Tangles, Trees and Flowers

<p>A tangle of order k in a connectivity function λ may be thought of as a "k-connected component" of λ. For a connectivity function λ and a tangle in λ of order k that satisfies a certain robustness condition, we describe a tree decomposition of λ that displays, up to a certain natural equivalence, all of the k-separations of λ that are non-trivial with respect to the tangle. In particular, for a tangle in a matroid or graph of order k that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or graph that displays, up to a certain natural equivalence, all of the k- separations of the matroid or graph that are non-trivial with respect to the tangle.</p>

Tangles, Trees and Flowers

<p>A tangle of order k in a connectivity function λ may be thought of as a "k-connected component" of λ. For a connectivity function λ and a tangle in λ of order k that satisfies a certain robustness condition, we describe a tree decomposition of λ that displays, up to a certain natural equivalence, all of the k-separations of λ that are non-trivial with respect to the tangle. In particular, for a tangle in a matroid or graph of order k that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or graph that displays, up to a certain natural equivalence, all of the k- separations of the matroid or graph that are non-trivial with respect to the tangle.</p>

Kerelevanan Optimum dalam Terjemahan Sifat Allah dalam Al-Quran ke Bahasa Inggeris: Analisis Baseer dalam Surah Al-Isra’

Al-Quran adalah teks suci yang mengandungi ciri-ciri linguistik dan retorik yang adakalanya menjangkau kemampuan aspek semantik untuk menerangkannya. Sifat Allah merupakan istilah khusus Al-Quran yang tidak dapat diterjemahkan hanya dengan melihat makna semantik semata-mata, kerana mengandungi makna khusus yang merujuk kepada sifat Ketuhanan. Dalam fenomena seperti ini, penterjemah disaran menggunakan pendekatan pragmatik yang menafsirkan makna ujaran dan menyampaikan maklumat berdasarkan konteks ayat untuk memberi pemahaman yang jelas tentang makna sebenar ujaran tersebut kepada pembaca bahasa sasaran, agar dapat mencapai tahap kerelevanan optimum. Kajian ini bertujuan mengkaji terjemahan salah satu sifat Allah dalam Al-Quran, Baseer untuk melihat sama ada terjemahannya mencapai kerelevanan optimum atau sebaliknya. Teori Relevan (TR) yang lazimnya diaplikasi dalam bidang komunikasi dijadikan landasan teori untuk menganalisis terjemahan perkataan Baseer dalam surah al-Isra’ untuk melihat bagaimana makna Baseer dinyatakan dalam teks terjemahan al-Quran yang dipilih iaitu teks terjemahan Al-Quran terkenal dalam bahasa Inggeris oleh Abdel Haleem, Pickthall dan George Sale. Kajian ini menunjukkan bahawa terdapat perbezaan dalam terjemahan sifat Baseer antara tiga teks sasaran. Sekiranya dilihat dari sudut Teori Relevan pula, kebanyakan terjemahannya tidak mencapai kerelevanan optimum, kerana makna yang diberikan oleh penterjemah tidak menyokong makna yang dimaksudkan oleh sifat Allah. This study aims to investigate the translation of one of the Devine Attributes in Quran, the word Baseer whether its translation achieves optimal relevance or otherwise. It employed a qualitative method using content analysis technique based on Relevance Theory, which is basically applied in communication field, in analyzing the translation of the word 'baseer' in surah al-Isra’ to investigate how the meaning of Baseer is rendered in in three well-known Quran translations in English language by Abdel Haleem, Pickthall and George Sale. This study shows that there are differences in the translation of Baseer among the three target texts. If viewed from the point of Relevance Theory, most of its translations do not achieve optimal relevance, because the meanings given by the translators do not support the meaning meant by the nature of Allah. This indicates that absolute equivalence could never been achieved when translating specific words of the Quran and in this study, the translation of Devine Attributes sometimes does not achieve optimal relevance. Therefore, the translator may select the closest natural equivalence of the meaning of the attribute.

On the finiteness of the number of elliptic fields with given degrees of S-units and periodic expansion of √f

For a field k of characteristic 0, up to a natural equivalence relation, it is proved that the number of nontrivial elliptic fields k(x)(f) with a periodic expansion of f ∈ k((x)), for which the corresponding elliptic curve contains a k-point of even order less or equal than 18 or k-point of odd order less or equal than 11, is finite. In case k is a quadratic extension of Q, all such fields are found.

Representations of the fundamental group and the torsor of deformations. Local study

This chapter focuses on representations of the fundamental group and the torsor of deformations. It considers the case of an affine scheme of a particular type, qualified also as small by Faltings. It introduces the notion of Dolbeault generalized representation and the companion notion of solvable Higgs module, and then constructs a natural equivalence between these two categories. It proves that this approach generalizes simultaneously Faltings' construction for small generalized representations and Hyodo's theory of p-adic variations of Hodge–Tate structures. The discussion covers the relevant notation and conventions, results on continuous cohomology of profinite groups, objects with group actions, logarithmic geometry lexicon, Faltings' almost purity theorem, Faltings extension, Galois cohomology, Fontaine p-adic infinitesimal thickenings, Higgs–Tate torsors and algebras, Dolbeault representations, and small representations. The chapter also describes the descent of small representations and applications and concludes with an analysis of Hodge–Tate representations.

Representations of the fundamental group and the torsor of deformations. An overview

This chapter provides an overview of a new approach to the p-adic Simpson correspondence, focusing on representations of the fundamental group and the torsor of deformations. The discussion covers the notation and conventions, small generalized representations, the torsor of deformations, Faltings ringed topos, and Dolbeault modules. The chapter begins with a short aside on small generalized representations in the affine case, which will be used as intermediary for the study of Dolbeault representations. It then introduces the notion of generalized Dolbeault representation for a small affine scheme and the companion notion of solvable Higgs module, and constructs a natural equivalence between these two categories. It establishes links between these notions and Faltings smallness conditions and relates this to Hyodo's theory. It also describes the Higgs–Tate algebras and concludes with an analysis of the logical links for a Higgs bundle, between smallness and solvability.

Linear equations over multiplicative groups, recurrences, and mixing III

Given an algebraic $\mathbf{Z}^{d}$-action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order $n+1$ of non-mixing. It is known that this is determined by the non-mixing sets of size $n+1$, and we show how to find these in an effective way. When the underlying characteristic is positive and $n\geq 2$, we prove that there are at most finitely many classes under a natural equivalence relation. We work out two examples, the first with five classes and the second with 134 classes.

FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

We introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

A categorical invariant of flow equivalence of shifts

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.