In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number. For instance, every function that has bounded first derivatives is Lipschitz. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed point theorem. We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous = continuouswhere 0 < α ≤ 1.
We have Lipschitz continuous ⊂ continuous ⊂ bounded variation ⊂ differentiable everywhere Given two metric spaces and, where dX denotes the metric on the set X and dY is the metric on set Y, a function f: X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X, d Y ≤ K d X. Any such K is referred to as a Lipschitz constant for the function f; the smallest constant is sometimes called the Lipschitz constant. If K = 1 the function is called a short map, if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction. In particular, a real-valued function f: R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x1 and x2, | f − f | ≤ K | x 1 − x 2 |. In this case, Y is the set of real numbers R with the standard metric dY = |y1 − y2|, X is a subset of R. In general, the inequality is satisfied if x1 = x2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x1 ≠ x2, d Y d X ≤ K.
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by K. The set of lines of slope K passing through a point on the graph of the function forms a circular cone, a function is Lipschitz if and only if the graph of the function everywhere lies outside of this cone. A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition. More a function f defined on X is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on X if there exists a constant M > 0 such that d Y ≤ M d X α for all x and y in X. Sometimes a Hölder condition of order α is called a uniform Lipschitz condition of order α > 0.
If there exists a K ≥ 1 with 1 K d X ≤ d Y ≤ K d X f is called bilipschitz. A bilipschitz mapping is injective, is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is Lipschitz. Lipschitz continuous functions Lipschitz continuous functions that are not everywhere differentiable Lipschitz continuous functions that are everywhere differentia
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise and in mathematics and physics as infinite-dimensional function spaces; the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, ergodic theory. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.
The success of Hilbert space methods ushered in a fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes, in analogy with Cartesian coordinates in the plane; when that set of axes is countably infinite, the Hilbert space can be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is in the older literature referred to as the Hilbert space.
Linear operators on a Hilbert space are fairly concrete objects: in good cases, they are transformations that stretch the space by different factors in mutually perpendicular directions in a sense, made precise by the study of their spectrum. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ3, equipped with the dot product; the dot product takes two vectors x and y, produces a real number x · y. If x and y are represented in Cartesian coordinates the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3; the dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: · y = ax1 · y + bx2 · y for any scalars a, b, vectors x1, x2, y, it is positive definite: for all vectors x, x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a inner product. A vector space equipped with such an inner product is known as a inner product space.
Every finite-dimensional inner product space is a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted ||x||, to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ. Multivariable calculus in Euclidean space relies on the ability to compute limits, to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n consisting of vectors in ℝ3 is convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞. Just as with a series of scalars, a series of vectors that converges also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞; this property expresses the completeness of
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Iterated function system
In mathematics, iterated function systems are a method of constructing fractals. IFS fractals are more related to set theory than fractal geometry, they were introduced in 1981. IFS fractals, as they are called, can be of any number of dimensions, but are computed and drawn in 2D; the fractal is made up of the union of several copies of itself, each copy being transformed by a function. The canonical example is the Sierpiński triangle; the functions are contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of, made up of copies of itself, ad infinitum; this is the source of its self-similar fractal nature. Formally, an iterated function system is a finite set of contraction mappings on a complete metric space. Symbolically, N ∈ N is an iterated function system if each f i is a contraction on the complete metric space X. Hutchinson showed that, for the metric space R n, such a system of functions has a unique nonempty compact fixed set S.
One way of constructing a fixed set is to start with an initial point of set S0 and iterate the actions of the fi, taking Sn+1 to be the union of the images of Sn under the fi. Symbolically, the unique fixed set S ⊆ X has the property; the set S is thus the fixed set of the Hutchinson operator H = ⋃ i = 1 N f i. The existence and uniqueness of S is a consequence of the contraction mapping principle, as is the fact that lim n → ∞ H ∘ n = S for any nonempty compact set A in X.. Random elements arbitrarily close to S may be obtained by the "chaos game," described below, it was shown that the IFSs of noncontractive type can yield attractors. These arise in projective spaces, though classical irrational rotation on the circle can be adapted too; the collection of functions f i generates a monoid under composition. If there are only two such functions, the monoid can be visualized as a binary tree, where, at each node of the tree, one may compose with the one or the other function. In general, if there are k functions one may visualize the monoid as a full k-ary tree known as a Cayley tree.
Sometimes each function f i is required to be a linear, or more an affine and hence represented by a matrix. However, IFSs may be built from non-linear functions, including projective transformations and Möbius transformations; the Fractal flame is an example of an IFS with nonlinear functions. The most common algorithm to compute IFS fractals is called the "chaos game", it consists of picking a random point in the plane iteratively applying one of the functions chosen at random from the function system to transform the point to get a next point. An alternative algorithm is to generate each possible sequence of functions up to a given maximum length, to plot the results of applying each of these sequences of functions to an initial point or shape; each of these algorithms provides a global construction which generates points distributed across the whole fractal. If a small area of the fractal is being drawn, many of these points will fall outside of the screen boundaries; this makes zooming into an IFS construction drawn in this manner impractical.
Although the theory of IFS requires each function to be contractive, in practice software that implements IFS only require that the whole system be contractive on average. PIFS called local iterated function systems, give good image compression for photographs that don't seem to have the kinds of self-similar structure shown by simple IFS factals. Fast algorithms exist to generate an image from a set of IFS or PIFS parameters, it is faster and requires much less storage space to store a description of how it was created, transmit that description to a destination device, regenerate that image anew on the destination device, than to store and transmit the color of each pixel in the image. The inverse problem is more difficult: given some original arbitrary digital image such as a digital p