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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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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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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not always guarantee to be lower bounds of υ. In other words, the minimum **word** length estimated from υ f , υ 1 or υ s may not always maintain stability. The measure υ r is not surely a lower bound of υ either, because υ r only provides a statistical **word** length guaranteeing stability with probability no less than 0 . 9777. The measure υ l based on l 1 **theory** [11] is a lower bound of υ . However, due to the lack of efficient **computational** tool for l 1 **theory**, costly numerical methods have to be used to solve the non-convex problem of maximizing υ l in order to obtain an optimal realization. Structured singular value (SSV) analysis [12],[13] is an important approach of studying stability robustness and linear matrix inequality (LMI) tech- niques are powerful **computational** tools for SSV analysis. We propose an SSV-based FWL stability measure υ µ , which is guaranteed to be a lower bound of υ. The optimal realization problem of optimizing υ µ can be easily solved using LMI toolboxes of MATLAB. A numerical example is given to illustrate the proposed design method.

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This paper presents a novel method for understanding and **solving** a general class of arithmetic **word** prob- lems. Our approach can solve all **problems** whose so- lution can be expressed by a read-once arithmetic ex- pression, where each quantity from the problem text appears at most once in the expression. We develop a novel theoretical framework, centered around the no- tion of monotone expression trees, and showed how this representation can be used to get a unique decom- position of the problem. This **theory** naturally leads to a **computational** solution that we have shown to uniquely determine the solution - determine the arithmetic oper- ation between any two quantities identified in the text. This **theory** underlies our algorithmic solution - we de- velop classifiers and a constrained inference approach that exploits redundancy in the information, and show that this yields strong performance on several bench- mark collections. In particular, our approach achieves state of the art performance on two publicly available arithmetic problem datasets and can support natural generalizations. Specifically, our approach performs competitively on multistep **problems**, even when it has never observed the particular problem type before.

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Since 2001, as an Arab Muslim, a lot of negativity has been channeled at our community and our students. Lack of research for our particular culture is disheartening. As I research this topic, very few studies, if any, have considered the Arabic student population. Most of the available research has focused on the effect of language on minority groups’ problem **solving** skills, mainly Hispanics. In my role as a math educator, I have chosen to focus my dissertation on Arab students in hopes that my research can fill part of the gap present in the current literature and ignite the interest of more researchers to listen to the voices of the Arabic students and be able to better address their academic needs. I also hope to provide essential information to better educate teachers in both Islamic and non-Islamic schools about the important role language plays in students’ overall academic development and the need to use their culture and language background to the students’ advantage.

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Examples of such approaches include creation of **word** lists for targeted vocabulary instruction at various grade levels that were compiled by educa- tional experts, such as Nation (1993) or Biemiller (2008). Such **word** difficulty assignments are also implicitly present in some readability formulas that estimate difficulty of texts, such as Lexiles (Stenner, 1996), which include a lexical difficulty component based on the frequency of occurrence of words in a representative corpus, on the as- sumption that **word** difficulty is inversely correlat- ed to corpus frequency. Additionally, research in psycholinguistics has attempted to outline and measure psycholinguistic dimensions of words such as age-of-acquisition and familiarity, which aim to track when certain words become known and how familiar they appear to an average per- son.

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step problem has more than two constant quanti- ties with signs of “+1” or “−1”. We report accu- racy and F 1 score in Table 2. According to em- pirical results illustrated in Table 2, our approach is able to give more accurate answers to multi-step **problems**, while the accuracy of single-step prob- lems is lower. On the other hand, three models have similar patterns in terms of performance for three types of signs. The F1 scores for signs of “+1” and “−1” are higher than scores of “0”. Af- ter examining outputs, we found that problem texts of single-step **problems** often contain more than two constant quantities, among which only two of them are supposed to be labeled as “+1” or “−1” and the rest should be tagged as “0”. However, in- correctly labeling an irrelevant quantity with “+1” or “−1” leads to wrong solutions to single-step **problems**. This also reveals that one main chal- lenge for automatically **solving** arithmetic **word** **problems** is to recognize the irrelevant quantities. Failures in identifying irrelevant information may due to implicit information of problem text or the external tool issues.

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We can distinguish between the SLU systems that work with the 1-best transcription and those that take a representation of the n-best (Hakkani-T¨ur et al., 2006; Tur et al., 2002). The use of a **word** graph as the input of the SLU module makes this task more difficult, as the search space becomes larger. On the other hand, the advantage of using them is that there is more information that could help to find the cor- rect semantic interpretation, rather than just taking the best sentence given by the ASR.

Although almost all **graphs** are non-**representable** (as discussed in [1]) and even though a criteria in terms of semi-transitive orientations is given in [5] for a graph to be **representable**, essentially only two explicit construc- tions of non-**representable** **graphs** are known. Apart from the so-called graph whose non-representability is proved in [2] in connection with **solving** an open problem in [1], the known constructions of non-re- presentable **graphs** can be described as follows. Note that the property of being **representable** is hereditary, i.e., it is inherited by all induced subgraphs, thus adding addi- tional nodes to a non-**representable** graph and connecting them in an arbitrary way to the original nodes will also result in a non-**representable** graph.

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Since the trial of 10 resulted in an answer that is too low, we should increase the number in the next trial. Pay close attention to the result of each trial. Each result helps determine the next trial as you narrow down the possible trials to reach the answer. Note: As students get more experience with using the 5-D process, they learn to make better-educated trials from one step to the next to solve **problems** quickly or to establish the pattern they need to write an equation.

With the advent of computer graphics, Bernstein polyno- mial restricted to the interval x ∈ [ ] 0 1 , becomes important in the form of Bezier curves [11,12]. Bernstein polynomials have many constructive properties such as the positivity, the con- tinuity, recursive relation, symmetry and unity partition of the basis set over the interval. For this reason, Bernstein opera- tional matrix method is a new and rising area in applied mathematical research which has gained considerable atten- tion in dealing with differential equations. Optimal stability of the Bernstein basis was discussed by Farouki and Goodman [13]. Bhatta and Bhatti [14] have been used modified Bern- stein polynomials for **solving** Korteweg-de Vries (KdV) equation. Chakrabarti and Martha [15] described a method for Fredholm integral equations. Bhattacharya and Mandal [16] presented Bernstein polynomials method for Volterra integral equa- tions. Yousefi and Behroozifar [17] found the operational matrices of integration and product of B-polynomials. The same authors [18] introduced the Bernstein operational matrix method (BOMM) for **solving** the parabolic type partial differ- ential equations (PDEs). Isik et al. [19] have demonstrated a new method to solve high order linear differential equations with initial and boundary conditions. Ordokhani et al. [20] intro- duced Bernstein polynomial for **solving** differential equations. Singh et al. [21] established the Bernstein operational matrix of integration for **solving** differential equations. Doha et al. [22,23] have implemented and proved new formulas about de- rivatives and integrals of Bernstein polynomials and **solving** high even-order differential equations by Bernstein polyno- mial based method. Yousefi et al. [24] described the Ritz- Galerkin method for **solving** an initial boundary value problem

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An intuitive approach to learning **word**-object mappings is to imagine a pool of words and a pool of objects. **Word** learning then consists in establishing links between an element in the **word** pool and an element in the object pool (see Figure 3). Pioneered by Miikkulainen (1993, 1997), this approach has been directly instantiated in a number of models based on self-organizing feature maps where units on one map become linked to units on the other through Hebbian learning and has since been adopted by others (e.g., Mayor & Plunkett, 2010). The most advanced developmental **word** learning model based on linked feature maps so far is DEVLEX (Li, Farkas, & MacWhinney, 2004) and its extension DEVLEX II (Li, Zhao, & Mac Whinney, 2007). The DEVLEX models explored the effects of the detailed statistical properties of the input heard by children on their lexical development. DEVLEX II consisted of three linked SOMs. A phonological map received **word** forms that were based on phonetic feature vector representations. A semantic map contained semantic concepts derived from large corpora of language. Finally, an output sequence map learned to generate sequences of phonemes to produce words. The model was trained on **word**- object pairings so that representations formed on the respective maps. In parallel, links between the maps were trained with Hebbian learning to strengthen for co- occurring words, semantic concepts and phoneme sequences.

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32 properties are determined by the information provided. In the case of the random network we need to specify the number of nodes and the probability for joining pairs of nodes. As we have seen in the previous section most of the structural properties of these networks are determined by this information. In contrast, to describe the structure of one of the networks representing a real-world system we need an awful amount of information, such as: number of nodes and links, degree distribution, degree-degree correlation, diameter, clustering, presence of communities, patterns of communicability, and other properties that we will study in this section. However, even in this case a complete description of the system is still far away. Thus, the network representation of these systems deserves the title of complex networks because their topological structures cannot be trivially described like in the cases of random or regular **graphs**. In closing, when referring to complex networks we are making implicit allusion to the topological or structural complexity of the **graphs** representing a complex system. We will consider some general topological and dynamical properties of these networks in the following sections and the reader is recommended to consult the Further Reading section at the end of this Chapter for more details and examples of applications.

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KBCO-SP model is a significant improvement with model in [24]. It reflects more realistic about knowledge of human. This model has been used Sample **Problems** as experience of human for **solving** practical **problems**. Thereby helping the inference of system is faster and simulate the way of human thinking better.

Importantly, along the way we showed implications of these results for CSN on other network connectivity prob- lems that are commonly used in PPI analysis—such as Shortest Path, Steiner Tree, Prize-Collecting Steiner Tree— when conditions are added. We showed that for each of these **problems**, we cannot guarantee (in polynomial time) a solution with a value below C − ǫ times the optimal value. These lower bounds are quite strict, in the sense that naively approximating the problem separately in every condition, and taking the union of those solutions, already gives an approximation ratio of O(C). At the same time, by relating the various condition Steiner **problems** to one another, we also obtained some positive results: the condition versions of Shortest Path and Steiner Tree admit good approxima- tions when the conditions are monotonic. Moreover, all of the condition **problems** (with the exception of Prize-Col- lecting Steiner Tree) can be solved using a natural integer programming framework that works well in practice. Proofs of main theorems

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South African learners struggle to achieve in both international and national Mathematics assessments. This has inevitably become a serious concern to many South Africans and people in the education arena. An algebraic **word** problem holds high preference among the topics and determines success in Mathematics, yet it remains a challenge to learners. Previous studies show there is a connection between learners’ low performance in Mathematics and errors they commit. In addition, others relate this low performance to English language inproficiency. This has encouraged the researcher to investigate the errors Grade 11 learners make when they solve algebraic **word** **problems**. The researcher used a sequential explanatory mixed approach to investigate Grade 11 learners from Gauteng, South Africa when they solve algebraic **word** **problems**. Accordingly, a convenient sampling helped to select three schools, and purposive sampling to choose the learners. In this study, the researcher employed a quantitative analysis by conducting a test named MSWPT with 150 learners. In addition, the researcher used qualitative analyses by conducting the Newman (1977) interview format with 8 learners to find out areas where errors are made and what kind of errors they are. Findings discovered that 90 learners demonstrated unfitness due to poor linguistic proficiency, while the remaining 60 learners fall into three main categories, namely those who benefitted from researcher unpacking of meaning; those who lack transition skills from arithmetic to algebra; and those who lack comprehension and calculation knowledge. Conclusively, the researcher found linguistic, comprehension, semantic and calculation errors. The reasons learners make these errors are due to (i) a lack of sufficient proficiency in English and algebraic terminology (ii) the gap between arithmetic and algebra. Keywords: Algebra, Algebraic **word**, Problem, Error, MSWPT Mathematics Strategic **Word** Problem

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The basic difficulty in translating the algorithm R onto a real-time Turing machine is in dealing with backtracking. Any time a reduction rule is applied to reduce a **word**, it is necessary afterwards to rescan a portion of the **word** starting just before the point at which the substitution took place, since a reducible subword may have been introduced which consists of some symbols before the substitution point and some of the substituted symbols. A single reduction may provoke several backtracks. As we backtrack we need to stack the backtracked symbols, and to continue to read from the input tape while we process the stacked symbols. We need to control the speed at which we process stacked input in relation to our reading speed so that we do not run out of symbols to read from the input tape prematurely during processing. The function f controls the total extent of backtracking, in relation to unread input from a **word** representing the identity, and so it is the properties of f which allow us to relate our processing and reading speeds so that the algorithm works in real time.

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