 3.1: Henry, Tom, and Fred are dividing among themselves a set of common ...
 3.2: Alice, Bob, and Carlos are dividing among themselves a set of commo...
 3.3: Angie, Bev, Ceci, and Dina are dividing among themselves a set of c...
 3.4: Mark, Tim, Maia, and Kelly are dividing among themselves a set of c...
 3.5: Allen, Brady, Cody, and Diane are sharing a cake. The cake had prev...
 3.6: Carlos, Sonya, Tanner, and Wen are sharing a cake. The cake had pre...
 3.7: Alex, Betty, and Cindy are sharing a cake. The cake had previously ...
 3.8: Alex, Betty, and Cindy are sharing a cake. The cake had previously ...
 3.9: Four partners (Adams, Benson, Cagle, and Duncan) jointly own a piec...
 3.10: Four players (Abe, Betty, Cory, and Dana) are sharing a cake. Suppo...
 3.11: Angelina and Brad jointly buy the chocolatestrawberry mousse cake ...
 3.12: Brad and Angelina jointly buy the chocolatestrawberry mousse cake ...
 3.13: Karla and five other friends jointly buy the chocolatestrawberryva...
 3.14: Marla and five other friends jointly buy the chocolatestrawberryva...
 3.15: Suppose that they flip a coin and Jared ends up being the divider. ...
 3.16: Suppose they flip a coin and Karla ends up being the divider. (a) D...
 3.17: Suppose that they flip a coin and Martha ends up being the divider....
 3.18: Suppose that they flip a coin and Nick ends up being the divider. (...
 3.19: Suppose that they flip a coin and David ends up being the divider. ...
 3.20: Suppose that they flip a coin and Paula ends up being the divider. ...
 3.21: Three partners are dividing a plot of land among themselves using t...
 3.22: Three partners are dividing a plot of land among themselves using t...
 3.23: Four partners are dividing a plot of land among themselves using th...
 3.24: Four partners are dividing a plot of land among themselves using th...
 3.25: Mark, Tim, Maia, and Kelly are dividing a cake among themselves usi...
 3.26: Allen, Brady, Cody, and Diane are sharing a cake valued at $20 usin...
 3.27: Four partners are dividing a plot of land among themselves using th...
 3.28: Four partners are dividing a plot of land among themselves using th...
 3.29: Five players are dividing a cake among themselves using the lonedi...
 3.30: Five players are dividing a cake among themselves using the lonedi...
 3.31: Four partners (Egan, Fine, Gong, and Hart) jointly own a piece of l...
 3.32: Four players (Abe, Betty, Cory, and Dana) are dividing a pizza wort...
 3.33: Suppose that Jared is the divider. (a) Describe how Jared should cu...
 3.34: Suppose that Lori ends up being the divider. (a) Describe how Lori ...
 3.35: Suppose that Angela and Boris are the dividers and Carlos is the ch...
 3.36: Suppose that Carlos and Angela are the dividers and Boris is the ch...
 3.37: Suppose that Angela and Boris are the dividers and Carlos is the ch...
 3.38: Suppose that Carlos and Angela are the dividers and Boris is the ch...
 3.39: Suppose that Arthur and Brian are the dividers and Carl is the choo...
 3.40: Suppose that Carl and Arthur are the dividers and Brian is the choo...
 3.41: Jared, Karla, and Lori are dividing the footlong half meatballhalf...
 3.42: Jared, Karla, and Lori are dividing the footlong half meatballhalf...
 3.43: Ana, Belle, and Chloe are dividing four pieces of furniture using t...
 3.44: Andre, Bea, and Chad are dividing an estate consisting of a house, ...
 3.45: Five heirs (A, B, C, D, and E) are dividing an estate consisting of...
 3.46: Oscar, Bert, and Ernie are using the method of sealed bids to divid...
 3.47: Anne, Bette, and Chia jointly own a flower shop. They cant get alon...
 3.48: Al, Ben, and Cal jointly own a fruit stand. They cant get along any...
 3.49: Ali, Briana, and Caren are roommates planning to move out of their ...
 3.50: Anne, Bess, and Cindy are roommates planning to move out of their a...
 3.51: Three players (A, B, and C) are dividing the array of 13 items show...
 3.52: Three players (A, B, and C) are dividing the array of 13 items show...
 3.53: Three players (A, B, and C) are dividing the array of 12 items show...
 3.54: Three players (A, B, and C) are dividing the array of 12 items show...
 3.55: Five players (A, B, C, D, and E) are dividing the array of 20 items...
 3.56: Four players (A, B, C, and D) are dividing the array of 15 items sh...
 3.57: Quintin, Ramon, Stephone, and Tim are dividing a collection of 18 c...
 3.58: Queenie, Roxy, and Sophie are dividing a set of 15 CDs 3 Beach Boys...
 3.59: Ana, Belle, and Chloe are dividing 3 Snickers bars, 3 Milky Way bar...
 3.60: Arne, Bruno, Chloe, and Daphne are dividing 3 Snickers bars, 3 Milk...
 3.61: Consider the following method for dividing a continuous asset S amo...
 3.62: Consider the following method for dividing a continuous asset S amo...
 3.63: Two partners (A and B) jointly own a business but wish to dissolve ...
 3.64: Three partners (A, B, and C) jointly own a business but wish to dis...
 3.65: Three players (A, B, and C) are sharing the chocolatestrawberryvan...
 3.66: Angelina and Brad are planning to divide the chocolatestrawberry ca...
 3.67: Standoffs in the lonedivider method. In the lonedivider method, a...
 3.68: Efficient and envyfree fair divisions. A fair division is called e...
 3.69: Suppose that N players bid on M items using the method of sealed bi...
 3.70: Asymmetric method of sealed bids. Suppose that an estate consisting...
 3.71: Lonechooser is a fairdivision method. Suppose that N players divi...
Solutions for Chapter 3: The Mathematics of Sharing
Full solutions for Excursions in Modern Mathematics  8th Edition
ISBN: 9781292022048
Solutions for Chapter 3: The Mathematics of Sharing
Get Full SolutionsThis textbook survival guide was created for the textbook: Excursions in Modern Mathematics, edition: 8. Excursions in Modern Mathematics was written by and is associated to the ISBN: 9781292022048. Chapter 3: The Mathematics of Sharing includes 71 full stepbystep solutions. Since 71 problems in chapter 3: The Mathematics of Sharing have been answered, more than 11845 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.