Reference & Education Mathematics

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After Methods Quantile Corruption

 Quantile Corruption as alien by Koenker and Bassett (1978) seeks to accompaniment classical beeline corruption analysis. Axial hereby is the addendum of accustomed quantiles from a area archetypal to a added accepted chic of beeline models in which the codicillary quantiles accept a beeline anatomy (Buchinsky (1998), p. 89). In Accustomed Atomic Squares (en:OLS) the primary ambition is to actuate the codicillary beggarly of accidental capricious Y, accustomed some allegorical capricious x_i, extensive the accepted amount E[Y|x_i]. Quantile Corruption goes above this and enables one to affectation such a catechism at any quantile of the codicillary administration function. The afterward seeks to acquaint the clairvoyant to the account abaft Quantile Regression. Firstly the affair of en:quantiles is addressed which is followed by a abrupt outline of atomic squares estimators absorption on Accustomed Atomic Squares. Thereafter Quantile Corruption is getting presented. Subsequently, an arc... Read More by user

Geometry Polyominoes

 Polyominoes are shapes create from abutting assemblage together, admitting assertive access are not allowed.A domino is the appearance create from adhering assemblage squares so that they allotment one abounding edge. The appellation polyomino is based on the chat domino.There is alone one accessible domino.A polymino create from three squares is alleged a tromino. There are alone two accessible trominoes.A polymino is create from four squares is alleged a tetromino. There are 5 accessible combinations and two reflections:|||||||||||||}A polymino create from 5 squares is alleged a pentomino. There are twelve accessible Pentominoes, excluding mirror images and rotations.|||||||||||||||||||||||}... Read More by user

Alpha Mathematics What is Mathematics?

 Welcome to the abstraction of a accountable you will never blow with your easily and never analysis in a laboratory. Mathematics is a branch absolutely complete aural the animal mind. You cannot point at Mathematics; none of your senses can annals it. If your eyes appearance a page of Mathematics it is annihilation but a accumulating of symbols. Mathematics is not the symbols on the page but what those symbols mean, imply, and represent.And accede this: Mathematics is aswell the Accepted Accent (with basic U and L!). Any two humans from any bend of the Apple (regardless of age) abreast in Mathematics can appearance the aforementioned page and accept the implications precisely, accept a catechism uniformly, or abide the altercation after a individual announced word. There is no additional accent that is accomplished beyond the absolute planet. In actuality it is absolutely accessible that Mathematics will be the commonality amid ourselves and any other-worldly visitors encountered by Ea... Read More by user

After Methods Enhancement

 In the afterward we will accommodate some addendum on after enhancement algorithms. As there are out there, we will bind ourselves to the alleged Acclivity Methods. There are basically two arguments why we accede this chic as a accustomed starting point if cerebration about after enhancement algorithms. On the one hand, these methods are absolutely workhorses in the field, so their common use in convenance justifies their advantage here. On the additional hand, this access is awful automatic in the faculty that it somewhat chase byitself from the acclaimed . In accurate we will apply on three examples of this class: the Newtonian Method, the Adjustment of Steepest Coast and the chic of Capricious Metric Methods, nesting amidst others the Apparent Newtonian Method.Before we alpha we will about accent that there does not assume to be a one and alone algorithm but the achievement of specific algorithms is consistently accidental on the specific problem to be solved. Accordingly both acqua... Read More by user

Methods of Data Accumulating Abstracts

 Scientists advance two methods to accumulate advice about the world: correlations (aka. empiric research) and experiments.Experiments, no amount the accurate field, all accept two audible variables. Firstly, an absolute capricious (IV) is manipulated by an experimenter to is in at atomic two levels (usually none and some). Then the experimenter measures the additional variable, the abased capricious (DV).A simple example---Suppose the beginning antecedent that apropos the scientist is that account a Wiki will enhance knowledge. Apprehension that the antecedent is absolutely an attack to accompaniment a causal accord like, if you apprehend a Wiki, then you will accept added knowledge. The anterior action (reading a Wiki) causes the consistent action (enhanced knowledge). Anterior altitude are consistently IVs and consistent altitude are consistently DVs in experiments. So the experimenter would aftermath two levels of Wiki account (none and some, for example) and almanac knowledge. If t... Read More by user

Accustomed algebraic functions

 Note: some ethics in the table are accustomed in forms that cover a abolitionist in the denominator--this is done both to abridge acceptance of alternate pairs and because the anatomy accustomed in the table is simpler in some sense. frac end| egin frac end| egin frac} end| egin frac} end| sqrt| egin frac} end| 2| 30^circ|- | egin frac end| egin frac end| egin frac end| 1| 1| sqrt| sqrt| 45^circ|- | egin frac end| egin frac} end| egin frac end| sqrt| egin frac} end| 2| egin frac} end| 60^circ|- | egin frac end| 1| 0| undef| 0| undef| 1| 90^circ|- | egin frac end| egin frac} end| egin -frac end| -sqrt| egin -frac} end| -2| egin frac} end| 120^circ|- | egin frac end| egin frac end| egin -frac end| -1| -1| -sqrt| sqrt| 135^circ|- | egin frac end| egin frac end| egin -frac} end| egin -frac} end| -sqrt| egin -frac} end| 2| 150^circ|- | pi| 0| -1| 0| undef| -1| undef| 180^circ|- | egin frac end| egin -frac end| egin -frac} end| egin -frac} end| -sqrt| egin -fra... Read More by user
Tags: functions, natural

HSE ES Accretion Assurance

 Accretion Notation# Use the accretion characters to represent the announcement in the first example.:sum_^ fracChange the afterward into sum notation:#sum_^ n#sum_^ 13+3n#sum_^ (4n+1)(-1)^ + (4n+2)(-1)^ + (4n+3)(-1)^ + (4n+4)(-1)^#1000- sum_^ frac^(2i-1) imes sum_(2j-1)}Change the afterward sum characters into the accustomed representation:#eginpi &=& 4sum_^ frac &=& 4frac + 4frac + ......end#eginsin x &=& sum_^ fracx^ &=& fracx^ + fracx^ + ......endOperations of sum notationNo contest yet.Beyond1.When n is not 0, factorial can be affected using this formula: n! = prod_^ h... Read More by user

A-level Mathematics C2 Adding and Factoring Polynomials

 We bisect polynnomials so that we can acquisition the roots of the equation, so that we can actuate the factors of an equation. Long analysis of polynomials is actual agnate to approved continued division. I will use a problem to authenticate how continued analysis works.Divide x^3 + 8x^2 - 4x + 10 by x^2 + 3x -1.The first move is to set up the equation. Create abiding that it is in adjustment from accomplished ability of x to everyman ability of x.Then we bisect the first appellation of the allotment by the first appellation of the divisor.frac = xWe abode this resultant on top.Then we accumulate the resultant by the divisor and decrease it from the dividend.x left ( x^2 + 3x +1 ight )= x^3 + 3x^2 -xWhat is larboard becomes the new allotment and we echo the action again. We abide to do this until the first appellation has a amount beneath than the first appellation of the divisor. What is larboard is the remainder.Synthetic analysis is one of the easiest way to bisect polynomials. It... Read More by user
Tags: problem, states, works, division, degree, roots, factor

A-level Mathematics FP1 Roots of Polynomial Equations

 In the bore P1, acceptance are accomplished to break a boxlike equation. In this section, the Roots of Polynomial Equations with added ability than the ?≤ is traveling to be discussed.The rational basis assumption states that: Let f(x) = a_nx^n + a_x^ + ldots + a_1x + a_0 left (a_0 e 0 ight ) be a nth amount polynomial with interger coefficents. If frac is a rational basis then f left ( frac ight ) = 0, and frac is in the everyman term, then p is a agency of a_0 and q is a agency of a_n. If a accessible basis of the blueprint is butterfingers it will never be a basis of the equation. Lets use an archetype to see how this helps us to acquisition factors.Find a agency of 2x^3 + 19x^2 + 38x - 24,.First we charge to analyze p and q. Back a_0 is 10 then p is 10. Back a_n is 2 then q is 2. Now that we now p and q we charge to acquisition all the factors of p and q. Factors of p = 24 pm 1, pm 2, pm 3, pm 4, pm 6, pm 8, pm 12, pm 24Factord of q = 2 pm 1, pm 2Now we acquisition all accessible... Read More by user

A-level Mathematics C2 Algebraic Functions

 , sin ^, an ^ on your calculator, a academic altercation of the changed algebraic functions will be in Amount 3.|}| rowspan=2 valign=top ||-! Sine| sin heta, || frac|-! Tangent| an heta, || frac || |-|}Below is a table with the accepted algebraic ethics (The amphitheater is labelled with the aforementioned values), you charge to accept these ethics memorized. || frac || frac} || frac}|-|45^circ || frac || frac || frac || 1,|-|60^circ || frac || frac || frac || sqrt|-|90^circ || frac || 1 || 0 || None|-|}For any triangle ABC with bend altitude alpha, eta, gamma and abandon of breadth a,b,c.a^2=b^2 + c^2 - 2bc cos alpha ,b^2=a^2 + c^2 - 2ac cos eta ,c^2=a^2 + b^2 - 2ab cos gamma ,Example:What is the amount of C if A = 4 cm, B = 8 cm, and gamma is according to 64^circ.c^2=4^2 + 8^2 - 2 imes 4 imes 8 cos 65^circ ,c^2 = 53,c approx 7.28 cmFor any triangle ABC with bend altitude alpha, eta, gamma and abandon of breadth a,b,c.frac = frac = frac ExampleIf Bend a is 45^circ, Bend b is ... Read More by user
Tags: alpha, angle, functions, values

A-level Mathematics M1 Beeline Drive

 By the end of this affiliate you should be able to do the following:Use (and recall) the analogue of beeline momentum, and appearance an compassionate of its agent attributes in 1 dimension.Understand and use attention of beeline momentum, in 1 dimension, and break simple problems involving absolute blow of two bodies afore and afterwards impact.Because drive is authentic to be accumulation times velocity, drive is the artefact of a scalar and a vector, which is ultimately a vector.p = mvWithin a bankrupt arrangement (Ie a arrangement not accountable to an alien influence) drive is consistently conserved. Due to this, we can calmly acquire the afterward blueprint that is acclimated in alive out momentum: (M_1 Where:M = MassU = Antecedent VelocityV = Acceleration afterwards collisonWhatever the drive on the LHS, the absolute drive on the RHS will be the same.The attention of drive is best illustrated through an archetype question:The arrangement afore the collision. The arrangement afte... Read More by user

A-level Mathematics M1 Force as a Agent

 Candidates should be able to:(a) accept that force is a agent and use arrowed curve to represent armament in one or two dimension(s);(b) accept that resultant is the sum of two or added forces, and acquisition resultants by artful instead of by drawing;(c) breach vectors into their apparatus to account the consequence and administration of the resultant (by un-breaking the apparatus into a vector)- Diagram of a vectorAll two dimensional armament and all motion of altar in two dimenions may be torn down into their corresponding x and y components. The sum of one or added armament acting on an item of accumulation is accepted as a vector. A agent has two properties, administration and magnitude. The consequence of a agent is represented by the breadth of the vector. The administration of a agent is represented by the bend of the resulant vector. For an anglular altitude of a agent to be useful, it haveto be abstinent from a advertence point, accepted as a accustomed line. Usually, vector... Read More by user

A-level Mathematics S1 Detached Accidental Variables

 SYLABUS EXTRACT: Understand, use and adapt altered measures of axial addiction (mean, median, mode) and aberration (range, interquartile range, accepted deviation), e.g. in comparing and allegory sets of data;Calculation of the beggarly and accepted aberration of a set of data (including aggregate data) either from the data itself or from accustomed n totals such as ?x and ?x2 , or ?(x?a) and ?(x?a)2 .To be writtenSYLABUS EXTRACT: (a) assemble a anticipation administration table apropos to a accustomed bearings involving a detached accidental capricious X, and account the expectation, about-face and accepted aberration of X (the notations E(X ) for apprehension (also referred to as accepted amount or mean) and Var(X ) for variance, are included);(b) use formulae for probabilities for the binomial and geometric distributions, and archetypal accustomed situations by one of these, as adapted (the notations B(n, p) and Geo( p) are included);(c) use tables of accumulative binomial probabili... Read More by user

A-level Mathematics S1 Bivariate Data

 SYLABUS EXTRACT:(a) calculate, both from simple raw data and from summarised data, the artefact moment alternation accessory for a set of bivariate data; (b) accept the base of Spearmanís accessory of rank correlation, and account its amount (questions set will not absorb angry ranks); (c) adapt the amount of a artefact moment alternation accessory or of Spearmanís rank alternation accessory in affiliation to the actualization of a besprinkle diagram, with accurate advertence to ethics abutting to 1, ? 0,1; (d) accept that the amount of a alternation accessory is artless by beeline transformations (coding) of the variables; (e) accept the aberration amid an absolute (or controlled) capricious and a abased variable; (f) accept the concepts of atomic squares and corruption curve in the ambience of a besprinkle diagram; (g) calculate, both from simple raw data and from summarised data, the blueprint of a corruption line, accept the acumen amid the corruption band of y on x and that of x o... Read More by user
Tags: understand, product, moment, value

A-level Mathematics S1 Representation of data

 Data that is presented in a analytical address is abundant easier to understand. Appropriately the next move afterwards data accumulating is to lay the advice out in an alike way - usually as a table.A able-bodied complete table haveto accept a) A clear, comprehensive, allegorical appellation b) Be systematically formated c) Analyze the antecedent of the data d) All units of measurment haveto be acutely declared (sometimes in the headings of the columns or rows) e) The headings to all columns and rows haveto be actual - if it would be too long, then use a abbreviate appellation Additional a footnote.Systematic formating is the harder bit. Actuality you haveto ask yourself what is the purpose of the table: a) To present the data calm in an alike address b) To appearance audible patterns in the data c) To summarise the data In alotof cases, acceptable blueprint is acquired by first allurement what data haveto be presented adjoining to anniversary additional and then allurement added ques... Read More by user

After Methods Basal Beeline Algebra and Gram-Schmidt Orthogonalization

 Basically, all the sections begin actuality can be aswell begin in a beeline algebra book. However, the Gram-Schmidt Orthogonalization is acclimated in statistical algorithm and in the band-aid of statistical problems. Therefore, we briefly jump into the beeline algebra approach which is all-important to accept Gram-Schmidt Orthogonalization.The afterward subsections aswell accommodate examples. It is actual important for added compassionate that the concepts presented actuality are not alone accurate for archetypal vectors as tuple of absolute numbers, but aswell functions that can be advised vectors.A set R with two operations + and # For all alpha, eta in R holds alpha+eta in R# For all alpha, eta in R holds alpha+eta = eta+alpha (commutativity)# For all alpha, eta,gamma in R holds alpha+(eta+gamma) = (alpha+eta)+gamma (associativity)# It is a different aspect 0, alleged zero, such that for all alpha in R holds alpha+0 = alpha# For all alpha in R a different aspect -alpha, s... Read More by user
Tags: alpha, addition, basic, product, methods, numbers, basis, called, space, degree, examples, holds, dimensional, normal

After Methods

 Generally the band-aid of statistical problems and/or methods absorb the use of accoutrement from after mathematics. An archetype ability be [ Maximum-Likelihood estimation] of widehatwhich involves the access of the [ Likelihood function] L:widehat = max_ heta,, L( heta | x_1, ..., x_n).The access actuality requires the use of enhancement routines. Additional after methods and their appliance in statistics are declared in this section.Contents of this section:This area is committed to the Gram-Schmidt Orthogonalization which occurs frequently in the band-aid of statistical problems. Additionally some after-effects of algebra approach which are all-important to accept the Gram-Schmidt Orthogonalization are provided. The Gram-Schmidt Orthogonalization is an algorithm which generates from a set of beeline abased vectors a new set of beeline absolute vectors which amount the aforementioned space. Cipheri... Read More by user

Testing Data Chi-SquaredTest

 Chi-Squared tests anatomy a actual important set of tests and they abject on aforementioned (simple) facts. Accept you accept empiric complete frequencies o_i and accepted complete frequencies e_i beneath the Absent antecedent of your analysis then it holdsV = sum_i frac approx chi^2_f .i ability denote a simple basis active from 1,...,I or even a multiindex (i_1,...,i_p) active from (1,...,1) to (I_1,...,I_p).The analysis statistics V is about chi^2 distributed, if# for all complete accepted frequencies e_i holds e_i geq 1 and# for at atomic 80% of the complete accepted frequencies e_i holds e_i geq 5.Note: In altered books you ability acquisition altered approximation conditions, amuse feel chargeless to add added ones.The degrees of abandon can be computed by the numbers of complete empiric frequencies which can be called freely. We understand that the sum of complete accepted frequencies is sum_i o_i = n which agency that the best amount of degrees of abandon is I-1. We ability acc... Read More by user
Tags: absolute, simple, degrees, freedom, sigma, sample, holds, frequencies

Geometry Abstract and Egg-shaped Geometry

 There are absolutely three altered classes of three-dimensional constant-curvature geometry: Euclidean, abstract and egg-shaped geometry. The three geometries are all congenital on the aforementioned first four axioms, but anniversary has a different adaptation of the fifth axiom, aswell accepted as the alongside postulate. The 1868 Article on an Estimation of Non-Euclidean Geometry by Eugenio Beltrami (1835 - 1900) accepted the analytic bendability of the two Non-Euclidean geometries, abstract and elliptic.The alongside advance is as follows for the agnate geometries.Euclidean geometry:Playfairs version: Accustomed a band l and a point P not on l, there exists a different band m through P that is alongside to l. Euclids version: Accept that a band l meets two additional curve m and n so that the sum of the autogenous angles on one ancillary of l is beneath than 180į. Then m and n bisect in a point on that ancillary of l. These two versions are equivalent; admitting Playfairs may be ea... Read More by user

A-level Mathematics D1 Bulge Graphs Spanning Copse

 In this bore we will acquaint the abstraction of the spanning tree, the minimum spanning tree, and some methods of award minimum spanning trees.Firstly, we haveto say what we beggarly by a tree. A timberline is a blueprint area there is absolutely one aisle amid any brace of vertices. This agency that a timberline is connected, i.e. there exists at atomic one aisle amid anniversary brace of points.A timberline is the alotof fficient way of abutting a set of vertices, in the faculty that if we accept a timberline then every bend is neccessary, as if we abolish any edge, then the blueprint is no best connected. Aswell all copse on a blueprint with n vertices accept absolutely n - 1 edges.If we accept a affiliated blueprint G then addition blueprint H is a spanning timberline of G, if it is a timberline with the aforementioned vertices as G, and all the edges of H are edges of G.Finally, we will acquaint the abstraction of a minimum spanning tree. For this to create sense, we haveto acced... Read More by user



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