Hamiltonian mechanicsHamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
Banach–Alaoglu theoremIn functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.
Lp spaceDISPLAYTITLE:Lp space In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces.
Hamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system.
Bolza surfaceIn mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus with the highest possible order of the conformal automorphism group in this genus, namely of order 48 (the general linear group of matrices over the finite field ). The full automorphism group (including reflections) is the semi-direct product of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation in .
Form (botany)In botanical nomenclature, a form (forma, plural formae) is one of the "secondary" taxonomic ranks, below that of variety, which in turn is below that of species; it is an infraspecific taxon. If more than three ranks are listed in describing a taxon, the "classification" is being specified, but only three parts make up the "name" of the taxon: a genus name, a specific epithet, and an infraspecific epithet. The abbreviation "f." or the full "forma" should be put before the infraspecific epithet to indicate the rank.
Normed vector spaceIn mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms: Non-negativity: for every ,. Positive definiteness: for every , if and only if is the zero vector.
Schwarz minimal surfaceIn differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz. In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces. The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the boundary lines also produce valid minimal surfaces that can be continuously joined to the original solution.
Reflexive spaceIn the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space.
SlopeIn mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "y = mx + c". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line.
Triply periodic minimal surfaceIn differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in R3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise. TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc.
Differential geometry of surfacesIn mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.
