In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883.
Rhythm generally means a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recurrence or pattern in time can apply to a wide variety of cyclical natural phenomena having a periodicity or frequency of anything from microseconds to several seconds ; to several minutes or hours, or, at the most extreme, even over many years.
A time signature is a convention in Western music notation that specifies how many note values of a particular type are contained in each measure (bar). The time signature indicates the meter of a musical movement.
In music, metre or meter refers to regularly recurring patterns and accents such as bars and beats. Unlike rhythm, metric onsets are not necessarily sounded, but are nevertheless implied by the performer and expected by the listener.
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped.
In music and music theory, the beat is the basic unit of time, the pulse, of the mensural level. The beat is often defined as the rhythm listeners would tap their toes to when listening to a piece of music, or the numbers a musician counts while performing, though in practice this may be technically incorrect. In popular use, beat can refer to a variety of related concepts, including pulse, tempo, meter, specific rhythms, and groove.
In musical notation, a bar is a segment of music bounded by vertical lines, known as bar lines, usually indicating one of more recurring beats. The length of the bar, measured by the number of note values it contains, is normally indicated by the time signature.
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
In music notation, a tie is a curved line connecting the heads of two notes of the same pitch, indicating that they are to be played as a single note with a duration equal to the sum of the individual notes' values. A tie is similar in appearance to a slur; however, slurs join notes of different pitches which need to be played independently, but seamlessly (legato).
In music, a tuplet is "any rhythm that involves dividing the beat into a different number of equal subdivisions from that usually permitted by the time-signature " This is indicated by a number, or sometimes two indicating the fraction involved. The notes involved are also often grouped with a bracket or a slur.
In Western musical notation, a dotted note is a note with a small dot written after it. In modern practice, the first dot increases the duration of the basic note by half of its original value. This means that a dotted note is equivalent to writing the basic note tied to a note of half the value – for instance, a dotted half note is equivalent to a half note tied to a quarter note. Subsequent dots add progressively halved value, as shown in the example to the right.
In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911.
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.
In popular music, half-time is a type of meter and tempo that alters the rhythmic feel by essentially doubling the tempo resolution or metric division/level in comparison to common-time. Thus, two measures of 4
4 approximate a single measure of 8
8, while a single measure of 4/4 emulates 2/2. Half-time is not to be confused with alla breve or odd time. Though notes usually get the same value relative to the tempo, the way the beats are divided is altered. While much music typically has a backbeat on quarter note (crotchet) beats two and four, half time would increase the interval between backbeats to double, thus making it hit on beats three and seven, or the third beat of each measure :
1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 1 2 3 4
In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are:
In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is zero if and only if the random vectors are independent. Thus, distance correlation measures both linear and nonlinear association between two random variables or random vectors. This is in contrast to Pearson's correlation, which can only detect linear association between two random variables.
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), in a piece by J.S. Bach. Slow introduction followed by an allegro traditionally taken at double the speed. Sixteenth notes in the old tempo prepare for eighth notes in the new tempo. 
