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Wednesday, September 21, 2022

# What Is Variation In Math

## Recognizing Direct Or Indirect Variation

Direct and inverse variation | Rational expressions | Algebra II | Khan Academy

You might be asked to look at functions s to unique \s well discuss later in theAlgebraic Functions section) and determine if they are direct, inverse, or neither:

 Function Direct, Inverse, or Neither Variation \ Neither: Direct Variation line must go through \\). . \ Direct: This is the same as \. \ 8 16

Inverse8InverseNeither

 \ 6 8

Neither

## What Is Made Possible To Learn When Using The Variation Theory Of Learning In Teaching Mathematics

The variation theory of learning emphasizes variation as a necessary condition for learners to be able to discern new aspects of an object of learning. In a substantial number of studies, the theory has been used to analyze teaching and students learning in classrooms. In mathematics education, variation theory has also been used to explore variation in sets of instructional examples. For example, it has been reported how teachers, by using variation and invariance within and between examples, can help learners to engage with mathematical structure.

• Math examples and question always vary – we assume that practice with varied examples bring fluency.
• It may bring fluency, but in a machine like way.
• Controlling variation in examples and question can bring understanding.
• We notice what varies when other things are held stable.
• We notice what is constant when everything else varies.
• If everything varies we don’t know what to focus on.

” Mathematics is about relationships when see the effects of one thing varying we can generalize about a relationship when we generalize about a relationship we are learning a concept. ”

Date:13th of April’ 2021

Director of Matho Logics

## Modern Definition And Relation To Measure Theory

Classically right-continuous functions of bounded variations can be mapped one-to-one to signed measures. More precisely, consider a signed measure $\mu$ on $\mathbb R$ with finite total variation . We then define the function\begin\labelF_\mu := \mu \, .\end

Theorem 2

• For every signed measure $\mu$ with finite total variation, $F_\mu$ is a right-continuous function of bounded variation such that $\lim_ F_\mu = 0$ and $TV$ equals the total variation of $\mu$ )$. • For every right-continuous function$f:\mathbb R\to \mathbb R$of bounded variation with$\lim_ f = 0$there is a unique signed measure$\mu$such that$f=F_\mu$. Moreover, the total variation of$f$equals the total variation of the measure$\mu$. For a proof see Section 4 of Chapter 4 in . Obvious generalizations hold in the case of different domains of definition. Recommended Reading: What Is Bachelor Of Psychology ## Help Others Learning Math Just Like You âââââ We hope you found this Math math tutorial “Variation. Types of Variation” useful. If you did it would be great if you could spare the time to rate this math tutorial and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines. ## Variation Types Of Variation Ratio and Proportion Learning Material Tutorial ID In this Math tutorial, you will learn: • What is variation? • What are the three types of variation? • Where does variation differ from proportion? • Where to use proportional and variation approach in graph problems? • What is the constant of proportionality used in variation? • What is the shape of the graph that illustrates direct/inverse variation? • Which are the properties of variation? You May Like: What Is Earth Movement In Geography ## Module : Rational And Radical Functions • Solve inverse variation problems. • Solve problems involving joint variation. A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for$4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate. ## How To Solve Direct Variation To solve direct variation, plug in matching values of variables to find k. Then, if necessary, use the value of k with a value of x or y to find the other variables value. #### Example 1: Solving A Direct Variation For k Lets say that there is a direct variation between x and y. That means: • y = kx for some constant k. Lets assume we know that y = 15 when x = 3. Plugging in these values of x and y, we get: • y = kx So, y = 42 when x = 7. Also Check: What Is Gradualism In Biology ## A General Note: Direct Variation If x and y are related by an equation of the form y=k^ then we say that the relationship is direct variation and y varies directly with the nth power of x. In direct variation relationships, there is a nonzero constant ratio k=\dfrac^}, where k is called the constant of variation, which help defines the relationship between the variables. ## How To Tell If A Graph Is A Direct Variation SAT Math Part 16 – Direct Variation Problems To tell if a graph is a direct variation, look for a line that passes through the origin. In particular, this means: • The graph is a line . • The slope is constant . • The line goes through the origin ). #### Example 1: A Graph That Is A Direct Variation Consider the graph: This graph is a direct variation, since: • The graph is a line . • The slope is constant . • The line goes through the origin ). #### Example 2: A Graph That Is Not A Direct Variation Consider the graph: This graph is not a direct variation, since: • The graph is not a line . #### Example 3: A Graph That Is Not A Direct Variation Consider the graph: This graph is not a direct variation, since: • The line does not pass through the origin . Read Also: Select Operation In Relational Algebra In Dbms ## Example: Solving A Direct Variation Problem The quantity y varies directly with the cube of x. If y=25 when x=2, find y when x is 6. The general formula for direct variation with a cube is y=k^. The constant can be found by dividing y by the cube of x. \begin k& =\dfrac^} \\ & =\dfrac^}\\ & =\dfrac\end Now use the constant to write an equation that represents this relationship. y=\dfrac^ Substitute x=6 and solve for y. \beginy& =\dfrac^ \\ & =675\hfill \end #### Analysis of the Solution ## What Is Direct Variation Formula Direct variation formula helps relate two quantities, having a mathematical relationship such that one of the variables is a constant multiple of the other. When two quantities are directly proportional to each other or are in direct variation they are represented using the symbol “\”. Suppose there are two quantities x and y that are in direct variation then they are expressed as follows: y \ x When the proportionality sign is removed then the direct variation formula is given as follows. Read Also: What Is Displacement In Chemistry ## How To: Given A Description Of A Direct Variation Problem Solve For An Unknown • Identify the input, x, and the output, y. • Determine the constant of variation. You may need to divide y by the specified power of x to determine the constant of variation. • Use the constant of variation to write an equation for the relationship. • Substitute known values into the equation to find the unknown. • ## Math Review Of Direct Inverse Joint And Combined Variation Overview: Equations that describe the relationship between two variables in a sentence express the variation between those variables. Examples of types of variation include direct, inverse, joint, and combined variation. What Is Direct Variation? In direct variation, as one variable is multiplied by a constant and increases, another variable also increases. Suppose Jane works at McDonalds and is paid$9.35 per hour. Her total wages vary directly with the amount of hours she works. If she works 1 hour, her gross wages are $9.35, and if she works 2 hours her gross wages are$18.70. If she works 8 hours, her wages are $74.80. If the hours she works are plotted on the x axis against her wages on the y axis, the resulting line is a straight line showing direct variation. What Is Inverse Variation? Inverse variation is a relationship between variables so that as one variable decreases the other variable increases. The equations expressing inverse variation take the form xy = k, where k is a constant, as well as y = k/x.. For example, the current c varies inversely with the resistance in ohms r. When the current is 40 amps, the resistance is 12 ohms. When the current is 24 amps, the resistance is 20 ohms. If the current is plotted on the x axis, and the resistance is plotted on the y axis, the graph is in 2 separate curves called a hyperbola. What Is Joint Variation? What Is Combined Variation? Don’t Miss: What Is Association In Chemistry ## What Do These Variation The keywords’ meanings are: • directly: things are working together you are multiplying everything. • inversely: things working against each other you are dividing something somewhere. • jointly: many things are working together you are multiplying lots of stuff. An example of a variation equation would be the formula for the area of the circle: A = r2 In the language of variation, this equation means: the area A varies directly with the square of the radius r …and the constant of variation is k = . This formula is an example of “direct” variation.”Direct variation” means that, in the one term of the formula, the variable is “on top”. Affiliate On the other hand, “inverse variation” means that the variable is underneath, in the bottom of a fraction. Suppose, for instance, that you inherit a money market account containing$100,000, and you wonder how much money your rich uncle initially invested eight years ago. Depending on the average interest rate “r“, the formula you would use would be:

…where P is the principal that your uncle invested. In the language of variation, this formula reads as:

the principal P varies inversely with

…with the constant of variation being k = 100,000.

The other case of variation is “jointly”. “Joint variation” means “directly, but with two or more variables”. An example would be the formula for the area of a triangle with base “b” and height “h“:

In words:

the area A varies jointly with b and h

• “F varies as x” means F = kx

## Origins Of Variation Theory

The study of Variation Theory arises mainly from work across two groups of researchers who arrived at similar, but not exactly the same conclusions.

One was Swedish researcher Ference Marton and colleagues who worked on the concept of Variation Theory the other was Gu Ling Yuan and his team in Shanghai who called it Bianchi . As one would expect, the emergence of two similar theories has caused confusion in some areas of the sector. In the context of the United Kingdom, the distinction between them is rarely made and so aspects of Bianchi are used under Variation Theory.

They both agree that the central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features. . This blog will mainly draw on the learning study of Gu and his team.

The application of Variation Theory is much wider than just the domain of mathematics teaching, but rather as a theory that underpins how students learn anything. The principle that underpins the theory is that new learning occurs when they understand how a critical aspect of a new concept changes against something that does not change .

It is unlikely that a worksheet on its own will bring about the required cognitive process in students needed to benefit from the advantages that Variation Theory offers students in their conceptualisation of mathematical structures.

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## What Is Variation Theory A Guide For Primary Maths Teachers

##### Neil Almond

This blog will look at the concept of Variation Theory and attempt to demystify the confusing language surrounding it. It is hoped that this will make Variation Theory an accessible and exciting tool that you can begin to use in your maths lessons, or give you an understanding of what is happening if you recognise the principles, despite never having heard of Variation Theory before!

If you are a premium member of Third Space Learning, you can access a whole-school CPD presentation that covers these points in our Maths Hub.

## What Is Variation Theory In Primary Maths

CSEC Maths – Direct and Indirect Variation

Variation Theory is a general theory of learning that is used predominantly in mathematics teaching and pedagogy. It emphasises a concepts essential features by focusing on what is kept the same and what changes, which offers the opportunity to make meaningful connections.

FREE Crib Sheet for Primary School Staff Meetings

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## Example: Solving An Inverse Variation Problem

A quantity y varies inversely with the cube of x. If y=25 when x=2, find y when x is 6.

The general formula for inverse variation with a cube is y=\dfrac^}. The constant can be found by multiplying y by the cube of x.

\begink& =^y \\ & =^\cdot 25 \\ & =200 \end

Now we use the constant to write an equation that represents this relationship.

\beginy& =\dfrac^},\hspacek=200 \\ y& =\dfrac^} \end

Substitute x=6 and solve for y.

\beginy& =\dfrac^} \\ & =\dfrac \end

#### Analysis of the Solution

\dfrac

The following video presents a short lesson on inverse variation and includes more worked examples.

## *footnote: Why Square The Differences

If we just add up the differences from the mean … the negatives cancel the positives:

 4 + 4 4 44 = 0

So that won’t work. How about we use absolute values?

 |4| + |4| + |4| + |4|4 = 4 + 4 + 4 + 44 = 4

 |7| + |1| + |6| + |2|4 = 7 + 1 + 6 + 24 = 4

Oh No! It also gives a value of 4,Even though the differences are more spread out.

So let us try squaring each difference :

 2 + 24) = = 4 2 + 24) = = 4.74…

That is nice! The Standard Deviation is bigger when the differences are more spread out … just what we want.

In fact this method is a similar idea to distance between points, just applied in a different way.

And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.

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## Differences In What Was Made Possible To Discern

There were general similarities between Lesson 1 and Lesson 2 . For instance, the lessons were about the same length and the same method for solving equations with one unknown was taught. In both lessons, the students worked with creating equations with one unknown for a fellow student. We focused on similarities and differences in enacted example spaces in order to explore, on a micro-level, effects of the intervention on teaching practice. We analyzed the examples introduced by the teacher during whole-class discussion using a variation-theory framework . Our analysis shows that the enacted object of learning, and thereby what was made possible to learn in the lesson, had changed. In L1, the object of learning entailed primarily the method and procedure for solving, by finding answers to equations with one unknown. When the same teacher taught L2 on the same topic after the intervention, the answers to the equations used in the lesson were already known, and the object of learning was instead about understanding the structure of an equation: how an equation can be created and the operations used in solving it.

## What Is Direct Variation

Direct variation exists between any two variables when one quantity is directly dependent on the other i.e. if one quantity increases with respect to the other quantity and vice versa. It is the relationship between two variables where one of the variables is a constant multiple of the other. Since the two variables are directly related to each other it is also termed as directly proportional.

Direct variation and inverse variation are two types of proportionalities. Proportionality refers to a relationship where two quantities are multiplicatively connected by a constant. In a direct variation, the ratio of the two quantities remains the same whereas in an inverse variation the product of the two quantities remains constant. Here we shall check in detail the definition and examples of direct variation.

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## Can Direct Variation Be Negative

Direct variation can be negative. To be more precise, the constant of variation k can be less than zero .

In practical terms, k < 0 means that y decreases as x increases. Similarly, y increases as x decreases.

For example, if k = -5, then we have the direct variation y = -5x.

When x increases by 1, y will decrease by 5.

When x decreases by 1, y will increase by 5.

The graph will be a line that goes through the origin and moves down as we move from left to right. The slope is -5 = -5 / 1 = rise / run, so we move down 5 units for every unit we move right.

## Wiener’s And Young’s Generalizations

One sometimes also considers classes $BV_\Phi$ defined as follows. Let $\Phi: \to \mathbb R$, we let $TV_$ be the least upper bound of sums of the type\where $a\leq x_1 < \ldots < x_< b$ is an arbitrary family of points. The quantity $TV_\Phi$ is called the $\Phi$-variation of $f$ on $$. If TV_\Phi < \infty one says that f has bounded \Phi-variation on$$, while the class of such functions is denoted by $BV_\Phi$ . If $\Phi =u$, one obtains Jordan’s class $BV$, while if $\Phi =u^p$, one obtains Wiener’s classes $BV_p$ . The definition of the class $BV_\Phi$ was proposed by L.C. Young in .

If\then\)\subset BV_ \subset BV_q \subset BV_)} \subset BV_)} \, .\]for $1\leq p < q$ and $0< \alpha< \beta< \infty$, these being proper inclusions.

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In this installment, I will introduce the concept of Boolean algebra, one of the main stars of this series, and relate it to concepts introduced in previous lectures (distributive lattice, Heyting algebra, and so on). Boolean algebra is the algebra of classical propositional calculus, and so has an abstract logical provenance; but one of our eventual goals is to show how any Boolean algebra can also be represented in concrete set-theoretic (or topological) terms, as part of a powerful categorical duality due to Stone.

There are lots of ways to define Boolean algebras. Some definitions were for a long time difficult conjectures (like the Robbins conjecture, established only in the last ten years or so with the help of computers) — testament to the richness of the concept. Here we’ll discuss just a few definitions. The first is a traditional one, and one which is pretty snappy:

A Boolean algebra is a distributive lattice in which every element has a complement.

(If $X$ is a lattice and $x \in X$, a complement of $x$ is an element $y$ such that $x \wedge y = 0$ and $x \vee y = 1$. A lattice is said to be complemented if every element has a complement. Observe that the notions of complement and complemented lattice are manifestly self-dual. Since the notion of distributive lattice is self-dual, so therefore is the notion of Boolean algebra.)

• Example: Probably almost everyone reading this knows the archetypal example of a Boolean algebra: a power set $PX$, ordered by subset inclusion. As we know, this is a distributive lattice, and the complement $S^c$ of a subset $S \subseteq X$ satisfies $S \cap S^c = \emptyset$ and $S \cup S^c = X$.
• Example: Also well known is that the Boolean algebra axioms mirror the usual interactions between conjunction $\wedge$, disjunction $\vee$, and negation $\neg$ in ordinary classical logic. In particular, given a theory $\mathbf{T}$, there is a preorder whose elements are sentences (closed formulas) $p$ of $\mathbf{T}$, ordered by $p \leq q$ if the entailment $p \to q$ is provable in $\mathbf{T}$ using classical logic. By passing to logical equivalence classes ($p \equiv q$ iff $p \leftrightarrow q$ in $\mathbf{T}$), we get a poset with meets, joins, and complements satisfying the Boolean algebra axioms. This is called the Lindenbaum algebra of the theory $\mathbf{T}$.

Exercise: Give an example of a complemented lattice which is not distributive.

As a possible leading hint for the previous exercise, here is a first order of business:

Proposition: In a distributive lattice, complements of elements are unique when they exist.

Proof: If both $b$ and $c$ are complementary to $a$, then $b = b \wedge 1 = b \wedge (a \vee c) = (b \wedge a) \vee (b \wedge c) = 0 \vee (b \wedge c) = b \wedge c$. Since $b = b \wedge c$, we have $b \leq c$. Similarly $c = b \wedge c$, so $b = c. \Box$

The definition of Boolean algebra we have just given underscores its self-dual nature, but we gain more insight by packaging it in a way which stresses adjoint relationships — Boolean algebras are the same things as special types of Heyting algebras (recall that a Heyting algebra is a lattice which admits an implication operator satisfying an adjoint relationship with the meet operator).

Theorem: A lattice is a Boolean algebra if and only if it is a Heyting algebra in which either of the following properties holds:

1. $(a \wedge x \leq y)$ if and only if $(a \leq \neg x \vee y)$
2. $\neg \neg x = x$ for all elements $x$

Proof: First let $X$ be a Boolean algebra, and let $x^c$ denote the complement of an element $x \in X$. Then I claim that $a \wedge x \leq y$ if and only if $a \leq x^c \vee y$, proving that $X$ admits an implication $x \Rightarrow y = x^c \vee y$. Then, taking $y = 0$, it follows that $\neg x := (x \Rightarrow 0) = x^c \vee 0 = x^c$, whence 1. follows. Also, since (by definition of complement) $x$ is the complement of $y$ if and only if $y$ is the complement of $x$, we have $x^{c c} = x$, whence 2. follows.

[Proof of claim: if $a \leq x^c \vee y$, then $x \wedge a \leq x \wedge (x^c \vee y) = (x \wedge x^c) \vee (x \wedge y) \leq 0 \vee y = y$. On the other hand, if $x \wedge a \leq y$, then $a = 1 \wedge a \leq (x^c \vee x) \wedge (x^c \vee a) = x^c \vee (x \wedge a) \leq x^c \vee y$. This completes the proof of the claim and of the forward implication.]

In the other direction, given a lattice which satisfies 1., it is automatically a Heyting algebra (with implication $\neg x \vee y$). In particular, it is distributive. From $\neg x \leq \neg x \vee 0$, we have (from 1.) $x \wedge \neg x \leq 0$; since $0 \leq x \wedge \neg x$ is automatic by definition of $0 = \bot$, we get $0 = x \wedge \neg x$. From $1 \wedge x \leq x$, we have also (from 1.) that $1 \leq \neg x \vee x$; since $\neg x \vee x \leq 1$ is automatic by definition of $1$, we have $\neg x \vee x = 1$. Thus under 1., every element $x$ has a complement $\neg x$.

On the other hand, suppose $X$ is a Heyting algebra satisfying 2.: $\neg \neg x = x$. As above, we know $x \wedge \neg x = 0$. By the corollary below, we also know the function $\neg: X \to X$ takes 0 to 1 and joins to meets (De Morgan law); since condition 2. is that $\neg$ is its own inverse, it follows that $\neg$ also takes meets to joins. Hence $\neg x \vee x = \neg x \vee \neg \neg x = \neg(x \wedge \neg x) = \neg 0 = 1$. Thus for a Heyting algebra which satisfies 2., every element $x$ has a complement $\neg x$. This completes the proof. $\Box$

• Exercise: Show that Boolean algebras can also be characterized as meet-semilattices $X$ equipped with an operation $\neg: X \to X$ for which $a \wedge x \leq y$ if and only if $a \leq \neg(x \wedge \neg y)$.

The proof above invoked the De Morgan law $\neg(x \vee y) = \neg x \wedge \neg y$. The claim is that this De Morgan law (not the other $\neg(x \wedge y) = \neg x \vee \neg y$!) holds in a general Heyting algebra — the relevant result was actually posed as an exercise from the previous lecture:

Lemma: For any element $c$ of a Heyting algebra $X$, the function $- \Rightarrow c: X \to X$ is an order-reversing map (equivalently, an order-preserving map $X^{op} \to X$, or an order-preserving map $X \to X^{op}$). It is adjoint to itself, in the sense that $- \Rightarrow c: X^{op} \to X$ is right adjoint to $- \Rightarrow c: X \to X^{op}$.

Proof: First, we show that $a \leq b$ in $X$ (equivalently, $b \leq a$ in $X^{op}$) implies $(b \Rightarrow c) \leq (a \Rightarrow c)$. But this conclusion holds iff $(b \Rightarrow c) \wedge a \leq c$, which is clear from $(b \Rightarrow c) \wedge a \leq (b \Rightarrow c) \wedge b \leq c$. Second, the adjunction holds because

$(b \Rightarrow c) \leq a$ in $X^{op}$ if and only if

$a \leq (b \Rightarrow c)$ in $X$ if and only if

$a \wedge b \leq c$ in $X$ if and only if

$b \wedge a \leq c$ in $X$ if and only if

$b \leq (a \Rightarrow c)$ in $X. \Box$

Corollary: $- \Rightarrow c: X^{op} \to X$ takes any inf which exists in $X^{op}$ to the corresponding inf in $X$. Equivalently, it takes any sup in $X$ to the corresponding inf in $X$, i.e., $(\bigvee_{s \in S} s) \Rightarrow c = \bigwedge_{s \in S} (s \Rightarrow c)$. (In particular, this applies to finite joins in $X$, and in particular, it applies to the case $c = 0$, where we conclude, e.g., the De Morgan law $\neg(x \vee y) = \neg x \wedge \neg y$.)

• Remark: If we think of sups as sums and infs as products, then we can think of implications $x \Rightarrow y$ as behaving like exponentials $y^x$. Indeed, our earlier result that $x \Rightarrow (-)$ preserves infs $\bigwedge_{s \in S} y_s$ can then be recast in exponential notation as saying $(\prod_{s \in S} y_s)^x = \prod_{s \in S} (y_s)^x$, and our present corollary that $(- \Rightarrow y)$ takes sups to infs can then be recast as saying $y^{\sum_{s \in S} x_s} = \prod_{s \in S} y^{x_s}$. Later we will state another exponential law for implication. It is correct to assume that this is no notational accident!

Let me reprise part of the lemma (in the case $c = 0$), because it illustrates a situation which comes up over and over again in mathematics. In part it asserts that $\neg = (-)\Rightarrow 0: X \to X$ is order-reversing, and that there is a three-way equivalence:

$a \leq \neg b$ if and only if $a \wedge b = 0$ if and only if $b \leq \neg a$.

This situation is an instance of what is called a “Galois connection” in mathematics. If $X$ and $Y$ are posets (or even preorders), a Galois connection between them consists of two order-reversing functions $f: X \to Y$, $g: Y \to X$ such that for all $x \in X, y \in Y$, we have $y \leq f(x)$ if and only if $x \leq g(y)$. (It’s actually an instance of an adjoint pair: if we consider $f$ as an order-preserving map $X \to Y^{op}$ and $g$ an order-preserving map $Y^{op} \to X$, then $f(x) \leq y$ in $Y^{op}$ if and only if $x \leq g(y)$ in $X$.)

Here are some examples:

1. The original example arises of course in Galois theory. If $k$ is a field and $k \subseteq E$ is a finite Galois extension with Galois group $G = Gal(E/k)$ (of field automorphisms $g: E \to E$ which fix the elements belonging to $k$), then there is a Galois connection consisting of maps $Aut_{(-)}(E): PE \to PG$ and $Fix: PG \to PE$. This works as follows: to each subset $S \subseteq E$, define $Aut_S(E)$ to be $\{g \in G: g(s) = s \mbox{ for all } s \in S \}$. In the other direction, to each subset $T \subseteq G$, define $Fix(T)$ to be $\{x \in E: g(x) = x \mbox{ for all } g \in T\}$. Both $Aut_{(-)}(E)$ and $Fix(-)$ are order-reversing (for example, the larger the subset $T \subseteq G$, the more stringent the conditions for an element $x \in E$ to belong to $Fix(T)$). Moreover, we have

$S \subseteq Fix(T)$ iff ($g(x) = x$ for all $x \in S, g \in T$) iff $T \subseteq Aut_S(E)$

so we do get a Galois connection. It is moreover clear that for any $T \subseteq G$, $Fix(T)$ is an intermediate subfield between $k$ and $E$, and for any $S \subseteq E$, $Aut_S(E)$ is a subgroup of $G$. A principal result of Galois theory is that $Fix(-)$ and $Aut_{(-)}(E)$ are inverse to one another when restricted to the lattice of subgroups of $G$ and the lattice of fields intermediate between $k$ and $E$. Such a bijective correspondence induced by a Galois connection is called a Galois correspondence.

2. Another basic Galois connection arises in algebraic geometry, between subsets $J \subseteq k[x_1, \ldots, x_n]$ (of a polynomial algebra over a field $k$) and subsets $V \subseteq k^n$. Given $J$, define $Z(J)$ (the zero locus of $J$) to be $\{(a_1, \ldots, a_n): f(a_1, \ldots, a_n) = 0 \mbox{ for each polynomial } f \in J\}$. On the other hand, define $I(V)$ (the ideal of $V$) to be $\{f \in k[x_1, \ldots, x_n]: f(a) = 0 \mbox{ for all } a = (a_1, \ldots, a_n) \in V\}$. As in the case of Galois theory above, we clearly have a three-way equivalence

$V \subseteq Z(J)$ iff ($f(a) = 0$ for all $a \in V, f \in J$) iff $J \subseteq I(V)$

so that $Z(-)$, $I(-)$ define a Galois connection between power sets (of the $n$-variable polynomial algebra and of $n$-dimensional affine space $k^n$). One defines an (affine algebraic) variety $V \subseteq k^n$ to be a zero locus of some set. Then, on very general grounds (see below), any variety is the zero locus of its ideal. On the other hand, notice that $I(V)$ is an ideal of the polynomial algebra. Not every ideal $I$ of the polynomial algebra is the ideal of its zero locus, but according to the famous Hilbert Nullstellensatz, those ideals $I$ equal to their radical $rad(I) = \{f \in k[x_1, \ldots, x_n]: f^n \in I \mbox{ for some } n \geq 1\}$ are. Thus, $Z(-)$ and $I(-)$ become inverse to one another when restricted to the lattice of varieties and the lattice of radical ideals, by the Nullstellensatz: there is a Galois correspondence between these objects.

3. Both of the examples above are particular cases of a very general construction. Let $X, Y$ be sets and let $R \subseteq X \times Y$ be any relation between them. Then set up a Galois connection which in one direction takes a subset $S \subseteq X$ to $S \backslash R := \{y \in Y: (x, y) \in R \mbox{ for all } x \in S\}$, and in the other takes $T \subseteq Y$ to $R/T := \{x \in X: (x, y) \in R \mbox{ for all } y \in T\}$. Once again we have a three-way equivalence

$S \subseteq R/T$ iff $S \times T \subseteq R$ iff $T \subseteq S \backslash R$.

There are tons of examples of this flavor.

As indicated above, a Galois connection between posets $X, Y$ is essentially the same thing as an adjoint pair between the posets $X, Y^{op}$ (or between $X^{op}, Y$ if you prefer; Galois connections are after all symmetric in $X, Y$). I would like to record a few basic results about Galois connections/adjoint pairs.

Proposition:

1. Given order-reversing maps $f: X \to Y$, $g: Y \to X$ which form a Galois connection, we have $x \leq g f(x)$ for all $x \in X$ and $y \leq f g(y)$ for all $y \in Y$. (Given poset maps $f, g$ which form an adjoint pair $f \dashv g$, we have $x \leq g f(x)$ for all $x \in X$ and $f g(y) \leq y$ for all $y \in Y$.)
2. Given a Galois connection as above, $f(x) = f g f(x)$ for all $x \in X$ and $g(y) = g f g(y)$ for all $y \in Y$. (Given an adjoint pair $f \dashv g$ as above, the same equations hold.) Therefore a Galois connection $(f, g)$ induces a Galois correspondence between the elements of the form $f(x)$ and the elements of the form $g(y)$.

Proof: (1.) It suffices to prove the statements for adjoint pairs. But under the assumption $f \dashv g$, $x \leq g f(x)$ if and only if $f(x) \leq f(x)$, which is certainly true. The other statement is dual.

(2.) Again it suffices to prove the equations for the adjoint pair. Applying the order-preserving map $f$
to $x \leq g f(x)$ from 1. gives $f(x) \leq f g f(x)$. Applying $f g(y) \leq y$ from 1. to $y = f(x)$ gives $f g f(x) \leq f(x)$. Hence $f(x) = f g f(x)$. The other equation is dual. $\Box$

Incidentally, the equations of 2. show why an algebraic variety $V$ is the zero locus of its ideal (see example 2. above): if $V = Z(S)$ for some set of polynomials $S$, then $V = Z(S) = Z I Z(S) = Z I(V)$. They also show that for any element $x$ in a Heyting algebra, we have $\neg \neg \neg x = \neg x$, even though $\neg \neg y = y$ is in general false.

Let $(f, g)$ be a Galois connection (or $f \dashv g$ an adjoint pair). By the proposition, $c = gf: X \to X$ is an order-preserving map with the following properties:

$x \leq c(x)$ for all $x \in X$

$c c(x) = c(x)$ for all $x \in X$.

Poset maps $c: X \to X$ with these properties are called closure operators. We have earlier discussed examples of closure operators: if for instance $G$ is a group, then the operator $c: PG \to PG$ which takes a subset $S \subseteq G$ to the subgroup generated by $S$ is a closure operator. Or, if $X$ is a topological space, then the operator $c: PX \to PX$ which takes a subset $S \subset X$ to its topological closure $\bar{S}$ is a closure operator. Or, if $X$ is a poset, then the operator $c: PX \to PX$ which takes $S \subseteq X$ to $\{a \in X: a \leq s \mbox{ for some } s \in S\}$ is a closure operator. Examples like these can be multiplied at will.

One virtue of closure operators is that they give a useful means of constructing new posets from old. Specifically, if $c: X \to X$ is a closure operator, then a fixed point of $c$ (or a $c$-closed element of $X$) is an element $x$ such that $c(x) = x$. The collection $Fix(c)$ of fixed points is partially ordered by the order in $X$. For example, the lattice of fixed points of the operator $c: PG \to PG$ above is the lattice of subgroups of $G$. For any closure operator $c$, notice that $Fix(c)$ is the same as the image $c(X)$ of $c$.

One particular use is that the fixed points of the double negation closure $\neg \neg: X \to X$ on a Heyting algebra $X$ form a Boolean algebra $Fix(\neg\neg)$, and the map $\neg \neg: X \to Fix(\neg \neg)$ is a Heyting algebra map. This is not trivial! And it gives a means of constructing some rather exotic Boolean algebras (“atomless Boolean algebras”) which may not be so familiar to many readers.

The following exercises are in view of proving these results. If no one else does, I will probably give solutions next time or sometime soon.

Exercise: If $X$ is a Heyting algebra and $x, y, z \in X$, prove the “exponential law” $((x \wedge y) \Rightarrow z) = (x \Rightarrow (y \Rightarrow z))$. Conclude that $\neg(x \wedge y) = (y \Rightarrow \neg x) = (x \Rightarrow \neg y)$.

Exercise: We have seen that $(x \Rightarrow y) \wedge x \leq y$ in a Heyting algebra. Use this to prove $(x \Rightarrow y) \wedge (y \Rightarrow z) \leq (x \Rightarrow z)$.

Exercise: Show that double negation $\neg \neg: X \to X$ on a Heyting algebra preserves finite meets. (The inequality $\neg \neg(x \wedge y) \leq \neg \neg x \wedge \neg \neg y$ is easy. The reverse inequality takes more work; try using the previous two exercises.)

Exercise: If $c: X \to X$ is a closure operator, show that the inclusion map $i: Fix(c) \hookrightarrow X$ is right adjoint to the projection $c: X \to Fix(c)$ to the image of $c$. Conclude that meets of elements in $Fix(\neg \neg)$ are calculated as they would be as elements in $X$, and also that $\neg \neg: X \to Fix(\neg \neg)$ preserves joins.

Exercise: Show that the fixed points of the double negation operator on a topology (as Heyting algebra) are the regular open sets, i.e., those open sets equal to the interior of their closure. Give some examples of non-regular open sets. Incidentally, is the lattice you get by taking the opposite of a topology also a Heyting algebra?

format: plain


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Categories

# Finding smallest positive Integer | AIME I, 1996 Problem 10

Try this beautiful problem from the American Invitational Mathematics Examination, AIME I, 1996 based on Finding the smallest positive Integer.

Try this beautiful problem from the American Invitational Mathematics Examination, AIME I, 1996 based on Finding the smallest positive Integer.

## Finding smallest positive Integer – AIME I, 1996

Find the smallest positive integer solution to $tan19x=\frac{cos96+sin96}{cos96-sin96}$.

• is 107
• is 159
• is 840
• cannot be determined from the given information

Functions

Trigonometry

Integers

## Check the Answer

AIME I, 1996, Question 10

Plane Trigonometry by Loney

## Try with Hints

$\frac{cos96+sin96}{cos96-sin96}$

=$\frac{sin(90+96)+sin96}{sin(90+96)-sin96}$

=$\frac{sin186+sin96}{sin186-sin96}$

=$\frac{sin(141+45)+sin(141-45)}{sin(141+45)-sin(141-45)}$

=$\frac{2sin141cos45}{2cos141sin45}$

=tan141

here $tan(180+\theta)$=$tan\theta$

$\Rightarrow 19x=141+180n$ for some integer n is first equation

multiplying equation with 19 gives

$x \equiv 141\times 19 \equiv 2679 \equiv 159(mod180)$ [since 2679 divided by 180 gives remainder 159]

$\Rightarrow x=159$.


text:
# How do we know photons have spin 1?

Electrons have spin 1/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in spectroscopic measurements.

On the other hand, a photon is neutral - how can one measure its spin if there's no magnetic moment? How do we know it has spin 1?

• Possible duplicates: physics.stackexchange.com/q/11197/2451 and links therein. Aug 12, 2013 at 8:39
• Not really a duplicate, I think. The other answer only mentions neutrinos and charged particles, so the photon question is still open. Aug 12, 2013 at 11:29
• Have you considered conservation of spin in interactions?
– Will
Aug 12, 2013 at 13:20
• Of course, there is also the fact that the Maxwell Lagrangian comes out identical to the Lagrangian for a massless spin-1 particle with U(1) gauge symmetry. Jan 11, 2014 at 18:23
• It might also be worth understanding the original 1931 "Experimental proof of the spin of the photon" by Raman and Bhagavantam: dspace.rri.res.in/bitstream/2289/2123/1/… Jan 1, 2020 at 12:15

Oddly, polarizing sunglasses provide a quite solid proof that photons are spin 1.

That's because if you rotate polarizers by only 90$^\circ$, you will find that you can break photons down into two mutually exclusive populations of photons. That is geometrically possible only if the particle in question is a vector boson, that is, a spin 1 particle.

In contrast, if you were worried instead about "electron glare" from oncoming spin 1/2 electron radiation (please don't try this at home, yes?), you would instead have to rotate your "electron polarizers" by 180$^\circ$ to observe and fully isolate the two distinct electron populations. Both of these polarization detector angles -- 180$^\circ$ for spin 1/2 particles, and 90$^\circ$ for spin 1 particles -- are deeply linked to the underlying symmetries of the particle spins, and so uniquely identify the spins of those particles.

However, I should also note that there is a profound strangeness in how such photon polarization states operate. You might call it the "missing state" problem.

By way of contrast, silver atoms are also spin 1, and conveniently have magnetic moments that allow "easy" separation. For example, by using three sequential Stern-Gerlach devices you can in principle divide up a population of silver atoms into six (yes, I said six, not three) populations that correspond geometrically to atoms with spin axes oriented along $\pm$X, $\pm$Y, and $\pm$Z. For a single silver atom, excruciatingly carefully use of this same arrangement can in principle create spatially isolated fractions of the wave function of the silver atom. When this done, though, you wind up with no more than three of the six "holders" being used, one along each axis. You might for example end up with 71% of the silver atom wave function in the +X Stern-Gerlach holder, 55% in -Y, and 45% in +Z. (It's a vector sum, so those percentages add as vector components.)

But where is the comparable experiment for photons? You can break them up along horizontal and vertical (or X and Y) polarizations, sure. But since you can't stop along their direction of propagation, how can you handle that aspect of the separation process?

(Hmm, odd thought: Actually, nowadays, there are a few labs that can bring photons to a full halt now by using specially tuned metal atom Bose condensates. So, have any of those folks thought to devise a way to look more closely at the "hidden photon state" issue, I wonder? Might that be related to some of the experimentation others mentioned in answers?)

(Second odd thought: Only photons in a vacuum travel at $c$. Shouldn't spin 1 photons traveling through refractive media at less than $c$ thus have some sort of explicit, accessible version of vector states along their propagation axes? After all, such slowing could I think be viewed (maybe?) as a mix of $c$ and "stopped" photon states, with the latter presumably showing explicit propagation axis states.)

Finally, at least in comparison to spin 1 atoms, photons also don't behave very nicely with respect to expressing a singular spin axis direction. For example, a single X-axis Stern-Gerlach divides silver atoms up into three groups: +X, -X, and "other" (that is, zero X spin). Later Stern-Gerlach devices can then further subdivide the "other" group into the remaining $\pm$Y and $\pm$Z populations.

But for photons, you only get one X axis group labeled "horizontal polarization." Where does the spin axis point in that case? It doesn't. The unique $\pm$X groups that are easily seen with silver atoms don't really have an analog with photons, at least none that I'm aware of. Perhaps someone else may have more insights?

So, my apologies for all the "added extras," but the original point remains: While odd in many ways, photons readily and provably give away their spin 1 natures by the way they can be broken up into two unique and isolated populations via a 90$^\circ$ rotation of a photon detector (polarizer). Beyond that, photon spins get pretty weird and not nearly so simple.

• The sunglasses and electron glare example is wonderful. Believe it or not I was having trouble coming up with a good, simple example of how the "$SO(3)$ double cover ribbon trick" shows up in fundamental particles and this definitely IS IT- I've found that even small children in science demonstrations at my daughter's school are fascinated by the belt trick and the idea that not everything comes back to its beginning state when rotated through $360^o$ - you even get questions like "can you make fancier ribbons so they need to spin three times to get untangled?" from children as young as 7. Aug 13, 2013 at 23:54
• Sx,Sy,Sz are non-commute, how can you get six? And how those six distributed? Aug 14, 2013 at 1:44
• Hmm, right or wrong, here was my thought: Using Feynman's hypothetical accelerate-then-stop S-Gs oriented along X, Y, and Z axes: (1) Apply S-G(X) to isolate {+1 X, 0 X, -1 X}; (2) Feed 0 X (only, oops?) into S-G(Y) (did I err there? maybe!) to isolate {+1 Y, 0 Y, -1 Y}; (3) Feed 0 X 0 Y into S-G(Z) to isolate {+1 Z, 0 Z, -1 Z}, with the assumption that 0 Z will be empty. So: The entire process definitely should be reversible, and you will definitely get either 6 or 7 groups... but what did I do with that process? In particular, will there be 6 or 7 groups? (Might make a nice question...) Aug 14, 2013 at 17:35
• "That's because if you rotate polarizers by only 90∘, you will find that you can break photons down into two mutually exclusive populations of photons. That is geometrically possible only if the particle in question is a vector boson, that is, a spin 1 particle." I don't understand why this is the case. Aug 15, 2013 at 13:11
• Although the polaroid argument is valid, the statement “silver atoms are spin 1” is incorrect. The spin of such object as a silver atom is undefined; see physics.stackexchange.com/questions/11197/… for details. Aug 20, 2014 at 8:35

One method is based on the conservation of angular momentum.

The electronic transition must follow the selection rule $\Delta l=\pm 1$. So the first thing to do is to choose an atom with zero total angular momentum, then let the atom absorb a photon and make a transition to $l=1$ state.

Secondly, we use the Stern-Gerlach experiment to detect the magnetic moment of this atom, which are $m=0,\pm 1$ in our case. Repeating experiments with random photons, we should see that there are mainly three bright spots on the screen: (1) non-deflected, (2) up and (3) down. The distance of outer spot from the center can also be calculated from the theory. Make sure the half-life of the excited state is long enough for the experiment.

In this way you can prove that the photon has spin=1. (That is what I thought, no reference for the actual experiment.)

• Does anybody have a reference for such an experiment? Aug 15, 2013 at 13:09
• This can be a viable idea for atoms with the nuclear spin 0; otherwise the confusion between angular momentum of the atom and one of its electron shells will disrupt conservation arguments; see physics.stackexchange.com/questions/11197/… for details. Also, it isn’t clear whether the author refers to the total angular momentum j or to orbital angular momentum ℓ; only the former is a conserved quantity. Aug 20, 2014 at 9:05
• @yippy_yay Yes, here is the link: opg.optica.org/DirectPDFAccess/… Aug 12 at 8:22

In Richard Beth's Mechanical Detection and Measurement of the Angular Momentum of Light, bright light from a mercury arc lamp was circularly polarized and passed through a half-wave plate (which reverses the sense of circular polarization) attached to a torsion pendulum. A bit of clever experimental design sent the light through the half-wave plate twice so that each photon would exchange angular momentum $4\hbar$ with the fiber. Beth confirmed this prediction, and also showed it was consistent with angular momentum stored in the fields of classical electromagnetism.

Spin of the photon is an ongoing theoretical research. For classical electromagnetic field the total angular momentum is $\vec{J}=\vec{r}\times <\vec{E}\times \vec{B}>$ (all vectors). In field theory this quantity is $\vec{J}=\vec{L}+\vec{S}$, it is gauge invariant and possible to observe. But! Nobody found a correct way to represent it in the sum of two gauge invariant operators for angular orbital momentum and spin (L and S). This concludes the theoretical knowledge - operator of photon's spin is not known.

It is possible to measure the Z-component of the angular momentum - helicity.

If you are thinking of photon's spin as a MAGNETIC quantity then you are definitely not correct (not wrong either). There is no double slit experiment for photons :) Unfortunately.

You can read further here : http://arxiv.org/abs/arXiv:1006.3876

Frankly speaking photon is not as simple as people are trying to pretend. Cracking it's nature will need more efforts.

Thank you for thinking.

I would say this is an empirical fact. In atomic physics you don't observe optical transitions (e.g. induced by a laser) without angular momentum transfer. The change in angular momentum is always $\pm$ 1, that's what the photon can transmit. See http://en.wikipedia.org/wiki/Selection_rule The atomic states with different angular momenta are identified via their different energy levels due to the Zeeman effect in a magnetic field.

• You've just restated the fact that the OP asked about. Say how it is known that there is always an angular momentum transfer... Mar 21, 2014 at 20:28
• @dmckee this answer basically explains it in the same was as accepted answer.
– mip
Dec 11, 2016 at 23:21

A perhaps not completely rigorous, but easy to understand derivation:

It can be shown that the magnitude of angular momentum of circularly polarized classical radiation of frequency $\omega$ and energy $E$ is given by $E/\omega$. If we now assume that radiation is quantized in packets of energy $E = \hbar \omega$, we arrive at the conclusion that the angular momentum of light is quantized in packets of $\hbar$.


text: In this paper, we study tensor products of Demazure modules for the current algebra $\lie{sl}_2[t]$. We establish a set of generators and relations for the tensor product of two local Weyl modules (which are also level 1 Demazure modules). We also establish a character formula for the tensor product of a level 2 Demazure module and a local Weyl module. To complete the proof of this character formula we also prove a short exact sequence of $V(\xi)$ modules. We further conjecture a character formula for the tensor product of any level $\ell$ Demazure module and a local Weyl module and provide a proof assuming the analogous short exact sequence holds.


text:
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# Math Help - multiplying/dividing fractions with exponents?

1. ## multiplying/dividing fractions with exponents?

Hello,

This is my first time posting here but im sure it wont be the last :P

Anyways im in my first math class in about 2-3 years and i have done this stuff before but i cant remember very much. These 3 in particular have got me stumped and im not even sure what you would call this in order to look it up.

$(3a^2x/4ay^2)/(6ax/5a^3x^2)$

Im not sure how to type the other two without it looking like gibberish so here is a picture of it. i hope this is legible, any help would be much appreciated.


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# William is the owner of a small pizza shop and is thinking of increasing products and lowering... 1 answer below »

William is the owner of a small pizza shop and is thinking of increasing products and lowering costs. William’s pizza shop owns four ovens and the cost of the four ovens is $1,000. Each worker is paid$500 per week.

 Workers employed Qty of pizzas produced per week 012345678 075180360600900114012601360
Show all of your calculations and processes. Describe your answer for each question in complete sentences, whenever it is necessary.
1. Which inputs are fixed and which are variable in the production function of William’s pizza shop? Over what ranges do there appear to be increasing, constant, and/or diminishing returns to the number of workers employed?
2. What number of workers appears to be most efficient in terms of pizza product per worker?
3. What number of workers appears to minimize the marginal cost of pizza production assuming that each pizza worker is paid \$500 per week?
4. Why would marginal productivity decline when you hire more workers in the short run after a certain level?
5. How would expanding the business affect the economies of scale? When would you have constant returns to scale or diseconomies of scale? Describe your answer.

Attachments:


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on

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Stuff I have been up to

I have been busy everywhere except on this blog. Here are a few highlights, mostly my public outreach:

Long term survival

On BBC Future I have an essay concluding their amazing season on long term thinking where I go really long-term: The greatest long term threats facing humanity.

The approach I take there is to look at the question “if we have survived X years into the future, what problems must we have overcome before that?” It is not so much the threats (or frankly, problems – threat seems to imply a bit more active maliciousness than the universe normally brings about) that are interesting as just how radically we need to change or grow in power to meet them.

The central paradox of survival is that it requires change, and long-term that means that what survives may be very alien. Not so much a problem for me, but I think many disagree. A solid state civilization powered by black holes in a starless universe close to absolute zero, planning billions of years ahead may sound like a great continuation of us, or something too alien to matter.

Debunking doom

Climate doom is in the air, and I am frankly disturbed by how many think that we are facing an existential threat to our survival in the next decade or so – both because it is based on a misunderstanding of the science (which is not entirely easy to read), and how it breeds fatalism. As a response to a youngster question I wrote this piece on the Conversation: Will climate change cause humans to go extinct?

Robots in space

In Quartz, I have an essay about the next 50 years of space exploration and whether we should send robots instead: We should stop sending humans into space to do a robot’s job.

As often the title (not chosen by me, I prefer “A small step for machinekind”) makes it seem I am arguing for something different than I am actually arguing. As I see it, sending machines to space makes much more sense than sending humans… but given the very human desire to be the ones exploring, we will send humans in any case. Long-term we should also become multiplanetary if only to reduce extinction risks, but that might require sending robots ahead – except that in order to do that we need a lot of cheap, low-threshold experimentation and testing.

Good versus evil, Moloch versus Maria

Last year I participated in the Nexus Instituut “intellectual opera” in Amsterdam, enjoying myself immensely. I ended up writing an essay AI, Good and Evil… and Moloch (Official version Sandberg, A. (2019) Kunstmatige intelligentie en Moloch. Tijdschrift Nexus 81: De strijd tussen goed en kwaad. Nexus Instituut, Amsterdam (Tr. Laura Weeda)).

My main point is that evil is usually seen as active maliciousness or neglecting others, suffering itself, or meaningless/removal of meaning. Bad AI is unlikely to be actively malicious and making machines that can experience suffering is likely tricky, but automation that perform bad actions without caring is all too simple. The big risk is getting systems that are effective at implementing pointless goals too efficiently, destroying value (human or other) for no gain, not even to themselves. A further problem is also that these systems are systems, not individuals. We tend to think of AI as robots, “the AI” and other individual entities, when it just as well can be an ambient functionality of the wider techno-social world – impossible to pull the plug, with everybody complicit. We need better ways of debugging adaptive technological systems.

Life extension

On Humanity 2.0 I discussed/debated digital afterlives with Steve Fuller, Sr. Mary Christa Nutt, James Madden and Matthew Harvey Sanders. Got a bit intense at some points, but there is an interesting issue in untangling exactly what we want from an extended life. Not all forms of continuity count for all people: a continuity of consciousness is very different from a continuity of memory, a continuity of social interactions or functions, or leaving the right life projects in order.

Other stuff

Hacking the Brain: Dimensions of Cognitive Enhancement. A paper on cognitive enhancement, the final fruit of the “comparing apples to oranges” Volkswagen foundation project I participated in.

The GoCAS existential risk project final outputs arrived in the journal Foresight. I have two pieces, There is plenty of time at the bottom: The economics, risk and ethics of time compression and the group-written Long-term trajectories of human civilization.

I have also helped a bit with an Oxford project on sensitive intervention points for a post-carbon society. Not all tipping points are bad, and sometimes cartoon heroes may help.

Grand futures

Behind the scenes, my book is expanding… whether that is progress remains to be seen.

I have given various talks about some contents, but there is so much more. I think I have to do a proper lecture series this fall.

A bit of existential hope for Christmas (and beyond)

Existential hope is in the air. The term was coined by my collegues Toby and Owen to denote the opposite of an existential catastrophe: the chance that things could turn out much better than expected.

Recently I had the chance to attend a visioning weekend with the Foresight Institute where we discussed ways of turning dystopias into utopias. It had a clear existential hope message, largely because  it was organised by Allison Duettman who is writing a book on the topic. I must admit that I got a bit nervous when I found out since I am also writing my own grand futures book, but I am glad to say we are dealing with largely separate domains and reasons for hope.

Now I extra am glad to add a podcast to the list of hopeful messages: the Future of Life Institute had me on the podcast Existential Hope in 2019 and beyond. It includes not just me and Allison, but also Max Tegmark, Anthony Aguirre, Gaia Dempsey, and Josh Clark (who also interviewed me for his podcast series End of the World).

I also participated in the Nexus Instituut event “The Battle between Good and Evil”. I assume the good guys won. I certainly had fun. I ended up arguing that good is only weak compared to evil like how water is weak compared to solid object – in small amounts it will deform and splash. In larger amounts it is like the tide or a tsunami: you better get out of the way. In retrospect that analogy might have been particularly powerful in the Netherlands. They know their water and how many hands (and windmills) can reshape a country.

Do we really have grounds for existential hope?

A useful analysis of the concept of hope can be found in Jayne M. Waterworth’s A Philosophical Analysis of Hope. He defines that hoping for something requires (1) a conception of an uncertain possibility, (2) a desire for an objective, (3) a desire that one’s desire be satisfied, and (4) that one takes an anticipatory stance towards the objective.

One can hope for things that have a certain or uncertain probability, but also for things that are merely possible. Waterworth calls the first category “hope because of reality” or probability hope, while the second category is “hope in spite of reality” or possibility hope. I might have probability hope in fixing climate change, but possibility hope in humanity one day resurrecting the dead – in the first case we have some ideas of how it might happen and what might be involved, in the second case we have no idea even where to begin.

Outcomes can also be of different importance: hoping for a nice Christmas present is what Waterworth calls an ordinary hope, while hoping for a solution of climate change or death is an extraordinary hope.

We may speak of existential hope in the sense that “existential eucatastrophes” can occur, or that our actions can make them happen. This would represent the most extraordinary kind of hope possible.

But note that this kind of hope is potentially “hope because of reality” rather than “hope in spite of reality”. We can affect the future to some extent (there is an interesting issue of how much). There doesn’t seem to be any law of nature dooming us to early existential risk or a necessary collapse of civilization. We have in the past changed the rules for our species in very positive ways, and may do so again. We may discover facts about the world that greatly expand the size and value of our future – we have already done so in the past. These are good reasons to hope.

Hope is a mental state. The reason hope is a virtue in Christian theology is that it is the antidote to despair.

Hope is different from optimism, the view that good things are likely to happen. First, optimism is a general disposition rather than directed at particular hoped for occurrences. Second, hope can be a very small and unspecific thing: rather than being optimistic about everything going the right way, a hopeful person can see the overwhelming problems and risks and yet hope that something will happen to get us through. Even a small grain of hope might be enough to fend of despair.

Still, there may be a psychological disposition towards being hopeful. As defined by Snyder in regarding motivations towards goals this involves a sense of agency (chosen goals can be achieved) and pathways (successful plans and strategies for those goals can be generated). This trait predicts academic achievement in students beyond intelligence, personality, and past achievement. Indeed, in law students hope but not optimism was predictive for achievement (but both contributed to life satisfaction). This trait may be more about being motivated to seek out good future states than actually being hopeful about many things, but the more possibilities are seen, the more likely something worth hoping for will show up.

If there is something I wish for everybody in 2019 and beyond it is having this kind of disposition relative to existential hope. Yes, there are monumental problems ahead. But we can figure out ways around/through/over them. There are opportunities to be grabbed. There are new values to be forged.

The winter solstice has just passed and the days will become brighter and longer for the next months. Cheers!

Blueberry Earth

[Update: I have a paper version of this essay on arXiv:1807.10553, extending and correcting some of the results.]

Supposing that the entire Earth was instantaneously replaced with an equal volume of closely packed, but uncompressed blueberries, what would happen from the perspective of a person on the surface?

Unfortunately the site tends to frown on fun questions like this, so it was in my opinion prematurely closed while I was working out the answer. So here it is, with some extra extensions:

The density of blueberries has been estimated to 625.56 kg/m3, WillO on Stackexchange estimated it to 13% of Earth’s density (5510*0.13=716.3 kg/m3), so assuming it to be around $\rho_{berries}=700$ kg/m3 appears to be reasonable. Blueberry pulp has a density similar to water,  980 to 1050 kg per m3 although this is both temperature dependent and depends on how much solids there are. The difference to the whole berries is due to the air between the berries. Note that these are likely the big, thick-skinned “American” blueberries rather than the small wild thin-skinned blueberries (bilberries) I grew up with; the latter would have higher density due to their smaller size and break far more easily.

So instantaneously turning Earth into blueberries will reduce its mass to 0.1274 of what it was. Gravity will become correspondingly weaker, $g_{BE}=0.1274 g$.

However, blueberries are not particularly sturdy. While there is a literature on blueberry mechanics (of course!), I did not manage to find a great source on their compressive strength. A rough estimate is possible: stacking a sugar cube (1 g) on a berry will not break it, while a milk carton (1 kg) will; 100 g has a decent but not certain chance. So if we assume the blueberry area to be one square centimetre the breaking pressure is on the order of $P_{break}=0.1 g / 10^{-4} \approx 10,000$ N/m2. This allows us to estimate at what depth the berries will start to break: $z=P_{break}/g_{BE}\rho_{berries} = 11.4188$ m. So while the surface will be free blueberries they will start pulping within a few meters of the surface.

This pulping has an important effect: the pulp separates from the air, coalescing into a smaller sphere. If we assume pulp to be an incompressible fluid, then a sphere of pulp with the same mass as the initial berries will be $\rho_{pulp} r_{pulp}^3 = \rho_{berries}r_{earth}^3$, or $r_{pulp} = (\rho_{berries}/ \rho_{pulp} )^{1/3}r_{earth}$. In this case we end up with a planet with 0.8879 times smaller radius (5,657 km), surrounded by a vast atmosphere.

The freefall timescale for the planet is initially 41 minutes, but relatively shortly the pulping interactions, the air convection etc will slow things down in a complicated way. I expect that the the actual coalescence will take hours, with some late bubbles from the deep interior erupting fairly late.

The gravity on the pulp surface is just 1.5833 m/s2, 16% of normal gravity – almost exactly lunar gravity. This weakens convection currents and the speed with which bubbles move up. The scale height of the atmosphere, assuming the same composition and temperature as on Earth, will be 6.2 times higher. This means that pressure will decline much less with altitude, allowing far thicker clouds and weather systems. As we will see, the atmosphere will puff up more.

The separation has big consequences. Enormous amounts of air will be pushing out from the pulp as bubbles and jets, producing spectacular geysers (especially since the gravity is low). Even more dramatic is the heating: a lot of gravitational energy is released as the mass is compacted. The total gravitational energy of a constant density sphere of radius R is

$\int_0^R G [4\pi r^2 \rho] [4 \pi r^3 \rho/3] / r dr$ $= (16\pi^2 G\rho^2/3) \int_0^R r^4 dr$
$=(16\pi^2 G/15)\rho^2 R^5$

(the first factor in the integral is the mass of a spherical shell of radius r, the second the mass of the stuff inside, and the third the 1/r gravitational potential). If we ignore the mass of the air since it is small and we just want an order of magnitude estimate,  the compression of the berry mass gives energy

$E=(16\pi^2 G/15)(\rho_{berries}^2 r_{earth}^5 - \rho_{pulp}^2R_{pulp}^5)$ $\approx 4.3594\times 10^{29}$ J.

This is the energy output of the sun over half an hour, nothing to sneeze at: blueberry earth will become hot. There is about 573,000 J per kg, enough to heat the blueberries from freezing to boiling.

The result is that blueberry earth will turn into a roaring ocean of boiling jam, with the geysers of released air and steam likely ejecting at least a few berries into orbit (escape velocity is just 4.234 km/s, and berries at the initial surface will be even higher up in the potential). As the planet evolves a thick atmosphere of released steam will add to the already considerable air from the berries. It is not inconceivable that the planet may heat up further due to a water vapour greenhouse effect, turning into a very odd Venusian world.

Meanwhile the jam ocean is very deep, and the pressure at depth will be enough to cause the formation of high pressure ice even if it is warm. If the formation process is slow there will be some separation of water into ice and a concentration of other chemicals in the jam ocean, but I suspect the rapid collapse will instead make some kind of composite pulp ice. Ice VII forms above 9 GPa, so if we just use constant gravity this happens at a depth $z_{ice}=P_{VII}/g_{BE}\rho_{pulp}\approx 1,909$ km, about two-thirds of the radius. This would make up most of the interior. However, gravity is a bit weaker in the interior, so we need to take that into account. The pressure from all the matter above radius r is $P(r) =(3GM^2/8\pi R^4)(1-(r/R)^2)$, and the ice core will have radius $r_{ice}=\sqrt{1-P_{VII}/P(0)}$  $\approx$ 3,258 km. This is smaller, about 57% of the radius, and just 20% of the total volume.

The coalescence will also speed up rotation. The original blueberry earth would of course make one rotation every 24 hours, but the smaller result would have a smaller moment of inertia. The angular momentum conservation gives $(2/5)MR_1^2(2\pi/T_1) = (2/5)MR_2^2(2\pi/T_2)$, or $T_2 = (R_2/R_1)^2 T_1$, in this case 18.9210 hours. This in turn will increase the oblateness a bit, to approximately 0.038 – an 8.8 times increase over Earth.

Another effect is the orbit of the Moon. Now the two bodies have about equal mass. Is the Moon bound to blueberry earth? A kilogram of lunar material has potential energy $GM_{BE}/r_{moon} \approx$ 1.6925 $\times 10^{5}$ J, while the kinetic energy is $2.6442\times 10^5$ J – more than enough to escape. Had it remained the jam ocean would have made an excellent tidal dissipation mechanism that would have slowed down rotation and moved blueberry earth towards tidal lock with the moon much earlier than the 50 billion years it would otherwise have taken.

So, to sum up, to a person standing on the surface of the Earth when it turns into blueberries, the first effect would be a drastic reduction of gravity. Standing on the blueberries might be possible in theory, except that almost immediately they begin to compress rapidly and air starts erupting everywhere. The effect is basically the worst earthquake ever, and it keeps on going until everything has fallen 714 km. While this is going on everything heats up drastically until the entire environment is boiling jam and steam. The end result is a world that has a steam atmosphere covering an ocean of jam on top of warm blueberry granita.

Ten years closer to the future

It is the best place in the world to work.

Why Cherry 2000 should not be banned, Terminator should, and what this has to do with Oscar Wilde

[This is what happens when I blog after two glasses of wine. Trigger warning for possibly stupid cultural criticism and misuse of Oscar Wilde.]

From robots to artificiality

On practical ethics I discuss what kind of robots we ought to campaign against. I have signed up against autonomous military robots, but I think sex robots are fine. The dividing line is that the harm done (if any) is indirect and victimless, and best handled through sociocultural means rather than legislation.

I think the campaign against sex robots has a point in that there are some pretty creepy ideas floating around in the world of current sex bots. But I also think it assumes these ideas are the only possible motivations. As I pointed out in my comments on another practical ethics post, there are likely people turned on by pure artificiality – human sexuality can be far queerer than most think.

Going off on a tangent, I am reminded of Oscar Wilde’s epigram

“The first duty in life is to be as artificial as possible. What the second duty is no one has as yet discovered.”

Being artificial is not the same thing as being an object. As noted by Barris, Wilde’s artificiality actually fits in with pluralism and liberalism. Things could be different. Yes, in the artificial world nothing is absolutely given, everything is the result of some design choices. But assuming some eternal Essence/Law/God is necessary for meaning or moral exposes one to a fruitless search for that Thing (or worse, a premature assumption one has found It, typically when looking in the mirror). Indeed, as Dorian Gray muses, “Is insincerity such a terrible thing? I think not. It is merely a method by which we can multiply our personalities.” We are not single personas with unitary identities and well defined destinies, and this is most clearly visible in our social plays.

Sex, power and robots

Continuing on my Wildean binge, I encountered another epigram:

“Everything in the world is about sex except sex. Sex is about power.”

I think this cuts close to the Terminator vs. Cherry 2000 debate. Most modern theorists of gender and sex are of course power-obsessed (let’s blame Foucault). The campaign against sex robots clearly see the problem as the robots embodying and perpetuating a problematic unequal power structure. I detect a whiff of paternalism there, where women and children – rather than people – seem to be assumed to be the victims and in the need of being saved from this new technology (at least it is not going as far as some other campaigns that fully assume they are also suffering from false consciousness and must be saved from themselves, the poor things). But sometimes a cigar is just a cigar… I mean sex is sex: it is important to recognize that one of the reasons for sex robots (and indeed prostitution) is the desire for sex and the sometimes awkward social or biological constraints of experiencing it.

The problem with autonomous weapons is that power really comes out of a gun. (Must resist making a Zardoz reference…) It might be wielded arbitrarily by an autonomous system with unclear or bad orders, or it might wielded far too efficiently by an automated armed force perfectly obedient to its commanders – removing the constraint that soldiers might turn against their rulers if being aimed against their citizenry. Terminator is far more about unequal and dangerous power than sex (although I still have fond memories of seeing a naked Arnie back in 1984). The cultural critic may argue that the power games in the bedroom are more insidious and affect more of our lives than some remote gleaming gun-metal threat, but I think I’d rather have sexism than killing and automated totalitarianism. The uniforms of the killer robots are not even going to look sexy.

It is for your own good

Trying to ban sex robots is about trying to shape society into an appealing way – the goal of the campaign is to support “development of ethical technologies that reflect human principles of dignity, mutuality and freedom” and the right for everybody to have their subjectivity recognized without coercion. But while these are liberal principles when stated like this, I suspect the campaign or groups like it will have a hard time keeping out of our bedrooms. After all, they need to ensure that there is no lack of mutuality or creepy sex robots there. The liberal respect for mutuality can become a very non-liberal worship of Mutuality, embodied in requiring partners to sign consent forms, demanding trigger warnings, and treating everybody who is not responding right to its keywords as suspects of future crimes. The fact that this absolutism comes from a very well-meaning impulse to protect something fine makes it even more vicious, since any criticism is easily mistaken as an attack on the very core Dignity/Mutuality/Autonomy of humanity (and hence any means of defence are OK). And now we have all the ingredients for a nicely self-indulgent power trip.

This is why Wilde’s pluralism is healthy. Superficiality, accepting the contrived and artificial nature of our created relationships, means that we become humble in asserting their truth and value. Yes, absolute relativism is stupid and self defeating. Yes, we need to treat each other decently, but I think it is better to start from the Lockean liberalism that allows people to have independent projects rather than assume that society and its technology must be designed to embody the Good Values. Replacing “human dignity” with the word “respect” usually makes ethics clearer.

Instead of assuming we can a priori figure out how technology will change us and then select the right technology, we try and learn. We can make some predictions with reasonable accuracy, which is why trying to rein in autonomous weapons makes sense (the probability that they lead to a world of stability and peace seems remote). But predicting cultural responses to technology is not something we have any good track record of: most deliberate improvements of our culture have come from social means and institutions, not banning technology.

“The fact is, that civilisation requires slaves. The Greeks were quite right there. Unless there are slaves to do the ugly, horrible, uninteresting work, culture and contemplation become almost impossible. Human slavery is wrong, insecure, and demoralising. On mechanical slavery, on the slavery of the machine, the future of the world depends.”

AMA on Reddit

I have been doing an AMA on Reddit. Everything from transhumanist activism and the end of the Earth to why we haven’t fixed baldness yet.

I like the format, although it stresses me out.

Awesome blogs

I recently discovered Alex Wellerstein’s excellent blog Restricted data: the nuclear secrecy blog. I found it while looking for nuclear stockpiles data, but was drawn in by a post on the evolution of nuclear yield to mass. Then I started reading the rest of it. And finally, when reading this post about the logo of the IAEA I realized I needed to mention to the world how good it is. Be sure to test the critical assembly simulator to learn just why critical mass is not the right concept.

Another awesome blog is Almost looks like work by Jasmcole. I originally found it through a wonderfully over the top approach to positioning a wifi router (solving Maxwell’s equations turns out to be easier than the Helmholz equation!). But there are many other fascinating blog essays on physics, mathematics, data visualisation, and how to figure out propeller speeds from camera distortion.

Quantifying busyness

Tempus fugit

If I have one piece of advice to give to people, it is that they typically have way more time now than they will ever have in the future. Do not procrastinate, take chances when you see them – you might never have the time to do it later.

One reason is the gradual speeding up of subjective time as we age: one day is less time for a 40 year old than for a 20 year old, and way less than the eon it is to a 5 year old. Another is that there is a finite risk that opportunities will go away (including our own finite lifespans). The main reason is of course the planning fallacy: since we underestimate how long our tasks will take, our lives tend to crowd up. Accepting to give a paper in several months time is easy, since there seems to be a lot of time to do it in between… which mysteriously disappears until you sit there doing an all-nighter. There is also the likely effect that as you grow in skill, reputation and career there will be more demands on your time. All in all, expect your time to grow in preciousness!

Mining my calendar

I recently noted that my calendar had filled up several weeks in advance, something I think did not happen to this extent a few years back. A sign of a career taking off, worsening time management, or just bad memory? I decided to do some self-quantification using my Google calendar. I exported the calendar as an .ics file and made a simple parser in Matlab.

It is pretty clear from a scatter plot that most entries are for the near future – a few days or weeks ahead. Looking at a histogram shows that most are within a month (a few are in the past – I sometimes use my calendar to note when I have done something like an interview that I may want to remember later).

Plotting it as a log-log diagram suggests it is lighter-tailed than a power-law: there is a characteristic scale. And there are a few wobbles suggesting 1-week, 2-week and 3-week periodicities.

Am I getting busier? Plotting the mean and median distance to scheduled events, and the number of events per year, suggests yes. The median distance to the things I schedule seems to be creeping downwards, while the number of events per year has clearly doubled from about 400 in 2008 to 800 in 2014 (and extrapolating 2015 suggests about 1000 scheduled events).

Plotting the number of events I had per 14-day period also suggests that I have way more going on now than a few years ago. The peaks are getting higher and the mean period is more intense.

When am I free?

A good measure of busyness would be the time horizon: how far ahead should you ask me for a meeting if you want to have a high chance of getting it?

One approach would be to look for the probability $Q(t)$ that a day $t$ days ahead is entirely empty. If the probability that I will fill in something $i$ days ahead is $P(i)$, then the chance for an empty day is $Q(t) = \prod_{i=t}^\infty (1-P(i))$. We can estimate $P(i)$ by doing a curve-fit (a second degree curve works well), but we can of course just estimate from the histogram counts: $\hat{P}(i)=N(i)/N$.

However, this method is slightly wrong. Some days are free, others have many different events. If I schedule twice as many events the chance of a free day should be lower. A better way of estimating $Q(t)$ is to think in terms of the rate of scheduling. We can view this as a Poisson process, where the rate of scheduling $\lambda(i)$ tells us how often I schedule something $i$ days ahead. An approximation is $\hat{\lambda}(i)=N(i)/T$, where $T$ is the time interval we base our estimate on. This way $Q(t) = \prod_{i=t}^\infty e^{-\lambda(i)}$.

If we slice the data by year, then there seems to be a fairly clear trend towards the planning horizon growing – I have more and more events far into future, and I have more to do. Oh, those halcyon days in 2007 when I presumably just lazed around…

If we plot when I have 50%, 75% and 90% chance of being free, the trend is even clearer. At present you need to ask about three weeks in advance to have a 50% chance of grabbing me, and 187 days in advance to be 90% certain (if you want an entire working week with 50% chance, this is close to where you should go). Back in 2008 the 50% point was about a week and the 90% point 1.5 months ahead. I have become around 3 times busier.

Conclusions

So, I have become busier. This is of course no evidence of getting more done – a lot of events are pointless meetings, and who knows if I am doing anything helpful at the other events. Plus, I might actually be wasting my time doing statistics and blogging instead of working.

But the exercise shows that it is possible to automatically estimate necessary planning horizons. Maybe we should add this to calendar apps to help scheduling: my contact page or virtual secretary might give you an automatically updated estimate of how far ahead you need to schedule things to have a good chance of getting me. It doesn’t have to tell you my detailed schedule (in principle one could do a privacy attack on the schedule by asking for very specific dates and seeing if they were blocked).

We can also use this method to look at levels of busyness across organisations. Who have flexibility in their schedules, who are so overloaded that they cannot be effectively involved in projects? In the past, tasks tended to be simple and the issue was just the amount of time people had. But today we work individually yet as part of teams, and coordination (meetings, seminars, lectures) are the key links: figuring out how to schedule them right is important for effectivity.

If team member $j$ has scheduling rates $\lambda_j(i)$ and they are are uncorrelated (yeah, right), then $Q(t)=\prod_{i=t}^\infty e^{-\sum_j\lambda_j(i)}$. The most important lesson is that the chance of everybody being able to make it to any given meeting day declines exponentially with the number of people. If the $\lambda_j(i)$ decline exponentially with time (plausible in at least my case) then scheduling a meeting requires the time ahead to be proportional to the number of people involved: double the meeting size, at least double the planning horizon. So if you want nimble meetings, make them tiny.

In the end, I prefer to live by the advice my German teacher Ulla Landvik once gave me, glancing at the school clock: “I see we have 30 seconds left of the lesson. Let’s do this excercise – we have plenty of time!” Time not only flies, it can be stretched too.