Pterophyllum

Pterophyllum is a small genus of freshwater fish from the family Cichlidae known to most aquarists as angelfish. All Pterophyllum species originate from the Amazon Basin, Orinoco Basin and various rivers in the Guiana Shield in tropical South America; the three species of Pterophyllum are unusually shaped for cichlids being laterally compressed, with round bodies and elongated triangular dorsal and anal fins. This body shape allows them to hide among roots and plants on a vertical surface. Occurring angelfish are striped longitudinally, colouration which provides additional camouflage. Angelfish prey on small fish and macroinvertebrates. All Pterophyllum species form monogamous pairs. Eggs are laid on a submerged log or a flattened leaf; as is the case for other cichlids, brood care is developed. Pterophyllum should not be confused with marine angelfish, perciform fish found on shallow ocean reefs; the recognized species in this genus are: Pterophyllum altum Pellegrin, 1903 Pterophyllum leopoldi Pterophyllum scalare The freshwater angelfish was described in 1824 by F. Schultze.

Pterophyllum is derived from phyllon. In 1906, J. Pellegrin described P. altum. In 1963, P. leopoldi was described by J. P. Gosse. Undescribed species may still exist in the Amazon Basin. New species of fish are discovered with increasing frequency, like P. scalare and P. leopoldi, the differences may be subtle. Scientific notations describe the P. leopoldi as having 29–35 scales in a lateral row and straight predorsal contour, whereas, P. scalare is described as having 35–45 scales in a lateral row and a notched predorsal contour. P. leopoldi shows the same coloration as P. scalare, but a faint stripe shows between the eye stripe and the first complete body stripe and a third incomplete body stripe exists between the two main body stripes that extends three-fourths the length of the body. P. scalare's body does not show the stripe between the eye stripe and first complete body stripe at all, the third stripe between the two main body stripes extends downward more than a half inch, if present.

P. leopoldi fry develop three to eight body stripes, with all but one to five fading away as they mature, whereas P. scalare only has two in true wild form throughout life. Angelfish were bred in captivity for at least 30 years prior to P. leopoldi being described. Angelfish are one of the most kept freshwater aquarium fish, as well as the most kept cichlid, they are prized for their unique shape and behavior. It was not until the late 1920s to early 1930s that the angelfish was bred in captivity in the United States; the most kept species in the aquarium is Pterophyllum scalare. Most of the individuals in the aquarium trade are captive-bred. Sometimes, captive-bred Pterophyllum altum is available. Pterophyllum leopoldi is the hardest to find in the trade. Angelfish are kept in a warm aquarium, ideally around 80 °F. Though angelfish are members of the cichlid family, they are peaceful when not mating. P. scalare is easy to breed in the aquarium, although one of the results of generations of inbreeding is that many breeds have completely lost their rearing instincts, resulting in the tendency of the parents to eat their young.

In addition, it is difficult to identify the sex of any individual until it is nearly ready to breed. Angelfish pairs form long-term relationships where each individual will protect the other from threats and potential suitors. Upon the death or removal of one of the mated pair, breeders have experienced the total refusal of the remaining mate to pair up with any other angelfish and breed with subsequent mates. Depending upon aquarium conditions, P. scalare reaches sexual maturity at the age of six to 12 months or more. In situations where the eggs are removed from the aquarium after spawning, the pair is capable of spawning every seven to 10 days. Around the age of three years, spawning frequency decreases and ceases; when the pair is ready to spawn, they choose an appropriate medium upon which to lay the eggs, spend one to two days picking off detritus and algae from the surface. This medium may be a broad-leaf plant in the aquarium, a flat surface such as a piece of slate placed vertically in the aquarium, a length of pipe, or the glass sides of the aquarium.

The female deposits a line of eggs on the spawning substrate, followed by the male, which fertilizes the eggs. This process is repeated until a total of 100 to more than 1,200 eggs are laid, depending on the size and health of the female fish; as both parents care for the offspring throughout development, the pair takes turns maintaining a high rate of water circulation around the eggs by swimming close to the eggs and fanning them with their pectoral fins. In a few days, the eggs hatch and the fry remain attached to the spawning substrate. During this period, the fry survive by consuming the remains of their yolk sacs. At one week, the fry become free-swimming. Successful parents keep close watch on the eggs until then. At the free-swimming stage, the fry can be fed suitably sized live food. P. altum is notably difficult to breed in an aquarium environment. In pet stores, the freshwater angelfish is placed in the semiaggressive category; some tetras and barbs are compatible with angelfish, but ones small enough to fit in the mouth of the angelfish may be eaten.

Generous portions of food should be available so the angelfish do not get hungry and turn on their tank mates. Most stra

Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space – physical space. The scalar may either be a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, spin-zero quantum fields, such as the Higgs field; these fields are the subject of scalar field theory. Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U; the region U may be a set in some Euclidean space, Minkowski space, or more a subset of a manifold, it is typical in mathematics to impose further conditions on the field, such that it be continuous or continuously differentiable to some order.

A scalar field is a tensor field of order zero, the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. Physically, a scalar field is additionally distinguished by having units of measurement associated with it. In this context, a scalar field should be independent of the coordinate system used to describe the physical system—that is, any two observers using the same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as vector fields, which associate a vector to every point of a region, as well as tensor fields and spinor fields. More subtly, scalar fields are contrasted with pseudoscalar fields. In physics, scalar fields describe the potential energy associated with a particular force; the force is a vector field, which can be obtained as the gradient of the potential energy scalar field. Examples include: Potential fields, such as the Newtonian gravitational potential, or the electric potential in electrostatics, are scalar fields which describe the more familiar forces.

A temperature, humidity or pressure field, such as those used in meteorology. In quantum field theory, a scalar field is associated with spin-0 particles; the scalar field may be complex valued. Complex scalar fields represent charged particles; these include the charged Higgs field of the Standard Model, as well as the charged pions mediating the strong nuclear interaction. In the Standard Model of elementary particles, a scalar Higgs field is used to give the leptons and massive vector bosons their mass, via a combination of the Yukawa interaction and the spontaneous symmetry breaking; this mechanism is known as the Higgs mechanism. A candidate for the Higgs boson was first detected at CERN in 2012. In scalar theories of gravitation scalar fields are used to describe the gravitational field. Scalar-tensor theories represent the gravitational interaction through a scalar; such attempts are for example the Jordan theory as a generalization of the Kaluza–Klein theory and the Brans–Dicke theory. Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.

This field interacts Yukawa-like with the particles that get mass through it. Scalar fields are found within superstring theories as dilaton fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor. Scalar fields are supposed to cause the accelerated expansion of the universe, helping to solve the horizon problem and giving a hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless scalar fields in this context are known as inflatons. Massive scalar fields are proposed, using for example Higgs-like fields. Vector fields; some examples of vector fields include the electromagnetic field and the Newtonian gravitational field. Tensor fields, which associate a tensor to every point in space. For example, in general relativity gravitation is associated with the tensor field called Einstein tensor. In Kaluza–Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".

The dilaton scalar is found among the massless bosonic fields in string theory. Scalar field theory Vector-valued function

Lorentz scalar

In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g. the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged. A Lorentz scalar is not always seen to be an invariant scalar in the mathematical sense, but the resulting scalar value is invariant under any basis transformation applied to the vector space, on which the considered theory is based. A simple Lorentz scalar in Minkowski spacetime is the spacetime distance of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation. Other examples of Lorentz scalars are the "length" of 4-velocities, or the Ricci curvature in a point in spacetime from General relativity, a contraction of the Riemann curvature tensor there.

In special relativity the location of a particle in 4-dimensional spacetime is given by x μ = where x = v t is the position in 3-dimensional space of the particle, v is the velocity in 3-dimensional space and c is the speed of light. The "length" of the vector is a Lorentz scalar and is given by x μ x μ = η μ ν x μ x ν = 2 − x ⋅ x = d e f 2 where τ is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by η μ ν = η μ ν =; this is a time-like metric. The alternate signature of the Minkowski metric is used in which the signs of the ones are reversed. Η μ ν = η μ ν =. This is a space-like metric. In the Minkowski metric the space-like interval s is defined as x μ x μ = η μ ν x μ x ν = x ⋅ x − 2 = d e f s 2. We use the space-like Minkowski metric in the rest of this article; the velocity in spacetime is defined as v μ = d e f d x μ d τ = = = γ where γ = d e f 1 1 − v ⋅ v c 2. The magnitude of

Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors, they provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, are studied in functional analysis; the first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product induces an associated norm, thus an inner product space is a normed vector space. A complete space with an inner product is called a Hilbert space. An space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space.

Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. Formally, an inner product space is a vector space V over the field F together with an inner product, i.e. with a map ⟨ ⋅, ⋅ ⟩: V × V → F that satisfies the following three axioms for all vectors x, y, z ∈ V and all scalars a ∈ F: Conjugate symmetry: ⟨ x, y ⟩ = ⟨ y, x ⟩ ¯ Linearity in the first argument: ⟨ a x, y ⟩ = a ⟨ x, y ⟩ ⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩ Positive-definite: ⟨ x, x ⟩ > 0, x ∈ V ∖. Positive-definiteness and linearity ensure that: ⟨ x, x ⟩ = 0 ⇒ x = 0 ⟨ 0, 0 ⟩ = ⟨ 0 x, 0 x ⟩ = 0 ⟨ x, 0 x ⟩ = 0 Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, since we have: ⟨ x, x ⟩ = ⟨ x, x ⟩ ¯. Conjugate symmetry and linearity in the first variable imply ⟨ x, a y ⟩ = ⟨ a y, x ⟩ ¯ = a ¯ ⟨ y, x ⟩ ¯ = a ¯ ⟨ x, y ⟩ ⟨ x, y + z ⟩ = ⟨ y + z, x ⟩ ¯ = ⟨ y, x ⟩ ¯ + ⟨ z, x ⟩ ¯ = ⟨ x, y ⟩ + ⟨ x, z ⟩.

Scale (music)

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, a scale ordered by decreasing pitch is a descending scale; some scales contain different pitches when ascending than when descending, for example, the melodic minor scale. In the context of the common practice period, most or all of the melody and harmony of a musical work is built using the notes of a single scale, which can be conveniently represented on a staff with a standard key signature. Due to the principle of octave equivalence, scales are considered to span a single octave, with higher or lower octaves repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance between two successive notes of the scale. However, there is no need for scale steps to be equal within any scale and as demonstrated by microtonal music, there is no limit to how many notes can be injected within any given musical interval.

A measure of the width of each scale step provides a method to classify scales. For instance, in a chromatic scale each scale step represents a semitone interval, while a major scale is defined by the interval pattern T–T–S–T–T–T–S, where T stands for whole tone, S stands for semitone. Based on their interval patterns, scales are put into categories including diatonic, major and others. A specific scale is defined by its characteristic interval pattern and by a special note, known as its first degree; the tonic of a scale is the note selected as the beginning of the octave, therefore as the beginning of the adopted interval pattern. The name of the scale specifies both its tonic and its interval pattern. For example, C major indicates a major scale with a C tonic. Scales are listed from low to high pitch. Most scales are octave-repeating. An octave-repeating scale can be represented as a circular arrangement of pitch classes, ordered by increasing pitch class. For instance, the increasing C major scale is C–D–E–F–G–A–B–, with the bracket indicating that the last note is an octave higher than the first note, the decreasing C major scale is C–B–A–G–F–E–D–, with the bracket indicating an octave lower than the first note in the scale.

The distance between two successive notes in a scale is called a scale step. The notes of a scale are numbered by their steps from the root of the scale. For example, in a C major scale the first note is the second D, the third E and so on. Two notes can be numbered in relation to each other: C and E create an interval of a third. A single scale can be manifested at many different pitch levels. For example, a C major scale can be started at C4 and ascending an octave to C5; as long as all the notes can be played, the octave they take on can be altered. Scales may be described according to the number of different pitch classes they contain: Chromatic, or dodecatonic Octatonic: used in jazz and modern classical music Heptatonic: the most common modern Western scale Hexatonic: common in Western folk music Pentatonic: the anhemitonic form is common in folk music in Asian music. Many music theorists concur that the constituent intervals of a scale have a large role in the cognitive perception of its sonority, or tonal character.

"The number of the notes that make up a scale as well as the quality of the intervals between successive notes of the scale help to give the music of a culture area its peculiar sound quality." "The pitch distances or intervals among the notes of a scale tell us more about the sound of the music than does the mere number of tones."Scales may be described by their symmetry, such as being palindromic, chiral, or having rotational symmetry as in Messiaen's modes of limited transposition. Scales can be described by their distribution patterns and possibilities for notation. For example, a heliotonic scale is one that can be notated with one note head on each line and space, using only single and double alterations, thus all heliotonic scales are heptatonic. Since heliotonia is a metric of a scale's tone distribution pattern, is related to evenness, spectra variation, the Myhill Property; the notes of a scale form intervals with each of the other notes of the chord in combination. A 5-note scale has 10 of these harmonic intervals, a 6-note scale has 15, a 7-note scale has 21, an 8-note scale has 28.

Though the scale is not a chord, might never be heard more than one note at a time, still the absence and placement of certain key intervals plays a large part in the sound of the scale, the natural movement of melody within the scale, the selection of chords taken from the scale. A musical scale that contains tritones is called tritonic (though the expression is used for any sca

Scaler (video game)

Scaler known as Scaler: The Shapeshifting Chameleon, is a video game released in 2004 by Global Star Software for the GameCube and PlayStation 2 video game consoles. The Xbox version is not compatible with Xbox 360. Scaler follows the story of a lizard-loving 12-year-old boy named Bobby "Scaler" Jenkins who accidentately stumbles across an evil plot to dominate the world through use of mutated lizards. Prior to the start of the game, lizard-loving animal activist Bobby "Scaler" Jenkins discovers that five extra-dimensional humanoid reptilian creatures - the leader and his henchmen Jazz, Rhombus and Turbine - have disguised themselves as humans and intend to conquer the multiverse. Looger and his subordinates discover that Scaler kidnap him. During the torture, frustrated by Scaler's taunts, accidentally opens an extra-dimensional portal, transforming Scaler into a blue-skinned reptilian humanoid and releasing him from his restraints. Scaler escapes through the portal, Looger and his henchmen follow after him.

Scaler finds himself in an parallel universe where he encounters another reptilian man named Leon, who Scaler notes as having the same name as his estranged father. Leon challenges Scaler to retrieve a lizard egg being incubated in a mysterious mechanism; when Scaler does so and returns the egg to Leon, he persuades Leon to let him help retrieve the rest of the eggs scattered in the multiverse. Leon grudgingly agrees, reveals that Scaler can exchange Klokkies with his pet repadactyl Reppy to improve Scaler's abilities, including sharper claws and the ability to discharge static electricity from his body. Leon reveals that Scaler can obtain the ability to transform into other creatures by defeating enough of that creature. Among the transformations Scaler obtains during the game are Bakuldan, Doozum and Swoom; as they travel into this strange world of isolated and variegated islands filled with vicious reptiles with the mysterious Leon, aboard Reppy, Scaler discovers more of Looger's secrets.

He learns that Looger controls a network of unstable portals that are the only connections between the different dimensions in the "multiverse" through the use of a mysterious device. Any being in control of these portals would have the ability to move between the different worlds and capture them. And, what Looger plans to do, by mutating and cloning en masse lizards in the form of horrible mutants. All of this is done in order to conquer every plane of the multiverse, it becomes clear that Scaler must help Leon to rescue all the eggs and stop Looger, or lose all of the universes to darkness. Meanwhile, due to the time spent with Bobby, the long-lost memories of Leon began resurfacing and Scaler discovers that Leon is in fact his father. Years prior, he was a scientist, while performing an experiment with his invention, the portal compass, he was dragged accidentally by his device into Looger's dimension through a portal, imprisoned by Looger for years; the tortures inflicted on him by Looger in attempt to make him reveal more about his technology left Leon an amnesiac, stripping him from most of his memories.

Only now is he able to escape the lonely island. So Leon never, as Bobby thought, abandoned his mother. Leon is overjoyed to not only remember who he is, but to see his son again, who has become a great hero. However, Scaler struggles a little to accept the truth and accept a father who for so long time he have thought being "a loser freak," who forced his mother to take two jobs to scrape by. In the end, however, he forgives his father. After defeating Jazz, Turbine, Bootcamp and a few of the mutant monsters, all the while rescuing the 20 remaining lizard eggs and Leon arrive at Looger's stronghold. After defeating him and reclaiming the portal compass, they rush to a last portal meant to bring them home but Leon, having returned behind to save an egg fallen from a hole in Scaler's sack, remains on the other side of the portal while Bobby crosses over it. Bobby is back in Looger's basement, it at last occurs to him that his father did not make it in time; the portal closes, leaving Bobby screaming in horror for having, lost his father.

By unlocking the secret ending, an unexpected event takes place: as Leon said, if the player alters the multiverse by any means, it can produce unpredictable effects the merging of different versions of history. So, when Bobby screams in horror from the second loss of his father, the basement door opens behind him, his father stands in its frame, concerned by the sudden scream. By defeating Looger, Bobby did in fact alter time, Leon was never captured by the evil reptile overlord, meaning that he technically never left his family. Quite the opposite has happened in this timeframe, the Jenkins family are living together in that same house that Looger would have owned. Bobby is overjoyed but soon discovers that he still has his chameleon tongue, the reflex to eat flies, much to his own disgust. Not much Leon returns to the basement again, revealing by a monologue to the player, that he in fact maintained the memories of his journey with