Stars and bars (combinatorics)In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins. The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics concerning the number of solutions to an equation.
CombinatoricsCombinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas.
TupleIn mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called respectively a singleton and an ordered pair. Tuple may be formally defined from ordered pairs by recurrence by starting from ordered pairs; indeed, a n-tuple can be identified with the ordered pair of its (n − 1) first elements and its nth element.
Number theoryNumber theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).
Discrete mathematicsDiscrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry.
FormulaIn science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be either formulas (from the most common English plural noun form) or, under the influence of scientific Latin, formulae (from the original Latin). In mathematics, a formula generally refers to an equation relating one mathematical expression to another, with the most important ones being mathematical theorems.
Rhythm and bluesRhythm and blues, frequently abbreviated as R&B or R'n'B, is a genre of popular music that originated in African-American communities in the 1940s. The term was originally used by record companies to describe recordings marketed predominantly to African Americans, at a time when "rocking, jazz based music ... [with a] heavy, insistent beat" was becoming more popular. In the commercial rhythm and blues music typical of the 1950s through the 1970s, the bands usually consisted of a piano, one or two guitars, bass, drums, one or more saxophones, and sometimes background vocalists.
JazzJazz is a music genre that originated in the African-American communities of New Orleans, Louisiana, in the late 19th and early 20th centuries, with its roots in blues and ragtime. Since the 1920s Jazz Age, it has been recognized as a major form of musical expression in traditional and popular music. Jazz is characterized by swing and blue notes, complex chords, call and response vocals, polyrhythms and improvisation. Jazz has roots in European harmony and African rhythmic rituals.
Time signatureThe time signature (also known as meter, metre, and measure signature) is a convention in Western music notation to specify how many of a particular note value are contained in each measure (bar). The time signature is a notational device representing the meter, an auditory feature of the music. In a music score the time signature appears at the beginning as a time symbol or stacked numerals, such as or (read common time or four-four time, respectively), immediately following the key signature (or immediately following the clef symbol if the key signature is empty).
Aerobic exerciseAerobic exercise (also known as endurance activities, cardio or cardio-respiratory exercise) is physical exercise of low to high intensity that depends primarily on the aerobic energy-generating process. "Aerobic" is defined as "relating to, involving, or requiring oxygen", and refers to the use of oxygen to meet energy demands during exercise via aerobic metabolism adequately. Aerobic exercise is performed by repeating sequences of light-to-moderate intensity activities for extended periods of time.
Clave (rhythm)The clave (ˈklɑːveɪ,_kleɪv; ˈklaβe) is a rhythmic pattern used as a tool for temporal organization in Cuban music. In Spanish, clave literally means key, clef, code, or keystone. It is present in a variety of genres such as Abakuá music, rumba, conga, son, mambo, salsa, songo, timba and Afro-Cuban jazz. The five-stroke clave pattern represents the structural core of many Cuban rhythms. The clave pattern originated in sub-Saharan African music traditions, where it serves essentially the same function as it does in Cuba.
Neurobiological effects of physical exerciseThe neurobiological effects of physical exercise are numerous and involve a wide range of interrelated effects on brain structure, brain function, and cognition. A large body of research in humans has demonstrated that consistent aerobic exercise (e.g., 30 minutes every day) induces persistent improvements in certain cognitive functions, healthy alterations in gene expression in the brain, and beneficial forms of neuroplasticity and behavioral plasticity; some of these long-term effects include: increased neuron growth, increased neurological activity (e.
