The Projective Line Over the Finite Quotient Ring GF(2)[xx]/<x3−x>< x^{3} - x> and Quantum Entanglement I. Theoretical Background

The paper deals with the projective line over the finite factor ring $R\_{\clubsuit} \equiv$ GF(2)[$x$]/$$. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As $R\_{\clubsuit}$ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. There are nine neighbour points to any point of the line, forming three disjoint families under the reduction modulo either of two maximal ideals of the ring. Two of the families contain four points each and they swap their roles when switching from one ideal to the other; the points of the one family merge with (the image of) the point in question, while the points of the other family go in pairs into the remaining two points of the associated ordinary projective line of order two. The single point of the remaining family is sent to the reference point under both the mappings and its existence stems from a non-trivial character of the Jacobson radical, ${\cal J}\_{\clubsuit}$, of the ring. The factor ring $\widetilde{R}\_{\clubsuit} \equiv R\_{\clubsuit}/ {\cal J}\_{\clubsuit}$ is isomorphic to GF(2) $\otimes$ GF(2). The projective line over $\widetilde{R}\_{\clubsuit}$ features nine points, each of them being surrounded by four neighbour and the same number of distant points, and any two distant points share two neighbours. These remarkable ring geometries are surmised to be of relevance for modelling entangled qubit states, to be discussed in detail in Part II of the paper.Comment: 8 pages, 2 figure

An Algorithm for constructing Hjelmslev planes

Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv planes can be constructed in this way. As a corollary it is shown that all 2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective Hjelmselv planes.Comment: 15 pages. Algebraic Design Theory and Hadamard matrices, 2014, Springer Proceedings in Mathematics & Statistics 13

Hjelmslev Geometry of Mutually Unbiased Bases

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p^2 and rank r. The q vectors of a basis of H\_q correspond to the q points of a (so-called) neighbour class and the q+1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.Comment: 4 pages, 1 figure; extended list of references, figure made more illustrative and in colour; v3 - one more figure and section added, paper made easier to follow, references update

The ‘Galilean Style in Science’ and the Inconsistency of Linguistic Theorising

Chomsky’s principle of epistemological tolerance says that in theoretical linguistics contradictions between the data and the hypotheses may be temporarily tolerated in order to protect the explanatory power of the theory. The paper raises the following problem: What kinds of contradictions may be tolerated between the data and the hypotheses in theoretical linguistics? First a model of paraconsistent logic is introduced which differentiates between week and strong contradiction. As a second step, a case study is carried out which exemplifies that the principle of epistemological tolerance may be interpreted as the tolerance of week contradiction. The third step of the argumentation focuses on another case study which exemplifies that the principle of epistemological tolerance must not be interpreted as the tolerance of strong contradiction. The reason for the latter insight is the unreliability and the uncertainty of introspective data. From this finding the author draws the conclusion that it is the integration of different data types that may lead to the improvement of current theoretical linguistics and that the integration of different data types requires a novel methodology which, for the time being, is not available

The ‘Galilean Style in Science’ and the Inconsistency of Linguistic Theorising

Chomsky’s principle of epistemological tolerance says that in theoretical linguistics contradictions between the data and the hypotheses may be temporarily tolerated in order to protect the explanatory power of the theory. The paper raises the following problem: What kinds of contradictions may be tolerated between the data and the hypotheses in theoretical linguistics? First a model of paraconsistent logic is introduced which differentiates between week and strong contradiction. As a second step, a case study is carried out which exemplifies that the principle of epistemological tolerance may be interpreted as the tolerance of week contradiction. The third step of the argumentation focuses on another case study which exemplifies that the principle of epistemological tolerance must not be interpreted as the tolerance of strong contradiction. The reason for the latter insight is the unreliability and the uncertainty of introspective data. From this finding the author draws the conclusion that it is the integration of different data types that may lead to the improvement of current theoretical linguistics and that the integration of different data types requires a novel methodology which, for the time being, is not available

Rhetorical Transformations in Multimodal Advertising Texts: From General to Local Degree Zero

The use of rhetoric in advertising research has been steadily gaining momentum since the 1980’s. Coupled with an increased interest in multimodality and the multiple interactions among verbal, pictorial and auditory registers, as structural components of an ad filmic text, the hermeneutic tools furnished by traditional rhetoric have been expanded and elaborated. This paper addresses the fundamental question of how ad filmic texts assume signification from a multimodal rhetorical point of view, by engaging in a fruitful dialogue with various research streams within the wider semiotic discipline and consumer research. By critically addressing the context of analysis of a multimodal ad text in the course of the argumentation deployed by different approaches, such as Social Semiotics (Kress/Leeuwen 2001), Film Semiotics (i.e. Metz 1982, Carroll 1980, Branigan 1982), Visual Semiotics (i.e. Sonesson 2008; 2010, Eco 1972;1976;1986, Groupe " 1992), Consumer Research (i.e. Mick/McQuarrie 1999; 2004, Philips 2003, Scott 1994), the relative merits of a structuralist approach that prioritizes the distinction between local and general degree zero, as put forward by Groupe " (1992), are highlighted. Furthermore, the modes whereby rhetorical transformations are enacted are outlined, with view to deepening the conceptual tackling of degree zero of signification, while addressing its applicability to branding discourse and multimodal ad texts

The semiology of changing brand image

This article considers the attempted change to the image of an established brand by studying the semiotics within the brand’s historical advertising campaigns. The use of semiotics to study the interpretation of messages is discussed, and the link between interpretation of messages and advertising effectiveness in changing brand image is explored. The authors deconstruct advertisements of a brand to provide a model containing opposing dialectics that may aid managers by highlighting alternative symbolic messages contained in advertisements. Oncwe identified, these alternative symbolic messages may be used to help change brand image and influence advertising effectiveness. Although the study focuses upon a major brand of beer, this is an industry in which there are numerous small firms, and many of those have constrained marketing budgets, and thus need to make sure that their advertising is effective. Equally, entrepreneurial marketing is not to found only in the small firm, and the case study discusses a radical and imaginative brand repositioning of a well established product

Historical ‘signposts’ and other temporal indicators in the Czech lexicon

This article posits that the Czechs employ a great many historical markers, previously applied to other events of national importance, which help to shape collective memory and right the ‘wrongs’ of the past. It is argued that these temporal indicators share a number of clearly defined characteristics, and that their use is too systematic and calculated to be merely a function of the constraints of the lexicon. The first part of the study considers in detail questions of semantics (especially the distinction between denotation and connotation), the lexicographical sources available to the researcher, and the lexical ‘signpost’ in context, while the second part focuses on practical examples of lexical re-appropriation since 1918, with particular reference to dictionaries and the Czech National Corpus.University of Wolverhampto