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As usually presented, octagons of opposition are rather complex objects and can be difficult to assimilate at a glance. We show how, under suitable conditions that are satisfied by most historical examples, different display conventions can simplify the diagrams, making them easier for readers to grasp without the loss of information. Moreover, those conditions help reveal the conceptual structure behind the visual display. [ABSTRACT FROM AUTHOR]

Today's typical application of geometric morphometrics to a quantitative comparison of organismal anatomies begins by standardizing samples of homologously labelled point configurations for location, orientation, and scale, and then renders the ensuing comparisons graphically by thin-plate spline as applied to group averages, principal components, regression predictions, or canonical variates. The scale-standardization step has recently come under criticism as unnecessary and indeed inappropriate, at least for growth studies. This essay argues for a similar rethinking of the centering and rotation, and then the replacement of the thin-plate spline interpolant of the resulting configurations by a different strategy that leaves unexplained residuals at every landmark individually in order to simplify the interpretation of the displayed grid as a whole, the "transformation grid" that has been highlighted as the true underlying topic ever since D'Arcy Thompson's celebrated exposition of 1917. For analyses of comparisons involving gradients at large geometric scale, this paper argues for replacement of all three of the Procrustes conventions by a version of my two-point registration of 1986 [originally Galton's of 1907 (Nature 76:617–618, 1907)]. The choice of the two points interacts with another non-Procrustes concern, interpretability of the grid lines of a coordinate system deformed according to a fitted polynomial trend rather than an interpolating thin-plate spline. The paper works two examples using previously published midsagittal cranial data; there result new findings pertinent to the interpretation of both of these classic data sets. A concluding discussion suggests that the current toolkit of geometric morphometrics, centered on Procrustes shape coordinates and thin-plate splines, is too restricted to suit many of the interpretive purposes of evolutionary and developmental biology. [ABSTRACT FROM AUTHOR]

As usually presented, octagons of opposition are rather complex objects and can be difficult to assimilate at a glance. We show how, under suitable conditions that are satisfied by most historical examples, different display conventions can simplify the diagrams, making them easier for readers to grasp without the loss of information. Moreover, those conditions help reveal the conceptual structure behind the visual display.

Polygons are an important topic in geometry. We propose and analyse activities for the exploration of their properties, with extensive usage of a dynamical geometry system. They are based on the analysis of monuments, connecting students' cultural backgrounds with their mathematical learning. Moreover, two opposite directions for work are presented: an analytic way beginning from the monument and exploring its geometric features, and a more synthetic way, using the output of the analysis to build a model with the software. We show how these STEAM activities, based on existing monuments of various shapes, enhance the 4 C's of twenty-first century skills. [ABSTRACT FROM AUTHOR]

Formalizing categorical propositions of traditional logic in the language of quantifiers and propositional functions is no straightforward matter, especially when modalities get involved. Starting with the formulas for non-modal categoricals, we consider various ways of modalizing the formulas and semantic criteria of their evaluation that can be derived from Buridan. In addition to the logical relations included in the octagon of divided modal propositions, three interrelated aspects are taken into account—existential import, sensitivity to ampliation of terms in modal contexts, and quantification over possibilia. We end by suggesting a representation of Buridan's divided modal propositions that relies on the use of actualist quantification over variable domains. The formulas adequately capture the truth conditions given by Buridan, and they preserve all relations of the octagon, as well as permissible conversions in modal S5. [ABSTRACT FROM AUTHOR]

During the interview with the carpenters of Dong nationality, the research team found that there were misunderstandings about the "octagon" in the sorting of survey and mapping data of variable angle Drum Tower in the past, which means not all the octagonal roof projections in the variable angle Drum Tower are regular octagons. With the oral statements of Yang Yingqi and Lu Wenli, two inheritors of intangible cultural heritage, this paper analyzes the relationship between the standard frame and the building curve, explains that in order to create a more beautiful building curve and to solve the modeling problem of "roof corner punching edge", carpenters break through the inherent thinking that the plane graphics of the drum tower should be positive octagons, and produce the so-called "crooked octagons" technique through adjusting the "eave purlin length" or "short column spacing". This paper also compares the differences and characteristics of the construction of variable angle Drum Tower caused by these two frame design methods. The discovery of this phenomenon reflects the diversified characteristics of the drum tower construction technology and the personal wisdom of the ink master. It also highlights that the oral history method is an important research means to establish the correlation between craftsman, architecture and technique. [ABSTRACT FROM AUTHOR]

A matrix manipulation new to the quantitative study of develomental stability reveals unexpected morphometric patterns in a classic data set of landmark-based calvarial growth. There are implications for evolutionary studies. Among organismal biology's fundamental postulates is the assumption that most aspects of any higher animal's growth trajectories are dynamically stable, resilient against the types of small but functionally pertinent transient perturbations that may have originated in genotype, morphogenesis, or ecophenotypy. We need an operationalization of this axiom for landmark data sets arising from longitudinal data designs. The present paper introduces a multivariate approach toward that goal: a method for identification and interpretation of patterns of dynamical stability in longitudinally collected landmark data. The new method is based in an application of eigenanalysis unfamiliar to most organismal biologists: analysis of a covariance matrix of Boas coordinates (Procrustes coordinates without the size standardization) against their changes over time. These eigenanalyses may yield complex eigenvalues and eigenvectors (terms involving i = - 1 ); the paper carefully explains how these are to be scattered, gridded, and interpreted by their real and imaginary canonical vectors. For the Vilmann neurocranial octagons, the classic morphometric data set used as the running example here, there result new empirical findings that offer a pattern analysis of the ways perturbations of growth are attenuated or otherwise modified over the course of developmental time. The main finding, dominance of a generalized version of dynamical stability (negative autoregressions, as announced by the negative real parts of their eigenvalues, often combined with shearing and rotation in a helpful canonical plane), is surprising in its strength and consistency. A closing discussion explores some implications of this novel pattern analysis of growth regulation. It differs in many respects from the usual way covariance matrices are wielded in geometric morphometrics, differences relevant to a variety of study designs for comparisons of development across species. [ABSTRACT FROM AUTHOR]

Copyright of Revista Nutrición Clínica y Dietética Hospitalaria is the property of Sociedad Espanola de Dietetica y Ciencias de la Alimentacion and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article attempts to deal with two sets of logical relations, namely Aristotelian relations and duality relations. Through ciassical diagrammatic representations assigned to them, we get the Aristotelian square of oppositions and the duality square which seem to have a sort of isomorphism. As a matter of fact, many interesting Aristotelian squares turn out to also be duality squares, and vice versa. The alm of this article, however is to show that the two sets of relations in question are neither equivalent nor isomorphic. They not only have a variety of different logical properties, but are also mutually independent both in essence and conceptually. By addmg more formulas, the diagrams exhibiting the logical relations ieetween these formulas became more complicated and the differences between the Aristotelian diagrams and the duality diagrams have become more perspticuous. One of these complicated diagrams that has been chosen for this article is the octagon provided by Buridan, the great 14lh century logician. This octagon is both an extension of the opposition square and the duality square. As a result, the nonnsomorphism between Anstotclian reiations and duality relations embodied n this one and single representation is more easily and clearly perceived. [ABSTRACT FROM AUTHOR]

Based on high symmetric structure, we propose the arbitrary-input and ultra-compact 1 × 2 and 1 × 3 power splitters by utilizing inverse design method. These devices can realize the functionality of power splitting, when the optical field is launched from arbitrary port. The shapes of their structures are 3.8 μm-wide regular hexagon and 4.0 μm-wide regular octagon, respectively. By utilizing 3D fine difference time domain solutions, the simulated results indicate that the excess loss of the 1 × 2 power splitter is less than 1.5 dB from 1,500 to 1,600 nm, and the excess loss and crosstalk of the 1 × 3 power splitter are less than 1.9 dB and lower than − 15.5 dB over 100 nm bandwidth at the centered wavelength of 1,550 nm respectively. In addition, the tolerances to fabrication errors are also investigated. [ABSTRACT FROM AUTHOR]

Nanotechnology has played an influential role in improving the energetic content without subsiding the performance of high energy materials in the current era. In this work, HMX (octahydro-1,3,5,7-tetranitro- 1,3,5,7-tetrazocine) nanoparticles were prepared by sonochemically assisted solvent-antisolvent spray technique focussing the reduction in its size so as to improve its energetic properties. In order to fabricate nano HMX various parameters such as different solvents and temperature were investigated. Sonication is one of the strategies recently explored in this regard; so time dependent study of sonication using probe sonicator was performed. It has been postulated that bubble formed during sonication when collapses generate high temperature and many nucleation sites which leads to the formation of uniform spherical particles with small size and fast transition phase. XRD studies depicted phase transformation from a to b as a result of sonication. The TEM images revealed that the rise in the sonication time resulted into decrease in the particle size from 300 to 10 nm. Differential scanning calorimetry (DSC) was employed to determine the heat release of the samples and enhancement in the heat release with the decrease in the particle size. A decrease in the spark sensitivity was observed from 2J (regular HMX) to 50 mJ (nano HMX). [ABSTRACT FROM AUTHOR]

Copyright of Revista Nutrición Clínica y Dietética Hospitalaria is the property of Sociedad Espanola de Dietetica y Ciencias de la Alimentacion and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations into account, then the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework. [ABSTRACT FROM AUTHOR]

The octagon abstract domain is a widely used numeric abstract domain expressing relational information between variables whilst being both computationally efficient and simple to implement. Each element of the domain is a system of constraints where each constraint takes the restricted form ± x i ± x j ≤ c . A key family of operations for the octagon domain are closure algorithms, which check satisfiability and provide a normal form for octagonal constraint systems. We present new quadratic incremental algorithms for closure, strong closure and integer closure and proofs of their correctness. We highlight the benefits and measure the performance of these new algorithms. [ABSTRACT FROM AUTHOR]

It is known that every three dimensional Gorenstein toric singularity has a crepant resolution. Although it is not unique, all crepant resolutions are connected by repeating the operation "flop". On the other hand, this singularity also has a non-commutative crepant resolution (NCCR) which is constructed from a consistent dimer model. Such an NCCR is given as the endomorphism ring of a certain module which we call splitting maximal modifying module. In this article, we show that all splitting maximal modifying modules are connected by repeating the operation "mutation" of splitting maximal modifying modules for the case of toric singularities associated with reflexive polygons. [ABSTRACT FROM AUTHOR]

Abstract Let θ 3 , 4 denote the graph consisting of three internally disjoint paths of four edges with the same pair of endpoints. In this paper, we give a lower bound of order n 5 / 4 on the greatest number of edges of any n -vertex θ 3 , 4 -free graph, matching an earlier upper bound by Faudree and Simonovits up to an absolute constant factor. The construction is algebraic in nature, arising from equations over finite fields, and is perhaps some evidence that the Turán Number for the octagon is also of order n 5 / 4. [ABSTRACT FROM AUTHOR]

The article discusses Zagreb, Croatia from a mathematician's perspective. It mentions that octagon is found in baptisteries and churches for its spiritual connotations. It also reveals economic reasons to build octagonal buildings such as octagons admitting more natural light and allowing a panoramic view.

As modern video coding standards are not designed to code panoramic videos, the pixels on the sphere need to be sampled onto a rectangle, and this process is called mapping. The mapping schemes generally include two steps: sampling points on sphere and arranging points into one compression-friendly rectangular frame. Traditional mapping schemes including equirectangular and cubic mapping have high sampling density on some sampling areas, which result in wasted pixels. In this letter, we propose a novel octagonal mapping scheme, which can decrease the oversampling areas and arrange points into an octagon. The octagon can be reshaped and rearranged into a rectangle before encoding. Experimental results demonstrate that the proposed mapping scheme saves more bitrates, compared to the existing mapping schemes. [ABSTRACT FROM AUTHOR]

Tree-like octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. An octagonal chain is a cata-condensed octagonal system with no branchings. In this paper, the extremal octagonal chains with n octagons having the minimum and maximum coefficients sum of the permanental polynomial are identified, respectively. [ABSTRACT FROM AUTHOR]

The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed. [ABSTRACT FROM AUTHOR]

Copyright of Revista Española de Filosofía Medieval is the property of Sociedad Espanola de Filosofia Medieval and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We propose a new numerical abstract domain for inferring linear invariants based on parallelotopes. The domain may encode any linear constraint, as the polyhedra abstract domain, while maintaining the efficiency of weakly relational abstract domains, such as intervals and octagons. We provide the full set of abstract operators, define a reduced product with intervals and present an experimental comparison with polyhedra and octagons. According to these experiments, the reduced product we propose is much more precise than both polyhedra and octagons in inferring interval constraints. [ABSTRACT FROM AUTHOR]

Perceptual grouping has been extensively studied, but some areas are still unexplored-in particular, the figural organizations that emerge when bundles of intersecting lines are drawn. Here, we will describe some figural organizations that emerge after the superimposition of bundles of lines forming the profile of regular triangular waves. By manipulating the lines' jaggedness and junction geometry (regular or irregular X junction) we could generate the following organizations: (a) a grid, or a figural configuration in which both the lines and closed contours are perceived, (b) a figure-ground organization composed of figures separated by portions of the background, and (c) a corrugated surface appearing as a multifaceted polyhedral shell crossed by ridges and valleys. An experiment was conducted with the aim at testing the role of the good-continuation and closure Gestalt factors. Good continuation prevails when the lines are straight or close to straightness, but its role is questionable in the appearance of a corrugated surface. This perceptual organization occurs despite the violation of the good-continuation rule and consists of a structure of such complexity so as to challenge algorithms of computer vision and stimulate a deeper understanding of the perceptual interpretation of groups of lines. [ABSTRACT FROM AUTHOR]

We investigate common developments that can fold into plural incongruent orthogonal boxes. Recently, it was shown that there are infinitely many orthogonal polygons that fold into three boxes of different size. However, the smallest one that folds into three boxes consists of 532 unit squares. From the necessary condition, the smallest possible surface area that can fold into two boxes is 22, and the smallest possible surface area for three different boxes is 46. For the area 22, it has been shown that there are 2,263 common developments of two boxes by exhaustive search. However, the area 46 is too huge to search. In this paper, we focus on the polygons of area 30, which is the second smallest area of two boxes that admits to fold into two boxes of size 1 × 1 × 7 and 1 × 3 × 3 . Moreover, when we fold along diagonal lines of rectangles of size 1 × 2 , this area 30 may admit to fold into a box of size 5 × 5 × 5 . The results are summarized as follows. There exist 1,080 common developments of two boxes of size 1 × 1 × 7 and 1 × 3 × 3 . Among them, there are nine common developments of three boxes of size 1 × 1 × 7 , 1 × 3 × 3 , and 5 × 5 × 5 . Interestingly, one of nine such polygons folds into three different boxes 1 × 1 × 7 , 1 × 3 × 3 , and 5 × 5 × 5 in four different ways. [ABSTRACT FROM AUTHOR]

In recent work, we constructed a new near octagon $$\mathcal {G}$$ from certain involutions of the finite simple group $$G_2(4)$$ and showed a correspondence between the Suzuki tower of finite simple groups, $$L_3(2)< U_3(3)< J_2< G_2(4) < Suz$$ , and the tower of near polygons, $$\mathrm {H}(2,1) \subset \mathrm {H}(2)^D \subset \mathsf {HJ} \subset \mathcal {G}$$ . Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon $$\mathsf {HJ}$$ is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters (2, 4; 0, 3), but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, $$\mathrm {H}(2)^D$$ . We also give a complete classification of near hexagons of order (2, 2) and use it to prove the uniqueness result for $$\mathrm {H}(2)^D$$ . [ABSTRACT FROM AUTHOR]

The only known examples of finite generalised hexagons and octagons arise from the finite almost simple groups of Lie type G 2 , D 4 3 , and F 4 2 . These groups act transitively on flags, primitively on points, and primitively on lines. The best converse result prior to the writing of this paper was that of Schneider and Van Maldeghem (2008): if a group G acts flag-transitively, point-primitively, and line-primitively on a finite generalised hexagon or octagon, then G is an almost simple group of Lie type. We strengthen this result by showing that the same conclusion holds under the sole assumption of point-primitivity. [ABSTRACT FROM AUTHOR]

One of the large jali screens adorning the mausoleum of Muhammad Ghaus in Gwalior (N India), built in 1565, contains panels composed of disordered composite octagons and Salomon stars. These elements show a rotational disorder with some interdependence. Analysis of these partially disordered patterns with rotatable configurations of the above elements suggested that they may be approximants of a quasiperiodic octagonal tiling based on a new type of composite tiles. Comparisons with the Amman's quasiperiodic tiling were made. Instances of similar or related periodic ornamental patterns at other northern Indian localities are analyzed as well. [ABSTRACT FROM AUTHOR]

This paper deals with the design of a microstrip series power divider for circularly polarized sequential rotational antenna array. The theoretical description of the design is firstly proposed, comprising the cases of nonuniform weighted antenna arrays. A more flexible open octagon shape instead of the classical open ring is suggested, highlighting benefits in the case of nonuniform power distribution. A design example of an ultra-high frequency (UHF) band 4×4 sequentially rotated Tschebischeff antenna array finally demonstrates the effectiveness of the proposed implementation. [ABSTRACT FROM AUTHOR]

Let d*(1,w)² = R² with the octagonal norm of weight w. It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on d*(1,w)². We also show that the unit sphere of the space of symmetric bilinear forms on d*(1,w)² is the disjoint union of the sets of smooth points, extreme points and the set A as follows: ... [ABSTRACT FROM AUTHOR]

The present paper discusses design and development of a force transducer of octagonal ring shape, which is a modified version of ring shaped force transducer. Nominal capacity of the force transducer designed and developed is 25 kN. This force transducer has been investigated for its utilization as force transfer standard for static force measurement and related applications. This force transducer is studied using dial gauge and strain gauges for its metrological characterization according to the calibration procedure based on standard ISO 376:2011/IS 4169:1988 (reaffirmed 2003) and is found to have good metrological results as compared to the existing ring shaped force transducers. [ABSTRACT FROM AUTHOR]

We give a simple formula for the looping rate of a loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice, the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental). [ABSTRACT FROM AUTHOR]

This paper presents a novel multiple splits octagon-shaped metamaterial which exhibits double left-handed frequency bands roughly from 9.25 GHz to 9.68 GHz and from 15.28 GHz to 15.36 GHz derived from the transmission characteristics by homogenization retrieval method. Two conventional microstrip patch antennas whose resonance frequency bands are located at the first left-handed frequency band and the second one respectively are designed to further investigate the influence by loading the proposed metamaterial as its superstrate. HFSS simulation numerical results indicate that the far field radiation directivity half-power bandwidth(HPBW) of the E-plane and H-plane have reduced by 64%, 63.7% and 59.4%, 59%, and the maximal gain increases from 8.251 dB to 11.024 dB and from 8.5432 dB to 11.9653 dB respectively after loading with the multiple splits octagonal metamaterial superstrate. [ABSTRACT FROM AUTHOR]

In the theory of generalized polygons, the question whether there exists a unique generalized octagon of order ( 2 , 4 ) is still open. In this paper, we show the uniqueness of such a generalized octagon under an extra assumption. We give a computer-assisted proof for the fact that the Ree–Tits octagon of order ( 2 , 4 ) is, up to isomorphism, the unique generalized octagon of order ( 2 , 4 ) containing at least one suboctagon of order ( 2 , 1 ) . [ABSTRACT FROM AUTHOR]

An-labelling of a graphGis an assignment of nonnegative integers, called labels, to the vertices ofGsuch that two adjacent vertices receive labels that differ by at leasthand those of distance two receive labels that differ by at leastk. Among the span of all-labellings ofG,is the smallest one. In this paper for the octagonal grid, we show that the upper bound foriswhen. And when, we show that. [ABSTRACT FROM AUTHOR]

Inthe present study, we investigate different types of 1D peanut-shapedfullerene polymers (PSFPs) using density functional theory to understandthe electronic states and the energetic stability of curved carbonnanomaterials. We generated 53 different models of the 1D PSFPs bymeans of the generalized Stone–Wales transformations and performedstructural optimization for each model. Band structures of the 1DPSFPs exhibit either metallic or semiconducting property accordingto the geometrical structures. We find that the energetic stabilityof the 1D PSFPs depends on the geometry: the more octagon and pentagon–octagonpairs (heptagons and hexagon–heptagon pairs) in their geometricalstructures, the more stable (unstable) the 1D PSFPs. [ABSTRACT FROM AUTHOR]

The article presents answers and questions related to polygons including necessary condition required for the sides and angles of the pentagon to form the octagon, finding the areas of the pentagon and octagon, and finding the median length.

LetKm,nandC8denote the complete bipartite graph with partite sets of cardinalities m and n and octagon, respectively. We give the necessary and sufficient conditions for the existence of aC8-decomposition ofKm,n, maximumC8-packings and minimumC8-coverings ofKm,n. [ABSTRACT FROM PUBLISHER]

The smallest known thick generalized octagon has order ( 2 , 4 ) and can be constructed from the parabolic subgroups of the Ree group F 4 2 ( 2 ) . It is not known whether this generalized octagon is unique up to isomorphism. We show that it is unique up to isomorphism among those having a point a whose stabilizer in the automorphism group both fixes setwise every line on a and contains a subgroup that is regular on the set of 1024 points at maximal distance to a . Our proof uses extensively the classification of the groups of order dividing 2 9 . [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mosaic Research is the property of Uludag University, Mosaic Research Center and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The traditional Square of Opposition consists of four sentence types. Two are universal and two particular; two are affirmative and two negative. Examples, where ‘S’ and ‘P’ designate the subject and the predicate, are: ‘every S is P’, ‘no S is P’, ‘some S is P’ and ‘some S is not P’. Taking the usual sentences of the square of opposition, quantifying over their predicates exhibits non-standard sentence forms. These sentences may be combined into non-standard Squares of Opposition (an Octagon in this case), and they reveal a new relationship not found in the usual Square. Medieval logicians termed ‘disparatae’ pairs of sentences like ‘every S is some P’ and ‘some S is every P’, which are neither subaltern nor contrary, neither contradictory nor subcontrary. Walter Redmond has designed a special language L to express the logical form of these sentences in a precise way. I will use this language to show how Squares of Opposition, standard and non-standard, form a complex network of relations which bring to light the subtleties contained in this traditional doctrine. [ABSTRACT FROM PUBLISHER]

A low-discrepancy cubic variant of β-Mn is presented exhibiting local octagonal symmetry upon projection along any of the three mutually perpendicular 〈100〉 axes. Ideal structural parameters are derived to be and for the P4132 enantiomorph. A comparison of the actual and ideal structure models of β-Mn is made in terms of the newly devised concept of geometrical discrepancy maps. Two-dimensional maps of both the geometrical star discrepancy D* and the minimal interatomic distance dmin are calculated over the combined structural parameter range and of generalized β-Mn type structures, showing that the `octagonal' variant of β-Mn is almost optimal in terms of globally minimizing D* while at the same time globally maximizing dmin. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies. [ABSTRACT FROM AUTHOR]