Maximum principleIn the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study.
Elliptic partial differential equationIn mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic and hyperbolic PDEs, which generally model phenomena that change in time. The canonical examples of elliptic PDEs are Laplace's equation and Poisson's equation.
Open setIn general topology and mathematical analysis, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).
Eberhard HopfEberhard Frederich Ferdinand Hopf (April 4, 1902, in Salzburg, Austria-Hungary – July 24, 1983, in Bloomington, Indiana, USA) was a German mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential geometry.
Harmonic functionIn mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb {R} } , where U {\displaystyle U} is an open subset of R n {\displaystyle \mathbb {R} ^{n}} , that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + ⋯ + ∂ 2 f ∂ x n 2 = 0 {\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0} everywhere on U {\displaystyle U} .
Symmetric matrixIn linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric⟺A=AT.{\displaystyle A{\text{ is symmetric}}\iff A=A^{\textsf {T}}.}Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal.
Richard CourantRichard Courant (German:; January 8, 1888 – January 27, 1972) was a German-American mathematician. He is best known by the general public for the book What is Mathematics?, co-written with Herbert Robbins. His research focused on the areas of real analysis, mathematical physics, the calculus of variations and partial differential equations. He wrote textbooks widely used by generations of students of physics and mathematics.
Bounded setIn mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric.Methoden der mathematischen PhysikMethoden der mathematischen Physik (translated into English with the title Methods of Mathematical Physics) is a 1924 book, in two volumes totalling around 1000 pages, published under the names of Richard Courant and David Hilbert. It was a comprehensive treatment of the "methods of mathematical physics" of the time. The second volume is devoted to the theory of partial differential equations.
David HilbertDavid Hilbert (; German:; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of all time.
Maximum and minimumIn mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.Differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator (which is an operation that accepts a function and returns another function, i.e. a higher-order function). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
Carl Friedrich GaussJohann Carl Friedrich Gauss ( ; German: Gauß; 30 April 1777 – 23 February 1855) was a German polymath active as a mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. His mathematical contributions spanned the branches of number theory, algebra, analysis, geometry, statistics, and probability. Gauss was director of the Göttingen Observatory in Germany and professor of astronomy from 1807 until his death in 1855.*As an Amazon Associate I earn from qualifying purchases.