Geometric transformationIn mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations. Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations).
History of quantum mechanicsThe history of quantum mechanics is a fundamental part of the history of modern physics. The major chapters of this history begin with the emergence of quantum ideas to explain individual phenomena -- blackbody radiation, the photoelectric effect, solar emission spectra -- an era called the Old or Older quantum theories. The invention of wave mechanics by Schrodinger and expanded by many others triggers the "modern" era beginning around 1925.
Transformation geometryIn mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geometry, that focuses on proving theorems. For example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line.
Rigid transformationIn mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space.
Cloud chamberA cloud chamber, also known as a Wilson cloud chamber, is a particle detector used for visualizing the passage of ionizing radiation. A cloud chamber consists of a sealed environment containing a supersaturated vapor of water or alcohol. An energetic charged particle (for example, an alpha or beta particle) interacts with the gaseous mixture by knocking electrons off gas molecules via electrostatic forces during collisions, resulting in a trail of ionized gas particles.
Periodic functionA periodic function or cyclic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic. A function f is said to be periodic if, for some nonzero constant P, it is the case that for all values of x in the domain.
Sine waveA sine wave, sinusoidal wave, or sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is: where: A, amplitude, the peak deviation of the function from zero. f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
Delta rayA delta ray is a secondary electron with enough energy to escape a significant distance away from the primary radiation beam and produce further ionization, and is sometimes used to describe any recoil particle caused by secondary ionization. The term was coined by J. J. Thomson. A delta ray is characterized by very fast electrons produced in quantity by alpha particles or other fast energetic charged particles knocking orbiting electrons out of atoms.
Yo (Cyrillic)Yo, Jo or Io (Ё ё; italics: Ё ё; jɵ) is a letter of the Cyrillic script. In Unicode, the letter is named CYRILLIC CAPITAL/SMALL LETTER IO. In English, the letter Yo is romanized using the Latin ë (according to the ALA–LC and British Standards), ë (yë word-initially) (BGN/PCGN) or yo/jo (orthographic transcription) for Russian, and as i͡o (ALA–LC), yo (BGN/PCGN), or ë (BSI) for Belarusian. In international systems, Yo is romanized as ë (ISO 9). It was derived from the Cyrillic letter Ye (Е е).
Cyrillic alphabetsNumerous Cyrillic alphabets are based on the Cyrillic script. The early Cyrillic alphabet was developed in the 9th century AD and replaced the earlier Glagolitic script developed by the Byzantine theologians Cyril and Methodius. It is the basis of alphabets used in various languages, past and present, Slavic origin, and non-Slavic languages influenced by Russian. As of 2011, around 252 million people in Eurasia use it as the official alphabet for their national languages. About half of them are in Russia.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Error functionIn mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.
