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# Drawing straws

Drawing straws is a selection method that is used by a group to choose one member of the group to perform a task after none has volunteered for it. The same practice can be used also to choose one of several volunteers, should an agreement not be reached.

## Process

The group leader takes a number of straws (or similarly long cylindrical objects) and ensures that one of them is physically shorter than the others. The leader then grabs all of the straws in his fist, such that all of them appear to be of the same length.

The group leader offers the clenched fist to the group. Each member of the group draws a straw from the fist of the group leader. At the end of the offering, the group member who has drawn the shortest straw is the one who must perform the task.

This practice is epitomized in the cliché "drawing the short straw"—to mean being randomly, unluckily, or unfairly selected to perform a task, at the risk of suffering a penalty.

## Probability

Drawing straws is a fair game; every participant has a ${\displaystyle {\tfrac {1}{N}}}$ chance to draw a short straw and ${\displaystyle {\tfrac {N-1}{N}}}$ chance NOT to draw it, where ${\displaystyle {N}}$ is the initial number of straws before the drawing starts.

For example, the chances of the second person to draw the short straw are the product of the chances of the short straw NOT to be pulled out by the first person: ${\displaystyle {\tfrac {N-1}{N}}}$, and the chances of the second person to pull it out of the smaller (by one) bunch of straws: ${\displaystyle {\tfrac {1}{N-1}}}$. The multiplication gives: ${\displaystyle {\tfrac {N-1}{N}}\times {\tfrac {1}{N-1}}={\tfrac {1}{N}}}$.

For the fourth person the chances to draw the short straw are a product of chances of the first three people consecutively NOT pulling the straw out and the fourth person to draw it afterwards (the size of the bunch gets progressively smaller by one for every next person): ${\displaystyle {\tfrac {N-1}{N}}\times {\tfrac {N-2}{N-1}}\times {\tfrac {N-3}{N-2}}\times {\tfrac {1}{N-3}}={\tfrac {1}{N}}}$.

The fact that it's more likely to draw the short straw from a smaller bunch is perfectly compensated by the chance of somebody else pulling the straw out before you.

## Politics

### United Kingdom

In the United Kingdom, if a local or national election has resulted in a tie in which candidates receive exactly the same number of votes after three recounts, the winner can be decided either by drawing straws/lots, coin flipping,[1] or drawing the high card in a pack of cards.[citation needed]

### USA

On 20 November 2015, a Mississippi state election was settled by 'drawing straws' after both candidates received 4,589 votes. This resulted in Blaine Eaton being re-elected to the Mississippi House of Representatives.[2]


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# Denotational semantics of the Actor model

The denotational semantics of the Actor model is the subject of denotational domain theory for Actors. The historical development of this subject is recounted in [Hewitt 2008b].

## Actor fixed point semantics

The denotational theory of computational system semantics is concerned with finding mathematical objects that represent what systems do. Collections of such objects are called domains. The Actor uses the domain of event diagram scenarios. It is usual to assume some properties of the domain, such as the existence of limits of chains (see cpo) and a bottom element. Various additional properties are often reasonable and helpful: the article on domain theory has more details.

A domain is typically a partial order, which can be understood as an order of definedness. For instance, given event diagram scenarios x and y, one might let "x≤y" mean that "y extends the computations x".

The mathematical denotation for a system S is found by constructing increasingly better approximations from an initial empty denotation called S using some denotation approximating function progressionS to construct a denotation (meaning ) for S as follows:

${\displaystyle \mathbf {Denote} _{\mathtt {S}}\equiv \lim _{i\to \infty }\mathbf {progression} _{{\mathtt {S}}^{i}}(\bot _{\mathtt {S}})}$

It would be expected that progressionS would be monotone, i.e., if x≤y then progressionS(x)≤progressionS(y). More generally, we would expect that

If ∀i∈ω xixi+1, then ${\displaystyle \mathbf {progression} _{\mathtt {S}}(\lim _{i\to \infty }{\mathtt {x}}_{i})=\lim _{i\to \infty }\mathbf {progression} _{\mathtt {S}}({\mathtt {x}}_{i})}$

This last stated property of progressionS is called ω-continuity.

A central question of denotational semantics is to characterize when it is possible to create denotations (meanings) according to the equation for DenoteS. A fundamental theorem of computational domain theory is that if progressionS is ω-continuous then DenoteS will exist.

It follows from the ω-continuity of progressionS that

progressionS(DenoteS) = DenoteS

The above equation motivates the terminology that DenoteS is a fixed point of progressionS.

Furthermore this fixed point is least among all fixed points of progressionS.

## Compositionality in programming languages

An important aspect of denotational semantics of programming languages is compositionality, by which the denotation of a program is constructed from denotations of its parts. For example consider the expression "<expression1> + <expression2>". Compositionality in this case is to provide a meaning for "<expression1> + <expression2>" in terms of the meanings of <expression1> and <expression2>.

The Actor model provides a modern and very general way the compositionality of programs can be analyzed. Scott and Strachey [1971] proposed that the semantics of programming languages be reduced to the semantics of the lambda calculus and thus inherit the denotational semantics of the lambda calculus. However, it turned out that concurrent computation could not be implemented in the lambda calculus (see Indeterminacy in concurrent computation). Thus there arose the problem of how to provide modular denotational semantics for concurrent programming languages. One solution to this problem is to use the Actor model of computation. In Actor model, programs are Actors that are sent Eval messages with the address of an environment (explained below) so that programs inherit their denotational semantics from the denotational semantics of the Actor model (an idea published in Hewitt [2006]).

### Environments

Environments hold the bindings of identifiers. When an environment is sent a Lookup message with the address of an identifier x, it returns the latest (lexical) binding of x.

As an example of how this works consider the lambda expression <L> below which implements a tree data structure when supplied with parameters for a leftSubTree and rightSubTree. When such a tree is given a parameter message "getLeft", it return leftSubTree and likewise when given the message "getRight" it returns rightSubTree.

 λ(leftSubTree, rightSubTree)
λ(message)
if (message == "getLeft") then leftSubTree
else if (message == "getRight") then rightSubTree

Consider what happens when an expression of the form "(<L> 1 2)" is sent an Eval message with environment E. One semantics for application expressions such as this one is the following: <L>, 1 and 2 are each sent Eval messages with environment E. The integers 1 and 2 immediately reply to the Eval message with themselves.

However, <L> responds to the Eval message by creating a closure Actor (process) C that has an address (called body) for <L> and an address (called environment) for E. The Actor "(<L> 1 2)" then sends C the message [1 2].

When C receives the message [1 2], it creates a new environment Actor F which behaves as follows:

1. When it receives a Lookup message for the identifier leftSubTree, it responds with 1
2. When it receives a Lookup message for the identifier rightSubTree, it responds with 2
3. When it receives a Lookup message for any other identifier, it forwards the Lookup message to E

The Actor (process) C then sends an Eval message with environment F to the following actor (process):

   λ(message)
if (message == "getLeft") then leftSubTree
else if (message == "getRight") then rightSubTree

### Arithmetic expressions

For another example consider the Actor for the expression "<expression1> + <expression2>" which has addresses for two other actors (processes) <expression1> and <expression2>. When the composite expression Actor (process) receives an Eval message with addresses for an environment Actor E and a customer C, it sends Eval messages to <expression1> and <expression2> with environment E and sends C a new Actor (process) C0. When C0 has received back two values N1 and N2, it sends C the value N1 + N2. In this way, the denotational semantics for process calculi and the Actor model provide a denotational semantics for "<expression1> + <expression2>" in terms of the semantics for <expression1> and <expression2>.

### Other programming language constructs

The denotational compositional semantics presented above is very general and can be used for functional, imperative, concurrent, logic, etc. programs (see [Hewitt 2008a]). For example it easily provides denotation semantics for constructs that are difficult to formalize using other approaches such as delays and futures.

## Clinger's Model

In his doctoral dissertation, Will Clinger developed the first denotation semantics for the Actor model.

### The domain of Actor computations

Clinger [1981] explained the domain of Actor computations as follows:

The augmented Actor event diagrams [see Actor model theory] form a partially ordered set < Diagrams,   > from which to construct the power domain P[Diagrams] (see the section on Denotations below). The augmented diagrams are partial computation histories representing "snapshots" [relative to some frame of reference] of a computation on its way to being completed. For x,yDiagrams, x≤y means x is a stage the computation could go through on its way to y. The completed elements of Diagrams represent computations that have terminated and nonterminating computations that have become infinite. The completed elements may be characterized abstractly as the maximal elements of Diagrams [see William Wadge 1979]. Concretely, the completed elements are those having non pending events. Intuitively, Diagrams is not ω-complete because there exist increasing sequences of finite partial computations
${\displaystyle x_{0}\leq x_{1}\leq x_{2}\leq x_{3}\leq ...}$
in which some pending event remains pending forever while the number of realized events grows without bound, contrary to the requirement of finite [arrival] delay. Such a sequence cannot have a limit, because any limit would represent a completed nonterminating computation in which an event is still pending.
To repeat, the actor event diagram domain Diagrams is incomplete because of the requirement of finite arrival delay, which allows any finite delay between an event and an event it activates but rules out infinite delay.

### Denotations

In his doctoral dissertation, Will Clinger explained how power domains are obtained from incomplete domains as follows:

From the article on Power domains: P[D] is the collection of downward-closed subsets of domain D that are also closed under existing least upper bounds of directed sets in D. Note that while the ordering on P[D] is given by the subset relation, least upper bounds do not in general coincide with unions.

For the actor event diagram domain Diagrams, an element of P[Diagrams] represents a list of possible initial histories of a computation. Since for elements x and y of Diagrams, x≤y means that x is an initial segment of the initial history y, the requirement that elements of P[Diagrams] be downward-closed has a clear basis in intuition.
...
Usually the partial order from which the power domain is constructed is required to be ω-complete. There are two reasons for this. The first reason is that most power domains are simply generalizations of domains that have been used as semantic domains for conventional sequential programs, and such domains are all complete because of the need to compute fixed points in the sequential case. The second reason is that ω-completeness permits the solution of recursive domain equations involving the power domain such as
${\displaystyle R\approx S\rightarrow P[S+(S\times R)]}$
which defines a domain of resumptions [Gordon Plotkin 1976]. However, power domains can be defined for any domain whatsoever. Furthermore the power domain of a domain is essentially the power domain of its ω-completion, so recursive equations involving the power domain of an incomplete domain can still be solved, provide the domains to which the usual constructors (+, ×, →, and *) are applied are ω-complete. It happens that defining Actor semantics as in Clinger [1981] does not require solving any recursive equations involving the power domain.
In short, there is no technical impediment to building power domains from incomplete domains. But why should one want to do so?
In behavioral semantics, developed by Irene Greif, the meaning of program is a specification of the computations that may be performed by the program. The computations are represented formally by Actor event diagrams. Greif specified the event diagrams by means of causal axioms governing the behaviors of individual Actors [Greif 1975].
Henry Baker has presented a nondeterministic interpreter generating instantaneous schedules which then map onto event diagrams. He suggested that a corresponding deterministic interpreter operating on sets of instantaneous schedules could be defined using power domain semantics [Baker 1978].
The semantics presented in [Clinger 1981] is a version of behavioral semantics. A program denotes a set of Actor event diagrams. The set is defined extensionally using power domain semantics rather than intensionally using causal axioms. The behaviors of individual Actors is defined functionally. It is shown, however, that the resulting set of Actor event diagrams consists of exactly those diagrams that satisfy causal axioms expressing the functional behaviors of Actors. Thus Greif's behavioral semantics is compatible with a denotational power domain semantics.
Baker's instantaneous schedules introduced the notion of pending events, which represent messages on the way to their targets. Each pending event must become an actual (realized) arrival event sooner or later, a requirement referred to as finite delay. Augmenting Actor event diagrams with sets of pending events helps to express the finite delay property, which is characteristic of true concurrency [Schwartz 1979].

### Sequential computations form an ω-complete subdomain of the domain of Actor computations

In his 1981 dissertation, Clinger showed how sequential computations form a subdomain of concurrent computations:

Instead of beginning with a semantics for sequential programs and then trying to extend it for concurrency, Actor semantics views concurrency as primary and obtains the semantics of sequential programs as a special case.
...
The fact that there exist increasing sequences without least upper bounds may seem strange to those accustomed to thinking about the semantics of sequential programs. It may help to point out that the increasing sequences produced by sequential programs all have least upper bounds. Indeed, the partial computations that can be produced by sequential computation form an ω-complete subdomain of the domain of Actor computations Diagrams. An informal proof follows.
From the Actor point of view, sequential computations are a special case of concurrent computations, distinguishable by their event diagrams. The event diagram of a sequential computation has an initial event, and no event activates more than one event. In other words, the activation ordering of a sequential computation is linear; the event diagram is essentially a conventional execution sequence. This means that the finite elements of Diagrams
${\displaystyle x_{0}\leq x_{1}\leq x_{2}\leq x_{3}\leq ...}$
corresponding to the finite initial segments of a sequential execution sequence all have exactly one pending event, excepting the largest, completed element if the computation terminates. One property of the augmented event diagrams domain < Diagrams,   > is that if x≤y and x≠y, then some pending event of x is realized in y. Since in this case each xi has at most one pending event, every pending event in the sequence becomes realized. Hence the sequence
${\displaystyle x_{0}\leq x_{1}\leq x_{2}\leq x_{3}\leq ...}$
has a least upper bound in Diagrams in accord with intuition.
The above proof applies to all sequential programs, even those with choice points such as guarded commands. Thus Actor semantics includes sequential programs as a special case, and agrees with conventional semantics of such programs.

## The Timed Diagrams Model

Hewitt [2006b] published a new denotational semantics for Actors based on Timed Diagrams. The Timed Diagrams model stands in contrast to Clinger [1981] which constructed an ω-complete power domain from an underlying incomplete diagrammatic domain, which did not include time. The advantage of the domain Timed Diagrams model is that it is physically motivated and the resulting computations have the desired property of ω-completeness (therefore unbounded nondeterminism) which provides guarantee of service.

### Domain of Timed Actor Computations

Timed Diagrams denotational semantics constructs an ω-complete computational domain for Actor computations. In the domain, for each event in an Actor computation, there is a delivery time which represents the time at which the message is delivered such that each delivery time satisfies the following conditions:

1. The delivery time is a positive rational number that is not the same as the delivery time of any other message.
2. The delivery time is more than a fixed δ greater than the time of its activating event. It will later turn out that the value of δ doesn't matter. In fact the value of δ can even be allowed to decrease linearly with time to accommodate Moore's Law.

The Actor event timed diagrams form a partially ordered set <TimedDiagrams, ≤>. The diagrams are partial computation histories representing "snapshots" (relative to some frame of reference) of a computation on its way to being completed. For d1,d2εTimedDiagrams, d1≤d2 means d1 is a stage the computation could go through on its way to d2 The completed elements of TimedDiagrams represent computations that have terminated and nonterminating computations that have become infinite. The completed elements may be characterized abstractly as the maximal elements of TimedDiagrams. Concretely, the completed elements are those having no pending events.

Theorem: TimedDiagrams is an ω-complete domain of Actor computations i.e.,

1. If D⊆TimedDiagrams is directed, the least upper bound ⊔D exists; furthermore ⊔D obeys all the laws of Actor model theory.
2. The finite elements of TimedDiagrams are countable where an element xεTimedDiagrams is finite (isolated) if and only if D⊆TimedDiagrams is directed and x≤VD, there exists dεD with x≤d. In other words, x is finite if one must go through x in order to get up to or above x via the limit process.
3. Every element of TimedDiagrams is the least upper bound of a countable increasing sequence of finite elements.

### Power domains

• Definition: The domain <Power[TimedDiagrams], ⊆> is the set of possible initial histories M of a computation such that
1. M is downward-closed, i.e., if dεM, then ∀d’εTimedDiagrams d’≤d ⇒ d’εM
2. M is closed under least upper bounds of directed sets, i.e. if D⊆M is directed, then VDεM
• Note: Although Power[TimedDiagrams] is ordered by ⊆, limits are not given by U. I.e.,
(∀i∈ω Mi≤Mi+1) ⇒ Ui∈ω Mi ⊆ ⊔i∈ω Mi
E.g., If ∀i diεTimedDiagrams and di≤di+1 and Mi= {dk | k ≤i} then
i∈ω Mi = Ui∈ωMi U{ ⊔i∈ω di }
• Theorem: Power [TimedDiagrams] is an ω-complete domain.

### Concurrency Representation Theorem

An Actor computation can progress in many ways. Let d be a diagram with next scheduled event e and X ≡ {e’ | e ─≈→1-message e’} (see Actor model theory), Flow(d) is defined to be the set of all timed diagrams with d and extensions of d by X such that

1. the arrival all of the events of X has been scheduled where
2. the events of X are scheduled in all possible orderings among the scheduled future events of d
3. subject to the constraint that each event in X is scheduled at least δ after e and every event in X is scheduled at least once in every δ interval after that.

(Recall that δ is the minimum amount of time to deliver a message.)

Flow(d) ≡ {d} if d is complete.

Let S be an Actor system, ProgressionS is a mapping

Power[TimedDiagrams]→Power[TimedDiagrams]
ProgressionS(M) ≡ UdεM Flow(d)

Theorem: ProgressionS is ω-continuous.

I.e., if ∀i Mi⊆Mi+1 then ProgressionS(⊔iεω Mi) = ⊔iεω ProgressionS(Mi)

Furthermore the least fixed point of ProgressionS is given by the Concurrency Representation Theorem as follows:

iεω ProgressionSi(⊥S)

where ⊥S is the initial configuration of S.

The denotation DenoteS of an Actor system S is the set of all computations of S.

Define the time abstraction of a timed diagram to be the diagram with the time annotations removed.

Representation Theorem: The denotation DenoteS of an Actor system S is the time abstraction of

iεω ProgressionSi (⊥S)

Using the domain TimedDiagrams, which is ω-complete, is important because it provides for the direct expression of the above representation theorem for the denotations of Actor systems by directly constructing a minimal fixed point.

The criterion of continuity for the graphs of functions that Scott used to initially develop the denotational semantics of functions can be derived as a consequence of the Actor laws for computation as shown in the next section.

## References

• Dana Scott and Christopher Strachey. Toward a mathematical semantics for computer languages Oxford Programming Research Group Technical Monograph. PRG-6. 1971.
• Irene Greif. Semantics of Communicating Parallel Professes MIT EECS Doctoral Dissertation. August 1975.
• Joseph E. Stoy, Denotational Semantics: The Scott-Strachey Approach to Programming Language Semantics. MIT Press, Cambridge, Massachusetts, 1977. (A classic if dated textbook.)
• Gordon Plotkin. A powerdomain construction SIAM Journal on Computing September 1976.
• Edsger Dijkstra. A Discipline of Programming Prentice Hall. 1976.
• Krzysztof R. Apt, J. W. de Bakker. Exercises in Denotational Semantics MFCS 1976: 1-11
• J. W. de Bakker. Least Fixed Points Revisited Theor. Comput. Sci. 2(2): 155-181 (1976)
• Carl Hewitt and Henry Baker Actors and Continuous Functionals Proceeding of IFIP Working Conference on Formal Description of Programming Concepts. August 1–5, 1977.
• Henry Baker. Actor Systems for Real-Time Computation MIT EECS Doctoral Dissertation. January 1978.
• Michael Smyth. Power domains Journal of Computer and System Sciences. 1978.
• C.A.R. Hoare. Communicating Sequential Processes CACM. August, 1978.
• George Milne and Robin Milner. Concurrent processes and their syntax JACM. April, 1979.
• Nissim Francez, C.A.R. Hoare, Daniel Lehmann, and Willem-Paul de Roever. Semantics of nondeterminism, concurrency, and communication Journal of Computer and System Sciences. December 1979.
• Nancy Lynch and Michael J. Fischer. On describing the behavior of distributed systems in Semantics of Concurrent Computation. Springer-Verlag. 1979.
• Jerald Schwartz Denotational semantics of parallelism in Semantics of Concurrent Computation. Springer-Verlag. 1979.
• William Wadge. An extensional treatment of dataflow deadlock Semantics of Concurrent Computation. Springer-Verlag. 1979.
• Ralph-Johan Back. Semantics of Unbounded Nondeterminism ICALP 1980.
• David Park. On the semantics of fair parallelism Proceedings of the Winter School on Formal Software Specification. Springer-Verlag. 1980.
• Will Clinger, Foundations of Actor Semantics. MIT Mathematics Doctoral Dissertation, June 1981. (Quoted by permission of author.)
• Carl Hewitt What is Commitment? Physical, Organizational, and Social Pablo Noriega et al. editors. LNAI 4386. Springer-Verlag. 2007.

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# Airport problem

Assume that there are $$m$$ different types of aircrafts. For each type $$l$$, we know the number of movements, $$n_{l}$$, and the overall runway costs, $$c_{l}$$, associated with type $$l$$ aircrafts. Since we want to find out how much each individual movement should be charged, we interpret each of them as a single player. $$N_{l}$$ be the set of "players" associated with movements of type $$l$$ aircrafts.

We assume that types are arranged in an increasing order with respect to the costs, i.e $$0. Let $$N$$ be the (finite) set of all movements, namely $$N:= \displaystyle\bigcup_{l=1}^{m}{N_l }$$.

The airport game associated with this is the TU-game with player $$N$$ and characteristic function $$c$$ defined, for each $$S\subset N$$,$$S\ne \varnothing$$ by $$v(S):=- max\{c_{l}:S\cap N_{l}\ne \varnothing \}$$.

Questions

1) Could you explain why the negative sign?.

2) Could you give me some help to verify that airport games is superadditive and convex?

Superadditive. If $$A,B\subset N$$, $$A\cap B=\varnothing$$, $$v(A\cup B)\ge v(A)+v(B)$$.

Convex.If $$A,B\subset N$$, $$v(A\cup B)\ge v(A)+v(B)-v(A\cap B)$$

Thank you.

Edit: The negative sign is because the cost function is not superadditive but its negative if it is. But why?

• Hi Franciscolli, this looks like a homework problem. The policy of this forum is to not just provide an answer key to homework problems. If you want you can add more details as to what have you tried doing and what is confusing. People in the forum will be more akin to help you that way. Also, it looks like something is off with your notation. The characteristic function is defined for any subset of movements, $S$, but seems to depend on a particular aircraft, since it depends on $N_l$. If that is the case, what is he value of $v(S)$ if $S\cap N_l=\emptyset$? – Regio Jul 3 at 14:31
• Notation is of book "An introductory course on mathematical game theory".I also find it confusing. – Franciscolli Jul 6 at 3:42
• If $S∩N_{l}=∅?$ I think that $v(S)=0$ – Franciscolli Jul 7 at 1:56


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# Spherical Earth: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

Medieval artistic representation of a spherical Earth - with compartments representing earth, air, and water (c.1400).

The concept of a spherical Earth dates back to ancient Greek philosophy from around the 6th century BCE,[1] but remained a matter of philosophical speculation until the 3rd century BC when Hellenistic astronomy established the spherical shape of the earth as a physical given. The Greek paradigm was gradually adopted, either directly or via later European science, by all major civilizations and world regions, becoming the dominant concept of modern times.[2][3][4][5] The final, practical evidence of the spherical shape of the earth provided its circumnavigation by the Portuguese explorer Ferdinand Magellan (1519−1521).

The concept of a spherical Earth displaced earlier beliefs in a flat Earth: In early Mesopotamian mythology, the world was portrayed as a flat disk floating in the ocean and surrounded by a spherical sky,[6] and this forms the premise for early world maps like those of Anaximander and Hecataeus of Miletus. Other speculations on the shape of Earth include a seven-layered ziggurat or cosmic mountain, alluded to in the Avesta and ancient Persian writings (see seven climes), or a wheel, bowl, or four-cornered plane alluded to in the Rigveda.[7]

As determined by modern instruments, a sphere approximates the Earth's shape to within one part in 300. An oblate ellipsoid shape with a flattening of 1/300 matches even more precisely. Recent measurements from satellites suggest that the Earth is, in fact, slightly pear-shaped.[8]

## History

### Early Greek and Hellenistic development

Though the earliest evidence of a spherical Earth comes from ancient Greek sources, there is no account of how the sphericity of the Earth was discovered. A plausible explanation is that it was "the experience of travellers that suggested such an explanation for the variation in the observable altitude and the change in the area of circumpolar stars, a change that was quite drastic between Greek settlements" around the eastern Mediterranean Sea, particularly those between the Nile Delta and the Crimea.[9]

Pythagoras

Early Greek philosophers alluded to a spherical Earth, though with some ambiguity.[10] This idea influenced Pythagoras (b. 570 BCE), who saw harmony in the universe and sought to explain it. He reasoned that Earth and the other planets must be spheres, since the most harmonious geometric solid form is a sphere.[1] After the fifth century BCE, no Greek writer of repute thought the world was anything but round.[10]

Herodotus

In The Histories, written 431 BCE - 425 BCE, Herodotus dismisses a report of the sun observed shining from the north. This arises when discussing the circumnavigation of Africa undertaken c. 615-595 BCE. (The Histories, 4.43) His dismissive comment attests to a widespread ignorance of the ecliptic's inverted declination in a southern hemisphere.

Plato

Plato (427 BCE - 347 BCE) travelled to southern Italy to study Pythagorean mathematics. When he returned to Athens and established his school, Plato also taught his students that Earth was a sphere. If man could soar high above the clouds, Earth would resemble "one of those balls which have leather coverings in twelve pieces, and is decked with various colours, of which the colours used by painters on earth are in a manner samples." (Phaedo, 110b)

Aristotle
When a ship is at the horizon its lower part is invisible due to Earth's curvature. This was one of the first arguments favoring a round-Earth model.

Aristotle (384 BCE - 322 BCE) was Plato's prize student and "the mind of the school." Aristotle observed "there are stars seen in Egypt and [...] Cyprus which are not seen in the northerly regions." Since this could only happen on a curved surface, he too believed Earth was a sphere "of no great size, for otherwise the effect of so slight a change of place would not be quickly apparent." (De caelo, 298a2-10)

Aristotle provided physical and observational arguments supporting the idea of a spherical Earth:

• Every portion of the Earth tends toward the center until by compression and convergence they form a sphere. (De caelo, 297a9-21)
• Travelers going south see southern constellations rise higher above the horizon; and
• The shadow of Earth on the Moon during a lunar eclipse is round. (De caelo, 297b31-298a10)

The concepts of symmetry, equilibrium and cyclic repetition permeated Aristotle's work. In his Meteorology he divided the world into five climatic zones: two temperate areas separated by a torrid zone near the equator, and two cold inhospitable regions, "one near our upper or northern pole and the other near the ... southern pole," both impenetrable and girdled with ice (Meteorologica, 362a31-35). Although no humans could survive in the frigid zones, inhabitants in the southern temperate regions could exist.

Eratosthenes

Eratosthenes (276 BCE - 194 BCE) estimated Earth's circumference around 240 BCE. He had heard that in Syene the Sun was directly overhead at the summer solstice whereas in Alexandria it still cast a shadow. Using the differing angles the shadows made as the basis of his trigonometric calculations he estimated a circumference of around 250,000 stades. The length of a 'stade' is not precisely known, but Eratosthenes' figure only has an error of around five to ten percent.[11][12]

Seleucus of Seleucia

Seleucus of Seleucia (c. 190 BC), who lived in the Seleucia region of Mesopotamia, stated that the Earth is spherical (and actually orbits the Sun, influenced by the heliocentric theory of Aristarchus of Samos).

Strabo

It has been suggested that seafarers probably provided the first observational evidence that the Earth was not flat, based on observations of the horizon. This argument was put forward by the geographer Strabo (c. 64 BC – AD 24), who suggested that the spherical shape of the Earth was probably known to seafarers around the Mediterranean Sea since at least the time of Homer.[13]

Claudius Ptolemy

Claudius Ptolemy (AD 90 - 168) lived in Alexandria, the centre of scholarship in the second century. Around 150, he produced his eight-volume Geographia.

The first part of the Geographia is a discussion of the data and of the methods he used. As with the model of the solar system in the Almagest, Ptolemy put all this information into a grand scheme. He assigned coordinates to all the places and geographic features he knew, in a grid that spanned the globe. Latitude was measured from the equator, as it is today, but Ptolemy preferred to express it as the length of the longest day rather than degrees of arc (the length of the midsummer day increases from 12h to 24h as you go from the equator to the polar circle). He put the meridian of 0 longitude at the most western land he knew, the Canary Islands.

Geographia indicated the countries of "Serica" and "Sinae" (China) at the extreme right, beyond the island of "Taprobane" (Sri Lanka, oversized) and the "Aurea Chersonesus" (Southeast Asian peninsula).

Ptolemy also devised and provided instructions on how to create maps both of the whole inhabited world (oikoumenè) and of the Roman provinces. In the second part of the Geographia he provided the necessary topographic lists, and captions for the maps. His oikoumenè spanned 180 degrees of longitude from the Canary Islands in the Atlantic Ocean to China, and about 81 degrees of latitude from the Arctic to the East Indies and deep into Africa. Ptolemy was well aware that he knew about only a quarter of the globe.

### Global diffusion

Spherical earth with the four seasons. Illustration in 12th century book Liber Divinorum Operum by Hildegard of Bingen

From its Greek origins, the idea of a spherical earth, along with much of Greek astronomical thought, slowly spread across the globe and ultimately became the adopted view in all major astronomical traditions:[2][3][4][5] In the west, the idea came naturally to the Romans through the lengthy process of cross-fertilization with Greek civilization. Many Roman authors such as Cicero and Pliny refer in their works to the rotundity of the earth as a matter of course.[14]

Treated as self-evident by early Christian authorities like Aurelius Augustinus, the idea survived into the medieval corpus of knowledge and became increasingly traceable with the rise of scholasticism and medieval learning.[14] A non-exhaustive list of more than a hundred Latin and vernacular writers from late antiquity to the 15th century, who were aware that the earth was spherical, has been compiled by Reinhard Krüger, professor for Romance literature at the University of Stuttgart:[14]

Portuguese exploration of Africa and Asia, Columbus voyage to the Americas (1492) and finally Ferdinand Magellan's circumnavigation of the earth (1519-21) provided the final, practical proofs for the global shape of the earth, while European colonists planted the idea in the American colonies.

With the rise of Greek culture in the east, Hellenistic astronomy filtered eastwards to ancient India where its profound influence became apparent in the early centuries AD.[15] The Greek concept of a spherical earth surrounded by the spheres of planets, vehemently supported by astronomers like Varahamihira and Brahmagupta, supplanted the long-standing Indian cosmological belief into a flat and circular earth disk.[15][16]

Islamic astronomy inherited the idea of a spherical earth from the Greek astronomical tradition.[17] The Islamic theoretical framework largely relied on the fundamental contributions of Aristotle (De caelo) and Ptolemy (Almagest), both of which worked with the premise that the earth was spherical and at the center of the universe (geocentric model).[17]

In the 17th century, the idea of a spherical earth, now considerably advanced by Western Astronomy, ultimately spread to Ming China, when Jesuit missionaries, who held high positions as astronomers at the imperial court, successfully challenged the Chinese belief that the earth was flat and square.[18][19][20]

### Other ancient views

Rigveda

While Indian cosmology long maintained the belief in a disk-shaped earth, K. V. Sarma mentions in passing that the Rigveda contains speculations on the idea of a spherical self-supporting earth, but does not give more details nor references.[21] It is impossible to date when any particular piece of the Rigveda was composed. While the core of the text probably dates to around 1400 BCE, many revisions and redactions are thought to have been incorporated around 600 BCE.[22] Nothing was written down until the 3rd century BCE at earliest, and possibly much later. See Dating and historical context of the Rigveda.

Aryabhata

The works of the classical Indian astronomer and mathematician, Aryabhata (476-550 AD), deal with the sphericity of the Earth and the motion of the planets. The final two parts of his Sanskrit magnum opus, the Aryabhatiya, which were named the Kalakriya ("reckoning of time") and the Gola ("sphere"), state that the Earth is spherical and that its circumference is 4,967 yojanas, which in modern units is 39,968 km, which is only 62 km less than the current value of 40,030 km.[23][24] He also stated that the apparent rotation of the celestial objects was due to the actual rotation of the Earth, calculating the length of the sidereal day to be 23 hours, 56 minutes and 4.1 seconds, which is also surprisingly accurate. It is likely that Aryabhata's results influenced European astronomy, because the 8th century Arabic version of the Aryabhatiya was translated into Latin in the 13th century.

Anania Shirakatsi

Anania Shirakatsi (Armenian: Անանիա Շիրակացի), also known as Ananias of Sirak, (610–685) was an Armenian scholar, mathematician, and geographer. His most famous works are Geography Guide (‘Ashharatsuyts’ in Armenian), and Cosmography (Tiezeragitutiun). He described the world as "being like an egg with a spherical yolk (the globe) surrounded by a layer of white (the atmosphere) and covered with a hard shell (the sky)." [25]

### Medieval Islamic scholars

Early Islamic scholars recognized earth's sphericity[26], leading Muslim mathematicians to develop spherical trigonometry[27] in order to further mensuration and to calculate the distance and direction from any given point on the Earth to Mecca. This determined the Qibla, or Muslim direction of prayer.

Al-Ma'mun

Around 830 CE, Caliph Al-Ma'mun commissioned a group of Muslim astronomers and Muslim geographers to measure the distance from Tadmur (Palmyra) to al-Raqqah, in modern Syria. They found the cities to be separated by one degree of latitude and the distance between them to be 66 2/3 miles and thus calculated the Earth's circumference to be 24,000 miles.[28]

Another estimate given by his astronomers was 56 2/3 Arabic miles per degree, which corresponds to 111.8 km per degree and a circumference of 40,248 km, very close to the currently modern values of 111.3 km per degree and 40,068 km circumference, respectively.[29]

Al-Farghānī

Al-Farghānī (Latinized as Alfraganus) was a Persian astronomer of the 9th century involved in measuring the diameter of the Earth, and commissioned by Al-Ma'mun. His estimate given above for a degree (56 2/3 Arabic miles) was much more accurate than the 60 2/3 Roman miles (89.7 km) given by Ptolemy. Christopher Columbus uncritically used Alfraganus's figure as if it were in Roman miles instead of in Arabic miles, in order to prove a smaller size of the Earth than that propounded by Ptolemy.[30]

Biruni

Abu Rayhan Biruni (973-1048) solved a complex geodesic equation in order to accurately compute the Earth's circumference, which was close to modern values of the Earth's circumference.[31][32] His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.[33][34][35] Biruni's method was intended to avoid "walking across hot, dusty deserts" and the idea came to him when he was on top of a tall mountain in India.[36] From the top of the mountain, he sighted the dip angle which, along with the mountain's height (which he calculated beforehand), he applied to the law of sines formula.[37][36] He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.[38]

John J. O'Connor and Edmund F. Robertson write in the MacTutor History of Mathematics archive:

"Important contributions to geodesy and geography were also made by Biruni. He introduced techniques to measure the earth and distances on it using triangulation. He found the radius of the earth to be 6339.6 km, a value not obtained in the West until the 16th century. His Masudic canon contains a table giving the coordinates of six hundred places, almost all of which he had direct knowledge."[39]

## Geodesy

Geodesy, also called geodetics, is the scientific discipline that deals with the measurement and representation of the Earth, its gravitational field and geodynamic phenomena (polar motion, Earth tides, and crustal motion) in three-dimensional time-varying space.

Geodesy is primarily concerned with positioning and the gravity field and geometrical aspects of their temporal variations, although it can also include the study of Earth's magnetic field. Especially in the German speaking world, geodesy is divided into geomensuration ("Erdmessung" or "höhere Geodäsie"), which is concerned with measuring the Earth on a global scale, and surveying ("Ingenieurgeodäsie"), which is concerned with measuring parts of the surface.

The Earth's shape can be thought of in at least two ways;

• as the shape of the geoid, the mean sea level of the world ocean; or
• as the shape of Earth's land surface as it rises above and falls below the sea.

As the science of geodesy measured Earth more accurately, the shape of the geoid was first found not to be a perfect sphere but to approximate an oblate spheroid, a specific type of ellipsoid. More recent measurements have measured the geoid to unprecedented accuracy, revealing mass concentrations beneath Earth's surface.

## Spherical models

The Earth as seen from the Apollo 17 mission.

There are several reasonable ways to approximate Earth's shape as a sphere. Most preserve a different feature of an ellipsoid that closely models the real Earth in order to compute the radius of the spherical model. All examples in this section assume the WGS 84 datum, with an equatorial radius a of 6,378.137 km and a polar radius b of 6,356.752 km. A sphere being a gross approximation of the spheroid, which itself is an approximation of the geoid, units are given here in kilometers rather than the millimeter resolution appropriate for geodesy.

• Preserve the equatorial circumference. This is simplest, being a sphere with circumference identical to the equatorial circumference of the ellipsoidal model. Since the circumference is the same, so is the radius, at 6,378.137 km.
• Preserve the lengths of meridians. This requires an elliptic integral to find, given the polar and equatorial radii: $\frac{2a}{\pi}\int_{0}^{\frac{\pi}{2}}\sqrt{\cos^2\phi + \frac{b^2}{a^2}\sin^2 \phi}\,d\phi$. A sphere preserving the lengths of meridians has a rectifying radius of 6,367.449 km. This can be approximated using the elliptical quadratic mean: $\sqrt{\frac{a^2+b^2}{2}}\,\!$, about 6,367.454 km; or even just the mean of the two axes: $\frac{a+b}{2}\,\!$, about 6,367.445 km.
• Preserve the surface area of the ellipsoidal model. This gives the authalic radius (denoted R2 by the International Union of Geodesy and Geophysics): $\sqrt{\frac{a^2+\frac{ab^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}{2}}\,\!$, or 6,371.007 km.
• Preserve the volume of the ellipsoidal model. This volumetric radius (denoted R3 by the IUGG) is computed as: $\sqrt[3]{a^2b}$, or 6,371.001 km.
• Synthesize some mean radius. The IUGG defines the mean radius (denoted R1) to be $\frac{2a+b}{3}\,\!$, giving 6,371.009 km. There are other ways to define the mean.

## References

1. ^ a b Dicks, D.R. (1970). Early Greek Astronomy to Aristotle. Ithaca, N.Y.: Cornell University Press. pp. 72–198. ISBN 9780801405617.
2. ^ a b Continuation into Roman and medieval thought: Reinhard Krüger: "Materialien und Dokumente zur mittelalterlichen Erdkugeltheorie von der Spätantike bis zur Kolumbusfahrt (1492)"
3. ^ a b Direct adoption of the Greek concept by Islam: Ragep, F. Jamil: "Astronomy", in: Krämer, Gudrun (ed.) et al.: Encyclopaedia of Islam, THREE, Brill 2010, without page numbers
4. ^ a b Direct adoption by India: D. Pingree: "History of Mathematical Astronomy in India", Dictionary of Scientific Biography, Vol. 15 (1978), pp. 533−633 (554f.); Glick, Thomas F., Livesey, Steven John, Wallis, Faith (eds.): "Medieval Science, Technology, and Medicine: An Encyclopedia", Routledge, New York 2005, ISBN0-415-96930-1, p. 463
5. ^ a b Adoption by China via European science: Jean-Claude Martzloff, “Space and Time in Chinese Texts of Astronomy and of Mathematical Astronomy in the Seventeenth and Eighteenth Centuries”, Chinese Science 11 (1993-94): 66-92 (69) and Christopher Cullen, "A Chinese Eratosthenes of the Flat Earth: A Study of a Fragment of Cosmology in Huai Nan tzu 淮 南 子", Bulletin of the School of Oriental and African Studies, Vol. 39, No. 1 (1976), pp. 106-127 (107)
6. ^ Otto E. Neugebauer (1975), A History of Ancient Mathematical Astronomy, Birkhäuser, p. 577, ISBN 354006995X
7. ^ Menon, CPS. Early Astronomy and Cosmology. Whitegishm MT, USA: Kessinger Publishing. p. 68. [1]
8. ^ Hugh Thurston, Early Astronomy, (New York: Springer-Verlag), p. 119. ISBN 0-387-94107-X.
9. ^ Otto E. Neugebauer (1975), A History of Ancient Mathematical Astronomy, Birkhäuser, pp. 575-6, ISBN 354006995X
10. ^ a b Dicks, D.R. (1970). Early Greek Astronomy to Aristotle. Ithaca, N.Y.: Cornell University Press. pp. 68. ISBN 9780801405617.
11. ^ "JSC NES School Measures Up". NASA. 11 April 2006. Retrieved 24 January 2008.
12. ^ "The Round Earth". NASA. 12 December 2004. Retrieved 24 January 2008.
13. ^ Hugh Thurston, Early Astronomy, (New York: Springer-Verlag), p. 118. ISBN 0-387-94107-X.
14. ^ a b c Reinhard Krüger: "Materialien und Dokumente zur mittelalterlichen Erdkugeltheorie von der Spätantike bis zur Kolumbusfahrt (1492)"
15. ^ a b D. Pingree: "History of Mathematical Astronomy in India", Dictionary of Scientific Biography, Vol. 15 (1978), pp. 533−633 (533, 554f.)
16. ^ Glick, Thomas F., Livesey, Steven John, Wallis, Faith (eds.): "Medieval Science, Technology, and Medicine: An Encyclopedia", Routledge, New York 2005, ISBN0-415-96930-1, p. 463
17. ^ a b Ragep, F. Jamil: "Astronomy", in: Krämer, Gudrun (ed.) et al.: Encyclopaedia of Islam, THREE, Brill 2010, without page numbers
18. ^
19. ^ Christopher Cullen, “Joseph Needham on Chinese Astronomy”, Past and Present, No. 87. (May, 1980), pp. 39-53 (42 & 49)
20. ^ Christopher Cullen, "A Chinese Eratosthenes of the Flat Earth: A Study of a Fragment of Cosmology in Huai Nan tzu 淮 南 子", Bulletin of the School of Oriental and African Studies, Vol. 39, No. 1 (1976), pp. 106-127 (107-109)
21. ^ Sarma, K.V. (2008), "Astronomy in India", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition) edited by Helaine Selin, pp. 317–321, Springer, ISBN 9781402045592.
22. ^ Oldenberg (p. 379) places it near the end of the Brahmana period, seeing that the older Brahmanas still contain pre-normalized Rigvedic citations. The Brahmana period is later than the composition of the samhitas of the other Vedas, stretching for about the 10th to 6th centuries. This would mean that the redaction of the texts as preserved was completed in roughly the 6th century BC. The EIEC (p. 306) gives a 7th century date.
23. ^ Aryabhata_I biography
24. ^ http://www.gongol.com/research/math/aryabhatiya The Aryabhatiya: Foundations of Indian Mathematics
25. ^ Hewson, Robert H. "Science in Seventh-Century Armenia: Ananias of Sirak, Isis, Vol. 59, No. 1, (Spring, 1968), pp. 32-45
26. ^ Muhammad Hamidullah. L'Islam et son impulsion scientifique originelle, Tiers-Monde, 1982, vol. 23, n° 92, p. 789.
27. ^ David A. King, Astronomy in the Service of Islam, (Aldershot (U.K.): Variorum), 1993.
28. ^ Gharā'ib al-funūn wa-mulah al-`uyūn (The Book of Curiosities of the Sciences and Marvels for the Eyes), 2.1 "On the mensuration of the Earth and its division into seven climes, as related by Ptolemy and others," (ff. 22b-23a)[2]
29. ^ Edward S. Kennedy, Mathematical Geography, pp=187-8, in (Rashed & Morelon 1996, pp. 185-201)
30. ^ Felipe Fernández-Armesto, Columbus and the conquest of the impossible, pp. 20-1, Phoenix Press, 1974.
31. ^ Khwarizm, Foundation for Science Technology and Civilisation.
32. ^ James S. Aber (2003). Alberuni calculated the Earth's circumference at a small town of Pind Dadan Khan, District Jhelum, Punjab, Pakistan.Abu Rayhan al-Biruni, Emporia State University.
33. ^ Lenn Evan Goodman (1992), Avicenna, p. 31, Routledge, ISBN 041501929X.
34. ^ Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom", Teaching Mathematics And Its Applications (Oxford University Press) 26 (1): 45-50, doi:10.1093/teamat/hrl009  (cf.  )
35. ^ Beatrice Lumpkin (1997), Geometry Activities from Many Cultures, Walch Publishing, pp. 60 & 112-3, ISBN 0825132851  [3]
36. ^ a b Beatrice Lumpkin (1997), Geometry Activities from Many Cultures, Walch Publishing, pp. 60 & 112-3, ISBN 0825132851  [4]
37. ^ Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom", Teaching Mathematics And Its Applications (Oxford University Press) 26 (1): 45-50, doi:10.1093/teamat/hrl009  (cf.  )
38. ^ Jim Al-Khalili, The Empire of Reason 2/6 (Science and Islam - Episode 2 of 3) at YouTube, BBC
39. ^  .


text:
# Maths summation algebra

1. Jan 3, 2010

### Homewoodm01

1. The problem statement, all variables and given/known data

Ok I have the answer to a question, all the working is given, however, I'm having trouble following it.

2. Relevant equations

3. The attempt at a solution

I am completely lost, could someone please explain the steps that have been taken, it would really help me.

Last edited by a moderator: May 4, 2017
2. Jan 3, 2010

First line to second: use the definition of $$\binom x y$$, then factor out terms that do not depend on x.

$$\binom x y= \frac{x!}{y! (x-y)!}$$

and factor from the sum any term that does not depend on the index of summation, $$x$$

Second line to third: shift the origin of summation from $$y$$ to 0 by replacing the index of summation by $$x = y$$. After this the sum becomes

$$\sum_{x=0}^\infty \lambda^{x+y} \frac{0.9^x}{x!} = \lambda^y \sum_{x=0}^\infty \frac{(0.9\, \lambda)^x}{x!}$$

You should be able to fill in the final step yourself.

3. Jan 3, 2010

### Homewoodm01

Thankyou very much for your explanation, I've pretty much got my head around it. Am I right i thinking the second line to the third all of the $$x$$ change to $$x+y$$ and is that a sort of rule when using summation? Thanks

4. Jan 3, 2010

I wouldn't say a rule, but a common bit of work. It's similar to making a substitution in a definite integral.

The original sum in line 2 is

$$\sum_{x=y}^\infty \frac{\lambda^x (0.9)^{x-y}}{(x-y)!}$$

The form of the summand is similar to the infinite series for an exponential, but the starting value isn't zero. Suppose I use a new index
of summation, defined as

$$t = x - y \quad \text{ so } \quad x = t+y$$

Since the original sum begins at $$x = y$$, the rewritten form begins at $$t = x - x = 0$$. In terms of the new variable the sum looks like

$$\sum_{t = 0}^\infty \frac{\lambda^{t+y} (0.9)^t}{t!} = \lambda^y \sum_{t=0}^\infty \frac{\lambda^t (0.9)^t}{t!}$$

Writing this new form with summation index equal to $$x$$ gives the form mentioned above.

Last edited: Jan 3, 2010
5. Jan 3, 2010

### Homewoodm01

Thanks for your time, you've really helped me, I'm confident I understand this now. Thanks again!!!


text:
# AVML 2012: ggplot2

–Josef Fruehwald, AVML, University of York, 2012

## Plotting principles

We are here to learn the basics of ggplot2. ggplot2 will be useful for producing complex graphics relatively simply. It won't be of any use for figuring out what is a sensible, useful, or accurate plot. To get a good handle on those, I'd advise simply reading a lot about data visualization.

## Installation

If you haven't already, take this opportunity to install the latest version of ggplot2.

install.packages("ggplot2")

library(ggplot2)

Right now, I'm running version 0.9.1, and some of the default behaviors of ggplot2 changed with version 0.9.0.

## Grammar of Graphics

ggplot2 is meant to be an implementation of the Grammar of Graphics, hence gg plot. The basic notion is that there is a grammar to the composition of graphical components in statistical graphics. By direcly controlling that grammar, you can generate a large set of carefully constructed graphics from a relatively small set of operations. As Wickham (2010), the author of ggplot2 said,

A good grammar will allow us to gain insight into the composition of complicated graphics, and reveal unexpected connections between seemingly different graphics

A good example of an unexpected connection would be that pie charts are just filled bar charts…

pie <- ggplot(mtcars, aes(x = factor(1), fill = factor(cyl))) +
geom_bar(width = 1, position = "fill", color = "black")
pie

…in polar coordinates.

pie + coord_polar(theta = "y")

## ggplot2 Materials

There are quite a few ggplot2 materials out there to guide you when you're not sure what to do next. First and foremost are the internal help pages for every piece of ggplot2 we'll cover here. They tend to be pretty useful.

?geom_jitter

## ggplot2 Basic Concepts

There are a few basic concepts to wrap your mind around for using ggplot2. First, we construct plots out of layers. Every component of the graph, from the underlying data it's plotting, to the coordinate system it's plotted on, to the statistical summaries overlaid on top, to the axis labels, are layers in the plot. The consequence of this is that your use of ggplot2 will probably involve iterative addition of layer upon layer until you're pleased with the results.

Next, the graphical properties which encode the data you're presenting are the aesthetics of the plot. These include things like

• x position
• y position
• size of elements
• shape of elements
• color of elements

The actual graphical elements utilized in a plot are the geometries, like

• points
• lines
• line segments
• bars
• text

Some of these geometries have their own specific aesthetic settings. For example,

• points
• point shape
• text
• text labels
• lines
• line weight
• line type

You'll also frequently want to plot statistics overlaid on top of, or instead of the raw data. Some of these include

• Smoothing and regression lines
• One and two dimensional binning
• Mean and medians with confidence intervals.

The aesthetics, geometries and statistics constitute the most important layers of a plot, but for fine tuning a plot for publication, there are a number of other things you'll want to adjust. The most common one of these are the scales, which encompass things like

• A logarithmic x or y axis
• Customized color scales
• Customized point shapes, or linetypes

The following sections are devoted to some of these basic elements in ggplot2.

### Layers

We'll be constructing plots with ggplot2 by building up “layers”. The layering of plot elements on top of each other is perhaps the most powerful aspect of the ggplot2 system. It means that relatively complex plots are built up of modular parts, which you can iteratively add or remove. For example, take this figure, which plots the relationship between vowel duration and F1 for 394 tokens of the lexical item “I”.

I_jean <- read.delim("http://bit.ly/avml_ggplot2_data")

This plot is composed of nine layers, which can be subdivided into five layer types.

#### The data layer

Every ggplot2 plot has a data layer, which defines the data set to plot, and the basic mappings of data to aesthetic elements. The data layer created with the functions ggplot() and aes(), and looks like this

ggplot(data, aes(...))

The first argument to ggplot() is a data frame (it must be a data frame), and its second argument is aes(). You're never going to use aes() in any other context except for inside of other ggplot2 functions, so it might be best not to think of aes() as its own function, but rather as a special way of defining data-to-aesthetic mappings.

For the plot from above, we'll be using data from the I_jean data frame, which looks like this:

head(I_jean)

##    Name Age Sex Word FolSegTrans Dur_msec     F1   F2
## 1 Jean   61   f  I'M           M      130  861.7 1336
## 2 Jean   61   f    I           N      140 1010.4 1349
## 3 Jean   61   f I'LL           L      110  670.1 1293
## 4 Jean   61   f  I'M           M      180  869.8 1307
## 5 Jean   61   f    I           R       80  743.0 1419
## 6 Jean   61   f I'VE           V      120  918.2 1581
##     F1.n    F2.n
## 1 1.6609 -0.8855
## 2 2.6883 -0.8536
## 3 0.3370 -0.9873
## 4 1.7168 -0.9536
## 5 0.8407 -0.6897
## 6 2.0512 -0.3068

I've decided that an interesting relationship in this data is between the vowel duration (Dur_msec) and the normalized F1 of the vowel (F1.n). Specifically, I'd like to map Dur_msec to the x-axis, and F1.n to the y-axis. Here's the ggplot2 code.

p <- ggplot(I_jean, aes(x = Dur_msec, y = F1.n))

Right off the bat, we can see one way in which ggplot2 is similar to lattice, and different from base graphics. If you've only ever used plot() in R, then you might be surprised to see me assigning the output of ggplot2 to an object, because plot() just creates a plot, not an object. ggplot2 plots, on the other hand, are objects, which you can assign, save, and manipulate.

If you try to print p right now though, you'll get an error. Right now, p is a ggplot2 plot that's all data, but no graphical elements. Adding graphical elements, or geoms is the next step.

#### The geometries layer

The next step, after defining the basic data-to-aesthetic mappings, is to add geometries to the data. We'll discuss geometries in more detail below, but for now, we'll add one of the simplest: points.

p <- p + geom_point()
p

There are a few things to take away from this step. First and foremost, the way you add new layers, of any kind, to a plot is with the + operator. And, as we'll see in a moment, there's no need to only add them one at a time. You can string together any number of layers to add to a plot, separated by +.

The next thing to notice is that all layers you add to a plot are, technically, functions. We didn't pass any arguments to geom_point(), so the resulting plot represents the default behavior: solid black circular points.

If for no good reason at all we wanted to use a different point shape in the plot, we could specify it inside of geom_point().

ggplot(I_jean, aes(x=Dur_msec, y=F1.n)) +
geom_point(shape = 3)

Or, if we wanted to use larger, red points, we could specify that in geom_point() as well.

ggplot(I_jean, aes(x=Dur_msec, y=F1.n)) +
geom_point(color = "red", size = 3)

Speaking of defaults, we can see a few of the default setting of ggplot2 on display here. Most striking is the light grey background, with white grid lines. Opinion varies on whether or not this is aesthetically or technically pleasing, but don't worry, it's adjustable.

Another default is to label the x and y axes with the column names from the data frame. I'll inject a bit of best practice advice here, and tell you to always change the axis names. It's nearly guaranteed that your data frame column names will make for very poor axis labels. We'll cover how to do that shortly.

Finally, note that we didn't need to tell geom_point() about the x and y axes. This may seem trivial, but it's a really important, and powerful aspect of ggplot2. When you add any layer at all to a plot, it will inherit the data-to-aesthetic mappings which were defined in the data layer. We'll discuss inheritance, and how to override, or define new data-to-aesthetic mappings within any geom.

#### The statistics layer

The final figure also includes a smoothing line, which is one of many possible statistical layers we can add to a plot.

p <- p + stat_smooth()
p

We'll go over the default behavior of stat_smooth() below, but in this plot, the smoothing line represents a loess smooth, and the semi-transparent ribbon surrounding the solid line is the 95% confidence interval.

One important thing to realize is that it's not necessary to include the points in order to add a smoothing line. Here's what the plot would look like with the points omitted.

ggplot(I_jean, aes(x = Dur_msec, y = F1.n))+
stat_smooth()

Notice how the y-axis has zoomed in to just include the range of the smoothing line and standard error.

#### Scale transformations

I also wanted to make some alterations to the default x and y axis scales. For example, the y-axis is currently running in reverse to the intuitive direction of F1. Higher vowels have lower F1 values, so we want to flip the y-axis. Additionally, durations are typically best displayed along a logarithmic scale, so we should convert the x-axis as well.

p <- p + scale_x_log10(breaks = c(50, 100,200,300,400))+
scale_y_reverse()
p

It's worth noting that the smoothing line here is calculated over the transformed data.

#### Cosmetic alterations

Finally, I wanted to make some cosmetic adjustments to the plot. For example, the x-axis label “Dur_msec” is not quite as useful as “Vowel duration (msec)” would be. I also added a title to the plot, and changed the color theme to black and white.

p <- p + ylab("Normalized F1")+
xlab("Vowel duration (msec)")+
theme_bw()+
opts(title = "394 tokens of 'I' from one speaker")
p

Here's all the layers, put together all at the same time.

ggplot(I_jean, aes(x=Dur_msec, y=F1.n))+
geom_point()+
stat_smooth()+
scale_x_log10(breaks = c(50, 100,200,300,400))+
scale_y_reverse()+
ylab("Normalized F1")+
xlab("Vowel duration (msec)")+
theme_bw()+
opts(title = "394 tokens of 'I' from one speaker")

### Aesthetics

In ggplot2, aesthetics are the graphical elements which are mapped to data, and they are defined with aes(). To some extent, the aesthetics you need to define are dependent on the geometries you want to use, because line segments have different geometric properties than points, for example. However, there is also a great deal of uniformity in the aesthetics used across geometries. Here is a list of the most common aesthetics you'll want to define.

• x
• x-axis location
• y
• y-axis location
• color
• The color of lines, points, and the outside borders of two dimensional geometries (polygons, bars, etc.). Hadley Wickham, the primary ggplot2 developer, is from New Zealand, so colour is also supported!
• fill
• The fill color of two dimensional geometries.
• size
• The size of points, or the weight of lines and borders of two dimensional geometries.
• shape
• This is specific to points, and defines the point shape. This is one of the few aesthetics to which you can't map a continuous variable.
• linetype
• This defines the line type of any kind of line, path, or border of a two dimensional geometry. This is another aesthetic which cannot be mapped to a continuous variable.
• alpha
• This defines the opacity of any geometric property. It's less commonly mapped to data, and more often hard coded to a single value as a solution for overplotting.
• xend, yend
• You'll use these more rarely, usually when plotting a line segment, or arrow. The beginning of the line segment will be located at x, y, and the end of the line segment will be located at xend,yend.
• ymin, ymax, (xmin, xmax)
• ymin and ymax are reserved for geometries which are devoted to representing ranges of data, like error bars, and ribbons. For the most part, these will be expressed along the y-axis, but xmin and xmax are utilized for some geometries as well.

The most important thing to keep in mind about aesthetics is not what they're called, though, but how they are inherited by the layers. Let's start by mapping the Word to color. 88% of the tokens are just “I”, so lets create a subset of the data that excludes “I” so it doesn't visually swamp the plot.

I_subset <- subset(I_jean, Word != "I")

ggplot(I_subset, aes(Dur_msec, F1.n, color = Word))+
geom_point()

Each point is now colored according to the word it corresponds to. ggplot2 has automatically generated a color palette of the right type and size, based on the data mapped to color, and created a legend to the side. As with everything, the specific color palette we use is adjustable, which will be discussed in more detail below under Scales. The default ggplot2 color palette is rather clever, however. Every color is equidistant around an HSL color circle, and have equal luminance. The idea is that no category should be accidentally visually emphasized, but they can be hard for some colorblind readers, and they will all print to the same shade of grey!

But on to more pressing matters. The only possible way to map the Word data to the color of points in the plot is to do it within aes(). If you are used to working with base graphics, then a lot of your instincts are wrong for ggplot2. For example, you might think the following would work:

I_subset$Color <- c("black", "red","blue", "green","goldenrod")[I_subset$Word]

ggplot(I_subset, aes(Dur_msec, F1))+
geom_point(color = I_subset$Color)  ## Error: When _setting_ aesthetics, they may only take one ## value. Problems: colour  The error is yelling at us about the distinction between setting an aesthetic, and mapping an aesthetic. Recall from above that we were able to set the color of all points to red by saying so inside of aes(), ggplot(I_subset, aes(Dur_msec, F1))+ geom_point(color = "red")  But that's as far as you can go with that. Everything beyond setting a single color for all points in a plot constitutes mapping colors, and you have to use aes() for that. But mapping data to specific color values is still not as simple as you might initially think. For example, mapping Color to color inside of aes() produces this stroop test. ggplot(I_subset, aes(Dur_msec, F1, color = Color))+ geom_point()  The lesson here is not to try this way of constructing your own custom color palettes. We'll go over how to construct custom palettes under Scales. #### Inheritance If we add one more geometry (a line), we see that it also inherits the mapping of Word to color. ggplot(I_subset, aes(Dur_msec, F1.n, color = Word))+ geom_point()+ geom_line()  There are a few important things to take note of in this plot. First, you can see that we have actually added four lines to the plot, one for each color. In most cases, when you map categorical data to an aesthetic like color, you are also defining sub-groupings of the data, and ggplot2 will draw a lines, calculate statistics, etc. separately for every sub-grouping of the data. The second important thing to notice is that geom_line() joins up points as they are ordered along the x-axis, not according to their order in the original data frame. There is a geom which will join up points that way called geom_path(). They point here, though, is that it is possible to define data-to-aesthetic mappings inside of geom functions, also by using aes(). Here, instead of mapping Word to color inside of ggplot(), we'll do it inside of geom_point(). ggplot(I_subset, aes(Dur_msec, F1.n))+ geom_point(aes(color = Word))+ geom_line()  The points are still colored according to the word, but there is only one, black line. We can also try passing aes(color = Word) to geom_line(). ggplot(I_subset, aes(Dur_msec, F1.n))+ geom_point()+ geom_line( aes(color = Word))+ scale_color_hue(direction = -1)  Now, the lines are colored according to the word, but the points are all black. This brings up the all important point about aesthetics: Geoms inherit aesthetic mappings from the ggplot() data layer, and not from any other layer. #### Grouping Let's look at the effect of mapping Word to color on the calculation of statistics, like smoothing lines. Note, inside of stat_smooth() I've said se = F to turn off the display of standard errors. ggplot(I_subset, aes(Dur_msec, F1.n, color=Word))+ geom_point()+ stat_smooth(se = F)  Just like separate lines were drawn for each group as defined by color=Word, ggplot2 has calculated separate smoothers for each subset. If we had only passed color=Word to geom_point(), though, stat_smooth() would not have inherited that mapping, resulting in a single smoother being calculated.  ggplot(I_subset, aes(Dur_msec, F1.n))+ geom_point(aes(color=Word))+ stat_smooth(se = F)  It's important to understand that when you map categorical variables to an aesthetic that you're also defining sub-groupings. For example, if we map Word to shape, instead of color, the point shapes will now represent the word. ggplot(I_subset, aes(Dur_msec, F1.n, shape=Word))+ geom_point()  Now if we add a smoother to this plot, even though shape isn't defined for lines, the smoother will still plot a different smoothing curve for each sub-grouping. ggplot(I_subset, aes(Dur_msec, F1.n, shape=Word))+ geom_point()+ stat_smooth(se = F)  If you really only wanted a single smoother line for all of the data in this case, one solution would be to move the shape=Word mapping from the data layer to the geom_point() layer. But in most cases, it's actually more desirable to override the aesthetic mapping. We can do this with the special aesthetic group. group does exactly what it sounds like it ought to: it defines groups of data. When you want to override groups defined in the data layer, you can do so by saying group=1. ggplot(I_subset, aes(Dur_msec, F1.n, shape=Word))+ geom_point()+ stat_smooth(se = F, aes(group = 1))  The effect it has on stat_smooth() is that just a single smoother is calculated. If we come back to color = Word, and then draw a line with group = 1, the effect is that we draw one line that varies in color. ggplot(I_subset, aes(Dur_msec, F1.n, color=Word))+ geom_line(aes(group = 1))  #### More aesthetics and their use. So far, we've only mapped categorical variables to color, but it's also possible to map continuous variables to color. Here we'll redundantly map F1.n to both y and color. ggplot(I_jean, aes(Dur_msec, F1.n, color = F1.n))+ geom_point()  Another important aesthetics distinction is between color and fill. If we wanted to create a bar chart of word frequencies, we could do so by mapping Word to the x-axis, and adding geom_bar() without any y-axis variable defined. ggplot(I_jean, aes(Word))+ geom_bar()  If you also wanted to color the bars according to the word, your first instinct would probably be to map color = Word. But the result is that only the colors of the bars' outlines are mapped to Word. ggplot(I_jean, aes(Word, color = Word))+ geom_bar()  What is probably more advisable is to map Word to fill, which control the filling color of two dimensional geoms. ggplot(I_jean, aes(Word, fill = Word))+ geom_bar()  As you might have figured out now, it's technically possible to map the fill color of bars to one variable, and the outline color to different variable. My advice is to never do such a thing, because the results almost always come out a jumbled mess. Instead, I would suggest setting the color of the bars to black. I find it more pleasing to the eye, and helps to emphasize the divisions between bars when they're stacked. Compare this plot: ggplot(I_subset, aes(Name, fill = Word))+ geom_bar()  to this one. ggplot(I_subset, aes(Name, fill = Word))+ geom_bar(color = "black")  ### Geometries So far, we've used the following geometries: • geom_point() • geom_line() • geom_bar() All geometries begin with geom_, meaning you can get a full list using apropos(). apropos("^geom_")  ## [1] "geom_abline" "geom_area" "geom_bar" ## [4] "geom_bin2d" "geom_blank" "geom_boxplot" ## [7] "geom_contour" "geom_crossbar" "geom_density" ## [10] "geom_density2d" "geom_dotplot" "geom_errorbar" ## [13] "geom_errorbarh" "geom_freqpoly" "geom_hex" ## [16] "geom_histogram" "geom_hline" "geom_jitter" ## [19] "geom_line" "geom_linerange" "geom_map" ## [22] "geom_path" "geom_point" "geom_pointrange" ## [25] "geom_polygon" "geom_quantile" "geom_raster" ## [28] "geom_rect" "geom_ribbon" "geom_rug" ## [31] "geom_segment" "geom_smooth" "geom_step" ## [34] "geom_text" "geom_tile" "geom_violin" ## [37] "geom_vline"  This is a quite extensive list, and we won't be able to cover them all today. Many of them are actually convenience functions for special settings of other geoms. For example, geom_histogram() is really just geom_bar() with special settings. ggplot(I_jean, aes(F1.n))+ geom_histogram()  Other geoms are just convenience functions for statistical layers. For example, you'll notice geom_smooth(), which if you add it to a plot will have the same behavior of stat_smooth(), which we've already been using extensively. ggplot(I_jean, aes(Dur_msec, F1.n))+ geom_smooth()  #### Some special geoms Some geoms are both unique and common enough in their usage to warrant special mention. ##### geom_line() vs geom_path() As I said above, when you add geom_line() to a plot, it connects points up according to their order along the x-axis. If you want to connect points according to their order in a data frame (say, to illustrate a trajectory through two-dimensional space over time), you should use geom_path(). mod_F1 <- loess(F1.n ~ Dur_msec, data = I_jean) mod_F2 <- loess(F2.n ~ Dur_msec, data = I_jean) pred <- data.frame(Dur_msec = seq(50, 400, length = 100)) pred$F1.n <- predict(mod_F1, newdata = pred)
pred\$F2.n <- predict(mod_F2, newdata = pred)

ggplot(pred, aes(-F2.n, -F1.n, color = Dur_msec))+
geom_path()+
geom_point()


text:
## Consistency of the total least squares estimator in the linear errors-in-variables regression. (arXiv:1810.09667v1 [math.PR])

This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.


text:
# Stopping Distance

## Stopping Distance

When an object is moving with a certain velocity then suddenly brakes are applied, and then object stops completely after covering a certain distance. This is called as stopping distance. It is the distance travelled between the time when an object deiced to stop a moving vehicle and the time when the vehicle stops completely. It depends on the factors including road surface and reflexes of the car’s driver and the stopping distance is denoted by d.

$$Stopping\text{ }Distance\left( d \right)=\frac{{{V}^{2}}}{2\mu g}$$ (Or) $$Stopping\text{ }Distance\left( d \right)=k{{V}^{2}}$$

Where,

V = Velocity $$\left( m/\sec \right)$$

$$\mu =$$ Co-efficient of friction

$$g=$$Acceleration due to gravity$$\left( m/{{\sec }^{2}} \right)$$

$$k=$$ Constant of Proportionality

How to calculate the Stopping Distance:

Problem: A vehicle moves with a velocity $$20m/\sec$$ and applies a brake. Calculate the stopping distance if constant of proportionality is 0.5.

Solution:

Given,

Velocity$$\left( V \right)=20m/\sec$$

Constant of proportionality$$\left( k \right)=0.5$$

Stopping distance (d) =?

We know that:

Stopping distance (d)$$=k{{V}^{2}}$$

$$\Rightarrow Stopping\text{ }distance(d)=0.5\times {{\left( 20m/\sec \right)}^{2}}=200m.$$


text:
Jump to content

# let's play figure out the radix!!

## Recommended Posts

Find the value of x if (23)x = 1111000102

I've converted it into decimal, which is 482.

But that's where I get stuck.

Any ideas?

##### Share on other sites

You need to break down 23 or any number in a place-holder system. The number 576 in decimal is interpreted without thinking - but what does it represent?

$576_{10} = (5*10^2) + (7*10^1) + (6 *10^0)$

in general

$576_x = (5*x^2) + (7*x^1) + (6 *x^0)$

You can then just rearrange your question into a nice polynomial which you solve for x. That said I must be missing something cos I cannot get yours to work - maybe I am just having a brain fade.

##### Share on other sites

Imfataal, the reason you're having trouble is:

Assuming his radix is an integer.

$23_x = 2x + 3 = 2(x+1) + 1 = 2n+1$ Is odd.

$111100010_2$ is even.

Either his radix is not an integer (rational solution to $2x+3 = 482$) or there's a transcription error somewhere.

Edited by Schrödinger's hat

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text:
## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 9 - Section 9.3 - The Parabola - Exercise Set - Page 997: 27

#### Answer

$x=\dfrac{1}{8}(y-2)^2+1$

#### Work Step by Step

General form of a horizontal parabola is given as: $(y-k)^2=4p(x-h)$...(1) Here, $\text{Vertex}=(h,k)$ and focus is: $(h+p, k)$ General form of a vertical parabola is given as: $(x-h)^2=4p(y-k)$...(2) Here, $\text{Vertex}=(h,k)$ and focus is: $(h, k+p)$ As we are given focus: $(3,2)$ Thus, from both forms of a parabola , we have two equations$h-p=-1$ and $h+p=3$ Add these two equations, we get: $h=1$ Thus, $p=3-h=3-1=2$ From equation (1), we have $(y-2)^2=4(2)(x-1)$ or, $(y-2)^2=8(x-1)$ or, $x-1=\dfrac{(y-2)^2}{8}$ or, $x=\dfrac{1}{8}(y-2)^2+1$

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


text:
# FindFit algorithm?

Does anyone know how to program a mathematica algorithm that does the same thing that FindFit does? Is there documentation of it somewhere? I assume it is a least squares algorithm but with very generalized arguments. I ask because I'm trying to use 'weighted' least squares and it basically involves just one more factor.

• NonlinearModelFit[] can handle weighted nonlinear least squares. – J. M. will be back soon Oct 31 '15 at 14:04

Fit works using singular value decomposition. FindFit uses the same method for the linear least-squares case, the Levenberg–Marquardt method for nonlinear least-squares, and general FindMinimum methods for other norms.

- source

NonlinearModelFit allows fitting of weighted data, as J.M. commented

Edit:

The best fit parameters are a property of the model:

p = Table[Prime[x], {x, 20}];

nlm = NonlinearModelFit[p, a x Log[b + c x], {a, b, c}, x];
nlm["BestFitParameters"]

{a -> 1.42076, b -> 1.65558, c -> 0.534645}

• How do I get the list parameters from NonlinearModel? FindFit outputs such a thing – minusatwelfth Oct 31 '15 at 16:56
• @minusatwelfth see my edit. – paw Oct 31 '15 at 17:12
• @minus, it's all in the docs, if you'd look for them. – J. M. will be back soon Nov 1 '15 at 2:04


text:
# Sum of the given series will be

Calculus Level 3

$\large \frac 7{2^3} + \frac {19}{6^3} + \frac {37}{12^3} + \frac {61}{20^3} + \cdots = \ ?$

Note that

• In the sequence $7,19,37,61,\ldots$, the difference between each consecutive terms follows an arithmetic progression.

• In the sequence $2,6,12,20, \ldots$,the difference between each consecutive terms follows an arithmetic progression as well.

×


text:
• Create Account

### #ActualShippou

Posted 29 June 2013 - 07:51 AM

If your dealing with Windows API, wouldn't it be a lot easier to use one of the "Visual Studio" suites ?

msgbox("Can't we all just get along?")

### #1Shippou

Posted 29 June 2013 - 07:44 AM

msgbox("Can't we all just get along ?")

PARTNERS


text:
# no of moles in 10g of calcium metal in finland

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