Quadratic reciprocityIn number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Then: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form for an odd prime ; that is, to determine the "perfect squares" modulo .
Frequency synthesizerA frequency synthesizer is an electronic circuit that generates a range of frequencies from a single reference frequency. Frequency synthesizers are used in many modern devices such as radio receivers, televisions, mobile telephones, radiotelephones, walkie-talkies, CB radios, cable television converter boxes, satellite receivers, and GPS systems. A frequency synthesizer may use the techniques of frequency multiplication, frequency division, direct digital synthesis, frequency mixing, and phase-locked loops to generate its frequencies.
Exponential sumIn mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function Therefore, a typical exponential sum may take the form summed over a finite sequence of real numbers xn. If we allow some real coefficients an, to get the form it is the same as allowing exponents that are complex numbers. Both forms are certainly useful in applications.
Gauss sumIn algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R+ into the unit circle, and χ is a group homomorphism of the unit group R× into the unit circle, extended to non-unit r, where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function. Such sums are ubiquitous in number theory.
Partition of sums of squaresThe partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion (also called variability). When scaled for the number of degrees of freedom, it estimates the variance, or spread of the observations about their mean value.
