# Top PDF Solving computational problems in the theory of word-representable graphs ### Solving computational problems in the theory of word-representable graphs

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 . 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 . 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 ### A comprehensive introduction to the theory of word-representable graphs

joint research with Steven Seif  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  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  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  is often of great help in dealing with word-representation of graphs. ### Solving computational problems in the theory of word representable graphs

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 . 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 . 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 ### Semi-transitive orientations and word-representable graphs

Related work. The notion of directed word-representable graphs was in- troduced in  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 , numerous properties of word-representable graphs were derived and several types of word-representable and non-word-representable graphs pinpointed. Some open questions from  were resolved recently in , including the representability of the Petersen graph. ### Solving finite word length realization problems in the framework of structured singular value

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  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 , 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. ### Solving General Arithmetic Word Problems

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. ### Exploring Bilingual Arab-American Students' Performance in Solving Mathematics Word Problems in Arabic and English

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. ### Word Maturity: Computational Modeling of Word Knowledge

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. ### Natural Language Programing with Automatic Code Generation towards Solving Addition Subtraction Word Problems

Solving mathematical word problems by understanding natural language texts and by representing them in the form of equa- tions to generate the final answers has been gaining importance in recent days. At the same time, automatic code genera- tion from natural language text input (nat- ural language programming) in the field of software engineering and natural lan- guage processing (NLP) is drawing the at- tention of researchers. Representing natu- ral language texts consisting of mathemat- ical or logical information into such pro- grammable event driven scenario to find a conclusion has immense effect in auto- matic code generation in software engi- neering, e-learning education, financial re- port generation, etc. In this paper, we pro- pose a model that extracts relevant infor- mation from mathematical word problem (MWP) texts, stores them in predefined templates, models them in object oriented paradigm, and finally map into an object oriented programming (OOP) 1 language ### Quantity Tagger: A Latent Variable Sequence Labeling Approach to Solving Addition Subtraction Word Problems

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. ### A Methodology for Obtaining Concept Graphs from Word Graphs

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. ### Word Representability of Line Graphs

Although almost all graphs are non-representable (as discussed in ) and even though a criteria in terms of semi-transitive orientations is given in  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  in connection with solving an open problem in , 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. ### Math 7 - 5.3.4 to 5.3.5 solving word problems using 5d process NOTES.pdf

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. ### A computational method for solving a class of singular boundary value problems arising in science and engineering

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 . Bhatta and Bhatti  have been used modified Bern- stein polynomials for solving Korteweg-de Vries (KdV) equation. Chakrabarti and Martha  described a method for Fredholm integral equations. Bhattacharya and Mandal  presented Bernstein polynomials method for Volterra integral equa- tions. Yousefi and Behroozifar  found the operational matrices of integration and product of B-polynomials. The same authors  introduced the Bernstein operational matrix method (BOMM) for solving the parabolic type partial differ- ential equations (PDEs). Isik et al.  have demonstrated a new method to solve high order linear differential equations with initial and boundary conditions. Ordokhani et al.  intro- duced Bernstein polynomial for solving differential equations. Singh et al.  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.  described the Ritz- Galerkin method for solving an initial boundary value problem ### Computational models of word learning

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. ### Graphs and networks theory

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. ### A Reasoning Method on Knowledge Base of Computational Objects and Designing a System for Automatically Solving Plane Geometry Problems

KBCO-SP model is a significant improvement with model in . 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. ### Connectivity problems on heterogeneous graphs

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 ### An investigation grade 11 learners errors when solving algebraic word problems in Gauteng, South Africa

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 