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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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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
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
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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
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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
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by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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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
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
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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
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
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Topics: Maths, Linear Algebra and Geometry, Geometry, Differential Geometry, Inverse Function Theorem,...
Source: http://www.flooved.com/reader/1105
Flooved
by Jeff Viaclovsky
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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
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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
by 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
by 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
by 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
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Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1513
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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
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
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
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
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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
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The definition of the variance of f is equivalent to its interpretation as the mean square of f's deviation from its mean. However, it can be used to put an upper limit on the probability that a random variable takes on values with deviations greater than x� for any x greater than 1. The simplest and worst such bound is called Tchebyshev's inequality.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1873
We will describe an algorithm (discovered by V.Strassen) and usually called �Strassen�s Algorithm) that allows us to multiply two n by n matrices A and B, with a number of multiplications (and additions) which is a small multiple of ... , when n is of the form 2^k.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1865
Flooved
by Daniel Kleitman;Peter Shor
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To do a 2^k FFT mod a prime p you need to choose a prime p whose remainders include 2^k-th roots of unity, and you need to find one such root that is not a 2^(k-1)-th root of unity
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1888
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1 Multi�ows and Disjoint Paths - Let G = (V,E) be a graph and let...
Topics: Maths, Optimization and Control, Optimization, Mathematics
Source: http://www.flooved.com/reader/1936
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Hartshorne only treats �atness after cohomology (so see III.9) and doesn�t talk about descent at all. The EGA reference for �atness is EGA IV, part 2, �2. I�m not sure if descent is discussed at all in EGA, so I gave references to SGA 1 instead.
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2060
This course requires linear algebra, so here is a quick review of the facts we will use frequently.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2023
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Topics: Physics, Special Relativity, General Relativity and Gravitation, Gravitational Waves, Wave...
Source: http://www.flooved.com/reader/2043
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2008
This section of the lecture examines some of the contemporary methods used to visualize computer networks. More information can be found at www.cybergeography.org which contains links to numberous internet-related visualization projects.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2009
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2066
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1. K�hler Geometry
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2108
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Sard�s Theorem: An extremely important notion in differential topology is that of general posi_tion or genercity. A particular map may have some horrible pathologies but often a nearby map has much nicer properties
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2129
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1. Symplectic Manifolds...
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2121
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In this unit, we describe a more intricate version of the sieve of Eratosthenes, introduced by Viggo Brun in order to study the Goldbach conjecture and the twin prime conjecture. It is most useful for providing lower bounds; for upper bounds, the Selberg sieve (to be introduced in the following unit) is much less painful.
Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2079
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1. Homeomorphism Classification of Simply Connected Compact 4-Manifolds...
Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Geometry, Geometry of Manifolds, Complex...
Source: http://www.flooved.com/reader/2119
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In the early sixties Smales realized that many of the ideas of differential topology can be applied to aid in the study of PDEs and as part of this program he showed how to generalize Sard�s theorem to the in�nite dimensional case. First we need to introduce the correct kind of mappings of Banach manifolds.
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2135
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Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2155
In the previous class, we have introduced various concepts necessary for the study of EM waves in photonic crystal structures. We shall now use these concepts to explain various results such as: � Reconstruction of the permittivity pro�le. � The band diagrams for rectangular and triangular lattices. � k-surfaces for various eigenvalues. In particular, we will show an example of how a periodic structure can exhibit k-surfaces typicalof a negative refraction material (the concept of...
Topics: Physics, Acoustics, Optics and Waves, Electromagnetism and Electromagnetic Radiation, Waves,...
Source: http://www.flooved.com/reader/2572
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2203
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2206
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2207
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2204
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Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2158
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Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2175
In this lecture, we will revisit the relationship between unstable modules over the (mod 2) Steenrod algebra A and analytic functors from the category of F2 vector spaces to itself.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/2212
Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2606
Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2600
Flooved
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A plasma is a gas in which an important fraction of the atoms is ionized, so that the electrons and ions are separately free. When does this ionization occur? When the temperature is hot enough. ...
Topics: Physics, Physics of Gases, Plasmas, and Electric Discharges, Physics of Plasmas and Electric...
Source: http://www.flooved.com/reader/2664
Flooved
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Objectives: The GR �eld equations
Topics: Physics, Special Relativity, General Relativity and Gravitation, Classical General Relativity,...
Source: http://www.flooved.com/reader/2823
Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism�, Physics
Source: http://www.flooved.com/reader/2846
Flooved
by Lawrence Evans;Mr. J. Edward Ladenburger
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Standing waves in a string. We consider again two harmonic waves with the same amplitude, wavelength and frequency, but now moving in opposite directions.
Topic: Maths
Source: http://www.flooved.com/reader/2903
Flooved
by Mr. Travis Byington;Lawrence Evans;Mr. Ryan Magee;Hao Zhang
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Superposition of Harmonic Waves- The essential characteristic of energy transport by waves is that waves obey the superposition principle. This means that two waves in the same spatial region can interfere, rearranging the energy in space in a pattern often quite different from that of either wave alone. Since light propagates as a wave, we will analyze this phenomenon.We begin with a mathematical problem: How do we �nd the wave function for the combined wave resulting from interference of...
Topics: Physics, Acoustics, Optics and Waves, Optics�, Waves, Wave Optics, Interference and Coherence,...
Source: http://www.flooved.com/reader/2918
These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the Seiberg-Witten invariants. Much of the necessary background material is given, including a crash course in topological �eld theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold theory. My main...
Topic: Maths
Source: http://www.flooved.com/reader/2970
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With the foregoing preparation, we are now in a position to apply the classical analogy orcanonical quantization program to achieve the second quantization of theelectromagnetic field. As our starting point and for reference, we, once again, set forth the vacuum or microscopic Maxwell's equations in the time domain...
Topics: Physics, Acoustics, Optics and Waves, Quantum Physics, Optics�, Quantum Optics�, Physics
Source: http://www.flooved.com/reader/2990
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We begin our discussion of optical pulse propagation 35 with a derivation of the nonlinear Schr�dinger (NLS) equation. To that end, we recall Equations [ VII-23 ] and [ VII-23 ] from the early lecture set entitled Nonlinear Optics I
Topics: Physics, Acoustics, Optics and Waves, Quantum Physics, Optics�, Quantum Optics�, Physics
Source: http://www.flooved.com/reader/2997
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Rotation Operators: Consider a scalar �eld f(r).Example 1: The temperature at point r After a rotation of the coordinate system, the same �eld is described by...
Topics: Physics, Mathematical Methods in Physics, Quantum Physics, Group Theory, Physics
Source: http://www.flooved.com/reader/2937
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The purpose of this course is to answer these questions. We shall see that the second viewpoint above is the most useful: the �eld is primary and particles are derived concepts, appearing only after quantization. We will show how photons arise from the quantization of the electromagnetic �eld and how massive, charged particles such as electrons arise from the quantization of matter �elds. We will learn that in order to describe the fundamental laws of Nature, we must not only introduce...
Topic: Maths
Source: http://www.flooved.com/reader/3072
Flooved
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The aim of this lecture is to explore the nature of the ground state and the character of the elementary excitation spectrum in the condensed phase.
Topics: Physics, Condensed Matter, Particle Physics and Fields, Quantum Physics, General Theory of Fields...
Source: http://www.flooved.com/reader/3069
Flooved
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We have already used Newton�s Second Law or Conservation of Energy to analyze systems like the bloc-spring system that oscillate. We shall now use torque and the rotational equation of motion to study oscillating systems like pendulums or torsional springs.
Topics: Physics, Classical Mechanics, Classical Mechanics of Discrete Systems, Fundamental Concepts, Simple...
Source: http://www.flooved.com/reader/3302
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Our goal in this section is to use the techniques of statistical mechanics to describe the dynamics of the simplest system: a gas. This means a bunch of particles, �ying around in a box. Although much of the last section was formulated in the language of quantum mechanics, here we will revert back to classical mechanics. Nonetheless, a recurrent theme will be that the quantum world is never far behind: we�ll see several puzzles, both theoretical and experimental, which can only truly be...
Topic: Maths
Source: http://www.flooved.com/reader/3130
Flooved
by Alan Guth;Barton Zwiebach
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Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes,...
Source: http://www.flooved.com/reader/3178
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by Peter Ouwehand
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Topic: Maths
Source: http://www.flooved.com/reader/3372
Flooved
by James S. Milne
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These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne).
Topics: Maths, Algebra, Mathematics
Source: http://www.flooved.com/reader/3429
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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
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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/1122
Flooved
by 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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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
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
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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
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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
by 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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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1489