In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.
https://en.wikipedia.org/wiki/Scalar_potential
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
https://en.wikipedia.org/wiki/Ring_(mathematics)
Non-ringsThe set of natural numbers N
with the usual operations is not a ring, since ( N , + ) 
is not even a group (not all the elements are invertible with respect to addition – for instance, there is no natural number which can be added to 3 to get 0 as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers Z . 
The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).Let R be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution:
( f ∗ g ) ( x ) = ∫ − ∞ ∞ f ( y ) g ( x − y ) d y .
Then R is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of R.https://en.wikipedia.org/wiki/Ring_(mathematics)
R.
An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Nilpotent
https://en.wikipedia.org/wiki/Nilpotent_matrix
https://en.wikipedia.org/wiki/Restriction_(mathematics)
https://en.wikipedia.org/wiki/Ring_homomorphism
https://en.wikipedia.org/wiki/Bijection
https://en.wikipedia.org/wiki/Inner_automorphism
https://en.wikipedia.org/wiki/Coset
https://en.wikipedia.org/wiki/Kernel_(algebra)#Ring_homomorphisms
https://en.wikipedia.org/wiki/Semiring
https://en.wikipedia.org/wiki/If_and_only_if
https://en.wikipedia.org/wiki/Logical_connective
https://en.wikipedia.org/wiki/Truth_function
https://en.wikipedia.org/wiki/Classical_logic
https://en.wikipedia.org/wiki/Double_negation#Elimination_and_introduction
https://en.wikipedia.org/wiki/Involution_(mathematics)
https://en.wikipedia.org/wiki/Reflection_(mathematics)
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
https://en.wikipedia.org/wiki/Triple_point
https://en.wikipedia.org/wiki/Helium-4
In mathematics, an involution, involutory function, or self-inverse function[1] is a function f that is its own inverse, f(f(x)) = x
for all x in the domain of f.[2] Equivalently, applying f twice produces the original value.
General properties
Any involution is a bijection.
The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation ( x ↦ − x
), reciprocation ( x ↦ 1 / x 
), and complex conjugation ( z ↦ z ¯ 
) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition g ∘ f of two involutions f and g is an involution if and only if they commute: g ∘ f = f ∘ g.[3]
https://en.wikipedia.org/wiki/Involution_(mathematics)
In mathematics, a reflection (also spelled reflexion)[1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
The term reflection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.
https://en.wikipedia.org/wiki/Reflection_(mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function.
In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point.
https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.[1] It is that temperature and pressure at which the sublimation, fusion, and vaporisation curves meet. For example, the triple point of mercury occurs at a temperature of −38.8 °C (−37.8 °F) and a pressure of 0.165 mPa.
In addition to the triple point for solid, liquid, and gas phases, a triple point may involve more than one solid phase, for substances with multiple polymorphs. Helium-4 is unusual in that it has no sublimation/deposition curve and therefore no triple points where its solid phase meets its gas phase. Instead, it has a vapor-liquid-superfluid point, a solid-liquid-superfluid point, a solid-solid-liquid point, and a solid-solid-superfluid point. None of these should be confused with the Lambda Point, which is not any kind of triple point.
The triple point of water was used to define the kelvin, the base unit of thermodynamic temperature in the International System of Units (SI).[2] The value of the triple point of water was fixed by definition, rather than by measurement, but that changed with the 2019 redefinition of SI base units. The triple points of several substances are used to define points in the ITS-90 international temperature scale, ranging from the triple point of hydrogen (13.8033 K) to the triple point of water (273.16 K, 0.01 °C, or 32.018 °F).
The term "triple point" was coined in 1873 by James Thomson, brother of Lord Kelvin.[3]
https://en.wikipedia.org/wiki/Triple_point
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