Character theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves.
Modular representation theoryModular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Weyl character formulaIn mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation.
Representation theory of finite groupsThe representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero.
Displacement mappingDisplacement mapping is an alternative computer graphics technique in contrast to bump, normal, and parallax mapping, using a texture or height map to cause an effect where the actual geometric position of points over the textured surface are displaced, often along the local surface normal, according to the value the texture function evaluates to at each point on the surface. It gives surfaces a great sense of depth and detail, permitting in particular self-occlusion, self-shadowing and silhouettes; on the other hand, it is the most costly of this class of techniques owing to the large amount of additional geometry.
Character (mathematics)In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. multiplicative character A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers.
HeightmapIn computer graphics, a heightmap or heightfield is a raster image used mainly as Discrete Global Grid in secondary elevation modeling. Each pixel stores values, such as surface elevation data, for display in 3D computer graphics. A heightmap can be used in bump mapping to calculate where this 3D data would create shadow in a material, in displacement mapping to displace the actual geometric position of points over the textured surface, or for terrain where the heightmap is converted into a 3D mesh.
Bump mappingBump mapping is a texture mapping technique in computer graphics for simulating bumps and wrinkles on the surface of an object. This is achieved by perturbing the surface normals of the object and using the perturbed normal during lighting calculations. The result is an apparently bumpy surface rather than a smooth surface although the surface of the underlying object is not changed. Bump mapping was introduced by James Blinn in 1978. Normal mapping is the most common variation of bump mapping used.
Texture mappingTexture mapping is a method for mapping a texture on a . Texture here can be high frequency detail, surface texture, or color. The original technique was pioneered by Edwin Catmull in 1974. Texture mapping originally referred to diffuse mapping, a method that simply mapped pixels from a texture to a 3D surface ("wrapping" the image around the object).
AttentionAttention is the concentration of awareness on some phenomenon to the exclusion of other stimuli. It is a process of selectively concentrating on a discrete aspect of information, whether considered subjective or objective. William James (1890) wrote that "Attention is the taking possession by the mind, in clear and vivid form, of one out of what seem several simultaneously possible objects or trains of thought. Focalization, concentration, of consciousness are of its essence.
Character encodingCharacter encoding is the process of assigning numbers to graphical characters, especially the written characters of human language, allowing them to be stored, transmitted, and transformed using digital computers. The numerical values that make up a character encoding are known as "code points" and collectively comprise a "code space", a "code page", or a "character map". Early character codes associated with the optical or electrical telegraph could only represent a subset of the characters used in written languages, sometimes restricted to upper case letters, numerals and some punctuation only.
TrajectoryA trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously. The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass. In control theory, a trajectory is a time-ordered set of states of a dynamical system (see e.
Eye movementEye movement includes the voluntary or involuntary movement of the eyes. Eye movements are used by a number of organisms (e.g. primates, rodents, flies, birds, fish, cats, crabs, octopus) to fixate, inspect and track visual objects of interests. A special type of eye movement, rapid eye movement, occurs during REM sleep. The eyes are the visual organs of the human body, and move using a system of six muscles. The retina, a specialised type of tissue containing photoreceptors, senses light.
Hyperbolic trajectoryIn astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.
DreamWorks AnimationDreamWorks Animation LLC (DWA, also known as DreamWorks Animation Studios and simply known as DreamWorks) is an American animation studio that produces animated films and television programs and is a subsidiary of Universal Pictures, a division of NBCUniversal, which is itself a division of Comcast. The studio has released a total of 46 feature films as of June 2023, 44 of which were theatrically released. Their catalogue includes several of the highest-grossing animated films of all time, with Shrek 2 (2004) having been the highest at the time of its release.
Eye movement in readingEye movement in reading involves the visual processing of written text. This was described by the French ophthalmologist Louis Émile Javal in the late 19th century. He reported that eyes do not move continuously along a line of text, but make short, rapid movements (saccades) intermingled with short stops (fixations). Javal's observations were characterised by a reliance on naked-eye observation of eye movement in the absence of technology.
Parabolic trajectoryIn astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy). Under standard assumptions a body traveling along an escape orbit will coast along a parabolic trajectory to infinity, with velocity relative to the central body tending to zero, and therefore will never return.
Traditional animationTraditional animation (or classical animation, cel animation, or hand-drawn animation) is an animation technique in which each frame is drawn by hand. The technique was the dominant form of animation in cinema until the end of the 20th century, when there was a shift to computer animation in the industry, specifically 3D computer animation. Animation production usually begins after a story is converted into an animation film script, from which a storyboard is derived.
Radial trajectoryIn astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line. There are three types of radial trajectories (orbits). Radial elliptic trajectory: an orbit corresponding to the part of a degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. The relative speed of the two objects is less than the escape velocity.
