France the French Republic, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, from the Rhine to the Atlantic Ocean, it is bordered by Belgium and Germany to the northeast and Italy to the east, Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic and Indian oceans; the country's 18 integral regions span a combined area of 643,801 square kilometres and a total population of 67.3 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Toulouse, Bordeaux and Nice. During the Iron Age, what is now metropolitan France was inhabited by a Celtic people. Rome annexed the area in 51 BC, holding it until the arrival of Germanic Franks in 476, who formed the Kingdom of Francia.
The Treaty of Verdun of 843 partitioned Francia into Middle Francia and West Francia. West Francia which became the Kingdom of France in 987 emerged as a major European power in the Late Middle Ages following its victory in the Hundred Years' War. During the Renaissance, French culture flourished and a global colonial empire was established, which by the 20th century would become the second largest in the world; the 16th century was dominated by religious civil wars between Protestants. France became Europe's dominant cultural and military power in the 17th century under Louis XIV. In the late 18th century, the French Revolution overthrew the absolute monarchy, established one of modern history's earliest republics, saw the drafting of the Declaration of the Rights of Man and of the Citizen, which expresses the nation's ideals to this day. In the 19th century, Napoleon established the First French Empire, his subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a tumultuous succession of governments culminating with the establishment of the French Third Republic in 1870.
France was a major participant in World War I, from which it emerged victorious, was one of the Allies in World War II, but came under occupation by the Axis powers in 1940. Following liberation in 1944, a Fourth Republic was established and dissolved in the course of the Algerian War; the Fifth Republic, led by Charles de Gaulle, remains today. Algeria and nearly all the other colonies became independent in the 1960s and retained close economic and military connections with France. France has long been a global centre of art and philosophy, it hosts the world's fourth-largest number of UNESCO World Heritage Sites and is the leading tourist destination, receiving around 83 million foreign visitors annually. France is a developed country with the world's sixth-largest economy by nominal GDP, tenth-largest by purchasing power parity. In terms of aggregate household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, human development.
France is considered a great power in global affairs, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a leading member state of the European Union and the Eurozone, a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, La Francophonie. Applied to the whole Frankish Empire, the name "France" comes from the Latin "Francia", or "country of the Franks". Modern France is still named today "Francia" in Italian and Spanish, "Frankreich" in German and "Frankrijk" in Dutch, all of which have more or less the same historical meaning. There are various theories as to the origin of the name Frank. Following the precedents of Edward Gibbon and Jacob Grimm, the name of the Franks has been linked with the word frank in English, it has been suggested that the meaning of "free" was adopted because, after the conquest of Gaul, only Franks were free of taxation.
Another theory is that it is derived from the Proto-Germanic word frankon, which translates as javelin or lance as the throwing axe of the Franks was known as a francisca. However, it has been determined that these weapons were named because of their use by the Franks, not the other way around; the oldest traces of human life in what is now France date from 1.8 million years ago. Over the ensuing millennia, Humans were confronted by a harsh and variable climate, marked by several glacial eras. Early hominids led a nomadic hunter-gatherer life. France has a large number of decorated caves from the upper Palaeolithic era, including one of the most famous and best preserved, Lascaux. At the end of the last glacial period, the climate became milder. After strong demographic and agricultural development between the 4th and 3rd millennia, metallurgy appeared at the end of the 3rd millennium working gold and bronze, iron. France has numerous megalithic sites from the Neolithic period, including the exceptiona
In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet, it was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. The construction of a Menger sponge can be described. Divide every face of the cube into 9 squares, like a Rubik's Cube; this will sub-divide the cube into 27 smaller cubes. Remove the smaller cube in the middle of each face, remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes; this is a level-1 Menger sponge. Repeat steps 2 and 3 for each of the remaining smaller cubes, continue to iterate ad infinitum; the second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, so on. The Menger sponge; the nth stage of the Menger sponge, Mn, is made up of 20n smaller cubes, each with a side length of n. The total volume of Mn is thus n; the total surface area of Mn is given by the expression 2n + 4n.
Therefore the construction's volume approaches zero. Yet any chosen surface in the construction will be punctured as the construction continues, so that the limit is neither a solid nor a surface; each face of the construction becomes a Sierpinski carpet, the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry; the number of these hexagrams, in descending size, is given by a n = 9 a n − 1 − 12 a n − 2, with a 0 = 1, a 1 = 6. The sponge's Hausdorff dimension is log 20/log 3 ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every curve is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one.
In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, might be embedded in any number of dimensions; the Menger sponge is a closed set. It has Lebesgue measure 0; because it contains continuous paths, it is an uncountable set. Formally, a Menger sponge can be defined as follows: M:= ⋂ n ∈ N M n where M0 is the unit cube and M n + 1:=. MegaMenger is a project aiming to build the largest fractal model, pioneered by Matt Parker of Queen Mary University of London and Laura Taalman of James Madison University; each small cube is made from 6 interlocking folded business cards, giving a total of 960 000 for a level-four sponge. The outer surfaces are covered with paper or cardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing. In 2014, twenty level-three Menger sponges were constructed, which
Lebesgue covering dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue. A modern definition is. An open cover of a topological space X is a family of open sets whose union contains X; the ply or order of a cover is the smallest number n such that each point of the space belongs to at most n sets in the cover. A refinement of a cover C is another cover, each of whose sets is a subset of a set in C; the covering dimension of a topological space X is defined to be the minimum value of n, such that every open cover C of X has an open refinement with ply n + 1 or below. If no such minimal n exists, the space is said to be of infinite covering dimension; as a special case, a topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in one open set of this refinement.
Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most two open arcs; that is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps. Any open cover of the unit disk in the two-dimensional plane can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient; the covering dimension of the disk is thus two. More the n-dimensional Euclidean space E n has covering dimension n. A non-technical illustration of these examples is given below. Homeomorphic spaces have the same covering dimension; that is, the covering dimension is a topological invariant. The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex.
The covering dimension of a normal space is equal to the large inductive dimension. Covering dimension of a normal space X is ≤ n if and only if for any closed subset A of X, if f: A → S n is continuous there is an extension of f to g: X → S n. Here, S n is the n dimensional sphere. A normal space X satisfies the inequality 0 ≤ dim X ≤ n if and only if for every locally finite open cover U = α ∈ A of the space X there exists an open cover V of the space X which can be represented as the union of n + 1 families V 1, V 2, …, V n + 1, where V i = α ∈ A, such that each V i contains disjoint sets and V i, α ⊂ U α for each i and α; the covering dimension of a paracompact Hausdorff space X is greater or equal to its cohomological dimension, one has H i = 0 for every sheaf A of abelian groups on X and every i larger than the covering dimension of X. Carathéodory's extension theorem Geometric set cover problem Dimension theory Metacompact space Point-finite collection Godement, Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092 Munkres, James R..
Topology. Prentice-Hall. ISBN 0-13-181629-2. Karl Menger, General Spaces and Cartesian Spaces, Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A. Edgar, Addison-Wesley ISBN 0-201-58701-7 Karl Menger, Dimensionstheorie, B. G Teubner Publishers, Leipzig. A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press. ISBN 0-521-20515-8 V. V. Fedorchuk, The Fundamentals of Dimension Th
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p: this means f gets closer and closer to L as x moves closer and closer to p. More when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist; the notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: a function is continuous if all of its limits agree with the values of the function, it appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime. Cauchy discussed variable quantities and limits and defined continuity of y = f by saying that an infinitesimal change in x produces an infinitesimal change in y in his 1821 book Cours d'analyse, while claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is written today, he introduced the notations lim and limx→x0. The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f, her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system.
Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x; as she gets closer and closer to it, she notices that her altitude approaches L. If asked about the altitude of x = p, she would answer L. What does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within ten meters of L, she reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, her altitude is always ten meters or less from L. The accuracy goal is changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p her altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.
The initial informal statement can now be explicated: The limit of a function f as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f remain within the target distance. This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, but not equal, to p; the following definitions are the accepted ones for the limit of a function in various contexts. Suppose f: R → R is defined on the real line and p,L ∈ R, it is said the limit of f, as x approaches p, is L and written lim x → p f = L, if the following property holds: For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f − L | < ε. The value of the limit does not depend on the value of f, nor that p be in the domain of f. A more general definition applies for functions defined on subsets of the real line.
Let be an open interval in R, p a point of. Let f be a real-valued function defined on all of except at p itself, it is said that the limit of f, as x approaches p, is L if, for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ implies | f − L | < ε. Here again the limit does not depend on f being well-defined; the letters ε and δ can be understood as "error" and "distance", in fact Cauchy used ε as an abbreviation for "error" in some of his work, though in his definition of continuity he used an infinitesimal α rather than either ε or δ. In these terms, the error in the measurement of the value at the limit can be made as small as desired by reducing the distance to the limit point; as discussed below this definition works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations. Alternatively x may approach p from
In mathematics, a self-similar object is or similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of artificial fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object, similar to the whole. For instance, a side of the Koch snowflake is both scale-invariant; the non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity f measured at different times are different but the corresponding dimensionless quantity at given value of x / t z remain invariant, it happens. The idea is just an extension of the idea of similarity of two triangles.
Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide. In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x- and y-directions; this means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation. A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms for which X = ⋃ s ∈ S f s If X ⊂ Y, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for. We call L = a self-similar structure; the homeomorphisms may be iterated. The composition of functions creates the algebraic structure of a monoid; when the set S has only two elements, the monoid is known as the dyadic monoid. The dyadic monoid can be visualized as an infinite binary tree; the automorphisms of the dyadic monoid is the modular group.
A more general notion than self-similarity is Self-affinity. The Mandelbrot set is self-similar around Misiurewicz points. Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically self-similar; this property means that simple models using a Poisson distribution are inaccurate, networks designed without taking self-similarity into account are to function in unexpected ways. Stock market movements are described as displaying self-affinity, i.e. they appear self-similar when transformed via an appropriate affine transformation for the level of detail being shown. Andrew Lo describes stock market log return self-similarity in econometrics. Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle; the Viable System Model of Stafford Beer is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, for whom the elements of its System One are viable systems one recursive level lower down.
Self-similarity can be found in nature, as well. To the right is a mathematically generated self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli, exhibit strong self-similarity. Strict canons display various amounts of self-similarity, as do sections of fugues. A Shepard tone is self-similar in the wavelength domains; the Danish composer Per Nørgård has made use of a self-similar integer sequence named the'infinity series' in much of his music. In the research field of music information retrieval, self-similarity refers to the fact that music consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than under scaling. "Copperplate Chevrons" — a self-similar fractal zoom movie "Self-Similarity" — New articles about Self-Similarity. Waltz Algorithm Mandelbrot, Benoit B.. "Self-affinity and fractal dimension". Physica Scripta. 32: 257–260. Bibcode:1985PhyS...32..257M. Doi:10.1088/0031-8949/32/4/001.
The Koch snowflake is a mathematical curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch; as the fractal evolves, the area of the snowflake converges to 8/5 the area of the original triangle, while the perimeter of the snowflake diverges to infinity. The snowflake has a finite area bounded by an infinitely long line; the Koch snowflake can be constructed by starting with an equilateral triangle recursively altering each line segment as follows: divide the line segment into three segments of equal length. Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. Remove the line segment, the base of the triangle from step 2; the first iteration of this process produces the outline of a hexagram. The Koch snowflake is the limit approached; the Koch curve described by Helge von Koch is constructed using only one of the three sides of the original triangle.
In other words, three Koch curves make a Koch snowflake. A Koch curve–based representation of a nominally flat surface can be created by segmenting each line in a sawtooth pattern of segments with a given angle; each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after n iterations is given by: N n = N n − 1 ⋅ 4 = 3 ⋅ 4 n. If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is: S n = S n − 1 3 = s 3 n; the perimeter of the snowflake after n iterations is: P n = N n ⋅ S n = 3 ⋅ s ⋅ n. The Koch curve has an infinite length, because the total length of the curve increases by a factor of 4/3 with each iteration; each iteration creates four times as many line segments as in the previous iteration, with the length of each one being 1/3 the length of the segments in the previous stage. Hence, the length of the curve after n iterations will be n times the original triangle perimeter and is unbounded, as n tends to infinity.
As the number of iterations tends to infinity, the limit of the perimeter is: lim n → ∞ P n = lim n → ∞ 3 ⋅ s ⋅ n = ∞, since |4/3| > 1. An ln 4/ln 3-dimensional measure has not been calculated so far. Only upper and lower bounds have been invented. In each iteration a new triangle is added on each side of the previous iteration, so the number of new triangles added in iteration n is: T n = N n − 1 = 3 ⋅ 4 n − 1 = 3 4 ⋅ 4 n; the area of each new triangle added in an iteration is 1/9 of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration n is: a n = a n − 1 9 = a 0 9 n. where a0 is the area of the original triangle. The total new area added in iteration n is therefore: b n = T n ⋅ a n = 3 4 ⋅ n ⋅ a 0 The total area of the snowflake after n iterations is: A n = a 0 + ∑ k = 1 n b k = a 0 = a 0 ( 1 + 1 3 ∑ k = 0 n − 1
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work to solve mathematical problems. Mathematics is concerned with numbers, quantity, space and change. One of the earliest known mathematicians was Thales of Miletus, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number", it was the Pythagoreans who coined the term "mathematics", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria, she succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells.
Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages was ongoing throughout the reign of certain caliphs, it turned out that certain scholars became experts in the works they translated and in turn received further support for continuing to develop certain sciences; as these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were polymaths. Examples include the work on optics and astronomy of Ibn al-Haytham; the Renaissance brought an increased emphasis on science to Europe.
During this period of transition from a feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli. As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in the seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle, at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.” In 1810, Humboldt convinced the King of Prussia to build a university in Berlin based on Friedrich Schleiermacher’s liberal ideas. Thus and laboratories started to evolve. British universities of this period adopted some approaches familiar to the Italian and German universities, but as they enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment, the same influences that inspired Humboldt.
The Universities of Oxford and Cambridge emphasized the importance of research, arguably more authentically implementing Humboldt’s idea of a university than German universities, which were subject to state authority. Overall, science became the focus of universities in the 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge; the German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of the kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of “freedom of scientific research and study.” Mathematicians cover a breadth of topics within mathematics in their undergraduate education, proceed to specialize in topics of their own choice at the graduate level.
In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics. Mathematicians involved with solving problems with applications in real life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, localized constructs, applied mathematicians work in the study and formulation of mathematical models. Mathematicians and applied mathematicians are considered to be two of the STEM careers; the discipline of applied mathematics concerns