The early study of **reverse** **mathematics** has led to the observation that most of the theorems happen to be equivalent to five main sub- systems of second-order arithmetic that Montalban [13] called the “Big Five”. However, Ramsey’s theory provides many statements escaping this observation. Perhaps the most well-known example is Ramsey’s theorem for pairs ( RT 2 2 ). The effective analysis of Ramsey’s theorem was started by Jockusch [10]. In the framework of **reverse** mathemat- ics, Simpson (see [21]), building on Jockusch results, proved that when- ever n ≥ 3, RT n k is equivalent to ACA 0 over RCA 0 . The case of RT 2 2 has

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The plan of the paper is as follows. In Section 2 we give some background concerning **reverse** **mathematics** in general, the **reverse** **mathematics** of well-quasi-orders, and Dorais’s coding of count- able second-countable topological spaces. Section 3 covers the details of expressing the notion of Noetherian space in second-order arithmetic, both in the countable second-countable case and in the uncountable case. In this section we also show that ACA 0 proves statements (ii)-(vi) of Theorem 4.7. The reversals of these implications are proved in Section 4, using the construction mentioned in the previous paragraph.

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Abstract: Using the techniques of **reverse** **mathematics**, we characterize subsets X ⊆ [0, 1] in terms of the strength of HB(X), the Heine-Borel Theorem for the subset. We introduce W(X), formalizing the notion that the Heine-Borel Theorem for X is weak, and S(X), formalizing the notion that the theorem is strong. Using these, we can prove the following three results: RCA 0 ` W(X) → HB(X),

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We introduce several new weakenings of the anti-Specker property and explore their role in con- structive **reverse** **mathematics**, identifying implication relationships that they stand in to other notable principles. These include, but are not limited to: variations upon Brouwer’s fan theorem, certain compactness properties, and so-called zero-stability properties. We also give similar classifi- cation results for principles arising directly from Specker’s theorem itself, and present new, direct proofs of related fan-theoretic results.

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The non-constructive nature of dichotomy can also be deduced from the well-known fact that dichotomy for reals is equivalent to Bishop’s lesser limited principle of omniscience. Discussions of this can be found in Bridges and Richman [2] and in Bridges and Vˆıt¸˘a [3]. For completeness we present a version of this fact, formulated in the fashion of a **reverse** **mathematics** result.

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Ishihara [3]. In this paper, we provide a classification of Brouwer’s fan theorem. A suitable framework for carrying out constructive **reverse** **mathematics** is a function- based, intuitionistic formal system, like HA ω . In this system, if A(n) is equivalent to a quantifier-free formula, one may prove:

Abstract: In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by ‘conditionalizations’ of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of **reverse** **mathematics**, suffices for the bulk of classical **mathematics**, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see eg the introduction of Simpson [47]. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of **mathematics**, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn’s lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model.

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Even though the proof above is very similar to the one given in [5] and [20], there are differences. Working with classical logic the law of excluded middle is available and used in Simpson’s system. On the other hand the proof above uses countable choice, which is avoided in Simpson’s framework. This is also a good time to point out that in [5] it is implicitly shown that—working in classical **reverse** **mathematics**—Vitali’s covering theorem is equivalent to WWKL. Even though this indicates, that Vitali’s covering theorem is not provable in RUSS, the construction above gives an explicit counterexample to that theorem.

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2. **Mathematics** Self-Efficacy Scale (2015) : This scale developed by the researcher consists of two parts. In the first part, general beliefs of students about their confidence in learning **mathematics** are measured using 15 items. In the second part, a student’s confidence about using **mathematics** in daily life using 10 items is measured. Its reliability was found to be 0.90 (Cronbach’s Alpha) and 0.81 (Test-Retest). All items in Part I were measured on a 4-point Likert-type scale (1 = strongly disagree, 2 = disagree, 3 = agree, 4 = strongly agree). In Part II, items were measured on a 4-point Likert-type scale (1 = very confident, 2 = confident, 3 = somewhat confident and 4 = not at all confident).

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The research was conducted only at the eighth-grade level and a more comprehensive and in-depth study could be conducted by selecting a mixed method design with students at secondary and higher education levels (economics, medicine, architecture, engineering, etc.). Thus, the change in anxiety and motivation of students and the change in **mathematics** achievement after HSEE can be observed because the process of career selection begins after this exam; and therefore, HSEE is at the intersection of academic development. In the study, anxiety levels of female students towards **mathematics** lesson were found to be lower than male students; motivation levels were higher than male students. While female students were closer to the desired student profile, male students were more distant. Therefore, this is in favor of female students and requires more in-depth research for male students. Whether variables such as gender perceptions and psychological differences have an effect on male students can be investigated in a separate study. The low motivation levels of students who received pre- school education compared to their peers who did not receive pre-school education is one of the interesting results of this study. Investigating the reasons of this result is important. Finally, the relationship between **mathematics** anxiety and motivation towards **mathematics** and mathematical skills (reasoning, problem-building and problem-solving skills, etc.) can also be investigated.

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This course provides a rigorous study of the concepts of effective problem solving strategies. Strategies will be applied to various problems taken from critical areas of algebra, number concepts, geometry, probability, statistics, measurement, and logic. The scope and sequence will be formative in nature and use a discovery approach to allow students to scaffold their critical thinking skills into a mathematical problem solving rubric. Logical reasoning will be emphasized in all strategies to distinguish the importance of the process of problem solving rather than just finding the answer. Appropriate computer software and hand held technologies will be utilized. With pre-service elementary teachers in mind, this course will also integrate the pedagogy of modeling these skills to elementary **mathematics** students. Prerequisite: MATH 1314 and MATH 1350 and MATH 1351 with a C or better.

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Our findings on the relationship between the computer and **mathematics**, engineering, **mathematics** and even economics with **mathematics** shows that the traditional view of society who believe that **mathematics** is an exact science that stands alone is not up to date, and always constant indicates the opposite situation, where when one study **mathematics** then by inadvertently then that person is actually looking into other sciences. Vice versa, when one studies other disciplines such as computer science, economics and even the actual techniques he is continually linking the activities of the higher **mathematics** and the use of mathematical algorithms increasingly complex.

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various web servers. In this case every web server has its individual application area. As a result, the **reverse** proxy server may have to alter the URL’s in each web page. By that we can achieve load balancing in the network. Intrusion detection system service helps to increase the security in the cloud system by using two methods i.e. behavior based and knowledge based service. We refer to the previous section for further information. The audited data is sent to the IDS service core, which analyzes the behavior using artificial intelligence to detect deviations. It has two sub systems namely analyzer and alert systems. Analyzer system the analyzer gets audit data and examines whether a heuristic in the database is being broken, after which it sends the outcome to the IDS service. For these outcomes, IDS estimates the attack probability and if probability ratio is high then it alerts the other nodes.

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Other researchers suggest that mathematical knowledge held by an individual is either procedural or conceptual. (Heibert & Leiverve, 1986). These researchers define procedural knowledge as competence in carrying out mathematical tasks while conceptual knowledge is knowledge that is rich in connections. This classification of knowledge is similar to Skemps’ notions of types of understanding. Nickerson (1985) suggests some characteristics that show that a learner has understood which are, agreement with experts, being able to see deeper characteristics of a concept, looking for specific information quickly and the ability to see connections within several concepts. It may not be easy to have one type of understanding without the other or a certain type of knowledge without the other, but most researchers seem to prefer instruction that promote why procedures work and to show connections between concepts as this promotes deeper understanding of **mathematics** concept and future learning. Basic notions in real analysis require the learners to have deep insights about them. The concepts of differentiability and continuity are also very abstract in nature and students should have correct concept images and concept definitions. A concept definition gives precisely instances and non-instances of a concept while a concept image is an individuals’ perception of the concept. The correct concept image may be formed if the learner is given several experiences of the concept. Dreyfus & Vinner (1989) suggests that concept images are not formed by definitions alone but through experiences.

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Line weight, reverse lines: .75 Line weight, reverse lines: .5 Line weight, reverse lines: .75 File format: Vector or. Raster[r]

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9 a Historical focus of the mathematics extension programme: on including the history of mathematics in the teaching of mathematics and on integrating humanities and sciences; b Learning[r]

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Studies involving researcher observations of home practices, however, suggest variability in both the quantity and quality of mathematical interactions between adult and child. In their US study, Tudge & Doucet (2004) investigated naturally occurring mathematical activities engaged in by three year olds. Conducting observations over the course of a single week in such a way as to cover the equivalent of a complete day, including time in the home, others’ homes, the childcare centre and public places, they found that many of the children in their study were ‘little or never involved in explicit **mathematics** activities, whether in the course of lessons or play with artifacts designed to encourage mathematical experiences’ (p. 34). Despite the considerable individual variation in the number of observed mathematical experiences, it was noted that parents consistently focused more on helping their children learn to read than on their development of mathematical understanding. Tudge & Doucet (2004, p. 36) argued that ‘at least with some of the parents whose children [they] studied ... there is certainly room for parents and other important people in children’s lives to enhance children’s opportunities for mathematical experiences’.

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Interview This curriculum promotes understanding and applying **mathematics** concepts. Interviewing a student allows the teacher to confi rm that learning has taken place beyond simple factual recall. Discussion allows a student to display an ability to use information and clarify understanding. Interviews may be a brief discussion between teacher and student or they may be more extensive. Such conferences allow students to be proactive in displaying understanding. It is helpful for students to know which criteria will be used to assess formal interviews. This assessment technique provides an opportunity to students whose verbal presentation skills are stronger than their written skills.

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Our work is connected with the results in artificial intelligence (Garrido, 2017). An important subject in the artificial intelligence is the automated (or mechanical) theorem proving, and, in particular, mechanical geometry theorem proving (Chou, 1988). The development of this last area has showed the evidence that in order to carry out proofs of geometry theorems mechanically, we have to strictly follow some rules and axioms (Chou, 1988). This is the main goal of our article, to show how important it is in **mathematics**, not only in artificial intelligence, to strictly follow axioms, definitions and theorems, that is, to read them semantically.

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