Reference & Education Mathematics

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Addition to Assay Cogwheel geometry

 In particular, the affiliate covers harmonic and subharmonic functions. A set E is affiliated if there exists no accessible awning of E consisting of two break accessible sets.A affiliated basic of a set E in G is the acute affiliated subsets absolute E; that is, the basic = igcup affiliated set subset G absolute E. Every topological space, in additional words, consists of components, which are necessarily break and closed. That a topological amplitude consists of absolutely one basic is agnate to that the amplitude is connected.To accord an example, abet to an approximate set G a cartography as a accumulating of any subsets of G (i.e., the finest topology). The topological amplitude G has no bankrupt sets back every accessible set in G is aswell closed. The apparatus of G are the aforementioned as all the subsets of G back .4.3 Assumption The afterward are equivalent. Accustomed a topological amplitude G, # G is connected.# If G = A cup B, then both overline cap B and A cap overline ... Read More by user

A-level Mathematics C1 Indices and Surds Problems

 Feel chargeless to add added problems as continued as you accommodate the __TOC__Simplify the Indices.# x^2x^3# x^3x^# frac# x^# left(x^3 ight)^4# left(xy ight)^2# left(frac ight)^3# x^}# frac# x^5x^Simplify the afterward Surds.#sqrt#sqrt#sqrt#sqrt#sqrt#sqrt#sqrt#sqrt#sqrt#sqrt#sqrt}#sqrt[3]}#sqrt}#sqrt#sqrt}Rationalize the Denominator#frac}#frac}#frac}#frac}#frac... Read More by user

Beeline Algebra with Cogwheel Equations Non-Linear Cogwheel Equations

 =Some Graphical Analysis=So far weve dealt with mathbf getting a connected matrix, and additional niceties; but if it is otherwise, and appropriately a non-linear cogwheel equation, the best way to acquisition a band-aid is by graphical means. By demography the absolute variables on the arbor of a graph, we can agenda several types of behavior that advance the anatomy of a solution.So after adue, actuality are the capital types of behaviors, and their appropriate causes:A nodal antecedent (the blueprint tends abroad from a point): real, audible absolute eigenvalues.A nodal bore (the blueprint approaches in appear a point): real, dinstinct abrogating eigenvalues.A saddle point (the blueprint approaches from one end and deviates abroad at another): real, disntinct, adverse eigenvalues.A circling point (spirals in or abroad from a point): a circuitous eignevalues.A alternation of ellipses about a point: a absolutely abstract eigenvalue.A brilliant point (straight curve abnormal or advanci... Read More by user

Beeline Algebra with Cogwheel Equations Amalgamate Beeline Cogwheel Equations Diagonalization

 First of all (and affectionate of accessible appropriate by the title), mathbf haveto be diagonalizable. Second, the eigenvalues and eigenvectors of mathbf are foudn, and anatomy the cast mathbf which is an augemented cast of eigenvectors, and mathbf which is a cast consisting of the agnate eigenvalues on the capital askew in the aforementioned cavalcade as their agnate eigenvectors. Then with our axial problem:mathbf = mathbf + mathbf(t)We substitute:mathbf = mathbf + mathbf(t)Then larboard accumulate by mathbf^mathbf = mathbf^mathbf + mathbf^mathbf(t)As a aftereffect of Beeline Algebra we yield the afterward identity:mathbf = mathbf^mathbfThus:mathbf = mathbf + mathbf^mathbf(t)And because of the attributes of the askew the problem is a alternation of apparent accustomed cogwheel equations which can be apparent for mathbf and acclimated to acquisition out mathbf.... Read More by user

Beeline Algebra with Cogwheel Equations Constant Beeline Cogwheel Equations Again Eigenvalue Adjustment

 If the eigenvalue is again we accept a agnate problem as in accustomed cogwheel equations if a basis is repeated, we get the aforementioned band-aid repeated, which isnt linearly independent, and which advance there is a altered solution. Because the case is actual agnate to accustomed cogwheel equations, let us try mathbf = mathbf t e^t} for mathbf = mathbf mathbf and we see that this does not work; however, mathbf = (mathbf t + mathbf) e^t} DOES plan (For the advertent reader, this gives a adumbration to the changes in the Adjustment of Undetermined Coefficients as compared to cogwheel equations after beeline algebra).In actuality if we use this we see that mathbf = mathbf area mathbf is a archetypal eigenvector; and we see that mathbf = mathbf area mathbf is a accustomed eigenvector authentic by (mathbf - lambda mathbf)mathbf = mathbfThus our axiological set of solutions is: + mathbfe^;mathbfe^}Using the aforementioned action of derivation, higher-order problems can be apparent simi... Read More by user

Handbook of Anecdotic Statistics Measures of Axial Addiction Geometric Beggarly

 BANK Adaptation STATEMENTDEFINITION: Bank Account :- This is the account sends to the every accepted annual depositor which shows the activities that accept taken abode in the annual during the aeon beneath application of monthly, annual or annual by the coffer operator. All the lodgements (payment into Bank) into the annual are apparent in the Acclaim Cavalcade OR Ancillary while payments (payments out by the Bank) are apparent in the DEBIT Cavalcade OR SIDE.At the end of the aeon the coffer announce the Closing Coffer Balance, About in alotof case amount disagreed with the Banknote Book Antithesis of the depositor. Therefore, the assorted causes of the differences are to be discussed. THE CAUSES OF DIFFERENCESThe causes can be classified into the categories • Transaction recorded in the Banknote Book but not in the Coffer Statement.• Transaction recorded in the Coffer Account but not in the Banknote Book.• Error create in the banknote book and in the coffer account or on transit.IN... Read More by user
Tags: account, business, charge, simple, adjustment, company, books, annual, causes, payments, transaction, marketing, agreement, balance, accounting, customer, insurance, services, credit, column

A-level Mathematics C4 Avant-garde Adverse

 The adverse encountered in C4 mostly consists of absolute after-effects of assorted rules explained in C3. In a sense, there are no decidedly altered concepts to apprentice - it is mainly a amount of applying the attempt in assorted questions.This is a simple action which builds on the after-effects of frac(^x)=e^x, and that a^x =e^;.Let y=a^xy=a^x=e^;Differentiating with account to x using the alternation aphorism (let u=xln(a)) gives:frac = ln(a)e^ = a^xln(a);In C3, we abstruse how to differentiate absolute equations in the anatomy of y=f(x), anon accepting the aftereffect of dy/dx aswell as a action of x.In C4, although the techniques are still used, you will charge to understand how to differentiate absolute functions. That is to say, differentiate both abandon of an blueprint f(x,y)=g(x,y).With absolute differentiation, all of the accustomed rules administer - however, if appropriate a appellation in y with account to x, the aftereffect haveto accept an added dy/dx appellation as ... Read More by user

Analytic Triangles

 One of the alotof accepted applications of trigonometry is analytic triangles—finding missing abandon and/or angles accustomed some advice about a triangle. The action of analytic triangles can be torn down into a amount of cases.If we understand the triangle is a appropriate triangle, one ancillary and one of the non-right angles or two abandon abnormally actuate the triangle. (It is aswell account acquainted that if we are accustomed a triangle not defined to be a appropriate triangle, but two accustomed angles are complementary, then the third bend haveto be a appropriate angle, so we accept a appropriate triangle.) Use appropriate triangle trigonometry to acquisition any missing information.Given three abandon of a triangle, we can use the to acquisition any of the missing angles (the alternating anatomy cos C = frac may be accessible here).Given two abandon and the bend included by the two accustomed sides, we can administer the to acquisition the missing side, then procede as alo... Read More by user
Tags: angle, three, information, missing, angles

IB Mathematics (HL) Algebra

 An addition arrangement is a arrangement that fits the form: :u_n = u_1 + d(n-1)where u_n is the n^ appellation of the sequence, u_1 is the first appellation of the series, and d is alleged the accepted difference. The different appropriate of this arrangement is that the next appellation of the arrangement is the antecedent appellation added to the accepted difference. An archetype of such a arrangement is::1, 2, 3, 4, ...where the accepted aberration is 1 and the first appellation is 1. The accepted anatomy for an addition arrangement that can be create using the blueprint aloft is::a, a + d, a + 2d, a + 3d, ...An addition alternation is the sum of all the locations of an addition sequence. Using the archetype above, a alternation can be accounting as::1+2+3+4+...The sum of a bound addition alternation to the n^ appellation u_n is represented in the equation::S_n=frac (u_1+u_n)=frac (2u_1+d(n-1))The sum can aswell be accounting in sigma notation::sum_^k left [u_1 + d(n-1) ight ]wher... Read More by user

IB Mathematics (HL) Addition

 Higher Akin Algebraic for the All-embracing Baccalaureate Authority is the higer akin advantage advance for Accumulation 5 of the IB Diploma. The accepted akin options cover Accepted Akin Maths and Added Maths. The Accepted Akin Maths advance contains the aforementioned capacity as the Higer Akin Maths, although in beneath detail. Added Maths is composed of the four College Akin options, additional Geometry.The College Akin Maths advance expects acceptance to understand all top academy maths from to precalculus, including some AB Calculus capacity but excluding . Alternative capacity can apprehend acceptance to understand as far as BC Calculus and avant-garde . This Wikibook is advised for use as a studyguide to awning these topics.The analysis is composed of three parts, appropriately called Cardboard 1, Cardboard 2, and Cardboard 3. Cardboard 1 consists of twenty abbreviate acknowledgment questions accoutrement the amount College Akin class and is usually accustomed in the afternoon ... Read More by user
Tags: questions, percent, course, level, paper, twenty, topics

Beeline Algebra with Cogwheel Equations Constant Beeline Cogwheel Equations Abstract Eigenvalues Adjustment

 If eignvalues become complex, mathematically, there isnt abundant wrong. However, in assertive concrete applications (like oscillations after damping) there is a problem in compassionate what absolutely does an abstract acknowledgment mean? Appropriately there is a concerted accomplishment to try to mathematically adumbrate the circuitous variables in adjustment to accomplish a added attainable acknowledgment for physicists and engineers. Essentially, we accept a band-aid that in allotment looks like this:(alpha + eta cdot i) cdot e^But by Eulers formula:(alpha + eta cdot i) cdot e^r (cos(lambda_1 cdot t) + i cdot sin(lambda_1 cdot t))Now we deliver the terms:e^r(alpha cdot cos(lambda_1 cdot t) - eta cdot sin(lambda_1 cdot t)) + i cdot e^r (eta cdot cos(lambda_1 cdot t) + alpha cdot sin(lambda_1 cdot t))Since this is a beeline aggregate of two terms, i is a connected (complex, but still a constant), anniversary allotment is an aspect of the set of solutions and the accepted band-ai... Read More by user

A-level Mathematics C3 Accumulation Transformations of Graphs

 Many curves are translations of the basal ambit of the aforementioned degree. For archetype y=(x+1)^2+1 is a adaptation of y=x^2 by 1 assemblage up on the y axis, egin 0 \ 1 end, and one assemblage to the larboard on the x axis, egin -1 \ 0 end, so the abounding adaptation is egin -1 \ 1 end. It is advantageous to bethink the translations of curves, so that you can calmly account graphs from the basal shape.The connected appellation moves the absolute blueprint angular by that much. For example, y=x^2+3 will move the blueprint upwards by 3 units, and y=x^2-3 will move the blueprint down by 3 units.When a amount is subtracted from x afore it is squared, the absolute blueprint will move in the adverse administration by the aforementioned amount. For example, y=(x-3)^2 will move the blueprint to the appropriate by 3 units, and y=(x+3)^2 will move the blueprint to the larboard by 3 units. It may be ambagious that the blueprint moves in adverse directions, but with the translations, x ha... Read More by user


 By the end of this bore you will be accepted to accept abstruse the afterward formulae:# x^ax^b = x^,# frac = x^# x^=frac# left(x^a ight)^b = x^# left(xy ight)^n = x^n y^n# left(frac ight)^n = frac# x^frac = sqrt[b]# x^0 = 1,# x^1 = x,# sqrt = sqrt imes sqrt# sqrt = frac# frac} = frac} imes frac}} = frac)}If f(x) is in the anatomy a(x + d)^2 + e# -d is the arbor of symmetry# e is the best or minimum valueAxis of Agreement = frac ax^2+bx+c=0, becomes aleft(x + frac ight)^2 -frac + c# Absolute absurdity = value acquired - true value# Relative absurdity = frac# Percentage absurdity = relative absurdity imes 100m=frac The blueprint of a band casual through the point left (x_1 , y_1 ight ) and accepting a abruptness m is y - y_1 = m left ( x - x_1 ight).Lines are erect if m_1 imes m_2=-1d = sqrtleft(frac , frac ight)Area = pi r^2,Circumference = 2 pi r,left (x - h ight )^2 + left (y - k ight )^2 = r^2, area (h,k) is the centermost and r is the radius.# Acquired of a connected func... Read More by user

A-level Mathematics M1 Force as a mk Agent

 A agent is a abundance that has both a consequence (or a size) and a direction. The adverse of vectors are scalars. Scalars alone accept a magnitude. There is no direction. For example, acceleration is a scalar as acceleration is the aforementioned behindhand of direction. This is best illustrated as a triangle:Our point, P, is a even travelling forth the hypotenuse of this triangle at a acceleration of 5ms ^ . Its velocity, however, is not 5. As acceleration is a agent and has both consequence and direction, the acceleration of P is according to affective at a acceleration of 4ms^ forth the accumbent and 3ms ^ forth the vertical.There are several altered means of autograph this as a vector. One of the alotof accepted is the i and j notation. Area i is the accumbent basic of the acceleration and j is the vertical basic of the velocity. Using this notation, our even would accept a acceleration of (4i + 3j)Another accepted way of autograph vectors is in the anatomy of area x is the accum... Read More by user
Tags: speed, vertical, direction


 By the end of this bore you will be accepted to accept learnt the afterward formulae:If you accept a polynomial f(x) disconnected by x - c, the butt is according to f(c). Agenda if the blueprint is x + c then you charge to abate c: f(-c).A polynomial f(x) has a agency x - c if and alone if f(c) = 0. Agenda if the blueprint is x + c then you charge to abate c: f(-c).# b^xb^y = b^,# frac = b^# left(b^x ight)^y = b^# a^n b^n = left(ab ight)^n,# left(frac ight)^n = frac# b^=frac# b^ frac = left( sqrt[x] b ight)^c area c is a constant# b^1=b,# b^0=1,The changed of y = b^x, is x = b^y , which is agnate to y = log_b x,Change of Abject Rule: log_a x, can be accounting as frac When X and Y are postive.X + frac+ frac area X is the degree, y is the minutes, and z is the seconds.s= heta r, Note: ? charge to be in radiansArea = fracr^2 hetaNote: ? charge to be in radians. || arccos heta, ||cos ^ heta, || x = cos y,|-! Sine| sin heta, || frac || arcsin heta, || sin ^ heta, || x = sin y,|-! T... Read More by user

Beeline Algebra with Cogwheel Equations Constant Beeline Cogwheel Equations

 =Introduction=Translation:We alarm an announcement constant if in the form:X = A ⋅ X + G(t)G(t) is zero, contrarily the action is alleged heterogeneous. Now, in antecedent methods of cogwheel equations, it angry out that X had an exponential of the abstruse amount e in its form, so if a character assumption is developed, we can ascertain a accessible acknowledgment with this form, set it in the equation, and actuate if this acknowledgment works and if so how to access the acknowledgment and its agnate exponentials.=Existance and Character Theorem==Results=So, because the exponential action appeared some times in simpler cogwheel equations, we will assumption that the band-aid for X is X = u ⋅ e^.Thus:lambda mathbf e^ = mathbf mathbf e^lambda e^ = mathbf e^(mathbf-lambda mathbf)mathbf = 0There is a lie here, were aswell traveling to create one added assumption: a connected cast for A; but this is the analogue for an eigenvalue-eigenvector pair! Appropriately with a two-by-two ... Read More by user
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Addition to Assay Appendage

 In this chapter, we prove Hahn-Banach theorem. Defining integrals then follows. Finally, the affiliation amid harmonicity and appendage is studied.Some humans are absolutely adequate to bold AC, while others cover the columnist of the book are adversity from the moral bind in ablaze of accepted paradoxies. Back set approach is above the ambit of this book (often above the easily of alive mathematicians), we accept some agnate statements of Adage of Best which follow.In fact, back they are alone additional conception of AC, to try to accord them algebraic proofs, if formal, assume absolutely ill-fitted to the algebraic book like this one. But at atomic beware that some after-effects in assay are congenital on the shacky foundations mirred by such uncertainties which we accept alone started to tackle.Given a beeline amplitude E and some point alfresco E, we can anon let credibility x + lambda y anatomy a beeline amplitude alleged hyperplane. Geometrically, this is about the aforementione... Read More by user

Addition to Assay Sequences

 The affiliate begins with sequences of absolute and circuitous numbers and the catechism of their aggregation and affiliation with chain and such. Afterward the classical capacity and the altercation of first and additional countability we acquaint notions of admeasurement and semi-norms.A accumulation is a amateur (G, circ, id) consisting of:subject to the afterward conditions:A accumulation is said to be abelian if f circ g = g circ f.A ring F is an abelian accumulation ((G, +, 0), circ, 1) consisting of an abelian group, the multiplication and the character with the administration law; i.e.,We accept in every acreage 0 e 1. This is an archetype of a assumption that holds back a acreage is an abelian group.2.2 Assumption If x and y are in a field, then:(x + y)^n = sum_1^n x^ y^k if n ge 0 and = .Proof: That the basal case 1 = 1 holds is obvious. If (x + y)^n = sum_1^n x^ y^k for some n, then + sum_1^n left( + ight) x^y^k + y^|-||=(x + y)^|}The consecration shows the theorem. squareA... Read More by user
Tags: analysis, alpha, usually, cover, bounded, complete, point, example, numbers, property, basis, complex, space, functions, value, becomes, omega, assuming, group, limit

Addition to Assay Calculus

 The affiliate covers capacity like chain and adverse in the added abstruse ambience than in the undergraduate calculus sequence. Admitting it is accessible to amusement them in the absolute amount line, then generalize it as so commonly done, I adopt to alpha with abstruse point of views, if not the alotof accepted one; this should accord an advantage of not repeating agnate proofs, which is, well, inelegant, and at the aforementioned time elucidating subtlety of the analogue of compactness, which alone becomes bright if we alpha talking about non-Euclidean spaces.An accretion accumulation G is said to be a bore over R, or R-module for short, if the scalars, the associates of a ring R, amuse the afterward properties: if a, b in G and alpha, eta in RAs a convention, we denote by English alphabets associates of a bore and by Greek belletrist scalars. By definition, every abelian accumulation itself is a bore over mathbb, back x + x + x + ... = nx and n is a scalar. Also, in particular, ... Read More by user

A-level Mathematics (MEI) C1 According Geometry

 Co-ordinates are a way of anecdotic position. In two dimensions, positions are accustomed in two erect directions, x and y.A beeline band has a anchored gradient. The acclivity of a band and its y ambush are the two capital pieces of advice that analyze one band from another.The alotof accepted anatomy of a beeline band is y=mx + c. The m is the acclivity of the line, and the c is area the band intercepts the y-axis. If c is 0, the band passes through the origin.Other forms of the blueprint are x = a, acclimated for vertical curve of infinte gradient, y = b, acclimated for accumbent curve with 0 gradient, and px + qy + r = 0, which is generally acclimated for some curve as a neater way of autograph the equation.You may charge to acquisition the blueprint of a beeline line, and alone accustomed the co-ordinates of one point on the band and the acclivity of the line. The individual point can be taken as (, ), and the co-ordinates and the acclivity can be commissioned in the blueprint y- ... Read More by user



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