.69; and the SEM was 4.23 (55.1%). Conclusions: The LEI was obtained through rigorous recommended methodology. The results showed adequate metric properties.

Tools for social policy management: the SiSo Scale for measuring situations of social difficulty

The design and evaluation of social policies requires information systems that enable social intervention with the people targeted by the programmes and services and that also offer indicators for the follow-up and monitoring of the policies adopted. The article presents the process of validation of a tool for diagnosing situations of social difficulty arising from social exclusion. The scale has been implemented in one of Spain’s seventeen Autonomous Communities and has been selected on the basis of Good Practice under the European Social Fund. Expert judges were consulted for content validity; the metric properties of the scores obtained by the scale were examined and an exploratory factorial analysis (EFA) was performed to study the internal structure. The results show that the scale has adequate levels of content validity, construct validity and internal consistency. The SiSo Scale supplies a synthetic index of Social Position, providing professionals with the technical tools needed to carry out social diagnoses and simultaneously giving valid and reliable information on the social condition of people in a situation of social exclusion, which can guide social policy decision-making.

Nonmetric ANOVA: a generic framework for analysis of variance on dissimilarity measures

Classic analysis of variance (ANOVA; cA) tests the explanatory power of a partitioning on a set of objects. Nonparametric ANOVA (npA) extends to a case where instead of the object values themselves, their mutual distances are available. While considerably widening the applicability of the cA, the npA does not provide a statistical framework for the cases where the mutual dissimilarity measurements between objects are nonmetric. Based on the central limit theorem (CLT), we introduce nonmetric ANOVA (nmA) as an extension of the cA and npA models where metric properties (identity, symmetry, and subadditivity) are relaxed. Our model allows any dissimilarity measures to be defined between objects where a distinctiveness of a specific partitioning imposed on those are of interest. This derivation accommodates an ANOVA-like framework of judgment, indicative of significant dispersion of the partitioned outputs in nonmetric space. We present a statistic which under the null hypothesis of no differences between the mean of the imposed partitioning, follows an exact F-distribution allowing to obtain the consequential p-value. Three biological examples are provided and the performance of our method in relation to the cA and npA is discussed.

Symmetry group of the equiangular cubed sphere

The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.

Metric properties of Cayley graphs of alternating groups

A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

A journey through mapping space: characterising the statistical and metric properties of reduced representations of macromolecules

A mapping of a macromolecule is a prescription to construct a simplified representation of the system in which only a subset of its constituent atoms is retained. As the specific choice of the mapping affects the analysis of all-atom simulations as well as the construction of coarse-grained models, the characterisation of the mapping space has recently attracted increasing attention. We here introduce a notion of scalar product and distance between reduced representations, which allows the study of the metric and topological properties of their space in a quantitative manner. Making use of a Wang–Landau enhanced sampling algorithm, we exhaustively explore such space, and examine the qualitative features of mappings in terms of their squared norm. A one-to-one correspondence with an interacting lattice gas on a finite volume leads to the emergence of discontinuous phase transitions in mapping space, which mark the boundaries between qualitatively different reduced representations of the same molecule. Graphicabstract

Hyperspace of finite unions of convergent sequences

The symbol S(X) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(X). Then we consider some cardinal invariants on S(X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on S(X), and give a space X with a rank 2-diagonal such that S(X) does not Gδ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S(X). Finally, we pose some questions about the hyperspace S(X).

113Social gradient in health literacy among primary healthcare users in Cyprus

Background From a notion concentrated on the ability to understand health information, health literacy (HL) has become a broad concept, considered a critical determinant of community health. The HLS-EU-measures perceived HL based on a theoretical model of the concept. Methods This study explored the metric properties of the tool in a new European setting among a convenience sample of 300 healthcare users in a state General Hospital, including the construct and known-group validity by social position and health-related behaviours. Results While factor analysis did not reveal the 12 theoretical subscales, there was a meaningful 3-factor structure (52.1% variance): “access to information”, “prevention and health promotion” and “user-provider interaction”. The postulated four cognitive skills (access, understand, appraise, apply) were evident within each domain (healthcare, prevention, health promotion), and vice versa. Overall, HL was problematic in 50.7% of participants with a steep gradient by social position. Alcohol consumption and physical activity were associated with HL, but not being overweight (mean BMI 26.8, SD: 5.2) or smoking (45.6% current or past smokers), which were generally prevalent. Conclusions HLS-EU-Q47 supports at least partly the theoretical construct of HL. The social gradient supports the criterion validity of the tool and highlights an important aspect of health inequality. Key messages HLS-EU-Q47 is a valid measure of perceived health literacy There was a steep gradient in low health literacy by social position