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NTUT Academic Writing Week 3: Mathematical Analysis

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Basic Analytical Structure Analytical chapter usually includes: 1. Main topics; (“consider…”) 2. Hypothesis, condition, or definition; (“Let…”) 3. Equations; 4. Analyses and results; 5. Conclusion based on discussion of the results. Analytical chapter usually includes: 1. Main topics; (“consider…”) 2. Hypothesis, condition, or definition; (“Let…”) 3. Equations; 4. Analyses and results; 5. Conclusion based on discussion of the results.

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Examples

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Introducing Mathematical Models Basic ingredients: 1. Theoretical bases of the model (citation may be needed); 2. Analytical approaches adopted in the paper; 3. Elaboration on the model’s characteristics and application; 4. Figures rendered from the model. Two examples: 1st e.g.: the author provides: A). Background of the model; B). Elaborate on the hypotheses or conditions of the model; C). Presenting the proposed model; D). (not listed in the e.g.) application of the model in certain aspects and further discussions on the theories involved in them. Basic ingredients: 1. Theoretical bases of the model (citation may be needed); 2. Analytical approaches adopted in the paper; 3. Elaboration on the model’s characteristics and application; 4. Figures rendered from the model. Two examples: 1st e.g.: the author provides: A). Background of the model; B). Elaborate on the hypotheses or conditions of the model; C). Presenting the proposed model; D). (not listed in the e.g.) application of the model in certain aspects and further discussions on the theories involved in them.

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1 st Example

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2nd e.g.: the author: A). Introduces a basic equation; B). Details some basic hypotheses on the equation; C). Produces a model based on the application of the equation and other data; D). (not listed in the e.g.) further discussions on the features of the model. 2nd e.g.: the author: A). Introduces a basic equation; B). Details some basic hypotheses on the equation; C). Produces a model based on the application of the equation and other data; D). (not listed in the e.g.) further discussions on the features of the model.

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2 nd Example

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Punctuation in Equation There is no strict rule as to the addition of punctuation in mathematical analyses involving equations and formulas. Some believe punctuation helps clarify grammatical structure in an article that contains equations. For instance:

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Others, however, believe that adding punctuation may overcrowd the page where there already exist too many signs and symbols. (Notice that one of the aforementioned examples dispenses with punctuation.) It is preferable that you use punctuation in your papers. Be sure to check with guidelines of the journals for your prospective publication. Position of the Equation: 1. Should leave the space of (at least) one line both above and below the equation. 2. Indentation: at least be identical with (but preferably more than) the first indented line of a new paragraph. (There are journals that prefer the “justified alignment” for the equation. Better check with the guidelines of the journals.) Others, however, believe that adding punctuation may overcrowd the page where there already exist too many signs and symbols. (Notice that one of the aforementioned examples dispenses with punctuation.) It is preferable that you use punctuation in your papers. Be sure to check with guidelines of the journals for your prospective publication. Position of the Equation: 1. Should leave the space of (at least) one line both above and below the equation. 2. Indentation: at least be identical with (but preferably more than) the first indented line of a new paragraph. (There are journals that prefer the “justified alignment” for the equation. Better check with the guidelines of the journals.)

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Tenses This part of the paper is mostly dedicated to presenting a general truth that is unaffected by time.

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There are writers who use future tense to introduce a new topic in a new paragraph or section. This tense can also be used in the 1st conditional to describe or show a possible outcome.

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While engaging in model description or mathematical analyses, you (sometimes) need to refer back to your previous experiments which were conducted IN THE PAST. In this case, you need to use past tense to describe the past deeds.

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In-Class Drills: Tenses

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Noteworthy Points About Style (in Analytical Chapters) 1. Do not hesitate to use simple words, phrases, or sentence structures. Clarity outweighs everything. 2. Sometimes, using pronouns such as “we” can be more effective than expressions like “it is…,” “there is…,” or “there are.” e.g. It is assumed here that the social planner controls the rate of production of all energy resources. (stiff) We assume that the social planner controls the rate of production of all energy resources. Let us assume that the social planner controls the rate of production of all energy resources. Assume that the social planner controls the rate or production of all energy resources. 1. Do not hesitate to use simple words, phrases, or sentence structures. Clarity outweighs everything. 2. Sometimes, using pronouns such as “we” can be more effective than expressions like “it is…,” “there is…,” or “there are.” e.g. It is assumed here that the social planner controls the rate of production of all energy resources. (stiff) We assume that the social planner controls the rate of production of all energy resources. Let us assume that the social planner controls the rate of production of all energy resources. Assume that the social planner controls the rate or production of all energy resources.

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3. Misplaced modifiers: (Doers can be very important!) e.g. Before describing the adaptive controller in detail, the ideal case in which dynamic model is exactly known will be considered. (lacking modifiable object) Before describing the adaptive controller in detail, we/we will/let us consider the ideal case in which the dynamic model is exactly known. The relationship between these two variables is/will now be considered. (vague doer identity; awkward tense) Now let us consider the relationship between these two variables. Consider the relationship between these two variables. These two variables are related in the following ways. 3. Misplaced modifiers: (Doers can be very important!) e.g. Before describing the adaptive controller in detail, the ideal case in which dynamic model is exactly known will be considered. (lacking modifiable object) Before describing the adaptive controller in detail, we/we will/let us consider the ideal case in which the dynamic model is exactly known. The relationship between these two variables is/will now be considered. (vague doer identity; awkward tense) Now let us consider the relationship between these two variables. Consider the relationship between these two variables. These two variables are related in the following ways.

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Sentences That Come in Handy for Analysis Consider the case in which A is equal to B. As an example, consider the case in which A = B. Let us consider the case in which A = B. We assume that A = B. Let A be equal to B. If A = B, then we have the following equation: Given that A = B, we obtain … (your equations) This problem can be written/expressed as/as follows … (your equations) Substituting A into Equation 2, we obtain … (your equations) We will now integrate Equation 3 in order to derive the solution. A is inversely proportional to B, as shown below. The relationship between A and B is as follows: … (your equations) Consider the case in which A is equal to B. As an example, consider the case in which A = B. Let us consider the case in which A = B. We assume that A = B. Let A be equal to B. If A = B, then we have the following equation: Given that A = B, we obtain … (your equations) This problem can be written/expressed as/as follows … (your equations) Substituting A into Equation 2, we obtain … (your equations) We will now integrate Equation 3 in order to derive the solution. A is inversely proportional to B, as shown below. The relationship between A and B is as follows: … (your equations)

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