1.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
2.
Graph theory
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In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, Graphs are one of the prime objects of study in discrete mathematics. Refer to the glossary of graph theory for basic definitions in graph theory, the following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more generalized notion, V is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, Many authors call this type of object a multigraph or pseudograph. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is |V|, its number of vertices, the size of a graph is |E|, its number of edges. The degree or valency of a vertex is the number of edges that connect to it, for an edge, graph theorists usually use the somewhat shorter notation xy. Graphs can be used to model many types of relations and processes in physical, biological, social, Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the network is sometimes defined to mean a graph in which attributes are associated with the nodes and/or edges. In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the structure of a website can be represented by a directed graph, in which the vertices represent web pages. A similar approach can be taken to problems in media, travel, biology, computer chip design. The development of algorithms to handle graphs is therefore of major interest in computer science, the transformation of graphs is often formalized and represented by graph rewrite systems. Graph-theoretic methods, in forms, have proven particularly useful in linguistics. Traditionally, syntax and compositional semantics follow tree-based structures, whose power lies in the principle of compositionality
3.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
4.
Kingdom of Prussia
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It was the driving force behind the unification of Germany in 1871 and was the leading state of the German Empire until its dissolution in 1918. Although it took its name from the region called Prussia, it was based in the Margraviate of Brandenburg, the kings of Prussia were from the House of Hohenzollern. Prussia was a power from the time it became a kingdom, through its predecessor, Brandenburg-Prussia. Prussia continued its rise to power under the guidance of Frederick II, more known as Frederick the Great. After the might of Prussia was revealed it was considered as a power among the German states. Throughout the next hundred years Prussia went on to win many battles and it was because of its power that Prussia continuously tried to unify all the German states under its rule. Attempts at creation of a federation remained unsuccessful and the German Confederation collapsed in 1866 when war ensued between its two most powerful states, Prussia and Austria. The North German Confederation which lasted from 1867–1871, created a union between the Prussian-aligned states while Austria and most of Southern Germany remained independent. The North German Confederation was seen as more of an alliance of military strength in the aftermath of the Austro-Prussian War, the German Empire lasted from 1871–1918 with the successful unification of all the German states under Prussian hegemony. This was due to the defeat of Napoleon III in the Franco-Prussian War of 1870–71, in 1871, Germany unified into a single country, minus Austria and Switzerland, with Prussia the dominant power. Prussia is considered the predecessor of the unified German Reich. The Kingdom left a significant cultural legacy, today notably promoted by the Prussian Cultural Heritage Foundation, in 1415 a Hohenzollern Burgrave came from the south to the March of Brandenburg and took control of the area as elector. In 1417 the Hohenzollern was made an elector of the Holy Roman Empire, after the Polish wars, the newly established Baltic towns of the German states including Prussia, suffered many economic setbacks. Many of the Prussian towns could not even afford to attend political meetings outside of Prussia, the towns were poverty stricken, with even the largest town, Danzig, having to borrow money from elsewhere to pay for trade. Poverty in these towns was partly caused by Prussias neighbors, who had established and developed such a monopoly on trading that these new towns simply could not compete and these issues led to feuds, wars, trade competition and invasions. However, the fall of these gave rise to the nobility, separated the east and the west. It was clear in 1440 how different Brandenburg was from the other German territories, not only did it face partition from within but also the threat of its neighbors. It prevented the issue of partition by enacting the Dispositio Achillea which instilled the principle of primogeniture to both the Brandenburg and Franconian territories, the second issue was solved through expansion
5.
Kaliningrad
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Kaliningrad is a seaport city and the administrative center of Kaliningrad Oblast, a Russian exclave between Poland and Lithuania on the Baltic Sea. In the Middle Ages, the locality was the site of the ancient Old Prussian settlement, the town was successively part of the monastic State of the Teutonic Order, enfeoffed to the Polish–Lithuanian Commonwealth, then part of Prussia and Germany. The city was damaged during World War II. Its ruins were occupied by the Red Army on 9 April 1945 and it was renamed Kaliningrad on July 4,1946, in honor of Soviet luminary Mikhail Kalinin, who died in the previous month. In 2005 the city marked 750 years of existence as Königsberg/Kaliningrad, according to the 2010 Census, its population was 431,902 – an increase from 430,003 recorded in the 2002 Census. Kaliningrad is at the mouth of the navigable Pregolya River, which empties into the Vistula Lagoon, Sea vessels can access Gdańsk Bay/Bay of Danzig and the Baltic Sea by way of the Vistula Lagoon and the Strait of Baltiysk. Until around 1900, ships drawing more than 2 meters of water could not pass the bar and come into town, larger vessels had to anchor at Pillau, where cargo was transferred to smaller vessels. In 1901 a ship canal between Königsberg and Pillau, completed at a cost of 13 million German marks, enabled vessels of a 6.5 meters draught to moor alongside the town. Khrabrovo Airport,24 kilometers north of Kaliningrad, has a few scheduled, there is the smaller Kaliningrad Devau Airport for general aviation. Kaliningrad is also home to Kaliningrad Chkalovsk naval air base, Königsberg was preceded by a Sambian fort called Twangste, meaning Oak Forest. During the conquest of the Sambians by the Teutonic Knights in 1255, the declining Old Prussian culture finally became extinct around the 17th century, after the surviving Old Prussians were integrated through assimilation and Germanization. Kaliningrad was the East Prussian provincial capital Königsberg, founded in 1255 by the Teutonic Knights, the city was named in honor of the Bohemian King Ottokar II. Through immigration and development over the seven centuries, the area became predominantly German, though having Polish. During World War II the city of Königsberg was heavily damaged by a British bombing attack in 1944, the President of the United States and the British Prime Minister have declared that they will support the proposal of the Conference at the forthcoming peace settlement. Königsberg was renamed Kaliningrad in 1946 after the death of Chairman of the Presidium of the Supreme Soviet of the USSR, Mikhail Kalinin, the survivors of the German population were forcibly expelled in 1946-1949, and the city was repopulated with Soviet citizens. The German language was replaced by the Russian language, the city was rebuilt, and, as the westernmost territory of the USSR, the Kaliningrad Oblast became a strategically important area during the Cold War. The Soviet Baltic Fleet was headquartered in the city in the 1950s, because of its strategic importance, Kaliningrad was closed to foreign visitors. In 1957 an agreement was signed and later came into force which delimited the border between Poland and the Soviet Union, due to the collapse of the Soviet Union in 1991, the Kaliningrad Oblast became an exclave, geographically separated from the rest of Russia
6.
Russia
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Russia, also officially the Russian Federation, is a country in Eurasia. The European western part of the country is more populated and urbanised than the eastern. Russias capital Moscow is one of the largest cities in the world, other urban centers include Saint Petersburg, Novosibirsk, Yekaterinburg, Nizhny Novgorod. Extending across the entirety of Northern Asia and much of Eastern Europe, Russia spans eleven time zones and incorporates a range of environments. It shares maritime borders with Japan by the Sea of Okhotsk, the East Slavs emerged as a recognizable group in Europe between the 3rd and 8th centuries AD. Founded and ruled by a Varangian warrior elite and their descendants, in 988 it adopted Orthodox Christianity from the Byzantine Empire, beginning the synthesis of Byzantine and Slavic cultures that defined Russian culture for the next millennium. Rus ultimately disintegrated into a number of states, most of the Rus lands were overrun by the Mongol invasion. The Soviet Union played a role in the Allied victory in World War II. The Soviet era saw some of the most significant technological achievements of the 20th century, including the worlds first human-made satellite and the launching of the first humans in space. By the end of 1990, the Soviet Union had the second largest economy, largest standing military in the world. It is governed as a federal semi-presidential republic, the Russian economy ranks as the twelfth largest by nominal GDP and sixth largest by purchasing power parity in 2015. Russias extensive mineral and energy resources are the largest such reserves in the world, making it one of the producers of oil. The country is one of the five recognized nuclear weapons states and possesses the largest stockpile of weapons of mass destruction, Russia is a great power as well as a regional power and has been characterised as a potential superpower. The name Russia is derived from Rus, a state populated mostly by the East Slavs. However, this name became more prominent in the later history, and the country typically was called by its inhabitants Русская Земля. In order to distinguish this state from other states derived from it, it is denoted as Kievan Rus by modern historiography, an old Latin version of the name Rus was Ruthenia, mostly applied to the western and southern regions of Rus that were adjacent to Catholic Europe. The current name of the country, Россия, comes from the Byzantine Greek designation of the Kievan Rus, the standard way to refer to citizens of Russia is Russians in English and rossiyane in Russian. There are two Russian words which are translated into English as Russians
7.
Pregolya River
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The Pregolya or Pregola is a river in the Russian Kaliningrad Oblast exclave. It starts as a confluence of the Instruch and the Angrapa and its length under the name of Pregolya is 123 km,292 km including the Angrapa. The basin has an area of 15,500 km², the average flow is 90 m³/s. Eulers Seven Bridges of Königsberg problem was based on the bridges in Königsberg. The ancient name by Ptolemy of the Pregolya River is Chronos, chernyakhovsk Znamensk Gvardeysk Kaliningrad Pissa Lava/Łyna Angrapa Instruch List of rivers of Russia
8.
Vertex (graph theory)
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In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices, a vertex w is said to be adjacent to another vertex v if the graph contains an edge. The neighborhood of a v is an induced subgraph of the graph. The degree of a vertex, denoted
9.
Edge (graph theory)
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This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges, for instance, α is the independence number of a graph, α′ is the matching number of the graph, which equals the independence number of its line graph. Similarly, χ is the number of a graph, χ ′ is the chromatic index of the graph. Achromatic The achromatic number of a graph is the number of colors in a complete coloring. A graph is acyclic if it has no cycles, an acyclic undirected graph is the same thing as a forest. Acyclic directed graphs are often called directed acyclic graphs. An acyclic coloring of a graph is a proper coloring in which every two color classes induce a forest. Adjacent The relation between two vertices that are both endpoints of the same edge, α For a graph G, α is its independence number, and α′ is its matching number. Alternating In a graph with a matching, a path is a path whose edges alternate between matched and unmatched edges. An alternating cycle is, similarly, a cycle whose edges alternate between matched and unmatched edges, an augmenting path is an alternating path that starts and ends at unsaturated vertices. A larger matching can be found as the difference of the matching and the augmenting path. Anti-edge Synonym for non-edge, a pair of non-adjacent vertices, anti-triangle A three-vertex independent set, the complement of a triangle. An apex graph is a graph in which one vertex can be removed, the removed vertex is called the apex. A k-apex graph is a graph that can be made planar by the removal of k vertices, Synonym for universal vertex, a vertex adjacent to all other vertices. Arborescence Synonym for a rooted and directed tree, see tree, arrow An ordered pair of vertices, such as an edge in a directed graph. An arrow has a x, a head y. The arrow is the arrow of the arrow. Articulation point A vertex in a graph whose removal would disconnect the graph
10.
Graph (discrete mathematics)
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In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices and each of the pairs of vertices is called an edge. Typically, a graph is depicted in form as a set of dots for the vertices. Graphs are one of the objects of study in discrete mathematics, the edges may be directed or undirected. In contrast, if any edge from a person A to a person B corresponds to As admiring B, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called a graph and the edges are called undirected edges while the latter type of graph is called a directed graph. Graphs are the subject studied by graph theory. The word graph was first used in this sense by J. J. Sylvester in 1878, the following are some of the more basic ways of defining graphs and related mathematical structures. In one very common sense of the term, a graph is an ordered pair G = comprising a set V of vertices, nodes or points together with a set E of edges, arcs or lines, which are 2-element subsets of V. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more general conception, E is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, many authors call these types of object multigraphs or pseudographs. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set, the order of a graph is |V|, its number of vertices. The size of a graph is |E|, its number of edges, the degree or valency of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends is counted twice. For an edge, graph theorists usually use the shorter notation xy. As stated above, in different contexts it may be useful to refine the term graph with different degrees of generality, whenever it is necessary to draw a strict distinction, the following terms are used
11.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
12.
Degree (graph theory)
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In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a v is denoted deg or deg v. The maximum degree of a graph G, denoted by Δ, in the graph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, all degrees are the same, the degree sum formula states that, given a graph G =, ∑ v ∈ V deg =2 | E |. The formula implies that in any graph, the number of vertices with odd degree is even and this statement is known as the handshaking lemma. The latter name comes from a mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even. The degree sequence of a graph is the non-increasing sequence of its vertex degrees. The degree sequence is a graph invariant so isomorphic graphs have the degree sequence. However, the sequence does not, in general, uniquely identify a graph, in some cases. The degree sequence problem is the problem of finding some or all graphs with the sequence being a given non-increasing sequence of positive integers. A sequence which is the sequence of some graph, i. e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the sum formula, any sequence with an odd sum, such as. The converse is true, if a sequence has an even sum. The construction of such a graph is straightforward, connect vertices with odd degrees in pairs by a matching, the question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm, the problem of finding or estimating the number of graphs with a given degree sequence is a problem from the field of graph enumeration. A vertex with degree 0 is called an isolated vertex, a vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, is a pendant edge and this terminology is common in the study of trees in graph theory and especially trees as data structures. A vertex with degree n −1 in a graph on n vertices is called a dominating vertex, if each vertex of the graph has the same degree k the graph is called a k-regular graph and the graph itself is said to have degree k
13.
Connectivity (graph theory)
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It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network, a graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices, a graph that is not connected is disconnected. A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints, a graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected, in an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are connected by a path of length 1, i. e. by a single edge. A graph is said to be connected if every pair of vertices in the graph is connected, a connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge, a directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected graph. It is connected if it contains a path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected, diconnected, or simply strong if it contains a path from u to v. The strong components are the maximal strongly connected subgraphs, a cut, vertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. The connectivity or vertex connectivity κ is the size of a minimal vertex cut, a graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater. In particular, a graph with n vertices, denoted Kn, has no vertex cuts at all. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. The local connectivity κ is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs, a graph G which is connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges, in the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, a cut of G is a set of edges whose removal renders the graph disconnected. A graph is called k-edge-connected if its edge connectivity is k or greater, if u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex
14.
Eulerian path
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In graph theory, an Eulerian trail is a trail in a graph which visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail which starts and they were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this, Given the graph in the image, is it possible to construct a path which visits each edge exactly once, the first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. The term Eulerian graph has two meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree and these definitions coincide for connected graphs. For the existence of Eulerian trails it is necessary that zero or two vertices have an odd degree, this means the Königsberg graph is not Eulerian, if there are no vertices of odd degree, all Eulerian trails are circuits. If there are two vertices of odd degree, all Eulerian trails start at one of them and end at the other. A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian, an Eulerian trail, or Euler walk in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian, an Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal, the term Eulerian graph is also sometimes used in a weaker sense to denote a graph where every vertex has even degree. For finite connected graphs the two definitions are equivalent, while a possibly unconnected graph is Eulerian in the sense if. For directed graphs, path has to be replaced with directed path, the definition and properties of Eulerian trails, cycles and graphs are valid for multigraphs as well. An undirected graph has an Eulerian cycle if and only if every vertex has even degree, an undirected graph can be decomposed into edge-disjoint cycles if and only if all of its vertices have even degree. So, a graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint cycles and its nonzero-degree vertices belong to a single connected component. An undirected graph has an Eulerian trail if and only if exactly zero or two vertices have odd degree, and if all of its vertices with nonzero degree belong to a connected component. A directed graph has an Eulerian cycle if and only if every vertex has equal in degree and out degree, Fleurys algorithm is an elegant but inefficient algorithm which dates to 1883. Consider a graph known to have all edges in the same component, the algorithm starts at a vertex of odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. It then moves to the endpoint of that edge and deletes the edge
15.
History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind Mathematical Papyrus, All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
16.
Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
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Aristotelianism
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Aristotelianism is a tradition of philosophy that takes its defining inspiration from the work of Aristotle. The works of Aristotle were initially defended by the members of the Peripatetic school, and, later on, by the Neoplatonists, who produced many commentaries on Aristotles writings. Moses Maimonides adopted Aristotelianism from the Islamic scholars and based his famous Guide for the Perplexed on it, scholars such as Albertus Magnus and Thomas Aquinas interpreted and systematized Aristotles works in accordance with Christian theology. Although this project was criticized by Trendelenburg and Brentano as non-Aristotelian, postmodernists, in contrast, reject Aristotelianisms claim to reveal important theoretical truths. In this, they follow Heideggers critique of Aristotle as the greatest source of the tradition of Western philosophy. Recent Aristotelian ethical and practical philosophy, such as that of Gadamer, the most famous contemporary Aristotelian philosopher is Alasdair MacIntyre. Politically and socially, it has been characterized as a newly revolutionary Aristotelianism and this may be contrasted with the more conventional, apolitical and effectively conservative uses of Aristotle by, for example, Gadamer and McDowell. Other important contemporary Aristotelian theorists include Fred D. Miller, Jr. in politics, the original followers of Aristotle were the members of the Peripatetic school. The most prominent members of the school after Aristotle were Theophrastus and Strato of Lampsacus, during the Roman era the school concentrated on preserving and defending his work. The most important figure in this regard was Alexander of Aphrodisias who commentated on Aristotles writings, in the Abbasid Empire, many foreign works were translated into Arabic, large libraries were constructed, and scholars were welcomed. Under the caliphs Harun al-Rashid and his son Al-Mamun, the House of Wisdom in Baghdad flourished, Christian scholar Hunayn ibn Ishaq was placed in charge of the translation work by the caliph. In his lifetime, Ishaq translated 116 writings, including works by Plato and Aristotle, into Syriac, al-Kindi was the first of the Muslim Peripatetic philosophers, and is known for his efforts to introduce Greek and Hellenistic philosophy to the Arab world. He incorporated Aristotelian and Neoplatonist thought into an Islamic philosophical framework and this was an important factor in the introduction and popularization of Greek philosophy in the Muslim intellectual world. The philosopher Al-Farabi had great influence on science and philosophy for several centuries and his work, aimed at synthesis of philosophy and Sufism, paved the way for the work of Avicenna. Avicenna was one of the interpreters of Aristotle. The school of thought he founded became known as Avicennism, which was built on ingredients and conceptual building blocks that are largely Aristotelian and Neoplatonist. At the western end of the Mediterranean Sea, during the reign of Al-Hakam II in Córdoba, a massive effort was undertaken. Averroes, who spent much of his life in Cordoba and Seville, was distinguished as a commentator of Aristotle
18.
Gordian Knot
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The Gordian Knot is a legend of Phrygian Gordium associated with Alexander the Great. It is often used as a metaphor for a problem solved easily by loophole or thinking outside the box. An oracle at Telmissus decreed that the man to enter the city driving an ox-cart should become their king. A peasant farmer named Gordias drove into town on an ox-cart and his position had also been predicted earlier by an eagle landing on his cart, a sign to him from the gods, and, on entering the city, Gordias was declared king. Out of gratitude, his son Midas dedicated the ox-cart to the Phrygian god Sabazios, so, in 333 BCE, while wintering at Gordium, Alexander the Great attempted to untie the knot. When he could not find the end to the knot to unbind it, he sliced it in half with a stroke of his sword and that night there was a violent thunderstorm. Alexanders prophet Aristander took this as a sign that Zeus was pleased, once Alexander had sliced the knot with a sword-stroke, his biographers claimed in retrospect that an oracle further prophesied that the one to untie the knot would become the king of Asia. Alexander is a figure of outstanding celebrity and the episode with the Gordian Knot remains widely known. Literary sources are Alexanders propagandist Arrian Quintus Curtius, Justins epitome of Pompeius Trogus, while sources from antiquity agree that Alexander was confronted with the challenge of the knot, the means by which he solved the problem are disputed. Some classical scholars regard this as more plausible than the popular account, Alexander later went on to conquer Asia as far as the Indus and the Oxus thus, for Callisthenes, fulfilling the prophecy. The knot may have been a religious knot-cipher guarded by Gordian/Midass priests, unlike fable, true myth has few completely arbitrary elements. This myth taken as a whole seems designed to confer legitimacy to change in this central Anatolian kingdom. The ox-cart suggests a longer voyage, rather than a journey, perhaps linking Gordias/Midas with an attested origin-myth in Macedon. The Gordian knot is alluded to in the motto of Ferdinand II of Aragon, Tanto monta and in the yoke representing Isabella in the emblem of the yoke and arrows. Brian Coless has suggested that Donald Wiseman cut the Gordian knot of the problem of identifying King Darius the Mede in the Book of Daniel. W. G. Sebald in The Rings of Saturn recounts the episode of Joseph Conrad who was shot or shot himself in the chest allowing him to Cut the Gordian Knot of, in Sebalds telling, a stormy love affair. Henry Purcell, an English composer, wrote a piece of music entitled The Gordion Knot Untyd and he called for the newborn artists, the anti-Alexanders, to heal the wound and repair the knot, Yes, the rebirth is in the hands of all of us. It is up to us if the West is to bring forth any anti-Alexanders to tie together the Gordian Knot of civilization cut by the sword, for this purpose, we must assume all the risks and labors of freedom
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World view
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A comprehensive world view or worldview is the fundamental cognitive orientation of an individual or society encompassing the entirety of the individual or societys knowledge and point of view. A world view can include natural philosophy, fundamental, existential, and normative postulates, or themes, values, emotions, the term is a calque of the German word Weltanschauung, composed of Welt and Anschauung. The German word is used in English. It is a fundamental to German philosophy and epistemology and refers to a wide world perception. Additionally, it refers to the framework of ideas and beliefs forming a global description through which an individual, group or culture watches and interprets the world, Worldview remains a confused and confusing concept in English, used very differently by linguists and sociologists. It is for this reason that Underhill suggests five subcategories, world-perceiving, world-conceiving, cultural mindset, personal world, and perspective. However, core worldview beliefs are often rooted, and so are only rarely reflected on by individuals. The founder of the idea that language and worldview are inextricable is the Prussian philologist, Humboldt argued that language was part of the creative adventure of mankind. Culture, language and linguistic communities developed simultaneously, he argued, human beings take their place in speech and continue to modify language and thought by their creative exchanges. Edward Sapir also gives an account of the relationship between thinking and speaking in English, the hypothesis was well received in the late 1940s, but declined in prominence after a decade. The theory has gained attention through the work of Lera Boroditsky at Stanford University. One of the most important concepts in philosophy and cognitive sciences is the German concept of Weltanschauung. The term Weltanschauung is often attributed to Wilhelm von Humboldt the founder of German ethnolinguistics. As Jürgen Trabant points out, however, and as Underhll reminds us in his Humboldt, Worldview and Language, Weltansicht was used by Humboldt to refer to the overarching conceptual and sensorial apprehension of reality shared by a linguistic community. But Humboldt maintained that the human being was the core of language. Worldviews are not prisons which contain and constrain us, they are the spaces we develop within, create and resist creatively in speaking together. A worldview can be expressed as the cognitive, affective, and evaluative presuppositions a group of people make about the nature of things. If the Sapir–Whorf hypothesis is correct, the map of the world would be similar to the linguistic map of the world
20.
University of Canterbury
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The University of Canterbury in Christchurch is New Zealands second oldest university. It was founded in 1873 as Canterbury College, the first constituent college of the University of New Zealand, the University originated in 1873 in the centre of Christchurch as Canterbury College, the first constituent college of the University of New Zealand. It became the institution in New Zealand providing tertiary-level education. Its foundation professors arrived in 1874, namely, Charles Cook, Alexander Bickerton, in 1933, the name changed from Canterbury College to Canterbury University College. In 1957 the name changed again to the present University of Canterbury, until 1961, the University formed part of the University of New Zealand, and issued degrees in its name. That year saw the dissolution of the system of tertiary education in New Zealand. Upon the UNZs demise, Canterbury Agricultural College became a constituent college of the University of Canterbury, Lincoln College became independent in 1990 as a full university in its own right. Over the period from 1961 to 1974, the university relocated from the centre of the city to its much larger current site in the suburb of Ilam. The neo-gothic buildings of the old campus became the site of the Christchurch Arts Centre, in 2004, the University underwent restructuring into four Colleges and a School of Law, administering a number of schools and departments. For many years the university worked closely with the Christchurch College of Education, leading to a merger in 2007. In September 2011, plans were announced to demolish some University buildings that were damaged from an earthquake, in the months following the earthquake, the University lost 25 per cent of its first-year students and 8 per cent of continuing students. The number of students, who pay much higher fees and are a major source of revenue. By 2013, the University had lost 22 per cent of its students, however, a record number of 886 PhD students are enrolled at the University of Canterbury as of 2013. In October 2011, staff were encouraged to take voluntary redundancies, the university was first governed by a board of governors, then by a college council, and since 1957 by a university council. The council is chaired by a chancellor, the Council includes representatives from the faculties, students and general staff, as well as local industry, employer and trade union representatives. Initially, the board was given power to fill their own vacancies, Professor Roy Sharp assumed the position of Vice-Chancellor on 1 March 2003. In May 2008 he announced his imminent resignation from the position, the then current Deputy Vice-Chancellor, Professor Ian Town, assumed the role of acting Vice-Chancellor on 1 July 2008. On 15 October 2008 the University announced that Dr Rod Carr, a former banker, under Carrs leadership, UCs position in the QS World University Rankings has steadily declined by about 30%, as of September 2014
21.
Christchurch
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Christchurch is the largest city in the South Island of New Zealand and the seat of the Canterbury Region. The Christchurch urban area lies on the South Islands east coast, just north of Banks Peninsula and it is home to 389,700 residents, making it New Zealands third most-populous urban area behind Auckland and Wellington. The city was named by the Canterbury Association, which settled the province of Canterbury. The name of Christchurch was agreed on at the first meeting of the association on 27 March 1848 and it was suggested by John Robert Godley, who had attended Christ Church, Oxford. Some early writers called the town Christ Church, but it was recorded as Christchurch in the minutes of the management committee of the association, Christchurch became a city by Royal Charter on 31 July 1856, making it officially the oldest established city in New Zealand. The Avon River flows through the centre of the city, with a park located along its banks. At the request of the Deans brothers, the river was named after the River Avon in Scotland, the usual Māori name for Christchurch is Ōtautahi. This was originally the name of a site by the Avon River near present-day Kilmore Street. The site was a dwelling of Ngāi Tahu chief Te Potiki Tautahi. The Ōtautahi name was adopted in the 1930s, prior to that the Ngāi Tahu generally referred to the Christchurch area as Karaitiana, a transliteration of the English word Christian. The citys name is abbreviated by New Zealanders to Chch. In New Zealand Sign Language, the name is the fingerspelled letter C signed twice, with the second to the right of the first. Archaeological evidence found in a cave at Redcliffs in 1876 has indicated that the Christchurch area was first settled by moa-hunting tribes about 1250 CE. These first inhabitants were thought to have followed by the Waitaha tribe. Following tribal warfare, the Waitaha were dispossessed by the Ngati Mamoe tribe and they were in turn subjugated by the Ngāi Tahu tribe, who remained in control until the arrival of European settlers. Their abandoned holdings were taken over by the Deans brothers in 1843 who stayed, the First Four Ships were chartered by the Canterbury Association and brought the first 792 of the Canterbury Pilgrims to Lyttelton Harbour. These sailing vessels were the Randolph, Charlotte Jane, Sir George Seymour, the Charlotte Jane was the first to arrive on 16 December 1850. The Canterbury Pilgrims had aspirations of building a city around a cathedral and college, the name Christ Church was decided prior to the ships arrival, at the Associations first meeting, on 27 March 1848
22.
New Zealand
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New Zealand /njuːˈziːlənd/ is an island nation in the southwestern Pacific Ocean. The country geographically comprises two main landmasses—the North Island, or Te Ika-a-Māui, and the South Island, or Te Waipounamu—and around 600 smaller islands. New Zealand is situated some 1,500 kilometres east of Australia across the Tasman Sea and roughly 1,000 kilometres south of the Pacific island areas of New Caledonia, Fiji, because of its remoteness, it was one of the last lands to be settled by humans. During its long period of isolation, New Zealand developed a distinct biodiversity of animal, fungal, the countrys varied topography and its sharp mountain peaks, such as the Southern Alps, owe much to the tectonic uplift of land and volcanic eruptions. New Zealands capital city is Wellington, while its most populous city is Auckland, sometime between 1250 and 1300 CE, Polynesians settled in the islands that later were named New Zealand and developed a distinctive Māori culture. In 1642, Dutch explorer Abel Tasman became the first European to sight New Zealand, in 1840, representatives of Britain and Māori chiefs signed the Treaty of Waitangi, which declared British sovereignty over the islands. In 1841, New Zealand became a colony within the British Empire, today, the majority of New Zealands population of 4.7 million is of European descent, the indigenous Māori are the largest minority, followed by Asians and Pacific Islanders. Reflecting this, New Zealands culture is derived from Māori and early British settlers. The official languages are English, Māori and New Zealand Sign Language, New Zealand is a developed country and ranks highly in international comparisons of national performance, such as health, education, economic freedom and quality of life. Since the 1980s, New Zealand has transformed from an agrarian, Queen Elizabeth II is the countrys head of state and is represented by a governor-general. In addition, New Zealand is organised into 11 regional councils and 67 territorial authorities for local government purposes, the Realm of New Zealand also includes Tokelau, the Cook Islands and Niue, and the Ross Dependency, which is New Zealands territorial claim in Antarctica. New Zealand is a member of the United Nations, Commonwealth of Nations, ANZUS, Organisation for Economic Co-operation and Development, Pacific Islands Forum, and Asia-Pacific Economic Cooperation. Dutch explorer Abel Tasman sighted New Zealand in 1642 and called it Staten Landt, in 1645, Dutch cartographers renamed the land Nova Zeelandia after the Dutch province of Zeeland. British explorer James Cook subsequently anglicised the name to New Zealand, Aotearoa is the current Māori name for New Zealand. It is unknown whether Māori had a name for the country before the arrival of Europeans. Māori had several names for the two main islands, including Te Ika-a-Māui for the North Island and Te Waipounamu or Te Waka o Aoraki for the South Island. Early European maps labelled the islands North, Middle and South, in 1830, maps began to use North and South to distinguish the two largest islands and by 1907, this was the accepted norm. The New Zealand Geographic Board discovered in 2009 that the names of the North Island and South Island had never been formalised and this set the names as North Island or Te Ika-a-Māui, and South Island or Te Waipounamu
23.
Rochester Institute of Technology
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Rochester Institute of Technology is a private doctoral university within the town of Henrietta in the Rochester, New York metropolitan area. RIT is composed of nine colleges, including National Technical Institute for the Deaf. It is most widely known for its arts, computing, engineering. The name of the institution at the time was called Rochester Athenæum. In 1944, the changed its name to Rochester Institute of Technology. The Institute originally resided within the city of Rochester, New York, proper, on a block bounded by the Erie Canal, South Plymouth Avenue, Spring Street and its art department was originally located in the Bevier Memorial Building. In 1961, a donation of $3. Upon completion in 1968, the Institute moved to the new suburban campus, in 1966, RIT was selected by the Federal government to be the site of the newly founded National Technical Institute for the Deaf. NTID admitted its first students in 1968, concurrent with RITs transition to the Henrietta campus, in 1979, RIT took over Eisenhower College, a liberal arts college located in Seneca Falls, New York. Despite making a 5-year commitment to keep Eisenhower open, RIT announced in July 1982 that the college would close immediately, one final year of operation by Eisenhowers academic program took place in the 1982–83 school year on the Henrietta campus. The final Eisenhower graduation took place in May 1983 back in Seneca Falls, in 1996, RIT also became the first college in the U. S to offer a Software Engineering degree at the undergraduate level. The current campus is housed on a 1,300 acres property and this property is largely covered with woodland and fresh-water swamp making it a very diverse wetland which is home to a number of somewhat rare plant species. The campus comprises 237 buildings and 5.1 million square feet of building space, though the buildings erected in the first few decades of the campuss existence reflected the architectural style known as brutalism, the warm color of the bricks softened the impact somewhat. More recent additions to the campus have diversified the architecture while still incorporating the traditional brick colors, in 2009, the campus was named a Campus Sustainability Leader by the Sustainable Endowments Institute. The residence halls and the side of campus are connected with a walkway called the Quarter Mile. Along the Quarter Mile, between the academic and residence hall side are various administration and support buildings and these symbols represent time to infinity. The Quarter Mile is actually 0.41 miles long when measured between the sculpture and the sundial. The name comes from a student fundraiser, where quarters were lined up from the sundial to the Infinity Sculpture, standing near the Administration Building and the Student Alumni Union is The Sentinel, a steel structure created by the acclaimed metal sculptor, Albert Paley
24.
Gene Polisseni Center
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The Gene Polisseni Center is an ice arena on the Rochester Institute of Technology campus in Henrietta, New York. Ground was broken for the project on October 19,2012, the arena is the home of the varsity ice hockey teams at RIT, replacing the Frank Ritter Memorial Ice Arena. Ritter Arena continues to be used as an ice arena, the Polisseni Center serves primarily as a hockey arena, and it will also be a multi-purpose venue. The Polisseni Center is built not far from Ritter Arena on the RIT campus, south of the Student Alumni Union, fundraising was started with a $1 million USD donation from Stephen and Vicki Schultz. Naming rights were based on a $4.5 million USD donation from the Polisseni Foundation and were announced on November 11,2011 during the mens hockey game against Air Force. With arena construction being an estimated $30 million, the funding of the project has divided into two components. The first half will be borrowed against RITs endowment fund, and paid back with sponsorship, the second half of the cost is a grassroots fundraising effort called the Tiger Power Play. The Tiger Power Play is an effort to bring in big name donations as well as garner support from students and alumni. Small donations could be made by cell phone text message, larger donations have been working directly with school administration. Sales of nameplates for seats is available, initially costing $1,000 per nameplate. J. M. Allain, CEO of Trans-Lux and an RIT graduate, donated $1 million for a new video scoreboard
25.
Glossary of graph theory
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This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges, for instance, α is the independence number of a graph, α′ is the matching number of the graph, which equals the independence number of its line graph. Similarly, χ is the number of a graph, χ ′ is the chromatic index of the graph. Achromatic The achromatic number of a graph is the number of colors in a complete coloring. A graph is acyclic if it has no cycles, an acyclic undirected graph is the same thing as a forest. Acyclic directed graphs are often called directed acyclic graphs. An acyclic coloring of a graph is a proper coloring in which every two color classes induce a forest. Adjacent The relation between two vertices that are both endpoints of the same edge, α For a graph G, α is its independence number, and α′ is its matching number. Alternating In a graph with a matching, a path is a path whose edges alternate between matched and unmatched edges. An alternating cycle is, similarly, a cycle whose edges alternate between matched and unmatched edges, an augmenting path is an alternating path that starts and ends at unsaturated vertices. A larger matching can be found as the difference of the matching and the augmenting path. Anti-edge Synonym for non-edge, a pair of non-adjacent vertices, anti-triangle A three-vertex independent set, the complement of a triangle. An apex graph is a graph in which one vertex can be removed, the removed vertex is called the apex. A k-apex graph is a graph that can be made planar by the removal of k vertices, Synonym for universal vertex, a vertex adjacent to all other vertices. Arborescence Synonym for a rooted and directed tree, see tree, arrow An ordered pair of vertices, such as an edge in a directed graph. An arrow has a x, a head y. The arrow is the arrow of the arrow. Articulation point A vertex in a graph whose removal would disconnect the graph
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Hamiltonian path
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In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a Hamiltonian path that is a cycle, determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. Hamilton solved this problem using the calculus, an algebraic structure based on roots of unity with many similarities to the quaternions. This solution does not generalize to arbitrary graphs, in 18th century Europe, knights tours were published by Abraham de Moivre and Leonhard Euler. A Hamiltonian path or traceable path is a path that each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph, a graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph, similar notions may be defined for directed graphs, where each edge of a path or cycle can only be traced in a single direction. A Hamiltonian decomposition is a decomposition of a graph into Hamiltonian circuits. All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian, an Eulerian graph G necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L, a tournament is Hamiltonian if and only if it is strongly connected. The number of different Hamiltonian cycles in an undirected graph on n vertices is. /2 and in a directed graph on n vertices is. These counts assume that cycles that are the same apart from their starting point are not counted separately, the best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac and Øystein Ore. Both Diracs and Ores theorems can also be derived from Pósas theorem, hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. Dirac and Ores theorems basically state that a graph is Hamiltonian if it has enough edges, Bondy–Chvátal theorem A graph is Hamiltonian if and only if its closure is Hamiltonian. Ore A graph with n vertices is Hamiltonian if, for pair of non-adjacent vertices. Meyniel A strongly connected directed graph with n vertices is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to 2n −1
27.
Icosian game
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The icosian game is a mathematical game invented in 1857 by William Rowan Hamilton. The games object is finding a Hamiltonian cycle along the edges of a such that every vertex is visited a single time. The puzzle was distributed commercially as a pegboard with holes at the nodes of the graph and was subsequently marketed in Europe in many forms. The motivation for Hamilton was the problem of symmetries of an icosahedron, the solution of the puzzle is a cycle containing twenty edges. Dual polyhedron Seven Bridges of Königsberg Weisstein, Eric W. Icosian game, puzzle Museum article with pictures Icosian game for Android
28.
Three utilities problem
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It is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces. In more formal terms, the problem asks whether the complete bipartite graph K3,3 is planar. This graph is often referred to as the utility graph in reference to the problem, a review of the problems history is given by Kullman who states that most published references to the problem characterize it as very ancient. In the earliest publication found by Kullman, Henry Dudeney names it water, gas, however, Dudeney states that the problem is as old as the hills. much older than electric lighting, or even gas. Dudeney also published the same puzzle previously, in The Strand Magazine in 1913, another early version of the problem involves connecting three houses to three wells. K3,3 makes an appearance as a mathematical graph in Henneberg. As it is presented, the solution to the utility puzzle is no -- meaning. The problem may be formalized mathematically as asking whether the complete bipartite graph K3,3 is planar, kazimierz Kuratowski stated in 1930 that K3,3 is nonplanar, from which it follows that the problem has no solution. Kullman, however, states that Interestingly enough, Kuratowski did not publish a proof that non-planar. One proof of the impossibility of finding a planar embedding of K3,3 uses a case involving the Jordan curve theorem. In this solution, one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph, in the utility graph, E =9 and 2V −4 =8, violating this inequality, so the utility graph cannot be planar. K3,3 is equivalent to the circulant graph Ci6 and it is toroidal, which means it can be embedded on a torus. In terms of the three cottage problem this means the problem can be solved by punching two holes through the plane and connecting them with a tube and this changes the topological properties of the surface and using the tube we can connect the three cottages without crossing lines. An equivalent statement is that the genus of the utility graph is one. A surface of one is equivalent to a torus. Another way of changing the rules of the puzzle is to allow utility lines to pass through the cottages or utilities, the utility graph K3,3 may be drawn with only one crossing, but not with zero crossings, so its crossing number is one. The utility graph K3,3 is the -cage, the smallest triangle-free cubic graph, like all other complete bipartite graphs, it is a well-covered graph, meaning that every maximal independent set has the same size. In this graph, the only two independent sets are the two sides of the bipartition, and obviously they are equal
29.
Twitter
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Twitter is an online news and social networking service where users post and interact with messages, tweets, restricted to 140 characters. Registered users can post tweets, but those who are unregistered can only read them, users access Twitter through its website interface, SMS or a mobile device app. Twitter Inc. is based in San Francisco, California, United States, Twitter was created in March 2006 by Jack Dorsey, Noah Glass, Biz Stone, and Evan Williams and launched in July. The service rapidly gained worldwide popularity, in 2012, more than 100 million users posted 340 million tweets a day, and the service handled an average of 1.6 billion search queries per day. In 2013, it was one of the ten most-visited websites and has described as the SMS of the Internet. As of 2016, Twitter had more than 319 million monthly active users. On the day of the 2016 U. S. presidential election, Twitter proved to be the largest source of breaking news, Twitters origins lie in a daylong brainstorming session held by board members of the podcasting company Odeo. Jack Dorsey, then a student at New York University. The original project name for the service was twttr, an idea that Williams later ascribed to Noah Glass, inspired by Flickr. The developers initially considered 10958 as a code, but later changed it to 40404 for ease of use. Work on the project started on March 21,2006, when Dorsey published the first Twitter message at 9,50 PM Pacific Standard Time, Dorsey has explained the origin of the Twitter title. we came across the word twitter, and it was just perfect. The definition was a short burst of inconsequential information, and chirps from birds, and thats exactly what the product was. The first Twitter prototype, developed by Dorsey and contractor Florian Weber, was used as a service for Odeo employees. Williams fired Glass, who was silent about his part in Twitters startup until 2011, Twitter spun off into its own company in April 2007. Williams provided insight into the ambiguity that defined this early period in a 2013 interview, With Twitter and they called it a social network, they called it microblogging, but it was hard to define, because it didnt replace anything. There was this path of discovery with something like that, where over time you figure out what it is, Twitter actually changed from what we thought it was in the beginning, which we described as status updates and a social utility. It is that, in part, but the insight we eventually came to was Twitter was really more of an information network than it is a social network, the tipping point for Twitters popularity was the 2007 South by Southwest Interactive conference. During the event, Twitter usage increased from 20,000 tweets per day to 60,000, the Twitter people cleverly placed two 60-inch plasma screens in the conference hallways, exclusively streaming Twitter messages, remarked Newsweeks Steven Levy
30.
Geographic coordinate system
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A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are chosen such that one of the numbers represents a vertical position. A common choice of coordinates is latitude, longitude and elevation, to specify a location on a two-dimensional map requires a map projection. The invention of a coordinate system is generally credited to Eratosthenes of Cyrene. Ptolemy credited him with the adoption of longitude and latitude. Ptolemys 2nd-century Geography used the prime meridian but measured latitude from the equator instead. Mathematical cartography resumed in Europe following Maximus Planudes recovery of Ptolemys text a little before 1300, in 1884, the United States hosted the International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt the longitude of the Royal Observatory in Greenwich, the Dominican Republic voted against the motion, while France and Brazil abstained. France adopted Greenwich Mean Time in place of local determinations by the Paris Observatory in 1911, the latitude of a point on Earths surface is the angle between the equatorial plane and the straight line that passes through that point and through the center of the Earth. Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they are parallel to the equator, the north pole is 90° N, the south pole is 90° S. The 0° parallel of latitude is designated the equator, the plane of all geographic coordinate systems. The equator divides the globe into Northern and Southern Hemispheres, the longitude of a point on Earths surface is the angle east or west of a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses, which converge at the north and south poles, the prime meridian determines the proper Eastern and Western Hemispheres, although maps often divide these hemispheres further west in order to keep the Old World on a single side. The antipodal meridian of Greenwich is both 180°W and 180°E, the combination of these two components specifies the position of any location on the surface of Earth, without consideration of altitude or depth. The grid formed by lines of latitude and longitude is known as a graticule, the origin/zero point of this system is located in the Gulf of Guinea about 625 km south of Tema, Ghana. To completely specify a location of a feature on, in, or above Earth. Earth is not a sphere, but a shape approximating a biaxial ellipsoid. It is nearly spherical, but has an equatorial bulge making the radius at the equator about 0. 3% larger than the radius measured through the poles, the shorter axis approximately coincides with the axis of rotation
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