In mathematics convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. The term convolution refers to the process of computing it, it is defined as the integral of the product of the two functions after one is shifted. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f or g is reflected about the y-axis. For continuous functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, computer vision, natural language processing and signal processing and differential equations; the convolution can be defined for functions on Euclidean space, other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. A discrete convolution can be defined for functions on the set of integers.

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution; the convolution of f and g is written f∗g, denoting the operator with the symbol ∗. It is defined as the integral of the product of the two functions after one is shifted; as such, it is a particular kind of integral transform: ≜ ∫ − ∞ ∞ f g d τ. An equivalent definition is: ≜ ∫ − ∞ ∞ f g d τ. While the symbol t is used above, it need not represent the time domain, but in that context, the convolution formula can be described as a weighted average of the function f at the moment t where the weighting is given by g shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function. For functions f, g supported on only [0, ∞), the integration limits can be truncated, resulting in: = ∫ 0 t f g d τ for f, g: [ 0, ∞ ) → R.

For the multi-dimensional formulation of convolution, see domain of definition. A common engineering convention is: f ∗ g ≜ ∫ − ∞ ∞ f g d τ ⏟, which has to be interpreted to avoid confusion. For instance, f ∗ g is equivalent to. Convolution describes the output of an important class of operations known as linear time-invariant. See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created; the existing ones are only modified. In other words, the output transform is the pointwise product of the input transform with a third transform. See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754.

An expression of the type: ∫ f ⋅ g d u is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Pa

Olbracht Łaski was a Polish nobleman, an alchemist and courtier during the reign of Stephen Batory. Łaski was suspected of plotting to seize the Polish throne in 1575, following the brief reign of Henry Valois. This episode is featured in the opéra-comique Le roi malgré lui by Emmanuel Chabrier.Łaski was notable for his fine beard, which Holinshed noted was of "such length and breadth, as that lying in his bed, parting it with his hands, the same overspread his breasts and shoulders, himself delighting therein, reputing it an ornament."Łaski arrived in England in April 1583. The French ambassador, Michel de Castelnau suggested that his visit was prompted by a desire to persuade the Muscovy Company to refrain from selling arms to Ivan the Terrible, he was provided with lodgings at Southwark. Łaski convinced John Dee to visit Poland. Count Albert Laski Ryszard Zieliński, Roman Żelewski Olbracht Łaski: Od Kieżmarku do Londynu, Warsaw: Czytelnik, ISBN 83-07-00656-2 Kraushar, Aleksander. Olbracht Łaski wojewoda sieradzki: wizerunek historyczny na tle dziejów Polski XVI wieku, Volumes 1, 2.

Gebethner i Wolff. OCLC 749958706. OCLC 749958746

The smooth-fronted caiman known as Schneider's dwarf caiman or Schneider's smooth-fronted caiman, is a crocodilian from South America, where it is native to the Amazon and Orinoco Basins. It is the second-smallest species of the family Alligatoridae, the smallest being Cuvier's dwarf caiman from tropical South America and in the same genus. An adult grows to around 1.2 to 1.6 m in length and weighs between 9 and 20 kg. Exceptionally large males can reach 36 kg in weight; the smooth-fronted caiman was first described by the German classicist and naturalist Johann Gottlob Schneider in 1801. The genus name Paleosuchus is derived from the Greek palaios meaning "ancient" and soukhos meaning "crocodile"; this refers to the belief that this crocodile comes from an ancient lineage that diverged from other species of caimans some 30 million years ago. The specific name trigonatus is derived from the Greek trigonos meaning "three-cornered" and Latin atus meaning "provided with" and refers to the triangular shape of the head.

The head of the smooth-fronted caiman is similar in appearance to that of the spectacled caiman, but no bony ridge or "spectacle" occurs between the eyes. The scutes on the back of the neck and the tail are large and sharp, it has ossified body armour on both its dorsal and ventral surfaces. The short tail is broad at its base and flattened dorsoventrally in contrast to most species of crocodilians which have laterally flattened tails; the bony scutes on the tail have sideways projections. This caiman is a dark greyish-brown with mid-brown eyes. Males grow to about 1.7 to 2.3 m long, with the largest recorded specimen being 2.6 m. Females do not exceed 1.4 m. It is a robust crocodilian, strong for its size, tends to carry its head high with its neck angled upwards; the smooth-fronted caiman is native to the Amazon and Orinoco Basins in South America and is found in Bolivia, Colombia, French Guiana, Peru and Venezuela. It inhabits small streams in forested areas where in some cases, the water may be insufficiently deep for it to submerge itself.

It is seen in open areas and does not bask in the sun in captivity. The adult smooth-fronted caiman has cryptic habits and is observed by day because it hides in underwater burrows or may spend much of its time up to 100 m away from water, concealed in dense undergrowth, in hollow logs, or under fallen trees. Males are territorial and females have small home ranges. Adults are semiterrestrial and feed on such animals as porcupines, snakes and lizards, consuming few fish or molluscs. Hatchlings feed on insects in their first few weeks, graduating to larger prey as they grow. Juvenile mortality is high, but adult mortality is low, although large carnivores such as the jaguar sometimes prey on them. Females become mature and start to breed at about 11 years and males at about 20; the female builds a large mound nest out of leaf litter and soil at the end of the dry season or may use a pre-existing nest. A clutch of 10 to 15 eggs is covered with further nesting material; some heat is generated by the decaying vegetation and good insulation helps to retain this.

The nests are built against the sides of termite mounds and metabolic heat generated by the termites helps to maintain the clutch at a near constant temperature. The eggs need to be maintained at a temperature of 31 to 32 °C for the production of male offspring; the incubation period is about 115 days and the female caiman remains near the nest for at least the earlier part of this time, providing protection against predators. During incubation, roots may grow through the nest, soil from the termite mound may cement the eggs together; this means parental assistance is necessary when the eggs hatch to enable the hatchlings to escape from the nest chamber. Having transported the newly emerged juveniles to a nursery area, the female stays with them for a few weeks after which time they disperse; the female may miss a year before breeding again. The smooth-fronted caiman is not extensively hunted because its skin contains many bony scutes, which make it of little use for leather; the animals are collected in Guyana, for the pet trade.

The main threats to this species are destruction of its forest habitat and pollution of its environment by gold mining activities. Over one million individuals are estimated to remain in the wild, the species is rated by the IUCN as being of least concern, it is listed on Appendix II of CITES, designed to limit overexploitation through international trade. Data related to Paleosuchus trigonatus at Wikispecies Media related to Paleosuchus trigonatus at Wikimedia Commons