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# Solutions for Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories

## Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition

ISBN: 9780387908069

Solutions for Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories

Solutions for Chapter 1.8
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##### ISBN: 9780387908069

This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Since 19 problems in chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories have been answered, more than 6125 students have viewed full step-by-step solutions from this chapter. Chapter 1.8: The dynamics of tumor growth, mixing problems and orthogonal trajectories includes 19 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Cholesky factorization

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

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Hankel matrix H.

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

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• Left inverse A+.

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

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= 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.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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