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Flooved is an online education platform founded in 2011, that seeks to provide free education to a global audience by providing lecture notes, handouts, and study guides online, beginning with undergraduate mathematics and physics content.



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Apr 6, 2015 Peter Dourmashkin
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Topics: Physics, Classical Mechanics, Physics
Source: http://www.flooved.com/reader/3310
Flooved
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Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1041
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Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1006
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Topic: Maths
Source: http://www.flooved.com/reader/1034
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Topics: Maths, Linear Algebra and Geometry, Geometry, Differential Geometry, Inverse Function Theorem,...
Source: http://www.flooved.com/reader/1105
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Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Geometry, Differential Geometry,...
Source: http://www.flooved.com/reader/1106
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Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1095
Flooved
texts

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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/1122
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Until Weierstrass published his shocking paper in 1872, most of the mathematical world (including luminaries like Gauss) believed that a continuous function could only fail to be differentiable at some collection of isolated points. In fact, it turns out that �most� continuous functions are non-differentiable at all points. (To understand what this statement could mean, you should take courses in topology and measure theory.) However, Weierstrass was not, in fact, the �rst to construct...
Topics: Maths, Analysis and Calculus, Mathematics
Source: http://www.flooved.com/reader/1197
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Arithmetic functions, the Mobius�function
Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/1119
Flooved
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Topics: Maths, Logic, Numbers and Set Theory, Analysis and Calculus, Countability and Uncountability,...
Source: http://www.flooved.com/reader/1273
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Elliptic regularity: Hitherto we have always assumed our solutions already lie in the appropriate C^(k,_) space and then showed estimates on their norms in those spaces. Now we will avoid this a priori assumption and show that they do hold a posteriori. This is important for the consistency of our discussion
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1332
Flooved
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In this lecture, we introduce the complementary slackness conditions and use them to obtain a primal-dual method for solving linear programming.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1321
Flooved
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De�nition: An anitchain A of a poset P is a subset of elements of P such that for all x, y _ A, x _� y and y _� x. We denote the levels of a graded poset P as Pi where Pi = {x _ P : rank(x) = i}.
Topics: Maths, Algebra, Statistics and Probability, Probability, Combinatorics, Mathematics
Source: http://www.flooved.com/reader/1265
Flooved
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This book begins with four special families of matrices�simple and useful, absolutely basic. We look �rst at the properties of these particular matrices Kn,Cn, Tn,and Bn. (Some properties are obvious, others are hidden.) It is terri�c to practice linear algebra by working with genuinely important matrices.
Topics: Maths, Linear Algebra and Geometry, Numerical Analysis, Linear Algebra, Linear Algebraic Systems,...
Source: http://www.flooved.com/reader/1323
Flooved
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Lemma. MF is closed under �*(X _ X)�
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1494
Flooved
- John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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In session on Phase Portraits, we described how to sketch the trajectories of a linear system x' = ax + by, y' = cx + dy where a, b, c, d constants.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1431
Flooved
- Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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This is meant as a follow-up on the review of vectors and matrices in the previous session.
Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Linear Algebra, Linear Independence,...
Source: http://www.flooved.com/reader/1458
Flooved
- John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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In summary, the procedure of sketching trajectories of the 2 _ 2 linear homogeneous system x_ = Ax, where A is a constant matrix
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1439
Flooved
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We will be discussing �compact operators� later, so we need some sort of idea of com_pactness. This is what the following theorem provides.
Topics: Maths, Topology and Metric Spaces, Differential Equations (ODEs & PDEs), Compact Spaces,...
Source: http://www.flooved.com/reader/1481
Flooved
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Density of the Schwartz Functions: To prove this we need to show the existence of a sequence of functions fn _ Sn for each f _ L2(R) such that fn f.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1479
Flooved
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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1489
Flooved
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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1488
Flooved
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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1491
Flooved
- Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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The fundamental matrix _(t) also provides a very compact and ef�cient integral formula for a particular solution to the inhomogeneous equation x' = A(t)x + F(t). (presupposing of course that one can solve the homogeneous equation x' = A(t)x �rst to get _.) In this short note we give the formula (with proof!) and one example.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Linear...
Source: http://www.flooved.com/reader/1400
Flooved
- Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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eye 102

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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1402
Flooved
- John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Poincare-Bendixson Theorem: Suppose R is the �nite region of the plane lying between two simple closed curves D1 and D2, and F is the velocity vector �eld for the system (1).
Topics: Maths, Differential Equations (ODEs & PDEs), Dynamics and Relativity, Dynamics and Relativity,...
Source: http://www.flooved.com/reader/1427
Flooved
- John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Separable Equations We will now learn our �rst technique for solving differential equation. An equation is called separable when you can use algebra to separate the two variables, so that each is completely on one side of the equation. We illustrate with some examples.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Nonlinear...
Source: http://www.flooved.com/reader/1426
Flooved
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Now say that X is a set, R is a ring of subsets and we have a function � : R _ [0, �). This is the �measure�, what we�re looking for. One of the properties we need for this function is additivity : �(A _ B) = �(A) + �(B), if A, B _ R, A _ B = _ From just this property we can derive a number of properties about �....
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1498
Flooved
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Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1513
Flooved
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Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1516
Flooved
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we show that the spherical representation z _ Z is conformal. This means that if l and m are two lines in the plane intersecting in z at an angle _, then the corresponding circles C and D through N and Z intersect Z at the same angle _
Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1504
In this course, we will mainly consider the case of free particles, in which V = 0 (i.e., the homogeneous Schr�odinger equation). In the case of free particles, there is an important family of solutions to (1.0.1), namely the free waves. The free wave solutions provide some important intuition about how solutions to the homogeneous Schr�odinger equation behave.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1607
Flooved
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De�nition: Conditional probability of Event A given Event B: ...
Topics: Maths, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1637
Flooved
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We introduce another powerful method of solving PDEs. First, we need to consider some preliminary de�nitions and ideas.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1671
We will now study the Laplace and Poisson equations on a domain (i.e. open connected subset) _ _ R^n
Topic: Maths
Source: http://www.flooved.com/reader/1613
_(_), parameter _ > 0, random variable takes values: {0, 1, 2, ...} p.f. ... Moment generating function:
Topics: Maths, Statistics and Probability, Probability, Statistics, Random Variables, Useful Distributions,...
Source: http://www.flooved.com/reader/1642
Flooved
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Let V be a Euclidean space, i.e. a real �nite dimensional linear space with a symmetric positive de�nite inner product _,_.
Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics
Source: http://www.flooved.com/reader/1666
Flooved
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A cycle in a digraph is de�ned by a path that begins and ends at the same vertex. This includes the cycle of length zero that begins and ends at the vertex. A directed acyclic graph (DAG) is a directed graph with no positive length cycles.
Topics: Maths, Logic, Numbers and Set Theory, Graph Theory, Posets and Zorn�s Lemma, Set Theory,...
Source: http://www.flooved.com/reader/1725
Let�s discuss how the wave equation arises as an approximation to the equations of �uid mechanics. For simplicity, let�s only discuss the case of 1 spatial dimension.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1600
Flooved
- Prof. Peter J. Cameron
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The original course was largely based on continued fractions: this technique is very amenable to hand calculation, and can be used to solve Pell�s equation, to write an integer as a sum of squares where this is possible, and to classify the inde�nite binary quadratic forms. This is still the centrepiece of the course, but I have given alternate treatment of sums of squares.
Topic: Mathematics
Source: http://www.flooved.com/reader/3516
Flooved
- William F. Trench
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The book is designed to �ll the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics.
Topics: Real Numbers, Analysis, Mathematics
Source: http://www.flooved.com/reader/3549
Flooved
- Prof. Jim Hefferon
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on functions involving a single independent variable and a single dependent variable. For
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/3490
Flooved
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I will present the basic de�nitions and properties of non-crossing partitions and free cumulants and outline its relations with freeness and random matrices. As examples, I will consider the problems of calculating the eigenvalue distribution of the sum of randomly rotated matrices and of the compression (upper left corner) of a randomly rotated matrix.
Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Mathematics
Source: http://www.flooved.com/reader/976
Flooved
- Prof. Dave Witte Morris;Prof. Joy Morris
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well as in the application of mathematics to the rest of the world involve many variables
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/3499
In this lecture and in the next, we�ll briefly review second-order PDEs. We�ll begin with one of the simplest of such PDEs: the Laplace equation.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1340
Flooved
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In this unit, we introduce (without proof for now) a formula which relates the distribution of primes to the zeroes of the Riemann zeta function. Given a suitable zero-free region for _(s) in the critical strip, this can be used to prove the prime number theorem with an estimate for the error term.
Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2083
Our goal in this lecture is to prove the following result: Theorem 1. Let n and k be nonnegative integers. Then the tensor product K(n) _ J(k) is an injective object in the category of unstable A-modules.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/2216
Flooved
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Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2622
The main goal of this section is to give a formula in terms of _A(t) for r(A) and b(A) when K = R (Theorem 2.5). We �rst establish recurrences for these two quantities.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/982
Flooved
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Theorem (Schensted): Let � be a permutation of {1, 2, . . . , n} written in one-line notation. Let P and Q be the Standard Young Tableaux (SYT) in the image of the (Robinson-Schensted-Knuth) RSK algorithm, i.e. RSK(�) = (P, Q), with shapes sh(P) = sh(Q) = _. Then the length of the longest increasing subsequence in � equals the length of the �rst row of _ and the length of the longest decreasing subsequence in � equals the length of the �rst column of _.
Topics: Maths, Algebra, Statistics and Probability, Probability, Combinatorics, Mathematics
Source: http://www.flooved.com/reader/1269
Flooved
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We have seen that a central question about representations of quivers is whether a certain connected quiver has only �nitely many indecomposable representations. In the previous subsection it is shown that only those quivers whose underlying undirected graph is a Dynkin diagram may have this property. To see if they actually do have this property, we �rst explicitly decompose representations of certain easy quivers.
Topics: Maths, Algebra, Representation Theory, Mathematics
Source: http://www.flooved.com/reader/1658
Flooved
- Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
texts

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In this note we state Green�s formula and look at some examples. We will prove it in the next note.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1397
Flooved
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To get around the ambiguity of English, mathematicians have devised a special mini-language for talking about logical relationships. This language mostly uses ordinary English words and phrases such as �or�, �implies�, and �for all�. But mathematicians endow these words with de�nitions more precise than those found in an ordinary dictionary.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1720
Flooved
- Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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In this section we will look at the ODE for an RC circuit. The �rst reason for doing this is to introduce circuits into the course. They will be a fruitful source of interesting examples. When we study second order equations we will be able to model and study circuits that include inductors.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1443
Flooved
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Theorems about homogeneous and inhomogeneous systems: On the basis of our work so far, we can formulate a few general results about square systems of linear equations. They are the theorems most frequently referred to in the applications.
Topics: Maths, Analysis and Calculus, Calculus, Mathematics
Source: http://www.flooved.com/reader/1809
We�ll see now how Schur�s lemma allows us to classify subrepresentations in �nite dimensional semisimple representations.
Topics: Maths, Algebra, Representation Theory, Mathematics
Source: http://www.flooved.com/reader/1655
Flooved
- N.M.J. Woodhouse
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This course has two aims. Aim 1 - To explain how Einstein�s theory models gravitation as the curvature of space-time. This involves setting up some new mathematical machinery, notably tensor calculus. The mathematical background will be developed in parallel with the theory. In textbooks, it is often done the other way: mathematics �rst, theory after. But the textbooks do not have to be read in linear order. The approach here should make it clearer at each stage where we are going. Aim 2 -...
Topic: Maths
Source: http://www.flooved.com/reader/2919
Flooved
- Michael Vaughan-Lee
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In this course we are mainly concerned with �eld extensions, and �nding roots of polynomials, but before we get on to this we need to investigate the notion of characteristic.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1071
Flooved
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Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1150
Flooved
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We discuss the existence and uniqueness for second-order linear differential equations with constant coef�cients by means of general principles which are valid for equations with variable coef�cients. We develop techniques of the Wronskian and apply to study the oscillatory behavior. We also give qualitative results which depend on conditions at an interior maximum and minimum and apply to second-order equations.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1561
Flooved
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Thus we seek a geometric depiction of the trajectories in a lower-dimensional space�in essence, a view of phase space without all the detail.
Topics: Maths, Dynamics and Relativity, Dynamical Systems, Mathematics
Source: http://www.flooved.com/reader/1845
Flooved
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Topics: Maths, Algebra, Coding and Cryptography, Codes, BCH Codes, Mathematics
Source: http://www.flooved.com/reader/1853
Flooved
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1. Spin Structures...
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2120
Flooved
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2145
Flooved
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Topics: Maths, Statistics and Probability, Statistics, Estimation Techniques, Mathematics
Source: http://www.flooved.com/reader/2166
Flooved
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The Strong Whitney Embedding Theorem - Whitney proved a stronger version of this theorem...
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2138
Flooved
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3.1 Binary collisions between charged particles - Reduced-mass for binary collisions: Two particles interacting with each other have forces F12 force on 1 from 2. F21 force on 2 from 1. By Newton�s 3rd law, F12= _F21. Equations of motion: ...
Topics: Physics, Physics
Source: http://www.flooved.com/reader/2693
We now present a version of the inversion theorem for symmetric matrices. If the matrix is not symmetric, the statement looks quite di_erent.
Topics: Maths, Numerical Analysis, Mathematics
Source: http://www.flooved.com/reader/1596
Flooved
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In particular, when we make an estimate by repeated sampling, we need to know how much con�dence we should have that our estimate is OK. Technically, this reduces to �nding the probability that an estimate deviates a lot from its expected value. This topic of deviation from the mean is the focus of this �nal chapter.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1719
Flooved
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In this chapter, we�ll put sequences in angle brackets to more clearly distinguish them from the many other mathematical expressions �oating around.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1714
Flooved
- Louis Scharf
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This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual.
Topics: Maths, Linear Algebra and Geometry, Algebra, Vectors and Matrices, Linear Algebra, Mathematics
Source: http://www.flooved.com/reader/1669
Flooved
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Topics: Maths, Linear Algebra and Geometry, Differential Equations (ODEs & PDEs), Vectors and Matrices,...
Source: http://www.flooved.com/reader/1580
Flooved
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Bayes Estimators: used when intuitive model can be used in describing the data.
Topics: Maths, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1627
Flooved
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Suppose that we �ip two fair coins simultaneously on opposite sides of a room. Intuitively, the way one coin lands does not affect the way the other coin lands. The mathematical concept that captures this intuition is called independence:
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1749