joint research with Steven Seif [20] on the celebrated Perkins semi- group, which has played a central role in semigroup **theory** since 1960, particularly as a source of examples and counterexamples. However, the first systematic study of **word**-**representable** **graphs** was not under- taken until the appearance in 2008 of the paper [18] by the author and Artem Pyatkin, which started the development of the **theory**. One of the most significant contributors to the area is Magn´ us M. Halld´ orsson. Up to date, nearly 20 papers have been written on the subject, and the core of the book [17] by the author and Vadim Lozin is devoted to the **theory** of **word**-**representable** **graphs**. It should also be mentioned that the software produced by Marc Glen [7] is often of great help in dealing with **word**-representation of **graphs**.

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The model we used is shown in full in Figure 6. There are three points of detail about the model which deserve mention. First, this model neither check **graphs** for being connected, nor for being non-isomorphic to each other. This is not easy to do very efficiently in constraints, so instead we constructed a list of all connected undirected **graphs** with no two **graphs** being isomorphic, using the program geng [24]. Second, we originally modelled an undirected graph as an input to the constraint model, which was then checked for **word**-representability. However, this proved to be very inefficient as the vast majority of the constraint modelling processes was the same for each graph. Instead, we provide the constraint model with a list of **graphs** produced by geng and insist that the solution is one of those **graphs**. This is achieved in constraints using the ‘table’ constraint, which can be propagated very efficiently [2]. As well as saving work at the modelling stage, it also provides the capability to save work at the solving stage. For example, if all **graphs** remaining for consideration contain a certain undirected edge ij, the variable u ij can be set true immediately. A major advantage of this

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The tool reads a graph and then tries to find a k-representation for k = 2, 3, 4, . . . by building the formula as presented above and then calling an SMT solver. As soon as a satisfying assignment is found, the computation stops and the resulting values are transformed to the corresponding k-uniform **word**-representation, which is returned to the user. The tool is available both for Windows (calling the SMT solver Z3) and for Linux (calling the SMT solver YICES), together with several examples. Typically, for **graphs** like the cube, the prism on the triangle, Petersen graph, and G 4 (see below), the k-uniform **word** representing

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Recently, a number of (fundamental) results on **word**-**representable** **graphs** were obtained in the literature; for example, see [1], [3], [5], [7], [9], [11], and [12]. In particular, Halld´ orsson et al. [7] have shown that a graph is **word**- **representable** if and only if it admits a semi-transitive orientation (to be de- fined in Section 2), which, among other important corollaries, implies that all 3-colorable **graphs** are **word**-**representable**. The **theory** of **word**-**representable** **graphs** is the main subject of the upcoming book [8].

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Related work. The notion of directed **word**-**representable** **graphs** was in- troduced in [13] to obtain asymptotic bounds on the free spectrum of the widely-studied Perkins semigroup, which has played central role in semi- group **theory** since 1960, particularly as a source of examples and coun- terexamples. In [12], numerous properties of **word**-**representable** **graphs** were derived and several types of **word**-**representable** and non-**word**-**representable** **graphs** pinpointed. Some open questions from [12] were resolved recently in [7], including the representability of the Petersen graph.

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In this paper we provide an alternative, purely algebraic, proof of this result. We will define the variety of lower semilattice-ordered residuated semigroups using finitely many equations. The subclass of **representable** algebras is given by the isomorphs of families of binary relations. Using a step-by-step construction we show that the free algebra of the variety of lower semilattice-ordered residuated semigroups is **representable**. On the other hand, there might be algebras in this variety that are not **representable**; we leave this as an open problem. Hopefully the technique we use for the representation of the free algebra could be used in other cases as well when

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below shows a two-dimensional representation of all 21 languages obtained from averaging the BBA and CCA distance matrices in the All21 setup, together with a k-means cluster assignment for k = 6. We note a grouping together of es, pt, fr, en, it; nl, da, de, sv; fi, et; ro, bg, el; hu, pl, cs, sk, sl; and lt, lv. In particular, {es, pt, fr, it, en} appear to form a homogeneous group with, consequently, similar semantic associations, as captured by **word** embeddings. Observing that fi is relatively similar to sv, which is at odds with genealogical/structural language classifications, we test another question, namely, whether the resulting semantic distance matrix is more similar to a distance matrix based on genealogical/structural relationships or to a distance matrix based on geographic relations. To this end, we determine the degree of structural similarity between two languages as the number of agreeing features (a feature is, e.g., Num- ber of Cases) in the WALS 7 database of structural properties of languages divided by the number of total

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Poly-constructionism (at least as construed by me) is also a prominent feature of the ethos of work in WG, rather than a defining characteristic of the **theory**, but unlike im plementationism it does in part characterize the spirit of the present work. WG’s ‘poly constructionist’ trait means, to generalize, that WG has a goal of rather strict observat ional adequacy: the method is more to come up with an analysis for all constructions and then refine the result, rather than refine an analysis for a select few constructions and then seek to apply the result to the constructions thus far neglected. Thus, in WG a highly stipulative analysis of a construction hitherto not analysed in WG would count as progress because it forms a basis for further refinements. An example of this would be Hudson (1990)’s analysis of prepositional passives, which establishes that WG can cope with the construction but, it would be fair to say, lacks the elegance of the analysis of other areas of grammar, such as extraction. I emphasize this point (at greater length than Hudson 1990 does) because it contributes to the ethos of if not WG itself then W G ’s theoreticians, when compared to other major formalist theories. In this thesis there are no analyses I’d be content to describe as highly stipulative, but there is an in terest in grammatical detail, which is a reflection of the same ethos.

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Let LE,F be the space of all bounded linear operators from E into F and BE* the unit ball of E*, the dual of E The completion of the injective tensor product of E and F is denoted by E F[r]

above discussion knowing whether (Ref(Res), SAT ∗ ) is **representable** in resolu- tion would answer this question. Atserias and Bonet [1] proved that resolution does not have the reflection property. By Proposition 10 this means that the disjointness of (Ref(Res), SAT ∗ ) is not provable in resolution with respect to the standard representation. However, we cannot exclude the possibility that we have short resolution proofs of the disjointness of (Ref(Res), SAT ∗ ) with respect to some other representation. At least we can remark that, unless the canonical pair of resolution is p-separable, these proofs would have to be essentially non-uniform.

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Akrobotu, Kitaev and Mas´ arov´ a have shown that a triangulation of a grid graph is **word**-**representable** if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder **graphs**; indeed, there are such **word**-**representable** **graphs** with chromatic number 4. In this paper we show that **word**-representability of triangulations of grid-covered cylinder **graphs** with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel **graphs** W 5 and W 7 ).

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Graph representation of text, by its natural structure, defines relationships between graph nodes. Each graph node represents a term, which can be defined in various ways including words, sentences, and n-grams. The node connections can define the ”closeness” of terms to each other in a richer way than the ”one-hot” vector representa- tion including lexical and semantic relations, con- textual overlap, etc. A potential advantage of us- ing a directed graph model over context-dependent representations, such as word2vec, is preserving information about the **word** order in the input text. Graph representation is not new in the world of text processing (Schenker et al., 2005; Son- awane and Kulkarni, 2014). Graph representa- tions outperformed the classical vector represen- tations of text documents, such as TF-IDF, on several NLP tasks including document classifi- cation (Markov et al., 2008), text summariza- tion (Garc´ıa-Hern´andez et al., 2009), **word** sense disambiguation (Agirre and Soroa, 2009), and keyword extraction (Litvak and Last, 2008). 2.2 **Word** Embedding

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The elaborated position on the above stages shows that we first describe the available information. Then, we make an attempt to find out possibly unknown and doubtful facts or feel uncertain about certain matters. Finally, we formulate the overall purpose of the research (aim) and the research hypothesis/es and/or questions. However, this cannot be considered to be true in all situations. Subject to the type, purpose and research idea, the formulation of research hypothesis and/or questions is not considered an absolute attribute. A valid point is that when writing the **introduction**, we move from general data (i.e. context, common information on the investigated field, topic or a problem being studied) to more specific facts and figures. This is the method of substantiating the research problem thus claiming this is the issue to be solved. Some examples of the problem under investigation are presented below (Table 1).

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dia, corporations, and other powerful public actors guide the collective memory and knowledge by its very conventions of representing the world in still and in moving images. Three studies in this collection of papers on Critical Visual **Theory** address this topic area with evidence from three very different world regions, India, Israel, and the United States. The first, by Keval J. Kumar, deals with “The ‘Bollywoodization’ of Popular Indian Visual Cul- ture” from a critical perspective. Kumar starts with a concise characterization of Indian visual culture as a historical battlefield of Hollywood, Bollywood, and new “fundamentalist forces.” The author deals with the visual standards of storytelling in Bollywood cinema, in which he sees a hegemonic discourse dominating not only conventions of visual narratives in Indian movies but also in other media, such as the press, television, advertising, the worldwide web, or social media. As Kumar concludes, the dominance of Bollywood film industries and their standards in representing not only fictional worlds but also reality masks many counterhege- monic trends in film art and media practices: “Indian visual culture may have been bolly- woodized over the last two decades or so, but this development has not gone unchallenged. The resistance to Bollywood’s many attempts to take over has been led by filmmakers and other visual artists from different parts of the country as well as from the diaspora. This coun- terhegemonic movement, still in its nascent stages, offers varied and alternative works of art, which celebrate subaltern perspectives.” Kumar’s contribution does not only address the question of collective knowledge and common sense, as it is shaped and represented by collective imagery, but also focuses on collective memory, when the author considers Bolly- wood’s hegemonic visual standards in the history of Indian visual culture. “Only time will tell whether the vibrant visual culture of India, known for its diversity and openness, will survive Bollywood and the lurking threats from fascist and fundamentalist forces” (cf. Nair and Tanvir 2014).

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This paper aims to provide an **introduction** to a basic form of the Q-tensor approach to modelling liquid crystals, which has seen increased interest in recent years. The increase in interest in this type of modelling approach has been driven by investigations into the fundamental nature of defects and new applications of liquid crystals such as bistable displays and colloidal systems for which a description of defects and disorder is essential. The work in this paper is not new research, rather it is an introductory guide for anyone wishing to model a system using such a **theory**. A more complete mathematical description of this **theory**, including a description of flow effects, can be found in numerous sources but the books by Virga and Sonnet and Virga are recommended. More information can be obtained from the plethora of papers using such approaches, although a general **introduction** for the novice is lacking. The first few sections of this paper will detail the development of the Q-tensor approach for nematic liquid crystalline systems and construct the free energy and governing equations for the mesoscopic dependent variables. A number of device surface treatments are considered and theoretical boundary conditions are specified for each instance. Finally, an example of a real device is demonstrated.

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using the NLM-based graph over the NLM dataset. However, the NLM-based does not perform as well in the cross-testing scenario, i.e. when applied to the Acronym datasets. This may be due to a greater specificity of the Acronym corpus, in which the different CUIs among which the dis- ambiguation algorithm has to choose (representing extended forms of the acronyms), correspond to more specific concepts. On the other hand, terms in the NLM-WSD corpus are much more general. Hence, it is possible that some of the target CUIs of the Acronym corpus do not even appear in the graph created from NLM-related abstracts. Also, it is likely that any graph created from a large enough set of abstracts (such as the one created with acronym-based abstracts) con- tains enough information about CUIs representing the general concepts of the “NLM” dataset to perform a good disambiguation. Finally, we can observe that results obtained with the joint graph improve those obtained with simpler **graphs** for all but one of the datasets. This suggests that the combined information that can be found inside the joint graph is useful to better represent the connections between concepts and hence help to improve the overall disambiguation. We have conducted some additional experiments comparing the accuracy obtained using either the NLM- based Graph or the Joint Graph, both built with the same number of documents, and the achieved results confirm this intuition: 75.42% of accuracy of the Joint Graph against 69.12% of the NLM- based Graph for 10,000 documents, 77.45% against 71.48% for 20,000 documents, and 77.78% against 72.93% for 30,000 documents.

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We first pause to recast the Halting Problem in terms of Turing machines: Is there a procedure for deciding whether a given Turing machine eventually reaches its halt state.. The same ar[r]

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The purpose of this paper is to initiate the exploration of maximal deviations of complex random structures from their typical behavior. We introduce a model for a high-dimensional random graph process and ask analogous questions to those of Vapnik and Chervonenkis for deviations of averages: how “rich” does the process have to be so that one sees atypical behavior. In particular, we study a process of Erd˝ os-R´enyi random **graphs**. In the G(n, p) model introduced by Erd˝ os and R´enyi (1959, 1960), a graph on n vertices is obtained by connecting each pair of vertices with probability p, independently, at random. The G(n, p) model has been thoroughly studied and many of its properties are well understood—see, e.g., the monographs of Bollob´ as (2001) and Janson et al. (2000).

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DqDp e i(S+J q+Kp) h0; out|q(∞); ∞i hq(−∞); −∞|0; ini (2.28) Thus we have to evaluate the ground state wave functions in the infinite past and future. We consider two concrete simple examples of free field theories, a quantum **theory** which reduces to the quantum harmonic oscillator in the absence of interactions and a real scalar field, but before discussing them we first introduce the Lagrangian version of the path integral.

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Local syntactic **graphs** perform better than logical **graphs** in Table 12. However, when we run other experiments using Frequency of co-occurrences, local logical **graphs** gave superior performance in comparison with other contexts (Sentence context window, syntactic **graphs**). We noted two issues in the Senseval experiment. On one side, the syntactic parsing was very noisy, due to improper punctuation (see the example above) and syntactic errors in the dependency parsing. The syntactic links between the words had many erroneous or high-level underspecified dependencies. On the other side, we obtained very incomplete logical representations using our logical analyzer on this set of data. As previously said, this might indicate that the analyzer may not cover enough syntactic structures, but we also noticed that noisy syntactic relationships had a big impact over our analyzer, which looks for specific well-defined syntactic structures. Despite these two issues, local syntactic and logical **graphs** contexts raised the performance of some WSD algorithms. Improving the syntactic and logical analysis might have an impact over the accuracy of WSD. This also raises the question of the kind of corpora that should be made available to the research community which is committed to full parsing for WSD. Obtaining better quality texts is probably one of the requirements of such approaches. Further work will tackle the enhancement of the logical analyzer, but also the manual definition of logical contexts in order to avoid any impact of bad syntactic and logical analyses on the WSD and to test our logical contexts on clean and non-noisy data.

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