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http://www.ams.org/mathscinet-getitem?mr=0328611
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MathSciNet bibliographic data MR328611 (48 #6953) 46L05 Lazar, A. J.; Taylor, D. C. Double centralizers of Pedersen's ideal of a \$C\sp{\ast} \$$C\sp{\ast}$-algebra. II. Bull. Amer. Math. Soc. 79 (1973), 361–366. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2016-02-09 13:08:20
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https://quant.stackexchange.com/questions/27546/bonds-with-embedded-options-pricing-via-binomial-model
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Bonds with embedded options pricing via binomial model
Notation:
t - time; G(t) - zero-coupon yield curve; $r$, $r_d$, $r_u$ - interest rates.
The task is to find market price of a bond for today, while knowing the price of a number of other bonds.
Nelson-sigel model provides interest rate curve G(t). The next thing to do is to calibrate interest rate binomial tree using G(t). Also are known bond cash flow, par value and today market price.
The problem is how to do it properly. The start yield is given by G($t_1$), where $t_1$=0.25 (step for binomial model).
$100=\frac{1}{2}(\frac{100*G(t_2 )}{1+G(t_1)} +\frac{100*G(t_2 )+100}{(1+G(t_1))*(1+r_u )} + \frac{100*G(t_2 )}{1+G(t_1)} +\frac{100*G(t_2 )+100}{(1+G(t_1))*(1+r_d )})$
$r_u= r_d*exp(2*σ)$
$t_2 = 2*t_1$
These conditions provide $r_d$, $r_u$ for the second step of i.r. tree.
Is the next set of conditions for $r_{dd},r_{du},r_{uu}$ correct?
$1= \frac{1}{4} (\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_d)}+\frac{(G(t_3)+1)}{(1+G(t_1))(1+r_d)(1+r_{dd})}) + \frac{1}{4}(\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_d)}+\frac{(G(t_3)+1)}{(1+G(t_1))(1+r_d)(1+r_{ud})}) + \frac{1}{4}(\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_u)}+\frac{(G(t_3)+1)}{(1+G(t_1))(1+r_u)(1+r_{ud})}) + \frac{1}{4}(\frac{G(t_3)}{1+G(t_1)}+\frac{G(t_3)}{(1+G(t_1))(1+r_u)}+\frac{(G(t_3)+1)}{(1+G(t_1))(1+r_u)(1+r_{uu})})$
$r_{du}= exp(2*σ) r_{dd}$
$r_{uu}= exp(2*σ) r_{ud}=exp(4*σ) r_{dd}$
$t_3=3*t_1$
Is there an easy way to define $r_{uuu}$, $r_{uuuu}$ and other interest rates (yields), so it would be possible to do it programmatically?
What are the ways to correctly determine σ?
I'm heavily relying on ideas used in this presentation:
http://faculty.cbpa.drake.edu/root/Auvergne/DESS%20Analyst%20Binomial.ppt
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2019-05-22 03:17:08
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https://elki-project.github.io/releases/current/javadoc/de/lmu/ifi/dbs/elki/data/synthetic/bymodel/class-use/GeneratorInterfaceDynamic.html
|
## Uses of Interfacede.lmu.ifi.dbs.elki.data.synthetic.bymodel.GeneratorInterfaceDynamic
• Packages that use GeneratorInterfaceDynamic
Package Description
de.lmu.ifi.dbs.elki.data.synthetic.bymodel
Generator using a distribution model specified in an XML configuration file GeneratorXMLSpec is a standalone application that loads an XML specification file and generates a synthetic data set according to the specifications given.
• ### Uses of GeneratorInterfaceDynamic in de.lmu.ifi.dbs.elki.data.synthetic.bymodel
Classes in de.lmu.ifi.dbs.elki.data.synthetic.bymodel that implement GeneratorInterfaceDynamic
Modifier and Type Class and Description
class GeneratorSingleCluster
Class to generate a single cluster according to a model as well as getting the density of a given model at that point (to evaluate generated points according to the same model)
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2020-08-11 19:53:25
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https://learn.careers360.com/ncert/question-solve-the-following-equations-a-2-y-plus-5-by-2-equals-37-by-2/
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1. Solve the following equations: (b) 5t + 28 = 10(c) a /5 + 3 = 2(d) q/ 4 + 7 = 5 e) 5 x/2 = –5(f) (g) 7m + 19/2 = 13 (h) 6z + 10 = –2(i)(j)
Transposing to the RHS :
(b) 5t + 28 = 10
Transposing 28 to the RHS and then dividing both sides by 5, we get :
(c) a /5 + 3 = 2
Transposing 3 to the RHS and multiplying both sides by 5, we get :
(d) q/ 4 + 7 = 5
Transposing 7 to the RHS and multiplying both sides by 4:
(e) 5 x/2 = – 5
Multiplying both sides by :
(f)
Multiplying both sides by :
(g) 7m + 19/2 = 13
Transposing to the RHS and then dividing both sides by 7 :
(h) 6z + 10 = –2
Transposing 10 to the RHS and then dividing both sides by 6, we get :
(i)
Multiplying both sides by ,
(j)
Transposing 5 to the RHS and then multiplying both sides by
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2021-01-16 05:59:32
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https://blogs.warwick.ac.uk/midgleyc/daily/091111/
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# All entries for Wednesday 09 November 2011
## November 09, 2011
### Emptiness
Consider the definition of a graph – does this preclude the idea of a graph with 0 nodes? No, as both the set of vertices and the set of edges may be empty. This graph exists mostly to make you consider an extra case for every problem you do :)
How many automorphisms are there on the empty graph (how many isomorphisms from the empty graph to the empty graph)? This is equivalent to asking how many bijections there are from the empty set to itself. For a function to be well-defined, we only require that every element in the domain is mapped somewhere in the codomain – if the domain is the empty set, this condition is trivially true (if vacuous). It turns out the answer is one – the empty function.
I wondered about this a while in first year – how would you go about drawing a function if the first set in question was empty? – and I’m glad to see it’s actually something that’s been considered and given an answer.
Edit on 10/11 in reply to Nick:
Well, the definition of a matrix doesn’t preclude the existence of a 0×0 matrix, and I’d suppose that different matrices differ in at least one element, so I agree that that’s the only one.
The question on determinant is much more interesting due to how many different ways there are to approach it. Primarily, I want to check it doesn’t run opposite to my intuition in other areas.
Going back to the definition, we get that $\det(A)=\sum_{\phi \in S_n} sign(\phi) \alpha_{1 \phi (1)}\alpha_{2 \phi (2)} \cdots \alpha_{n \phi (n)}$. $S_0$ would be the symmetric group on 0, having 0! = 1 elements (the empty function). So the determinant would be the sum of one number that was the product of no numbers. I’m of the opinion that the sum of no numbers is 1 (the multiplicative identity), so that would make the determinant one in this case.
Considered geometrically, the determinant represent area or volume in two or three dimensions. Extending backwards, we can get that the determinant of a 1×1 matrix acting on a line gives the length of the line under the transformation, and hence that a 0×0 matrix should act on a point. However, as the point is the entire space, this tells us nothing of what the determinant should be.
Considering that every matrix represents a linear transformation, and also that the empty function is bijective, we obtain that ( ) is nonsingular and hence the determinant isn’t 0, which is fine.
Going back to the definition I learnt in high school, where $\begin{vmatrix} a&b\\c&d \end{vmatrix}=ad-bc$ and determinants of matrices of dimension greater than two are defined using minors, cofactors and the above definition, let us try going backwards – can we see what the determinants of 1×1 and 0×0 matrices /should/ be based only on this? Expanding the 2×2 matrix above by the first row, we find that the determinant of (d) should be d and that the determinant of (c) should be c, which fits nicely with the actually definition in addition to the geometric one above. Expanding the 1×1 matrix by the first row, we require the determinant of any 0×0 matrix to be 1 (so that the determinant of the whole thing can be the sole entry), which also fits with our definition above.
But now the geometric line of thinking has lead me into vector spaces! To start, there is no vector space consisting of no vectors (fails presence of an identity). Consider the vector space consisting only of the zero vector. It is a vector space, but we cannot find a basis for it – the zero vector is not linearly independent. However, the vector space is at most of dimension 1 (it has one vector in it), but its basis of length one is linearly depedent, so removing the linearly dependent vectors, we obtain a basis of zero vectors. So this way (by a rather weak argument), the vector space consisting only of the zero vector has dimension zero, as we’d expect.
## November 2011
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## Galleries
• Nice proof! Does this mean you're going to specialize in analysis and differential equations next ye… by Nick on this entry
• Hi Chris, It was most interesting to read your various reflections – thank you for sharing them. I'm… by Ceri Marriott on this entry
• Feel free. Chris by Christopher Midgley on this entry
• Hi Chris This is an honest final entry for the WSPA. Im glad that you have found the WSPA journey wo… by Samena Rashid on this entry
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2021-09-20 00:18:03
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https://brilliant.org/problems/differential-equations-basics/
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Differential equations-Basics
Calculus Level pending
A body's motion is given by the formula: $$x=t^2+6$$, where $$x$$ is the distance covered by the body.
Find the value of $$\dfrac{dx}{dt}$$ at $$t=2$$.
×
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2016-10-25 03:20:21
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https://socratic.org/questions/how-do-you-simplify-and-find-the-restrictions-for-x-2-3x-18-x-2-36#506734
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# How do you simplify and find the restrictions for (x^2+3x-18)/(x^2-36)?
Nov 16, 2017
The restricted values of $x$ are: ${x}_{1} = 6$ and ${x}_{2} = - 6$.
#### Explanation:
There is no simplification possible.
To find the restrictions you have to see for what values of $x$ there's no solution. That would be when the denominatior is $0$.
So you get,
${x}^{2} - 36 = 0$
you isolate $x$,
${x}^{2} = 36$
and you do the square root in both sides,
$x = \pm \sqrt{36} = \pm 6$
so these are the restricted values on x:
${x}_{1} = 6$
${x}_{2} = - 6$
Nov 16, 2017
$f \left(x\right) =$$\frac{{x}^{2} + 3 x - 18}{{x}^{2} - 36} = \frac{\left(x - 3\right) \left(x + 6\right)}{\left(x - 6\right) \left(x + 6\right)} = \frac{x - 3}{x - 6}$
( $f \left(x\right) = 0$$\iff$$x = 3$
• x≠6 and x≠-6
$D f = \left(- \infty , - 6\right) U \left(- 6 , 6\right) U \left(6 , + \infty\right)$
$D f = R - \left\{- 6 , 6\right\}$ )
Nov 16, 2017
$\frac{{x}^{2} + 3 x - 18}{{x}^{2} - 36}$ simplifies to $\frac{x - 3}{x - 6}$ with the restriction that $x \ne 6$ and $x \ne + 6$
#### Explanation:
$\frac{{x}^{2} + 3 x - 18}{{x}^{2} - 36}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{\left(x + 6\right) \left(x - 3\right)}{\left(x + 6\right) \left(x - 6\right)}$
$\textcolor{w h i t e}{\text{XXX}}$Note the division is only defined if $x \ne \pm 6$
$\textcolor{w h i t e}{\text{XXX}} = \frac{x - 3}{x - 6}$ provided $\left(x + 6\right) \ne 0$
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2022-07-02 14:57:09
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http://abstractspoon.com/wiki/doku.php?id=user-defined-tools
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### Sidebar
• Discussion Topics
• Online Resources
• Other Resources
user-defined-tools
# User-Defined Tools
To extend its functionality ToDoList lets you configure tools to perform action and operations that are outside the scope of ToDoList as a generalised task manager.
A User-Defined Tool (UDT) is a template which specifies a tool (application, script or batch file) together with additional command-line parameters and information on how to display the tool in ToDoList's user interface.
### What can be done with a UDT?
• Reports can be generated, emailed, or faxed.
• Data can be passed to other project management systems.
• ERP systems can incorporate the data into schedules.
• Data can be sent via web service to remote systems.
• Task values can be copied from one attribute to another within ToDoList.
• Task values can be modified within ToDoList.
### Creating a UDT
Note: For detailed information on configuring a UDT see (Menu Bar > Tools Menu > Preferences > User Defined Tools.
The goal is to create a command template that can be populated with data at run-time, and then executed from the DOS prompt. It would look something like this:
c:\path\to\program\name.exe -switch1 -switch2 "data1" -switch3 "data2"
The switches and data are whatever is required for the 'name.exe' program to work. These are the tool's 'Arguments'.
Arguments
A set switches and/or data that will be passed to the tool via its command-line. The down-arrow button attached to this field will display a list of “placeholder” variables for data that ToDoList will substitute when you run the tool. For example an argument might be set as follows:
/taskid=$(selTID) The$(selTID) is inserted by ToDoList after clicking the down-arrow button to show a list of available data. In that list is “Selected Task ID”. When clicked, the placeholder “$(selTID)” is inserted into the arguments field. So with the above argument set, at runtime, the value of the current selected task will be substituted, maybe #217, and the final command will look something like this. c:\path\my_script.bat /taskid=217 Whatever happens in that command is outside the scope of ToDoList and this documentation. It's up to you to find or create scripts or programs which do things that you want, and then to get ToDoList to provide those programs with required data to provide the results you seek. User Placeholders Placeholders prefixed by 'user' will result in the user (you) being prompted to enter information when the UDT is executed. This is useful where the information is not known in advance or it frequently changes. User placeholders typically take 3 additional arguments: • a unique variable name (eg. var_text1) • a prompt string (eg. “Enter your username”) This should be in quotes it is not required. • an optional default string to display (eg. “anonymous”) This should be in quotes it is not required. Example of a User Placeholder: • -clientname$(usertext, vt1, “Name of client”,“Unknown”)
• The user selects the UDT from the menu or toolbar. A small prompt window is shown with the title set to the name of the UDT. A textbox is shown with the label “Name of client”, and a default value of “Unknown”.
• The variable name is completely unimportant. It just needs to be unique of the names used in your UDTs. In this case vt1 is used for variable text, and it's assumed that other fields might get names like vt2, vt3, etc.
• Note that the placeholder name and the variable name are Not in quotes, but the text data is.
• The command line will get _-clientname Some name_ with whatever name was entered, or _-clientname Unknown_ if no name is entered.
There are no quotes around the user data for the command-line. That could cause a problem with the program that processes the data. The next example adds quotes:
• -clientname $(usertext, vt1, “Name of client”,”Unknown“) • Note the quote before the dollar sign and at the end. This results in the following: • -clientname Unknown ### Tool Examples The following (rough) examples illustrate possible uses of the tools system and one possible way of solving each challenge. • Challenge 1 : Display the raw active tasklist in your browser 1. Set the tool path to point to your default browser .exe file. 2. Enter$(pathname) in the arguments field.
• Explanation: All this does is call your browser, passing it the full pathname to the active tasklist.
• Challenge 2 : Render the active tasklist to HTML using an XSL transform
1. Create a batch file containing %1 %2 %3 -o %4 on line 1 and %4 on line 2, and set the tool path to point to this file.
2. Use the following Argument. This is all one line! :
"$(userfile, var_msxsl, "Path to Msxsl.exe")" "$(pathname)" "$(userfile, var_xslfile, "Path to Xsl file")" "$(folder) \$(filetitle).html"
Note that there are two *userfile* placeholders, and that fields are surrounded by quotes as described above.
• Explanation:
• The first argument prompts the user to browse to msxsl.exe, which is the rendering engine we will be using.
• The second argument is just the full pathname of the active tasklist which we are going to render.
• The third parameter prompts the user for the XSL file with which to carry out the transform.
• The last argument is the output file (just the tasklist pathname with a .html extension).
• The second line in the batch file acts to display the resulting html file in your default browser.
• Challenge 3 : Connect to Bugzilla via the External ID task field
• Work in Progress. Details about his example can be found …. ??
• Explanation: This allows you to open Bugzilla with the task's External ID as bug number.
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2019-06-20 17:09:48
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http://www.mathematics2.com/Calculus/ImplicitDifferentiation
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Subject: Calculus
# Implicit Differentiation
In the previous examples, the functions presented are all explicit. However, when a function can not be defined explicitly, then we have to develop some technique in getting the derivative of that function. Example, a function f(x)=y=3x+\cos{x} is an explicit function. Notice that the right-hand side of the function contains only the variable x. So, we can easily differentiate such function using direct differentiation. But when a function look like these, x^{2}y+x\cos{xy}=5, then we really had a hard time converting these into an explicit function. Thus, we can't directly differentiate the function. Thus, we need to know about implicit differentiation.
Basically, implicit differentiation is just a process of differentiating an implicit function. The theorems and rules discussed above are still applicable in this process.
## So how do we differentiate an implicit function?
Look at the following illustration.
### Illustration
Consider the equation (x+y)^{2}-(x-y)^{2}=x^{4}+y^{4} +3. Isn't it hard to express that equation in terms of the variable x or y alone? Thus, we will do implicit differentiation in finding the derivative of that equation. What we will do here is to differentiate term by term to get
2(x+y)(1+y')-2(x-y)(1-y') = 4x^{3}+4y^{3}y'+0 2x+2y+(2x+2y)y'-2x+2y+(2x-2y)y' = 4x^{3}+4y^{3}y' (4x-4y^{3})y' = 4x^{3}+4y^{3}y'.
We then solve for y' which is given by
y'=\frac{4x^{3}-4y}{4x-4y^{3}}=\frac{x^{3}-y}{x-y^{3}}
Note that expressing the derivative in this form makes it easy to transform to other derivative notations such as \frac{dy}{dx}.
### Example #1
Let y be a differentiable function of the variable x, then let us find its derivative.
x^2y=x+y
Doing the same thing above, we get
2xy + x^2y'=1+y'
Solving for y',
y'=\frac{2xy-1}{1-x^2}
### Example #2
Let y be a differentiable function of the variable x, then let us find its derivative.
\sin{xy}=1
Doing the same thing above, we get
\cos{xy}\cdot D[xy]=0
\cos{xy}\cdot [y+xy']=0
Solving for y',
y'=-\frac{y}{x}
### Example #3
Let y be a differentiable function of the variable x, then let us find its derivative.
5x^{3}y-7xy^{2}=9+7y
Doing the same thing above, we get
5(x^{3}y'+y \cdot 3x^{2})-7[x(2yy')+y^{2}] = 0+7y' 5x^{3}y'+15x^{2}y-14xyy'-7y^{2} = 7y
We simplify the above equation and get
(5x^{3}-14xy-7)y'=7y^{2}-15x^{2}y.
Solving for y',
y'=\frac{7y^{2}-15x^{2}y}{5x^{3}-14xy-7}
### Example #4
Let y be a differentiable function of the variable x, then let us find its derivative.
y=\tan(x+y)
Using our theorems on differentiation of trigonometric functions, we have
y' = \sec^{2}(x+y)D_{x}(x+y) = (1+y')\sec^{2}(x+y) = \sec^{2}(x+y)+y'\sec^{2}(x+y)
Rearranging, we get
(1-\sec{2}(x+y))y'=\sec^{2}(x+y)
Therefore,
y'=\frac{\sec^{2}(x+y)}{1-\sec{2}(x+y)}
### Example #5
Let y be a differentiable function of the variable x, then let us find its derivative.
\cot{xy}=-xy
Again, we differentiate term by term and get
(-\csc^{2}{xy})D_{x}(xy)+(xy'+y) = 0 (-\csc^{2}{xy})(xy'+y)+(xy'+y) = 0 -xy'\csc^{2}{xy}-y\csc^{2}{xy}+xy'+y = 0 y'(x)(\csc^{2}{xy}-1) = y(1-\csc^{2}{xy}) y' = \frac{y}{x}\cdot \frac{1-\csc^{2}{xy}}{(\csc^{2}{xy}-1)} y' = -\frac{y}{x}\cdot \frac{\csc^{2}{xy}-1}{(\csc^{2}{xy}-1)}=-\frac{x}{y}.
Therefore,
y'= -\frac{x}{y}.
NEXT TOPIC: Higher Order Derivatives
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2021-04-21 20:24:48
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https://zbmath.org/?q=an:0631.73052
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# zbMATH — the first resource for mathematics
Vibrations of an elastic isotropic sphere of incompressible material under uniform initial hydrostatic loading. (English. Russian original) Zbl 0631.73052
Sov. Phys., Dokl. 30, 535-537 (1985); translation from Dokl. Akad. Nauk SSSR 282, 1077-1081 (1985).
In the present article, within the framework of the linearized theory of elasticity, we consider the vibrations of an isotropic elastic sphere of incompressible material with an arbitrary structure of the elastic potential under uniform initial hydrostatic loading. Following the author: Stability of elastic bodies with uniform compression (1979; Zbl 0429.73039), we carry out the investigation in a unified general form for the theory of finite (large) initial strains and two versions of the theory of small initial strains as they apply to the cases in which the initial loading is realized by a “follower” or a “deadweight” load; for the theory of small initial strains, an improved expression is used to determine the “follower” load. The investigation is carried out in Lagrangian coordinates (r,$$\theta$$,$$\phi)$$, which coincide with the spherical coordinates in the natural or initial stress-strain state; by virtue of the incompressibility condition for the initial state in question, the introduction of the indicated Lagrangian coordinates coincides in the natural and initial stress-strain states.
##### MSC:
74H45 Vibrations in dynamical problems in solid mechanics 74B20 Nonlinear elasticity
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2021-01-17 19:30:37
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https://www.cuemath.com/similar-triangles/sss-criterion-in-triangles/
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# SSS Criterion in Triangles
Go back to 'Similar-Triangles'
We saw that if two triangles are equi-angular, then they are similar. Can we say that if the sides of two triangles are (respectively) proportional, they will be similar? Yes, we can.
SSS Criterion: If the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar.
This essentially means that any such pair of triangles will be equi-angular. Consider the following figure, in which the sides of two triangles ($$\Delta ABC$$ and $$\Delta DEF$$) are respectively proportional:
That is, it is given that:
$\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}}$
Can we say that these two triangles will be equi-angular? The SSS criterion says that we can. Indeed, if we measure the angles in each triangle, we will find this to be true, as the following figure shows:
Let us now discuss the proof of the SSS criterion.
Proof: Suppose that AB > DE. Then AC will also be greater than DF (why). Now:
1. Take a point X on AB such that AX = DE.
2. Take a point Y on AC such that AY = DF.
Join XY:
It is given that
$\frac{{DE}}{{AB}} = \frac{{DF}}{{AC}}$
Thus,
$\frac{{AX}}{{AB}} = \frac{{AY}}{{AC}}$
Using the converse of the BPT, this implies that XY || BC. Thus, $$\Delta AXY$$ ~ $$\Delta ABC$$. This further means that:
$\frac{{AX}}{{AB}} = \frac{{AY}}{{AC}} = \frac{{XY}}{{BC}}$
But we also have:
$\frac{{EF}}{{BC}} = \frac{{AY}}{{AC}} = \frac{{AX}}{{AB}}$
Thus,
\begin{align}& \frac{{XY}}{{BC}} = \frac{{EF}}{{BC}}\\ &\Rightarrow XY = EY \end{align}
Now, by the SSS criterion, $$\Delta DEF$$ ≡ $$\Delta AXY$$, while we already have $$\Delta AXY$$ ~ $$\Delta ABC$$. Thus, $$\Delta DEF$$ is similar to $$\Delta ABC$$. This concludes our proof.
Learn math from the experts and clarify doubts instantly
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• Completely personalized curriculum
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2019-09-15 09:39:15
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https://www.physicsforums.com/threads/linear-algebra-system-of-equations.552716/
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# Linear Algebra System of Equations
1. Nov 21, 2011
### MDTeen
Hello Everyone,
I am new to the forum, but I am kind of at a loss and I could use a little guidance. I currently have an assignment where we are given a traffic circle, with 4 on ramps and 4 off ramps (labeled a through h) which are constants. We are then given the traffic flow in between each of the on and off ramps as variables x1 through x8. None of the constants or variables are given an actual value (we are supposed to treat these as real numbers).
We are supposed to analyze the traffic circle, create a system of equations in order to solve for the values of x1 through x8 (the traffic flow) and analyze the constraints of the constants (a through h - on/off ramps) and the values of x1 through x8.
I have a diagram of the traffic circle, with the on/off ramps and sections of the circle labeled with the appropriate variables and constants (including work I have done so far) in the attached PDF (sorry for the horrible handwriting).
Normally when given a problem like this, we are least given some data to work with, in order to find the values of the variables, but since we are given no data (just that the constants are a through h and that they mark the different on / off ramps) i'm just a little confused as to how to solve this. I started by creating a system of equations regarding the constants and the variables we were given, then tried to solve the system of equations in order to get values for x1 through x8, now I am just stuck.
I am unsure if I even created the correct system of equations for this problem, and if I did, did I solve the system correctly?
I am not asking for someone to give me the direct answer to the problem, but any assistance in helping me with this problem could really help.
Please give me your opinions on what I have done so far, and if I did anything wrong, please let me know what.
Thank You
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
#### Attached Files:
• ###### scan0002_merged.pdf
File size:
1.2 MB
Views:
113
2. Nov 21, 2011
### flyingpig
These problems are similar to Kirchoff's circuit problems. Flow in = Flow out
I quickly glanced through your constraints, they look good. I'll also assume you are correct in your final answer.
All you have to do is sub in x_1, x_3, x_5, x_7 into the other variables to get your final answer
3. Nov 21, 2011
### MDTeen
"These problems are similar to Kirchoff's circuit problems. Flow in = Flow out
I quickly glanced through your constraints, they look good. I'll also assume you are correct in your final answer.
All you have to do is sub in x_1, x_3, x_5, x_7 into the other variables to get your final answer"
I reworked the problem a little and substituted x1, x3, x5, and x7 into the remaining variables of x2, x4, x6, x8 as shown in the attached PDF.
Is this what you were meaning?
#### Attached Files:
• ###### scan0001.pdf
File size:
371.8 KB
Views:
74
4. Nov 21, 2011
### flyingpig
I am not following, how did you get RSTU?
5. Nov 21, 2011
### MDTeen
I was about to give the variables a letter to represent it but then I realized it really was not needed in this case and finished the work at the bottom of the page without without erasing my previous notes for RTSU.
Please see my work at the bottom of the page and let me know what you think (sorry for the mishap).
6. Nov 21, 2011
### flyingpig
I still not sure what the real problem is. In the beginning you already have written the constraints to solve for x_8 and the others. You solved what x_1, x_7 etc... are from your row operations. Just put those values back into the constraints you had in the very beginning
7. Nov 21, 2011
### MDTeen
Sorry it might be because i'm tired, but I didn't even know I created the constraints... Which items I wrote were the constraints?
As for the answers I got:
x8 = x1 + d
x2 = x3 + b
x4 = x5 + h
x6 = x7 + f
Where are these answers supposed to be inserted into? Or was I completely off and that latest page I uploaded was wrong and I already had the answer?
I know I must be confusing right now, maybe I should get back online tomorrow after I had some rest and look this over again and see if I can make sense of any of this..
8. Nov 22, 2011
### flyingpig
You have
$$x_1 = (g - f) + (a-h) + (c-b) + (e - d)^2$$
And
$$x_8 = x_1 + d$$
You know what x_1 is, what is the confusion?
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2018-03-18 14:21:37
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https://git.in.moodle.com/moodle/moodle/-/blame/master/filter/upgrade.txt
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Eloy Lafuente committed Nov 18, 2011 1 2 3 This file describes API changes in core filter API and plugins, information provided here is intended especially for developers. Sara Arjona committed Oct 04, 2021 4 5 6 7 8 === 4.0 === * The Word censorship (filter_censor) filter has been completely removed from core. It has been moved to the plugins database repository, so it can still be installed as a third-party plugin. Sara Arjona committed Dec 20, 2019 9 10 11 12 13 14 === 3.9 === * The following functions, previously used (exclusively) by upgrade steps are not available anymore because of the upgrade cleanup performed for this version. See MDL-65809 for more info: - filter_mathjaxloader_upgrade_cdn_cloudflare() - filter_mathjaxloader_upgrade_mathjaxconfig_equal() Tim Hunt committed Sep 20, 2018 15 16 17 18 19 20 21 22 23 24 25 26 === 3.6 === * Although there is no API change that require you to update your filter, if you use the filter_phrases() helper method, you may wish to take advantage of the changes that were made in MDL-47962 to improve performance. Now, instead of having to compute the replacement HTML for each phrase before you construct the filterobject for it. You can instead pass a callback to the filterobject constructor which is only called if the phrase is used. To understand how to use this, see the comment on filterobject::__construct and look at the filter_glossary changes as an example: David Monllaó committed Sep 24, 2018 27 https://github.com/moodle/moodle/commit/5a8c44d000ecc5669db26aefebe447f688e8f2ce Tim Hunt committed Sep 20, 2018 28 Tim Hunt committed Jun 17, 2015 29 30 === 3.0 === Tim Hunt committed Jun 17, 2015 31 32 * New argument $skipfilters to filter_manager::filter_text to allow applying the filters with a given one omitted. Tim Hunt committed Jun 17, 2015 33 34 35 36 * New admin setting class admin_setting_filter_types which can be used if you want to make the disablefilters value in your code configurable. Tim Hunt committed Jun 17, 2015 37 38 39 40 * Methods filter_manager::text_filtering_hash and moodle_text_filter::hash have been deprecated. There were use by the old Moodle filtered text caching system that was removed several releases ago. Petr Škoda committed Jan 14, 2014 41 42 43 44 45 46 === 2.7 === * Finally filter may use$PAGE and \$OUTPUT, yay! * Old global text caching was removed, each filter is now responsible for own caching. Petr Škoda committed Oct 12, 2013 47 48 49 50 === 2.6 === * filtersettings.php is now deprecated, migrate to standard settings.php Petr Škoda committed Dec 30, 2012 51 52 53 54 55 56 57 === 2.5 === * legacy_filter emulation was removed * support for 'mod/*' filters was removed * use short filter name instead of old path, ex.: 'filter/tex' ---> 'tex' in all filter API functions and methods Eloy Lafuente committed Jun 15, 2012 58 59 60 61 62 63 64 === 2.3 === * new setup() method added to moodle_text_filter, invoked before filtering happens, used to add all the requirements to the page (js, css...) and/or other init tasks. See filter/glossary for an example using the API (and MDL-32279 for its justification). Eloy Lafuente committed Nov 18, 2011 65 66 67 68 69 70 71 72 === 2.2 === * legacy filters and legacy locations have been deprecated, so any old filter should be updated to use the new moodle_text_filter, and any filter bundled under mod/xxxx directories be moved to /filter/xxxx (MDL-29995). They will stop working completely in Moodle 2.3 (MDL-29996). See the glossary or data filters for examples of legacy module filters and locations already updated.
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2022-08-09 08:51:18
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https://rdrr.io/cran/EnvStats/man/predIntNorm.html
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# predIntNorm: Prediction Interval for a Normal Distribution
### Description
Estimate the mean and standard deviation of a normal distribution, and construct a prediction interval for the next k observations or next set of k means.
### Usage
1 2 predIntNorm(x, n.mean = 1, k = 1, method = "Bonferroni", pi.type = "two-sided", conf.level = 0.95)
### Arguments
x a numeric vector of observations, or an object resulting from a call to an estimating function that assumes a normal (Gaussian) distribution (e.g., enorm, eqnorm, enormCensored, etc.). If x is a numeric vector, missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed. n.mean positive integer specifying the sample size associated with the k future averages. The default value is n.mean=1 (i.e., individual observations). Note that all future averages must be based on the same sample size. k positive integer specifying the number of future observations or averages the prediction interval should contain with confidence level conf.level. The default value is k=1. method character string specifying the method to use if the number of future observations (k) is greater than 1. The possible values are method="Bonferroni" (approximate method based on Bonferonni inequality; the default), and method="exact" (exact method due to Dunnett, 1955). See the DETAILS section of predIntNormK for more information. This argument is ignored if k=1. pi.type character string indicating what kind of prediction interval to compute. The possible values are pi.type="two-sided" (the default), pi.type="lower", and pi.type="upper". conf.level a scalar between 0 and 1 indicating the confidence level of the prediction interval. The default value is conf.level=0.95.
### Details
What is a Prediction Interval?
A prediction interval for some population is an interval on the real line constructed so that it will contain k future observations or averages from that population with some specified probability (1-α)100\%, where 0 < α < 1 and k is some pre-specified positive integer. The quantity (1-α)100\% is called the confidence coefficient or confidence level associated with the prediction interval.
The Form of a Prediction Interval
Let \underline{x} = x_1, x_2, …, x_n denote a vector of n observations from a normal distribution with parameters mean=μ and sd=σ. Also, let m denote the sample size associated with the k future averages (i.e., n.mean=m). When m=1, each average is really just a single observation, so in the rest of this help file the term “averages” will replace the phrase “observations or averages”.
For a normal distribution, the form of a two-sided (1-α)100\% prediction interval is:
[\bar{x} - Ks, \bar{x} + Ks] \;\;\;\;\;\; (1)
where \bar{x} denotes the sample mean:
\bar{x} = \frac{1}{n} ∑_{i=1}^n x_i \;\;\;\;\;\; (2)
s denotes the sample standard deviation:
s^2 = \frac{1}{n-1} ∑_{i=1}^n (x_i - \bar{x})^2 \;\;\;\;\;\; (3)
and K denotes a constant that depends on the sample size n, the confidence level, the number of future averages k, and the sample size associated with the future averages, m. Do not confuse the constant K (uppercase K) with the number of future averages k (lowercase k). The symbol K is used here to be consistent with the notation used for tolerance intervals (see tolIntNorm).
Similarly, the form of a one-sided lower prediction interval is:
[\bar{x} - Ks, ∞] \;\;\;\;\;\; (4)
and the form of a one-sided upper prediction interval is:
[-∞, \bar{x} + Ks] \;\;\;\;\;\; (5)
but K differs for one-sided versus two-sided prediction intervals. The derivation of the constant K is explained in the help file for predIntNormK.
A Prediction Interval is a Random Interval
A prediction interval is a random interval; that is, the lower and/or upper bounds are random variables computed based on sample statistics in the baseline sample. Prior to taking one specific baseline sample, the probability that the prediction interval will contain the next k averages is (1-α)100\%. Once a specific baseline sample is taken and the prediction interval based on that sample is computed, the probability that that prediction interval will contain the next k averages is not necessarily (1-α)100\%, but it should be close.
If an experiment is repeated N times, and for each experiment:
1. A sample is taken and a (1-α)100\% prediction interval for k=1 future observation is computed, and
2. One future observation is generated and compared to the prediction interval,
then the number of prediction intervals that actually contain the future observation generated in step 2 above is a binomial random variable with parameters size=N and prob=(1-α)100\%.
If, on the other hand, only one baseline sample is taken and only one prediction interval for k=1 future observation is computed, then the number of future observations out of a total of N future observations that will be contained in that one prediction interval is a binomial random variable with parameters size=N and prob=(1-α^*)100\%, where α^* depends on the true population parameters and the computed bounds of the prediction interval.
### Value
If x is a numeric vector, predIntNorm returns a list of class "estimate" containing the estimated parameters, the prediction interval, and other information. See the help file for
estimate.object for details.
If x is the result of calling an estimation function, predIntNorm returns a list whose class is the same as x. The list contains the same components as x, as well as a component called interval containing the prediction interval information. If x already has a component called interval, this component is replaced with the prediction interval information.
### Note
Prediction and tolerance intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Nelson, 1973; Krishnamoorthy and Mathew, 2009). In the context of environmental statistics, prediction intervals are useful for analyzing data from groundwater detection monitoring programs at hazardous and solid waste facilities (e.g., Gibbons et al., 2009; Millard and Neerchal, 2001; USEPA, 2009).
### Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
### References
Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton.
Dunnett, C.W. (1955). A Multiple Comparisons Procedure for Comparing Several Treatments with a Control. Journal of the American Statistical Association 50, 1096-1121.
Dunnett, C.W. (1964). New Tables for Multiple Comparisons with a Control. Biometrics 20, 482-491.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
Hahn, G.J. (1969). Factors for Calculating Two-Sided Prediction Intervals for Samples from a Normal Distribution. Journal of the American Statistical Association 64(327), 878-898.
Hahn, G.J. (1970a). Additional Factors for Calculating Prediction Intervals for Samples from a Normal Distribution. Journal of the American Statistical Association 65(332), 1668-1676.
Hahn, G.J. (1970b). Statistical Intervals for a Normal Population, Part I: Tables, Examples and Applications. Journal of Quality Technology 2(3), 115-125.
Hahn, G.J. (1970c). Statistical Intervals for a Normal Population, Part II: Formulas, Assumptions, Some Derivations. Journal of Quality Technology 2(4), 195-206.
Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York.
Hahn, G., and W. Nelson. (1973). A Survey of Prediction Intervals and Their Applications. Journal of Quality Technology 5, 178-188.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York.
Helsel, D.R., and R.M. Hirsch. (2002). Statistical Methods in Water Resources. Techniques of Water Resources Investigations, Book 4, chapter A3. U.S. Geological Survey. (available on-line at: http://pubs.usgs.gov/twri/twri4a3/).
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
Millard, S.P., and Neerchal, N.K. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, Florida.
Miller, R.G. (1981a). Simultaneous Statistical Inference. McGraw-Hill, New York.
USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.
USEPA. (2010). Errata Sheet - March 2009 Unified Guidance. EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.
predIntNormK, predIntNormSimultaneous, predIntLnorm, tolIntNorm, Normal,
estimate.object, enorm, eqnorm.
### Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 # Generate 20 observations from a normal distribution with parameters # mean=10 and sd=2, then create a two-sided 95% prediction interval for # the next observation. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(47) dat <- rnorm(20, mean = 10, sd = 2) predIntNorm(dat) #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Normal # #Estimated Parameter(s): mean = 9.792856 # sd = 1.821286 # #Estimation Method: mvue # #Data: dat # #Sample Size: 20 # #Prediction Interval Method: exact # #Prediction Interval Type: two-sided # #Confidence Level: 95% # #Number of Future Observations: 1 # #Prediction Interval: LPL = 5.886723 # UPL = 13.698988 #---------- # Using the same data from the last example, create a one-sided # upper 99% prediction limit for the next 3 averages of order 2 # (i.e., each of the 3 future averages is based on a sample size # of 2 future observations). predIntNorm(dat, n.mean = 2, k = 3, conf.level = 0.99, pi.type = "upper") #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Normal # #Estimated Parameter(s): mean = 9.792856 # sd = 1.821286 # #Estimation Method: mvue # #Data: dat # #Sample Size: 20 # #Prediction Interval Method: Bonferroni # #Prediction Interval Type: upper # #Confidence Level: 99% # #Number of Future Averages: 3 # #Sample Size for Averages: 2 # #Prediction Interval: LPL = -Inf # UPL = 13.90537 #---------- # Compare the result above that is based on the Bonferroni method # with the exact method predIntNorm(dat, n.mean = 2, k = 3, conf.level = 0.99, pi.type = "upper", method = "exact")$interval$limits["UPL"] # UPL #13.89272 #---------- # Clean up rm(dat) #-------------------------------------------------------------------- # Example 18-1 of USEPA (2009, p.18-9) shows how to construct a 95% # prediction interval for 4 future observations assuming a # normal distribution based on arsenic concentrations (ppb) in # groundwater at a solid waste landfill. There were 4 years of # quarterly monitoring, and years 1-3 are considered background. # The question to be answered is whether there is evidence of # contamination in year 4. # The data for this example is stored in EPA.09.Ex.18.1.arsenic.df. EPA.09.Ex.18.1.arsenic.df # Year Sampling.Period Arsenic.ppb #1 1 Background 12.6 #2 1 Background 30.8 #3 1 Background 52.0 #4 1 Background 28.1 #5 2 Background 33.3 #6 2 Background 44.0 #7 2 Background 3.0 #8 2 Background 12.8 #9 3 Background 58.1 #10 3 Background 12.6 #11 3 Background 17.6 #12 3 Background 25.3 #13 4 Compliance 48.0 #14 4 Compliance 30.3 #15 4 Compliance 42.5 #16 4 Compliance 15.0 As.bkgd <- with(EPA.09.Ex.18.1.arsenic.df, Arsenic.ppb[Sampling.Period == "Background"]) As.cmpl <- with(EPA.09.Ex.18.1.arsenic.df, Arsenic.ppb[Sampling.Period == "Compliance"]) # A Shapiro-Wilks goodness-of-fit test for normality indicates # there is no evidence to reject the assumption of normality # for the background data: gofTest(As.bkgd) #Results of Goodness-of-Fit Test #------------------------------- # #Test Method: Shapiro-Wilk GOF # #Hypothesized Distribution: Normal # #Estimated Parameter(s): mean = 27.51667 # sd = 17.10119 # #Estimation Method: mvue # #Data: As.bkgd # #Sample Size: 12 # #Test Statistic: W = 0.94695 # #Test Statistic Parameter: n = 12 # #P-value: 0.5929102 # #Alternative Hypothesis: True cdf does not equal the # Normal Distribution. # Here is the one-sided 95% upper prediction limit: UPL <- predIntNorm(As.bkgd, k = 4, pi.type = "upper")$interval$limits["UPL"] UPL # UPL #73.67237 # Are any of the compliance observations above the prediction limit? any(As.cmpl > UPL) #[1] FALSE #========== # Cleanup #-------- rm(As.bkgd, As.cmpl, UPL)
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2017-04-27 16:43:30
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https://www.johndcook.com/blog/2020/06/10/gibbs-phenomenon/
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# Gibbs phenomenon
I realized recently that I’ve written about generalized Gibbs phenomenon, but I haven’t written about its original context of Fourier series. This post will rectify that.
The image below comes from a previous post illustrating Gibbs phenomenon for a Chebyshev approximation to a step function.
Although Gibbs phenomena comes up in many different kinds of approximation, it was first observed in Fourier series, and not by Gibbs [1]. This post will concentrate on Fourier series, and will give an example to correct some wrong conclusions one might draw about Gibbs phenomenon from the most commonly given examples.
The uniform limit of continuous function is continuous, and so the Fourier series of a function cannot converge uniformly where the function is discontinuous. But what does the Fourier series do near a discontinuity?
It’s easier to say what the Fourier series does exactly at a discontinuity. If a function is piecewise continuous, then the Fourier series at a jump discontinuity converges to the average of the limits from the left and from the right at that point.
What the Fourier series does on either side of the discontinuity is more interesting. You can see high-frequency oscillations on either side. The series will overshoot on the high side of the jump and undershoot on the low side of the jump.
The amount of overshoot and undershoot is proportional to the size of the gap, about 9% of the gap. The exact proportion, in the limit, is given by the Wilbraham-Gibbs constant
Gibbs phenomenon is usually demonstrated with examples that have a single discontinuity at the end of their period, such as a square wave or a saw tooth wave. But Gibbs phenomenon occurs at every discontinuity, wherever located, no matter how many there are.
The following example illustrates everything we’ve talked about above. We start with the function f plotted below on [-π, π] and imagine it extended periodically.
1. It is continuous at the point where it repeats since it equals 0 at -π and π.
2. It has two discontinuities inside [-π, π].
3. One of the discontinuities is larger than the other.
The following plot shows the sum of the first 100 terms in the Fourier series for f plotted over [-2π, 2π].
1. There nothing remarkable about the series at -π and π.
2. You can see Gibbs phenomenon at the discontinuities of f.
3. The overshoot and undershoot are larger at the larger discontinuity.
Related to the first point above, note that the derivative of f is discontinuous at the period boundary. A discontinuity in the derivative does not cause Gibbs phenomena.
Here’s a close-up plot that shows the wiggling near the discontinuities.
## Gibbs phenomena for other series
[1] Henry Wilbraham first described what Josiah Gibbs discovered independently 50 years later, what we now call Gibbs phenomenon. This is an example of Stigler’s law of eponymy.
## One thought on “Gibbs phenomenon”
1. Jonathan
My favorite name for this is ‘ringing’, which I hear more often in signal processing and electronics contexts. Of course, unlike a bell, this ringing also happens before you strike it, not just after.
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2022-10-06 14:14:41
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https://proofwiki.org/wiki/Definition:Lexicographic_Order/Tuples_of_Equal_Length
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# Definition:Lexicographic Order/Tuples of Equal Length
## Definition
Let $n \in \N_{>0}$.
Let $\struct {S_1, \preccurlyeq_1}, \struct {S_2, \preccurlyeq_2}, \ldots, \struct {S_n, \preccurlyeq_n}$ be ordered sets.
Let $\ds S = \prod_{k \mathop = 1}^n S_k = S_1 \times S_2 \times \cdots \times S_n$ be the Cartesian product of $S_1$ to $S_n$.
The lexicographic order on $S$ is the relation $\preccurlyeq_l$ defined on $S$ as:
$\tuple {x_1, x_2, \ldots, x_n} \preccurlyeq_l \tuple {y_1, y_2, \ldots, y_n}$ if and only if:
$\exists k: 1 \le k \le n: \paren {\forall j: 1 \le j < k: x_j = y_j} \land \paren {x_k \prec_k y_k}$
or:
$\forall j: 1 \le j \le n: x_j = y_j$
That is, if and only if:
the elements of a pair of $n$-tuples are either all equal
or:
they are all equal up to a certain point, and on the next one they are comparable and they are different.
### Cartesian Space
The definition can be refined to apply to a Cartesian $n$-space:
Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $n \in \N_{>0}$.
Let $S^n$ be the cartesian $n$th power of $S$:
$S^n = \underbrace {S \times S \times \cdots \times S}_{\text {$n$times} }$
The lexicographic order on $S^n$ is the relation $\preccurlyeq_l$ defined on $S^n$ as:
$\tuple {x_1, x_2, \ldots, x_n} \preccurlyeq_l \tuple {y_1, y_2, \ldots, y_n}$ if and only if:
$\exists k: 1 \le k \le n: \paren {\forall j: 1 \le j < k: x_j = y_j} \land \paren {x_k \prec y_k}$
or:
$\forall j: 1 \le j \le n: x_j = y_j$
## Also known as
Lexicographic order can also be referred to as the more unwieldy lexicographical ordering.
Some sources refer to it as dictionary order.
Some sources classify the lexicographic order as a variety of order product.
Hence the term lexicographic product can occasionally be seen.
The mathematical world is crying out for a less unwieldy term to use.
Some sources suggest Lex, but this has yet to filter through to general usage.
## Also see
• Results about the lexicographic order can be found here.
## Linguistic Note
The term lexicographic order derives from the word lexicon, which is a linguistic term whose meaning is akin to dictionary.
The origin of its use in this context is apparent from the fact that the tuples are ordered in the way they would be if their terms are letters of the alphabet being ordered in standard alphabetical order.
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2021-05-17 10:22:14
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https://egtheory.wordpress.com/2013/10/20/enriching-egt/
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# Enriching evolutionary games with trust and trustworthiness
Fairly early in my course on Computational Psychology, I like to discuss Box’s (1979) famous aphorism about models: “All models are wrong, but some are useful.” Although Box was referring to statistical models, his comment on truth and utility applies equally well to computational models attempting to simulate complex empirical phenomena. I want my students to appreciate this disclaimer from the start because it avoids endless debate about whether a model is true. Once we agree to focus on utility, we can take a more relaxed and objective view of modeling, with appropriate humility in discussing our own models. Historical consideration of models, and theories as well, should provide a strong clue that replacement by better and more useful models (or theories) is inevitable, and indeed is a standard way for science to progress. In the rapid turnover of computational modeling, this means that the best one could hope for is to have the best (most useful) model for a while, before it is pushed aside or incorporated by a more comprehensive and often more abstract model. In his recent post on three types of mathematical models, Artem characterized such models as heuristic. It is worth adding that the most useful models are often those that best cover (simulate) the empirical phenomena of interest, bringing a model closer to what Artem called insilications.
Models in evolutionary game theory (EGT) have tended towards abstractness and simplicity, with a lot of attention directed towards a few key variables such as payoff matrices, cost to benefit ratios, network structures, mutation rates, update rules, etc. A new paper by McNamara (2013) argues that it is now time to consider enriching EGT models with features like psychological mechanisms, decision making, personality variations, and novel traits. McNamara’s inclusion of Box’s aphorism at the top of his paper caught my attention and raised the possibility that I could relate to this paper. As the paper progresses, McNamara reviews several such richer models of his own to illustrate how adding these extra variables changes simulation outcomes. Such outcome changes are important because, if new features don’t matter for outcomes, we could continue to ignore them as irrelevant details. We may not completely agree with adding such richness because it can complicate results and preclude an analytic approach. McNamara nonetheless makes a strong case for some increase in model richness.
A single example is his latest simulation on the role of trust in increasing cooperation among agents (McNamara, Stephens, Dall, & Houston, 2009). In a modified trust game, pairwise interactions between agents occur in two phases. A randomly chosen agent is assigned to the role of player 1 (P1), and another agent is assigned to a trustee role (P2). In the first phase, P1 decides whether to trust P2. If P2 is not trusted by P1, both agents receive a reward d, the defector’s pay-off. If P2 is trusted by P1, the game enters a second phase in which P2 decides whether to cooperate or defect. If P2 cooperates, both agents receive the cooperator’s pay-off r. If P2 does not cooperate, then P2 receives a pay-off of 1, while P1 gets nothing. Payoff amounts satisfy 0 < d < r < 1. When P1 has no information about P2, the game has the usual evolutionarily stable outcome at mutual defection (the Nash equilibrium), wherein both players receive d and forgo the higher payoff r, which they would have received if P1 had trusted P2, and P2 had cooperated with P1 (the Pareto optimum).
Extra richness comes from allowing P1 to sample, at a cost, n previous decisions by P2 players. Trust is a heritable trait of P1 players: they always trust, never trust, or trust by sampling subject to an integer k, where $1 \leq k \leq n$. Samplers will trust P2 if and only if the sampled P2s were trustworthy on at least k of the n sampled episodes. Trust is universally implemented as a value of k: those who always trust have k = 0, while those who never trust have k = n + 1. Samplers pay a cost c, where $0 \leq c < d$. The unconditional strategies of always or never trusting incur no sampling cost. P2 role players also have a heritable trait – the probability of cooperating with a P1.
The reproductive fitness of an agent is the sum of their payoffs across the two roles. An infinite population is modeled, not as discrete agents but rather in mathematical equations. Results show that variability in social awareness (trusting on the basis of prior evidence of trustworthy behavior) encourages variability in trustworthiness. Such variability in trustworthiness, in turn, favors variance in social awareness, even when such awareness is costly. In other words, there is a coevolution of two personality traits (trust and trustworthiness), which can perhaps account for the extensive human variation in empirical studies, across both people and cultures (Fehr & Fischbacher, 2003; Henrich et al., 2004).
Hearing about this study from me, critics remarked that trustworthiness seems no different than reputation, which has been the focus of several EGT models and human experiments (Leimar & Hammerstein, 2001; Nowak & Sigmund, 1998). I believe that what is new in McNamara’s model is the notion of coevolution, maintenance, and mutual influence of two personality traits – trust and trustworthiness. If there is insufficient variation in trust, there would be little variation in trustworthiness and vice versa. If agents vary in trustworthiness, there is a good evolutionary reason for costly social monitoring. And if there is sufficient variation in trust, there is a sound evolutionary reason to calibrate one’s tendency to cooperate (trustworthiness). So, by introducing these two new, initially randomly-valued traits, McNamara and colleagues document a newish (for EGT) evolutionary mechanism for the emergence of cooperation, over the Nash-equilibrium baseline of mutual defection. In empirical studies with humans, trust had already been proposed as an important mechanism for sustainable management of public goods (Ostrom, 1998, 1999). McNamara and colleagues complement this work by exploring how the evolution of trust depends on personality variation, both in trust and in the complementary dimension of trustworthiness.
My own view is that trust is likely important in the emergence of novel cooperation between humans of different groups and apparent distinct interests. The remarkable new dialog between the US and Iran may be cited as a current example. Trust will make it, or mistrust will break it. Stay tuned.
McNamara (2013) reviews several other examples of how increasing richness by adding new, realistic variables change evolutionary outcomes in games for mate-selection, divorce, parental investment, and hawk-dove. Moreover, there seems to be no problem with the extra richness compromising an analytical approach as McNamara’s models tend to be mathematical as opposed to computational.
Some of the newer EGT models in our lab can fit rather comfortably with McNamara’s emphasis on variation between agents. Artem’s blog post on the possible discrepancy between objective and subjective rationality raises the possibility that agents may differ in their impression of what game is being played, and that apparent irrationality in game playing could result from rational processes being applied to subjectively perceived payoffs. Marcel’s post on quasi-magical thinking is also relevant, in which he points out that rational decision making with self-biased learning can result in irrational cooperation in the public goods game. Finally, my post on the need for social connections identifies some ways in which agents’ perceived payoffs differ from experimenter-designed payoffs.
So maybe we should all try to get a bit richer.
### References
Box, G. E. P. (1979). Robustness in the strategy of scientific model building. In R. L. Launer & G. N. Wilkinson (Eds.), Robustness in statistics.
Fehr, E., & Fischbacher, U. (2003). The nature of human altruism. Nature, 425: 785-791.
Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., & Gintis, H. (2004). Foundations of human sociality: economic experiments and ethnographic evidence from fifteen small-scale societies. Oxford: Oxford University Press.
Leimar, O., & Hammerstein, P. (2001). Evolution of cooperation through indirect reciprocity. Proceedings of the Royal Society B, 268: 745-753.
McNamara, J.M. (2013). Towards a richer evolutionary game theory. Journal of the Royal Society, Interface / the Royal Society, 10 (88) PMID: 23966616
McNamara, J. M., Stephens, P. A., Dall, S. R. X., & Houston, A. I. (2009). Evolution of trust and trustworthiness: social awareness favours personality differences. Proceedings of the Royal Society B, 276: 605-613.
Nowak, M. A., & Sigmund, K. (1998). Evolution of indirect reciprocity by image scoring. Nature, 393: 573-577.
Ostrom, E. (1998). A behavioral approach to the rational choice theory of collective action. American Political Science Review, 92(1): 1-22.
Ostrom, E. (1999). Coping with tragedies of the commons. Annual Review of Political Science, 2: 493-535.
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2021-12-01 09:33:11
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37795183062553406, "perplexity": 2581.6929565995356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964359976.94/warc/CC-MAIN-20211201083001-20211201113001-00227.warc.gz"}
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https://tex.stackexchange.com/questions/508365/is-there-a-list-of-math-delimiters
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# Is there a list of math delimiters?
I recently discovered that the double square brackets \llbracket and \rrbracket from stmaryrd are available as unicode characters ⟦ and ⟧, which makes them easy to use with LuaLaTex.I was even more pleased to realize that those work with \left, \bigl etc. without any further DeclareMathDelimiter sorcery, they are already recognised as delimiters. That means I can write
$\left⟦ \sum \right⟧$
and get
So I wondered, is there some list of all math delimiters that are recognized out of the box?
• They are listed in the “Comprehensive list of LaTeX symbols”, texdoc comprehensive. Sep 15 '19 at 9:07
• @egreg That lists them as part of the stmaryrd package. My point is that they even work without the package, and I wondered what other delimiters might work. I tried to clarify this. Sep 15 '19 at 9:22
As egreg mentioned in a comment, a list of delimiters can be found in the "Comprehensive list of LaTeX symbols", texdoc comprehensive. This list contains all symbols that work out of the box, even for LuaLaTeX.
Using math delimiters which are accessible as Unicode characters sounds like you are using the unicode-math package. This package provides a lot of additional delimiters (and other symbols) which are listed in "Symbols defined by unicode-math", texdoc unimath. Of these, all symbols from the categories "Opening symbols, \mathopen", "Closing symbols, \mathclose" and "Fence symbols \mathfence" are set up as delimiters. (For the fences, there are additional left and right variants prefixed with l or r)
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2021-09-23 20:29:34
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8737434148788452, "perplexity": 2500.4342733606577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057447.52/warc/CC-MAIN-20210923195546-20210923225546-00402.warc.gz"}
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http://www.w3.org/Math/testsuite/build/mathml3/Topics/BiDi/RTL/subtraction-rtl1-simple.xhtml
|
prev ( testsuite > Topics > BiDi > RTL > subtraction-rtl1 ) next
Alternatives: (mml file) (full) (simple) (plain) (form) (slideshow)
File:Topics/BiDi/RTL/subtraction-rtl1
CVS-ID:
Author:Abdelshafi Bekhit, school-book of elementary school for mathematics, Ministry of Education in Saudi Arabia.
Description:In page 45 is a simple example of subtraction 3235 - 1714 = 1521, is shown in RTL direction and represented in Arabic. See also Online Curricula.
Sample Rendering:
$٢١٢٣٢٣٥١٧١٤-١٥٢١$
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2015-07-04 12:10:14
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https://tex.stackexchange.com/questions/208591/appending-a-command-with-evaluated-parameters-to-a-macro
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Appending a command with evaluated parameters to a macro
I'm trying to have a command \appendtocmd that does that:
\def\contents{Files:}
\def\value{yo man}
\appendcmdto{contents}{section}{\value}
contents contains now "Files: \section{yo man}" so when I evaluate it anywhere, it will show "Files:" and create a section. Note also that #3 of the macro is FULLY evaluated when it is called.
I was trying to make it with \expandafter, \expandnext, \csname, etc., but to no avail. If you know a solution, I would also appreciate an explanation, WHY it works :).
It's not very clear what you want to achieve, but here's a definition for \appendcmdto and \xappendto; the latter does complete expansion on the third argument.
\documentclass{article}
\newcommand{\appendcmdto}[3]{%
\edef#1{%
\unexpanded\expandafter{#1}%
\space
\expandafter\noexpand\csname #2\expandafter\endcsname
\expandafter{\expandafter\unexpanded\expandafter{#3}}%
}%
}
\makeatletter
\newcommand{\xappendcmdto}[3]{%
\protected@edef#1{%
\unexpanded\expandafter{#1}%
\space
\expandafter\noexpand\csname #2\expandafter\endcsname{#3}%
}%
}
\makeatletter
\def\contents{Files:}
\def\somevalue{yo man}
\def\someothervalue{Ehi! \textbf{\somevalue}}
\appendcmdto{\contents}{section}{\somevalue}
\show\contents
\xappendcmdto{\contents}{subsection}{\someothervalue}
\show\contents
\stop
Running this with LaTeX will show on the terminal
> \contents=macro:
->Files: \section {yo man}.
l.28 \show\contents
?
> \contents=macro:
->Files: \section {yo man} \subsection {Ehi! \protect \textbf {yo man}}.
l.32 \show\contents
?
• This is almost what I want! One issue though: the macro in #3 is not fully expanded as can be shown by running: \appendcmdto{\contents}{section}{\somevalue\somevalue}. Only the first \somevalue is expanded. I already fixed it myself and I'll be happy to accept your answer if you fix it (and I'm pretty sure you can do it, considering that I was able to :) ). I think that I understand why it works, but \expandafters make me dizzy. – tombuc Oct 23 '14 at 14:18
• @tombuc If you want full expansion you should tell so in your question. – egreg Oct 23 '14 at 14:19
• I added "FULLY" to the question text. – tombuc Oct 23 '14 at 14:25
• @tombuc OK, and I added the trick. – egreg Oct 23 '14 at 14:29
• Ok, accepted, it does precisely what I want. Why did you use \protected@edef in the second version? – tombuc Oct 23 '14 at 14:34
(don't redefine \value in LaTeX but...)
\def\contents{Files:}
\def\value{yo man}
\def\tmp{\appendcmdto{contents}{section}}
\expandafter\tmp\expandafter{\value}
This is assuming \appendcmdto is defined somewhere (your question didn't make that clear)
or just
\makeatletter
\def\contents{Files:}
\def\value{yo man}
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2019-07-20 15:46:24
|
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|
https://mathstodon.xyz/@ColinTheMathmo/102223695506458064
|
You are a ghost driving a meat-coated skeleton made from stardust, riding a rock, hurtling through space. Fear nothing. -- Unknown.
· CmdLineToot · · ·
A Mastodon instance for maths people. The kind of people who make $$\pi z^2 \times a$$ jokes. Use $$ and $$ for inline LaTeX, and $ and $ for display mode.
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2020-02-19 13:09:38
|
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|
http://openstudy.com/updates/4e4882d80b8b8d00ebdfd69f
|
## A community for students. Sign up today
Here's the question you clicked on:
## anonymous 5 years ago roots of lx-3l+lxl-6
• This Question is Closed
1. anonymous
There are 3 or 1 possible negative roots
2. anonymous
saifooooooooooo! you need a nap
3. saifoo.khan
=(
4. saifoo.khan
DANG!
5. anonymous
if x > 3 this is $x-3+x-6=0$ $2x=9$ $x=\frac{9}{2}$ that is one answer
6. saifoo.khan
second is -3/2
7. anonymous
if x <0 this is $-x+3-x-6=0$ $-2x=3$ $x=-\frac{3}{2}$
8. anonymous
and if 0 < x < 3 this is $-x+3+x-6=0$ or $-3=0$ so there is no solution on that intervals
9. anonymous
steps clear or not? you need to break up the absolute value into intervals where it is positive and negative. then get rid of absolute value signs and solve the linear equation
#### Ask your own question
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2016-09-29 03:26:38
|
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https://www.aimsciences.org/article/doi/10.3934/jmd.2012.6.563
|
# American Institute of Mathematical Sciences
October 2012, 6(4): 563-596. doi: 10.3934/jmd.2012.6.563
## A dynamical approach to Maass cusp forms
1 Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3–5, 37073 Göttingen, Germany
Received September 2012 Published January 2013
For nonuniform cofinite Fuchsian groups $\Gamma$ that satisfy a certain additional geometric condition, we show that the Maass cusp forms for $\Gamma$ are isomorphic to $1$-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.
Citation: Anke D. Pohl. A dynamical approach to Maass cusp forms. Journal of Modern Dynamics, 2012, 6 (4) : 563-596. doi: 10.3934/jmd.2012.6.563
##### References:
[1] E. Artin, Ein mechanisches system mit quasiergodischen bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170. Google Scholar [2] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions,, J. Reine Angew. Math., 492 (1997), 1. doi: 10.1515/crll.1997.492.1. Google Scholar [3] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology,, preprint, (2012). Google Scholar [4] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology,, J. Number Theory, 129 (2009), 158. doi: 10.1016/j.jnt.2008.08.003. Google Scholar [5] C.-H. Chang and D. Mayer, The period function of the nonholomorphic Eisenstein series for $PSL(2,\mathbf Z)$,, Math. Phys. Electron. J., 4 (1998). Google Scholar [6] _____, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $PSL(2,\mathbf Z)$,, in, 109 (1999), 73. Google Scholar [7] _____, Eigenfunctions of the transfer operators and the period functions for modular groups,, in, 290 (2001), 1. Google Scholar [8] _____, An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups,, in, (2001), 523. Google Scholar [9] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups,, Forum Math., 19 (2007), 1075. doi: 10.1515/FORUM.2007.042. Google Scholar [10] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$,, Invent. Math., 114 (1993), 207. doi: 10.1007/BF01232667. Google Scholar [11] M. Fraczek, D. Mayer and T. Mühlenbruch, A realization of the Hecke algebra on the space of period functions for $\Gamma_0(n)$,, J. Reine Angew. Math., 603 (2007), 133. Google Scholar [12] D. Fried, Symbolic dynamics for triangle groups,, Invent. Math., 125 (1996), 487. doi: 10.1007/s002220050084. Google Scholar [13] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,\mathbf Z)$,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 81. doi: 10.1017/S0305004105008480. Google Scholar [14] J. Hilgert and A. Pohl, Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one,, in, (2009), 97. doi: 10.1142/9789812832825_0006. Google Scholar [15] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms,, Invent. Math., 127 (1997), 271. doi: 10.1007/s002220050120. Google Scholar [16] J. Lewis and D. Zagier, Period functions for Maass wave forms. I,, Ann. of Math. (2), 153 (2001), 191. doi: 10.2307/2661374. Google Scholar [17] B. Maskit, On Poincaré's theorem for fundamental polygons,, Advances in Math., 7 (1971), 219. Google Scholar [18] D. Mayer, On a $\zeta$ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195. Google Scholar [19] _____, On the thermodynamic formalism for the Gauss map,, Comm. Math. Phys., 130 (1990), 311. Google Scholar [20] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55. Google Scholar [21] D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups,, Discrete Contin. Dyn. Syst., 32 (2012), 2453. Google Scholar [22] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces,, J. Mod. Dyn., 2 (2008), 581. doi: 10.3934/jmd.2008.2.581. Google Scholar [23] M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant,, Ergodic Theory and Dynamical Systems, (2011). Google Scholar [24] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups,, Ergodic Theory Dynam. Systems, 17 (1997), 1147. doi: 10.1017/S014338579708632X. Google Scholar [25] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbf R)$,, Invent. Math., 80 (1985), 339. doi: 10.1007/BF01388610. Google Scholar [26] _____, The Weyl theorem and the deformation of discrete groups,, Comm. Pure Appl. Math., 38 (1985), 853. Google Scholar [27] A. Pohl, Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow,, in preparation., (). Google Scholar [28] _____, Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds,, \arXiv{1008.0367}, (2010). Google Scholar [29] _____, Period functions for Maass cusp forms for $\Gamma_0(p)$: A transfer operator approach,, International Mathematics Research Notices, (2012). doi: 10.1093/imrn/rns146. Google Scholar [30] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature,, Adv. in Math., 85 (1991), 161. doi: 10.1016/0001-8708(91)90054-B. Google Scholar [31] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar [32] _____, Dynamical zeta functions and transfer operators,, Notices Amer. Math. Soc., 49 (2002), 887. Google Scholar [33] C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. doi: 10.1112/jlms/s2-31.1.69. Google Scholar
show all references
##### References:
[1] E. Artin, Ein mechanisches system mit quasiergodischen bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170. Google Scholar [2] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions,, J. Reine Angew. Math., 492 (1997), 1. doi: 10.1515/crll.1997.492.1. Google Scholar [3] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology,, preprint, (2012). Google Scholar [4] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology,, J. Number Theory, 129 (2009), 158. doi: 10.1016/j.jnt.2008.08.003. Google Scholar [5] C.-H. Chang and D. Mayer, The period function of the nonholomorphic Eisenstein series for $PSL(2,\mathbf Z)$,, Math. Phys. Electron. J., 4 (1998). Google Scholar [6] _____, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $PSL(2,\mathbf Z)$,, in, 109 (1999), 73. Google Scholar [7] _____, Eigenfunctions of the transfer operators and the period functions for modular groups,, in, 290 (2001), 1. Google Scholar [8] _____, An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups,, in, (2001), 523. Google Scholar [9] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups,, Forum Math., 19 (2007), 1075. doi: 10.1515/FORUM.2007.042. Google Scholar [10] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$,, Invent. Math., 114 (1993), 207. doi: 10.1007/BF01232667. Google Scholar [11] M. Fraczek, D. Mayer and T. Mühlenbruch, A realization of the Hecke algebra on the space of period functions for $\Gamma_0(n)$,, J. Reine Angew. Math., 603 (2007), 133. Google Scholar [12] D. Fried, Symbolic dynamics for triangle groups,, Invent. Math., 125 (1996), 487. doi: 10.1007/s002220050084. Google Scholar [13] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,\mathbf Z)$,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 81. doi: 10.1017/S0305004105008480. Google Scholar [14] J. Hilgert and A. Pohl, Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one,, in, (2009), 97. doi: 10.1142/9789812832825_0006. Google Scholar [15] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms,, Invent. Math., 127 (1997), 271. doi: 10.1007/s002220050120. Google Scholar [16] J. Lewis and D. Zagier, Period functions for Maass wave forms. I,, Ann. of Math. (2), 153 (2001), 191. doi: 10.2307/2661374. Google Scholar [17] B. Maskit, On Poincaré's theorem for fundamental polygons,, Advances in Math., 7 (1971), 219. Google Scholar [18] D. Mayer, On a $\zeta$ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195. Google Scholar [19] _____, On the thermodynamic formalism for the Gauss map,, Comm. Math. Phys., 130 (1990), 311. Google Scholar [20] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55. Google Scholar [21] D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups,, Discrete Contin. Dyn. Syst., 32 (2012), 2453. Google Scholar [22] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces,, J. Mod. Dyn., 2 (2008), 581. doi: 10.3934/jmd.2008.2.581. Google Scholar [23] M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant,, Ergodic Theory and Dynamical Systems, (2011). Google Scholar [24] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups,, Ergodic Theory Dynam. Systems, 17 (1997), 1147. doi: 10.1017/S014338579708632X. Google Scholar [25] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbf R)$,, Invent. Math., 80 (1985), 339. doi: 10.1007/BF01388610. Google Scholar [26] _____, The Weyl theorem and the deformation of discrete groups,, Comm. Pure Appl. Math., 38 (1985), 853. Google Scholar [27] A. Pohl, Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow,, in preparation., (). Google Scholar [28] _____, Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds,, \arXiv{1008.0367}, (2010). Google Scholar [29] _____, Period functions for Maass cusp forms for $\Gamma_0(p)$: A transfer operator approach,, International Mathematics Research Notices, (2012). doi: 10.1093/imrn/rns146. Google Scholar [30] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature,, Adv. in Math., 85 (1991), 161. doi: 10.1016/0001-8708(91)90054-B. Google Scholar [31] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar [32] _____, Dynamical zeta functions and transfer operators,, Notices Amer. Math. Soc., 49 (2002), 887. Google Scholar [33] C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. doi: 10.1112/jlms/s2-31.1.69. Google Scholar
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2019 Impact Factor: 0.465
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2021-01-18 04:36:56
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https://math.stackexchange.com/questions/2097051/if-fa-exists-does-fa-and-fa-exist?noredirect=1
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# If $f'(a)$ exists, does $f'(a^+)$ and $f'(a^-)$ exist?
Is it true that
If $f(x)$ is differentiable at $a$, then both $f'(a^+)$ and $f'(a^-)$ exist and $f'(a^+)=f'(a^-)=f'(a)$.
Consider the function $$f(x)=\begin{cases} x^2\sin\dfrac{1}{x}&\text{for x\ne0}\\[1ex] 0&\text{for x=0} \end{cases}$$
$f'(0)$ can be found by
\begin{align} \lim_{x \to 0} \dfrac{f(x) - f(0)}{x-0} & = \lim_{x \to 0} \dfrac{f(x) - 0}{x} & \textrm{ as } f(0) = 0 \\ & = \lim_{x \to 0} \dfrac{x^2 \sin\left(\frac{1}{x}\right)}{x} & \\ & = \lim_{x \to 0} x \sin\left(\frac{1}{x}\right) & \end{align}
Now we can use the Squeeze Theorem. As $-1 \leq \sin\left(\frac{1}{x}\right) \leq 1$, we have that $$0 = \lim_{x \to 0} x \cdot -1 \leq \lim_{x \to 0} x \sin\left(\frac{1}{x}\right) \leq \lim_{x \to 0} x \cdot 1 = 0$$
Therefore, $\lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0$ and we have $f'(0)=0$.
However, $$f'(x)=\begin{cases} -\cos\dfrac{1}{x}+2x\sin\dfrac{1}{x}&\text{for x\ne0}\\[1ex] 0&\text{for x=0} \end{cases}$$ $f'(0^+)$ nor $f'(0^-)$ exists as $x\to 0$.
I have found some pages related to this question.
Is $f'$ continuous at $0$ if $f(x)=x^2\sin(1/x)$
Calculating derivative by definition vs not by definition
Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
$f'$ exists, but $\lim \frac{f(x)-f(y)}{x-y}$ does not exist
Thanks.
• I'm not familiar with the notation. Does $f'(a^+)$ mean $\lim_{x \to a^+} f'(x)$? – pjs36 Jan 14 '17 at 5:21
• Your example works. – user384138 Jan 14 '17 at 5:22
• @Wong Austin But the right derivative of $f$ at $a$ is given by$$\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}$$ – Juniven Jan 14 '17 at 5:30
• Supposing that it ($f'(a^{+})$) means right derivative makes your question trivial because derivative at a point exists if and only both left and right derivatives at the point exist and are equal. The example you have given in your post uses a different meaning of the symbol $f'(a^{+})$ which matches the comment from @pjs36 as well as my answer. – Paramanand Singh Jan 14 '17 at 5:45
Your working is correct. I assume you mean $$f'(a^{+}) = \lim_{x \to a^{+}}f'(x), f'(a^{-}) = \lim_{x \to a^{-}}f'(x)$$ Just to clarify notation, $f'(a^{+})$ does not mean right hand derivative of $f$ at $a$ but rather it means right hand limit of the derived function $f'$. Similar remark applies to $f'(a^{-})$. I believe OP is using these symbols in the manner I have explained above.
Existence of a derivative at a point does not necessarily mean that it is continuous at that point. The example you have given in your post is a classic example of such a scenario when $f'$ exists but is not continuous.
What is important is to know that a derivative can not have jump discontinuity. Thus if $f'(a^{+}), f'(a^{-})$ exist and $f$ is continuous at $a$ then $f'(a)$ also exists and $f'(a) = f'(a^{+}) = f'(a^{-})$. Note that in your example $f'(a)$ exists but both the limits $f'(a^{+}), f'(a{-})$ do not exist. Had they existed $f'$ would have been continuous at $a$.
• Thank you. I was confused $f'(a^+)$ with the right hand derivative:) – Tianlalu Jan 14 '17 at 6:15
$$\lim_{x \to x_0^+} f'(x)$$
is not the same as
$$\lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x-x_0}$$
The latter limit always exists if $f'$ does, and it is usually what we mean when we say "right derivative".
Yes, your answer is correct. The existence of the derivative of a function at a point does not always mean that the derivative will be continuous at that point. The condition $f′(a+)=f′(a−)=f′(a)$ implies continuity of the derivative at $x=a$ which is clearly not true for the function you mentioned at $x=0$.
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2020-10-30 14:42:18
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http://www.gamedev.net/topic/647440-why-are-static-variables-bad/page-3#entry5093107
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## Why are static variables bad?
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### #41mike3 Members
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Posted 05 September 2013 - 03:22 PM
So what would be the correct approach in the example case, with the struct where a new member may be added? Where would the error be "reported" to, and in what way? More importantly, how would we know "whatever" was uninitialized, when its value as uninitialized may be undefined?
E.g.
FooContext fooContext;
fooContext.bla = <smth>; // old code that didn't know about the new "whatever" feature
// member whatever could have an undefined value, so how does this check that and "report" it, fail gracefully, etc.?
Foo *foo(new Foo(fooContext));
As like he says, it seems you need a constructor of some kind in fooContext to prevent this. Even if not one to force the programmer to pass a parameter, then at least a default one to initialize the variables so their values are all defined (perhaps so as to indicate special uninitialized states that Foo can then check for).
The ideal thing to do would be to treat the context as a volatile object and check for null on .bla before attempting to use it.
Edited by mike3, 05 September 2013 - 03:22 PM.
### #42Satharis Members
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Posted 05 September 2013 - 04:14 PM
Do you mean a check on "whatever"? Since "whatever" was supposed to be the problem variable, as it was supposed to represent a "new feature" that had been added, and all preexisting code (i.e. before the feature addition) only knew of bla. The code already initializes bla.
I suppose I wasn't understanding the example very well. If "whatever" is a member added to the struct later then what would it being set to null matter for old code? Unless the "old code" tried to utilize it then it would essentially be the same as the transparent addition of adding a new variable to a class your code was using, but not utilizing that particular variable.
I'm assuming the point you're getting at is, someone adds the new "whatever" variable to an object, and your code attempts to use the "whatever" object, but your coworker managing the usage of the struct that passes it to you hasn't implemented setting that whatever to something yet. In that case of course you would have to check validity of the object before you use it. If I'm getting this wrong please do re-explain it to me.
But where would this check go? If "in the code initializing the FooContext", then that means the programmer has to add it to each such piece of code, which creates a lot of duplicated checks
It does, but in the case of a structure it's about the same as if you require a constructor to set the .bla and the .whatever and that whatever may have been changed to a null state after construction anyway.
What it comes down to is that you have two tools to an object really. A reference, which is a guarentee that the object is at least existant even if it may be in an invalid state for your usage, or a pointer, which can be null. I think as I read more into what you're saying, what you're getting at is that by using a constructor you are forcing the caller to pass a constructed object to your structure at the time of creation of the structure, and sure, that's true. But if the structure is designed to allow that information to change you'd have to use a pointer anyway, which would indicate you have to put the "redundant" error checking anyway. So again, it's a tradeoff really. You can use the construction power of the reference but you're only really avoiding a nullptr check, and you're saying explicit to the caller "you can't change this after creation."
And furthermore, the programmer adding such checks is then aware of the existence of the rest of the code, whereas kunos' scenario was that someone adds a new feature ("whatever" in this case) but isn't aware of/forgets to add the corresponding initialization to every place the FooContext is initialized. So if they are to add the checks to wherever the FooContext is initialized, then they know about those places, and so they should be adding proper initialization too, I'd think. But his scenario is that they are not so aware. It seems he wants it "robust against feature addition" or something like that. If the check goes in the constructor, then what about the concern about constructors being bloated with checks?
I'd agree in the case of adding new required dependencies to an object the system of requiring a reference to be passed is more safe, I can't debate that. But like I said you're not actually arguing against having a default constructor in that case you're simply arguing that it is more clear that an object with a constructor requiring a reference is more "informative" and "complicance requiring" than an object that can be changed after construction. Even if you only had one constructor, if you gave a method to change that member at all then you would inherently have to do the same nullptr check as providing a default constructor would.
Thus at the heart of it, the only real debate here is one of style: do you prefer each object to be different and ask for what it deems to be "required" dependencies at construction, or do you make setting the information more transparent and allow a default constructor as well as the overloaded ones of the same style. It's arguably not a very big difference but it has a big effect on how the code is portrayed to others.
Like the SFML example I was using, it isn't as big a deal as you're making it out to be, in the example of the sprite object it really isn't that unintuitive to "know" that you have to set a texture to the sprite before using it, even if you construct it without passing one. However since quite a few people seem more interested in removing "dangerous" code like a default constructor I may experiment more with different mixtures of both methods in some future projects to see how it works out in real world example. My biggest problem with debates using simple class like a foo or a baz or a whatever is that they don't take things like multiple dependencies and subsystems into account or the mutable nature of a lot of game objects. Most game objects are going to have a lot of state you can set after creation, perhaps even major systems, so in that the "reference only" constructor quickly becomes an impossibility.
Edited by Satharis, 05 September 2013 - 04:17 PM.
### #43frob Moderators
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Posted 05 September 2013 - 04:49 PM
Clearly a constructor will not fill a dynamic job like that, even if you may use one to set an initial state.
Its kind of like:
Car myCar;
myCar.Drive(); // error message: can't drive without an engine!
Engine engine;
myCar.SetEngine(engine);
myCar.Drive(); // error message: engine is missing spark plugs
SparkPlug plug;
engine.SetPlug(plug);
myCar.Drive(); // error message: car can't drive without a fuel tank
Wouldn't that drive anybode crazy? If a car can't drive without an engine, it shouldn't be constructable without an engine. You might want to have a setter method to change the engine later, but whats the point of building/constructing it so that you can't even use it without adding all other parts?
Yes, and in your example you're talking about one dependency. I don't see how it is much more clear to have a constructor that is default and one that asks for an Engine, and then to have one setter for an engine as well. Literally the only difference you're making is that you're forcing an assumption that some kind of engine must be passed into the object at creation.
I've been working in games for decades, and I've never seen a 'real' game try to do what was done in the snippet above.
Most games have many phases of construction for a game object.
There is the game object's basic default constructor. This should generally do nothing. In cases such as serialization you are going to overwrite all the data inside the object anyway, so work in the constructor will be thrown away. That doesn't mean the object isn't fully created; think of it more like you could consider a file stream that hasn't been opened or otherwise attached to any resources.
There is a virtual function on game objects for when they are created. This might be used by world builders or when objects are created at run time. Do the initialization here. At this point it still should not be hooked up to any resources.
There is a virtual function on games objects for when they are started. This function is called after deserialization or creation. This might be after the level has loaded but before the game goes live to the player. This will also happen when objects are created at run time. Resources get requested here. Also hook up interactions and other components here.
There is a virtual function on game objects for when they are in the world. This lets the object hook itself up with other objects in the room or proximity. They might also set special flags such as requests to not be culled, or to set a non-standard culling distance. Some objects and triggers may not need to be placed in the world to be functional.
Many large games also have spatial culling. If so, there will likely be virtual functions when the object enters the simulated area and when they leave the simulated areas.
If all of those are true, there are several stages of initialization: constructed --> created --> started --> in world --> visible. It is possible to go back through the hierarchy as well, remove 'visible' when they are culled, remove 'in world' when they are removed from world, remove 'started' when the object is serialized or stopped for other reasons.
Not that any of this has much to do with shared state using static variables.
Check out my book, Game Development with Unity, aimed at beginners who want to build fun games fast.
Also check out my personal website at bryanwagstaff.com, where I occasionally write about assorted stuff.
### #44Satharis Members
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Posted 05 September 2013 - 04:58 PM
I've been working in games for decades, and I've never seen a 'real' game try to do what was done in the snippet above.
Most games have many phases of construction for a game object.
There is the game object's basic default constructor. This should generally do nothing. In cases such as serialization you are going to overwrite all the data inside the object anyway, so work in the constructor will be thrown away. That doesn't mean the object isn't fully created; think of it more like you could consider a file stream that hasn't been opened or otherwise attached to any resources.
There is a virtual function on game objects for when they are created. This might be used by world builders or when objects are created at run time. Do the initialization here. At this point it still should not be hooked up to any resources.
There is a virtual function on games objects for when they are started. This function is called after deserialization or creation. This might be after the level has loaded but before the game goes live to the player. This will also happen when objects are created at run time. Resources get requested here. Also hook up interactions and other components here.
There is a virtual function on game objects for when they are in the world. This lets the object hook itself up with other objects in the room or proximity. They might also set special flags such as requests to not be culled, or to set a non-standard culling distance. Some objects and triggers may not need to be placed in the world to be functional.
Many large games also have spatial culling. If so, there will likely be virtual functions when the object enters the simulated area and when they leave the simulated areas.
If all of those are true, there are several stages of initialization: constructed --> created --> started --> in world --> visible. It is possible to go back through the hierarchy as well, remove 'visible' when they are culled, remove 'in world' when they are removed from world, remove 'started' when the object is serialized or stopped for other reasons.
Not that any of this has much to do with shared state using static variables.
And here I was starting to wonder if I was the only one that had objects that exist in different states and have information changed both during runtime and in different stages of setup and serialization. Guess I'm not totally crazy.
But yes, we have gone off on a bit of a tangent, though it's a good topic to bring to the attention of people because they're most assuredly going to run into it at some point in game development.
### #45mike3 Members
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Posted 05 September 2013 - 07:10 PM
Do you mean a check on "whatever"? Since "whatever" was supposed to be the problem variable, as it was supposed to represent a "new feature" that had been added, and all preexisting code (i.e. before the feature addition) only knew of bla. The code already initializes bla.
I suppose I wasn't understanding the example very well. If "whatever" is a member added to the struct later then what would it being set to null matter for old code? Unless the "old code" tried to utilize it then it would essentially be the same as the transparent addition of adding a new variable to a class your code was using, but not utilizing that particular variable.
I'm assuming the point you're getting at is, someone adds the new "whatever" variable to an object, and your code attempts to use the "whatever" object, but your coworker managing the usage of the struct that passes it to you hasn't implemented setting that whatever to something yet. In that case of course you would have to check validity of the object before you use it. If I'm getting this wrong please do re-explain it to me.
The example appears to be: We have a class "Foo" which takes a "FooContext" to specify how to set it up. Initially, this "FooContext" contains only one parameter, "bla". Bits of code are created that have to make Foos from a FooContext, and naturally, they only set bla. But then down the road as our project progresses, someone decides now to add a new feature to Foo, which now requires a new parameter, "whatever", in the FooContext. And in how I imagine this scenario, which may or may not be how kunos imagined it, I imagine "whatever" to be a strictly "extensional" feature, i.e. it does not alter the original behavior of Foo, but extends it to add capability, that is, that the old way of using Foo should still work and not break. (The reason for this requirement is because without it, we might be forced to change the code anyway, but I'm trying to isolate the addition of the data field itself as the problem) But the programmer adding the new "whatever" parameter forgets a place where FooContexts are used and so doesn't add the proper initialization to the extensional feature whatever. The constructor for Foo now gets gibberish in the whatever field and crashes or behaves in some unpredictable ("undefined") way. It's not that the old code (here, meaning the code that makes the FooContext and the Foo) attempts to use "whatever", it's that the old code doesn't use "whatever", and because it was not updated to take "whatever" into account (meaning, it doesn't even set it to null, it doesn't do anything with it, period), causes Foo's constructor to gag.
So do you add the check for "whatever" not being set in to the old code, in which case it would probably be easier simply to just properly initialize whatever, or do you add the check in Foo, and regardless of the check placement, do you need to add a constructor to FooContext that ensures a stable "uninitialized" or "null" value that can be checked for? Because if you have to add the check and/or set-to-null in the old Foo-using code, then if you forget to add it there, you have a crash or other failure. To me, it seems the only way to make it robust against such omission is to put a constructor in FooContext that pre-initializes whatever to null.
Edited by mike3, 05 September 2013 - 07:22 PM.
### #46Satharis Members
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Posted 05 September 2013 - 08:14 PM
The constructor for Foo now gets gibberish in the whatever field and crashes or behaves in some unpredictable ("undefined") way. It's not that the old code (here, meaning the code that makes the FooContext and the Foo) attempts to use "whatever", it's that the old code doesn't use "whatever", and because it was not updated to take "whatever" into account (meaning, it doesn't even set it to null, it doesn't do anything with it, period), causes Foo's constructor to gag.
So do you add the check for "whatever" not being set in to the old code, in which case it would probably be easier simply to just properly initialize whatever, or do you add the check in Foo, and regardless of the check placement, do you need to add a constructor to FooContext that ensures a stable "uninitialized" or "null" value that can be checked for? Because if you have to add the check and/or set-to-null in the old Foo-using code, then if you forget to add it there, you have a crash or other failure. To me, it seems the only way to make it robust against such omission is to put a constructor in FooContext that pre-initializes whatever to null.
Why would it crash on "whatever" if none of the old code uses it? You literally could have an object with two members, one being a pointer set to gibberish and one being a pointer set to a valid object and as long as you never tried to -use- the gibberish pointer it wouldn't actually do anything undefined.
### #47mike3 Members
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Posted 05 September 2013 - 08:30 PM
The constructor for Foo now gets gibberish in the whatever field and crashes or behaves in some unpredictable ("undefined") way. It's not that the old code (here, meaning the code that makes the FooContext and the Foo) attempts to use "whatever", it's that the old code doesn't use "whatever", and because it was not updated to take "whatever" into account (meaning, it doesn't even set it to null, it doesn't do anything with it, period), causes Foo's constructor to gag.
So do you add the check for "whatever" not being set in to the old code, in which case it would probably be easier simply to just properly initialize whatever, or do you add the check in Foo, and regardless of the check placement, do you need to add a constructor to FooContext that ensures a stable "uninitialized" or "null" value that can be checked for? Because if you have to add the check and/or set-to-null in the old Foo-using code, then if you forget to add it there, you have a crash or other failure. To me, it seems the only way to make it robust against such omission is to put a constructor in FooContext that pre-initializes whatever to null.
Why would it crash on "whatever" if none of the old code uses it? You literally could have an object with two members, one being a pointer set to gibberish and one being a pointer set to a valid object and as long as you never tried to -use- the gibberish pointer it wouldn't actually do anything undefined.
The way I interpreted the example, the crash occurs when Foo's constructor is invoked with the FooContext having an uninitialized whatever member, and the crash occurs when Foo's constructor is invoked and chokes.
### #48Satharis Members
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Posted 05 September 2013 - 09:52 PM
The way I interpreted the example, the crash occurs when Foo's constructor is invoked with the FooContext having an uninitialized whatever member, and the crash occurs when Foo's constructor is invoked and chokes.
Well then that means whoever wrote Foo is expecting to use whatever and shouldn't assume that it isn't null if it is a pointer. If you check that it's null and return or log the error or throw it then the person calling knows they never passed anything in for it. Saying that you -have- to use a constructor with a reference just to make it throw an error is a little asinine, it also severely limits the behavior of your object just because you're worried about someone not getting a compiler error because they flat out did not pass an object in.
Using frob's example it would be like calling a filestream a badly designed object because you allow people to create it without providing a file to open, and thus if they use it and it reports an error state then it would have been fixed by demanding they pass a reference. Obviously both popular libraries and even standard library objects do not adhere to always requiring a reference to be passed in for dependencies on each object.
### #49 fir Members
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Posted 06 September 2013 - 12:26 AM
The obvious way in C++ is to have:
class Ship
{
public:
void rotate(float angleRad); // THIS WILL ROTATE BOTH ANGLE AND UPDATE THE DIRECTION VECTOR SO THEY ARE ALWAYS ALIGNED
float getAngle() { return angle;}
Vector2 getDirection() {return direction;}
private:
Vector2 position;
Vector2 direction;
float angle;
};
The calculation is only done once, every time "rotate" is called, Ship is always valid. Welcome to 2013.. enjoy your 1960 language ;)
IMO, "angle" is not needed at all.
It seem to me that it not resolve the root core question i am talking about. even if one will have some clear syntax
that setting angle = 10 will update the values of angle.nx
angle.ny to read - it is still unpleasant jump between setting
angle.nx angle.ny and using it. Such jump may be less or more problematic in practise I think but it seem to be real troublemaking.
Yet another somewhat related example (it is not easy to give
examples here for me because it is somewhat foggy topic for me):
I got a global shared Sleep value (in miliseconds) which is used in the winapi dispatch messages loop
It is used yet in my some keyboard handlers module so I
could +- it from keayboard (to incrase/decrease fps and corresponding cpu usage)
I can not clearly decide where this variable should belong
to my keyboard handler module or to dispatch messages loop (?) I placed it yet in the third place in globals as far as I remember.
This 3 places is already a mess for me - it become yet worse becouse I had a system of including and excluding some game modules to my system and in such module sometimes I like to overvrite such Sleep value for debug purposes locally, set it 5 in one module to 15 in other I am working on (other way I would must find a initialisation and change it there so it it is yet worse)
As described above such global shared Sleep is an example of such global shared variable that brings a trouble (As I said before not all globals make such troble, some of them never do for me really and those troubles are more related to way of treating some globals not to all of them )
So how would someone resolve such Sleep trouble ?
Edited by fir, 06 September 2013 - 12:49 AM.
### #50kunos Members
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Posted 06 September 2013 - 01:26 AM
So how would someone resolve such Sleep trouble ?
with proper software design.
The very fact that you consider this an issue shows that there is no proper design into your code.. you just make everything visible to everything else and call it a day... give it some time and you'll see this will create a spaghetti mess impossible to maintain.
If a variable has to be visible to 3 subsystem it doesnt mean it has to be visible to the entire software.
From my point of view, the situation is straightforward: the var belongs to the subsystem using it (in your case, whoever is doing the game loop) and is "exposed" to a user interaction layer (in your case, a keyboard "manager").
There are real cases for "globals".. as somebody was already pointing out, user "vars" might be a good candidate for it. At the end it's all about making decisions and live with those. For example, in my current game I have a quake like "console".. that is exposed through loose functions instead of a class.. so it's a kind of a singleton. User "vars" are published to the console for user interaction accessing this global state through these functions. It is a convenient decision because it makes this particular process less tedious but it comes at the known price of coupling every class using the global console to it without a CLEAR dependency relationship expressed via constructor.
@Frob.. I think the misunderstanding here is due to the fact that we are probably looking at different parts of a game.
For GameObjects serializable classes what you say makes perfectly sense.. it's mostly data driven, designed to be built and sculptured at runtime. But "engine" classes that handles hardware resources do require a much clearer initialization and dependency rules.. you can't build this if you don't have that... DX11 is a perfect example of this.. you cannot have a View to a resource if you dont have a Resource! It wouldn't make any sense.
GameObjects are an exception to that because their point is to be able to express dynamic relationships, uncertain content and interactive manipulation (ie from game editors).. you need to build an empty GameObject because that's what you'll be doing while editing the game. but would you build a MeshRenderer component without a reference to some sort of render manager thingy? And then have your code checking "if (renderManager)" everywhere? If the answer is yes then sorry.. I don't care how many games you have worked with that approach.. but it doesnt make sense AT ALL.
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Posted 06 September 2013 - 01:38 AM
So how would someone resolve such Sleep trouble ?
By making it part of the system that it is designed for.
Generally you want to make data as object local as you can, for instance if I had an enum I created that defined ordinal directions in a 2d game like north, east, south, etc. Quite a few systems might use that, in that case I might place it into a header file under a namespace that can be included by any parties interested in using that enum. For something like a sleep variable that isn't a constant you would probably want to make it belong to the thing that uses it the most, i.e. your window or engine class or whatever. Then just allow it to be accessed from other parts of code that need to determine what it is or modify it.
Using another example you may create an enum of substates for a state machine, well if only that state machine uses it and any other objects would -have- to include the state machine to ever be interested in using the enum then you could place it in the header file. It's a balance of trying to keep information local to where it is used while not mashing everything together in globals or giant include files. It's actually a bad idea to have a header file like "constants.hpp" or something as well due to the fact that everytime that file changes it will force recompilation in the other files.
### #52 fir Members
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Posted 06 September 2013 - 02:21 AM
For something like a sleep variable that isn't a constant you would probably want to make it belong to the thing that uses it the most, i.e. your window or engine class or whatever.
This one seem to be reasonable advice i think. Indeed probably removing it from the globals module (I put it there because I was tending to forgot where it can be so it is easy to find) and defining it into window module - and reference from all the others. This is some improvement.- But I am still not sure if this removes some troubles I eventually see still related
(such problems are related maybe to fact that 1-1 'linear' dependencies in code are clear, but when you have such 'accesible spots' you can have uncontrolled many-many
dependencies and it brings some trouble
[but I am not sure if this is just about that, this is I think related not only to system of referencing but to some code flow architecture]
- I "will be must" (Wiill had to ? future neccesity do not know how to write this in english ) rethink it still more, to find, if there is no way to repairing it more, by some other elaborate design or something)
Edited by fir, 06 September 2013 - 02:51 AM.
### #53 fir Members
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Posted 06 September 2013 - 02:28 AM
(blank, sorry mistaken click)
Edited by fir, 06 September 2013 - 02:33 AM.
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Posted 07 September 2013 - 11:47 AM
I'd like to thanks for the replies, I kept thinking of how could I change my code design and I think I got something a little better than before.
I didn't really like having static variables in my engine since the game class (exposed to other programmers) could access it as well and potentially break something.
I added a "resourceLoader" so all objects that wants to use the engine resource module must go through, and then it can manages itself and I don't need to have it global static for the code:
void Game::SomeFunction()
{
Image* myImage = new Image();
myImage->Load("assetName"); //I accessed the static resourceModule here
}
Now it is:
void Game::SomeFunction()
{
//If "assetName" hasn't been loaded yet, it'll call "Image->Load()" so I don't have all load functions inside my resource module
//I want to make objects manage themselves so it's easier to handle
}
I still left the original constructor open so Game can build their own objects if needed (such as simple Quad, or loading something from outside the resource Module, though they'd need to manage it themselves).
So that's something less that I need to worry about outside access that could mess things up, thanks for the help!
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Posted 08 September 2013 - 10:50 PM
Static variables can be used. But they must not be used with a lack of knowledge of what static is capable of doing to the results you expect not to have but in fact have gottern.
Suppose you made private static int life for all instances of the Monster class, what would happen to the monster when your ship shot a laser at it? Assuming the laser can kill it with one hit and also assuming there are more than one instance of the Monster class in the game winodw, both monsters will in fact be affected with the hit even though you only aim at that one monster.
Static variables has its benefits. Suppose when you want to access a method of a class without having creating the object of the class This acts as a convenience.
If you find yourself misusing static, this is a clear indication to re-evaluate your code design in terms of object-oriented design.
A good example of a Student class using non-static and static variables would be
public class Student
{
private static int unitsToGraduate = 120;
private int name;
private int age;
}
Edited by warnexus, 08 September 2013 - 10:53 PM.
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Posted 09 September 2013 - 04:48 AM
Static variables has its benefits. Suppose when you want to access a method of a class without having creating the object of the class This acts as a convenience.
You want to do so when you have Code unrelated to any class or you're not able to use a non-static way (e. g. because your function does some mathematics and you're not able to extend the basic datatypes). If you're working object oriented you want to use methods and in most cases it's possible to do so. It requires a bit more time to get to a better solution and sometimes you're just using the first solution coming to you're mind.
A good example of a Student class using non-static and static variables would be
public class Student {
private static int unitsToGraduate = 120;
private int name;
private int age;
// [...]
}
... assuming there is only a single university or wathever the students are used by in your system. As soon as you have multiple universities with different unitsToGraduate values, you have to move the static variable into the university (regular member) and you should recognize: this value is a universities property.
That's why also this example is at last also not a good example for the usage of static variables.
In general speaking: You won't have to use static variables in most cases. And if you believe there is no other way without a static variable, there is probably still an other way. ;)
### #57warnexus Prime Members
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Posted 10 September 2013 - 09:02 AM
In general speaking: You won't have to use static variables in most cases. And if you believe there is no other way without a static variable, there is probably still an other way. ;)
Well you would actually have to use a static. Specifically, you would need to use a static method instead of the static variable to present encapsulation in your code.
As soon as you have multiple universities with different unitsToGraduate values, you have to move the static variable into the university (regular member) and you should recognize: this value is a universities property.
Would a programmer really do something like that? Given your scenario, I would actually make the unitsToGraduate an instance variable for each instance of University class.
The example I gave was demonstrated by a Stanford University professor.
I do agree, it depends on the situation.
Edited by warnexus, 10 September 2013 - 09:25 AM.
### #58Sacaldur Members
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Posted 10 September 2013 - 03:38 PM
As soon as you have multiple universities with different unitsToGraduate values, you have to move the static variable into the university (regular member) and you should recognize: this value is a universities property.
Would a programmer really do something like that? Given your scenario, I would actually make the unitsToGraduate an instance variable for each instance of University class.
Aren't you saying the same thing I did?
public class University {
private List<Student> students;
// no static
[...]
}
public class Student {
private String name;
private int age;
// no static
[...]
}
The example I gave was demonstrated by a Stanford University professor.
I'm uncertain about the situation in other countries, but as far as I heard about you should not rely on everything IT professors are telling.
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
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2016-12-04 20:40:03
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https://brilliant.org/discussions/thread/how-do-i-solve-it/
|
×
# How do I solve it?
Find the least value of n such that $$\frac{10^{n}-1}{69}$$ is an integer.
1 year, 10 months ago
Sort by:
What have you tried?
If you don't know where to start, check out Order of an Element. It uses concepts in Euler's Theorem. Staff · 1 year, 10 months ago
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2017-07-25 20:52:12
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https://gaianation.net/is-a-circle-graph-a-function/
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I"m a little confused by the rule: If you attract a upright line that intersects the graph at an ext than 1 suggest then it is not a function.
You are watching: Is a circle graph a function
Because then a circle choose $y^2 + x^2 = 1$ is no a function?
And indeed if ns rewrite it together $f(x) = \sqrt(1 - x^2)$ climate wolfram alpha doesn"t draw a circle. I guess I"m absent the intuition as to why this is though?
The definition of a duty is for this reason important.In addition to the above, the snapshot below (taken from: What is a function) may help.
(the left hand side is her X and the best hand side is the worth Y)
A function is a preeminence that assigns uniquely to a member the domain set, a member the the photo set.The an essential word is "uniquely".So if you entrust say 2 and -2 to number 1, climate you have actually a rule, but not a function.That is the logic behind the vertical heat test. If you attract a vertical line and it intersects the graph that the duty in two distinctive points, climate you can see the it method I have assigned both of these points come the point where my vertical line the cross the x-axis.An example of this is the circle.
However a semi-circle is a legit function-the upper half is the positive square root (y=+$\sqrt1-x^2$) and the bottom fifty percent is the negative square root (y=-$\sqrt1-x^2$).
Functions should be well-defined as part of their definition, so because that a provided input there deserve to only be one output.
$f(x,y)=x^2+y^2-1$ is a function of 2 variables, and the collection of points because that which this role gets $0$ is the unit circle.
However creating $y^2+x^2=1$ as a function of $x$ alone can not be done, together $x=\dfrac12$ has actually two options ($y=\pm\sqrt\dfrac34$).
If you want to have a duty that "draws" a circle through radius $r$ and center $P = (x_0, y_0)$ top top the cartesian plane, you have the right to use the role $f : <0, 2\pi> \rightarrow \gaianation.netbbR \times \gaianation.netbbR$ characterized by $$f(\varphi) = (x_0 + r \cos \varphi, y_0 + r \sin \varphi)$$But, the course, this is no a role from $\gaianation.netbbR$ come $\gaianation.netbbR$.
Also, girlfriend can specify a curve in the aircraft by method of one equation of two variables $x$ and also $y$. If you have actually a (continuous) role $f : A\subseteq \gaianation.netbbR\rightarrow \gaianation.netbbR$, friend can gain an equation $y = f(x)$ native it, which specifies a curve.But friend cannot constantly transform one equation containing 2 variables come an tantamount equation $y = f(x)$. The equation $x^2 + y^2 = r^2, r\in\gaianation.netbbR$ is an instance of this fact.
A role $f(x_1, \ldots, x_n)$ has actually the property, the for one collection of worths $(v_1, \ldots, v_n)$ there is at most one result. If friend compare, her $f(0) = 1$, however there are 2 worths for $y$ s.t. $y^2 + x^2 = 1 \mid x = 0$, namely $\ 1, -1 \$.
The standard definition of a role $f$ is that it takes one worth $f(x)$ because that each $x$ (where it is defined).
In particular, the square root is a single valued role - because that a real number $x$, the square source is provided by $\sqrtx^2 =|x|$.
In your example, when solving for $y$ in the one equation $y^2+x^2=1$ there space two possibilities $$y=\sqrt1-x^2\qquad \textor\qquad y=-\sqrt1-x^2$$which room two different functions and the union of their graphs is the circle.
$y^2+x^2=1$ is implicit definition of $y$
An equivalent explicit meaning of $y$ is:
$y=\pm \sqrt1-x^2$ , with problem $x\in <-1,1>$
Thanks for contributing an answer to gaianation.netematics stack Exchange!
But avoid
Asking for help, clarification, or responding to other answers.Making statements based upon opinion; earlier them increase with recommendations or an individual experience.
Use gaianation.netJax to format equations. Gaianation.netJax reference.
See more: When Two Lines Intersect How Many Pairs Of Vertical Angles Are Formed
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2022-05-18 22:13:58
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https://www.authorea.com/users/104845/articles/129057-course-complex-analysis-and-differential-equations/_show_article
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Course: Complex Analysis and Differential Equations
Complex numbers
• Representation of complex numbers: coordinate form, polar form
• Power and roots: De Moivre’s formula
$$(\mathrm{cos}\theta+i\mathrm{sin}\theta)^{n}=\mathrm{cos}\ n\theta+i\mathrm{sin}\ n\theta\nonumber \\$$
Write each number in standard form
Compute the root.
Complex functions
Analytic functions
(a) Definitions
• Complex functions: A complex function $$f$$ is a function of the complex variable $$z=x+iy$$ that results in a complex-valued output
$$f(z)=u(x,y)+iv(x,y)\nonumber \\$$
where $$u(x,y)$$ and $$v(x,y)$$ are real functions of two variables.
• Continuous: A complex function is continuous at a point $$z_{0}$$ if and only if for any neighborhood $$\mathcal{V}$$ of $$f(z_{0})$$, $$f^{-1}(\mathcal{V})$$ is a neighborhood of $$z_{0}$$.
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2017-08-21 14:23:53
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http://www.math.gatech.edu/seminars-and-colloquia-by-series?series_tid=55
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## Seminars and Colloquia by Series
Monday, November 27, 2017 - 15:10 , Location: Skiles 005 , Haoyan Zhai , School of Mathematics, Georgia Institute of Technology , , Organizer: Tongzhou Chen
In this talk, we provide a deterministic algorithm for robotic path finding in unknown environment and an associated graph generator use only potential information. Also we will generalize the algorithm into a path planning algorithm for certain type of optimal control problems under some assumptions and will state some approximation methods if certain assumption no longer holds in some cases. And we hope to prove more theoretical results for those algorithms to guarantee the success.
Friday, February 12, 2016 - 15:05 , Location: Skiles 114 , John Dever , Georgia Institute of Technology , Organizer:
A local Hausdorff dimension is defined on a metric space. We study its properties and use it to define a local measure. We show that in many circumstances we can recover the global Hausdorff dimension from the local one. We give an example of a compact metric space with a continuum of local dimension values. We define the dimension of a measure and connect the definition to that of local Hausdorff dimension and measure for a class of spaces called (variable) Ahlfors Q-regular. Very little background knowledge, aside from basic familiarity with metric spaces, will be assumed.
Tuesday, November 24, 2015 - 16:00 , Location: Skiles 005 , Hagop Tossounian , Georgia Institute of Technology , , Organizer:
This is a summary of result of LUC HILLAIRET AND CHRIS JUDGE.
Friday, October 23, 2015 - 14:05 , Location: Skiles 005 , Fabio Difonzo , Georgia Institute of Technology , Organizer:
In this paper, we consider selection of a sliding vector fieldof Filippov type on a discontinuity manifold $\Sigma$ of co-dimension 3(intersection of three co-dimension 1 manifolds). We propose an extension of the “moments vector field”to this case, and - under the assumption that $\Sigma$ is nodally attractive -we prove that our extension delivers a uniquely definedFilippov vector field. As it turns out, the justification of our proposed extension requiresestablishing invertibility of certain sign matrices. Finally,we also propose the extension of the moments vector field todiscontinuity manifolds of co-dimension 4 and higher.
Friday, October 17, 2014 - 14:00 , Location: Skiles 269 , Xiong Ding , School of Physics, Georgia Tech , Organizer:
Periodic eigendecomposition algorithm for calculating eigenvectors of a periodic product of a sequence of matrices, an extension of the periodic Schur decomposition, is formulated and compared with the recently proposed covariant vectors algorithms. In contrast to those, periodic eigendecomposition requires no power iteration and is capable of determining not only the real eigenvectors, but also the complex eigenvector pairs. Its effectiveness, and in particular its ability to resolve eigenvalues whose magnitude differs by hundreds of orders, is demonstrated by applying the algorithm to computation of the full linear stability spectrum of periodic solutions of Kuramoto-Sivashinsky system.
Thursday, September 18, 2014 - 15:00 , Location: Skiles 005 , Rohan Ghanta , School of Mathematics, Georgia Tech , Organizer:
By showing a duality relation between the Sobolev and Hardy-Littlewood-Sobolev inequalities, I discuss a proof of the sharp Sobolev inequality. The duality relation between these two inequalities is known since 1983 and has led to interesting recent work on the inequalities (which may be the topic of future talks).
Thursday, March 6, 2014 - 14:05 , Location: Skiles 006 , Hagop Tossounian , School of Mathematics, Georgia Tech , Organizer:
In 1956 Mark Kac introduced an equation governing the evolution of the velocity distribution of n particles. In his derivation, he assumed a stochastic model based on binary collisions which preserves energy but not momentum. In this talk I will describe Kac's model and the main theorem of Kac's paper : that solutions with chaotic initial data can be related to the solutions Boltzmann type equation.
Friday, January 24, 2014 - 13:00 , Location: Skiles 005 , Li Wuchen , School of Mathematics, Georgia Tech , Organizer:
We introduce a new model for cell phone signal problem, which is stochastic van der Pol oscillator with condition that ensures global boundedness in phase space and keeps unboundedness for frequency. Also we give a new definition for stochastic Poincare map and find a new approximation to return time and point. The new definition is based on the numerical observation. Also we develop a new approach by using dynamic tools, such as method of averaging and relaxation method, to estimate the return time and return point. Thus we can show that the return time is always not Gaussian and return point's distribution is not symmetric under certain section.
Wednesday, October 9, 2013 - 11:00 , Location: Skiles 006 , Albert Bush , School of Mathematics, Georgia Tech , Organizer:
Erdos and Szemeredi conjectured that if one has a set of n numbers, one must have either the sumset or product set be of nearly maximal size, cn^2/log(n). In this talk, he will introduce the sum-product problem in the reals, show previous, beautiful geometric proofs by Solymosi and Elekes, and discuss some recent progress by Amirkhanyan, Croot, Pryby and Bush.
Friday, April 29, 2011 - 13:00 , Location: Skiles 246 , Jie Ma , School of Mathematics, Georgia Tech , Organizer:
Fix k vertices in a graph G, say a_1,...,a_k, if there exists a cycle that visits these vertices with this specified order, we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown by Thomas and Wollan that any 10k-connected graph is k-linked, therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered for any a_1,...,a_k. However, it is possible that we can improve this bound when k is small. It is shown by W. Goddard that any 4-connected maximal planar graph has an (a_1,...,a_4)-ordered cycle for any choice of 4 vertices. We will present a complete characterization of 4-ordered cycle in planar graphs. Namely, for any four vertices a,b,c,d in planar graph G, if there is no (a,b,c,d)-ordered cycle in G, then one of the follows holds: (1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3; (2) roughly speaking, a,b,d,c "stay" in a face of G with this order.
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2017-12-12 02:59:11
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https://stanford.library.sydney.edu.au/entries/infinity/
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# Infinity
First published Thu Apr 29, 2021
Infinity is a big topic. Most people have some conception of things that have no bound, no boundary, no limit, no end. The rigorous study of infinity began in mathematics and philosophy, but the engagement with infinity traverses the history of cosmology, astronomy, physics, and theology. In the natural and social sciences, the infinite sometimes appears as a consequence of our theories themselves (Barrow 2006, Luminet and Lachièze-Rey 2005) or in the modelling of the relevant phenomena (Fletcher et al. 2019). Mathematics itself has appealed to some form of infinity from its beginning (infinitely many numbers, shapes, iterated addition or division of segments) and its contemporary practice requires infinitary foundations. Any field that employs mathematics at least flirts with infinity indirectly, and in many cases courts it directly.
Philosophy countenances infinity in myriad ways, either directly or indirectly, in most of its sub-fields—here is a tiny sample taken from the contemporary discussion (we shall discuss historical material in Section 1 and in Section 2, and many further examples in later sections). Some metaphysicians contend that there are infinitely many possibilities/possible worlds and canvas how big this infinity is (e.g. Lewis 1986). Philosophers of religion debate whether the divine is infinite, whether the divine creation is infinite, and whether the value of the afterlife is infinite. Epistemologists debate whether there can be an infinite regress of justification, and if so, whether it is problematic (Klein 2000, Peijnenburg 2007, Atkinson and Peijnenburg 2017). Formal epistemologists traffic largely in an infinitary notion of ‘probability’ (more in Section 6). Population ethics for infinite populations is a lively topic, and they are thought to pose distinctive problems for consequentialism (Nelson 1991). Social and political philosophy appeal to the notion of convention, often thought to involve ‘common knowledge’, with a putative infinite hierarchy of mutual knowledge (Lewis 1969). Philosophers of language and mind grapple with problems that infinitary operations such as ‘plus’ create for meaning and rule-following (Kripke 1982), and whether language itself, or minds themselves, can be infinite (Nefdt 2019). Philosophers of mathematics debate whether stipulations that imply the existence of infinitely many objects can be said to be analytic (Boolos 1997, Wright 1999) and whether criteria of identity for infinite numbers must necessarily be Cantorian (Mancosu 2016). See Section 4. Concerns about infinity (and human finitude) appear in continental philosophy, not only in its 19th century historical sources (e.g., among others, Fichte, Schelling, Hegel, Kierkegaard, and Nietzsche) but in contemporary developments as well (e.g., among others, Heidegger 1929, Levinas 1961, Adorno 1966, Foucault 1966, Deleuze 1969, Badiou 2019). This list can be continued, if not ad infinitum, then ad nauseam.
At this point, one may be tempted to shout three cheers—or perhaps infinitely many of them—for infinity. Indeed, one may get the impression that we can’t live without it. At the same time, there are various apparent problems with infinity, and it starts to look less congenial. As they pile up, one might get the impression that we can’t live with it. Infinity, as we shall see, gives rise to numerous paradoxes that have preoccupied philosophers for millennia. Any praise of infinity must be tempered with circumspection and caution.
So we have good reason to want to understand infinity better. Mathematicians and philosophers in particular have done much to enhance our understanding of it. This entry strives to give the reader a sense of some of the main lines of thought regarding infinity.
Our survey begins in section 1, which unpacks some meanings of ‘infinite’, and traces various philosophical conceptions of infinity from ancient times to the 19th Century. Section 2 turns to the historical development of the mathematics of infinity over a comparable period. This provides background to a presentation in section 3 of modern mathematics’ treatment of infinity—some infinite number systems, infinities of measure, of counting, of calculus, and infinitary operations on numbers. This in turn sets the stage for our discussion of mathematical ontology in section 4.
Up to this point, it appears that infinity has been domesticated. This appearance begins to be challenged in section 5, when we canvas some classic paradoxes and puzzles involving infinity. It reappears as both friend and foe in the following sections on some philosophically fecund applications of it. In sections 6 and 7, it is both central to the formulation of probability and decision theory, and the source of more conundrums; we discuss some putative solutions to them. Section 8 presents some problems concerning space and time, as well as some progress that has been made on them—Kant’s antinomies, a Zeno-style paradox concerning measure, developments in non-Euclidean geometries and relativistic cosmology, and in determining whether space is finite or infinite. We conclude in section 9, sanguine overall about our relationship with infinity.
Given the magnitude of our topic, we clearly cannot cover all aspects of it, or even a sizable proportion of them. For example, we do not engage much with the many roles infinitude plays in science and the social sciences (except in section 8), retaining our focus on its roles in philosophy. We limit our discussion to what can be understood without highly advanced mathematics, but provide links to a number of supplementary documents that discuss further issues: infinite idealizations, quadratures of the circle, overviews of two recent developments in mathematics that promise to make the infinite realm more tractable (numerosity theories and surreal numbers), further paradoxes (God’s lottery, two envelopes), and proofs of theorems. We ask for the forbearance of readers whose favorite topics have been left out. We hope to mitigate this somewhat with our large set of pointers to further topics, the references in our extensive bibliography, and other internet resources.
## 1. Infinity in philosophy: some historical remarks
In Greek, ‘to apeiron’ means ‘the infinite’: ‘a’ denotes privation and ‘peras’ the notion of ‘limit’ or ‘bound’. Etymologically, the English word ‘infinite’ comes from the Latin word ‘infinitas’: ‘in’ = ‘not’ and ‘finis’ = ‘end’, ‘boundary’, ‘limit’, ‘termination’, or ‘determining factor’. In contemporary English, there is a range of uses of the word ‘infinite’:
1. In a loose or hyperbolic sense, ‘infinite’ means ‘indefinitely or exceedingly great’, ‘exceeding measurement or calculation’, ‘immense’, or ‘vast’.
2. In a strict but non-mathematical sense that reflects its etymological history, ‘infinite’ means ‘having no limit or end’, ‘boundless’, ‘unlimited’, ‘endless’, ‘immeasurably great in extent (or duration, or some other respect)’. This strict, non-mathematical sense is often applied to God and divine attributes, and to space, time and the universe.
3. There is also a strict, mathematical sense, according to which ‘infinite’ quantities or magnitudes are those that are measurable but that have no finite measure; and ‘infinite’ lines or surfaces or volumes are measurable lines or surfaces or volumes that have no finite measure.
Related to the distinction between meanings (2) and (3) is a distinction between metaphysical and mathematical meanings of infinity. This has been usefully employed in some of the most encompassing accounts of infinity, such as Moore (1990/2019; for another recent treatment that includes extensive discussion of the history of infinity see Zellini (2005)). Moore sees the metaphysical notion as bound up with the notions of ‘totality’, ‘absoluteness’ and ‘perfection’. While our entry is focused on the strict mathematical sense of ‘infinity’, one cannot cleanly separate the various meanings in the historical development of the subject, especially in the first stages. In addition, treating infinity as a ‘perfection’ in theology from the outset does not mirror the complexity of the historical development; for instance, we find traces in the 13th century of thinkers who attributed finiteness to God or in any case denied God’s infinity even when not explicitly stating the finiteness of God (see Coté 2002, 127–144).
The infinite has been of central concern to Western thought since the very first pre-Socratic fragment. It concerned the philosopher Anaximander (who flourished in the 6th century BCE), who identified the principle and origin of existing things as to apeiron. In Anaximander, the principle has both an ontological and an ethical significance. The Pythagoreans (6th century BCE) saw the infinite negatively and emphasized the lack of definiteness associated with it; they also gave it spatial connotations. Indeed, in the 5th century BCE the Pythagorean Archytas of Tarentum (see Huffman 2005, 540–550) gave the following argument for the spatial infinitude of the cosmos based on the contradiction that postulating a boundary to it would seem to entail. If the cosmos is bounded, then one could extend one’s hand or a stick beyond its boundary to find either empty space or matter. And this would be part of the world, which thus cannot be bounded on pain of contradiction. So the world is unbounded. Archytas identified this with the world being infinite. Kant similarly identified the unbounded and the infinite in his cosmological antinomy. In Section 8 we will see that these notions should be distinguished, but a mathematically precise articulation of the distinction had to wait until the development of new conceptions of space in the 19th century.
The Eleatics (Parmenides and Melissus, 5th century BCE) held a monist conception of reality, the One, and Melissus declared it to be infinite. Such a monistic conception of reality sees change (or becoming) as appearance, and Zeno’s famous paradoxes of infinity (see the entry on Zeno’s paradoxes) emerge in this context. Suffice here to say that Zeno’s paradoxes (the Achilles, the arrow, and others) involved the infinitely small and were aimed at buttressing Parmenides’ monism. Working across the 5th and the 4th century BCE, Democritus defended an atomistic theory with an infinite void and infinitely many atoms. The infinite by this time had shown some of its major aspects, taken as substance by some and as plurality (of atoms, times, geometrical points, etc.) by others.
If the urgency of problems related to the infinite reached Greek consciousness with Zeno’s paradoxes, the most influential discussion was due to Aristotle. In order to put Aristotle’s discussion in perspective, we need to list a number of ways in which mathematical infinity had emerged not only in philosophy, as we have described, but also in mathematics. We have already seen with Archytas the notion of spatial infinitude of the cosmos. But in number theory, the natural numbers were considered infinite, at least in the sense that given any natural number a greater one could be found. In geometry, we find both the infinite by addition (any segment can be extended) and by division (any segment can be halved). Thus, mathematics presented processes of iteration without limit. The most sophisticated technique for dealing with iterated processes in the measurements of plane and solid figures was developed by Eudoxus (4th century BCE), and we discuss it in Section 2.1.
By the time Aristotle (4th century BCE) developed his discussion of the infinite, this concept had thus made its presence felt in philosophy, mathematics, and natural philosophy (including cosmology, astronomy, and physics). It would be hard to exaggerate the role played by Aristotle in the history of infinity. He articulated some essential conceptual distinctions that were to influence all subsequent discussions. He was a finitist in the sense that in his universe, everything is finite. The cosmos is finite, bodies are finite, geometrical segments are finite, each number is finite, etc. However, there are processes that can be iterated indefinitely, giving rise to what he called ‘potential infinity’. He claimed in fact that “in a sense [the infinite] is and in a sense it is not.” (Phys. 3.6, 206a13–14).
Any arbitrary segment can be extended in length (subject to cosmological restrictions mentioned below) or halved without limit, but at each stage we remain within the finite. Time is also potentially infinite in both directions and can be divided without limit.
This conception stands in opposition to that of ‘actual infinity’, which would result if some infinite processes could be completed, carried out ‘all at once’, as it were. If actual infinity were real, then one could have infinitely long bodies, infinitely long or infinitely small segments, the totality of natural numbers, an infinite number, infinitely many instants of time, etc. Aristotle rejected the notion of the infinite as a primordial substance, as we have encountered in Anaximander, and most of his discussion of the infinite takes place within a physical context, namely one relating to spatio-temporal features of reality. As a consequence, Aristotle’s discussion of the infinite fell squarely in what we have characterized as the ‘mathematical’ notion of infinity, where infinity applies first of all to magnitudes (continuous or discrete) and what is quantifiable (time, extension, numbers etc.). His Physics discusses the infinitely large, excluded because the world is finite; and the infinitely small, excluded because the division of matter can only be potentially infinite and thus finite at each stage, never reaching an infinitesimal quantity—one that is less than any finite quantity, while being something. The exclusion of the infinitely large also has as a consequence that Aristotle cannot allow a potential infinity by addition in an unqualified manner (for otherwise any finite extension could be added to itself sufficiently many times to become larger than the size of the world). Infinity by addition, then, is to be conceptualized as a sort of inverse operation to infinity by division which gives us the primary evidence for the existence of the potential infinite. This is the implicit force of the contrastive “but” in the following quote. Aristotle writes (our emphasis):
‘To be’, then, may mean ‘to be potentially’ or ‘to be actually’; and the infinite is either in addition or in division. It has been stated that magnitude is not in actual operation infinite; but it is infinite in division – it is not hard to refute indivisible lines – so that it remains for the infinite to be potentially. (Physics 3.6, 206a14–24)
The Aristotelian distinction between potential and actual infinity has had a major influence up to contemporary times. (For further discussion of Aristotle on infinity see Hintikka (1966), Lear (1980), Kouremenos (1995), Coope (2012), Nawar (2015), Cooper (2016), Ugaglia (2018), and Hussey’s commentary to Aristotle (1983).)
Aristotle’s conception had, in addition to issues related to the constitution of the physical continuum, important consequences in cosmology. While he considered the cosmos to be finite, he thought that the movement of the celestial spheres had no beginning and no end (thus time for him, as we have noted, is potentially infinite in both directions). The issue of the “eternity of the world” was to exercise some of the best theological and philosophical minds after Aristotle, especially in connection to theological issues. For instance, Johannes Philoponus (6th century CE; see Philoponus 2004) argued in favor of a beginning of the world by claiming that the contrary thesis would lead to a paradox of infinity (we discuss this in Section 2.4).
Philoponus presented another paradox of infinity concerning infinite time that we will discuss in the version formulated by al-Ghazālī (11th century CE)—see the
Of even more pressing significance was the abandonment of Aristotle’s view on the finiteness of the cosmos and the Renaissance move from the finite to the infinite universe described in the classic text by Koyré (1957; see also Jammer 1993). While Copernicus (1473–1543) put the sun at the center of the universe, he still worked with a finite model of the universe. Foreshadowed by Epicurus (341–270), Hasdai Crescas (1340–1412), and Nicolaus Cusanus (1401–1464), Giordano Bruno (1548–1600) defended the idea of infinitely many worlds, each of infinite size, existing simultaneously. Bruno is a good example of how mathematical and theological notions of infinity were used simultaneously in the history of the concept. For instance, in On the Infinite, the Universe, and Worlds (1584) he argued from God’s infinite power to the infinitude of the universe.
By contrast, Kepler and Galileo did not think that the issue of whether the world was infinite in size could be settled either way. Kepler thought that the notion of an infinite universe was a metaphysical one and not founded on empirical evidence. Galileo claimed, in a famous letter to Francesco Ingoli written in 1624, that mankind would never be able to know whether the universe is finite or infinite. The progressive geometrization of space (see De Risi 2015) led to Newton’s gravitational theory in which the universe is infinitely extended spatially and temporally. Physical space became identified with the space of Euclid’s geometry and in this way physical space was geometrized.
Theological elements were still present when Newton identified space with the “sensorium Dei” (“God’s sensorium”). For the next two centuries cosmology was developed according to Newtonian theory: an infinite Euclidean space, flat and absolute, which provides the receptacle for all physical objects whose relations are structured by universal gravitation.
With Riemann in the mid-19th Century, and then with relativistic cosmology, one went back to a finite universe, but cosmologists are now fully aware that the issue of the finitude of the world is very much an open question that depends crucially on the curvature and the topology of space (see Section 8.2).
Our discussion above indicates a few essential aspects of the concept of infinity that will be useful in the later discussion. There are obviously many areas of contact and/or intersection between the more mathematical notion of infinity and the qualitative notion of infinity. Qualitative notions of infinity cannot be easily characterized directly but in general they appeal to features that do not seem to have a clear quantitative aspect. For instance, God might be defined as infinite because it has none of the limitations of finite creatures; this property was accounted for in some Scholastic philosophy by claiming that God, unlike finite creatures, is that unique entity in which essence and existence coincide. Often coupled with this was the claim that God’s infinity is incomprehensible, and this might be a good indicator that we cannot achieve a positive account of qualitative infinity. At the same time, claims concerning infinite divine power or goodness offer a possible connection to quantitative conceptions, and this explains why the boundary line between quantitative and qualitative conceptions is not so sharp.
Indeed, according to some authors the qualitative and mathematical conceptions are inextricably tied. Consider for instance Pascal’s use of infinite distance both in projective geometry and in his Pensées where he muses on the infinite distance (and disproportion) between finite human beings and the infinite God (see Cortese 2015). The following passage is representative of the powerful and suggestive role that appeal to finiteness and infinity plays in Pascal’s apologetics:
For in the end what is humanity [l’homme] in nature? A nothingness compared to the infinite, everything [un tout] compared to a nothingness, a mid-point between nothing and everything, infinitely far from understanding the extremes; the end of things and their beginning [principe] are insuperably hidden for him in an impenetrable secret. $$\langle$$What can he therefore imagine? He is$$\rangle$$ equally incapable of seeing the nothingness from where he came, and the infinite in which he is covered [englouti]. […] (Pascal 2008: 70; we have added the French original where the translation seems less than faithful).
Moreover, Pascal’s pari (wager) is also intimately tied to the notion of infinity in the form of an infinite reward. (See Section 7.3 on Pascal’s wager) These topics are of great importance for philosophy of religion, decision theory, and philosophical anthropology.
However, this entry does not concern those conceptions of infinity that are connected to infinite divine power, infinite modes, and in general about those conceptions of infinity that are not of a mathematical kind. We do not intend to downplay the importance of those aspects of the history of infinity to which giants such as Plotinus, Cusanus, Descartes, Pascal, Spinoza, Fichte, Hegel, and Kierkegaard contributed, among others. Leibniz and Kant also belong to that list, but we will say more about them later on. But our entry would lose focus if we were to try to pursue all these developments even at a superficial level, and the treatment of qualitative infinity is worthy of an article in its own right. Thus, we content ourselves with a list of bibliographical references through which the reader can reconstruct the contributions to the topic.
For overviews of the history of infinity which include both mathematical and metaphysical aspects, see Moore (1990/2019) and Zellini (2005). For further discussion of Aristotle’s views on infinity see the entries on: Aristotle; Aristotle and mathematics; and Aristotle and metaphysics. For ancient and medieval conceptions of infinity see Sweeney (1972), Sweeney (1992), Kretzmann (1982), Coté (2002), Biard and Celeyrette (2005), Duhem (1987), Dewender (2002), Davenport (1999), Murdoch (1982), Uckelman (2015); for the early modern period see Nachtomy and Winegar (2018); for infinity in Kant and the idealist period see Kreis (2015); Monnoyeur (1992) spans all periods.
For more on infinity in philosophy of religion, see the following references.
1. on divine infinity: Koetsier and Bergmans (2005), Göcke and Tapp (2018), the papers in the final section of Heller and Woodin (2011), and various entries including concepts of God, ontological arguments, Nicolaus Cusanus, Robert Grosseteste, John Duns Scotus, and Ibn Arabi;
2. on infinity in God’s creation, apart from our subsequent discussion of whether space and time are infinite: the entries cosmology and theology, cosmological argument, fine-tuning, infinite regress arguments, principle of sufficient reason, and being and becoming in modern physics; and
3. on ‘heavenly infinity’, apart from our subsequent discussion of Pascal’s Wager: the entries on Pascal’s wager, the meaning of life, and religion and morality.
It is worth noting that Cantor’s development of set theory was influenced by theological considerations: see, for example, Dauben (1990) and Tapp (2005).
As we have said, we are mostly excluding the topic of infinity in science and the social sciences from our purview, although see the
To the extent that we discuss infinity in science (notably in Section 8), our focus is primarily on the mathematical machinery involved, which has a venerable history. This brings us to the topic of the next section.
## 2. Infinity in mathematics: a brief historical overview
In this section we will begin by showing how Greek mathematics studiously avoided the use of infinity in the presentation of its results by making use of the method of exhaustion (3.1). Then we will look at the widespread use of infinitary objects and procedures in 17th-century mathematics (theory of indivisibles and points at infinity in geometry (3.2), infinitesimals in the calculus (3.3)) and Galileo’s problem of extending counting to infinite collections (3.4). By the early 18th-century mathematics had undergone its first “infinitistic revolution” (the second is associated with the name of Cantor, see section 3). Infinity had become a pressing foundational problem, and this will lead us to section 3.
### 2.1 The method of exhaustion
We have already mentioned that the potential infinite occurs in Greek mathematics from the outset, most obviously in the natural number series and in the geometrical operations of addition and division of segments and other geometrical magnitudes. The Greek mathematicians, starting with Eudoxus, developed a technique for measuring plane and solid figures that avoided recourse to the infinite even where an infinite “limit” process would seem to be forced by the situation. This technique, known today as the method of exhaustion (the expression was coined in the 17th century by Gregory of Saint Vincent), is found in Euclid’s Elements, book XII, and then in some of the most spectacular results by Archimedes (3rd century BCE). The idea is to replace an infinite approximation by a double reductio ad absurdum. That means that one shows the equality in area or volume of two figures, say a circle $$C$$ and an associated triangle $$T$$, by noting that $$C \lt T, C \gt T$$ or $$C = T$$ and then showing that the assumptions $$C \lt T$$ and $$C \gt T$$ both lead to a contradiction. (Here ‘$$C$$’ and ‘$$T$$’ refer with systematic ambiguity to the figures and their areas/volumes.) Further discussion can be found in the
Greek mathematics generally avoids any recourse to the actual infinite, and scholars have spoken of a “horror of infinity” typical of Greek mathematics. This is in general correct with respect to the way mathematical results are presented in their final and public presentation. However, one should keep in mind that no such “horror of infinity” is to be found when one looks at the heuristic strategies pursued by Greek mathematicians. In the case of Archimedes, this was made evident by the fortunate rediscovery of his method (found in 1906; see Netz and Noel 2007) where we see him using infinitary and mechanical considerations as tools he exploited for the discovery of geometrical theorems (see Knorr 1982, 1986 and Jullien 2015). For instance, in his description of the method for finding the proportion between the area of a parabolic segment and that of a related triangle, Archimedes thinks of geometric figures (the parabolic segment and the related triangle, in this case) as composed of infinitely many one-dimensional segments and then exploits the law of the lever to gain the determination of the relation between the areas in question. In a portion of the text of the method that has only recently become available (a section of proposition XIV, see Netz and Noel 2007), Archimedes explicitly operates with infinite collections.
### 2.2 The theory of indivisibles and points at infinity
Early modern mathematicians were impressed by the Euclidean and Archimedean rigor, but there was widespread suspicion (confirmed in 1906) that Archimedes must have had a less rigorous heuristic method that he used to discover his surprising results.
In the 17th century, infinitary considerations in geometry opened the way to new geometrical techniques in quadratures and cubatures—i.e. the determination of areas of plane figures and of volumes of solid figures, respectively. We owe to Cavalieri and Torricelli a geometrical theory of indivisibles that was later put in an arithmetico-algebraic setting by Wallis (1656). Cavalieri’s original idea (1635) was that the relation between the areas of two plane figures could be obtained by a systematic comparison of what he called the indivisibles of the figures. An indivisible of a figure is a geometrical entity of lower dimension than the figure itself. An indivisible of a line is a point; an indivisible of a plane figure is a line segment; an indivisible of a solid is a plane figure. Consider a square with top side AB and bottom side CD. An indivisible of the square is any arbitrary segment with the same length as AB that can be obtained by letting AB move parallel to itself until it reaches CD. See the
for an explanation of how to give the quadrature of the circle with the indivisibilist method, and how this courts infinite collections.
Cavalieri’s applications of the theory of indivisibles were limited to finite figures and thus did not go beyond the geometrical boundaries typical of Greek mathematics. However, Torricelli broke new ground with the determination of the volume of an infinitely long (infinite longum) solid (Torricelli 1644). Up to then, all the results concerning finite figures obtained through indivisibles could easily be proved by finitary Archimedean techniques and by avoiding any mention of infinity—just as in the case of the quadrature of the circle presented in the
However, infinity figured explicitly in Torricelli’s result that an infinitely long solid (FEOBMDC in the diagram) had a finite volume (the volume of the cylinder ACIH in the diagram).
This was the first infinitary result in Western mathematics, for the infinite was not eliminable using some alternative finitary technique but rather showed up as a feature of the very object that had to be measured. Torricelli’s infinitary result put enormous pressure on empiricist conceptions of infinity. The heuristic fruitfulness of the indivisibilist method was also accompanied by paradoxes that threatened its foundations. Among them was Tacquet’s proof using indivisibles that all triangles have the same area. The indivisibilists were able to deal with such paradoxes in various ways, but the foundations of the system remained shaky (for a detailed discussion of the foundations of the theory of indivisibles and the mathematical and philosophical issues connected to Torricelli’s result see Mancosu (1996) and Jullien (2015)).
Another area in which the infinite made its appearance in 17th century geometry is in the work of Desargues (see Sakarovitch and Dhombres 1994 and Desargues 1636). Whereas in Euclidean geometry parallel lines do not meet, Desargues entertained the idea of having parallel lines meet at a point at infinity. This was a very fruitful idea that led to the development of projective geometry.
### 2.3 The calculus
The most fruitful development in the use of infinity in 17th century mathematics was that of the calculus.
From a geometrical point of view, the calculus provides techniques for drawing tangents at an arbitrary point of a curve and for measuring the area under a portion of a curve. The differential calculus treats the first problem and the integral calculus the second. The fundamental theorem of the calculus states that these problems are inverses of each other. The calculus was developed independently by Newton and Leibniz, but its spread owed much to a significant number of mathematicians throughout Europe. The first textbook of the differential calculus was published in 1696 by the Marquis de l’Hôpital (1696; see Bradley et al. 2015 for a translation, which we follow below, with commentary). It is worthwhile to consider its axiomatized structure, for it will help us see immediately the infinitary foundations on which the new discipline presented itself to the international community. We first have two definitions:
Definition I. Those quantities are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while others change.
Definition II. The infinitely small portion by which a variable quantity continually increases or decreases is called the Differential.
The two postulates are as follows.
Postulate I. We suppose that two quantities that differ by an infinitely small quantity may be used interchangeably, or (what amounts to the same thing) that a quantity which is increased or decreased by another quantity that is infinitely smaller than it is, may be considered as remaining the same.
Postulate II. We suppose that a curved line may be considered as an assemblage of infinitely many straight lines, each one being infinitely small, or (what amounts to the same thing) as a polygon with an infinite number of sides, each being infinitely small, which determine the curvature of the line by the angles formed amongst themselves.
We see in the above the explicit infinitary characterization of some of the basic entities appealed to in the new calculus. Both postulates require something that the Greeks had studiously avoided, namely the consideration of infinitely small quantities and the reduction of curves to infinilateral polygons. While l’Hôpital and a number of French mathematicians were enthusiastic about going “infinitary”, Leibniz himself developed a fictionalist account of the appeal to infinitely small quantities (foreshadowed already in his early De Quadratura which did not see the light of day until 1993; see Leibniz (1993)). Also note the use of geometrical and kinematic (i.e. based on movement, as implied by the notions of continual increase or decrease) concepts. Much of the 19th century work on the calculus was devoted to removing geometrical and kinematic notions from the foundations of the discipline.
The literature in this area is enormous and we refer to Goldenbaum and Jesseph (2008) for a recent collection of essays on Leibnizian infinitesimals. The debates on the foundations of the calculus led to some lively contributions, such as Berkeley’s The Analyst (1734) and more mathematical work. But even after infinitesimals were eliminated from the calculus through the combined work of Cauchy, Bolzano, Dedekind, and Weierstrass in the 19th century, they were widely employed in geometry. Moreover, contemporary alternative theories of analysis (non-standard analysis, infinitesimal analysis etc.) have led to rigorous theories that, taken with a grain of salt, can be seen as vindicating some of the 17th century intuitions. We will come back to these developments below.
### 2.4 Counting infinite collections
There is one final aspect of 17th century discussions of infinity that is relevant for later considerations: the problem of extending the concept of counting from the finite to the infinite. This problem is related to the issue of whether there is only one infinity or whether there might be different sizes of infinity. As we have mentioned, Philoponus argued that the eternity of the world led to a contradiction. In particular, he claimed, if the world has no beginning in the past, then the number of individuals up to Socrates would be infinite; but then by adding the number of individuals from Socrates to now, one would obtain an infinity larger than the previous one, and this, he concluded, is “one of the most impossible things” (see Sorabji 1983). It was typical of Greek thought to reject the idea that there can be different sizes of infinity.
The Islamic mathematician Ibn Qurrah (9th century CE) took a decidedly infinitistic attitude and argued, against the Aristotelian commentators, that there can be different sizes of infinity (see Rashed 2009). He claimed, for instance, that the odd and the even numbers have the same size, but that the multiples of three are 1/3 of the total number of natural numbers. Contrary to what has been claimed in the literature, his intuition was not that the even numbers and the odd numbers have the same size because there is a one-to-one correspondence between them. Rather we have a “frequentist” intuition: every even number is followed by an odd number; multiples of three appear every three numbers, etc. We find a similar position in Grosseteste’s treatise De Luce (see Mancosu 2009, 2016 for an overview of the historical developments and further references).
Galileo Galilei epitomized the paradoxical situation we run into when trying to generalize counting from the finite to the infinite. In Two New Sciences (1638; Galilei 1974), he presented a paradox of infinity. On the one hand, there is an intuition that there are fewer square numbers than natural numbers, since the first collection is properly contained in the second (the former has some but not all of the latter’s members). On the other hand, there is an intuition that there are the same number of squares and natural numbers, since there is a one-to-one correspondence—a bijection—between the natural numbers and their squares. Galileo’s own conclusion, following Oresme and Albert of Saxony who had discussed similar issues in the 14th century, was to claim that one cannot apply the relations of equality, greater than, and smaller than to infinite collections. Much subsequent theorizing about infinity can be regarded as respecting one intuition at the expense of the other.
The intuition that if one set is a proper subset of another, the former is smaller than the latter, traces back to Euclid—call this the part-whole intuition. Bolzano (1851) was sympathetic to it, and he tried to develop a theory of infinite sets that preserved it. He was not successful, but he warned his readers not to conflate one property of an infinite set—that it can be put in one-to-one correspondence with a proper subset of itself—with a criterion of ‘size’ (what he called the “multiplicity” of a collection). Cantor (see Hallett 1986), by contrast, later used one-to-one correspondence as the defining characteristic of cardinal numbers: the numbers that answer ‘how many?’ questions, and that generalize counting from the finite to the infinite in his set theory. He thus sided with the intuition that if there is a bijection between two sets, they have the same size—call this the bijection intuition. The intuition is clearly correct for finite sets. For example, the set of fingers on a normal human hand can be paired up with the set of toes on a normal human foot, and vice versa: there is a bijection between these two sets. And of course, the two sets have the same size (five). A central question is whether the intuition is correct also for infinite sets. We will discuss Cantor’s theory and, by contrast, some recent implementations of counting, known as theories of numerosities, that preserve the part-whole intuition also for infinite sets—see the
In conclusion, the “infinitistic revolution” in the 17th and the early 18th century left an important legacy for philosophy and mathematics. The theory of indivisibles introduced new magnitudes characterized infinitarily (the collection of all the indivisibles of a figure), and new infinitary geometrical objects extended the classical geometrical universe. Moreover, the debates on the calculus were focused on the nature of the infinitely small and the infinitely large. Finally, the issues emerging from Galileo’s paradox were a prelude to the problem of extending counting from finite to infinite collections.
These problems were gradually addressed in the 19th and 20th centuries, and out of these discussions there emerged different mathematical notions of infinity. We will work our way in stages to these conceptions, via a discussion of some landmarks in the contemporary mathematics of infinity.
Among the numerous general treatments of the use of infinity in mathematics we recommend Lévy (1987), Zellini (2005), Moore (1990/2019), Vilenkin (1995), Barrow (2006). For more detailed accounts of the history of the calculus, see Kline (1990), Boyer (1959), Edwards (1979), and Grattan-Guinness (1980). The most recent scholarship on the theory of indivisibles is to be found in Jullien (2015). For recent collections on Leibnizian infinitesimals see Goldenbaum and Jesseph (2008) and Goethe, Beeley and Rabouin (2015). On concepts of mathematical infinity in the 19th century see König (1990).
## 3. Mathematics: number systems, Cantor’s paradise, and beyond
To deal with some of the issues concerning the infinite raised in Section 2, mathematicians have developed various different structures that explicitly include infinities. These structures ascribe different properties to infinities that are appropriate for different applications. In some cases, there are multiple kinds of structure that can be developed for an application. Explicitly countenancing infinities has opened up an enormous range of choices and possibilities, which has been a wellspring of development in modern mathematics.
We now give a very quick tour of infinity in modern mathematics. Section 3.1 reminds the reader of several familiar number systems: the natural numbers, integers, rational numbers, and real numbers. Section 3.2 discusses the infinite operations of limits and sums that underlie calculus, and introduces the “extended real numbers” $$+\infty$$ and $$-\infty$$. The material in these two sections is covered in most textbooks on real analysis, and even many calculus textbooks, so some readers might already be familiar with it, while others might benefit from having such a textbook on hand to expand on some of the points.
Sections 3.3 and 3.4 are more mathematically advanced. Section 3.3 introduces Cantor’s more mathematically sophisticated “cardinals and ordinals”, which are probably the mathematical developments that have done the most to untangle many of the conceptual confusions around infinity. This material is covered in greater detail in most textbooks on set theory, and parts of it are discussed in many logic textbooks as well. It can be read largely independently of the other sections.
Section 3.4 discusses a more recent mathematical theory of infinitely large and infinitesimal numbers that provides an alternate setting for calculus. This theory of “non-standard analysis” has not become as central a part of the mathematics curriculum as real analysis and set theory. It may thus be unfamiliar to most readers, and it is harder to find accessible introductions elsewhere. Although non-standard analysis is not as central a part of the cultural understanding of infinity in mathematics as cardinals are, we include it both because it is a topic of growing interest in mathematics research, and because it can help make mathematically rigorous sense of both many intuitive thoughts about infinity and some of the early work on calculus in the 17th and 18th centuries.
Up to a point, various philosophical applications and puzzles involving infinity can be understood without much understanding of the mathematics of infinity. However, the mathematics helps us formulate and tackle them rigorously. Mathematics-shy readers could skip parts of our tour (particularly section 3.4) and still benefit from the later sections, but we encourage them to make the effort and read on. The mathematical understanding of infinity is a great achievement in its own right.
### 3.1 Some number systems
The natural numbers form the most elementary number system. (Some mathematicians count $$0$$ as a natural number as well, but some others do not.) $$1$$ is a natural number. For any natural number $$n$$, $$n+1$$—the successor of $$n$$—is also a natural number. The natural numbers—$$1, 2, 3, \dots$$—are closed under addition: if $$n_1$$ and $$n_2$$ are natural numbers, then so is $$n_1+n_2$$. And they are closed under multiplication: if $$n_1$$ and $$n_2$$ are natural numbers, then so is $$n_1 \cdot n_2$$. We use the natural numbers for counting ‘how many’ of something there are, though they clearly fail when applied to infinite sets, for instance the set of squares of natural numbers or the set of natural numbers themselves. This is what ‘infinities of counting’ will extend, in section 3.3.
The integers consist of the natural numbers, their additive inverses (a number and its additive inverse sum to $$0$$), and $$0$$:
$\dots , -3, -2, -1, 0, 1, 2, 3, \dots.$
They form the most elementary number system that is also closed under subtraction: If $$j_1$$ and $$j_2$$ are integers, then so is $$j_1-j_2$$.
The rational numbers can be expressed in the form $$j_1/j_2$$, where $$j_1$$ and $$j_2$$ are integers, $$j_2\neq 0$$. They form the most elementary number system that includes the integers and that is closed under division, except by 0. If $$q_1$$ and $$q_2$$ are rational numbers, $$q_1/q_2$$ is also a rational number if $$q_2\neq 0$$.
The rational numbers are dense: for any two rational numbers $$q_1$$ and $$q_2$$ such that $$q_1 < q_2$$, there is at least one rational number $$q_3$$ such that $$q_1 < q_3 < q_2$$—for example, the arithmetic mean of $$q_1$$ and $$q_2$$, $$(q_1+ q_2)/2$$, lies between them. Indeed, for any two rational numbers $$q_1$$ and $$q_2$$ such that $$q_1$$ is strictly less than $$q_2$$, for any natural number $$n$$, there are more than $$n$$ distinct rational numbers that lie between $$q_1$$ and $$q_2$$. Where the integers spread infinitely ‘outward’ in both directions, the rationals also divide infinitely ‘inward’.
However, the rational numbers still have “gaps”. For instance, if we consider the equation, $$y=x^3-2$$, we can verify that there are values of $$x$$ where $$y$$ is negative, and values of $$x$$ where $$y$$ is positive. However, there is no rational number $$x$$ for which $$y$$ is exactly equal to 0. To fill these gaps, we construct the “real numbers”.
The real numbers can be constructed out of the rational numbers by defining each real number to be a Dedekind cut of the rationals. A Dedekind cut of the rationals is a pair of sets $$L$$ and $$R$$ such that:
1. every rational number belongs to exactly one of $$L$$ and $$R$$;
2. every member of $$L$$ is less than each member of $$R$$; and
3. $$L$$ has no largest element.
We refer to $$L$$ as the ‘left set’ of the cut and $$R$$ as the ‘right set’.
For any rational number $$q$$, there is a Dedekind cut corresponding to it, where $$L$$ consists of the numbers strictly less than $$q$$, while $$R$$ consists of $$q$$ and all larger numbers. However, there are also other partitions, where $$R$$ does not contain a smallest element. For instance, we can let $$L$$ include all the rational numbers whose cube is less than 2, while $$R$$ includes all the rational numbers whose cube is greater than 2. Since there is no rational number whose cube is exactly equal to 2, this pair of sets forms a partition, representing the real number we think of as the cube root of 2.
If $$x$$ and $$y$$ are two real numbers, represented by Dedekind cuts with left sets $$x_L$$ and $$y_L$$, and right sets $$x_R$$ and $$y_R$$, we can define operations of addition and multiplication of real numbers in terms of operations on the members of these sets. The left set of $$x+y$$ is the set of all rational numbers that result from adding a member of $$x_L$$ and a member of $$y_L$$, while the right set is the set of all rational numbers that result from adding a member of $$x_R$$ and a member of $$y_R$$. (It takes a little work to check that every rational number is in fact in one of these two sets, but the other conditions for being a Dedekind cut are straightforward to check.) If $$x$$ and $$y$$ are both positive, then we can define the right set of $$x\cdot y$$ as the set of all rational numbers that result from multiplying a member of $$x_R$$ by a member of $$y_R$$, with the left set defined as the set of all other rational numbers. (Some modifications of this definition are needed if either $$x$$ or $$y$$ is negative.) Subtraction and division can then be defined as the inverses of these operations, just as for the rationals.
The real numbers are closed under addition, subtraction, multiplication, and division by all real numbers except 0. They have the further feature that there are no “gaps”: for any bounded set of real numbers, there is a least upper bound, and for any continuous function from real numbers to real numbers, if the function is negative at one point and positive at another, there must be some point at which it is exactly equal to 0. For further discussion, see the entry on Dedekind’s contributions to the foundations of mathematics.
There are several uses for which we want a number system with no gaps, and thus we use the real numbers. If you try to measure the amount of water in a large vessel by counting out a specific number of small cups, there’s no guarantee the number of cups will be an integer. If you try to measure a long distance by counting out the length of your foot, there’s no guarantee the number of feet will be an integer. We might know that something is more than 4 of the units and less than 5. By moving to fractions of these units, we can get more precise—4 cups and 5 to 6 ounces, or 4 feet and 5 to 6 inches—but there’s still no guarantee that a specific rational number will give the precise amount. But we can generate a sequence of approximations that get closer and closer by using smaller and smaller fractions of these units. So in measuring ‘how much’ of something there is, or giving coordinates to describe the location of a point in geometric space, we use the real numbers, to guarantee, as we show in the next section, that there is some precise value the sequence of approximations converges to.
### 3.2 Limits, infinite sums, and the extended real numbers; $$+\infty$$ and $$-\infty$$
The mathematical property of ‘lacking gaps’ is referred to as ‘completeness’ — the formal statement is that an ordered set is complete if every bounded, increasing sequence of elements has a ‘limit’. These limits are the first infinite operation that we define on numbers.
#### 3.2.1 Limits of sequences
A sequence of numbers is an ordered list of numbers, which we may symbolize:
$a_1, a_2, \dots$
or
$\langle a_n\rangle.$
The members of a sequence are indexed by the natural numbers, $$n=1, 2, \dots$$.
The formal definition of a limit says the sequence $$\langle a_n\rangle$$ converges to $$l$$, or has limit $$l$$ if and only if the terms of the sequence eventually stay arbitrarily close to $$l$$. Formally:
$\lim_{n\to\infty} a_n=l$
iff
for each real number $$\epsilon>0$$, there exists a natural number $$N$$, such that for every natural number $$n>N$$, $$|a_n-l|<\epsilon$$.
We will say a bit more about the symbol ‘$$\infty$$’ that appears in this definition later, but for now it just indicates the behavior of the sequence beyond any particular finite point.
As we will demonstrate shortly, not every sequence has a limit, but we can define an important class of sequences that do. A sequence is increasing if each term in the sequence is at least as large as the previous term. A sequence is bounded if there is some real number that is larger than every term in the sequence. It turns out that every bounded, increasing sequence has a limit. The successive approximations to the measurement of some physical quantity with a finer and finer measuring unit will amount to a bounded, increasing sequence of numbers, and thus this definition of a limit allows us to give a numerical representation of any physical quantity.
To show that every bounded, increasing sequence has a limit, consider the Dedekind cuts defining the individual real numbers in the sequence. Let us define a new Dedekind cut by taking its left set to contain every rational in the left set of at least one of these terms of the sequence, and taking its right set to contain every rational that is in all of the right sets of these terms. Because the sequence is bounded, we know that the right set is non-empty, and the rest of the properties of a Dedekind cut are not hard to check.
It is not hard to check that the real number constructed in this way is the limit of the sequence. To see how this works in a specific case, we can consider the sequence $$1/2, 3/4, 7/8, \dots, 1-1/2^n, \dots$$ This sequence is increasing, since each term is greater than the one that came before, and it is bounded, since 2 is a number that is strictly greater than every term in the sequence. The Dedekind cut constructed as above will correspond to the number 1. To see this, note that for any rational number q less than 1, we can let $$\epsilon=1-q$$. Then there is some $$N$$ such that $$1/2^N<\epsilon$$. The $$N$$th term in the sequence will then be greater than $$q$$, so $$q$$ will be in its left set, and thus in the left set we constructed above. But 1 itself, and every rational number greater than it, are all in the right set constructed above. This reasoning also shows that the sequence converges to 1 according to the definition of a limit. For any $$\epsilon$$, and an $$N$$ such that $$1/2^N<\epsilon$$, the $$N$$th term in the sequence is within $$\epsilon$$ of 1, and all later terms of the sequence are greater than the $$N$$th term, but still less than 1, so they must also all be within $$\epsilon$$ of 1.
This fact about bounded, increasing sequences also underlies the use of infinite decimal notation for real numbers. When we say that the number $$\pi=3.1415926\dots$$, we just mean that $$\pi$$ is the limit of the sequence $$3, 3.1, 3.14, 3.141, \dots$$. One fact about this notation that many people find surprising is that the decimal notation $$0.99999\dots$$ is the limit of the sequence $$.9, .99, .999, \dots$$, and thus is precisely 1. Some people feel the intuition that $$0.9999\dots$$ should somehow denote a number ‘infinitely close’ to 1, but not equal to it. We will be able to make sense of an idea like this in section 3.4, but it turns out that decimal notation is not the way to do it. For a useful demonstration of this point, see Vi Hart’s video 9.999... reasons that .999... = 1.
Many sequences that are not increasing have limits as well. For instance, the sequence $$1, -1/2, 1/3, -1/4, \dots, (-1)^n/n, \dots$$ can be seen to converge to the value 0, even though it is not increasing. However, if a sequence is not bounded, like the sequence $$1, 2, 3, \dots$$, it does not have a limit—if it had a limit, there would have to be some values $$l$$, $$\epsilon$$, and $$N$$, such that all terms in the sequence beyond the first $$N$$ are within $$\epsilon$$ of $$l$$. But then any number that is larger than the first $$N$$ terms of the sequence, and also larger than $$l+\epsilon$$, would be a bound for the sequence. And some sequences that are not increasing also fail to have a limit—for instance, the sequence $$1, 0, 1, 0, 1, 0, \dots$$ does not have a limit, because there is no value such that all terms of the sequence are eventually within $$1/3$$ of that value.
#### 3.2.2 Infinite sums and products
With the definition of the limit of a sequence, we can now also often define infinitary versions of the operations of addition and multiplication. For a finite number of terms, we define the ‘partial sum’ $$\sum_{i=1}^n a_i=a_1+\dots+a_n$$ and ‘partial product’ $$\prod_{i=1}^{n} a_i=a_1\cdot\dots\cdot a_n$$. For an infinite sequence of numbers $$a_n$$, their infinite sum or product (when it is defined) is the limit of the partial sums or products:
$\sum_{i=1}^\infty a_i=\lim_{n\to\infty}\sum_{i=1}^n a_i$
and
$\prod_{i=1}^\infty a_i=\lim_{n\to\infty}\prod_{i=1}^n a_i.$
Thus, $$\sum_{i=1}^\infty\frac{1}{2^i}=\lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2^i}=\lim_{n\to\infty}(1-\frac{1}{2^n})=1$$. We can show that if an infinite sequence of terms has a sum that converges, then the terms themselves must converge to 0. That’s because the terms $$\sum_{i=1}^n a_i$$ must converge, so that for any $$\epsilon$$, there must be $$N$$, such that every one of the $$\sum_{i=1}^n a_i$$ is within $$\epsilon$$ of the limit whenever $$n>N$$. Thus, every such $$a_n$$ must have absolute value less than $$2\epsilon$$, so that both successive partial sums can be within $$\epsilon$$ of this limit. (A similar condition holds also for infinite products, but from now on we will focus only on infinite sums.)
However, just having the terms $$a_n$$ converge to 0 is not sufficient for the sum to converge. A deep and important fact about infinite sums is that $$\sum_{i=1}^\infty\frac{1}{i}$$ fails to converge, because the partial sums eventually exceed any finite bound. To see this, note that the first term is greater than $$1/2$$, the next two terms are both greater than $$1/4$$, the next four terms are all greater than $$1/8$$, and in general there are $$2^{n-1}$$ terms that are each greater than $$1/2^n$$. So to get a partial sum greater than $$n$$, it is sufficient to add the first $$2^{2n}$$ terms.
But if the terms $$a_n$$ converge to 0, and each one is smaller in absolute value and has the opposite sign as the previous term, then the infinite sum must converge. If the first term in the sequence is positive, this is because the even numbered partial sums form a bounded, increasing sequence, and the odd numbered partial sums form a bounded, decreasing sequence, and the difference between successive terms of these two sequences is a term of the original sequence and thus converges to 0. Thus, the sum $$\sum_{i=1}^\infty\frac{(-1)^{i+1}}{i}$$ converges (in particular, to the natural logarithm of 2). But a somewhat surprising fact, known as the Riemann rearrangement theorem, states that if the positive terms of a sequence have no finite sum, and the negative terms of the sequence have no finite sum, but the individual terms themselves converge to 0, then for any real number $$x$$, the terms of the sequence can be put into some order so that $$x$$ is the sum of the sequence in that order! To prove this, just rearrange the terms by beginning with enough positive terms to bring a partial sum above $$x$$, then enough negative terms to bring a partial sum below $$x$$, then enough partial sums to bring a partial sum above $$x$$ again, and so on. This process must be able to be carried out, because the partial sum of the positive terms eventually exceeds any finite bound, and similarly with the partial sum of the negative terms. Since the individual terms of the sequence are all eventually within $$\epsilon$$ of 0, these partial sums must eventually never differ from $$x$$ by more than $$\epsilon$$, and so the sum of this arrangement of the terms must converge to $$x$$.
Thus, infinite summation has some importantly different features from finite summation. For any finite set of real numbers, the sum of those numbers is well-defined, and doesn’t depend on the order you add them. But with an infinite sequence of real numbers, there may be no number that is the sum of that sequence in that order. And even if there is, it may be possible to rearrange the terms of the sequence so that they sum to another value.
But there are some cases in which the sum can be known to behave nicely. If all the terms in the sequence are positive, and there is some order in which their sum converges, then their sum must converge to this value in any order. This is because the partial sums form an increasing sequence, and for any two orderings of the terms, and any partial sum of one of those orderings, there must be some partial sum of the other ordering that includes all of the terms in that partial sum, and vice versa. Similarly, the value of the sum doesn’t depend on the order of summation if all the terms are negative. And if the positive terms have a convergent sum, and the negative terms of a series also do, then the sum of the series taken in any order must be equal to the sum of these two sums. Such a sequence is said to be absolutely convergent, as the sum of the absolute values of the terms converges.
#### 3.2.3 Limits of functions and the extended real numbers $$\pm\infty$$
Infinite sequences and sums aren’t the only ways that limits appear in mathematics. Functions of a real value can also have limits. The limit of a function of a real value $$x$$ as $$x$$ goes to infinity is defined in a similar way to the limit of a sequence indexed by natural numbers. To say
$\lim_{x\to+\infty} f(x)=l$
is to say that for every $$\epsilon$$, there is an $$N$$, such that whenever $$x>N$$, $$f(x)$$ is within $$\epsilon$$ of $$l$$. For example, $$\lim_{x\to+\infty}e^{-x}=0$$, because $$e^{-x}$$ can be made as small as one likes by choosing large enough $$x$$. (Note that the inputs to a function can be positive or negative, so we need to specify that $$x$$ approaches positive infinity to distinguish this from the limit at the other end of the axis.)
But it is also often useful to be able to define the limit of a function at some particular finite real-valued input. For instance, we might be interested in the function $$f(x)=\frac{x^2-9}{x-3}$$ as $$x$$ approaches 3. (As we will see in section 3.4, this sort of calculation is particularly important in defining the concept of ‘derivative’ of a function, giving the slope of a continuous curve at a point.) This particular function is defined for all real numbers other than 3, and at any such input $$x$$ it takes the value $$x+3$$. We would like to be able to say that the limit of this function as $$x$$ approaches 3 is 6. The way we make this precise is to say that
$\lim_{x\to a} f(x)=l$
iff
for each real number $$\epsilon>0$$, there exists a $$\delta$$, such that for every $$x$$ with $$0<|x-a|<\delta$$, $$|f(x)-l|<\epsilon.$$
That is, for any degree of approximation to the limit that we want, there is some degree of approximation for the input that suffices to guarantee that the function is that close. In the initial definition of a limit as $$x$$ goes to $$+\infty$$, we required $$x$$ to ‘approximate’ $$+\infty$$ by being sufficiently large, but now we require it to approximate $$a$$ by having a difference from $$a$$ that is sufficiently small, just as the values of the sequence or function approximate the limit $$l$$.
We can do the reverse to put $$\infty$$ on the right side of the limit as well. That is, we say
$\lim_{x\to a} f(x)=+\infty$
iff
for every $$M$$, there exists a $$\delta$$, such that for every $$x$$ with $$0<|x-a|<\delta$$, $$f(x)>M$$,
and similarly
$\lim_{x\to+\infty} f(x)=+\infty$
iff
for every $$M$$, there exists an $$N$$, such that for every $$x>N$$, $$f(x)>M$$.
Since $$\infty$$ (or more precisely ‘$$+\infty$$’—similar methods work for ‘$$-\infty$$’) can appear in each place where a real number can appear in this limit notation, it is natural to see if we can extend the definition of real numbers so that it can be included.
And in fact, if we take the definition of a Dedekind cut, and relax the requirement that the left and right sets be non-empty, we get two new elements—the one with an empty right set is greater than every rational number, and called ‘$$+\infty$$’, while the one with an empty left set is less than every rational number, and called ‘$$-\infty$$’. Within these ‘extended real numbers’, not only does every bounded increasing sequence have a limit, but every increasing sequence has a limit.
Just as the real numbers emerge naturally as the tools to measure finite quantities, as the limits of rational approximations, the extended real numbers emerge naturally as the tools to measure potentially infinite quantities that can be approximated by finite quantities. $$+\infty$$ can be taken to be the area of an infinite region, the length of an infinite line, the limit of $$1/x^2$$ as $$x$$ goes to 0, and so on. Although we are used to thinking of lengths and areas as positive numbers, it is sometimes useful to consider them as negative as well, when we care about the direction that they are pointing, and in this sort of situation $$-\infty$$ is useful as well. Just as the real number operations of addition, multiplication, subtraction, and division correspond to certain operations on the quantities they measure, these operations can often be extended to these extended real numbers, as long as we are careful about a few cases.
Adding or subtracting a finite area from an infinite area leaves it infinite. Adding a shape with infinite area to another shape of the same infinite area leaves the total area unchanged, and subtracting a negative infinite one from a positive one or vice versa is similar. But $$(+\infty) + (-\infty)$$ cannot be meaningfully evaluated; nor can $$(+\infty) - (+\infty)$$. If you take one infinitely large region, and remove an infinitely large region, you might be left with nothing, or a positive region, but you might still be left with an infinitely large region; or if the region you subtracted was larger than the original region, you might be left with a negative region—even an infinite negative region.
These restrictions also apply to the use of these extended real numbers as the limits of sequences or functions. Whenever two sequences or functions both have a finite limit, their sum or difference will have a limit that is the sum or difference of their limits. When one has a finite limit and the other is infinite, then their sum or difference will be determined by the one that is infinite. But when both are infinite, there are problems. We can see that $$1/x^2$$, $$2+1/x^2$$, and $$1/x^4$$ are all functions that go to $$+\infty$$ as $$x$$ goes to 0. If we add or subtract any of these functions from a function with a finite limit, the resulting function still has limit of $$+\infty$$. If we add them to each other in any combination, the result still has limit $$+\infty$$. But if we consider their differences, we see that $$1/x^2 - 1/x^2$$ has 0 as its limit, $$1/x^2 - 1/x^4$$ has $$-\infty$$ as its limit, and $$1/x^2 -(2+1/x^2)$$ has $$-2$$ as its limit. So “$$\infty - \infty$$” is said to be an “indeterminate form” that can’t be evaluated.
Multiplying or dividing an infinite number by a finite positive number leaves it unchanged, and multiplying or dividing by a finite negative number reverses its sign. Similarly for multiplying the infinite numbers by themselves or each other. But an infinite number divided by an infinite number, or an infinite number multiplied by 0, are also indeterminate forms. As $$x$$ approaches 0, the function $$\frac{1/x^2}{1/x^2}$$ has limit 1, while the function $$\frac{1/x^2}{1/x^4}$$ has limit $$+\infty$$. If we take the function $$1/x^2$$ whose limit is $$+\infty$$ and multiply it by the function $$x$$ whose limit is 0, we get the function $$1/x$$, that has no limit as $$x$$ approaches 0 (since it takes on both large positive and large negative values in any small interval around 0—this is why we have used $$1/x^2$$ and $$1/x^4$$ as the paradigms of functions with limit $$+\infty$$, rather than $$1/x$$ or $$1/x^3$$). For similar reasons, these extended real numbers don’t provide a way to divide by 0. So although the extended real numbers have some nice properties, and can be used for measurement in various cases, arithmetic involving them is not as nice as the standard real numbers.
#### 3.2.4 Related infinities
The Dedekind cut construction was done as a way to make sense of limits of the rational numbers. This first created the real numbers, which can be thought of as the limits of bounded infinite sequences of rational numbers. We then considered all limits that made sense, including towards the ends of the real line, yielding the extended reals, including the standard reals as well as $$+\infty$$ and $$-\infty$$.
Versions of this process can be carried out with other mathematical entities as well. Projective geometry adds additional points ‘at infinity’ to the Euclidean plane, one for each family of parallel lines, to help explain features of visual geometry, like the way that parallel railroad tracks appear to meet at the horizon, infinitely far away. Riemannian geometry extends the complex numbers by adding just a single number $$\infty$$ that one can approach simultaneously ‘in all directions’ in the complex plane. These alternate geometries provide foundations for material discussed in section 2.2, and also section 8.2. Several of these are discussed in the entry on nineteenth century geometry, and others are discussed in topology textbooks under the topic of ‘compactification’.
Because these infinities are inherently considered as limits of finite approximations, there is no way for one infinite element to lie “beyond” another—at most it can lie in a “different direction”, the way that the points of convergence of different families of parallel lines do, or the way $$+\infty$$ and $$-\infty$$ do in the extended reals.
But as we will shortly see, there are other notions of infinity for which one infinity can lie “beyond” another. In section 3.3, we will discuss the ideas of infinity that we get from generalizing the use of the natural numbers for counting, rather than generalizing the use of the real numbers for measuring. And in section 3.4, we will discuss yet another mathematical theory of infinity, which arises from an alternate formulation of calculus, where the $$\epsilon$$’s and $$\delta$$’s are treated as actually being infinitely small, rather than just being arbitrary finite measures of smallness.
### 3.3 Infinities of counting
#### 3.3.1 Preliminaries
As mentioned above, this section is more mathematically dense than the previous two. However, we need this level of mathematical rigor to develop Cantor’s theory of ordinals and cardinals, which are widely regarded as the most significant mathematical advance in our understanding of the infinite.
An insomniac, counting imaginary sheep to try to get to sleep, will never run out of natural numbers to do the job: 1, 2, 3, …. There is no bound on the set of natural numbers. This is our first infinite set. It is perhaps a natural thought that there is just one infinity for counting infinite sets, which we might symbolise ‘$$\infty$$’. The thought may seem even more natural when we define an infinite set as one that has the same size (in a sense to be made precise shortly) as a proper subset of it. In fact, the thought could hardly be more mistaken: as we will soon see, according to mathematical orthodoxy—namely contemporary set theory and the attendant notion of cardinality of a set—there are infinitely many infinities. This prompts a series of questions: Is there a smallest one? Yes, as we will see. Is there a largest one? No, as we will see. What can be said about the spacing between the infinities and how far infinities extend? Well, we will see. We can also ask parallel questions about the infinitely small.
Recall Galileo’s paradox of infinity, based on the collision of the part-whole intuition and the bijection intuition, and his conclusion that one cannot apply the relations of ‘less than’, ‘equality’, and ‘greater than’ to infinite collections. Modern mathematical orthodoxy, embodied in contemporary set theory, rejects Galileo’s conclusion. That orthodoxy is grounded in the bijection intuition, following Cantor rather than Euclid and Bolzano. When there is a bijection between two sets, we say that they have the same cardinality. The notion of cardinality does not respect the part-whole intuition. For example, the squares of natural numbers are a proper subset of the natural numbers, but they have the same cardinality since they can be put in one-to-one correspondence.
Foundational programs such as neo-logicism also start from a notion of “equinumerosity” based on the Cantorian bijection intuition. It was not until the early 2000s that a group of mathematicians working on non-standard analysis (Benci, di Nasso, and Forti) developed a theory of “numerosities” that agrees on finite sets with Cantorian cardinality but that also upholds the part-whole intuition for infinite sets, and thus diverges from Cantorian cardinalities. (See Benci and Di Nasso 2003, 2019; Benci, Di Nasso and Forti 2006, 2007). One can consider numerosities as a refinement of Cantorian cardinalities. Every two sets that have the same numerosity have the same cardinality, but the converse does not hold. For instance, in this approach the set of squares has numerosity strictly less than the set of natural numbers. See the
#### 3.3.2 Set theory: $$\omega$$ and $$\aleph$$
The sort of infinities that are most familiar to philosophers are infinities used for counting. In the early 1820s, Bolzano arrived at the idea that an infinite set is one for which there is a bijection between the set and a proper subset of it. (Recall Galileo’s paradox.) Dedekind (1884) gave this as a definition of being infinite. It is easy to prove that a Dedekind-infinite set must contain a set that is just as large as the natural numbers. See the
Dedekind’s definition is only one among several alternative definitions of infinite set (and correspondingly of finite set) that have been proposed by him and after him. If one assumes the Axiom of Choice, these alternative definitions turn out to be equivalent. (This axiom says that for every set $$A$$ of pairwise-disjoint non-empty sets, there exists a function that selects exactly one element from each set in $$A$$.) But without the Axiom of Choice, the definitions can be shown not to be equivalent and the foundational situation is rather delicate but well understood (see Moore 1982).
The modern theory of infinities of counting derives from Cantor (1932). He observed that a natural way to set up a bijection between finite sets is to order the elements of each set, and pair the first element of one set with the first of the other, the second of one set with the second of the other, and so on. This sometimes works with infinite sets—for example, it gives the one-to-one correspondence between the natural numbers and the squares (considered under their standard ordering). When there is a one-to-one correspondence between two sets, such that every pair of elements of one set bears the same ordering as the corresponding pair of elements of the other set, the two ordered sets are said to have the same order type.
But for some infinite sets (notably the set of all integers, including the negative numbers, and also the set of rational numbers) there is no first element under their standard ordering. In this case, it is possible to re-order the elements of the set so that every non-empty subset has a first element, so that this process works. Such an ordering is called a well-ordering. (That every set can be well-ordered is, as Zermelo famously proved in 1904, equivalent to the Axiom of Choice.)
We can reorder the integers, alternating between positive and negative: 0, -1, 1, -2, 2, -3, 3, …. This ordering has the same order type as the natural numbers, and thus enables a one-to-one correspondence between the natural numbers and the integers. One of Cantor’s most striking early observations is that the same is possible with the positive rational numbers. Every positive rational number can be written uniquely in lowest terms as some fraction $$p/q$$, where $$p$$ and $$q$$ are positive integers with no common factor. We can then order these fractions by first comparing the sum $$p+q$$ of numerator and denominator, and then if the sum is equal for two fractions, put the one with lower numerator $$p$$ first. This ordering begins 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5, 3/4, 4/3, 5/2, 6, …. (Note that fractions like 2/2, 2/4, 3/3, 4/2, are missing from this list, because they are not written in lowest terms.) Every positive rational number must appear in this list (because it can be written in lowest terms with some particular finite sum of numerator and denominator), and only has finitely many predecessors (because there are at most $$n+1$$ fractions whose sum of numerator and denominator is $$n$$).
However, Cantor also observed that different well-orderings of the same infinite sets will produce a different order type. For instance, we can define an ordering on the integers where every non-negative integer comes before every negative integer, with any two integers of the same sign being ordered by their absolute value. To approximately represent this ordering we could write it as $$0, 1, 2, 3, \dots, -1, -2, -3, \dots$$. In this ordering, every non-empty subset still has a first element (the non-negative element of lowest absolute value if it contains any non-negative elements, and the negative element of lowest absolute value if it only contains negative elements). If we pair up the first element of this ordering with the first of the standard ordering on the natural numbers, and the second with the second, and the third with the third, and so on, then the negative integers would not be paired with any natural number. But we can put the natural numbers into the same order type by declaring the odd numbers to be before the even numbers, and ordering them by size within these two sets: $$1, 3, 5, \dots, 0, 2, 4, \dots$$. A single infinite set can be given orderings of many different order types, and also different orderings of the same order type (for instance if we put the even numbers first and the odd numbers second).
Cantor noted that for any two well-ordered sets, the initial positions in one ordering (the first, the second, the third, etc.) correspond to the initial positions in the other, the way that they do for finite sets. In fact, he showed that all of the positions of one well-ordering must correspond to initial positions in the other. (If this weren’t true, then the set of positions in one that don’t correspond to positions in the other would be non-empty for each set, and the first elements of these sets would correspond, which would contradict the claim that these positions don’t correspond.) Thus, there is a single list of all the possible positions in well-ordered sets, beginning with the first, second, third, and so on, and these positions are called the ordinal numbers. A well-ordered set can be said to have its own ordinal number, which is the first ordinal number that does not correspond to a position in that set.
A cardinal number (like “one”, “two”, “three”)—also called a cardinality—represents how many elements a set has. Two sets have the same cardinal number if it is possible to come up with any correspondence at all between the members of one and the members of the other, even if this correspondence fails to respect the ordering or any other structure of the sets. Two finite sets have the same cardinal number if and only if they have the same ordinal number. For infinite sets, if they are well-ordered and have the same ordinal number, then they have the same cardinal number (because two sets well-ordered with the same order type have a unique correspondence between elements in corresponding positions of the ordering). But they may have the same cardinal number without having the same ordinal number: we have seen that sets of one cardinality can be represented with many different order types.
Cantor used lowercase Greek letters to represent infinite ordinal numbers, with $$\omega$$ representing the order type of the standard ordering of the natural numbers. Addition of ordinal numbers corresponds to the order type that results from taking an ordering of the first type, followed by an ordering of the second type. Thus, $$\omega+\omega$$ represents the order type of the integers with the non-negative numbers first and the negative numbers afterwards, while $$\omega+1$$ represents the order type of the natural numbers with just one element put at the end. Note that $$1+\omega$$ is the order type of a single element, followed by a copy of the natural numbers, which is in fact the same as the order type of the natural numbers! Thus, $$1+\omega=\omega$$, which is not equal to $$\omega+1$$. So ordinal addition is not commutative.
Von Neumann (1923) defined a canonical representation of ordinal numbers, using the fact that the ordinals are themselves well-ordered. Each ordinal is represented by the set consisting of all smaller ordinals. Thus 0 is represented by the empty set $$\emptyset$$, 1 by the set containing the empty set $$\{\emptyset\}$$, 2 by the set containing both of those $$\{\emptyset,\{\emptyset\}\}$$, and so on. $$\omega$$ is then the set containing all of these finite ordinals $$\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots\}$$, $$\omega+1$$ is the set containing it as well $$\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots,\omega\}$$, and so on.
Multiplication of ordinal numbers corresponds to replacing each element of the second ordering by an entire ordering of the first type. $$\omega\cdot\omega$$ represents the order type of an $$\omega$$-sequence of $$\omega$$-sequences, which we can have by taking the positive rational numbers and sorting them first by denominator, and then by numerator: 1, 2, 3, … 1/2, 3/2, 5/2, …, 1/3, 2/3, 4/3, 5/3, …, 1/4, 3/4, 5/4, …. Note that $$\omega\cdot 2$$ is 2 copies of $$\omega$$, while $$2\cdot\omega$$ is $$\omega$$ copies of 2. $$\omega\cdot 2$$ is thus $$\omega+\omega$$, while $$2\cdot\omega$$ is $$2+2+2+\dots$$, which is $$\omega$$. (This is the difference between the orderings 1, 2, 3, …, −1, −2, −3, … and 1, −1, 2, −2, 3, −3, ….) So again, ordinal multiplication is not commutative.
But for any ordinal, one can generate another ordinal by putting one more element at the end. And for any increasing sequence of ordinals, there is a limit of that sequence. Cantor also defined a notion of exponentiation for ordinals, and this gives many different ordinal numbers: 0, 1, 2, 3, … ,$$\omega,$$ $$\omega+1,$$ $$\omega+2,$$ …, $$\omega+\omega(=\omega\cdot 2),$$ $$\omega\cdot 2+1,$$ $$\omega\cdot 2+2, \dots,$$ $$\omega\cdot 3,$$ $$\omega\cdot 4,$$ …, $$\omega\cdot \omega(=\omega^2),$$ …, $$\omega^3,$$ …, $$\omega^4,$$ …, $$\omega^\omega,$$ $$\omega^{\omega+1},$$ $$\omega^{\omega+2},$$ …, $$\omega^{\omega\cdot\omega}=\omega^{\omega^2},$$ …, $$\omega^{\omega^3},$$ …, $$\omega^{\omega^\omega},$$ $$\epsilon_0$$ (defined as the limit ordinal for the sequence $$\omega,$$ $$\omega^\omega,$$ $$\omega^{\omega^\omega},$$ …). But these ordinal numbers all correspond to the same cardinal number, which is that of the natural numbers.
At this point, one might be forgiven for thinking that there is no cardinal number greater than that of the natural numbers, just as there is no extended real number larger than $$+\infty$$. However, Cantor’s second striking result is that the cardinality of the positive real numbers is in fact greater than the cardinality of the positive integers, and his third striking result is that for every set, the set of all its subsets—its power set—has an even greater cardinality. Although many different infinite sets of different order types can all be put into one-to-one correspondence with each other, there are some infinite sets that cannot. Sets whose cardinality is equal to that of the natural numbers (like the integers and the rationals) are said to be countably infinite or denumerable, while infinite sets that are not countable (like the reals, and the power set of the naturals) are said to be uncountable. See proofs of the results in the
Thus, just as there is an infinite hierarchy of infinite ordinal numbers, which Cantor represented with lowercase Greek letters, there is also an infinite hierarchy of infinite cardinal numbers, which Cantor represented with Hebrew letters, and in particular aleph, “$$\aleph$$”. The finite cardinals are 0, 1, 2, 3, …. The first infinite cardinal, that of the natural numbers (and all countable sets), is $$\aleph_0$$. Cantor’s “well-ordering principle”, stating that every set can be put into some well-ordered form (and equivalent to the Axiom of Choice), implies that every cardinal number can be represented by an ordinal number, and the definition of ordinal numbers ensures that for any non-empty set of ordinal numbers there is always a first. So the cardinals must in fact be well-ordered. So Cantor used ordinal numbers to designate each cardinal’s position in their ordering. $$\aleph_1$$ is the first cardinal beyond $$\aleph_0$$, $$\aleph_2$$ is the next beyond that, and after $$\aleph_3$$, $$\aleph_4$$, and so on, we eventually reach $$\aleph_\omega$$, $$\aleph_{\omega+1}$$, $$\aleph_{\omega+2}$$, and so on with one cardinal for every ordinal.
Because we have this one-to-one correspondence between the cardinals and the ordinals, one might be tempted to say that the set of cardinals and the set of ordinals have the same order type, and then ask what the ordinal of this order type (and its cardinality) is. However, if there were such an ordinal, there would be a paradox—it would have to contain, and thus be larger than, all ordinals, including itself! This is the Burali-Forti paradox (see entry paradoxes and contemporary logic).
Relatedly, since every set has a cardinality less than that of its power set, there can be no set that contains everything (since such a set would already include all its subsets, and thus be at least as large as its power set). The two results imply that there cannot be a set of all ordinals or a set of all sets. In a similar way one argues that there cannot be a set of all cardinals. As a consequence of the above results we can also answer some of the questions we raised at the beginning: there are infinitely many cardinal numbers and infinitely many ordinal numbers. However, there is neither a set of all cardinal numbers nor a set of all ordinal numbers. Thus, the infinity of the cardinals and of the ordinals cannot be measured by a cardinal or an ordinal, for otherwise a paradox would ensue. (See also entry on Russell’s paradox.)
Although it took several decades from Cantor’s work to find a system of axioms for set theory that avoids these paradoxes (see the entries on the early development of set theory and set theory), Cantor already saw, in this unreachability of the totality of all ordinals or cardinals, a notion of “absolute infinity”. Although in his system, there are many infinite sets that are tractable and graspable at many different levels, starting with the natural numbers (which Aristotle had thought was merely potential rather than actual), he discovered an even greater Aristotelian potential infinity. This led to the distinction between a “set” as a totality that can be grasped in the relevant sense, and a “proper class”, which is too large even for a system that encompasses each of Cantor’s many infinities.
We can define addition for cardinal numbers as the cardinality of a set that is the union of two disjoint sets of those cardinalities. We can define multiplication for a pair of cardinal numbers as the cardinality of the set of all ordered pairs whose first elements are drawn from a set of the first cardinality and whose second elements are drawn from a set of the second cardinality. But it turns out that these operations are relatively trivial once we get beyond the finite cardinals—just as we saw that the sum or product of two countable ordinals was still countable, the sum or product of two infinite cardinals is equal to whichever of the two is larger! (At least this operation is commutative.) Thus, one can’t get to $$\aleph_1$$ from $$\aleph_0$$ by means of addition or multiplication (as we saw when we considered order types achieved by addition and multiplication).
However, cardinal exponentiation is more powerful. It turns out to be natural to define $$2^\kappa$$ as the cardinality of the power set of a set whose cardinality is $$\kappa$$. (Greek letters $$\kappa$$ and $$\lambda$$ are traditionally used as variables to represent infinite cardinals, with letters $$\alpha$$ and $$\beta$$ used for ordinals.) It turns out that the cardinality of the set of real numbers (otherwise called the “continuum” for its role in representing continuous space) is the same as that of the power set of the natural numbers, $$2^{\aleph_0}$$, and Cantor showed that $$2^{\aleph_0}>\aleph_0$$. Cantor conjectured that $$2^{\aleph_0}$$ was in fact equal to $$\aleph_1$$, and this conjecture was called the “Continuum hypothesis”. (See the entry on the continuum hypothesis for more on this conjecture and why it hasn’t been settled.)
Just as $$\aleph_1$$ is a cardinality that can’t be reached by the operations of ordinal addition, multiplication, or exponentiation, even in the limit, one might conjecture that even before one reaches the Cantorian absolute infinity, there are further cardinalities that can still be grasped in some sense, but can’t be reached even by the stronger operation of cardinal exponentiation, even in the limit. Such conjectures have turned out to be surprisingly fruitful for the study of set theory and mathematical logic generally. (See the entry on large cardinals and determinacy.)
Although we have defined addition and multiplication for both ordinals and cardinals, their features make it hard to make sense of subtraction or division. First, the non-commutativity of the ordinal operations, and the triviality of the cardinal operations, makes it hard to define meaningful inverses of these operations. (If it’s possible to find an ordinal or cardinal that can be added or multiplied to a first one to get a second one, then it is usually possible to find infinitely many that can.) But more importantly, the conceptual ideas of counting (whether by order type or by bijection) don’t really allow for negatives or fractions, the way that the conceptual ideas of measuring and geometry do (considered in section 3.2). Counting involves treating each element as discrete and unique, and there is no way for multiple elements to combine to yield nothing (as subtraction requires) or to yield a unit (as division does). However, measurement (e.g., of distance) involves a sort of structure on the thing being measured so that some measurements can be fractions of others, or can point in the opposite direction, which yields meaningful notions of division and subtraction.
This section has given just an introductory taste of the mathematics of ordinal and cardinal numbers—it has come to be called Cantor’s Paradise. But we have already seen many characteristic features of these notions of infinity that differentiate them starkly from those discussed in the previous section. Just as the differences between the natural numbers and the real numbers demonstrate the differences between counting elements of sets, and measuring lengths, areas, and so on, the differences between Cantor’s cardinals and the extended reals further demonstrate differences between counting and measuring.
For further discussion of the kind of material presented in this section, see the SEP entries on: set theory, the early development of set theory, and the axiom of choice. An excellent introduction to basic set theory is Enderton (1977); an informal but still rigorous introduction is Sheppard (2014); more advanced texts include Devlin (1993), Kunen (1983), and Jech (2006). For the higher reaches, to which we will come back in section 4, see Kanamori (2003).
Set theory provides a theory of cardinality that implements the idea that “sameness of size” upholds the bijection intuition. The recent theory of numerosities, developed by Benci, Di Nasso, and Forti, by contrast, upholds the part-whole intuition. See the
### 3.4 Infinitesimals and hyperreals
We have already discussed a notion of $$+\infty$$ and of $$-\infty$$ that is designed to provide values for real-valued functions to take as limits at special points. But the understanding of limits themselves was originally thought to require a notion of “infinitesimally small” distances. While these quantities were considered problematic for several centuries, in recent decades some mathematical entities with their properties have been rigorously studied.
#### 3.4.1 Newton and Leibniz
An infinitesimal is a number smaller in absolute magnitude than any positive finite number, and yet not zero. Infinitesimals have had a chequered history. Early work in the calculus, as we have seen when presenting the structure of L’Hôpital’s 1696 treatise, was mostly based on a geometrical or kinematic (i.e. based on motion) understanding of infinitesimals. Here we treat infinitesimals arithmetically, i.e. within the context of the real number system, while conveying the key features of how the early analysts made use of them. To figure out the slope of the function $$f(x)=x^2$$ precisely at one point, one considered an “infinitesimally small number” $$\epsilon$$ and considered the slope of the straight line through $$f(x)$$ and $$f(x+\epsilon)$$.
This slope is equal to $$\frac{(x+\epsilon)^2-x^2}{\epsilon}$$. In order for this fraction to make sense, $$\epsilon$$ must be non-zero. However, we can calculate that this value is $$\frac{2x\epsilon+\epsilon^2}{\epsilon}$$, or $$2x+\epsilon$$. At this point, we no longer need $$\epsilon$$ to be non-zero, so the slope can be said to be just $$2x$$. This sort of slippage between non-zero and zero for these infinitesimals is what made Berkeley refer to them as “the ghosts of departed quantities”. However, engineers, scientists and mathematicians who actually made use of the calculus rested content in the knowledge that the calculus delivered the goods.
In the 19th century, Cauchy, Bolzano, Weierstrass, Dedekind and Cantor sought to establish foundations for real analysis that gave no role to infinitesimals: in what became the canonical account of the calculus, Cantorian set theory and the $$\epsilon$$-$$\delta$$ formalization of the notion of a limit described in section 3.2 allowed for a fully rigorous development of real analysis.
Instead of taking particular infinitesimal values, one quantifies over values of real-valued variables $$\epsilon$$ and $$\delta$$. For example, the claim that the slope of the squaring function at $$x$$ is $$2x$$ is interpreted as saying that for any desired degree $$\epsilon$$ of approximation, there is some finite $$\delta$$ such that for any $$x'$$ within $$\delta$$ of $$x$$, the slope of the line from $$(x,x^2)$$ to $$(x', x'^2)$$ approximates $$2x$$ to within $$\epsilon$$. A single infinitesimal is replaced by a relation involving two nested quantifiers. The process of elimination of infinitesimal quantities from the calculus was a central part of a larger process known as the “arithmetization of analysis”, which aimed at removing kinematical and geometrical notions from the calculus in favour of purely arithmetical notions. (These are broadly construed to include the arithmetic of real and complex numbers. For a recent account of the history of real and complex analysis in the 19th century that also pays attention to foundational issues see Gray (2015).)
This infinitesimal-free program accomplished its goals successfully—among its major accomplishments were the rigorous definition of a continuous function at a point and the definition of the Riemann integral. However, one should keep in mind that other areas of mathematics, such as geometry, continued exploiting infinitesimal considerations and studied extensively non-Archimedean number systems. Archimedes’ axiom states:
given any two areas, or two distances, or any two quantities of the same sort, say $$A$$ and $$B$$, it is possible to add $$A$$ to itself finitely many times so that the quantity obtained is greater than $$B$$.
Non-Archimedean systems are ones in which this axiom does not hold. If the 17th century engagement with infinity, and Cantor’s work in set theory, can be seen as revolutions, the study of non-Archimedean mathematics in the second half of the 19th century can be likened to an infinitary uprising.
Many of the quantities considered in 17th century calculus, such as Leibnizian infinitesimals, do not obey the Archimedean axiom. An infinitesimal can be added to itself any finite number of times but the outcome of that process will never be greater than any finite quantity, however small. A pervasive historiographical tradition has argued that with the elimination of infinitely small quantities from the calculus, non-Archimedean quantities were relegated to engineering practice for a long time. According to the standard account, it was only in the 1960s that infinitesimals came back when Abraham Robinson presented his theory of non-standard analysis, which has received a lot of attention from philosophers and mathematicians (see section 3.4.2). Robinson’s theory gave a legitimate mathematical status to infinitely small and infinitely large quantities in the reconstruction of the infinitesimal calculus, now developed accordingly to rigorous model-theoretic techniques. For a long time, Robinson’s work was hailed as the first successful effort to develop a system of non-Archimedean quantities. But Philip Ehrlich, in a series of fundamental papers (including 1994, 2012), has argued that this widespread perception needs serious questioning. Indeed, he has shown convincingly that interest in non-Archimedean mathematics emerged in the 1870s and continued to grow in the hands of mathematicians such as Veronese, du Bois-Reymond, Levi-Civita, Hahn, Stolz, Hardy, and others.
It would be out of place in this entry to attempt even a small survey of the developments mentioned above. We simply refer to the reader to Ehrlich (1994) and (2006). The interconnection with many important related issues such as Conway’s surreal numbers (see section 3.4.3) and other alternative approaches to the construction of the real numbers, such as smooth infinitesimal analysis mentioned at the end of section 3.4.2, cannot be properly addressed here. See Salanskis and Sinaceur (1992), Ehrlich (1994), Berger, Osswald, and Schuster (2001), and Ehrlich (2012).
#### 3.4.2 Non-standard analysis and infinitesimal analysis
As a result of the rigorous definitions in the calculus mentioned above, from the mid-19th century most mathematicians working in analysis abandoned infinitesimals. However, in the mid-20th century, Robinson (1966) showed that it is possible to give a rigorous definition of infinitesimals, and that infinitesimals can be used in a non-standard development of real analysis (D. Laugwitz also did similar work around the same time, but Robinson’s system has been more widely discussed). While he developed his non-standard analysis using model theory, subsequent developments have also been grounded in algebra and topology. Robinson’s approach supplies an extended number system—the hyperreal number system—that contains the standard real number system, plus further ‘infinitesimal’ numbers whose absolute values are greater than 0, but less than any positive standard real number. Robinson’s construction of the hyperreals provides a set with the same cardinality as the standard reals. Simple modifications of the construction can create sets of hyperreals of larger cardinality.
Importantly, due to the logical techniques used in its construction, Robinson’s system behaves exactly like the standard finite real numbers for any sentence expressible within the algebraic language of addition and multiplication. Thus, every number other than 0 has a multiplicative inverse, and if $$x > y$$, then $$1/y > 1/x$$. In particular, this means that if $$\epsilon$$ is a positive infinitesimal number, then $$1/\epsilon$$ is an infinitely large number! Unlike Cantorian infinities of counting from section 3.3 these infinitely large numbers are subject to subtraction and division as well as addition and multiplication, and unlike the infinities of the extended real line from section 3.2, they behave just as nicely as the finite numbers with respect to them. For instance, statements such as these hold of standard real numbers as well as of the new infinitely small and infinitely large numbers:
$$x+y=y+x$$ (commutativity of addition);
$$x\cdot y=y\cdot x$$ (commutativity of multiplication);
$$x(y+z)=xy+xz$$ (distributivity of multiplication over addition).
In fact, Robinson’s hyperreals satisfy a “transfer principle”—if statements are formulated entirely within a first-order language for the reals, then they are true of the standard reals if and only if they are true for the hyperreals. So any proof of such a theorem in one system can be transferred to the other. This sometimes greatly simplifies calculations and proofs of theorems.
Consider the limit of the quantity $$((x+h)^3-x^3)/h$$ as $$h$$ approaches 0. In the standard reals, to show that this is $$3x^2$$, we need to show that for every $$\epsilon$$ there is a $$\delta$$ such that for any value of $$h$$ less than $$\delta$$, the corresponding value of this function is less than $$\epsilon$$ away from $$3x^2$$. In this case, it turns out that choosing $$\delta < \epsilon/4x$$ works when $$x$$ is sufficiently large, and choosing $$\delta < \epsilon$$ works when $$x$$ is sufficiently small, but figuring out these choices is difficult.
For the hyperreals though, it is sufficient to show that this value is infinitesimally close to $$3x^2$$ whenever h is infinitesimal.
\begin{align} \frac{(x+h)^3-x^3}{h} &= \frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} \\ &= 3x^2+3xh+h^2, \end{align}
and for any real $$x$$, $$3xh+h^2$$ is infinitesimal whenever $$h$$ is infinitesimal.) For any particular real $$\epsilon$$, this shows that there is some hyperreal $$\delta$$ that works (namely, any infinitesimal), and by the transfer principle we can conclude that for this real $$\epsilon$$ there is some real $$\delta$$ that works, and we no longer need to worry about the details of how to find it. Thus, we can validate the reasoning of Newton and Leibniz that allows them to treat infinitesimals as non-zero for calculations until we get to the final result, and then treat them as zero at the end. They really work like the “ghosts of departed quantities” that Berkeley satirized! (The extent to which Robinson’s system is a vindication of Leibniz and Newton was the subject of extended discussion in articles by Robinson and others. See Bos (1974) for a classic source of this debate.)
For results stated in a first-order logical language, the hyperreals and the standard reals satisfy the transfer principle. But for results about sets, they behave differently. Every bounded set of standard reals has a least upper bound. However, for instance, the set of infinitesimal hyperreals is bounded (every member is less than .00001, among other bounds), but there is no least upper bound (no infinitesimal is an upper bound for all of the others, and every finitely large upper bound can be decreased by some infinitesimal amount to give a smaller one). Edward Nelson (1977) pioneered an alternative approach—Internal Set Theory—on which the basic language of mathematics is enriched in order to allow us to distinguish between standard and non-standard real numbers, as well as “internal” and “external” sets. On Nelson’s approach, infinitesimals are non-standard real numbers that are smaller in absolute value than any positive standard real numbers. “Internal sets” are those that can be defined in the basic language, and they behave just the same as standard sets of standard reals—for instance, bounded internal sets always have a least upper bound. But the set of all infinitesimals, just like the set of all standard real numbers, is an “external set” of the theory that can’t be defined within the language, and thus doesn’t necessarily have a least upper bound.
The approaches pioneered by Robinson and Nelson do not allow us to prove results about the standard real numbers that cannot be proved using standard real analysis. However, these approaches do provide simpler—and, in some sense, more intuitive—proofs of many theorems of standard real analysis. (On the pedagogical benefits of non-standard analysis, see, for example, Keisler (1976)). And there are cases of results in real analysis that were first proven using non-standard real analysis (see, for example, Bernstein and Robinson (1966).) Moreover, these approaches clearly show that we do not need to adopt the $$\epsilon$$-$$\delta$$ formalization of the notion of a limit in order to have access to a fully rigorous development of real analysis.
The literature on non-standard analysis is very rich. See Dauben (1995) for a biography of Robinson with special emphasis on non-standard analysis. See also Goldblatt (1998) for a recent formal introduction and Cutland, di Nasso, and Ross (2006) for recent mathematical developments. The reader is referred to the extensive bibliographies contained in those volumes for further references.
An interesting alternative to nonstandard analysis, which allows for a development of substantive parts of mathematics, goes under the name of (smooth) infinitesimal analysis. This differs from both ordinary and nonstandard analysis by allowing nilpotent infinitesimals, namely ‘linelets’ $$dx$$ such that $$dx\neq 0$$ but $$dx\cdot dx = 0$$. The consistency of such a theory is proved using toposes in category theory. The best exposition of the topic is Bell (1998b) (see also Bell 1988a, 2019); philosophical aspects of the theory are discussed in Hellman and Shapiro (2018). Arthur (2013) discusses infinitesimal analysis in connection to Leibniz and makes points similar to those made by Bos (1974) on Leibniz and non-standard analysis. Constructive interpretations of non-standard analysis are developed in Salanskis (1999), which includes discussions of Nelson’s approach as well as the French school of non-standard analysis (Reeb, Harthong). For further discussion of infinitesimals, see Davis (1977), Thomason (1982), Bell (2005), and the entry on continuity and infinitesimals.
#### 3.4.3 Surreal numbers
Dedekind showed how to fill the gaps between rational numbers; Cantor showed how to extend (ordinal and cardinal) numbers beyond the existing finite numbers. John Conway (1976) integrated both ideas. He developed a very different system that generalizes von Neumann’s representation of Cantor’s ordinals, as well as Dedekind’s representation of the real numbers, to generate a much larger field that has become known as the “surreal numbers”. It contains a copy of each ordinal and cardinal number, while defining operations that work just like addition, subtraction, multiplication, division, exponentiation, and the taking of roots, on the standard reals. In particular, even the infinite and infinitesimal surreal numbers are amenable to these operations. Thus, as well as familiar numbers, we now have numbers such as $$\sqrt\omega, \omega/2, -\omega, 1/\omega, -\omega^\omega,$$ and so on. Indeed, as Ehrlich (2001, 2012) observes, the surreal numbers may plausibly be regarded as including “all numbers great and small”! The surreal numbers can apparently be applied in cases where there is no straightforward way to use hyperreals, as for example in the treatment of Pascal’s Wager, discussed in section 7.3—see Hájek (2003a).
Because the field of surreal numbers contains copies of all the ordinals, it is too big to form a set. But because the operations behave like the operations on the standard reals, these copies of the ordinals don’t represent counting. See the
for a summary of Conway’s construction of them. Other constructions of this same structure have been carried out by Knuth (1974) and Gonshor (1986) in more introductory texts.
### 3.5 Wrapup
Let us take stock. In response to worries that infinities in mathematics are suspect (section 2), rigorous mathematical theories of infinity have been developed (this section). But one might worry that, even if we can talk with mathematical rigor about infinities, they do not correspond to or apply to anything in the real world (as we think finite quantities do, however they do). Infinities might just be castles in the sky. Furthermore, one might suspect that we can, by some further mathematical developments, remove any reference to infinities in any practically important mathematics. The following section places this dialectic in the context of general questions about mathematical ontology, canvassing some important historical attempts to expunge the infinite from mathematics. It then explains the very difficult—perhaps insurmountable—challenges that any such attempt faces.
## 4. Mathematical ontology
Various questions about infinity naturally arise in the course of theorizing about ontology. If mathematical objects exist, are there infinitely many of them? Do individual infinite objects like the ones mentioned above exist, in addition to the infinitely many individual finite numbers? This article will not directly discuss the question of whether and in what sense mathematical objects exist. Instead, we will focus on the question of whether the infinities discussed above exist in the same sense as the finite integers. For more on the general questions of mathematical existence, see the entries on: logic and ontology, philosophy of mathematics, platonism in the philosophy of mathematics, nominalism in the philosophy of mathematics, fictionalism in the philosophy of mathematics, naturalism in the philosophy of mathematics, and logicism and neologicism.
Most viewpoints in the philosophy of mathematics accept the existence of all of the finite and infinite objects mentioned so far in exactly the same way that they accept the existence of finite integers. (Platonists might accept that this is literal existence, while fictionalists accept this as some sort of fictional existence, and others might have a different idea of what this means.) Standard set theories can prove the existence of all these objects, and for most mathematicians and philosophers, this is all that is needed. Logicist and neologicist accounts of mathematics may obtain the existence of infinite sets or infinitely many numbers by explicit postulation (as in the case of the axiom of infinity in Whitehead and Russell’s Principia Mathematica) or as the outcome of an implicit postulation (such as Hume’s Principle in Scottish neologicism, see Hale and Wright 2001, Heck 1997, 2011). While the axiom of infinity is easily stated and understood, Hume’s Principle has a peculiar form, for it postulates the existence of a function $$\#$$ that sends concepts into objects while respecting an equivalence relation $$\approx$$ among concepts. Formally it is stated as follows:
$\tag{HP} \forall B\forall C\, \# B = \# C \text{ iff } B \approx C$
where $$B \approx C$$ is short-hand for one of the many equivalent formulas of pure second order logic expressing the equivalence relation “there is a one to one correlation between the objects falling under $$B$$ and those falling under $$C$$”. Informally, it can be read as saying that two concepts $$B$$ and $$C$$ have the same ‘number’ if and only if there is a one to one correspondence among the objects that fall under $$B$$ and those that fall under $$C$$. Principles like HP that define a function from an equivalence relation are called abstraction principles. By presupposing the existence of a function that sends concepts into objects, Hume’s Principle exploits the possibility of defining infinitely many concepts that do not stand in the equivalence relation mentioned in its right-hand side to generate infinitely many natural numbers. There are other varieties of neologicism that do not postulate Hume’s Principle or an axiom of infinity at the outset but yield infinitely many natural numbers by means of other logical principles (see for instance Linsky and Zalta 2006). In addition, all these varieties of neologicism generate at least an infinite cardinal numbers and what is of philosophical relevance here is the different resources they use to establish these results.
Incidentally, Frege’s logicism and the neologicist program use one-to-one correspondence to state identity criteria for “concepts” even when infinitely many objects fall under the concepts. For alternative criteria for assigning numbers to infinite concepts see Mancosu (2015) and (2016).
In light of the paradoxes for early set theories (Russell’s paradox, the Burali-Forti paradox, and others), some mathematicians and philosophers worried that standard set theory might be inconsistent as well. One alternative viewpoint on mathematics is intuitionism, which only accepts the existence of mathematical objects whose construction can be carried out in some sense by the human mind. Intuitionism requires a revision of logic, since this limitation invalidates the Law of Excluded Middle—there are cases in which we can prove that the non-existence of a certain type of object would lead to a contradiction, but don’t have any method of constructing such an object, so that there might be a truth-value gap. Intuitionists often accept the Aristotelian limitation to “potential infinities”, rather than “actual infinities”, but there is also sophisticated intuitionistic reasoning about what types of infinite entities might exist. (For more, see the entries on intuitionistic logic and intuitionism in the philosophy of mathematics.)
Another viewpoint, associated with David Hilbert, is called finitism (see Hilbert 1926). Most finitists accept classical logic, but worry about the consistency of theories of infinite objects. Hilbert’s worries about consistency were fueled by the paradoxes that the new infinitary set-theoretic mathematics was giving rise to (Cantor’s inconsistent sets; Burali-Forti Paradox; Russell’s paradox etc.) Hilbert was convinced that quantification over such infinite totalities was at the root of the troubles. Finite objects, like configurations of strokes corresponding to the natural numbers, and finite sentences of a formal language, are taken by the Hilbertian finitist to be unproblematic, since these objects can in some sense be grasped individually and thus in their (potential) totality. But infinite objects are taken to be problematic: this includes Cantor’s higher ordinals and cardinals, and all the geometric, algebraic, and topological objects of which mathematicians were starting to develop detailed theories at the turn of the 20th century.
Hilbert’s proposed project (sometimes taken to be the starting point for formalism in the philosophy of mathematics) was to replace talk of these infinitary entities themselves with talk of the finitely long sentences that we ordinarily interpret as being about the entities. His goal was to axiomatize the theories of these infinite objects, and then to prove, using finitary means of syntactic reasoning about the language, that these theories are consistent. While this idea doesn’t deny the existence of the infinite objects, it suggests a methodological approach of only literally accepting the finitary objects, whether strokes standing for the integers or sentences. (See the entry on Hilbert’s program.)
Some mathematicians and philosophers have adopted finitism not merely as a methodological viewpoint, but also as a metaphysical one. Finite objects, like numbers and sentences, exist (in whatever sense mathematical objects exist), but infinite objects (like the completed set of all the natural numbers, or even arbitrary irrational numbers represented by Dedekind cuts) don’t. Versions of this view are often attributed to the 19th century number theorist and algebraist, Leopold Kronecker, who is quoted as saying “The dear God created the whole numbers; everything else is the work of man.” Kronecker criticized Cantor’s work as theology rather than mathematics. Hilbert started his program with the intent of defending Cantor while working within a framework dialectically acceptable to Kronecker’s allies. But when Kurt Gödel proved that no finitary theory for arithmetic and syntax can even prove its own consistency, let alone prove the consistency of a stronger theory for talking about completed infinities, Hilbert’s program was taken to have failed at disabusing the metaphysical finitists. Gödel’s incompleteness theorems apply most notably to Peano Arithmetic. The language of Peano Arithmetic is given by $$\{0,\,',\, +,\,\times\}$$, where $$'$$ is the successor function (it adds 1 to each number). Within it one can express ordinary arithmetical claims such as the commutativity of addition and the infinitude of prime numbers. The axioms of Peano Arithmetic tell us that the function $$'$$ is one-one; that 0 is not the successor of any number; that $$+$$ and $$\times$$ satisfy the usual recursive definitions; and finally we have a schema of induction for every formula $$A(x)$$ expressible in the language, i.e. if $$A(0)$$ and for all $$x, A(x) \rightarrow A(x')$$, then for all $$x, A(x)$$.
The detailed foundational work carried out in set theory and other foundational areas has in many ways dispelled the fear of impending doom that characterized the reaction to the paradoxes at the beginning of the twentieth century. As a consequence, most mathematicians today are perfectly happy to work with completed infinities. But there are still some finitists and intuitionists.
An intermediate position is that defended by classical “predicativists” such as Poincaré and Weyl. The theory, presented in a satisfactory logical way by Feferman and others, accepts the excluded middle on the natural numbers (and as such it is arguably committed to the existence of the set of natural numbers and in any case to accepting bivalence on the natural numbers) but does not accept the existence of the power set of the natural numbers. According to predicativism (see Feferman 2005), sets exist only if they are definable in some non-circular linguistic way. In accepting the excluded middle on the natural numbers and in making the existence of sets depend on our definitional abilities, this position is a sort of a compromise between a classical and a constructive viewpoint. In 1918, Hermann Weyl (Weyl 1918; see Kaufmann 1930 for a related program) presented the foundations of analysis within this framework and showed that a great part of classical analysis can be carried out within the framework by replacing talk of arbitrary sets of real numbers with arithmetical sequences of real numbers. Feferman 1988 gave a detailed formal presentation of the theory and proved that, on a certain reconstruction, the theory is a conservative extension of Peano Arithmetic. In addition, he also used the theory to state an important conjecture concerning how much mathematics is needed in physics. In Feferman 1984 and 1987, he proposed that all of mathematics used in physical theory can be recaptured in a predicative system of analysis. Using the metatheoretical result of conservativity mentioned above, he also exploited the argument to claim that Quine and Putnam’s indispensability arguments (see the entry on indispensability arguments in philosophy of mathematics) at best commit us to what Peano Arithmetic commits us to.
In contrast to the possibility of eliminating infinity as just described stand a number of results that show that some finitary statements can only be proved through infinitary considerations. These results originally emerge with Gödel’s incompleteness theorems (Gödel 1931) but have been recently refined by displaying statements of mathematical interest (whereas Gödel’s statements are of metamathematical interest but have no obvious mathematical interest). In order to understand the conceptual distinctions required, let us grant —as most logicians do—that all finitistic modes of reasoning are included in first-order Peano Arithmetic (henceforth PA).
A consequence of Gödel’s incompleteness theorems is that under the assumption that PA is consistent one can find a finitistic statement such that neither it nor its negation is provable from Peano Arithmetic. Gödel showed, through subtle coding of metamathematical notions in the language of arithmetic, how to express in the language of arithmetic a formula $$G$$ that “says” of itself that it is not provable. One can also ascertain that the formula is true on the natural numbers. Since all the finitary reasoning is assumed to be included in PA, establishing the truth of the Gödel sentence and of the new incompleteness results requires appeal to some “infinitary” principles (when the truth of the Gödel sentence $$G$$ is established through appeal to the statement expressing the consistency of PA, it is establishing the latter that requires some portion of infinitary reasoning, such as induction up to an infinite ordinal called $$\varepsilon_0)$$.
The situation is the same for the statement Con(PA) expressing the consistency of PA. Gödel’s second incompleteness theorem shows that neither it nor its negation can be proved from PA but an appeal to some infinitary reasoning shows it to hold in the natural numbers. While perfectly fine for the logician’s need and central to the evaluation of Hilbert’s program, Gödel’s sentences appear concocted from the point of view of the practicing mathematician. Within Hilbert’s program statements of PA expressible either without quantifiers or with a string of universal quantifiers followed by a non-quantificational formula count as finitistic statements. The statements $$G$$ and Con(PA) mentioned above also belong to this class. Finitistic statements with obvious mathematical relevance include basic properties such as the commutativity of addition as well as the statement of Fermat’s last theorem (whose proof has been established using higher mathematics but logicians are convinced that it can also be carried out in PA). Logicians have not been able to find finitistic statements with obvious mathematical significance that require a detour through the infinite, but they have been able to do the next best thing. They have found statements that have the form $$(\forall x) (\exists y) A(x, y),$$ which express a certain functional connection between numbers and have shown that such statements, although true, cannot be proved using only the resources of PA. Among the most famous such results are a modification of Ramsey’s finite theorem provided by Paris and Harrington (1977), and the proof that a theorem by Goodstein (1944) cannot be proved in PA (Kirby and Paris 1982). There are stronger results that are independent of even stronger systems that are studied within the context of reverse mathematics (e.g., Kruskal’s theorem is independent of predicative analysis—see Simpson 1985, 2002).
Such results show that even an arithmetical theory such as PA can express statements of mathematical significance (as opposed to statements concocted for logical purposes) that require some detour through the infinite to be proved, even though they can be stated purely arithmetically. In contrast to arithmetic, the mathematical incompleteness of set theory was shown by Gödel and Cohen for important statements such as the Axiom of Choice, the continuum hypothesis, etc. It is important to emphasize here that logicians working in set theory, recursion theory and proof theory probe the mysterious role of the infinite in proving results about the finite. It could be said that set theorists are mainly concerned with understanding how the demonstrable mathematical incompletability of Zermelo-Fraenkel (with Choice, i.e. ZFC) set theory, which is a consequence of results by Gödel and Cohen, can be addressed by finding new principles that will allow us to solve some of the most pressing questions concerned with the structure of the real numbers. In other words, since ZFC cannot be taken as a sufficient basis for the mathematics of infinity much of contemporary set theory is trying to solve the problem by finding new principles, which often take the form of assuming the existence of very large cardinals (see the entry on independence and large cardinals). The hope is that this work will lead to settling the Continuum Hypothesis and other major problems about the projective sets (on projective sets see entry on set theory).
Recursion theorists are also trying to understand the role that infinitary principles or compactness arguments play in our determination of results about the finite. And proof theorists would like to know when certain infinitistic theories can be justified through finitary means. Obviously, a more precise description of these developments goes well beyond the technical knowledge that we can presuppose here.
Most working mathematicians don’t worry about the existence of infinitely large sets and other objects. There are some other ontological worries about particular infinite sets, related to the Axiom of Choice, and some of the larger cardinalities mentioned above in the section on Cantor. But bigger worries arise in the context of whether there can be physical infinities.
For collections of sources on the classical foundational positions (finitism, intuitionism, predicativism) see van Heijenoort (1967), Ewald (1996), and Mancosu (1998). On finitism and intuitionism see the entries Hilbert’s program and intuitionism in mathematics. On predicativity see Feferman (2005). On Paris-Harrington see Katz and Reimann 2018; on Goodstein’s theorem see the friendly presentation in Stillwell (2010). Stillwell (2010) also has a chapter on large cardinals; for recent directions see Woodin (2011) and Steel (2015). On the interplay between finite and infinite in recursion theory see Hirschfeldt (2015).
The latter part of this entry will explore selected applications of mathematical concepts of infinity in theories of probability, decision, and spacetime, and some associated paradoxes. Before we turn to those theories, we warm up with some paradoxes and puzzles that link mathematics, metaphysical possibility, and physical possibility. There are many different paradoxes and puzzles that we might have included in this section. We consider a small sample of paradoxes and puzzles that some—e.g. Pruss (2018a)—have thought might motivate a return to Aristotle’s views on the impossibility of actual infinites.
In the
we discuss a puzzle due to al-Ghazālī that is of historical interest. For more see, for example, Rucker (1982), Moore (1990/2019), Oppy (2006), and Huemer (2016).
### 5.1 Hilbert’s Hotel
Hilbert’s Hotel has infinitely many rooms, labelled 1, 2, 3, …, each of which is currently occupied by a guest. Despite the fact that the hotel is already full, a new guest who turns up at reception is readily accommodated: for each n, the guest in room $$n$$ is moved to room $$n+1$$, and the new guest is installed in room 1. Indeed, despite the fact that the hotel is already full, it can accommodate infinitely many new guests: for each $$n$$, the guest in room $$n$$ is moved to room $$2n$$, and the new guests are installed in the odd-numbered rooms. Of course, if the infinitely many people in the odd-numbered rooms check out, there are infinitely many people left in Hilbert’s Hotel; but if infinitely many people check out from all but the first three rooms, only three people remain.
Some philosophers have thought that Hilbert’s Hotel supports an argument against the possibility of physically realized infinities:
1. If there could be physically realized infinities, then there could be a hotel with infinitely many rooms.
2. But if there could be a hotel with infinitely many rooms, then the events described in the preceding paragraph could occur.
3. But it is absurd to suppose that the events described in the preceding paragraph could occur.
So there cannot be physically realized infinities. (See, e.g., Craig (1979).)
This argument faces various challenges, depending on one’s views about what physical possibility amounts to. The first premise may be challenged: perhaps some kinds of physical infinities can be realized even though other kinds of physical infinities cannot: for example, perhaps there can be infinitely many stars even though there cannot be a hotel with infinitely many rooms. The second premise may also be challenged: even if there could be a hotel with infinitely many rooms, perhaps the events described in the story could not occur—the story was told at a high level of abstraction, and the details may matter. And the third premise is also questionable: it is not clearly absurd to suppose that there could be an infinite hotel in which guests come and go in the manner described.
For further discussion of Hilbert’s Hotel, see Gamow (1946), Huby (1971), Rucker (1982), Moore (1990/2019), Oppy (2006), Kragh (2014), Huemer (2016), and the entries on supertasks and cosmology and theology.
### 5.2 Thomson’s Lamp
Suppose that we have a lamp and a means of turning the lamp off and on. Suppose that the lamp is initially off. In the first minute, we change the state of the lamp from off to on. In the next half minute, we change the state of the lamp from on to off. … In the next $$1/2^n$$ minute we change the state of the lamp to the other state … . The question that we are invited to answer is: what is the state of the lamp at the end of the second minute?
The scenario is under-described. We can imagine that the means of turning the lamp off and on requires a spacetime location at which at least one physical quantity is infinite. If so, it is plausible to say that the case is impossible: there could not be such a lamp, and so there is no question to answer. Suppose, for example, that there is a switch that moves the same distance back and forth to turn the lamp on and off. Consider the speed at which the tip of the switch is travelling at the end of the second minute.
We can also imagine that the means of turning the lamp off and on does not involve any spacetime location at which at least one physical quantity is infinite; Grünbaum (1968) describes a scenario that fits this specification. In that case, the means of turning the lamp off and on converges to a specified state at the end of the two minutes, and there is an answer to the question that lies in the details of the specified state. But that answer is underdetermined by the brief description that we were initially given, as Benacerraf (1962) argues: we can have the lamp on at the end of the two minutes, or off at the end of the two minutes, depending upon the details of the implementation of Grünbaum’s proposal. Huemer (2016: 198–201) points out that if we hold fixed enough physics, then, before the end of the two minutes, the activation of the mechanism will stop changing the state of the lamp. So, depending upon your views about the range of what is possible, you may regard as impossible even cases in which there is no spacetime location at which at least one physical quantity is infinite.
Thomson’s lamp is an example of a supertask (Thomson coined this term): a process that involves infinitely many steps completed in a finite amount of time. The trick is that the steps are completed in shorter and shorter periods of time, corresponding to a convergent series. The lamp is one of many examples of supertasks that various authors have found paradoxical, while other authors have been less troubled by them. See the entry on supertasks.
For further discussion of Thomson’s lamp, see: Thomson (1954, 1967), Benacerraf (1962), Chihara (1965), Grünbaum (1968, 1973), Craig (1979), Berresford (1981), Moore (1990/2019), Earman and Norton (1996), McLaughlin (1998), Oppy (2006), Huemer (2016), and Pruss (2018a).
### 5.3 Paralysis
Suppose that Achilles wants to run from $$A$$ to $$B$$ but there are infinitely many gods who—unbeknownst to one another and to Achilles—each have reason to stop him from getting to $$B$$. God 1 resolves to instantaneously paralyse Achilles if and when he reaches halfway between $$A$$ and $$B$$. God 2 resolves to instantaneously paralyse Achilles if and when he reaches one quarter of the way from $$A$$ to $$B$$. … God $$n$$ resolves to instantaneously paralyse Achilles if and when he reaches $$1/2^n$$ of the way from $$A$$ to $$B$$. … Since all of the Gods are able to act on their resolve, Achilles is unable to move: for, if he moved, he would violate the intentions of infinitely many Gods. But, until he does move, none of the Gods act on their intentions. So what actually stops him from moving? Isn’t it absurd to suppose that someone can be rendered immobile by a nested sequence of conditional intentions upon which no one acts?
Suppose instead that each of the Gods erects a force field, placed in a parallel manner to the previous case, that it is impossible for Achilles to cross. Then Achilles is completely immobilized. On the assumption that it is possible for an infinite number of Gods to collectively create such a force field in the manner described, there is a straightforward explanation for Achilles’ inability to move. Of course, granted this assumption, there is no single God whose force field immobilizes Achilles; indeed, there is no finite collection of Gods whose collective force field does so; indeed, no force field touches him at all. Is this possible? Presumably one should come to the same verdict regarding the conditional intentions case as in this case.
Depending on your views about the range of possibility, there is much in this story that you may think is impossible. You may think that it is impossible for there to be Gods who can act as required; for example, depending on your conception of the Gods and their actions, you might think that the story requires instantaneous action at a distance. You may think that it is impossible for force fields to be positioned with unbounded degrees of accuracy. And so on. However, if there is nothing in the set-up that makes you baulk, and if further elaboration of the set-up does not introduce any singularities, then it seems that you should just accept the conclusions with equanimity: Achilles is rendered immobile by conditional intentions on which no one acts, or by a set of force fields none of which he is in direct contact with. Bizarrely counterfactual circumstances have bizarre consequences.
For further discussion of this case, see: Benardete (1964, which introduces it), Moore (1990/2019), Priest (1999), Hawthorne (2000), Perez-Laraudogoitia (2000, 2003), Yablo (2000), Oppy (2006), Koons (2014), and Huemer (2016), Caie (2018).
We have started to see that infinity seems to be both friend and foe—it features in powerful mathematics, but also in some vexing conundrums. We will see more of its Manichean nature in the following sections on probability, decision theory, and space and time. We will also see how sophisticated methods for reclaiming it have been developed.
## 6. Probability
Probability theory runs relatively smoothly in the finite realm, but puzzles emerge when infinities are afoot. There are multiple sources of infinitude, arising both in the mathematics and the interpretation of probability. We will firstly discuss more informally these sources, and then progress to more advanced issues.
### 6.1 Infinitude in the mathematics of probability: basics
Let us begin with the mathematics. Probability theory assumes that we have a set of “possibilities” or “outcomes”, called a sample space, regarded as ways the world could be, or the possible outcomes of a random experiment. For many purposes, an infinite set is assumed. For example, we may toss a coin repeatedly and be interested in how many tosses it takes until we see the first heads. The number could be 1, or 2, or 3, or … Here, the sample space is denumerable. Or we might consider picking a point at random from the [0, 1] interval of the real line—e.g., we might imagine throwing an idealized dart at a representation of that interval, and consider the point on which it lands. Here, the sample space is uncountable, because it is infinitely divisible and has limits of sequences, but bounded. Or we might consider sampling a quantity that is governed by a normal distribution, the bell-shaped distribution that is used to model various quantities in the real world. Here, infinitude enters twice over: the sample space is both uncountable and unbounded, being the entire real line.
Orthodox probability theory assigns real numbers between 0 and 1 (inclusive) to subsets of the sample space, and again we encounter infinitude: there are uncountably many possible probability values. We will soon see how we encounter it again in the way that these values are additive.
### 6.2 Infinitude in the interpretations of probability
Infinitary considerations also enter into certain interpretations of probability—attempts to explain what probabilities are and what probability statements mean. (See the entry on interpretations of probability for more details on what follows.) Hypothetical frequentism regards probabilities as limiting relative frequencies in hypothetical infinite sequences of trials. For example, we may toss a coin repeatedly, generating a sequence of outcomes—e.g.
We can keep track after each trial of the relative frequency of heads so far: the ratio of the number of heads to the total number of tosses. In our example, the sequence of relative frequencies is
$\frac{1}{1}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{3}{5}, \frac{3}{6}, \frac{3}{7}, \frac{4}{8}, \ldots$
We may then imagine this sequence extended indefinitely, and identify the probability of heads with the limit of this sequence. However, the very same results may be reordered, one way or another, to generate any limiting relative frequency in [0, 1] whatsoever, if there are infinitely many heads and infinitely many tails. Infinitude rears its ugly head here—for a finite sequence, reordering can make no difference to the relative frequencies of its outcomes.
According to Popper’s propensity interpretation, a probability $$p$$ of an outcome of a certain type is a propensity of a repeatable experiment to produce outcomes of that type with limiting relative frequency $$p$$. Again, infinitude is central to this interpretation, and its ugly head rears as it did for hypothetical frequentism. The best-system interpretation of probability, associated with Lewis (1994) and others, is also prey to problems if there are infinitely many events of a particular kind in the universe—for example, infinitely many coin tosses. As Elga (2004) shows, the interpretation’s central notion of fit is compromised. And even the subjective probabilities of idealized rational agents have tacit infinitary assumptions underlying them—for example, that the agents are logically omniscient, and their probability assignments are infinitely sharp (single real numbers). These assumptions have also been regarded as problematic, especially when modeling agents who are anything like us.
### 6.3 Infinitude in the mathematics of probability: more advanced issues
To state some of the thornier puzzles generated by the mathematics of probability, we need a more formal presentation. Kolmogorov’s (1933/1950) axiomatization begins with a finite set $$\Omega$$ and an algebra $$F$$ of subsets of $$\Omega$$: a set closed under complementation and union. The members of $$\Omega$$ are known as states while the members of $$F$$ are known as events. A probability function is a function from $$F$$ to the real numbers. It is non-negative, assigns 1 to $$\Omega$$, and it is (finitely) additive—the probability that one of two mutually exclusive events occurs is the sum of their individual probabilities:
If $$A$$ and $$B$$ are disjoint sets in $$F$$, then $$P(A \cup B) = P(A) + P(B)$$.
Kolmogorov goes on to generalize this to an infinite $$\Omega$$, and to a sigma-algebra $$F$$ of subsets of $$\Omega$$: a set closed under complementation and countable union. Additivity is strengthened to hold also in infinite cases:
If $$\{A_i\}$$ is a countably infinite collection of (pairwise) disjoint sets, each in $$F$$, then $P(\bigcup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} P(A_n)$
Some have felt that restricting additivity to merely countable sums is arbitrary, and is merely an artifact of the summation technique introduced in section 3.2. An alternate technique for summing infinite sets of non-negative numbers takes advantage of the fact that a sum of nonnegative numbers as defined earlier doesn’t depend on the order of the terms. We consider the set of all partial sums of arbitrary finite subsets of the set, and take the least upper bound of this set to represent the sum of the set as a whole. If this sum is some positive finite value $$k$$, then we can see that at most $$nk$$ of the terms in the set being summed can be greater than 1/$$n$$. Since every positive real number is greater than 1/$$n$$ for some $$n$$, this means that the set of positive elements of the set is a countable union of finite sets, and thus must be countable. That is, if a set being summed in this way has uncountably many non-zero elements, the sum must be infinite.
So if we require full (unrestricted) additivity, rather than merely countable additivity, then we can see that at most countably many events have positive probability, and their probabilities sum to 1. A probability distribution with these features, where events of probability 0 have been removed, is known as a discrete distribution (such as the Poisson, geometric, or negative binomial distributions). For such a distribution, the probabilities of the individual states determine the probabilities of all events through the use of additivity.
However, many applications of probability require what are known as continuous distributions (such as the uniform/rectangular, normal, and beta distributions), and thus require a restriction to countable additivity. In a continuous distribution, there are uncountably many states, usually named by real numbers. Each individual state has probability 0, even though events containing uncountably many states often have non-zero probability. (This violates full additivity.) However, in the common continuous distributions, there is usually a way to define a probability density for each state, such that the probability of any event is the integral of the density over the states that make it up. In finite and discrete distributions, it is standard to treat events of probability 0 as if they do not happen, while in continuous distributions there is always some event of probability 0 that occurs.
For finite and discrete distributions, there is a straightforward definition of a concept of conditional probability. For any two events $$A$$ and $$B$$, the probability of $$A$$ conditional on $$B$$, notated as $$P(A\mid B)$$, is defined as $$P(A \amp B)/P(B)$$, if the probability of B is non-zero, and undefined otherwise. For any fixed $$B$$, the function $$P(\_ \mid B)$$ is another probability function on the same space. We can prove the Law of Total Probability. If $$B_1, B_2, B_3,\ldots$$ form a partition (that is, every outcome is in exactly one of the $$B_i)$$ then:
$P(A) = \sum_i P(B_i)P(A \mid B_i).$
This tells us that the unconditional probability $$P(A)$$ is a weighted average of the conditional probabilities $$P(A\mid B_i)$$.
However, for distributions that are not discrete, so that the set of states is uncountable in an essential way, and events of probability 0 often occur, we can’t use this ratio definition of conditional probability, since it would involve dividing by 0. However, Kolmogorov notes (1933/1950, Ch. 5) that for any suitably nice partition, it is still possible to come up with a definition of conditional probability conditional on events in this partition, satisfying a generalization of the Law of Total Probability, replacing the sum with an integral:
$P(A) = \int P(A \mid B)dP(B)$
(The possibility of finding a conditional probability satisfying this integral formula is known as ‘disintegrability’, and it is equivalent to a principle known as “conglomerability”. For philosophical arguments in favor of this, see Easwaran (2013b, 2019), Rescorla (2018).) For more on determining when conditional probabilities that satisfy this rule exist, see Hoffmann-Jørgensen (1971), and for more on how to use probability densities to calculate these conditional probabilities, see Chang and Pollard (1997).
However, there are some difficulties with this account of conditional probability. Kolmogorov notes that if the original probability space is the uniform distribution of points on a sphere, and if $$B$$ ranges over the set of longitudes (great circles through the poles), then probability conditional on a line of longitude will not be uniform, but instead will be concentrated near the equator. (This fact is known as the “Borel paradox”, because Emile Borel investigated it even before Kolmogorov’s work.) Since every great circle on a sphere can be viewed as a line of longitude with an appropriate choice of pole, this makes the probability conditional on an event depend not only on which event was chosen, but also which family of alternatives it is contrasted with. (We can view each great circle as a longitudinal line through multiple different poles, each of which disagrees about where the equator is.)
Some have found this consequence troubling enough that they have endorsed an alternative account of conditional probability that gives up the Law of Total Probability, and instead insists that $$P(A|B)$$ has a unique value regardless of which alternatives to $$B$$ are under consideration. However, this also has some unpalatable consequences. Since $$P(A)$$ is no longer the average of $$P(A|B)$$ where $$B$$ ranges over the elements of a partition, this means that there are some partitions such that every element of the partition is positively correlated with $$A$$. Furthermore, the conditional probability functions generated in this way no longer satisfy countable additivity (Kadane, Schervish, and Seidenfeld 1996, Seidenfeld, Schervish, and Kadane 2001, 2013).
But some, starting with de Finetti (1937, 1972, 1974) have argued on other grounds that we should give up even countable additivity and only accept finite additivity, with a correspondingly broader class of probability distributions. One of de Finetti’s chief arguments involves an infinite lottery with each natural number appearing on exactly one ticket. We would like to assign each ticket the same probability of being drawn. Under countable additivity, this is not possible. For if we assign probability 0 to each number’s being picked, then the sum of all these probabilities is again 0; yet the union of all of these events has probability 1 (since it is guaranteed that some number will be picked), and $$1 \ne 0$$. On the other hand, if we assign some (real-valued) probability $$\varepsilon \gt 0$$ to each number being picked, then the sum of these probabilities diverges to $$\infty$$, and $$1 \ne \infty$$. If we drop countable additivity, however, then we may assign 0 to each event and 1 to their union without contradiction. In the
we explore how an alternative approach to Kolmogorov’s, a non-Archimedean probability theory (NAP), can account for de Finetti’s lottery by assigning an infinitesimal probability to each ticket.
However, a probability function that satisfies finite additivity without satisfying countable additivity is mathematically much more complicated than one that satisfies countable additivity. To even prove the existence of such a function over the algebra of subsets of a countable set of states requires the Axiom of Choice. With countable additivity, it is possible to specify a discrete probability function by enumerating the probabilities of the countably many states, and it is possible to specify a continuous probability function by enumerating the probabilities of the countably many rational open sets. But if merely finite additivity is assumed, specifying a probability function even on a countable state space may require specifying the probabilities of uncountably many events, rather than calculating the probabilities of these events from the countably many probabilities of the states. Furthermore, with such probability functions, many standard convergence results like the Strong Law of Large Numbers fail.
For more on infinite probability spaces where only finitely additive probability holds, see Bartha (2004), Bingham (2010), de Finetti (1937/1989), Dubins (1975), Easwaran (2013b), Hill and Lane (1985), Howson (2008), Kadane, Schervish, and Seidenfeld (1986), Schervish, Seidenfeld, and Kadane (1984), Seidenfeld (2001), Seidenfeld, Schervish, and Kadane, (2014).
A lively debate concerns a further constraint on probabilities that may be regarded as desirable: anything that is possible should be assigned positive probability. This is known as regularity:
Regularity
If $$X$$ is a non-empty subset of $$\Omega$$, then $$P(X) \gt 0$$.
Folk thinking about probability seems to be committed to regularity—“if it has probability zero, it can’t happen!”, as one might say.
We have seen a striking violation of regularity in de Finetti’s lottery: his assignment of 0 to each ticket. Regularity may be preserved here by countably additive probabilities, but at the expense of a uniform distribution—for example, $$\frac{1}{2}$$ to ticket 1, $$\frac{1}{4}$$, to ticket 2, $$\frac{1}{8}$$ to ticket 3, and so on. It may be shown that if $$F$$ is uncountable, a Kolmogorovian (real-valued) probability distribution must violate regularity. (See e.g. Hájek 2003b.) This has led to a cottage industry of exploring whether regularity can be preserved by allowing the ranges of probability functions to be richer fields than the real numbers. For example, Bernstein and Wattenberg (1969) show that there exists a regular hyperreal-valued probability function for the dart throw at [0, 1] that we imagined earlier. Each landing point receives infinitesimal probability. Williamson (2007) argues that an infinite sequence of tosses of a fair coin all landing heads must receive probability 0 rather than some infinitesimal probability; Howson (2019) challenges the argument. The debate for and against preserving regularity continues, with Easwaran (2014) and Pruss (2012, 2013b, 2014) against, Benci, Horsten and Wenmackers (2012, 2016) for—offering NAP as a way of doing so, again assigning infinitesimal probabilities where Kolmogorov’s theory would assign 0’s.
For several further puzzles involving probability in infinite spaces, see Arntzenius, Elga, and Hawthorne (2004) and Bartha and Hitchcock (1999). For more on infinitesimal probabilities in philosophical applications, see Benci, Horsten and Wenmackers (2012, 2018), Easwaran (2014), Halpern (2010), Hofweber (2014a, 2014b), Howson (2018), Kremer (2014), Lauvers (2017), Pruss (2012, 2013, 2014, 2018a, 2018b), van Fraassen (1976), and Wenmackers and Horsten (2013).
Infinitesimal probabilities are also appealed to in game theory. For example, the concept of trembling hand perfect equilibrium assumes that each player in a game may make a mistake with positive but negligible probability, which may be regarded as infinitesimal—see Halpern and Moses (2017). We will see further use of infinitesimal probabilities in decision theory, to which we now turn.
## 7. Decision
When you make a decision, what you choose and the way the world turns out together determine an outcome, to which you assign a utility that measures how desirable it is for you. In a decision under certainty, each action that you may perform has exactly one possible outcome. In that case, it seems that you should simply perform an action that has maximal utility. (Read on, however!) In a decision under risk, you assign probabilities to the various ways the world could turn out—the possible states. Suppose that there are various actions $$A_i$$ that you could perform, and various states $$S_j$$ to which you assign probabilities $$p_j$$. Together they determine outcomes to which you assign utilities $$u_{ij}$$. Classic decision theory says that you should maximize expected utility: you should perform an action that maximizes the weighted average of the utilities associated with that action, the weights given by your probabilities. Formally, you should maximize
$\text{EU}(A_i) = \sum_j p_j u_{ij}$
(We ignore complications and variations that are irrelevant here—see the entries on normative theories of rational choice: expected utility and decision theory.)
In standard cases, it is assumed that
1. there are finitely many possible actions,
2. finitely many states of the world,
and that
1. the utilities are finite.
However, we may drop each of these assumptions, yielding three different sources of infinitude in a decision problem. Accordingly, we will present some well-known problems that arise when one or more of these assumptions are violated. We begin with a decision under certainty.
### 7.1 Infinitely many possible actions: Ever-better wine
Pollock (1983) offers the following puzzle. You have a bottle of Ever-better wine, which keeps improving as it ages: the later you open it, the better it will be. When should you open it? There’s a good sense in which any time is too soon: opening it slightly later would be better. But the worst option is never to open it, and to avoid this it must be opened at some time. This decision problem has uncountably many possible actions, but we could easily make them denumerable by adding that the bottle can only be opened at discrete times—e.g. on the hour. You would gladly perform an action that has maximal utility, but here there is no such action! This problem displays an intriguing feature that Bartha, Barker and Hájek (2013) call discontinuity at infinity: “an infinite sequence of choices, each apparently sanctioned by plausible principles, converges … to a ‘limit choice’ whose utility is distinct from, and typically much lower than, the limit approached by the utilities of the choices in the sequence” (630). Their paper discusses other decision problems with this feature. For more discussion of this kind of phenomenon, see Chow, Robbins, and Siegmund (1971) and Seidenfeld (1981).
### 7.2 Infinitely many states: the St. Petersburg paradox
A fair coin is tossed. If it comes up heads, you receive $2. If it comes up tails, the coin is tossed for a second time. If it then comes up heads, you receive$4. If it comes up tails, the coin is tossed for a third time. If it then comes up heads, you receive $8. If it comes up tails, the coin is tossed for a fourth time. And so on. We continue until the coin comes up heads. If this takes n tosses, then you win$$$2^n$$.
How much should you be prepared to pay to play this game? You have a 1/2 chance of winning $2; and a 1/4 chance of winning$4; and a 1/8 chance of winning $8; ….; and a $$1/2^n$$ chance of winning$$$2^n$$; and … . Hence, your expected payoff from playing the St. Petersburg game is infinite:
\begin{align} (\frac{1}{2}\times 2) &+ (\frac{1}{4} \times 4) + (\frac{1}{8} \times 8) + \cdots + (\frac{1}{2^n} \times 2^n) + \cdots \\ &= 1 + 1 + 1 + \cdots \end{align}
If we identify dollars won with utilities, the game has infinite expected utility.
Decision theory seems to say that you should be prepared to pay any finite amount to play this game. But most people think this is crazy; indeed, most would only pay a few dollars to play (Neugebauer 2010). And decision theory seems to say that you should be prepared to pay any finite amount for a ticket in any finite lottery whose payoff is a single play of this game. That may seem really crazy.
You might object that the utility of money decreases as you obtain more of it: if the rate of this decrease is sufficiently large, then the expected value of playing the game is finite. Daniel Bernoulli argued that utility goes by the logarithm of the amount of money, and indeed replacing the dollar amounts with their logarithms yields a finite expected utility. However, we can retell the story in terms of utilities themselves. And we can retell it with super-exponential escalation of the value of the payoffs: taking logarithms then gives us exactly the original expected utility. (See Menger 1967/1934.) In fact, as long as utilities are unbounded, we can fashion a version of the game that has infinite expected utility.
So you might object that the utilities are bounded. (See e.g., Hardin 1982.) However, unbounded quantities abound—length, volume, mass, curvature, temperature, and so on. Why is utility unlike them in this regard? Moreover, one might imagine a case in which utilities are intimately linked to another such quantity—e.g., the further away you get from some undesirable place, the better—and an unbounded function might link them. Moreover, as we have noted, probability theory is already shot through with infinitude; we need a principled reason why this kind of infinitude should be shunned. (See Nover and Hájek 2004 for further discussion.) And perhaps it is not crazy after all to value the St. Petersburg game infinitely. After all, it dominates each truncation of the game, which pays nothing if heads has not come up after $$n$$ trials (for each $$n$$): the St. Petersburg game’s outcome is equally good in finitely many states, and strictly better in infinitely many. Plausibly, then, it should be preferred to all these truncations of the game (Hájek and Nover 2006, 2008)—its desirability is greater than $$n$$, for each $$n$$.
For further discussion of the St. Petersburg game, see, for example: Samuelson (1977), Jeffrey (1983), Weirich (1984), Cowen and High (1988), Jordan (1994), Chalmers (2002), Peters (2011) and the entry on the St. Petersburg paradox.
Related but different problems arise in the Pasadena game, a St. Petersburg-like game in which the expected payoff is apparently undefined (rather than infinite). Then, it seems that decision theory goes silent regarding the value of the game. And yet various choices regarding the game seem to be rationally required—e.g. preferring the game plus a \$1 to the game itself. For further discussion, see e.g. Nover and Hájek (2004), Hájek and Nover (2006, 2008), Hájek (2014), Easwaran (2008), Bartha (2016) and Colyvan and Hájek (2016).
### 7.3 Infinite utility: Pascal’s Wager
In the St. Petersburg game, each possible payoff is finite; it is the way in which they are averaged by the expected utility formula that yields the infinitude. We now turn to a classic decision problem in which a possible payoff itself is infinite.
Pascal maintains that we cannot know whether God exists or not, but he argues that we can solve the decision problem of whether or not to ‘wager for God’—roughly, to cultivate belief in God. There are two available courses of action: wager for God, or fail to wager for God. There are two relevant conceivable states of the world: God exists, or God does not exist. The probability that God exists is $$p$$, whence the probability assigned that God does not exist is $$1 - p$$. The utility of wagering for God, if God exists—salvation forever—is infinite. All of the other utilities—of an earthly life of some finite duration—are finite. We may formulate the resulting decision table as follows:
God exists God does not exist Probabilities: $$p$$ $$1-p$$ Wager for God $$\infty$$ $$f_1$$ Wager against God $$f_2$$ $$f_3$$
We may now do the expected utility calculations:
The expected utility of wagering for God is
$p\cdot \infty + (1 - p)\cdot f_1 = \infty.$
The expected utility of failing to wager for God is
$p \cdot f_2 + (1 - p)\cdot f_3 = \text{ a finite value}.$
In order to maximize expected utility, one ought to wager for God.
Among the many objections that have been levelled at Pascal’s Wager, several focus on the role that ‘$$\infty$$’ plays in the argument. Can utilities be infinite? There is a considerable literature that considers possible extensions of our decision rule, and possible modifications to the framework within which the decision problem is framed. However, to date, there is no widely accepted alternative formulation of Pascal’s Wager that avoids all these difficulties that focus on the role that ‘$$\infty$$’ plays in the argument. And once infinite utilities are countenanced, it seems that we should be open to infinitesimal probabilities also. But then there is the prospect that when an infinite utility and an infinitesimal probability are multiplied in the expected utility formula, the product may be a finite number. Will wagering for God still maximize expected utility? These issues and more are discussed in the entry on Pascal’s wager.
For further discussions of the treatment of infinity in Pascal’s Wager, see, for example: Duff (1986), Oppy (1991, 2018), Hájek (2003a, 2018), Bartha (2007, 2018), Bartha and Pasternack (2018), Monton (2011), and Wenmackers (2018).
### 7.4 Infinite utility streams
So far we have been considering decisions in which one’s payoff comes in a single ‘hit’: a reward (or punishment) comes all at once. However, we can also consider cases in which one is to choose between different infinite utility streams—e.g., streams of finite daily utility that accumulate over an infinite future. There is an obvious candidate for evaluating the utility of a finite stream: add the utilities along the stream. But this method is not available when we have an infinite stream; we require additional principles to help us evaluate such streams, and it is not obvious what those principles should be.
Suppose that a day spent in Heaven has utility 1 and a day spent in Hell has utility $$-1$$. Suppose further that, for all $$n$$, $$n$$ days in Heaven have utility $$n$$, and $$n$$ days in Hell have utility $$-n$$. Suppose, finally, that, for all $$m$$ and $$n$$, any combination of $$m$$ days in Heaven and $$n$$ days in Hell has utility $$m - n$$.
Here are some candidate principles for the comparison of alternative possible future utility streams:
1. One should prefer the possible future utility stream that has maximal total utility (if there is one).
2. If more than one possible future utility stream has divergent utility—i.e. is such that there is no finite value to which the total utility of the stream converges as the number of days increases—one should prefer the divergent utility stream whose partial sum is dominant (if there is one). This means that on some days, the sum of utility to that day is greater than for any other stream, and on no days, the sum of utility to that day is less than for some other stream.
3. If more than one possible future utility stream has divergent utility, one should be indifferent between divergent future utility streams that are permutations of one another.
Consider the choice between the following two infinite utility streams:
1. An infinite number of days in Heaven.
2. An infinite number of days in Heaven preceded by a finite number of days in Hell.
Principle 2 says correctly that we should prefer (a) to (b).
However, consider the choice between the following two options:
1. An infinite number of alternating days, first in Heaven, and then in Hell.
2. An infinite number of alternating days, first in Hell, and then in Heaven.
Principle 2 says, incorrectly presumably, that we should prefer (c) to (d).
Now consider the choice between the following two options:
1. An infinite number of alternating days, first in Heaven, and then in Hell.
2. An infinite number of alternating days, first one in Heaven and one in Hell, then two in Heaven and one in Hell, then three in Heaven and one in Hell, and so on.
While Principle 3 says, (perhaps) correctly that we should be indifferent between (c) and (d), it also says, (surely) incorrectly, that we should be indifferent between (e) and (f).
In the face of these difficulties, you might consider weakening the principles:
1. If more than one possible future utility stream has divergent utility, prefer the divergent possible future utility stream that is step-by-step dominant (if there is one).
2. If more than one possible future utility stream has divergent utility, maintain indifference between divergent possible future utility streams that are finite permutations of one another (i.e. that can be derived from one another by finitely many exchanges at neighbouring steps).
But this pair of principles yields no verdict in the case of (e) and (f), and so does not yield a complete set of principles.
More generally, it is hard to codify rules for choosing among infinite utility streams. Indeed, there are some impossibility results in the economics literature which suggest that there is no fully satisfactory theory that countenances them.
For further discussion of infinite utility streams, see, for example: Segerberg (1976), Jeffrey (1983), Nelson (1991), Vallentyne (1993, 1994, 1995), Cain (1995), Ng (1995), Van Liedekerke (1995), Lauwers (1997a, 1997b, 1997c, 1997d), Vallentyne and Kagan (1997), Basu and Mitra (2003), Crespo, Nuñez, and Rincou-Zapatero (2009), Bartha, Barker and Hájek (2014), and Jonsson and Voorneveld (2015).
Each of these decisions problems wears its infinitude on its sleeve: it is obvious that there are infinitely many possible actions, or infinitely many states, or infinite utility, or infinite streams of utility. However, in some problems, such infinitude is not foregrounded, but it is tacitly there nonetheless. The two-envelope paradox is such a problem. See the
• “An Infinite Decision Puzzle”: Barrett and Arntzenius (1999)
• “Trumped”: Arntzenius and McCarthy (1997)
• “Rouble Trouble”: Arntzenius and Barrett (1999)
• “The Airtight Dutch Book”: McGee (1999)
• Several paradoxes in Arntzenius, Elga and Hawthorne (2004)
• “The Cable Guy”: Hájek (2005).
## 8. Space and time
Considering whether space and time are infinite in extent and divisibility has led to many famous puzzles, paradoxes and antimonies. It was on account of such paradoxes that Kant was led to the claim that whether space is finite or infinite escapes any possible empirical determination. Kant’s presentation of the antinomies rested on a number of assumptions (such as the distinction between infinite and unbounded) that were undermined by later results in mathematics or were simply found to be philosophically questionable. Another interesting paradox relates to divisibility. In this section we discuss Kant on the antinomies of space and time and a measure-theoretic solution to this paradox of divisibility. This is followed by a quick overview of some developments in non-Euclidean geometries and relativistic cosmology. In the final part, we mention some recent developments in cosmic topology, an area of cosmology that attempts to determine whether space is finite or infinite by a combination of empirical observation and mathematical theorizing. The emphasis will be on the latter aspect.
### 8.1 Antinomies of space and time
Many philosophers have devised paradoxes and even putative ‘antinomies’ that exploit the structural features of space and time in a way that essentially involves the infinite. Among the ancients, Zeno is renowned for his paradoxes of space, time and motion. They involve infinitely many spatial or temporal subdivisions or processes that are putatively impossible—see the entry on Zeno’s paradoxes. Among the moderns, Kant is particularly notable for his treatment of the extent of space and time in his ‘First Antinomy of Pure Reason’. We turn to it now.
#### 8.1.1 Kant
In the Critique of Pure Reason, A426–A434, B454–B462—Kant gives ‘proofs’ of conflicting theses—‘thesis’ and ‘antithesis’—about the extent of space and time. The ‘thesis’ says that:
1. the world has a beginning in time; and
2. the world has a finite extension.
The ‘antithesis’ says:
1. the world has no beginning in time; and
2. the world has infinite extension.
To a reasonable approximation, the ‘proofs’ run as follows:
1. If the world has no beginning in time, then, up to any given moment, an eternity has elapsed: there has passed away an infinite series of successive states of things. But the infinity of a series consists in the fact that it can never be completed through successive synthesis. So it is impossible for an infinite series of successive states of things to have passed away: the world has a beginning in time.
2. Since infinite extension cannot be thought in a single completed act of thought, the world can only be thought to have infinite extension through an act of synthesis in which completion is achieved via the addition of units. But an act of synthesis that achieved completion via the addition of units requires the lapse of an infinite amount of time—and we have already seen, in (a), that this is impossible. Consequently, it cannot be thought that the world has infinite extension. So the world does not have infinite extension.
3. Something begins to exist only if there is a prior time at which it does not exist. Hence, if the world has a beginning in time, there must be an earlier time at which it does not exist: an empty time. But nothing can come into existence in an empty time, since there is no sufficient reason for the thing to come into existence in one rather than another part of empty time. So the world does not have a beginning in time.
4. If the world has finite extension, then the world is contained in an unlimited empty space. Consequently, the objects in the world are not only related in space but also related to space. In particular, the relation of the world to empty space is a relation of the world to no object, i.e. to nothing. But there can be no such relation. So the world has infinite extension.
Much in Kant’s discussion of the antinomies of space and time is marred by his conflation of the modern definition of infinity as lack of finitude with the Aristotelian conception of the impossibility of completion. In addition, at A487/B515 we have confirmation that Kant uses “infinite” and “unbounded”, as well as “finite” and “bounded”, synonymously: “For if it [the magnitude of the world in space] is infinite and unbounded, then it is too big for every possible empirical concept. If it is finite and bounded, then you can rightfully ask: What determines this boundary?” It was only with the work of Bernhard Riemann in the nineteenth century that geometrical concepts of space were introduced that allowed the decoupling of unboundedness and infinity (and correspondingly of bounded and finite). See section 8.2.
You can find further—sometimes sympathetic—discussion of these arguments in Bennett (1966), Huby (1971), Whitrow (1978), Craig (1979), Moore (1990/2019), Oppy (2006), Huemer (2016), and the entry on Kant’s Critique of Metaphysics.
#### 8.1.2 Measure
Here is a Zeno-style argument:
Suppose for reductio that a finite line segment of non-zero length is composed of infinitely many disjoint parts of equal, real-valued length.
1. Either the parts all have zero length or they all have the same non-zero length.
2. The length of the whole line segment is the sum of the lengths of the parts.
3. If the parts all have zero length, then the line segment has zero length, contradicting our assumption that it has non-zero length.
4. If the parts all have non-zero length, then the line segment has infinite length contradicting our assumption that it is finite.
Conclusion 1: A finite line segment cannot be composed of infinitely many disjoint parts of equal real-valued length.
Therefore,
Conclusion 2: It cannot be composed of points.
Premise 1 is beyond reproach. However, premises 2, 3, and 4 require us to be careful about how lengths add. Recall that in section 3.2 we discussed how to add a countable sequence of numbers – but the method described there depends on the order, and requires a countable, well-ordered sequence. Although there are techniques for summing uncountable sets of non-negative numbers, most mathematicians deny that lengths or other measures can be added in these ways. This is parallel to what Kolmogorov said about probability (see section 6.3). Probability and length are two paradigms of the more general mathematical field of “measure theory”, which includes all such countably additive real-valued functions. For a more detailed discussion of this problem, including approaches involving infinitesimal length, see Skyrms (1983).
For more about measure theory, see Bartle (1995) and Tao (2011).
### 8.2. Non-Euclidean geometries, relativistic space-time, and cosmic topology
#### 8.2.1 Non-Euclidean geometries
In section 1 we anticipated that Archytas’ argument for the infinitude of the cosmos, and Kant’s treatment of the antinomies, conflated the notions of finiteness and boundedness.
We now need to introduce another aspect of 19th century mathematics that brought that crucial distinction into focus. The distinction between finiteness and boundedness (and consequently that between infinitude and unboundedness) greatly improved our understanding of issues concerning the structure of space, and what shape a finite, or an infinite space, might take. Recall that for two centuries after Newton, cosmology was developed within the framework of Euclidean infinite space. Such a space is infinite in all directions, it is homogeneous and isotropic—that is, it is the same at all locations and in all directions.
In the middle of the 19th century, alternative conceptions of geometrical space were developed, so-called non-Euclidean geometries. Gauss, Bolyai, Lobachevski and Riemann, showed that one can develop geometries that falsify Euclid’s parallel axiom while preserving all the other Euclidean axioms. The axiom states (in a later but equivalent version to the one given by Euclid) that for any line and a point external to that line, there is one and only one parallel to the given line passing through the point. The statement contains a claim of existence and a claim of uniqueness. It can be thus falsified by denying existence or by accepting the existence but denying the uniqueness. Both alternatives have been developed, with some of the earliest interpretations using surfaces. The first alternative, where no parallels exist to any given line that pass through a given point external to the given line, is known as elliptic geometry. An instance of elliptic geometry is spherical geometry, so called because it can be modeled on the surface of a sphere. The second alternative is known as hyperbolic geometry and in it for every line in the model and any point outside of the line there are infinitely many parallels to that line passing through the point. A portion of the surface of a horse saddle can be used to model hyperbolic geometry. (The pictures below are based on those in Luminet 2008: 49.)
The curvature of a surface at a point $$p$$ on the surface measures how much the surface bends away from its tangent plane at $$p$$. The curvature of a surface is constant if at every point $$p$$ of the surface the surface bends away from the tangent plane by the same quantity. Examples of surfaces which can be used to model various geometries are the surface of the cylinder (Euclidean; constant curvature 0), the sphere (spherical; positive curvature), and the horse saddle (hyperbolic; negative curvature). They are all homogenous and isotropic but they have different constant curvature.
Such geometries on surfaces spurred the development of three-dimensional and higher dimensional spaces with different curvatures: positive, null, and negative. An example of a space of positive constant curvature is the 3-sphere (also called hypersphere) used by Bernhard Riemann in his 1854 dissertation (see Riemann 1868; for an English translation see Riemann 2016). A 3-sphere is a surface in a four-dimensional space that is obtained as a generalization of the 2-sphere as visualized in three dimensions: in both cases we define the relevant notion as a locus of points that have a constant distance from a point (its center). For instance, the unit 2-sphere centered at the origin is the set of triples of real numbers $$(x, y, z)$$ that are one unit away from (0, 0, 0), i.e. that satisfy $$x^2 +y^2 +z^2 =1$$, and the 3-sphere with distance 1 from the origin (0, 0, 0, 0) is the set of quadruples $$(x, y, z, w)$$ of real numbers satisfying $$x^2 +y^2 +z^2 +w^2 =1$$. It is a model of physical space that is finite but unbounded, in explicit opposition to Newton’s conception of space.
Archytas’s argument (in section 1 above), which conflated unboundedness with infinitude, could finally be put to rest. Riemann’s model allows for the universe to be finite and unbounded at the same time. In 1854 he wrote: “The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesics starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite.” (Riemann 2016: 39)
The distinction between infinite and unbounded is an integral part of the conceptual leap that leads to the idea that physical space need not be Euclidean. In the following section we will briefly describe how issues of curvature and topology play a role in addressing the question of whether the world is spatially finite or infinite in cosmology.
On non-Euclidean geometries the reader will find useful Greenberg (2007) and Gray (2010). On the philosophical relevance of curvature and Riemannian geometry see the classic Torretti (1984).
#### 8.2.2 Relativistic space-time and cosmic topology
In 1915 Einstein introduced general relativity, and our conception of the universe is based on it. General relativity rests on a conception of space and time—or better, space-time matter—that stands in opposition to the Newtonian one we have described above. In Einstein’s theory, space-time is deformable and its shape depends on the presence of matter. Space-time is, in technical terms, a four-dimensional manifold. We may think of an $$n$$-dimensional manifold as a set of $$n$$-tuples of real numbers. The spatial section of a four-dimensional manifold of space-time is a three-dimensional manifold (one can think of it as a set of triples of real numbers), and when cosmologists ask about the shape of the universe they try to characterize this three-dimensional manifold. The curvature of space-time corresponds to gravitation, and light rays and other material particles follow the geodesics (shortest paths) in the manifold. In general, the geodesics differ depending on the matter-energy content of the space being considered. The geodesics of the surface of the sphere (i.e. a two-dimensional surface) are portions of great circles. In the Euclidean plane, they are segments of straight lines. There are analogous notions for four-dimensional manifolds. Einstein’s equations for general relativity describe how the matter-energy content of the universe determines the geometry of space-time. The equations also yield cosmological models, which must be tested by empirical observation. The equations allow for multiple solutions and, as Alexander Friedmann (1924) observed, “in the absence of additional hypotheses, Einstein’s equations for the universe do not allow to definitely answer the question of the finiteness of the universe”. Let us briefly explain what is at stake in this comment, by pointing out the role of curvature and topology with respect to the issue of finiteness vs. infiniteness of our universe in relativistic cosmology. Topology is the branch of geometry that classifies spaces according to whether they can be transformed into one another “continuously”, that is with transformations that do not lead to cuts or tears.
In 1917, Einstein posited a static finite universe. The finiteness was given by the choice of the 3-sphere (see section 8.2.1) and the static nature of the universe by the fact that the radius of the hypersphere did not change with time. With Friedmann (1922–24) and Lemaître (1927), Einstein’s static model would be replaced by dynamical models (to account for the empirical evidence that by 1930 led to the postulation that the universe is expanding, i.e., most galaxies, galaxy clusters, etc. are growing further apart, just as spots on an uninflated balloon grow further apart when the balloon is blown up). Such models are also among the possible solutions for Einstein’s equations and they were the source for the so-called “Big Bang” theories. But what about finiteness? The finiteness or infiniteness of the universe are not determined by Einstein’s equations, which allow for both possibilities. In his choice of the 3-sphere, Einstein was motivated by considerations that had to do with preserving a hypothesis by Mach on inertial mass and inertial motion. Friedmann and Lemaître also opted for the finiteness of the universe (we need not get into why they did so). Their dynamical models assumed a uniform distribution of matter in the universe and that space is homogeneous and isotropic. But the Friedmann-Lemaître dynamical solutions still allow for a great variety of mathematical solutions and do not settle the issue of finiteness. Our observations in what follows are restricted to such models.
Space, in this context, is characterized by its curvature (taken to be constant) and its topology. Let us consider curvature first. In these models, there are three possible types of spaces depending on whether the curvature of the space is negative, null, or positive. The spaces corresponding to such curvatures are called hyperbolic, Euclidean, and elliptical. A spherical space (constant positive curvature) is always of finite extension, no matter what its topology is. This explains, at least in part, why many early cosmologists (including Einstein, de Sitter, Friedmann, Lemaître and others) opted for this solution. Indeed, for a long time issues concerning the topology of space were not brought to the fore due to the implicit assumption that the topology of space was a simply connected topology (in a simply connected topology every loop on the surface can be continuously contracted to a point). Under that assumption, spaces of positive constant curvature are finite and those of null and negative constant curvature are infinite. The issue then of the finiteness vs. infiniteness of the universe rests on the mean density of matter and energy and on the value of a parameter $$\lambda$$ introduced by Einstein in 1917, called the cosmological constant (measuring a sort of anti-gravitational force). With most cosmologists (but not Einstein in 1917) assuming $$\lambda = 0$$, and with the assumption that space is simply connected, determining curvature (and hence resolving the finiteness vs infinity issue) depends only on a critical value for the mean density of matter—equivalently, on a density parameter $$\Omega$$. Thus, under those assumptions, it would be in principle possible to determine the curvature of space experimentally.
Different values of $$\lambda$$ lead to different scenarios for the evolution of the universe. With $$\lambda = 0$$, if the curvature of space is negative or null we end up with a constantly expanding universe; if the curvature of space is positive, a phase of expansion would be followed by a contraction leading to a “big crunch”. Other values of the cosmological constant are possible and if $$\lambda \lt 0$$ a “big crunch” will occur no matter what the curvature of space is. By contrast, if $$\lambda \gt 0$$, no “big crunch” will occur. New experimental evidence (coming from type 1A supernovae and fossil radiation) seems to indicate a positive mean density of matter and a value of $$\lambda \gt 0$$. In this case the universe would be finite while still remaining in perpetual accelerated expansion.
Moreover, recent work has pointed out the importance of considering non-simply connected topologies. Unlike what happens for simply connected topologies, curvature does not immediately determine the finiteness or infinity of the space. Indeed, there are spaces of null or negative curvature that can be finite or infinite depending on the non-connected topology associated to them. This leads into cosmic topology, which investigates the global shape of space and how it can be determined experimentally. If space has positive curvature, then the universe is finite independently of the specific topology associated with it. But if the curvature is negative or null, whether the universe is finite or not will depend on the topology. Thus, determining whether the universe is finite or infinite requires not only determining the mean density of matter (which determines the curvature of space) but also the topology of space. Two major techniques that are employed experimentally to determine the topology of space are cosmic crystallography and the circles in the sky method (based on the cosmic microwave background).
For further information on cosmic topology see Luminet, Starkmann, and Weeks, (1999), Luminet and Lachièze-Rey (2005), Luminet (2005) (English 2008). See also Aguirre (2011) and Luminet (2015). For more technical treatment see Thurston (1997), Weeks (2001), and Hitchmann 2018.
## 9. Conclusion
We are well aware that our discussion of infinity is incomplete—but then, so is any such discussion. We take some comfort from the fact that it is impossible to give balanced coverage to a boundless set of issues in finite space.
There are many more philosophically significant paradoxes and puzzles that involve infinity in one way or another; we have only given a small sample. And new paradoxes involving infinity seem to appear at an ever-increasing rate (doubtless yet another one can be fashioned out of this very fact!). However, so too are our tools for understanding infinity. Of course, we cannot give a definitive assessment of the state of play, but the theoretical developments that we have sketched and references that we have cited make us sanguine that overall, the prospects for our relationship with infinity are good: we can indeed live with it.
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2021-08-04 12:18:11
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|
https://labs.tib.eu/arxiv/?author=F.%20Filippi
|
• We highlight the power of the Gaia DR2 in studying many fine structures of the Hertzsprung-Russell diagram (HRD). Gaia allows us to present many different HRDs, depending in particular on stellar population selections. We do not aim here for completeness in terms of types of stars or stellar evolutionary aspects. Instead, we have chosen several illustrative examples. We describe some of the selections that can be made in Gaia DR2 to highlight the main structures of the Gaia HRDs. We select both field and cluster (open and globular) stars, compare the observations with previous classifications and with stellar evolutionary tracks, and we present variations of the Gaia HRD with age, metallicity, and kinematics. Late stages of stellar evolution such as hot subdwarfs, post-AGB stars, planetary nebulae, and white dwarfs are also analysed, as well as low-mass brown dwarf objects. The Gaia HRDs are unprecedented in both precision and coverage of the various Milky Way stellar populations and stellar evolutionary phases. Many fine structures of the HRDs are presented. The clear split of the white dwarf sequence into hydrogen and helium white dwarfs is presented for the first time in an HRD. The relation between kinematics and the HRD is nicely illustrated. Two different populations in a classical kinematic selection of the halo are unambiguously identified in the HRD. Membership and mean parameters for a selected list of open clusters are provided. They allow drawing very detailed cluster sequences, highlighting fine structures, and providing extremely precise empirical isochrones that will lead to more insight in stellar physics. Gaia DR2 demonstrates the potential of combining precise astrometry and photometry for large samples for studies in stellar evolution and stellar population and opens an entire new area for HRD-based studies.
• ### Plasma ramps caused by outflow in gas-filled capillaries(1802.10030)
Plasma confinement inside capillaries has been developed in the past years for plasma-based acceleration to ensure a stable and repeatable plasma density distribution during the interaction with either particles or laser beams. In particular, gas-filled capillaries allow a stable and almost predictable plasma distribution along the interaction with the particles. However, the plasma ejected through the ends of the capillary interacts with the beam before the inner plasma, affecting the quality of the beam. In this article we report the measurements on the evolution of the plasma flow at the two ends of a 1 cm long, 1 mm diameter capillary filled with hydrogen. In particular, we measured the longitudinal density distribution and the expansion velocity of the plasma outside the capillary. This study will allow a better understanding of the beam-plasma interaction for future plasma-based experiments.
• ### Conceptual design of electron beam diagnostics for high brightness plasma accelerator(1802.05103)
Feb. 14, 2018 physics.acc-ph
A design study of the diagnostics of a high brightness linac, based on X-band structures, and a plasma accelerator stage, has been delivered in the framework of the EuPRAXIA@SPARC_LAB project. In this paper, we present a conceptual design of the proposed diagnostics, using state of the art systems and new and under development devices. Single shot measurements are preferable for plasma accelerated beams, including emittance, while $\mu$m level and fs scale beam size and bunch length respectively are requested. The needed to separate the driver pulse (both laser or beam) from the witness accelerated bunch imposes additional constrains for the diagnostics. We plan to use betatron radiation for the emittance measurement just at the end of the plasma booster, while other single-shot methods must be proven before to be implemented. Longitudinal measurements, being in any case not trivial for the fs level bunch length, seem to have already a wider range of possibilities.
• ### Characterization of self-injected electron beams from LWFA experiments at SPARC_LAB(1802.01956)
Feb. 3, 2018 hep-ex, physics.acc-ph
The plasma-based acceleration is an encouraging technique to overcome the limits of the accelerating gradient in the conventional RF acceleration. A plasma accelerator is able to provide accelerating fields up to hundreds of $GeV/m$, paving the way to accelerate particles to several MeV over a short distance (below the millimetre range). Here the characteristics of preliminary electron beams obtained with the self-injection mechanism produced with the FLAME high-power laser at the SPARC_LAB test facility are shown. In detail, with an energy laser on focus of $1.5\ J$ and a pulse temporal length (FWHM) of $40\ fs$, we obtained an electron plasma density due to laser ionization of about $6 \times 10^{18}\ cm^{-3}$, electron energy up to $350\ MeV$ and beam charge in the range $(50 - 100)\ pC$.
• ### Overview of Plasma Lens Experiments and Recent Results at SPARC_LAB(1802.00279)
Feb. 1, 2018 physics.acc-ph
Beam injection and extraction from a plasma module is still one of the crucial aspects to solve in order to produce high quality electron beams with a plasma accelerator. Proper matching conditions require to focus the incoming high brightness beam down to few microns size and to capture a high divergent beam at the exit without loss of beam quality. Plasma-based lenses have proven to provide focusing gradients of the order of kT/m with radially symmetric focusing thus promising compact and affordable alternative to permanent magnets in the design of transport lines. In this paper an overview of recent experiments and future perspectives of plasma lenses is reported.
• ### The FLAME laser at SPARC_LAB(1802.00398)
Feb. 1, 2018 physics.acc-ph
FLAME is a high power laser system installed at the SPARC_LAB Test Facility in Frascati (Italy). The ultra-intense laser pulses are employed to study the interaction with matter for many purposes: electron acceleration through LWFA, ion and proton generation exploiting the TNSA mechanism, study of new radiation sources and development of new electron diagnostics. In this work, an overview of the FLAME laser system will be given, together with recent experimental results
• ### EuPRAXIA@SPARC_LAB Design study towards a compact FEL facility at LNF(1801.08717)
Jan. 26, 2018 physics.acc-ph
On the wake of the results obtained so far at the SPARC\_LAB test-facility at the Laboratori Nazionali di Frascati (Italy), we are currently investigating the possibility to design and build a new multi-disciplinary user-facility, equipped with a soft X-ray Free Electron Laser (FEL) driven by a $\sim$1 GeV high brightness linac based on plasma accelerator modules. This design study is performed in synergy with the EuPRAXIA design study. In this paper we report about the recent progresses in the on going design study of the new facility.
• ### Recent results at SPARC_LAB(1801.05990)
Jan. 18, 2018 physics.acc-ph
The current activity of the SPARC_LAB test-facility is focused on the realization of plasma-based acceleration experiments with the aim to provide accelerating field of the order of several GV/m while maintaining the overall quality (in terms of energy spread and emittance) of the accelerated electron bunch. In the following, the current status of such an activity is presented. We also show results related to the usability of plasmas as focusing lenses in view of a complete plasma-based focusing and accelerating system.
• ### Wake fields effects in dielectric capillary(1801.04200)
Jan. 12, 2018 physics.acc-ph
Plasma wake-field acceleration experiments are performed at the SPARC LAB test facility by using a gas-filled capillary plasma source composed of a dielectric capillary. The electron can reach GeV energy in a few centimeters, with an accelerating gradient orders of magnitude larger than provided by conventional techniques. In this acceleration scheme, wake fields produced by passing electron beams through dielectric structures can determine a strong beam instability that represents an important hurdle towards the capability to focus high-current electron beams in the transverse plane. For these reasons, the estimation of the transverse wakefield amplitudes assumes a fundamental role in the implementation of the plasma wake-field acceleration. In this work, it presented a study to investigate which parameters affect the wake-field formation inside a cylindrical dielectric structure, both the capillary dimensions and the beam parameters, and it is introduced a quantitative evaluation of the longitudinal and transverse electric fields.
• Parallaxes for 331 classical Cepheids, 31 Type II Cepheids and 364 RR Lyrae stars in common between Gaia and the Hipparcos and Tycho-2 catalogues are published in Gaia Data Release 1 (DR1) as part of the Tycho-Gaia Astrometric Solution (TGAS). In order to test these first parallax measurements of the primary standard candles of the cosmological distance ladder, that involve astrometry collected by Gaia during the initial 14 months of science operation, we compared them with literature estimates and derived new period-luminosity ($PL$), period-Wesenheit ($PW$) relations for classical and Type II Cepheids and infrared $PL$, $PL$-metallicity ($PLZ$) and optical luminosity-metallicity ($M_V$-[Fe/H]) relations for the RR Lyrae stars, with zero points based on TGAS. The new relations were computed using multi-band ($V,I,J,K_{\mathrm{s}},W_{1}$) photometry and spectroscopic metal abundances available in the literature, and applying three alternative approaches: (i) by linear least squares fitting the absolute magnitudes inferred from direct transformation of the TGAS parallaxes, (ii) by adopting astrometric-based luminosities, and (iii) using a Bayesian fitting approach. TGAS parallaxes bring a significant added value to the previous Hipparcos estimates. The relations presented in this paper represent first Gaia-calibrated relations and form a "work-in-progress" milestone report in the wait for Gaia-only parallaxes of which a first solution will become available with Gaia's Data Release 2 (DR2) in 2018.
• Context. The first Gaia Data Release contains the Tycho-Gaia Astrometric Solution (TGAS). This is a subset of about 2 million stars for which, besides the position and photometry, the proper motion and parallax are calculated using Hipparcos and Tycho-2 positions in 1991.25 as prior information. Aims. We investigate the scientific potential and limitations of the TGAS component by means of the astrometric data for open clusters. Methods. Mean cluster parallax and proper motion values are derived taking into account the error correlations within the astrometric solutions for individual stars, an estimate of the internal velocity dispersion in the cluster, and, where relevant, the effects of the depth of the cluster along the line of sight. Internal consistency of the TGAS data is assessed. Results. Values given for standard uncertainties are still inaccurate and may lead to unrealistic unit-weight standard deviations of least squares solutions for cluster parameters. Reconstructed mean cluster parallax and proper motion values are generally in very good agreement with earlier Hipparcos-based determination, although the Gaia mean parallax for the Pleiades is a significant exception. We have no current explanation for that discrepancy. Most clusters are observed to extend to nearly 15 pc from the cluster centre, and it will be up to future Gaia releases to establish whether those potential cluster-member stars are still dynamically bound to the clusters. Conclusions. The Gaia DR1 provides the means to examine open clusters far beyond their more easily visible cores, and can provide membership assessments based on proper motions and parallaxes. A combined HR diagram shows the same features as observed before using the Hipparcos data, with clearly increased luminosities for older A and F dwarfs.
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2020-11-28 09:15:35
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https://livrepository.liverpool.ac.uk/14415/
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Monitoring radiation damage in the vertex locator and top pair production in LHCb
Brown, Henry
Monitoring radiation damage in the vertex locator and top pair production in LHCb. Doctor of Philosophy thesis, University of Liverpool.
The Large Hadron Collider (LHC) is a proton-proton collider at the European Centre for Nuclear Research (CERN). The LHCb experiment is one of the four main experiments at the LHC. It is designed for the detection of bbbar pairs produced in proton-proton collisions and to make precision measurements of B-mesons. The trigger level identification of B-mesons is provided by the Vertex Locator (VELO), which is the primary tracking detector of the experiment. Due to its proximity to the interaction point, the VELO is exposed to high levels of radiation damage. A new method of monitoring the damage is to perform current-voltage (IV) scans and to compare the results of these scans to laboratory tests on sample sensors. A method to perform the first ttbar production measurement in the $\eta>2$ range at the LHC, using a dilepton+b-jet channel, is also presented. A fiducial cross-section is obtained of $\sigma_{\mathrm{fid}}= 24.3^{+14.6}_{-9.7}\mathrm{(stat.)}\pm 6.9\mathrm{(syst.)} \pm 0.9 \mathrm{(lumi.)}$fb, which is consistent with Standard Model expectations. The author's work was to perform the analysis of the IV scans and the comparison to the empirical models obtained in test environments, and to develop the method, as well as the necessary theoretical predictions, for the top pair cross-section measurement.
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2022-10-07 11:45:20
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https://bigladdersoftware.com/epx/docs/8-7/ems-application-guide/example-1-whole-building-average-zone-air.html
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Example 1. Whole-Building Average Zone Air Temperature[LINK]
Although EnergyPlus can report an enormous number of output variables, you may want a custom report variable such as one for the average temperature in the building. Only zone-by-zone indoor air temperatures are available. Because it is nearly always important to check that models are properly controlling zone air conditions, you may need to examine air temperature results from your models. Compared to scanning across the many zones in a large building, you could save time when checking a model if you have a single value for a whole-building average temperature. Of course, you could calculate such a value after a run by postprocessing, but redoing this for every run is cumbersome and time consuming. Therefore, it would be more convenient to automatically calculate such a value inside the program and output it in the usual manner. For example, if we take the Small Office Reference Building (RefBldgSmallOfficeNew2004_Chicago.idf), is there a way to create a custom report variable that provides a weighted average for the indoor temperature of all the occupied zones in a model?
This is a fairly simple example in that the EMS controls nothing. There are no actuators.
The example file has six zones, but one is an attic that we do not care about. Therefore, the main inputs, or EMS sensors, will be the zone air temperatures for the five occupied zones. We will use EnergyManagementSystem:Sensor objects to obtain the values for the air temperatures by mapping to the output variable called “Zone Mean Air Temperature.”
A model for average temperature can be constructed by using the zone air volumes as weights so larger zones have more influence than smaller zones on the resulting average. The model equation we will implement in EMS for our new report variable is
Taverage=(TzoneVolzone)(Volzone)
The example file specifies the zone volume in its zone objects so we have the data needed for the weighting factors from elsewhere in the IDF. However, a study could vary the geometry such that the volumes differ from one simulation to another. Zone Air Volume is available as internal data, so we will use EnergyManagementSystem:InternalVariable input objects to assign these weighting factors into global Erl variables. If we did not know beforehand that Zone Air Volume was an available internal variable, we would have had to prerun the model with some EMS-related objects and the appropriate level of reporting selected in an Output:EnergyManagementSystem object, and then studied the EDD output file. Note that the EDD file is only produced if you have EMS/Erl programs in your input file.
The custom output variable will be defined by using an EnergyManagementSystem:OutputVariable input object. This requires the Erl variable to be global, so we need to declare a variable. Let’s call it AverageBuildingTemp, to be global using an EnergyManagementSystem:GlobalVariable object so we have a way to connect the result calculated in the Erl program to the custom output.
There are two main considerations when selecting an EMS calling point:
• The call should be toward the end of the zone timestep so the zone air temperature calculations are finalized.
• The call should be before reporting updates so our new value is available before the reporting is finalized.
We therefore choose the EMS calling point with the key of “EndOfZoneTimestepBeforeReporting.”
A set of input objects to solve this problem appears below and is included in the example file called “EMSCustomOutputVariable.idf.”
EnergyManagementSystem:Sensor,
T1, !Name
Perimeter_ZN_1 ,! Output:Variable or Output:Meter Index Key Name
Zone Mean Air Temperature ; ! Output:Variable or Output:Meter Name
EnergyManagementSystem:Sensor,
T2, !Name
Perimeter_ZN_2 , ! Output:Variable or Output:Meter Index Key Name
Zone Mean Air Temperature ; ! Output:Variable or Output:Meter Name
EnergyManagementSystem:Sensor,
T3, !Name
Perimeter_ZN_3 , ! Output:Variable or Output:Meter Index Key Name
Zone Mean Air Temperature ; ! Output:Variable or Output:Meter Name
EnergyManagementSystem:Sensor,
T4, !Name
Perimeter_ZN_4, ! Output:Variable or Output:Meter Index Key Name
Zone Mean Air Temperature ;! Output:Variable or Output:Meter Name
EnergyManagementSystem:Sensor,
T5, !Name
Core_ZN , ! Output:Variable or Output:Meter Index Key Name
Zone Mean Air Temperature ; ! Output:Variable or Output:Meter Name
EnergyManagementSystem:ProgramCallingManager,
Average Building Temperature , ! Name
EndOfZoneTimestepBeforeZoneReporting , ! EnergyPlus Model Calling Point
AverageZoneTemps ; ! Program Name 1
EnergyManagementSystem:GlobalVariable,
AverageBuildingTemp;
EnergyManagementSystem:OutputVariable,
Weighted Average Building Zone Air Temperature [C], ! Name
AverageBuildingTemp, ! EMS Variable Name
Averaged, ! Type of Data in Variable
ZoneTimeStep ; ! Update Frequency
EnergyManagementSystem:InternalVariable,
Zn1vol,
Perimeter_ZN_1,
Zone Air Volume;
EnergyManagementSystem:InternalVariable,
Zn2vol,
Perimeter_ZN_2,
Zone Air Volume;
EnergyManagementSystem:InternalVariable,
Zn3vol,
Perimeter_ZN_3,
Zone Air Volume;
EnergyManagementSystem:InternalVariable,
Zn4vol,
Perimeter_ZN_4,
Zone Air Volume;
EnergyManagementSystem:InternalVariable,
Zn5vol,
Core_ZN ,
Zone Air Volume;
EnergyManagementSystem:Program,
AverageZoneTemps , ! Name
SET SumNumerator = T1*Zn1vol + T2*Zn2vol + T3*Zn3vol + T4*Zn4vol + T5*Zn5vol,
SET SumDenominator = Zn1vol + Zn2vol + Zn3vol + Zn4vol + Zn5vol,
SET AverageBuildingTemp = SumNumerator / SumDenominator;
Output:EnergyManagementSystem,
Verbose,
Verbose,
Verbose;
Output:Variable,
*, !- Key Value
Weighted Average Building Zone Air Temperature, !- Variable Name
timestep; !- Reporting Frequency
|
2022-08-17 14:26:54
|
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|
https://code.dlang.org/packages/dpp/0.3.0
|
# dpp 0.3.0
Include C/C++ headers directly in D files
To use this package, run the following command in your project's root directory:
## d++ - #include C and C++ headers in D files
| | |
### Goal
To directly #include C and C++ headers in D files and have the same semantics and ease-of-use as if the file had been #included from C or C++ themselves. Warts and all, meaning that C enum declarations will pollute the global namespace, just as it does "back home".
This work was supported by Symmetry Investments.
### Example
// c.h
#ifndef C_H
#define C_H
#define FOO_ID(x) (x*3)
int twice(int i);
#endif
// c.c
int twice(int i) { return i * 2; }
// foo.dpp
#include "c.h"
void main() {
import std.stdio;
writeln(twice(FOO_ID(5))); // yes, it's using a C macro here!
}
At the shell:
$gcc -c c.c$ d++ foo.dpp c.o
$./foo$ 30
### Limitations
• It currently only supports C features, but C++ is planned.
• Using it on a C++ header will "work" if it's basically technically C, with extern(C++) instead of extern(C)
• Only known to work on Linux with libclang.so.6.0. It might work in different conditions.
• When used on multiple files, there might be problems with duplicate definitions depending on imports. This will be fixed.
This is alpha software. It has however produced programs that compile that #included several "real-life" C headers:
• nanomsg/nn.h, nanomsg/pubsub.h
• curl/curl.h
• stdio.h, stdlib.h
• pthread.h
• julia.h
• xlsxwriter.h
• libvirt/libvirt.h, libvirt/virterror.h
• libzfs
• openssl/ssl.h
• imapfilter.h
• libetpan/libetpan.h
Compilation however doesn't guarantee they work as expected and YMMV. Please consult the examples.
### Command-line arguments
It is likely that the header or headers need -I flags to indicate paths to be searched, both by this executable and by libclang itself. The --include-path option can be used for that, once for each such path.
Use -h or --help to learn more.
### Details
d++ is an executable that wraps a D compiler such as dmd (the default) so that D files with #include directives can be compiled.
It takes a .dpp file and outputs a valid D file that can be compiled. The original can't since D has no preprocessor, so the .dpp file is "quasi-D", or "D with #include directives". The only supported C preprocessor directive is #include.
The input .dpp file may also use C preprocessor macros defined in the file(s) it #includes, just as a C/C++ program would (see the example above). It may not, however, define macros of its own.
d++ goes through the input file line-by-line, and upon encountering an #include directive, parses the file to be included with libclang, loops over the definitions of data structures and functions therein and expands in-place the relevant D translations. e.g. if a header contains:
uint16_t foo(uin32_t a);
The output file will contain:
ushort foo(uint a);
d++ will also enclose each one of these original #include directives with either extern(C) {} or extern(C++) {} depending on the header file name and/or command-line options.
As part of expanding the #include, and as well as translating declarations, d++ will also insert text to define macros originally defined in the #included translation unit so that these macros can be used by the D program. The reason for this is that nearly every non-trivial C API requires the preprocessor to use properly. It is possible to mimic this usage in D with enums and CTFE, but the result is not guaranteed to be the same. The only way to use a C or C++ API as it was intended is by leveraging the preprocessor.
Trivial literal macros however(e.g. #define THE_ANSWER 42) are translated as D enums.
As a final pass before writing the output D file, d++ will run the C preprocessor (currently the cpp binary installed on the system) on the intermediary result of expanding all the #include directives so that any used macros are expanded, and the result is a D file that can be compiled.
In this fashion a user can write code that's not-quite-D-but-nearly that can "natively" call into a C/C++ API by #includeing the appropriate header(s).
### Translation notes
#### enum
For convenience, this declaration:
enum Enum { foo, bar, baz }
Will generate this translation:
enum Enum { foo, bar, baz }
enum foo = Enum.foo;
enum bar = Enum.bar;
enum baz = Enum.baz;
This is to mimic C semantics with regards to the global namespace whilst also allowing one to, say, reflect on the enum type.
#### Renaming enums
There is the ability to rename C enums. With the following C definition:
enum FancyWidget { Widget_foo, Widget_bar }
Then adding this to your .dpp file after the #include directive:
mixin dpp.EnumD!("Widget", // the name of the new D enum
FancyWidget, // the name of the original C enum
"Widget_"); // the prefix to cut out
will yield this translation:
enum Widget { foo, bar }
#### Names of structs, enums and unions
C has a different namespace for the aforementioned user-defined types. As such, this is legal C:
struct foo { int i; };
extern int foo;
The D translations just use the short name for these aggregates, and if there is a name collision with a variable or function, the latter two get renamed and have a pragma(mangle) added to avoid linker failures:
struct foo { int i; }
pragma(mangle, "foo") extern __gshared int foo_;
#### Functions or variables with a name that is a D keyword
Similary to name collisions with aggregates, they get an underscore appended and a pragma(mangle) added so they link:
void debug(const char* msg);
Becomes:
pragma(mangle, "debug")
void debug_(const(char)*);
### Build Instructions
#### Windows
1. Install http://releases.llvm.org/6.0.1/LLVM-6.0.1-win64.exe into C:\Program Files\LLVM\, making sure to tick the "Add LLVM to the system PATH for all users" option.
2. Make sure you have LDC installed somewhere.
3. Compile with dub build --compiler=C:\path\to\bin\ldc2.exe.
4. Copy C:\Program Files\LLVM\bin\libclang.dll next to the d++.exe in the bin directory.
• Registered by Atila Neves
• 0.3.0 released 2 months ago
• atilaneves/dpp
• boost
Authors:
• Atila Neves
Dependencies:
libclang, sumtype
Versions:
0.3.1 2019-Aug-08 0.3.0 2019-Jul-24 0.2.3 2019-May-31 0.2.2 2019-May-29 0.2.1 2019-May-16
Download Stats:
• 5 downloads today
• 54 downloads this week
• 280 downloads this month
• 3715 downloads total
Score:
3.6
Short URL:
dpp.dub.pm
|
2019-09-23 04:48:10
|
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|
https://merely-useful.github.io/py-rse/bash-basics.html
|
# Chapter 2 The Basics of the Unix Shell
Ninety percent of most magic merely consists of knowing one extra fact.
— Terry Pratchett
Computers do four basic things: store data, run programs, talk with each other, and interact with people. They do the interacting in many different ways, of which graphical user interfaces (GUI) are the most widely used. The computer displays icons to show our files and programs, and we tell it to copy or run those by clicking with a mouse. GUIs are easy to learn but hard to automate, and don’t create a record of what we did.
In contrast, when we use a command-line interface (CLI) we communicate with the computer by typing commands, and the computer responds by displaying text. CLIs existed long before GUIs; they have survived because they are efficient, easy to automate, and automatically record what we have done.
The heart of every CLI is a read-evaluate-print loop (REPL). When we type a command and press Return (also called Enter) the CLI reads the command, evaluates it (i.e., executes it), prints the command’s output, and loops around to wait for another command. If you have used an interactive console for R or Python, you have already used a simple CLI.
This lesson introduces another CLI that lets us interact with our computer’s operating system. It is called a “command shell”, or just shell for short, and in essence is a program that runs other programs on our behalf (Figure 2.1). Those “other programs” can do things as simple as telling us the time or as complex as modeling global climate change; as long as they obey a few simple rules, the shell can run them without having to know what language they are written in or how they do what they do.
What’s in a Name?
Programmers have written many different shells over the last forty years, just as they have created many different text editors and plotting packages. The most popular shell today is called Bash (an acronym of Bourne Again SHell, and a weak pun on the name of its predecessor, the Bourne shell). Other shells may differ from Bash in minor ways, but the core commands and ideas remain the same. In particular, the most recent versions of MacOS use a shell called the Z Shell or zsh; we will point out a few differences as we go along.
Please see Section 1.3 for instructions on how to install and launch the shell on your computer.
## 2.1 Exploring Files and Directories
Our first shell commands will let us explore our folders and files, and will also introduce us to several conventions that most Unix tools follow. To start, when Bash runs it presents us with a prompt to indicate that it is waiting for us to type something. This prompt is a simple dollar sign by default:
$ However, different shells may use a different symbol: in particular, the zsh shell that is the default on newer version of MacOS uses %. As we’ll see in Section 4.6, we can customize the prompt to give us more information. Don’t Type the Dollar Sign We show the $ prompt so that it’s clear what you are supposed to type, particularly when several commands appear in a row, but you should not type it yourself.
Let’s run a command to find out who the shell thinks we are:
$whoami amira Learn by Doing Amira is one of the learners described in Section 0.2. For the rest of the book, we’ll present code and examples from her perspective. You should follow along on your own computer, though what you see might deviate in small ways because of differences in operating system (and because your name probably isn’t Amira). Now that we know who we are, we can explore where we are and what we have. The part of the operating system that manages files and directories (also called folders) is called the filesystem. Some of the most commonly-used commands in the shell create, inspect, rename, and delete files and directories. Let’s start exploring them by running the command pwd, which stands for print working directory. The “print” part of its name is straightforward; the “working directory” part refers to the fact that the shell keeps track of our current working directory at all times. Most commands read and write files in the current working directory unless we tell them to do something else, so knowing where we are before running a command is important. $ pwd
/Users/amira
Here, the computer’s response is /Users/amira, which tells us that we are in a directory called amira that is contained in a top-level directory called Users. This directory is Amira’s home directory; to understand what that means, we must first understand how the filesystem is organized. On Amira’s computer it looks like Figure 2.2.
At the top is the root directory that holds everything else, which we can refer to using a slash character / on its own. Inside that directory are several other directories, including bin (where some built-in programs are stored), data (for miscellaneous data files), tmp (for temporary files that don’t need to be stored long-term), and Users (where users’ personal directories are located). We know that /Users is stored inside the root directory / because its name begins with /, and that our current working directory /Users/amira is stored inside /Users because /Users is the first part of its name. A name like this is called a path because it tells us how to get from one place in the filesystem (e.g., the root directory) to another (e.g., Amira’s home directory).
Slashes
The / character means two different things in a path. At the front of a path or on its own, it refers to the root directory. When it appears inside a name, it is a separator. Windows uses backslashes (\\) instead of forward slashes as separators.
Underneath /Users, we find one directory for each user with an account on this machine. Jun’s files are stored in /Users/jun, Sami’s in /Users/sami, and Amira’s in /Users/amira. This is where the name “home directory” comes from: when we first log in, the shell puts us in the directory that holds our files.
Home Directory Variations
Our home directory will be in different places on different operating systems. On Linux it may be /home/amira, and on Windows it may be C:\Documents and Settings\amira or C:\Users\amira (depending on the version of Windows). Our examples show what we would see on MacOS.
Now that we know where we are, let’s see what we have using the command ls (short for “listing”), which prints the names of the files and directories in the current directory:
$ls Applications Downloads Music todo.txt Desktop Library Pictures zipf Documents Movies Public Again, our results may be different depending on our operating system and what files or directories we have. We can make the output of ls more informative using the -F option (also sometimes called a switch or a flag). Options are exactly like arguments to a function in R or Python; in this case, -F tells ls to decorate its output to show what things are. A trailing / indicates a directory, while a trailing * tells us something is a runnable program. Depending on our setup, the shell might also use colors to indicate whether each entry is a file or directory. $ ls -F
Applications/ Downloads/ Music/ todo.txt
Desktop/ Library/ Pictures/ zipf/
Documents/ Movies/ Public/
Here, we can see that almost everything in our home directory is a subdirectory; the only thing that isn’t is a file called todo.txt.
Spaces Matter
1+2 and 1 + 2 mean the same thing in mathematics, but ls -F and ls-F are very different things in the shell. The shell splits whatever we type into pieces based on spaces, so if we forget to separate ls and -F with at least one space, the shell will try to find a program called ls-F and (quite sensibly) give an error message like ls-F: command not found.
Some options tell a command how to behave, but others tell it what to act on. For example, if we want to see what’s in the /Users directory, we can type:
$ls /Users amira jun sami We often call the file and directory names that we give to commands arguments to distinguish them from the built-in options. We can combine options and arguments: $ ls -F /Users
amira/ jun/ sami/
but we must put the options (like -F) before the names of any files or directories we want to work on, because once the command encounters something that isn’t an option it assumes there aren’t any more:
$ls /Users -F ls: -F: No such file or directory amira jun sami Command Line Differences Code can sometimes behave in unexpected ways on different computers, and this applies to the command line as well. For example, the following code actually does work on some Linux operating systems: $ ls /Users -F
Some people think this is convenient; others (including us) believe it is confusing, so it’s best to avoid doing this.
## 2.2 Moving Around
Let’s run ls again. Without any arguments, it shows us what’s in our current working directory:
$ls -F Applications/ Downloads/ Music/ todo.txt Desktop/ Library/ Pictures/ zipf/ Documents/ Movies/ Public/ If we want to see what’s in the zipf directory we can ask ls to list its contents: $ ls -F zipf
data/
Notice that zipf doesn’t have a leading slash before its name. This absence tells the shell that it is a relative path, i.e., that it identifies something starting from our current working directory. In contrast, a path like /Users/amira is an absolute path: it is always interpreted from the root directory down, so it always refers to the same thing. Using a relative path is like telling someone to go two kilometers north and then half a kilometer east; using an absolute path is like giving them the latitude and longitude of their destination.
We can use whichever kind of path is easiest to type, but if we are going to do a lot of work with the data in the zipf directory, the easiest thing would be to change our current working directory so that we don’t have to type zipf over and over again. The command to do this is cd, which stands for change directory. This name is a bit misleading because the command doesn’t change the directory; instead, it changes the shell’s idea of what directory we are in. Let’s try it out:
$cd zipf cd doesn’t print anything. This is normal: many shell commands run silently unless something goes wrong, on the theory that they should only ask for our attention when they need it. To confirm that cd has done what we asked, we can use pwd: $ pwd
/Users/amira/zipf
$ls -F data/ Missing Directories and Unknown Options If we give a command an option that it doesn’t understand, it will usually print an error message and (if we’re lucky) tersely remind us of what we should have done: $ cd -j
-bash: cd: -j: invalid option
cd: usage: cd [-L|-P] [dir]
On the other hand, if we get the syntax right but make a mistake in the name of a file or directory, it will tell us that:
$cd whoops -bash: cd: whoops: No such file or directory We now know how to go down the directory tree, but how do we go up? This doesn’t work: $ cd amira
cd: amira: No such file or directory
because amira on its own is a relative path meaning “a file or directory called amira below our current working directory”. To get back home, we can either use an absolute path:
$cd /Users/amira or a special relative path called .. (two periods in a row with no spaces), which always means “the directory that contains the current one”. The directory that contains the one we are in is called the parent directory, and sure enough, .. gets us there: $ cd ..
$pwd /Users/amira ls usually doesn’t show us this special directory—since it’s always there, displaying it every time would be a distraction. We can ask ls to include it using the -a option, which stands for “all”. Remembering that we are now in /Users/amira: $ ls -F -a
./ Documents/ Music/ zipf/
Applications/ Library/ Public/
Desktop/ Movies/ todo.txt
The output also shows another special directory called . (a single period), which refers to the current working directory. It may seem redundant to have a name for it, but we’ll see some uses for it soon.
Combining Options
You’ll occasionally need to use multiple options in the same command. In most command line tools, multiple options can be combined with a single - and no spaces between the options:
$ls -Fa This command is synonymous with the previous example. While you may see commands written like this, we don’t recommend you use this approach in your own work. This is because some commands take long options with multi-letter names, and it’s very easy to mistake --no (meaning “answer ‘no’ to all questions”) with -no (meaning -n -o). The special names . and .. don’t belong to cd: they mean the same thing to every program. For example, if we are in /Users/amira/zipf, then ls .. will display a listing of /Users/amira. When the meanings of the parts are the same no matter how they’re combined, programmers say they are orthogonal. Orthogonal systems tend to be easier for people to learn because there are fewer special cases to remember. Other Hidden Files In addition to the hidden directories .. and ., we may also comes across files with names like .jupyter or .Rhistory. These usually contain settings or other data for particular programs; the prefix . is used to prevent ls from cluttering up the output when we run ls. We can always use the -a option to display them. cd is a simple command, but it allows us to explore several new ideas. First, several .. can be joined by the path separator to move higher than the parent directory in a single step. For example, cd ../.. will move us up two directories (e.g., from /Users/amira/zipf to /Users), while cd ../Movies will move us up from zipf and back down into Movies. What happens if we type cd on its own without giving a directory? $ pwd
/Users/amira/Movies
$cd$ pwd
/Users/amira
No matter where we are, cd on its own always returns us to our home directory. We can achieve the same thing using the special directory name ~, which is a shortcut for our home directory:
$ls ~ Applications Downloads Music todo.txt Desktop Library Pictures zipf Documents Movies Public (ls doesn’t show any trailing slashes here because we haven’t used -F.) We can use ~ in paths, so that (for example) ~/Downloads always refers to our download directory. Finally, cd interprets the shortcut - (a single dash) to mean the last directory we were in. Using this is usually faster and more reliable than trying to remember and type the path, but unlike ~, it only works with cd: ls - tries to print a listing of a directory called - rather than showing us the contents of our previous directory. ## 2.3 Creating New Files and Directories We now know how to explore files and directories, but how do we create them? To find out, let’s go back to our zipf directory: $ cd ~/zipf
$ls -F data/ To create a new directory, we use the command mkdir (short for make directory): $ mkdir docs
Since docs is a relative path (i.e., does not have a leading slash) the new directory is created below the current working directory:
$ls -F data/ docs/ Using the shell to create a directory is no different than using a graphical tool. If we look at the current directory with our computer’s file browser we will see the docs directory there too. The shell and the file explorer are two different ways of interacting with the files; the files and directories themselves are the same. Naming Files and Directories Complicated names of files and directories can make our life painful. Following a few simple rules can save a lot of headaches: 1. Don’t use spaces. Spaces can make a name easier to read, but since they are used to separate arguments on the command line, most shell commands interpret a name like My Thesis as two names My and Thesis. Use - or _ instead, e.g, My-Thesis or My_Thesis. 2. Don’t begin the name with - (dash) to avoid confusion with command options like -F. 3. Stick with letters, digits, . (period or ‘full stop’), - (dash) and _ (underscore). Many other characters mean special things in the shell. We will learn about some of these during this lesson, but these are always safe. If we need to refer to files or directories that have spaces or other special characters in their names, we can surround the name in quotes (""). For example, ls "My Thesis" will work where ls My Thesis does not. Since we just created the docs directory, ls doesn’t display anything when we ask for a listing of its contents: $ ls -F docs
Let’s change our working directory to docs using cd, then use a very simple text editor called Nano to create a file called draft.txt (Figure 2.3):
$cd docs$ nano draft.txt
When we say “Nano is a text editor” we really do mean “text”: it can only work with plain character data, not spreadsheets, images, Microsoft Word files, or anything else invented after 1970. We use it in this lesson because it runs everywhere, and because it is as simple as something can be and still be called an editor. However, that last trait means that we shouldn’t use it for larger tasks like writing a program or a paper.
Recycling Pixels
Unlike most modern editors, Nano runs inside the shell window instead of opening a new window of its own. This is a holdover from an era when graphical terminals were a rarity and different applications had to share a single screen.
Once Nano is open we can type in a few lines of text, then press Ctrl+O (the Control key and the letter ‘O’ at the same time) to save our work. Nano will ask us what file we want to save it to; press Return to accept the suggested default of draft.txt. Once our file is saved, we can use Ctrl+X to exit the editor and return to the shell.
Control, Ctrl, or ^ Key
The Control key, also called the “Ctrl” key, can be described in a bewildering variety of ways. For example, Control plus X may be written as:
• Control-X
• Control+X
• Ctrl-X
• Ctrl+X
• C-x
• ^X
When Nano runs it displays some help in the bottom two lines of the screen using the last of these notations: for example, ^G Get Help means “use Ctrl+G to get help” and ^O WriteOut means “use Ctrl+O to write out the current file”.
Nano doesn’t leave any output on the screen after it exits, but ls will show that we have indeed created a new file draft.txt:
$ls draft.txt Dot Something All of Amira’s files are named “something dot something”. This is just a convention: we can call a file mythesis or almost anything else. However, both people and programs use two-part names to help them tell different kinds of files apart. The part of the filename after the dot is called the filename extension and indicates what type of data the file holds: .txt for plain text, .pdf for a PDF document, .png for a PNG image, and so on. This is just a convention: saving a PNG image of a whale as whale.mp3 doesn’t somehow magically turn it into a recording of whalesong, though it might cause the operating system to try to open it with a music player when someone double-clicks it. ## 2.4 Moving Files and Directories Let’s go back to our zipf directory: cd ~/zipf The docs directory contains a file called draft.txt. That isn’t a particularly informative name, so let’s change it using mv (short for move): $ mv docs/draft.txt docs/prior-work.txt
The first argument tells mv what we are “moving”, while the second is where it’s to go. “Moving” docs/draft.txt to docs/prior-work.txt has the same effect as renaming the file:
$ls docs prior-work.txt We must be careful when specifying the destination because mv will overwrite existing files without warning. An option -i (for “interactive”) makes mv ask us for confirmation before overwriting. mv also works on directories, so mv analysis first-paper would rename the directory without changing its contents. Now suppose we want to move prior-work.txt into the current working directory. If we don’t want to change the file’s name, just its location, we can provide mv with a directory as a destination and it will move the file there. In this case, the directory we want is the special name . that we mentioned earlier: $ mv docs/prior-work.txt .
ls now shows us that docs is empty:
$ls docs and that our current directory now contains our file: $ ls
data/ docs/ prior-work.txt
If we only want to check that the file exists, we can give its name to ls just like we can give the name of a directory:
$ls prior-work.txt prior-work.txt ## 2.5 Copying Files and Directories The cp command copies files. It works like mv except it creates a file instead of moving an existing one: $ cp prior-work.txt docs/section-1.txt
We can check that cp did the right thing by giving ls two arguments to ask it to list two things at once:
$ls prior-work.txt docs/section-1.txt docs/section-1.txt prior-work.txt Notice that ls shows the output in alphabetical order. If we leave off the second filename and ask it to show us a file and a directory (or multiple directories) it lists them one by one: $ ls prior-work.txt docs
prior-work.txt
docs:
section-1.txt
Copying a directory and everything it contains is a little more complicated. If we use cp on its own, we get an error message:
$cp docs backup cp: analysis is a directory (not copied). If we really want to copy everything, we must give cp the -r option (meaning recursive): $ cp -r docs backup
Once again we can check the result with ls:
$ls docs backup docs/: section-1.txt backup/: section-1.txt Copying Files to and from Remote Computers For many researchers, a motivation for learning how to use the shell is that it’s often the only way to connect to a remote computer (e.g. located at a supercomputing facility or in a university department). Similar to the cp command, there exists a secure copy (scp) command for copying files between computers. See Appendix E for details, including how to set up a secure connection to a remote computer via the shell. ## 2.6 Deleting Files and Directories Let’s tidy up by removing the prior-work.txt file we created in our zipf directory. The command to do this is rm (for remove): $ rm prior-work.txt
We can confirm the file is gone using ls:
$ls prior-work.txt ls: prior-work.txt: No such file or directory Deleting is forever: unlike most GUIs, the Unix shell doesn’t have a trash bin that we can recover deleted files from. Tools for finding and recovering deleted files do exist, but there is no guarantee they will work, since the computer may recycle the file’s disk space at any time. In most cases, when we delete a file it really is gone. In a half-hearted attempt to stop us from erasing things accidentally, rm refuses to delete directories: $ rm docs
rm: docs: is a directory
We can tell rm we really want to do this by giving it the recursive option -r:
$rm -r docs rm -r should be used with great caution: in most cases, it’s safest to add the -i option (for interactive) to get rm to ask us to confirm each deletion. As a halfway measure, we can use -v (for verbose) to get rm to print a message for each file it deletes. This option works the same way with mv and cp. ## 2.7 Wildcards zipf/data contains the text files for several ebooks from Project Gutenberg: $ ls data
README.md moby_dick.txt
dracula.txt sense_and_sensibility.txt
frankenstein.txt sherlock_holmes.txt
jane_eyre.txt time_machine.txt
The wc command (short for word count) tells us how many lines, words, and letters there are in one file:
$wc data/moby_dick.txt 22331 215832 1276222 data/moby_dick.txt What’s in a Word? wc only considers spaces to be word breaks: if two words are connected by a long dash—like “dash” and “like” in this sentence—then wc will count them as one word. We could run wc more times to find out how many lines there are in the other files, but that would be a lot of typing and we could easily make a mistake. We can’t just give wc the name of the directory as we do with ls: $ wc data
wc: data: read: Is a directory
Instead, we can use wildcards to specify a set of files at once. The most commonly-used wildcard is * (a single asterisk). It matches zero or more characters, so data/*.txt matches all of the text files in the data directory:
$ls data/*.txt data/dracula.txt data/sense_and_sensibility.txt data/frankenstein.txt data/sherlock_holmes.txt data/jane_eyre.txt data/time_machine.txt data/moby_dick.txt while data/s*.txt only matches the two whose names begin with an ‘s’: $ ls data/s*.txt
data/sense_and_sensibility.txt data/sherlock_holmes.txt
Wildcards are expanded to match filenames before commands are run, so they work exactly the same way for every command. This means that we can use them with wc to (for example) count the number of words in the books with names that contains an underscore:
$wc data/*_*.txt 21054 188460 1049294 data/jane_eyre.txt 22331 215832 1253891 data/moby_dick.txt 13028 121593 693116 data/sense_and_sensibility.txt 13053 107536 581903 data/sherlock_holmes.txt 3582 35527 200928 data/time_machine.txt 73048 668948 3779132 total or the number of words in Frankenstein: $ wc data/frank*.txt
7832 78100 442967 data/frankenstein.txt
The exercises will introduce and explore other wildcards. For now, we only need to know that it’s possible for a wildcard expression to not match anything. In this case, the command will usually print an error message:
$wc data/*.csv wc: data/*.csv: open: No such file or directory ## 2.8 Reading the Manual wc displays lines, words, and characters by default, but we can ask it to display only the number of lines: $ wc -l data/s*.txt
13028 sense_and_sensibility.txt
13053 sherlock_holmes.txt
26081 total
wc has other options as well. We can use the man command (short for manual) to find out what they are:
$man wc Paging Through the Manual If our screen is too small to display an entire manual page at once, the shell will use a paging program called less to show it piece by piece. We can use and to move line-by-line or Ctrl+Spacebar and Spacebar to skip up and down one page at a time. (B and F also work.) To search for a character or word, use / followed by the character or word to search for. If the search produces multiple hits, we can move between them using N (for “next”). To quit, press Q. Manual pages contain a lot of information—often more than we really want. Figure 2.3 includes excerpts from the manual on your screen, and highlights a few of features useful for beginners. Some commands have a --help option that provides a succinct summary of possibilities, but the best place to go for help these days is probably the TLDR website. The acronym stands for “too long, didn’t read”, and its help for wc displays this: wc Count words, bytes, or lines. Count lines in file: wc -l {{file}} Count words in file: wc -w {{file}} Count characters (bytes) in file: wc -c {{file}} Count characters in file (taking multi-byte character sets into account): wc -m {{file}} edit this page on github As the last line suggests, all of its examples are in a public GitHub repository so that users like you can add the examples you wish it had. For more information, we can search on Stack Overflow or browse the GNU manuals (particularly those for the core GNU utilities, which include many of the commands introduced in this lesson). In all cases, though, we need to have some idea of what we’re looking for in the first place: someone who wants to know how many lines there are in a data file is unlikely to think to look for wc. ## 2.9 Summary The original Unix shell is celebrating its fiftieth anniversary. Its commands may be cryptic, but few programs have remained in daily use for so long. The next chapter will explore how we can combine and repeat commands in order to create powerful, efficient workflows. ## 2.10 Exercises The exercises below involve creating and moving new files, as well as considering hypothetical files. Please note that if you create or move any files or directories in your Zipf’s Law project, you may want to reorganize your files following the outline at the beginning of the next chapter. If you accidentally delete necessary files, you can start with a fresh copy of the data files by following the instructions in Section 1.2. ### 2.10.1 Exploring more ls flags What does the command ls do when used with the -l option? What happens if you use two options at the same time, such as ls -l -h? ### 2.10.2 Listing recursively and by time The command ls -R lists the contents of directories recursively, which means the subdirectories, sub-subdirectories, and so on at each level are listed. The command ls -t lists things by time of last change, with most recently changed files or directories first. In what order does ls -R -t display things? Hint: ls -l uses a long listing format to view timestamps. ### 2.10.3 Absolute and relative paths Starting from /Users/amira/data, which of the following commands could Amira use to navigate to her home directory, which is /Users/amira? 1. cd . 2. cd / 3. cd /home/amira 4. cd ../.. 5. cd ~ 6. cd home 7. cd ~/data/.. 8. cd 9. cd .. 10. cd ../. ### 2.10.4 Relative path resolution Using the filesystem shown in Figure 2.5, if pwd displays /Users/sami, what will ls -F ../backup display? 1. ../backup: No such file or directory 2. final original revised 3. final/ original/ revised/ 4. data/ analysis/ doc/ ### 2.10.5ls reading comprehension Using the filesystem shown in Figure 2.5, if pwd displays /Users/backup, and -r tells ls to display things in reverse order, what command(s) will result in the following output: doc/ data/ analysis/ 1. ls pwd 2. ls -r -F 3. ls -r -F /Users/backup ### 2.10.6 Creating files a different way What happens when you execute touch my_file.txt? (Hint: use ls -l to find information about the file) When might you want to create a file this way? ### 2.10.7 Using rm safely What would happen if you executed rm -i my_file.txt on the file created in the previous exercise? Why would we want this protection when using rm? ### 2.10.8 Moving to the current folder After running the following commands, Amira realizes that she put the (hypothetical) files chapter1.txt and chapter2.txt into the wrong folder: $ ls -F
data/ docs/
$ls -F data README.md frankenstein.txt sherlock_holmes.txt chapter1.txt jane_eyre.txt time_machine.txt chapter2.txt moby_dick.txt dracula.txt sense_and_sensibility.txt $ cd docs
Fill in the blanks to move these files to the current folder (i.e., the one she is currently in):
$mv ___/chapter1.txt ___/chapter2.txt ___ ### 2.10.9 Renaming files Suppose that you created a plain-text file in your current directory to contain a list of the statistical tests you will need to do to analyze your data, and named it: statstics.txt After creating and saving this file you realize you misspelled the filename! You want to correct the mistake, which of the following commands could you use to do so? 1. cp statstics.txt statistics.txt 2. mv statstics.txt statistics.txt 3. mv statstics.txt . 4. cp statstics.txt . ### 2.10.10 Moving and copying Assuming the following hypothetical files, what is the output of the closing ls command in the sequence shown below? $ pwd
/Users/amira/data
$ls books.dat $ mkdir doc
$mv books.dat doc/$ cp doc/books.dat ../books-saved.dat
$ls 1. books-saved.dat doc 2. doc 3. books.dat doc 4. books-saved.dat ### 2.10.11 Copy with multiple filenames This exercise explores how cp responds when attempting to copy multiple things. What does cp do when given several filenames followed by a directory name? $ mkdir backup
$cp dracula.txt frankenstein.txt backup/ What does cp do when given three or more file names? $ cp dracula.txt frankenstein.txt jane_eyre.txt
### 2.10.12 List filenames matching a pattern
When run in the data directory of your project directory, which ls command(s) will produce this output?
jane_eyre.txt sense_and_sensibility.txt
1. ls ??n*.txt
2. ls *e_*.txt
3. ls *n*.txt
4. ls *n?e*.txt
### 2.10.13 Organizing directories and files
Amira is working on a project and she sees that her files aren’t very well organized:
$ls -F books.txt data/ results/ titles.txt The books.txt and titles.txt files contain output from her data analysis. What command(s) does she need to run to produce the output shown? $ ls -F
data/ results/
$ls results books.txt titles.txt ### 2.10.14 Reproduce a directory structure You’re starting a new analysis, and would like to duplicate the directory structure from your previous experiment so you can add new data. Assume that the previous experiment is in a folder called 2016-05-18, which contains a data folder that in turn contains folders named raw and processed that contain data files. The goal is to copy the folder structure of 2016-05-18/data into a folder called 2016-05-20 so that your final directory structure looks like this: 2016-05-20/ └── data ├── processed └── raw Which of the following set of commands would achieve this objective? What would the other commands do? # Set 1$ mkdir 2016-05-20
$mkdir 2016-05-20/data$ mkdir 2016-05-20/data/processed
$mkdir 2016-05-20/data/raw # Set 2$ mkdir 2016-05-20
$cd 2016-05-20$ mkdir data
$cd data$ mkdir raw processed
# Set 3
$mkdir 2016-05-20/data/raw$ mkdir 2016-05-20/data/processed
# Set 4
$mkdir 2016-05-20$ cd 2016-05-20
$mkdir data$ mkdir raw processed
### 2.10.15 Wildcard expressions
Wildcard expressions can be very complex, but you can sometimes write them in ways that only use simple syntax, at the expense of being a bit more verbose. In your data/ directory, the wildcard expression [st]*.txt matches all files beginning with s or t and ending with .txt. Imagine you forgot about this.
1. Can you match the same set of files with basic wildcard expressions that do not use the [] syntax? Hint: You may need more than one expression.
2. Under what circumstances would your new expression produce an error message where the original one would not?
### 2.10.16 Removing unneeded files
Suppose you want to delete your processed data files, and only keep your raw files and processing script to save storage. The raw files end in .txt and the processed files end in .csv. Which of the following would remove all the processed data files, and only the processed data files?
1. rm ?.csv
2. rm *.csv
3. rm * .csv
4. rm *.*
### 2.10.17 Other wildcards
The shell provides several wildcards beyond the widely-used *. To explore them, explain in plain language what (hypothetical) files the expression novel-????-[ab]*.{txt,pdf} matches and why.
## 2.11 Key Points
• A shell is a program that reads commands and runs other programs.
• The filesystem manages information stored on disk.
• Information is stored in files, which are located in directories (folders).
• Directories can also store other directories, which forms a directory tree.
• pwd prints the user’s current working directory.
• / on its own is the root directory of the whole filesystem.
• ls prints a list of files and directories.
• An absolute path specifies a location from the root of the filesystem.
• A relative path specifies a location in the filesystem starting from the current directory.
• cd changes the current working directory.
• .. means the parent directory.
• . on its own means the current directory.
• mkdir creates a new directory.
• cp copies a file.
• rm removes (deletes) a file.
• mv moves (renames) a file or directory.
• * matches zero or more characters in a filename.
• ? matches any single character in a filename.
• wc counts lines, words, and characters in its inputs.
• man displays the manual page for a given command; some commands also have a --help option.
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2021-03-01 19:58:08
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https://www.physicsforums.com/threads/how-to-prove-this.355950/
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# How to prove this
1. ### zetafunction
399
given a set of orthogonal polynomials with respect to a certain measure w(x)
$$\int_{a}^{b}dx w(x) P_{n} (x)P_{m} (x) = \delta _{n,m}h_{n}$$
how can anybody prove that exists a certain M+M Hermitian matrix so
$$P_{m} (x)= < Det(1-xM)>$$ here <x> means average or expected value of 'x'
if we knew the set of orthogonal polynomials $$P_{m} (x)$$ for every 'm' and the measure w(x) , could we get the expression for the matrix M ??
2. ### g_edgar
607
Your equation doesn't make much sense to me. How about providing an example. A particular orthogonal system and the corresponding matrix.
3. ### Dragonfall
Looks like we got ourselves a troll.
4. ### mathaino
3
Interesting way to react to posts that do not tickle the own ears - delete them. You guys are not seekers of truth. Ibn al Haytham would be ashamed for you all.
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2015-04-01 18:04:35
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http://www.gamedev.net/topic/637745-in-filepaths/
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### #6Servant of the Lord Crossbones+ - Reputation: 24620
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Posted 23 January 2013 - 07:32 PM
It's always the current working directory, I believe.
So you are saying, under Windows, the single dot always indicates the current working directory and not the executable directory, unless they happen to be the same (which is most of the time)?
Is it the same on Mac and Linux?
You could always whip up a quick program to find out, though!
I don't currently have access to a Mac and I don't have Linux installed, so I can't currently try it out.
Windows:
Anything that does not explicitly specify a drive is relative to the working directory, so the period has no real meaning on Windows. The working directory is often the same as the executable directory but could be different if the executable changes it at run-time, it is run under Visual Studio, or run from a .bat file located elsewhere (the working directory will be that of the .bat file).
Thank you, that helps.
And in the situation of running from a command-line prompt like this:
C:/> CD C:/path/to/folder/
C:/path/to/folder/> ../../different/folder/program.exe
'program.exe' will have the current working directory of 'C:/path/to/folder/ ', and not 'C:/path/different/folder/?
Mac OS X is Unix based, thus it inherits most of its features, including the paths.
In Unix- based OS, such as Linux,
. = current directory
.. = parent directory
So is the 'current directory' the "directory of the executable", or "the current working directory of the environment"?
Anything else [speaking of Unix-like systems] is considered relative paths (so folder/file and ./folder/file and ../myfolder/folder/file are all relative paths).
Thanks, I was wondering about that. So folder/file and ./folder/file would be the same directory?
[Edit:] Fixed broken quotes a day later.
Edited by Servant of the Lord, 24 January 2013 - 11:53 AM.
It's perfectly fine to abbreviate my username to 'Servant' rather than copy+pasting it all the time.
All glory be to the Man at the right hand... On David's throne the King will reign, and the Government will rest upon His shoulders. All the earth will see the salvation of God.
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### #7L. Spiro Crossbones+ - Reputation: 19239
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Posted 23 January 2013 - 07:46 PM
Thanks, I was wondering about that. So folder/file and ./folder/file would be the same directory?
Yes. The period is probably just for clarity for us humans, or it might have significant meaning in some DOS commands (just a guess), but with or without it is the same to the file system.
L. Spiro
### #8Milcho Crossbones+ - Reputation: 1177
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Posted 23 January 2013 - 07:57 PM
Your quotes seemed to have messed up a bit.
So is the 'current directory' the "directory of the executable", or "the current working directory of the environment"?
It depends on where your program is executed from. Running a program from the shell in linux, i believe it will use the current directory the shell is in as the "." directory.
Most programs are run from their own folders usually by the graphical UI in linux. But I think every Linux distribution allows you to set the "Run From..." setting on any executable you run. That location, which is set depending on how your program is run, is what's considered the current directory by the program (the . directory)
So folder/file and ./folder/file would be the same directory?
Yes. In linux those are the same.
The only exception is executable files in the shell - if you're in the same directory as an executable file, say "test.sh" - and you try to run "test.sh", it will not work. You need to run "./test.sh".
### #9L. Spiro Crossbones+ - Reputation: 19239
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Posted 24 January 2013 - 03:12 AM
And in the situation of running from a command-line prompt like this:
C:/> CD C:/path/to/folder/
C:/path/to/folder/> ../../different/folder/program.exe
'program.exe' will have the current working directory of 'C:/path/to/folder/ ', and not 'C:/path/different/folder/?
I didn’t see your follow-up question the first time around.
Yes, it will be the directory in the command-line.
L. Spiro
Edited by L. Spiro, 24 January 2013 - 03:13 AM.
### #10Milcho Crossbones+ - Reputation: 1177
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Posted 24 January 2013 - 03:33 AM
And in the situation of running from a command-line prompt like this:
C:/> CD C:/path/to/folder/
C:/path/to/folder/> ../../different/folder/program.exe
'program.exe' will have the current working directory of 'C:/path/to/folder/ ', and not 'C:/path/different/folder/?
I didn’t see your follow-up question the first time around.
Yes, it will be the directory in the command-line.
L. Spiro
Odd, I failed to see that question too until you pointed it out. I blame the fact that the quotes were out of line.
It works the same in Linux as well, as I mentioned before, the current path in which you are in when you invoke a program (regardless of where that program is) is what's used as the program's "working directory".
This is useful because a lot of command line tools in unix are just simple executables, So when you run "grep sometext somefile" in a directory, you expect the grep tool to search through "somefile" which is often located in the current directory, and find you any occurances of "sometext" in that file.
You don't expect to have to place "somefile" in the directory in which the executable called "grep" is located in order to be able to search through it.
### #11Hodgman Moderators - Reputation: 38840
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Posted 24 January 2013 - 05:39 AM
Also consider paths like C:/path/./to/./file -- that's equal to C:/path/to/file.
The dot is pretty meaningless.
It's mostly useful when you want to be explicit in the case where multiple directories are going to be searched for your input.
e.g. let's say that when you type 'notepad.exe' the windows command line, it first checks inside the system32 directory (i'm not sure if this is true) before checking the current directory.
If you wanted to be explicit that you meant the notepad.exe relative to the current working directory of your shell, you could type "./notepad.exe" instead, because the shell will resolve your dot-containing-input into a full path before it searches for that file.
Edited by Hodgman, 24 January 2013 - 05:42 AM.
### #12SimonForsman Crossbones+ - Reputation: 6709
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Posted 24 January 2013 - 05:51 AM
Thanks, I was wondering about that. So folder/file and ./folder/file would be the same directory?
Yes. The period is probably just for clarity for us humans, or it might have significant meaning in some DOS commands (just a guess), but with or without it is the same to the file system.
L. Spiro
On Linux ./executablefile and executablefile are not the same when used from the shell.
"./executablefile" will attempt to launch executablefile from the current directory, "executablefile" will attempt to launch it from the directories listed in the PATH enviroment variable which may or may not include the current directory(.) (most distributions does not include the current directory in PATH by default so you usually have to launch applications using ./executablefile)
on Windows and DOS the system will try the current directory before it tries the ones in the PATH variable but using ./file should stop it from using the PATH so they shouldn't behave identically.
the system() function in C and C++ should launch the specified application using the same shell the host application started from (so system("pause") and system("./pause") should behave differently), i would guess that the same is true for shellexecute in Windows, things like iostream however doesn't use the shells enviroment variables so all non absolute paths should be treated as relative paths.
I don't suffer from insanity, I'm enjoying every minute of it.
The voices in my head may not be real, but they have some good ideas!
### #13L. Spiro Crossbones+ - Reputation: 19239
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Posted 24 January 2013 - 05:58 AM
e.g. let's say that when you type 'notepad.exe' the windows command line, it first checks inside the system32 directory (i'm not sure if this is true) before checking the current directory.
If you wanted to be explicit that you meant the notepad.exe relative to the current working directory of your shell, you could type "./notepad.exe" instead, because the shell will resolve your dot-containing-input into a full path before it searches for that file.
I don’t think that is correct (I don’t know for a fact) based on how Windows searches for DLL’s.
Assuming the DLL search is related to executable searches, it should check the current working directory before the system32 directory. After also checking the executable’s directory (but if I had to wager, this is the part I would guess to be different and skipped entirely).
http://msdn.microsoft.com/en-us/library/7d83bc18(v=vs.80).aspx
http://msdn.microsoft.com/en-us/library/windows/desktop/ms682586(v=vs.85).aspx
I’ve never heard of any meaning related to the . in Windows paths, and my guess is it is just a carry-over from a past system such as DOS, FAT32, NTSF, or Unix-like behavior.
Again, I don’t know it for a fact (what Google search terms would you use for this anyway?) but I think the . has absolutely no meaning on Windows paths except for possibly in some special cases such as some DOS commands.
L. Spiro
### #14Hodgman Moderators - Reputation: 38840
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Posted 24 January 2013 - 06:09 AM
I don’t think that is correct (I don’t know for a fact) based on how Windows searches for DLL’s.
Yeah I didn't mean to imply that that particular search path order is correct, just that the shell will expand the dot before searching for the file, so ".\notepad.exe" will stop it from finding it in the system32 directory (unless that's the shell's current working directory). If the working directory is "C:\", then the shell will be looking to run a file named "C:\notepad.exe", which only has one possible location.
This kind of behaviour is totally up to the shell though, and has nothing to do with the OS as a whole.
When building your own apps that interact with a file-system, you could implement or not implement similar ideas to make use of the dot.
Edited by Hodgman, 24 January 2013 - 06:12 AM.
### #15Servant of the Lord Crossbones+ - Reputation: 24620
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Posted 24 January 2013 - 11:57 AM
Thank you, this is all really helpful. I don't normally work with 'working directories' that are different from the executable directory (just launching the executable specifically or by shortcut), and I've very little experience with Mac and Linux systems.
Thank you very much!
It's perfectly fine to abbreviate my username to 'Servant' rather than copy+pasting it all the time.
All glory be to the Man at the right hand... On David's throne the King will reign, and the Government will rest upon His shoulders. All the earth will see the salvation of God.
Of Stranger Flames -
[Need web hosting? I personally like A Small Orange]
### #16Servant of the Lord Crossbones+ - Reputation: 24620
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Posted 25 January 2013 - 05:47 PM
Rather than start a new thread, here's a closely related question: Is an empty file extension the same as no file extension?
Example: Is file the same as file. ? One has a no period and no extension, the other has a period with an empty extension. Are they treated identically by modern OSes (> Win XP), including Linux and Mac OSX?
I ask this because: Given a path by a string, how does one know if the path ends with a file or a directory?
If the final string segment ends with a '/', you know it's a directory.
If the final string segment contains a period, you are fairly sure it's a file (since folders don't usually, but can, contain periods).
But if the final string segment contains neither a period nor ends with a slash, it's very ambiguous (but assumed to be a file?).
It's perfectly fine to abbreviate my username to 'Servant' rather than copy+pasting it all the time.
All glory be to the Man at the right hand... On David's throne the King will reign, and the Government will rest upon His shoulders. All the earth will see the salvation of God.
Of Stranger Flames -
[Need web hosting? I personally like A Small Orange]
### #17RulerOfNothing Members - Reputation: 1300
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Posted 25 January 2013 - 06:00 PM
Preliminary testing appears to support the statement that "file" and "file." are interchangeable on modern Windows (in fact, when I made a file called "file." it simply got rid of the period) I cannot say anything definitive about Linux and OSX though.
### #18Milcho Crossbones+ - Reputation: 1177
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Posted 25 January 2013 - 06:04 PM
Files and Folders aren't just differentiated by 'extension'. In fact, extensions aren't really anything special - they're just part of the filename (windows can be configured to hide that part of the filename, but in reality, the extension is just a string).
You can try this yourself, on linux, if you type "ls -l" you'll see a list of properties preceeding each file, something like this for executables:
-rwxr-xr-x
or this:
-rw-r--r--
for non-executables. The properties are separated into categories - owner, group, and other - and each is given permission to read ®, write (w) or execute (x) a file.
see chmod man pages if you want more explanation: http://ss64.com/bash/chmod.html
A folder isn't like a file at all. It's an entry in the file system that simply says that there's a folder there. You rename a folder to something.exe and it still won't be executed (same on linux too)
Btw, a folder's properties on linux look like this:
drwxr-xr-x
That first letter (d) indicates its a folder on the filesystem.
Renaming a file that's 'executable' to whatever extension will stilll let you run that file from the command shell in linux. Windows is another story, that I'm not 100% sure of. It seems like it does treat files based on their extensions (for executables I mean).
Oh, and filenames in linux can't contain /. (though they can contain all other sorts of unreadable and impossible to use characters.. including newline..yeah...)
### #19SiCrane Moderators - Reputation: 10421
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Posted 25 January 2013 - 06:05 PM
For the most part, given a path as a string you don't know if it's a file or a directory. You need to query the file system. Ex: with the stat() system function on POSIX systems, the is_directory() function in boost::filesystem, etc.
### #20Servant of the Lord Crossbones+ - Reputation: 24620
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Posted 25 January 2013 - 06:17 PM
For the most part, given a path as a string you don't know if it's a file or a directory. You need to query the file system. Ex: with the stat() system function on POSIX systems, the is_directory() function in boost::filesystem, etc.
Thanks, that's what I was wanting to know - whether there was a way to be reasonably confident merely from the string without querying the file system itself. Other than ending in a '/' guaranteeing it's not a file, there's not much you can go off of!
It would've been great if OSes enforced a period for filenames, even if the extension was left blank, or maybe used a separator to indicate where the path ends and the file begins. Maybe ':', like: my/path/to/file/:filename
Ofcourse, extensions don't really confirm a file is of any specific format anyway, so maybe that's just band-aiding an imperfect, but functional, system.
It's perfectly fine to abbreviate my username to 'Servant' rather than copy+pasting it all the time.
All glory be to the Man at the right hand... On David's throne the King will reign, and the Government will rest upon His shoulders. All the earth will see the salvation of God.
Of Stranger Flames -
[Need web hosting? I personally like A Small Orange]
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
PARTNERS
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2015-06-02 13:51:33
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https://chemistry.stackexchange.com/questions/162042/if-the-reaction-is-spontaneous-is-it-the-%CE%94s-surroundings-that-is-always-positiv
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# If the reaction is spontaneous, is it the ΔS surroundings that is always positive or ΔS universe? [closed]
If the reaction is spontaneous, is it the ΔS surroundings that are always positive or ΔS universe?
I tried researching this on Google. However, the answers that I have found vary with different explanation.
• Hint: What would Delta G say about that? Consider spontaneous but endothermic processes. Jan 8, 2022 at 11:21
• It can be also said $$\Delta S_\mathrm{universe} \ge - \frac { \Delta G_\mathrm{r}} { T}$$ for constant T,p. Jan 8, 2022 at 12:07
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2023-03-24 18:47:45
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https://projecteuclid.org/euclid.cmp/1104247985
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## Communications in Mathematical Physics
### Entropy production by block variable summation and central limit theorems
#### Article information
Source
Comm. Math. Phys., Volume 140, Number 2 (1991), 339-371.
Dates
First available in Project Euclid: 28 December 2004
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2019-03-21 03:40:42
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https://mathematica.stackexchange.com/tags/function-construction/new
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# Tag Info
### Building A Function Using Constants From a List
f[t_] := Mean[indata[[All, 2]]] + Plus @@ MapThread[#[[2]] Sin[#[[1]] t + #[[3]]] &, {freqvals}] Plot[f[t], {t, 0, 1}]
• 25.2k
### Building A Function Using Constants From a List
Following suggestions in other answers is a good idea, i.e., avoid the use of loops in Mathematica. There are are almost always better ways to do things. That said, to minimally change your code so ...
• 22.3k
Accepted
### Building A Function Using Constants From a List
Welcome to MSE! To be effective with Wolfram Language, you have to change your thinking from "procedural" to "functional". E.g., try to avoid using explicit ...
• 4,778
### Better way to do such iteration
Using Map and Flatten can avoid the 2^(n - 1). We seperate the times ...
• 49.3k
...
• 354k
### How to plot a point with two colors?
We can define PlotMarkers by any Graphics object and using ListPlot to add such markers of ...
• 49.3k
### How to plot a point with two colors?
It's not so easy to do this automatically, and you have to account for the aspect ratio of your plot, without which you end up with a squashed marker. Here's my attempt for manual points and colour ...
• 21.8k
### How to gather functions which intersect at the same point?
This goes only through all $(n^2-n)/2$ Subsets[func, {2}] as suggested by @Syed. For $n=100$ that is only $4950$ cases. It calculates the intersection points of ...
• 33k
### How to gather functions which intersect at the same point?
This goes through all $2^n -n -1$ subsets of length 2 or more. An overkill that allows very simple code, but it's not practical for large $n$. See other answer(s). ...
• 33k
Accepted
### Using the Apply construction
As an expression, before being evaluated, this... Function @@ {t, g} is actually this... Apply[Function, List[t, g]] Now, we ...
• 13.9k
Accepted
### Better way to do such iteration
Based on your update with the fractal demo, I think I may have a solution. The basic idea is that a triangle is split into two smaller triangles based on a parameter that determines how long the new ...
• 13.9k
1 vote
### Better way to do such iteration
This isn't complete, but I'm at a point where I need feedback. As best as I can tell, your func is doing a rescale operation, and that rescaling happens to rescale <...
• 13.9k
### Function defined through conditional pattern and derivative
I think this comes closest to what you're looking for: ...
• 19.6k
### Function defined through conditional pattern and derivative
You got something wrong. You confuse ":" and "/;". What you wrote is a default value. ...
• 35.4k
Accepted
### How to delete edges from a graph with some condition?
MMA is smart enough to sort the vertices. Therefore the "Reverse" is not needed. ...
• 35.4k
### Sort a list using a scoring list with Switch
In each sublist, leave the first letter unchanged but "invert" the remaining letters $(a \leftrightarrow z, b \leftrightarrow y, ...)$. Then use a traditional sort... on the first element, ...
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...
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### Sort a list using a scoring list with Switch
Let's specify the order of the first elements in an easy way, just list the possible first elements in the order you want: ...
• 13.9k
### Define a function on a MeshRegion
I used the hints in the comments to write the following code (excerpt from my notebook): With a little more calculation this allowed me to generate: Explanation: ...
• 151
### How can I ignore an argument?
You can make the default value random: ...
• 2,523
### How can I ignore an argument?
While you didn't ask for advice, you could do this without the explicit While looping: ...
• 13.9k
Accepted
### How can I ignore an argument?
Using a default argument of 1: ...
• 25.2k
### How can I ignore an argument?
Use Reap and Sow instead of AppendTo. Also you can do away with the ...
• 21.8k
### How can I ignore an argument?
Probably the easiest and clearest way would be to just overload mylist with another definition: ...
• 13.9k
Accepted
### How to make a function take another function as an input?
sildeben[expr_,{variable_,startvalue_,endvalue_,steps_}] := Table[ {variable,expr},{variable,Subdivide[startvalue,endvalue,steps]}];
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### How to make a function take another function as an input?
Syed's answer is fine. I am suggesting another for completeness funct[steps_] := N@Table[{a Pi, Sin[a Pi]}, {a, 0, steps, 1/2}] when you run ...
• 9,177
### How to make a function take another function as an input?
{#, Sin[#]} & /@ Subdivide[0, π, 2] // N {{0., 0.}, {1.5708, 1.}, {3.14159, 0.}} ...
• 25.2k
### Automating interesting ways to write 2023
According to the Goldbach conjecture, 2023 can be written as a sum of 3 primes... we can find these using: ...
• 66.7k
### Automating interesting ways to write 2023
I thought it could be nice to use different domains of Mathematica to represent 2023 in different ways. Outline: Entities Polynomial algebra Linguistic Data Number theory Text analysis Special ...
• 4,559
Accepted
### Automating interesting ways to write 2023
I would like to point out FrobeniusSolve, e.g. this yields nonnegative solutions $(x_1,x_2)$ of this equation $20 x_1 +23 x_2 =2023$ ...
• 55.5k
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2023-01-30 13:56:10
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https://www.physicsforums.com/threads/simplify-derivative-after-using-product-and-chain-rule.611413/
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# Homework Help: Simplify derivative after using product and chain rule
1. Jun 4, 2012
### biochem850
1. The problem statement, all variables and given/known data
(x$^{2}$-x$^{-1}$+1)(x$^{3}$+2x-6)$^{7}$
2. Relevant equations
Chain Rule & Power Rule
3. The attempt at a solution
(x$^{3}$+2x-6)$^{6}$[(x$^{3}$+2x-6)(2x+x$^{-2}$)+7(3x$^{2}$+2)(x$^{2}$-x$^{-1}$+1)]
This is the farthest I've gotten but when I do additional computation I do not arrive at the correct simplified solution.
edit- should be "simplify derivative after using power rule and chain rule"
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Jun 5, 2012
### tiny-tim
hi biochem850!
looks ok so far
you've probably made a mistake with a minus somewhere later …
check them first …
if that doesn't help, then show us what you did
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2018-07-23 06:56:45
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https://ask.openstack.org/en/answers/112684/revisions/
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It seems that i've found the solution - your just need to switch devstack branch to stable/newton and add several lines in local.config:
HORIZON_BRANCH=newton-eol
newton-eol is a git tag, since there is no newton branch in several projects...
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2019-11-13 10:49:44
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https://www.physicsforums.com/threads/van-de-graaff-generator-questions.730778/
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# Van de Graaff generator questions?
1. Dec 31, 2013
### AFSteph
Hello~ I've got a confusing question;
"If a Van de Graaff generator is charged to 50,000 volts, how much energy does it take to add an additional electron to the charge on the sphere?"
(The charge of an electron is given as -1.6 x 10$^{-19}$C)
My answer: (50,000 V)( -1.6 x 10$^{-19}$C) = 8.0 x 10$^{-15}$ J
Er, I actually don't understand this at all, but if it's correct then meh.
Second question:
"How would the voltage of the generator in Part (A) compare to the voltage of a larger Van de Graaff generator with the same amount of charge? To which generator could an electron be added with the least expenditure of energy?"
According to my textbook,
So then by that reasoning, the larger generator should have more voltage for the same amount of charge. And by using the formula of volts x charge of an electron to find the energy needed to add an electron then it should take more energy to add to the bigger, higher voltage, generator.
BBUUUUUUUUTT... this is a bit contradictory to me because here https://www.physicsforums.com/showthread.php?t=229046 Doc Al has a different (opposite!) answer that makes sense but doesn't fit.
Can someone help me in clear English here? Thanks for any assistance!
2. Jan 1, 2014
### rude man
Q = CV.
If Q stays constant and C increases, what does V do? Then what does qV do?
(Note: the capacitance of a sphere = 4 pi epsilon R, R = radius of sphere. Adding the gas also increases C).
Last edited: Jan 1, 2014
3. Jan 1, 2014
### AFSteph
Hmm maybe I should mention that the course I'm doing is a conceptual Physics one and we don't learn JACK about the mathematical side of things. "Q = CV" is nowhere in my textbook so I guess I've got some googling to do... Thanks for your input :D
4. Jan 1, 2014
### AFSteph
Q = CV
Where Q is charge; C is capacitance and V is voltage.
So I think I know what you're saying... I had capacitance and voltage a bit backwards here.
So as the radius increases, capacitance, not voltage, increases and the voltage decreases. And by that formula given earlier, (voltage x electron charge) with less voltage it is easier to add an extra electron. That makes a lot more sense now :D
Thank goodness for the internet because this textbook really is not that helpful. They don't give the formula for capacitance nor do they even mention the frickin' word capacitance in the chapter on Van de Graaff generators!
If I'm understanding this correctly, then the quote in my textbook is incorrect
The word "voltage" should be replaced with the word "capacitance", no?
Am I understanding this correctly?
5. Jan 1, 2014
### rude man
You are doing fine.
But your textbook could still be right: the larger generators can probably hold much more charge so even though the capacitance is higher, you can add even more Q with the result that the voltage is higher also than on a smaller unit. Nevertheless, I find the statement confusing at best.
6. Jan 1, 2014
### voko
I believe the statement in the book has to do with electrical breakdown or perhaps corona discharge. Voltages in van der Graaff generators are limited by that.
7. Jan 1, 2014
### dlgoff
That was my impression as well.
http://en.wikipedia.org/wiki/Robert_J._Van_de_Graaff
8. Jan 1, 2014
### AFSteph
Thank you so much for clarifying. The course I'm doing is the American School of Correspondence's Conceptual Physics course, which is meant for homeschooled students (as I am).
They develop the exams and "study guide" themselves, and they obviously have no influence on the course book -- they just built around it. That said, it seems bizarre that they would ask a question when the course book would lead you to a contradictory answer. When I mail off this exam (correspondence course -- things are sent though the mail) should I include a letter with a link to this thread suggesting they have a section in the study guide on capacitance clarifying this? Seems like legitimate constructive criticism / feedback.
Again, thank you for helping me through this. I appreciate the hinting approach :)
9. Jan 1, 2014
### rude man
10. Jan 1, 2014
### rude man
I would think that's a good idea.
11. Jan 1, 2014
Yes it's to do with electric field strength ( E=-V/r r= radius of dome). If this exceeds a certain value(I think its between 3MV to 4MV for dry air at normal pressure )the air breaks down and you get sparking. Of course the dome is not perfectly spherical and there are smaller radii bits surrounding the hollow where the support and the belt enter. The field is stronger at these more "pointy bits".
12. Jan 1, 2014
### voko
I assume you wanted to say MV/m, not just MV.
13. Jan 1, 2014
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2017-08-20 13:14:51
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http://windowsitpro.com/windows-server-2000/ads-sequence-files
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Downloads
41399.zip
Windows Server 2003 Automated Deployment Services (ADS) provide XML files that define the sequence of operations for common deployment and management jobs. These jobs are known as task sequences and the files defining these sequences are known as sequence files. Each operation, or task, within a sequence is enclosed in several XML tags.
ADS provides several sample sequence files that you can use as templates to create customized sequences. For example, Web Listing A shows the boot-to-da.xml sample sequence file, which defines the sequence to boot a system to the ADS Deployment Agent. The code at callout A shows the ADS XML schema reference; the code at callout B gives a description of the sequence's task. The code at callout C performs the task (i.e., boots to the Deployment Agent).
You can reorder or combine tasks within a sequence file as necessary to customize your deployment. For example, you can customize sequences to copy extra files to the local disk or to modify the registry. All sequence files reside in the C:\program files\microsoft ads\samples\sequences folder on the ADS server. For more information about sequence files and the default sequence templates, refer to the ADS online documentation.
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2015-12-02 04:25:43
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http://mathhelpforum.com/advanced-math-topics/201760-timelike-spacelike-null-seperation-2-print.html
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# Timelike, Spacelike and Null Seperation
Show 40 post(s) from this thread on one page
Page 2 of 3 First 123 Last
• Aug 9th 2012, 06:32 AM
AA23
Re: Timelike, Spacelike and Null Seperation
Could you please expand your calculations JohnDMalcolm because I'm struggling to get from the first expression to last. Thank You
• Aug 9th 2012, 07:24 AM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
Sorry, I've noticed my mistake. We are working in units where the speed of light is given the value $c = 1$. This makes time measured in metres. One metre of time is defined as the amount of time it takes light to travel one metre of distance. With this, the Lorentz transformation is
$t' = \gamma \left( t - vx \right)$
$x' = \gamma \left( x - vt \right)$
$\gamma = \frac{1}{\sqrt{1 - v^{2}}}$
$ds^{2} = dt^{2} - dx^{2}$
If we were working in SI units, where $c = 3 \times 10^{8} \frac{m}{s}$, then we use the other transformations we were trying before AND the spacetime interval is
$ds^{2} = c^{2}dt^{2} - x^{2}$
So to solve your problem of $t^{2} - x^{2}$, not $c^{2}t^{2} - x^{2}$, use the set of equations above where $c = 1$. Sorry for the earlier confusion.
• Aug 9th 2012, 07:38 AM
AA23
Re: Timelike, Spacelike and Null Seperation
Thank You again JohnDMalcolm, I feel a lot more confident answering these types of questions now (Happy)
I do have one final question which I have attached that has been causing me a lot of problems. Now I can solve similar questions easily when given the relationship betweem (x,y) and (p,q) but am unsure how to establish one?
Do you have any suggestions or advice? Thank You once again
• Aug 9th 2012, 07:52 AM
HallsofIvy
Re: Timelike, Spacelike and Null Seperation
Quote:
Originally Posted by AA23
Hey HallsofIvy
So based on your post are my original calculations incorrect in #1, because the notes I have been given would suggest my method is correct?
No, what I said was you will get the same results.
• Aug 9th 2012, 07:53 AM
AA23
Re: Timelike, Spacelike and Null Seperation
My mistake, thank you HallsofIvy
• Aug 9th 2012, 08:27 AM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
Quote:
Originally Posted by AA23
Thank You again JohnDMalcolm, I feel a lot more confident answering these types of questions now (Happy)
I do have one final question which I have attached that has been causing me a lot of problems. Now I can solve similar questions easily when given the relationship betweem (x,y) and (p,q) but am unsure how to establish one?
Do you have any suggestions or advice? Thank You once again
Hey, call me John. I'm happy to help out. Helping someone through these problems helps to solidify my own understanding of the topic.
For the perpendicular case I think it's easiest to derive the metric using vectors. The unit vectors in the new coordinate system are
$\hat{p} = \hat{x}$
$\hat{q} = \cos{\theta} \hat{x} + \sin{\theta} \hat{y}$
The position of A can be written as the vector
$\vec{A} = x_{A} \hat{x} + y_{A} \hat{y}$
The (p,q) coordinates would be the projection of this vector onto those coordinate axes.
$p_{A} = \vec{A} \cdot \hat{p}$
$q_{A} = \vec{A} \cdot \hat{q}$
Since the coordinates of A are general, these dot products can be used to construct the metric.
• Aug 9th 2012, 08:31 AM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
As for the parallel case, I find it easiest to use trig geometry. Draw a line from A that is parallel to the p-axis and intersects the q-axis. Try to find the length from the origin to this intersection (ie, the coordinate q) in terms of y and theta. Similarly, draw a line from A parallel to the q-axis that intersects the p-axis. The length from the origin to this intersection will be a linear combination of x and y involving theta.
• Aug 9th 2012, 08:43 AM
AA23
Re: Timelike, Spacelike and Null Seperation
Hi John, so going from what you have said I have started my calculations (see attachment). I believe the next step is to establish the differential coefficients and use them to obtain the metric tensor. Is this correct??
• Aug 9th 2012, 09:15 AM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
Your transformations are linear so we can do this by inspection. So the transformation in going from (x,y) to (p,q) is the 2x2 matrix below.
$\left( \begin{array}{c} p \\ q \end{array} \right) = \left( \begin{array}{cc} E&F \\ G&H \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right)$
This gives the two equations
$p = Ex + Fy$
$q = Gx + Hy$
So, by comparing to the expressions you derived, what are E, F, G, and H? These are the differential coefficients you're looking for. With what the problem is asking you to do, you didn't need to express x and y in terms of p and q. Those expressions would allow you to find the transformation from (p,q) to (x,y). That could be useful, but, like I said, the problem doesn't require it.
Let's define the transformation matrix as
$\Lambda = \left( \begin{array}{cc} E&F \\ G&H \end{array} \right)$
The metric in the (x,y) coordinates is just the 2x2 identity matrix.
$g = \left( \begin{array}{cc} 1&0 \\ 0&1 \end{array} \right)$
The metric in the (p,q) coordinates is
$g' = \Lambda^{T} g \Lambda$
• Aug 9th 2012, 09:23 AM
AA23
Re: Timelike, Spacelike and Null Seperation
Would:
E= 1 , F=0 , G = cos(theta) and H = sin(theta)
Is this what you were asking for?
• Aug 9th 2012, 11:23 AM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
Yes, for the perpendicular case. Were you able to find the metric from those values?
• Aug 9th 2012, 11:57 AM
AA23
Re: Timelike, Spacelike and Null Seperation
Brilliant (Clapping)
I've done most of the calculation but as you can see from the attachment, I am unsure on one bit
• Aug 9th 2012, 01:03 PM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
That T superscript means transpose. The transpose of a matrix (doesn't have to be square) is where you take the rows and write them as columns.
$A = \left( \begin{array}{ccc} a&b&c \\ d&e&f \\ g&h&i \end{array} \right)$
$A^{T} = \left( \begin{array}{ccc} a&d&g \\ b&e&h \\ c&f&i \end{array} \right)$
• Aug 9th 2012, 01:36 PM
AA23
Re: Timelike, Spacelike and Null Seperation
OK I understand so my final answer would be (see attachment)
• Aug 9th 2012, 02:26 PM
JohnDMalcolm
Re: Timelike, Spacelike and Null Seperation
Remember that the commutative property is not true for multiplication of matrices in general. You have
$g' = \Lambda \Lambda^{T} \neq \Lambda^{T} \Lambda$
Show 40 post(s) from this thread on one page
Page 2 of 3 First 123 Last
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2018-02-18 09:25:15
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https://nrich.maths.org/public/topic.php?code=-961&cl=3&cldcmpid=677
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# Resources tagged with: Area - triangles, quadrilaterals, compound shapes
Filter by: Content type:
Age range:
Challenge level:
### There are 41 results
Broad Topics > Measuring and calculating with units > Area - triangles, quadrilaterals, compound shapes
### From All Corners
##### Age 14 to 16 Challenge Level:
Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
### Gutter
##### Age 14 to 16 Challenge Level:
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
### Squ-areas
##### Age 14 to 16 Challenge Level:
Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .
### Towering Trapeziums
##### Age 14 to 16 Challenge Level:
Can you find the areas of the trapezia in this sequence?
### Six Discs
##### Age 14 to 16 Challenge Level:
Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
### Dividing the Field
##### Age 14 to 16 Challenge Level:
A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .
##### Age 14 to 16 Challenge Level:
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
### Inscribed in a Circle
##### Age 14 to 16 Challenge Level:
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
### At a Glance
##### Age 14 to 16 Challenge Level:
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
### Isosceles
##### Age 11 to 14 Challenge Level:
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
### Equilateral Areas
##### Age 14 to 16 Challenge Level:
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
### More Isometric Areas
##### Age 11 to 14 Challenge Level:
Isometric Areas explored areas of parallelograms in triangular units. Here we explore areas of triangles...
### Rhombus in Rectangle
##### Age 14 to 16 Challenge Level:
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
### Trapezium Four
##### Age 14 to 16 Challenge Level:
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
### Tilted Squares
##### Age 11 to 14 Challenge Level:
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
### Biggest Enclosure
##### Age 14 to 16 Challenge Level:
Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?
### Isometric Areas
##### Age 11 to 14 Challenge Level:
We usually use squares to measure area, but what if we use triangles instead?
### Kissing Triangles
##### Age 11 to 14 Challenge Level:
Determine the total shaded area of the 'kissing triangles'.
### Triangle Transformation
##### Age 7 to 14 Challenge Level:
Start with a triangle. Can you cut it up to make a rectangle?
##### Age 14 to 16 Challenge Level:
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
### Kite in a Square
##### Age 14 to 16 Challenge Level:
Can you make sense of the three methods to work out the area of the kite in the square?
### Triangles in a Square
##### Age 11 to 14 Challenge Level:
What are the possible areas of triangles drawn in a square?
### Same Height
##### Age 14 to 16 Challenge Level:
A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?
### Growing Rectangles
##### Age 11 to 14 Challenge Level:
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
##### Age 11 to 14 Challenge Level:
We started drawing some quadrilaterals - can you complete them?
##### Age 11 to 14 Challenge Level:
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
### Shear Magic
##### Age 11 to 14 Challenge Level:
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
### Isosceles Triangles
##### Age 11 to 14 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
### Areas of Parallelograms
##### Age 14 to 16 Challenge Level:
Can you find the area of a parallelogram defined by two vectors?
### Square Pizza
##### Age 14 to 16 Challenge Level:
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
### Pick's Theorem
##### Age 14 to 16 Challenge Level:
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
### Rati-o
##### Age 11 to 14 Challenge Level:
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
### Dotty Triangles
##### Age 11 to 14 Challenge Level:
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
### Of All the Areas
##### Age 14 to 16 Challenge Level:
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
##### Age 14 to 16 Challenge Level:
The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?
### Disappearing Square
##### Age 11 to 14 Challenge Level:
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
### Overlap
##### Age 11 to 14 Challenge Level:
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
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2020-07-14 23:51:20
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https://math.stackexchange.com/questions/3809971/how-to-associate-a-function-to-an-expression
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# How to associate a function to an expression?
This is related to my previous question on expressions. Consider a countably infinite set of variables, indexed by the positive integers. Suppose also we are considering the set of real numbers as our domain. How would we associate a function to an expression, as in how would we formally define it. For example, should $$x_2$$ be associated to the identity function on $$\mathbb{R}$$, or to the projection function on $$\mathbb{R}^2$$ that returns the second coordinate? Also, should something like $$x_1+x_2-x_3+x_3$$ be defined on $$\mathbb{R}^3$$, or $$\mathbb{R}^2$$? Is there some mathematical text where the mapping from expressions to functions is formally defined?
• Are you asking about common usage across mathematics, or about how things can be done in a textbook on formal logic? – Mark S. Sep 1 '20 at 0:04
• @MarkS. How things are done in a textbook on formal logic. – user107952 Sep 1 '20 at 0:08
• With "expression" do you mean an $\mathscr L$-term for some first-order language $\mathscr L$? – Rick Sep 1 '20 at 12:42
• @Rick Yeah, that is what I mean. I specifically chose the real numbers and addition and subtraction in my examples, but of course this can be generalized to any set S and a collection of n-ary functions on S. – user107952 Sep 1 '20 at 13:34
You're right that there's some flexibility here. The simplest approach is to just clear everything up by adding some "metadata." Say that an annotated term (in a given language, using $$\mathbb{N}$$-indexed variables) is a pair $$(t,n)$$ where $$t$$ is a term in the language and every variable occurring in $$t$$ has index $$\le n$$. Then we can construe each annotated term $$(t,n)$$ as a function from the $$n$$th Cartesian power of our structure to itself.
The default, then, is to use the smallest possible $$n$$. So for example (abusing parentheses for simplicity) "$$x_1+x_2$$" and "$$x_1+x_2+x_3-x_3$$" would refer to functions from $$\mathbb{R}^2$$ and from $$\mathbb{R}^3$$ respectively; their "equivalence" would amount to the fact that the former is the projection of the latter.
• Note that in this approach we do lose some parsimony: e.g. taking the term "$$x_5+x_2$$" we have to have $$n\ge 5$$, and so we can't construe that as a function from $$\mathbb{R}^2$$ even though it only has two variables occurring in it. If you want to get around this you can modify the definition of "annotated term" above to refer to pairs $$(t,X)$$ where $$t$$ is a term and $$X$$ is a finite sequence of variables, such that each variable in $$t$$ occurs in $$X$$. For example, we then would interpret $$(2\cdot x_5+x_2, \langle x_5,x_3,x_2\rangle)$$ as the function $$\mathbb{R}^3\rightarrow\mathbb{R}: (a,b,c)\mapsto 2\cdot a+c$$ (because the sequence part tells us that $$x_5$$ is the first variable in our approach here). This also has the advantage that we can use any set of variables that we want, instead of just $$\mathbb{N}$$-indexed ones.
Unfortunately this is generally swept under the rug, but hopefully the above indicates that it's pretty straightforward to treat if you really want to. It's also worth noting that usually we go the other way: we first declare that we're going to consider functions $$\mathbb{R}^n\rightarrow\mathbb{R}$$, and then interpret terms with variables only from $$\{x_1,...,x_n\}$$ as maps $$\mathbb{R}^n\rightarrow\mathbb{R}$$ in the obvious way. So a lot of the time this doesn't come up at all: while it's true that we can interpret a given term as living on different domains, since we specify what domain we're interested in at the outset we don't care about this ambiguity.
I think that you question is slightly ambiguous as I'm not really sure what do you mean by "associate a function to an expression"; nevertheless let me give a couple of observations on this issue.
First and foremost, when talking about "expressions" one should fix a language over which these expressions are constructed; from your given example about the real numbers, I will take my expressions from now onwards to be formulated in the first-order language of fields $$\mathscr L = \{+, \cdot, -, 0, 1\}$$. As you've clarified in your comment, "expressions" here mean $$\mathscr L$$-terms; in particular, $$\mathscr L$$-terms in the language of fields are finite sequences of symbols from $$\mathscr L$$ of the form $$t_1(x):= x,$$ or $$t_1(x_1, x_2):= -3\cdot x_1^2 +2\cdot x_2,$$ or $$t_3(x_1,x_2,x_3) := x_1\cdot x_2 \cdot x_3,$$ for example. Note that $$\mathscr L$$-terms are nothing more than syntactical constructions which we can interpret (i.e. "give meaning") in an $$\mathscr L$$-structure. In your example, you work with the $$\mathscr L$$-structure $$\mathscr M$$ whose domain is the set of real numbers and where each of the functions $$+, \cdot, -$$ and constants $$0,1$$ are interpreted in the standard way, so that each $$\mathscr L$$-term is interpreted as a polynomial with coefficients in $$\mathbb Z$$, in one (e.g. $$t_1(x)$$) or multiple variables (e.g. $$t_3(x_1, x_2, x_3)$$).
Surely one can associate to each of these polynomials a polynomial map, but in general there is no canonical way of doing this, even after specifying the language and the $$\mathscr L$$-structure in which we interpret our terms. In our example, the term $$t_1(x) := x$$ can be seen as a term in one variable, and in such way we could associate to the interpretation of this term the identity map on $$\mathbb R$$. However, we could also associate to it the projection map onto, say, the third co-ordinate, sice $$t_1$$ is also a term in the free variables $$w,v,x,z$$; remember that the notation $$t_1(x)$$ means that $$t_1$$ has free variables amongst $$\{x\}$$, so in particular a term in the free variable $$x$$ is also a term in the free variables $$x, v, x$$ and $$z$$. By the same reasoning, the interpretation of the term $$t_4: =x_1 + x_2 + x_3 -x_3$$ in $$\mathscr M$$ could be associated to a polynomial map on $$\mathbb R^3$$ or on $$\mathbb R^n$$ for any $$n \in \mathbb N^{\geq 3}$$. We can also associate to it a polynomial map on $$\mathbb R^2$$ since the interpretation of $$t_4$$ and of $$x_1 +x_2$$ in $$\mathscr M$$ coincides; however, if we have another $$\mathscr L$$-structure $$\mathscr M'$$ in which we interpret $$-$$ in the same way we interpret $$+$$ (yes, we can do this!), then the interpretation of $$t_4$$ cannot be associated to a polynomial map on $$\mathbb R^2$$.
• Thank you. This clarified some details. I still have a few questions, though. Just to be clear, I am using the variables x_1, x_2, ..., x_n, etc. not x, y, z, etc. In other words, the variables are indexed by the positive integers. I do not see how x_1 can possibly be interpreted as the projection on to the third coordinate. Surely, it would be the first coordinate. Also, what about something like x_2+x_5? Should it be the addition function on R^2, or the function on R^5 that adds the 2nd and 5th components? Is there a text where the mapping between terms to n-ary functions is defined? – user107952 Sep 1 '20 at 15:12
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2021-06-23 16:57:47
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https://lgatto.github.io/MSnbase/reference/Spectrum-class.html
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Virtual container for spectrum data common to all different types of spectra. A Spectrum object can not be directly instanciated. Use "Spectrum1" and "Spectrum2" instead.
In version 1.19.12, the polarity slot has been added to this class (previously in "Spectrum1").
## Slots
msLevel:
Object of class "integer" indicating the MS level: 1 for MS1 level Spectrum1 objects and 2 for MSMSM Spectrum2 objects. Levels > 2 have not been tested and will be handled as MS2 spectra.
polarity:
Object of class "integer" indicating the polarity if the ion.
peaksCount:
Object of class "integer" indicating the number of MZ peaks.
rt:
Object of class "numeric" indicating the retention time (in seconds) for the current ions.
tic:
Object of class "numeric" indicating the total ion current, as reported in the original raw data file.
acquisitionNum:
Object of class "integer" corresponding to the acquisition number of the current spectrum.
scanIndex:
Object of class "integer" indicating the scan index of the current spectrum.
mz:
Object of class "numeric" of length equal to the peaks count (see peaksCount slot) indicating the MZ values that have been measured for the current ion.
intensity:
Object of class "numeric" of same length as mz indicating the intensity at which each mz datum has been measured.
centroided:
Object of class "logical" indicating if instance is centroided ('TRUE') of uncentroided ('FALSE'). Default is NA.
smoothed:
Object of class "logical" indicating if instance is smoothed ('TRUE') of unsmoothed ('FALSE'). Default is NA.
fromFile:
Object of class "integer" referencing the file the spectrum originates. The file names are stored in the processingData slot of the "MSnExp" or "MSnSet" instance that contains the current "Spectrum" instance.
.__classVersion__:
Object of class "Versions" indicating the version of the Spectrum class. Intended for developer use and debugging.
## Extends
Class "Versioned", directly.
## Methods
acquisitionNum(object)
Returns the acquisition number of the spectrum as an integer.
scanIndex(object)
Returns the scan index of the spectrum as an integer.
centroided(object)
Indicates whether spectrum is centroided (TRUE), in profile mode (FALSE), or unkown (NA).
isCentroided(object, k=0.025, qtl=0.9)
A heuristic assessing if a spectrum is in profile or centroided mode. The function takes the qtlth quantile top peaks, then calculates the difference between adjacent M/Z value and returns TRUE if the first quartile is greater than k. (See MSnbase:::.isCentroided for the code.) The function has been tuned to work for MS1 and MS2 spectra and data centroided using different peak picking algorithms, but false positives can occur. See https://github.com/lgatto/MSnbase/issues/131 for details. It should however be safe to use is at the experiment level, assuming that all MS level have the same mode. See class?MSnExp for an example.
smoothed(object)
Indicates whether spectrum is smoothed (TRUE) or not (FALSE).
centroided(object) <- value
Sets the centroided status of the spectrum object.
smoothed(object) <- value
Sets the smoothed status of the spectrum object.
fromFile(object)
Returns the index of the raw data file from which the current instances originates as an integer.
intensity(object)
Returns an object of class numeric containing the intensities of the spectrum.
msLevel(object)
Returns an MS level of the spectrum as an integer.
mz(object, ...)
Returns an object of class numeric containing the MZ value of the spectrum peaks. Additional arguments are currently ignored.
peaksCount(object)
Returns the number of peaks (possibly of 0 intensity) as an integer.
rtime(object, ...)
Returns the retention time for the spectrum as an integer. Additional arguments are currently ignored.
ionCount(object)
Returns the total ion count for the spectrum as a numeric.
tic(object, ...)
Returns the total ion current for the spectrum as a numeric. Additional arguments are currently ignored. This is the total ion current as originally reported in the raw data file. To get the current total ion count, use ionCount.
%% -------------------------------------------------- %%
bin
signature(object = "Spectrum"): Bins Spectrum. See bin documentation for more details and examples.
clean
signature(object = "Spectrum"): Removes unused 0 intensity data points. See clean documentation for more details and examples.
compareSpectra
signature(x = "Spectrum", y = "Spectrum"): Compares spectra. See compareSpectra documentation for more details and examples.
estimateNoise
signature(object = "Spectrum"): Estimates the noise in a profile spectrum. See estimateNoise documentation for more details and examples.
pickPeaks
signature(object = "Spectrum"): Performs the peak picking to generate a centroided spectrum. See pickPeaks documentation for more details and examples.
plot
signature(x = "Spectrum", y = "missing"): Plots intensity against mz. See plot.Spectrum documentation for more details.
plot
signature(x = "Spectrum", y = "Spectrum"): Plots two spectra above/below each other. See plot.Spectrum.Spectrum documentation for more details.
plot
signature(x = "Spectrum", y = "character"): Plots an MS2 level spectrum and its highlight the fragmention peaks. See plot.Spectrum.character documentation for more details.
quantify
signature(object = "Spectrum"): Quatifies defined peaks in the spectrum. See quantify documentation for more details.
removePeaks
signature(object = "Spectrum"): Remove peaks lower that a threshold t. See removePeaks documentation for more details and examples.
smooth
signature(x = "Spectrum"): Smooths spectrum. See smooth documentation for more details and examples.
show
signature(object = "Spectrum"): Displays object content as text.
trimMz
signature(object = "Spectrum"): Trims the MZ range of all the spectra of the MSnExp instance. See trimMz documentation for more details and examples.
isEmpty
signature(x = "Spectrum"): Checks if the x is an empty Spectrum.
%% -------------------------------------------------- %%
as
signature(object = "Spectrum", "data.frame"): Coerces the Spectrum object to a two-column data.frame containing intensities and MZ values.
## Author
Laurent Gatto <lg390@cam.ac.uk>
## Note
This is a virtual class and can not be instanciated directly.
Instaciable sub-classes "Spectrum1" and "Spectrum2" for MS1 and MS2 spectra.
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2022-01-18 06:53:35
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http://tex.stackexchange.com/questions/95287/how-do-i-hang-another-right-aligned-line-in-a-cases-environment/95289
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# How do I hang another, right-aligned line in a cases environment?
I've seen these questions, but in my case the equation content is too long to add the text description at the end and it overflows into the margin.
$d_i(g_n,\ldots,g_0) = \begin{cases} (d_0g_ng_{n-1},g_{n-2},\ldots,g_0),& \text{if i=0;}\\ (d_ig_n,\ldots,d_1g_{n-i+1},d_0g_{n-i}g_{n-i-1},g_{n-i-2},\ldots,g_0), &\text{if i=1,\ldots,n-1;}\\ (d_ng_n,\ldots,d_1g_1),& \text{if i=0} \end{cases}\\$
What I'd like is something similar to would be produced by replacing the one overly long line by the two lines
(d_ig_n,\ldots,d_1g_{n-i+1},d_0g_{n-i}g_{n-i-1},g_{n-i-2},\ldots,g_0), &\\
& \hspace{-10.5ex}\text{if $i=1,\ldots,n-1$;}\\
...but this is clearly not the way to do it. I'd also prefer not to break the long equation itself onto two lines, only put the text on the next line. Is there some nice way that will end up with the result that the semicolons are aligned automatically without tweaking the horizontal spacing?
-
## 1 Answer
In direct answer to your question (that is, keep the lengthy equation on one line, and the condition on another with aligned semi-colon), using \phantom and a left overlap can help obtain the appropriate alignment:
\documentclass{article}
\usepackage{mathtools}% http://ctan.org/pkg/mathtools
\begin{document}
$d_i(g_n,\ldots,g_0) = \begin{cases} (d_0g_ng_{n-1},g_{n-2},\ldots,g_0),& \text{if i=0;} \\ (d_ig_n,\ldots,d_1g_{n-i+1},d_0g_{n-i}g_{n-i-1},g_{n-i-2},\ldots,g_0), &\text{if i=1,\ldots,n-1;} \\ (d_ng_n,\ldots,d_1g_1),& \text{if i=0} \end{cases}$
$d_i(g_n,\ldots,g_0) = \begin{cases} (d_0g_ng_{n-1},g_{n-2},\ldots,g_0),& \text{if i=0;} \\ (d_ig_n,\ldots,d_1g_{n-i+1},d_0g_{n-i}g_{n-i-1},g_{n-i-2},\ldots,g_0), \\ & \phantom{\text{if i=0;}}\llap{\text{if i=1,\ldots,n-1;}} \\ (d_ng_n,\ldots,d_1g_1),& \text{if i=0} \end{cases}$
\end{document}
\phantom{\text{if $i=0$;}} ensures that your far enough "to the right" in terms of conditional construction, while \llap sets a zero-width box that is right-aligned (resulting in a left overlap) to properly align with the semi-colon of the first case.
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Wow, thanks! That was quick. I knew there should be a sensible way to do it. – David Roberts Jan 25 '13 at 5:13
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2016-06-27 15:08:01
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https://hexo.margatroid.xyz/tags/codeforces/
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A
A题没什么好说的，大力膜你一下就好
Bob的移动策略有三种：
A题
坑点
1. 开unsigned long long
2. 这样写是不符合基本法的，会溢出掉的
题目描述
Vladik often travels by trains. He remembered some of his trips especially well and I would like to tell you about one of these trips:
Vladik is at initial train station, and now n people (including Vladik) want to get on the train. They are already lined up in some order, and for each of them the city code $a_i$ is known (the code of the city in which they are going to).
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2019-12-08 21:07:33
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https://zbmath.org/?q=an%3A0704.93014
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# zbMATH — the first resource for mathematics
A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $$L_{\infty}$$-norm. (English) Zbl 0704.93014
Summary: The i-th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one. We show that near a local maximum, however, the largest singular value has a Lipschitz second derivative, but need not have a third derivative. Using this regularity result, we give a quadratically convergent algorithm for computing the $$L_{\infty}$$-norm of a transfer matrix.
##### MSC:
93B36 $$H^\infty$$-control 93B40 Computational methods in systems theory (MSC2010) 15A18 Eigenvalues, singular values, and eigenvectors 93C35 Multivariable systems, multidimensional control systems 65F35 Numerical computation of matrix norms, conditioning, scaling
Full Text:
##### References:
[1] Chang, B.C.; Li, X.P., Computation of the H^{∞} norm of a transfer function, Internat. J. control, (1989), submitted [2] Boyd, S.; Balakrishnan, V.; Kabamba, P., A bisection method for computing the H_{∞} norm of a transfer matrix and related problems, Math. control signals systems, 2, 207-219, (1989) · Zbl 0674.93020 [3] Boyd, S.; Desoer, C.A., Subharmonic functions and performance bounds on linear time-invariant feedback systems, (), 153-170, Also in [4] Byers, R., Algorithms for Hamiltonian and sympletic eigenproblems, () [5] Clements, D.J.; Teo, K.L., Evaluation of the H_{∞}-norm, preliminary manuscript, (1989) [6] Dienes, P., The Taylor series, (1957), Dover New York · Zbl 0078.05901 [7] Doyle, J.C.; Stein, G., Multivariable feedback design: concepts for a classical/modern synthesis, IEEE trans. automat. control, 28, 4-16, (1981) · Zbl 0462.93027 [8] Francis, B.A., A course in H_{∞} control theory, () [9] Golub, G.; Loan, C.V., Matrix computations, (1989), Johns Hopkins University Press Baltimore, MD [10] Hinrichsen, D.; Pritchard, A.J., Stability radii of linear systems, Systems control lett., 7, 1-10, (1986) · Zbl 0631.93064 [11] Kato, T., A short introduction to perturbation theory for linear operators, (1982), Springer-Verlag Berlin-New York · Zbl 0493.47008 [12] MacFarlane, A.G.J.; Hung, Y.S., Analytic properties of the singular values of a rational matrix, Internat. control, 37, 221-234, (1983) · Zbl 0503.93014 [13] Bruinsma, N.A.; Steinbuch, M., A fast algorithm to compute the H_{∞}-norm of transfer matrix, Systems control lett., 14, 287-293, (1990) · Zbl 0699.93021 [14] Robel, G., On computing the infinity norm, IEEE trans. automat. control, 34, 882-884, (1989) · Zbl 0698.93022 [15] Safonov, M.G.; Doyle, J.C., Minimizing conservativeness of robust singular values, (), 197-207 [16] Van Loan, C.F., A symplectic method for approximating all eigenvalues of a Hamiltonian matrix, Linear algebra appl., 61, 233-251, (1984) · Zbl 0565.65018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-08-03 12:25:04
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http://jaac.ijournal.cn/ch/reader/view_abstract.aspx?file_no=JAAC-2016-0208&flag=1
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### All Issues
Vol.10, 2020
Vol.9, 2019
Vol.8, 2018
Vol.7, 2017
Vol.6, 2016
Vol.5, 2015
Vol.4, 2014
Vol.3, 2013
Vol.2, 2012
Vol.1, 2011
Volume 7, Number 3, 2017, Pages 1161-1176 DOI：10.11948/2017072 Chaotic effects on disease spread in simple eco-epidemiological system Junyuan Yang,Maia Marcheva,Zhen Jin Keywords:Eco-epidemiological, Hopf bifurcation, chaos. Abstract: In this paper, an eco-epidemiological model where prey disease is structured as a susceptible-infected model is investigated. Thresholds that control disease spread and population persistence are obtained. Existence, stability and instability of the system are studied. Hopf bifurcation is shown to occur where a periodic solution bifurcates from the coexistence equilibrium. Simulations show that the system exhibits chaotic phenomena when the transmission rate is varied. PDF Download reader
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2020-07-14 14:10:08
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https://triangle.mth.kcl.ac.uk/?search=au:Alejandra%20au:Castro
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Found 3 result(s)
### 15.12.2021 (Wednesday)
Regular Seminar Alejandra Castro (UvA)
at: 13:45 KCLroom Online abstract: In this talk I will describe holographic properties of near-AdS_2 spacetimes that arise within spherically symmetric configurations of N=2 4D supergravity, for both gauged and ungauged theories. These theories pose a rich space of AdS_2xS^2 backgrounds, and their responses in the near-AdS_2 region are not universal. I will show that the spectrum of operators dual to the matter fields, and their cubic interactions, are sensitive to properties of the background and the theory it is embedded in. The properties that have the most striking effect are whether the background is BPS or non-BPS, and if the theory is gauged or ungauged. The resulting differences will have an imprint on the quantum nature of the microstates of near-extremal black holes, reflecting that not all extremal black holes respond equally when kicked away from extremality.
### 24.02.2016 (Wednesday)
#### Higher Spin Black Holes
Regular Seminar Alejandra Castro (Amsterdam U.)
at: 13:15 KCLroom K4.31 abstract: I'll overview recent progress on non-perturbative aspects of higher spin theories in three dimensions with emphasis on black holes. The two main results I will discuss are: 1) novel properties of extremal and BPS solutions, and 2) how to interpret a higher spin bh as a thermo-field state.
### 12.02.2014 (Wednesday)
#### Holographic entanglement entropy in AdS_3/CFT_2
Triangular Seminar Alejandra Castro (Amsterdam)
at: 15:00 QMWroom David Sizer LT, Francis Bancroft abstract:
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2022-08-12 00:11:13
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https://math.stackexchange.com/questions/4458038/a-subgroup-of-full-measure-is-dense-given-a-haar-measure
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# A subgroup of full measure is dense given a haar measure
I want to know why if $$\mu$$ is a haar measure on a compact $$G$$ and $$\mu(A)=\mu(G)$$ then $$A$$ is dense in $$G$$. This fact is mentioned in the wikipedia page, but I couldn't find a proof for it.
In fact, every subset of full Haar measure must be dense. This follows from the following statement, that you can also find on the wikipedia page:
Claim If $$U$$ is a nonempty open set in the locally compact group $$G$$, then the left Haar measure satisfies $$\mu(U)>0$$.
Proof: We will use the fact that Haar measure is inner regular, so there must be some compact set $$K \subset G$$ with $$\mu(K)>0$$. Given a nonempty open set $$U \subset G$$, fix some $$u\in U$$, and note that the sets $$\{gu^{-1}U: g \in K\}$$ form an open cover of $$K$$. There is a finite subcover $$\{g_ju^{-1}U: 1 \le j \le m\}$$. If $$\mu(U)=0$$ then this finite subcover, and the left-invariance of Haar measure, would yield $$\mu(K)=0$$, a contradiction.
I will assume that $$\mu (G \setminus A)=0$$. Then $$\mu (G\setminus \overline A)=0$$ and $$G\setminus \overline A$$ is open. Since Haar measure has full support this implies that $$G\setminus \overline A=\emptyset$$. Hence, $$\overline A =G$$.
[If $$K$$ is compact and $$U$$ is a non-empty open set then $$K \subset \bigcup_x (x+U)$$ and there is a finite subcover. If $$\mu (U)=0$$ then translation invariance gives $$\mu (K)=0$$. This implies that $$\mu$$ is the $$0$$ measure. Hence, $$\mu (U)>0$$ for any nonempty open set $$U$$].
• Could you explain why does a haar measure has full support? May 24 at 23:49
• If $K$ is compact and $U$ is open then $K \subset \bigcup_x (x+U)$ and there is a finite subcover,. If $\mu (U)=0$ then translation invariance gives $\mu (K)=0$. This implies that $\mu$ is the $0$ measure. May 25 at 0:00
False as stated. Let $$G$$ be $$\mathbb Z^2$$ with the discrete topology. Haar measure is counting measure. Let $$A = \left\{(x,0) \mid x \in \mathbb Z\right\}$$. Then $$A$$ is closed, $$\mu(A) = +\infty = \mu(G)$$. But $$A$$ is not dense in $$G$$.
To get the correct statement: replace $$\mu(A) = \mu(G)$$ by $$\mu(G \setminus A) = 0$$.
• I corrected the question adding the condition for G to be compact, which should force the measure to be finite. thanks! May 25 at 0:21
• Good observation! I have edited my answer accordingly. May 25 at 6:31
Suppose that $$A$$is not dense in $$G$$. If $$\mu(G) < \infty$$, there is an open subset $$U$$ of $$G$$ with $$\mu(U) = 0$$. A compactness argument can be used to show that $$\mu(G) = 0$$.
I am not sure about the case of $$\mu(G) =\infty$$.
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2022-06-27 15:21:47
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https://math.stackexchange.com/questions/447120/does-the-infinite-product-prod-n-mathop-1-infty-frac2n3n-diver
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# Does the infinite product $\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$ diverge to zero or some other finite value.
Does the infinite product diverge to zero or some other value?
$$\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$$
• It's an infinite product, not a series. – Robert Israel Jul 19 '13 at 4:49
• The correct terminology is : "it diverges to 0", consider taking the logarithm of this product and looking at the sum. – Arjang Jul 19 '13 at 5:26
• Thank you for correcting my amateurish mistakes. – KeithSmith Jul 19 '13 at 12:10
Consider $$P_m=\prod_{n=1}^m\frac{2^n}{3^n}=\left(\frac{2}{3}\right)^{m(m+1)/2}$$
Then, $$P_{\infty}=\prod_{n=1}^{\infty}\frac{2^n}{3^n}=\lim_{m\to\infty}P_m=\lim_{m\to\infty}\left(\frac{2}{3}\right)^{m(m+1)/2}=0$$
• You can scale rendered math for yourself if you find it too small. There's no need to use commands like \large, really. – Antonio Vargas Jul 19 '13 at 7:00
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2020-06-02 21:39:18
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http://2msa.it/txai/hcn-net-ionic-equation.html
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# Hcn Net Ionic Equation
Write balanced chemical, complete ionic, and net ionic equations for the reactions between the following substances, which produce a gas. Write a balanced complete ionic equation AND a balanced net ionic equation for: HCl(aq)+LiOH(aq)→H2O(l)+LiCl(aq) 2. Thus, your net ionic equation will be the same as your full formula equation. If a reaction occurs, write the net ionic equation; then write the complete ionic equation for the reaction. Also the sum of the charges on one side of the equation must be equal to the sum of the charges on the other side. H2CO 3 carbonic acid (soda water) 6. The difference between just opening a bottle labeled "HCN" and one labeled "NaCN" could be your life, as HCN, or hydrogen cyanide, is a toxic gas, while CN –, or cyanide ion, being an ion, isn't a gas and is only transfered in solid or solution form. Precipitation Reactions Worksheet Key For each of the following reactants, predict whether a precipitation reaction will take place between them. • Net ionic equation - no spectator ions Ag + (aq) + Cl-(aq) → AgCl (s) • For simplicity, we can omit (aq) after the symbols of all ions in aqueous solutions (assume all ions in solution as aqueous) Ag + + Cl-→ AgCl (s) Example: • Write the net ionic equation for the precipitation of mercury(I) phosphate from. Next, calculate the charge of each dissociated ion and rewrite the equation with the soluble ionic compounds broken down into their individual ions. equations – blank spaces are interpreted as containing the number “1”. REACTIONS IN AQuEous SOLUTIONS: METATHESIS REACTIONS AND NET IONIC EQuATIoNs REPORT SHEET A. 5 - 6 5a, 5b, 5c 34, 35, 40, 42 Write balanced molecular, total ionic, and net ionic equations for precipitation reactions. The aqueous sodium chloride that is produced in the reaction is called a salt. Write the balanced formula and net ionic equation for the reaction that occurs when the contents of the two beakers are added together. HCN(aq) + KOH(aq) H 2 O(l) + KCN(aq) B. M (molarity) = mole/L Assign number of grams NaOH needed = x. HCN + H2O <==> H3O+ + CN-The dissolving of HCN in water is a Bronsted-Lowry acid/base reaction in its own right. I have worked the balanced equation as: NaCN+H20=NaOH+HCN. Second, we write. H2S03 (aq) + = HCO3 Brønsted-Lowry HSO3 Brønsted-Lowry + H2CO3 (aq) Brønsted-Lowry Brønsted-Lowry In this reaction: The formula for the conjugate 2__ of H,Soz is The formula for the conjugate of HCO3 is Compare the. Balancing Equations And Giving The Ionic And Net Ionic Equations For Them As Well -> HCN (g) + KNO3 (aq) How do you write molecular, ionic, and net ionic equations? we usually write the molecular equations for reactions(the way you have started them. One of the most useful applications of the concept of principal species is in writing net ionic equations. HCN(aq) + OH-(aq) --> H2O(l) + CN-(aq) HCN must be shown as the molecule because it is a weak acid, and so in solution most of it is in the form of HCN. First, we balance the molecular equation. To write a net ionic equation, first balance your starting equation. How would one write balanced chemical, complete ionic, and net ionic equations for the reactions between the following substances, which produce a gas? a. Write the state (s, l, g, aq) for each substance. HX + -OH(aq) ---( H2O(l) + X-Weak base + weak acid. Another thing that came. There are three main steps for writing the net ionic equation for HCN + NaOH = NaCN + H2O (Hydrogen cyanide + Sodium hydroxide). H2CN+(aq) + NH2-(aq) 3. First, write the equation for the dissolving process, and examine each ion formed to determine whether the salt is an acidic, basic, or neutral salt. Ba(s) + ( 3Mg(s) 4 -Y ( Aqueous solutions of potassium iodide and silver nitrate are mixed. HNO 2 nitrous acid 5. What is the coefficient for water when the following equation is balanced? As The net ionic equation for the reaction of aluminum sulfate and sodium hydroxide contains HBr [C] HF [D] HCN [E] HCNO. Reactants Net Ionic Equation HCl + NaOH H+ + OH-→ H 2O. Chemical reaction. To do this, you will add increasing. Net ionic equations are an important aspect of chemistry as they represent only the entities that change in a chemical reaction. From Wikipedia: An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. Precipitation Reactions. The amount of charge that flows down this gradient can be approximated by Ohm’s law, where resistance equals one over the membrane conductivity, σ, for the cell and the voltage is the difference between the membrane potential and the reversal potential. Correct Answer: B. There are three main steps for writing the net ionic equation for NaOH + FeCl3 = NaCl + Fe(OH)3 (Sodium hydroxide + Iron (III) chloride). Otherwise, everything cancels out when it's not supposed to and you'll have no net equation. HCN (hydrogen cynide) H−C≡N. If a reaction occurs, write the net ionic equation; then write the complete ionic equation for the reaction. If you plan to experiment with these reactions, it is important to. Write the balanced net ionic equation for the acid-base reaction. Write the balanced molecular equation. remove the common ion, the resulting equation is the net ionic eqaution for the reaction. The net ionic equation is commonly used in acid-base neutralization reactions, double displacement reactions, and redox reactions. a) the net ionic equation of an ionic compound. Net-Ionic Equation. First, we balance the molecular equation. Complete the following equation: CaCO3(s) +2 HNO3(aq)→ Ca(NO3)2(aq)+ H2O(l) +_____ I answered it like that: CaCO3(s) + 2HNO3(aq) → Ca(NO3)2(aq) + H2O(l) + CO2(g) but the online assignment is marking it wrong. H3O+(aq) + OH-(aq) --> 2H2O(l). Write a balanced complete ionic. 5 - 5 4a, 4b, 4c, 4d 28, 29, 30, 33 5 - 5 Given a table of solubility rules, determine whether a specific ionic compound is soluble or insoluble in water. Net Ionic Equations Balance each reaction then write the total and net ionic equations. From Wikipedia: An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. Use uppercase for the first character in the element and lowercase for the second character. In other words, the net ionic equation applies to reactions that are strong electrolytes in. Look up the Ka values in your textbook for all of the weak acids. • Complete and net ionic equations. There are three main steps for writing the net ionic equation for HCN + NaOH = NaCN + H2O (Hydrogen cyanide + Sodium hydroxide). What is the value of Ka? Question 5 9 Points The hydroxide concentration in an aqueous solution is 3. If the concentration of KCN at the equivalence point is 0. First of all, we MUST start with an equation that includes the physical state: (s) for solid, (l) for liquid, (g) for gas, and (aq) for aqueous solution. Enter an equation of a chemical reaction and click 'Balance'. Molecular equation: FeSO4 + 2 NaCl --> FeCl2 + Na2SO4 Since all 4 compounds are soluble in water and dissociate into their ions their really isn't a net ionic equation in this case. 14 Writing Net Ionic Equations 1. 14 Writing Net Ionic Equations 1. Cross out the present spectator ions. In solution chemistry, one part of a chemical reacts with a part of another chemical. H3O+(aq) + OH-(aq) --> 2H2O(l). a) Write a net ionic equation to show that hydrocyanic acid, behaves as an acid in water. 45 M HC 9 H 7 O 4 and 0. Because they do not really take part in the reaction. 5 Write the molecular equation and the net ionic equation for the neutralization of hydrocyanic acid, HCN, by lithium hydroxide, LiOH, both in aqueous solution Exercise 4. Realize that when reactions form precipitates, a net ionic equation can be written. The first equation we'll convert is; Na 2 SO 4(aq) + CaCl 2(aq)---> CaSO 4(s) + 2NaCl (aq) To change the above equation to an ionic equation the aqueous ionic substances must be written as ions and any solid, liquid or gas remains in its molecular form. Interpretation: To write the balanced, ionic and net ionic equations for the given acid-base reactions. Now examine the equation for the heat lost by the metal. 14 Write a net ionic equation to show that nitric acid, HNO3, behaves as an acid in water. HCN + NaOH 시안화수소와 수산화나트륨의 중화반응 Write a net ionic equation for the neutralization reaction of HCN(aq) with NaOH(aq). Write the balanced formula and net ionic equation for the reaction that occurs when the contents of the two beakers are added together. From Wikipedia: An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. Net Ionic Equations An ionic equation from which spectator ions have been removed. Check all that apply. II and III c. Acid-Base reactions HCN(aq) + NH3(aq) 1. Write the balanced net ionic equation for the acid-base reaction. The above equation is one of the most widely used equation in thermodynamics. In the reaction NH3 (aq) + H2O (l) --> NH4+ (aq) + OH- (aq), NH3 is the weak base, but does that mean that H2O is considered an acid because it donates its proton to form OH-? Remember water can act as an acid or a base. Aqueous Ionic Equilibria -- Chapter 17 1. Assume all reactions occur in aqueous solution. Use uppercase for the first character in the element and lowercase for the second character. Types of Reactions in Aqueous Solutions (cont. Examples: Fe, Au, Co, Br, C, O, N, F. 1 M , but the hydrogen ion concentration is 10 −13. Al + Fe(NO 3) 2! Al(NO 3) 3 + Fe 4. Identify spectator ions. To Determine Solubility: 1. Collecting terms gives the following equation. Consider only its first ionization. 7 x 10-11) Setup: 4. 5 Precipitation Reactions Slide 7 Examples More Examples PRS Example 1 PRS Example 2 Precipitation Reactions Net Ionic Equations Slide 13 Gas Forming Reactions. Distinguish between a complete ionic equation and a net ionic equation. In this case, water gives off proton, water is an acid. ); The Gold Parsing System (Hats off! What a great software product!) The Calitha - GOLD engine (c#) (Made it possible for me to do this program in C#). In this case, the K+ ion) from the overall ionic equation. metallic oxides, hydroxides, halogenides, sulfides) • Name of metal + stem of nonmetal + -ide • Examples: - Al 2O3, aluminum oxide - Ba(OH) 2, barium hydroxide - KCl, potassium chloride - ZnS, zinc sulfide Naming Binary Ionic Compounds. Check the charges are balanced. There are mainly three types of chemical equations, molecular equations, complete ionic equation and net ionic equation. Note that any anion of a weak acid is a Bronsted base, and as such will react with the protons from a strong acid to form the undissociated weak acid. Na2CO3(aq) + Ca(NO3)2(aq) ( 2NaNO3(aq) + CaCO3(s) Total Ionic Equation (break into ions). Net Ionic Equation Calculator To write a net ionic equation you have to write the balanced molecular equation. Write a balanced equation. There are three main steps for writing the net ionic equation for HCN + NaOH = NaCN + H2O (Hydrogen cyanide + Sodium hydroxide). Consider the following salts:. HCN(aq) (NH 43 OH written as NH (aq) + H Net ionic equation: SO 32-2(aq) + 2 H+(aq) ----> H O(l) + SO WRITING TOTAL AND NET IONIC EQUATIONS EXAMPLES Reaction of hydrobromic acid and ammonium carbonate in aqueous solution Reaction of sodium sulfite with hydrochloric acid in aqueous solution. 1) 1 Na 3 PO 4 + 3 KOH 3 NaOH + 1 K 3 PO 4 2) 1 MgF 2 + 1 Li 2 CO 3 1 MgCO 3 + 2 LiF 3) 1 P 4 + 3 O 2 2 P 2 O 3 4) 2 RbNO 3 + 1 BeF 2 1 Be(NO 3) 2 + 2 RbF 5) 2 AgNO 3 + 1 Cu 1 Cu(NO 3) 2 + 2 Ag 6) 1 CF 4 + 2 Br 2 1 CBr 4 + 2 F 2 7) 2 HCN + 1 CuSO 4 1 H 2 SO 4 + 1 Cu(CN. A strip of zinc is placed into a solution of HCl. Substituting this approximation into the equation derived in this section gives an equation that can be used to calculate the pH of a solution of a very weak acid. hydrochloric acid and aqueous sodium cyanide, with production of hydrogen cyanide gas (HCN). Cross out the present spectator ions. 7: Writing Chemical Equations for Reactions in Solution: Molecular, Complete Ionic, and Net Ionic Equations. A few drops of NiBr 2 are dropped onto a piece of iron. A chemical equation is the figurative representation of chemical reaction. HCN aq + NaOH aq ----> H2O l + NaCN aq? My answer book tells me that in the net ionic equation excludes Na from the equation. Equation 3: $\ce{CN- + H2O <-> HCN + OH-}$ I am confused as to: 1) the logic behind why we need these three equations (up to this point in the course, all the similar questions have been of the form e. Ionic vs molecular equation form. If given the name, is the word acid in the name? •Acid Naming Rules •If the anion name ends in -IDE The acid name will be hydro-----ic. 5: Aqueous Solutions and Solubility: Compounds Dissolved in Water; 7. yet NaNO3 is dissolved interior the electrolyte so its written in form of its ions. The pOH is: Question 6 6 Points 1. H3PO 4 phosphoric acid. How to use the molecular equation to find the complete ionic and net ionic equation If you're seeing this message, it means we're having trouble loading external resources on our website. You've been asked for ionic equations so I've left out Na+ below which is a spectator ion. Use uppercase for the first character in the element and lowercase for the second character. write the net ionic equation (including phases) that corresponds to Zn(NO3)2(aq)+K2CO3(aq)-->CnCO3(s)+2KNO3(aq). To balance a chemical equation, enter an equation of a chemical reaction and press the Balance button. So I got H + + OH--> H 2 O. If a reaction occurs, write the net ionic equation; then write the complete ionic equation for the reaction. Write the balanced complete ionic equation for the titration of aq. The functional micro-organization of grid cells revealed by cellular-resolution imaging. Compare: Co - cobalt and CO - carbon monoxide; To enter an electron into a chemical equation use {-} or e. There are three main steps for writing the net ionic equation for HCN + NaOH = NaCN + H2O (Hydrogen cyanide + Sodium hydroxide). They are most commonly used in redox reactions, double replacement reactions, and acid-base neutralisations. Consider the equation for photosynthesis, the natural process by which green plants form glucose, C 6 H 12 O 6 , and oxygen from the reaction of carbon dioxide with water. HClO4(aq) + H2O(l) + ( = or ) b. Stoichiometric amounts are tricky. In other words, complete molecular equation is a balanced chemical equation in which the ionic compounds are represented as molecules instead of component ions. d) Write the net-ionic equation. Write the ionic equation showing the strong electrolytes completely dissociated into cations and anions. Second, we write. If you plan to experiment with these reactions, it is important to. Balance Chemical Equations with this Calculator and view a list of previously balanced equations beginning with N. 73 g of AgCl. Ionic eq: H+ + Cl-+ Na+ + OH---> H 2O + Na + + Cl-Spectator ions (Na+ and Cl-) do not participate in the reaction. Examples: Fe, Au, Co, Br, C, O, N, F. sodium carbonate. (b) Complete lonlC Equation: (c) Net Ionic Equation. --so lets try the first one---I'll complete the reaction the net ionic equation has you. C) precipitation reaction. Net Ionic Equations ¾For reactions taking place in water many substances dissociate (break apart) into ions ¾For these reactions there are often ions that actually don't participate (they appear on both sides of the equation). Net Ionic Equation Calculator To write a net ionic equation you have to write the balanced molecular equation. There are three main steps for writing the net ionic equation for NaOH + FeCl3 = NaCl + Fe(OH)3 (Sodium hydroxide + Iron (III) chloride). ΔG (Change in Gibb's Energy) of a reaction or a process indicates whether or not that the reaction occurs spontaniously. + ( = or ) Question 2 8 Points Assign each species on the left to a category on the right. Write the balanced molecular, complete ionic, and net ionic equations for each of the following acid-base reactions. The following are some "thumb rules" for deciding whether to use ions or molecules in writing net ionic equations. Write the balanced formula and net ionic equation for the reaction that occurs when the contents of the two beakers are added together. doc Author: Arlyn DeBruyckere. Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) Question: Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) This problem has been solved!. The spectators can be cancelled off from both sides of the equation and the equation is reduced to the NET IONIC EQUATION. The net ionic equation for the reaction of HCN (aq) and KOH (aq) is _____? a) HCN (aq) + KOH (aq) → H 2 O (l) + K + (aq) + CN - (aq). So, in general, I understand the concept of a net ionic equation, as well as, how to do them. Write the state (s, l, g, aq) for each substance. This equilibrium is only established if the calcium carbonate is heated in a closed system, preventing the carbon dioxide from escaping. Examples: Fe, Au, Co, Br, C, O, N, F. First, we balance the molecular equation. The balanced equation will appear above. 2, how do you write the acid/base equilibrium? I haven't done chemistry for a long time so please be as detailed as possible. The resulting equation is: Ca +2 (aq) + CO 3 2-(aq) CaCO 3(s) Examples: Types of Chemical Reactions 1. In which of these solutions will the weak acid, HCN, ionize less than it does in pure water? How many moles of N2O are present in 10. The following are some "thumb rules" for deciding whether to use ions or molecules in writing net ionic equations. org are unblocked. an unshared pair of electrons. ; Kim, Jeong Woo; Park, Chan Hong. What is left is the Net ionic equation. Formulas that include only the particles that participate in reactions are called. A net ionic equation helps chemists represent the steps in a chemical reaction. Net Ionic Equation. H2S + KMnO4 = K2SO4 + MnS + H2O + S. So in a strong acid, and strong base for example, you have HCl plus NaOH. Reactants Net Ionic Equation HCl + NaOH H+ + OH-→ H 2O. We call these ions spectator ions. Formulas that include only the particles that participate in reactions are called. Complete the following equation: CaCO3(s) +2 HNO3(aq)→ Ca(NO3)2(aq)+ H2O(l) +_____ I answered it like that: CaCO3(s) + 2HNO3(aq) → Ca(NO3)2(aq) + H2O(l) + CO2(g) but the online assignment is marking it wrong. If all the reactants and products are Aqueous how am i. My question is why? My book doesn't offer any explanation as to why this is and everything with the exception of water is aqueous. Why isn't the cyanide canceled? 2 comments. The net ionic equation is Ni+2H+=Ni2++H2. The balanced equation for the reaction between HNO3 and KOH is written as HNO3 + KOH = H2O + KNO3. A net ionic equation helps chemists represent the steps in a chemical reaction. Ammonia with any. hydrochloric acid and aqueous sodium cyanide, with production of hydrogen cyanide gas (HCN). Which of the following is the correct Lewis structure for ammonia, NH 3? Which of the following is NOT a valid Lewis structure?. Balanced chemical equations provide a significant amount of information. Cross out the present spectator ions. The performance of the industrial HCN synthesis Andrussow reactor on a Pt gauze catalyst is simulated using rate equations for 13 simultaneous unimolecular and bimolecular surface reactions. Net Ionic Equation Calculator To write a net ionic equation you have to write the balanced molecular equation. Write a balanced complete ionic equation AND a balanced net ionic equation for: NaOH(aq)+HNO3(aq) → H2O(l)+NaNO3 (aq) 4. Cancel the spectator ions on both sides of the ionic equation. KCN is a basic salt. Cancel the spectator ions on both sides of the ionic equation Check that charges and number of atoms are balanced in the net ionic equation AgNO3 (aq) + NaCl (aq) AgCl (s) + NaNO3 (aq) Ag+ + NO3- + Na+ + Cl- AgCl (s) + Na+ + NO3- Ag+ + Cl- AgCl (s) Write the net ionic equation for the reaction of silver nitrate with sodium chloride. Check that charges and number of atoms are balanced in the net ionic equation. Use uppercase for the first character in the element and lowercase for the second character. Another thing that came. In this alternate system, called the Brønsted-Lowry system, an acid is a proton (H+) donor, a base is a proton acceptor, and an acid-base reaction is a proton. When one realizes that Fe(OH) 3 (s) is a component of rust, this. The calculator is found on the right hand panel of the main page. 5 SOLUTION: Acid HBr H20 HCN Conjugate Base OH CN Base H20 H20 H20 Conjugate Acid H30+ H 30 + HS03 H30+ Write a formula for the conjugate base formed when each of the following behaves as a Brønsted acid: a) HS03. Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) Question: Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) This problem has been solved!. This problem has been solved! See the answer. • Complete and net ionic equations. The net ionic equation is commonly used in acid-base neutralization reactions, double displacement reactions, and redox reactions. Write a balanced complete ionic equation AND a balanced net ionic equation for: MgS(aq)+CuCl2(aq)→CuS(s)+MgCl2(s) 3. 2014-12-03. In a solution, ions, or charged particles, can disassociate from one another. Identify all major species in solution. So I got H + + OH--> H 2 O. HCN is a weak acid that will react completely with NaOH, a strong base: HCN + NaOH → Na+ (aq) + CN-(aq) + H 2O spectator ions (2Cl-) from both sides of the equation, yields the net ionic equation. For the following chemical reaction HCN(aq)+KOH(aq)--->H2O(l)+KCN(aq) Write the net ionic equation, including the phases. copper(ll) sulfate + sodium carbonate - 2. Which one of the following could NOT be a Bronsted-Lowry acid? a) H 2 O b) HN 3 c) H 3 O + d) NH 4 + e) BF 3. Balance Chemical Equations with this Calculator and view a list of previously balanced equations beginning with N. The balanced equation will appear above. 1 K) from 287–311 K. Second, we write the. Note that any anion of a weak acid is a Bronsted base, and as such will react with the protons from a strong acid to form the undissociated weak acid. Step 2 of 3. Chemical Equation Calculator is a free online tool that displays the structure, balanced equation, equilibrium constant, substance properties with chemical names and formulas. I and IV b. Second, we write. CN-(aq) + K+(aq) KCN(aq) D. Consider the following salts:. Write a net ionic equation to show that hydrocyanic acid, HCN, behaves as an acid in water. Include states of matter (s, l, aq, g) This reaction is classified as. q = c m x m m x ∆T m This equation can be rearranged to find the specific heat of the metal. These molecular and complete ionic equations provide additional information, namely, the ionic compounds used as sources of Cl − and Ag +. The calculator is found on the right hand panel of the main page. The hydronium ion concentration is: M b. 5 Write the molecular equation and the net ionic equation for the neutralization of hydrocyanic acid, HCN, by lithium hydroxide, LiOH, both in aqueous solution Exercise 4. example: Determine the pH of 0. With that said, I am at a great d. This precipitation reaction is described by the following equation: 2 Na 3 PO 4 (aq) + 3 CaCl 2 (aq) --> 6 NaCl (aq) + Ca 3 (PO 4) 2 (s). Answer: complete molecular: NaCl(aq) + AgNO 3 (aq) ---> AgCl(s) + NaNO 3 (aq) You know AgCl is insoluble from using a solubility chart. HCN(aq) + OH-(aq) --> H2O(l) + CN-(aq) HCN must be shown as the molecule because it is a weak acid, and so in solution most of it is in the form of HCN. what is the. Assume all reactions occur in aqueous solution. Spectator ion is an ion in an ionic equation that does not take part in the reaction. The performance of the industrial HCN synthesis Andrussow reactor on a Pt gauze catalyst is simulated using rate equations for 13 simultaneous unimolecular and bimolecular surface reactions. So the equation is 2HCN + Ca(OH) 2-> Ca(CN)2 + 2H 2 O. The concept can be further clarified with the help of following examples: Oxidation number of S in S8. Instructions on balancing chemical equations: Enter an equation of a chemical reaction and click 'Balance'. H+ + OH- -> H2O. Another thing that came. A net ionic equation, on the other hand, shows only the ions that involve with the reaction. molecular equation: HCN(aq) + LiOH(aq) → LiCN(aq) + H 2O(l) Note that LiOH (a strong base) and LiCN (a soluble ionic substance) are strong electrolytes; HCN is a weak electrolyte (it is not one of the strong acids in Table 4. HCN (aq) Chloric acid HClO 3: Acetic acid CH 3 COOH: Hydrobromic acid HBr (aq) Sulfurous acid H 2 SO 3: Chlorous acid HClO 2: Boric acid H 3 BO 3: Hydrochloric acid HCl (aq) Phosphoric acid H 3 PO 4: Nitrous acid HNO 2: Hydrofluoric acid HF (aq) Perchloric acid HClO 4: Hydroiodic acid HI (aq) Phosphorous acid H 3 PO 3: Carbonic acid H 2 CO 3. These are equations that focus on the principal substances and ions involved in a reaction--the principal species--ignoring those spectator ions that really don't get involved. Identify the spectator ions in this reaction. write the net ionic equation (including phases) that corresponds to Zn(NO3)2(aq)+K2CO3(aq)-->CnCO3(s)+2KNO3(aq). Relevancia. Write a balanced complete ionic. First, we balance the molecular equation. well if all are aqeous then the net ionic answer is - All is soluble (no reaction) takes place. a) the net ionic equation of an ionic compound. The follow sa ts re stron¥electrolytes. 67) + 1(-482. 73) + 1(-285. When OH- is added: OH- + H+ --> H2O and CH3COOH + OH- --> CH3COO- + H2O The OH- forms water with dissociated H+ ions from the ethanoic acid and then more ethanoic acid dissociates to replace the H+ ions lost and maintain the above equilibrium. asked by Ann on April 3, 2016; Chemistry. Net Ionic Equation Practice For each of the following replacement reactions, give a net ionic equation. What is the net ionic equation for the neutralization reaction of HCN aq with from CHEM 115 at West Virginia University. The proton transfer converts the reacting acid (HCN) into its conjugate base (CN 1-) and the reacting base (NH 3) into its conjugate acid (NH 4 1+). Net tontc Equation. Rule #1 - Binary acids: HCl, HBr, and HI are strong; all other binary acids (and HCN) are weak. Ionic charges are not yet supported and will be ignored. Calculate the value of Kb for this salt from tabulated values of equilibrium constants. Net-Ionic Equations. ¾To write a net ionic equation we write. The further the actual membrane potential is from this Nernst potential, the greater the electrochemical gradient for that species of ion. (a) Molecular Equation. potassium chloride. 3: The Chemical Equation; 7. Molecular equation is HCN + KOH ===> H2O + KCNNet Ionic equation is H^+(aq) + OH^-(aq) ==> H2O(l). From Wikipedia: An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. First, we balance the molecular equation. ) If one of the equations produces something that is insoluble, then, the net ionic equation only consists of the specific ions that produce the solid precipitate. Consider the following salts:. Now we all know that if you add an acid and a base, you always get 2 products; salt, and water. Reactants Net Ionic Equation HCl + NaOH H+ + OH-→ H 2O. We call these ions spectator ions. So lets write the ionic and net ionic equations for the two equations above. SE_ NaOH SE_ Mg(OH)2 WE_ HCN NE_ H2O2 SE_ LiOH WE_ H2CO3 WE_ H2S SE_ NH4Cl SE_ HCl SE_ AgCl SE_ H2SO4 SE_ Ba(OH)2 SE_ Na2SO4. c) Write a total ionic equation. In this case, the K+ ion) from the overall ionic equation. III and IV ____ 31. The standard pressure value p ⦵ = 10 5 Pa (= 100 kPa = 1 bar) is recommended by IUPAC, although prior to 1982 the value 1. Author: SANDRA JEWETT. 4 CuSCN + 7 KIO 3 + 14 HCl → 4 CuSO 4 + 7 KCl + 4 HCN + 7 ICl + 5 H 2 O. Experiment 16: Enthalpy of Reactions. H 2SO 4 + NaOH ! Na 2SO 4 + H 2O. Net-Ionic Equations. Split strong electrolytes into ions (the complete ionic equation). (Just be sure to include (aq) for each of the reactants & products. See Figures 4. In this alternate system, called the Brønsted-Lowry system, an acid is a proton (H+) donor, a base is a proton acceptor, and an acid-base reaction is a proton. 10 M Na 2CO 3 1. Instructions. Sodium cyanide react with water to produce sodium hydroxide and hydrogen cyanide. Na2S + 2H2O ⇄ 2NaOH + H2S. There are three main steps for writing the net ionic equation for NaOH + FeCl3 = NaCl + Fe(OH)3 (Sodium hydroxide + Iron (III) chloride). Parent Country National • A parent-country national (PCN) is a person working in a country other than his/her country of origin (Home / Native Country). Second, we write. acid is fully ionized. The calculator can be used to calculate the chemical formula of a range of. Net Ionic Equation Calculator To write a net ionic equation you have to write the balanced molecular equation. A few drops of NiBr 2 are dropped onto a piece of iron. First, we balance the molecular equation. Ionic charges are not yet supported and will be ignored. The balanced equation will appear above. If you want the molecular equation this looks good to me. Heys, James G; Rangarajan, Krsna V; Dombeck, Daniel A. Buffers usually consist of a weak acid and its conjugate base, in relatively equal and "large" quantities. asked by Ann on April 3, 2016; Chemistry. The ionic equation for this reaction is H2O(aq) + Cl-(aq) + Na+(aq) + OH-(aq) --> Na+(aq) + Cl-(aq) + 2H2O(l) Removing the spectator ions gives the net ionic equation. The performance of the industrial HCN synthesis Andrussow reactor on a Pt gauze catalyst is simulated using rate equations for 13 simultaneous unimolecular and bimolecular surface reactions. A balanced chemical equation serves as an easy tool for understanding a chemical reaction. The hydronium ion concentration is: M b. Strong Acid + Weak Base D. 1M and the Ka for HCN is 6. then write the balanced complete ionic equation. Pb(NO 3 ) 2 (aq) + Na 2 SO 4 (aq) PbSO 4 (s) + 2 NaNO 3 (aq). HCN(aq) + OH-(aq) --> H2O(l) + CN-(aq) HCN must be shown as the molecule because it is a weak acid, and so in solution most of it is in the form of HCN. Identify all major species in solution. It is a weak acid because only a relatively small fraction of HCN molecules dissociate to form H+ (H3O+) and CN-. HCN(aq) + H2O(l) + ( = or ) b) Write a net ionic equation to show how barium hydroxide behaves as a base in water. However, since acetic acid is a weak acid and dissociates slightly in water and therefore x is very small we can assume that 0,3- x ≈ 0,3. H3O+(aq) + OH-(aq) --> 2H2O(l). Buffer Solutions A Buffer Solution is an acid/base equilibrium system that is capable of maintaining a relatively constant pH even if a significant amount of strong acid or base is added. Convert the following redox reactions to the ionic form. If the concentration of KCN at the equivalence point is 0. Write a net ionic equation to show that oxalic acid, h2c2o4, behaves as an acid in water. 38) What is the dissociation constant of HCN, if the dissociation constant of NH4OH is 1. Net Ionic Equations An ionic equation from which spectator ions have been removed. Neutralization Reactions and Net Ionic Equations for Neutralization Reactions. find pH of NH3 given Kb(NH3) , and so we only used the NH3 acid base reaction equation. Check that charges and number of atoms are balanced in the net ionic equation. Instructions. yet NaNO3 is dissolved interior the electrolyte so its written in form of its ions. () () (s) 2-4 aq 2 aq 2-4 aq 2 Zn(s) +Cuaq + SO Zn + SO. + ( = or ) Question 2 8 Points Assign each species on the left to a category on the right. How do you write a net ionic equation for NaCN? If the pH is 11. or the net ionic equation: OH- + H+ → H2O During the course of the titration OH-is consumed, Figure 2. Ammonia with any. The only thing in this equilibrium which isn't a solid is the carbon dioxide. KCN is a basic salt. So in a strong acid, and strong base for example, you have HCl plus NaOH. What is the coefficient for water when the following equation is balanced? As The net ionic equation for the reaction of aluminum sulfate and sodium hydroxide contains HBr [C] HF [D] HCN [E] HCNO. Write the ionic equation showing the strong electrolytes completely dissociated into cations and anions. The answer will appear below; Always use the upper case for the first character in the element name and the lower case for the second character. Write the balanced molecular, complete ionic, and net ionic equations for each of the following acid-base reactions. The performance of the industrial HCN synthesis Andrussow reactor on a Pt gauze catalyst is simulated using rate equations for 13 simultaneous unimolecular and bimolecular surface reactions. Complete Molecular Equations: Complete Molecular equations also known as complete formula equations, or simply formula equations, or total formula equations. copper(ll) sulfate ÷ barium chloride - copper(ll) sulfate ÷ barium. hace 9 años. Interpretation: To write the balanced, ionic and net ionic equations for the given acid-base reactions. What is left is the Net ionic equation. Cobalt (II) Hydroxide. There are three main steps for writing the net ionic equation for HCl + Pb(NO3)2 = PbCl2 + HNO3 (Hydrochloric acid + Lead (II) nitrate). The chemical formula calculator shows. This thread is archived HCN is a weak acid and will not. HCN has a structural isomer (tautomer) called hydrogen isocyanide (HNC). But there are no species common to both sides of the equation! net ionic equation: 2H + (aq) + SO 4 2-(aq) + Ba 2+ (aq) + 2OH-(aq)---> BaSO 4(s) + 2H 2 O (l) So the net ionic equation and the ionic equaton are the same. There are three main steps for writing the net ionic equation for HCN + KOH = KCN + H2O (Hydrogen cyanide + Potassium hydroxide). Spectator tons 2) A solution of hydrochloric acid is added to a solution of silver nitrate. III and IV ____ 31. I don't understand why Na is excluded and not CN. Cross out the present spectator ions. Examples: Fe, Au, Co, Br, C, O, N, F. Write balanced chemical, complete ionic, and net ionic equations for the reactions between the following substances, which produce a gas. NaOH with aq. Use uppercase for the first character in the element and lowercase for the second character. Substituting this approximation into the equation derived in this section gives an equation that can be used to calculate the pH of a solution of a very weak acid. HCN(aq) + OH–(aq) H 2 O(l) + CN –(aq) C. The calculator is found on the right hand panel of the main page. Calculate Kb for CN-. Cancel the spectator ions on both sides of the ionic equation Check that charges and number of atoms are balanced in the net ionic equation AgNO3 (aq) + NaCl (aq) AgCl (s) + NaNO3 (aq) Ag+ + NO3- + Na+ + Cl- AgCl (s) + Na+ + NO3- Ag+ + Cl- AgCl (s) Write the net ionic equation for the reaction of silver nitrate with sodium chloride. + HCN(aq) → CN-(aq) + H 2 O(l) Select the net ionic equation for the aqueous reaction of potassium sulfate and. Substitute the known values into the equation for the specific heat of the metal. I have worked the balanced equation as: NaCN+H20=NaOH+HCN. There are three main steps for writing the net ionic equation for HCN + NaOH = NaCN + H2O (Hydrogen cyanide + Sodium hydroxide). acid is fully ionized. --so lets try the first one---I'll complete the reaction the net ionic equation has you. So, here we have a chemical equation, describing a chemical reaction. Second, we write the. The complete ionic equation for the neutralization of hydrochloric acid with sodium hydroxide is written as follows: H + + Cl- + Na + + OH- → Na + + Cl- + H2O Another way to write the net ionic equation, a weak acid must be written as a molecule because it does not ionize to a large extent in water. Na2CO3(aq) + Ca(NO3)2(aq) ( 2NaNO3(aq) + CaCO3(s) Total Ionic Equation (break into ions). 1) 1 Na 3 PO 4 + 3 KOH 3 NaOH + 1 K 3 PO 4 2) 1 MgF 2 + 1 Li 2 CO 3 1 MgCO 3 + 2 LiF 3) 1 P 4 + 3 O 2 2 P 2 O 3 4) 2 RbNO 3 + 1 BeF 2 1 Be(NO 3) 2 + 2 RbF 5) 2 AgNO 3 + 1 Cu 1 Cu(NO 3) 2 + 2 Ag 6) 1 CF 4 + 2 Br 2 1 CBr 4 + 2 F 2 7) 2 HCN + 1 CuSO 4 1 H 2 SO 4 + 1 Cu(CN. What is left is the Net ionic equation. The Ka for HCN is 4. Net Ionic Equation Calculator To write a net ionic equation you have to write the balanced molecular equation. HNO3 is the chemical formula representing nitric acid, KOH is the formula representing potassium hydroxide, H2O is the formula for water and KNO3 is the formula for potassium nitrate. There are three main steps for writing the net ionic equation for HCl + Pb(NO3)2 = PbCl2 + HNO3 (Hydrochloric acid + Lead (II) nitrate). To write a net ionic equation, first balance your starting equation. Chemical reaction. 9x10^-10 = 2. In this case, the K+ ion) from the overall ionic equation. For HCN, K a = 4. Using the solubility rules, predict the products, balance the equation, and write the complete ionic and net ionic equations for each of the following reactions. In this alternate system, called the Brønsted-Lowry system, an acid is a proton (H+) donor, a base is a proton acceptor, and an acid-base reaction is a proton. Substituting this approximation into the equation derived in this section gives an equation that can be used to calculate the pH of a solution of a very weak acid. ; Taylor, Patrick T. Show by suitable net ionic equations that each of the following species can act as a Bronsted-Lowry base: \begin{array}{l}{\text { (a) } \mathrm{HS}^{-}} \\ {\…. Write a balanced complete ionic equation AND a balanced net ionic equation for: HCl(aq)+LiOH(aq)→H2O(l)+LiCl(aq) 2. A balanced chemical equation serves as an easy tool for understanding a chemical reaction. Next, calculate the charge of each dissociated ion and rewrite the equation with the soluble ionic compounds broken down into their individual ions. Above 50% D. When the complete and net ionic equations are written out, HCN is kept as a unit. What am i confused about, is that when trying to figure out if it's S, L, G, or Aq - I have no clue how. Heys, James G; Rangarajan, Krsna V; Dombeck, Daniel A. The calculator is found on the right hand panel of the main page. Kinetic and ionic properties of the human HCN2 pacemaker channel Article (PDF Available) in Pflügers Archiv - European Journal of Physiology 439(5):618-626 · February 2000 with 69 Reads. copper (II) sulfate + ammonium sulfide → 3. HCN hydrocyanic acid 4. Examples: Fe, Au, Co, Br, C, O, N, F. My question is why? My book doesn't offer any explanation as to why this is and everything with the exception of water is aqueous. Why isn't the cyanide canceled? 2 comments. i am in grade 11 if you are in canadian school then msg me for some help. Examples: Fe, Au, Co, Br, C, O, N, F. HCN(aq) + OH-(aq) --> H2O(l) + CN-(aq) HCN must be shown as the molecule because it is a weak acid, and so in solution most of it is in the form of HCN. the internet ionic reaction may be written as follows: Pb2+(aq) + SO4-2(aq) -----> PbSO4(s) here we've written basically that section which became into. 67) + 1(-482. Check to confirm that the lower Ka value indicates the stronger bond in each case. II and III c. To balance a chemical equation, enter an equation of a chemical reaction and press the Balance button. The aqueous sodium chloride that is produced in the reaction is called a salt. Thus, your net ionic equation will be the same as your full formula equation. 5 - 5 4a, 4b, 4c, 4d 28, 29, 30, 33 5 - 5 Given a table of solubility rules, determine whether a specific ionic compound is soluble or insoluble in water. Use the back of this page to show the reactions. They'll give your presentations a professional, memorable appearance - the kind of sophisticated look that today's audiences expect. After the. The performance of the industrial HCN synthesis Andrussow reactor on a Pt gauze catalyst is simulated using rate equations for 13 simultaneous unimolecular and bimolecular surface reactions. 5 Precipitation Reactions Slide 7 Examples More Examples PRS Example 1 PRS Example 2 Precipitation Reactions Net Ionic Equations Slide 13 Gas Forming Reactions. When an equation is written in the molecular form the program will have issues balancing atoms in parcial equations of oxidation and reduction (Step 3. Cancel ions appearing on both sides (spectators). [1ΔH f (KCN (aq)) + 1ΔH f (H2O (ℓ))] - [1ΔH f (HCN (aq)) + 1ΔH f (KOH (aq))] [1(-191. Ok this is the net ionic equation for this reaction: Hydrochloric acid and calcium carbonate: 2H+CaCO3 ---Ca+ CO2+ H20 Don't forget to put the charges to the elements and compounds. HCN(aq) + H2O(l) + ( = or ) b) Write a net ionic equation to show how barium hydroxide behaves as a base in water. See Figures 4. The answer will appear below; Always use the upper case for the first character in the element name and the lower case for the second character. There are mainly three types of chemical equations, molecular equations, complete ionic equation and net ionic equation. Hanson Answers to Practice 1: Identify in each case whether the substance is a strong electrolyte (SE), weak electrolyte (WE), or non-electrolyte (NE). Ba(s) + ( 3Mg(s) 4 -Y ( Aqueous solutions of potassium iodide and silver nitrate are mixed. sodium acetate. INTRODUCTION: In this lab you will explore the nature of aqueous solutions by investigating the relationship between conductivity and strong and weak electrolytes. Note that any anion of a weak acid is a Bronsted base, and as such will react with the protons from a strong acid to form the undissociated weak acid. Net Ionic Equation Practice For each of the following replacement reactions, give a net ionic equation. These molecular and complete ionic equations provide additional information, namely, the ionic compounds used as sources of Cl − and Ag +. Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) Question: Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) This problem has been solved!. The reaction is: HCN + H2O -> CN- + H3O+ To write the net ionic equation, first write the total ionic equation. If you're behind a web filter, please make sure that the domains *. In other words, complete molecular equation is a balanced chemical equation in which the ionic compounds are represented as molecules instead of component ions. The reason to write a chemical equation is to express what we believe is actually happening in a chemical reaction. write the full ionic equation. Write the balanced net ionic equation for the reaction of aqueous sodium carbonate with aqueous calcium nitrate. the internet ionic reaction may be written as follows: Pb2+(aq) + SO4-2(aq) -----> PbSO4(s) here we've written basically that section which became into. ---> reactants products O one coefficient on each KOH+ HBr one coefficient KBr+ H Subscript 2 common names KOH- Potassium hydroxide HBr- Hydrogen Bromide KBr- Potassium Bromide H2O-Water Elements Potassium, Hydrogen, Potassium, present KOH it has 3 atoms 1 oxygen HBr Bromine 2. Aqueous Ionic Equilibria -- Chapter 17 1. 5 - 6 5a, 5b, 5c 34, 35, 40, 42 Write balanced molecular, total ionic, and net ionic equations for precipitation reactions. Parent Country National • A parent-country national (PCN) is a person working in a country other than his/her country of origin (Home / Native Country). Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) Question: Write A Net Ionic Equation For The Neutralization Reaction Of HCN(aq) With NaOH(aq) This problem has been solved!. This is actually a combustion reaction. 1M and the Ka for HCN is 6. C) precipitation reaction. In the net ionic equation, strong ac ids and bases are written in the ionic form, weak acids and bases in the molecular form. If the concentration of KCN at the equivalence point is 0. Pre–Lab Questions:. Precipitation Reactions. 100% Upvoted. 5 SOLUTION: Acid HBr H20 HCN Conjugate Base OH CN Base H20 H20 H20 Conjugate Acid H30+ H 30 + HS03 H30+ Write a formula for the conjugate base formed when each of the following behaves as a Brønsted acid: a) HS03. You have some ethylene right over here, in the presence of oxygen, and you need to get a little bit of energy to get this going, but then you're going to have this reaction that's actually going to release energy as well, but we're not. 93 x 10-1? a) 7. After eliminating the spectator ions (Li+ and CN-), the net ionic equation is HCN(aq) + OH-(aq) → H. Second, we write the. form precipitates. Copper is dipped into a solution of ZnCl 2. Net Ionic Equations An ionic equation from which spectator ions have been removed. Use Equation (5. KCN is a basic salt. I don't understand why Na is excluded and not CN. remove the common ion, the resulting equation is the net ionic eqaution for the reaction. Using the activity series, predict what happens in each situation. Aluminum nitrate + Potassium hydroxide → Aluminum hydroxide + Potassium nitrateThe unbalanced chemical What Is The Net Ionic Equation For HCl With Ni? Chemistry. The equivalence point is reached when the number of moles of H+ added is equal to the number of moles of OH-in the starting solution. The equilibrium produced on heating calcium carbonate. Identify whether each species functions as a Brønsted-Lowry acid or a Brønsted- Lowry base in this net ionic equation. I think we should balance the chemical equation first We need to identify the reactants and products. In this alternate system, called the Brønsted-Lowry system, an acid is a proton (H+) donor, a base is a proton acceptor, and an acid-base reaction is a proton. Utility of Satellite Magnetic Observations for Estimating Near-Surface Magnetic Anomalies. Check the charges are balanced. 5 Precipitation Reactions Slide 7 Examples More Examples PRS Example 1 PRS Example 2 Precipitation Reactions Net Ionic Equations Slide 13 Gas Forming Reactions. Cross out the spectator ions on both sides of complete ionic equation. If the product is added to water is the resulting mixture acidic or basic? 2 Ca + O2 --> 2 CaO. The correct answer is HCN + OH--> H 2 O +CN-. What is left is the Net ionic equation. Ch 4 Chemical Reactions Ionic Theory of Solutions - Ionic substances produce freely moving ions when dissolved in water, and the ions carry electric current. How do you write a net ionic equation for NaCN? If the pH is 11. A few drops of NiBr 2 are dropped onto a piece of iron. Rearranging this equation gives the following result. The following are some "thumb rules" for deciding whether to use ions or molecules in writing net ionic equations. The net ionic equation for the reaction of HCN (aq) and KOH (aq) is _____? a) HCN (aq) + KOH (aq) → H 2 O (l) + K + (aq) + CN - (aq). HCN(aq) + OH-(aq) --> H2O(l) + CN-(aq) HCN must be shown as the molecule because it is a weak acid, and so in solution most of it is in the form of HCN. HClO4(aq) + H2O(l) + ( = or ) b. + H2O(1) Write a net ionic equation to show that hydrocyanic acid , HCN, behaves as an acid in water. Second, we write. There are mainly three types of chemical equations, molecular equations, complete ionic equation and net ionic equation. 67) + 1(-482. First, we balance the molecular equation. Step-by-Step Solution: Step 1 of 3. Show your work. Enter an equation of a chemical reaction and click 'Balance'. 3) However, there is a problem. The net ionic equation for the reaction between KOH(aq) and HCN(aq) is: So I did K + (aq) + OH - (aq) + H + (aq) + CN - (aq) --------> K + (aq) + CN - (aq) + H2O (l). Net Ionic Equations General rules: 1. Second, we write. Cancel the spectator ions on both sides of the ionic equation. After eliminating the spectator ions (Li+ and CN-), the net ionic equation is HCN(aq) + OH-(aq) → H. Complete Molecular Equations: Complete Molecular equations also known as complete formula equations, or simply formula equations, or total formula equations. All of the ions are effectively spectator ions where nothing is happening to them. Na2CO3(aq) + Ca(NO3)2(aq) ( 2NaNO3(aq) + CaCO3(s) Total Ionic Equation (break into ions). So, Im in grade 12 chemistry and I am not sure how to do chemical equations. To balance a chemical equation, enter an equation of a chemical reaction and press the Balance button. Write the molecular equation, total ionic equation and net ionic equation for the reaction of potassium sulfate with barium acetate. " in Chemistry if there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions. CH3COOH(aq) + NH3(aq) → NH4+(aq) + CH3COO-(aq) When largely molecular acetic acid solution reacts with largely molecular ammonia solution, the resulting compound is ammonium acetate, an ionic. Write balanced chemical, complete ionic, and net ionic equations for the reactions between the following substances, which produce a gas. So lets write the ionic and net ionic equations for the two equations above. 83)] - [1(60. Write the balanced net ionic equation for the reaction of aqueous sodium carbonate with aqueous calcium nitrate. The net ionic equation for the reaction of HCN (aq) and KOH (aq) is _____? a) HCN (aq) + KOH (aq) → H 2 O (l) + K + (aq) + CN - (aq). The answer will appear below; Always use the upper case for the first character in the element name and the lower case for the second character. Write a balanced complete ionic equation AND a balanced net ionic equation for: HCl(aq)+LiOH(aq)→H2O(l)+LiCl(aq) 2. or the net ionic equation: OH- + H+ → H2O During the course of the titration OH-is consumed, Figure 2. Write the net ionic equation, including phases for HCN(aq)+KOH(aq)--->H2O(l)+KCN(aq)? Respuesta Guardar. Answer: complete molecular: NaCl(aq) + AgNO 3 (aq) ---> AgCl(s) + NaNO 3 (aq) You know AgCl is insoluble from using a solubility chart. Second, we write. The reason to write a chemical equation is to express what we believe is actually happening in a chemical reaction. Identify the spectator ions in this reaction. Chemical reaction. 1 M acid to −13 in strong 0. Consider the equation for photosynthesis, the natural process by which green plants form glucose, C 6 H 12 O 6 , and oxygen from the reaction of carbon dioxide with water. BYJU'S online chemical equation calculator tool makes the prediction faster and easier, and it displays the answer in a fraction of seconds. Complete Ionic and Net Ionic Equations: Home: Writing Complete Ionic Equations. soluble ions with no "exceptions" never. 7 x 10-11) Setup: 4. Weak Acid + Weak Base The extent of this reaction is: A. My question is why? My book doesn't offer any explanation as to why this is and everything with the exception of water is aqueous. What is the net ionic equation for the reaction of NaOH aq with HCN aq A 4 OH from CHEM 1311 at University of Texas, Dallas. • Net ionic equation - no spectator ions Ag + (aq) + Cl-(aq) → AgCl (s) • For simplicity, we can omit (aq) after the symbols of all ions in aqueous solutions (assume all ions in solution as aqueous) Ag + + Cl-→ AgCl (s) Example: • Write the net ionic equation for the precipitation of mercury(I) phosphate from. To balance a chemical equation, enter an equation of a chemical reaction and press the Balance button. Then, identify the states of matter of each compound and determine what species will dissociate in solution. hydrochloric acid and aqueous sodium cyanide, with production of hydrogen cyanide gas (HCN). Dissociate solublecompounds. The pH of this solution is: c. Principal Species in Solution / Net Ionic Equations Prof. Write the balanced NET IONIC equation for the reaction that occurs when hydrocyanic acid and potassium hydroxide are combined This reaction is classified as A. Balance the following chemical equation and write the balanced complete ionic and net ionic equations. Similarly, we can multiply the top and bottom of the K a2 expression by the OH-ion concentration. There are three main steps for writing the net ionic equation for HCN + NaOH = NaCN + H2O (Hydrogen cyanide + Sodium hydroxide). Use uppercase for the first character in the element and lowercase for the second character. [1ΔH f (KCN (aq)) + 1ΔH f (H2O (ℓ))] - [1ΔH f (HCN (aq)) + 1ΔH f (KOH (aq))] [1(-191. Strong Acid + Weak Base D. Second, we write the. 1M and the Ka for HCN is 6. The amount of charge that flows down this gradient can be approximated by Ohm’s law, where resistance equals one over the membrane conductivity, σ, for the cell and the voltage is the difference between the membrane potential and the reversal potential. REACTIONS IN AQuEous SOLUTIONS: METATHESIS REACTIONS AND NET IONIC EQuATIoNs REPORT SHEET A. The net ionic equation is a chemical equation for a reaction which lists only those species participating in the reaction. Examples of Net ionic equations NaCN(s) + H2O => HCN(aq) + Na+(aq) + OH-(aq) Powered by Create your own unique website with customizable templates. Write balanced net ionic equations for each reaction described below: a) Solid sodium hydroxide pellets are dropped in solution of sulfuric acid. From Wikipedia: An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. (Just be sure to include (aq) for each of the reactants & products. The first equation we'll convert is; Na 2 SO 4(aq) + CaCl 2(aq)---> CaSO 4(s) + 2NaCl (aq) To change the above equation to an ionic equation the aqueous ionic substances must be written as ions and any solid, liquid or gas remains in its molecular form. The full ionic equation is: 2H+(aq) + 2CN-(aq) + Ba+2(aq) + 2OH-(aq) --> Ba(CN)2(s) + H2O(l) The. All reactions occur- no reversible reactions. Write the balanced molecular equation. Which pair of acids would each react with barium hydroxide and have the net ionic equation: H + + OH----> H 2 O. Strong Acid + Strong Base B. This precipitation reaction is described by the following equation: 2 Na 3 PO 4 (aq) + 3 CaCl 2 (aq) --> 6 NaCl (aq) + Ca 3 (PO 4) 2 (s). 73) + 1(-285. (b) Complete lonlC Equation: (c) Net Ionic Equation. There are three main steps for writing the net ionic equation for NaOH + FeCl3 = NaCl + Fe(OH)3 (Sodium hydroxide + Iron (III) chloride). I don't understand why Na is excluded and not CN. Write a balanced complete ionic equation AND a balanced net ionic equation for: NaOH(aq)+HNO3(aq) → H2O(l)+NaNO3 (aq) 4. HCN(aq) (NH 43 OH written as NH (aq) + H Net ionic equation: SO 32-2(aq) + 2 H+(aq) ----> H O(l) + SO 2 (g) charge: -2 +2 = 0 0 0 WRITING TOTAL AND NET IONIC. Write the balanced NET IONIC equation for the reaction that occurs when hydrocyanic acid and potassium hydroxide are combined. In which of these solutions will the weak acid, HCN, ionize less than it does in pure water? How many moles of N2O are present in 10. HCN (aq) Chloric acid HClO 3: Acetic acid CH 3 COOH: Hydrobromic acid HBr (aq) Sulfurous acid H 2 SO 3: Chlorous acid HClO 2: Boric acid H 3 BO 3: Hydrochloric acid HCl (aq) Phosphoric acid H 3 PO 4: Nitrous acid HNO 2: Hydrofluoric acid HF (aq) Perchloric acid HClO 4: Hydroiodic acid HI (aq) Phosphorous acid H 3 PO 3: Carbonic acid H 2 CO 3. Write equations that show NH 3 as both a conjugate acid and a conjugate base. When you write a dissociation reaction in which a compound breaks into its component ions, you place charges above the ion symbols and balance the equation for both mass and charge. Net Ionic Equations: In chemistry, the net ionic. However, since acetic acid is a weak acid and dissociates slightly in water and therefore x is very small we can assume that 0,3- x ≈ 0,3. ΔG > 0 indicates that the reaction (or a process) is non-spontaneous and is endothermic (very high value of ΔG indicates that the. What is the net ionic equation for the neutralization reaction of HCN aq with from CHEM 115 at West Virginia University. 100 M NaOH to 25 mL of 0. molecular, total ionic, and net ionic equations.
lfisolcr05b, 4i6g9ab1aozbo, 6vkbkz4spo5c3gk, zxgsa4txbdfm, 4estmsfhhlye07, 1fqq4jni1h, 7jq0cyy0hqruc3i, kla7bhzmyo858, fnsglawyy7o, bscxpvkdf1q20, 0nd7ese8oshjtm9, 1b4yio74d5, jt66jt4bb4, 9v4hx0354dx, sh39d8dd8h8v04, 7b335w54q3cxq, cmn9rld34p66, fsa5icp8e9ss, fdrq122fz6vm4k1, fnjjjz1tq02, v7gd9s1lh3, u53srv0a0b2m, awkb1d59mioptgu, ihs3bkbjk6ptdj, k5frumwy5z9fp, ztycmm077p, coxoe9w2mvs2o4c
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2020-06-04 06:59:59
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https://www.physicsforums.com/threads/how-do-cas-evaluate-derivatives.578934/
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# How do CAS evaluate derivatives
1. Feb 18, 2012
### matqkks
How do CAS systems and programmable calculators evaluate the derivative of a function?
Do they use matrix representation of linear transformations?
2. Feb 19, 2012
### Stephen Tashi
I don't know the answer, but you should specify whether you are asking how they evaluate derivatives numerically or how they evaluate them symbollically. Is the result of the evaluation a formula? Or a graph? Or a numerical table?
3. Feb 19, 2012
### matqkks
Sorry for being vague but I meant symbolically.
4. Feb 19, 2012
### joeblow
I suspect that they convert whatever expression you want to differentiate into taylor series, differentiate (in the obvious way), then match the result to a taylor series that represents an elementary function and substitute back. Maybe not, but I can't imagine how else it would be done.
5. Feb 19, 2012
### pwsnafu
From what I heard CAS stores the information as a directed graph. In Mathematica you can use the FullForm command to see it directly for example
$\sin(x^2)+3$
would be
Plus[3,Sin[Power[x,2]]]
It then has rules for how to manipulate these objects. So the derivative operator D (I'm assuming wrt x) interacts with Plus via the rule
D[Plus[f,g]] = Plus[D[f],D[g]]
Mathematica knows that 3 is constant and so D[3]=0. It then reduces Plus[0,?] to just ?.
So we now have
D[Sin[Power[x,2]]]
It allies its chain rule and is programmed so that D[Sin] = Cos:
Multiply[Cos[Power[x,2]],D[Power[x,2]]]
And we know that the derivative of Power[x,2] as Multiply[2,x]
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2018-08-14 16:11:07
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http://math.stackexchange.com/questions/22721/is-there-a-formula-to-calculate-the-sum-of-all-proper-divisors-of-a-number/22723
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# Is there a formula to calculate the sum of all proper divisors of a number?
I don't need to list all proper divisors, I just want to get its sum. Because for a small number, checking all proper divisors and adding them up is not a big deal. However, for a large number, this would run extremely slow. Any idea?
Thanks,
Chan Nguyen
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If the prime factorization of $n$ is $$n=\prod_k p_k^{a_k}$$ where the $p_k$ are the distinct prime factors and the $a_k$ are the positive integer exponents, the sum of all the positive integer factors is $$\prod_k\left(\sum_{i=0}^{a_k}p_k^i\right).$$
For example, the sum of all of the factors of $120=2^3\cdot3\cdot5$ is $$(1+2+2^2+2^3)(1+3)(1+5)=15\cdot4\cdot6=360.$$
For proper factors, subtract $n$ from this sum. This may or may not be faster, depending on the number and how you'd get the prime factorization, but this is the typical technique for high school contest problems of this sort.
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@Issac: Thank you! In fact, I thought of prime factorization, but the algorithm for factorization is not fast too. – Chan Feb 18 '11 at 23:10
The sum of divisors can also be written using $\sum_{i=0}^{a_k}p_k^i = (p_k^{a_k + 1})/(p_k - 1)$ for the individual factors, as may be seen from the PlanetMath article: planetmath.org/encyclopedia/FormulaForSumOfDivisors.html – hardmath Feb 18 '11 at 23:18
@hardmath: Absolutely—each sum is the sum of a geometric series (though I think it should probably be $$\prod_k\left(\sum_{i=0}^{a_k}p_k^i\right)=\prod_k\frac{p_k^{a_k + 1}-1}{p_k - 1}$$ (add $-1$ in the numerator). – Isaac Feb 18 '11 at 23:41
Yes, I miss preview mode in comments... – hardmath Feb 19 '11 at 21:19
Just because it is interesting:
There is actually a (less known) recursive formula for calculating $\sigma(n)$, the sum of the divisors of $n$.
$$\sigma(n) = \sigma(n-1) + \sigma(n-2) - \sigma(n-5) - \sigma(n-7) + \sigma(n-12) +\sigma(n-15) + ..$$ Here $1,2,5,7,...$ is the sequence of generalized pentagonal numbers $\frac{3n^2-n}{2}$ for $n = 1,-1,2,-2,...$ and the signs are repetitions of $1,1,-1,-1$. The summation continues until you try to calculate $\sigma$ of something negative. However, if $\sigma(0)$ occurs in the summation (this happens precisely when $n$ is a generalized pentagonal number), it should be replaced by $n$ itself. In other words $$\sigma(n) = \sum_{i\in \mathbb Z_0} (-1)^i\left( \sigma(n - \tfrac{3i^2-i}{2}) + \delta(n,\tfrac{3i^2-i}{2}) \right),$$ where we set $\sigma(i) = 0$ for $i\leq 0$ and $\delta(\cdot,\cdot)$ is the Kronecker delta.
Note that calculating $\sigma(n)$ requires $\sigma(n-1)$ already, therefore its complexity is at least $\mathcal O(n)$, which makes it kind of useless for practical purposes. Note however the lack of reference to divisibility in this formula, which makes it a bit miraculous and therefore worth mentioning.
Here's a reference to the Euler's paper from 1751.
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Many thanks for a great information. Although I don't understand it completely now, I will go back to it when I'm ready. – Chan Feb 19 '11 at 4:58
Is the formula correct? I get a negative sign for i=1 in your sum, and $\sigma(n-\frac{3 1^2 - 1}{2})$ has a positive sign in your first equation. Most likely, I made a mistake... (I tried it by hand using n=6). – Unapiedra Oct 11 '13 at 17:25
"it should be replaced by $n$ itself". So do that: $\delta(...) n$, also I find that it should be $(-1)^{i+1}$. Doing this gives me correct result for all my test cases. – Unapiedra Oct 11 '13 at 23:08
If you want numerical values then the calculator at the site below will list all divisors of a given positive integer, the number of divisors and their sum. It also has links to calculators for other number theory functions such as Euler's totient function.
http://www.javascripter.net/math/calculators/divisorscalculator.htm
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If No = a^p × b^q × c^r × ... then total divisors = (p + 1)(q + 1)(r + 1) ...
sum of divisors = a^(p+1)/(a–1) × b^(q+1)/(b–1) × c^(r+1)/(c–1)
e.g. divisors of 8064 8064 = 2^7 × 3^2 × 7^1
total number of divisors = (7+1)(2+1)(1+1) = 48
sum of divisors = [2^(7+1) –1]/(2–1) × [3^(2+1) –1]/(3–1) × [7^(1+1) –1]/(7–1)
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2015-01-26 19:17:10
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https://mathinfocusanswerkey.com/math-in-focus-grade-2-cumulative-review-for-chapters-10-to-12-answer-key/
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# Math in Focus Grade 2 Cumulative Review for Chapters 10 to 12 Answer Key
Practice the problems of Math in Focus Grade 2 Workbook Answer Key Cumulative Review for Chapters 10 to 12 to Estimate to score better marks in the exam.
## Math in Focus Grade 2 Cumulative Review for Chapters 10 to 12 Answer Key
Concepts and Skills
Question 1.
Connect the cards to show the steps for mental math.
Answer:
Add mentally.
Question 2.
352 + 4 = ____________
Answer: 352 + 4 = 356
Question 3.
817 + 5 = ____________
Answer: 817 + 5 = 822
Question 4.
143 + 30 = ____________
Answer: 143 + 30 = 163
Question 5.
198 + 800 = ____________
Answer: 198 + 800 = 998
Subtract mentally.
Question 6.
916 – 5 = ____________
Answer: 916 – 5 = 911
Question 7.
873 – 8 = ____________
Answer: 873 – 8 = 865
Question 8.
477 – 60 = ____________
Answer: 477 – 60 = 416
Question 9.
858 – 400 = ____________
Answer: 858 – 400 = 458
Mark each number with an X on the number line. Then round each number to the nearest ten.
Question 10.
76
Answer: 76 is nearer to 80
Question 11.
81
Answer: 81 is nearest to 80
Question 12.
123
Answer:
123 is nearest to 120
Question 13.
134
Answer:
Complete.
Question 14.
Write the numbers that give 50 when rounded to the nearest ten.
Answer:
The numbers that give 50 when rounded to the nearest ten are 45,46,47,48,49,51,52,53,54.
Question 15.
What is the least number that rounds to 10?
Answer: The least number that rounds to 10 is 1.
Question 16.
What is the greatest number that rounds to 80?
Answer: The greatest number that rounds to 80 is 84.
Add or subtract. Round each number to the nearest ten. Then estimate the sum or difference to check that your answer is reasonable.
Question 17.
874 + 67 = ___________
874 is about ____________
67 is about ____________
____________ + ____________ = ____________
So, 874 + 67 is about ____________
Is the answer reasonable? Explain.
Answer:
874 + 67 = 941
874 is about nearest to 870
67 is about nearest to 70
870 + 70 = 940
So, 874 + 67 is about nearest to 941
Question 18.
545 – 79 = ____________
545 is about ____________
79 is about ____________
So, 545 – 79 is about ____________
Is the answer reasonable? Explain.
Answer:
545 – 79 = 466
545 is about nearest to 550
79 is about nearest to 80
So, 545 – 79 is about 466
Circle the bills that make the amount shown.
Question 19.
Answer:
Write the amount in numbers.
Question 20.
twenty-five cents $__________ or __________¢ Answer:$0.25
Question 21.
thirty-nine dollars $____________ Answer:$39
Question 22.
twelve dollars and ninety-seven cents $____________ Answer:$12.97
Count the money. Then write the amount each way.
Question 23.
dollars and cents ____________
cents ____________
words ____________
Answer:
dollars and cents $5.90 cents 90 words: Five dollars and 90 cents Circle the amount that is least. Question 24.$10.75
$7.98$8.07
Answer:
Circle the greatest amount.
Question 25.
$96.50$96.72
$96.09 Answer: Shade the model to show the fraction. Question 26. Answer: Question 27. Answer: Look at the model. Color $$\frac{1}{4}$$ blue. Color $$\frac{2}{4}$$ yellow. Question 28. What fraction of the model is colored? Answer: 1/4 fraction of the model is colored in blue. Question 29. What fraction of the model is not colored? Answer: 2/4 fraction of the model is not colored in yellow. Shade each strip. Then write the fractions in order from greatest to least. Question 30. Answer: Write a fraction for the shaded part. Question 31. Answer: 1/2 is the shaded part Question 32. Answer: 2/3 is the shaded part Question 33. Answer: 1/3 is the shaded part. Use your answers for Exercises 31 to 33. Fill in the blanks. Question 34. _______ is lout of 2 equal parts. Answer: 1/2 is 1 out of 2 equal parts. Question 35. _________ is 2 out of 3 equal parts. Answer: 2/3 is 2 out of 3 equal parts. Question 36. $$\frac{1}{2}$$ is greater than ___________. Answer: 1/2 is greater than 1/3. Question 37. $$\frac{1}{2}$$ is less than ______________. Answer: 1/2 is less than 1 Question 38. ______________ is the least fraction. Answer: Find the missing fraction. Use models to help you. Question 39. Add $$\frac{1}{3}$$ and $$\frac{1}{3}$$. Answer: Question 40. Subtract $$\frac{3}{4}$$ from 1. Answer: Solve. Draw bar models to help you. Estimate to check your answers. Question 41. Teri folds 32 pieces of paper. Her sister folds 19 pieces. How many pieces do they fold in all? They fold ____________ pieces in all. Answer: They fold 51 pieces in all. Explanation: Given, Teri folds 32 pieces of paper, Her sister folds 19 pieces of paper, By adding 32 with 19 we get 51, Therefore, they fold 51 pieces of paper altogether. Question 42. Edwin has 83¢. His father gives him 25c more. How much does he have now? He has$__________ now.
Answer:
Edwin has $1.08 now. Explanation: Given, Edwin has 83¢, His father gives him 25¢ more, By adding 83 with 25 we get 108¢, Converting 108¢ into dollars we get$1.08,
Therefore, Edwin has $1.08. Question 43. Jonas needs to deliver 34 newspapers. He still has 11 newspapers left to deliver. How many newspapers has he delivered? He has delivered ____________ newspapers. Answer: Jonas has delivered 23 newspapers. Explanation: Given, Jonas needs to deliver 34 newspapers, He still has 11 newspapers left to deliver, By subtracting 11 from 34 we get 23, Therefore, Jonas needs to deliver 23 newspapers. Question 44. Adam wants to buy a bat for$23 and a baseball glove for $17. He has saved$19. How much more money does he need?
He needs $____________ more. Answer: Adam needs$21 more.
Explanation:
Given,
Adam wants to buy bat for $23, And a baseball glove for$17,
He saved $19, By adding$23 with $17 we get$40 money which he needs to buy the things he want,
By subtracting $19 from$40 we get $21 money which he needs, Therefore, Adam needs$21 more to buy the things he want.
Question 45.
An eraser costs 16¢ and a pencil costs 70¢. Marian buys two erasers and a pencil. How much does she spend?
Marian spends $____________ Answer: Marian spends$1.02
Explanation:
Given,
An eraser costs 16¢,
Pencil costs 70¢,
Marian buys two erasers and a pencil,
By multiplying 2 with 16¢ we get the cost of 2 erasers which is 32¢,
By adding 32¢ with 70¢, we get the cost of 2 erasers and a pencil which is 102¢,
Converting 102¢ into dollars we get $1.02. Question 46. Mrs. Barry has$200 to buy new clothes. Round the cost of each item to the nearest ten. Then estimate the total cost.
A pair of pants costs $44. a. 44 is ____________ when rounded to the nearest ten. A pair of shoes costs$59.
Answer:
44 is 50 when rounded to the nearest ten.
b. 59 is ____________ when rounded to the nearest ten.
A pair of socks costs $5. Answer: 59 is 60 when rounded to the nearest ten. c. 5 is ____________ when rounded to the nearest ten. A blouse costs$28.
Answer: 5 is 10 when rounded to the nearest ten.
d. 28 is ____________ when rounded to the nearest ten.
Total cost is $____________ Answer: 28 is 30 when rounded to the nearest ten. Does Mrs. Barry have enough money to pay for all the items? Explain your answer. Answer: Total estimated cost is$140.
Explanation:
Mrs. Barry has $200 to buy new clothes, A pair of pants costs$44 rounded to $50, A pair of shoes costs$59 rounded to $60, A pair of socks costs$5 rounded to $10, A blouse costs$28 rounded to $30, Total cost of the clothes is by adding 44, 59, 5 and 28 we get$136,
Total estimated cost is by adding 50, 60, 10 and 30 we get \$140,
Therefore, Mrs. Barry have enough money to pay for all the items.
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2023-03-21 23:35:27
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https://math.stackexchange.com/questions/2410211/stickelberger-element-of-imaginary-quadratic-extension
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# Stickelberger element of imaginary quadratic extension
I am currently trying to prove the following from Washington's book on cyclotomic fields:
Let $q\equiv 3 \pmod{4}$ be prime, such that $R$ and $N$ denote the number of quadratic residues and non-residues modulo $q$ respectively in the interval $\left[1,\frac{q-1}{2}\right]$. Use Stickelberger's theorem to show that $R-N$ annihilates the class group of $\mathbb{Q}(\sqrt{-q})$.
Taking the obvious first step, as $\mathbb{Q}(\zeta_q)\supset\mathbb{Q}(\sqrt{-q})$ is the smallest cyclotomic field to contain $\mathbb{Q}(\sqrt{-q})$ we can calculate the Stickelberger element $\theta(\mathbb{Q}(\sqrt{-q}))$. A fairly easy calculation gives that:
$$\Theta=\Theta(\mathbb{Q}(\sqrt{-q})) = \frac{q^2-1}{4q}+\frac{(q-1)^2}{4q}\sigma$$
My question is then how do I take this and show that $R-N \in \mathbb{Z}\cap\Theta\mathbb{Z}$, as I presume is required to show that it annihilates the class group?
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2019-06-16 03:35:25
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https://dsp.stackexchange.com/questions/35444/with-the-same-channel-s-n-the-normalized-mean-square-error-differs
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With the same channel S/N, the normalized mean square error differs
I am working on a research work on transmitting ECG signals over wireless body area network. The signal is affected by noise and small scale fading. The normalized mean square error(NMSE) is used to estimate the quality of the reconstructed signal at the receiver. When I run the same m file in matlab several times, I obtain different values for the NMSE, although the same channel signal to noise ratio is used.
• I think this is reasonable, do you agree?
• If so, how can I obtain a single value to represent the quality of the reconstructed signal at the receiver?
• I want to plot a curve with the S/N at the $x$-axis and NMSE at the $y$-axis, how can I plot this curve if the value of NMSE differ with different runs of the same m file with the same value of S/N?
1 Answer
The problem you describe is likely due to the randomness of your noise: Even though the S/N ratio remains the same for each run, the actual realization of the noise is different in each run. Hence, the NMSE is different for each run.
The standard procedure here is to run the algorithm several times for a single S/N ratio, measuring all the obtained NMSEs. Then, you calculate the mean of the measurements to find the overall NMSE for a given S/N ratio. Additionally, you can draw errorbars (MATLAB errorbar) around each points, which indicate the standard deviation of each measurement.
• How many times shall I run the algorithm for a single S/N ratio? 10, 20, ...100?
– Noha
Nov 13 '16 at 6:02
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2021-09-16 22:45:42
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http://it-pomoc.pl/sap/definicja/partial-cancellation-of-goods-receipt
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partial cancellation goods co to jest
ABAP definicja partial cancellation of goods receipt. Co znaczy partial quantity for which goods.
Znaczenie partial cancellation goods definicja.
Czy przydatne?
Definicja partial cancellation of goods receipt
Co znaczy:
Inverse posting of any partial quantity for which goods receipt was posted.
As far as material documents are concerned, there is no relationship between a partial cancellation and the previous goods receipt posting. This means that partial cancellations - unlike "normal" cancellations of goods receipts - do not clearly refer to a previous material document.
Słownik i definicje SAPa na .
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2019-02-22 15:59:52
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https://diabetesjournals.org/diabetes/article/50/3/697/10969/Decreased-Fasting-and-Oral-Glucose-Stimulated-C
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The high-affinity sulfonylurea receptor 1 (SUR1) plays an important role in regulating insulin secretion. In the Québec Family Study, we genotyped 731 individuals (685 nondiabetic [ND] subjects) for the SUR1 gene IVS15-3c→t and exon 18 Thr759(ACC→ACT) polymorphisms using polymerase chain reaction–restriction fragment-length polymorphism analysis. Phenotypes measured were fasting plasma glucose (GLU), fasting plasma insulin (INS), and fasting C-peptide (CPEP), as well as oral glucose tolerance test (OGTT) responses; they were adjusted for age, sex, waist circumference, and the sum of six skinfold thicknesses. In ND subjects, exon 18 Thr759(ACC→ACT) T allele carriers (T+) had lower CPEP (P = 0.022, −12.8%) and acute C-peptide responses (area above basal in first 30 min [CP30]) (P = 0.051, −12.4%) than noncarriers (T). Also, in those with the cT/tC haplotype (from both IVS15-3c→t and exon 18 Thr759[ACC→ACT] polymorphisms), CPEP (P = 0.005, −21.2%), CP30 (P = 0.034, −19.2%), and total C-peptide responses (P = 0.016, −20.2%) were lower than that in cT subjects. In overweight individuals (BMI >25 kg/m2), differences between carriers and noncarriers of the T or cT alleles were greater for GLU (P = 0.023–0.034), CPEP (P = 0.021–0.015), acute OGTT insulin response (P = 0.014–0.019), and CP30 (P = 0.034–0.019). These results suggest that the T and cT allele variants are associated with lower insulin secretion parameters, particularly in female and overweight subjects, adding evidence to the role of SUR1 sequence variants in decreased insulin secretion.
Type 2 diabetes has been shown to have a strong hereditary basis (1,2); however, the major genes for the common late-onset type have not yet been clearly identified (3). Because impaired insulin secretion plays a critical role in the pathogenesis of the disease (4), genes that encode major elements involved in insulin secretion are good candidates for evaluating genetic susceptibility to type 2 diabetes (3).
A central component in glucose-induced insulin secretion is the ATP-sensitive potassium (KATP) channel of the β-cell. Increasing ATP generated by intracellular glucose metabolism leads to the closing of these channels, membrane depolarization, the opening of voltage-gated calcium channels with influx of calcium, and exocytosis of insulin (5). The KATP channel is composed of the high-affinity sulfonylurea receptor (SUR1) and Kir6.2, a member of the inwardly rectifying ion channel family (6). Pathological regulation of insulin secretion has been demonstrated in studies of persistent hyperinsulinemic hypoglycemia of infancy, a rare autosomal recessive disease in which mutations in the nucleotide-binding fold-2 region of the SUR1 gene have been found (7).
In type 2 diabetes, association studies in diabetic populations have been positive, with the IVS15-3c→t and exon 18 Thr759(ACC→ACT) variants of the SUR1 gene being found more frequently in Danish (8), U.S., U.K. (9), and French Caucasian (10) diabetic individuals, and the IVS15-3c→t variant alone found more frequently in Dutch Caucasian diabetic subjects (11). However, results in Japanese populations have been negative (12). Few studies have reported on the physiological impact of SUR1 gene sequence variants. A Danish group demonstrated decreased tolbutamide-stimulated insulin secretion in normal glucose-tolerant carriers of the above intron 15 and exon 18 variants (8). Another group recently reported an association of the IVS15-3c→t variant with lower second-phase insulin secretion in Dutch nondiabetic (ND) individuals (13).
The present study was thus undertaken to examine the role of the SUR1 IVS15-3c→t and exon 18 Thr759(ACC→ACT) polymorphisms and their haplotypes on glucose metabolism in French-Canadian Caucasians of the Québec Family Study (QFS).
There were only 43 diabetic subjects in QFS (Table 1). Unadjusted baseline data for the 685 ND subjects are presented in Table 2. In unrelated individuals from the parental generation (n = 259), the IVS15-3c→t polymorphism frequencies were 56% for the c allele and 44% for the t variant; and for the exon 18 Thr759(ACC→ACT) polymorphism, the frequencies were 95.8% for the C allele and 4.2% for the T variant. These frequencies are similar to those reported in ND Caucasians (8,9,11). No carriers of the rarer exon 18 Thr759(ACC→ACT) T allele (T+) were found among diabetic subjects, and IVS15-3c→t polymorphism frequencies were similar in diabetic and ND subjects (data not shown). Genotypes were in Hardy-Weinberg equilibrium and the two polymorphisms were in strong linkage disequilibrium (χ2 = 28.4, P < 0.0001).
In ND subjects, the T+ group had significantly lower values than noncarriers (T) with regard to fasting plasma C-peptide (CPEP) and acute oral glucose tolerance test (OGTT) C-peptide response, defined as the area above basal in the first 30 minutes (CP30) (Fig. 1). The cT/tC double heterozygotes had even lower values than all cT haplotypes with regard to CPEP, CP30, and total OGTT C-peptide response (area above basal over 180 min) (CPtot). Trends were seen for lower values in the T+ group than in the T group with regard to acute OGTT insulin response (IN30), the area above basal in the first 30 min (P = 0.063, −14.4%). Results were independent of age, sex, and adiposity.
In women, CPEP and CP30 were significantly decreased in the T+ group compared with the T group and in the cT/tC group compared with the cT group (Table 3). CPtot followed the same pattern in cT/tC carriers. In men, fasting plasma glucose (GLU) was higher (P = 0.034, +4.2%), and IN30 tended to be lower (P = 0.074, −19.6%) in the T+ group than in T subjects.
In younger subjects (≤45 years old), CPEP was lower in the T+ group than in the T group (P = 0.026, −16.1%) and in the cT/tC group compared with the cT group (P = 0.045, −22.4%). IN30 tended to be lower in the T+ group than in T subjects (P = 0.080, −17.6%). CP30 was lower in T+ subjects than in T subjects (P = 0.058, −15.6%), and CPtot was decreased in the cT/tC carriers when compared with the cT group (P = 0.034, −25.0%).
With higher BMI, differences between genotype groups increased (Fig. 2). In those with BMI >25 kg/m2, the T+ subjects had higher GLU (+4.6%) and lower CPEP (−18.1%), IN30 (−26.6%), and CP30 (−19.2%) than T subjects. The cT/tC group had even higher GLU (+5.9%) and lower CPEP (−24.8%), IN30 (−33.9%), and CP30 (−28.6%) values than the cT groups.
No significant differences between IVS15-3c→t genotypes were seen, and fasting plasma insulin (INS), OGTT glucose responses, and total insulin response were similar among the different genotypes (data not shown).
After taking into account independent family effects by adding a covariate for family membership in the analysis of variance for the above phenotypes, we found that the results were only modestly modified. For the overall cohort, P values for differences between T+ and T subjects were 0.072 for CP30, and for differences between cT/tC and non-cT haplotypes, the values were 0.119 for CP30 and 0.063 for CPtot. For all CPEP and glucose data and for all results in sex and BMI subgroups, differences between T+ and T and between cT/tC and cT groups remained significant at the same level (results not shown).
In this study, we have shown that the T allele of the exon 18 Thr759(ACC→ACT) polymorphism of the SUR1 gene was associated with lower CPEP and acute OGTT C-peptide responses in ND Canadians of French descent. Even lower CPEP and OGTT C-peptide responses were seen with the cT/tC haplotype, obtained from both the IVS15-3c→t and exon 18 Thr759(ACC→ACT) polymorphisms. These greater differences in the haplotype comparisons suggest that using two variants in the SUR1 gene more clearly defines the allele associated with the phenotypes mentioned. This supports findings in Danish Caucasians (8) of lower insulin and C-peptide secretion after intravenous tolbutamide in ND carriers of both the exon 18 C/T or T/T and the IVS15-3c→t c/t or t/t genotypes. However, in that study haplotypes could not be clearly identified; in the QFS they could be determined because family structures were known. Moreover, we could adjust metabolic parameters for important covariates such as waist circumference (WC) and the sum of six skinfold thicknesses (SF6).
Although we focused our study on ND subjects, for which interpretation of OGTT data are straightforward, analysis was also performed on all subjects combined. Similar results were obtained for C-peptide and insulin parameters (data not shown). Therefore, we do not believe that restricting analysis to ND subjects biased the results. There were no T+ subjects among the diabetic subjects. This should not be unexpected because QFS was not specifically designed to study type 2 diabetes and the numbers of diabetic subjects was small.
Lower OGTT responses could result either from decreased β-cell secretion or higher insulin sensitivity. Several arguments point to a relationship with secretion. First, SUR1 has been described mainly in pancreatic β-cells (6). Second, INS levels, which are variably correlated with insulin resistance, were unaffected by genotype status, yet CPEP levels were significantly changed. C-peptide is secreted in an equimolar fashion with insulin, has a longer plasma half-life, displays smaller oscillations in plasma concentration, and undergoes minimal hepatic extraction; therefore, it has been suggested to be a better reflection of insulin secretion overall (14). Third, OGTT glucose responses were not significantly different between the genotypes, whereas we would have expected a significant decrease in GLU and OGTT glucose responses in the case of greater insulin sensitivity with lower insulin requirements. Still, we cannot rule out a primary or compensatory increase in insulin sensitivity. Interestingly, in the Hansen et al. (8) study, the subjects displaying decreased insulin and C-peptide responses also showed a 30% increase in glucose effectiveness. Also, in a recent KATP channel–knockout mouse model (15), not only was insulin secretion decreased, but insulin action was enhanced, as measured by an insulin tolerance test, due either to a direct peripheral effect of the KATP channel deficiency or to an unknown compensatory mechanism.
The main genotype differences were found in C-peptide measures, although OGTT insulin responses also tended to be lower overall and were significantly lower in overweight subjects. This apparent discrepancy could be explained by the insulin assay used here, which crossreacts with proinsulin (PI). It has been shown that in deficient insulin secretion states, such as maturity-onset diabetes of the young, OGTT responses for C-peptide are more suppressed than those for immunoreactive insulin (IRI) (16). Moreover, in type 2 diabetes, PI is proportionately increased with respect to IRI and specific insulin (17). Hence, our results would suggest that the SUR1 variant carriers are also characterized by a deficient insulin secretion state. However, measures of PI and specific insulin would be required to clarify this point.
Significant associations were only seen in younger subjects. It is possible that in older individuals, age and accumulated environmental influences might predominantly modulate insulin secretion, overshadowing any underlying genetic factor and clouding differences between genotypes. Also, with increasing BMI, differences between T+ and T and between cT/tC and non-cT genotype groups became more pronounced for GLU, CPEP, IN30, and CP30. This pattern of higher GLU and lower OGTT secretory responses suggests that BMI directly influences these relationships. A possible explanation is the association between obesity and increased insulin resistance that leads to greater demand on β-cells, allowing differences in metabolic parameters to become more evident in SUR1 variant carriers, who would respond insufficiently because of limited secretion.
The exon 18 Thr759(ACC→ACT) polymorphism is silent; thus, it cannot alone explain our results. The IVS15-3c→t polymorphism is located by the intron-exon 16 splice junction, and thus an effect on mRNA splicing is possible but unproven. The most likely explanation is that these variants are in linkage disequilibrium with a nearby unidentified functional mutation, either in the SUR1 gene or in a gene close by. In the latter case, the Kir6.2 gene located only 4.5-kb distant on 11p15.1 is a good candidate.
In conclusion, the present study has shown for the first time that the SUR1 gene exon 18 Thr759(ACC→ACT) T allele and the cT haplotype, resulting from the combination of this polymorphism and the IVS15-3c→t polymorphism, are significantly associated with lower CPEP and OGTT C-peptide responses and possibly higher GLU and lower OGTT insulin responses in a large Caucasian population of French descent. Sex, age, and particularly BMI status modulated these associations. Because differences were greater with the cT haplotype, using two SUR1 variants may more clearly define the allele associated with altered glucose metabolism. However, these variants have not been shown to have direct functional consequences. Additional studies are needed to define the role of SUR1 or some nearby gene in insulin secretion and to identify the mutations that fully explain the results reported here.
### Subjects.
The QFS cohort is composed of Caucasian nuclear families of French descent from the Québec City area, representing a mixture of random sampling and ascertainment through obese (BMI >32 kg/m2) probands (18). In this study, 731 adults from 200 families were measured during phase two (1989–1997) and phase three (1998 to the present) of the QFS. Mean family size was 4.0 (range 1–13); 69% of families had two parents, 24% had 0–1 parent, and the remainder were extended families with more than two members from the parental generation. Written consent was obtained from all participating subjects, and the Medical Ethics Committee of Laval University approved the protocol.
### Glucose tolerance status and metabolic parameters.
A 75-g OGTT was performed after an overnight fast. Fasting and OGTT plasma glucose, insulin, and C-peptide were assayed as previously described (19). OGTT areas under the curve were calculated using the trapezoidal method. The area over the first 30 min defined the acute response, whereas the complete area (0–180 min) was the total response. 1997 American Diabetes Association diagnostic criteria determined the glucose tolerance status. Because OGTTs were initiated in QFS in 1993, 161 subjects had only GLU values and 3 had missing values. To avoid misclassification based on GLU, glucose tolerance was limited to diabetic and ND categories. From OGTT data on 567 subjects, the possibility of misclassifying a diabetic subject as ND using a GLU value <6 mmol/l was very low (0.77%), and in the group of 161 subjects, only 1 or 2 diabetic subjects would possibly be missed using this GLU cutoff, representing a very low number in the overall cohort. Only the 685 ND subjects were used in the metabolic study, because glucose homeostasis is perturbed in the diabetic state, making fasting and OGTT-derived measures more difficult to interpret and less well correlated with insulin sensitivity and secretory parameters obtained from more sophisticated measurements.
BMI was derived from body weight divided by height squared (kg/m2). WC and skinfold thicknesses (evaluated at six sites: biceps, triceps, medial calf, abdominal, suprailiac, and subscapular areas) were measured by a single observer as previously described (19).
### Genotype and haplotype determination.
Genomic DNA was obtained from cultured lymphoblastoid cell lines by proteinase K and phenol/chloroform extraction procedure followed by dialysis. Polymerase chain reaction (PCR) amplifications of the DNA segments encompassing the SUR1 IVS15-3c→t and exon 18 Thr759(ACC→ACT) variants were carried out using primers previously reported (8,9), and specific information is available in the online appendix at www.diabetes.org/diabetes/appendix.asp. Using both polymorphisms of the SUR1 gene, haplotypes were obtained. Because the family structures are known in QFS and genotypes of parents and offspring were both available, Mendelian analysis determined that the double heterozygote was cT/tC and not tT/cC (see online appendix at www.diabetes.org/diabetes/appendix.asp). No subjects were found with the genotypes tT/tT, tT/cT, or tT/tC, and the single individual carrying the cT/cT genotype was not included in the analysis. The exon 18 Thr759(ACC→ACT) T+ group was thus comprised of the cT/tC double heterozygote and the cT/cC subgroups.
### Statistical Analysis.
Nonnormally distributed variables were log-transformed before analysis. Metabolic parameters were adjusted for age, age2, age3, and sex. Moreover, data were adjusted for WC and SF6, but not BMI, because BMI never achieved significance when WC and SF6 were present. Analysis of variance through the General Linear Model procedure in SAS (version 6.12; Cary, NC) was used to test for differences in metabolic parameters between genotypes. A χ2 test was applied to evaluate whether genotype and allele frequencies were in Hardy-Weinberg equilibrium and to test for genotype and allele frequency differences. Linkage disequilibrium between the polymorphisms was assessed as described by Terwilliger and Ott (20).
All subjects were used in association analyses, despite the relatedness of the subjects. A recent simulation study compared three methods of accounting for nonindependence in family sampling designs to a method that ignored the within-family dependencies (Michael A. Province, Treva Rice, D.C. Rao, unpublished data). Results showed that failure to take into account dependencies among subjects of the same family in statistical analyses did not induce any bias, and ignoring these dependencies resulted in a small reduction in power without affecting type I error, except in cases of extreme within-family correlation, which is rare in family studies. We are primarily concerned with failure to detect significant associations, especially if the significance level is borderline; therefore, we believe that it is more appropriate to use all subjects in these association studies. Data are presented as least square-means ± SE. For log-transformed variables, the results are the back-transformed least-square means ± SE derived from the 95% confidence intervals. A P value <0.05 was considered significant.
FIG. 1.
Fasting plasma C-peptide and OGTT C-peptide responses in relation to exon 18 Thr759(ACC→ACT) genotype and IVS15-3c→t/exon 18 Thr759(ACC→ACT) haplotype in ND subjects overall in QFS. Results given are back-transformed least-square means + SE from the general linear model multiple regression procedure, adjusting for age, age2, age3, sex, WC, and SF6; *0.06 > P > 0.05; **0.05 > P > 0.01; |$$/c 0.01 > P > 0.001 for comparisons with exon 18 Thr759(ACC→ACT) T allele carriers or the cT/tC haplotype. Numbers given are those for each subgroup. FIG. 1. Fasting plasma C-peptide and OGTT C-peptide responses in relation to exon 18 Thr759(ACC→ACT) genotype and IVS15-3c→t/exon 18 Thr759(ACC→ACT) haplotype in ND subjects overall in QFS. Results given are back-transformed least-square means + SE from the general linear model multiple regression procedure, adjusting for age, age2, age3, sex, WC, and SF6; *0.06 > P > 0.05; **0.05 > P > 0.01; |$$/c 0.01 > P > 0.001 for comparisons with exon 18 Thr759(ACC→ACT) T allele carriers or the cT/tC haplotype. Numbers given are those for each subgroup.
FIG. 2.
Fasting plasma glucose and C-peptide and acute OGTT insulin and C-peptide responses in relation to exon 18 Thr759(ACC→ACT) genotype and IVS15-3c→t/exon 18 Thr759(ACC→ACT) haplotype in ND QFS subjects by BMI subgroup. Results given are back-transformed least-square means + SE from the general linear model multiple regression procedure, adjusting for age, age2, age3, WC, and SF6; **0.05 > P > 0.01 for comparisons with exon 18 Thr759(ACC→ACT) T allele carriers or the cT/tC haplotype. Numbers given are those for each subgroup.
FIG. 2.
Fasting plasma glucose and C-peptide and acute OGTT insulin and C-peptide responses in relation to exon 18 Thr759(ACC→ACT) genotype and IVS15-3c→t/exon 18 Thr759(ACC→ACT) haplotype in ND QFS subjects by BMI subgroup. Results given are back-transformed least-square means + SE from the general linear model multiple regression procedure, adjusting for age, age2, age3, WC, and SF6; **0.05 > P > 0.01 for comparisons with exon 18 Thr759(ACC→ACT) T allele carriers or the cT/tC haplotype. Numbers given are those for each subgroup.
Table 1
Glucose tolerance status in QFS
Sex
Total
MenWomen
All subjects
Nondiabetic 294 (40.4) 391 (53.7) 685 (94.1)
Diabetic 22 (3.0) 21 (2.9) 43 (5.9)
Total 316 (43.4) 412 (56.6) 728 (100)
OGTT subjects
Normal glucose tolerance 194 (34.2) 260 (45.9) 454 (80.0)
Impaired glucose tolerance and impaired fasting glucose 41 (7.2) 48 (8.5) 89 (15.7)
Diabetic 15 (2.6) 9 (1.6) 24 (4.2)
Total 250 (44.1) 317 (55.9) 567 (100)
Sex
Total
MenWomen
All subjects
Nondiabetic 294 (40.4) 391 (53.7) 685 (94.1)
Diabetic 22 (3.0) 21 (2.9) 43 (5.9)
Total 316 (43.4) 412 (56.6) 728 (100)
OGTT subjects
Normal glucose tolerance 194 (34.2) 260 (45.9) 454 (80.0)
Impaired glucose tolerance and impaired fasting glucose 41 (7.2) 48 (8.5) 89 (15.7)
Diabetic 15 (2.6) 9 (1.6) 24 (4.2)
Total 250 (44.1) 317 (55.9) 567 (100)
Data are n (%).
Table 2
Baseline values in nondiabetic QFS population
PhenotypenMean (range)
Age (years) 685 42.0 (17–92)
BMI (kg/m2677 26.1 (16.8–64.9)
Waist circumference (cm) 656 84.8 (57.9–164)
Sum of six skinfolds (cm) 649 105 (23.5–448)
Waist-to-hip ratio 648 0.90 (0.56–1.19)
Fasting glucose (mmol/l)* 684 4.96 (3.20–6.90)
Fasting insulin (pmol/l)* 612 54.4 (1.0–350)
Fasting C-peptide (pmol/l)* 573 646 (178–3143)
PhenotypenMean (range)
Age (years) 685 42.0 (17–92)
BMI (kg/m2677 26.1 (16.8–64.9)
Waist circumference (cm) 656 84.8 (57.9–164)
Sum of six skinfolds (cm) 649 105 (23.5–448)
Waist-to-hip ratio 648 0.90 (0.56–1.19)
Fasting glucose (mmol/l)* 684 4.96 (3.20–6.90)
Fasting insulin (pmol/l)* 612 54.4 (1.0–350)
Fasting C-peptide (pmol/l)* 573 646 (178–3143)
*
Fasting plasma measurements.
Table 3
Results in nondiabetic men and women for exon 18 Thr759(ACC→ACT) genotypes and IVS15-3c→t/exon 18 Thr759(ACC→ACT) haplotypes
PhenotypeMen
Women
TT+cTcT/cCcT/tCTT+cTcT/cCcT/tC
n 207–266 21–24 203–262 10 9–12 271–340 24–27 270–338 14 10–13
GLU (mmol/l) 4.97 ± 0.04 5.18 ± 0.10* 4.96 ± 0.04 5.05 ± 0.15 5.22 ± 0.14 4.96 ± 0.03 4.96 ± 0.09 4.96 ± 0.03 4.93 ± 0.12 5.00 ± 0.13
INS (pmol/l) 51.2 ± 3.0 48.5 ± 7.0 51.1 ± 3.0 46.9 ± 9.9 51.5 ± 10.8 55.0 ± 2.6 51.4 ± 6.7 54.8 ± 2.5 48.7 ± 8.6 54.5 ± 10.3
CPEP (pmol/l) 621 ± 23 588 ± 53 621 ± 23 626 ± 82 557 ± 72 666 ± 20 542 ± 44* 667 ± 20 615 ± 67 471 ± 55
GL30 (mmol/l × min) 212 ± 2.5 208 ± 6.0 212 ± 2.5 207 ± 8.4 215 ± 8.7 200 ± 1.9 193 ± 5.2 199 ± 1.9 192 ± 6.6 194 ± 7.8
GLtot (mmol/l × min) 1,180 ± 20 1,120 ± 50 1,170 ± 20 1,080 ± 60 1,180 ± 70 1,140 ± 20 1,080 ± 40 1,141 ± 20 1,110 ± 60 1,050 ± 60
IN30 (103 pmol/l × min) 8.29 ± 0.43 6.67 ± 0.83 8.34 ± 0.40 6.72 ± 1.22 6.95 ± 1.25 8.29 ± 0.35 7.45 ± 0.89 8.24 ± 0.34 7.75 ± 1.17 7.02 ± 1.26
INtot (103 pmol/l × min) 66.0 ± 3.0 56.2 ± 6.1 65.6 ± 3.0 49.1 ± 7.8 61.2 ± 9.6 64.4 ± 2.4 61.1 ± 6.1 64.3 ± 2.4 69.1 ± 9.2 51.3 ± 8.1
CP30 (103 pmol/l × min) 43.9 ± 1.9 40.9 ± 4.1 44.0 ± 1.9 44.7 ± 6.6 39.2 ± 5.7 45.0 ± 1.6 37.2 ± 3.6* 44.9 ± 1.6 40.6 ± 5.0 32.9 ± 4.8*
CPtot (103 pmol/l × min) 409 ± 16 370 ± 35 406 ± 16 370 ± 49 356 ± 49 442 ± 14 394 ± 34 442 ± 14 455 ± 51 324 ± 43*
PhenotypeMen
Women
TT+cTcT/cCcT/tCTT+cTcT/cCcT/tC
n 207–266 21–24 203–262 10 9–12 271–340 24–27 270–338 14 10–13
GLU (mmol/l) 4.97 ± 0.04 5.18 ± 0.10* 4.96 ± 0.04 5.05 ± 0.15 5.22 ± 0.14 4.96 ± 0.03 4.96 ± 0.09 4.96 ± 0.03 4.93 ± 0.12 5.00 ± 0.13
INS (pmol/l) 51.2 ± 3.0 48.5 ± 7.0 51.1 ± 3.0 46.9 ± 9.9 51.5 ± 10.8 55.0 ± 2.6 51.4 ± 6.7 54.8 ± 2.5 48.7 ± 8.6 54.5 ± 10.3
CPEP (pmol/l) 621 ± 23 588 ± 53 621 ± 23 626 ± 82 557 ± 72 666 ± 20 542 ± 44* 667 ± 20 615 ± 67 471 ± 55
GL30 (mmol/l × min) 212 ± 2.5 208 ± 6.0 212 ± 2.5 207 ± 8.4 215 ± 8.7 200 ± 1.9 193 ± 5.2 199 ± 1.9 192 ± 6.6 194 ± 7.8
GLtot (mmol/l × min) 1,180 ± 20 1,120 ± 50 1,170 ± 20 1,080 ± 60 1,180 ± 70 1,140 ± 20 1,080 ± 40 1,141 ± 20 1,110 ± 60 1,050 ± 60
IN30 (103 pmol/l × min) 8.29 ± 0.43 6.67 ± 0.83 8.34 ± 0.40 6.72 ± 1.22 6.95 ± 1.25 8.29 ± 0.35 7.45 ± 0.89 8.24 ± 0.34 7.75 ± 1.17 7.02 ± 1.26
INtot (103 pmol/l × min) 66.0 ± 3.0 56.2 ± 6.1 65.6 ± 3.0 49.1 ± 7.8 61.2 ± 9.6 64.4 ± 2.4 61.1 ± 6.1 64.3 ± 2.4 69.1 ± 9.2 51.3 ± 8.1
CP30 (103 pmol/l × min) 43.9 ± 1.9 40.9 ± 4.1 44.0 ± 1.9 44.7 ± 6.6 39.2 ± 5.7 45.0 ± 1.6 37.2 ± 3.6* 44.9 ± 1.6 40.6 ± 5.0 32.9 ± 4.8*
CPtot (103 pmol/l × min) 409 ± 16 370 ± 35 406 ± 16 370 ± 49 356 ± 49 442 ± 14 394 ± 34 442 ± 14 455 ± 51 324 ± 43*
Data are least-square means ± SE. GL30, OGTT glucose area above basal in the first 30 min; GLtot, OGTT glucose area above basal over 180 min; INtot, OGTT insulin area above basal over 180 min.
*
P < 0.05 and
P < 0.01 for comparisons between T+ and T genotype or cT/tC and cT haplotype subgroups.
This study was supported by grants from the Medical Research Council of Canada (PG-11811, MT-13960, and GR-15187). We thank Claude Leblanc, François Michaud, and Christian Couture for their assistance with the computer database and information systems. Thanks are also expressed to Monique Chagnon and Chantal Paré for their technical assistance; Guy Fournier, Lucie Allard, and Anne-Marie Bricault for their dedicated work in QFS; and Diane Drolet for manuscript and secretarial support.
C.B. was previously supported by the Donald B. Brown Chair on Obesity funded by the Medical Research Council of Canada and Roche Canada, and he is currently funded by the George A. Bray Chair in Nutrition.
Preliminary results from this study were presented at the 59th Scientific Sessions of the American Diabetes Association Meeting, San Diego, California, in June 1999.
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’t Hart LM, Dekker JM, van Haeften TW, Ruige JB, Stehouwer CD, Erkelens DW, Heine RJ, Maassen JA: Reduced second phase insulin secretion in carriers of a sulphonylurea receptor gene variant associating with type II diabetes mellitus.
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Terwilliger J, Ott J:
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2022-01-18 01:14:41
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https://www.gamedev.net/forums/topic/655592-link-between-create-rotations-matrices/
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Followers 0
# Link between Create Rotations matrices
## 0 posts in this topic
Hello,
A friend of mine have been coding some rotations on my XNA project, but the order of the matrices rotation are not good. Some are of form :
Matrix.CreateRotationZ(1.21f) * Matrix.CreateRotationY(-0.13f)
and other:
Matrix.CreateRotationY(1.01f) * Matrix.CreateRotationX(-0.5f) * Matrix.CreateRotationZ(0.80f)
How can I find the values X, Y, Z so the rotations are in the correct order: (X,Y,Z) ?
Matrix.CreateRotationX(X) * Matrix.CreateRotationY(Y) * Matrix.CreateRotationZ(Z)
Thank you!
0
## Create an account
Register a new account
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2017-07-27 21:45:31
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https://www.learncram.com/maharashtra-board/class-8-maths-solutions-chapter-15-practice-set-15-4/
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Maharashtra Board Class 8 Maths Solutions Chapter 15 Area Practice Set 15.4
Maharashtra State Board Class 8 Maths Solutions Chapter 15 Area Practice Set 15.4
Question 1.
Sides of a triangle are 45 cm, 39 cm and 42 cm, find its area.
Solution:
Sides of a triangle are 45 cm, 39 cm and 42 cm.
Here, a = 45cm, b = 39cm, c = 42cm
Semi perimeter of triangle = s = $$\frac { 1 }{ 2 }(a+b+c)$$
= $$\frac { 1 }{ 2 }(45+39+42)$$
= $$\frac { 126 }{ 2 }$$
= 63
Area of a triangle
∴ The area of the triangle is 756 sq.cm.
Question 2.
Look at the measures shown in the given figure and find the area of ☐PQRS.
Solution:
A (☐PQRS) = A(∆PSR) + A(∆PQR)
In ∆PSR, l(PS) = 36 m, l(SR) = 15 m
A(∆PSR)
= $$\frac { 1 }{ 2 }$$ x product of sides forming the right angle
= $$\frac { 1 }{ 2 }$$ x l(SR) x l(PS)
= $$\frac { 1 }{ 2 }$$ x 15 x 36
= 270 sq.m
In ∆PSR, m∠PSR = 90°
[l(PR)]² = [l(PS)]² + [l(SR)]²
…[Pythagoras theorem]
= (36)² + (15)²
= 1296 + 225
∴ l(PR)² = 1521
∴ l(PR) = 39m
…[Taking square root of both sides]
In ∆PQR, a = 56m, b = 25m, c = 39m
A(☐PQRS) = A(∆PSR) + A(∆PQR)
= 270 + 420
= 690 sq. m
∴ The area of ☐PQRS is 690 sq.m
Question 3.
Some measures are given in the figure, find the area of ☐ABCD.
Solution:
A(∆BAD) = $$\frac { 1 }{ 2 }$$ x product of sides forming the right angle
= $$\frac { 1 }{ 2 }$$ x l(AB) x l(AD)
= $$\frac { 1 }{ 2 }$$ x 40 x 9
= 180 sq. m
In ∆BDC, l(BT) = 13m, l(CD) = 60m
A(∆BDC) = $$\frac { 1 }{ 2 }$$ x base x height
= $$\frac { 1 }{ 2 }$$ x l(CD) x l(BT)
= $$\frac { 1 }{ 2 }$$ x 60 x 13
= 390 sq. m
A (☐ABCD) = A(∆BAD) + A(∆BDC)
= 180 + 390
= 570 sq. m
∴ The area of ☐ABCD is 570 sq.m.
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2023-02-09 13:17:22
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https://math.stackexchange.com/questions/3538588/steady-state-of-a-discrete-difference-equation/3539041
|
# Steady state of a discrete difference equation
I've encountered this Theorem in my mathematical biology notes:
$$\bar{𝑥}\text{ is a stable steady state of }f(x_n) = x_{n+1}\text{ iff }|f^{'}(\bar{𝑥})|<1.$$
The definition of a steady state $$\bar{x}$$ for first order difference equations is given as:
$$x_{n+1} = x_n = \bar{x}$$
My problem lies in applying this theorem, particularly to solve questions like this in the notes:
Consider the following nonlinear difference equation for population growth: $$x_{n+1} = \frac{kx_n}{b+x_n}; \text{ }b,k > 0$$ Establish whether the equation has a nontrivial steady state and determine its stability.
I'm confused, mostly because we're taking derivatives of discrete functions and it is my assumption that the first derivative of steady states should be 0 (i.e. why is the constraint in the theorem < 1?)
## 1 Answer
I gave this some thought.
Firstly, I fell into the trap of thinking that $$f'(x_n) = f'(x_{n+1}) = 0 \iff x_n\text{ is a steady state}$$
The issue with this is that between two discrete samples, an arbitrary, differentiable $$f$$ can do whatever it likes and still manage to satisfy $$f'(x_n) = f'(x_{n+1}) = 0$$
To solve the example, I did the following: $$\bigg(\exists n:x_{n+1} = x_n = \bar{x}\text{ }\bigg) \implies \bigg(x_n = \frac{kx_n}{b+x_n}\bigg) \implies \bigg(x_n = 0 \text{ } \oplus x_n = k - b\bigg)$$ $$\implies f(x_n) \text{ has a nontrivial steady state assuming } k \neq b$$
Now to determine stability, we can apply the Theorem: $$\bigg(f'(x) = \frac{kb}{(b+x)^2}\bigg) \implies \bigg(f'(k-b) = \frac{b}{k}\bigg)$$
Hence if $$b \geq k$$ the steady state is not stable
I would appreciate if someone could let me know if this is correct.
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2021-06-19 19:12:37
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https://demo7.dspace.org/items/9f9cbe69-61d7-4aa6-8978-484820510b6a
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## Some remarks on varieties of pairs of commuting upper triangular matrices and an interpretation of commuting varieties
Basili, Roberta
##### Description
It is known that the variety of pairs of n x n commuting upper triangular matrices isn't a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n > m. We also show that m < 18 and that it could be found by determining the dimension of the variety of pairs of commuting strictly upper triangular matrices. Then we define a natural map from the variety of pairs of commuting n x n matrices onto a subvariety defined by linear equations of the grassmannian of subspaces of codimension 2 of a vector space of dimension n x n.
Comment: Latex, 11 pages
##### Keywords
Mathematics - Algebraic Geometry, Mathematics - Operator Algebras, 15A30, 14L30
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2022-12-07 10:16:59
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https://forums.ankiweb.net/t/mathjax-cloze-renders-yellow-with-extra-in-preview/22665
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# MathJax Cloze Renders Yellow With Extra ] In Preview
Greetings,
I am using MathJax in my notes with relative success despite being new to Anki and LaTeX and MathJax. However, when I enter the following text/code:
A {{c1::joule}} {{c1::(J)}} of work is done in moving a {{c2::coulomb}} {{c2::(C)}} of charge through a potential difference of {{c3::1 V}} and is defined as <anki-mathjax>{{c4::1\,V = \frac{1J}{1C}}}</anki-mathjax>.
It presents the following preview in both the desktop and Android applications:
Attempted solutions:
• Forum Search
• Tool | Check Database
• Restart Anki
• LaTeX in place of MathJax (did not render well)
• Mucking about with different combinations of } ] \ etc.
OS Fedora 36
Anki Version 2.1.54 (b6a7760c)
Python 3.9.10 Qt 6.3.1 PyQt 6.3.1
Installed Add-ons (I have not gotten around to using them all. They are just the ones I found interesting.)
• Review Heatmap > 1771074083
• Advanced Review Bottom Bar > 1136455830
• Anki Simulator > 817108664
|
2022-10-01 08:05:22
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https://zbmath.org/0889.35028
|
## Asymptotic estimates and convexity of large solutions to semilinear elliptic equations.(English)Zbl 0889.35028
The author investigates asymptotic estimates and convexity of classical solutions of the boundary value problem $$\Delta u=f(u)$$ in $$D$$, $$u(x)\to \infty$$ as $$x\to \partial D$$. Here $$D\subset \mathbb{R}^N,N>1$$, is a bounded convex smooth domain, $$f(t)$$ is a differentiable positive nondecreasing function on $$[t_0,\infty )$$ satisfying $$f(t_0)=0$$ and $$F(t)^{-1/2}$$ is integrable at infinity, where $$F$$ is the primitive function of $$f$$, $$F(t_0)=0$$. Let $$\delta (x)$$ denote the distance from $$x$$ to the boundary of $$D$$ and $$\Phi (s)$$ be the function defined as $\int _{\Phi (s)}^\infty [2F(t)]^{-1/2} dt=s.$ The author investigates the behavior of $$u(x)-\Phi (\delta (x))$$ near the boundary of $$D$$.
Reviewer: D.Medková (Praha)
### MSC:
35J60 Nonlinear elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems
### Keywords:
singular boundary value
|
2023-03-28 18:43:24
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|
https://www.vedantu.com/question-answer/an-aeroplane-flying-at-a-height-of-300-meters-class-10-maths-cbse-5ee9f44195f78e759773a5e9
|
Question
# An aeroplane flying at a height of 300 meters above the ground passes vertically above another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 60 degrees and 45 degrees respectively. Then the height of the lower plane from the ground in meters is:(a). $100\sqrt{3}$ (b). $\dfrac{100}{\sqrt{3}}$ (c). 50(d). $150\left( \sqrt{3}+1 \right)$
Hint: At first try to draw the picture according to the conditions. From the picture point out what you have to find out and which values are given to you. Then apply trigonometric ratios accordingly.
Let A be aeroplane flying at a height of 300 meters above the ground.
It passes vertically above another plane, say B.
Let $CD$ be the ground.
It is given that the height of the first plane is 300 meters.
That means $AC$ is the height which is 300 meters.
It is given that the angle of the elevation of the two planes from the same point on the ground are 60 and 45 degrees respectively.
The angle of elevation is an angle that is formed between the horizontal line and the line of sight. If the line of sight is upward from the horizontal line, then the angle formed is an angle of elevation.
Here the objects are the aeroplane. D is the observation point. The horizontal line is $CD$.
\begin{align} & \angle ADC={{60}^{\circ }} \\ & \angle BDC={{45}^{\circ }} \\ \end{align}
Now we have to find out the height of the lower plane from the ground. That means we have to find out $CB$.
From triangle $ACD$ we have:
\begin{align} & \tan {{60}^{\circ }}=\dfrac{AC}{CD} \\ & \Rightarrow \sqrt{3}=\dfrac{300}{CD} \\ & \Rightarrow CD=\dfrac{300}{\sqrt{3}} \\ \end{align}
From triangle $BCD$ we have:
\begin{align} & \tan {{45}^{\circ }}=\dfrac{BC}{CD} \\ & \Rightarrow 1=\dfrac{BC}{CD} \\ & \Rightarrow BC=CD=\dfrac{300}{\sqrt{3}}=\dfrac{3\times 100}{\sqrt{3}}=\dfrac{{{\left( \sqrt{3} \right)}^{2}}\times 100}{\sqrt{3}}=100\sqrt{3} \\ \end{align}
The height of the lower plane from the ground is $100\sqrt{3}$ meters.
Hence, option (a) is correct.
Note: Since we have to find out the height here and in both the triangles the base is the same, try to use the height and base ratio. If we go for height and hypotenuse ratio it will become more lengthy.
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2021-05-09 16:28:28
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http://math.stackexchange.com/questions/401959/subrings-and-homomorphisms-of-unitary-rings/401961
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# Subrings and homomorphisms of unitary rings
Let $(R,+,\cdot)$ be a ring (in the definition i use multiplication is associative operation and it's not assumed that there is unity in the ring).
I've seen two definitons of subring.
1) non-empty subset $S \subset R$ is called subring of ring $(R,+,\cdot)$ iff $(S,+,\cdot)$ is a ring
2) Let $(R,+,\cdot)$ be unitary ring with unity $e$. Non-empty subset $S \subset R$ is called subring of ring $(R,+,\cdot)$ iff $(S,+,\cdot)$ is a ring and $e \in S$.
I'm looking for an example of such ring R and its subset S that $(S,+,\cdot)$ is a ring but not a unitary ring.
I will be also very grateful for an example of such UNITARY rings $(R_1,+_1,\cdot_1)$, $(R_2,+_2,\cdot_2)$ and function $f: ~~ R_1 \longrightarrow R_2$ that
$(1) \forall a,b \in R_1 ~~~ f(a+_1 b) = f(a) +_2 f(b)$
$(2) \forall a,b \in R_1 ~~~ f(a\cdot_1 b) = f(a) \cdot_2 f(b)$
$(3) f(e_1) \neq e_2$
where $e_1$ is unity in $R_1$; $e_2$ - in $R_2$.
Thanks in advance.
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This question pops up here almost every week or so ... what about using the search function. – Martin Brandenburg May 25 '13 at 12:54
## 3 Answers
1. $R = \mathbb Z$ and $S = 2\mathbb Z$
2. $R_1 = R_2 = \mathbb Z$ and $f(a) = 0$.
More generally let $R$ be a unitary ring. Any proper ideal $S$ provides an example for 1. For arbitrary unitary rings $R_1$ and $R_2\neq\{0\}$, the zero map is always an example for 2.
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1. Take any $S$ without $1$ and join to it extra unity $1$, so you get the unitary ring $R=\{a+n\cdot 1|a\in S, n\in \mathbb{Z}\}$ in which $S$ is a subring.
2. Let $M$ be a monoid with an idempotent $e\ne 1$, $\mathbb{Z}M$ its semigroup ring. Then the embedding $\mathbb{Z}e\to \mathbb{Z}M$ is what you want.
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Possibly the simplest example of the second kind is provided by the rings of matrices $$R_1 = \left\{ \begin{bmatrix}a&0\\0&0\end{bmatrix} : a \in \Bbb{Z} \right\}, \qquad R_2 = \left\{ \begin{bmatrix}a&0\\0&b\end{bmatrix} : a, b \in \Bbb{Z} \right\},$$ with $$f\left(\begin{bmatrix}a&0\\0&0\end{bmatrix}\right) = \begin{bmatrix}a&0\\0&0\end{bmatrix}.$$
The identity of $R_1$ is $$e_1 = \begin{bmatrix}1&0\\0&0\end{bmatrix},$$ that of $R_2$ is $$e_2 = \begin{bmatrix}1&0\\0&1\end{bmatrix},$$ and $f(e_1) = e_1 \ne e_2$.
The same example can also be seen via $f : \Bbb{Z} \to \Bbb{Z} \times \Bbb{Z}$ given by $f(a) = (a, 0)$, but using matrices makes it slightly more elementary.
Also, $R_1$ is a subring of $R_2$, both rings are unitary, but the identity elements are different.
-
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2016-05-27 08:28:17
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http://math.univ-lyon1.fr/~aubrun/recherche/
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Areas of interest
I'm interested in the following topics
• Convex geometry, especially high-dimensional. This field is sometimes called Asymptotic Geometric Analysis.
• Quantum information theory and its interplay with the former.
• Random matrices.
Alice and Bob meet Banach (book written with Stanislaw Szarek)
Publications and preprints
1. Two-point symmetrization and convexity (with Mathieu Fradelizi), Archiv der Mathematik 82 (2004), 282-288.
2. A sharp small deviation inequality for the largest eigenvalue of a random matrix, Séminaire de probabilités (2005), volume XXXVIII (LNM 1857).
3. Tensor product of convex sets and the volume of separable states on N qudits (with Stanislaw Szarek), Physical Review A 73 (2006).
4. Random points in the unit ball of \ell_p^n, Positivity 10, (2006) 755-759.
5. Sampling convex bodies: a random matrix approach, Proceedings AMS 135 (2007), 1293-1303.
6. Catalytic majorization and \ell_p norms (with Ion Nechita), Communications in Mathematical Physics 278 (2008), 133-144.
7. Stochastic ordering for iterated convolutions and catalytic majorization (with Ion Nechita), Annales de l'Institut Henri Poincaré (probabilité et statistiques) 45 (3), 611-625 (2009).
8. On almost randomizing channels with a short Kraus decomposition, Communications in Mathematical Physics 288, 1103-1116 (2009).
9. Maximal inequality for high-dimensional cubes, Confluentes Mathematici 1, 169-179 (2009).
10. Non-additivity of Rényi entropy and Dvoretzky's Theorem, (with Stanislaw Szarek and Elisabeth Werner), Journal of Mathematical Physics 51, 022102 (2010).
11. Hastings's additivity counterexample via Dvoretzky's theorem, (with Stanislaw Szarek and Elisabeth Werner), Communications in Mathematical Physics 305, 85-97 (2011)
12. Partial transposition of random states and non-centered semicircular distributions, Random Matrices: Theory and Applications 1, 1250001 (2012).
13. The multiplicative property characterizes l_p and L_p norms, (with Ion Nechita), Confluentes mathematici 3, 637 (2011).
14. Entanglement thresholds for random induced states, (with Stanislaw Szarek and Deping Ye), Communications in Pure and Applied Mathematics 67, 129-171 (2013?) ; see also the non-technical overview Phase transitions for random states and a semi-circle law for the partial transpose, Physical Review A (Rapid Communications) 85, 030302 (2012) and the proceeding from ICMP 2012: Is a random state entangled?
15. Realigning random states, (with Ion Nechita), Journal of Mathematical Physics 53, 102210 (2012).
16. Zonoids and sparsification of quantum measurements, (with Cécilia Lancien), to appear in Positivity (2015?).
17. Locally restricted measurements on a multipartite quantum system: data hiding is generic, (with Cécilia Lancien), Quantum Information and Computation 15, no. 5-6, 513--540. (2015).
18. Catalysis in the trace class and weak trace class ideals, (with Fedor Sukochev and Dmitriy Zanin), to appears in Proceedigs AMS (2015?).
19. Dvoretzky's theorem and the complexity of entanglement detection, (with Stanislaw Szarek), preprint.
Here is a version(dvi ps pdf) of my PhD thesis.
And there is also a curriculum vitae (dvi).
To reach me:
aubrun (arrobas) math. univ-lyon1. fr
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2015-10-10 16:01:34
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http://www.birs.ca/events/2008/summer-schools/08ss045/videos/watch/200808141330-Labesse.html
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## Video From 08ss045: The stable trace formula, automorphic forms, and Galois representations
Thursday, August 14, 2008 13:30 - 14:33
The stable trace formula part II
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2017-01-17 17:11:38
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https://stats.stackexchange.com/questions/487497/random-forest-with-train-auc-1-and-test-auc-58
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# Random Forest with train AUC = 1 and test AUC = 58%
I'm trying to understand why my train AUC = 1 while my test AUC is near 58% using random forest.
• Context: You are trying to sell a product, and you have historic data about the purchases/noPurchases of such product, from March 2020 to August 2020. You leave August for test data, and the rest for training data. It results approximately in 80% train and 20% test. Each purchase/noPurchase is related to one client, and the data available for that client is the month before the purchase. For example: If a client purchase/noPurchase on July, the data available for that client in that month is June.
• Data (train+test): 75 columns and 10k rows, target variable is binary with 90%-10% imbalance. All data is numeric.
• Modeling: The scoring is ROC_AUC, and all predictions should be probabilities (to plot roc curve)
Code: train = X_train + y_train; test = X_test + y_test
# STANDARDIZE AND IMPUTE TRAIN AND TEST SEPARATED
scaler = StandardScaler()
imputer = KNNImputer()
X_train_scaled = scaler.fit_transform(X_train)
X_train_scaled = imputer.fit_transform(X_train_scaled)
X_test_scaled = scaler.fit_transform(X_test)
X_test_scaled = imputer.fit_transform(X_test_scaled)
# MODEL
model = RandomForestClassifier() # we instantiate the model
model.fit(X_train_scaled, y_train) # fit
y_train_predictions = model.predict_proba(X_train_scaled) # predict
# EXTRACT TRAIN CLASSES TO PLOT
for i,k in enumerate(model.classes_ == 1):
print(i,k)
if k == True:
y_train_predictions = y_train_predictions[:,i]
y_test_predictions = model.predict_proba(X_test_scaled) # predict
# EXTRACT TEST CLASSES TO PLOT
for i,k in enumerate(model.classes_ == 1):
print(i,k)
if k == True:
y_test_predictions = y_test_predictions[:,i]
# PLOT
fpr, tpr, _ = roc_curve(y_train, y_train_predictions)
plt.plot(fpr, tpr)
fpr, tpr, _ = roc_curve(y_test, y_test_predictions)
plt.plot(fpr, tpr)
And the problem is that train AUC = 1 and test AUC = 58% approx. I thought on these possible causes:
• The dependent variable is included in the X_train matrix. (Checked, it isn't)
• There is some explanatory variable (e.g. credit card charges of the product, if I have credit card charges data) that is included in X_train
However, I discard both possible causes above because I checked meticulously the first one, and the second one is not possible given that the maximum feature importance is 0.0429
But still, is there any other possible explanation for train AUC = 1 and test AUC = 58%, or something I'm missing?
PS: Here is a very similar question, but all the answers are general given that the OP doesn't specify how is building the model (that's why I put all the code)
• You are saying using two separate scalers and imputers is misleading, but you also say that test data should be scaled/imputed using a separated scaler ("test data should be scaled/imputed using test data scaler"), which is what I just did. On another point, why there is data leakage if you use separated scalers/imputers? – Chris Sep 15 '20 at 12:36
• Sorry, typo! My bad. Using two separate scalers (and imputers) one for the training and one of the test data is misleading. Test data should be scaled/imputed using the train data scaler otherwise there is leakage. Regularise more; for example, increase the minimum number of instances per leaf and/or reduced the maximum tree depth. Leakage will exist because we do not know beforehand anything about the mean or variance of our test data. Think the scenario we have a single test point, what would be the variance then? – usεr11852 Sep 15 '20 at 12:48
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2021-02-26 01:46:56
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http://umgspotlight.com/kindle/an-introduction-to-metric-spaces-and-fixed-point-theory
|
# Download PDF by Mohamed A. Khamsi: An Introduction to Metric Spaces and Fixed Point Theory
By Mohamed A. Khamsi
ISBN-10: 0471418250
ISBN-13: 9780471418252
Provides updated Banach area results.
* positive aspects an in depth bibliography for outdoor reading.
* offers distinct workouts that elucidate extra introductory fabric.
Similar linear books
The assumption of optimization runs via so much components of regulate conception. the easiest optimum controls are preplanned (programmed) ones. the matter of making optimum preplanned controls has been greatly labored out in literature (see, e. g. , the Pontrjagin greatest precept giving useful stipulations of preplanned keep watch over optimality).
Download e-book for kindle: Theory of dimensions, finite and infinite by Ryszard Engelking
Magnet hyperlink : magnet:? xt=urn:btih:a4076e60212dc4c9bc12b13c40941552da202bd6&dn=Theory%5Fof%5FDimensions%5FFinite%5Fand%5FInfinite-Ryszard%5FEngelking-Vol. 10%5FHeldermann%5FVerlag-1995. djvu&tr=udp%3A%2F%2Ftracker. openbittorrent. com%3A80%2Fannounce&tr=udp%3A%2F%2Ftracker. publicbt. com%3A80&tr=udp%3A%2F%2Ftracker.
Extra info for An Introduction to Metric Spaces and Fixed Point Theory
Example text
TTien there exists x € M suc/i ί/ιαί g(x) = x. (**) 58 CHAPTER 3. METRIC CONTRACTION PRINCIPLES Proof. Introduce the partial order > in M as follows. For x,y € M say that y > χ <=> max{d(x, y),cd(f(x), f(y))} < φ(/(χ)) - ), and for α,β € / set β > a <=> Χβ > xa. ι is a nonincreasing net in R + so there exists r > 0 such that = r. \imip(f(xa)) a Let ε > 0. Then there exists ao 6 / such that a > ao implies r < tp(f(xa)) a > ao, m a x i d ^ ^ ^ ) ^ ^ / ^ ) , / ^ ) ) } <
METRIC CONTRACTION PRINCIPLES Proof. Introduce the partial order > in M as follows. For x,y € M say that y > χ <=> max{d(x, y),cd(f(x), f(y))} < φ(/(χ)) - ), and for α,β € / set β > a <=> Χβ > xa. ι is a nonincreasing net in R + so there exists r > 0 such that = r. \imip(f(xa)) a Let ε > 0. Then there exists ao 6 / such that a > ao implies r < tp(f(xa)) a > ao, m a x i d ^ ^ ^ ) ^ ^ / ^ ) , / ^ ) ) } <
Uniqueness of z follows from the contractive condition on T. , rj J. r > 0 => ^ ( r j ) ~* VK1"))· This extension of Banach's Principle is due to Browder [25]. 48 CHAPTER 3. 2) d(T(x),T(y))(d(x,y)). Then T has a unique fixed point z, and {Tn(x)} converges to z, for each x 6 M. Proof. This theorem is actually a special case of the previous theorem. First introduce the function φ : R —» [0,1) by setting ^>(0) = φ(0) and φ(ί) = Ά for t > 0. To see that φ is in the class S suppose φ{ίη) —» 1. Then {i„} must be bounded (otherwise, lim inf φ(ίη) = 0 ) .
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2018-11-19 20:18:14
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https://socratic.org/questions/56cd3a5611ef6b7a8a723db6
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# Question 23db6
Jun 9, 2016
If this is a kinetics problem, you need to know the order of the reaction, the integrated rate law, and the rate constant at 420 °C
Since you don’t state the specific problem, let's arbitrarily assume that the reaction is first order, the initial concentration of ${\text{SO"_2"Cl}}_{2}$ is 0.0225 mol/L and that the rate constant is 2.90 × 10^"-4"color(white)(l) "s"^"-1" at 420 °C.
Whenever a question asks, "How much is left after an amount of time?", that is a clue for you to use an integrated rate law.
The integrated rate law for a first order reaction is
$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \ln \left({A}_{0} / {A}_{t}\right) = k t \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$
where
$\text{A"_0 = "concentration at time 0}$
$\text{A"_t = "concentration at time} \textcolor{w h i t e}{l} t$
$k = \text{rate constant}$
$t = \text{time}$
t = 16.2 color(red)(cancel(color(black)("h"))) × (60 color(red)(cancel(color(black)("min"))))/(1 color(red)(cancel(color(black)("h")))) × "60 s"/(1 color(red)(cancel(color(black)("min")))) = "58 320 s"
Then,
ln("0.0225 mol/L"/"A"_t) = 2.90 × 10^"-4" color(red)(cancel(color(black)("s"^"-1"))) × "58 320" color(red)(cancel(color(black)("s"))) = 16.91
"0.0225 mol/L"/"A"_t= e^16.91 = 2.21× 10^7#
$\text{A"_t = "0.0225 mol/L"/(2.21× 10^7) = 1.02 × 10^"-9" color(white)(l)"mol/L}$
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2019-08-18 06:49:57
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https://gateoverflow.in/313742/asymptotic_notations-self_doubt
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186 views
Please give an example case for which all the three conditions
$f(n)\neq O(g(n))$,
$f(n)\neq \Theta (g(n))$ and
$f(n)\neq \Omega (g(n))$
holds true.
retagged | 186 views
+2
Please do not use '#' in GO -- it'll lead to duplicates as we do not use it anywhere. Meanwhile anyone who answers this correctly is strong in asymptotic notations.
+1
When there will be no equal , means either strictly grater than or strictly lesser than, in that cases all these three will be false.
Say $o\left ( g\left ( n \right ) \right )=${$f(n)$: there exists constant $c$ and $n_{0}$ such that $0< f(n)< cg\left ( n \right )$}
or
$\omega \left ( g\left ( n \right ) \right )=${$f(n)$: there exists constant $c$ and $n_{0}$ such that $0< cg\left ( n \right )<f(n)$}
0
Can you give an example @srestha
0
@Satbir
Suppose $2n+10=O(n)$ but $\\=o(n^{2})\\ \text{or}\\ o(n^{3})$
0
@Satbir You meant "none of them holds" rt? It is better you can rewrite the sentence using $\neq$ instead of $=$ and none. And I suppose you know the answer here.
0
@Arjun Sir ,updated the question. Yes you are correct.
@srestha
so you are saying that f(n) = 2n+10 and g(n) = $n^2$ right ?
+2
@Satbir
f(n) = sin n
g(n) = cos n
0
@Hirak
can you explain in detail.
+1
See, first of all the condition that u have mentioned is not possible in case of any polynomial..so my thought process shifted to curve, and the first thing that came to my mind is sin and cosine curve..
Lets take some examples--> say we have sin 30 which is 0.5, now for the the same value of n, we have cos 30 = 0.87
so at this instance sin n = O( cos n)
but lets take another case.. sin 45 = cos 45 , so at this point sin n = theta (cos n)
now for sin 60 we have 0.86 but cos 60 =0.5
so here cos n= O(sin n)
and it will be more clear if u look at the sin and cosine graph,
so here we can say that-->
f(n)≠O(g(n)),
f(n)≠Θ(g(n)) and
f(n)≠Ω(g(n))
all holds true here as none of the condition is static.
0
@Hirak What about the remaining conditions for asymptotic notations? They are not merely <, > and =. What about the constants used in the definitions?
+1
@Arjun sir
Θ(g(n)) = {f(n): there exist positive constants c1, c2 and n0 such
that 0 <= c1*g(n) <= f(n) <= c2*g(n) for all n >= n0}
going by the definition of theta for my function where i have taken f(n) = sin n and g(n)= cos n the condition of theta will never hold as for a certain value of n except n=(X*pi+45) [X= any whole number], sin and cos curve deviates from each other except these points .
O(g(n)) = { f(n): there exist positive constants c and
n0 such that 0 <= f(n) <= c*g(n) for
all n >= n0}
This definition also wont hold true even if we assign constant or even if we take whatever $n_0$ we want, same reason here also as sin n becomes negative for certain values and positive for certain in which cos tends to do the reverse..
Similar case for Omega..
That is why f(x) = sin x and g(x) =cos x works fine for this example. Even if we assign f(x)= cos x and g(x)= sin x it will not matter.
0
Nice. So, $\sin$ and $\cos$ becoming $0$ and negative are important here.
Is there some general behaviour for such functions which satisfy the asked conditions?
0
@Arjun sir
general behaviour means?
0
For certain type of functions the given condition always holds -- for which type of functions?
0
I think only oscillating curves intersecting at multiple points will exhibit such feature, other than sin and cos i am unable to think of other doing the same.
0
Is oscillation mandatory? Is intersection mandatory?
0
@Arjun Sir
I thought of another possibilty, but i doubt its practicality
say we have function of O(n), say f(n)= O(n)
Now say we have another function g(n) = 1+2+3 +4+ 5+..upto n terms
so g(n) is O($n^2$).
so, f(n) =O(g(n)
But what about if n is infinity?
according to Ramanujan 1+2+3 +4+ 5+...... = -1/12
so g(n) = -1/12 , but f(n)= positive infinity
so this time , g(n) = O(f(n))
and it can be easily stated that they f(n) is not theta of g(n).
So this case also does satisfies-->
f(n)≠O(g(n)),
f(n)≠Θ(g(n)) and
f(n)≠Ω(g(n))
Doubt its practicality, but i think mathematicaly it is sound..
0
summation strictly increasing func., has asymptotically upper bound
isnot it??
0
0
Actually , that is $O\left ( 1 \right )$, right??
oscillation is more appropriate
0
as it is O(1) hence g(n)=O(f(n)) here..
Mathematically it is correct..
0
how??
'=' is not a violation of condition
and isnot it satisfying $\Omega (n)$
0
Moreover, to find tight bound, when n is sufficiently large, f(n) should be non negative
0
g(n) is showing upperbound as well as lowerbound for same function but not theta bound..
That is why it is satisfying the condition of the question
N must be non negative and here it is so..
0
What is upper bound, lower bound and tight bound for an A.P.??
0
This question is nothing but, it is satisfying irreflexive property of the relation
0
0
I think what he is trying to say is that time taken to calculate them will be O(1) for both f(n) and g(n), and i cant deny with him as well.
But strictly going by definition and (even from the graphs) of asymptotic notation we can see that at certain point sin x upperbounds cosx and vice versa.
+1
@Hirak
0
How does sin and cos functions satisfy those 3 conditions? If I take a constant then one function will be completely above the other function so it doesn't satisfy the conditions.
n^(1+sin n) and n^(1+cos n) these two functions are incomparable so satisfy all three properties given . Correct me if wrong
by (93 points)
+1 vote
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2019-12-05 21:09:28
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http://vlasisku.lojban.org/ci'ai'u
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ci'ai'u VUhU experimental cmavo
unary mex operator: n-set; maps a nonnegative integer 'a' to the set \1, \dots ,a\ (the intersection of the set of all natural numbers with the closed ordered interval [1,a] such that a geq 1).
0 maps to the empty set. Inputting infinity produces the set of all natural numbers, N.
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2018-07-23 15:53:15
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https://mailman.ntg.nl/pipermail/ntg-context/2016/085945.html
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# [NTG-context] Strange result with formulaframed
Hans Hagen pragma at wxs.nl
Wed Jun 22 13:36:21 CEST 2016
On 6/22/2016 12:05 PM, Otared Kavian wrote:
> Dear Hans,
>
> I just did some experiments with the new \setupformulaframed, which you added recently after a demand by Aditya.
> However it seems to me that it does not do exactly what one might expect, at least in the example below.
>
> Would it be possible to have a mechanism which gives somehow what Aditya was asking?
well, afaik he was asking for framed formulas but it will probably take
a bit of time to test it with all kind of interferences
> % begin framed-frormula.tex
> \setupformulaframed[framecolor=red,frame=on,offset=1ex]
> \setupformula[option=frame,color=blue]
>
> \starttext
>
> \startplaceformula
> \startformula
> f(x) := \left\{
> \startalign
> \NC = -1\NC\quad \mbox{if }\, x < 0 \NR
> \NC = +1\NC\quad \mbox{if }\, x > 0 \NR
> \stopalign\right.
> \stopformula
> \stopplaceformula
>
> \stoptext
> % end framed-frormula.tex
it's not related to framing ... you can put this in cont-new.mkiv after
the \unprotect command there
\def\math_halign_checked
{\halign
\ifnum\c_strc_formulas_mode=\plustwo
% currently there is no need for width juggling
\else
\ifcase\eqalignmode \or to \checkeddisplaywidth \fi
\fi}
\def\math_both_eqalign_no_normal#1#2%
{\ifmmode
\the\mathdisplayaligntweaks
\vcenter\bgroup
\let\math_finish_eqalign_no\egroup
\else
\let\math_finish_eqalign_no\relax
\fi
#1%
\math_halign_checked\expandafter\bgroup\the\scratchtoks\crcr#2\crcr\egroup
\math_finish_eqalign_no}
\def\math_both_eqalign_no_aligned#1%
{\ifmmode
\the\mathdisplayaligntweaks
\global\mathnumberstatus\plusone
\ifcase\mathraggedstatus
\def\math_finish_eqalign_no{\crcr\egroup}%
\else
% we're in a mathbox
\vcenter\bgroup
\def\math_finish_eqalign_no{\crcr\egroup\egroup}%
\fi
\fi
#1%
\math_halign_checked\expandafter\bgroup\the\scratchtoks\crcr}
-----------------------------------------------------------------
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2021-11-30 06:17:11
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https://proofwiki.org/wiki/Non-Square_Positive_Integers_not_Sum_of_Square_and_Prime/Examples/10
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# Non-Square Positive Integers not Sum of Square and Prime/Examples/10
$10$ cannot be expressed as the sum of a square and a prime.
Testing each $m \in \Z_{>0}$ such that $m^2 < 10$ it is established that there is no solution to $10 - m^2 = p$ where $p$ is prime:
$\displaystyle 10 - 1^2$ $=$ $\displaystyle 9$ which is composite: $9 = 3^2$ $\displaystyle 10 - 2^2$ $=$ $\displaystyle 6$ which is composite: $6 = 2 \times 3$ $\displaystyle 10 - 3^2$ $=$ $\displaystyle 1$ $1$ is not Prime
$\blacksquare$
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2020-02-20 17:56:42
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http://construccioneslavid.com/cksa-soccer-tgik/herald-standard-apartments-for-rent-021736
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1000 - 100. Query group information is required for ranking tasks by either using the group parameter or qid parameter in fit method. You can sort data according to their scores in their own group. from xgboost import xgbClassifier model = xgbClassifier() model.fit(train) Thanks. If the weight in some query group is large, then XGBoost will try to make the ranking correct for this group first. 1600 Girls - 200. Easily Portable. Before running XGBoost, we must set three types of parameters: general parameters, booster parameters and task parameters. By clicking “Sign up for GitHub”, you agree to our terms of service and Does it mean that the optimization will be performed only on a per query basis, all other features specified will be considered as document features and cross-query learning won't happen? My whipped cream can has run out of nitrous. (Think of this as an Elo ranking where only winning matters.) This information might be not exhaustive (not all possible pairs of objects are labeled in such a way). Here’s a link to XGBoost 's open source repository on GitHub Successfully merging a pull request may close this issue. A total of 7302 radiomic features and 17 radiological features were extracted by a … @xd-kevin. From our literature review we saw that other teams achieved their best performance using this library, and our data exploration suggested that tree models would work well to handle the non-linear sales patterns and also be able to group … Follow asked Mar 9 '17 at 5:13. jimmy15923 jimmy15923. We could stop … Or just use different groups. r python xgboost. In XGBoost documentation it's said that for ranking applications we can specify query group ID's qid in the training dataset as in the following snippet: I have a couple of questions regarding qid's (standard LTR setup set of search queries and documents, they are represented by query, document and query-document features): 1) Let's say we have qid's in our training file. XGBoost lets you use a wide range of applications for solving user-defined prediction, ranking, classification, and regression problems. groupId - ID to identify a group within a match. Why do wet plates stick together with a relatively high force? winPoints - Win-based external ranking of player. It runs smoothly on OSX, Linux, and Windows. We’ll occasionally send you account related emails. A rank profile can inherit another rank profile. We are using XGBoost in the enterprise to automate repetitive human tasks. Surprisingly, RandomForest didn’t work as well , might be because I didn’t tune that well. with labels or group_info? 勾配ブースティングのとある実装ライブラリ（C++で書かれた）。イメージ的にはランダムフォレストを賢くした（誤答への学習を重視する）アルゴリズム。RとPythonでライブラリがあるが、ここではRライブラリとしてのXGBoostについて説明する。 XGBoostのアルゴリズム自体の詳細な説明はこれらを参照。 1. https://zaburo-ch.github.io/post/xgboost/ 2. https://tjo.hatenablog.com/entry/2015/05/15/190000 3. If there is a value other than -1 in rankPoints, then any 0 in winPoints should be treated as a “None”. And there is a early issue here may answer this: Sign up for a free GitHub account to open an issue and contact its maintainers and the community. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Or just use different groups. VIRGINIA BEACH, Va. (AP) — Virginia Marine Police and a group of volunteers are continuing to search for the driver whose truck plunged over the side of … with labels or group_info? redspark-xgboost 0.72.3 Jul 9, 2018 XGBoost Python Package. Thank very much~. XGBoost had the highest AUC value, followed by Random Forest, KNN, Neural Network, SVM, and Naïve Bayes. 23 1 1 silver badge 3 3 bronze badges $\endgroup$ add a comment | 1 Answer Active Oldest Votes. Girls Long Jump - 90. If you have models that are trained in XGBoost, Vespa can import the models and use them directly. (In Python). Use MathJax to format equations. This procedure firstly filters a set of relative important features based on XGBoost, and then permutes to find an optimal subset from the filtered features using Recursive Feature Elimination (RFE), as illustrated in Algorithm 2. A two-step hybrid method is developed to rank and select key features by machine learning. The AUC of XGBoost using the Group 2 predictors was up to 92%, which was the highest among all models . Within each group, we can use machine learning to determine the ranking. Already on GitHub? How likely it is that a nobleman of the eighteenth century would give written instructions to his maids? which one make's more sence?Maybe it's not clear. (Think of this as an Elo ranking where only winning matters.) I also have a set of features that are likely to work pretty well for more traditional models, so I went with XGBoost for an initial iteration simply because it is fairly easy to interpret the results and extremely easy to score for new languages with multi-class models. 2 predictors was much higher than that of the group size get the group 1 predictors nobleman the! We still have qid 's specified in the Python Build Tools category of a tech stack Network,,. The size of each query group take place, but how to do a stratified nfold take. As well, might be not exhaustive ( not all possible pairs of objects are labeled in such way. On OSX, Linux, and Windows still have qid 's specified in the Build... Depth, Minimum Child Weight, Gamma ) laurae: this post, we to. Answer this: # 270 or we should just list query, document and query-document features with fun! Message, Maybe it 's not clear ) Thanks has to happen within each,! To Choose parameters, which helps me to Build new models quicker parameters depend which! The scores: general parameters relate to which booster we are using XGBoost for. ) is a tool in the life of a high-pass filter not 0 when input! K-Folds instead run out of nitrous we ’ ll occasionally send you related. Own group method of doing so was updated successfully, but how to do stratified. A huge amount of data that well bags for both XGBoost and GBM did... And minimize the ranking among instances within a match stratified K-Folds instead xgboost ranking group created. Booster you have chosen do wet plates stick together with a relatively HIGH force eighteenth! Labour Party push for proportional representation as well, might be because I didn ’ t tune well! And use them directly total of 7302 radiomic features and 17 radiological were! ; user contributions licensed under cc by-sa, I don ’ t have a huge of... - more TBD the first obvious choice is to use the following configuration settings: Choose winPoints... Work as well, might be not exhaustive ( not all possible pairs of objects labeled... Least destructive method of doing so with xgb.cv'nfold fun hepatocellular carcinoma ( HCC ) patients can. ) machine learning correct for this group first models for ranking.. Exporting models from.! Elo ranking where only winning matters., a stratified nfold a free GitHub account to open an issue contact! Is 0 pairs of objects are labeled in such a way ) we... Have that, then any 0 in winPoints should be parallelized as much possible. Train, some group for train, some group for train, some group for train, some for. We can use machine learning Community ( DMLC ) group not get the group 2 predictors much! We discuss leveraging the large number of cores available on the GPU to massively parallelize computations... And 17 radiological features were extracted by a … model Building although Neural! 4, 2020 xgboost-ray 0.0.2 Jan 12, 2021 a Ray backend for Distributed XGBoost GitHub account to an... Port be reused concurrently for multiple destinations a valuable predictor of survival in hepatocellular carcinoma ( HCC ) patients should! Let me know which site is a value other than -1 in rankPoints, then any 0 winPoints... To ranking, you need to have qid 's and during inference we do need! Query, document and query-document features century would give written instructions to his maids clarification, or responding to answers. Errors were encountered: may the cv function can not get the 1. Information might be not exhaustive ( not all possible pairs of objects are labeled in such way. Of survival in hepatocellular carcinoma ( HCC ) patients any 0 in winPoints should be treated as a “ ”! Active Oldest Votes during inference we do n't need them as input the life a. 'S and during inference we do n't need them as input stratified nfold should take place, but how do... You account related emails a way ) started with XGBoost, use the library. Helps me to Build new models quicker we need to have qid 's and during inference we do n't them... Why do wet plates stick together with a relatively HIGH force you can sort data according to their in... Extension for easy ranking & TreeFeature a valuable predictor of survival in carcinoma! Can Shor ‘ s code correct two- or three-qubit errors multiple destinations why does n't the UK Labour Party for.: this post is about tuning the regularization in the Python XGBoost interface of nitrous ranking. Reused concurrently for multiple destinations group only - Win-based external ranking of player is an open tool! Personal experience, you can iteratively sample these pairs and minimize the ranking correct for this group first Answer,! Ranking correct for this group first was up to 92 %, helps... Xgboost ’ s JSON model dump ( E.g of this as an Elo ranking where winning! Error between any pair - more TBD the first obvious choice is to use the following configuration:... ' spherically symmetric TCP port be reused concurrently for multiple destinations of using XGBoost in particular directly! New models quicker Python Build Tools category of a high-pass filter not 0 when the input is 0 's in. Inference we do n't need them as input use machine learning Community ( DMLC ) group predictive models using gradient. The training file or we should just list query, document and query-document features with gradient boosted trees and.... Can use machine learning one group are comparable © 2021 stack Exchange Inc ; user contributions under... Using XGBoost models for ranking.. Exporting models from XGBoost comment | 1 Answer Active Oldest Votes Shor... If there is a value other than -1 in rankPoints, then any 0 winPoints! Will try to directly use sklearn 's stratified K-Folds instead the ranking so training. Training we need to be sorted by query group stack Exchange Inc ; user contributions licensed under by-sa... Determine the ranking error between any pair the text was updated successfully but. The life of a universe each group, I don ’ t have huge... Repetitive human tasks each group easy ranking & TreeFeature port be reused concurrently for multiple?. Relatively HIGH force stages in the Python Build Tools category of a high-pass filter not 0 the! Ltr Algorithms from XGBoost import xgbClassifier model = xgbClassifier ( ) model.fit ( train ) Thanks ( DMLC group! A “ None ” ranking.. Exporting models from XGBoost objects are labeled in such a way.! Of parameters: general parameters, which helps me to Build new models quicker # 270 using... Not clear inherit another rank profile can inherit another rank profile ensemble of the group 2 was. T tune that well an issue and contact its maintainers and the Community can use machine to! Among all models Inc ; user contributions licensed under cc by-sa gradient boosted trees and XGBoost than of... Automate repetitive human tasks initially maintained by the Distributed ( Deep ) machine learning to rank for examples using. Auc value, followed by Random Forest, KNN, Neural Network approach may work better in theory, created. Trained in XGBoost, I work with gradient boosted trees and XGBoost your RSS reader does the! Model = xgbClassifier ( ) method in the Python Build Tools category of a high-pass filter not 0 the... Work as well, might be not exhaustive ( not all possible pairs xgboost ranking group... Gbm and did a final rank average ensemble of the group 2 predictors was higher. Port be reused concurrently for multiple destinations importing XGBoost ’ s JSON model dump ( E.g choice. Model Building ) machine learning to rank and select key features by machine learning Community ( )! Or linear model with a relatively HIGH force training file or we should just list query, document query-document... Each group to predict MVI preoperatively 5:13. jimmy15923 jimmy15923 for our final model your... A pull request may close this issue objects are labeled in such a way.... Final model, your data need to have qid 's specified in the Python Build Tools of. By Tianqi Chen and initially maintained by the Distributed ( Deep ) machine learning Community ( )! Add a comment | 1 Answer Active Oldest Votes ( HCC ) patients choice is to use XGBoost! A universe XGBoost in the life of a universe their scores in one group are comparable gradient boosting XGBoost! Can sort data according to their scores in one group are comparable Vespa supports importing XGBoost s... A way ) GBM and did a final rank average ensemble of the eighteenth century would written! Ll occasionally send you account related emails a “ None ”, SVM and. Often quoted as 'especially ' spherically symmetric RSS reader query group? Maybe it not... Xgboost will try to directly use sklearn 's stratified K-Folds instead 's in! Can inherit another rank profile can inherit another rank profile method of doing so pattern to parameters... Other answers features and 17 radiological features were extracted by a … Building. ( DMLC ) group feed, copy and paste this URL into your RSS reader XGBoost will try to use! In their own group using the group 2 predictors was up to 92,. Were extracted by a … model Building ( Maximum Depth, Minimum Child Weight, ). Features by machine learning Community ( DMLC ) group model Building them as input array that contains the of... That queries are represented by query group is large, then any 0 in winPoints should treated... Likely it is that a nobleman of the eighteenth century would give instructions. We discuss leveraging the large number of cores available on the GPU to massively parallelize these computations ranking error any... An open source tool with 20.4K GitHub stars and 7.9K GitHub forks XGBoost!
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2021-10-20 10:36:44
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https://math.stackexchange.com/questions/624102/how-to-find-the-range-of-the-function-fx-sqrtx-12-sqrt3-x
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# How to find the range of the function: $f(x) = \sqrt{x-1}+2\sqrt{3-x}$
Problem :
Find the range of the function: $f(x) = \sqrt{x-1}+2\sqrt{3-x}$
Solution :
Domain of this function can be determined as :
$x - 1 >0 ; 3-x >0 \Rightarrow x >0 ; x <3 ;$
$\therefore$ domain of $x \in [1,3]$
Now if I put the values of this domain in my function then it gives the following values :
at 1 ; the value of the function is $2\sqrt{2}$
at 2 : the value of the function is $1+2 = 3$
at 3 : the value of the function is $2$
Can we say that the maximum value of the function is 3 and minimum value of the function is 2;
Therefore the range of this function is [2,3] but this answer is wrong. please suggest..
Also suggest that how can we use differentiation method to find the range... thanks.
We get $$f^\prime (x)=\frac{1}{2\sqrt{x-1}}-\frac{2}{2\sqrt{3-x}}=\frac{\sqrt{3-x}-2\sqrt{x-1}}{2\sqrt{(x-1)(3-x)}}.$$
So, $$f^\prime(x)\ge0\iff \sqrt{3-x}\ge2\sqrt{x-1}\iff 3-x\ge4(x-1)\iff x\le\frac{7}{5}.$$ Now we know that $f(x)$ is increasing in $1\le x\lt 7/5$ and that $f(x)$ is decreasing in $7/5\lt x\le 3$.
So, we know that the max is $f(7/5)$, and that the min is $\min(f(1),f(3)).$
This function is only defined on $[1,3]$. We can differentiate it, and search for points where the derivative is $0$. Since it is a continuous (on $[1,3]$) and differentiable (on $(1,3)$) function, extreme values must happen either at $1$, $3$ or at such points with derivative $0$.
Since the function is continuous, the image of the closed interval $[1, 3]$, i.e. $f([1, 3])$ is also a closed interval.
Since the function is also differentiable, we can find the maximum and minimum values (which the function attains) by comparing all the critical points (set $f'(x) = 0$) and the end points ($f(1)$ and $f(3)$).
I try to give another solution without using derivative. Here, we only use the intermediate value theorem (https://en.wikipedia.org/wiki/Intermediate_value_theorem) for continuous functions and Cauchy Schwarz inequality (https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality).
Clearly, the domain of the function is [1;3]. For every $$x\in[1,3]$$, we have \begin{align} f(x)-f(3)&=\sqrt{x-1}-\sqrt{2}+2\sqrt{3-x}\\ &=\frac{x-3}{\sqrt{x-1}+\sqrt{2}}+2\sqrt{3-x}\\ &=\frac{\sqrt{3-x}(2\sqrt{x-1}+2\sqrt{2}-\sqrt{3-x})}{\sqrt{x-1}+\sqrt{2}}\\ &\geq 0. \end{align} By the Cauchy-Schwarz inequality we have \begin{align} f(x)-f(7/5)&\leq \sqrt{(1+4)(x-1+3-x)}-(\sqrt{2}/\sqrt{5}+2\sqrt{8}/\sqrt{5})=0 \end{align} Hence, $$f([1,3])\subset [f(3),f(5/7)]$$. On the other hand, by the intermediate value theorem, we have $$f([1,3])\supset f([7/5,3])\supset [f(3),f(7/5)].$$ The range of $$f$$ is $$[f(3),f(5/7)]$$.
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2021-09-28 00:41:58
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http://umj.imath.kiev.ua/authors/name/?lang=en&author_id=3202
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2019
Том 71
№ 6
### Finiteness Properties of Minimax and $\mathfrak{a}$-Minimax Generalized Local Cohomology Modules
Let $R$ be a commutative Noetherian ring with nonzero identity, let $\mathfrak{a}$ be an ideal of $R$, and let $M$ and $N$ be two (finitely generated) $R$-modules. We prove that $H_{\mathfrak{a}}^i\left( {M,N} \right)$ is a minimax $\mathfrak{a}$-cofinite $R$-module for all $i < t, t ∈ {{\mathbb{N}}_0}$, if and only if $H_{\mathfrak{a}}^i\left( {M,N} \right)$ is a minimax ${R_{\mathfrak{p}}}$ -module for all $i < t$. We also show that, under certain conditions, $\mathrm{Ho}{{\mathrm{m}}_R}\left( {\frac{R}{\mathfrak{a}},H_{\mathfrak{a}}^t\left( {M,N} \right)} \right)$ is minimax $(t ∈ {{\mathbb{N}}_0})$. Finally, we study necessary conditions for $H_{\mathfrak{a}}^i\left( {M,N} \right)$ to be $\mathfrak{a}$-minimax.
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2019-06-26 13:48:01
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https://www.physicsforums.com/threads/acceleration-on-incline.135250/
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# Homework Help: Acceleration on incline
1. Oct 7, 2006
### Cole07
Problem:
Soccer ball released from the top of a smooth incline after 4.58 seconds the ball travels 10 meters 1 second later it has reached the bottom of the incline, the balls acceleration is constant and determine its value
(m/s^2) also need to know the length of the incline
I am posting for the first time and I hope this is ok
Tried to answer by taking 10 meters and dividing by 4.58 seconds but I was told this was wrong with no explanation
To find the incline I would use d= 1/2 a(t)^2
I am just a beginner in Physics and I am definetly confused any help would be greatly appreciated to get me on the right path to solving this problem I have tried other expressions to no avail thanks in advance
2. Oct 7, 2006
The equation of displacement along the incline is, as you stated d = 1/2*a*t^2. Now, you know what distance the ball traveled during the period of 4.58 seconds. Plug these values into the equation, and solve to retrieve the acceleration a. Now, which acceleration is causing a ball roll down an incline? Further on, which component of this acceleration is directed along the incline? You can find the angle of the incline from a simple equation based on these facts.
3. Oct 7, 2006
### arildno
First of all SPLIT up the information neatly, and give names to the quantities you think is relevant!
So, names:
a-acceleration. We know of this it is a constant
d-length of incline
These are the quantities you need to find!!
Info:
1. After time $t_{1}=4.58$ seconds, the ball has travelled 10 meters
2. After time $t_{2}=5.58$ seconds, the ball has travelled d meters (reached the end of the incline)
So, what equations to use??
"To find the incline I would use d= 1/2 a(t)^2"
This is perfectly okay!
But remember that that equation has TWO unknown quantities, namely d and a! (the time is known to be 5.58)
But do you agree that you equally well could use the very same equation with 10 meters substituted for d, that is the distance travelled in 4.58 seconds?
4. Oct 7, 2006
### Cole07
ok some of this makes since but i still don't understand if i'm doing the right thing to get the acceleration.
5. Oct 7, 2006
### arildno
Well, let uss use the distance&acceleration of yours when the distance is 10, and time is 4.58!
Plug this in, and you get the equation:
$$10=\frac{1}{2}*a*(4.58)^{2}$$
Do you agree with that?
6. Oct 7, 2006
### Cole07
ok that works great so how do i find how long the incline is now
7. Oct 7, 2006
### arildno
Well, now that you know acceleration "a", how many unknows do you have in your distance formula when the distance is the as yet unkown length of incline?
8. Oct 7, 2006
### Cole07
the distance formula is d=1/2a(t)^2 right
9. Oct 7, 2006
### arildno
Right, so how many unknowns do you have to tackle now?
10. Oct 7, 2006
### Cole07
ok i plugged this in to the equation d=1/2*(0.953452451)(4.58)^2 and i get 9.999999997 but the comes up incorrect and it can't be anyway since you know you have already gone 10m isn't this right?
11. Oct 7, 2006
### arildno
But is the time 4.58 when the ball has reached the end of the incline?
Think again!
12. Oct 7, 2006
### Cole07
ok i got it thank you so much you have been a huge help !!!
13. Oct 7, 2006
### arildno
As you can see, you had most of it inside your head already before I answered you.
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2018-09-23 18:18:46
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https://leanprover-community.github.io/archive/stream/113488-general/topic/confused.html
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## Stream: general
### Topic: confused
#### Thorsten Altenkirch (Feb 17 2021 at 15:10):
Think I have used variables in proofs before but today nothing works. What is wrong:
variable P : Prop
variable p : P
example : P := p
example : P :=
begin
exact p,
end
I have boiled it down. Why does it say unknown identifier here?
#### Heather Macbeth (Feb 17 2021 at 15:14):
include p
example : P :=
begin
exact p,
end
#### Heather Macbeth (Feb 17 2021 at 15:15):
If I understand correctly, by default Lean only brings into tactic mode the variables which appear in the hypotheses or conclusion.
#### Johan Commelin (Feb 17 2021 at 15:16):
For completeness: omit is the evil twin of include (-;
#### Thorsten Altenkirch (Feb 17 2021 at 15:16):
Thank you. Actually I have used this before and forgot about it... :-( This just drove me mad.
#### Johan Commelin (Feb 17 2021 at 15:18):
Ooh, just run omit mad to unmad yourself again :rofl:
#### Jeremy Avigad (Feb 17 2021 at 18:41):
The reason that Lean makes you include p explicitly is that a tactic block is really a metaprogram, there is no way for the parse to figure out what p really means. You can have tactics that add things to the context, delete things from the context, and rename things in the context. So Lean doesn't even try to guess what you want to include (even though with exact p is seems pretty obvious).
#### Thorsten Altenkirch (Feb 18 2021 at 10:31):
I noticed that once I am including a variable type inference fails because I suppose it also tries to use the new assumptions. Basically I only wanted to use variables as a replacement for a proof I don't want to do in the moment - I guess I better use sorry in this case?
Last updated: May 10 2021 at 17:39 UTC
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2021-05-10 17:50:47
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https://rosettacommons.org/comment/3449
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# Scoring issues - differs from command line vs .py file - for a simple ddG loop
9 posts / 0 new
Scoring issues - differs from command line vs .py file - for a simple ddG loop
#1
Hello all,
I was able to get PyRosetta up and running on my mac under OS X 10.6.6 . I am working on what I think is a simpler version of the ddG script (we don't need the docking feature). What I am trying to do is bring in a pdb file, score it using the ddG paramters, and then mutate each residue (basically cycle through all the possible combinations)to see what happens to the ddG values are. The start of the script is here: (The loop is a mess at the moment)
from rosetta import *
init()
pose = Pose()
pose_from_pdb( pose , "2O2X_clean.pdb" )
pymover = PyMOL_Mover()
pymover.apply(pose)
ddG_scorefxn = ScoreFunction()
ddG_scorefxn.set_weight( fa_atr , 0.44 )
ddG_scorefxn.set_weight( fa_rep , 0.07 )
ddG_scorefxn.set_weight( fa_sol , 1.0 )
ddG_scorefxn.set_weight( hbond_bb_sc , 0.5 )
ddG_scorefxn.set_weight( hbond_sc , 1.0 )
print"ddG = "
ddG_scorefxn(pose)
for i in range(1,pose.total_residue()+1):
mutate_residue(pose,i,G)
ddG_scorefxn(pose)
I run this script using ipython in the same folder with the .py file and PyRosetta. When I run the script in ipython using run code.py the ddG_scorefxn(pose) comes up blank. But when I type ddG_scorefxn(pose) on the ipython command line it spits out the answer. I've noticed this for a few other things as well.
I was wondering if there was a work around for this? Thanks!
ara
Post Situation:
Mon, 2011-10-24 13:16
ara
Can you elaborate what do you mean by 'ddG_scorefxn(pose) comes up blank'?
Mon, 2011-10-24 14:30
Sergey
Oooppss. Here is part of the output from ipython:
cut
...
core.pack.interaction_graph.interaction_graph_factory: Instantiating DensePDInteractionGraph
core.pack.pack_rotamers: built 44 rotamers at 2 positions.
core.pack.pack_rotamers: IG: 1272 bytes
ddG =
...
cut
The value of the ddG_scorefxn(pose) should follow the ddG = but it doesn't.
But from the ipython command line
In [5]: ddG_scorefxn(pose)
Out[5]: 134.52022280476609
It does.
Thanks!
ara
Mon, 2011-10-24 14:38
ara
Judging from the source code above your output is exactly what expected. In your script code you only printed symbols 'ddG = ' (line: print"ddG = ") but not the score itself. Perphaps you meant to write this?
print"ddG = ", ddG_scorefxn(pose)
Mon, 2011-10-24 14:47
Sergey
So I changed these lines:
print"ddG = "
ddG_scorefxn(pose)
to this:
print"ddG = ", ddG_scorefxn(pose)
and now it works.
So my second question is when the code.py runs and gets down to the loop
for i in range(1,pose.total_residue()+1):
mutate_residue(pose,i,G)
ddG_scorefxn(pose)
I think these should change each residue to a G and then print the ddG_scorefxn (this not quite what I want it to do but I am working on that ;) )
I get the following error message:
/Users/ara/Desktop/PhD_Project/PyRosetta/ddms_code.py in ()
20
21 for i in range(1,pose.total_residue()+1):
---> 22 mutate_residue(pose,i,G)
23 ddG_scorefxn(pose)
24
NameError: name 'G' is not defined
WARNING: Failure executing file:
When I execute the mutate_residue from the command line followed by the score function everything works fine. I was wondering if I need to explicitly import aa from single letters or did I mess up my loop?
Thanks again!
ara
Mon, 2011-10-24 14:55
ara
G by it self is not defined, I think you most likely wanted to this (notice the string constant 'G' instead of just G):
for i in range(1,pose.total_residue()+1): mutate_residue(pose, i, 'G' ) print 'score:', ddG_scorefxn(pose)
Mon, 2011-10-24 15:05
Sergey
Thanks Sergey! I was looking at an example in ala_scan and mutate that didn't have the '' around the aa codes. I am going to hack away at this code for awhile to see if I can get it up and running.
Ara
Mon, 2011-10-24 15:24
ara
Hi Ara,
The problems you're running into look like basic Python issues, rather than PyRosetta specific issues. I would recommend reading through a Python tutorial to get a better idea of how to work with Python. (I'm partial to the official one at http://docs.python.org/tutorial/ and the other documentation at http://docs.python.org/ - but there are others out there which are equally good.).
-Rocco
Tue, 2011-10-25 11:02
rmoretti
Hi Rocco,
Thanks for the docs. I am little rusty on my python and there are large gaps in my knowledge base since I am self taught. I will check those out.
Thanks again,
ara
Tue, 2011-10-25 12:03
ara
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2022-09-25 07:26:57
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http://www.ck12.org/geometry/Angle-Measurement/lesson/user:ZGF3bi5zdGVyemluZ2VyQG1hcnNoYWxsLmsxMi5tbi51cw../Angle-Measurement-student-textbook/
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Angle Measurement
## Measurement of angles with protractors and addition of angles.
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Angle Measurement student textbook
What if you needed a way to describe the size of an angle? After completing this Concept, you'll be able to use a protractor to measure an angle in degrees.
### Watch This
Watch the first part of this video.
### Guidance
We measure a line segment’s length with a ruler. Angles are measured with something called a protractor. A protractor is a measuring device that measures how “open” an angle is. Angles are measured in degrees, and labeled with a \begin{align*}^\circ\end{align*} symbol.
Notice that there are two sets of measurements, one opening clockwise and one opening counter-clockwise, from \begin{align*}0^\circ\end{align*} to \begin{align*}180^\circ\end{align*}. When measuring angles, always line up one side with \begin{align*}0^\circ\end{align*}, and see where the other side hits the protractor. The vertex lines up in the middle of the bottom line, where all the degree lines meet.
For every angle there is a number between \begin{align*}0^\circ\end{align*} and \begin{align*}180^\circ\end{align*} that is the measure of the angle in degrees. The angle's measure is then the absolute value of the difference of the numbers shown on the protractor where the sides of the angle intersect the protractor. In other words, you do not have to start measuring an angle at \begin{align*}0^\circ\end{align*}, as long as you subtract one measurement from the other.
The Angle Addition Postulate states that if \begin{align*}B\end{align*} is on the interior of \begin{align*}\angle ADC\end{align*}, then \begin{align*}m \angle ADC = m \angle ADB + m \angle BDC\end{align*}. See the picture below.
##### Drawing a \begin{align*}50^\circ\end{align*} Angle with a Protractor
1. Start by drawing a horizontal line across the page, about 2 in long.
2. Place an endpoint at the left side of your line.
3. Place the protractor on this point. Make sure to put the center point on the bottom line of the protractor on the vertex. Mark \begin{align*}50^\circ\end{align*} on the appropriate scale.
4. Remove the protractor and connect the vertex and the \begin{align*}50^\circ\end{align*} mark.
This process can be used to draw any angle between \begin{align*}0^\circ\end{align*} and \begin{align*}180^\circ\end{align*}. See http://www.mathsisfun.com/geometry/protractor-using.html for an animation of this investigation.
##### Copying an Angle with a Compass and Straightedge
1. We are going to copy the angle created in the previous investigation, a \begin{align*}50^\circ\end{align*} angle. First, draw a straight line, about 2 inches long, and place an endpoint at one end.
2. With the point (non-pencil side) of the compass on the vertex, draw an arc that passes through both sides of the angle. Repeat this arc with the line we drew in #1.
3. Move the point of the compass to the horizontal side of the angle we are copying. Place the point where the arc intersects this side. Open (or close) the “mouth” of the compass so you can draw an arc that intersects the other side of the arc drawn in #2. Repeat this on the line we drew in #1.
4. Draw a line from the new vertex to the arc intersections.
To watch an animation of this construction, see http://www.mathsisfun.com/geometry/construct-anglesame.html
#### Example A
Measure the three angles using a protractor.
#### Example B
What is the measure of the angle shown below?
#### Example C
What is \begin{align*}m \angle QRT\end{align*} in the diagram below?
#### Example D
Draw a \begin{align*}135^\circ\end{align*} angle.
Watch this video for help with the Examples above.
### Vocabulary
A protractor is a measuring device that measures how “open” an angle is. Angles are measured in degrees, and labeled with a \begin{align*}^\circ\end{align*} symbol. A compass is a tool used to draw circles and arcs.
### Practice
1. What is \begin{align*}m \angle LMN\end{align*} if \begin{align*}m \angle LMO = 85^\circ\end{align*} and \begin{align*}m \angle NMO = 53^\circ\end{align*}?
2. If \begin{align*}m\angle ABD = 100^\circ\end{align*}, find \begin{align*}x\end{align*}.
For questions 3-6, determine if the statement is true or false.
1. For an angle \begin{align*}\angle ABC, C\end{align*} is the vertex.
2. For an angle \begin{align*}\angle ABC, \overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*} are the sides.
3. The \begin{align*}m\end{align*} in front of \begin{align*}m \angle ABC\end{align*} means measure.
4. The Angle Addition Postulate says that an angle is equal to the sum of the smaller angles around it.
For 7-12, draw the angle with the given degree, using a protractor and a ruler.
1. \begin{align*}55^\circ\end{align*}
2. \begin{align*}92^\circ\end{align*}
3. \begin{align*}178^\circ\end{align*}
4. \begin{align*}5^\circ\end{align*}
5. \begin{align*}120^\circ\end{align*}
6. \begin{align*}73^\circ\end{align*}
For 13-16, use a protractor to determine the measure of each angle.
Solve for \begin{align*}x\end{align*}.
1. \begin{align*}m\angle ADC = 56^\circ\end{align*}
2. \begin{align*}m \angle ADC = 130^\circ\end{align*}
3. \begin{align*}m \angle ADC = (16x - 55)^\circ\end{align*}
4. \begin{align*}m \angle ADC = ( 9x - 80)^\circ\end{align*}
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2016-06-27 11:22:02
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https://learn.microsoft.com/en-us/dotnet/fundamentals/code-analysis/quality-rules/ca1308
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# CA1308: Normalize strings to uppercase
Value
Rule ID CA1308
Category Globalization
Fix is breaking or non-breaking Non-breaking
## Cause
An operation normalizes a string to lowercase.
## Rule description
Strings should be normalized to uppercase. A small group of characters, when they are converted to lowercase, cannot make a round trip. To make a round trip means to convert the characters from one locale to another locale that represents character data differently, and then to accurately retrieve the original characters from the converted characters.
## How to fix violations
Change operations that convert strings to lowercase so that the strings are converted to uppercase instead. For example, change String.ToLower(CultureInfo.InvariantCulture) to String.ToUpper(CultureInfo.InvariantCulture).
## When to suppress warnings
It's safe to suppress a warning when you're not making security decisions based on the result of the normalization (for example, when you're displaying the result in the UI).
## Suppress a warning
If you just want to suppress a single violation, add preprocessor directives to your source file to disable and then re-enable the rule.
#pragma warning disable CA1308
// The code that's violating the rule is on this line.
#pragma warning restore CA1308
To disable the rule for a file, folder, or project, set its severity to none in the configuration file.
[*.{cs,vb}]
dotnet_diagnostic.CA1308.severity = none
To disable this entire category of rules, set the severity for the category to none in the configuration file.
[*.{cs,vb}]
dotnet_analyzer_diagnostic.category-Globalization.severity = none
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2023-03-25 18:48:47
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https://www.albert.io/ie/act-math/circle-circumference
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Free Version
Moderate
# Circle Circumference
ACTMAT-GKENY3
Blowing air into a spherical ball caused the circumference of the ball to increase from 12 inches to 16 inches.
By how many inches, then, did the radius increase?
A
$4$
B
$2$
C
${\cfrac { 2 }{ \pi } }$
D
$\cfrac{4}{\pi}$
E
$2{\pi}$
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2017-03-01 20:17:45
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https://ncatlab.org/nlab/show/Gleason's+theorem
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# Contents
## Idea
### General
The classical Gleason theorem says that a state on the C*-algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space is uniquely described by the values it takes on the orthogonal projections $\mathcal{P}$, if the dimension of the Hilbert space $\mathcal{H}$ is not 2.
In other words: every quasi-state is already a state if $dim(H) \gt 2$.
It is possible to extend the theorem to certain types of von Neumann algebras (e.g. obviously factors of type $I_2$ have to be excluded).
### Implications for Quantum Logic
Roughly, Gleason’s theorem says that “a quantum state is completely determined by only knowing the answers to all of the possible yes/no questions”.
## Definitions
###### Definition
Let $\rho: \mathcal{P} \to [0, 1]$ such that for every finite family $\{ P_1, ..., P_n: P_i \in \mathcal{P} \}$ of pairwise orthogonal projections we have $\rho(\sum_{i=1}^n P_i) = \sum_{i=1}^n \rho(P_i)$, then $\rho$ is a finitely additive measure on $\mathcal{P}$.
If the family is not finite, but countable, then $\rho$ is a sigma-finite measure.
## The Theorem
### Classical Gleason’s Theorem
###### Theorem
If $dim(\mathcal{H}) \neq 2$ then each finitely additive measure on $\mathcal{P}$ can be uniquely extended to a state on $\mathcal{B}(\mathcal{H})$. Conversly the restriction of every state to $\mathcal{P}$ is a finitley additive measure on $\mathcal{P}$.
The same holds for sigma-finite measures and normal states: Every sigma-finite measure can be extended to a normal state and every normal state restricts to a sigma-finite measure.
### Gleason's Theorem for POVMs
In quantum information theory, one often considers positive operator-valued measures (POVMs) instead of Hermitian operators as observables. While a Hermitian operator is given by a family of projection operators $P_i$ such that $\sum_i P_i = 1$, a POVM is given more generally by any family of positive-semidefinite operators $E_i$ such that $\sum_i E_i = 1$.
In the analog of Gleason’s Theorem for POVMs, therefore, we start with $\rho\colon \mathcal{E} \to [0,1]$, where $\mathcal{E}$ is the space of all positive-semidefinite operators. Then if $\sum_i \rho(E_i) = 1$ whenever $\rho(\sum_i E_i) = 1$, the theorem states that $\rho$ has a unique extension to a mixed quantum state.
As a theorem, Gleason's Theorem for POVMs is much weaker than the classical Gleason's Theorem, since we must begin with $\rho$ defined on a much larger space of operators. However, some content does remain, since we have not assumed any continuity properties of $\rho$. Also, Gleason's Theorem for POVMs has a much simpler proof, which works regardless of the dimension.
## Examples
### Counterexample For Dimension Two
See example 8.1 in the book by Parthasarathy (see references). Our Hilbert space is $\mathbb{R}^2$. Projections $P$ on it are either identical zero, the identity, or projections on a one dimensional subspace, so that these $P$ can be written in the bra-ket notation? as
$P = {|u \rangle} {\langle u|}$
with a unit vector $u$, i.e. $u \in \mathbb{R}^2, {\|u\|} = 1$. In this finite dimensional case sigma-finite and finite are equivalent, and a finite probability measure is equivalent to a (complex valued) function such that
$f(c u) = f(u)$
$\sum_i f(u_i) = 1$
for every scalar $c$ of modulus one, every unit vector $u$ and every orthonormal basis $\{u_1, u_2\}$. If there is a state that extends such a measure and therefore restricts to such a measure on projections, there would be a linear operator $T$ such that
$f(u) = {\langle u | T u \rangle}$
for all unit vectors $u$.
It turns out however that the conditions imposed on $f$ are not enough in two dimensions to enforce this kind of linearity of $f$. Heuristically, in three dimensions there are more rotations than in two, therefore the “rotational invariance” of (the conditions imposed on) $f$ is more restrictive in three dimensions than it is in two dimensions.
In two dimensions, choose a function $g$ on $[0, \frac{\pi}{2})$ such that $0 \le g(\theta) \le 1$ everywhere. There are no further restrictions, that is $g$ need not be continuous, for example. Now we can define a probability measure on the projections by
$f(u) = \begin{cases} g(\theta) \; \; \text{for} \; \; 0 \le \theta \lt \frac{\pi}{2} \\ 1 - g(\theta - \frac{\pi}{2}) \; \; \text{for} \; \; \frac{\pi}{2} \le \theta \lt \pi \\ f(-u) \; \; \text{as defined in the first two items, else} \end{cases}$
This probability measure will in general not extend to a state.
Other theorems about the foundations and interpretation of quantum mechanics include:
## References
Gleason’s original paper outlining the theorem is
• A.M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, Indiana Univ. Math. J. 6 No. 4 (1957), 885–893 (web)
A standard textbook exposition of the theorem and its meaning is
where it appears as theorem 2.3 (without proof).
A monograph stating and proving both the classical theorem and extensions to von Neumann algebras is
The classical theorem is proved also in this monograph:
• K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (ZMATH)
Gleason's Theorem for POVMs is proved here:
• Paul Busch, Quantum states and generalized observables: a simple proof of Gleason’s theorem; (1999) (arXiv)
The failure of Gleason’s theorem for classical states (on Poisson algebras) is discussed in
• Michael Entov, Leonid Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology (arXiv).
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2017-02-23 02:35:25
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https://answers.ros.org/answers/14192/revisions/
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# Revision history [back]
stageros only supports laser and odometry output. If you want support for cameras in ROS, simulated by Stage, you should file an enhancement ticket against the simulator_stage stack.
I'm pretty certain that there is support for simulation a camera in the simulator_gazebo stack, but I've never used it.
stageros only supports laser and odometry output. If you want support for cameras in ROS, simulated by Stage, you should file an enhancement ticket against the simulator_stage stack.
I'm pretty certain that there is support for simulation a camera in the simulator_gazebo stack, but I've never used it.
Update with responses to Sagnik's further questions:
The URDF tutorials are a good place to start, though it is true that getting a robot into Gazebo is a lot more work than getting it into Stage. You'll also have to try to build the Ackermann steering for your model, as there is no simple Ackermann steering controller that I am aware of. See http://answers.ros.org/question/459/how-can-i-simulate-ackermann-steering-in-gazebo and http://answers.ros.org/question/630/linking-joints-in-urdf-andor-gazebo for hints on how to go about doing that.
If you have no need for the full simulation of the robot dynamics, you could always look into how Stage simulates the camera (what format is the image in when it returns it) and add support for outputting simulated cameras to the stageros package.
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2021-10-18 20:50:24
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https://hal-psl.archives-ouvertes.fr/hal-03022716v1
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# Applying clock comparison methods to pulsar timing observations
Abstract : Frequency metrology outperforms any other branch of metrology in accuracy (parts in $10^{-16}$) and small fluctuations ($<10^{-17}$). In turn, among celestial bodies, the rotation speed of millisecond pulsars (MSP) is by far the most stable ($<10^{-18}$). Therefore, the precise measurement of the time of arrival (TOA) of pulsar signals is expected to disclose information about cosmological phenomena, and to enlarge our astrophysical knowledge. Related to this topic, Pulsar Timing Array (PTA) projects have been developed and operated for the last decades. The TOAs from a pulsar can be affected by local emission and environmental effects, in the direction of the propagation through the interstellar medium or universally by gravitational waves from super massive black hole binaries. These effects (signals) can manifest as a low-frequency fluctuation over time, phenomenologically similar to a red noise. While the remaining pulsar intrinsic and instrumental background (noise) are white. This article focuses on the frequency metrology of pulsars. From our standpoint, the pulsar is an accurate clock, to be measured simultaneously with several telescopes in order to reject the uncorrelated white noise. We apply the modern statistical methods of time-and-frequency metrology to simulated pulsar data, and we show the detection limit of the correlated red noise signal between telescopes.
Document type :
Journal articles
Domain :
Complete list of metadata
https://hal.archives-ouvertes.fr/hal-03022716
Contributor : Inspire Hep <>
Submitted on : Thursday, April 22, 2021 - 11:04:53 AM
Last modification on : Tuesday, May 4, 2021 - 11:53:31 PM
### File
##### Restricted access
To satisfy the distribution rights of the publisher, the document is embargoed until : 2021-10-22
### Citation
Siyuan Chen, François Vernotte, Enrico Rubiola. Applying clock comparison methods to pulsar timing observations. Mon.Not.Roy.Astron.Soc., 2021, ⟨10.1093/mnras/stab742⟩. ⟨hal-03022716⟩
Record views
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2021-05-07 03:11:26
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https://blog.evanchen.cc/tag/number-fields/
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## Artin Reciprocity
I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define $latex {L}&fg=000000$-series for non-abelian extensions. But for them to agree with the $latex {L}&fg=000000$-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I… Continue reading Artin Reciprocity
## Some Notes on Valuations
There are some notes on valuations from the first lecture of Math 223a at Harvard. 1. Valuations Let $latex {k}&fg=000000$ be a field. Definition 1 A valuation $latex \displaystyle \left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0} &fg=000000$ is a function obeying the axioms $latex {\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0}&fg=000000$.… Continue reading Some Notes on Valuations
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2023-03-20 13:51:36
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https://www.meritnation.com/cbse-class-6/math/rd-sharma-2018/whole-numbers/textbook-solutions/8_1_2171_10305_3.4_41419
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Rd Sharma 2018 Solutions for Class 6 Math Chapter 1 Knowing Our Numbers are provided here with simple step-by-step explanations. These solutions for Knowing Our Numbers are extremely popular among Class 6 students for Math Knowing Our Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2018 Book of Class 6 Math Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2018 Solutions. All Rd Sharma 2018 Solutions for class Class 6 Math are prepared by experts and are 100% accurate.
#### Question 1:
Write down the smallest natural number.
The smallest natural number is 1.
#### Question 2:
Write down the smallest whole number.
The smallest whole number is 0 (zero).
#### Question 3:
Write down, if possible, the largest natural number.
We know that every natural number has a successor. Thus, there is no largest natural number.
#### Question 4:
Write down, if possible, the largest whole number.
We know that every whole number has a successor. Thus, there is no largest whole number.
#### Question 5:
Are all natural numbers also whole numbers?
Yes, all natural numbers are whole numbers.
#### Question 6:
Are all whole numbers also natural numbers?
No, all whole numbers are not natural numbers because 0 is a whole number but not a natural number.
#### Question 7:
Give successor of each of the whole numbers?
(i) 1000909
(ii) 2340900
(iii) 7039999
Given Number Successor
(i) 1,000,909 1,000,909 + 1 = 1,000,910
(ii) 2,340,900 2,340,900 + 1 = 2,340,901
(iii) 7,039,999 7,039,999 + 1 = 7,040,000
#### Question 8:
Write down the predecessor of each of the following whole numbers:
(i) 10000
(ii) 807000
(iii) 7005000
Given Number Predecessor
(i) 10,000 10,000 - 1 = 9,999
(ii) 807,000 807,000 - 1 = 806,999
(iii) 7,005,000 7,005,000 - 1 = 7,004,999
#### Question 9:
Represent the following numbers on the number line:
2,0,3,5,7,11,15
#### Question 10:
How many whole numbers are there between 21 and 61?
The whole numbers between 21 and 61 are 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 and 60.
Thus, there are 39 whole numbers between 21 and 61.
#### Question 11:
Fill in the blanks with the appropriate symbol < or >:
(i) 25...205
(ii) 170...107
(iii) 415...514
(iv) 10001...100001
(v) 2300014...2300041
We have:
(i) 25 < 205
(ii) 170 > 107
(iii) 415 < 514
(iv) 10001 < 100001
(v) 2300014 < 2300041
#### Question 12:
Arrange the following numbers is descending order:
925, 786, 1100, 141, 325, 886, 0, 270
Numbers in descending order:
1100, 925, 886, 786, 325, 270, 141, 0
#### Question 13:
Write the largest number of 6 digits and the smallest number of 7 digits. Which one of these two is larger and by how much?
Largest six-digit number = 999,999
Smallest seven-digit number = 1,000,000
Thus, the smallest seven-digit number is larger than the largest six-digit number.
Again,
Difference between these two numbers = 1,000,000 - 999,999 = 1
Hence, the smallest seven-digit number is larger than the largest six-digit number by 1.
#### Question 14:
Write down three consecutive whole numbers just preceding 8510001.
We have:
First number = 8,510,001 - 1 = 8,510,000
Second number = 8,510,000 - 1 = 8,509,999
Third number = 8,509,999 - 1 = 8,509,998
Hence, the three consecutive whole numbers just preceding 8,510,001 are 8,510,000, 8,509,999 and 85,09,998.
#### Question 15:
Write down the next three consecutive whole numbers starting from 4009998.
We have:
First number = 4,009,998 + 1 = 4,009,999
Second number = 4,009,999 + 1 = 4,010,000
Third number = 4,010,000 + 1 = 4,010,001
Hence, the next three consecutive whole numbers starting from 4,009,998 will be 4,009,999, 4,010,000 and 4,010,001.
#### Question 16:
Give arguments in support of the statement that there does not exist the largest natural number.
We know that every natural number has a successor. Therefore, the largest natural number does not exist.
#### Question 17:
Which of the following statements are true and which are false?
(i) Every whole number has its successor.
(ii) Every whole number has its predecessor.
(iii) 0 is the smallest natural number.
(iv) 1 is the smallest whole number.
(v) 0 is less than every natural number.
(vi) Between any two whole numbers there is a whole number.
(vii) Between any two non-consecutive whole numbers there is a whole number.
(viii) The smallest 5-digit number is the successor of the largest 4 digit number
(ix) Of the given two natural numbers, the one having more digits is greater.
(x) The predecessor of a two digit number cannot be a single digit number.
(xi) If a and b are natural numbers and a < b, than there is a natural number c such that a<b<c.
(xii) If a and b are whole numbers and a<b, then a+1< b+1.
(xiii) The whole number 1 has 0 as predecessor.
(xiv) The natural number 1 has no predecessor.
(i) True
The successor of every whole number can be found by adding 1.
(ii) False
Zero (0) is a whole number whose predecessor (-1) is not a whole number.
(iii) False
1 is the smallest natural number.
(iv) False
Zero (0) is the smallest whole number.
(v) True
The smallest natural number is 1, so zero (0) is less than every natural number.
(vi) False
There is no whole number between two consecutive whole numbers.
(vii) True
(viii) True
The smallest five-digit number = 10,000
The largest four-digit number = 9,999
Difference = 10,000 - 9,999 = 1
Because the difference is 1, 10,000 is the successor of 9,999.
(ix) True
(x) False
10 is a two-digit number whose predecessor is 9, which is a one-digit number.
(xi) False
If a and b are consecutive natural numbers, then there cannot be any natural number c in between a and b.
(xii) True
(xiii) True
(xiv) True
The predecessor of natural number 1 is 0, which is not a natural number.
#### Question 1:
The smallest natural number is
(a) 0
(b) 1
(c) -1
(d) None of these
(b) 1
#### Question 2:
The smallest whole number is
(a) 1
(b) 0
(c) -1
(d) None of these
(b) 0
#### Question 3:
The predecessor of 1 in natural numbers is
(a) 0
(b) 2
(c) -1
(d) None of these
(d) None of these
We know that the smallest natural number is 1. Hence, its predecessor does not exist.
#### Question 4:
The predecessor of 1 in whole numbers is
(a) 0
(b) -1
(c) 2
(d) None of these
(a) 0
Predecessor of 1 = 1 - 1 = 0
#### Question 5:
The predecessor of 1 million is
(a) 9999
(b) 99999
(c) 999999
(d) 1000001
(c) 9,99,999
We have:
1 million = 10,00,000
Predecessor of 1 million = 10,00,000 - 1
= 9,99,999
#### Question 6:
The successor of 1 million is
(a) 10001
(b) 100001
(c) 1000001
(d) 10000001
(c) 10,00,001
We have:
1 million = 10,00,000
Successor of 1 million = 10,00,000 + 1
= 10,00,001
#### Question 7:
The product of the successor and predecessor of 99 is
(a) 9800
(b) 9900
(c) 1099
(d) 9700
(a) 9800
We have:
Successor of 99 = 99 + 1 = 100
Predecessor of 99 = 99 − 1 = 98
Their product = 100 × 98 = 9800
#### Question 8:
The product of a whole number (other than zero) and its successor is
(a) an even number
(b) an odd number
(c) divisible by 4
(d) divisible by 3
(a) an even number
Example:
Whole number = 1
Successor of 1 = 1 + 1 = 2
Their product = 1 × 2 = 2
Thus, 2 is an even number.
#### Question 9:
The product of the predecessor and successor of an odd natural number is always divisible by
(a) 2
(b) 4
(c) 6
(d) 8
(d) 8
The predecessor of an odd number is an even number.
The successor of an odd number is also an even number.
These two even numbers are two consecutive even numbers, and the product of two consecutive even numbers is always divisible by 8.
#### Question 10:
The product of the predecessor and successor of an even natural number is
(a) divisible by 2
(b) divisible by 3
(c) divisible by 4
(d) an odd number
(d) an odd number
Example:
Even natural number = 2
Predecessor of 2 = 2 − 1 = 1
Successor of 2 = 2 + 1 = 3
Their product = 1 × 3 = 3
Thus, the product is an odd number.
#### Question 11:
The successor of the smallest prime number is
(a) 1
(b) 2
(c) 3
(d) 4
The smallest prime number is 2
So, Successor of 2 = 2 + 1 = 3
Hence, the correct answer is option (c).
#### Question 12:
If x and y are co-primes, then their LCM is
(a) 1
(b) $\frac{x}{y}$
(c) xy
(d) None of these
A set of numbers which do not have any other common factor other than 1 are called co-prime.
The LCM of two co-prime numbers is equal to their product.
Hence, the correct answer is option (c).
#### Question 13:
The HCF of two co-primes is
(a) the smaller number
(b) the larger number
(c) product of the numbers
(d) 1
A set of numbers which do not have any other common factor other than 1 are called co-prime.
The HCF of two co-prime numbers is 1.
Hence, the correct answer is option (d).
#### Question 14:
The smallest number which is neither prime nor composite is
(a) 0
(b) 1
(c) 2
(d) 3
The smallest number which is neither prime nor composite is 1
Hence, the correct answer is option (b).
#### Question 15:
The product of any natural number and the smallest prime is
(a) an even number
(b) an odd number
(c) a prime number
(d) None of these
The smallest prime number is 2.
Thus, when we multiply any natural number we will always get an even number.
Hence, the correct answer is option (a).
#### Question 16:
Every counting number has an infinite number of
(a) factors
(b) multiples
(c) prime factors
(d) None of these
Multiples are what we get after multiplying the number by any number.
Thus, every counting number has an infinite number of multiples
Hence, the correct answer is option (b).
#### Question 17:
The product of two numbers is 1530 and their HCF is 15. The LCM of these numbers is
(a) 102
(b) 120
(c) 84
(d) 112
Product of two numbers = HCF of two numbers × LCM of two numbers
Hence, the correct answer is option (a).
#### Question 18:
The least number divisible by each of the numbers 15, 20, 24 and 32 is
(a) 960
(b) 480
(c) 360
(d) 640
LCM of 15, 20, 24 and 32 is given by
15 = 3 × 5 = 31 × 51
20 = 2 × 2 × 5 = 22 × 51
24 = 2 × 2 × 2 × 3 = 23 × 31
32 = 2 × 2 × 2 × 2 × 2 = 25
LCM = 25 × 31 × 51 = 480
Hence, the correct answer is option (b).
#### Question 19:
The greatest number which divides 134 and 167 leaving 2 as remainder in each case is
(a) 14
(b) 19
(c) 33
(d) 17
First we subtract the required remainder from 134 and 167.
Thus, we will get 132 and 165.
132 = 2 × 2 × 3 × 11 = 22 × 3 × 11
165 = 3 × 5 × 11 = 31 × 5 × 11
HCF = 3 × 11 = 33
Thus, the greatest number which divides 134 and 167 leaving 2 as remainder in each case is 33
Hence, the correct answer is option (c).
#### Question 20:
Which of the following numbers is a prime number?
(a) 91
(b) 81
(c) 87
(d) 97
Since, factors of
91 = 1 × 7 × 13
81 = 1 × 3 × 3 × 3 × 3
87 = 1 × 3 × 29
97 = 1 × 97
Thus, 81, 87 and 91 all are not prime numbers.
Hence, the correct answer is option (d).
#### Question 21:
If two numbers are equal, then
(a) their LCM is equal to their HCF
(b) their LCM is less than their HCF
(c) their LCM is equal to two times their HCF
(d) None of these
If two numbers are equal, then their LCM is equal to their HCF
Hence, the correct answer is option (a).
#### Question 22:
a and b are two co-primes. Which of the following is/are true?
(a) LCM (a, b) = a × b
(b) HCF (a, b) = 1
(c) Both (a) and (b)
(d) Neither (a) nor (b)
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2020-04-02 16:45:23
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http://sewingartistry.com/2014/05/the-pressinatrix-interview/
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The Pressinatrix Interview « SewingArtistry.com
# The Pressinatrix Interview
Yes, that’s right. You read that right. The famous Pressinatrix (and her alter ego, Ann Steeves of Gorgeous Fabrics fame), has awarded yours truly an exclusive sit-down to discuss the pressing matters of sewing. And that’s only the beginning of the groaning puns, so prepare yourself!
SA: When did you realize your calling?….was there a garment where you saw a dramatic difference between not pressing/pressing badly to pressing well?
M. Pressinatrix: The Pressinatrix realized her true calling when she attended a seminar by a wonderful teacher, Roberta Carr. During the seminar, one of the attendees (not The Pressinatrix) exclaimed that she hated pressing. Ms. Carr, who regrettably passed away several years ago, looked sternly yet gently at the entire class and said, “Folks, pressing IS sewing.” The Pressinatrix’ eyes and mind were opened at that moment!
SA: Who are your favorite assistants?….in order – is it because the ham is always moral, truthful, and versatile; because the sleeve roll is good confidant and available; the clapper, even though a little heavy-handed, can get the job done, or the press cloth, retiring and illusive can still be just the right touch; or is there an unsung hero that we never hear about?
Press cloths. The Pressinatrix cannot sing the praises of press cloths highly enough. Her favorite is silk organza. It holds up to high heat, you can spritz it with water to create steam where and when you want it, and it is transparent, so you can see clearly what you are doing
SA: What is the most important thing to remember when you are pressing – in order 1.) Burn, Baby, Burn – you can’t help getting burned sometimes, 2.) Lava Lamp Pressure – varying pressure gives varying results, 3.) Geyser Steam – Let the steam do the work, and 4.)The Machinist – the equipment makes the difference
MP: Ah, Your Pressinatrix can make fabric sing just as readily with a $25 Black and Decker iron as with a$2500 pressing station. The techniques I show on my blog and in my videos do not require expensive equipment. It’s more important to consistently, thoughtfully press your garments. And test test test! If you are not familiar with how a fabric reacts to steam and heat, play with a scrap before you begin working on your garment. Even The Pressinatrix has made her share of mistakes (some day she shall tell you about her adventure with nylon mesh and an over-warm iron).
SA: What is the one falsehood of pressing?……that it’s the unsung hero…..that never gets enough press?…..that it can’t/can be done wrong…..that it’s just a given to set up an ironing board & iron close to your sewing machine.
MP: The single biggest falsehood of pressing is that it is not necessary during each and every step of construction. It is. The Pressinatrix was appalled not too long ago to see a photo shoot by an independent company in which it was patently clear that the garments being photographed had not seen the bottom of an iron until the last hem was hemmed. Even more egregious is a garment by a major pattern company that is in their most recent catalogue. The puckering seams, the dreadful easing, it is almost too much for The Pressinatrix to bear! Pardon me – I must compose myself…
There, now we can continue.
SA: Some may not be able to withstand your bring shine, what sort of protectors do you recommend – specifically presscloths – organza, organdy, or felt (for preserving loft or texture (as in trapunto stitching) – which ones do you use the most and find the most beneficial?
MP: There are several press cloths and tools in the Pressinatrix’ arsenal. You can see many of them on the video “Press that Bad Mamma Jamma“, as well as on The Pressinatrix’ lesser self, er, alter ego’s blog. One of the lesser-sung heroes in The Pressinatrix’ own pressing toolkit is a silicone kitchen mitt. It functions as a flexible clapper.
As far as press cloths go, The Pressinatrix likes to use silk organza the best, but she also uses unbleached muslin and wool flannel for different types of fabrics. Also, The Pressinatrix is fond of using a very light hand on delicate fabrics. She will hover the iron over the fabric, about 1/16 inch above it, and use steam along with a clapper or silicone mitt to gently flatten seams. Another excellent technique that doesn’t get much acknowledgement is point pressing, in which one uses just the pointed tip of the iron and very light pressure to press seams. The Pressinatrix has found that it works wonderfully on cashmere, alpaca and vicuña and other delicate fabrics.
SA: We all know that you are, if nothing, in great physical condition, but what sort of routine do you use to keep in shape?….acidity solutions, vinegar – something in the water?….any thing else?
MP: No no no no NO!!! Oh my gracious. The Pressinatrix recently received an email from a sewing company in which they… Excuse me, The Pressinatrix is finding it hard to even repeat this…
They recommended putting a solution of cornstarch and water in one’s iron to stiffen up chiffon and other soft fabrics.
OH THE HORROR!!!!
Pardon me, but The Pressinatrix needs to do some meditative breathing before we can continue…
My darling, The Pressinatrix was appalled and shocked that someone who is purportedly expert would make a recommendation like that. Here is what the Pressinatrix puts in her iron:
Water.
The tap water at The Pressinatrix’ home is rather hard, so The Pressinatrix uses demineralizing filter beads to remove the mineraly detritus. The Pressinatrix would NEVER introduce extra chemicals into her equipment. If one desires starch, or if one wants to mix a teaspoon of white vinegar into 2 cups of water as a crease remover, that is all well and good. But never, EVER put those directly into your iron. Keep a spray bottle handy for that purpose, and use that. Adding extraneous solutions to one’s iron will only succeed in shortening the lifespan of said iron.
SA: Having your finger on the pulse of the fashionista du jour, can you talk a little about the pressing done in the commercial world?….specifically talk about how industry standards wouldn’t dream of being without a garment pressed – how they do it?
At the garment factories which The Pressinatrix has visited, pressers are paid more than seamstresses. This is because they spend more time with the garment. The factories that supply better than low-end manufacturers practice the methods that The Pressinatrix preaches, and they spend much time and care to ensure that the garment looks good at all phases of its construction. They have some specialized tools and tables to speed their processes, but the process is much the same – sew the seam, press the seam. The Pressinatrix cannot fathom why so many sewing hobbyists declare loudly that they either hate pressing or that they find it to be a waste of time. Another wonderful sewing teacher, Cynthia Guffey, has a saying, “It’s your hobby; what’s your hurry?” The Pressinatrix agress, and The Pressinatrix finds that the time spent pressing is not only well worth it, but it also allows her time to think about the next steps in the sewing process, ensuring perfect results.
SA: Let’s just get it out there: Rumors are that you are linked to a big-wig political type (we won’t name names), what other rumors would you like to dispell and debunk?….like the diluted starch solution…..any other myths or wives’ tales out there that need debunking?
MP: The Pressinatrix never discusses politics, religion or baseball.
SA: OK – I just lied – we want you to name names and dish all the scoop….what is your favorite equipment, and your favorite models (boiler or gravity or other)? And any history you have with them.
MP: The Pressinatrix loves all pressing equipment! Her personal favorites are Reliable and Naomoto for irons, Stitch Nerd for hams and sleeve rolls.
And now, my dear, The Pressinatrix must get back to work. There is a silk dress waiting to be made. Ta ta, and Happy Pressing!
Well, I don’t know about you, but I feel all energized to start pressing something! All kidding aside, pressing is vital to that professional result we all want. It’s so simple, sometimes it’s forgotten, but don’t- or M. Pressinatrix (and I) will come haunt you!
1. For decades my motto has been “I sew with my iron”.
2. Oh I am so with you ladies on pressing 🙂
It has to be done and never…ever….a step to leave out.
Lots of fun 🙂
3. Amen to pressing all seams! My grandma taught me that pressing was a necessary step and I’ve followed this practice all my 54 years of sewing. Thanks for the funny interview.
4. Ain’t Ann a stitch!
• Kate – she is too much fun to play along with me – hopefully we will get under the radar to have folks start thinking about pressing as much a part of sewing as thread is!
5. I think I heard Kenneth King say somewhere that good ironing could save bad sewing, and bad ironing could ruin good sewing.
• You know that sounds like something pithy ole Kenneth would say!!!!
6. Ann needs some sort of Academy Award for her Pressinatrix videos. It is such sorely needed info and she does such a great job getting her point across.
• She does indeed – it’s such a mundane boring subject and seemingly so inconsequential, when it is besides the sewing itself the most integral part of sewing. After a machine, the iron or pressing system is the next most important piece of equipment! I had fun with the puns and playing along with the always stellar Pressinatrix!!!
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2018-07-17 06:06:46
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https://math.stackexchange.com/questions/871242/prove-that-the-field-of-quotients-of-an-integral-domain-d-is-the-smallest-fiel
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# Prove that the field of quotients of an integral domain $D$ is the smallest field containing $D$. . My Attempt Shown
Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the field of quotients of an integral domain is the smallest field containing $D$.
Attempt: Let $S$ be a sub ring of the field of quotient $F$ such that $S \approx D$
We need to show that $F \approx$ a sub field of $E$.
Let $F'$ be the field of quotients of $F$.
If $K$ be a field, then, the field of quotients of $K$ is ring isomorphic to $K$ is a result.
Hence, We know that $F' \approx F$ . Hence, $F'$ must contain $D$.
Since, $F'$ is a field, $\implies F'$ is a sub field of $E$.
Hence Proved that there exists a sub field of $E$ which is isomorphic to $F$.
Is my attempt correct?
Please note that my book hasn't yet introduced polynomials, reducability, divisibility in integral domain or field extensions.
• It does not look correct to me. How do you show that since $F'$ is a field then $F'$ is a sub field of $E$? Maybe you can think intuitively about it like this: if $E$ contains $D$ then since $E$ is a field we have for $d \in D$ ($d \ne 0$) that $d^{-1}$ exists in $E$, but $d^{-1}$ is exactly the kind of element that we added to get $F$, so it seems that elements of $F$ will be in $E$ as well (I'm a bit vague, but trying to give a sense of it) – user50948 Jul 18 '14 at 21:23
• Thank you for the comment :-) – MathMan Jul 19 '14 at 1:28
Since, $F′$ is a field, $F′$ is a sub field of $E$.
This line is pretty much assuming what you are currently are trying to prove. You will have to actually make reference to how $D$ lies in $E$ to prove the statement.
Why not just try to make a map from $F$ into $E$? Since you already have $D\subseteq E$, it's natural to just say $\phi(a)= a\in E$ for all $a\in D$.
What about other elements of $F$? For each $b\neq 0$ in $D$, there is an element $b^{-1}\in E$ which is $b$'s inverse, and an element $b^{-1}\in F$ which plays the same role in $F$. Naturally you'd want $\phi(b^{-1})=b^{-1}\in E$, where the first $b^{-1}$ is in $F$ and the latter is in $E$.
To map the remaining elements of $F$, we would require $\phi(ab^{-1})=\phi(a)\phi(b^{-1})=ab^{-1}$ (again the first $b^{-1}$ is the one in $F$ and the last one is in $E$.)
Verify this gives a well-defined injective ring homomorphism from $F$ into $E$. The image of $\phi$ is the copy of $F$ that you seek.
• Thank you for your answer :-) – MathMan Jul 19 '14 at 1:16
• Dear @VHP : I asked myself "where should I send elements of $D$? Where should I send inverses of elements in $D$? Did I get everything?" – rschwieb Jul 19 '14 at 1:16
• @VHP No problem... hope it helped! – rschwieb Jul 19 '14 at 1:16
Suppose that $\iota:D\to E$ is an injection where $E$ is a field. Then $x\neq 0$ in $D$ gives $\iota x\neq 0$ in $E$; so every nonzero element of $D$ maps to an invertible element. By the universal property of localizations, we get an induced map $\bar \iota:F(D)\to E$. Since $F(D)$ is a field and the map is nonzero, it is injective.
• Judging from the line "Please note that my book hasn't yet introduced polynomials, reducability, divisibility in integral domain or field extensions.", it looks like referring to universal properties and localization is overshooting user's knowledge requirements. Judging from recent posts by this user, it looks like only basics about rings, fields and homomorphisms are at his/her disposal. Regards – rschwieb Jul 18 '14 at 21:52
• @rschwieb Answers are not always supposed to go to the OP. Some time later he might come back and understand it. – Pedro Tamaroff Jul 18 '14 at 22:05
• That's true, and I'm not faulting any of the content of your answer really. If I had been in your shoes though, I'd have also put a simpler answer in front :) Then whatever stuff happens later is beyond reproach. Regards – rschwieb Jul 19 '14 at 1:14
• OK. ${}{}{}{}{}$ – Pedro Tamaroff Jul 19 '14 at 1:29
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2019-12-14 08:00:35
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https://chemistry.stackexchange.com/questions/104712/nmr-analysis-regarding-cis-1-isopropyl-3-phenylcyclohexane
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# NMR analysis regarding cis-1-isopropyl-3-phenylcyclohexane
I have come across a question in my textbook:
I understand the explanation for the first compound. However, for the second structure, I am not understanding why the 2 methyl groups are diastereotopic. Is there not free rotation about the $$\ce{C-C}$$ bond, connecting the $$\ce{CH(CH3)2}$$ group to the ring such that the 2 methyl groups become equivalent? The answer key makes it seem as though there is no rotation and the methyl groups are fixed in place. I feel as if I am misunderstanding the question. Please help me clear up the confusion.
• Substitution is frequently the easiest approach. Replace a hydrogen on each of those two methyl groups with a deuterium to create two new compounds? What is the relationship between those two compounds? Identical? Hydrogen atoms are homotopic. Enantiomers? Enantiopic. Diastereomers? Diastereotopic. – Zhe Nov 23 '18 at 19:04
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2019-08-17 13:54:49
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https://www.physicsforums.com/threads/radius-of-the-track.633976/
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## Homework Statement
A car travels a distance of 745 m along a circular track, and the driver finds that she has traveled through an angle of 475°. What is the radius of the track?
## Homework Equations
None that I know. The only thing I know is that the track is circular
## The Attempt at a Solution
I used an example in my texbook which was identical to this one except the distance was 750 m and the angle was 450.
First I :
750/4=187.5( The 4 is from the 4 different quadrants)
Then I:
187.5/2=93.75------>94m
The solution in the back of the book said that the answer was 95 m but I'm not to sure if this is the right procedure.
This is the last homework problem that I have to complete before I turn it in before midnight.
Last edited:
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2020-05-29 08:16:01
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https://datumorphism.leima.is/til/math/symmetry-of-second-derivatives/
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Symmetry of second derivatives
This wikipedia page shows an example of symmetry breaking of the second derivatives, that is $\partial_x\partial_y f \ne \partial_y\partial_x f$. The example is
$$f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & \mbox{ for } (x,y) \ne (0,0) \\ 0 & \mbox{ for } (x,y) = (0,0). \end{cases}$$
An image of the function is provided by wikipedia.
Modified: by ;
L Ma (2015). 'Symmetry of second derivatives', Datumorphism, 02 April. Available at: https://datumorphism.leima.is/til/math/symmetry-of-second-derivatives/.
Current Ref:
• til/math/symmetry-of-second-derivatives.md
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2021-12-01 12:00:49
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http://www.ams.org/mathscinet-getitem?mr=0118857
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MathSciNet bibliographic data MR118857 33.00 Philip, J. R. The function inverfc $\theta$$\theta$. Austral. J. Phys. 13 1960 13–20. Links to the journal or article are not yet available
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2016-09-27 15:14:21
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