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Section 4.3 Linear Recurrence Relations. 3. In this lecture, we will discuss the Recurrence Relation in Discrete Structure. A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation.The method represents one of the oldest and best-known pseudorandom number generator algorithms. 3. But there is a di culty: 2 ts into the format of which is a solution of the homogeneous problem. Thus the solution of the homogeneous recurrence relation is a n (h) = 2 n a_n^{(h)}=\\alpha \\cdot 2^n a n (h) = 2 n. Solution nonhomogeneous linear recurrence relation \\textbf{Solution nonhomogeneous linear recurrence relation} Solution nonhomogeneous linear recurrence relation a n = 4 a n 1 + 4 a n 2. In this case, since 3 was the 0 th term, the formula is a n = 3*2 n. These recurrence relations are called linear homogeneous recurrence relations with constant coefficients. For example. For second-order and higher order recurrence relations, trying to guess the formula or use iteration will usually result in a lot of frustration.

Examples Case 1: If sis not a characteristic root of the associated linear homogeneous recurrence relation Linear means that the previous terms in the definition are only multiplied by a constant (possibly zero) and nothing else. Find the sequence (hn) satisfying the recurrence relation hn = 4hn1 4hn2, n 2 and the initial conditions h0 = a and h1 = b. ., are real numbers and F(n) is a function not identically zero depending only on n. 17 : ch. ., ar, f with a 0, ar 6 0 such that 8n 2N, arxn+r + a r 1x n+r + + a 0xn = f The denition is malleable: in particular n satises the linear nonhomogeneous recurrence relation with constant coefcients: a n = c 1a n 1 +c 2a n 2 +:::+c ka n k +F(n) with F(n) of the form: F(n) = (b tnt +b t 1nt 1 +:::+b 1n+b 0)sn where b 0;b 1;:::;b t and sare real numbers. Assumptions. Find a recurrence relation for the number of ways to give someone n dollars if you have 1 dollar coins, 2 dollar coins, 2 dollar bills, and 4 dollar bills where the order in which the coins and bills are paid matters. A linear recurrence equation is a recurrence equation on a sequence of numbers expressing as a first-degree polynomial in with . 4. For linear recurrence relations the technique demonstrated here will always work. has the general solution un=A 2n +B (-3)n for n 0 because the associated characteristic equation 2+ -6 =0 has 2 distinct roots 1=2 and 2=-3. For a linear recurrence relation, you can use matrices and vectors to generate values. Types of recurrence relations. solving a non linear (log-linear) recurrence relation. The unknown (to be solved for) is y n, the nth term of the sequence.

In solving the rst order homogeneous recurrence linear relation xn = axn1; it is clear that the general solution is xn = anx0: This means that xn = an is a solution. (a) Set up a recurrence relation for Bn. Let P and Q be two non- empty sets. A binary relation R is defined to be a subset of P x Q from a set P to Q. Here logb (a) = log2 (2) = 1 = k. Therefore, the complexity will be Eg. Their most common form is x n+1 + ax n + bx n-1 = f(n); we will analyse the simpler cases where the right-hand side is a constant.

Our DAA Tutorial is designed for beginners and professionals both. The left hand variables don't appear on the right side and vice versa. 5. Subsection 4.3.1 Linear Recurrence Realtions. Second-order linear homogeneous recurrence relations De nition A second-order linear homogeneous recurrence relation with constant coe cients is a recurrence relation of the form a k = Aa k 1 + Ba k 2 for all integers k greater than some xed integer, where A and B are xed real numbers with B 6= 0. Imagine a recurrence relation takin the form a n = 1a n 1 + 2a n 2 + + ka n k, where the i are This relation is a well-known formula for finding the numbers of the Fibonacci series. DAA Tutorial. Problem. A relation is a relationship between sets of values. Find the sequence (hn) satisfying the recurrence relation hn = 2hn1 +hn2 2hn3, n 3 and the initial conditions h0 = 1,h1 = 2, and h2 = 0. Suppose that r c 1 r c 2 = 0 has two distinct roots r 1 and r 2. Does the second recurrence have the same property?

Solve for any unknowns depending on how the sequence was initialized. Find the general term of the Fibonacci sequence. For recurrence relation T (n) = 2T (n/2) + cn, the values of a = 2, b = 2 and k =1. First step is to write the above recurrence relation in a characteristic equation form. Also, these recurrence relations will usually not telescope to a simple sum. Note that our characteristic polynomial computed as (4) is the same as the one we referred to as the characteristic polyomial of Hot Network Questions Pairs at every distance Weighted coin flip strings How to respond politely to a client who is wrong about a small detail How I Definition. Recall that the recurrence relation is a recursive definition without the initial conditions. Solution First we observe that the homogeneous problem. 2. Solution. T ( n) T ( n 1) T ( n 2) = 0.

A recurrence relation is an equation that recursively defines a sequence. an = 4an1+4an2. Find the sequence (hn) satisfying the recurrence relation hn = 4hn1 4hn2, n 2 and the initial conditions h0 = a and h1 = b. linear recurrence relations with constant coefficients A rr of the form (5) ay n+2 +by n+1 +cy n =f n is called a linear second order rr with constant coefficients . Linear recurrences of the first order with variable coefficients . If no initial conditions are given, obtain n linear equations in n unknowns and solve them, if possible to get total solutions. Due to the right side of the equation, it must be a inhomogeneous equation. 4. The procedure for finding the terms of a sequence in a recursive manner is called recurrence relation.We study the theory Concept of Recurrence Relation2. Recurrence relation Difference equation. A linear recurrence equation of degree k or order k is a recurrence equation which is in the format (An is a constant and Ak0) on a sequence of numbers as a first-degree polynomial. Most of the recurrence relations that you are likely to encounter in the future are classified as finite order linear recurrence relations with constant coefficients.

Since the r.h.s. The order of the recurrence relation is determined by k. We say a recurrence relation is Linear Recurrence Relations | Brilliant Math & Science Wiki The first two problems are [Problem 1] The basics about the subspace of sequences satisfying a linear recurrence relations. In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms.. Linear Recurrence Relations. This class is the one that we will spend most of our time with in this chapter. The function f n is called the forcing function. Initially these disks are plased on the 1 st peg in order of size, with the lagest in the bottom. A recurrence relation is also called a difference equation, and we will use these two terms interchangeably. This is the last problem of three problems about a linear recurrence relation and linear algebra. Find the sequence (hn) satisfying the recurrence relation hn = 2hn1 +hn2 2hn3, n 3 and the initial conditions h0 = 1,h1 = 2, and h2 = 0. Then the sequence {a. n First order Recurrence relation :- A recurrence relation of the form : an = can-1 + f (n) for n>=1. What is a Relation?

While it is possible to produce a function that provides the n n th term, this is generally not easy. A linear recurrence relation is a recurrence relation that only contains linear multiples of previous terms. Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$. Definition 8.3.3. Second degree linear homogeneous recurrence relations. Depression. Explore conditions on f and g such that the sequence generated obeys Benfords Law for all initial values. 5.7: Linear Recurrence Relations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch. Much research on recurrence of depression has relied on the criteria for major depressive disorder (MDD) in the Diagnostic and Statistical Manual of Mental Disorders (DSM, American Psychiatric Association, 1980, 1987, 1994, 2000), or on similar diagnostic approaches that served as the precursor for DSM-III (e.g., Feighner et al., 1972; Spitzer, Williams, & Gibbon, If a set of linear equations can be expressed as let's say. a1 = 5, a2 = 24, an+2 = 4an+1+4an. Which is exactly what I got in my post, we get the "base case" for $$b_n$$ from $$a_0 = \sqrt{3}$$ What I did was correct. Problems: 1. Given a function f(x) defined as a linear recurrence relation. a 1 = 5, a 2 = 24, a n + 2 = 4 a n + 1 + 4 a n. Denition 4.1. recurrence relation a n= f(a n 1;:::;a n k). The solution of second order recurrence relations to obtain a closed form First order recurrence relations, proof by induction of closed forms. if the initial terms have a common factor g then so do all the terms in the seriesthere is an easy method of producing a formula for sn in terms of n.For a given linear recurrence, the k series with initial conditions 1,0,0,,0 0,1,0,0,0 (b) If 100 bacteria are used to begin a new colony, how man Prove or disprove that there exists a bijection from (0, 1] to (0, 1]^2 Prove or disprove that there exists a bijection from (0, 1] to [0, )^2. Binary Relation. T (n) = 2T (n/2) + cn T (n) = 2T (n/2) + n. Then for each positive integer $n$ find $a_n$ and $b_n$ such that In mathematics (including combinatorics, linear algebra, and dynamical systems ), a linear recurrence with constant coefficients: ch.

Find a recursive formula for the number of ways he could end up at step Note he starts at step 0 (not on the stairs). The Fibonacci sequence is an example of a linear recurrence relations. Solve an+2+an+1-6an=2n for n 0 . Linear First-Order Recurrence Relations Expand, Guess, and Verify One technique for solving recurrence relations is an "expand, guess, and verify" approach that repeatedly uses the recurrence relation to expand the expression for the $$n_{th}$$ term until the general pattern can be guessed. Note : To know the time The above is an illustration of the 7 steps in which a tower of three stones stacked onto a rod in ascending order of size is moved to the target rod in the middle, using a vacant rod on the right. When the order is 1, parametric coefficients are allowed. Solution 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable that is, in the values of the elements of a sequence. First solve the closed form of the sequence (a n), then Problem 323. Recognize that any recurrence of the form an = r * an-1 is a geometric sequence.

If f n is 0 then the rr is called homogeneous. In case of the Fibonacci sequence, with exception of the first two elements, all other elements of the sequence depend on two previous elements. The general form of linear recurrence relation with constant coefficient is. 1.

Discrete Mathematics - Recurrence Relation Definition. You can define the Fibonacci matrix to be the 2 x 2 matrix with values {0 1, 1 1}. in which some agents' actions depend on lagged variables. Let us now consider linear homogeneous recurrence relations of degree two. For relations in a particular form: $$a_n$$ is given as a linear combination of some number of previous terms. The basis of the recursive denition is also called initial conditions of the recurrence. For any , this defines a unique sequence Then, x, y and z can take values of any combination and are called free variables. . Now solve for $$b_n$$ and utilize the method for finding a closed-form solution of the linear homogenous recurrence relation. Alex Jordan. Example 2 (Non-examples). The recurrence relation F n = F n 1 + F n 2 is a linear homogeneous recurrence relation of degree two. a n = a n 1 + 2 a n 2 + a n 4. A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. In this chapter, we will discuss how recursive techniques can derive sequences and be used for solving counting problems. Example: (The Tower of Hanoi) A puzzel consists of 3 pegs mounted on a board together with disks of different size. The space of tempered growth solutions to the first recurrence relation should be spanned (as a vector space) by the delta function and some linear combinations of its partial derivatives. T(n) = T(n-1) + n for n>0 and T(0) = 1 These types of recurrence relations can be easily solved using substitution method. Degree of recurrence relation. What is Linear Recurrence Relations? Linear Recurrence Relations of Degree 2 a n+1 = f(n)a n +g(n)a n 1 with non-constant coefcients f(n) and g(n). Theorem: 2Let c 1 and c 2 be real numbers. The recurrence relation a, = 3a,-, + 2n is an example of a linear nonhomogeneous recurrence relation with constant coefficients, that is, a recurrence relation of the form a,C,an-1 tCa,-2t + ca,-+ Ea) where C. C2.. In this example, we generate a second-order linear recurrence relation. 3. second degree linear homogeneous recurrence relation has only one root r 1, then all solutions are of the form an = b 1r1 n + b 2nr1 n for n 0, where b 1 and b 2 are constants. This suggests that, for the second order homogeneous recurrence linear relation (2), we may have the solutions of the form xn = rn: In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) (1) A quotient-difference table eventually yields a line of 0s iff the starting sequence is defined by a linear recurrence equation. In math, the relation is between the x-values and y-values of ordered pairs.The set of On the left side, there is only one variable. The idea here is to solve the characteristic polynomial equation associated with the homogeneous recurrence relation. 4.1 Linear Recurrence Relations The general theory of linear recurrences is analogous to that of linear differential equations. 2. We will now take a look at second order linear recurrence relations, named so because, as you may have guessed, the terms in the sequence are written as an equation of the 2 preceding terms. For example,

4. In general for linear recurrence relations the size of the matrix and vectors involved in the matrix form will be identied by the order of the relation. a = 3x + 4y + 5z - 12. b = 2x + 8y + z - 11. c = 9x + 7y -z - 15. where. Compute f(N),where N is a large number . n 1 is a linear homogeneous recurrence relation of degree one. These types of recurrence relations can be easily solved using Master Method. Given $$\alpha _1, \ldots, \alpha _k\in \mathbb C$$ , it is immediate to verify (by induction, for instance) that there is exactly one linear recurrent sequence ( a n ) n 1 satisfying ( 21.1 ) and such that a j = j for Linear recurrence relations Remember that a recurrence relation is a sequence that gives you a connection between two consecutive terms. From these conditions, we can write the following relation x = x + x. Problems: 1. The recurrence relation a n = a n 1a n 2 is not linear. of the nonhomogeneous recurrence relation is 2 , if we formally follow the strategy in the previous lecture, we would try = 2 for a particular solution. So, this sequence f(i) = f(i-1) * f(i-2) is not a linear recurrence. In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) Here both a and b must be non-zero: b Finally the guess is verified by mathematical induction. A linear recurrence relation is homogeneous if f(n) = 0. The degree of recurrence relation is K if the highest term of the numeric function is expressed in terms of its previous K terms. That is, there can be no terms in the recurrence relation such as $a_{n-1}^2$ or $a_{n-1}a_{n-2}$. Output : 0 15. Recurrence Relations Can easily describe the runtime of recursive algorithms Can then be expressed in a closed form (not defined in terms of itself) Consider the linear search: Kurt Schmidt Drexel University Eg. Example an = 6a n-1 9a n-2, a 0=1 and a 1=6 Characteristic equation r 2 6r + 9 = 0 with only one root 3 2 6 0 2 1 Doing so is called solving a recurrence relation. Degree = highest coefficient - lowest coefficient Linear recurrence relation with constant coefficients. Linear recurrence relations can be subdivided into homogeneous and non-homogeneous relations depending on whether or not {eq}f (n)=0 {/eq}. The recurrence rela-tion m So I think the way should be to find a general solution for the homogeneous equation first. This is basically done with an algorithmic process that can be summarized in three steps:Find the linear recurrence characteristic equationNumerically solve the characteristic equation finding the k roots of the characteristic equationAccording to the k initial values of the sequence and the k roots of the characteristic equation, compute the k solution coefficients Checkpoint. In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements.The number of instances given for each element is called the multiplicity of that element in the multiset. The theory behind them is relatively easy to understand, and they are easily implemented and fast, 1. Paper 9FM0/4B Further Statistics . These two terms are Example. We say a recurrence relation is linear if fis a linear function or in other words, a n = f(a n 1;:::;a n k) = s 1a n 1 + +s ka n k+f(n) where s i;f(n) are real numbers. Example: Find a recurrence relation for C n the number of ways to parenthesize the product of n + 1 numbers x 0, x 1, x 2, , x n to specify the order of multiplication. currence linear relation is also a solution. If (a, b) R and R P x Q then a is related to b by R i.e., aRb. 1 - Linear Search Recursively Look at an element (constant work, c), then search the remaining elements T(n) = T( n-1 ) + c The cost of searching n elements is the cost of looking at 1 element, plus the cost of searching n-1 elements Kurt The homogeneous refers to the fact that there is no additional term in the recurrence relation other than a multiple of $$a_j$$ terms. RSolve not reducing for a certain recurrence relation. A recurrence relation is a functional relation between the independent variable x, dependent variable f (x) and the differences of various order of f (x). Write the closed-form formula for a geometric sequence, possibly with unknowns as shown. Best case : Order of growth will be constant because in the best case we are assuming that (n) is even Average case : In this case we will assume that even and odd are equally likely, therefore Order of growth will be linear Worst case : Order of growth will be linear because in this case we are assuming that (n) is always odd. If f (n) = 0, the relation is homogeneous otherwise non-homogeneous. where c is a constant and f (n) is a known function is called linear recurrence relation of first order with constant coefficient. Last time we worked through solving linear, homogeneous, recurrence relations with constant coefficients of degree 2 Solving Linear Recurrence Relations (8.2) The recurrence is linear because the all the a n terms are just the terms (not raised to some power nor are they part of some function). Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics. un+2 + un+1 -6un=0. PURRS is a C++ library for the (possibly approximate) solution of recurrence relations . The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. in which some agents' actions depend on lagged variables. currence linear relation is also a solution. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form = (,) >, where : is a function, where X is a set to which the elements of a sequence must belong. +Ck xnk = bn, where C0 6= 0. If bn = 0 the recurrence relation is called homogeneous. A sequence (xn) n=1 satises a linear recurrence relation of order r 2N if there exist a 0,. . We set A = 1, B = 1, and specify initial values equal to 0 and 1. $$n^{th}$$ Order Linear Recurrence Relation. This suggests that, for the second order homogeneous recurrence linear relation (2), we may have the solutions of the form xn = rn: Linear recurrence relations and matrix iteration. Find the general term of the Fibonacci sequence. Sequences satisfying linear recurrence relation form a subspace Linear Recurrence Relations To do this, we compute the eigenvectors of Aby nding the characteristic polynomial: c A( ) = det(A I) = det 2 3 1 = (2 ) ( ) 1 3 = 2 2 3 (4) = ( 3)( + 1) which has roots 3 and 1.

Solving Recurrence Relations The solutions of this equation are called the characteristic roots of the recurrence relation. Otherwise it is called non-homogeneous. Our DAA Tutorial includes all topics of algorithm, asymptotic analysis, algorithm control structure, recurrence, master method, recursion tree method, simple sorting algorithm, bubble sort, selection sort, insertion sort, divide and conquer, binary search, merge sort, counting sort, lower


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# Exponential Function

A function of the form ${f{{\left({x}\right)}}}={{a}}^{{x}}$, where ${a}>{0}$, is called an exponential function.

Do not confuse it with the power function ${f{{\left({x}\right)}}}={{x}}^{{a}}$ in which the variable is the base.

The domain of the exponential function is ${\left(-\infty,\infty\right)}$, and the range is ${\left({0},\infty\right)}$, provided ${a}\ne{1}$.

If ${a}={1}$, we have that ${f{{\left({x}\right)}}}={{1}}^{{x}}={1}$.

Exponential functions are useful for modeling many natural phenomena such as population growth (if ${a}>{1}$) and radioactive decay (if ${0}<{a}<{1}$).

Let's see what is meant under ${f{{\left({x}\right)}}}={{a}}^{{x}}$.

If ${x}={n}$, where ${n}$ is a postive integer, we have that ${{a}}^{{n}}=\underbrace{\left({a}\cdot{a}\cdot\ldots\cdot{a}\right)}_{{{n}}}$.

If ${x}={0}$, it can be stated that ${{a}}^{{0}}={1}$, and if ${x}=-{n}$, where ${n}$ is a positive integer, we have that ${{a}}^{{-{n}}}=\frac{{1}}{{{{a}}^{{n}}}}$.

If ${x}$ is a rational number, ${x}=\frac{{p}}{{q}}$, where ${p}$ and ${q}$ are integers, it can be stated that ${{a}}^{{x}}={{a}}^{{\frac{{p}}{{q}}}}={\sqrt[{{q}}]{{{{a}}^{{p}}}}}={{\left({\sqrt[{{q}}]{{a}}}\right)}}^{{p}}$.

There are three kinds of exponential functions:

1. If ${0}<{a}<{1}$, the exponential function decreases.
2. If ${a}={1}$, the exponential function is constant.
3. If ${a}>{1}$, it increases (the bigger ${a}$, the more rapidly it increases).

Properties of exponents.

If ${a}$ and ${b}$ are positive numbers, and ${x}$ and ${y}$ are any real numbers, it can be stated that:

1. ${{a}}^{{{x}+{y}}}={{a}}^{{x}}{{a}}^{{y}}$
2. ${{a}}^{{{x}-{y}}}=\frac{{{{a}}^{{x}}}}{{{{a}}^{{y}}}}$
3. ${{\left({{a}}^{{x}}\right)}}^{{{y}}}={{a}}^{{{x}{y}}}$
4. ${{\left({a}{b}\right)}}^{{x}}={{a}}^{{x}}{{b}}^{{x}}$

Applications of exponential functions.

As stated above, exponential functions are widely used in growth/decay problems.

Example 1. Suppose that the initial population of some bacteria is 100. It is known that every 1 hour the population triples. Find the population after 10 hours.

If the number of bacteria at a time ${t}$ is ${p}{\left({t}\right)}$, where ${t}$ is measured in hours, and ${p}{\left({0}\right)}={100}$ is the initial population, we have that

${p}{\left({1}\right)}={3}{p}{\left({0}\right)}={300}$

${p}{\left({2}\right)}={3}{p}{\left({1}\right)}={3}\cdot{3}\cdot{p}{\left({0}\right)}={{3}}^{{2}}{p}{\left({0}\right)}={900}$

${p}{\left({3}\right)}={3}{p}{\left({2}\right)}={3}\cdot{3}\cdot{3}\cdot{p}{\left({0}\right)}={{3}}^{{3}}{p}{\left({0}\right)}={2700}$

${p}{\left({4}\right)}={3}{p}{\left({3}\right)}={3}\cdot{3}\cdot{3}\cdot{3}\cdot{p}{\left({0}\right)}={{3}}^{{4}}{p}{\left({0}\right)}={8100}$.

Do you see the pattern? It seems that ${p}{\left({t}\right)}={100}\cdot{{3}}^{{n}}$.

So, ${p}{\left({10}\right)}={100}\cdot{{3}}^{{{10}}}={5904900}$.

Now, let's proceed to another example.

Example 2. Suppose that the half-life of some radioactive element is 3 years (recall that the half-life is the amount of time needed to disintegrate a half of any quantity initially presented). Suppose that the initially presented mass is 1000 mg. Find the mass after 10 years.

If ${m}{\left({t}\right)}$ is the mass of the element at any time ${t}$ measured in years, we have that

${m}{\left({0}\right)}={1000}$

${m}{\left({3}\right)}=\frac{{1}}{{2}}{m}{\left({0}\right)}={500}$

${m}{\left({6}\right)}=\frac{{1}}{{2}}{m}{\left({3}\right)}=\frac{{1}}{{{2}}^{{2}}}{m}{\left({0}\right)}={250}$

From this pattern, we can conclude that

${m}{\left({t}\right)}=\frac{{1}}{{{{2}}^{{\frac{{t}}{{3}}}}}}\cdot{1000}={1000}\cdot{{2}}^{{-\frac{{t}}{{3}}}}$.

So, in 10 years, there will be ${m}{\left({10}\right)}={1000}\cdot{{2}}^{{-\frac{{10}}{{3}}}}\approx{99.21}$ mg.

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# Equality of Towers in Set

## Theorem

Let $X$ be a non-empty set.

Let $\struct {T_1, \preccurlyeq_1}$ and $\struct {T_2, \preccurlyeq_2}$ be towers in $X$.

Then either:

$\struct {T_1, \preccurlyeq_1} = \struct {T_2, \preccurlyeq_2}$

or:

$\struct {T_1, \preccurlyeq_1}$ is an initial segment of $\struct {T_2, \preccurlyeq_2}$

or:

$\struct {T_2, \preccurlyeq_2}$ is an initial segment of $\struct {T_1, \preccurlyeq_1}$

## Proof

By the definition of tower:

$\struct {T_1, \preccurlyeq_1}$ and $\struct {T_2, \preccurlyeq_2}$ are well-ordered sets.
$(1): \quad$ the towers are order isomorphic to each other

or:

$(2): \quad$ one is order isomorphic to an initial segment in the other.

Without loss of generality, in case $(2)$, assume that $\struct {T_1, \preccurlyeq_1}$ is order isomorphic to an initial segment in $\struct {T_2, \preccurlyeq_2}$.

$i: \struct {T_1, \preccurlyeq_1} \to \struct {T_2, \preccurlyeq_2}$

such that $i \sqbrk {T_1}$, the image set of $T_1$ under $i$, is equal to $T_2$ or to an initial segment of $T_2$.

From Characterization of Strictly Increasing Mapping on Woset, we can characterize $i$ as follows:

$\forall t \in T_1: \map i t = \map \min {T_2 \setminus i \sqbrk {S_t} }$

and:

$i \sqbrk {S_t} = S_{\map i t}$

Define:

$Y = \set {y \in T_1: \map i y = y}$

Then for any $t \in T_1$ and $S_t \subseteq Y$:

 $\ds t$ $=$ $\ds \map c {X \setminus S_t}$ Definition of Tower in Set $\ds$ $=$ $\ds \map c {X \setminus i \sqbrk {S_t} }$ by definition of $Y$ $\ds$ $=$ $\ds \map c {X \setminus S_{\map i t} }$ as $i \sqbrk {S_t} = S_{\map i t}$ $\ds$ $=$ $\ds \map i t$ Definition of Tower in Set

So $\map i y = y$ for any initial segment $S_t \subseteq Y$.

By Induction on Well-Ordered Set, $Y = T$.

Thus $i$ is the identity mapping from $T_1$ onto its image.

Recall that the image set of $i$ is $T_2$ or an initial segment of $T_2$.

So $T_1 = T_2$ or an initial segment of $T_2$.

$\blacksquare$


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## Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole

# Chapter 1 - Section 1.1 - Real Numbers - 1.1 Exercises: 52

#### Answer

{$x | x \lt 1$}

#### Work Step by Step

This set includes from negative infinity up to but not including one. This means that $x$ can be anything "less than" one, or $x \lt 1$. When indicating this on a number line, you start with an open point at one to show it is not included, and then shade towards the negative end of the number line, and put an arrow at the end to show it continues to negative infinity.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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Special Perspectives

# Managing Forests for Climate Change Mitigation

See allHide authors and affiliations

Science  13 Jun 2008:
Vol. 320, Issue 5882, pp. 1456-1457
DOI: 10.1126/science.1155458

## Abstract

Forests currently absorb billions of tons of CO2 globally every year, an economic subsidy worth hundreds of billions of dollars if an equivalent sink had to be created in other ways. Concerns about the permanency of forest carbon stocks, difficulties in quantifying stock changes, and the threat of environmental and socioeconomic impacts of large-scale reforestation programs have limited the uptake of forestry activities in climate policies. With political will and the involvement of tropical regions, forests can contribute to climate change protection through carbon sequestration as well as offering economic, environmental, and sociocultural benefits. A key opportunity in tropical regions is the reduction of carbon emissions from deforestation and degradation.

Forest ecosystems are important components of the global carbon cycle in at least two ways. First, terrestrial ecosystems remove nearly 3 billion tons of anthropogenic carbon every year (3 Pg C year–1) through net growth, absorbing about 30% of all CO2 emissions from fossil fuel burning and net deforestation (1, 2). Forests are major contributors to this terrestrial carbon sink and its associated economic benefits (1). Second, 4 billion hectares of forest ecosystems (4 × 103 Mha; about 30% of the global land area) store large reservoirs of carbon, together holding more than double the amount of carbon in the atmosphere (3, 4). Although the climate protection role of forests is in no doubt, it is complex to determine how much of the forest carbon sink and reservoir can be managed to mitigate atmospheric CO2 buildup, and in what way.

A first approximation to the upper limit of carbon sequestration on land is the carbon emitted from historical land transformation, about 200 Pg C, mostly from the conversion of forests to nonforest land cover. Assuming that three-fourths of this carbon came from forest conversion and can be returned by reforestation over the next 100 years, the resulting potential sequestration of about 1.5 Pg C year–1 would reduce the atmospheric CO2 concentration by 40 to 70 parts per million by 2100 (5). However, the achievable sequestration is only a fraction of this potential because of competing land needs (agriculture, bioenergy, urbanization, and conservation) and sociocultural considerations.

Four major strategies are available to mitigate carbon emissions through forestry activities: (i) to increase forested land area through reforestation (6), (ii) to increase the carbon density of existing forests at both stand and landscape scales, (iii) to expand the use of forest products that sustainably replace fossil-fuel CO2 emissions, and (iv) to reduce emissions from deforestation and degradation.

Estimates covering a range of carbon prices suggest that reforestation could average 0.16 to 1.1 Pg C year–1 to 2100 (79) with land requirements up to 231 Mha. In one of the most comprehensive synthesis efforts undertaken so far, the Fourth Assessment of the Intergovernmental Panel on Climate Change estimated that an economic potential of 0.12 Pg C year–1 could be reached by 2030 at U.S. $20 per ton of CO2, and more than 0.24 Pg C year–1 at U.S.$100 per ton of CO2 (10, 11). Land transformation requirements are large; for example, China has used 24 Mha of new forest plantations and natural forest regrowth to transform a century of net carbon emissions in the forest sector to net gains of 0.19 Pg C year–1 (3, 12), offsetting 21% of Chinese fossil fuel emissions in 2000.

Net carbon sequestration can also be achieved by increased forest carbon density, through both stand-scale management and landscape-scale strategies such as longer harvesting cycles or reduced disturbances. Fire suppression and harvest exclusion in U.S. forests during the 20th century, although not implemented for the purpose of carbon sequestration, led to a 15% (8.1 Pg C) increase in forest biomass between 1927 and 1990 (13). The overall biophysical potential of management activities to increase carbon density can be substantial and comparable to that of reforestation (10).

Joint use of carbon sequestration and the provision of forest-derived products (e.g., timber and biomass for energy) will optimize the contribution of forestry in climate mitigation. Such options are particularly attractive in temperate regions where land availability is limited by high prices and strong competition with other land uses (Fig. 1). Although complexities in quantifying the net carbon benefits of some of these activities may limit their role in global carbon markets, they will have a place in national mitigation strategies, particularly when used synergistically with goals and policies other than climate mitigation. For instance, fire reduction policies that require the removal of undergrowth and occasional thinning can contribute to production of bioenergy.

Finally, reducing deforestation has high potential for cost-effective contributions to climate protection. Currently, 13 Mha year–1 are deforested, almost exclusively in tropical regions, with net emissions of 1.5 Pg C year–1 (2, 3). Reducing rates of deforestation by 50% by 2050, and stopping deforestation when countries reach 50% of their current forested area, would avoid emissions equivalent to 50 Pg C (14). This “50:50:50:50” estimate shows that even with continuing deforestation over the next 40 years, the mitigation potential is large, in addition to protecting the sink capacity of forest for continued removal of atmospheric CO2.

Combining all forestry activities together, there is economic potential to achieve 0.4 Pg C year–1 by 2030 using carbon sequestration and avoidance at U.S. $20 per ton of CO2, and double this amount for prices under U.S.$100 per ton of CO2 (10). These levels of carbon sequestration, of which one-third to one-half would be through avoided deforestation, could offset 2 to 4% of the 20 Pg C year–1 of projected emissions by 2030 on the basis of current growth rates (2, 15). Tropical regions would account for 65% of the total offset (10).

Climate mitigation through forestry carries the risk that carbon stores may return to the atmosphere by disturbances such as fire and insect outbreaks, exacerbated by climate extremes and climate change. A recent increase in areas affected by wildfires and insect outbreaks has driven Canadian forests from a CO2 sink (before 2000) to a source expected to continue for at least the next two to three decades (16). Similarly, increased forest biomass in the western United States caused by fire suppression and reduced harvesting rates over the past century is now threatened by a factor of 4 increase in fire frequency due to longer and hotter dry seasons (17). These new patterns of disturbances are reshaping the view held in the past that vast forest resources anywhere would always play a major role in climate mitigation.

There is indeed uncertainty about the future size and stability of the terrestrial carbon sink and stock. Most global coupled climate-carbon models show carbon accumulation during this century, largely aided by the fertilization effect of increasing atmospheric CO2 (18). However, there are large uncertainties in the magnitude of the CO2 fertilization effect (19), and vulnerable regions with large carbon stores have been identified that could lead to the release of hundreds of Pg C by the end of this century (20); these include peat swamp forests in Southeast Asia where climate models uniquely agree on a future drying trend (21), further stressing the need for conservation and reduced human impacts.

Although sequestering carbon in forests is good for the climate, forests also affect biophysical properties of the land surface such as sunlight reflectivity (albedo) and evaporation, with further implications for radiative forcing of climate. Climate models suggest that large reforestation programs in boreal regions would have limited climate benefits because of the substitution of bright snow-dominated regions for dark forest canopies (22, 23). Conversely, the climate benefits of reforestation in the tropics are enhanced by positive biophysical changes such as cloud formation, which further reflects sunlight. These patterns of full radiative forcing reinforce the large potential of tropical regions in climate mitigation, discourage major land use changes in boreal regions, and suggest avoiding large albedo changes in temperate regions to maximize the climate benefits of carbon sequestration.

Forestry, and reforestation in particular—like any large-scale transformation of land use patterns—can lead to unintended environmental and socioeconomic impacts that could jeopardize the overall value of carbon mitigation projects. Concerns include decreased food security, reduced stream flows, and loss of biodiversity and local incomes (24). However, well-directed carbon sequestration projects, along with the provision of sustainably produced timber, fiber, and energy, will yield numerous benefits, including additional income for rural development, prospects for conservation and other environmental services, and support for indigenous communities (10, 25). Principles of sustainability must govern the resolution of trade-offs that may arise from ancillary effects in order to simultaneously maximize climate change protection and sustainable development.

The challenges facing sustainable mitigation through forestry activities, anywhere but particularly in the tropics, are surmountable but large. They include the development of appropriate governance institutions to manage the transition to new sustainable development pathways. An example of this difficulty is the lack of a sustainable tropical timber industry despite two decades of national and international efforts. Currently, only 7% of all tropical timber trade comes from sustainably managed forests (26).

The potential of carbon sequestration will depend on the degree to which climate protection and ancillary benefits are aligned. The magnitude of this potential will be increased by high carbon prices driven by aggressive emission reduction targets, and by the political will to include forestry activities as part of mitigation portfolios. Sustainable involvement of tropical regions is essential to take up the full global potential for climate change mitigation through forestry.

View Abstract


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# Math Help - Help with checking Vector Question

1. ## Help with checking Vector Question

Hey guys I am not sure if this is the right thread (can't seem to find one for vectors) but i am not sure if i'm on the right track here so i need some advice

Question is:

if $\vec{a}=3i-2j$ , $\vec{b}=-4i+4j$ , and $\vec{c}=6i+-9j$ express the following vectors in their simplest form:

(i) $-5\vec{b}$
(ii) $2\vec{a}-\frac{1}{2}\vec{c}$
(iii) $\frac{2}{3}\vec{a}-\frac{1}{2}\vec{b}-\frac{1}{4}\vec{c}$

My Solution:

(i) $-5\left(-4i+4j \right)=20i+\left(-20j \right)$
(ii) $2(3i-2j)-\frac{1}{2}(6i+(-9)j)$
$= 6i-4j-3i+(-\frac{9}{2})j$
$= (6-3=3i) -4j-\frac{9}{2}$
$= -\frac{4}{1}-\frac{9}{2}$
$=-\frac{8}{2}-\frac{9}{2}$
$=-\frac{17}{2}$

Answer? $3i-\frac{17}{2}j$

(iii) $\frac{2}{3}(3i-2j)-\frac{1}{2}(-4i+4j)-\frac{1}{4}(6i+(-9)j)$
$(2i-\frac{4}{3}j)-(-2i+2j)-(\frac{3}{2}i+(-\frac{9}{4}j)$
$2i+2i-\frac{3}{2}i$
$= \frac{4}{1}i+\frac{3}{2}i$
$= \frac{8}{2}+\frac{3}{2}$
$=\frac{11}{2}i$

$-\frac{4}{3}j+\frac{2}{1}j-\frac{9}{4}j$
$= -\frac{16}{12}j+\frac{24}{12}j-\frac{27}{12}j$
$= -\frac{19}{12}j$

Answer? $\frac{11}{2}i-\frac{19}{12}j$

2. Looks Ok to me but you're presentation was hard to follow.

3. Originally Posted by ojones
Looks Ok to me but you're presentation was hard to follow.
ah thank you! i'll write it better in my maths book and write it better next time i need help on here

4. Actually, there is an arithmetic error in (ii): answer should be $3{\bf i}+(-4+\frac{9}{2}){\bf j}=3{\bf i}+\frac{1}{2}{\bf j}$

(iii) also has an error: you changed $-\frac{3}{2}{\bf i}$ to $\frac{3}{2}{\bf i}$ by mistake.

5. Originally Posted by ojones
Actually, there is an arithmetic error in (ii): answer should be $3{\bf i}+(-4+\frac{9}{2}){\bf j}=3{\bf i}+\frac{1}{2}{\bf j}$

(iii) also has an error: you changed $-\frac{3}{2}{\bf i}$ to $\frac{3}{2}{\bf i}$ by mistake.
ah yea thanks! Great pickup!


text:
# Revision history [back]

Hi

Yes, with Combo Box and ListBox you can define the Source cell range in the Data tab (design mode):

Note: as you can see in the screenshot linked cell (B1 in this example) is a text, even if the source data range contains numbers.

Regards

Hi

Yes, with Combo Box and ListBox you can define the Source cell range in the Data tab (design mode):

Note: as you can see in the screenshot linked cell (B1 in this example) is a text, even if the source data range contains numbers.

[EDIT]

I join the Combo.ods example so you can check the form.

Regards


text:
### Home > INT2 > Chapter 12 > Lesson 12.1.3 > Problem12-35

12-35.

For the each of the following, write down a factorial expression and then compute the value. Use the $_nC_r$ and $_nP_r$ functions on your calculator to make the computation.

1. $_{10}P_8$

$\frac{10!}{2!} = 1{,}814{,}400$

1. $_{10}C_8$

$\frac{10!}{8!2!} = 45$

1. $_6C_1$

$\frac{6!}{1!5!}=6$

Explore using the eTool below.
Click the link to the right for full version. Int2 12-35 HW eTool.


text:
# Chapter 8 - Right Triangles and Trigonometry - 8-1 The Pythagorean Theorem and It's Converse - Practice and Problem-Solving Exercises: 12

$17=x$

#### Work Step by Step

The Pythagorean Theorem states that if $a$ and $b$ are the legs of a right triangle and $c$ is the hypotenuse, then $a^2+b^2=c^2$. We plug in the values for $a$, $b$, and $c$, using the legs and the hypotenuse of the right triangle shown, and then solve the equation using the properties of algebra and equality: $$8^2+15^2=x^2$$$$64+225=x^2$$$$289=x^2$$$$\sqrt {289}=\sqrt {x^2}$$$$17=x$$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Thread: Solve x⁴+ x + x + x + 1 = 0

1. ## Solve x⁴+ x + x + x + 1 = 0

Solve x⁴+ x + x + x + 1 = 0

2. $\displaystyle f(x) = x^4 + x^3 + x^2 + x + 1$ has no real roots.

3. Originally Posted by feyomi
Solve x⁴+ x + x + x + 1 = 0
$\displaystyle x^5-1 = (x-1)(x^4+x^3+x^2+x+1)$

All roots of $\displaystyle x^5-1$ are $\displaystyle 5^{th}$ roots of unity...

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# 1 x x x x⁴=0 find the value of x

Click on a term to search for related topics.


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# Questions Asked onMay 14, 2013

1. ## 8need a function rh grade math

a 154 lb person burns 420 calories per hr riding an exercise bike at 15 mph. Need a function rule to represent the total calories burned over time.

2. ## trigonometry

find acute angle A and B satisfying secAcotB-secA-2cotB+2=0

3. ## Math Help? Check my answers.

1. You mix the letters: S E M I T R O P I C A and L thoroughly. Without looking you draw one letter. Find the probability that you select a vowel. From a fraction to simplest form. 12/5 5/12 1/3 7/12 2. You roll a standard cube once. Fine P(0) 7/6 1 1/2 0

4. ## Calculus

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5. ## chem

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7. ## chem

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8. ## chemistry

5. Meals-ready-to-eat (MREs) are military meals that can be heated on a flameless heater. The heat is produced by the following reaction: Mg(s) + 2H2O(l) --> Mg(OH)2(s) + H2(g) Calculate the number of grams of Mg needed for this reaction to release enough

9. ## Equation help?

From midnight to 6:00 am, the temperature rose 8°C. At 6:00 am, the temperature was –20°C. What was the temperature at midnight. I know it's -20-8 = -28 C temp at 12am, but how do I write it an equation?

10. ## Chemistry

Is Cl2 + 2e- -> 2Cl- an oxidation half-reaction?

11. ## english

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12. ## calculus (gr 12)

the productivity of a person at work (on a scale of 0-10) is modeled by a cosine function: 5cos (pie/2t) +5, where t is in hours. If the person starts work at t=0, being 8:00 am, at what times is the worker the least productive? a) 12pm B) 10 am and 2pm C)

13. ## physics

A guitar string is 92 cm long and has a mass of 3.7 g. The length L from the bridge to the support post is 65 cm, and the string is under a tension of 570 N. What is the frequency of the fundamental tone thanks!

14. ## physics

he index of refraction of crown glass is 1.53 for violet light, and it is 1.51 for red light. (a) What is the speed of violet light in crown glass? (b) What is the speed of red light in crown glass?

15. ## Physics

A roller coaster has a mass of 425 kg. It sits at the top of a hill with height 66 m. If it drops from this hill, how fast is it going when it reaches the bottom? (Assume there is no air resistance or friction.)

16. ## physics

A steel rod of length 1m is heated from 25°c to 75°c keeping its length constant.The longitudinal strain developed in the rod is:(steel:12 X 10^-6/°c) "6X10^-4" ; "-6X10^-4" ; "0" please help me as soon as possible by explain this.

17. ## Math~Polynomials

5)Find the degree of -5w-4w^2+7w+16(1point) -5 16 14*** 3 Ok I put stars next to my answer though I'm not sure if I'm correct.. or not so Steve could you tell me if my calculations are way off or if they are correct?....

28. ## math

a 10-foot ladder is placed against the side of a building as shown i figure i below.the bottom of the ladder is 8 feet from the base of the building .in the order to increase the reach of the ladder against the building ,it is moved 4 feet closer to the

29. ## physics

What is the magnitude of the electrostatic force on a +5 C harge 1.75 m to the left of a -2.5 C Charge? (show your work)

30. ## Physics helpp

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31. ## Social Studies

What happened to unions after the protest at Haymarket Square? My answer: Afterward, the Chicago police arrested hundreds of union leaders, socialists, and anarchists. Opposition to unions increased. The membership in the Knights of Labor dropped, even

32. ## physics

thus far in the course we have ignored that on the earths surface an object must have a slightly unbalanced force to rotate with the earth, pretending that the normal force precisely balances the objects weight. A student with a mass of 65.0kg stands at

33. ## math

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34. ## chemistry

3. When a mixture of 10.0 grams of acetylene (C2H2) and 10.0 grams of oxygen (O2) is ignited, the resulting combustion reaction produces CO2 and H2O. A) Write a balanced equation for this reaction. B) What is the limiting reactant? C) How many grams of C2,

35. ## math

A cable 20 feet long connects the top of a flagpole to apoint on the ground that is 16 feet from the base of the pole.how tall is the flagpole? (A) 26ft (B) 10ft (C)12ft (D)8ft..

36. ## mathematics

Consider the following eqution x2 + 4y2 + 9z2 = 14x + 28y+42z + 147 Then the value of x+2y+3z is

37. ## Physics to Elena

At the upper surface of the earth's atmosphere, the time-averaged magnitude of the Poynting vector is referred to as the solar constant and is given by =1.35×103W⋅m−2 . (a) If you assume that the sun's electromagnetic radiation is a plane sinusoidal

38. ## calculus

The marginal cost for a product is given by dC/dq = 1000/sqrt3q+20, in hundreds of dollars, where q is given in thousands of units. Determine the cost in dollars involved in increasing product from 10 to 15 thousand units. If R(0)= C(0)= 0, the marginal

39. ## math

What is The ratio of w's to consonants in this sentence? Wes and Willow went walking.

40. ## math

Rewrite this expression using the distributive property. 4 x (8 + m)

41. ## Informations

What does Current GPA mean? What does Cumulative GPA?

42. ## math

You deposit $1200 in an account that pays an APR of 7% compounded monthly. How long will it take for your balance to reach$150,000?

43. ## science

The scientific binomial name of a species is Canis vulpis. Which of the following is true? A. The species belongs in the family Cranis. B. the species belongs in the genus vulpis. C. the species belongs in the genus Canis. D. the species belongs in the

44. ## algegra

a person jumps off of a 6 foot high diving board with an initial velocity of 13 feet per second. how many seconds does it take the person to hit the water?

45. ## arithmetic

there are n arithmetic means between 5 and 32 such that the ratio between the 3rd and 7th means is 7:13,find the value of n.

46. ## Equation Ms Sue?

From midnight to 6:00 am, the temperature rose 8°C. At 6:00 am, the temperature was –20°C. What was the temperature at midnight. I know it's -20-8 = -28 C temp at 12am, but how do I write it an equation?

47. ## math

You and a friend are biking in opposite directions starting from the same point. You travel 108 feet every 8 seconds. Your friend travels 63 feet every 6 seconds. How far apart are you and your friend after 15 minutes?

48. ## Chemistry

Is Cl2 + 2e- -> 2Cl- an oxidation half-reaction?

49. ## math

A juice company decides to test six different brands of juice by comparing each brand with each of the other brands. How many different pairs of the six juices will be tested? 10 12 15 7 is it 15?

50. ## chemistry

4. A 35.0 mL sample of 1.00 M KBr and a 60.0-mL sample of 0.600 M KBr are mixed. The solution is then heated to evaporate water until the total volume is 50.0 mL. What is the molarity of the KBr in the final solution?

51. ## Math

A wheel on Greg's bicycle is 0.3 m in diameter. Greg races the bicycle for 31 m. How many times does the wheel turn as the bicycle travels this distance?

52. ## HISTORY

Was imperialism necessary in the 19th century?

53. ## algebra

A women has a ladder that is 13 feet long.if she sets the base of the ladder on level ground 5 feet from the side of a house,how many feet above the ground will the top of the ladder be when it rests against the house?

54. ## History

1. What major problem did the Roman Empire face in the mid to late Fourth Century? A. Weak leadership B. Immigration and invasion C. Lack of education within the empire D. Poor infrastructure 2. A staple food transplanted from Southeast Asia to the

55. ## chemistry

A 211 g sample of barium carbonate, BaCO3, reacts with a solution of nitric acid to give bar- ium nitrate, carbon dioxide and water. If the acid is present in excess, what mass and volume of dry carbon dioxide gas at STP will be produced

87. ## Functions - Math - 8

x: 2 | 4 | 6 y: 1 | 0 | -1 Please show me how to find the rule, thanks. such as... y= *equation*

88. ## math alggggggggggggebra

Find the largest possible integer n such that there exists a non-constant quadratic polynomial f(x) with integer coefficients satisfying f(1)∣f(2),f(2)∣f(3),…f(n−1)∣f(n).

89. ## math

A rectangular flower garden is surrounded by a grass border. the grass border is 5 meters wide on all sides of the flower garden and the outer dimensions are 50 meters by 40 meters. Determine the area of the grass border and show how you got this answer

90. ## chemistry

balance the net ionic equation for the overall reaction. Pt | Fe2+(aq), Fe3+(aq) || I-(aq) | AgI(s) | Ag(s) a) Fe3+(aq) + Ag(s) + I-(aq) → Fe2+(aq) + AgI(s) b) Fe2+(aq) + Ag(s) + I-(aq) → Fe3+(aq) + AgI(s) c) Fe3+(aq) + AgI(s) → Fe2+(aq) + Ag(s) +

91. ## question

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92. ## chemistry

If 15 grams of table salt is used how many moles of sodium sulfate are produced

93. ## Chemistry

0.1 mol of HCl is added to 1 L of water. What is [H+] and [OH-]?

94. ## Perform the indicated operation and simplify

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95. ## math alggggggggebra

Find the largest number of real numbers x1,...,xn such that for all i≠j, |xi−xj|>1/100(1+xixj).

96. ## physics

A person takes a trip, driving with a constant speed of 92.5 km/h except for a 20.0 min rest stop. If the person's average speed is 71.2 km/h, how much time is spent on the trip and how far does the person travel?

97. ## genetics

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98. ## physics

the space shuttle typically orbits 400 km above the earths surface. The earth has a mass of 5.98 * 10^24 kg and a radius of 6380km. A) How much would a 2000 kg part for the space station weigh when it has been lifted to that orbit in the shuttles cargo

99. ## Sigmund freud

Provide detailed descriptions of individuals based on careful observation or formal psychological testing sigmund freud is famous for

100. ## Social Studies

How did Carnegie's company break the union at the Homestead mills? My answer: It broke the union after the company responded to the union's refusal to accept reduced wages by locking out union workers from the mill and announcing that it would hire

101. ## science

The nonliving matrix of the connective tissue through which the blood cells freely flow is the? A.plasma. B. extracellular fluid. C. intracellular fluid. D. submucosa.

102. ## Social Studies

How did Carnegie's company break the union at the Homestead mills?

103. ## chemistry

how much of 0.1 M NaOH is needed to neutralize 5mL of 0.1 M of HCL to a ph 7(color on the ph scale)...Thank you

104. ## mechanics

Find the total kinetic energy of a 1500 g wheel, 700 mm in diameter, rolling across a level surface at 650 rpm. Assume that the wheel can be considered as a hoop. (b) For the wheel in part (a), calculate the torque required to decrease its rotational speed

105. ## phycis

A 12 g bullet is stopped in a block of wood (mw = 5.9 kg). The speed of the bullet- plus-wood combination immediately after the collision is 0.95 m/s. What was the original speed of the bullet? Answer in units of m/s

106. ## physics

A 12 g bullet is stopped in a block of wood (mw = 5.9 kg). The speed of the bullet- plus-wood combination immediately after the collision is 0.95 m/s. What was the original speed of the bullet? Answer in units of m/s

107. ## science

An attack helicopter is equipped with a 20- mm cannon that fires 61.6 g shells in the forward direction with a muzzle speed of 585 m/s. The fully loaded helicopter has a mass of 4890 kg. A burst of 208 shells is fired in a 2.32 s interval. What is the

108. ## Physics

The truss structure ABCD is loaded with a concentrated force P = 2 kN applied at a 45° angle at joint D as indicated in the figure. All bars in the truss have equal cross sectional area and are made of the same homogeneous linear elastic material. The

109. ## Literacy

Who publishes Pediatric Nursing

-10i(5+8i)

111. ## science

A 27 kg gun is standing on a frictionless sur- face. The gun fires a 56 g bullet with a muzzle velocity of 314 m/s. The positive direction is that of the bullet. Calculate the momentum of the bullet im- mediately after the gun was fired. Answer in units of

112. ## chmistry

Given that \rm S is the central atom, draw a Lewis structure of \rm OSF_4 in which the formal charges of all atoms are zero.

113. ## science

A 27 kg gun is standing on a frictionless sur- face. The gun fires a 56 g bullet with a muzzle velocity of 314 m/s. The positive direction is that of the bullet. Calculate the momentum of the bullet im- mediately after the gun was fired. Answer in units of

Horses, pigs and cows is really easy I can do it in my head but, my teacher told me I would have to start showing my work. I really have no clue how to do that I have always done it in my head. Can you pretty please help me I would gladly appreciate it.

115. ## science

A 27 kg gun is standing on a frictionless sur- face. The gun fires a 56 g bullet with a muzzle velocity of 314 m/s. The positive direction is that of the bullet. Calculate the momentum of the bullet im- mediately after the gun was fired. Answer in units of

116. ## life orientation

5 ways in which the air pollution problem on the community.

117. ## enlish literture

narnia the lion the the witch and the wabode two literary devives were used in chapter 10 repetition and smiple identify the line that features these device

118. ## Social Studies

Which of the following contributed to Britain's Industrial Revolution? A. Large Labor force C. powerful monarchy B. Decreasing food supplies D. decreasing money supply. The textile industry met its last major challenge to full mechanization with the

119. ## Social Studies

The spread of the factory system resulted in: A. an increase in home production B. increased exports of British cotton good C. poorer quality of iron D. improved working conditions in factories. The industrial Revolution created new jobs in all of the

120. ## math

The probability that a positive divisor of 60 is greater than 9 can be written as a/b, where a and b are coprime positive integers. What is the value of a+b?

Hi! This question says Joe is thinking of o function machine.He says that the output is 12 if the input is 4.what are 2 rules that he could be thinking of for his function machine?( I would really be grateful if you helped me) ;)thanks

122. ## maths

5x+6y=37 2x-3y=-4

123. ## math helllllllllllpppp plzzzzzzzzzzzzzzzzzzzzzzz

What are the last 2 digits of 43^2082?

124. ## language arts

1. Which of the following would be the least appropriate topic when drafting a piece of descriptive writing? (1 point) a report on the condition of women workers in Afghanistan an instruction manual for a microwave oven a call to action to clean up a

125. ## Science

what kind of energy is stored before the earthquake

126. ## algebra

The wheel in a bicycle has a diameter of 2 feet. if the wheel is turning at 50 rpm (revolutions per minute), how far, in feet, will the bicycle travel in 20 minutes?

127. ## 7th Grade Math ~Multiplying Polynomials

1. (2x)(10x) (1point) 20x 12x 20x^2** 5x 2. -3m(m+8) (1point) -3m^2+24 -3m^2-24m** -24m^2 -4m+5 3. 4d(2d-9) (1point) 8d-36d -d -28d^2** 8d^2-36d 4. 5b(6b-10) 11b-10 5b^2+6b-10 -b-10 30b^2-50b** Are these right

128. ## Health

Which fats are considered the "essential fatty acids"? A. saturated fats B. unsaturated fats C. trans fatty acids D. All the above I think it is B or D...? I know A and C are bad for you but they are necessary.

129. ## life orientation

6 ways in which earthquake can impact negatively on one's health

130. ## math

If a mail man could deliver to 112 homes on foot and to 168 homes by driving what percent productivity increase by driving

131. ## PHYSICS/CHEMISTRY

Can someone explain to me, what does Metastable isotope means? I already read the definitions on the internet but I really don't understand it. Thanks in advance.

132. ## Math

A bullet reaches a speed of 900 ft per sec in .2 sec. What is its acceleration.

133. ## Reiny or Steve could you check my Math?

1. (9m+6)+(-5m-6) my answer: 4m 2. (3r^2+7r+1)+(4r^2-8r-2) my answer: 7r^2-r-1 3. (6h-1)-(9h+4) my answer: -3h-3 4. (m^2-m-4)+(m-5) my answer: m^2-9 5. (4x+4)+(3x+2+1) my answer: 7x+7 Now this was book work for me so it doesn't have multiple choice, so are

134. ## Governmnet

which of the mass media has the most impact on american politics? A. television B. the internet

135. ## Justice system

1. __________ are the largest and most visible segment of the criminal justice system. A. The police B. Court clerks C. The courts D. Attorneys is it A 2. Structurally, policing in the United States is: A. centralized. B. decentralized. C. interdependent.

136. ## physics

A beam of light strikes the flat, glass side of a water-filled aquarium at an angle of 37.5° to the normal. For glass, n = 1.50. (a) At what angle does the beam enter the glass? (b) At what angle does the beam enter the water?

137. ## government

which provides evidence that political parties are not as strong as they once were? A. the increasing number of independents B. the lack of straight-ticket voting C. increased spending on campaigns D. increased numbers of registered Democrats

138. ## physics

A thick sheet of plastic, n = 1.500, is used as the side of an aquarium tank. Light reflected from a fish in the water has an angle of incidence of 30.0°. At what angle does the light enter the air?

139. ## math

what is the percent in the form specified of 48%ratio,1%decimal,99%fraction

140. ## math

0.2(x+30)-0.03(x-40)=17.4

141. ## physics

A ray of light in a tank of water has an angle of incidence of 23°. What is the angle of refraction in air? (Enter 0 if total internal reflection occurs.)

142. ## Literature (Ms. Sue)(or Writeacher)

The short story is called Chicoria by Rudolfo A. Anaya. How does the server respond when Chicoria says he expects to be fed well at the rancher's table? My answer: The servant becomes rather surprised by Chicoria's statement, causing him to explain to

143. ## physics

The speed of light in a certain piece of clear plastic is 1.95 108 m/s. A ray of light strikes the plastic at an angle of 21°. At what angle is the ray refracted?

144. ## Science (bobpursley)

Why is an extra day added to February every four years?

145. ## physics

A convex lens is needed to produce an image that is 0.70 times the size of the object and located 28 cm from the lens on the other side. What focal length should be specified?

146. ## Math

What is 4963rounded to the nearest hundred?

147. ## physics

Camera lenses are described in terms of their focal length. A 50.0 mm lens has a focal length of 50.0 mm. (a) A camera with a 50 mm lens is focused on an object 4.0 m away. Locate the image. (b) A 980 mm lens is focused on an object 125 m away. Locate the

148. ## physics

In order to clearly read a book 16 cm away, a farsighted girl needs an image to be 36 cm from her eyes. What focal length is needed for the lens in her eyeglasses?

149. ## physics

The convex lens of a copy machine has a focal length of 22.0 cm. A letter to be copied is placed 37.0 cm from the lens. (a) How far from the lens is the copy paper? (b) How much larger will the copy be?

150. ## Justice system

1. Customs and Border Protection (CBP) is part of the: A. Department of Homeland Security. B. Bureau of Dangerous Drugs. C. Drug Enforcement Administration (DEA). D. U.S. Marshals Service. is it A 2. The most visible private police agency during

151. ## math

Judy frog brings 6 socks on a trip. one third of the socks are red. the rest are green. how maby socks are red? how many are green?

152. ## math help

How do I write this in inequality? The value of r is at most 10 and would it be an open or closed on the number line? I think open?

153. ## math

julia studied math for 3 1/3 hours during the 4 days before her last math test what was the average amount of time she studied each day

154. ## Math

on Monday the temperature was -2 degrees. By 4:30 Tuesday the temperature dropped by 4 degrees. By 8am Tuesday, the temperature was 1 degree. therefore find: What was the temperature at 4:30am? What the temperature change between 4:30am and 8am?

155. ## government

to be president of the united states a person must have lived in the united states for at least this number of years

156. ## Science space

I am doing a project on the planet uranus and I need a map showiong the path to my planet that i have chosen. The picture of the path must include asteroids, comets or other interesting objects. I am having so much trouble looking for such a picture. I

157. ## math

1. change to improper fraction: 9 5/9 2. change to improper fraction: 6 23/39 3. change the improper fraction to a whole number:115/20 4. multiply 4 3/4 x 1 1/7 5,subtract- 20 11/15- 17 1/3 6 2(-6)(-8)(0)(-5): multiply 7. solve for the variable: 0.1p=7

158. ## chemistry

4. A 35.0 mL sample of 1.00 M KBr and a 60.0-mL sample of 0.600 M KBr are mixed. The solution is then heated to evaporate water until the total volume is 50.0 mL. What is the molarity of the KBr in the final solution?

159. ## calculus

Farmer Jones has 210 meters of fence. She wishes to construct a rectangular pen using the river as one side and dividing the pen into six separate rectangular pens as shown. What should the dimensions of the combined pens be to maximize the area enclosed?

160. ## Justice system

Detectives more typically focus their energies on homicide and rape cases, as opposed to burglary and robbery cases, because: A. there is political pressure to do so. B. the victims can more often provide clues that make the cases easier to solve. C.

161. ## Programming Logic

How do I write case logic pseudocode and visual logic nested if-then-else flow charts for atm check balances, making remote photo capture check deposits, and perform balance transfers?

162. ## math

Ralph, Mario, Jack, Laura, and Maria are in the math club. The club advisor will assign students to 3-person teams at the next math team competition. How many different 3-person teams can be formed from these five students? would this be 10, because of 5 *

163. ## physics

An organ pipe is 84 cm long and at a temperature of 20 degrees C. What is the fundamental (in Hertz) if the pipe is closed at one end thanks!

164. ## math

A major leaugue baseball player hits a ball 3 feet above the ground with a velocity of 103 feet per secon in the direction of a 10 foot wall that is 300 feet from home plate. If the hit is at an angle of elevation of 34 degrees and there is wind blowing 22

166. ## College Algebra! help!

Consider the following functions f(x)= (7x+8)/(x+3) and g(x)= (3x-8)/(7-x) (a) Find g(g(x)) (b) Find g(f(x)) (c) Determine whether the functions f and g are inverses of each other.

167. ## math

A major leaugue baseball player hits a ball 3 feet above the ground with a velocity of 103 feet per second in the direction of a 10 foot wall that is 300 feet from home plate. If the hit is at an angle of elevation of 34 degrees and there is wind blowing

225. ## english

You could not----- the enemy destroyer in the foy 1,maroon 2,circumvent 3,overlook 4,discern 5,reveal

226. ## Physics

Newtons second law What net force acting on a 14kg wagon produces an acceleration of 1.5 m/s^2?

227. ## helllllp MATHS

x and y are positive values satisfying x+y=1/4 . When ln(y^2+4xy+7) attains its maximum(subject to the given constraint), what is the value of 1/xy ?

228. ## life orientation

in 10-15 lines critically discuss 5 ways in which substance abuse impacts on the community.

229. ## english

the managers decision was not based on a consideration of fairness it was simply a case ----- 1 expediency 2 importance 4 trial 5 judgment 6 corruption

230. ## mathematics

Find the largest number of real numbers x1,...,xn such that for all i≠j, |xi−xj|>1/100(1+xi xj).


text:
## Lagrange multipliers revisited

I spent the last few days trying to decide whether to teach Lagrange multipliers in my Theoretical Mechanics course. Ultimately I decided to go ahead and do it and I wanted to get down my thoughts on why and what we ended up doing in class today.

I made a lot of progress on how to teach Lagrange multipliers the last time I taught this class, and I have to say that it was great being able to read my old thoughts when prepping this class. In that post I break down how to derive the needed result, so I won’t repeat that here (though I did find a slightly better approach for one of the steps that I’ll talk about below).

So why was I waffling? Every time I come around to this topic, I begin to realize that if my students are predominantly going to numerically integrate the ultimate equations of motion (or equation of motions as I usually pluralize the acronym – eoms) they can get the same information that Lagrange multipliers provide (typically the constraint forces) by simply plotting the total acceleration of the relevant particles minus the known forces. In fact that’s one of the beauties of the Lagrangian approach — you can ignore the constraint forces when figuring out the equations of motion. At the end of the day, you have access to the full accelerations which come about due to both the known external forces AND the constraint forces. What Lagrangian multipliers do is help you explicitly calculate those constraint forces, but sometimes I don’t really see the value of that.

So what tipped the scales towards teaching them this year? Well, I realized that actually doing what I suggest above is kind of a hassle. Consider the prototypical problem of when a sled will lose contact with the ground as it goes down a hill. If you have the equation of the hill, you can then reduce the problem to a one-dimensional one, typically the horizontal variable. Then you can do the Lagrangian approach where you force the sled to stay on the curve. When you’re done you can look at the total acceleration and subtract gravity from that vector. Then you have the normal force vector and you can investigate when that switches from a vector pointing out of the ground to a vector pointing into the ground. When that happens the sled will lose contact with the ground. Now, that all sounds great, but actually determining which way the vector points involves dotting it with a vector that’s normal to the curve. To get that you have to do some derivatives on the function that defines the curve in the first place. In other words, there’s a little bit of a hassle.

Contrast that with learning about Lagrange multipliers and applying them. Now, there’s no question that learning about them (the whole separation of variables thing in my other post) is not a cake walk. However, there’s some very cool calculus of variations in there and it helps me reinforce the original derivation of the Euler equation (which I always choose to teach). So teaching it isn’t really the big deal. What is the big deal is whether what you get after implementing it makes your life easier. And guess what? It does! For the sled problem, if you have access to the Lagrange multiplier as a function of time (which you will after implementing the approach) you just have to plot it and look for when it changes sign. That doesn’t require the multivariable calculus that’s necessary to define the normal direction that you need for the other way.

Here’s the screencasts that I made for my students today. We were tackling whether a sled would ever lose contact with the ground on a parabolic hill. I did it once without Lagrange multipliers and once with them. It’s my contention that the second way is a more clear way of finding when the sled leaves the ground (too long, didn’t watch: it never does for a parabolic hill). But I recognize it’s not a slam dunk case.

I was talking with a math professor buddy of mine today and he suggested that it might be a slam dunk case if you have a constraint that is hard to solve for one variable in terms of the other. I get what he was saying, but I don’t immediately have an idea of a sled/hill constraint where that would be the case.

Quick note about a change to the derivation: There’s a point in my other post where I say that the constraint should not be a function of the perturbation variable, $\alpha$. I then take a derivative and find a relationship between the two perturbation functions, $\eta_1$ and $\eta_2$. I realized this time around that there’s a different way to approach that. Basically we know that the constraint, g, has to be obeyed whether you’re on the best paths (x(t) and y(t)) or on nearby paths ($x_\text{best}+\alpha \eta_1$ and $y_\text{best}+\alpha \eta_2$):

$g(x_b, y_b)=g(x_b+\alpha \eta_1, y_b+\alpha \eta_2)$

$latex g(x_b+\alpha \eta_1, y_b+\alpha \eta_2) \approx g(x_b, y_b)+\frac{\partial g}{\partial x}(\alpha \eta_1)+\frac{\partial g}{\partial y}(\alpha \eta_1)$

or

$latex \frac{\partial g}{\partial x}(\alpha \eta_1)+\frac{\partial g}{\partial x}(\alpha \eta_1)=0&s=2$

which leads directly to the same result I have in the last page.

Update on how to do this in Mathematica: In the screencasts above you might notice that I deviate from what I suggest in my other post. The reason is that Mathematica has a new method for NDSolve that saves the day. Now you can simply tell NDSolve about the constraint and go, without having to do the differentiation that was my work around. The key is to use Method->{“IndexReduction”->Automatic} which Mathematica kindly suggests if you try to do NDSolve without it. It’s great, you even only have to give initial conditions for one of the variables. Mathematica will figure out the initial conditions for the other variable(s) by using the constraint. Awesome!

Your thoughts? Here are some starters for you:

• I like this, I’ve been looking for better motivation and I’m going to use this to . . .
• I hate this and all things having to do with Lagrange multipliers. Please stop posting about this.
• I’m in this class and I found today very useful. Here’s why . . .
• I’m in this class and I thought today was a waste of time. Here’s why . . .
• Of course you don’t leave the ground on a parabolic hill, everyone knows that.
• Why didn’t you type up all the stuff you said in the screencasts? I hate watching videos.
• Thanks for the screencasts, do your students find them useful?
• Mathematica had that new method two years ago, I just decided not to tell you about it
• Here’s a suggestion for a sled/hill problem with a constraint that’s hard to solve for one of the variables . . .