query: when was the four color map problem solved

pos: Four color theorem 1800s, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken, but only after many false proofs and counterexamples. It was the first major theorem to be proved using a computer. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. Since then the proof has gained wide acceptance, although some doubters remain. To dispel any remaining doubts about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas.

pos: Graph coloring a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London Mathematical Society. In 1890, Heawood pointed out that Kempe’s argument was wrong. However, in that paper he proved the five color theorem, saying that every planar map can be colored with no more than "five" colors, using ideas of Kempe. In the following century, a vast amount of work and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by Kenneth

pos: Wolfgang Haken Wolfgang Haken Wolfgang Haken (born June 21, 1928) is a mathematician who specializes in topology, in particular 3-manifolds. Haken was born in Berlin, Germany. In 1962 he left Germany to become a visiting professor at the University of Illinois at Urbana-Champaign. He became a full professor by 1965 and is now an emeritus professor. In 1976 together with colleague Kenneth Appel, also at the University of Illinois at Urbana-Champaign, Haken solved the four-color problem. They proved that any map can be filled in with four colors without any adjacent "countries" sharing the same color. Haken has introduced several ideas, including

pos: Wolfgang Haken in Helsinki. He was a recipient of the 1979 Fulkerson Prize of the American Mathematical Society for his solution with Appel of the four-color problem. Wolfgang Haken Wolfgang Haken (born June 21, 1928) is a mathematician who specializes in topology, in particular 3-manifolds. Haken was born in Berlin, Germany. In 1962 he left Germany to become a visiting professor at the University of Illinois at Urbana-Champaign. He became a full professor by 1965 and is now an emeritus professor. In 1976 together with colleague Kenneth Appel, also at the University of Illinois at Urbana-Champaign, Haken solved the four-color problem. They

pos: Kenneth Appel Kenneth Appel Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous problems in mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any adjacent "countries" sharing the same color. Appel was born in Brooklyn, New York, on October 8, 1932, and he died in Dover, New Hampshire, on April 19, 2013, after being diagnosed with esophageal cancer in October 2012. He grew up

pos: Kenneth Appel article Appel and Schupp introduced four theorems that are true about Coxeter groups and then proved them to be true for Artin groups. The proofs of these four theorems used the "results and methods of small cancellation theory." Kenneth Appel Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous problems in mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any adjacent "countries"

pos: Heawood conjecture formula. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained the conjecture. In other words, Ringel, Youngs and others had to construct extreme examples for every genus g = 1,2,3... If g = 12s + k, the genera fall into 12 cases according as k = 0,1,2,3,4,5,6,7,8,9,10,11.

pos: Francis Guthrie would be sufficient to colour any map. This became known as the Four Color Problem, and remained one of the most famous unsolved problems in topology for more than a century until it was eventually proven in 1976 using a lengthy computer-aided proof. Guthrie arrived in South Africa on 10 April 1861 and was met and entertained by Dr Dale (later Sir Langham Dale), who was instrumental in the establishing of the University of the Cape of Good Hope in June 1873. Guthrie took up the post of mathematics master at the Graaff-Reinet College. While there he gave a course

neg: Conjecture example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not. Möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23, 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late

neg: Map-coloring games game is partisan. The set of colors could also depend on the state of the game; for instance it could be required that the color used be different from the color used on the previous move. The map-based constraints on a move are usually based on the region to be colored and its neighbors, whereas in the map-coloring problem, regions are considered to be neighbors when they meet along a boundary longer than a single point. The classical map-coloring problem requires that no two neighboring regions be given the same color. The "classical" move constraint enforces this by prohibiting coloring

neg: Map coloring Map coloring Map coloring is the act of assigning different colors to different features on a map. There are two very different uses of this term. The first is in cartography, choosing the colors to be used when producing a map. The second is in mathematics, where the problem is to determine the minimum number of colors needed to color a map so that no two adjacent features have the same color. Color is a very useful attribute to depict different features on a map. Typical uses of color include displaying different political divisions, different elevations, or different kinds of

neg: Graph coloring of "maps". While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Guthrie’s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, who mentioned it in a letter to William Hamilton in 1852. Arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. The same year, Alfred Kempe published a paper that claimed to establish the result, and for

neg: Four fours for a much greater proportion of numbers; for example, 113 = ( + ( + 4!)%) ÷ ()%. This problem and its generalizations (like the five fives and the six sixes problem, both shown below) may be solved by a simple algorithm. The basic ingredients are hash tables that map rationals to strings. In these tables, the keys are the numbers being represented by some admissible combination of operators and the chosen digit "d", e.g. four, and the values are strings that contain the actual formula. There is one table for each number "n" of occurrences of "d". For example,

neg: Open problem and "closed" by researchers in the late twentieth century are Fermat's Last Theorem and the four color theorem. An important open mathematics problem solved in the early 21st century is the Poincaré conjecture. Important open problems exist in all scientific fields. For example, one of the most important open problems in biochemistry is the protein structure prediction problem – how to predict a protein's structure from its sequence. Open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable

neg: Watchman route problem solution whose length is within a polylogarithmic factor of optimal. Watchman route problem The Watchman Problem is an optimization problem in computational geometry where the objective is to compute the shortest route a watchman should take to guard an entire area with obstacles given only a map of the area. The challenge is to make sure the watchman peeks behind every corner and to determine the best order in which corners should be visited in. The problem may be solved in polynomial time when the area to be guarded is a simple polygon. The problem is NP-hard for polygons with

neg: Watchman route problem Watchman route problem The Watchman Problem is an optimization problem in computational geometry where the objective is to compute the shortest route a watchman should take to guard an entire area with obstacles given only a map of the area. The challenge is to make sure the watchman peeks behind every corner and to determine the best order in which corners should be visited in. The problem may be solved in polynomial time when the area to be guarded is a simple polygon. The problem is NP-hard for polygons with holes, but may be approximated in polynomial time by a

neg: Multi-Color Graphics Array mode and didn't attempt to use VGA-specific features such as multiple screen pages. Games lacking support for 256-color graphics were forced to fall back to four-color CGA mode (or not run at all) due to the incompatibility with EGA video modes (320×200 in 16 colors, as well as sometimes 640x200 or 640x350). Some games, including point-and-click adventures from Sierra On-line and Lucasfilm Games, as well as simulation and strategy titles from Microprose solved this problem for low-resolution titles by supporting MCGA in its 320×200 256-color mode and picking the colors most resembling the EGA 16-color RGB palette, while leaving the

neg: Four color theorem Four color theorem In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a "map", no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. "Adjacent" means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. Unlike the five color theorem, a theorem that states that five colors are enough to color a map, which was proved in the

neg: Four color theorem formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, "every planar graph is four-colorable". As far as is known, the conjecture was first proposed on October 23, 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. At the time, Guthrie's brother, Frederick, was a student of Augustus De Morgan (the former advisor of Francis)

neg: Graph theory theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by Francis Guthrie in 1852 and its first written record is in a letter of De Morgan addressed to Hamilton the same year. Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the

neg: Boolean Pythagorean triples problem Boolean Pythagorean triples problem The Boolean Pythagorean triples problem is a problem relating to Pythagorean triples which was solved using a computer-assisted proof in May 2016. This problem is from Ramsey theory and asks if it is possible to color each of the positive integers either red or blue, so that no Pythagorean triple of integers "a", "b", "c", satisfying formula_1 are all the same color. For example, in the Pythagorean triple 3, 4 and 5 (formula_2), if 3 and 4 are colored red, then 5 must be colored blue. Marijn Heule, Oliver Kullmann and Victor Marek investigated the problem,

neg: Hajdučka Republika Mijata Tomića in the middle is with blue color marked the picture of Mijat Tomić. Problems began when Vinko Vukoja wanted to solve a problem in connection with the main power supply net. As this area was on unclaimed land – three municipalities (Posušje, Tomislavgrad and Jablanica, couldn't agree for years under whose responsibility the area fell, and whenever the motel owner asked any of the municipalities to solve its problem, they rejected him, saying it is not their problem, and referred him to one of other two municipalities. Finally, he solved that with his money. So in that alleged "no one's

neg: OptimJ the variable. Constraints express conditions that must be true in any solution of the problem. A constraint can be any Java boolean expression involving decision variables. In the map coloring example, this set of constraints states that in any solution to the map coloring problem, the color of Belgium must be different from the color of Germany, and the color of Germany must be different from the color of Denmark. The operator codice_8 is the standard Java not-equal operator. Constraints typically come in batches and can be quantified with the codice_9 operator. For instance, instead of listing all countries and

neg: Exotic sphere to computing homotopy groups of spheres. The map is either an isomorphism (the image is the whole group), or an injective map with index 2. The latter is the case if and only if there exists an "n"-dimensional framed manifold with Kervaire invariant 1, which is known as the Kervaire invariant problem. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem. , the Kervaire invariant problem is almost completely solved, with only the case remaining open; see that article for details. This is primarily the work of , which proved that such

neg: Difference-map algorithm problems. Since these applications include NP-complete problems, the scope of the difference map is that of an incomplete algorithm. Whereas incomplete algorithms can efficiently verify solutions (once a candidate is found), they cannot prove that a solution does not exist. The difference-map algorithm is a generalization of two iterative methods: Fienup's Hybrid input output (HIO) algorithm for phase retrieval The problem to be solved must first be formulated as a set intersection problem in Euclidean space: find an formula_1 in the intersection of sets formula_2 and formula_3. Another prerequisite is an implementation of the projections formula_4 and formula_5 that, given

neg: Automatic label placement of labels into smaller sets that can be solved independently. Two labels are "rivals" if they can overlap in one of the possible placements. Transitive closure of this relation divides the set of labels into possibly much smaller sets. On uniformly and densely labelled maps, usually the single set will contain the majority of labels, and on maps for which the labelling is not uniform it may bring very big performance benefits. For example, when labelling a map of the world, America is labelled independently from Eurasia etc. If a map labeling problem can be reduced to a situation in

neg: ANTIC output pixels incorrectly and misalign the Mode 8-9 line. The bug is absent when narrow width playfield is used. Since the HSCROL values causing the problem are greater than 7, the problem can be averted when Mode 8 or 9 is NOT using Horizontal scrolling by limiting HSCROL shifts to 0 through 7 color clocks and updating the Display List LMS address to horizontally coarse scroll in smaller increments. This strategy resolves all cases in all the Text modes, and all Map modes from A through F which generate 8 or fewer color clocks per byte of screen memory. A

neg: Jakob Erhardt fellow missionaries Krapf and Rebmann. In November 1854 while talking about the problem to Rebmann, "at one and the same moment, the problem flashed on both of us solved by the simple supposition that where geographical hypothesis had hitherto supposed an enormous mountain-land, we must now look for an enormous valley and an inland sea." On the map that he and Rebmann drew the three lakes are shown as one very large S-shaped lake. In 1855 Erhardt was repatriated due to poor health, and took the map with him. It was first published in the "Calwer Missionsblatt" in 1855, and

neg: Five color theorem proved the five color theorem based on Kempe's work. First of all, one associates a simple planar graph formula_1 to the given map, namely one puts a vertex in each region of the map, then connects two vertices with an edge if and only if the corresponding regions share a common border. The problem is then translated into a graph coloring problem: one has to paint the vertices of the graph so that no edge has endpoints of the same color. Because formula_1 is a simple planar, i.e. it may be embedded in the plane without intersecting edges, and it

neg: Peg + Cat 2016, PBS Kids released the first part of the film on its YouTube channel. When a math problem is solved, Peg, Cat and others sing like this: "Problem solved, The problem is solved, We solved the problem, Problem solved." In Canada the show is broadcast on Treehouse TV. In late 2016, Fred Rogers Productions and 9 Story Media Group signed a deal with the Montreal-based toy manufacturer Imports Dragon to create an assortment of plush, arts-and-crafts sets, games and wooden puzzles based on the "Peg + Cat" characters. The toys join numerous books from Candlewick Press, as well as DVDs

neg: Rational variety group of order 47. A celebrated case is Lüroth's problem, which Jacob Lüroth solved in the nineteenth century. Lüroth's problem concerns subextensions "L" of "K"("X"), the rational functions in the single indeterminate "X". Any such field is either equal to "K" or is also rational, i.e. "L" = "K"("F") for some rational function "F". In geometrical terms this states that a non-constant rational map from the projective line to a curve "C" can only occur when "C" also has genus 0. That fact can be read off geometrically from the Riemann–Hurwitz formula. Even though Lüroth's theorem is often thought as

neg: Dowding system of Britain. The Dowding system was developed after tests demonstrated problems relaying information to the fighters before it was out of date. Air Chief Marshal Hugh Dowding, commander of RAF Fighter Command, solved the problem through the use of hierarchical reporting chains. Information was sent to Fighter Command Headquarters (FCHQ) central "filter room" at Bentley Priory and used to prepare a map of the battle. Details of the map were then relayed to the Group and Sector headquarters, where operators re-created the map at a scale covering their area of operations. Looking at the maps, commanders could make decisions on

neg: Conjecture and prior to its proof it was in the "Guinness Book of World Records" for "most difficult mathematical problems". In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a "map", no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called "adjacent" if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For

neg: Gerardus Mercator between the pole and an arbitrary position: he had solved the longitude problem—if his theory had been correct. Further comments on magnetism may be found in an earlier letter to Perrenot and on the later world map. In the Hogenberg portrait (below) his dividers are set on the position of the magnetic pole. In 1554 Mercator published the long-awaited wall map of Europe, dedicating it to his friend, now Cardinal, Antoine Perrenot. He had worked at it for more than twelve years, collecting, comparing, collating and rationalising a vast amount of data and the result was a map of unprecedented

neg: X Window System core protocol installed when the hardware shows colors according to it. A client can request the server to install a colormap. However, this may require the uninstalling of another colormap: the effect is that windows using the uninstalled colormap are not shown with the correct color, an effect dubbed "color flashing" or "technicolor". This problem can be solved using "standard colormaps", which are colormaps with a predictable association between pixelvalues and colors. Thanks to this property, standard colormaps can be used by different applications. The creation of colormaps is regulated by the ICCCM convention. Standard colormaps are regulated by the ICCCM and

neg: Longest path problem can be used to find paths of logarithmic length, if they exist, but this gives an approximation ratio of only formula_7. The longest path problem is fixed-parameter tractable when parameterized by the length of the path. For instance, it can be solved in time linear in the size of the input graph (but exponential in the length of the path), by an algorithm that performs the following steps: Since the output path has length at least as large as formula_8, the running time is also bounded by formula_13, where formula_14 is the length of the longest path. Using color-coding, the

neg: Instant Insanity if the two colors are on the opposite sides of the cube, and a loop at a vertex if the opposite sides have the same color. Trial and error is a slow way to solve this problem, as there are 41,472 possible arrangements of the four cubes, and only 8 of these give a valid solution. Every position of Instant Insanity can be solved in eight moves or less. The puzzle was created by Franz Owen Armbruster, also known as Frank Armbruster, and published by Parker Brothers in 1967. Over 12 million puzzles were sold. The puzzle is isomorphic to

neg: BAPC Under no circumstances will the jury give hints as to what caused the error, which tests failed, or where the Bug is. For each solved problem, one point is awarded. For each point scored within the first four hours of the contest, a helium balloon is attached to the team's computer. After four hours, the scoreboard is no longer updated, so that it remains a secret who won until the award ceremony. Ties are solved by a time penalty (the lower, the better). Every team starts out with a penalty time of 0 minutes. For every solved problem, the time

neg: Special cases of Apollonius' problem two new circles, "c" and "c". Thus, the problem becomes that of finding a solution line tangent to the two inverted circles, which was solved above. There are four such lines, and re-inversion transforms them into the four solution circles of the original Apollonius problem. CLL problems generally have 8 solutions. This special case is solved most easily using scaling. The given circle is shrunk to a point, and the radius of the solution circle is either decreased by the same amount (if an internally tangent solution) or increased (if an externally tangent circle). Depending on whether the solution circle

neg: Type III error Raiffa describes an occasion when he, too, "fell into the trap of working on the wrong problem" (1968, pp. 264–265). In 1974, Ian Mitroff and Tom Featheringham extended Kimball's category, arguing that ""one of the most important determinants of a problem's solution is how that problem has been represented or formulated in the first place"". They defined type III errors as either ""the error ... of having solved the wrong problem ... when one should have solved the right problem"" or ""the error ... [of] choosing the wrong problem representation ... when one should have ... chosen the right problem

neg: Nonblocker problem with at most formula_5 vertices. First, remove all isolated vertices from formula_1, as they cannot be part of any nonblocker. Once this has been done, the remaining graph must have a nonblocker that includes at least half of its vertices; for instance, if one 2-colors any spanning tree of the graph, each color class is a nonblocker and one of the two color classes includes at least half the vertices. Therefore, if the graph with isolated vertices removed still has formula_5 or more vertices, the problem can be solved immediately. Otherwise, the remaining graph is a kernel with at

neg: Minkowski problem Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere "S" to be the surface area measure of a convex body in formula_1. Here the surface area measure "S" of a convex body "K" is the pushforward of the "(n-1)"-dimensional Hausdorff measure restricted to the boundary of "K" via the Gauss map. The Minkowski problem was solved by Hermann Minkowski, Aleksandr Danilovich Aleksandrov, Werner Fenchel and Børge Jessen: a Borel measure "μ" on the unit sphere is the surface area measure of a convex body if and only if "μ" has centroid at

neg: History of Cyprus since 1878 the problem. It was now to be solved in conjunction with Greece and Turkey, the latter thoroughly alerted to the dangers of "enosis" for the Turkish Cypriot population. Violence was renewed in Cyprus by EOKA, but it increasingly drew in the Turkish community when a new plan for unitary self-government, of British Governor Sir Hugh Foot, incited Turkish Cypriot riots and produced a hostile response from the Turkish government. Violence between the two communities developed into a new and deadly feature of the situation. In 1957 the U.N. decided that the issue should be resolved according to its Statutory Map.

neg: Interactive Decision Maps gray scale copy of a color computer display for a real-life water quality problem involving five objectives. The decision map consists of four superimposed bi-objective differently colored slices. A palette shows the relation between the values of the "third" objective and colors. Two scroll-bars are related to the values of the fourth and the fifth objectives. A movement of a scroll-bar results in a change of the decision map. One can move the slider manually. However, the most effective form of displaying information to the DM is based on an automatic movement of the slider, that is, on a gradual

neg: Development of No Man's Sky to wrap around the planet's sphere to create the illusion they had left the spherical map, but this led to problems with overlap at the edges of the map and terrain generation problems. The team then looked at the problem of map projections that had already been solved by cartographers. Some projections like the Mercator projection would have fit well with the generation system but would be costly in computation time to present to the player. Others, like Peirce quincuncial projection, while being better suited for fast computations, would have limited the type of larger terrain features they could generate

neg: Empirical theory of perception that visual system uses to work around the inverse problem. Over millions of years individuals whose visual systems more successfully linked sensory stimulation with successful behavior won out. In this view the inverse problem is not actually solved—which would be analogous to computing the outputs of all three hoses—the result might be close enough to behave appropriately in response to stimuli. Color vision is dependent on activation of three cone cell types in the human retina, each of which is primarily responsive to a different spectrum of light frequencies. While these retinal mechanisms enable subsequent color processing, their properties alone

neg: Penetron signal was very similar to the B&W signal used on existing sets, which meant the millions of B&W televisions would be able to receive the new signal while newer colors sets could see these in either B&W or color if that additional signal was provided. This was a huge advantage over the CBS system, and a modified version was selected by the NTSC as the new color standard in 1953. The major disadvantage was the difficulty in correctly focussing the beam on the correct color, a problem RCA solved with their shadow mask system. The shadow mask is a thin

neg: Karel Šmirous because of all their common interests, they became firm friends, and he remained in contact with them till end of their lives. After gaining his Doctorate in Chemistry, Šmirous worked as assistant to Prof. Votočka. During the period 1913 – 1918, he made first experiments with color printing, and after the war, he was sent by the Czech Academy of Sciences to visit the laboratory of the Lumière brothers in France. He worked there several times in his career, and in 1931, he solved the problem of enlarging color photographs under artificial light. Šmirous' long research into color printing culminated

neg: Lake Uniamési Rebmann, whose account was published in Krapf's memoirs, Erhardt was struck by the fact that various travelers who had gone inland from different points on the east coast of Africa had all come to an inland sea, and made a map based on available information, including the findings of Krapf and Rebmann. In November 1854 while talking about the problem to Rebmann, "at one and the same moment, the problem flashed on both of us solved by the simple supposition that where geographical hypothesis had hitherto supposed an enormous mountain-land, we must now look for an enormous valley and an

neg: Where in the World is Carmen Sandiego? 3 - New Carmen Adventure digital map of Atlantis, it turns out that Carmen Sandiego had been intercepting ACME communications and planned this entire mission. Carmen Sandiego steals the map and heads to South America. The gameplay consists of point and click elements. To progress the player must choose dialogue options during conversations, solve puzzles, make use of the digital handheld ACME computer and select a travel destination on the world map until a case is solved. When a case is solved the player gains a higher ACME rank. A review of Where in the World Is Carmen Sandiego? 3 at PALGN was negative: "It's

neg: Two-body problem be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the "n"-body problem for "n" ≥ 3) cannot be solved in terms of first integrals, except in special cases. Let x and x be the vector positions of the two bodies, and "m" and "m" be their masses. The goal is to determine the trajectories x("t") and x("t") for all times "t", given the initial positions x("t" = 0) and x("t" = 0) and the initial velocities v("t" = 0) and v("t" = 0). When applied to the two masses, Newton's

neg: Transcomputational problem inputs. Since the number 2 is a transcomputational number (that is, a number greater than 10), the problem of testing such a system of integrated circuits is a transcomputational problem. This means that there is no way one can verify the correctness of the circuit for all combinations of inputs through brute force alone. Consider a "q"×"q" array of the chessboard type, each square of which can have one of "k" colors. Altogether there are "k" color patterns, where "n" = "q". The problem of determining the best classification of the patterns, according to some chosen criterion, may be solved

neg: Feedback vertex set NP-hard on graphs of maximum degree four. The feedback vertex set problem can be solved in polynomial time on graphs of maximum degree at most three. Note that the problem of deleting as few "edges" as possible to make the graph cycle-free is equivalent to finding a spanning tree, which can be done in polynomial time. In contrast, the problem of deleting edges from a directed graph to make it acyclic, the feedback arc set problem, is NP-complete. The corresponding NP optimization problem of finding the size of a minimum feedback vertex set can be solved in time "O"(1.7347), where

neg: Cube mapping the object. Problems occur when the mesh vertices are not dense enough, resulting in insufficient sampling of the specular lighting. This in turn results in highlights with brightness proportionate to the distance from mesh vertices, ultimately compromising the visual cues that indicate curvature. Unfortunately, this problem cannot be solved simply by creating a denser mesh, as this can greatly reduce the efficiency of object rendering. Cube maps provide a fairly straightforward and efficient solution to rendering stable specular highlights. Multiple specular highlights can be encoded into a cube map texture, which can then be accessed by interpolating across the surface's

neg: Structured support vector machine space is the predicted label. Solving for this maximizer is the so-called inference problem and similar to making a maximum a-posteriori (MAP) prediction in probabilistic models. Depending on the structure of the function formula_19, solving for the maximizer can be a hard problem. The above quadratic program involves a very large, possibly infinite number of linear inequality constraints. In general, the number of inequalities is too large to be optimized over explicitly. Instead the problem is solved by using delayed constraint generation where only a finite and small subset of the constraints is used. Optimizing over a subset of the

neg: John Kendrew However, the problem still remained insurmountable, until in 1953 Max Perutz discovered that the phase problem in analysis of the diffraction patterns could be solved by multiple isomorphous replacement — comparison of patterns from several crystals; one from the native protein, and others that had been soaked in solutions of heavy metals and had metal ions introduced in different well-defined positions. An electron density map at 6 angstrom (0.6 nanometre) resolution was obtained by 1957, and by 1959 an atomic model could be built at 2 angstrom (0.2 nm) resolution. In 1963 Kendrew became one of the founders of the

neg: Map coloring produce three-dimensional effects from two-dimensional maps, either by explicit color-coding of the two images intended for different eyes, or by using the characteristics of the human visual system to make the map look three-dimensional. In mathematics there is a very strong link between map coloring and graph coloring, since every map showing different areas has a corresponding graph. By far the most famous result in this area is the four color theorem, which states that any planar map can be colored with at most four colors. Map coloring Map coloring is the act of assigning different colors to different features

neg: Choropleth map situations. Broadly speaking, choropleths represent two types of data: spatially extensive or spatially intensive. Another common error in choropleths is the use of raw data values to represent magnitude rather than normalized values to produce a map of densities. This is problematic because the eye naturally integrates over areas of the same color, giving undue prominence to larger polygons of moderate magnitude and minimizing the significance of smaller polygons with high magnitudes. The problem with using data in total counts arises when the polygons are not all the same size (in area or total population), as in the figure at

neg: Change of variables be more easily solved when switching to polar coordinates. Consider for example the equation This may be a potential energy function for some physical problem. If one does not immediately see a solution, one might try the substitution Note that if formula_48 runs outside a formula_49-length interval, for example, formula_50, the map formula_26 is no longer bijective. Therefore formula_26 should be limited to, for example formula_53. Notice how formula_54 is excluded, for formula_26 is not bijective in the origin (formula_48 can take any value, the point will be mapped to (0, 0)). Then, replacing all occurrences of the original variables

neg: Equivariant topology these actions. The problem is solved if we can show the nonexistence of an equivariant map formula_44. Obstructions to the existence of such maps are often formulated algebraically from the topological data of formula_3 and formula_46. An archetypal example of such an obstruction can be derived having formula_42 a vector space and formula_48. In this case, a nonvanishing map would also induce a nonvanishing section formula_49 from the discussion above, so formula_50, the top Stiefel–Whitney class would need to vanish. Equivariant topology In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces,

neg: Light-emitting diode allows precise dynamic color control. As more effort is devoted to investigating this method, multicolor LEDs should have profound influence on the fundamental method that we use to produce and control light color. However, before this type of LED can play a role on the market, several technical problems must be solved. These include that this type of LED's emission power decays exponentially with rising temperature, resulting in a substantial change in color stability. Such problems inhibit and may preclude industrial use. Thus, many new package designs aimed at solving this problem have been proposed and their results are now

neg: Exidy Sorcerer graphics had to lie within a checkerboard pattern on the screen, and the system was generally less flexible than machines with "real" graphics. In addition, the high resolution was well beyond the capability of the average color TV, a problem they solved by not supporting color. In this respect the Sorcerer was similar to the PET and TRS-80 in that it had only "graphics characters" to draw with, but at least on the Sorcerer one could define a custom set. It was also possible to provide animation by character replacement or by redefining the character bitmap. Given these limitations, the

neg: Frank Stanton (executive) CBS color system. CBS color broadcasting only lasted for four months. CBS suspended it when the manufacture of color television receivers was halted by the US government as part of the Korean War effort. When the ban on color sets was rescinded in 1953, CBS announced that it had no plans to resume broadcasting using its field-sequential color system. A major problem with the CBS system was that the video was not "compatible" with existing black-and-white TV sets. A competing dot-sequential color system being developed by RCA was compatible, and in late 1953, the FCC switched its approval to an

neg: Mathematical problem compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically. Also provably unsolvable are so-called undecidable problems, such as the halting problem for Turing machines. Many abstract problems can be solved routinely, others have been solved with great effort, for some significant inroads have been made without having led yet to a full solution, and yet others have withstood all attempts, such as Goldbach's conjecture and the Collatz conjecture. Some well-known difficult abstract problems that have been solved relatively recently are the four-colour theorem, Fermat's Last Theorem, and the Poincaré conjecture. On the view of modern

neg: Kosaburo Hashiguchi Kosaburo Hashiguchi In 1988, he found the first algorithm to determine the star height of a regular language, a problem that had been open since 1963 when Lawrence Eggan solved the related star height problem of showing that there is no finite bound on the star height. Hashiguchi's algorithm for star height is extremely complex, and impractical on all but the smallest examples. A simpler method, showing also that the problem is PSPACE-complete, was provided in 2005 by Kirsten. Earlier, in 1979, Hashiguchi had also solved another open problem on regular languages, of deciding whether, for a given language formula_1,

neg: Necklace splitting problem Necklace splitting problem Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon and Douglas B. West. The basic setting involves a necklace with beads of different colors. The necklace should be divided between several partners, such that each partner receives the same amount of every color. Moreover, the number of "cuts" should be as small as possible (in order to waste as little as possible of the metal in the links between the beads). The following variants of the problem have been solved

neg: Treewidth shown to be equivalent to treewidth by . Later, several authors independently observed at the end of the 1980s that many algorithmic problems that are NP-complete for arbitrary graphs may be solved efficiently by dynamic programming for graphs of bounded treewidth, using the tree-decompositions of these graphs. As an example, the problem of coloring graph of treewidth "k" may be solved by using a dynamic programming algorithm on a tree decomposition of the graph. For each set "X" of the tree decomposition, and each partition of the vertices of "X" into color classes, the algorithm determines whether that coloring is

neg: Vehicle rescheduling problem Single Depot Vehicle Scheduling Problem and the Multi Depot Vehicle Scheduling Problem, there is a significant difference in runtime requirements, as VRSP need to be solved in near real-time to allow rescheduling during operations, while SDVSP and MDVSP are typically solved using long running linear programming methods. Another field where VRSP is used is in transportation of goods in order to reschedule the routes when demand substantially changes Vehicle rescheduling problem The vehicle rescheduling problem (VRSP) is a combinatorial optimization and integer programming problem seeking to service customers on a trip after change of schedule such as vehicle break down

neg: Unicode rendering engine and fonts are being used to display the characters. Similarly, underdots, as needed in the romanization of Indic, will often be placed incorrectly. Unicode characters that map to precomposed glyphs can be used in many cases, thus avoiding the problem, but where no precomposed character has been encoded the problem can often be solved by using a specialist Unicode font such as Charis SIL that uses Graphite, OpenType, or AAT technologies for advanced rendering features. The Unicode standard has imposed rules intended to guarantee stability. Depending on the strictness of a rule, a change can be prohibited or

neg: T-shirt is a direct-to-garment digital printing technology using full color artwork to transfer images to polyester and polymer-coated substrate based T-shirts. Dye-sublimation (also commonly referred to as all-over printing) came into widespread use in the 21st century, enabling some designs previously impossible. Printing with unlimited colors using large CMYK printers with special paper and ink is possible, unlike screen printing which requires screens for each color of the design. All-over print T-shirts have solved the problem with color fading and the vibrancy is higher than most standard printing methods, but requires synthetic fabrics for the ink to take hold. The key

neg: Bivariate map Bivariate map A bivariate map displays two variables on a single map by combining two different sets of graphic symbols or colors. Bivariate mapping is an important technique in cartography. It is a variation of simple choropleth map that portrays two separate phenomena simultaneously. The main objective is to accurately and graphically illustrate the relationship between two spatially distributed variables. It has potential to reveal relationships between variables more effectively than a side-by-side comparison of the corresponding univariate maps. Bivariate mapping is a comparatively recent graphical method. A bivariate choropleth map uses color to solve a problem of representation in

neg: Geer tube the mechanical methods of producing color television that were being experimented with in the 1940s, and decided that an electronically scanned system would be superior, if someone would only invent one. Mentioning it later to his wife, she replied "You'd better get busy and invent it yourself". Geer solved the display problem with the novel application of optics. Instead of trying to focus the electron beams onto tiny spots, he instead focused them onto larger areas and used simple optics to re-combine each individual primary color at any given place on the screen into a single pixel. The tube was

neg: Shōnen Hollywood is usually not assertive, accepting anything that comes before him. Later on, it is revealed that he cannot sing well; a problem which is eventually solved with the help of the other Shounen Hollywood members. His Hollywood color is red. Makki is an ex-Yankee who didn't finish his high school. Due to this, he has problems in reading kanji. He is now the center of the idol group. Later on, it is revealed that the reason he left high school was because he got into a fight with his fellow Yankees. His Hollywood color is green. Kira is a former

neg: Robbins' problem the motivations to study Robbins' problem is that with its solution all classical (four) secretary problems would be solved. But the major reason is to understand how to cope with full history dependence in a (deceptively easy-looking) problem. On the Ester's Book International Conference in Israel (2006) Robbins' problem was accordingly named one of the four most important problems in the field of optimal stopping and sequential analysis. Herbert Robbins presented the above described problem at the International Conference on Search and Selection in Real Time in Amherst, 1990. He concluded his address with the words "I should like to

neg: Discrete Fourier transform or −"i". Since there are only four distinct eigenvalues for this formula_159 matrix, they have some multiplicity. The multiplicity gives the number of linearly independent eigenvectors corresponding to each eigenvalue. (Note that there are "N" independent eigenvectors; a unitary matrix is never defective.) The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to have been equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the value of "N" modulo 4, and is given by the following table: Otherwise stated, the characteristic polynomial of formula_150 is: No

neg: Nikolaus Hofreiter assistant at the TU Vienna) where in 1929 he was assistant to Furtwängler and then habilitated in 1933. He was even then an excellent teacher, and gave lectures not only in Vienna but also in Graz. His dissertation and habilitation thesis dealt with the reduction theory of quadratic forms, which Gauss, Charles Hermite and Hermann Minkowski had worked on previously. Hofreiter treated the case of four variables of a problem of Minkowski (Minkowski had solved the problem for two variables, while Robert Remak had solved it for three variables) on the product of inhomogeneous linear forms and achieved significant progress.

neg: Asian option option pricing , the problem for geometric average can be solved via the Effective Classical potential Rogers and Shi solve the pricing problem with a PDE approach Variance Gamma model can be efficiently implemented when pricing Asian style options. Then using the Bondesson series representation for generating the variance gamma process shows to increase performance when pricing this type of option. Within Lévy models the pricing problem for geometrically Asian options can still be solved. For the arithmetic Asian option in Lévy models one can rely on numerical methods or on analytic bounds Asian option An Asian option (or "average

neg: TRS-80 Color Computer color television and allow the user access to near-real-time information useful to their day-to-day operations on the farm. Motorola's MC6847 Video Display Generator (VDG) chip was released about the same time as the joint venture started and it has been speculated that the VDG was actually designed for this project. At the core of the prototype "Green Thumb" terminal, the MC6847, along with the MC6809 microprocessor unit (MPU), made the prototype a reality by about 1978. Unfortunately, the prototype contained too many chips to be commercially viable. Motorola solved this problem by integrating all the functions of the many smaller

neg: Cramer–Castillon problem Cramer–Castillon problem In geometry, the Cramer–Castillon problem is a problem stated by the Swiss mathematician Gabriel Cramer solved by the italian mathematician, resident in Berlin, Jean de Castillon in 1776. The problem consists of (see the image): Given a circle formula_1 and three points formula_2 in the same plane and not on formula_1, to construct every possible triangle inscribed in formula_1 whose sides (or their elongations) pass through formula_2 respectively. Centuries before, Pappus of Alexandria had solved a special case: when the three points are collinear. But the general case had the reputation of being very difficult. After the geometrical

neg: New York City Subway map D’Adamo found a set of colors for subway lines that avoided clashes; Goldstein used D’Adamo's concept but invented his own color allocation; and finally Dante Calise at Diamond Packaging devised the color assignment that was used in the published map of 1967. Those colors were inherited by Unimark International and used in the famous 1972 Vignelli map. That color scheme was rearranged by Tauranac in 1979 to create a trunk-colored map. The colors used to denote services in the current iteration of the subway map are as follows: This is a table of when each service has existed (and been

neg: Gradient-domain image processing to add constraints so that only the pasted section is solved for, leaving the rest of the image unmodified. It is also worth noting that the gradient pictured above represents the derivative of only one color channel (red), and was rendered with colors representing the strength and direction of the gradient. In practice two grayscale gradient images are found per color channel, one representing the change in "x" and the other representing the change in "y". Each color channel is solved for independently when reconstructing the final image. Gradient-domain image processing Gradient domain image processing, also called Poisson image editing,

neg: Puget Sound faults show the velocity contrasts that would indicate contrasting rock types. Another problem with the SWIF/RMFZ as CRBF is that a large westward step is required to connect from the RMFZ to the Saint Helens Zone (SHZ; see map), whereas the RMFZ turns easterly to align with the OWL. This last problem is partly solved because there is a locus of seismicity, and presumably faulting, extending from the northern end of the SHZ to the northern end of the Western Rainier Zone (see Fig. 48), along the edge of a formation known as the "Southern Washington Cascades Conductor". However, gravity and

neg: Independent set (graph theory) label placement: given a set of locations in a map, find a maximum set of disjoint rectangular labels near these locations. Finding a maximum independent set in intersection graphs is still NP-complete, but it is easier to approximate than the general maximum independent set problem. A recent survey can be found in the introduction of . The problem of finding a maximal independent set can be solved in polynomial time by a trivial greedy algorithm. All maximal independent sets can be found in time O(3) = O(1.4423). Independent set (graph theory) In graph theory, an independent set, stable set or

neg: Problem Solved Ohio. While a spokesperson for Kmart's parent company Sears Holdings Corporation said that the shirt was intended to be "light hearted", Meijer reported that they had pulled the shirts immediately after a protest. Problem Solved Problem Solved is a slogan on a t-shirt designed by Route 66 Attitude, a clothing line sold by major U.S. department store Kmart. The shirt depicts a male stick figure and a female stick figure arguing in the first frame (captioned "Problem"), and in the second frame, the male pushes the female out of a window (captioned "Solved.") Due to its encouragement of violence against

neg: Branch and price to the LP relaxation as needed. The approach is based on the observation that for large problems most columns will be nonbasic and have their corresponding variable equal to zero in any optimal solution. Thus, the large majority of the columns are irrelevant for solving the problem. The algorithm typically begins by using a reformulation, such as Dantzig–Wolfe decomposition, to form what is known as the Master Problem. The decomposition is performed to obtain a problem formulation that gives better bounds when the relaxation is solved than when the relaxation of the original formulation is solved. But, the decomposition usually

neg: Permutation pattern result to intervals of separable permutations. Given a permutation formula_4 (called the "text") of length formula_5 and another permutation formula_6 (called the "pattern"), the "permutation pattern matching (PPM)" problem asks whether formula_6 is contained in formula_4. When both formula_5 and formula_10 are regarded as variables, the problem is known to be NP-complete, and the problem of counting the number of such matches is #P-complete. However, PPM can be solved in linear time when "k" is a constant. Indeed, Guillemot and Marx showed that PPM can be solved in time formula_11, meaning that it is fixed-parameter tractable with respect to formula_10.

neg: Rimhak Ree series', in which Max Zorn solved a problem about the convergence of certain power series with complex coefficients. In the paper, Zorn posed a question of whether the same result held for power series with real coefficients. Ree solved the problem and sent the solution to Max Zorn. When Zorn received Ree's solution, it was sent to the Bulletin of the American Mathematical Society to be published in 1949 with the title 'On a problem of Max Zorn' and become the first mathematical paper published by a Korean in an international journal. During the Korean War, he fled south to

neg: Etak Los Angeles area, for example, three to four tapes were required. When an edge of the map was reached, the driver needed to change cassette tapes to continue benefitting from the accuracy of map-matching. The system had a tape drive that was designed to be installed within easy reach of the driver, so this could be done while driving. The map moved on the screen as the car was driven, but instead of the color raster graphics display of today's systems it had a green vector display. The Navigator had address geocoding (the ability to convert a street address to

neg: Four color theorem four colors. Second, for the purpose of the theorem, every "country" has to be a connected region, or contiguous. In the real world, this is not true (e.g., the Cabinda Province as part of Angola, Nakhchivan as part of Azerbaijan, Kaliningrad as part of Russia, and Alaska as part of the United States are not contiguous). Because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map: In this map, the two regions labeled "A" belong to the same country, and must be the same color.

neg: Charles Sanders Peirce Edited by Patricia Ann Turisi, in print. 2010: "Philosophy of Mathematics: Selected Writings" collects important writings by Peirce on the subject, many not previously in print. Edited by Matthew E. Moore, in print. Peirce's most important work in pure mathematics was in logical and foundational areas. He also worked on linear algebra, matrices, various geometries, topology and Listing numbers, Bell numbers, graphs, the four-color problem, and the nature of continuity. He worked on applied mathematics in economics, engineering, and map projections (such as the Peirce quincuncial projection), and was especially active in probability and statistics. Peirce made a number of

neg: Problem solving and to write down any dreams that they then had. They were instructed to think about the problem again for 15 minutes when they awakened in the morning. The sequence OTTFF is the first letters of the numbers: one, two, three, four, five. The next five elements of the series are SSENT (six, seven, eight, nine, ten). Some of the students solved the puzzle by reflecting on their dreams. One example was a student who reported the following dream: With more than 500 undergraduate students, 87 dreams were judged to be related to the problems students were assigned (53 directly

neg: UEFA Euro 2008 used as the city's venue, but legal challenges delayed the plan to a point that would not have allowed the ground to be used in 2008. This created a problem, as the agreement between UEFA and the organisers stipulated that four venues would be used in each country. The problem was solved when the organisers proposed renovating Letzigrund instead; UEFA approved the revised plan in January 2005. The Letzigrund stadium hosted its first football match on 23 September 2007. Each team had access to a "team base camp" for its stay between the matches. The teams trained and resided in

neg: Taxonomy of Anopheles when the problem of nontransmission of malaria by "Anopheles gambiae" was solved by Falleroni, who recognised that "An. gambiae" was a complex of six species, of which only four could transmit malaria. This complex has subsequently been revised to a total of seven species of which five transmit malaria. Subgenus "Nyssorhynchus" has been divided in three sections: "Albimanus" (19 species), "Argyritarsis" (11 species) and "Myzorhynchella" (four species). The "Argyritarsis" section has been subdivided into "Albitarsis" and "Argyritarsis" groups. The "Anopheles" group was divided by Edwards into four series: "Anopheles" (worldwide), "Myzorhynchus" (Palearctic, Oriental, Australasian and Afrotropical), "Cycloleppteron" (Neotropical) and "Lophoscelomyia"

neg: United States Post Office, Courthouse, and Custom House (Spokane, Washington) with only minimal interruptions to patrons. Only four months after the post office opened, rats were discovered in the basement. To remedy the problem, seven cats were allowed to live in the building, and the postmaster received $18 annually for each cat's care and feeding. The cats lived in the post office for at least one year, although no one is sure when the rat problem was solved. As Spokane continued to grow and postal activity expanded, city officials realized the need to enlarge the building. The site directly north of the existing building was purchased and an addition was

neg: Alternating direction implicit method closed form using elliptic integrals, and can easily be computed numerically. More generally, if closed, disjoint sets formula_50 and formula_51, where formula_52 and formula_53, are known, the optimal shift parameter selection problem is approximately solved by finding an extremal rational function that attains the value formula_54 where the infimum is taken over all rational functions of degree formula_55. This approximation problem is related to several results in potential theory, and was solved by Zolotarev in 1877 for formula_50 = [a, b] and formula_57 The solution is also known when formula_58 and formula_59 are disjoint disks in the complex plane. When

neg: Two-body problem in general relativity solved only approximately. The earliest approximation method to be developed was the post-Newtonian expansion, an iterative method in which an initial solution is gradually corrected. More recently, it has become possible to solve Einstein's field equation using a computer instead of mathematical formulae. As the two bodies orbit each other, they will emit gravitational radiation; this causes them to lose energy and angular momentum gradually, as illustrated by the binary pulsar PSR B1913+16. For binary black holes numerical solution of the two body problem was achieved after four decades of research, in 2005, when three groups devised the breakthrough techniques.

neg: Kernelization edges and (because each edge has at most two endpoints and there are no isolated vertices) at most formula_22 vertices. This kernelization may be implemented in linear time. Once the kernel has been constructed, the vertex cover problem may be solved by a brute force search algorithm that tests whether each subset of the kernel is a cover of the kernel. Thus, the vertex cover problem can be solved in time formula_23 for a graph with formula_24 vertices and formula_25 edges, allowing it to be solved efficiently when formula_2 is small even if formula_24 and formula_25 are both large. Although

neg: Four color theorem region only has to be colored differently from regions it touches directly, not regions touching regions that it touches. If this were the restriction, planar graphs would require arbitrarily large numbers of colors. Other false disproofs violate the assumptions of the theorem in unexpected ways, such as using a region that consists of multiple disconnected parts, or disallowing regions of the same color from touching at a point. While every planar map can be colored with four colors, it is NP-complete in complexity to decide whether an arbitrary planar map can be colored with just three colors. The four-color theorem

neg: Fair item assignment a proportional allocation exists is NP-complete: it can be reduced from the partition problem. 2. Suppose the agents have ordinal rankings on items, with or without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial time: it can be reduced to the problem of checking whether a bipartite graph admits a feasible "b-matching" (a matching when the edges have capacities). For two agents, a simpler algorithm exists. 3. Suppose the agents have ordinal rankings on items, without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial

neg: Perturbation theory form of the original problem, which is "simple enough" to be solved exactly. In celestial mechanics, this is usually a Keplerian ellipse. Under non-relativistic gravity, an ellipse is exactly correct when there are only two gravitating bodies (say, the Earth and the Moon) but not quite correct when there are three or more objects (say, the Earth, Moon, Sun, and the rest of the solar system) and not quite correct when the gravitational interaction is stated using formulations from General relativity. The solved, but simplified problem is then ""perturbed"" to make the conditions that the perturbed solution actually satisfies closer

neg: Prepress proofing describes the simulation of the print result on a monitor. This is the cheapest solution for proofing since no additional equipment, except a monitor, is needed. Usually soft proof is done in software applications in combination with color management systems. The monitor used for soft proofing must be calibrated, so that the image on the screen looks like the image on the paper. The major problem is the difference of color spaces (RGB in monitor and CMYK in print), and this is solved by using ICC profiles for input and output devices. Moreover, colors on a monitor depend not only