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Vector Calculus
Vector Calculus

SuperCollider Tutorial
SuperCollider Tutorial

... This is because of something called scope. Variables only exist in the code block in which they are declared. Code blocks are zero or more lines of code surrounded by parenthesis, or by curly braces. This means that variables and arguments declared inside a function only exist inside that function. ...
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3.6 The Real Zeros of a Polynomial Function

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Discrete Mathematics

CIRCULAR (TRIGONOMETRIC) FUNCTIONS RECIPROCAL
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... The inverse function of the restricted cosine function: f : [0, π ] → [ −1, 1] , f ( x) = cos( x) is called the inverse cosine function or the arccosine function. It is denoted by: cos −1 ( x) , arcos( x ) or acos( x ) . It has a domain of − 1 ≤ x ≤ 1 and range of 0 ≤ y ≤ π . The inverse tangent fun ...
arXiv:math/0703236v1 [math.FA] 8 Mar 2007
arXiv:math/0703236v1 [math.FA] 8 Mar 2007

Minimal number of periodic points for C self
Minimal number of periodic points for C self

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ROOT NUMBERS OF HYPERELLIPTIC CURVES 1. Introduction

The Science of Proof - University of Arizona Math
The Science of Proof - University of Arizona Math

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JEE Main, Mathematics Volume I, Notes (Guide)

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Irrationality measures for some automatic real numbers

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A Survey On Euclidean Number Fields

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The Limit of a Sequence of Numbers

... {xn } may not itself be a subsequence of {an }, each xn may or may not be one of the numbers ak , so that there really is something to prove. In fact, this is the hard part of this lemma. To nish the proof of part (4), we must dene an increasing sequence {nk } of natural numbers for which the corr ...
Continued Fractions in Approximation and Number Theory
Continued Fractions in Approximation and Number Theory

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Polar Coordinates and Complex Numbers Infinite Series Vectors

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2.6 Rational Functions (slides, 4 to 1)

... As x gets large in absolute value, the quadratic terms x2 begin to dominate. For example, if x = 1, 000, 000 then the denominator x2 + x − 12 is equal to 1, 000, 000, 000, 000 + 1, 000, 000 − 12 = 1, 000, 000, 099, 988, which for all practical purposes can be approximated by 1, 000, 000, 000, 000. S ...
EXHAUSTIBLE SETS IN HIGHER
EXHAUSTIBLE SETS IN HIGHER

... We say that the set K is exhaustible if the above problem can be algorithmically solved for any continuous p defined on K, uniformly in p. The uniform dependency on p is formulated by giving the algorithm the type (D → B) → B, where D is a domain, K ⊆ D, and B is the domain of booleans. The main que ...
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Non-standard calculus

In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.Calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless.Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Jerome Keisler, ""Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century.""
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