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Mixing (mathematics)

From Wikipedia, the free encyclopedia

In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc.

The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity).

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  • ✪ Math Antics - Adding Mixed Numbers
  • ✪ Math Antics - Mixed Numbers
  • ✪ Math Antics - Subtracting Mixed Numbers
  • ✪ Multiplying Mixed numbers
  • ✪ Dividing mixed numbers

Transcription

Hi, I’m Rob. Welcome to Math Antics. In this lesson, we’re gonna learn how to add Mixed Numbers. If you’re not quite sure what Mixed Number are, then you should definitely watch our video called “Mixed Numbers” first. As you remember, a Mixed Number is a combination (or sum) of a Whole Number and a Proper Fraction. And for this lesson, it’s gonna be important to remember that even though the plus symbol isn’t usually shown between those two parts of a Mixed Number, they’re being ADDED together. 3 and 1/4 means 3 PLUS 1/4 2 and 5/8 means 2 PLUS 5/8. Here’s why that’s so important to remember… Let’s say you’re given a problem where you need to add the Whole Number 2 to the Mixed Number 3 and 1/4. If you know that 3 and 1/4 is the same as 3 PLUS 1 /4 then you can see that the problem is really 2 + 3 + 1/4 Well that’s easy… all you have to do is add the 2 and the 3 to get 5 and you’ll have 5 plus 1/4 which is the Mixed Number 5 and 1/4. So if you need to add a Whole Number to a Mixed Number, you can just add the Whole Number parts and you’re done! Okay, but what if you need to add a Mixed Number to a fraction? …like in the problem: 1 and 3/8 pus 1/8 Again, if you remember that 1 and 3/8 means 1 PLUS 3/8 then you can see that this problem is really 1 + 3/8 + 1/8 That looks pretty easy also. 3/8 and 1/8 are what we call ‘like’ fractions… they have the same denominator and can be added easily. 3/8 + 1/8 = 4/8 So our answer is simply the Mixed Number 1 and 4/8 Oh, but you might notice the fraction part can be simplified. 4/8 simplifies to 1/2, so we should write our answer as 1 and 1/2 instead. It’s not mathematically “wrong” if you don’t simplify a fraction, but teachers (and tests) usually require you to simplify whenever you can, so it’s a good habit to get into. Notice that in each of those examples, we just added Whole Numbers to Whole Numbers and fractions to fractions. And it work the same way when adding a Mixed Number to a Mixed Number, like 2 and 1/5 plus 4 and 2/5 Again, let’s show our Mixed Numbers with the plus signs so we can see the real problem: 2 + 1/5 + 4 + 2/5 Because all of these parts are being added, and addition has the commutative property, it really doesn’t matter what order we do the addition in. That means we can rearrange this problem to make it simpler: Now we have 2 + 4 + 1/5 + 2/5 Adding the Whole Numbers is easy: 2 + 4 = 6 and adding these ‘like’ fractions is easy too: 1/5 + 2/5 = 3/5 That leaves us with 6 + 3/5 which is the Mixed Number 6 and 3/5. So when you add Mixed Numbers, you can just add the Whole Number parts to get the Whole Number of the answer, and you add the fraction parts to get the fraction part of the answer. That’s why in a lot of math books, you’ll see addition of Mixed Numbers written in a stacked form like this. This is similar to the way you would stack multi-digit numbers up to add them, and it helps you remember that you can add the fraction parts and the Whole Number parts in two separate columns and write your answer below the answer line just like in multi-digit addition. And do you remember how in multi-digit addition, if a column of digits added up to 10 or more, you had to “carry” or “re-group” to the next column? Well, something similar to that can happen when adding Mixed Numbers. Sometimes adding the fraction parts of two Mixed Numbers actually effects the Whole Number part of the answer. To see what I mean by that, let’s say you hosted a massive pizza party for all your friends… Hey Man… this is a great party! You’ve got some really cool friends. Hey thanks! You should hang out with us more often. I think you’d really fit in. And after the party ended, you had 1 and 3/8 cheese pizzas left over and 1 and 5/8 pepperoni pizzas left over. What’s the total amount of leftover pizza? Well, we just need to add those two mixed numbers together. Let’s stack them like I just showed you and add them column by column. 3/8 + 5/8 is 8/8 and 1 + 1 = 2 So, the answer is 2 and 8/8. Ah, but do you see what happened? The fraction parts of the two Mixed Numbers combined to form what I call a “Whole Fraction” (8/8) And we know that 8/8 simplifies to 1. So having 2 + 8/8 is the same as having 2 + 1 which is 3. We added two Mixed Numbers together and ended up with the Whole Number 3. And our leftover pizza shows us that we got the answer right. Here’s another example that shows how the fraction parts can effect the Whole Number part of the answer when adding Mixed Numbers: 1 and 3/7 plus 2 and 5/7 This time we’ll use the commutative property to rearrange the addition and then we add the whole number parts: 1 + 2 = 3. And then we’ll add the fraction parts: 3/7 + 5/7 = 8/7. So the answer we get is 3 and 8/7 But do you notice something funny about the fraction part of that answer? It’s an Improper Fraction which means its value is greater than 1. And it’s really bad form to leave an Improper Fraction in a Mixed Number like this because, as we saw in the last video, the Improper Fraction ITSELF can be converted into a Mixed Number. 8/7 contains a ”Whole Fraction” that we can simplify out of it… it’s the same as 7/7 + 1/7 And since 7/7 equals 1, that gives us the Mixed Number 1 and 1/7 So just like in the last example, we can add that extra 1 to the Whole Number part of our answer which gives us 4 and 1/7. That’s a much less confusing answer than 3 and 8/7, which almost sounds like it’s less than 4 but it’s actually MORE than 4. Are you getting it so far? Adding Mixed Numbers is pretty easy when you realize that you can just add the whole number parts and fraction parts separately and then just watch for cases where the fraction parts add up to more than 1. But there are cases where adding Mixed Numbers can get a little bit tougher. All of the examples we’ve seen so far had fraction parts that were ‘like’ fractions. But what if you had to add two Mixed Numbers with ‘unlike’ fractions? …like this problem: 1 and 1/2 plus 2 and 1/4 If we re-arrange the problem as usual, we see that the Whole Numbers are still easy to add: 1 + 2 = 3 But the fractions have different denominators. We can’t add them until we change them so that the bottom numbers are the same. We cover how to change fractions so that they have the same bottom number (or a “common denominator”) in other videos that you may want to watch if the steps I’m about to do seem new to you. 4 is a multiple of 2, so 4 is going to be a good choice for a common denominator. To change 1/2 into fourths, we will multiply it by the “Whole Fraction” 2/2. On the top we have 1 times 2 which is 2, and on the bottom we have 2 times 2 which is 4, just like we want. So now we have 2/4 + 1/4 which equals 3/4. That means the answer to this problem is 3 and 3/4. That wasn’t so bad after all. Let’s try one more example where we need to change ‘unlike’ fractions into ‘like’ fractions in order to add the Mixed Numbers: 3 and 2/3 plus 4 and 3/4 After re-arranging the parts, we see that we need to add 3 and 4 which is 7, and we also need to add 2/3 and 3/4. Since these are ‘unlike’ fractions we need to change them. 3 and 4 are not multiples of each other, so it looks like using the “Easiest Common Denominator” will be our best option here. 3 x 4 = 12 so that will be our new denominator. To convert 2/3 we multiply it by 4/4 which gives us the new equivalent fraction 8/12 To convert 3/4 we need to multiply it by 3/3 which gives us the new equivalent fraction 9/12 Now that we have ‘like’ fractions we can add them easily: 8/12 + 9/12 = 17/12 That gives us 7 and 17/12 as our answer. But once again, the fraction part is Improper, so we have to simplify it because its value is greater than 1. 17/12 is the same as 12/12 + 5/12 Which is the mixed Number 1 and 5/12 We need to add that 1 to our Whole Number part. 7 + 1 = 8 Which means our final answer is 8 and 5/12 Alright, that should give you a pretty good idea of how to add Mixed Numbers. You can add the Whole Number parts and the fractions parts separately. But the fraction part of the answer may effect the Whole Number part if its value is 1 or greater. And remember, if the fraction parts have different denominators, you’ll need to change them to have a common denominator before you can add them. With complicated arithmetic like this, it’s important to practice what you’ve learned so it’ll really make sense. So be sure to try some exercise problems on your own. As always, thanks for watching Math Antics and I’ll see ya next time. Learn more at www.mathantics.com

Contents

Mixing in stochastic processes

Let be a stochastic process on a probability space . The sequence space into which the process maps can be endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology.

Define a function , called the strong mixing coefficient, as

for all . The symbol , with denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. the σ-algebra generated by .

The process is said to be strongly mixing if as . That is to say, a strongly mixing process is such that, in a way that is uniform over all times and all events, the events before time and the events after time tend towards being independent as ; more colloquially, the process, in a strong sense, forgets its history.

Types of mixing

Suppose were a stationary Markov process with stationary distribution and let denote the space of Borel-measurable functions that are square-integrable with respect to the measure . Also let

denote the conditional expectation operator on Finally, let

denote the space of square-integrable functions with mean zero.

The ρ-mixing coefficients of the process {xt} are

The process is called ρ-mixing if these coefficients converge to zero as t → ∞, and “ρ-mixing with exponential decay rate” if ρt < eδt for some δ > 0. For a stationary Markov process, the coefficients ρt may either decay at an exponential rate, or be always equal to one.[1]

The α-mixing coefficients of the process {xt} are

The process is called α-mixing if these coefficients converge to zero as t → ∞, it is “α-mixing with exponential decay rate” if αt < γeδt for some δ > 0, and it is α-mixing with a sub-exponential decay rate if αt < ξ(t) for some non-increasing function satisfying

as .[1]

The α-mixing coefficients are always smaller than the ρ-mixing ones: αtρt, therefore if the process is ρ-mixing, it will necessarily be α-mixing too. However, when ρt = 1, the process may still be α-mixing, with sub-exponential decay rate.

The β-mixing coefficients are given by

The process is called β-mixing if these coefficients converge to zero as t → ∞, it is β-mixing with an exponential decay rate if βt < γeδt for some δ > 0, and it is β-mixing with a sub-exponential decay rate if βtξ(t) → 0 as t → ∞ for some non-increasing function satisfying

as .[1]

A strictly stationary Markov process is β-mixing if and only if it is an aperiodic recurrent Harris chain. The β-mixing coefficients are always bigger than the α-mixing ones, so if a process is β-mixing it will also be α-mixing. There is no direct relationship between β-mixing and ρ-mixing: neither of them implies the other.

Mixing in dynamical systems

A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let be a dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any , one has

For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with replaced by with g being the continuous-time parameter.

To understand the above definition physically, consider a shaker full of an incompressible liquid, which consists of 20% wine and 80% water. If is the region originally occupied by the wine, then, for any region within the shaker, the percentage of wine in after repetitions of the act of stirring is

In such a situation, one would expect that after the liquid is sufficiently stirred (), every region of the shaker will contain approximately 20% wine. This leads to

where , because measure-preserving dynamical systems are defined on probability spaces, and hence the final expression implies the above definition of strong mixing.

A dynamical system is said to be weak mixing if one has

In other words, is strong mixing if in the usual sense, weak mixing if

in the Cesàro sense, and ergodic if in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing.

For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one.[citation needed] In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.

formulation

The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system is equivalent to the property that, for any function , the sequence converges strongly and in the sense of Cesàro to , i.e.,

A dynamical system is weakly mixing if, for any functions and

A dynamical system is strongly mixing if, for any function the sequence converges weakly to i.e., for any function

Since the system is assumed to be measure preserving, this last line is equivalent to saying that so that the random variables and become orthogonal as grows. Actually, since this works for any function one can informally see mixing as the property that the random variables and become independent as grows.

Products of dynamical systems

Given two measured dynamical systems and one can construct a dynamical system on the Cartesian product by defining We then have the following characterizations of weak mixing:

Proposition. A dynamical system is weakly mixing if and only if, for any ergodic dynamical system , the system is also ergodic.
Proposition. A dynamical system is weakly mixing if and only if is also ergodic. If this is the case, then is also weakly mixing.

Generalizations

The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which

holds for all measurable sets A, B, C. We can define strong k-mixing similarly. A system which is strong k-mixing for all k = 2,3,4,... is called mixing of all orders.

It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.

Examples

Irrational rotations of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.

Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, and the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature.

Topological mixing

A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map is said to be topologically transitive if, for every pair of non-empty open sets , there exists an integer n such that

where is the nth iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.

Lemma: If X is a complete metric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point , that is, a point x such that its orbit is dense in X.

A system is said to be topologically mixing if, given open sets and , there exists an integer N, such that, for all , one has

For a continuous-time system, is replaced by the flow , with g being the continuous parameter, with the requirement that a non-empty intersection hold for all .

A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.

Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.

References

  • Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence". Journal of Econometrics. 155 (2): 155–169. CiteSeerX 10.1.1.597.8777. doi:10.1016/j.jeconom.2009.10.001.
  • Achim Klenke, Probability Theory, (2006) Springer ISBN 978-1-84800-047-6
  • V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (1968) W. A. Benjamin, Inc.
This page was last edited on 28 August 2019, at 17:37
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