Combinatorics is a branch of mathematics that focuses on the permutation, enumeration, and combination of sets of elements. Sometimes, mathematicians use the word “combinatorics” to denote a larger subset of discrete math that includes graph theory. What is commonly referred to as combinatorics, in that case, is then called “enumeration’.
First combinatorial problems were studied by ancient Greek, Arabian and Indian mathematicians. Interest in combinatorics increased during the 19th and 20th century, with the development of graph theory and the four color theorem. Blaise Pascal, Jacob Bernoulli and Leonhard Euler are some of the leading mathematicians.
The most classical area of combinatorics is enumerative combinatorics. It focuses on counting the number of some combinatorial objects. While counting the number of components in a set is a quite broad mathematical problem, numerous problems that arise in applications boast of a relatively simple combinatorial description.
Analytic combinatorics focuses on the combinatorial structures enumeration using tools from probability theory and complex analysis. Compared to enumerative combinatorics that use clear generating functions and combinatorial formulae to describe the outcomes, the aim of analytic combinatorics is to obtain asymptotic formulae.
Other subfields and approaches of combinatorics are partition theory, design theory, finite geometry, order theory, matroid theory, extremal combinatorics, probabilistic combinatorics, and algebraic combinatorics, combinatorics on words, infinitary combinatorics, arithmetic combinatorics and topological combinatorics.
Combinatorics has numerous applications in other mathematics areas, including coding and cryptography, graph theory, and probability.
If you find your computer unresponsive at times, it’s time to get an in-depth check of your system. First, you need to identify the cause behind computer freezing to detect if it is hardware related or software specific. Identifying the cause helps users to resolve the problem quickly. They are many types of computer freeze issues, and each type is detected by system behavior. System freeze can be grouped into hang, generic freeze, random hang and single-app freeze
Hang— every time a specific set of a procedure is carried out on a PC, it hangs up and requests to be restarted to recover.
Generic freeze—the system becomes unresponsive and automatically goes to usual functional state without troubleshooting.
Random hang— occurs when the PC turns unresponsive often at regular intervals of time, and the user has to restart it.
Single-app Freeze —this occurs when PC freezes abnormally when the user attempts to start a particular program, a game or a heavy browsing website.
The main reasons (software as well as hardware) that cause a computer to hang are too many apps running, driver issues, operating system issues, excess heating up, hardware misconfiguration, insufficient RAM, BIOS settings, power issues, external devices, and hard drive malfunction.
Geometry is a branch of mathematics that focuses on questions of size, shape, the properties of space, and relative position of figures. Mathematicians who work in the geometry field are referred to as geometers.
In several early cultures, geometry arose as a practical approach for dealing with volumes, areas, and lengths. While geometry has changed significantly over the years, there are several concepts that are more or less basic to geometry. These concepts include points, surfaces, angles, curves, and lines.
Geometry is applied in many fields, including physics, architecture, art, as well as other branches of mathematics. Modern geometry has several subfields:
Euclidean geometry—this is geometry in its classical sense. It has applications in crystallography, computer science, and various branches of contemporary mathematics.
Differential geometry—this subfield uses techniques of linear algebra and calculus to study problems in geometry.
Topology –This subfield deals with the geometric object’s properties that are unaffected by continuous mappings.
Convex geometry –examines convex shapes in the space of Euclidean and it’s more abstract analogs.
Algebraic geometry– studies geometry using multivariate polynomials or other algebraic methods.
Discrete geometry—it mainly focuses on relative position questions of geometric objects, such as lines, points, and circles.
You often hear the word ‘cybercrime’ discussed nowadays. Due to the high number of connected devices and people, cybercrime is very common. But what is it exactly?
Cybercrime is a crime that involves a network and a computer. The computer may be the target, or it may be used in a commission of a crime. Cybercrime can also be defined as an offense that is committed against an individual or a group of individuals with a motive to intentionally cause mental or physical harm, using modern telecommunication networks such as mobile phones and the internet.
Cybercrimes may threaten a nation or person’s financial health and security. Types of cybercrimes include hacking, unwarranted mass-surveillance, copyright infringement, child grooming, and child pornography. Other common types of cybercrime are online scams and fraud, attacks on computer systems, identity theft and prohibited or illegal online content. Globally, both government and non-state actors are involved in cybercrimes, including financial theft, espionage, and other cross-border crimes. Sometimes, an activity that involves the interest of at least one country and cross-international borders is referred to as cyber warfare.
In the past, computer-related crimes were committed mainly by individuals. Nowadays, we are seeing very complex cybercriminal networks bringing together people from various parts of the world in real time to commit offenses on an unprecedented scale.
Mathematics is important to the understanding of the world. Often referred to as the language of the universe, mathematics has made an impact in almost all place, from the satellite that beams your TV to the faucet in your kitchen to your home. Based on their contributions to mathematics, the following are top three greatest mathematicians:
Living from 1707 to 1783, Euler is the greatest mathematician to have ever walked on earth. It is believed that all mathematical formulas were named after the next person after being discovered by Euler. His primary contribution to the field of mathematics includes the introduction of mathematical notation such as the concept of a function, shorthand trigonometric functions, The Euler Constant, the letter ‘/i’ for imaginary units and the symbol pi for the ratio of a circumference of a circle to its diameter.
Carl Friedrich Gauss
Also referred to as the ‘Prince of Mathematics,’ Gauss is known for his outstanding mental ability. He made several important contributions to mathematics. For instance, he introduced the Gaussian gravitational constant and proved the fundamental theorem of algebra.
G. F. Bernhard Riemann
Born to a poor family in 1826, Bernhard Riemann rose to become one of the most prominent mathematicians in the world in the 19th Century. He has many theorems bearing his name including Riemannian geometry, the Riemann Integral, and Riemannian Surfaces.
A spreadsheet is a computer program or application that allows the user to arrange and collate data. Also referred to as a worksheet, a spreadsheet is made up of columns and rows that help to sort data, organize data, and calculate numerical data.
The ability to calculate values using the data in cells and mathematical formulas is what makes a spreadsheet program unique. Spreadsheet enables users to adjust any stored value and note the effects on calculated values. For example, a spreadsheet may be used to create an overview of an individual’s bank’s balance.
Spreadsheets have completely replaced paper-based systems in the business world. Even though they were first developed for bookkeeping or accounting tasks, today they are used widely in any setting where tabular lists are made, sorted, and shared.
Besides performing arithmetic and mathematical functions, spreadsheets offer built-in functions for common statistical and financial operations. Below are several uses of spreadsheets:
Finance– spreadsheets are best for financial data, such as checking account information, transactions, and budgets.
Forms–form templates are created to handle evaluation, inventory, quizzes, performance reviews, and surveys.
School and grades—teacher use spreadsheets to calculate grade, track students, and identify any relevant data.
In mathematics and computer science, graph theory is the study of graphs, mathematical structures that show the relation between variable quantities. A graph is made up of vertices, points, or nodes which are connected by arcs, lines, or edges. In discrete mathematics, graphs are one of the main objects of study.
Graphs are used to model numerous types of processes and relations in biological, physical, social and information systems. Graphs can be used to represent many real-world problems. Emphasizing their application to practical systems, the term network is occasionally defined to mean graphs in which attributes are associated with the edges and nodes.
In computer science, graphs are mainly used to represent data organization, networks of communication, the flow of computation, computational devices, etc. For example, a website link structure can be represented by a directed graph. In this case, directed edges will represent links from one page to another, and the vertices will represent web pages. A similar approach can be used to represent problems in biology, social media, computer chip design, travel, and many other fields. Therefore, the development of algorithms that handle graphs is of major interest in computer science.
World Maths Day is an online international mathematics competition that is powered by educational resource provider 3P Learning. It is an annual event that encourages thousands of students to take a break from standard mathematics lessons; instead, they participate in competitive math-themed games. Students who take part in the games have a chance to win prizes and certificates. The aim is to help raise numeracy standards.
Since it was first held on March 14, 2007, World Maths Day has been held the 1st Wednesday in March in every year. Despite the origins, the phrases “World Math Day” and “World Maths Day” are trademarks and should not be confused with other days such as Pi Day or competitions such as the International Mathematical Olympiad. World Maths Day made a Guinness World Record for the Largest Online Maths Competition in 2010.
In 2011, the team behind World Maths Day added a second event known as World Spelling Day. The event was later rebranded the World Education Games. World Science Day, the third event, was added in 2012. World Education Games and World Maths Day are now sponsored by Samsung.
Since it was founded, the World Maths Day has seen its popularity grow fast. It links schools from many parts of the world. Children as young as five years participate, competing against others from overseas in real time. This year, the event took place on 14th March 2017.
Probability is the study of the likelihood of an event happening. Simply, it is how likely something is to happen. Probability plays a role in all activities, directly or indirectly. For example, we may say that tomorrow will probably be sunny because most of the days we have observed were sunny days.
Many events cannot be predicted with certainty. When people are unsure about the outcome of an event, they can talk about the probabilities of specific outcomes—their likelihood. The analysis of events that are governed by probability is known as statistics.
To understand probability, the best example is flipping a coin. There are two possible outcomes: tails (T) and heads (H). What is the probability of the coin landing on Tails or Head? The probability of the coin landing T is ½, and the probability of the coin landing H is ½. Another example that can help you understand probability is throwing a dice. When a die is thrown, there are six possible outcomes. The probability of any one of them is 1/6.
In general, the probability of an event = (Number of ways it can occur) / (total number of outcomes).
These lessons on probability often include the following topics: the probability of events, samples in probability, theoretical probability, probability problems, experimental probability, tree diagrams, independent events, mutually exclusive events, dependent events, permutations, factorial, combinations, probability, probability in statistics, and combinatorics.
Combinatorial game theory is a branch of theoretical computer science and mathematics that studies sequential games. The study has been mainly confined to two-player games with a position in which players take turn changing. Traditionally, the combinational game theory has not studied games of chance or games that use incomplete or imperfect information, favoring games that provide complete information in which set of available moves and state of the game is always known by both players. As mathematical techniques advance, however, the types of games that can be analyzed mathematically expands, therefore, the boundaries of the field are constantly changing.
Examples of combinatorial games are Go, Checkers, Chess, and Tic-tac-toe. The first three games are categorized as non-trivial while the last one is categorized as trivia. In combinatorial game theory, the moves in these games are represented as a game tree. Some one-player combinatorial puzzles (such as Sudoku) and no-player automata (such as Conway’s Game of Life) are also categorized as combinatorial games.
In general, game theory includes games of imperfect knowledge, games of chance, and games in which player’s moves simultaneously, tending to represent actual decision-making situations.
An important notion in combinatorial game theory is that of solved game. Tic-tac-toe, for example, is considered a solved game because it can be established that if both players play optimally, a game will result in a draw.