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Bendixson–Dulac Theorem
In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a C^1 function \varphi(x, y) (called the Dulac function) such that the expression :\frac + \frac has the same sign (\neq 0) almost everywhere in a simply connected region of the plane, then the plane autonomous system : \frac = f(x,y), : \frac = g(x,y) has no nonconstant periodic solutions lying entirely within the region. "Almost everywhere" means everywhere except possibly in a set of measure 0, such as a point or line. The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac in 1923 using Green's theorem. Proof Without loss of generality, let there exist a function \varphi(x, y) such that :\frac +\frac >0 in simply connected region R. Let C be a closed trajectory of the plane autonomous system in R. Let D be the interior of C. Then by Green's theorem, : \begin & \iint_D \left( \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, a dynamical system has a State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dulac in the ''Shrek'' film series
{{disambig, surname ...
Dulac can refer to: People * Bill DuLac, American football player * Catherine Dulac, a professor for molecular biology * Edmund Dulac, French book illustrator * Germaine Dulac, French film director and early film theorist * Henri Dulac, French mathematician Places * Dulac, Louisiana, United States See also * Duloc, the kingdom formerly ruled by Lord Farquaad Lord Maximus Farquaad is the main antagonist of the 2001 animated feature film ''Shrek'', as well as ''Shrek 4-D'' and the musical. He is voiced by John Lithgow. He does not appear in William Steig's original picture book of the same name. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of ''almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Autonomous System
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Solution
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi 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. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A function with period will repeat on intervals of length , and these intervals are sometimes also referred to as periods of the function. Geometrically, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivar Bendixson
Ivar Otto Bendixson (1 August 1861 – 29 November 1935) was a Swedish mathematician. Biography Bendixson was born on 1 August 1861 at Villa Bergshyddan, Djurgården, Oscar Parish, Stockholm, Sweden, to a middle-class family. His father Vilhelm Emanuel Bendixson was a merchant, and his mother was Tony Amelia Warburg. On completing secondary education in Stockholm, he obtained his school certificate on 25 May 1878. On 13 September 1878 he enrolled to the Royal Institute of Technology in Stockholm. In 1879 Bendixson went to Uppsala University and graduated with the equivalent of a Master's degree on 27 January 1881. Graduating from Uppsala, he went on to study at the newly opened Stockholm University College after which he was awarded a doctorate by Uppsala University on 29 May 1890. On 10 June 1890 Bendixson was appointed as a docent at Stockholm University College. He then worked as an assistant to the professor of mathematical analysis from 5 March 1891 until 31 May 1892. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Dulac
Henri Claudius Rosarius Dulac (3 October 1870, Fayence – 2 September 1955, Fayence) was a French mathematician. Life Born in Fayence, France, Dulac graduated from École Polytechnique (Paris, class of 1892) and obtained a Doctorate in Mathematics. He started to teach a class of mathematic analysis at University, in Grenoble (France), Algiers (today Algeria) and Poitiers (France). Holder of a pulpit in pure mathematics in the Sciences University of Lyon (France) in 1911, his teaching was suspended during the first world war (1914 – 1918) and he had to serve as officer in the French army. After the war, he became holder of a pulpit of differential and integral calculus and also taught in École Centrale Lyon. He became examiner at École Polytechnique (Paris) and President of the admission jury. Awarded Officer of Legion d'honneur, the French order established by Napoleon and associate member of the French Academy of Sciences, he published part of Euler's works and contribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriented, piecewise smooth, simple closed curve in a plane, and let be the region bounded by . If and are functions of defined on an open region containing and have continuous partial derivatives there, then \oint_C (L\, dx + M\, dy) = \iint_ \left(\frac - \frac\right) dx\, dy where the path of integration along is anticlockwise. In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter. Proof when ''D'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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