See the disclaimer on the previous page.
- Questions from theoretical computer science inspired much interest in the combinatorics community, and for many of its leaders became a primary scientific goal. This collaboration has been extremely beneficial to both the discrete math and theoretical computer science communities, with healthy exchange of ideas, problems and techniques (see also Discrete Mathematics: Past, Present and Future).
- The mathematical challenges which arise from (mainly complexity) questions in theoretical computer science (see Special Year on Computational Complexity 2000-2001, topic page), seem to demand in certain cases the use of techniques in other branches of math, like algebra, topology and analysis, and these occurrences are becoming more frequent. More importantly, the fundamental problems of theoretical computer science, like the P vs. NP problem, have gained the appropriate prominence as central problems of mathematics, and drawn pure mathematicians to tackle them.
- It is extremely important that this conversation between mathematics and theoretical computer science is two-way. More and more mathematicians are considering "computational" aspects of their areas, following theorems establishing the existence of some objects with an investigation of the efficient constructibility of such objects. This approach (which has deep roots in the work of figures such as Euclid and Hilbert) typically reveals further structural questions, and a combination of math and algorithmic techniques have fostered active research areas like computational number theory, computational algebra and computational group theory.
- Scientists' use of computers has grown tremendously in the last two decades, mostly for the analysis of massive data sets. But the currently available resources of even the newest computer systems are far from being sufficient for solving all problems of interest. Efficient algorithms have yet to be (and are being) developed. Among these are e.g. the convergence analysis of some Monte Carlo algorithms used by statistical physicists, and the growing work on computational biology. Yet another exciting area of collaboration, where foundational work on theoretical computer science side goes hand-in-hand with experimental work in physics, is quantum computing.
- A whole new type of algorithmic problems from natural sciences are challenging theoretical computer science: namely, problems in which the required output is not "well defined in advance." Typical data might be a picture, a sonogram, readings from the Hubble Space Telescope, stock-market share values, DNA sequences, neuron recordings of animals reacting to stimuli, or any other sample of "natural phenomena". The algorithm (like the scientist), is "trying to make sense" of the data, "explain it", "predict future values of it", etc. The models, problems and algorithms here fall into the research area of computational learning and its extensions.
- Economics is also playing a growing role as a source of problems and paradigms for theoretical computer science, beyond the analysis and prediction of the stock market. Historically, economic and decision making problems initiated one of the first grand achievements in algorithm design - the simplex method (and its successors) for linear programming. But now the roles are reversed and economic theories are called to solve computer science problems, as multitudes of autonomous robots, or of independent programs on the Web, have to be programmed to function in adversarial (or at least selfish) environments, and best achieve their goals. Exciting beginnings of such models and solutions are now budding.
Some related external links and references
Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. Discrete means individual, separate, distinguishable implying discontinuous or not continuous, so integers are discrete in this sense even though they are countable in the sense that you can use them to count. 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.
Almost all middle or junior high schools and high schools across the country closely follow a standard mathematics curriculum with a focus on “Continuous Mathematics.” The typical sequence includes:
Pre-Algebra Algebra 1 Geometry Algebra 2/Trigonometry Precalculus Calculus Multivariable Calculus/Differential Equations
Discrete mathematics has not yet been considered a separate strand in middle and high school mathematics curricula. Discrete mathematics has never been included in middle and high school high-stakes standardized tests in the USA. The two major standardized college entrance tests: the SAT and ACT, do not cover discrete mathematics topics.
Discrete mathematics grew out of the mathematical sciences’ response to the need for a better understanding of the combinatorial bases of the mathematics used in the real world. It has become increasingly emphasized in the current educational climate due to following reasons:
Many problems in middle and high school math competitions focus on discrete math
Approximately 30-40% of questions in premier national middle and high school mathematics competitions, such as the AMC (American Mathematics Competitions), focus on discrete mathematics. More than half of the problems in the high level math contests, such as the AIME (American Invitational Mathematics Examination), are associated with discrete mathematics. Students not having enough knowledge and skills in discrete mathematics can’t do well on these competitions. Our AMC prep course curriculum always includes at least one-third of the studies in discrete mathematics, such as number theory, combinatorics, and graph theory, due to the significance of these topics in the AMC contests
Discrete Mathematics is the backbone of Computer Science
Discrete mathematics has become popular in recent decades because of its applications to computer science. Discrete mathematics is the mathematical language of computer science. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in all branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are tremendously significant in applying ideas from discrete mathematics to real-world applications, such as in operations research.
The set of objects studied in discrete mathematics can be finite or infinite. In real-world applications, the set of objects of interest are mainly finite, the study of which is often called finite mathematics. In some mathematics curricula, the term “finite mathematics” refers to courses that cover discrete mathematical concepts for business, while “discrete mathematics” courses emphasize discrete mathematical concepts for computer science majors.
Discretemathplays the significant rolein big data analytics.
The Big Data era poses a critically difficult challenge and striking development opportunities: how to efficiently turn massively large data into valuable information and meaningful knowledge. Discrete mathematics produces a significant collection of powerful methods, including mathematical tools for understanding and managing very high-dimensional data, inference systems for drawing sound conclusions from large and noisy data sets, and algorithms for scaling computations up to very large sizes. Discrete mathematics is the mathematical language of data science, and as such, its importance has increased dramatically in recent decades.
IN SUMMARY, discrete mathematics is an exciting and appropriate vehicle for working toward and achieving the goal of educating informed citizens who are better able to function in our increasingly technological society; have better reasoning power and problem-solving skills; are aware of the importance of mathematics in our society; and are prepared for future careers which will require new and more sophisticated analytical and technical tools. It is an excellent tool for improving reasoning and problem-solving abilities.
We highly suggest that starting from the 6th grade, students should some effort into studying fundamental discrete math, especially combinatorics, graph theory, discrete geometry, number theory, and discrete probability. Students, even possessing very little knowledge and skills in elementary arithmetic and algebra, can join our competitive mathematics classes to begin learning and studying discrete mathematics.
More articles about math competitions:
This entry was posted in Math Competitions. Bookmark the permalink.