 23.6.23.1.92: Find flow augmenting paths:
 23.6.23.1.93: Find flow augmenting paths:
 23.6.23.1.94: Find flow augmenting paths:
 23.6.23.1.95: Find flow augmenting paths:
 23.6.23.1.96: Find the maximum flow by inspection: In Prob. 1.
 23.6.23.1.97: Find the maximum flow by inspection:In Prob. 2.
 23.6.23.1.98: Find the maximum flow by inspection:In Prob. 3.
 23.6.23.1.99: Find the maximum flow by inspection:In Prob. 4.
 23.6.23.1.100: In Fig. 495 find T and cap (5. T) if 5 equals[1,2.31
 23.6.23.1.101: In Fig. 495 find T and cap (5. T) if 5 equals[I. 2.4.51
 23.6.23.1.102: In Fig. 495 find T and cap (5. T) if 5 equals [1, 3, 51
 23.6.23.1.103: Find a minimum cut set in Fig. 495 and verify that its capacity equ...
 23.6.23.1.104: Find examples of flow augmenting paths and the maximum flow in the ...
 23.6.23.1.105: In Fig. 498 find T and cap (5. T) if 5 equals
 23.6.23.1.106: In Fig. 498 find T and cap (5. T) if 5 equals
 23.6.23.1.107: In Fig. 498 find T and cap (5. T) if 5 equals
 23.6.23.1.108: In Fig. 498 find a minimum cut set and its capacity.
 23.6.23.1.109: Why are backward edge~ not considered III the definition of the cap...
 23.6.23.1.110: In which case can an edge U, j) be used as a forward as well as a b...
 23.6.23.1.111: (Incremental network) Sketch the network in Fig. 498, and on each e...
Solutions for Chapter 23.6: Flows in Networks
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 23.6: Flows in Networks
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 20 problems in chapter 23.6: Flows in Networks have been answered, more than 44508 students have viewed full stepbystep solutions from this chapter. Chapter 23.6: Flows in Networks includes 20 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.