The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse Fourier transform:

An important simplification occurs when the sequences are of finite length, N. In terms of the DFT and inverse DFT, it can be written as follows:

A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the ( forward ) DFT, via several well-known " tricks ".

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

In the same way, one defines the binary operation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of " reciprocal " ( multiplicative inverse ).</ ref >

The inverse transform, known as Fourier series, is a representation of s < sub > P </ sub >( t ) in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:

In terms of S < sub > k </ sub >, the inverse transform is given by:

We note that an inverse system in a category C admits an alternative description in terms of functors.

However, this distribution of meanings between the two terms used to be precisely inverse at the time they were coined: The Oxford English Dictionary defined “ racialism ” as “ belief in the superiority of a particular race ” and gives a 1907 quote as the first recorded use.

where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space, that transforms u into r. That kind of equation can be used to constrain the output function u in terms of the forcing function r. The transfer function, written as an operator, is the right inverse of L, since.

The introduction of simple auxiliary terms, due to C. F. Gauss ( Dioptrische Untersuchungen, Göttingen, 1841 ), named the focal lengths and focal planes, permits the determination of the image of any object for any system ( see lens ).

A rendered image can be understood in terms of a number of visible features.

In simpler terms, this expresses the idea that an image cannot display details, peaks or troughs in color or intensity, that are smaller than one pixel.

( Form of Content, that Louis Hjelmslev distinguished from Form of Expression ) than how the word " house " may be tied to a certain image of a traditional house ( i. e. the relationship between signifier and signified ) with each term being established in reciprocal determination with the other terms than by an ostensive description or definition: when can we talk about a " house " or a " mansion " or a " shed "?

Whilst the terms are often used interchangeably, branding is more strictly related to the identifying mark or trade name for a product or service, whereas corporate identity can have a broader meaning relating to the structure and ethos of a company, as well as to the company's external image.

Sherry Salman considers the image of the Horned God in Jungian terms, as an archetypal protector and mediator of the outside world to the objective psyche.

: knowingly ( A ) uses an interactive computer service to send to a specific person or persons under 18 years of age, or ( B ) uses any interactive computer service to display in a manner available to a person under 18 years of age, any comment, request, suggestion, proposal, image, or other communication that, in context, depicts or describes, in terms patently offensive as measured by contemporary community standards, sexual or excretory activities or organs.

The Z < sub > i </ sub > terms represent image impedance and it is assumed that connections are between matching image impedances.

The terms are impedance scaling terms and their use is explained in the image impedance article.

Both of these terms became outmoded in the late 20th century except in specialist usage, as most fax machines now use the digital ITU-T fax standards, which encode the image digitally over a QAM-modulated signal.

* André Béguin's dictionary ; enormous dictionary of terms, relating more to the printing than the creation of the image

The terms scanning aperture and sampling aperture are often used to refer to the opening through which an image is sampled, or scanned, for example in a Drum scanner, an image sensor, or a television pickup apparatus.

Time and again through the years, Berlioz would distill the favorite image of a play and distill it into musical terms.

The visual image should be accurate in terms of dimensions and proportions, and should provide " an overall impression of what an object is or does, to enhance the viewer ’ s interest and understanding ".

For this reason, people will often distinguish the meanings of the two terms in this context, using " image " for the abstract categorical concept and " range " for the elementary function-theoretic concept.

Another difference, in comparison with JPEG, is in terms of visual artifacts: JPEG 2000 produces ringing artifacts, manifested as blur and rings near edges in the image, while JPEG produces ringing artifacts and ' blocking ' artifacts, due to its 8 × 8 blocks.

As this atom approaches the surface of a metal and forms adsorption, this potential energy V < sub > a </ sub > will be modified due to the image charges by additional potential terms which are quadratic in the displacements:

Within the image, a double arrow indicates two inverted terms of the first proportion.

In terms of naming, a hard disk image for a certain Virtual Machine monitor has a specific file type extension, e. g.,. vmdk for VMware VMDK,. vhd for Xen and Microsoft Hyper-V,. vdi for Oracle VM VirtualBox, etc.

The major benefit of quantifying a lens in terms of its optical power rather than its focal length is because the lens-makers equation has the object distance, image distance, and lens focal length all as reciprocals.

Talbot's original contributions included the concept of a negative from which many positive prints can be made ( although the terms negative and positive were coined by Herschel ), and the use of gallic acid for developing the latent image.

Other researchers have discussed the phenomena of the OBE in terms of a distortion of the body image ( Horowitz, 1970 ) and depersonalization ( Whitlock, 1978 ).

In a pre-abelian category, exact functors can be described in particularly simple terms.

We note that a direct system in a category admits an alternative description in terms of functors.

A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme ( over a base category ), abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.

of functors from rings to abelian groups, for all integers n. For historical reasons, the lower K-groups K < sub > 0 </ sub > and K < sub > 1 </ sub > are thought of in somewhat different terms from the higher algebraic K-groups K < sub > n </ sub > for n ≥ 2.

Such functors give representations of an abstract category in terms of known structures ( i. e. sets and functions ) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

These collections of inverse image functors provide a more intuitive view on fibred categories ; and indeed, it was in terms of such compatible inverse image functors that fibred categories were introduced in Grothendieck ( 1959 ).

There are several different stability factors, such as the Stern stability factor and the Linvil stability factor, which specify a condition that must be met for the absolute stability of an amplifier in terms of its two-port parameters.

Public finances were in so dire a condition at the time that the terms of the loan were that it was to be serviced at a rate of 8 % per annum, and there was also a service charge of £ 4, 000 per annum for the management of the loan.

In terms of its value as a collector's item, a coin is generally made more or less valuable by its condition, specific historical significance, rarity, quality / beauty of the design and general popularity with collectors.

As it is more value-neutral than terms like disease, the term medical condition is sometimes preferred by people with health issues that they do not consider to be deleterious.

As terms of the form λ < sub > i </ sub > – ∑ ( multiples of other eigenvalues ) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.

There is, however, no finite set of statements that are couched in purely sensory terms and can express the satisfaction of the condition of the presence of a normal observer.

" It cannot be supposed that hypothetical contractors they should intend, had they a power so to do, to give any one or more an absolute arbitrary power over their persons and estates, and put a force into the magistrate's hand to execute his unlimited will arbitrarily upon them ; this were to put themselves into a worse condition than the state of nature, wherein they had a liberty to defend their right against the injuries of others, and were upon equal terms of force to maintain it, whether invaded by a single man or many in combination.

At Mantes on 9 July 1193, Philip came to terms with Richard's ministers who agreed that Philip could keep his gains and would be given some extra territories if he ceased all further aggressive actions in Normandy, along with the condition that Philip would hand back the captured territory if Richard would pay homage to Philip.

In other terms, this condition says that x and F can be separated by a continuous function.

The definitions that we have given here are the ones usually used today ; however, some authors switch the meanings of the two kinds of terms, or use all terms synonymously for only one condition.

Grant gave generous terms ; Confederate troops surrendered their weapons and were allowed to return to their homes, with their mounts, on the condition that they would not take up arms against the United States.

In the first case, the condition of bandlimitation of the sampled signal can be accomplished by assuming a model of the signal which can be analysed in terms of the frequency components it contains ; for example, sounds that are made by a speaking human normally contain very small frequency components at or above 10 kHz and it is then sufficient to sample such an audio signal with a sampling frequency of at least 20 kHz.

The inclusion of interpersonality amnesia helped to distinguish DID from dissociative disorder not otherwise specified, but the condition retains an inherent subjectivity due to difficulty in defining terms such as personality, identity, ego-state and even amnesia.

In terms of the primary line coefficients this is the condition ;

However, the complete theory of the Abbe sine condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius f centered about the focal point, ..." In this sense, the traditional thin-lens definition and illustration of f-number is misleading, and defining it in terms of numerical aperture may be more meaningful.

In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition ;

Rock rust and desert patina are other terms which are also used for the condition, but less often.

When this condition is expressed in terms of probability densities, the result is called the Chapman – Kolmogorov equation.

In the most general and simple terms, feminist literary criticism before the 1970s — in the first and second waves of feminism — was concerned with the politics of women's authorship and the representation of women's condition within literature.

However, definitions are usually still phrased in terms of regularity, since this condition is more well known than any weaker one.

As Bohm and Hiley worded it, " the Schrodinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...] the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles ".

In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set.

Dermatitis comes from the Greek word for skinand both terms refer to exactly the same skin condition.