Orthogonality - Orthogonality In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. It means at right angles. It comes from the Greek "ortho", meaning "right" and "gonia", meaning "angle". Two streets that cross each other at a right angle are orthogonal to each other. Two vectors in an inner product space are orthogonal if their inner product is zero. The word normal is sometimes also used for this concept by mathematicians, although that word is rather overburdened. For example, in a 2- or 3-dimensional Euclidean space, two vectors are orthogonal if their dot product is zero, i.e., they make an angle of 90 degrees or π/2 radians. Hence orthogonality is a generalization.
Hermitian - square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinite-dimensional Hilbert spaces. The spectrum of any Hermitian operator is real; in particular all its eigenvalues are real. A version of the spectral theorem also applies to Hermitian operators; while the eigenvectors to different eigenvalues are orthogonal, in general it is not true that the Hilbert space H admits an orthonormal basis consisting only of eigenvectors of the operator. In fact, Hermitian operators need not have any eigenvalues or eigenvectors at all. In the mathematical formulation of quantum mechanics, one considers even more general Hermitian operators: they are only defined on a dense subspace of a Hilbert space and don't have to be continuous. For example, consider the complex Hilbert space.
Hermite polynomials - of contents showTocToggle("show","hide") 1 Definition 2 Orthogonality 3 Various properties 4 Generalization 5 "Negative variance" 6 Eigenfunctions of the Fourier transform 7 Combinatorial interpretation of the coefficients Definition In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced "air MEET"), compose a polynomial sequence defined either by or sometimes by which is not equivalent. These are Hermite polynomial sequences of different variances; see the material on variances below. Below, we follow the first convention. That convention is sometimes preferred by probabilists because is the probability density function for the normal distribution with expected value 0 and standard deviation 1. The other convention is often followed by physicists. The first several Hermite polynomials are: Orthogonality The nth function in this list is an nth-degree polynomial for n = 0, 1,.
Fourier series - be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f. A partial answer is that if f is square-integrable then (this is convergence in the norm of the space ). That much was proved in the 19th century, as was the fact that if f is piecewise continuous then the series converges at each point of continuity. Perhaps surprisingly, it was not shown until the 1960s that if f is quadratically integrable then the series converges for every value of x except those in some set of measure zero. General formulation The useful properties of Fourier series are largely derived from the orthogonality of the functions ei n x. Other sequences of orthogonal functions.
Atomicity - operations are completed. See concurrent programming. Atomicity in the component-based paradigm context Atomicity is an element of orthogonality guaranteeing hermetic interfaces among components of a component-based system. This avoids that malfunctions in one component would result into side-effects in another. In addition it guarantees that services provided by a component are either offered as a whole, or not at all. Systems build according to this design principle localise the side-effects of changes within the module that such change was manifested. Changes affect the emergent behaviour of such a system only if their technical effect is explicitly described by the formal definition of its logic..
Bessel function - chosen to be real-valued for real arguments x. They are the two linearly independent solutions to the modified Bessel's equation: Unlike the ordinary Bessel functions, which are oscillating, Iα and Kα are exponentially growing and decaying functions, respectively. Like the ordinary Bessel function Jα, the function Iα goes to zero at x=0 for α > 0 and is finite at x=0 for α=0. Analogously, Kα diverges at x=0. Spherical Bessel functions When solving for separable solutions of Laplace's equation in spherical coordinates, the radial equation has the form: The two linearly independent solutions to this equation are called the spherical Bessel functions jn and yn (also denoted nn), and are related to the ordinary Bessel functions Jα and Yα by: There are also spherical analogues of the Hankel functions: In fact,.
Code division multiple access - different spreading codes (or the same spreading code but a different timing offset) appear as wideband noise reduced by the process gain. The way this works is that each station is assigned a spreading code or chip sequence. Such chip sequences are expressed as a sequence of -1 and +1 values. The dot product of each chip sequence with itself is 1 (and the dot product with its complement is -1), whereas the dot product of two different chip sequences is 0. E.g. if C1 = (-1,-1,-1,-1) and C2 = (+1,-1,+1,-1) C1 . C1 = (-1,-1,-1,-1) . (-1,-1,-1,-1) = +1 C1 . -C1 = (-1,-1,-1,-1) . (+1,+1,+1,+1) = -1 C1 . C2 = (-1,-1,-1,-1) . (+1,-1,+1,-1) = 0 C1 . -C2 = (-1,-1,-1,-1) . (-1,+1,-1,+1) = 0 This property is called.
Component-based paradigm - the production scheme. This article is a stub. You can help Wikipedia by fixing it.'' related articles mechatronics automata automation robot orthogonality atomicity external link http://www.lboro.ac.uk/departments/mm/research/manufacturing-systems/.
CPU design - economical and practical. Around 1971, the first calculator and clock chips began to show that very small computers might be possible. The first microprocessor was the 4004, designed in 1971 for a calculator company, and produced by Intel. The 4004 is the direct ancestor of the Intel 80386, even now maintaining some code compatibility. Just a few years later, the word size of the 4004 was doubled to form the 8008. By the mid-1970s, the use of integrated circuits in computers was commonplace. The whole decade consists of unheavals caused by the shrinking price of transistors. It became possible to put an entire CPU on a single printed circuit board. The result was that minicomputers, usually with 16-bit words, and 4k to 64K of memory, came to be commonplace. CISCs were.
RISC - even though this seriously limited speed. Others dedicated their registers to specific tasks in order to reduce complexity; for instance, one might be able to apply math only to one or two of the registers, while storing the result in any of them. In the microcomputer world of the 1970s this was even more of an issue because the CPUs were very slow - in fact they tended to be slower than the memory they talked to. In these cases it made sense to eliminate almost all of the registers and instead provide the programmer with a number of ways of dealing with the external memory to make their task easier. Given the addition example, most CPU designs strove to create a command that would do all of the work automatically:.
Pythagorean theorem - cosines which is a generalization of the Pythagorean theorem applying to all (Euclidean) triangles, not just right-angled ones. Another generalization of the Pythagorean theorem was already given by Euclid in his Elements: If one erects similar figures (see geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one. The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane -- if (a, b) and (c, d) are points in the plane, then the distance between them is given by This distance formula generalises to inner product spaces, and the version of the Pythagorean Theorem in inner product spaces is known as Parseval's identity. The Pythagorean theorem also generalizes.
Normalizing constant - the whole space, i.e., to get a probability measure. In a simple discrete case we have where P(H0) is the prior probability that the hypothesis is true; P(DH0) is the likelihood of the data given that the hypothesis is true; and P(H0D) is the posterior probability that the hypothesis is true given the data. P(D) should be the probability of producing the data, but on its own is difficult to calculate, so an alternative way to describe this relationship is as one of proportionality: . Since P(HD) is a probability, the sum over all possible (mutually exclusive) hypotheses should be 1, leading to the conclusion that In this case, the value is the normalizing constant. It can be extended from countably many hypotheses to uncountably many by replacing the sum by.
Motorola 68000 - hardware prices would fall. To address the perceived markets, the actual 68000 was designed in three forms. The base-form had a 24-bit address, and a 16-bit data bus. The short form, the 68008, had an 18-bit address (possibly 19 or 20 bits, at least one firm addressed 512KBytes with 68008s), and an 8-bit data bus. A planned future form (later the 68020) had a 32-bit data and address bus. Internal registers The CPU had 8 general-purpose data registers (D0-D7), and 8 address registers (A0-A7). The last address register was also the standard stack pointer, and could be called either A7 or SP. This was a good number of registers in many ways. It was small enough to make the 68000 respond quickly to interrupts (because only 15 or 16 had to.
Linear independence - and consider the following elements in V: e1 = (1,0,0,...,0) e2 = (0,1,0,...,0) ... en = (0,0,0,...,1) Then e1,e2,...,en are linearly independent. Proof: Suppose that a1, a2, ,an are elements of Rn such that a1e1 + a2e2 + ... + anen = 0 Since a1e1 + a2e2 + .. + anen = (a1,a2,..,an) then ai = 0 for all i in {1,..,n}. Example III: (Calculus required) Let V be the vector space of all functions of a real variable t. Then the functions et and e2t in V are linearly independent. Proof: Suppose a and b are two real numbers such that aet + be2t = 0 (1) for all values of t. We need to show that a=0 and b=0. In order to do this, we differentiate equation (1) to get.
List of linear algebra topics - transformation Column space Row space Null space, nullity Rank-nullity theorem Dual space Linear function Linear functional Orthogonality Category of vector spaces Multilinear algebra Tensor Classical treatment of tensors Intermediate treatment of tensors Component-free treatment of tensors Affine space Affine transformation Affine group Affine geometry.
