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Stanford UniversitySTATS 231hw0-solutions
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Homework 0 solutions 
CS229T/STATS231 
1. Linear algebra (0 points) 
a (dual norm of L1 norm) The L1 norm k · k1 of a vector v 2 Rn is defined as 
kvk1 = 
nX i 
=1 
jvij: (1) 
The dual norm k · k∗ of a norm k · k is defined as 
kvk∗ = sup 
kwk≤1 
(v · w): (2) 
Compute the dual norm of the L1 norm. (Here v · w denotes the inner product between v and w: v · w , 
Pn i=1 viwi) 
Solution: 
We will prove that 
sup 
kwk1≤1 
(v · w) = max 
i2[n] 
vi = kvk1 (3) 
which implies that the dua...
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Exam (elaborations) TEST BANK FOR Understanding Analysis 2nd Edition By Stephen Abbott (Instructors' Solution Manual)
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1 The Real Numbers 1 
1.1 Discussion: The Irrationality of 
p 
2 . . . . . . . . . . . . . . . . . 1 
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1 
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . 6 
1.4 Consequences of Completeness . . . . . . . . . . . . . . . . . . . 8 
1.5 Cantor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14 
2 Sequences and Series 19 
2.1 Discussion: Rearrangements of Infinite Series . . . . . . . ....
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Exam (elaborations) TEST BANK FOR Fundamentals of Digital Communication By Upamanyu Madhow (Solution Manual)
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Exam (elaborations) TEST BANK FOR Fundamentals of Digital Communication By Upamanyu Madhow (Solution Manual) 
Solutions to Chapter 2 Problems 
Fundamentals of Digital Communication 
Problem 2.1: Rather than doing the details of the convolution, we simply sketch the shapes of 
the waveforms. For a signal s = sc + jss and a filter h = hc + jhs, the convolution 
y = s ∗ h = (sc ∗ hc − ss ∗ hs) + j(sc ∗ hs + ss ∗ hc) 
For h(t) = smf (t) = s(−t), rough sketches of Re(y), Im(y) and |y|...
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Exam (elaborations) TEST BANK FOR Fundamentals of Digital Communication By Upamanyu Madhow (Solution Manual)
- Prüfung • 223 Seiten • 2021
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Problem 2.1: Rather than doing the details of the convolution, we simply sketch the shapes of 
the waveforms. For a signal s = sc + jss and a filter h = hc + jhs, the convolution 
y = s ∗ h = (sc ∗ hc − ss ∗ hs) + j(sc ∗ hs + ss ∗ hc) 
For h(t) = smf (t) = s(−t), rough sketches of Re(y), Im(y) and |y| are shown in Figure 1. 
Clearly, the maximum occurs at t = 0. 
= 
c*hc −ss*hs 
Re(s* h) 
sc*hs ss*hc Im(s* h) 
|s* h| 
+ = 
+ 
s 
Figure 1: The convolution of a signal with its mat...
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Differential Equations
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Differential equation for Mathematics and calculus for engineering students covering topics of cauchy homogeneous linear equation, partial integral, complementary function etc.
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Exam (elaborations) TEST BANK FOR Advanced Engineering Mathematics [Volume 2] By Herbert Kreyszig and Erwin Kreyszig (Student Solutions Manual and Study Guide)
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Exam (elaborations) TEST BANK FOR Advanced Engineering Mathematics [Volume 2] By Herbert Kreyszig and Erwin Kreyszig (Student Solutions Manual and Study Guide) 
P A R T D 
Complex 
Analysis 
Chap. 13 Complex Numbers and Functions. 
Complex Differentiation 
Complex numbers appeared in the textbook before in different topics. Solving linear homogeneous ODEs 
led to characteristic equations, (3), p. 54 in Sec. 2.2, with complex numbers in Example 5, p. 57, and 
Case III of the table on p. 58. Solv...
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Ordinary Differential Equations
- Notizen • 44 Seiten • 2024
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Course Objective:- To solve ordinary differential equations which arises in engineering and physics. 
 
It contains - 
 
First order linear ordinary differential equations - Solution and simple applications 
Higher order linear order ordinary differential equation with constant coefficients - Solution of homogenous and non-homogenous equations, Method of undetermined coefficients, Method of variation of parameters, Cauchy-Euler differential equations 
Applications of second order linear differen...
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TestBank for Introduction to Econometrics 3e Stock Watson SM completed Instructors Guide Completed for Exam preparation and Review
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TestBank for Introduction to Econometrics 3e Stock Watson SM 
For Instructors 
Solutions to End-of-Chapter Exercises©2011 Pearson Education, Inc. Publishing as Addison Wesley 
Chapter 2 
Review of Probability 
2.1. (a) Probability distribution function for Y 
Outcome (number of heads) Y 0 Y 1 Y 2 
Probability 0.25 0.50 0.25 
(b) Cumulative probability distribution function for Y 
Outcome (number of heads) Y 0 0 Y 1 1 Y 2 Y 2 
Probability 0 0.25 0.75 1.0 
(c) ...
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Exam (elaborations) TEST BANK FOR Advanced Engineering Mathematics [Volume 2] By Herbert Kreyszig and Erwin Kreyszig (Student Solutions Manual and Study Guide)
- Prüfung • 223 Seiten • 2021
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Complex numbers appeared in the textbook before in different topics. Solving linear homogeneous ODEs 
led to characteristic equations, (3), p. 54 in Sec. 2.2, with complex numbers in Example 5, p. 57, and 
Case III of the table on p. 58. Solving algebraic eigenvalue problems in Chap. 8 led to characteristic 
equations of matrices whose roots, the eigenvalues, could also be complex as shown in Example 4, p. 328. 
Whereas, in these type of problems, complex numbers appear almost naturally as compl...
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Sequences and series
- Zusammenfassung • 10 Seiten • 2023
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Notes on sequences and series from book 'Understanding Analysis by Abott'. This includes topics like convergence of a sequence, infinite series, convergence of series, monotone convergence theorem, harmonic series, bolzano weistrass theorem,cauchy sequence, conditional convergence etc. It also contains tests like ratio test, p series test, abel's test, dirichlet's test, comparison test, alternating series test and more. There are many theorems included but the proof are not provided.