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Asymptotic notation properties Let f .n/ and g.n/ be

Introduction to Algorithms | 3rd Edition | ISBN: 9780262033848 | Authors: Thomas H. Cormen ISBN: 9780262033848 130

Solution for problem 3-4 Chapter 3

Introduction to Algorithms | 3rd Edition

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Introduction to Algorithms | 3rd Edition | ISBN: 9780262033848 | Authors: Thomas H. Cormen

Introduction to Algorithms | 3rd Edition

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Problem 3-4

Asymptotic notation properties Let f .n/ and g.n/ be asymptotically positive functions. Prove or disprove each of the following conjectures. a. f .n/ D O.g.n// implies g.n/ D O.f .n//. b. f .n/ C g.n/ D .min.f .n/; g.n///. c. f .n/ D O.g.n// implies lg.f .n// D O.lg.g.n///, where lg.g.n// 1 and f .n/ 1 for all sufficiently large n. d. f .n/ D O.g.n// implies 2f .n/ D O 2g.n/ . e. f .n/ D O ..f .n//2/. f. f .n/ D O.g.n// implies g.n/ D .f .n//. g. f .n/ D .f .n=2//. h. f .n/ C o.f .n// D .f .n//.

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Module 12 Notes Functions of Language Written by: Krista Anstead March 2016 Learning Outcomes • articulate the communicative and relational aspects of language. • summarize different functions of language. • be cognizant of cultural differences of linguistic meaning and language use. • employ cultural analysis in lesson, whenever applicable. Readings/References Razfar, A. & Rumenapp, J. (2014). Functions of language. In Applying linguistics in the classroom: A sociocultural approach (pp. 224-­‐246). New York: Routledge. Chapter 11 Notes Introduction • linguistic universals: common features of phonology, morphology, and syntax o viable candidates for common language functions shared between the world’s languages o based on biological traits contained within an individual • cultural universals/functional universals: based on the idea that people from around the world have similar needs, problems, and purposes • Linguistic ladder: INSERT INFO HERE FROM CHAPTER 1 o Sixth rung: pragmatics § Way we use language to perform cultural functions and fulfill personal needs Speech Acts • Belief that when we use language, we are not only “saying something” but also “doing something” • Purposes of talking: asking questions, giving information, directing others • Pieces of speech acts o Supralinguistic cues: tone, pitch, gestures o Context • Searle’s common speech acts o Representatives: commits speaker to the truth § The grass is green o Directives: speaker attempts to get the hearer to do something § Can you close the window o Commissives: commits the speaker to some future action § I promise to love and cherish you o Expressives: speaker expresses a psychological state § Thank you for your time. o Declarations: world conforms to what the speaker says § I now pronounce you husband and wife o Categorized in terms of what they accomplish, not how they are grammatically structured o Suggests that all questions are actually directives, since they are attempts to get the hearer to do something Speech events • Norms and rules that tell us how to speak in a given situation • Identifying specific norms as a genre of speech • Can extend to greetings, business meetings, and other goal-­‐oriented norm-­‐governed activities • Speech events do change as relationships change • Range of language functions is considerable, and naming them helps ELLs better understand how speech events are organized Functions of language • Common language functions o Referential § Language is used to talk about something else § Used to identify objects in the world § One of the primary uses of language § Can be verbal or nonverbal § used to establish the presence of something within social interactions o Negation § Language is used to express the absence of something § Act of rejecting a proposition, taking an opposite stance or falsifying a claim § Act of negation is common to all languages § Double negatives • Spanish uses them for making negation more unequivocal • English uses technical logic so it does NOT use double negatives o Influenced by philosophy, logic, mathematics, and science • Also used by many dialects of English § Used worldwide, however, it operates on a variety of logics § One of the fundamental ways that children discover meaning through experience o Counting/quantifying § Language is used to talk about numbers and quantities § All languages of the world have some type of quantifying system, or number system • Used to categorize, group, and analyze ideas § While the underlying purpose is universal, the symbolic system is local § Piraha people • Perhaps only known language where the existence of quantification and numeracy practices is contested • Has grammatically-­‐built in citation system • Concerned only with that which is in their immediate context, and since counting and quantifying allows for comparison across time and space, they do not perceive a need to use this § Underlying value and purpose is precision § Found in systems of measurement § Mass nouns versus count nouns • Mass: not counted as individual or pieces, nut rather are quantified in block measurements o Never plural o Must not be measured and cannot be counted individually o Example: sugar, rice, milk o Used with: number, cup, some, little • Count nouns: can be counted individually o Can be pluralized o Used with: number, couple, a, few, multiple • Mass nouns are not always mass nouns due to cultural usages o Sugar: can also refer to sugar cubes or types of sugar instead of individual grains • Mass and count nouns are not simply discrete categories of nouns, but rather help us to culturally talk about significant amounts of objects that need to be measured o Greetings § Language is used to welcome or acknowledge someone § Normative language function found in all observed languages of the world § Devised in order to accomplish initial contact with each other § Depending on the relationship and situation, the greetings function has a wide range of implicit and explicit rules § Affected by gender norms in terms of physical contact, space, and what is considered appropriate practice § Also affected by age, status, and honorifics § Underlying relationship plays a significant role in how greeting is accomplished § Different depending on the region – urban or rural § Many Spanish-­‐speaking countries greet one another with a benediction o Terms of address § Language may be used to index social status or solidarity when addressing someone § How we address another person § One of the most pervasive ways terms of address differentiate status is through the variations in the second person pronoun: you • Spanish, French, and Farsi distinguish between formal and informal uses of you § Use of formal language or informal language can be used to mark difference or build solidarity in status • Decisions based on solidarity, not status, when there are conflicting levels of closeness and superiority § Decision to use formal title or a first name basis is based on convergence of interpersonal factors and societal expectations • Relationships also change over time § Understanding how language functions as a marker of solidarity and status is critical to English learning o Honorifics § Some languages have special forms for polite language, and language used to speak to someone of a higher status § Specialized group of words, or lexicon, that is used for the sole purpose of marking social indifference § Javanese, Japanese, and Korean have very highly developed honorific systems § Only some languages have an honorific system at the structural level § People use language functions to index, or point to, their collective ideas, values, and ideologies of respect and honor o Reproduction § Narratives and cultural ideas of reproduction are expressed through language o Affective § Language is used to express emotions o Ontology § Language is used to talk about the world as it really is o Epistemology § Language is used to talk about knowledge o Climate § Language is used to talk about the weather or environment o Time § language is used to tell and talk about times, dates, and/or historical events o Narration § Language is used to tell stories that reveal social identities and cultural values o Transactions § Language can be used to perform transactions o Aesthetics § Language can be used to talk about art and beauty • Metaphysical o Addresses people’s fundamental need to make sense of metaphysical experiences § The world that is just beyond our reach o People seek to relate beings, or higher powers, or ultimate realities in many different ways § Talk addressed to God: done using rituals, respect registers, special language, and pattern prayers o One of the most common language functions is how it is used to communicate with the afterlife, other worlds, and other dimensions o Function to interact with the metaphysical world is a common, if not universal, function of language, though the actual speech events vary drastically Conclusion • Languages have these functions because they arise from people’s needs to solve authentic problems and answer deep identity questions about who they are, where they’ve been, and where they are going Db Post • Submit your critical Q/A by Friday,11:59 pm, EST. • Provide your feedback on peers' Q/As (at least 2) by Sunday, 11:59 pm, EST.

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Textbook: Introduction to Algorithms
Edition: 3
Author: Thomas H. Cormen
ISBN: 9780262033848

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Asymptotic notation properties Let f .n/ and g.n/ be