Reference & Education Mathematics

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A-level Mathematics S1

 Statistics is the accurate access to after information. Statistical development has apparent that some aspects of advance depend aloft the actual assay of after advice - decidedly in science, economics and business.# # # # # Harper Statistics M&E Handbooks, 1981www.samer.comLangley Applied Statistics artlessly explained, Pan, 1979Larson Addition to anticipation approach and statistical inference, 2nd copy Wiley All-embracing 1974Oxford Cambridge and RSA Examinations - Syllabus[http://www.ocr.org.uk/OCR/WebSite/Data/Publication/Specifications%2c%20Syllabuses%20%26%20Tutors%20Handbooks/AS_A_Level86696.pdf]... Read More by user

A-level Mathematics C2 Logarithms and Exponentials

 An exponential action is a action area a connected abject (b) is aloft to a variable.Firstly, b^x imes b^ is b^, which is x^,. So if you accumulate a abject by the aforementioned abject you add the variables. To clarify, actuality is an archetype with numbers:Secondly frac is b^,(also y eq 0). So if a abject is disconnected by the aforementioned abject you decrease the variables. Here is an archetype with numbers:frac=frac=4=2^2.Thirdly left(b^ ight)^ is b^ which is b^. So if a abject with a capricious is raises to an capricious you accumulate the variables. Actuality is addition archetype with numbers: (when x = 1) left(2^2 ight)^3=4^4=64=2^6.Fourtly if a^2 imes b^2 = ab imes ab it is the aforementioned as left(ab ight)^2,. Actuality is an archetype with numbers: 2^2 imes 3^2 = 36 = 6^2. There is a agnate bearings with division: left(frac ight)^2 = frac imes frac = frac. So if you assorted or bisect two altered bases aloft to the aforementioned capricious you can multply or bisec... Read More by user

A-level Mathematics D1 Bulge Graphs Addition

 This area is anxious with bulge graphs, sometimes accepted as graphs, or networks.Firstly, we haveto say what we beggarly by a graph. A blueprint is a set of credibility (which we alarm nodes, or vertices), affiliated by curve (arcs, or edges). If cerebration about graphs, the breadth and blueprint of anniversary arc do not matter, alone which vertices are affiliated to which additional vertices.The Konigsberg Arch Problem, or the Seven Bridges of Konigsberg, is a archetypal problem, one of the first in blueprint theory. It was aggressive by the absolute blueprint of the city-limits of Konigsberg, Prussia (now Kaliningrad). Above is a map of the city, with a river coloured in blue, and the briges in green. The problem is this, is it accessible to alpha about in the city, and cantankerous anniversary arch one and alone once, and acknowledgment to the starting point. Mathematician Leonard Euler (1707 1783 managed to break this problem, by apperception on the important aspects. He reali... Read More by user

Affidavit of Assumption 5

 A timberline of adjustment n has n-1 edges.:For a blueprint G, the afterward are equivalent:For n=1, all graphs accept 0=n-1 edges. For n=2, there are two graphs, one with an bend and one without. The blueprint after an bend is disconnected, and accordingly is not a tree, admitting the one with an bend is affiliated and has no cycles and accordingly is a tree.For n=3, there are 4 graphs, one anniversary with 0, 1, 2 or 3 edges. The graphs with 0 and 1 edges are disconnected, while the blueprint with 3 edges has a cycle. The blueprint with 2 edges is affiliated and has no cycle, so the assumption is accepted for nle 3.Now accept we accept accepted this assumption for all n, and let T be a timberline of adjustment k. Abolish an bend e from T to anatomy a new blueprint T-e. By Assumption 4, T is a basal affiliated graph, and appropriately T-e is disconnected. Any affiliated basic of T-e (by the connectedness of T) haveto accept one of the vertices of e in it, and appropriately T-e has abs... Read More by user

After Methods Departure

 = Departure =A accepted problem in science and engineering is that of multivariateinterpolation of a action f whose ethics are accepted alone on a bound set of points. Accordingly let Omegasubsetmathbf^ be an accessible belted domain. Accustomed a set of audible credibility Xand absolute numbers, the assignment is to assemble a function s:mathbf^ ightarrowmathbf that satisfies the departure conditions: s(x_) = f_ In the endure decades, the access of adorable base action interpolationbecame more accepted for problems in which the credibility in Xare anyhow broadcast in space. In its basal form, adorable base function departure chooses a anchored function phi:mathbf} ightarrowmathbf and defines an interpolant by phi where the coefficients c_ are absolute numbers, for |cdot| one chooses usually the euclidian norm, and phi is a adorable base function.We see that the approximating functions is a beeline aggregate of radially symmetric functions, each centered on a audible point x_i. The cre... Read More by user

Affidavit of Assumption 4

 For a blueprint G, the afterward are equivalent:: A blueprint is a timberline if and alone if for every brace of audible vertices u,v, there is absolutely one u,v-path.Suppose G is a tree, and let uv be an bend not in G. By , there is absolutely one u,v-path. If this aisle is ux_1ldots x_kv, then abacus the bend uv will make the aeon ux_1ldots x_kvu. As a timberline is to be a affiliated backwoods and a backwoods is to be acyclic. A timberline is accordingly a acute acyclic graph.Now accept G is a tree, and let uv be an bend of G. By , there is absolutely one u,v-path. This aisle is acutely just the bend uv. Thus, in the blueprint G-uv, there is no u,v-path, and appropriately G-uv is disconnected. Back a timberline is to be a affiliated forest, a timberline is accordingly a basal affiliated graph.Now accept G is a acute acyclic graph, but is not a tree. Back an acyclic blueprint is a forest, and a timberline is to be a affiliated forest, G haveto be disconnected. Appropriately there ex... Read More by user
Tags: shows, connected, cycle, disconnected

Affidavit of Aftereffect 3

 A blueprint is a timberline if and alone if for every brace of audible vertices u,v, there is absolutely one u,v-path.: A blueprint is a backwoods if and alone if for every brace of audible vertices u,v, there is at alotof one u,v-path.A timberline is to be a affiliated forest. Furthermore, a blueprint is to be affiliated if and alone if for every brace of audible vertices u,v, there is at atomic one u,v-path. The affidavit should now be evident, but for completeness:If blueprint G is not a tree, it is either not a forest, or not connected. If G is not a forest, by , there exists a brace of vertices u,v with added than one u,v-path. If G is not connected, there exists a brace of vertices u,v with no u,v-path. Appropriately if G is not a tree, it is not the case that anniversary brace of vertices has absolutely one u,v-path.If blueprint G is a tree, fix a brace of vertices u,v. As G is a forest, by , there exists at alotof one u,v-path. Back G is connected, there is at atomic one u,v-pa... Read More by user

Affidavit of Assumption 2

 A blueprint is a backwoods if and alone if for every brace of audible vertices u,v, there is at alotof one u,v-path.Suppose you accept a blueprint G which is not a forest. Back a backwoods is to be a blueprint absolute no cycles, it is bright that G haveto accept a aeon C. All cycles accept at atomic three vertices. Let v_1, v_2, ldots, v_n (with n/ge 3) be the vertices on C in order. Then there are two audible v_1,v_2-paths, namely v_1v_2 and v_1v_nv_ldots v_3v_2.Now accept that G is not a forest, but has two audible u,v-paths, namely ua_1a_2ldots a_nv and ub_1b_2ldots b_mv. We would like to say that these make a aeon ua_1ldots a_nvb_mb_ldots b_1u. However, a aeon can alone cover anniversary acme once, and it is accessible that one of the b_i is according to one of the a_j. To that end, address a_0=b_0=u and a_=b_=v. Let i be the better amount of i such that a_i=b_j for some j. Then we acquisition a cycle, namely a_ia_ldots a_nvb_mb_ldots b_j. By our best of i, acutely no acme appears... Read More by user
Tags: cycle

Affidavit of Assumption 1

 The sum of the degrees of the vertices of a blueprint G is absolutely alert the admeasurement of G.Let S=\!,. We will calculation the amount of elements of S in two altered ways, using the adjustment of Bifold counting.The amount of a acme is to be the amount of edges adventure with that vertex. Accordingly acme v is the vertex-part of absolutely d(v) elements of S. Appropriately the amount of elements of S is the sum of the degrees of the vertices.The additional accessible way to calculation the amount of elements of S is in agreement of the edges. Acutely anniversary bend is adventure with absolutely two vertices. Accordingly bend e is the edge-part of absolutely 2 elements of S. Appropriately the amount of elements of S is alert the amount of edges.Since a set has the aforementioned amount of elements, behindhand of how you calculation them (unless, of course, you blow them), it follows that the sum of the degrees of the vertices is alert the amount of edges.In combinatorics, you ca... Read More by user

A-level Mathematics M3 Adaptable Strings and Springs

 The accustomed breadth is the breadth of an adaptable cord or bounce if it is not continued or compressed. An adaptable cord or bounce adventures a astriction if its breadth is greater than the accustomed length. In addition, a bounce adventures a compression if its breadth is beneath than its accustomed length. To abridge our analysis, we use the appellation astriction to accredit to both types of armament (i.e. astriction and compression). Also, we use the appellation addendum to accredit to the change in the breadth of the cord or spring. Thus, the addendum for a aeroembolism bounce is negative.According to Hookes Law, the addendum x is proportional to the astriction T activated to the adaptable cord or spring. Although Hookes Law holds alone up to the absolute of elasticity, we may cautiously accept its account unless contrarily told. We may address this accord in agreement of the accustomed breadth l and the modulus of animation lambda (which is a acreage of the adaptable cord or ... Read More by user
Tags: energy, force, elastic, natural, string, spring

A-level Mathematics M3 Annular Motion

 Consider a atom affective in a amphitheater with ambit r and centre at the agent O.Let mathbf denote the displacement of the particle. Using the angular displacement heta (as abstinent from the absolute x-axis) as a parameter, we havemathbf= r left[ egincos heta\ sin heta end ight]. To access the acceleration dotmathbf of the particle, we differentiate mathbf wrt time t. Applying the alternation rule, we have|-||=frac frac|-||=left( r frac left[ egincos heta\ sin heta end ight] ight) left( dot ight)|-||=r dot left[ egin-sin heta\ cos heta end ight]|}To actuate the dispatch ddotmathbf of the particle, we differentiate dotmathbf wrt t and get|-||=frac frac|-||=left( r frac left ight) left( dot ight)|-||=r dot left( dot left[ egin -cos heta\ -sin heta end ight] + ddot left[ egin-sin heta\ cos heta end ight] ight)|-||=-r dot^2 left[ egin cos heta\ sin heta end ight] + r dot ddot left[ egin-sin heta\ cos heta end ight]|}To abridge our expressions, we acquain... Read More by user

Top Academy Mathematics Extensions Added Adverse

 This area and the In calculus, adverse is a actual important operation activated to functions of absolute numbers. To differentiate a action f(x), we artlessly appraise the limit:lim_fracwhere the lim_ agency that we let h access 0. However, for now, we can artlessly anticipate of it as putting h to 0, i.e., absolution h = 0 at an adapted time. There are assorted notations for the aftereffect of adverse (called the derivative), for example:f(x) = lim_fracand: frac= lim_fracmean the aforementioned thing. We say, f(x) is the acquired of f(x). Adverse is advantageous for some purposes, but we shall not altercate why calculus was invented, but rather how we can administer calculus to the abstraction of breeding functions.It should be bright that if:g(x) = f(x)then:g(x) = f(x)the aloft law is important. If g(x) a closed-form of f(x), then it is accurate to differentiate both abandon to access a new breeding function.Also if:h(x) = g(x) + f(x)then:h(x) = g(x) + f(x)This can be absolute by se... Read More by user

Arbitrary courses

 On this page will be listed added mathematics courses for , which are not allotment of the binding University program but may be either analytic added to University program in the approaching either be accomplished in a university as advantageous arbitrary courses, as able-bodied as appropriate courses for accurate specializations.Additional courses in the acreage of class theory:Category of Endomorphisms (Currently not a allotment of university program but is anticipation to be recommended to be added even as an binding advance for assertive specializations.)[http://www.mathematics21.org/pseudomorphisms-category.xml Class of Endomorphisms]. Endomorphism of any class approach are class approach of assertive class theory. This produces a rather busy theory, abnormally in the case if the class approach of the accustomed class are fractional order.Study of these categories is important for specialists in algebraic logic, computer science, abstruse algebra, and some additional fields of ma... Read More by user

Geometry for elementary academy A affidavit of applesauce

 In mathematics, a rational amount is a absolute amount that can be accounting as the arrangement of two integers, i.e., it is of the anatomy :a/b area a and b are integers and b is not zero. An aberrant amount is a amount that cannot be accounting as a arrangement of two integers, i.e., it is not of the anatomy :a/b . The analysis of aberrant numbers is usually attributed to Pythagoras, added accurately to the Pythagorean Hippasus of Metapontum, who produced a affidavit of the applesauce of the sqrt. The adventure goes that Hippasus apparent aberrant numbers if aggravating to represent the aboveboard basis of 2 as a atom (proof below). However, Pythagoras believed in the absoluteness of numbers, and could not acquire the actuality of aberrant numbers. He could not belie their actuality through logic, but his behavior would not acquire the actuality of aberrant numbers and so he bedevilled Hippasus to afterlife by drowning.As you see, mathematics ability be dangerous.One affidavit of th... Read More by user
Tags: numbers

A-level Mathematics D1 Algorithms

 =Algorithms=The afterward is a skeleton for the agreeable of D1 algorithms, with the agreeable taken from AQA, OCR, OCR MEI and Edexcels specifications. Not all blueprint cover all of the afterward content.An algorithm is a absolute set of instructions which, if followed, will break a problem. They can be presented in assorted forms, e.g. words, pictures, breeze diagrams. Recipes, knitting patterns, instructions to set a VCR or body a board are all examples of algorithms we may appointment in accustomed life. The capital advantage of algorithms is that we can then use computational methods of analytic problems. Its rather simple for one to put the numbers 2, 5, 3, 1 and 4 in order, but it would yield much, abundant best for one to array a account of 1000 accidental numbers.You may charge to be able to chase or make breeze diagrams in adjustment to authenticate an algorithm. There are three basal shapes used: Given a accouter of 52 cards to be sorted according to their suit; their absol... Read More by user

Geometry for elementary academy Bisecting a articulation

 In this chapter, we will apprentice how to bifurcate a segment. Accustomed a articulation overline, we will bisect it to two according segments overline and overline. The architecture is based on [http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI10.html book I, hypothesis 10].# riangle ABD on overline.# on angle ADB using the articulation overline.# Let C be the circle point of overline and overline.# Both overline and overline are according to bisected of overline.# overline and overline are abandon of the boxlike triangle riangle ABD .# Hence, overline equals overline.# The articulation overline equals to itself.# Due to the architecture angle ADE and angle EDB are equal.# The segments overline and overline lie on anniversary other.# Hence, angle ADE equals to angle ADC and angle EDB equals to angle CDB .# Due to the triangles riangle ADC and riangle CDB congruent.# Hence, overline and overline are equal.# Back overline is the sum of overline and overline, anniversary of... Read More by user
Tags: angle, school, hence, equals

Geometry for elementary academy Bisecting an bend

 In this chapter, we will apprentice how to bifurcate an angle. Accustomed an bend angle ABC we will bisect it to two according angles. The architecture is based on [http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI9.html book I, hypothesis 9] # Accept an approximate point D on the articulation overline. # circ B,overline . # Let E be the circle point of overline and circ B,overline . # overline. # on overline with third acme F and get riangle DEF . # overline. # The angles angle ABF , angle FBC according to bisected of angle ABC .# overline is a articulation from the centermost to the ambit of circ B,overline and accordingly equals its radius.# Hence, overline equals overline.# overline and overline are abandon of the boxlike triangle riangle DEF .# Hence, overline equals overline.# The articulation overline equals to itself# Due to the triangles riangle ABF and riangle FBC congruent.# Hence, the angles angle ABF , angle FBC according to bisected of angle ABC .We showed a ... Read More by user

Testing Data types of tests

 A statistical analysis is consistently about one or added ambit of the anxious citizenry (distribution). The appropiate analysis depends on the blazon of absent and another antecedent about this (these) parameter(s) and the accessible advice from the sample.It is accepted that British accouchement accretion added weight lately. Appropriately the citizenry beggarly μ of the weight X of accouchement of lets say 12 years of age is the constant at stake. In the contempo accomplished the beggarly weight of this accumulation of accouchement angry out to be 45 kg. Appropriately the absent antecedent (of no change) is::,H_0: mu = 45.As we doubtable a accretion in weight, the another antecedent is::,H_1: mu > 45.A accidental sample of 100 accouchement shows an boilerplate weight of 47 kg with a accepted aberration of 8 kg.Because it is reasonable to accept that the weights are commonly distributed, the adapted analysis will be a t-test, with analysis statistic::T = frac - 45}sqrt.Under the a... Read More by user
Tags: types, children, testing, value, sample, hypothesis

Addition

 The prerequisites for this book are just as one would think, trigonometry, and some Calculus. As for the bulk of Calculus, if alotof classes acquaint the accepted adjustment of barter would be a acceptable time; but absolutely alone the abject ability of an basic is needed.The capital purpose of this argument is to explain a assertive adjustment ofsolving a ample chic of basic calculus problems, affectionallyknown as algebraic substitution.One of the causes why trigonometry has such a bad acceptability isprobably because there are a lot of algebraic identities, andthey assume to accept no credible actual application. Whiletrigonometric barter may or may not be advised as an answerto the latter, it absolutely puts the above in acceptable use.The abstraction abaft the algebraic barter is infact quitesimple: to abate complicated expressions involving aboveboard roots topolynomials of accepted algebraic functions. Integrals involvingpolynomial expressions (albeit in algebraic functions) ar... Read More by user

Geometry for elementary academy Amalgam a triangle

 In this chapter, we will appearance how to assemble a triangle from three segments. The architecture is based on [http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI22.html Book I, hypothesis 22] Given three band segments overline, overlineand overlinewe body a trianlge whose abandon according the segments.# overlineto point A. If you accept abandoned how to do this, chase the instructions in the antecedent section. Your architecture should attending like the blah curve in the account below. Alarm the new band overline Its a acceptable abstraction to abolish your architecture curve now, so all thats larboard are the four band segments apparent below. # overline to point B Your architecture should attending like the blah curve in the account below. Alarm the new band overline # circ A,overline , whose centermost is A and ambit is overline.# circ B,overline , whose centermost is B and ambit is overline.# Let J be an circle point of circ A,overline and circ B,overline . # overline.... Read More by user

 

 

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