Applications of Discrete Mathematics - Free Essay Example

Sample details Pages: 7 Words: 2027 Downloads: 1 Date added: 2017/09/16 Category Advertising Essay Did you like this example? DISCRETE MATHEMATICS Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying smoothly, the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in continuous mathematics such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the integers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term discrete mathematics. Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions. The set of objects studied in discrete mathematics can be finite or infinite. Don’t waste time! Our writers will create an original "Applications of Discrete Mathematics" essay for you Create order The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research. Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well. Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term discrete mathematics is therefore used in contrast with continuous mathematics, which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers. The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in discrete mathematics Complexity studies the time taken by algorithms, such as this sorting routine. Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of continuous computational topics such as analog computation, continuous computability such as computable analysis, continuous complexity such as information-based complexity, and continuous systems and models of computation such as analog VLSI, analog automata, differential petri nets, real time process algebra. Information theory The ASCII codes for the word Wikipedia, given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms. Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding, analog encryption. Logic Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirces law (((PQ)P)P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software. Logical formulas are discrete structures, as are proofs, which form finite trees[8] or, more generally, directed acyclic graph structures[9][10] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e. g. , fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied,[11] e. g. infinitary logic. Set theory Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantors work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics. Combinatorics Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects e. g. the twelvefold way provides a unified framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite. Graph theory Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A4. Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right. [12] Algebraic graph theory has close links with group theory. Graph theory has widespread applications in all areas of mathematics and science. There are even continuous graphs. Probability Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values {0, 1, 2, . On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards, calculat ing the probability of events is basically enumerative combinatorics. Number theory The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of prime numbers. Main article: Number theory Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography, cryptanalysis, and cryptology, particularly with regard to prime numbers and primality testing. Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields. Algebra Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages. Calculus of finite differences, discrete calculus or discrete analysis A function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation. Difference equations are similar to a differential equations, but replace differentiation by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance where there are integral transforms in harmonic analysis for studying continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces. Geometry Computational geometry applies computer algorithms to representations of geometrical objects. Main articles: discrete geometry and computational geometry Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems. Topology Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of continuous deformation of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space. Operations research Operations research provides techniques for solving practical problems in business and other fields — problems such as allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimization, queuing theory, scheduling theory, network theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory. Game theory, decision theory, utility theory, social choice theory | Cooperate| Defect| Cooperate| -1, -1| -10, 0| Defect| 0, -10| -5, -5| Payoff matrix for the Prisoners dilemma, a common example in game theory. One player chooses a row, the other a column; the resulting pair gives their payoffs| Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of various goods and services. Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory. Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex. There are even continuous games, see differential game. Topics include auction theory and fair division. Discretization Discretization concerns the process of transferring continuous models and quations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example. Discrete analogues of continuous mathematics There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, disc rete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, and discrete dynamical systems. In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relations. Hybrid discrete and continuous mathematics The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous data.

The Discovery Of America By Jan Van Der Straet - 1987 Words

The â€Å"discovery† of America was one that introduced a colonial discourse in Europe, which would shape the relationship between the Europeans of the Old World and the indigenous people of the New World. Exoticism, anxiety, and absurd speculation would fuel the European knowledge of the Americas during the late 15th and early 16th centuries. The drawing titled America by Jan Van der Straet, is a classic example of how Europeans used outlandish notions about indigenous Americans to distance themselves from these natives and thus, establish European superiority. Ultimately, Jan van der Straet’s image supports and justifies European colonialism by depicting the indigenous people as savage, primordial and in need of the paternal guidance of the†¦show more content†¦The continent would then be named after Amerigo Vespucci as America (Almagià  ). The image also makes use of the two figures in the center in an allegorical way. The woman is representative of the Ame ricas and the male is representative of Europe. Overall, the image makes use of a variety of iconographic elements to justify colonialism. Of these elements, one of the most significant is the indigenous woman in the center. The naked indigenous woman is a personification of America and is beckoning Vespucci, whom is a personification of Europe, to conquer and save her from the barbaric cannibals and creatures that are surrounding her in this unruly land. It is establishing the narrative that the Americas were consenting to colonization by Europe so that it may be rescued from the vicious and primitive people and creatures that inhabit it. The woman is the center piece of this image and is sitting upon a hammock while she is nude. As she is sitting down, she is drawing her hand towards Vespucci, who has clearly just arrived on the shores of the Americas as indicated by the caravel ships that are depicted behind him. As the two figures are meeting in the foreground of the image, cann ibals in the background of the image are cooking and eating a human leg around a fire. Furthermore, the image is littered with depictions of monstrous animals such as the tiger and wolf towards the

Food Crisis free essay sample

Food is the foundation of human live and people cannot survive without it. Food security is a vital issue facing the governments around the world. However, food shortage is becoming increasingly severe in this day and age. There are several reasons which led to the universal food shortage and they are interconnected to each other. Increasing world population, extreme weather and the wide spread use of biofuels are the main causes of worldwide shortage of food. These factors lead to food price rises rapidly. This essay will firstly discuss three major factors which have caused the world food crisis. Then it will examine how these factors affect the food price. Finally, it will suggest some solutions to tackle the food crisis. Global food crisis is being compounded by several causes such as growing population, extreme weather and famers switching out of cereals to grow agro-fuels crops. According to Vidal (2007), There is no one cause but a lot of things are coming together to lead to this. Its hard to separate out the factors. Global population is continuing growing nowadays; Eating habits are changed in many regions all over the world, especially in China and India; Extreme weather is caused by global climate nomaly which is related droughts and floods in many key production regions, such as a years-long drought in Australia; The wider use of agro-fuel led to divert food crops to grow biofuels crops; Agriculture costs are much higher, which are caused by soaring oil price, limited farmland and water. Global population growth is one of the most important causes of food shortage. According to Gritzner and Charles (2010, 60), By mid-century there will be some 9 billion people†an increase of 2 billion more people than today†eating at the global dining table. This means that demand for food will continue to increase over the oming decades. As rapid economic development and income growth among the middle class in developing countries, especially in densely populated countries of Asia, the demand of meat, dairy products and oils is experiencing rapid growth (HoJJat 2009, 421). In other words, people can afford better food and change their eating habits to high quality and the variety of food. Producing these products requires more crops to feed livestock. To produce one kilogram of meat, cows need eight kilograms grain; pigs require five kilograms and chickens need three kilograms (McPherson 2008). The second cause which has sparked much debate is the diversion of food for fuels. Due to the oil price increased and global environment worsen, scientists from all over the world realised to develop new energies to replace fossil fuels. Biofuels, which are made from crops, as a type of new energy sources are used widely in the United States, Europe, and many other countries. (Pereira 2008) However, it is a direct competition with the use of these crops for food. Take the United States for example, more than twenty percent of the crops were used to make biofuels in 2007. It is stimated that over a third of crops in the U. S. will be diverted to make biofuels over Extreme weather is another major responsible factor which causes food shortage. Global food production has been reduced by droughts, floods, and cyclones. Australia as a major wheat production country experienced a six-year-long drought (Vidal 2007). China, a self-sufficient country, started to import food from international market because of the unusual cold and snow weather during the winter in 2007 (Gritzner and Charles 2010, 62). A 2007 cyclone in Bangladesh destroyed approximately 600 million dollars worth of tis rice crop. Lavelle and Garber 2008). Food price growth is due to the imbalance between demand and supply. When agricultural production is rising slower than the increasing demand for food consumption, the food prices go up (HoJJat 2009, 421). On the demand side, global population growth puts a great deal of pressure on food demand. As income growth, consumers increase their intakes of higher quality food, such as fish, meat dairy products. The other side is supply side. Producing biofuels on a large scale reduces supply of particular crops significantly, which are used for food production. Moreover, global food supply is directly influenced by bad weather. Besides, there are other reasons which cause food prices higher than before. Higher oil price raises food prices through increasing cost of farming transport and shipping; seeds and fertilizer prices growth increase food production costs as well. To overcome worldwide food crisis, effective measures should be carried out by governments, international organizations and individuals. Firstly, Agricultural productivity and production should be increased. As HoJJat (2009, 424) notes: The future direction of world food prices will depend on whether research and evelopment increase agricultural productivity faster than growth in world food demand. We need much greater investment in agriculture: poor countries need more funds for infrastructure, irrigation, seeding, and fertilizers, tractors, new technologies, rural credit to enhance productivity of the agriculture industry and reduce dependency on corn. Secondly, governments should stop all subsidies for the production of biofuels and encourage developing alternative source of energy (Lavelle and Garber 2008). The United States and Europe, two main biofuels producers, need removal of policies promoting biofuels and increase investment in lternative energies research. Thirdly, international organizations such as United Nations agencies need to establish an international fund for emergencies, especially helping the developing countries to deal with the grain underproduction caused by extreme weather (Singh 2009, 30). Finally, individuals need to change eating habits and reduce food waste. Nutritionists and environmentalists claim that the protein of plant is more healthful and environmentally friendly than livestock (Lavelle and Garber 2008). As this essay has outlined, global food crisis is becoming one of the most severe issues nowadays. Food crisis spread widely due to three main factors: increasing world population, extreme weather and widely used biofuels. These factors put price increases during the past several years. The governments, international organizations and individuals need work together to alleviate the high food price through increasing agricultural productivity and production, developing alternate source of energy, establishing an international fund and changing eating habits. It is a big challenge for human to make a more resilient, sustainable and equitable worlds food production and realize the global food security.

HDI as a Measure of Human Development

Question: Discuss about the HDI as a Measure of Human Development? Answer: 1. Gross Domestic Product is known as GDP which is referred to as the money value of the total finished services as well as products within the territory of a country in a particular period of time (Blanchard and Johnson, 2013). GDP is calculated in quarterly basis as well as on yearly basis. The most important measure of standard of living is the measure of real per capita GDP. It can be said that the increased per capita GDP might be at the increased pollutions costs which will lead to a decline in the living standards of the rise in GDP. GDP is used to measure the Purchasing Power Parity. Thus, it assists to measure the real costs of living. Thus, it can be possible for the economists to compare the PPP of different countries (Goodhew, 2013). GDP is generally measured in dollars. Japanese measure GDP in Yen. Thus, the Japanese will convert the GDP to dollars to compare it with Yen. The conversion process will be done by Yen/Dollar. 2. GDP per capita may not be a good measure especially in the case of measuring living standards and the economic welfare. It is said to be an average and hence it fails to capture the picture of inequality, poverty and other economic activities (Gordon, 2012). It cannot measure the leisure value and also the longevity value. As an example, it can be said that a traffic jam will raise the GDP due to an increasing utilisation of gasoline but the quality of life or the living standards will be hampered by this. GDP per capita failed to capture this. The HDI can be used to measure the standard of living or countrys well-being (Ram and Ural, 2013). Table 1: Top ten countries (GDP level- 2010) It fails to determine the living standards and the well-being of the countries. Thus, to measure well-being and the living standards, it is better to use HDI, HPI etc. (Al-Hilani, 2012). Figure 1: Country-wise GDP Source: (Author) 3. To do a comparative analysis, the researcher can take India as developing country and US as developed country. Here, HDI is taken to show the standard of living trends between two countries. Table 2: UNDP report on HDI Source: (HumanDevelopmentReports, 2016) From table 2, it can be said that in the US, the HDI value had risen marginally in 2014 from 2013. Thus, a very high HDI can be seen in the US according to the UNDP report. In the case of India, the value is also increasing and in 2014, the value is 0.609. The rank is 131 according to the report of UNDP. It will imply that a medium HDI can be seen in India. Table 3: HDI growth (Dept, 2014) From table 3, it can be seen that the rate of HDI growth diminishes in 2010-2014. In the case of US, the rate has changed marginally but in the case of India, the rate has declined from 1.67 to 0.97. Figure 2: HDI in US and India Figure 2 has also shown the upward trend from the last 7 years. Table 4: Components (HumanDevelopmen ReportsComponents. 2016) There are some components include in the HDI Life expectancy, expected schooling years and mean schooling years, GNI per capita (Mankiw, 2012). HDI is said to be a composite index which measures the average achievement in basic 3 dimensions of development in human, these are a healthy as well as long life, standard of living and knowledge. Life expectancy is referred to as the number of period an infant newborn can expect to live. For knowledge, expected schooling years and mean schooling years will be taken. The first one is referred to as the entrance age of schooling which can be expected if age particularity enrolment is persisted. Mean schooling years refers to the average years of education which an individual can receive. GNI per capita is an economys aggregate income that is generated from its production and factors of productions ownership. It can be converted to global dollar value by utilising PPP divided by the population of midyear. Thus, by visualising table 4, it can be said that it will take more than 20 years to be doubled if there are no economic fluctuations in both of the countries. 4. The standard of living is referred to as the wealth level, level of comforts, necessities and material goods which are available in a general socioeconomic class under a specific geographical territory. The standards of living in the countries of NMSs have converged in a rapid manner towards the mean living standard of Europe in the first 10 years of this new century. In this region, many countries like Bulgaria, Baltic countries etc., per capita GDP has been adjusted for PPP. It has assisted in comparing the standard of living of different countries. It had raised greater than 100 percent in the year 1999-2008, but in case of euro areas Member States, the similar indicator had increased only by 30% on an average. However, the convergence rates of the standards of living of the NMS were not at all similar from 1999 to 2008. The countries were divided into two parts on the basis of convergence path. In the first group, Hungary, Poland and the Czech Republic were there and the highest standard of living was seen. In the second group, the rate of convergence was too slower over the 10 years. The countries present in the second group were Romania, Bulgaria, Baltic countries. The declination size of PPP-adjusted per capita GDP of the second groups countries in 1999 and many problems of them in the year 2009 has returned the convergence before the crisis had raised a question of imbalances. Latvia and Estonia had a CAD of 10% of the GDP levels whereas Bulgaria was present at 9% of the GDP level. The countries except Hungary belonging from the first group stood at CAD less than 5% of GDP. Thus, convergence and divergence had also arisen due to CAD effects in GDP (Moukarzel, 2013). The PPP adjusted per capita GDP extrapolation in 2010 had shown that living standard had resumed its increasing trend in most of the countries of NMSs in the year 2009-2010. However, it had faced a slower pace in between 1999 to 2008. This had confirmed that there were a lasting de-convergence did not occur. However, it can be said in this context that the global financial crisis adversely affected the process of convergence (Kokkoris, 2014). In 2010, there was another slowdown in the standard of living increment in comparison with the past decade which was relatively high pronounced for the countries belonging from the second group. The slowdown had arisen in the second group due to some macroeconomic imbalances. Thus, the countries belonging from the second group should perceive sustainable strategies of growth (Alonazi and Thomas, 2014). References Al-Hilani, H. (2012). HDI as a Measure of Human Development: A Better Index than the Income Approach?. IOSR Journal of Business and Management, 2(5), pp.24-28. Alonazi, W. and Thomas, S. (2014). Quality of Care and Quality of Life: Convergence or Divergence?. Health Services Insights, p.1. Blanchard, O. and Johnson, D. (2013). Macroeconomics. Boston (Mass.): Pearson. ConvergencePDF, (2016). [online] Tresor.economie.gouv.fr. Available at: https://www.tresor.economie.gouv.fr/file/326916 Dept, I. (2014). India. Washington: International Monetary Fund. Goodhew, P. (2013). Growth of what: GDP or quality of life?. European Journal of Engineering Education, 38(2), pp.119-120. Gordon, R. (2012). Macroeconomics. Boston: Addison-Wesley. HumanDevelopmen ReportsComponents. (2016). [online] Hdr.undp.org. Available at: https://hdr.undp.org/en/composite/HDI HumanDevelopmentReports. (2016). [online] Hdr.undp.org. Available at: https://hdr.undp.org/en/composite/trends Kokkoris, I. (2014). Introduction: EU and U.S. Competition Enforcement--Convergence or Divergence. The Antitrust Bulletin, 59(1), pp.1-8. Mankiw, N. (2012). Macroeconomics. New York: Worth. Moukarzel, C. (2013). Per-capita GDP and nonequilibrium wealth-concentration in a model for trade. J. Phys.: Conf. Ser., 475, p.012011. Ram, R. and Ural, S. (2013). Comparison of GDP Per Capita Data in Penn World Table and World Development Indicators. Soc Indic Res, 116(2), pp.639-646. Ryser, L. and Halseth, G. (2011). Informal Support Networks of Low-Income Senior Women Living Alone: Evidence from Fort St. John, BC. Journal of Women Aging, 23(3), pp.185-202.