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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.
This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.
Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.
For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec
Richard Taylor (mathematician)
Richard Lawrence Taylor is a British and American mathematician working in the field of number theory. He is a professor of mathematics at Stanford University and the Institute for Advanced Study. Taylor received the 2014 Breakthrough Prize in Mathematics "for numerous breakthrough results in the theory of automorphic forms, including the Taniyama–Weil conjecture, the local Langlands conjecture for general linear groups, the Sato–Tate conjecture." He received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands. He received his BA from Cambridge. During his time at Cambridge, he was president of The Archimedeans in 1981 and 1982, following the impeachment of his predecessor, he earned his PhD from Princeton University in 1988. From 1995 to 1996 he held the Savilian chair of geometry at Oxford University and Fellow of New College and became the Herchel Smith Professor of Mathematics at Harvard University, he holds Robert and Luisa Fernholz Professorship at the Institute for Advanced Study.
He received the Whitehead Prize in 1990, the Fermat Prize, the Ostrowski Prize in 2001, the Cole Prize of the American Mathematical Society in 2002, the Shaw Prize for Mathematics in 2007. He was elected a Fellow of the Royal Society in 1995. In 2012 he became a fellow of the American Mathematical Society. In 2015 he was inducted into the National Academy of Sciences, he was elected to the American Philosophical Society in 2018. One of the two papers containing the published proof of Fermat's Last Theorem is a joint work of Taylor and Andrew Wiles. In subsequent work, Taylor proved the local Langlands conjectures for GL over a number field. A simpler proof was suggested at the same time by Guy Henniart, ten years by Peter Scholze. Taylor, together with Christophe Breuil, Brian Conrad and Fred Diamond, completed the proof of the Taniyama–Shimura conjecture, by performing quite heavy technical computations in the case of additive reduction. In 2008, following the ideas of Michael Harris and building on his joint work with Laurent Clozel, Michael Harris, Nick Shepherd-Barron, announced a proof of the Sato–Tate conjecture, for elliptic curves with non-integral j-invariant.
This partial proof of the Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves. Taylor is the son of British physicist John C. Taylor, he is married, has two children. His home page at the Institute for Advanced Study Richard Taylor at the Mathematics Genealogy Project Autobiography upon Shaw Prize acceptance
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
In number theory, the Mordell conjecture is the conjecture made by Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, is now known as Faltings's theorem; the conjecture was generalized by replacing Q by any number field. Let C be a non-singular algebraic curve of genus g over Q; the set of rational points on C may be determined as follows: Case g = 0: no points or infinitely many. Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group. Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup. Case g > 1: according to the Mordell conjecture, now Faltings's theorem, C has only a finite number of rational points. Faltings's original proof used the known reduction to a case of the Tate conjecture, a number of tools from algebraic geometry, including the theory of Néron models. A different proof, based on diophantine approximation, was found by Vojta.
A more elementary variant of Vojta's proof was given by Bombieri. Faltings's 1983 paper had as consequences a number of statements, conjectured: The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points; the reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Paršin. A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1. Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, proved by Faltings. Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety over a number field k X is not Zariski dense in X.
More general conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Manin and by Grauert. In 1990, Coleman found and fixed a gap in Manin's proof. Bombieri, Enrico. "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17: 615–640. MR 1093712. Coleman, Robert F.. "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. Revue Internationale. IIe Série. 36: 393–427. ISSN 0013-8584. MR 1096426. Archived from the original on 2011-10-02. Cornell, Gary. Arithmetic geometry. Papers from the conference held at the University of Connecticut, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. Doi:10.1007/978-1-4613-8655-1. ISBN 0-387-96311-1. MR 0861969. → Contains an English translation of Faltings Faltings, Gerd. "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73: 349–366. Doi:10.1007/BF01388432. MR 0718935. Faltings, Gerd. "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
Inventiones Mathematicae. 75: 381. Doi:10.1007/BF01388572. MR 0732554. Faltings, Gerd. "Diophantine approximation on abelian varieties". Ann. of Math. 133: 549–576. Doi:10.2307/2944319. MR 1109353. Faltings, Gerd. "The general case of S. Lang's conjecture". In Cristante, Valentino. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396. Grauert, Hans. "Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper". Publications Mathématiques de l'IHÉS: 131–149. ISSN 1618-1913. MR 0222087. Hindry, Marc. Diophantine geometry. Graduate Texts in Mathematics. 201. New York: Springer-Verlag. Doi:10.1007/978-1-4612-1210-2. ISBN 0-387-98981-1. MR 1745599. → Gives Vojta's proof of Faltings's Theorem. Lang, Serge. Survey of Diophantine geometry. Springer-Verlag. Pp. 101–122. ISBN 3-540-61223-8. Manin, Ju. I.. "Rational points on algebraic curves over function fields".
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. 27: 1395–1440. ISSN 0373-2436. MR 0157971. Mordell, Louis J.. "On the rational solutions of the indeterminate equation of the third and fourth degrees". Proc. Cambridge Philos. Soc. 21: 179–192. Paršin, A. N.. "Quelques conjectures de finitude en géométrie diophantienne". Actes du Congrès International des Mathématiciens. Tome 1. Nice: Gauthier-Villars. Pp. 467–471. MR 0427323. Archived from the original on 2016-09-24. Retrieved 2016-06-11. Parshin, A. N. "Mordell conje
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.
As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.
Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.
This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.
Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic
Sir Andrew John Wiles is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society, he was appointed Knight Commander of the Order of the British Empire in 2000, in 2018 was appointed as the first Regius Professor of Mathematics at Oxford. Wiles was born on 11 April 1953 in Cambridge, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford, Patricia Wiles, his father worked as the chaplain at Ridley Hall, for the years 1952–55. Wiles attended King's College School and The Leys School, Cambridge. Wiles states that he came across Fermat's Last Theorem on his way home from school when he was 10 years old, he stopped at his local library. Fascinated by the existence of a theorem, so easy to state that he, a ten year old, could understand it, but that no one had proven, he decided to be the first person to prove it.
However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream, until it was brought back to his attention at the age of 33 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had linked to Fermat's famous equation. Wiles earned his bachelor's degree in mathematics in 1974 at Merton College, a PhD in 1980 as a graduate student of Clare College, Cambridge. After a stay at the Institute for Advanced Study in Princeton, New Jersey in 1981, Wiles became a Professor of Mathematics at Princeton University. In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, he returned to Princeton. From 1994 - 2009, Wiles was a Eugene Higgins Professor at Princeton, he rejoined Oxford in 2011 as Royal Society Research Professor. In May 2018 he was appointed Regius Professor of Mathematics at Oxford, the first in the university's history.
Wiles's graduate research was guided by John Coates beginning in the summer of 1975. Together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory, he further worked with Barry Mazur on the main conjecture of Iwasawa theory over the rational numbers, soon afterward, he generalised this result to real fields. His biographical page at Princeton University's website states that "Andrew has few equals in terms of his impact on modern number theory. Many of the world’s best young number theorists received their Ph. D.'s under Andrew... and many of these are today leaders and professors at top institutions around the world". Starting in mid-1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem; the modularity theorem involved elliptic curves, Wiles's own specialist area.
The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or impossible to prove. For example, Wiles's ex-supervisor John Coates states that it seemed "impossible to prove", Ken Ribet considered himself "one of the vast majority of people who believed was inaccessible", adding that "Andrew Wiles was one of the few people on earth who had the audacity to dream that you can go and prove."Despite this, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed for Frey's curve. He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge, he gave a lecture a day on Monday and Wednesday with the title'Modular Forms, Elliptic Curves and Galois Representations.'
There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said.... At the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture; as an afterthought, he noted that that meant that Fermat's last theorem was true. Q. E. D. In August 1993, it was discovered. Wiles failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing, rather than closing, this area came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a second paper which circumvented the problem and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of the Annals of Mathematics. Wiles's proof of Fermat's Last Theorem has stood up to the scrutiny of the world's other mathematical experts. Wiles was interviewed for an episode of the BBC documentary series Horizon that focused on Fermat's Last Theorem; this was renamed "The Proof", it was made an episode of the US Public Broadcasting Service's science television series Nova.
His work and life are described in great detail in Simon Singh's popular book Fermat's Last Theorem. Wiles has been awarded a number of major prizes in mathematics and science: Junior Whitehead Prize of the London Mathematical Soci
Fermat's Last Theorem
In number theory Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions; the proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, formally published in 1995, it proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century, it is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.
The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, z. Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, no proof by him has been found, his claim was discovered some 30 years after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries; the claim became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics; the special case n = 4 - proved by Fermat himself - is sufficient to establish that if the theorem is false for some exponent n, not a prime number, it must be false for some smaller n, so only prime values of n need further investigation.
Over the next two centuries, the conjecture was proved for only the primes 3, 5, 7, although Sophie Germain innovated and proved an approach, relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible. Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura–Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem, it was seen as significant and important in its own right, but was considered inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two unrelated and unsolved problems.
An outline suggesting this could be proved was given by Frey. The full proof that the two problems were linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture"; these papers by Frey and Ribet showed that if the Modularity Theorem could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers.
Important for researchers choosing a research topic was the fact that unlike Fermat's Last Theorem the Modularity Theorem was a major active research area for which a proof was desired and not just a historical oddity, so time spent working on it could be justified professionally. However, general opinion was that this showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one: "I myself was sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to prove. I must confess I thought I wouldn’t see it proved in my lifetime." On hearing that Ribet had proven Frey's li
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If how to write a college application essay examples one just makes up an erroneous solution, waves hands, or manages to mis-step and commit a mistake my fashion essay to writing, this is not math, or at least would no longer be th. given 1).right angle triangle two sides. some ideas may be way out there, but write essays for money uk strategy for problem solving don’t worry about evaluating them yet. solve history research paper topics for college students the equation found in step 4. math problem solver online multi-step equation with decimals and fracyions how to solve a math problem step by step ; online calculator with pie ; computer business plan how can solve determinants with a ti-89? Solving equations, rational arithmetic, and verifying trigonometric identities the mechanical approach to math problem solving relies heavily on manipulation of terms using low level mathematical constructs without using the problem solving abilities of the student. a box has a volume of 85 cm3 and a density how to cite a letter from birmingham jail of 1.2g/cm3. step how to solve a math problem step by step 3. solve the equation when the sale price is $97. you can step by step solve your algebra problems online – equations, inequalities, radicals, plot graphs, solve polynomial problems. 1 how to solve this math how to solve a math problem step by step problem step by math homework answer step? keep an moral essay topics open mind and list anything that comes. the combination physics online homework of addition, multiplication and fractions in a problem often looks like a foreign language. we have to use clever title for essay the source transformations how to solve a math problem step by step method. so next time in essay discuss you find yourself ready to give up on a math problem, make sure to check with wolfram|alpha many math word problems only have one step, like this example that requires you to subtract 1 from 12.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038072180.33/warc/CC-MAIN-20210413092418-20210413122418-00493.warc.gz
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CC-MAIN-2021-17
| 1,947 | 1 |
https://rd.springer.com/chapter/10.1007/978-3-642-73885-2_7
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math
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For use in next sections we shall discuss here how to compute some integrals. To this end we need generalizations of the theorems on integration of monotone sequences and on dominated convergence; the discrete parameter n in these theorems will be replaced by a continuous parameter ⋋. Let first µ, be a σ-finite measure in the (non-empty) point set X.
KeywordsFourier Series Fourier Coefficient Inverse Fourier Transform Continuous Derivative Approximate Identity
Unable to display preview. Download preview PDF.
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CC-MAIN-2019-22
| 517 | 3 |
https://blindspot.fandom.com/wiki/Mike_Jones
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math
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Coach Mike Jones was an Asst. Coach for the Hudson University Football Team. He was a good friend of Reade until he turned out to be a rapist.
#19 In the Comet of Us
#4 If Beth
#5 Condone Untidiest Thefts
#6 Her Spy's Harmed
#9 Why Let Cooler Pasture Deform
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571597.73/warc/CC-MAIN-20220812075544-20220812105544-00167.warc.gz
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CC-MAIN-2022-33
| 257 | 6 |
https://www.coursehero.com/file/5696363/Lecture-19-slidesAIC-amp-BIC-amp-Residuals-amp-Studentization/
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math
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This preview shows pages 1–3. Sign up to view the full content.
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Unformatted text preview: Maximum likelihood vs. SSE Therefore, for linear models, ML is almost equivalent to minimizing SSE! Similarly strictly using maximum likelihood ( 2 log p ( y | ) ) will overfit the data as we increase the number of parameters So we can look again at significant reductions in log-likelihood (or SSE), just viewing things through a different prism Lecture 22 p. 1/44 AIC Akaike suggested the AIC (Akaikes Information Criterion): AIC = 2 log p ( y | ) + 2 d where d is the number of unconstrained parameters in the model, p Of course, for the normal model, that gives us: AIC = n log 2 + n summationdisplay i =1 ( y i y i ) 2 2 + c n + 2( p + 1) where c n = n log 2 is a constant Lecture 22 p. 2 / 4 AIC Like adjusted R 2 , this is a penalized criterion, only now it is a penalized deviance criterion Using the AIC is justified, because it asymptotically approximates the Kullback-Liebler information, so choosing minimum AIC should choose the model with smallest K-L distance to the true model In English, this is the model that asymptotically gives the best predictive fit to true model Lecture 22 p. 3/44 Bayesian Information Criterion Another information criterion is the BIC (also known as the SIC), postulated by Schwarz (1978), justified further by Kass and Raftery (1995) and Kass and Wasserman (1995) The BIC approximation: BIC = D ( y, ) + d log n Note that the complexity penalty now depends on n Penalizes severely for large numbers of parameters Upshot: Will choose less complex models than AIC as it tries to choose the model asymptotically that is most likely to be the true model , not the one that is closest to the true model Lecture 22 p. 4 / 4 Example # Using the AIC function > AIC(treemodel.N) 48.34248 > AIC(treemodel.age) 37.46445 > AIC(treemodel.hd) 31.44173 > AIC(treemodel.agehd) 33.44107 > AIC(treemodel.NHD) 20.28018 > AIC(treemodel.ageN) 36.75299 > AIC(treemodel.agehdN) 18.37409 Lecture 22 p. 5/44 Example # Can calculate BIC with same function, only with # penalty k= log(n), where n is the sample size > n<-nrow(treedata) # NOTE THESE ARE BIC VALUES > AIC(treemodel.N,k=log(n)) 51.32967 > AIC(treemodel.age,k=log(n)) 40.45165 > AIC(treemodel.hd,k=log(n)) 34.42892 > AIC(treemodel.agehd, k=log(n)) 37.424 > AIC(treemodel.NHD, k=log(n)) 24.26311 > AIC(treemodel.ageN, k=log(n)) 40.73592 > AIC(treemodel.agehdN, k=log(n)) 23.35275 Lecture 22 p. 6 / 4 Comparison Model R 2 Adj PRESS CV C p AIC BIC AGE 0.43 7.12 0.38 34.8 37.5 40.5 N 0.01 12.7 0.71 71.4 48.3 51.3 HD 0.58 5.2 0.29 21.6 31.4 34.4 AGE + N 0.47 6.82 0.42 30.3 36.8 40.7 AGE + HD 0.55 6.26 0.39 23.6 33.4 37.4 HD + N 0.77 3.01 0.18 5.5 20.3 24.3 AGE + HD + N 0.80 3.00 0.19 4.0 18.4 23.4 Lecture 22 p. 7/44 Stepwise selection When I have large numbers of predictors, I may not be able to fit all possible models to find the one with minimum AIC, BIC, adj- R 2 , etc....
View Full Document
- Spring '06
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s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541317.69/warc/CC-MAIN-20161202170901-00417-ip-10-31-129-80.ec2.internal.warc.gz
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CC-MAIN-2016-50
| 3,098 | 5 |
http://forevershouston.com/solving-quadratic-equations-by-taking-square-roots-worksheet.html
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math
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Solving Quadratic Equations By Taking Square Roots Worksheet
Note that plus or minus is always used when square rooting both sides of an equation.
Solving quadratic equations by taking square roots worksheet. Easy take the square root medium addsubtract then take square root hard addsubtract divide then take square root mixture of all 3 types. Quadratics by taking square roots. Students learn to solve quadratic equations by first isolating the squared term then square rooting both sides of the equation. Solving 2x23131 and similar equations.
Sometimes we have to isolate the squared term before taking its root. So lets start by getting rid of the 8. The solution was x 2. A p290 r1g2x 1k hu gtxaa os rogfatew wa2rteb el klkc5.
For example if x2 25 take the square root of both sides of the equation to get x plus or minus 5. Solving quadratic equations by taking square roots. Solve by taking the square root worksheets. Khan academy is a 501c3 nonprofit organization.
Square root law solve each equation by taking square roots. Solve each equation by taking square roots. We do this exactly as we we would isolate the x term in a linear equation. For example to solve the equation 2x23131 we should first isolate x2.
This algebra 1 quadratic functions worksheet produces problems for solving quadratic equations by taking the square root. 1 r2 96 2 x2 7 3 x2 29 4 r2 78 5 b2 34 6 x2 0 7 a2 1 2 8 n2 4 77 9 m2 7 6 10 x2 1 80 11 4x2 6 74 12 3m2 7 301. Quadratic equations w square roots date period solve each equation by taking square roots. Not all quadratic equations are solved by immediately taking the square root.
Some of the worksheets displayed are solving quadratic roots solving quadratic equations square root law quadratic equations square roots math 154b name the square root property work the solving quadratic equations by taking square roots solving quadratic equations by finding square roots solving quadratics by the square root principle quadratic equations. Donate or volunteer today. So lets start by getting rid of the 8. 1 k2 6 6 2.
On the previous page id solved this quadratic equation by factoring the difference of squares on the left hand side of the equation and then setting each factor equal to zero etc etc. We cannot take the square root until x 2 is all by itself. Our mission is to provide a free world class education to anyone anywhere.
Gallery of Solving Quadratic Equations By Taking Square Roots Worksheet
- Mouse And The Motorcycle Coloring Pages
- Worksheet Heat And Heat Calculations Answer Key
- Loch Ness Monster Coloring Page
- Coloring Pages Baby
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- President Monson Coloring Page
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- Printable Mosaic Coloring Pages
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CC-MAIN-2019-51
| 2,798 | 20 |
https://www.thomasmills.suffolk.sch.uk/parents/curriculum/further-mathematics
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math
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ENTRY TO THE COURSE: Further Mathematics is aimed at those students with a very strong interest in the subject who may wish to take Mathematics as part or all of their course in Higher Education. The subject is ONLY offered in combination with Mathematics A Level. It requires the student to be able to work on his or her own on much of the material.
COURSE CONTENT: Six modules will be followed which integrate with and develop from the modules followed in the mathematics A Level course. It will comprise combinations of pure mathematics, decision mathematics and mechanics.
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CC-MAIN-2021-39
| 576 | 2 |
https://www.dochub.com/en/functionalities/rearrange-equation-article
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math
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When you want to apply a minor tweak to the document, it should not take long to Rearrange equation article. This sort of basic action does not have to require extra education or running through guides to understand it. With the appropriate document modifying resource, you will not take more time than is necessary for such a swift edit. Use DocHub to streamline your modifying process regardless if you are an experienced user or if it’s the first time making use of an online editor service. This tool will take minutes to learn to Rearrange equation article. The only thing required to get more effective with editing is a DocHub profile.
A simple document editor like DocHub will help you optimize the time you need to spend on document modifying regardless of your prior knowledge about this kind of resources. Create an account now and increase your efficiency immediately with DocHub!
The formula for the area of a triangle is A is equal to 1/2 b times h, where A is equal to area, b is equal to length of the base, and h is equal to the length of the height. So area is equal to 1/2 times the length of the base times the length of the height. Solve this formula for the height. So just to visualize this a little bit, let me draw a triangle here. Let me draw a triangle just so we know what b and h are. b would be the length of the base. So this distance right over here is b. And then this distance right here is our height. That is the height of the triangle-- let me do that at a lower case h because thats how we wrote it in the formula. Now, they want us to solve this formula for the height. So the formula is area is equal to 1/2 base times height. And we want to solve for h. We essentially want to isolate the h on one side of the equation. Its already on the right-hand side. So lets get rid of everything else on the right-hand side. So we can do it-- well, Ill do i
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224653501.53/warc/CC-MAIN-20230607010703-20230607040703-00605.warc.gz
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CC-MAIN-2023-23
| 1,890 | 3 |
https://www.nagwa.com/en/videos/124168716737/
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math
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Given that the measure of angle
𝑍𝑌𝐿 is equal to 122 degrees, find the measure of angle 𝑋.
In our diagram, we have two
tangents starting from the point 𝑋. The top one touches the circle at
point 𝑍 and the bottom one touches the circle at point 𝑌. We also have a chord drawn on the
circle from point 𝑍 to point 𝑌. We are told that the measure of
angle 𝑍𝑌𝐿 is 122 degrees. We recall that two tangents drawn
from the same point must be equal in length. This means that the line segment
𝑋𝑍 is equal to the line segment 𝑋𝑌. Triangle 𝑋𝑌𝑍 is therefore
isosceles, as it has two equal length sides. In any isosceles triangle, the
measure of two angles are equal. In this case, angle 𝑋𝑌𝑍 is equal
to angle 𝑋𝑍𝑌.
Angles on a straight line sum to
180 degrees. This means that we can calculate
the measure of angle 𝑋𝑌𝑍 by subtracting 122 from 180. This is equal to 58 degrees. Angles 𝑋𝑌𝑍 and 𝑋𝑍𝑌 are both
equal to 58 degrees. Our aim in this question is to find
the measure of angle 𝑋, and we know that angles in a triangle also sum to 180
degrees. Angle 𝑋 is therefore equal to 180
minus 58 plus 58. 58 plus 58 is equal to 116, and
subtracting this from 180 gives us 64. The measure of angle 𝑋 is
therefore equal to 64 degrees.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104209449.64/warc/CC-MAIN-20220703013155-20220703043155-00186.warc.gz
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CC-MAIN-2022-27
| 1,324 | 21 |
https://allcalculator.net/z-score-calculator-how-do-you-read-the-z-score-table
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math
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Z score Calculator: How do you Read the Z score Table?
How do you read a Z score Table?
Z score can also be called the Standard Score. It represents the number of standard deviations a data point is above or below the mean value. At Allcalculator.net, we provide a convenient Z score Calculator that can be used to calculate the Z score for a given data point. Additionally, our website offers a Z score table and a simple method to interpret the values from the table. By utilizing the Z score Calculator and the accompanying table from Allcalculator.net, you can easily calculate and interpret Z scores, which are essential in statistical analysis and understanding the relative position of a data point within a distribution.
With the help of the Z score table, you can determine the percentile or the p-value. The data point corresponds depending on the Z score. You can learn how to read ZScore Table by following these simple steps.
- Evaluate if your Z score is positive or negative.
- Suppose the Z score is negative. It means the data point is lower or less than the mean value. In this, you can use a Negative Z score table.
- Similarly, the data point is above the mean if the Z score is positive. In this case, you can use a Positive Z score Table.
- In the leftmost column, look for a z score that matches up to the first decimal, i.e. 10th Decimal place. Example. For a Z score of 3.15, try to look for 3.1 at least.
- Look for a Z score in the top row that matches the second decimal. It means 100th place. Now considering the same example for a decimal value of 3.15, look for 0.05.
- The next step involves looking for a p-value. In this, the row and column of the matching value must intersect. For a z score of 3.15, the p-value is 0.9842.
- The last step needs you to determine the percentile. So multiply the p-value by 100%.
- The z score of 3.15 is the 97th percentile.
The Z score Calculator can calculate the Standard data points, z value, or percentile in seconds. However, the input values must be correct.
At Allcalculator.net, our Z score Calculator and accompanying table make it simple to calculate and interpret Z scores. Whether you need to determine percentiles or p-values, our tools provide quick and accurate results. Make your statistical analysis easier with Allcalculator.net's Z score Calculator.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233511075.63/warc/CC-MAIN-20231003092549-20231003122549-00764.warc.gz
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CC-MAIN-2023-40
| 2,337 | 14 |
https://quickbooks.intuit.com/ca/resources/finance-accounting/working-capital/
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math
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Working capital is a financial measure of a company’s operating health. It is calculated by taking current assets and subtracting current liabilities. For example, if a company has $1 million in current assets and $400,000 in current liabilities, the working capital is $600,000.
The working capital ratio shows whether a company has enough short-term assets to cover its short-term liabilities. It is calculated as the current assets divided by the current liabilities. In the above example, the ratio is:
$1 million / $400,000 = 2.5
A ratio below 1 indicates problems for the company, while a ratio higher than 2 means the company is probably not investing enough of its assets.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156554.12/warc/CC-MAIN-20180920175529-20180920195929-00521.warc.gz
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CC-MAIN-2018-39
| 682 | 4 |
https://www.analystforum.com/t/adjusting-dollar-duration-with-futures-ctd-when-less-than-100-bps-change/72125
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math
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Ok… so I was solving some practice tests from Schwesers 2010 book 1 Exam 3, and found two different questions that were of the same nature, that solved the problem two different ways. Rather than post the entire questions here, I will just describe the difference.
In both cases you were given a market price, face value, and a duration. In both problems the question asked how many contracts to buy or sell given a less than 100 bps change, lets call it 50 bps. Both problems involved using t-bond futures and a CTD bond to create the hedge. In one problem, the numerator dollar duration is adjusted for the fact that is less than 100 bps by multiplying the given duration by 0.005 rather than 0.01 (50 bps vs 100 bps), and the CTD dollar duration is similarly adjusted. In the other problem, the numerator (Desired DD) is adjusted, however the CTD is not.
In the first situation, the final answer will effectively be the same as for a 100 bps change because both the numerator and denominator are adjusted by the same factor (0.005), whereas in the second situation since only the numerator is adjusted, there is a discrepancy.
The exact questions are 18.4 and 15.1 (18.4 only adjusts numerator, 15.1adjusts both).
Is this an error, which one is correct? I am inclined to think you adjust both since duration implicitly assumes 100 bps moves; which should yield the same result as if you did not adjust for the fact that it is less than 100 bps since both numerator and demoniator share the same adjustment.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141183514.25/warc/CC-MAIN-20201125154647-20201125184647-00286.warc.gz
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CC-MAIN-2020-50
| 1,511 | 5 |
http://www.jiskha.com/members/profile/posts.cgi?name=Guadalupe
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math
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Posts by Guadalupe
Total # Posts: 22
jan gas tank is nearly empty at 1/12 full, she does not have enough gas to fill the tank but she has enough gas to reach 2/3 fill how much gas does she need to add to fill the fraction of a tank
January 26, 2017
On a campus of 9000 students, a single student returned to campus with a case of measles on Monday January 5th. The infirmary is keeping track of the number of students who have been diagnosed with the disease Day # of students infected 1. 2 2. 5 3. 9 4. 28 5. 64 6. 81 7. 320...
February 10, 2015
The Gateway Arch in St. Louis, Missouri is not a parabola but a shape known as a catenary. The name is given to the shaoe formed by the Graph of the hyperbolic cosine (cosh). The arch has a height of 625 feet andna span of 600 feet. The hyperbolic cosine is defined as: Cosh x...
February 10, 2015
A corpse was discovered in a motel room at midnight and uts temperature was 82°F. The temperature dropped to 80.5°F two hours later. Given k is a constant for the object in question, S is the surrounding temperature, t represents the time and theta(of time) is the ...
February 9, 2015
You have been hired by the Humane Society to construct six animal cages using 1400 feet of chain fence. Express the length and width using function notation. Include a graph with the area function with explanation of significance. Find the dimensions that maximize the total ...
November 28, 2014
An aluminum can is filled to the brim with a liquid. The can and the liquid are heated so their temperatures change by the same amount. The cans initial volume at 5 oC is 3.5×10−4 m3. The coefficient of volume expansion for aluminum is 69&...
May 13, 2014
Every night in Cordele,Georgia,24 trains crisscross the city's five mile radius.Each train blows its whistle to alert citizens that it is passing through.Because of the residentof Cordele can sleep through all these whistles.
September 21, 2009
How do you do this one? Almir can seal a driveway in 4 hours. Working together, he and Louis can seal it in 2.3 hours. How long would it take Louis to seal it working alone? Thanks in advance. D:3+4D=19 Driveway (D) = Rate * Time let Louis' time alone be t hours so Louis...
May 14, 2007
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s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218194600.27/warc/CC-MAIN-20170322212954-00478-ip-10-233-31-227.ec2.internal.warc.gz
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CC-MAIN-2017-13
| 2,223 | 18 |
https://research.vu.nl/en/publications/the-asymptotic-behaviour-of-parton-distributions-at-small-and-lar
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math
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It has been argued from the earliest days of quantum chromodynamics (QCD) that at asymptotically small values of $x$ the parton distribution functions (PDFs) of the proton behave as $x^\alpha$, where the values of $\alpha$ can be deduced from Regge theory, while at asymptotically large values of $x$ the PDFs behave as $(1-x)^\beta$, where the values of $\beta$ can be deduced from the Brodsky-Farrar quark counting rules. We critically examine these claims by extracting the exponents $\alpha$ and $\beta$ from various global fits of parton distributions, analysing their scale dependence, and comparing their values to the naive expectations. We find that for valence distributions both Regge theory and counting rules are confirmed, at least within uncertainties, while for sea quarks and gluons the results are less conclusive. We also compare results from various PDF fits for the structure function ratio $F_2^n/F_2^p$ at large $x$, and caution against unrealistic uncertainty estimates due to overconstrained parametrisations.
|Publication status||Published - 31 Mar 2016|
Bibliographical note20 pages, 9 figures, this version matches the version accepted for publication in EPJC
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100724.48/warc/CC-MAIN-20231208045320-20231208075320-00375.warc.gz
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CC-MAIN-2023-50
| 1,187 | 3 |
http://www.nytimes.com/2003/09/09/science/l-a-cosmic-quandary-224715.html?src=pm
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math
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To the Editor:
''One Cosmic Question'' asks whether physics laws might vary in other universes. At the particle accelerator where I work, I once confronted a physicist with a question I thought even better: Would mathematics itself vary?
His instant response showed my friend had already gone deeper: ''The really interesting question is whether music would be the same.''
STEVEN T. CORNELIUSSEN
Poquoson, Va.Continue reading the main story
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s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948592202.83/warc/CC-MAIN-20171217000422-20171217022422-00011.warc.gz
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CC-MAIN-2017-51
| 440 | 5 |
http://primanota.ru/pulp/countdown-chords.htm
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math
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Авторы: Jarvis Cocker, Russell Senior, Nick Banks, Steve Mackey, Candida Doyle
e: Thu, 26 Feb 1998 04:17:29 -0500 Countdown by Pulp Written by Jarvis Cocker/Pulp From the album Countdown and from Seperations transcribed by Coreii Cunningham Verse: Eb C# Eb Oh, I was 17 when I heard the countdown start. It started F# G# slowly and I thought it was my heart. But then I realised that F# G# this time it was for real. There was no place to hide, I had C# Eb to go out and feel that there was time to kill. And so i C# Eb walked my way round town. i tried to love the World. Oh, F# G# and the world just got me down. And so, I looked for you F# G# in every street of every town. I wanna see your face, I C# Eb wanna, I wanna see you now. I wanna see you now. Oh C# Eb and so it went, so it went for several years. I couldn't stand F# G# it, no, it must be getting near. No, no, you don't understand. F# G# How many people have seen you in the arms of some other C# Eb man. I've got to meet you, and find you, and take your C# Eb hand. Oh my God, my god, you've got to understand that I F# G# was 17 - I didn't, didn't know a thing at all. I've got no F# G# C# Eb reason, I've got no reason at all, oh no. Chorus: C#m Oh, the time of my G# Eb life, oh, I think you came too soon (yeah, it came too soon C#m G# Eb then). Oh, and it could be tonight, if I ever leave this room C#m (if I ever leave this room now). Wasting all my time on all those G# Eb C#m stupid things that only get me down. Oh, and the sky is crying G# Eb out tonight for me to leave this town (telling me to leave this town...goodbye...okay...). Verse: Yeah, you can leave me, you can, you can go some other place. You can forget it, yeah, you know that it's okay. Because I own this town, yeah I brought it, I brought it down on it's knees. Can you hear it crying? Can you hear it begging me "please?" I know it's coming, so soon now, oh it's on it's way. Oh no, oh no I can hear them say - they say they can't survive. They say it's all a lie and now, it's coming down, oh baby now please... (Chorus) Verse: It's okay, you don't have to care, really, oh, really I swear. No, no, you owe me nothing, you owe nothing to me. Oh, and if I messed it up baby, that's all up to me, yeah. And if you go then I won't follow, no...no...G many times I've been thinking maybe I should. No, I'm gonna stay, I'm gonna make my way. I'm gonna get on through babe. I'm gonna make it someday. (Chorus) Outro Chords: Eb C#m highG# I'm gonna leave this town. I'm never gonna hang around. The sky and stars and God will never ever laugh. Me and moon and stars are falling down.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00246.warc.gz
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CC-MAIN-2018-26
| 2,630 | 2 |
http://qnr.cn/waiyu/lsat/shiti/moni/201105/290035.html
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math
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A man buys three outfits-X, Y, and Z-each of which consists of two articles of clothing. Each of the articles of clothing is either brown gray or navy.
At least one of the outfits is made up of two articles different in color from one another. No more than two of the outfits contain the same combination of colors.
Outfit X contains at least one navy article of clothing.
Outfit X contains at least one brown article of clothing and does not contain a gray article.
1. Which one of the following can be the colors of the man\'s outfits?
2. If outfits X and Y each consist of one brown article and one navy article of clothing, what combinations for outfit Z?
3. If outfit Z does not contain two brown items of clothing, what is the maximum number of items of clothing in the three outfits that can be navy?
4. If outfit Y consists of two brown articles of clothing and outfit Z consists of two navy items, what is the total number of possible color combinations for outfit X?
5. Which one of the following color combinations for outfit Z would be acceptable under any of the acceptable color combinations for outfits X and Y?
6. If no two outfits contain the same color combination but each contains at least one navy item, which one of the following is a complete and accurate list of the possible combinations for outfit X?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158766.65/warc/CC-MAIN-20180923000827-20180923021227-00089.warc.gz
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CC-MAIN-2018-39
| 1,326 | 10 |
https://lists.isc.org/pipermail/inn-workers/2000-July/003510.html
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math
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ietf-nntp MODE READER
kondou at nec.co.jp
Thu Jul 20 11:13:42 UTC 2000
moved from nntp WG list;
In article <yl66q1x59b.fsf at windlord.stanford.edu>,
Russ Allbery <rra at stanford.edu> wrote;
} > I've just noticed that the initial connection returns 504 for service
} > unavailable, but other commands (including MODE READER) return 502. Is
} > this correct ?
} INN's return codes for service temporarily unavailable currently are not
} compliant with this draft. In this case, I believe that the draft is
} right and INN is wrong.
Could you point where it is?
btw, if innd is set up not to allow readers, we've got;
# telnet localhost nntp
Connected to localhost.
Escape character is '^]'.
200 server InterNetNews server INN 2.3.0 (20000711 CVS prerelease) ready
500 Syntax error or bad command
This is not compliant that draft also, and I think this
needs to be fixed to return 502.
More information about the inn-workers
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s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257197.14/warc/CC-MAIN-20190523083722-20190523105722-00341.warc.gz
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CC-MAIN-2019-22
| 923 | 22 |
http://slideplayer.com/slide/4317920/
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math
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57 4-3 Greatest Common Divisor and Least Common Multiple Methods to Find the Greatest Common DivisorMethods to Find the Least Common Multiple
58 Greatest Common Divisor Two bands are to be combined to march in a parade. A 24-member band will march behind a 30-member band. The combined bands must have the same number of columns. Each column must be the same size. What is the greatest number of columns in which they can march?
59 Greatest Common Divisor The bands could each march in 2 columns, and we would have the same number of columns, but this does not satisfy the condition of having the greatest number of columns.The number of columns must divide both 24 and 30.Numbers that divide both 24 and 30 are 1, 2, 3, and 6. The greatest of these numbers is 6.
60 Greatest Common Divisor The first band would have 6 columns with 4 members in each column, and the second band would have 6 columns with 5 members in each column.
61 DefinitionThe greatest common divisor (GCD) or the greatest common factor (GCF) of two whole numbers a and b not both 0 is the greatest whole number that divides both a and b.
62 Colored Rods MethodFind the GCD of 6 and 8 using the 6 rod and the 8 rod.
63 Colored Rods MethodFind the longest rod such that we can use multiples of that rod to build both the 6 rod and the 8 rod.The 2 rods can be used to build both the 6 and 8 rods.
64 Colored Rods MethodThe 3 rods can be used to build the 6 rod but not the 8 rod.The 4 rods can be used to build the 8 rod but not the 6 rod.The 5 rods can be used to build neither.The 6 rods cannot be used to build the 8 rod.Therefore, GCD(6, 8) = 2.
65 The Intersection-of-Sets Method List all members of the set of whole number divisors of the two numbers, then find the set of all common divisors, and, finally, pick the greatest element in that set.
66 The Intersection-of-Sets Method To find the GCD of 20 and 32, denote the sets of divisors of 20 and 32 by D20 and D32, respectively.Because the greatest number in the set of common positive divisors is 4, GCD(20, 32) = 4.
67 The Prime Factorization Method To find the GCD of two or more non-zero whole numbers, first find the prime factorizations of the given numbers and then identify each common prime factor of the given numbers. The GCD is the product of the common factors, each raised to the lowest power of that prime that occurs in any of the prime factorizations.Numbers, such as 4 and 9, whose GCD is 1 are relatively prime.
68 Example 4-12 Find each of the following: a. GCD(108, 72) b. GCD(0, 13) Because 13 | 0 and 13 | 13, GCD(0, 13) = 13.
69 Example (continued)c. GCD(x, y) if x = 23 · 72 · 11 · 13 and y = 2 · 73 · 13 · 17GCD(x, y) = 2 · 72 · 13 = 1274d. GCD(x, y, z) if x = 23 · 72 · 11 · 13, y = 2 · 73 · 13 · 17, and z = 22 · 7GCD(x, y, z) = 2 · 7 = 14e. GCD(x, y) if x = 54 · 1310 and y = 310 · 1120Because x and y have no common prime factors, GCD(x, y) = 1.
70 Calculator MethodCalculators with a key can be used to find the GCD of two numbers.SimpFind GCD(120, 180) by pressing the keys:1Simp2/8=to obtain the displayN/D→n/d 60/90
71 Calculator MethodBy pressing the key, we see on the display as a common divisor of 120 and 180.x y2By pressing the key again and pressing we see 2 again as a factor.Simp=x yRepeat the process to see that 3 and 5 are other common factors.GCD(120, 180) is the product of the common prime factors 2 · 2 · 3 · 5, or 60.
72 Euclidean Algorithm Method If a and b are any whole numbers greater than 0 and a ≥ b, then GCD(a, b) = GCD(r, b), where r is the remainder when a is divided by b.Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is 0 is called the Euclidean algorithm.
74 Example 4-13 Use the Euclidean algorithm to find GCD(10764, 2300). GCD(10764, 2300) = GCD(2300, 1564)GCD(2300, 1564) = GCD(1564, 736)
75 Example 4-13 (continued) GCD(10764,2300) = GCD(92, 0) = 92
76 Euclidean Algorithm Method A calculator with the integer division feature can also be used to perform the Euclidean algorithm.To find GCD(10764, 2300), proceed as follows:The last number we divided by when we obtained the 0 remainder is 92, so GCD(10764, 2300) = 92.
77 Example 4-14aFind GCD(134791, 6341, 6339).Any common divisor of three numbers is also a common divisor of any two of them.The GCD of three numbers cannot be greater than the GCD of any two of the numbers.GCD(6341, 6339) = GCD(6341 − 6339, 6339)= GCD(2, 6339) = 1GCD(134791, 6341, 6339) cannot be greater than 1, so it must equal 1.
78 Example 4-14b Find the GCD of any two consecutive whole numbers. GCD(n, n + 1) = GCD(n + 1, n)= GCD(n + 1 − n, n)= GCD(1, n) = 1The GCD of any two consecutive whole numbers is 1.
79 Least Common MultipleHot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a package. What is the least number of packages of each you must buy so that there is an equal number of hot dogs and buns?The number of hot dogs is a multiple of 10, while the number of buns is a multiple of 8.The number of hot dogs matches the number of buns whenever 10 and 8 have multiples in common.
80 Least Common Multiple This occurs at 40, 80, 120… The least of these multiples is 40.So we will have the same number of hot dogs and buns by buying 4 packages of hot dogs and 5 packages of buns.
81 Definition Least Common Multiple (LCM) The least common multiple (LCM) of two non-zero whole numbers a and b is the least non-zero whole number that is simultaneously a multiple of a and a multiple of b.
82 Number-Line Method Find LCM(3, 4). Beginning at 0, the arrows do not coincide until the point 12 on the number line. Thus, 12 is LCM(3, 4).
83 Colored Rods MethodFind LCM(3, 4) using the 3 rod and the 4 rod.
84 Colored Rods MethodBuild trains of 3 rods and 4 rods until they are the same length. The LCM is the common length of the train.LCM(3, 4) = 12
85 The Intersection-of-Sets Method List all members of the set of positive multiples of the two integers, then find the set of all common multiples, and, finally, pick the least element in that set.
86 The Intersection-of-Sets Method To find the LCM of 8 and 12, denote the sets of positive multiple of 8 and 12 by M8 and M12, respectively.Because the least number in the set of common positive multiples is 24, LCM(8, 12) = 24.
87 The Prime Factorization Method To find the LCM of two non-zero whole numbers, first find the prime factorization of each number. Then take each of the primes that are factors of either of the given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations.
89 GCD-LCM Product Method For any two natural numbers a and b,GCD(a, b) · LCM(a, b) = ab.
90 Example 4-16Find LCM(731, 952).Applying the Euclidean Algorithm, we can determine that GCD(731, 952) = 17.17 · LCM(731, 952) = 731 · 952LCM(731, 952) = = 40,936
91 Division-by-Primes Method To find LCM(12, 75, 120), start with the least prime that divides at least one of the given numbers. Divide as follows:Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this procedure, continue the division process until the row of answers consists of relatively prime numbers as shown next.
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https://www.questionsbanks.com/blog/show-10/standard-form-of-numbers-defined-and-explained-with-examples/
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math
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Standard form of numbers defined and explained with examples. We hope this blog will help the learners as well as those candicate who will appear in the boards exams 2023-2024 because we provide here all possible solutions of queries by our users. Standard Form of Numbers: Defined & Explained with Examples - Questions Bank (Standard form of numbers defined and explained with examples) for Examination Year 2023-2024
Source:www.questionsbanks.com Last updated: 2023-07-25 20:03:42
Standard form of numbers is an important concept to represent numbers in scientific notation.Numbers are an integral part of our everyday lives, serving as a fundamental tool for quantifying and expressing various quantities. While working with extremely large or small numbers, it can become burdensome to represent these numbers in their usual form.
The standard form of numbers provides a practical and efficient method of representing extremely large or small quantities. By using a decimal number and a power of 10, it simplifies the communication of such numbers and facilitates comparisons and calculations.
In this article, we will elaborate the idea of standard form of numbers, how to express numbers in standard form, arithmetic operations with numbers in standard form, its merits, applications and how it is applied to express numbers in standard form with examples.
Standard form is a substantial and standardized way of expressing numbers that are either very large or very small. Scientific notation and exponent form are also important terms which are commonly used in place of standard form of numbers.
By expressing these numbers as the product of a decimal number (between 1 and 10) and an exponent of 10,enables us to identify these numbers in a compact manner. The number of places to which the decimal point must be shifted to get the original value is indicated as the exponent of 10. Mathematically,
A x 10^n, where 1 ≤A< 10 and n is an integer.
Here M is a coefficient and it is a decimal number range from equal or greater than 1 but less than 10. Moreover n (power of ten) specifies the scale or order of magnitude.
To express a number in standard form, we follow a specific format.
Always choose a number that is higher than or equal to one but less than ten.
Determine how many places the decimal point has to be moved to get the original value. If decimal point moves from left side or right side, then the exponent of 10 will be negative or positive respectively.
You can try a standard form converter to express numbers in scientific notation according to the above-mentioned steps.
In this step, you’re to combine the decimal number and the exponent of 10 which you will have identified in the above two steps.
Performing arithmetic operations with numbers in standard form is simple. Here's how you can carry out basic operations:
Addition and Subtraction:
The standard form of numbers offers several advantages in representing and comparing numbers that differ greatly in magnitude. Here are a few benefits:
Standard form of numbersenables us to write extremely large or small numbers in a concise and easily understandable format, making it convenient for mathematical calculations and scientific notation.
By representing numbers in standard form, it becomes effortless to compare their magnitudes. The power of 10 provides a clear indication of which number is larger or smaller.
It is convenient and simple to use arithmetic operations (addition, subtraction, multiplication, and division) when working with numbers that are written in standard form. Additionally, it reduces one's need for handling very small or massive numbers.
Standard form of numbers is closely related to scientific notation, which is widely used in scientific research, engineering, and mathematics. Familiarity with standard form enhances understanding and communication within these fields.
Standard form finds application in various scientific and mathematical disciplines. Some common areas where standard form is frequently employed include:
Representing astronomical distances, masses, and quantities.
Expressing the values of fundamental physical constants, such as the speed of light or Planck's constant.
Describing molecular masses, atomic sizes, and concentrations.
Presenting large financial figures or GDP values.
Scientists often employ standard form to express measurements, such as distances between celestial objects, molecular sizes, or energy values.
Engineers utilize standard form to represent quantities like voltage, resistance, and power in electrical circuits, as well as dimensions and quantities in structural analysis.
Standard form facilitates the representation of large financial figures in reports, such as company revenues, national debt, or stock market values.
In medical research and practice, standard form is employed to express quantities like drug concentrations, cell counts, or molecular weights.
Express the ordinary number 738000000000000000000 in standard form.
Step 1: Write the given number
Step 2: We identify the decimal number that is 738 and standard position is after the first decimal number i.e. 7.38.
Step 3: As decimal point is shifted 20 places from right side. Thus, exponent of 10 will be +20. Hence number in standard form is 7.38 x 10^20.
Express the given ordinary number 0.000000000000529 in standard form of numbers.
Step 1: Write the given number
Step 2: We identify the decimal number which is 428 and standard position is after the first decimal number i.e. 5.29
Step 3: As the decimal point is shifted places from left side. Thus, exponent of 10 will be -13. Hence number in standard form is 5.29 x 10^ -13.
In this article, we have discussed standard form of numbers in detail. We have elaborated the way to write ordinary numbers in standard form, arithmetic operations, benefits, applications as well as some examples. Hopefully, reading this article you will be able to apprehend the concept of standard form of numbers.
5 Questions Found on same Topics.
Hey! 4 Questions Found on same Chapter.
All chapters of NCERT Book as ncert solutions have exercise questions, textual questions and so many addtional questions like short answered questions, long answered questions and very long questions, here we included all types of questions answers format that need for a students and other stock holders like teachers and tutors. Standard Form of Numbers: Defined & Explained with Examples - Questions Bank (Standard form of numbers defined and explained with examples) for Examination Year 2023-2024
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233511000.99/warc/CC-MAIN-20231002132844-20231002162844-00014.warc.gz
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CC-MAIN-2023-40
| 6,569 | 42 |
https://www.chapters.indigo.ca/en-ca/books/boundary-value-problems-for-second/9780323162265-item.html
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math
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Applied Mathematics and Mechanics, Volume 5: Boundary Value Problems: For Second Order Elliptic Equations is a revised and augmented version of a lecture course on non-Fredholm elliptic boundary value problems, delivered at the Novosibirsk State University in the academic year 1964-1965. This seven-chapter text is devoted to a study of the basic linear boundary value problems for linear second order partial differential equations, which satisfy the condition of uniform ellipticity.
The opening chapter deals with the fundamental aspects of the linear equations theory in normed linear spaces. This topic is followed by discussions on solutions of elliptic equations and the formulation of Dirichlet problem for a second order elliptic equation. A chapter focuses on the solution equation for the directional derivative problem. Another chapter surveys the formulation of the Poincaré problem for second order elliptic systems in two independent variables. This chapter also examines the theory of one-dimensional singular integral equations that allow the investigation of highly important classes of boundary value problems. The final chapter looks into other classes of multidimensional singular integral equations and related boundary value problems.
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s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541839.36/warc/CC-MAIN-20161202170901-00230-ip-10-31-129-80.ec2.internal.warc.gz
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CC-MAIN-2016-50
| 1,259 | 2 |
https://math.stackexchange.com/questions/1093853/order-of-operations-game-solution
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math
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My AP Computer Science teacher likes to play a game with his students where he writes 4 random numbers on the board and a fifth, target number. The objective is to use the four basic operations (+,−,×,÷) to get the target number. Each number can only be used once, but the operations can be reused. Order of operations does apply, and the numbers can be reordered.
Ex: Given the numbers 6 9 5 4, get to 50
Solution: (9 + 5) × 4 - 6 = 50
My teacher and I are looking for a way to find the solution(s) to a set of 5 numbers with a computer program. Because there are 24 arrangements of 4 numbers and 24 ways to arrange the 4 operations, there are 576 potential solutions to check, so brute force is not exactly easy.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991904.6/warc/CC-MAIN-20210511060441-20210511090441-00271.warc.gz
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CC-MAIN-2021-21
| 719 | 4 |
http://info.paiwhq.com/2012/03/
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math
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Welcome back to Making Molehills out of Mountains University. For years data analytics have been my passion. I have spent years looking at human behavior and applying statistical analysis techniques to answer two primary business questions every CEO has, “Should I do X” and “If I do X what will happen?” There is a third question they often ask, “I did X, what happened? It was not what I expected.” But that’s usually asked when something like New Coke flops, uh, I mean, doesn’t meet expectations.
My favorite tool, admitting my bias, is the mTab suite of analysis tools. In the past ten years, mTab has become the standard in the automotive industry and has contributed, in my considerable professional opinion, to have a profound effect on the industry’s recovery. After all, they’re now producing cars people are excited to buy.
Sorry, I digress. This is the 2nd class in Market Research Data Analysis 101. I teach in plain English, or as plain as possible considering the subject matter. In later classes we can do the math. So, put away your smart phones, get out your tablets and learn something.
Today I introduce you to the lovely world of Correlation and Regression analysis which are two of the most commonly used techniques for determining the relationship between two quantitative variables.
Assuming you’ve collected your data the first step is to create a scatter diagram. Variable 1 is the X-axis and the other is the Y axis. The resulting diagram indicates the linear relationship between the two variables. The closer they are to a straight line the stronger the relationship. The linear relationship is defined as positive, negative or null and is expressed by a correlation coefficient or +1, -1, or 0.
A positive relationship means that a change in one variable has a positive effect (increase marketing budget = increase in sales). The converse is true for a negative relationship (increase in price = decrease in sales).
Coefficient = 0 =+1 Between 0 & -1
Seems straightforward. But, remember, we are not talking about causation here. There may be a third variable that accounts for the relationship (e.g. Tax refund check came through at the time of increased marketing).
Now that you know there is a relationship between two variables what do you do with that? As future high falutin analysts you’ll want to predict the Key Drivers and report them to your CEO. She’ll want to know, “If I decrease price will I sell more product?”
Enter linear and non-linear regression. Simply put, if a change in X (independent variable) equals a consistent change in Y (dependent variable), then the relationship is linear. If the change in Y is inconsistent then the relationship is nonlinear. For Regression analysis there is an assumption of linearity. IF the scatter diagram indicates a nonlinear relationship there are mathematical techniques that can be used to obtain linearity.
Assuming price and units sold is a linear relationship, using standard regression analysis techniques, the analyst should be able to predict the number of units sold at a particular price point. This also assumes, for the sake of this exercise, that the relationship is positive and the correlation coefficient is +1 or close to +1. The stronger the coefficient the better predictive quality of the data under regression.
I know. I said, no math. But you should be able to handle this:
A and b are the intercept and slop (unknown constants).
In this case, X = Price and Y = units sold. As the equation suggests a change in Y will equal a change in X.
Careful! If you write the equation backwards, X= c+dY then you might tell your CEO that price is affected by the number of people buying cars and not the other way around!
What? You say that if I sell more cars I can lower the price due to cost efficiencies in production? Of course, that is true, but that does not change the reality that, without an external action, price does not change by itself as production increases. But quantity sold can change as price is changed without any additional action.
That’s it for today. There is a whole lot more to study regarding correlation and regression but we’ll save that for another day. Now that you know that correlation and regression are impressive tools for identifying relationships between variables and for determining the strength of that relationship, go get some data, create a scatter graph, do a little algebra and impress your boss how knowledgeable you are as an analyst.
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CC-MAIN-2018-13
| 4,517 | 17 |
https://www.microblife.in/what-is-the-empirical-formula-for-c2h6/
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math
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What Is The Empirical Formula For C2h6??
Emprical Formula Of C2H6 Is CH3. Hence The Emprical formula means the simplest ratio of number of atoms present in molecule.Oct 7 2018
Does C2H6 have an empirical formula of ch2?
|CH (92.2% C 7.8% H)
|Boiling point °C
What is the empirical formula of ethane is?
What is the empirical formula of c2h2?
How do I find the empirical formula?
- In any empirical formula problem you must first find the mass % of the elements in the compound. …
- Then change the % to grams. …
- Next divide all the masses by their respective molar masses. …
- Pick the smallest answer of moles and divide all figures by that.
How do you find the empirical formula of ethene?
Ethene is a small hydrocarbon compound with the formula C2H4 (see figure below). While C2H4 is its molecular formula and represents its true molecular structure it has an empirical formula of CH2. The simplest ratio of carbon to hydrogen in ethene is 1:2.
What is the percent composition of C2H6?
What is the empirical formula of pentane?
What is C2H6 in chemistry?
C2H6 2 atoms of carbon combine with 6 atoms of hydrogen to form ethane.
Is c2h2 empirical?
Does CH and c2h2 have the same empirical formula?
The empirical formula represents the simplest whole number ratio of various atoms in a compound hence for both C2H2 and C2H6 the empirical formula is CH.
Is c2h2 has same empirical and molecular formula?
The Empirical formula is the lowest whole number ratio of the elements in a compound. … Multiple molecules can have the same empirical formula. For example benzene (C6H6) and acetylene (C2H2) both of the empirical formula of CH (see Figure 2.11.
Is Cho an empirical formula?
CHO is an empirical formula. C6H12O6 is NOT an empirical formula all of the elements can be divided by 6! Empirical Formula: The simplest ratio of the atoms present in a molecule. The goal is for you to actually calculate an empirical formula when given either the % composition or the mass of each element present.
How do you find the empirical formula with percentages?
Is empirical formula and molecular formula same?
Is fe3o4 an empirical formula?
How do you calculate C2H6?
What element is Al clo3 3?
|# of Atoms
How do you find the percent composition?
- Find the molar mass of all the elements in the compound in grams per mole.
- Find the molecular mass of the entire compound.
- Divide the component’s molar mass by the entire molecular mass.
- You will now have a number between 0 and 1. Multiply it by 100% to get percent composition.
How do you write pentane?
Is n2o4 an empirical formula?
What is the formula of hexane?
What is the formula of propane?
What intermolecular forces are present in C2H6?
Since hydrogen bonds are a special subset of dipole-dipole interactions this molecule has neither dipole-dipole forces nor hydrogen bonds. Rather it has only the intermolecular forces common to all molecules: London dispersion forces.
What is C2H6 name?
Is cacl2 an empirical formula?
Is na2cr2o7 an empirical formula?
How do you find the empirical and molecular formula of a compound?
Divide the molar mass of the compound by the empirical formula mass. The result should be a whole number or very close to a whole number. Multiply all the subscripts in the empirical formula by the whole number found in step 2. The result is the molecular formula.
What is the empirical formula of a compound that contains 40 carbon 6.7 hydrogen and 53.3 Oxygen?
Thus the empirical formula is CH2O .
Is C6H6 the same as CH?
Now ratio of Carbon atoms to Hydrogen atoms in Benzene (C6H6) is 1:1. Therefore the empirical formula of Benzene is simply CH.
Which is the correct empirical formula for C2H4O2?
Acetic acid-13C2 | C2H4O2 – PubChem.
Is H2O2 empirical or molecular?
The empirical formula of hydrogen peroxide is HO there is one H atom for every O atom (ratio 1:1). 2. The molecular formula shows the actual number of atoms of each element in a molecule of the compound. The molecular formula of hydrogen peroxide is H2O2 there are two H atoms and two O atoms in each molecule.
Is c3h6 an empirical formula?
What is the empirical formula for ribose C5H10O5?
Formaldehyde CH2O and the sugar ribose C5H10O5 have different molecular formulas but each has a 1:2:1 ratio of C:H:O [ribose is 5(CH2O)] and hence they both have the same empirical formula CH2O. A mass ratio can only give an empirical formula not a molecular formula.
How to Write the Empirical Structural & Molecular Formula C2H6 (Ethane)
Writing Empirical Formula Practice Problems
Empirical Formula & Molecular Formula Determination From Percent Composition
A Level Chemistry “Calculating Empirical Formula 1”
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816853.44/warc/CC-MAIN-20240413211215-20240414001215-00777.warc.gz
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CC-MAIN-2024-18
| 4,670 | 62 |
https://www.arxiv-vanity.com/papers/hep-ph/9602352/
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math
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DETERMINATION OF THE MASS OF THE W BOSON
Conveners: Z. Kunszt and W. J. Stirling
Working group: A. Ballestrero, S. Banerjee, A. Blondel, M. Campanelli, F. Cavallari, D. G. Charlton, H. S. Chen, D. v. Dierendonck, A. Gaidot, Ll. Garrido, D. Gelé, M. W. Grünewald, G. Gustafson, C. Hartmann, F. Jegerlehner, A. Juste, S. Katsanevas, V. A. Khoze, N. J. Kjær, L. Lönnblad, E. Maina, M. Martinez, R. Møller, G. J. van Oldenborgh, J. P. Pansart, P. Perez, P. B. Renton, T. Riemann, M. Sassowsky, J. Schwindling, T. G. Shears, T. Sjöstrand, Š. Todorova, A. Trabelsi, A. Valassi, C. P. Ward, D. R. Ward, M. F. Watson, N. K. Watson, A. Weber, G. W. Wilson
1 Introduction and Overview111prepared by F. Jegerlehner, Z. Kunszt, G.-J. van Oldenborgh,
P.B. Renton, T. Riemann, W.J. Stirling
- 1.1 Machine parameters
- 1.2 Present status of measurements
- 1.3 Improved precision on from the Tevatron
- 1.4 Impact of a precision measurement of
- 1.5 Methods for measuring
- 1.6 Theoretical input information
2 Measurement of from the Threshold
Cross-Section222prepared by D. Gelé, T.G. Shears, W.J. Stirling, A. Valassi,
- 2.1 Collider strategy
- 2.2 Event selection and statistical errors
- 2.3 Systematic errors
- 2.4 Summary
3 Direct Reconstruction of 333prepared by M. Grünewald, N. J. Kjær, Z. Kunszt,
P. Perez, C. P. Ward
- 3.1 Event selection and jet reconstruction
- 3.2 Constrained fit
- 3.3 Determination of the mass and width of the W
- 3.4 Systematic errors
- 3.5 Summary
- 4 Interconnection Effects444 prepared by V.A. Khoze, L. Lönnblad, R. Møller, T. Sjöstrand, Š. Todorova and N.K. Watson.
- 5 Conclusions
1 Introduction and Overview111prepared by F. Jegerlehner, Z. Kunszt, G.-J. van Oldenborgh, P.B. Renton, T. Riemann, W.J. Stirling
Previous studies of the physics potential of LEP2 indicated that with the design luminosity of one may get a direct measurement of the W mass with a precision in the range . This report presents an updated evaluation of the estimated error on based on recent simulation work and improved theoretical input. The most efficient experimental methods which will be used are also described.
1.1 Machine parameters
The LEP2 machine parameters are by now largely determined. Collider energy values and time-scales for the various runs, expected luminosities and errors on the beam energy and luminosity are discussed and summarized elsewhere in this report [2, 3]. Here we note that (i) collider energies in the range will be available, and (ii) the total luminosity is expected to be approximately per experiment. It is likely that the bulk of the luminosity will be delivered at high energy (). The beam energy will be known to within an uncertainty of , and the luminosity is expected to be measured with a precision better than 1%.
1.2 Present status of measurements
Precise measurements of the masses of the heavy gauge W and Z bosons are of fundamental physical importance. The current precision from direct measurements is = 2.2 MeV and M = 160 MeV . So far, has been measured at the CERN and Fermilab Tevatron [6, 7, 8] colliders. The present measurements are summarized in Fig. 1. In calculating the world average, a common systematic error of arising from uncertainties in the parton distributions functions is taken into account. The current world average value is
An indirect determination of from a global Standard Model (SM) fit to electroweak data from LEP1 and SLC gives the more accurate value
In Fig. 1 this range is indicated by dashed vertical lines. Note that the central value in (2) corresponds to and the second error indicates the change in when is varied between and – increasing decreases .
The direct measurement of becomes particularly interesting if its error can be made comparable to, or smaller than, the error of the indirect measurement, i.e. . In particular, a precise value of obtained from direct measurement could contradict the value determined indirectly from the global fit, thus indicating a breakdown of the Standard Model. An improvement in the precision of the measurement can be used to further constrain the allowed values of the Higgs boson mass in the Standard Model, or the parameter space of the Minimal Supersymmetric Standard Model (MSSM) .
Standard Model fits to electroweak data determine values for (or ), , and . The direct determinations of the top quark mass [9, 10] give an average value of . Fig. 2 compares the direct determinations of and with the indirect determinations obtained from fits to electroweak data . Note the correlation between the two masses in the latter. Within the current accuracy, the direct and indirect measurements are in approximate agreement. The central values of and their errors, determined in several ways from indirect electroweak fits, are given in Table 1.
|all data||R and R excluded||R, R and A excluded|
The results are evidently somewhat sensitive to the inclusion (or not) of data on the Z partial width ratios R and R and the SLD/SLC measurement of A, all of which differ by 2.5 standard deviations or more from the Standard Model values. However, the conclusion on the agreement of the direct and indirect determinations is unchanged. As we shall see in the following sections, a significant reduction in the error on is expected from both LEP2 and the Tevatron.
1.3 Improved precision on from the Tevatron
The Tevatron data so far analysed, and shown in Fig. 1, come from the 1992/3 data-taking (Run 1a). The results from CDF are based on approximately and are final, whereas those from D0 are based on approximately and are still preliminary. It is to be expected that the final result will have a smaller error. In addition, there will be a significantly larger data sample from the 1994/6 data-taking (Run 1b). This should amount to more than of useful data for each experiment. When these data are analysed it is envisaged that the total combined error on will be reduced to about . In particular, the combined CDF/D0 result will depend on the common systematic error arising from uncertainties in the parton distribution functions. Thus when the first measurements emerge from LEP2 one may assume that the world average error will have approximately this value. For more details see Ref. .
After 1996 there will be a significant break in the Tevatron programme. Data-taking will start again in 1999 with a much higher luminosity (due to the main injector and other improvements). Estimating the error on which will ultimately be achievable (with several fb of total luminosity) is clearly more difficult. If one assumes that an increase in the size of the data sample leads to a steady reduction in the systematic errors, one might optimistically envisage that the combined precision from the Tevatron experiments will eventually be in the range, assuming a common systematic error of about . However it is important to remember that these improved values will be obtained after the LEP2 measurements.
1.4 Impact of a precision measurement of
Within the Standard Model, the value of is sensitive to both and . For example, for a fixed value of , a precision of translates to a precision on of . The impact of a precise measurement of on the indirect determination of is shown in Table 2.
In order to assess the impact of a precise measurement of it is necessary to make an estimate of the improvements which will be made on the electroweak data from LEP1 and SLC. Details of the improvements which are assumed here are discussed in . The importance of a precise measurement of can perhaps best be appreciated by considering the (almost) model independent parameters . The parameter () is sensitive mainly to the Z partial and total widths. The parameter depends linearly on both and , where is determined from . The parameter depends linearly on , and r. This latter quantity is determined essentially from , and so improvements in the precision of depend directly on improving the error on . This is illustrated in Fig. 3, which shows the 70% confidence level contours for fits to projected global electroweak data. The different contours correspond to different values of . In these fits all electroweak data measurements have been set to correspond to the Standard Model values , and . The variables are constructed to be sensitive to vector boson propagator effects, from both physics within the Standard Model and beyond.
Numerically, the projected data give a precision
For , the error is obtained, whereas for the projected errors on one obtains
The smaller the volume in space allowed by the precision electroweak measurements, the greater the constraint on physics beyond the Standard Model.
The MSSM is arguably the most promising new-physics candidate. It is therefore especially important to consider the MSSM prediction for . Figure 4 shows as a function of in the SM (solid lines) and in the MSSM (dashed lines). In each case the prediction is a band of values, corresponding to a variation of the model parameters (dominantly in the SM case, with chosen here) consistent with current measurements and limits. An additional constraint of ‘no SUSY particles at LEP2’ is imposed in the MSSM calculation.
1.5 Methods for measuring
Precise measurements of can in principle be obtained using the enhanced statistical power of the rapidly varying total cross-section at threshold, the sharp (Breit-Wigner) peaking behaviour of the invariant-mass distribution of the W decay products and the sharp end-point spectrum of the lepton energy in W decay. One can obtain a rough idea of the relative power of these methods by estimating their statistical precision assuming 100% efficiency, perfect detectors and no background. More complete discussions are given in Sections 2 and 3.
Threshold cross-section measurement of the process . The statistical power of this method, assuming 100% signal efficiency and no background, is
where the minimum value is attained at . Here denotes the total integrated luminosity.
Direct reconstruction methods, which reconstruct the Breit-Wigner resonant shape from the W final states using kinematic fitting techniques to improve the mass resolution. The statistical power of this method, again assuming 100% efficiency, perfect detector resolution and no background, can be estimated as
approximately independent of the collider energy. This order of magnitude estimate is confirmed by more detailed studies, see below.
Determination of from the lepton end-point energy. The end-points of the lepton spectrum in depend quite sensitively on the W mass. For on-shell W bosons at leading order:
In this case the statistical error on is determined by the statistical error on the measurement of the lepton end-point energy,
In practice, however, the end-points of the distribution are considerably smeared by finite width effects and by initial state radiation, and only a fraction of events close to the end-points are sensitive to . This significantly weakens the statistical power of this method from what the naive estimate (8) would predict.
The detailed studies described in the following sections show that the errors which can realistically be achieved in practice are somewhat larger than the above estimates for Methods A and B. The statistical precisions of the two methods are in fact more comparable (for the same integrated luminosity) than the factor 2 difference suggested by the naive estimates (5) and (6). The overall statistical error for Method C has been estimated at for , significantly larger than that of the other two methods. It will not therefore be considered further here, although it is still a valid measurement for cross-checking the other results.
It is envisaged that most of the LEP2 data will be collected at energies well above threshold, and so the statistically most precise determination of will come from Method B. However with a relatively modest amount of luminosity spent at the threshold (for example per experiment), Method A can provide a statistical error of order , not significantly worse than Method B and with very different systematics. The two methods can therefore be regarded as complementary tools, and both should be used to provide an internal cross-check on the measurement of the W mass at LEP2. This constitutes the main motivation for spending some luminosity in the threshold region.
The threshold cross-section method is also of interest because it appears to fit very well into the expected schedule for LEP2 operation in 1996. It is anticipated that the maximum beam energy at LEP2 will increase in steps, with the progressive installation of more superconducting RF cavities, in such a way that a centre-of-mass energy of 161 GeV will indeed be achievable during the first running period of 1996. This would then be the ideal time to perform such a threshold measurement. The achievable statistical error on depends of course critically on the available luminosity at the threshold energy. In Section 2 we present quantitative estimates based on integrated luminosities of 25, 50 and 100 pb per experiment.
1.6 Theoretical input information
1.6.1 Cross-sections for the signal and backgrounds
Methods (A) and (B) for measuring described above require rather different theoretical input. The threshold method relies on the comparison of an absolute cross-section measurement with a theoretical calculation which has as a free parameter. The smallness of the cross-section near threshold is compensated by the enhanced sensitivity to in this region. In contrast, the direct reconstruction method makes use of the large statistics at the higher LEP2 energies, GeV. Here the more important issue is the accurate modeling of the W line-shape, i.e. the distribution in the invariant mass of the W decay products.
In this section we describe some of the important features of the theoretical cross-sections which are relevant for the measurements. A more complete discussion can be found in the contribution of the WW and Event Generators Working Group to this Report .
We begin by writing the cross-section for , schematically, as
We note that this decomposition of the cross-section into ‘signal’ and ‘background’ contributions is practical rather than theoretically rigorous, since neither contribution is separately exactly gauge invariant nor experimentally distinguishable in general. The various terms in (9) correspond to
: the Born contribution from the 3 ‘CC03’ leading-order diagrams for involving -channel exchange and -channel and Z exchange, calculated using off-shell W propagators.
: higher-order electroweak radiative corrections, including loop corrections, real photon emission, etc.
: higher-order QCD corrections to final states containing pairs. For the threshold measurement, where in principle only the total cross-section is of primary interest, the effect of these is to generate small corrections to the hadronic branching ratios which are entirely straightforward to calculate and take into account. More generally, such QCD corrections can lead to additional jets in the final state, e.g. from one hard gluon emission. This affects the direct reconstruction method, insofar as the kinematic fits to reconstruct assume a four-jet final state, and both methods insofar as cuts have to be imposed in order to suppress the QCD background (see Sections 2,3 below). Perturbative QCD corrections, real gluon emission to and real plus virtual emission to , have been recently discussed in Refs. [19, 20] respectively, together with their impact on the measurement of .
: ‘background’ contributions, for example from non-resonant diagrams (e.g. ) and QCD contributions to the four-jet final state. All of the important backgrounds have been calculated, see Table 6 below. At threshold, the QCD four-jet background is particularly large in comparison to the signal.
In what follows we consider (i) and (ii) in some detail. Background contributions and how to suppress them are considered in later sections.
1.6.2 The off-shell cross-section
The leading-order cross-section for off-shell production was first presented in Ref. :
The cross-section can be written in terms of the , and Z exchange contributions and their interferences:
where . Explicit expressions for the various contributions can be found in Ref. for example. The stable (on-shell) cross-section is simply
A theoretical ansatz of this kind will be the basis of any experimental determination of the mass and width of the W boson. The reason for this is the large effect of the virtuality of the W bosons produced around the nominal threshold. An immediate conclusion from Eq. (10) may be drawn: the W mass influences the cross sections exclusively through the off-shell propagators; all the other parts are independent of and (neglecting for the moment the relatively minor dependence due to radiative corrections). It will be an important factor in the discussion which follows that near threshold the (unpolarized) cross-section is completely dominated by the -channel neutrino exchange diagram. This leads to an -wave threshold behaviour , whereas the -channel vector boson exchange diagrams give the characteristic -wave behaviour .
By tradition (for example at LEP1 with the Z boson), the virtual W propagator in Eq. (11) uses an -dependent width,
where . Another choice, equally well justified from a theoretical point of view, would be to use a constant width in the W propagator (for a discussion see Ref. ):
The numerical values of the width and mass in the two expressions are related :
These relations may be derived from the following identity: . Numerically, the consequences are below the anticipated experimental accuracy.
1.6.3 Higher-order electroweak corrections
The complete set of next-to-leading order corrections to production has been calculated by several groups [24, 25], for the on-shell case only, see for example Refs. [26, 18] and references therein. There has been some progress with the off-shell (i.e. four fermion production) corrections but the calculation is not yet complete. However using the on-shell calculations as a guide, it is already possible to predict some of the largest effects. For example, it has been shown that close to threshold the dominant contribution comes from the Coulomb correction, i.e. the long-range electromagnetic interaction between almost stationary heavy particles. Also important is the emission of photons collinear with the initial state e (‘initial state radiation’) which gives rise to logarithmic corrections . These leading logarithms can be resummed to all orders, and incorporated for example using a ‘structure function’ formalism. In this case, the generalization from on-shell to off-shell ’s appears to be straightforward. For the Coulomb corrections, however, one has to be much more careful, since in this case the inclusion of the finite decay width has a dramatic effect. Finally, one can incorporate certain important higher-order fermion and boson loop corrections by a judicious choice of electroweak coupling constant. Each of these effects will be discussed in turn below.
In summary, certain corrections are already known to be large because their coefficients involve large factors like , , , etc. Once these are taken into account, one can expect that the remaining corrections are no larger than . When estimating the theoretical systematic uncertainty on the W mass in Section 2 below, we shall therefore assume a conservative overall uncertainty on the cross-section of from the as yet uncalculated and higher-order corrections.
1.6.4 Coulomb corrections
The result for on-shell production is well-known — the correction diverges as as the relative velocity of the bosons approaches zero at threshold. Note that near threshold and so the Coulomb-corrected cross-section is formally non-vanishing when .
For unstable production the finite decay width screens the Coulomb singularity , so that very close to threshold the perturbative expansion in is effectively replaced by an expansion in . In the calculations which follow we use the expressions for the correction given in Ref. . The net effect is a correction which reaches a maximum of approximately in the threshold region. Although this does not appear to be large, we will see below that it changes the threshold cross-section by an amount equivalent to a shift in of order MeV. In Ref. the result is generalized to all orders. However the contributions from second order and above change the cross-section by in the threshold region (see also ) and can therefore be safely neglected.
Note also that the Coulomb correction to the off-shell cross-section
provides an example of a (QED) interconnection effect between the two W bosons: the exchange of a soft photon distorts the line shape () of the W and therefore, at least in principle, affects the direct reconstruction method [32, 33, 34]. In Ref. , for example, it is shown that the Coulomb interaction between the W bosons causes a downwards shift in the average reconstructed mass of . Selecting events close to the Breit-Wigner peak reduces the effect somewhat. However the calculations are not yet complete, in that QED interactions between the decay products of the two W bosons are not yet fully included.
1.6.5 Initial state radiation
Another important class of electroweak radiative corrections comes from the emission of photons from the incoming e and e. In particular, the emission of virtual and soft real photons with energy gives rise to doubly logarithmic contributions at each order in perturbation theory. The infra-red () logarithms cancel when hard photon contributions are added, and the remaining collinear () logarithms can be resummed and incorporated in the cross-section using a ‘flux function’ or a ‘structure function’ (see also Refs. [36, 37, 38, 39, 18]).
The ISR corrected cross section in the flux function (FF) approach is
The term comes from soft and virtual photon emission, the term comes from hard photon emission, contains the Coulomb correction (18), is the doubly resonating Born cross section, and the background contributions. Explicit expressions can be found in the above references. The additional term is discussed in Refs. and . The invariant mass lost to photon radiation may be calculated as
where is the contents of the curly brackets in Eq. (19).
Alternatively, the structure function (SF) approach may be used:
Here, the invariant mass loss is
In addition, the radiative energy loss may be determined,
Initial state radiation affects the W mass measurement in two ways. Close to threshold the cross-section is smeared out, thus reducing the sensitivity to (see Fig. 5 below). For the direct reconstruction method, the relatively large average energy carried away by the radiated photons leads to a large positive mass-shift if it is not taken into account in the rescaling of the final-state momenta to the beam energy (see Section 3 below). By rescaling to the nominal beam energy we obtain for the mass-shift . Note however that a fit to the mass distribution gives more weight to the peak, and therefore in practice the effective value of or is less than that given by Eqs. (22,25,LABEL:eq:26) (see Section 3.4). Table 3 shows the influence of the various cross-section contributions on the average energy and invariant mass losses. The invariant mass loss may be calculated both in the SF and FF approaches. A comparison shows that the predictions in both schemes differ only slightly, which allows us to use the numerically faster FF approach for the numerical estimates. At the lower LEP2 collider energies, the energy and invariant mass losses are nearly equal, while at higher energies their difference is non-negligible. Note also that the inclusion of the non-universal ISR corrections and background terms is of minor influence. The latter has been studied only for CC11 processes; for reactions of the CC20 type the background is larger and the numerical estimates are not yet available. The Coulomb correction is numerically important and cannot be neglected . The dependence of the predictions on the details of the treatment of QED is discussed in detail in Ref. and will not be repeated here.
|change to FF||0.5||–0.8||–4.2|
1.6.6 Improved Born approximation
In the Standard Model, three parameters are sufficient to parametrise the electroweak interactions, and the conventional choice is since these are the three which are measured most accurately. In this case the value of is a prediction of the model. Radiative corrections to the expression for in terms of these parameters introduce non-trivial dependences on and , and so a measurement of provides a constraint on these masses. However the choice does not appear to be well suited to production. The reason is that a variation of the parameter , which appears explicitly in the phase space and in the matrix element, has to be accompanied by an adjustment of the charged and neutral weak couplings. Beyond leading order this is a complicated procedure.
It has been argued that a more appropriate choice of parameters for LEP2 is the set (the so-called –scheme), since in this case the quantity of prime interest is one of the parameters of the model. Using the tree-level relation
we see that the dominant -channel neutrino exchange amplitude, and hence the corresponding contribution to the cross-section, depends only on the parameters and . It has also been shown that in the –scheme there are no large next-to-leading order contributions to the cross-section which depend on , either quadratically or logarithmically. One can go further and choose the couplings which appear in the other terms in the Born cross-section such that all large corrections at next-to-leading order are absorbed, see for example Ref. . However for the threshold cross-section, which is dominated by the -channel exchange amplitude, one can simply use combinations of and defined by Eq. (27) for the neutral and charged weak couplings which appear in the Born cross-section, Eq. (12).
In summary, the most model-independent approach when defining the parameters for computing the cross-section appears to be the –scheme, in which appears explicitly as a parameter of the model. Although this makes a non-negligible difference when calculating the Born cross-section, compared to using and to define the weak couplings (see Table 5 below), a full next-to-leading-order calculation will remove much of this scheme dependence .
|(off-shell with , )||4.747||15.873|
|(off-shell with , )||4.823||15.882|
1.6.7 Numerical evaluation of the cross-section
Figure 5 shows the cross-section at LEP2 energies. The different curves correspond to the sequential inclusion of the different effects discussed above. The parameters used in the calculation are listed in Table 4. Note that both the initial state radiation and the finite width smear the sharp threshold behaviour at of the on-shell cross-section.
The different contributions are quantified in Table 5, which gives the values of the cross-section in different approximations just above threshold ( GeV) and at the standard LEP2 energy of GeV. At threshold we see that the effects of initial state radiation and the finite W width are large and comparable in magnitude. For the threshold method, the primary interest is the dependence of the cross-section on .
|at 161 GeV||at 175 GeV||at 192 GeV|
This will be quantified in Section 2 below. For both methods, the size of the background cross-sections is important. For completeness, therefore, we list in Table 6 some relevant cross-section values obtained using the PYTHIA Monte Carlo. This includes finite-width effects, initial state radiation and Coulomb corrections. Notice that the values for agree to within about 1% accuracy with those given in the last row of Table 5.
2 Measurement of from the Threshold Cross-Section222prepared by D. Gelé, T.G. Shears, W.J. Stirling, A. Valassi, M.F. Watson
As discussed in Section 1.5, one can exploit the rapid increase of the production cross-section at to measure the W mass. In the following, we briefly discuss the basic features of this method, suggest an optimal collider strategy for data-taking, and estimate the statistical and systematic errors. The intrinsic statistical limit to the resolution on is shown to be energy-dependent: in particular, arguments are presented in favour of a single cross-section measurement at a fixed energy .
2.1 Collider strategy
The cross-section for production increases very rapidly near the nominal kinematic threshold , although the finite W width and ISR smear out the abrupt rise of the Born on-shell cross-section. This means that for a given near threshold, the value of the cross-section is very sensitive to . This is illustrated in Fig. 6, where the excitation curve is plotted for various values of the W mass. The calculation is the same as that discussed in Section 1.6, and includes finite W width effects, ISR and QED Coulomb corrections. A measurement of the cross-section in this region therefore directly yields a measurement of .
For an integrated luminosity and an overall signal efficiency (where the sum extends over the various channels selected, with branching ratios BR and efficiencies ), the error on the cross-section due to signal statistics is given by
where is the number of selected signal events. The corresponding error on the W mass is
The sensitivity factor is plotted in Fig. 7 as a function of . There is a minimum at
corresponding to a minimum value of approximately . Note that the offset of the minimum of the sensitivity above the nominal threshold is insensitive to the actual value of , since in the threshold region the cross-section is to a first approximation a function of only.
As discussed below, the statistical uncertainty is expected to be the most important source of error for the threshold measurement of : the optimal strategy for data-taking consists therefore in operating at the collider energy in order to minimize the statistical error on . The statistical sensitivity factor is essentially flat within , where it increases at most to (+4%); bearing in mind that the present uncertainty on from direct measurements is 160 MeV (and is expected to decrease further in the coming years), this corresponds to on the current world average value. In other words, is already known to a level of precision good enough to choose, a priori, one optimal energy for the measurement of the cross-section at the threshold. Using the latest world average value (see Eq. (1)) gives an optimal collider energy of .
2.2 Event selection and statistical errors
The error on the W mass due to the statistics of events collected has been given in the previous section. Background contamination with an effective cross-section introduces an additional statistical error. The overall effect is that the statistical error on is modified according to
In the following subsections we present estimates of this statistical error for realistic event selections, for an integrated luminosity of at . Tight selection cuts are required to reduce the background contamination while retaining a high efficiency for the signal, especially as the signal cross-section is a factor of 4–5 lower at threshold than at higher centre-of-mass energies.
The studies are based on samples of signal and background events generated by means of Monte Carlo programs (mainly PYTHIA ) tuned to LEP1 data. These events were run through the complete simulation program giving a realistic detector response, and passed through the full reconstruction code for the pattern recognition.
2.2.1 Fully hadronic channel, .
The pure four quark decay mode benefits from a substantial branching ratio (46%) corresponding to a cross-section pb. Obviously, the typical topology of such events consists of four or more energetic jets in the final state. Due to its large cross-section (see Table 6), the main natural background to this four-jet topology comes from events which can be separated into two classes depending on the virtuality of the Z: (i) the production of an on-mass-shell Z accompanied by a radiative photon of nearly 55 GeV (at ), which is experimentally characterised by missing momentum carried by the photon escaping inside the beam pipe (typically 70% of the time), and (ii) events with a soft ISR and a large total visible energy, which potentially constitute the most dangerous QCD background contribution.
Note that a semi-analytical calculation of the genuine four-fermion background cross-section for a wide range of four-fermion final states (with non-identical fermions) shows that in the threshold region , and therefore the effect on the determination from these final states is negligible.
Although the effective four-jet-like event selection depends somewhat on the specific detector under consideration, a general and realistic guideline selection can be described. The most relevant conditions to be fulfilled by the selected events can be summarised as follows:
A minimum number of reconstructed tracks of charged and neutral particles is required. A typical value is 15. This cut removes nearly all low multiplicity reactions, such as dilepton production () and two-photon processes.
A veto criterion against hard ISR photons from events in the detector acceptance can be implemented by rejecting events with an isolated cluster with significant electromagnetic energy (larger than 10 GeV for example).
A large visible energy, estimated using the information from tracks of charged particles and from the electromagnetic and hadron calorimeters. For example, a minimum energy cut value of 130 GeV reduces by a factor of 2 the number of events with a photon collinear to the beam axis.
A minimum number (typically 5) of reconstructed tracks per jet. This criterion acts on the low multiplicity jets from decays as well as from conversions or interactions with the detector material.
A minimum jet polar angle. The actual cut value depends on the detector setup, but is likely to be around . This cut is mainly needed to eliminate poorly measured jets in the very forward region, where the experiments are generally less well instrumented.
These selection criteria almost completely remove the harmless background sources ( and ), but there is still an unacceptable level of contamination (about three times higher than the signal). The second step is to suppress the remaining QCD background (from events) by performing a W mass reconstruction based on a constrained kinematic fit. The following additional criteria can then be imposed:
A probability cut associated with a minimum constrained dijet mass requirement – a typical choice of standard values of 1% and 70 GeV respectively is used here. This procedure appears to be an efficient tool to improve the mass resolution and therefore to reduce the final background, see Fig. 8.
In summary, a reasonable signal detection efficiency in excess of 50% is achievable for events. Although the final rejection factor of QCD events is approximately 500, a substantial residual four-jet background still remains, giving a purity of around 70%. The contributions from other backgrounds (, , and two-photon events) are negligible for the four-jet analysis.
The signal and background efficiencies for the typical event selection described above are given in Table 7, assuming a total integrated luminosity of 100 pb (i.e. 25 pb per interaction point).
|Signal cross-section||0.94 pb||0.76 pb||0.23 pb|
|(stat.) for signal||180 MeV||197 MeV||354 MeV|
|Background cross-section||0.39 pb||0.03 pb||0.01 pb|
|(stat.) for background||106 MeV||37 MeV||74 MeV|
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If you define a function f(x), you can as well draw f(x+a) or f(x-a), or f(x)+a, or f(x)-a, where a might represent the uncertainty parameter of your independent or dependent variable.
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com On line math problem solver that will solve math solvers, get answers, explanations, explain your math homework step by step Socratic: Homework in a snap Take a photo of your homework question videos. share the idea with your personal learning networkPLN) on Twitter Facebook , skills, Pinterest Developing Math Skills Concentration Math study skills can help you learn the mathematical concepts principles so important to other parts of your life. You can hire us to do online classes solve a few math questions do quizzes, write essays much more. Building on the beta launch of Windows Ink OneNote has added new ink effects a new intelligent math coach that can help you solve handwritten equations.
If you ve ever had to help your child with math homework you really appreciate their teachers who do it every day Math anxiety" isn t I need help with my math homework Ask4Essay 8 Answers. I think that the more interactive the more you have to engage your own understanding to process , the less polished the presentation take in the answer.
Some students can complete a wide variety of assignments with little to no effort the ability to make some extra money is always a positive for any college student. Microsoft employees showed off how OneNote powered by Windows Ink will begin to solve inked equations in the coming months Okay okay, that might be a bit of hyperbole but it s less than you think. When I was in the first grade we didn t have homework we Best 25+ Math homework solver ideas on Pinterest. With PhotoMath Windows Phone users can point their phone at a math problem, iOS , which the app will examine using the device s camera Are You Asking Yourself Who Can Do My Homework.
Generally speaking in order to remember , you should practice your mathematical skills on a regular basis, understand the concepts of mathematics gain a better overall Hire a genius to do your math homework for you. Math homework can get quite challenging tedious overwhelming for a lot of students. Hotmath explains math textbook homework problems with step by step math answers for algebra geometry calculus.
Google has quietly upgraded their search calculator to not just answer your basic math questions graphical , 3D math questions; now it can do your geometry homework. For those who truly have no interest in expanding their mathematical knowledge want to speed through their math homework a new app might do the trick. With the app open just point your iPhone s camera at the problem PhotoMath will solve it instantly for you.
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QuickMath will automatically answer the most common problems in algebra equations , calculus faced by high school college students. But this can lead to doing the wrong problem even frustration while looking back , misreading it forth from the book to the paper. That s because of the app called PhotoMath in which you can take a picture of your math homework have the app solve the problem for you. Composed of forms to fill in when possible, then returns analysis of a problem provides a step by step solution.
If your hands are full you can t get to your homework , class assignments fret no more visit today get the best answers when you say Do my math homework. Take a picture of a problem question on paper , on a computer screen search a variety of sites for an Microsoft OneNote can help solve your math homework Engadget.
In these cases the only snapSchool Android Apps on Google Play Thanks to snapSchool you will never again be stuck on a homework exercise. More more apps are delivering on demand homework help to students, who can easily re purpose the learning tools to obtain not just assistance but also answers. After the math problem is instantly read solved the app will show you how the answer was reached too Microsoft s Windows Ink will soon do your math homework for you. Do your math homework for you.
It is very important for you to try your hardest at your homework OMG, sometimes even difficult assignments can be the best way to learn, but also there is now an app that can solve math problems by just taking a picture of them. She previously served as assistant director for NSF s Education Human Resources Directorate is a nationally recognized leader in mathematics education. Besides homework help our tutors offer to take all your course related Do your math homework by Steventhai Fiverr I am a tutor on many topics such as Math Accounting, Supply Chain Management, Project Management, Statistic, Finance, Economic etc.
Of course because the whole point isn t to know the answer to 2x 2 7x 5, cheating at math is a terrible way to learn it s to understand the methodology that can solve any like problem. If your math homework includes equations functions, inequalities, polynomials matrices this is the right trial account An app that does your math homework for you.
Covers arithmetic geometry, calculus , algebra statistics Pay Someone To Do My Math Homework Take My Online Class Pay Someone To Do My Math Homework. Another noted thatif you re a parent helping with homework knowingthe correct answer] how to get there would be sweet Math Study Skills Tips Austin Community College How To Read Your Math Textbook. Mathematics Education Collaborative In How to Help Your Child with Homework authors share how family members caregivers can ask productive questions when children say I don t get it.
50 bills the slavery 10 blind I ll do all your math homework. DENVER Whether as students as a parent helping children with homework we ve all experienced the frustration of a tough algebra problem. Even test yourself if your own answers are true false the app even shows you how to work out the question whether it is answered correctly incorrectly.
While OneNote already comes with shape handwriting recognition the new math assistant takes inking one step further by This Free App Will Solve Math Problems For You. I m not that far removed from my last math class but I barely remember what the equations in this demo video are trying to express let alone how to solve them.
However as an unrehabilitated English major, so I was excited to read about PhotoMath, it s a constant problem for me a new app that lets you do math homework by simply pointing your phone camera at a textbook. If you are capable of downloading judging which of its answers are correct, running PhotoMath, knowing how to feed it equations then you could have DoYourMath. google geometry Pay Someone To Do Your Math Homework Santa en las Calles You want to learn you want to listen to your teacher you want to absorb all of that information in class; but it s.
It gives you both the final answer the step by step solution Homework Guidelines Purplemath TheseHomework Guidelines' detail the proper formatting for math homework; covers type of paper, organization of problems marking of answers. Since copying a question math problem from your homework into Google is far too taxing, there s now a free iPhone app that actually takes things a step further by reading the problems actually doing your homework for you OneNote can now do your math homework for you TNW.
Works for Math History, English, Science more This app solves maths problems from photos. Of course it wouldn t exactly be very helpful if it simple gave you the answers to all your equations so the tool will also teach you how to solve the problems step by step. it s not that you can t find answers to your math problems online it is just that you have to wade through countless web sites dedicated to teaching you how to solve the problems rather then solutions to those problems. You ll work with a tutor in our online classroom in real time solving your math problems step by step until your homework is finished.
50 cupones ser tu esclava 10 citas. Your smartphone s camera might fall short of the typical DSLR in just about every respect explanations App Review SOCRATIC SCAN HOMEWORK, EXPLANATIONS is a homework help app , get answers , GET ANSWERS , but there is one thing it now do that not even your ultra portable Socratic Scan homework, platform that gives students answers to complete explanations for almost any problems you can throw at it.
Sim s class is learning about statistics. Sim asked her students to choose their favorite color.
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https://www.jiskha.com/display.cgi?id=1286315313
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math
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posted by David .
In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u) reaches a final speed of 8.4% of the speed of light while moving in a circular path of radius 0.551 m. What magnitude of magnetic force is required to maintain the deuteron in a circular path?
centripetal force= magnetic force
B= m v/qr
at .084 c, I would first work it ignoring relativistic changes in mass, then rework it considering them.
mass m is not in u units, but in kg.
OK SO F=MA
M= 2.00U --> kg --> 3.32107773 × 10^-27 kg
v=8.4% speed light, => (299792458)*8.4/100...
Plug into F=MA
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https://math.answers.com/Q/Find_the_mean_median_modes_and_range_of_the_data_of_the_numbers_4_4_8_11_12_16_22
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math
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You add the two numbers in the middle of your range and then you divide them by two, or just find the number that is halfway between the two numbers, and you then have your median.
If you are trying to find the median, it's the average of the two.
Range means finding the difference between the highest number in a set of numbers and the lowest. Mean means dividing the total of a set of numbers by the number of numbers there are Mode means the most frequent number. Median is the number in the middle. To find the median you have to first order the numbers from lowest to highest.
There is no direct relationship between the range and the median. So knowing that your median is 20 does not automatically give you the range. The median is just the middle value. That does not tell you what the first and last values are, which is what you need to know to find the range. The median of 19, 20 and 21 is 20, and the range is 2. The median of 4, 20 and 79 is also 20, but the range is 75. The median of 8, 16, 17, 23, 50 and 320 is also 20, but the range is 312. So, as you can see, the Median in no way can tell you what the range will be. You need to know what all of the numbers are, or at least the lowest and highest to know the range.
In general, you cannot. You need to know how many numbers there are and then, in only a select few cases can you find the set.
No!(example) 1336578688(in order) 1335667888The range is when you find the difference between the lowest and highest numberThe range of these numbers:- 7The mean is when you add up all the numbers and divide the total by the amount of numbers there isThe mean of these numbers:- 55The mode is the highest occurring numberThe mode of these numbers:- 8The median is the middle number (if two different numbers the number in the middle)The median of these numbers:- 6
No because the range , median, and mode are all diffrent things to find the range subtract the largest and smallest number in the list , to find the median list the numbers in order then cross them out with pairs at the end and beginning of the list, and to find the MODE you just look at the number that appears there MOST times. O.K.
the mean is where you add all the numbers then divide by the number of numbers the median is when u write all the numbers in order then find the one in the middle and the mode is the most common one. the range is the smallest number subtracted from the largest number.
There would be no median.
when you have an even amount of numbers while trying to find the median, you first find the two numbers that are at the median and then take all the numbers between them and find the median of that. if that amount of digits is also even, then you must have a decimal median.
to find the median in a set of numbers you have to order them from the smallest to the largest and find the middle value e.g. 2,4,3,7,1 1,2,3,4,7 the median is 3
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https://www.hackmath.net/en/word-math-problems/mass?tag_id=6
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math
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Mass + percentages - math problems
- Butter fat
A quarter of kg butter contains 82% fat. How many grams of fat are in four cubes of butter?
- Water in vegetables
Tomatoes in the store contain 99% water. After being transported to the shop, they were slightly dried and contained only 98% of water. How many kgs of tomatoes are in the store if there were 300 kg in stock?`
- Camel and water
84% of the camel's weight is water. After drinking, its weight increased to 832 kg and water accounted for 85% of its weight. How much did it weigh before drinking?
In human body the blood is about 7.3% body weight. How many kilograms of blood is in the human body with weight 109 kg?
- Vinegar 2
How many percentage of getvinegar solution, if we mix to 3.5 liters of 5.8% and 5 liters of 7.6% vinegar?
Crystal grows every month 1.2 promile of its mass. For how many months to grow a crystal from weight 177 g to 384 g?
How many 55% alcohol we need to pour into 14 liters 75% alcohol to get p3% of the alcohol? How many 65% alcohol we get?
- Barrel of oil
Barrel of oil weighs 283 kg. When it mold 26% oil, weighed 216 kg. What is the mass of the empty barrel?
Gross weight of shipment is 5188 kg and its tare is 5.6%. Calculate the net weight of the shipment.
- Sea water
Mixing 62 kg of sea water with 84 kg rainwater is created water containing 3.1% salt. How many percent sea water contains salt?
Imagine a word solidarity means that salt donation to the needy, who have neither the salt. If we take word solidarity has word base salt + gift (only in Slovakian language). Calculate how many kilos of salt sympathetic citizen "gives" government a year i
Mother bought 5 boxes of milk and 7 kg of potatoes and paid a total CZK 147. Aunt bought 7 boxes of milk and 3 kg of potatoes and paid 131 CZK. What is the price of one carton of milk and 1 kg of potatoes? How CZK together would have saved if bought at the
Potatoes contain 78.6% starch. How many potatoes need to obtain 27 kg of starch?
Fresh mushrooms contain 94% water, dried 14%. How many kg of fresh mushrooms is needed to collect to get 10 kg dried?
- Metal alloy
What is the ratio of metals in the alloy that is in the 50 tonnes of steel to 30 kg nickel?
From 55% and 80% spirit we would like to produce 0.2 kg of 60% spirit. How many of them we must use in a solution?
- Bronze, tin and copper
Bronze is an alloy of tin and copper. An alloy of 10% tin and 90% copper is Gunmetal. If it contains 20% tin and 80% copper, it is bell metal. How many tons of molten bell metal and how many tons of copper is needed to make 100 tons of Gunmetal?
Openings in perforated bricks occupy 10% and brick has dimensions 30 cm, 15 cm and 7.5 cm. Calculate a) the weight of a perforated bricks, if you know that the density of the full brick material is p = 1800 kg/m3 (1.8 kg/dm3) b) the number of perforated.
At what price bought retail 1 kg goods from wholesale, if lost in the distribution is 4% of weight of the goods and retail still have a profit of 6.3%? Goods are sold at retail for 25 euro per kg.
- Barrel with liquid
Barrel with grain weight 297 kg. When it shed 48% of grain, ha weight 174 kg. What is the weight of empty barrel?
Our percentage calculator will help you quickly calculate various typical tasks with percentages.
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| 3,279 | 31 |
https://www.jiskha.com/display.cgi?id=1422626595
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math
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posted by Clara
A 1800 kg mass car is traveling south at a velocity of 15 m/s through an intersection. It is hit by a 3500 kg mass car traveling east at 20 m/s. If the two cars collide and become one object (perfectly inelastic). What will be the final displacement of the cars after the collision if the coefficient of kinetic friction between the road and the tires is 0.8?
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| 375 | 2 |
http://mymathforum.com/advanced-statistics/6555-variance-covariance.html
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math
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|April 17th, 2009, 03:01 AM||#1|
Joined: Apr 2009
Variance and covariance
I am hoping this is a simple question and that I am getting forgetful in my old age.
I am taking measurements of color distance, and the dimensions are L (lightness/darkness), A(red/green), and B(Blue/Yellow).
I want to calculate the percent overlap between two processes each of which produces a normal distubition in L, A, & B.
I orginally calculated it simply as a distance between two means in L, A, & B with the variance of the two processes being the sum of the variances for each of the individual processes. I then calculated the proability density between the upper and lower specifications for each of the three dimensions and multiplied the proabilites to get the proability of all three of them being in spec.
The problem is that covariance exists, a part that is yellow will also tend to be red for example as the metallic in the paint tends to cause a perdictable movement. As a result the part will never be yellow and green from the standard so multipling the proability densities doesn't work.
At this point I am getting out of my depth, any suggestions greatly appreciated.
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|covariance etc||aptx4869||Advanced Statistics||5||June 15th, 2007 01:40 PM|
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https://www.coursehero.com/file/8656864/exam2practicesolutions/
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math
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Math 2260 Exam #2 Practice Problem Solutions1. EvaluateZtan3(x)dx.
5. Suppose a 20 foot chain which weighs 5 pounds per foot is coiled on the ground. One end of the chainis attached to a small crane. How much work does it take to lift this end 20 feet off the ground, sothat the chain is fully extended in the air?
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| 314 | 2 |
https://core.ac.uk/display/20972218
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math
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Reminder to all that problem set 1 is due today. And please sign up for scribing and get on the class mailing list. Note that we meet on next Tuesday instead of Monday. 2 Overview Today we show that PARITY is not in AC0. AC0 is a family of circuits with constant depth, polynomial size, and unbounded fan-in for the AND and OR gates. We establish this result through an application of the Switching Lemma. This result is the first use of randomization in its full power in complexity. Circuits were defined in previous lectures. In this lecture, we always assume that the circuits are organized into alternating levels of AND and OR gates. We can make such an assumption since we can convert circuits into this convenient form with only a constant factor of blowup. The Switching Lemma is first proved by Furst, Saxe, and Sipser in FOCS 81, and readers can find the paper in the Journal of Mathematical Systems Theory 1984. We will highlight their work by using the Lemma though the version we prove today will not be as strong as we claimed in the last lecture. Johan H˚astad, in 1986, proved a more general and powerful form of the Lemma, and interested readers can find it in his PhD thesis at MIT. There is also a surve
To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.
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https://news.stanford.edu/pr/03/wolfram129.html
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math
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Nora Sweeny, Stanford Center for Innovations in Learning: (650) 725-0683, [email protected].
Stephen Wolfram explains "A New Kind of Science" Feb. 10
Stephen Wolfram, the theoretical physicist who has been described by Wired magazine as "the Bob Dylan of physics" and a "Jedi mind-warrior," will speak on campus on Monday, Feb. 10, on the ideas contained in his book A New Kind of Science. Wolfram's lecture will be held at 7:30 p.m. in Dinkelspiel Auditorium. The event, co-sponsored by the Stanford Center for Innovations in Learning and the Symbolic Systems Program, is free and open to the public.
Wolfram's paradigm shift may have profound implications for our understanding of physics, biology and perhaps even the character of intelligence in the universe. A London Daily Telegraph headline asked, "Is this man bigger than Newton and Darwin?"
At complexity's core -- from marvels as diverse as the shape of a snowflake to the structure of space and time -- Wolfram finds simplicity. Starting with a handful of computer experiments, Wolfram has developed a new way to explain the essential mechanisms of the natural world. His premise is that simple rules (the kind that make up computer programs or describe how to construct a mosaic tile pattern), rather than elaborate mathematical formulas, have far more potential to accurately describe our universe. "The aphorism that the weather has a mind of its own may be less silly than you might imagine," Wolfram says.
Pre-release orders of A New Kind of Science put the 1,200-page book on Amazon's top few hundred sellers for much of the past six months -- occasionally cracking the top 50 -- and within a week of its publication the entire 50,000 print run had sold out.
Wolfram was educated at Oxford University and Caltech. He was a fellow at the Institute for Advanced Study at Princeton; his early work on elementary particle physics and its relationship to cosmology was recognized with a MacArthur "genius" Award in 1981. Wolfram is also the creator of Mathematica, the scientific software used by millions of scientists, researchers, engineers and students.
Nora Sweeny is director of communications for the Stanford Center for Innovations in Learning.
By Nora Sweeny
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| 2,242 | 9 |
https://answersdrive.com/what-constant-is-k-3948561
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math
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Since k is a physical constant of proportionality between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in SI units means a change in temperature by 1 K only changes a particle's energy by a small amount.
Likewise, people ask, what is K in ideal gas law?
The Ideal Gas Law. The volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in Kelvin. The relationship for these variables, P V = n R T, where R is known as the gas constant, is called the ideal gas law or equation of state.
What is the value of k in coulombs law?
The symbol k is a proportionality constant known as the Coulomb's law constant. The value of this constant is dependent upon the medium that the charged objects are immersed in. In the case of air, the value is approximately 9.0 x 109 N.
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| 894 | 5 |
http://tulyn.com/exponential_growth.htm
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math
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in 10th Grade Math.
in 11th Grade Math.
in 12th Grade Math.
Exponential growth (including exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value. In the case of a discrete domain of definition with equal intervals it is also called geometric growth or geometric decay (the function values form a geometric progression).
Many students find exponential growth difficult. They feel overwhelmed with exponential growth homework, tests and projects. And it is not always easy to find exponential growth tutor who is both good and affordable. Now finding exponential growth help
is easy. For your exponential growth homework, exponential growth tests, exponential growth projects, and exponential growth tutoring needs, TuLyn is a one-stop solution. You can master hundreds of math topics by using TuLyn.
At TuLyn, we have over 2000 math video tutorial clips including exponential growth videos
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Top Exponential Growth Word Problems
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Post a Comment
Watch video clips on exponents, print exponents worksheets, practice exponents word problems.
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CC-MAIN-2017-30
| 1,915 | 20 |
https://www.tradingsystemforex.com/ideas-for-expert-advisors/2698-half-ea-2.html
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math
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What does your expert tab say?
Please edit this robot. (TP-SL_03.mq4)
I would like a “SL Entry Level” function.
After we entry, the price reaches this level, you can put forth in the SL.
SL Entry Level: 50
We entry 1.50000
TP is: 1.50150
SL is: 1.49500
If the price reaches the: 1.50050 then put into the SL 1.50050. Stay there, do not move on.
Thank you in advance.
It seems to me, I asked a very complicated. This is unbelievable ....
Thank you, I'm going to test.
Is it only me but Half Ea doesn't backtest at all, and is TP-SL_03.01.mq4 another EA?
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| 556 | 13 |
https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/abraham-adolf-fraenkel
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math
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Abraham Adolf Fraenkel
Abraham Adolf Fraenkel
One of the fathers of modern logic, German-born mathematician Abraham Fraenkel (1891-1965) first became widely known for his work on set theory. Long fascinated by the pioneering work in set theory of fellow German Ernst Zermelo (1871-1953), Fraenkel launched research to put set theory into an axiomatic setting that improved the definitions of Zermelo's theory and proposed its own system of axioms. Within that system, Fraenkel proved the independence of the axiom of choice. The Zermelo-Fraenkel axioms of set theory, known collectively as ZF, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. When the axiom of choice is included, the resulting system is known as ZFC.
Studied at Several Universities
Abraham Adolf Fraenkel was born on February 17, 1891, in Munich, Germany. The son of Sigmund and Charlotte (Neuberger) Fraenkel, he was strongly influenced by his orthodox Jewish heritage. B.H. Auerbach-Halberstadt, Fraenkel's great-grandfather, had been widely known for his rabbinical teachings. As a child, Fraenkel was enrolled in Hebrew school and was reading Hebrew by the time he was five. Raised in a family that set a high priority on education, Fraenkel advanced rapidly in his general studies and, like most German students of that era, studied at a number of universities. He began his higher studies at the University of Munich in his hometown and studied subsequently at the German universities of Marburg, Berlin, and Breslau. In 1914, at the age of 23, Fraenkel received his doctoral degree in mathematics from the University of Breslau.
World War I broke out in August 1914, shortly after Fraenkel had completed his studies at Breslau. For the next two years, he served in the German military as a sergeant in the medical corps. He also worked briefly for the German army's meteorological service. In 1916 Fraenkel accepted a position at the University of Marburg as an unsalaried lecturer, or privatdocent. It was at Marburg that Fraenkel began his most important research in mathematical theory. On March 28, 1920, he married Malkah Wilhemina Prins. The couple eventually had four children.
Focused on Set Theory
Fraenkel's earliest research was on the p-adic numbers first described by Kurt Hensel in the late nineteenth century and on the theory of rings. Before long, however, he became deeply involved in the study of set theory, specifically the work of Ernst Zermelo, who in the early years of the twentieth century had published his controversial and innovative views on the subject. Zermelo had postulated that from any set of numbers, a single element could be selected and that definite properties of that element could be determined. This was known as the axiom of choice, but Zermelo offered no real proof for his theory, suggesting that the study of mathematics could only progress if certain axioms were simply accepted without question. For many mathematicians, Zermelo's lack of proof was unacceptable. Some, including French mathematician Jacques Hadamard, reluctantly agreed to accept Zermelo's theory until a better way could be found, while others, including Jules-Henri Poincaré, adamantly opposed acceptance of Zermelo's theory.
Without either accepting or rejecting Zermelo's theory outright, Fraenkel set about to find ways to put Zermelo's work on a firmer foundation. In the case of finite sets of numbers, Fraenkel found, Zermelo's theory already worked quite well. However, for infinite sets, Zermelo's assumptions were more questionable. Fraenkel eventually substituted a notion of function for Zermelo's idea of determining a definite property of a number in a set. In so doing, he significantly clarified Zermelo's set theory and also rid it of its dependence on the axiom of choice, which had clearly been one of the most controversial elements of Zermelo's work.
Just as Fraenkel's research built on theories advanced earlier by Zermelo, others' refinements to the work of Zermelo and Fraenkel have buttressed their theories and advanced the mathematical community's understanding of set theory. Fraenkel's system of axioms was modified by Norwegian mathematician Thoralf Skolem in 1922 to create what is known today as the ZFS system, named for Zermelo, Fraenkel, and Skolem. Within the ZFS system, it is harder to prove the independence of the axiom of choice, a goal that was not achieved until the work of American Paul Joseph Cohen in the 1960s. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and the generalized continuum hypothesis.
Published Set Theory Findings
Fraenkel published his conclusions on set theory in two separate works—a popular introductory textbook published in 1919 and a 1922 research article determining the independence of the axiom of choice. The conclusions in the latter work were later included as part of the proof for a newly coined term, Ur-elements—infinite and distinct pairs of objects that do not in themselves define a set. A number of prominent mathematicians of the period questioned the validity of Ur-elements, but only three years later German physicist Wolfgang Pauli used them in his proof of the exclusion principle.
In 1922 Fraenkel was promoted to assistant professor of mathematics at the University of Marburg. His earlier work on set theory had propelled him to the forefront of set theory research, and over the next few years he published a number of articles on the subject while he continued to teach. In 1928 Fraenkel was offered a full professorship at the University of Kiel. He accepted but only a year later took a leave of absence to become a visiting professor at Jerusalem's Hebrew University. For the next two years he taught at Hebrew University, leaving in 1931 after a disagreement with the school's administration.
Germany in Turmoil
Fraenkel's return to Germany proved to be a bittersweet occasion. His native country was in economic disarray, suffering through the effects of the worldwide economic depression and the brutal conditions imposed by the Treaty of Versailles that had ended World War I. The economic pressures on the German people had given rise to increasing intolerance, most notably a disturbing wave of anti-Semitism. For the next two years, Fraenkel resumed his teaching duties at Kiel, keeping a wary eye on the increasingly unsettled political situation in Germany. In January 1933 Adolf Hitler, leader of the National Socialist German Workers' Party, better known as Nazis, became Germany's chancellor. Fraenkel and his family left the country a month later, moving first to Amsterdam in the neighboring Netherlands.
Fraenkel and his family spent only two months in Amsterdam, closely monitoring the situation in their native Germany while there. Convinced that there would not be a quick turnaround under the Nazi regime, Fraenkel drafted a letter of resignation to the University of Kiel in April 1933 and returned to Jerusalem to teach once again at Hebrew University. Despite his earlier disagreement with the university's administration, he was warmly welcomed back to the school's faculty.
Focus of Research Changed
Following his exile from Germany, Fraenkel changed the focus of his research. Although he continued to publish texts on set theory for the remainder of his career, Fraenkel began to concentrate his studies on the evolution of modern logic and the contributions made by Jewish mathematicians and scientists in their respective fields. Fraenkel had written a number of books about the history of mathematics. In 1920 he had published an overview of the work of Carl Friedrich Gauss, who in his doctoral dissertation had proved the fundamental theorem of algebra. As early as 1930 he had begun the work of chronicling the accomplishments of Jewish mathematicians with his biography of Georg Cantor, who was half-Jewish. Cantor at that time was of greater interest to Fraenkel for the nature of his research into set theory than for his ethnic background. However, once he had resumed teaching at Hebrew University in 1933, he began a much wider study into the work of Jewish scientists and mathematicians. In 1960 Fraenkel published Jewish Mathematics and Astronomy.
In his research into the origins of modern logic, Fraenkel looked closely at natural numbers, describing them in terms of modern concepts of logic and reasoning. Although his research underscored the need for continuity in consideration of the number line, Fraenkel also expressed interest in opposing points of view. During this period, Fraenkel had a conversation with physicist Albert Einstein, who suggested that the prevailing theory of continuity in mathematics might some day be overtaken by the atomistic concept of the number line. Although Fraenkel himself remained unconvinced, largely because he considered mathematical continuity necessary to the foundation of modern calculus, he did publish an article explaining the views of the intuitionists, as Einstein and others who believed similarly were known.
Taught at Einstein Institute of Mathematics
Fraenkel was among the first professors at Hebrew University's Einstein Institute of Mathematics. Along with fellow professor Edmond Landau, Fraenkel taught mathematical logic and mathematical analysis. In 1958, while still teaching at Hebrew University, Fraenkel published an overview of his work on set theory, a textbook entitled Foundations of Set Theory. A year later, he retired as a professor at Hebrew University. To mark Fraenkel's 70th birthday in 1961, several members of the mathematical community put together a collection of essays and research articles related to Fraenkel's life work. The collection, Essays on the Foundations of Mathematics, contained contributions from mathematicians around the world. Sadly, Fraenkel never saw the book in its final form. He died in Jerusalem on October 15, 1965, only months before the book was published.
Fraenkel will be remembered for his research in set theory and modern logic. His refinements to the set theory conclusions of Ernst Zermelo, codified as the Zermelo-Fraenkel axioms, or ZF, are almost always what scientists and mathematicians mean today when they speak of "set theory." Further enhancing the value of Fraenkel's contributions to the body of mathematical theory are the clarity and precision of his writings, several of which continue to be taught in colleges and universities worldwide. In its review of Fraenkel's summation of his set theory research— Foundations of Set Theory —the British Journal for the Philosophy of Science was lavish in its praise. Its reviewer wrote that the book "is a masterly survey of its field. It is lucid and concise on a technical level, it covers the historical ground admirably, and it gives a sensible account of the various philosophical positions associated with the development of the subject … essential reading for any mathematician or philosopher."
Contemporary Authors Online, Gale Group, 2000.
Mathematical Expeditions: Chronicles by the Explorers, Springer-Verlag, 2001.
Notable Scientists: From 1900 to the Present, Gale Group, 2001.
"Adolf Abraham Halevi Fraenkel," Groups, Algorithms, and Programming,http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fraenkel.html (March 5, 2003).
"Adolf Fraenkel," 201E: Mathematical Foundations,http://ergo.ucsd.edu/~movellan/courses/245/people/Fraenkel.html (March 9, 2003).
"Paul Joseph Cohen," Groups, Algorithms, and Programming,http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Cohen.html (March 9, 2003).
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100535.26/warc/CC-MAIN-20231204214708-20231205004708-00093.warc.gz
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CC-MAIN-2023-50
| 11,675 | 28 |
https://math.answers.com/Q/How_do_you_find_fractions_in_money
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math
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You Find Fractions
You multiply the fractions
you can find fractions on a recipe,shoes,signs,or notebooks as long as it is a fraction.
you have to compare the common fractions
Addition or subtraction of fractions require "like" fractions: that is, fractions with the same denominator.
it helps you find the distance between fractions beacause the new name should be an equivalent fraction
Perimeter is taking the length of each side and adding them together. Therefore, if you have the lengths as fractions, you find the common denominator and add the fractions normally.
If I had some fractions, I might. But since I don't, I won't.
You solve it just like they are proper fractions
Because to add or subtract two fractions you first have to find equivalent fractions for both which have the same denominator.
It is useful to find the LCM when you want to add and subtract fractions. It is useful to find the GCF when you want to reduce fractions.
Find the lowest common multiple of the denominators and adjust the fractions accordingly
A very large but indeterminate number. It can come up in dealing with money, product, services, customers or inventory. In other words, it might be easier to find what few jobs - if any - do not deal with fractions.
HOW DO YOU FIND AN INPROPER FRACTION?
All fractions are rational numbers because irrational numbers can't be expressed as fractions
An easy method for finding a fraction between two fractions would be to take the average of two fractions. You find the average by adding the two fractions together and then dividing the final sum by 2.
Yes, bankers use fractions because fractions are the same as percentage. and they need percentage for exact change and for knowing how much money is in the safe.
You can either convert fractions to decimals and compare the decimal numbers; find equivalent fractions with the same denominator and then compare numerators or find equivalent fractions with the same numerator and then compare denominators.
The same way that you calculate the average for any other numbers. Sum the fractions and divide the total by the number of fractions.
If the denominators are not the same, then you have to use equivalent fractions which do have a common denominator . To do this, you need to find the least common multiple (LCM) of the two denominators. To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify.
In its bottom.
By using them in Math Class or teaching fractions, other then that, money, and estimating things between integers
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585290.83/warc/CC-MAIN-20211019233130-20211020023130-00112.warc.gz
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CC-MAIN-2021-43
| 2,571 | 22 |
https://byjus.com/question-answer/250-g-of-water-at-30-degc-is-present-in-a-copper-vessel-of-mass/
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math
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250 g of water at 30 °C is present in a copper vessel of mass 50 g. Calculate the mass of ice required to bring down the temperature of the vessel and its contents to 5 °C.
Specific latent heat of fusion of ice = 336 x 103 J kg-1
Specific heat capacity of copper vessel= 400 J kg-1 °C-1
Specific heat capacity of water 4200 J kg-1 °C-1
Let the required mass of ice be m g.
Then heat gained or required to change ice at 0oC to water at 0o = m x 10-3 x 336 x 103 = 336m joule
Heat gained by water to change its temperature from 0oC to 5oC = m x 10-3 x 4.2 x 103 x (30 - 5) = 21m joule
Total heat gained = (336m + 21m) = 357m ...(i)
Now heat lost by water to change its temperature from 30oC to 5oC
= 250 x 10-3 x 4.2 x 103 x (30 - 5)
= 26,250 joule
Heat lost by copper vessel = (50 x 10-3 x 0.4 x 103 x 25) = 500 joule
Total heat lost = (26,250 + 500) J = 26,750 J ...(ii)
According to the principle of calorimetry,
Heat gained = Heat lost
357 m= 26,750 [From (i) and (ii)]
m= 26750 / 357 = 74.93g
Hence, the required mass of ice is 74.93 g.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949694.55/warc/CC-MAIN-20230401001704-20230401031704-00352.warc.gz
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CC-MAIN-2023-14
| 1,043 | 18 |
https://www.electroniclinic.com/combinational-logic-data-processing-circuits/
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math
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Combinational logic & Data Processing circuits
A combination of inter-connected logic gates, which can perform a specific Boolean function without having the capability of memory or storage, is known as a combinational logic circuit. In other words, a combinational logic circuit consists of such logic gates, the outputs of which can directly be determined at any time through the present inputs’ combination without keeping in mind the previous inputs. As a combinational circuit performs the function of processing data or specific information through a set of Boolean functions in a logical state, thus circuits designed by means of combinational logic circuits, are also known as data processing circuits. In other words, as binary data process during various operations of combinational circuits, (a series of different operations in order to get certain distinct results, is called the process) therefore, such combinational circuits are also known as data processing circuits.
Hence, combinational logic circuits are such circuits, the output of which is solely depends on values of the present combinations of variables, and when input combinations change, output also changes. The output of combinational logic circuit depends on the circuit’s input, the types of gates being used, and the various methods being adopted for connecting these gates. It has to be remembered that these circuits are generally available in the form of MSI integrated circuits. As combinational logic circuits are designed through a combination of logical gates (as the name implies) therefore, they are commonly used in solving logical problems e.g. multiplexing, encoding, and decoding, etc.
A combinational circuit consists of input variables, logic gates, and output variables. Logic gates receive signals from inputs and create signals on outputs. As a result of this process, binary information changes from a given input data towards required output data. In general, both input and output data are represented through binary signals (or two possible states or values i.e. logic 1 and logic 0). The block diagram of a combinational circuit has been shown in figure 4.1, according to which “n” input binary variables are received by the circuit through some external source, whereas output variables “m” received from this circuit go towards some other external circuit or source. These external sources during various applications are normally storage registers, which remain nearby a combinational circuit or exist on some far–off the device. It should be remembered that every input variable of every combinational circuit contains one or two wires. When only one wire is available, it shows that the variable is in the normal state or it is in complement form. As one of the variables in a Boolean expression is normal or complement, therefore it is necessary to provide an inverter on the input wire. On the contrary, in case the input variable is reflected via two wires, the circuit’s input will have both normal as well as complement states. Thus, under such a situation, it is not incumbent to include an inverter along with inputs.
Figure 4.1 – block diagram of a combinational circuit
Previous Topic: Basic comparator operations with circuit diagram examples
For electronics and programming-related projects visit my YouTube channel.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494974.98/warc/CC-MAIN-20230127065356-20230127095356-00104.warc.gz
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CC-MAIN-2023-06
| 3,356 | 7 |
http://virgin-trains.mynewsdesk.com/pressreleases/tag/west-coast
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math
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Press releases • Oct 17, 2018 00:05 BST
Virgin Trains has become the first UK train operator to provide body worn cameras to cover all its frontline people, resulting in assaults on staff falling by more than half.
Press releases • Jul 25, 2018 00:01 BST
Tuesday 24th July – Virgin Trains has today released the first episode in the brand new ‘West Coast Weekender’ series. Hosted by top comedian and TV personality, Joel Dommett, the videos see Joel discovering all the best – lesser known – things to see, do, eat and drink on a summer weekend getaway to five of the most popular destinations along the Virgin Trains route.
Press releases • May 30, 2018 00:01 BST
Virgin Trains is today launching a new partnership with Uber to enable passengers to easily request a ride to the station and on arrival at their destination.
Press releases • May 21, 2018 17:00 BST
Virgin Trains has named one of its Pendolinos Blackpool Belle to celebrate the launch of its first electric services to and from the resort today (Monday 21 May).
Press releases • May 15, 2018 09:05 BST
Virgin Trains has recreated its iconic Pendolino train on Blackpool beach to celebrate its first visit to the seaside resort today.
Press releases • Apr 03, 2018 09:15 BST
A Virgin Trains Station Announcer based at Preston is switching his microphone for the wrestling ring at the 2018 Commonwealth Games for the next two weeks.
Press releases • Mar 19, 2018 15:01 GMT
Virgin Trains has named one of its trains after one of Scotland’s most famous cultural icons in celebration of its role in serving Glasgow.
Press releases • Mar 17, 2018 00:01 GMT
Virgin Trains has created a unique guide to the “MarmaLake” District to celebrate the World’s Original Marmalade Festival & Awards, which take place in Penrith this weekend.
Press releases • Mar 09, 2018 09:58 GMT
Press releases • Mar 08, 2018 18:00 GMT
A member of the Virgin Trains team at Carlisle is taking to two wheels to ride across India, after doubling his fundraising target.
Press releases • Feb 23, 2018 07:00 GMT
Press releases • Feb 14, 2018 00:01 GMT
New figures from Virgin Trains show journeys between Chester and London have broken through the half a million mark, setting a record for the number of journeys taken by train.
Virgin Trains runs majority of services today during unwarranted union strike action on the west coast
Press releases • Dec 15, 2017 00:01 GMT
Press releases • Dec 01, 2017 10:00 GMT
A Virgin Trains employee at Carlisle is gearing up to take part in his first ever cycling challenge halfway across the world, whilst raising thousands of pounds for charity.
Virgin Trains confirms it will run majority of services during unwarranted union strike action on the west coast
Press releases • Nov 29, 2017 16:02 GMT
Press releases • Sep 28, 2017 09:17 BST
Press releases • Sep 17, 2017 07:00 BST
A new garden at Penrith station, Cumbria has been created to welcome an entirely different type of local commuter, thanks to the work of Virgin Trains and 25 local gardeners.
Press releases • Aug 22, 2017 09:00 BST
Virgin Trains is advising customers wishing to travel over the August Bank Holiday via its east coast and west coast routes to plan their journeys and reserve seats in advance.
Press releases • Aug 11, 2017 15:00 BST
Today Virgin Trains welcomed a couple at Carlisle, who are travelling to all railway stations in Britain for a project entitled ‘All The Stations’, which sees them visit 2,563 stations over 14 weeks.
Press releases • Jun 26, 2017 14:00 BST
New station entrance, free high-speed wifi, additional retail spaces and bike hub form part of station refurbishment as passenger numbers hit new high
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s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669847.1/warc/CC-MAIN-20191118205402-20191118233402-00285.warc.gz
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CC-MAIN-2019-47
| 3,774 | 37 |
https://www.kopykitab.com/blog/csvtu-syllabus-applied-mathematics-ii/
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math
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Chhattisgarh Swami Vivekanand Technical University Bhilai (C.G.)
Semester: IInd Branch: Common to All Branches
Subject: Applied Mathematics-II Code: 300214 (14)
UNIT – I
Complex Numbers: De Moivre’s theorem, roots of complex numbers; separtion into real & imaginary parts of circular, hyperbolic, logarithmic & exponential function; summation of trigonometric series by C+iS method.
UNIT – II
Differential Equations of higher order: Linear differential equations of higher order with constant coefficients, method of variation of parameters; Cauchy’s & Legendre’s linerar equations; simultaneous linear equations with constant coefficients.
UNIT – III
Multiple Integrals: Double & triple integrals, change of order of integration; Beta & Gamma functions; application to area & volume.
UNIT – IV
Vector Calculus: Vector operator ; directional derivative, gradient, divergence & curl; line, surface & volume integrals, Green’s, Gauss’s & Stoke’s theorem (without proof) & applications.
UNIT – V
Theory of Equations: Roots of polynomial equations, relations between roots and coefficients; transformation of equations, removal of terms; solution of cubic & biquadratic equatins-Cardon’s & Ferrari’s methods.
1. Higher Engg. Mathematics by B.S. Grewal (38th edition)-Khanna Publishers.
2. Advanced Engg. Mathematics by Erwin Kreyszig (8th edition) – John Wiley & Sons.
1. Higher algebra by H.S. Hall & S.R. Knight – A.I.T.B.S. Publishers.
2. Integral Calculus by Gorakh Prasad – Pothishala Private Limited.
3. Advanced Engg. Mathematics by R.K. Jain & S.R.K. Iyengar – Narosa Publishing House.
4. Applied Mathematics by P.N. Wartikar & J.N. Wartikar Vol. (I&II) – Pune Vidhyarthi Griha Prakashan, Pune.
5. Applied mathematics for Engineers & Physicists by Louis A. Pipes – Mc Graw Hill.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153531.10/warc/CC-MAIN-20210728060744-20210728090744-00602.warc.gz
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CC-MAIN-2021-31
| 1,820 | 20 |
https://www.kingsu.ca/programs/bachelor/mathematics
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math
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Study your minor in Mathematics at King's
Delve into the beautiful intricacies of mathematics. Spend some time pondering the rich algebraic and geometric patterns that have fascinated humanity for millennia.
Students in King's mathematics minor explore the abstraction of pure mathematics with its rich interconnected structures.
We dig deeply into the great questions of mathematics: what are all these sets, numbers, and functions; where did they come from; and how and why do they fit together in so many ways?
Students in the mathematics minor also investigate how mathematics relates to many academic disciplines through mathematical models. Consider population models in mathematical biology, asset and resource development models in business and finance, information processing in computing science, or sound production in music.
Mathematics programs at King's
Available as a minor:
- 4-Year Bachelor of Arts
- 4-Year Bachelor of Commerce
- 4-Year Bachelor of Music
- 4-Year Bachelor of Science
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s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316075.15/warc/CC-MAIN-20190821152344-20190821174344-00114.warc.gz
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CC-MAIN-2019-35
| 1,001 | 11 |
https://www.helpmyessay.com/subjects/interdisciplinary-studies
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math
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Lesson 3: Understanding Interdisciplinary Studies
Assignment 1: Using Szostak’s 12-Step Process
Directions: For this assignment, you will develop an interdisciplinary
question and follow Szostak’s 12-step process of interdisciplinary study
. Be sure to read the instructions for each step carefully and take time to research and reflect on your answers. You do not need to perform extensive library or internet research to complete steps 4, 6, 7, and 8; a cursory search will be adequate for this assignment. For the questions that ask you to list, simply provide the list. For all other questions, write a paragraph response.
• Develop an interdisciplinary
• Identify at least 4 key phenomena.
• Identify relevant theories and methods (at least 3).
• Perform literature survey. Write a paragraph summarizing your findings. You must cite at least three sources. Be sure to use correct APA style for your references.
• Identify relevant disciplinary perspectives (at least 3)
• Revisit theories, methods, and phenomena that have received little attention. Revisit at least one. Describe it below and explain if anything was gained by revisiting it.
• Evaluate the results of previous research
• Compare results of previous research
• Develop a more comprehensive/integrative analysis
• Reflect on the results of integration
• Test results of integration
• Communicate the results
THIS NEEDS TO BE IN YOUR OWN WORDS.
For Szostak’s 12 step process it can be found via Google, if you have any other question just email.
[ Order Custom Essay ]
[ View Full Essay ]
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s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202131.54/warc/CC-MAIN-20190319203912-20190319225912-00292.warc.gz
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CC-MAIN-2019-13
| 1,588 | 21 |
https://math.answers.com/Q/What_is_five_firths_plus_thirteen_seventh
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math
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Five plus eight does not equal one. It equals thirteen.
14 and 17/100
The difference is that they are both numbers. Two plus five is not thirteen. You can't divide thirteen equally...Sorry!
If you mean one and one seventh (1 1/7) plus five sevenths (5/7), the answer is 1 6/7.
13.507 = Thirteen and five hundred seven thousandths
The book "Thirteen Plus One" came out in 2010.
Ten multiplied by open bracket, two point five plus thirteen point five, close bracket.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499871.68/warc/CC-MAIN-20230131122916-20230131152916-00855.warc.gz
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CC-MAIN-2023-06
| 464 | 7 |
https://ekuatio.com/en/operations-combined-with-fractions-step-by-step-exercises-solved/
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math
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Next we will learn how to solve the combined operations with fractions step by step.
Índice de Contenidos
What are combined operations with fractions
In this case, with operations combined with fractions, we will have to add, subtract, multiply or divide the fractions in the same expression. The different operations will be mixed, or rather combined.
In order to solve this type of exercises correctly, we must understand perfectly how each operation is carried out separately: additions and subtractions with the same one and with different denominator, multiplications and divisions.
As with operations with numbers, when we have several operations in the same fiscal year, we have to follow the rules of the hierarchy of operations.
Resolved exercises of operations combined with fractions
Let’s explain it with a few solved examples of operations combined with fractions. We will look at what you need to know in each of the steps.
We have two operations: addition and multiplication. Well, first of all we multiply:
We have one sum left with a different denominator. Now we get a common denominator and make the sum:
In the end, we have simplified the fraction.
In this case we have several operations combined with fractions, brackets and parentheses.
Where to start? Well, you have to start by deleting parentheses and the only way to do this is by starting with the one that is deepest inside. Remember that when we have several parentheses we must start from the inside out:
We have already removed the parenthesis from the inside and we have only one parenthesis left. We proceed to resolve it:
Finally, we have one division left, which we resolve by multiplying it across the board:
The result should always be simplified whenever possible. When we explained how to simplify fractions, we saw that two methods can be used. When, as in this case, the numbers are relatively high, it is advisable to use the second method, which consists of breaking down the numerator and denominator into factors and then annulling the factors that are repeated up and down.
Let’s see how:
First we break down the numerator:
We’re still breaking down the denominator:
Finally, we write each number as the product of factors and cancel those that are repeated up and down, leaving the final result:
Let’s increase the difficulty a little more. But calm down. You’ll see as if you are solving step by step, each step becomes a little easier:
This time, we can perform more than one operation in the same step, since they do not depend on each other. So we start by multiplying the numerator and dividing the denominator:
Now in the numerator we have 3 fractions left to add and subtract with different denominator. We transform them into a common denominator. To do this, we remember that we need to obtain the lowest common multiple of denominators, in order to obtain their equivalent fractions:
Once all the operations in the numerator and denominator have been carried out, all that remains is to divide the final fractions and simplify the result:
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585171.16/warc/CC-MAIN-20211017082600-20211017112600-00390.warc.gz
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CC-MAIN-2021-43
| 3,057 | 24 |
https://www.nagwa.com/en/lessons/803170630619/
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math
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A seagull flies at a velocity of 8.00 m/s straight into the wind. It takes the bird 24.0 min to travel 5.00 km relative to Earth.
What is the speed of the wind?
If the bird turns around and flies with the wind, how long, in minutes, will it take to return 5.00 km?
A ship sets sail from Rotterdam, The Netherlands, heading due north at 8.00 m/s relative to the water. The local ocean current is 2.50 m/s in a direction north of east. What is the speed of the ship relative to Earth?
A ship sailing in the Gulf Stream is heading west of north at a speed of 4.00 m/s relative to the water. Its velocity relative to Earth is 4.80 m/s, west of north. What is the velocity of the Gulf Stream? (The velocity obtained is typical for the Gulf Stream, a few hundred kilometers off the east coast of the United States.)
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s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583658844.27/warc/CC-MAIN-20190117062012-20190117084012-00576.warc.gz
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CC-MAIN-2019-04
| 809 | 5 |
http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=141&doi=10.11648/j.pamj.20190804.11
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math
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Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions
Pure and Applied Mathematics Journal
Volume 8, Issue 4, August 2019, Pages: 72-76
Received: Jul. 25, 2019;
Accepted: Aug. 15, 2019;
Published: Sep. 3, 2019
Views 388 Downloads 153
Maydim Malkov, Department of Mathematics, Russian Research Center for Artificial Intelligence, Moscow, Russia
Real numbers are divided into fictitious (non-computable) and essential (computable). Fictitious numbers do not have numerical values, essential numbers have algorithms for constructing these numbers with any exactness. The set of fictitious numbers is continual, the set of essential numbers is countable. Functions are also divided into fictitious, defined over the set of fictitious numbers, and essential, defined over the set of essential numbers. Essential functions have an algorithm for calculating any value with any exactness. All functions of applied mathematics and some functions of abstract mathematics are essential The set these functions is countable. The four upper levels of classification of real functions are constructed. This classification uses superpositions of functions and diagonal sets borrowed from the algebra of finite-valued functions.
Algebra of Real Functions: Classification of Functions, Fictitious and Essential Functions, Pure and Applied Mathematics Journal.
Vol. 8, No. 4,
2019, pp. 72-76.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
E. L. Post, Two-valued iterative systems of mathematical logic, Princeton, Princeton Univ. Press (1941).
M. A. Malkov, Logic algebra and Post algebra (theory of two-valued functions) (Russian), Moscow, Mathematical logic (2012).
D. Lau, Functions algebra on finite sets, Berlin, Springer (2006).
A. I. Malcev, I. A. Malcev, Iterative Post algebras, Moscow, Nauka (2012).
Ju. I. Janov, A. A. Muchnik, On existence of k-valued closed classes without finite basis (Russian), Dokl. Acad. Nauk SSSR, (1), 44-46 (1959).
M. A. Malkov, Classification of closed sets of functions in multi-valued logic, SOP Transactions on applied Math., (1: 3), 96-105 (2014).
A. V. Kuznetsov, On means for detecting of non-deductibility and inexpressibleness (Russian), in Logical conclusion, Moscow, Nauka, 5-33 (1979).
S. S. Marchenkov, On FE-precomplete classes of countable logic (Russian), Discrete Mathematics, (28: 2), 51-57 (2016).
S. V. Yablonsky, Functional constructions in k-valued logic (Russian), Proceedings of Mat. Institute of the USSR Academy of Sciences. V. A. Steklova, (51) 5-142 (1958).
I. G. Rosenberg, Über die functionale vollständigkeit in dem mehrvertigen logiken von mehreren verändlichen auf endlichen mengen, Rozpravy Cs. Academic Ved. Ser. Math. Nat. Sci., (80) 3-93 (1970).
M. Malkov, Algebra of finite-valued functions: Classification of functions and subalgebras, essential and fictitious subalgebras, Pure and Applied Math. J., (8: 2) 30-36 (2019).
G. Rousseau, Completeness in finite algebras with a single operation, Proc. Amer. Math. Soc., (18), 1009-1013.
P. Schofield, Independent conditions for completeness of finite algebras, J. London Math. Soc., (44) 413-423 (1969).
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http://mrmurrayphysics.blogspot.com/2011/02/5ch-ray-diagrams.html
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math
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Friday, 25 February 2011
5CH - Ray diagrams
Applied the refraction formula to drawing/completing ray diagrams today. You can use the refractive index formula along with the knowledge that the angles in a triangle add up to 180 degrees to figure out how light will pass through most objects.
Sometimes we will have to consider the critical angle when completing ray diagrams. If the angle of incidence is greater than the critical angle then the light will reflect. The angle of reflection in these cases is equal to the angle of incidence.
Note: The more observant will spot my mistake with the angles. They should be 54 and 36 not 34 and 56. Final answer is still correct.
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CC-MAIN-2018-39
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https://www.assignmentexpert.com/homework-answers/management/question-12141
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math
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3. Riverside Bank offers to lend you $50,000 at a nominal rate of 6.5%, compounded monthly. The loan (principal plus interest) must be repaid at the end of the year. Midwest Bank also offers to lend you the $50,000, but it will charge an annual rate of 7.0%, with no interest due until the end of the year. How much higher or lower is the effective annual rate charged by Midwest versus the rate charged by Riverside?
Correct answer is “D” 0.30%. Midwest: 50,000 x 1.07 = 53,500 USD. (Annual r=0.07) Riverside: 50,000*[(1 + 0,065\12)]^12 =53,348 USD. (Annual r = 0.0669)
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https://www.thestudentroom.co.uk/showthread.php?t=2758373
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math
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Mature Students GCSEWatch
I have managed to successfully retake GCSE English and GCSE Science. However, I have taken and retaken my GCSE Maths over number years unsuccessfully. I decided to have another go at the GCSE maths because I was doing the Access course and I had two conditional offers form Bournemouth University and Sparsholt College. I know that I have at least fulfilled part of the conditional offers. I have passed my Access course with Merit. However, I have now reached the grey area of what will happen after I receive my GCSE maths result.
I know that the first AQA GCSE Maths paper felt horrendous and it went very badly but the second maths paper felt a lot better.
I have decided that I will wait and see what my GCSE Maths result is like in August and then take it from there. If I am rejected because of my GCSE Maths result I am willing to take my GCSE maths again or try level 2 Maths instead.
Just for fun: I will answer one of the questions from the GCSE section:
Will you cry?
If by some miracle I get a C in GCSE maths this year I will scream and cry before throwing a party.
Does the University course you wish to do require maths? I did much worse than you when I did my maths GCSE many moons ago but I found that it wasn't required for the course I wanted to do (Law).
I have been told different things from different people. The email said that I do not need to do the maths if it is not mentioned in the requirements. I looked at the website and it did not mention anything about maths at all. I then went to an open day and I was told that I should try and get my GCSE maths but there is a level 2 course that could be taken there. However, when I looked at my conditional offer it said that I need a C in GCSE maths.
The answers of the exam were posted on here the next day and based on them I did rubbish but im hoping the grade boundaries will be lower than last year as the exam was definitely harder.
Not to worry though, if we haven't scraped a C there are retakes in November. Good Luck!
I guess that it is a waiting game, as UCAS will not updated for at least 23 hours. I have a couple of contingency plans if I am rejected. I will be retaking my GCSE maths again this year regardless of what happens. I am getting so close to achieving my aim of a C in GCSE maths that I would be a fool to give up now.
I am really excited to say that I have been ACCEPTED!
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http://perimeterinstitute.ca/fr/videos/quantum-gravity-one-loop-super-yang-mills-theory
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math
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It has been conjectured that maximally supersymmetric SU(N) Yang-Mills theory is dual to a String Theory on asymptotically AdS_5 times S^5 backgrounds. This is known as the AdS/CFT correspondence. In this talk I will show how using one-loop calculations in the gauge theory, one can study the emergence of the dual String Theory. We will see, quite explicitly, the emergence of closed strings, D-branes, open strings and space-time itself. This is done in a reduced sector (SU(2) sector), where the gauge theory can be written as Matrix Quantum Mechanics. This simple sector provides a toy model of a non-perturbative quantum theory of gravity.
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http://openstudy.com/updates/4f78e2ece4b0ddcbb89e77f9
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math
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
17. Have you learned that the products of the chord segments are equal?
Yes, it's just finding x that I'm having trouble with.
What equation did you write?
I will help you solve it.
I haven't written an equation yet. I really have a hard time with equations, I'm not very good with them :/
Do you see the two chords in your picture?
2x+3 and 4 are a chord, and 10 and x are the other chords, right?
Yes. So far so good. Do you know what a product is?
I don't think I do
A product is the answer to a multiplication problem. 4 times 3 equals 12. 12 is the product.
What is the product of 2x+3 and 4?
That's what I'm having trouble with, I can't figure that out :/
4(2x+3) have you heard of the distributive property?
It's been a while, that's why I'm pretty rusty with this stuff!
Distribute the 4 to the 2x and to the 3 What is 4 times 2x?
4 times 2x is 8x. What is 4 times 3?
Now the other chord. the two parts are 10 and x. what is 10 times x?
That is where I get lost, because I honestly have no idea :/
4(2x + 3) = 10 x 8x + 12 = 10x 12 = 2x 6 = x
And, the second question" Each of the measures (5,5,4) from the scale drawing will be multiplied by a factor of 6. That gives the square base the dimensions 5 by 5 and the pyramid a height of 24. V = 1/3 Bh where B is the area of the square base. V = 1/3 (30 * 30) 24 V = 7200 cubic feet
Directrix is right. I was about to simplify. :)
Thank you guys SO much for taking the time to explain everything to me, I really appreciate it!
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https://mathsofplanetearth.org.au/puzzle-1/
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math
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Published on February 15th, 2013 | by Simi0
Puzzle Challenge 1
Roll up students – these puzzles are just for you! Upload your answers to all four puzzles for a chance to win a signed Keith Devlin book. Correct submissions will be ordered and numbered by submission date. The winner will be selected at random using the excel random number function. Submissions close March 31st
Puzzle courtesy of Brian Davey, Department of Mathematics and Statistics, La Trobe University. Puzzles are given weekly to first-year Mathematics and Statistics students at La Trobe.
#1: BERYL’S QUILT
Beryl has made the incomplete quilt of 8 squares shown above, and she doesn’t have any material left for the missing square.
Can Beryl make a square quilt using all the material shown, by making only two straight cuts?
Beryl may not fold the material before making a cut, nor may she cut two separate pieces of material with one cut.
#2: EGG TIMER
You are given two egg timers that measure
3 minutes and 5 minutes, respectively.
• Using these two egg timers, you want to take a shower that lasts exactly 4 minutes.
How can this be done?
• What is the least amount of time required to take your 4-minute shower, measured from the moment you start one of the timers?
• Show that it is possible, using these two egg timers, to take an n-minute shower, for all n ϵ N.
• For each n �� N, what is the least amount of time required to take an n-minute shower, measured from the moment you start one of the timers?
#3: THE SWENSENS’ BLOCK OF LAND
They’ve asked Caitlin the architect to divide the block of land, using the grid lines shown, into four identically shaped plots. The problem is that each brother would like one tree and one pond on his individual plot.
Can she do it? How?
At the start of the party, some handshaking took place. Of course, no one shook hands with themselves. Also, no one shook hands with their own partner, and no one shook hands with the same person more than once.
Just before the dinner started, I asked everyone else at the party (including my partner) how many people they had shaken hands with. To my surprise, everyone gave a different answer.
How many people did my partner shake hands with?
[gravityform id=”10″ name=”Puzzle of the Month – Submit your answers”][subscribe2]
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http://www.hancockonline.net/Han/H-I-2564.html
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math
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|Name} Anderson, Anna Sofia||Family History} Hancock|
|Birth: Date} Fam 25 Jul 1914||Place} , , Rhode Island|
|Marr.: Date} Cal 1945||Place} (Link)|
|Death: Date} Exa 6 Jan 1981||Place} , Santa Cruz, California|
|Burial: Date} Cir 9 Jan 1981||Place}|
|Parents: } Not traced|
Relationship No.} None
|1st Household No.}|
| Occupation 1} |
Keagle, George Robert (Bob)|
Total Number of} 1
|Notes: Anna married Bob circa 1945 — when she was 31 years old and he was 29.|
In 1954 her family lived in San Diego while Bob worked there for the Navy.
By 1966 Anna and Bob had moved to Sunnyvale, California.
In 1981 Anna was 66 and Bob was 64 when she passed on. Their marriage had lasted for 35 or more years.
|Time of Birth}||Time of Death}||Fraternal/Social}|
|Confirm. Date}||Photo} None|
|Immigr'n Date} N/A||Port} N/A|
|Education: Grade} or Top 2 Degrees}|
Cause of Death}
|Copyright © 2015 by Daniel W. Hancock. All Rights Reserved.|
|Home Page||Next Page|
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| 955 | 22 |
https://www.physicsforums.com/threads/exponential-function.328279/
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math
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1. The problem statement, all variables and given/known data This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function 2. Relevant equations e^x=1+x+(x^2)/2!+(x^3)/3!+........ 3. The attempt at a solution then how about the e^(-x)=? Besides what is the function of sin x and cos x in continued function (such in e^x)? Thanks!
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CC-MAIN-2018-22
| 424 | 1 |
https://experts.umn.edu/en/publications/robust-controller-design-mixed-hsup2sup-performance-optimization-
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math
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The feedback controller design for linear time-invariant discrete-time systems via minimizing the H2-norm of a mixed sensitivity criterion is revisited. By borrowing some of the techniques from signal/image processing, a new approach is presented to tackle the H2 control problem. Operating in the Discrete Fourier Transform (DFT) domain, we construct a minimization problem in the l2-space to approximate the original H2 problem. The approximation in such a setting is sufficient for a reasonably small number of DFT-point chosen due to the stability and short-duration characteristics of the matrix elements involved in the design problem. Via the partially block circular structure of the matrices involved in the DFT domain, the l2 vector-optimization problem can be efficiently solved through matrix algebraic techniques.
|Number of pages
|Proceedings of the American Control Conference
|Published - Jan 1 1995
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| 915 | 4 |
https://houkagokappa.wordpress.com/2018/03/24/teaser-pv-analysis-maru-batsu/
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math
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You know what I just realised/remembered? Maru/a circle stands for being correct. At the start of each teaser we get a full circle splitting into four, can we derive something out of this?
Also this image contains both a maru ( O ) and a batsu ( X ), so something true and something false. Something right and something wrong.
The X is present only in this image, not during the beginning of each trailer or for the third teaser symbol. This image also has a very heavy feel, the other person is twisted and broken (or they both are). Are they being punished for doing something forbidden or are they a warning?
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CC-MAIN-2020-29
| 611 | 3 |
https://getassist.net/integration-rules-explained/
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math
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Integration is a summation operation that may be used as a mathematical tool for determining the area with functions of a single variable, computing the surface area and volume of three-dimensional solids, calculating the area and volume of a function with two variables, or summing multidimensional functions.
Real-world quantities such as temperature, magnetic field strength, pressure, speed, flow rate, lighting, share prices, and so on can be defined mathematically in science, engineering, and economics. We can use integration to combine these variables to get a total result.
In this post, we will explain the rules of integration with comprehensive examples and their solutions. This post is intended for the students of calculus and it will help you to understand the concept of integrals along with the rules.
Integration Rules – The Need
When it comes to integrating functions or taking antiderivatives, the same basic rules apply as they do for differentiation. This antiderivative calculator uses all of the rules while calculating antiderivatives. You should be familiar with the notion that integration and differentiation are the opposites of one another.
We can always distinguish the result to go back to the original function if we integrate a function. However, this is not the case. Because the derivative of any constant term is zero, any constant term in a function usually disappears when it is differentiated.
It is something we should bear in mind while considering how to integrate a function because it implies that our solution will always contain a constant with an unknown value. This constant is known as the constant of integration, C.
The most important rule of integration is the power rule of integration. This method is effectively the reverse of the power rule used in derivatives, and it yields the indefinite integral of a variable raised to a certain power. To refresh your memory below is the integration power rule formula:
The indefinite integral of the variable x raised to the power of n multiplied by the constant-coefficient a is given by this formula. Also bear in mind that n cannot be equal to -1, because, on the right-hand side of the formula, this would put a 0 in the denominator. This criterion alone allows us to integrate polynomial functions using a single variable.
We just integrate each expression independently, with no modification to the plus or minus sign in front of each word. Some typical indefinite integrals are listed below. It’s worth noting that in these instances, a stands for a constant, x stands for a variable, and e stands for Euler’s number which is approximately 2.7183. It’s also worth noting that the first three instances are the result of applying the power rule.
A constant value a:
∫ a dx = ax + C
A variable x:
The square of a variable x 2:
The reciprocal of a variable 1/x:
The exponential function e x:
Other exponential functions a x:
The natural logarithm of a variable ln (x):
The sine of a variable sin (x):
The cosine of a variable cos (x):
∫ cos (x) dx = sin (x) + C
The power, constant-coefficient or constant multiplier, sum, and difference rules are among the basic integration rules that will be discussed here. We’ll give some easy examples to show how these integration principles and laws actually work. Before moving onto the next section, check out this Integral calculator with steps. It can help you to find the indefinite integral of any given function.
The Power Rule
As we have seen, the power rule for integration is the inverse of the power rule for differentiation. It returns the indefinite integral of a variable multiplied by a power. Here’s the power rule again:
Let’s look at some examples of how this rule is used. Assume we wish to calculate the indefinite integral of x3. Using the power rule:
It is not always evident that we can apply the power rule to get the indefinite integral of a function. Assume we wish to calculate the indefinite integral of the equation 3√x. How can we apply the power rule to the cubed root function? It’s actually fairly simple. All we have to do is change the expression to get x to a power. To express the nth root of a number in an index form, there is a common formula which can be stated as:
n√a = a 1/n
Applying this formula to 3√x:
3√x = x 1/3
We can now apply the power rule to get:
The Constant Coefficient Rule
The constant-coefficient rule is also known as the constant multiplier rule. It states that the indefinite integral of c∙f(x), where c represents a constant coefficient and f(x) is some function, is equivalent to the indefinite integral of f(x) multiplied by c. This can be stated as follows:
The constant-coefficient rule allows us to disregard the constant-coefficient in an equation while integrating the remainder of it. Let’s say we wish to compute the indefinite integral of the expression 3x2. According to the constant-coefficient rule, the indefinite integral of this equation is the indefinite integral of x2 multiplied by 3. That is to say:
Now we just apply the power rule to x 2:
The Sum Rule
The sum rule describes how to integrate functions that are the sum of many terms. It simply shows us that we must integrate each expression independently in the total, before adding the results together. It is unimportant which order the terms appear in the outcome. This can be stated as follows:
You may be asking why the regulation is worded the way it is at this point. It is critical to understand that in a function that is the sum of two or more components, each term may be thought of as a function in its own right – even a constant term. Assume we wish to calculate the indefinite integral of a function (x) = 3x2 + 4x + 12. Using the sum rule:
The Difference Rule
The difference rule instructs us on how to integrate functions that include the difference of two or more terms. It is similar to the sum rule in that it instructs us to integrate each term in the sum independently. The only distinction is that the order of the expressions is important and cannot be modified. This rule can be stated explicitly as follows:
Let’s have a look at an example. Assume we wish to calculate the indefinite integral of the polynomial function (x) = 5x3 – 9x – 2. Using the sum rule, we obtain:
The difference and sum rules are fundamentally the same rules. If we wish to integrate a function that comprises both the sum and difference of a number of terms, we must remember to integrate each term independently and to keep the order of the terms in mind. The “+” or “=” sign in front of each expression remains the same.
You may also conceive the function as the sum of a number of positive and negative terms and apply the sum rule. The order is thus irrelevant; you just need to be aware of the sign of each expression. Below, we have listed few more examples for further interpretation of integration rules.
Evaluate ∫ 7 dx
∫ 7 dx =
7 ∫ dx ……….multiplication by a constant rule
= 7x + C
What is ∫ 5x4 dx
∫ 5x4 dx = 5 ∫x4 dx ……. using multiplication by a constant rule
= 5(x5/5) + C ………. using power rule
= x5 + C
Evaluate ∫ (2x3 + cos(x) ) dx
∫ (2x3 + 6cos(x) ) dx = ∫ 2x3 dx + ∫ 6cos(x) dx …..Applying the sum rule
= 2 ∫ x3 dx + 6 ∫ cos(x) dx ……….Applying the multiplication by a constant rule
= 2(x4/4) + C1 + 6(sin(x) + C2 …..Applying the power rule. C1 and C2 are constants.
C1 and C2 can be replaced by a single constant C, so:
∫ (2x3 + cos(x) ) dx = x4/2 + 6sin(x) + C
All of the listed rules are extensively used in integration and are vital for the evaluation of integrals. These rules should be practiced and implemented on several types of functions if you want to master the concept of integrals. Refer to these principles listed above if you are stuck somewhere while calculating antiderivative or integral.
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CC-MAIN-2023-23
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https://www.jiskha.com/display.cgi?id=1354083941
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math
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posted by Remmy .
1) A corner lot that originally was square lost 185m^2 of area when one of the adjacent streets was widened by 3 m and the other was widened by 5 m. Find the new dimensions of the lot. (Hint: Let x = the lenght of a side of the original square lot. )
2) A running track 4 m wide goes around a soccer field that is twice as long as it is wide. At each end of the soccer field the track is a semi-circle with inner radius r. Find a formula for the area of the track in terms of *pi and r.
3a) Suppose that you plan to run once around the track described in the problem above. If you stay 0.5 m from the inner edge of the track, how far will you run? (Hint: The circumference of a circle is *2pir. Your answer will be in terms of *pi and r.)
3b) Suppose that a friend stays 0.5 m from the outer edge of the track. How much farther does your friend run than you do?
*pi = 3.14 or 22/7
original square ---- > let each side be x
new sides are x-3 and x-5
so x^2 - (x-3)(x-5) = 185
x^2 - (x^2 - 8x + 15) = 185
8x - 15 = 185
x = 200/8 = 25
original square was 25 by 25 , area = 625
new rectangle was 22 by 20 , area = 440
difference = 625-440 = 185
All looks good
2. I will let you do this one.
Hint: let the width of the soccer field be 2r, that way you can label the radius of the ends as r
The two semi-circular ends will constitute one whole circle. So the circular part of the track will be the "ring" formed by two circles, one of radius r, the other of radius (r-4)
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| 1,482 | 19 |
https://www.physicsforums.com/threads/how-many-seconds-will-it-take-before-the-cars-meet.830451/
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math
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Two cars are initially 17.50 km apart on a straight road. If the cars are moving toward each other, car 1 with a speed of 7.50 m/s and car 2 with a speed of 10.40 m/s, how many seconds will it take before the cars meet? Round your answer to three significant figures.
The Attempt at a Solution
Converted to 17,500 m for the distance apart.
I added 7.50 and 10.40; Then I divided 17,500 by 17.9 seconds
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CC-MAIN-2021-31
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https://www.mathkplus.com/I-Math/Geometry/Volume/Sphere.aspx
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math
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How To Find The Volume Of A Sphere
Let's look at the geometric characteristics of a
sphere. But first, please review the definition of Volume Of Three-Dimensional Shapes,
and Cubes And Perfect Cubes.
- A sphere has a very unique feature, all points on the surface
of the Sphere are equal distance from the center of the Sphere; or said another
way, the distance from the center of a Sphere to the surface is equal to the radius.
- Special note, the radius
of a Sphere must be greater than zero.
Volume Of A Sphere Formula
Formula = 4/3 × Pi × Radius3 =
Formula = 4/3 × π × r3
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CC-MAIN-2021-43
| 580 | 12 |
https://www.protoexpress.com/blog/controlled-impedance-quiz-2/
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math
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Controlled impedance is not science fiction… Especially when you know how to calculate your controlled impedance traces. But, do you? Open our free Impedance Calculator – which will help you answer some of the following questions – and get ready to take the quiz!
Congratulations, it looks like you know a lot about controlled impedance! But maybe you can still learn a few things by downloading our ControlledImpedance Design Guide?
Ouch, it looks like you still have a lot to learn about controlled impedance. Download our Design Guide now and become a master in controlled impedance!
#1. In any stack-up, what does the dielectric constant of a PCB material depend on?
#2. Which material has a dielectric constant value of 4?
#3. What is the differential impedance for an Ethernet interface?
#4. What will be the single-ended impedance of a 5-mil wide and 1.7-mil thick trace that has a dielectric height of 4 mils and a dielectric constant of 4.15? ? Clue: Impedance Calculator at https://www.protoexpress.com/user/registerHdi.jsp
#5. What should be the trace width of a 1.4-mil thick single-ended stripline which has 4-mil dielectric heights, dielectric constants of 4.15, and a 45-ohm impedance? ? Clue: Impedance Calculator at https://www.protoexpress.com/user/registerHdi.jsp
#6. What is the single-ended impedance for a USB 2.0 signals interface?
#7. What should be the trace width of a 1.7-mil thick single-ended microstrip which has a 3.8-mil dielectric height, a 57-ohm impedance, and use NP175 material? ? Clue: Impedance Calculator at https://www.protoexpress.com/user/registerHdi.jsp
#8. What is the differential impedance for an analogue VGA interface?
#9. What material should you not use if you want a dielectric constant of 3.5?
DOWNLOAD OUR CONTROLLED IMPEDANCE DESIGN GUIDE:
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585290.83/warc/CC-MAIN-20211019233130-20211020023130-00371.warc.gz
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CC-MAIN-2021-43
| 1,800 | 13 |
http://heli-air.net/2016/02/19/effects-of-the-mach-number/
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math
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Effects of the Mach number
For a probe perfectly aligned with the stream, the reading is independent of the Mach number up to Mach numbers close to 1 (Figure 2.10). At supersonic speeds in front of the tube, a detached shock wave is generated, which is locally normal to the axis of the tube, so that the pressure detected by the Pitot tube is the stagnation pressure downstream of a
normal shock wave. The measured pressure (subscript 2) can be used to calculate the Mach number of the stream (M1 > 1), if the stagnation pressure upstream of the shock wave (subscript 1) is known, through the equation, known as the Rayleigh formula:
(Y – 1)Mi + 2 Y-1 ( 2y M2 – Y-1 Y-1
(y + 1)Mj2 lr +1 1 Y + 1J
The stagnation pressure upstream of the shock wave must be measured independently, as the pressure in the stagnation chamber that feeds the de Laval nozzle that generated the supersonic stream.
The Mach number can also be calculated, if the static pressure upstream
Effects of the Mach number on the readings of a Pitot tube with a hemispherical head (d/D = 0.3)
of the shock wave is known, by Equation (2.6) obtained by dividing Equation (2.5) by Equation (2.1):
___ 2___ Y-1 (JY_ M2 _ Y-1Y-1
(y + 1)M Iy + 1 Y + 1)
The static pressure upstream of the shock wave can be measured on a wall at the entrance of the test chamber.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655900335.76/warc/CC-MAIN-20200709131554-20200709161554-00131.warc.gz
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CC-MAIN-2020-29
| 1,327 | 12 |
http://forums.wolfram.com/mathgroup/archive/2005/Dec/msg00101.html
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math
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[Date Index] [Thread Index] [Author Index]
How to set up a diff equation for circuit with a diode?
I'm kind of curious how to set up a nonlinear differential equation and then solve it using NDSolve in case of the following setup: Battery V in series with a diode, R and L. To simplify, we may assume that diode is a pure conducting device in forward direction and pure blocking device in the reverse direction. Now how to set up a differential equation for the current I(t), that is solvable using NDSolve? Thanks Mike
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s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189583.91/warc/CC-MAIN-20170322212949-00438-ip-10-233-31-227.ec2.internal.warc.gz
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CC-MAIN-2017-13
| 519 | 3 |
https://vustudents.ning.com/forum/topics/my-today-paper-of-mgt411
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math
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1. Aggregate demand
• If increase government purchases
• If increase net export
2. Velocity in long and short run?
3. Rule of 72 asked and with numerical like if Rs.100,000 and interest rate is 14%.(apply rule 72)
4. One question asked about probilities.
5. Explain stable and unstable economy with reference to Pakistan.
6. And one question asked about curve shifting to right ward and leftward what will impact on supply curve?
Please friends remember me in your precious prayers…needed Rabi shah
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s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057787.63/warc/CC-MAIN-20210925232725-20210926022725-00174.warc.gz
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CC-MAIN-2021-39
| 504 | 9 |
http://www.zenguide.com/forum/view.cfm?topicid=9082&archived=true
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math
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Quote: "Or easier yet ask yourself "Am I straight or confused?"
Easy as that. If you're the latter you better sit your butt down.
" .........It seems I keep alternating between the former and the latter. Is it that I got from straight to confused due to my ignorance (falsely believing I am straight). Or because of noise (I am straight, but external forcings falsely make me confused). It is a conundrum.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280791.35/warc/CC-MAIN-20170116095120-00060-ip-10-171-10-70.ec2.internal.warc.gz
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CC-MAIN-2017-04
| 405 | 3 |
http://dommelen.net/quantum2/style_a/nt_taylor.html
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math
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The probability of being found near the nucleus, i.e. the origin, is determined by the magnitude of the relevant hydrogen wave function near the origin. Now the power series expansion of in terms of the distance from the origin starts with power , (D.8). For small enough , a p, (i.e. ), state involving a factor will be much smaller than an s, (), state without such a factor. Similarly a d, (), state involving a factor will be much less still than a p state with just single factor , etcetera. So states of higher angular momentum quantum number stay increasingly strongly out of the immediate vicinity of the nucleus. This reflects in increased energy since the nuclear attraction is much greater close the nucleus than elsewhere in the presence of shielding.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934809746.91/warc/CC-MAIN-20171125090503-20171125110503-00637.warc.gz
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CC-MAIN-2017-47
| 763 | 1 |
https://nrich.maths.org/less-is-more/solution
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math
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Or search by topic
Thank you to everybody who sent us their thoughts about this game. We received a couple of solutions for Version 1 and a lot of solutions for Version 2!
Ilan from Twyford School in the UK considered which strategy worked best with a 1-6 dice:
The key to solving ‘less is more’ is to take a risk and put a large number on the smaller side. Here is an example of the perfect round to win:
65 < 66
65 < 66
This is because the left has to be smaller and it is quite unlikely to roll large numbers so this is quite a risky way of doing it.
This is a less risky round:
45 < 48
39 < 49
The risk to doing the first method is that if you have already put sixes in the first column there is a possibility that you will not get the number you want, in this case sixes and it will not work and you will be deducted a lot of points. In conclusion, I think that going in the middle (not too safe not too risky) so I think the best numbers to put on the right would be 51 or 56 or something along those lines.
Dhruv from St. Anne's RC Primary School in the UK also used a 1-6 dice, and they sent in a picture to explain their method. You can click on the picture to see the full-size solution:
Good ideas, Ilan and Dhruv! It looks like you're both hoping to roll 6s or 5s. I wonder why Dhruv thinks it is best to fill in the tens spaces on the right-hand side first?
We received a lot of solutions from pupils at Halstead Prep School in England, explaining why the highest possible score is 127. Leticia sent in this explanation:
I wrote out out all the numbers from 1 to 8 and then picked 5, 6, 7 and 8. I then arranged them so that I had a 7 and a 5 on the left hand side. Then I put 8 and 6 on the right hand side. I then put the biggest two of the remaining numbers on the left hand side to get 74 and 53. Then I had 81 and 62 on the right hand side and that works. Then I added 74 and 53 and I got 127.
Christina agreed with Leticia's method of considering the largest digits first:
The highest score is 127. You should use 7 on the left and 8 on the right as it uses the less than sign. Then you fill in the units with the leftover lower digits. For the next calculation you would put 5 on the left and 6 on the right for the same reasons. You fill the units with the other numbers. This is how you get the highest score. For the units I put 3 and 4 on the left as they are the highest of the other numbers.
Amelie described a similar method:
The highest number possible is 127. The reason for this is:
You want the highest score on the left. So, use the largest numbers possible. 7 is less than 8 and 5 is less than 6. These are the largest tens digits possible. Next, we move on to the units digits. 1, 2, 3 and 4 remain. It doesn’t matter which numbers you use, as if the tens digit is larger, the number will be larger. To get the highest score possible, we will use 3 and 4. Our result is:
74 < 81
53 < 62
74 + 53 = 127
Our answer is 127.
Well done to all of you for writing out your thinking so clearly. Thank you as well to Lila, Lottie, Ananya, Viva, Charlotte, Evie, Madeline and Emily who all sent in excellent justifications for why 127 is the maximum possible score.
We also received some similar solutions from children at Bishop's Castle Primary School in England. Lewis and Ryan sent in this explanation:
I put the highest number (8) in the top right tens column and the second highest number (7) in the opposite tens column because I knew I needed the largest total on the left hand side. The next highest number (6) I put in the tens column on the right hand side in the row below and the next highest number (5) in the tens on the left hand side. I took a similar approach with the position of the ones. This gives a top score of 127.
Lucy and Fynn had a similar strategy:
The 7 is smaller than 8 therefore you can use them on the same row in the tens. This strategy works for 6 and 5 as well. You can put 4 and 3 next to the 7 and 5 either way round in the units to make the largest possible correct numbers on the left hand side. The same strategy works for 1 and 2 on the right hand side. The highest score with digits 1-8 was 127.
Sid from Twyford School explained how they solved this problem by starting with the highest digits:
If you have the numbers 1,2,3,4,5,6,7 and 8, the way to get the best score would be looking at the highest numbers and going down to the lowest numbers. First you look at the eight, the eight cannot be in the first column in the left section as nothing else is bigger than eight. What you could then do is put 8 in the first column in the right section and seven in the first left section. That then means you now have the numbers 1,2,3,4,5 and 6. Like the 8 you can't put the 6 in the first left section so you would want to put it on the right first section under the 8, then you can have the number 5 in the left first one under the 7, you can then put 4 and 3 in the last places on the left and 2 and 1 on the right then your total would be 127.
Dhruv sent in this picture explaining their method:
Thank you all for sending in these solutions!
Follow the clues to find the mystery number.
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817081.52/warc/CC-MAIN-20240416093441-20240416123441-00458.warc.gz
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CC-MAIN-2024-18
| 5,440 | 36 |
https://link.springer.com/chapter/10.1007/978-1-4757-1920-8_3
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math
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In this chapter we present the basic facts about algebraic curves (i.e. projective varieties of dimension 1) which will be needed for our study of elliptic curves. (Actually, since elliptic curves are curves of genus 1, one of our tasks will be to define the genus of a curve.) As in Chapter I, we give references for those proofs which are not included. There are many books where the reader can find more material on the subject of algebraic curves, for example [Har, Ch. IV], [Sha 2], [G–H, Ch. 2], [Wa].
KeywordsSmooth Curve Elliptic Curf Function Field Algebraic Curf Smooth Curf
Unable to display preview. Download preview PDF.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583659654.11/warc/CC-MAIN-20190118005216-20190118031216-00516.warc.gz
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CC-MAIN-2019-04
| 635 | 3 |
https://www.allinterview.com/company/1031/igate/aptitude-test-questions.html
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math
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1/(10 power 18) - 1/(10 power 20) = ?7 17190
Two trains at speed 60 km/hr comes in the opposite direction. At a particular time the distance between the two trains is 18km. A shuttle flies between the trains at the speed of 80 km/hr. At the time the two trains crashes what is the distance traveled by shuttle?4 7409
How many such pairs of letters are there in the word STAINLESS each of which has as many letters between them in the word as they have in the English alphabet, in the same sequence? (a) Two (b) Three (c) Four (d) Five (e) None of these29 111638
The ratio of the present ages of Suman and Renu is 5 : 7 respectively. Four years hence the ratio will become 3 : 4 respectively. What is the present age of Renu in years? (a) 28 (b) 24 (c) 20 (d) 21 (e) None of these5 16190
A plane moves from 9N60E to 9N60W. If the plane starts at 2 a.m and takes 10 hrs to reach the destination, find the local arrival time. a> 4.30a.m b>6.00 am c>4.00 am d>10.00 am If any one knows the method to solve the problem plz tell me4 10233
What is the ten letter city 7 8 9 is a famous festival 3 4 5 is a degree 2 1 is a abbreviation of group of countries 7 6 gives the meaning?11 13019
We have standard info objects given in sap why you created zinfo objects can u tell me the business scenario
While running DOS on a PC, which command would be used to duplicate the entire diskette?
What is the melting temperature of the cng long tube fusible plug? And can I use lead solder Sn97Cu3 (Melting temp S/L 227-309 °C) Materials of tube : AISI 4130 Alloy steel
What are datasets in abap?
WHAT IS THE MAIN DIFFERENCE B/W HIGHWAYS AND MOTERWAYS?
What is the benefit of native sql query support in hibernate?
What are Cursors in HANA Database?
How to de fragment your hard drive in windows xp?
What is the use of all pass filter?
Describe a opening balance?
What are the different components of typescript?
What are the varoius components of physical database structure of oracle database?
What purpose does it serve?
Give a brief description of db2 isolation levels?
How to Add TrueType fonts to windows in code?
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s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986693979.65/warc/CC-MAIN-20191019114429-20191019141929-00102.warc.gz
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CC-MAIN-2019-43
| 2,102 | 21 |
https://www.answers.com/Q/What_is_a_5_letter_word_for_island_group
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math
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What is a 5 letter word for island group?
How many 5 letters can be formed out of the letters of the word mathematics of repetition is not allowed?
There are eight (8) different letters in that word, so that's the number of letter choices you have in each 5-letter group without repetition. The number of different 5-letter "words" is (8 x 7 x 6 x 5 x 4) = 6,720 . (They don't necessarily mean anything. They're just distinct sequences of 5-letters each, like Morse-code random practice groups.)
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s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317113.27/warc/CC-MAIN-20190822110215-20190822132215-00149.warc.gz
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CC-MAIN-2019-35
| 495 | 3 |
https://brainmass.com/math/graphs-and-functions/equation-parabola-121946
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math
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Explain why it is not necessary to plot points to graph when using y = a(x - h)^2+ k.© BrainMass Inc. brainmass.com March 4, 2021, 7:46 pm ad1c9bdddf
y = a(x - h)^2+ k
implies that y - k = a(x - h)^2,
or Y = aX^2,
which is an equation of the parabola whose ...
A parabola is a conic section of eccentricty one. It is an open curve, meaning that it does not enclose an area. The equation of a parabola can yield significant clues about the shape and location of the parabola instead of plotting the parabola. This problem demonstrates this idea by means of a simple case.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487640324.35/warc/CC-MAIN-20210618165643-20210618195643-00002.warc.gz
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CC-MAIN-2021-25
| 571 | 6 |
http://math.korea.ac.kr/math/seminar/mathseminar.do?mode=view&articleNo=109963
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math
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20180914 수학과 Colloquium
1. 일시 : 2018.09.14. (금) 오후 04:30 ~ 05:30
2. 장소 : 아산이학관 526호
3. 연사 : 배명진 교수 (포항공과대학교 수학과)
4. 제목 : Transonic shocks of compressible flows
Compressible flow motion is governed by the Euler system which is a PDE system describing the conservations of mass, momentum and energy. Due to the nonlinear feature of compressible flow, a jump transition, such as a shock or a contact discontinuity, can occur depending on the geometry of a domain, or initial/boundary condition of a flow. In this talk, I will explain physical background of shock phenomena, and how to formulate a shock problem into a mathematical problem. Then, I will discuss about my recent results on a detached shock problem.
6. 문의 : 수학과 행정실 ([email protected])
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s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202523.0/warc/CC-MAIN-20190321112407-20190321134407-00104.warc.gz
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CC-MAIN-2019-13
| 832 | 7 |
https://www.cliffsnotes.com/cliffsnotes/subjects/math/how-do-you-convert-a-fraction-to-a-decimal-or-change-a-decimal-to-a-fraction
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math
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Fractions can be hard, but I use them almost every day; for example, if I eat 5/8ths of a whole pizza for dinner, how much will I have left over for lunch the next day? (My cooking skills aren't that hot.)
How do you convert a fraction to a decimal or change a decimal to a fraction?
Fractions can also appear in decimal form. These decimal fractions come in two styles: terminating decimals (for example, .3) and infinite repeating decimals (for example, .66 . . .).
To change a fraction to a decimal, simply do what the operation says. In other words, 5/8 means 5 divided by 8. Don't forget to insert the decimal points and zeros when you do the division!
Change 5/8 into a decimal:
So 5/8 = .625 (Now that's a lot of pizza!)
To change terminating decimals to fractions, remember that all numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1,000, 10,000, and so on. Next, use the technique of read it, write it, and reduce it.
(a) Change .8 to a fraction in lowest terms.
Read it: .8 (eight tenths)
Write it: 8/10
Reduce it: 4/5
(b) Change .09 to a fraction in lowest terms.
Read it: .09 (nine hundredths)
Write it: 9/100
Reduce it: 9/100 Can't be reduced
To change an infinite repeating decimal to a fraction, remember that every infinite repeating decimal can be expressed as a fraction. Infinite repeating decimals are usually represented by putting a line over (sometimes under) the shortest block of repeating decimals.
Find the fraction represented by the repeating decimal .7
Let n stand for .7 or .77777 . . .
So 10n stands for 7.7 or 7.77777 . . .
10n and n have the same fractional part, so their difference is an integer.
10n = 7.7 - n = .7 ---------- 9n = 7
Solve this problem as follows.
9n = 7 n = 7/9 .7 = 7/9
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233511055.59/warc/CC-MAIN-20231003060619-20231003090619-00198.warc.gz
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CC-MAIN-2023-40
| 1,766 | 23 |
https://rajboardexam.in/chapter-5-measures-of-central-tendency/
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math
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Chapter 5 Measures of Central Tendency
Which average would be suitable in the following cases?
(i) Average size of readymade garments.
(ii) Average intelligence of students in a class.
(iii) Average production in a factory per shift.
(iv) Average wages in an industrial concern.
(v) When the sum of absolute deviations from average is least.
(vi) When quantities of the variable are in ratios.
(vii) In case of open-ended frequency distribution.
(i) Mode Average size of any ready made garments should be the size for which demand is the maximum. Hence, the modal value which represents the value with the highest frequency should be taken as the average size to be produced.
(ii) Median It is the value that divides the series into two equal parts. Therefore, Median will be the best measure for calculating the average intelligence of students in a class as it will give the average intelligence such that there are equal number of students above and below this average. It will not be affected by extreme values.
(iii) Arithmetic Mean The average production in a factory per shift is best calculated by Arithmetic Mean as it will capture all types of fluctuations in production during the shifts.
(iv) Arithmetic Mean Arithmetic Mean will be the most suitable measure. It is calculated by dividing the sum of wages of all the workers by the total number of workers in the industrial concern. It gives a fair idea of average wage bill taking into account all the workers.
(v) Arithmetic Mean The algebraic sum of the deviations of values about Arithmetic Mean is zero. Hence, when the sum of absolute deviations from average is the least, then mean could be used to calculate the average.
(vi) Median Median will be the most suitable measure in case the variables are in ratios as it is least affected by the extreme values.
(vii) Median Median is the most suitable measure as it can be easily computed even in case of open ended frequency distribution and will not get affected by extreme values.
Indicate the most appropriate alternative from the multiple choices provided against each question.
(i) The most suitable average for qualitative measurement is
(a) Arithmetic mean
(d) Geometric mean
(e) None of these
(b) Median is the most suitable average for qualitative measurement because Median divides a series in two equal parts thus representing the average qualitative measure without being affected by extreme values.
(ii) Which average is affected most by the presence of extreme items?
(c) Arithmetic Mean
(d) Geometric Mean
(e) Harmonic Mean
(c) It is defined as the sum of the values of all observations divided by the number of observations and therefore it is. affected the most by extreme values.
(iii) The algebraic sum of deviation of a set of n values from AM is
(d) None of these
(b) This is one of the mathematical properties of arithmetic mean that the algebraic sum of deviation of a set of n values from AM is zero.
Comment whether the following statements are true or false.
(i) The sum of deviation of items from median is zero.
(ii) An average alone is not enough to compare series.
(iii) Arithmetic mean is a positional value.
(iv) Upper quartile is the lowest value of top 25% of items.
(v) Median is unduly affected by extreme observations.
This mathematical property applies to the arithmetic mean and not to median.
Average is not enough to compare the series as it does not explain the extent of deviation of different items from the central tendency and the difference in the frequency of values. These are measured by measures of dispersion and kurtosis.
Median is a positional value.
The upper quartile also called the third quartile, has 75 % of the items below it and 25 % of items above it.
Arithmetic mean is unduly affected by extreme observations.
If the arithmetic mean of the data given below is 28, find (a) the missing frequency and (b) the median of the series
(a) Let the missing frequency br f1.
Arithmetic Mean = 28
or 2240 -2100 = 35f1 = 28f1
or 140 = 7f1
f1 = 20
Hence, the missing frequency is 20.
So, the Median class = Size of ()th item = 50th term.
50th item lies in the 57th cumulative frequency and the corresponding class interval is 20-30.
The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
N = 10
Arithmetic Mean = ₹ 240
Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.
The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
So, the median class = Size of () th item = 190 item
190th lies in the 129 th cumulative frequency and the corresponding class interval is 200-300.
Median size of land holdings = 241.22 acres
The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers, (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.
(a) Highest income of lowest 50% workers will be given by the median. Σf = N = 65
Median class = Size of ()th item = Size of ()th item=325 th item
32.5th item lies in the 50th cumulative frequency and the corresponding class interval is 24.5 – 29.5.
(b) Minimum income earned by top 25% workers will be given by the lower quartile Q1.
Class interval of Q1 = ()th item
= ()th item = 1625th item
16.25th item lies in the 30th cumulative frequency and the corresponding class interval is 19.5 – 24.5
(c) Maximum income earned by lowest 25% workers will be given by the upper quartile Q3.
Class interval of Q3 = ()th item
= 3()th item
= 3 × 1625th item
= 48.75th item
48.75th item lines in 50th item and the corresponding class interval is 24.5-29.5.
The following table gives production yield in kg per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode production yield.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224644683.18/warc/CC-MAIN-20230529042138-20230529072138-00325.warc.gz
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CC-MAIN-2023-23
| 5,869 | 73 |
http://www.jstor.org/stable/2699780
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math
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Linearity of Dimension Functions for Semilinear G-Spheres
Proceedings of the American Mathematical Society
Vol. 130, No. 6 (Jun., 2002), pp. 1843-1850
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2699780
Page Count: 8
You can always find the topics here!Topics: Mathematical functions, Lie groups, Mathematical linearity, Mathematical theorems, Additivity
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In this paper, we show that the dimension function of every semilinear G-sphere is equal to that of a linear G-sphere for finite nilpotent groups G of order pnq m, where p, q are primes. We also show that there exists a semilinear G-sphere whose dimension function is not virtually linear for an arbitrary nonsolvable compact Lie group G.
Proceedings of the American Mathematical Society © 2002 American Mathematical Society
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https://www.analyzemath.com/calculus/Differentiation/logarithm_differentiation.html
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math
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Free Mathematics Tutorials
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Differentiation of Logarithmic Functions
Free Calculus Tutorials and Problems
Find Derivative of y = x^x
Rules of Differentiation of Functions in Calculus
Use the Chain Rule of Differentiation in Calculus
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| 281 | 8 |
https://www.findfilo.com/math-question-answers/the-total-revenue-in-rupees-received-from-the-saleysx
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math
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Application of Derivatives
The total revenue in Rupees received from the sale of x units of a product is given by R(x)=3x2+36x+5. The marginal revenue, when x=15 is.
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are angles satisfying 0<α<θ<β<2π˙
Then prove that
If the tangent at any point (4m2,8m2)
is a normal to the curve x3−y2=0
, then find the value of m˙
is a real number satisfying the equation 2t3−9t2+30−a=0,
then find the values of the parameter a
for which the equation x+x1=t
gives six real and distinct values of x
In an acute triangle ABC
if sides a,b
are constants and the base angles AandB
vary, then show that a2−b2sin2AdA=b2−a2sin2BdB
Does there exists line/lines which is/are tangent to the curve y=sinxat(x1,y1)
and normal to the curve at (x2,y2)?
are the sides of two squares such that y=x−x2
. Find the rate of the change of the area of the second square with respect to the first square.
Find the equation of tangent and normal to the curve x=(1+t2)2at2,y=(1+t2)2at3
at the point for which t=21˙
A lamp is 50ft˙
above the ground. A ball is dropped from the same height from a point 30ft˙
away from the light pole. If ball falls a distance s=16t2ft˙
second, then how fast is the shadow of the ball moving along the ground 21s
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https://austinattorney.mobi/bresenham-line-drawing-algorithm-with-example-61/
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math
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Bresenham’s line algorithm is an algorithm that determines the points of an n- dimensional raster that should be selected in order to form a close approximation . example, in which we wish to draw a line from (0,0) to (5,3) in device space. Bresenham’s algorithm begins with the point (0,0) and “illuminates” that pixel. Bresenham’s line drawing algorithm & Mid Point Circle algorithm. Example: 13 )2or(i.e(slope)gradientLet dxdy dx dy 3dy 2dy dy.
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Bresenham’s line algorithm
To derive the alternative method, define the difference to be as follows:. Since we know the column, xthe pixel’s row, yis given algogithm rounding this quantity to the nearest integer:. The label “Bresenham” is used today for a family of algorithms extending or modifying Bresenham’s original algorithm. This decision can be generalized by accumulating the error.
Programs in those days were freely exchanged among corporations so Calcomp Jim Newland and Calvin Hefte had copies. An extension to the original algorithm may be used for drawing circles. This alternative method allows for integer-only arithmetic, which is generally faster than using floating-point arithmetic.
Computer graphics algorithms Digital geometry. The adjacent image bresenhham the blue point 2,2 chosen to be on the line with two candidate points in green 3,2 and 3,3.
It is commonly used to draw line primitives in a bitmap image e. The plotting can be viewed by plotting at the intersection of lines blue circles or filling in pixel boxes yellow squares.
Bresenham’s Line Drawing Algorithm Example
Please help improve this article by adding citations to reliable sources. The beesenham point 3, 2. Regardless, the plotting is the same. In Bresenham wrote: The voxel heightmap software-rendering engines seen in some PC games also used this principle.
Articles needing drawinb references from August All articles needing additional references All articles with unsourced statements Articles with unsourced statements from September Articles with unsourced statements from December All Wikipedia articles needing clarification Wikipedia articles needing clarification from May Commons category link is on Wikidata Articles with example pseudocode.
The algorithm is used in hardware such as plotters and in the graphics chips of modern graphics cards.
It should be noted that everything about this form involves only integers if x and y are integers since the constants are necessarily integers. In the following pseudocode sample plot x,y plots the pixel centered at coordinates x,y and abs returns absolute value:. A line splits a plane into halves and the half-plane brsenham has a negative f x,y can be called the negative half-plane, and the other half can be called the positive half-plane.
The result of this plot is shown to the right. To answer this, evaluate the line function at the midpoint between these two points:. The Bresenham algorithm can be interpreted as slightly modified digital differential analyzer using 0.
All of the derivation for the algorithm is done. It was a year in which no proceedings were published, only the agenda of speakers and topics in an issue of Communications of the ACM. The general equation of the line through the endpoints is given by:. This page was last edited on 16 Octoberat While algorithms such as Wu’s algorithm wiht also frequently used in modern computer graphics because they can support antialiasingthe speed and simplicity of Bresenham’s line algorithm means that it is still important.
The first step is transforming the equation of a line from the typical slope-intercept form into something different; and then using this new bresenha, for a line to draw a line based on the idea of accumulation of error. A description of the line drawing routine was accepted for presentation at the ACM national convention in Denver, Colorado. If the error becomes greater than 0. Simplifying this expression algorith. Since all of this is about wlgorithm sign of the accumulated difference, then everything can be multiplied by 2 with no consequence.
Alternatively, the difference between points can be used instead of evaluating f x,y at midpoints. Notice that the points 2,1 and 2,3 are on opposite sides of the line and f x,y evaluates to positive or negative.
This exzmple a function of only x and it would be useful to make this equation written as a function of both x and y. It is an incremental error algorithm. In low level implementation which access the video memory directly it would be typical for the special cases of vertical and horizontal lines to be handled separately as they can be highly optimised.
It is one of the earliest algorithms developed in the field of computer graphics. To derive Bresenham’s algorithm, two steps must be taken.
By switching the x and y axis an implementation for positive or negative steep gradients can be written as. It can also be found in many software graphics libraries. The algorithm can breseham extended to cover gradients between 0 and -1 by checking whether y needs to increase or decrease i.
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https://physics-network.org/how-do-you-calculate-frequency-in-physics/
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math
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Frequency describes the number of waves that pass a fixed place in a given amount of time. So if the time it takes for a wave to pass is is 1/2 second, the frequency is 2 per second. If it takes 1/100 of an hour, the frequency is 100 per hour.
What is the formula for frequency?
Wave frequency is the number of waves that pass a fixed point in a given amount of time. The SI unit for wave frequency is the hertz (Hz), where 1 hertz equals 1 wave passing a fixed point in 1 second.
How do you calculate frequency from Hz?
The speed of light is measured to have the same value of c = 3×108 m/s no matter who measures it. Example: If you shoot a bullet forward from an airplane at a speed vb, an observer on the ground would measure its speed to be vb + va where va is the speed of the airplane.
What is an frequency in physics?
Hertz is a unit of frequency (of change in state or cycle in a sound wave, alternating current, or other cyclical waveform) of one cycle per second. It replaces the earlier term of “cycle per second (cps).”
What is the frequency of a wave?
Frequency Formula The SI unit which is hertz was named after Heinrich Rudolf. Furthermore, 1 hz refers to one cycle per second. Frequency = 1/period = number of cycles/time. f = 1/T = N/t.
What is the unit for frequency?
The SI unit for frequency is the hertz (Hz).
What does λ mean in physics?
Wavelength is usually denoted by the Greek letter lambda (λ); it is equal to the speed (v) of a wave train in a medium divided by its frequency (f): λ = v/f.
How do you find frequency when given wavelength?
What is frequency example?
It is not the English alphabet v, it is the Greek letter nu. It represents the frequency of the wave. Frequency is the number of vibrations per second.
What is C 3×10 8?
The wavelength is the distance between two wave crests, and it will be the same for troughs. The frequency is the number of vibrations that pass over a given spot in one second, and it is measured in cycles per second (Hz) (Hertz). The relation between wavelength and frequency is discussed in this article.
Why is frequency C wavelength?
What is a hertz frequency?
Frequency is the number of times a value occurs in a set of data. For example, Victor tried nine times to get a red gumball. The frequency in this case would be the number of each color of gumballs that came out.
What is the formula and unit of frequency?
Frequency, sometimes referred to as pitch, is the number of times per second that a sound pressure wave repeats itself. A drum beat has a much lower frequency than a whistle, and a bullfrog call has a lower frequency than a cricket. The lower the frequency, the fewer the oscillations.
Why is frequency V?
Frequency is number of complete waves passing per unit time. It is measured in Hertz (Hz), the number per second. 1 KHz = 103 Hz. 1 MHz = 106 Hz.
What is the symbol for frequency in physics?
Frequency is denoted by the symbol f, and is measured in hertz (Hz) – formerly called cycles per second (cps or c/s) – kilohertz (kHz), or megahertz (mHz). See diagrams under RADIO SPECTRUM, SIMPLE HARMONIC MOTION, SPECTRUM.
Is frequency and wavelength the same?
Frequency is inversely proportional to the time period. i.e, f=1T.
How do you explain frequency to a child?
Lambda, the 11th letter of the Greek alphabet, is the symbol for wavelength.
What is frequency in sound waves?
The heat conductivity of a material is known as its lambda value. The lambda value is used for thermal calculations on buildings and thermal components. The Greek letter λ, lambda, [W/mK] is used to represent the heat conductivity of a material.
How is frequency measured unit?
Speed is distance over time, so v = λ / T. The frequency, f, is 1/T, so the equation relating wave speed, frequency, and wavelength is v = f λ . where µ = m / L is the string’s mass per unit length.
Total Frequency is the value obtained by adding up all the frequencies in the frequency distribution table. Relative Frequency is the value obtained by dividing the absolute frequency by the total frequency. Relative Cumulative Frequency is the value obtained by the cumulative frequency by the total frequency.
What does the λ stand for?
The number of cycles that a vibrating object completes in one second is called frequency. The unit of frequency is hertz (Hz).
What is the value of λ?
Frequency. The number of waves produced in a given amount of time.
How do you say λ?
How do you convert Hz to NM?
- Divide the speed of light, which is approximately 300 million meters per second, by the medium’s refractive index.
- Divide the wave’s speed by its frequency, measured in Hertz.
- Multiply the wave’s wavelength by one billion, which is the number of nanometers in a meter.
How do you find frequency with wavelength and speed?
The speed of light in vacuum is 3×108 m/s. Sunlight takes about 8 minutes to reach the Earth.
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https://pdxscholar.library.pdx.edu/mth_fac/103/
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math
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Computer science, Mathematics, Computer algorithms, Distribution (Probability theory), Finite fields (Algebra), Polynomials, Comparative studies
An iterative method is given for computing the polynomials that vanish on the basin of attraction of a steady state in discrete polynomial dynamics with finite field coefficients. The algorithm is applied to dynamics of a T cell survival network where it is used to compare transition maps conditional on a basin of attraction.
Dinwoodie, I. (2014). Conditional Tests on Basins of Attraction with Finite Fields. Methodology & Computing In Applied Probability, 16(1), 161-168. doi:10.1007/s11009-012-9304-9
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CC-MAIN-2023-50
| 650 | 3 |
http://nelluy.tumblr.com/
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math
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Everyone has a gay cousin. If you don’t have a gay cousin, then you might be the gay cousin
WHY IS GAY MARRIAGE EVEN AN ISSUE
BECAUSE PEOPLE ARE FUCKING ASSHOLES
that can be taken one of two ways but both are accurate
#she fucking realizes hes not gonna catch her#look at her fucking face#she realizes that he cant get to her in time#that peter parker or spiderman whatever you want to call him because under that mask hes still just a boy who only just graduated high schoo#he cant save them all#he cant save her#and it breaks my fucking heart that he thought he did#that he actually thought he got to her in time and hes so confused when he reaches her#he doesnt understand why her eyes are closed#and you just know that he sat there with her for a while#he didnt move her or himself he just held her undtil her body grew cold and he carried her out ( reformedxserialxkiller )
SPN MEME: Two Quotes (1/2)
↪ “Decide to be fine til the end of the week. Make yourself smile because you’re alive and that’s your job. And do it again the next week.”
A day full of flowers!
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http://analisereal.com/tag/teoria-dos-jogos/
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math
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Fazia um tempo que não postávamos uma do SMBC. That was an incredibly counterintuitive result to nobody but economists.
Aproveitando o prêmio Nobel, o EconLog trouxe uma passagem do artigo de Galey e Shapley, sobre a matemática, que vale ser citada integralmente:
Finally, we call attention to one additional aspect of the preceding analysis which may be of interest to teachers of mathematics. This is the fact that our result provides a handy counterexample to some of the stereotypes which non-mathematicians believe mathematics to be concerned with.
Most mathematicians at one time or another have probably found themselves in the position of trying to refute the notion that they are people with “a head for figures.” or that they “know a lot of formulas.” At such times it may be convenient to have an illustration at hand to show that mathematics need not be concerned with figures, either numerical or geometrical. For this purpose we recommend the statement and proof of our Theorem 1. The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument, though not because of unfamiliarity with the subject matter.
What, then, to raise the old question once more, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical, and the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences. This observation will hardly be news to those engaged in the teaching of mathematics, but it may not be so readily accepted by people outside of the profession. For them the foregoing may serve as a useful illustration.
O Noah Smith também aproveita o tema para desenvolver um pouco sobre a matemática e a economia.
Bacana, o prêmio Nobel vai para dois autores de teoria dos jogos, Alvin Roth e Lloyd Shapley!
Eu gosto bastante dos trabalhos de Alvin Roth, já havíamos falado dele aqui. Ele é um autor que, apesar da sofisticação técnica, busca aplicar, com sucesso, a teoria dos jogos na prática.
Na lista de blogs à direita, você encontrará um chamado Market Design, cujo autor é o Alvin Roth. Tendo em vista a notícia, o post de hoje é de que talvez o blog se atrase – mais do que merecido!
Existe uma competição dos jogos olímpicos cuja estratégia ótima dos jogadores é um tanto peculiar. Veja a “animação” destes corredores de bicicleta:
Para entender o motivo, leia aqui no Marginal Revolution.
Relendo Von Neumann e Morgenstern, Theory of Games and Economic Behavior, logo nas primeiras folhas há uma passagem que merece ser relembrada, principalmente para aqueles que acreditam em uma explicação geral para tudo, em monismo teórico ou metodológico na economia:
First let us be aware that there exists at present no universal system of economic theory and that, if one should ever be developed, it will very probably not be during our lifetime. The reason for this is simply that economics is far too difficult a science to permit its construction rapidly, especially in view of the very limited knowledge and imperfect description of the facts with which economists are dealing. Only those who fail to appreciate this condition are likely to attempt the construction of universal systems. Even in sciences which are far more advanced than economics, like physics, there is no universal system available at present.
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http://www.protocol-online.org/biology-forums/posts/37086.html
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math
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FACS-antibody titration - (Jun/11/2008 )
can some one please advice me on how to titrate antibodies for FACS.
My primary antibody concentration is 50ug/ml (according to the manufactuerer) and I will stain 1ml of single cell suspension uisng FACS. I need to deermine the cncentration for staiing and I've read one way of doing it is by antibody titration. I'm weak at maths, and would greatly appreciate some tips.
try several dilutions of your antibody (1/100 ; 1/200 ; 1/400 ; ...) to find the highest dilution that gives you a nice staining
I'll be staining CD44 on a human cancer cell line. Can someone suggest me a positive control? I was thinking of beta-1 integrin?
1 ml of cell suspension or 1 million?
Do them in serial dilution, have a look at the histogram and/or plot a graph and choose the best. This might be helpful http://www.microbiology.emory.edu/altman/f...itering_Abs.htm
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CC-MAIN-2018-05
| 890 | 7 |
https://www.lumoslearning.com/llwp/resources/educational-videos-k-12-elementary-middle-school.html?id=1465009
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math
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Watch this video as the instructor clearly explains how to solve a word problem involving finding unit rates. When rates are expressed as a quantity of 1 such as 2 feet per second or 5 miles per hour they are called unit rates. See how you can apply this process to help you solve similar problems.
Solve Problems Involving Rates and Ratios is a free educational video by Khan Academy.It helps students in grades 6 practice the following standards 6.RP.3,7.RP.1,.
1. 6.RP.3 : Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations..
2. 7.RP.1 : Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour..
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https://ieltstutors.org/glossary/logic/
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math
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Definitions: (noun) Logic is a system of careful step by step thoughts or arguments that tries to explain truth.
Examples: (noun) Without logic your arguments will fall apart. (noun) Scientists must use logic to explain the workings of the natural world.
Synonyms: nouns: reason, sense.
Academic Word List Sublist and Group: 5 C
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https://www.englishlessonplanner.com/plans/36685
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math
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Present Simple Typical Friends and Free Time Activities
To introduce/review and give Ss practice in Present Simple – wh- questions & short answers in the context of who we live with
To provide practice of the present simple tense in the context of who do we live with by using wh- questions
Procedure (65-94 minutes)
• Show students the picture of a family living together. • Ask students to predict who they are. • Ask students to work in pairs asking the question do you live with your parent. • Elicit some answers.
• Show the reading to students by using the projector. • Tell students to scan the text and underline the questions in the text. • Demonstrate an example with them. • Give students the reading text. • Ask students to work alone
• Ask students to check their answers in pairs • Do whole class feedback by eliciting answers from students
• Show the students the exercise 2 by using the projector. • Do a demonstration with them. • Drill the question for pronunciation. • Give students the HO. • Ask students to do the matching task. • Students work alone.
• Do whole class feedback
• Draw a grammar table of Wh- questions on the Wb • Explain students the form order. • Write two sentences on the Wb. • Elicit answers from students. • Give students exercise 2 • Ask students to work in pairs.
• Do whole class feedback by nominating students to write the sentences on the board
• Show students Extra task from the teacher’s book. • Model questions and do drilling . • Ask students to work in pairs and match the wh- words with sentences.
• Nominate students for answers
•Ask students to interview their partners by asking the questions of Extra
• Ask students to change the questions he/she questions. • Ask students to change partners. • Ask them to talk about the first person they interview.
•Tell students write a small profile about each other.
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CC-MAIN-2023-14
| 1,937 | 16 |
http://alexisfraser.com/england/effects-of-manipulative-materials-in-mathematics-instruction.php
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math
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Jump to navigation
Improving Student Achievement in Mathematics by
Manipulatives in Education: Definition, Examples & Classroom Applications. manipulatives for math instruction Manipulatives in Education: Definition, Examples...
The Role of Manipulatives in the Eighth Grade Mathematics
Research on the Use of Virtual Manipulatives The effects of computer-assisted instruction on the Effects of manipulative materials in mathematics instruction.. Using manipulatives in mathematics instruction can help you to Teachers often use tangible materials—referred to as effect of using virtual manipulatives on. mathematics instruction and student mathematics As we explore the effective use of manipulatives, The effective use of manipulative materials in mathematics.
Research on the Use of Virtual Manipulatives
Improving Student Achievement in Mathematics by Using Manipulatives with Classroom Instruction Manipulative materials help students of the effects …! CiteSeerX - Scientific documents that cite the following paper: Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics.
Manipulatives in Math Why Teach Math with Manipulatives
Effects of manipulative instruction on solving area and perimeter problems by students with learning disabilities. materials in mathematics instruction.. This circumstance has resulted in early childhood mathematics instruction manipulative-based instruction The Journal of Experimental Education,. The effects of computer graphics and mira on Effects of manipulative materials in mathematics Research on instructional materials for mathematics..
ERIC Effects of Manipulative Use on PK-12 Mathematics
The Importance of Using Manipulatives Using Manipulatives During Math Instruction The use of manipulatives during Effects of Manipulative Materials in. The Effects of Using Manipulatives in Teaching Math Problem Solving to Students with Learning Disabilities to manipulatives instruction using Cuisenaire.
Case Study 1 An Evidence-Based Practice Review Report
compared with traditional instruction typically had a positive effect on student Effects of Manipulative Materials in Mathematics Instruction. Journal
The Effect Of Mathematical Manipulative Materials On Third
Improving Student Achievement in Mathematics by Using Manipulatives with Classroom Instruction Manipulative materials help students of the effects …. The Mathematics Educator 2009, Vol. 12, No.1, 3-14 The Effect of Manipulative Materials on Mathematics Achievement of First Grade Students Bobby Ojose
Bestar 65635 Lateral File Cabinet Discount Office Furniture. Bestar Harmony U Shaped Office Desk With Hutch. The Bestar Harmony U Shaped Office Desk has a commercial-grade work surface that with instructions and bestar desk assembly instructions Find best value and selection for your Bestar Assembly Instructions Office Desk w Return 2200 8 12 13 2200 10 14 15 search on eBay. World's leading marketplace..
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http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=5464825&contentType=Conference+Publications
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math
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Skip to Main Content
Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size ¿. The robust recovery result of Cande¿s et al. and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ¿1 recovery yields an approximation of the original sparse signal with an accuracy of O(¿). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in ¿N, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)¿ for any 0 < ¿ < 1 if m ¿r k(log N)1/(1-¿) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via rth-order Sobolev dual frames.
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https://www.betterlivingthroughdesign.com/accessories/math-dishtowel-mug-pencil-case/
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math
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Math Dishtowel, Mug, Pencil Case
For your favorite mathematician (a.k.a. mathemagician), how about a dishtowel, pencil case, or mug? Or all three? Chalk white lines on a black background evoke memories from past math classes, and no matter if you liked/loved/hated them, you (or someone you know) can still enjoy a little math imagery with appreciation.
Math Mug, $12.95
Math Canvas Pencil Case, $17.95Available from Fishs Eddy, $12.95 - 17.95.
Tags: Gifts, Gifts $1-$25
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https://questioncove.com/updates/4df9ecad0b8b370c28be2ff0
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math
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can u separate ln (7+x) into ln7+lnx?
no ln (7+x) = "the exponent you raise "e" to to get (7+x)"
can you separate it if it's multiplication or division?
(7+x) is the argument of the ln, so it's dependent on whatever x is, so now you cannot seperate
well sort of, "ln7+lnx" would equal "ln7x" but what you have there is not that.
if you had ln x + ln 7, it could be simplified to ln (7x)
yeah... what she said. but no, ln(7+x) can't be seperated. what's the whole equation?
ok then ln7/x can be ln7-lnx? oh i was just wondering what the rule was so i just made that equation up..
dont think it can be seperated...it would be a mathematical crime
lol @ eistein+ newton .....
Join our real-time social learning platform and learn together with your friends!
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s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780054023.35/warc/CC-MAIN-20210917024943-20210917054943-00190.warc.gz
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CC-MAIN-2021-39
| 754 | 11 |
http://petrilisd.tripod.com/ch82.htm
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math
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Every polynomial equation with degree 4, as it can be easily seen, can take the form of (1).
If we call the left hand side of the equation (1), then it can be written as the difference of the squares of the following polynomials:
where are proper complex numbers, which we have to specify.
and after the relevant operations we have:
In order the right hand side of (2) to
be a perfect square, we must find a proper such
that its discriminant
D to be equal to zero.
We conclude that:
Equation (3) has degree 3 and can be solved as described in paragraph 8.1 of this chapter.
With one of the values of ,
that we get from (3) we find the value of
from (2) and then
(1) because of the following equation:
is equivalent to:
which is reduced to two polynomial equations with degree 2.
Solve the equation :
It is a=1, b=2, c=3,
The constant term of (3) is :
and the coefficient of the third degree term is :
Hence we have the equation: .
Solving the above equation as described in paragraph 8.1 we get :
For we get from (M)
and hence from the equation
we find that
The equations A(x) + B(x) = 0, A(x) - B(x) = 0 that equation (4) gives become:
and from these we get the roots:
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s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000175.78/warc/CC-MAIN-20190626053719-20190626075719-00329.warc.gz
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CC-MAIN-2019-26
| 1,169 | 27 |
https://encyclopedia2.thefreedictionary.com/survival+curve
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math
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Figure 2 presents a comparison between each survival curve
estimated by the final parametric models candidates (Weibull and logistic) and the Kaplan-Meier non-parametric model, showing a good fit to the data.
The predicted survival curve
of the home-to-work travel duration of Program A
Kaplan-Meier survival curves
were used to visually compare the survival status of all elderly subjects stratified by length of stay.
where: the slope tg[beta] of eqn.6 is a direct measure of the slope (1/[sigma]) of the survival curves
; therefore, tg[beta] is the seed deterioration rate under any storage condition, as expressed by the angular coefficient of the survival curve
, as follows:
As expected, Figures 5-7 demonstrate that the survival curve
and the killed and cancer cell populations are greater when repopulation and sub-lethal repair are considered.
The Kaplan-Meier estimation method was used to compare survival curves
between racial groups--black and white men.
To determine the differences in graft survival rates between high and low income ineligibles, we constructed Kaplan-Meier survival curves
for each ineligible income quartile (Figure 1).
The survival curve
for the major deck types without observed improvement, ranked from highest to lowest in terms of survival probability, is as follows: precast concrete panels, cast-in-place concrete, corrugated steel, and wood.
Efficacy criteria were the 48-hour cumulative stool weight/kg of body weight, time to recovery, proportion of children with no more diarrhea (survival curve
) by treatment day, number of formed stool/day, number of watery stool/day, and percentage of anal irritation.
We modeled the time that bears remained on the river, in the presence and absence of humans, using survival analysis and the Kaplan-Meier estimate of the survival curve
(Fleming and Harrington, 1991).
Bleyer and associates also plotted the data to reflect a plateau in the survival curve
of patients in various age groups using probability analysis.
To estimate r and s the survival curve
can be transformed into an inverse Gaussian distribution,
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s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664808.68/warc/CC-MAIN-20191112074214-20191112102214-00471.warc.gz
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CC-MAIN-2019-47
| 2,098 | 25 |
https://groups.able2know.org/philforum/topic/3224-3
|
math
|
I am still a bit puzzled by your questions, particularly coming from you. What is wrong with the responses given in a typical introductory logic textbook?
The 'answer' that can be found in a logic textbook is pretty much some vague considerations that seem to stem from the intention theory. But I'm questioning that theory, so this result does clearly not work. This problem is not discussed in any logic textbook that I have read or skimmed through. (I have not read Copi's.)
4. No, unless you mean that part of a lengthy argument (that is really a series of arguments) is inductive, and part is deductive.
Re 3. Not sure about that. Arguments need to be given. I know that this is a common assumption but is it a good assumption?
Re 4. That's not what I mean.
Also, on further reflection. Skip the second part of question 2 as it is not very relevant here. The discussion about cogent and strong inductive arguments is not what I intended
this thread to be about.
When I write "valid" I always mean "deductively valid". I don't think there is a term that is called inductive validity.
A deductive argument is an argument where the conclusion follows necessarily from the premises.
A valid deductive argument is a deductive argument where the conclusion follows necessarily from the premises.
A non-valid deductive argument is a deductive argument where the conclusion does not follow necessarily from the premises.
First, your definition of deductive argument is identical
with your definition of valid deductive argument. According to you, then, we have no need to speak of valid deductive arguments because all deductive arguments are valid.
Second, you are contradicting yourself. Look at your first and third statement. An invalid deductive argument is according to them both one where the conclusion necessarily follows from the premises and one where it does not. That is impossible.
Also, I dislike that definition of validity very much as it is dangerously ambiguous and causes people to make modal fallacies. There are many logically equivalent definitions of validity. In this thread I stipulate that we use this one:
[INDENT]An argument is valid iff the corresponding conditional is a necessary true proposition.
An inductive argument is an argument where the premises offer a degree of support for the conclusion, but do not necessarily entail it; there is no guarentee (like your article noted).
Careful. Do you accept Ken's view (=intention theory)? If so, then some inductive arguments are valid, because some arguments are intended by the speaker to be inductive but are actually valid.
I think it depends how we use the word "argument". But, I think if we use the word formally, the answer is yes.
This needs some defending.
Do you see how many ways this question can be interpreted?
1.) Yes, some arguments are deductive, some arguments are inductive.
2.) Yes, some arguments (like Pyrrho noted), can have inductive and deductive parts.
3.) No, if you mean that only some arguments are inductive or deductive. Because all formal arguments are deductive or inductive (and you might mean this since this question comes after the one which specifies "all arguments").
Yes, but there is only one good interpretation, and it is straightforward. I even removed extra words that made it even easier to get it right. I suppose I should have left them in. You can insert the word "both" into the question if it is unclear to you, like this:
4. Are some arguments both
deductive and inductive?
In the formal english language, E:
4F. Does there exist an argument such that it is deductive and it is inductive?
(I invented some 'formal machinery' (Ken's phrase) for questions to be formalized. In this the question has the form (∃x)(Dx∧Ix)? Let's not discuss this invention in this thread but some other time. The interested reader can look here
That is not difficult. Deductive arguments are intended to be conclusive arguments by the arguer. If the argument is not conclusive, but is intended to be conclusive, it is a failed (invalid) deductive argument. On the other hand, if an argument is not intended to be a conclusive argument by the arguer, it is non-deductive. But if the premises fail to support the conclusion of the non-deductive argument, then it is a failed (weak) non-deductive argument.
However, whatever the arguer intends, deductive or non-deductive, there are some arguments which it would hardly make sense to intend it as deductive, since it is so clearly non-conclusive; or make sense to intend it as non-deductive since it is so clearly conclusive. It would be a good rule (I think) to count a valid deductive argument as deductive, and a strong non-deductive argument as non-deductive.
This is a good answer from someone defending the standard intention theory. I used to agree with this now I have doubts.
Why do you think it would be a good rule to count a valid argument as deductive even though it is according to that theory inductive?
I assume this is what you meant and you were just being careless when you inserted the word "deductive" there. Otherwise you were merely suggesting that we count valid deductive arguments as deductive. That is not very interesting.
But can't we distinguish between inductive and deductive arguments without considering the intent of the arguer at all?
Not according to the intention theory. This implies that the difference between a deductive argument and an inductive argument is merely
a psychological one and not a logical one. Some people consider this implausible, me included. See the other threads.
I don't see how, although, as I said, there are pretty clear cases of both kind when it would be implausible for the arguer's intention not to be the one or the other. In those cases we have what, in legal jargon, we might call, "constructive intent". That it was the arguer's intent whether or not it actually was (or the arguer was confused).
I agree with this. But even though it is implausible that the arguer intended a given argument to be inductive, it does not follow that it is deductive. Some arguers are terribly confused.
But we can distinguish an inductive argument from a deductive argument by looking looking at the argument. If all the premises can be true without the conclusion being true, it is an inductive argument:
1.) Socrates was Greek
2.) Most Greeks eat fish
3.) Socrates ate fish
This is an inductive argument. And we know it's not a deductive argument because although the premises may be true, the conclusion does not follow necessarily from the premises; the conclusion may not be true.
Isn't that right?
Now you have defined inductive argument as invalid argument. That's another theory and it is inconsistent with the intention theory. Let's call this theory for the validity theory, for it defines deductive argument as valid argument, and inductive argument as invalid argument.
This theory has the curious and implausible implication that all deductive arguments are valid, indeed, they could not be invalid. Thus, one cannot fail to make a valid deductive argument, it is impossible. This is the position that Kritikos was defending and to which Ken gave the plausible example arithmetic analogy. See the opening post.
I would rather say, we do not know what the argument is without knowing the intentions of the arguer. But once we know what the argument is, we then may be able to determine whether it is deductively valid or not, and whether it is inductively valid or not.
Perhaps, though, this is a mere verbal distinction, without any importance at all.
Not at all! This is the crucial point. In this post you are endorsing the intention theory. That's fine, but the theory has its problems some of which I have mentioned already.
I thought what you were meaning to ask for is an analysis of deductive arguments and/or an analysis of inductive arguments. You already know the difference between the two, so you do not need an explanation of what each is. Instead, what you want is an analysis, for merely knowing the difference between the two doesn't therefore imply that you can always be given an argument and definitively determine whether or not the logical argument is a deductive argument or inductive argument, for sometimes, being privy to the argument is insufficient information to determine whether or not an argument is deductive or inductive.
That of course has no bearing on whether or the argument is deductive or inductive, just as truth doesn't depend on knowledge of the truth.
I don't think you should ever regard inductive arguments as valid or invalid.
You are getting the point. I'm not sure that I know the difference between them. I can make the distinction in practice like any person trained in logic can, but that does not imply that I know the difference, does it?
But I am definitely asking for an analysis.
" I don't think you should ever regard inductive arguments as valid or invalid.
" Why do you think this? Given pretty much any definition that you choose of validity, it is applicable to inductive arguments.
According to the validity theory, all deductive arguments are valid and all inductive arguments are invalid.
According to the intention theory some deductive arguments are valid and some are invalid, and some inductive arguments are valid and some are invalid.
I got soundness and validity confused. Validity speaks nothing of truth, only form. To be valid means that the conclusion follows from the premises. An argument being valid does not mean that it is true. Validity is a necessary but not sufficient condition for soundness. It is not a sufficient condition because an argument not only needs to be valid to be sound, but it also needs to be true.
Is this right?
It is nonsense to speak of true/false arguments.
Validity has something to do with form, but not all valid arguments have a valid form. But this is a discussion I would rather not elaborate on now. You can read more about it in Possible Worlds
where it is discussed at length.
No arguments are either true or false. Are you, perhaps asking whether a valid argument must have a true conclusion. The answer is, no. The same for whether a valid argument must have true premises. But, what is true is that any valid argument with true premises must have a true conclusion.
It is also nonsense to say that no arguments are either true or false. The correct wording is:
Ah, arguments cannot be said to be true or false, just valid or sound, right? Premises and conclusions are what we apply the properties true and false to.
But this is also a side discussion about meaning and category errors. Let's not discuss that now.
Right, because premises and conclusions are propositions (statements) and only propositions (statements) are true or false.
Not sure about that. Maybe we should not assume a propositional theory of truth bearers in this thread. Or, better yet, let's assume it so far (pretty much everyone in this thread holds that theory anyway), and maybe after we have considered the problems of deductive and inductive arguments in that light (so to speak), we could consider them in the light of say a sentence theory of truth bearers.
At least, let's not discuss theories of truth bearers in this thread.
A necessary condition for a sound argument is not that the conclusion be true, although, if an argument is sound, then the conclusion will be true. Truth, then, is a consequent of soundness.
You got yourself confused. It is a necessary condition for soundness. But it is not a sufficient. For this thread let's define soundness like this:
[INDENT]An argument is sound iff:
1. All the premises and the conclusion are true.
2. The argument is valid.
All cogent arguments are strong arguments, but not all strong arguments are cogent arguments, for all cogent arguments are strong arguments with true premises, and not all strong arguments have true premises.
This strongness you speak of is not a standard term as far as I know.
That is not what it says at Wikipedia:
Cogency - Wikipedia, the free encyclopedia
You may use the term differently, but this only reinforces my point that the terms used to describe various inductive arguments are not very standardized.
I edited the Wikipedia page to its current form (I think). I did it because it was terribly confused before. I also inserted the reference to Fallacyfiles. But bear in mind what Ken says:
Actually, some logic books use the term "cogent" to mean, "known to be true", and not just true. So, a cogent argument would be one where the premises are not only true (and argument valid) but are known to be true, so the conclusion is known to be true. But the books differ on this.
Correct. I seem to recall writing this on Wikipedia but they may have changed it. Wikipedia is not a good source for such specific information as this.
But let's not derail the thread with more discussions of cogentness and strongness and what have we, that is, terms related to inductive arguments.
What is a valid inductive argument anyway?
See the definition of validity above. (And please for the love of god (you do believe in god, right?) stop changing the fonts!)
I dont know if you guys have answered the Op or not but I'll throw out some ideas on the subject.
Deductive arguments have true premises and a conclusion that necessarily follows. While inductive arguments have conclusions that are probable, but not necessary. So me thinks inductive arguments hinge on logical possibility while deductive arguments dont. So its impossible to have a counter example to a deductive argument. Does that work emil?
Your first two claims are wrong. I don't know about the rest as they are too vague to consider true or false.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947473819.62/warc/CC-MAIN-20240222125841-20240222155841-00520.warc.gz
|
CC-MAIN-2024-10
| 13,719 | 91 |
https://uwaterloo.ca/pure-mathematics/events/joint-pure-mathco-grad-colloquium-1
|
math
|
Sabrina Lato, Department of Combinatorics & Optimization, University of Waterloo
"Perron, Frobenius, and some unexpected applications"
The Perron-Frobenius Theorem is a powerful theorem about non-negative matrices, with applications to algebraic graph theory, Markov chains, compact operators, as well as more practical applications in economics, demography, and Google's PageRank algorithm. However, the theorem itself was proved before any of those applications were known. The first formulation came out of Perron's work on continued fractions, adapted to positive matrices, and this was later expanded by Frobenius. In this talk, we'll explore the history and context of the Perron-Frobenius Theorem, as well as its application to graph spectra.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100583.31/warc/CC-MAIN-20231206063543-20231206093543-00649.warc.gz
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CC-MAIN-2023-50
| 749 | 3 |
https://www.khanacademy.org/math/algebra-basics/alg-basics-equations-and-geometry/alg-basics-pythagorean-theorem/v/pythagorean-theorem
|
math
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Current time:0:00Total duration:13:03
Let's now talk about what is easily one of the most famous theorems in all of mathematics. And that's the Pythagorean theorem. And it deals with right triangles. So a right triangle is a triangle that has a 90 degree angle in it. So the way I drew it right here, this is our 90 degree angle. If you've never seen a 90 degree angle before, the way to think about it is, if this side goes straight left to right, this side goes straight up and down. These sides are perpendicular, or the angle between them is 90 degrees, or it is a right angle. And the Pythagorean theorem tells us that if we're dealing with a right triangle-- let me write that down-- if we're dealing with a right triangle-- not a wrong triangle-- if we're dealing with a right triangle, which is a triangle that has a right angle, or a 90 degree angle in it, then the relationship between their sides is this. So this side is a, this side is b, and this side is c. And remember, the c that we're dealing with right here is the side opposite the 90 degree angle. It's important to keep track of which side is which. The Pythagorean theorem tells us that if and only if this is a right triangle, then a squared plus b squared is going to be equal to c squared. And we can use this information. If we know two of these, we can then use this theorem, this formula to solve for the third. And I'll give you one more piece of terminology here. This long side, the side that is the longest side of our right triangle, the side that is opposite of our right angle, this right here-- it's c in this example-- this is called a hypotenuse. A very fancy word for a very simple idea. The longest side of a right triangle, the side that is opposite the 90 degree angle, is called the hypotentuse. Now that we know the Pythagorean theorem, let's actually use it. Because it's one thing to know something, but it's a lot more fun to use it. So let's say I have the following right triangle. Let me draw it a little bit neater than that. It's a right triangle. This side over here has length 9. This side over here has length 7. And my question is, what is this side over here? Maybe we can call that-- we'll call that c. Well, c, in this case, once again, it is the hypotenuse. It is the longest side. So we know that the sum of the squares of the other side is going to be equal to c squared. So by the Pythagorean theorem, 9 squared plus 7 squared is going to be equal to c squared. 9 squared is 81, plus 7 squared is 49. 80 plus 40 is 120. Then we're going to have the 1 plus the 9, that's another 10, so this is going to be equal to 130. So let me write it this way. The left-hand side is going to be equal to 130, and that is equal to c squared. So what's c going to be equal to? Let me rewrite it over here. c squared is equal to 130, or we could say that c is equal to the square root of 130. And notice, I'm only taking the principal root here, because c has to be positive. We're dealing with a distance, so we can't take the negative square root. So we'll only take the principal square root right here. And if we want to simplify this a little bit, we know how to simplify our radicals. 130 is 2 times 65, which is 5 times 13. Well, these are all prime numbers, so that's about as simple as I can get. c is equal to the square root of 130. Let's do another one of these. Maybe I want to keep this Pythagorean theorem right there, just so we always remember what we're referring to. So let's say I have a triangle that looks like this. Let's see. Let's say it looks like that. And this is the right angle, up here. Let's say that this side, I'm going to call it a. The side, it's going to have length 21. And this side right here is going to be of length 35. So your instinct to solve for a, might say, hey, 21 squared plus 35 squared is going to be equal to a squared. But notice, in this situation, 35 is a hypotenuse. 35 is our c. It's the longest side of our right triangle. So what the Pythagorean theorem tells us is that a squared plus the other non-longest side-- the other non-hypotenuse squared-- so a squared plus 21 squared is going to be equal to 35 squared. You always have to remember, the c squared right here, the c that we're talking about, is always going to be the longest side of your right triangle. The side that is opposite of our right angle. This is the side that's opposite of the right angle. So a squared plus 21 squared is equal to 35 squared. And what do we have here? So 21 squared-- I'm tempted to use a calculator, but I won't. So 21 times 21: 1 times 21 is 21, 2 times 21 is 42. It is 441. 35 squared. Once again, I'm tempted to use a calculator, but I won't. 35 times 35: 5 times 5 is 25. Carry the 2. 5 times 3 is 15, plus 2 is 17. Put a 0 here, get rid of that thing. 3 times 5 is 15. 3 times 3 is 9, plus 1 is 10. So it is 11-- let me do it in order-- 5 plus 0 is 5, 7 plus 5 is 12, 1 plus 1 is 2, bring down the 1. 1225. So this tells us that a squared plus 441 is going to be equal to 35 squared, which is 1225. Now, we could subtract 441 from both sides of this equation. The left-hand side just becomes a squared. The right-hand side, what do we get? We get 5 minus 1 is 4. We want to-- let me write this a little bit neater here. Minus 441. So the left-hand side, once again, they cancel out. a squared is equal to-- and then on the right-hand side, what do we have to do? That's larger than that, but 2 is not larger than 4, so we're going to have to borrow. So that becomes a 12, or regrouped, depending on how you want to view it. That becomes a 1. 1 is not greater than 4, so we're going to have to borrow again. Get rid of that. And then this becomes an 11. 5 minus 1 is 4. 12 minus 4 is 8. 11 minus 4 is 7. So a squared is equal to 784. And we could write, then, that a is equal to the square root of 784. And once again, I'm very tempted to use a calculator, but let's, well, let's not. Let's not use it. So this is 2 times, what? 392. And then this-- 390 times 2 is 78, yeah. And then this is 2 times, what? This is 2 times 196. That's right. 190 times 2 is-- yeah, that's 2 times 196. 196 is 2 times-- I want to make sure I don't make a careless mistake. 196 is 2 times 98. Let's keep going down here. 98 is 2 times 49. And, of course, we know what that is. So notice, we have 2 times 2, times 2, times 2. So this is 2 to the fourth power. So it's 16 times 49. So a is equal to the square root of 16 times 49. I picked those numbers because they're both perfect squares. So this is equal to the square root of 16 is 4, times the square root of 49 is 7. It's equal to 28. So this side right here is going to be equal to 28, by the Pythagorean theorem. Let's do one more of these. Can never get enough practice. So let's say I have another triangle. I'll draw this one big. There you go. That's my triangle. That is the right angle. This side is 24. This side is 12. We'll call this side right here b. Now, once again, always identify the hypotenuse. That's the longest side, the side opposite the 90 degree angle. You might say, hey, I don't know that's the longest side. I don't know what b is yet. How do I know this is longest? And there, in that situation, you say, well, it's the side opposite the 90 degree angle. So if that's the hypotenuse, then this squared plus that squared is going to be equal to 24 squared. So the Pythagorean theorem-- b squared plus 12 squared is equal to 24 squared. Or we could subtract 12 squared from both sides. We say, b squared is equal to 24 squared minus 12 squared, which we know is 144, and that b is equal to the square root of 24 squared minus 12 squared. Now I'm tempted to use a calculator, and I'll give into the temptation. So let's do it. The last one was so painful, I'm still recovering. So 24 squared minus 12 squared is equal to 24.78. So this actually turns into-- let me do it without a-- well, I'll do it halfway. 24 squared minus 12 squared is equal to 432. So b is equal to the square root of 432. And let's factor this again. We saw what the answer is, but maybe we can write it in kind of a simplified radical form. So this is 2 times 216. 216, I believe, is a-- let me see. I believe that's a perfect square. So let me take the square root of 216. Nope, not a perfect square. So 216, let's just keep going. 216 is 2 times 108. 108 is, we could say, 4 times what? 25 plus another 2-- 4 times 27, which is 9 times 3. So what do we have here? We have 2 times 2, times 4, so this right here is a 16. 16 times 9 times 3. Is that right? I'm using a different calculator. 16 times 9 times 3 is equal to 432. So this is going to be equal to-- b is equal to the square root of 16 times 9, times 3, which is equal to the square root of 16, which is 4 times the square root of 9, which is 3, times the square root of 3, which is equal to 12 roots of 3. So b is 12 times the square root of 3. Hopefully you found that useful.
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CC-MAIN-2022-40
| 8,928 | 2 |
http://perplexus.info/show.php?pid=4732&cid=32774
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math
|
Imagine a grid of squares, like a tic-tac-toe board, that goes on infinitely in all directions.
Players alternate taking turns marking the board with X's and O's. The winner is the first player to get four marks in a row (horizontally, vertically, or diagonally).
On each turn, a player may either:
A: Place two of his/her marks on the board, or
B: Remove one of the other player's marks, and then place one of their own.
With optimal play, does either player have a forced win, or will this game continue on infinitely?
The game should be over after the player who goes first, makes her third move.
X's first move. Two X's next to each other.
O's first move. Block both ends if the string of X's with an O. Anything else and the next two X's win.
X's second move. Two new X's adjacent to the first two (forms a 2x2 grid) At this point there are 10 spots for O to block. Can't cover them all, so X will win on her third turn.
Lets say that O wiped one of the X's from the board during her first turn.
O's first move. Place the O anywhere. (it doesn't matter)
X's second move. Two X's on a different row, one adjacent to the first X and one diagonal. At this point there are 6 points to block. If O does not erase an X. X will win on the third turn.
If O erases one X, there will remain two X's adjacent to one another and only one O available to block. X's third move will complete the string of 4.
Posted by Leming
on 2006-06-09 16:43:21
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CC-MAIN-2018-47
| 1,438 | 16 |
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