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Irrational Winding Of A Torus
In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the ''n''-dimensional torus :\mathbb^n = \underbrace_n which is represented by the following differential equations with respect to the standard angular coordinates (''θ''1, ''θ''2, ..., ''θ''''n''): :\frac=\omega_1, \quad \frac=\omega_2,\quad \ldots, \quad \frac=\omega_n. The solution of these equations can explicitly be expressed as :\Phi_\omega^t(\theta_1, \theta_2, \dots, \theta_n) = (\theta_1+\omega_1 t, \theta_2+\omega_2 t, \dots, \theta_n+\omega_n t) \bmod 2\pi. If we represent the torus as \mathbb = \mathbb^n / \mathbb^n we see that a starting point is moved by the flow in the direction ''ω'' = (''ω''1, ''ω''2, ..., ''ω''''n'') at constant speed and when it reaches the border of the unitary ''n''-cube it jumps to the opposite face of the cube. For a linear flow on the torus either all orbits are periodic or all orbits ar ... [...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] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) where the group operation on the left side of the equation is that of ''G'' and on the right side that of ''H''. From this property, one can deduce that ''h'' maps the identity element ''eG'' of ''G'' to the identity element ''eH'' of ''H'', : h(e_G) = e_H and it also maps inverses to inverses in the sense that : h\left(u^\right) = h(u)^. \, Hence one can say that ''h'' "is compatible with the group structure". Older notations for the homomorphism ''h''(''x'') may be ''x''''h'' or ''x''''h'', though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that ''h''(''x'') becomes simply xh. In areas of mathematics where one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace Topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced topology, or the trace topology). Definition Given a topological space (X, \tau) and a subset S of X, the subspace topology on S is defined by :\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace. That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X, \tau). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X, \tau). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map :\iota: S \hookrightarrow X is continuous. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class ''C''''r'' for a fixed , and all morphisms are differentiable of class ''C''''r''. Immersed submanifolds An immersed submanifold of a manifold ''M'' is the image ''S'' of an immersion map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset ''S'' together with a topology and differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category ''C'' is an epimorphism in the dual category ''C''op. Every section is a monomorphism, and every retraction is an epimorphism. Relation to invertibility Left-invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meaning ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Subspace
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Mapping
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
