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However, on the top step, we will not have any tread (because the final step is the decking itself!) In addition, we will show you how the stair stringer calculator works out the rise, run, angle and stringer length, allowing you to plan the layout of your intended staircase effectively. Our calculator is programmed to give accurate measurements within the guidance of the Council of American Building Officials and International Code Council. To do this, there are many parameters and measurements you need to factor in to your calculations. The unique KALK.PRO program is online tool with professional stair calculators: accurate drawings and layouts, all dimensions, interactive 3D models, templates, calculation of … Provides a step by step display and a print out option showing the finished stringer with cut dimensions and a materials list. Consider a single step, the total horizontal distance will be given by: $$Total\,Horizontal\,Distance = Run + Tread\,Overlap + Riser\,Thickness = 10.41\,in + 2\,in + 1\,in = 13.41\,in$$. Zoom to adjust screen scale to actual size (measure full scale diagram with ruler) and hit 'Set' to set scale on other pages. Air Change Rates in typical Rooms and Buildings - Fresh air - or make up air - requirements - recommended air change rates - ACH - for typical rooms and buildings - auditoriums, kitchens, churches and more This calculator will tell you: How Many Rises You'll Need Dimensions for each rise And the angle (or pitch!) After input of necessary initial data, you will get detailed drawings of the 90 degree turn stair with an interactive 3D visualisation, all stair dimensions you will need to … Stairs Calculator This calculator is intented as a guide. We simply remember that the final step takes us onto the decking. Council of American Building Officials and International Code Council both recommend the rise to be no more than 7.75 inches (17,78 cm) and runs to be no less than 10 inches (25,4 cm). This calculator can also be used for pales and pickets. Fig. We recommend that you are aware of all building regulations in your home country. Simply put in values for the total run and rise, along with some measurements for the individual steps to calculate the layout of your staircase! 2. These regulations recommend the stair rise to be no more than 7.75 inches (17.78 cm) and stair run to be no less than 10 inches (25.4 cm). The answer is very simple! Total Rise [A] - The total rise of a set of stairs is the vertical distance between the bottom of the first … Find a Store Near Me. Click Slider to select then use keyboard cursor keys, Create and Print Full Scale PDFs with diagrams on this page (templates), Copy all diagrams on this page to bottom of page - Make multiple copies to Print or Compare Results. Our online stair calculator can help make this task a breeze. Related Topics . Enter the thickness of the treads in the field labeled (E) in the stair … And the calculator figures out the spacing for the lengths as before: $$A_1 = Rise × sin\,\alpha_2 = 7.06\,in × sin\,55.86^o = 5.84\,in$$, $$A_2 = Stringer\,Width – A_1 = 12\,in – 5.84\,in = 6.16\,in$$, $$C_2 = \sqrt{Run^2 – {A_1}^2} = \sqrt{(10.41\,in)^2 – (5.84\,in)^2} = 8.62\,in$$, $$C_3 = A_1 × tan\,\alpha_2 = 5.84\,in × tan\,55.86^o = 8.62\,in$$, $$B_1 = \sqrt{Rise^2 – {A_1}^2} = \sqrt{(7.06\,in)^2- (5.84\,in)^2} = 3.96\,in$$, $$B_2 = A_2 × tan\,\alpha_1 = 6.16\,in × tan\,34.14^o = 4.17\,in$$, $$B_3 = (Rise -Tread\,Thickness) × cos\,\alpha_2 = (7.06\,in – 2\,in) × cos\,55.86^o = 2.84\,in$$, $$A_3 = (Rise – Tread\,Thickness) × cos\,\alpha_1 = (7.06\,in – 2\,in) × cos\,34.14^o = 4.19\,in$$, $$A_4 = Stringer\,Width – A_3 = 12\,in – 4.19\,in = 7.81\,in$$, $$C_4 = A_4 × tan\,\alpha_2 = 7.81\,in × tan\,55.86^o = 11.52\,in$$. dimensions on the layout plan. Explore a number of building and housing related calculators, as well as hundreds of other calculators involving topics such as finance, math, fitness, health, and more. Set the Stair direction. To use our stair calculator, you need to enter the following measurements as a minimum: Using these values, our calculator works out the vertical height rise of each individual step with the fraction: $$Rise = {Total\,Rise \over Number\,of\,Steps}$$. Calculate steps unit rise and run in stairs. Staircase calculator with landing should be considered in the minimum scale in a 3’ square landing. Example: Stair No 300-2 Bottom flight added to top flight has a 2.461m+2.641m = 5.102m total run around the wall. We will show you how this can be done easily. Warning! Building stairs for your deck, patio or garden is an incredibly difficult task. If you've benefited from this free service please consider supporting: Stair Geometry - Top Tread Under Upper Floor, All Inch inputs and dimensions are actual physical finished sizes, If the US ever goes Metric, you'll be able to use a Meter to measure Meters - A Meter Meter. 3 Example of a spiral stair drawing We would recommend this website as a great starting place: https://codes.iccsafe.org/content/IBC2018P4/chapter-10-means-of-egress#IBC2018P4_Ch10_Sec1011. How to calculate stairs To use our stair calculator, you need to enter the following measurements as a minimum: Total Run – The total horizontal length of the stair stringer Total Rise – The vertical height between the bottom of the first step and the top of the final step in the stair stringer As before, including the tread when considering the run of the steps gives: Thus, our calculator works out the total horizontal distance between the edge of each step. Posted on by . Math isn’t my strongest suit, so I always use a calculator to figure out stuff like the rise and run of stairs. Finally, the calculator works out the total stringer length as before: $$Stringer\,Length = C_3 + B_3 + (Number\,of\,Steps – 1) × (B_1 + C_2) = 8.62\,in + 2.84\,in + (17-1) × (3.96\,in + 8.62\,in) = 212.72\,in$$. No! To calculate a spiral staircase, a half spiral, quarter spiral, helical or even straight with a curvature, you just need to think of it as straight and apply the formulas to the pitch line The pitch line is 50cm from the interior handrail, or in the middle, if the clearance width is less than a meter. You will need approximately 4' of carpet to cover the width, to allow the carpet to flow through the posts and wrap around the outside of the stair. Calculate the number of spindles needed and the space in-between the spindles. Thus, our deck staircase calculator can figure out the total horizontal distance between the edge of each step (not forgetting that the final step is the decking! We shall take you through the example calculations for first a standard mount, and then a flush mount set of stairs. You can calculate and create detailed and precise stairs plans. As before, we calculate the vertical height rise: Unlike the standard mount measurement, where we subtracted the final step from the rise, with a flush mount we have that: $$Rise\,of\,steps = Total\,Rise = 120\,in$$. Calculating the stringer length is the same method as for the standard mount. However, a good guide is that any business or church, organization, office or commercial property should meet the given stair standards of your home country. Make sure you don’t round your answer before the final calculation. Now … Now, using a standard mount staircase, the vertical height rise of the steps will be one step less than with a flush mount because the final step is taken up by the decking. A stair that has one open side with railing posts on it is known as a cap stair. Before laying out your tools and getting started, you need to make sure each step will be identical in both size and shape. Order custom made glass balutrades for your stairs or landing. Re calculate the tow stairs in the sub floor height based on your plan. 1 General drawing of the staircase Y – the height of the opening (total rise) – usually determined by the distance between the floors in your house. See them in 2D from side to top. Number of Tread = Length of Stair/One number of Tread. Total area for all treads and rises 79.92ft² ~ 8.88y² To calculate and display upper floor opening and stair head-room, enter upper floor thickness, check the Show Head-room check-box and drag the Head slider to calculate and animate diagram head-room and floor opening. Link to Lowe's Home Improvement Home Page Lowe's Credit Cards Order Status … But if I used a standard calculator, I’d have to enter measurements as decimals and convert the answers back into fractions, a process guaranteed to generate some bizarre numbers. The calculator works out all these distances using the method given below: $$A_1 = Rise × sin\,\alpha_2 = 7.06\,in × sin\,57.46^o = 5.95\,in$$, $$A_2 = Stringer\,Width – A_1 = 12\,in – 5.95\,in = 6.05\,in$$, $$C_2 = \sqrt{Run^2 – {A_1}^2} = \sqrt{(11.06\,in)^2 – (5.95\,in)^2} = 9.32\,in$$, $$C_3 = A_1 × tan\,\alpha_2 = 5.95\,in × tan\,57.46^o = 9.33\,in$$, $$B_1 = \sqrt{Rise^2 – {A_1}^2} = \sqrt{(7.06\,in)^2- (5.95\,in)^2} = 3.8\,in$$, $$B_2 = A_2 × tan\,\alpha_1 = 6.05\,in × tan\,32.54^o = 3.86\,in$$, $$B_3 = (Rise -Tread\,Thickness) × cos\,\alpha_2 = (7.06\,in – 2\,in) × cos\,57.46^o = 2.72\,in$$, $$A_3 = (Rise – Tread\,Thickness) × cos\,\alpha_1 = (7.06\,in – 2\,in) × cos\,32.54^o = 4.26\,in$$, $$A_4 = Stringer\,Width – A_3 = 12\,in – 4.27\,in = 7.74\,in$$, $$C_4 = A_4 × tan\,\alpha_2 = 7.74\,in × tan\,57.46^o = 12.12\,in$$. Therefore, you are enabled to construct the stairs yourself and don’t worry if the dimensions are correct. Mar 17, 2017 - This free stair calculator determines stair parameters such as rise, total run, and angle, stringer length, based on height, run, tread, and headroom requirement. Method 1 : The walkine is located 50 cm (20") from the inner handrail or in the centre if the stair width is less than 1m (40"). ... You may calculate the length and height of the tread or rise of the staircase by using the following calculator. Effectively plan your project with the many helpful calculators at Lowes.com. Design & Buy Glass Balustrades. The calculator first figures out the angles: $$\alpha_2 = 90^o – \alpha_1 = 90^o – 34.14^o = 55.86^o$$, $$\alpha_3 = 90^o + \alpha_2 = 90^o + 55.86^o = 145.86^o$$, $$\alpha_5 = 90^o + \alpha_1 = 90^o + 34.14^o = 124.14^o$$. Real-time graphics. Headroom Check. In doing so, we shall calculate the dimensions of the steps and the staircase. The results returned are the number of steps … Built something using our calculators? Thus, we calculate: $$Rise\,of\,steps = Total\,Rise – Rise = 120\,in – 7.06\,in = 112.94\,in$$. Stair Calculator To Work Out Run and Rise Lengths, Free Online Tool Use our easy calculator to work out the rise and run (going) lengths of straight length stairs. Section 1011 has lots of information from the 2018 international building code. https://codes.iccsafe.org/content/IBC2018P4/chapter-10-means-of-egress#IBC2018P4_Ch10_Sec1011, Total Run – The total horizontal length of the stair stringer, Total Rise – The vertical height between the bottom of the first step and the top of the final step in the stair stringer, Stringer Width – The total diagonal measurement between the edge of the step and the back line, For both our examples below, we imagine we want to have 17 steps running up to a landing with a total vertical height rise of 120 inches and total horizontal run of 180 inches, Let’s say that we have a tread thickness of 2 inches. . X – width of the opening (total run) – depends on how much space you can allocate in … To calculate and display upper floor opening and stair head-room, enter upper floor thickness, check the Show Head-room check-box and drag the Head slider to … Download this app from Microsoft Store for Windows 10 Mobile, Windows Phone 8.1, Windows Phone 8. Online service Kalk.pro calculates L-shaped Stair with landing (quarter turn staircase) according to your stairwell parameters. Method 1 : The walkine is located 50 cm (20") from the inner handrail or in the centre if the stair width is less than 1m (40"). Then, build the landing working as the determination before. Calculator Design Analysis. Раз-два-раз, вовочка *******. between the horizontal and the staircase is calculated by: $$tan\,\alpha_1 = {Rise\,of\,steps \over Run\,of\,steps} → \alpha_1 = tan^{-1} {Rise\,of\,steps \over Run\,of\,steps} = tan^{-1} {112.94\,in \over 177\,in} = 32.54^o$$, Now, to find the stringer length of the staircase requires some more complicated Mathematics. The Pear Stairs StairCreator is a free, revolutionary staircase design tool that gives you the ability to design, plan and price a staircase in minutes. Deciding to use one mount over the other will depend on your individual situation and DIY project. With the KALK.PRO stairs calculators you can determine all- important stair dimensions - simply and quickly. The deck stair calculator thus works out: $$Run\,of\,steps = Total\,Run – (Tread\,Overlap + Riser\,Thickness) = 180\,in – (2\,in + 1\,in) = 177\,in$$. Как слышно? Loading… Call: 01938 553311 See more ideas about Stairs stringer, Building stairs, Stairs. Doing this work is much easier using metric measurements. You should check with the building regulations of your home country for an exact answer. The first stair start form the bottom to the landing and the second stair design form the landing to the top. Method 2 : The walkline is located half-way along the stair width Method 3 : The walkline is located 2/3 of the way along the stair width, measured towards the inner handrail Method 4 : The walkline is located 50cm (20") from the outer handrail Use our calculator, do some research and follow the above examples to make sure you are meeting them before getting started! Our calculator can work in many different units at the same time. See screenshots, read the latest customer reviews, and compare ratings for Stairs Calculator. Start by selecting your stair type Go to the Online Planner. Stair Calculator with Landings for 7 3/4″ Max Risers. Easy to use free stair calculator by EZ Stairs. 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My Math Forum Radius of Gyration and Standard Deviation

 November 12th, 2007, 12:20 PM #1 Newbie   Joined: Nov 2007 From: South America Posts: 6 Thanks: 0 Radius of Gyration and Standard Deviation In statics, the radius of gyration of a plane rectangular cross-section, for example, is the root-mean-square of the ratio of the distance, from the neutral axis, of each fiber to the distance of the extreme fiber, and in this case, ends up being 1.15 times as great as the mere mean. However, the meaning is not selected arbitrarily. The geometry of the section imposes that the distance to each fiber be factored in twice, hence the root-mean-square rather than the simple mean. So, anyway, it is the standard deviation of the distances. I see the standard deviation used as a measure of dispersion in statistics, which I've never really studied, for example in IQ distributions. However, this lends added weight to the data points that are farther away from the mean. But why should the farther points carry more weight? I've never been able to fathom what the standard deviation tells me that the mere mean deviation wouldn't tell me. In other words, using the standard deviation in this context has always seemed to me an arbitrary practise. Can anyone explain to me any considerations that compel the choice, as they do in the case of the radius of gyration.
 November 12th, 2007, 05:44 PM #2 Member   Joined: Aug 2007 Posts: 93 Thanks: 0 While the MAD is a useful statistic, it is difficult to use in analytical formulas describing probability distributions. I do not know how MAD could generate higher moments like continuing past squared to cubed deviations and so on. It is also debatable which better gives a proper weight to the tails of a distribution.
 November 14th, 2007, 06:06 AM #3 Newbie   Joined: Nov 2007 From: South America Posts: 6 Thanks: 0 Thank you. You seem to be agreeing that the use of the standard deviation is a somewhat arbitrary choice, being perhaps preferable without being absolutely mandatory by the logic of the thing.
 February 23rd, 2017, 12:16 AM #4 Newbie   Joined: Feb 2017 From: Mumbai Posts: 1 Thanks: 0 When we ask for standard deviation, we need how close/far are the points located (squared average: standard deviation) from the mean and not how balanced (simple average) the whole system is! Now why square and not cube/higher orders for that matter? That's simply because the distance formula mandates squares.
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 Originally Posted by Indraneel When we ask for standard deviation, we need how close/far are the points located (squared average: standard deviation) from the mean and not how balanced (simple average) the whole system is! Now why square and not cube/higher orders for that matter? That's simply because the distance formula mandates squares.
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## The Annals of Applied Statistics

### Hypothesis testing for high-dimensional multinomials: A selective review

#### Abstract

The statistical analysis of discrete data has been the subject of extensive statistical research dating back to the work of Pearson. In this survey we review some recently developed methods for testing hypotheses about high-dimensional multinomials. Traditional tests like the $\chi^{2}$-test and the likelihood ratio test can have poor power in the high-dimensional setting. Much of the research in this area has focused on finding tests with asymptotically normal limits and developing (stringent) conditions under which tests have normal limits. We argue that this perspective suffers from a significant deficiency: it can exclude many high-dimensional cases when—despite having non-normal null distributions—carefully designed tests can have high power. Finally, we illustrate that taking a minimax perspective and considering refinements of this perspective can lead naturally to powerful and practical tests.

#### Article information

Source
Ann. Appl. Stat., Volume 12, Number 2 (2018), 727-749.

Dates
Revised: February 2018
First available in Project Euclid: 28 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1532743474

Digital Object Identifier
doi:10.1214/18-AOAS1155SF

Mathematical Reviews number (MathSciNet)
MR3834283

#### Citation

Balakrishnan, Sivaraman; Wasserman, Larry. Hypothesis testing for high-dimensional multinomials: A selective review. Ann. Appl. Stat. 12 (2018), no. 2, 727--749. doi:10.1214/18-AOAS1155SF. https://projecteuclid.org/euclid.aoas/1532743474

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text:
## 😴 Sleep function in Go

##### April 12, 2021
introduction goroutine time

To pause the execution of a current program in Go, you just need to use Sleep() function defined in time package. As an argument, this function takes a variable of type time.Duration, which is the amount of time the program execution should be stopped for. It can be expressed as a number multiplied by a unit constant, for example 3*time.Second means that the execution will be stopped for 3 seconds. Available units are:

• time.Nanosecond
• time.Microsecond
• time.Millisecond
• time.Second
• time.Minute
• time.Hour
package main

import (
"fmt"
"time"
)

func main() {
fmt.Println("before Sleep()")

time.Sleep(3 * time.Second)

fmt.Println("waking up after Sleep()")
}

#### ✒ Write to a CSV file in Go

##### Learn how to write data to a CSV or TSV file
introduction file

#### 🙌 Case-insensitive string comparison in Go

##### Learn how to compare strings in a case-insensitive manner
introduction strings

#### 🏁 Check if a string starts with a substring in Go

##### Learn how to use strings.HasPrefix() function
introduction strings


text:
# Thread: Convergence Tests of Series

1. ## Convergence Tests of Series

My lecturer tends to skip a lot of steps in his model answers so I'm unsure of some things. (I've written out his model answers so it's tidier)

Where do the sqrt(4n) and 3sqrt(n) come from? I can't see any logical connection.

2. ## Re: Convergence Tests of Series

For large enough $n$ we have $4n > n+1$. Admittedly, $n$ doesn't have to be very large for this to be true.

And (for all positive $n$) $\sqrt{4n}=2\sqrt{n}$.

3. ## Re: Convergence Tests of Series

EDIT:// posted the same as Archie. Archie's answer is equivalent to mine (and better)

No skipped steps.

Compare sqrt(n+1) with sqrt(4n) for natural numbers n. Which one is greater? Using that inequality, you can deduce the inequality of line 3. I'll leave it to you as an exercise. Try it. Trust me. It's beneficial in the long run if you prove it yourself.

3sqrt(n) is the sum of sqrt(4n) + sqrt(n) = 2sqrt(n) + sqrt(n). The process of having a fraction such as 1/3sqrt(n) is that if the sum is greater than this fraction, and the fraction is greater than 0, then clearly the sum is greater than 0.


text:
# dots

0th

Percentile

##### Kernel Functions

The kernel generating functions provided in kernlab. The Gaussian RBF kernel $k(x,x') = \exp(-\sigma \|x - x'\|^2)$ The Polynomial kernel $k(x,x') = (scale <x, x'> + offset)^{degree}$ The Linear kernel $k(x,x') = <x, x'>$ The Hyperbolic tangent kernel $k(x, x') = \tanh(scale <x, x'> + offset)$ The Laplacian kernel $k(x,x') = \exp(-\sigma \|x - x'\|)$ The Bessel kernel $k(x,x') = (- Bessel_{(\nu+1)}^n \sigma \|x - x'\|^2)$ The ANOVA RBF kernel $k(x,x') = \sum_{1\leq i_1 \ldots < i_D \leq N} \prod_{d=1}^D k(x_{id}, {x'}_{id})$ where k(x,x) is a Gaussian RBF kernel. The Spline kernel $\prod_{d=1}^D 1 + x_i x_j + x_i x_j min(x_i, x_j) - \frac{x_i + x_j}{2} min(x_i,x_j)^2 + \frac{min(x_i,x_j)^3}{3}$ \ The String kernels (see stringdot.

Keywords
symbolmath
##### Usage
rbfdot(sigma = 1)polydot(degree = 1, scale = 1, offset = 1)tanhdot(scale = 1, offset = 1)vanilladot()laplacedot(sigma = 1)besseldot(sigma = 1, order = 1, degree = 1)anovadot(sigma = 1, degree = 1)splinedot()
##### Arguments
sigma

The inverse kernel width used by the Gaussian the Laplacian, the Bessel and the ANOVA kernel

degree

The degree of the polynomial, bessel or ANOVA kernel function. This has to be an positive integer.

scale

The scaling parameter of the polynomial and tangent kernel is a convenient way of normalizing patterns without the need to modify the data itself

offset

The offset used in a polynomial or hyperbolic tangent kernel

order

The order of the Bessel function to be used as a kernel

##### Details

The kernel generating functions are used to initialize a kernel function which calculates the dot (inner) product between two feature vectors in a Hilbert Space. These functions can be passed as a kernel argument on almost all functions in kernlab(e.g., ksvm, kpca etc).

Although using one of the existing kernel functions as a kernel argument in various functions in kernlab has the advantage that optimized code is used to calculate various kernel expressions, any other function implementing a dot product of class kernel can also be used as a kernel argument. This allows the user to use, test and develop special kernels for a given data set or algorithm. For details on the string kernels see stringdot.

##### Value

Return an S4 object of class kernel which extents the function class. The resulting function implements the given kernel calculating the inner (dot) product between two vectors.

kpar

a list containing the kernel parameters (hyperparameters) used.

The kernel parameters can be accessed by the kpar function.

##### Note

If the offset in the Polynomial kernel is set to $0$, we obtain homogeneous polynomial kernels, for positive values, we have inhomogeneous kernels. Note that for negative values the kernel does not satisfy Mercer's condition and thus the optimizers may fail.

In the Hyperbolic tangent kernel if the offset is negative the likelihood of obtaining a kernel matrix that is not positive definite is much higher (since then even some diagonal elements may be negative), hence if this kernel has to be used, the offset should always be positive. Note, however, that this is no guarantee that the kernel will be positive.

stringdot, kernelMatrix , kernelMult, kernelPol

##### Aliases
• dots
• kernels
• rbfdot
• polydot
• tanhdot
• laplacedot
• besseldot
library(kernlab) # NOT RUN { rbfkernel <- rbfdot(sigma = 0.1) rbfkernel kpar(rbfkernel) ## create two vectors x <- rnorm(10) y <- rnorm(10) ## calculate dot product rbfkernel(x,y) # }


text:
## File: pathfield.h

package info (click to toggle)
widelands 1:19+repack-3
 12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394 /* * Copyright (C) 2008-2010 by the Widelands Development Team * * This program is free software; you can redistribute it and/or * modify it under the terms of the GNU General Public License * as published by the Free Software Foundation; either version 2 * of the License, or (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. * */ #ifndef WL_LOGIC_PATHFIELD_H #define WL_LOGIC_PATHFIELD_H #include #include #include #include #include "logic/cookie_priority_queue.h" namespace Widelands { /** * Used in pathfinding. For better encapsulation, pathfinding structures * are separate from normal fields * * Costs are in milliseconds to walk. * * Note: member sizes chosen so that we get a 16byte (=nicely aligned) * structure */ struct Pathfield { struct LessCost { bool operator()(const Pathfield& a, const Pathfield& b) const { return a.cost() < b.cost(); } }; using Queue = CookiePriorityQueue; Queue::Cookie heap_cookie; int32_t real_cost; // true cost up to this field int32_t estim_cost; // estimated cost till goal uint16_t cycle; uint8_t backlink; // how we got here (WALK_*) int32_t cost() const { return real_cost + estim_cost; } Queue::Cookie& cookie() { return heap_cookie; } }; struct Pathfields { std::unique_ptr fields; uint16_t cycle; Pathfields(uint32_t nrfields); }; /** * Efficiently manages \ref Pathfields instances. * * This allows the use of more than one such structure at once, * which is required for pathfinding reentrancy. */ struct PathfieldManager { PathfieldManager(); void set_size(uint32_t nrfields); boost::shared_ptr allocate(); private: void clear(const boost::shared_ptr& pf); using List = std::vector>; uint32_t nrfields_; List list_; }; } #endif // end of include guard: WL_LOGIC_PATHFIELD_H


text:
Dihedral Groups

(Difference between revisions)
 Revision as of 15:55, 16 July 2011 (edit)← Previous diff Revision as of 15:58, 16 July 2011 (edit) (undo)Next diff → Line 326: Line 326: {{EquationRef2|Subgroup|Lemma 3.}}Given $d \mid n$, let $m = n/d$. For every $0 \leq i < n$ let $A(i,d) = \{ab^{i+km}: \ 0 \leq k < d \}$, where $a$ denotes a reflection and $b$ denotes a primitive rotation. Let $B(i,d) = A(i,d) \cup \langle b^m \rangle$. Then $B(i,d)$ is a subgroup of $D_{n}$ and $|B(i,d)|=2d$. We also have $|\{B(i,d) : \ 0 \leq i < n \}|=m=n/d.$ {{EquationRef2|Subgroup|Lemma 3.}}Given $d \mid n$, let $m = n/d$. For every $0 \leq i < n$ let $A(i,d) = \{ab^{i+km}: \ 0 \leq k < d \}$, where $a$ denotes a reflection and $b$ denotes a primitive rotation. Let $B(i,d) = A(i,d) \cup \langle b^m \rangle$. Then $B(i,d)$ is a subgroup of $D_{n}$ and $|B(i,d)|=2d$. We also have $|\{B(i,d) : \ 0 \leq i < n \}|=m=n/d.$ {{HideThis|1=The Proof|2= {{HideThis|1=The Proof|2= - '''Proof.''' If $ab^{i+km}=ab^{i+rm}$, for some $0 \leq k,r < d$, then $b^{(k-r)m}=1$ and thus $d \mid k-r$, because $o(b)=n=md$. + '''Proof.''' If $ab^{i+km}=ab^{i+rm}$, for some $0 \leq k,r < d$, then $b^{(k-r)m}=1$ and thus $d \mid k-r$, because $|b|=n=md$. Therefore $k=r$ because $0 \leq k,r < d$. Therefore $k=r$ because $0 \leq k,r < d$. Line 332: Line 332: So we have proved that $|A(i,d)|=d$. So we have proved that $|A(i,d)|=d$. - Clearly $A(i,d) \cap \langle b^m \rangle = \emptyset$ and $|\langle b^m \rangle = d$, because $o(b)=n$. + Clearly $A(i,d) \cap \langle b^m \rangle = \emptyset$ and $|\langle b^m \rangle = d$, because $|b|=n$. Thus $|B(i,d)|=|A(i,d)| + |\langle b^m \rangle|=2d$. Thus $|B(i,d)|=|A(i,d)| + |\langle b^m \rangle|=2d$.

Revision as of 15:58, 16 July 2011

Dihedral Symmetry of Order 12
Each snowflake in the main image has the dihedral symmetry of a natual regular hexagon. The group formed by these symmetries is also called the dihedral group of degree 6. Order refers to the number of elements in the group, and degree refers to the number of the sides or the number of rotations.

Basic Description

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

Dihedral groups arise frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups of symmetry. Chrysler’s logo has $D_5$ as a symmetry group, and that Mercedes-Benz has $D_3$. The ubiquitous five-pointed star has symmetry group $D_5$.

Notation

There are two different kinds of notation for a dihedral group associated to a polygon with $n$ sides.

In geometry, we usually call it $D_n$ or $Dih_n$, where $n$ indicates the number of the sides.

In algebra, we call it $D_2n$, where $2n$ indicates the number of elements in the group.

On this page, we will use the notation $D_n$ to describe a dihedral group. For $D_n$, we will call it the dihedral group of order $2n$ or the group of symmetries of a regular $n$-gon.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Basic Abstract Algebra

Elements

The $n^\text{th}$ dihedral group is the symmetry group of t [...]

Elements

The $n^\text{th}$ dihedral group is the symmetry group of the regular $n$-sided polygon. The group consists of $n$ reflections, $n-1$ rotations, and the identity transformation.

Here is an example of $D_6$. This group contains 12 elements, which are all rotations and reflections. The very first one is the identity transformation.

Image 1

If $n$ is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If $n$ is even there are $\frac{n}{2}$ axes of symmetry connecting the mid-points of opposite sides and $\frac{n}{2}$ axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and $2n$ elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. In Image 1, through $S_0$ to $S_5$ are the axes of symmetries. All the reflections can be described as reflections of the identity through six axes of symmetries.

Definition

There are several different way to define a Dihedral Group. We will introduce three of them.

We will use $R_0$ to represent the identity, $R_k, k=1,2,\cdots,n-1$, to represent the rotations, and $S_k, k=0,1, \cdots, n-1$, to represent the reflections.

Complex Plane Presentation

For $n \geqslant 3$, the dihedral group $D_n$ is defined as the rigid motions of the plane preserving a regular $n$-gon, with the operation of composition. Our model $n$-gon will be an $n$-gon centered at the origin, with vertices at the n-th roots of unity. $1$ is always an $n$-th root of unity, but $-1$ is such a root only if $n$ is even. In general, the roots of unity form a regular polygon with $n$ sides, and each vertex lies on the unit circle.

The $n$-th roots of unity are roots $\omega = e^{\frac{2 \pi i}{n}}$ of the cyclotomic equation $x^n=1$.

Since a vector on the complex plane can be described as $p=a+bi$, a vector with an angle $\theta$ counterclockwise from x-axis can be described as $p=r\cos \theta +i r\sin \theta$, where $r=\sqrt{a^2+b^2}$ is the magnitude of the vector.

Leonhard Euler's formula says that $e^{i \theta}=\cos \theta +i \sin \theta$, so any point on complex plane is $p=re^{i \theta}$.

For a regular $n$-gon, the first angle counterclockwise from the x-axis is $\frac{2 \pi}{n}$, so the primitive root of unity is $\omega = e^{\frac{2 \pi i}{n}}$.

Letting $\omega = \frac{2 \pi i}{n}$ denote a primitive $n^ \text{th}$ root of unity, and assuming the polygon is centered at the origin, the rotations $R_k$, $k=0, 1, 2, \cdots, n-1$(Note: $R_0$ denotes the identity), are given by

$R_k:z \mapsto \omega^k z,\quad z\in\cnums.$

Notation $\mapsto$ means a function. For example: $input \mapsto output$.

For the reflections, $S_k$, $k=0, 1, 2, \cdots, n-1$, the functions are given by

$S_k: z\mapsto \omega^k \bar{z},\quad z\in\cnums.$

Group Presentation

A presentation of a group is a description of a set $I$ and a subset $R$ of the free group $F(I)$ generated by $I$, written as $\left \langle (x_i)_{i \in I}|(r)_{r \in R} \right \rangle$, where the equation $r=1$ (the identity element) is often written in place of the element $r$. A group presentation defines the quotient group of the free group $F(I)$ by the normal subgroup generated by $R$, which is the group generated by the generators $x_i$ subject to the relations $r \in R$.

We can use the presentation:

$D_n = \left \langle r,s|r^n=1,s^2=1,srs=r^{-1} \right \rangle$, or

$D_n = \left \langle x,y|x^2=1,y^n=1,(xy)^2=1 \right \rangle$ to define a group, isomorphic to the dihedral group $D_n$ of finite order $2n$, which is the group of symmetries of a regular $n$-gon.

We will use the second presentation, in which $x$ refers to a reflection, and $y$ refer to a primitive rotation.

For $x^2=1$, $x$ means an arbitrary mirror image of the $n$-gon, and $1$ means the identity. This equation means that if we reflect the $n$-gon once, you get $x$. If reflect the $n$-gon twice, the result will return to the identity.
For $y^n=1$, the equation means that $y$ is a rotation, and its $n$th power $y^n$ equals the identity. That is, if we rotate the $n$-gon $n$ times, we get back to the identity.
For $(xy)^2=1$, $xy$ means the mirror image of $y$. Reflecting the $n$-gon through the axis of symmetry of $xy$ twice, the result is the identity.

Following the group presentation, we can label all the reflections and rotations in terms of $x$ and $y$.

• Identity: $1$
• Rotations: $y, y^2, y^3, \cdots, y^{n-1}$, and $y^n=1$, which is the identity.
• Reflections: $x, xy, xy^2, \cdots, xy^{n-1}$. There is not $xy^n$, because $y^n=1$, and so $xy^n=x$ is the reflection of the identity.

Matrix Representation

If we center the regular polygon at the origin, then the elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of $D_n$ as matrices, with composition being matrix multiplication. This is an example of a (2-dimensional) group representation.

For example, the elements of the group $D_4$ can be represented by the following eight matrices:

$\begin{matrix} R_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & R_1=\bigl(\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}\bigr), & R_2=\bigl(\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}\bigr), & R_3=\bigl(\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}\bigr), \\[1em] S_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}\bigr), & S_1=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr), & S_2=\bigl(\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & S_3=\bigl(\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}\bigr). \end{matrix}$

If we represent the columns of each matrix as basis vectors, we can observe directly all the rotations and reflections.

In general, we can write any dihedral group as:

$R_k = \begin{bmatrix} \cos \frac{2\pi k}{n} & -\sin \frac{2\pi k}{n} \\ \sin \frac{2\pi k}{n} & \cos \frac{2\pi k}{n} \end{bmatrix}$,

$S_k = \begin{bmatrix} \cos \frac{2\pi k}{n} & \sin \frac{2\pi k}{n} \\ \sin \frac{2\pi k}{n} & -\cos \frac{2\pi k}{n} \end{bmatrix}$,

where $R_k$ is a rotation matrix, expressing a counterclockwise rotation through an angle of $\frac{2 \pi k}{n}$, and $S_k$ is a reflection across a line that makes an angle of $\frac{\pi k}{n}$ with the x-axis.

Linear Transportation

Since the dihedral groups also act by linear transformations:

$(x,y)\to(\hat{x},\hat{y}),\quad (x,y)\in \reals^2$

there is a corresponding action on polynomials $p\to\hat{p}$ , defined by

$\hat{p}(\hat{x},\hat{y}) = p(x,y),\quad p\in \reals[x,y].$

Properties

Cayley Table

Cayley Table

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.

A Cayley table, named after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.

The Cayley Table on the right shows the effect of composition in the group $D_6$ (the symmetries of a hexagon). $R_0$ denotes the identity; $R_1$ to $R_5$ denote counterclockwise rotations by 60, 120, 180, 240,and 300 degrees; and $S_0$ to $S_5$ denote reflections across the six diagonals. In general, $ab$ denotes the entry at the intersection of the row with $a$ at the left and the column with $b$ at the top.

In the table, the same or different rotations and reflections work together and result in a new rotation or reflection. For example, look first at the vertical axis to find a element, $R_4$. Then look at the horizontal axis to get the second element for our composition. We choose $S_3$. Composing two elements is just the progression of a rotation or a reflection followed by another rotation or a reflection. In this case, our elements are $R_4$ and $S_3$. First we rotate the hexagon counterclockwise 240 degrees, and then reflect it along the axis of symmetry of $S_3$. The result is the same as reflecting the identity transformation through an angle of 60 degrees, which is $S_1$. See Example 1 below.

Now, look back to Cayley Table, you will find that the intersection of $R_4$ and $S_3$ is $S_1$. If you like, you can create your own Cayley table for a dihedral group of any order and find the natual rule for it.

Explore the Cayley Table

Perhaps the most important feature of this table is that it has been completely filled in without introducing any new motions.

• Closure: Algebraically, this says that if $A$ and $B$ are in $D_6$, then so is $AB$. This property is called closure, and it is one of the requirements for a mathematical system to be a group.
• identity: Notice that if $A$ is any element of $D_6$, then $AR_0=R_0 A=A$. Thus, combining any element $A$ on either side with $R_0$ yields $A$ back again. An element $R_0$ with this property is called an identity, and every group must have one.
• Inverse: We see that for each element $B$ in $D_6$, there exists an element $A$ such that $AB=BA=R_0$. In this case, $B$ is said to be the inverse of $A$ and vise versa. The term inverse is a descriptive one, for if $A$ and $B$ are inverses of each other, then $B$ "un-does" whatever $A$ "does", in the sense that $A$ and $B$ taken together in either order produce $R_0$, representing no change.
• Non-Abelian: Another property of $D_6$ deserves special comment. Obverse that $S_1 R_3=R_3 S_1$, but $S_3 R_4 \neq R_4 S_3$. Thus in a group $ab$ may or may not be the same as $ba$. If it happens that $ab=ba$ for all choices of group elements $a$ and $b$, we say the group is commutative or --better yet-- Abelian (in honor of the great Norwegian mathematician Niel Abel). Otherwise, we say the group is non-Abelian. All dihedral groups are non-Abelian, except $D_1$ and $D_2$.
• Associativity: For all dihedral groups, it holds true that $(ab)c=a(bc)$ for all a, b, c in the group.
Multiplication Table

If we want to know what is the composition of any two elements, it is convenient to use aCayley Map, because it tells us the result directly. But when we want to know the gradual change of the compositions, we will need another tool, a multi-colored Muliplication Table.

Muptiplication Table

In this multi-colored Muliplication Table, each color represents one rotation or reflection. In the image, pink colors represent rotations, and the deepest pink represents the identity transformation. Green colors represent all the six reflections.

From the changing of color, we can observe that the gradual change of the composition of two elements in $D_6$. However, we cannot easily determine the exact result of composition by observing directly this multi-colored Muliplication Table.

The abstract group structure is given by:

$R_k R_l=R_{k+l}$

$S_k S_l=R_{k-l}$

$R_k S_l=S_{k+l}$

$S_k R_l=S_{k-l}$

Uniqueness of the Identity

In a dihedral group $D_n$, there is only one identity element.

PROOF: Suppose both $R_0$ and $R_0'$ are identities of $D_n$. Then,

Eq. 1         $aR_0=a$ for all $a$ in $D_n$, and
Eq. 2         $R_0' a=a$ for all $a$ in $D_n$.

The choices of $a=R_0'$ in Eq. 1 and $a=R_0$ in Eq. 2 yield $R_0' R_0=R_0'$ and $R_0' R_0=R_0$.

Thus, $R_0$ and $R_0'$ are both equal to $R_0' R_0$ and so are equal to each other.

Cancellation

In a dihedral group $D_n$, the right and left cancellation laws hold; that is, $ba=ca$ implies $b=c$, and $ab=ac$ implies $b=c$.

PROOF: Suppose $ba=ca$. Let $a'$ be an inverse of $a$.

Then, muliplying on the right by $a'$ yields $(ba)a'=(ca)a'$.

Associativity yields $b(aa')=c(aa')$.

Then, $bR_0=cR_0$ and, therefore, $b=c$ as desired.

Similarly, one can prove that $ab=ac$ implies $b=c$ by multiplying by $a'$ on the left.

A consequence of the cancellation property is the fact that in a Cayley table for a dihedral group, each group element occurs exactly once in each row and column. Another consequence of the cancellation property is the uniqueness of inverses.

Uniqueness of Inverses

For each element $a$ in a dihedral group $D_n$, there is a unique element $b$ in $D_n$ such that $ab=ba=R_0$.

PROOF: Suppose $b$ and $c$ are both inverses of $a$.

Then $ab=R_0$ and $ac=R_0$, so that $ab=ac$.

Canceling the $a$ on both sides gives $b=c$, as desired.

Sock-Shoes property

For dihedral group elements $a$ and $b$, $(ab)^{-1}=b^{-1} a^{-1}$.

PROOF: Since $(ab)(ab)^{-1}=R_0$ and

$(ab)(b^{-1} a^{-1})=a(bb^{-1})a^{-1}=aR_0a^{-1}=aa^{-1}=R_0$,

we have by the Uniqueness of Inverses theorem that $(ab)$ has only one inverse $x$ such that $(ab)x=R_0$.

We get $x=(ab)^{-1}=(b^{-1} a^{-1})$.

3D Rotational Symmetry

An example of abstract group $D_n$, and a common way to visualize it, is the group $D_n$ of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. $Dn$ consists of $n$ rotations of multiples of $\frac{360^\circ}{n}$ about the origin, and reflections across $n$ lines through the origin, making angles of multiples of $\frac{180^\circ}{n}$ with each other. This is the symmetry group of a regular polygon with $n$ sides (for $n \geqslant 3$, and also for the degenerate case $n = 2$, where we have a line segment in the plane).

If we put a dihedral group in three dimensions, the reflections are also rotations of $180^\circ$

Rotation in 3D

The proper symmetry group of a regular polygon embedded in three-dimensional space (if $n \geqslant 3$). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.

Infinite Dihedral Groups

The infinite dihedral group $Dih(C_\infty)$ is denoted by $D_\infty$. The infinite dihedral group can be described as the group of symmetries of a circle, which has infinite symmetries.

We use the group presentation:

$D_\infty = \langle r, s \mid s^2 = 1, srs = r^{-1} \rangle$, or

$D_\infty = \langle x, y \mid x^2 = y^2 = 1 \rangle$ to represente the infinite dihedral group.

In the presentation, it says that because there are infinitely many symmetries, we can never rotate back to the identity, and so there are infinitely many rotations and reflections.

Subgroups

Definition: A subgroup is a subset $H$ of group elements of a group $G$ that satisfies the four group requirements. It must therefore contain the identity element. "$H$ is a subgroup of $G$" is written as $H \subseteq G$, or sometimes $H \leq G$.[1]

Now we want to know exactly how many subgroups for $D_n$, and what they are. Fortunately, mathematician Stephan A. Cavior had already proved this for us in 1975. In the theorem, for any dihedral group in order of $2n$, there are $\tau(n) + \sigma(n)$ subgroups in total, including $\{1\}$ and $D_{n}$. $\{1\}$ is just the identity itself.

Definitions of Terms

• $\tau(n)$: the number of divisors of $n$,
e.g. $\tau(12)=6$.
• $\sigma(n)$: the sum of divisors of $n$,
e.g. $\sigma(12)=1 + 2 + 3 + 4 + 6 + 12=28$.
• $\mathbb{Z}/n \mathbb{Z}$: the notation for the cyclic group of order $n$, can be also written as $\mathbb{Z}_n$. This is a quotient group presentation.
• $d \mid n$: $d$ is a divisor of $n$.
• $N=\langle b \rangle$: group $N$ is a subgroup generated by $b$. It means $N=\{1,b,b^2,b^3,\cdots,b^{n-1}\}$.
• $|HN|$: the order of $HN$.
• $[D_{n}:N]=2$: the index. $[D_{n}:N]=2$ means $\frac{|D_n|}{|N|}=2$.
• $\{ab^{i+km}: \ 0 \leq k < d \}$: a set with elements looking like $ab^{i+km}$.
• $G(i,d)$: this means a group. $i$ is the index, which labels all the elements; $d$ is the order of the group.

This proof is complicated.

After we know what kind of group can be a subgroup of a dihedral group $D_n$, the dihedral group of order $2n$, we will start to find all subgroups of $D_n$.

Lemma 1.        The number of subgroups of a cyclic group of order $n \geq 1$ is $\tau(n)$. [2]

A cyclic group of order $n$ is the group of all the rotations including the identity of the dihedral group of order $2n$.

Proof. Let $G$ be a cyclic group of order $n$. Then $G \cong \mathbb{Z}/n \mathbb{Z}$. A subgroup of $\mathbb{Z}/n \mathbb{Z}$ is in the form $d \mathbb{Z}/n \mathbb{Z}$ where $d \mathbb{Z} \supseteq n \mathbb{Z}$. The condition $d \mathbb{Z} \supseteq n \mathbb{Z}$ is obviously equivalent to $d \mid n. \$

Lemma 2.        Let $b$ be the element of order $n$ in $D_{n}$ and let $H$ be a subgroup of $D_{n}$. Then either $H \subseteq \langle b \rangle$ or $|H \cap \langle b \rangle| =d$ and $|H|=2d$ for some $d \mid n$.

Proof. Let $N=\langle b \rangle$. Clearly $N$ is a normal subgroup of $D_{n}$ because $[D_{n}:N]=2$. Thus $HN$ is a subgroup of $D_{n}$ and hence the order of dihedral group $HN$ is a divisor of $2n$, and we use the notation:

Eq. 1         $|HN| \mid 2n.$ to represent.

On the other hand,

Eq. 2         $|HN|=\frac{|H| \cdot |N|}{|H \cap N|}=\frac{n |H|}{|H \cap N|}$.
Therefore, by Eq. 1, and Eq. 2. Hence either $|H| = |H \cap N|$ or $|H|=2|H \cap N|$. If $|H|=|H \cap N|$, then $H = H \cap N$ and thus $H \subseteq N$. If $|H|=2|H \cap N|$, then let $|H \cap N|=d$ and so $|H|=2d$. Clearly $d \mid n$ because $H \cap N$ is a subgroup of $N$ and $|N|=n. \$

Lemma 3.        Given $d \mid n$, let $m = n/d$. For every $0 \leq i < n$ let $A(i,d) = \{ab^{i+km}: \ 0 \leq k < d \}$, where $a$ denotes a reflection and $b$ denotes a primitive rotation. Let $B(i,d) = A(i,d) \cup \langle b^m \rangle$. Then $B(i,d)$ is a subgroup of $D_{n}$ and $|B(i,d)|=2d$. We also have $|\{B(i,d) : \ 0 \leq i < n \}|=m=n/d.$

Proof. If $ab^{i+km}=ab^{i+rm}$, for some $0 \leq k,r < d$, then $b^{(k-r)m}=1$ and thus $d \mid k-r$, because $|b|=n=md$.

Therefore $k=r$ because $0 \leq k,r < d$.

So we have proved that $|A(i,d)|=d$.

Clearly $A(i,d) \cap \langle b^m \rangle = \emptyset$ and $|\langle b^m \rangle = d$, because $|b|=n$.

Thus $|B(i,d)|=|A(i,d)| + |\langle b^m \rangle|=2d$.

Proving that $B(i,d)$ is a subgroup of $D_{n}$ is very easy.

Just note that every element of $A(i,d)$ is the inverse of itself (because they all have order two) and also note that $ab^s = b^{-s}a$, for all $s$, because $ab=b^{-1}a$.

Finally, the set $\{B(i,d) : \ 0 \leq i < n \}$ has $m$ elements because clearly $B(i,d)=B(j,d)$ if and only if $A(i,d)=A(j,d)$ if and only if $i \equiv j \mod m. \$

Suppose that $H$ is a subgroup of $D_{n}$. There are two disjoint cases to consider.

Case 1. $H \subseteq \langle b \rangle$.

By Lemma 1. the number of these subgroups is $\tau(n)$.

Case 2. $H \nsubseteq \langle b \rangle$.

In this case, by Lemma 2. we have $|H|=2d$ and $|H \cap \langle b \rangle|=d$, for some $d \mid n$.
Let $n = md$. Since $H \cap \langle b \rangle$ is a subgroup of $\langle b \rangle$, which is a cyclic group of order $n$, we have
Eq. 1        $H \cap \langle b \rangle = \langle b^m \rangle.$
Let $A(i,d)$ and $B(i,d)$ be as were defined in Lemma 3.
Now, since $H$ is not contained in $\langle b \rangle$, there exists some $0 \leq i < n$ such that $ab^i \in H$.
Then, since $H$ is a subgroup, we must have $ab^ib^{km} \in H$, for all $k$.
Thus $ab^{i + km} \in H$ and so $A(i,d) \subseteq H$ and therefore, by Eq. 1, we have $B(i,d) \subseteq H$.
Thus, since $|H|=|B(i,d)|=2d$, we must have $H=B(i,d)$.
The converse is obvously true, i.e. given $d \mid n$ and $0 \leq i < n$, $B(i,d)$ is a subgroup of $D_{n}$, by Lemma 3., and $B(i,d) \nsubseteq \langle b \rangle$ because it contains $A(i,d)$.
So the subgroups in this case are exactly the ones in the form $B(i,d)$, where $0 \leq i < n$ and $d \mid n$.
Thus, by Lemma 3. the number of subgroups in this case is
$\sum_{d \mid n} | \{B(i,d) : \ 0 \leq i < n \} | = \sum_{d \mid n} n/d = \sum_{d \mid n} d = \sigma(n).$

So, by Case 1. and Case 2. the number of subgroups of $D_{n}$ is $\tau(n) + \sigma(n)\$.

Why It's Interesting

In Music

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords.[3]

Dihedral groups as a kind of special symmetric groups are studied in music. In music, we use operations Transportation and Inversion, which are denoted as $T$ and $I$, to represente rotations and reflections in dihedral groups.

Musicians usually study about $D_{12}$, because 12 is a normal cycle in music.C C D E E F F G G A B B, and then C again.

Based on this 12 elements cycle, $D_{12}$ is important in music theory. Musicians use Transportations and Inversions(rotations and reflections) of a simple note to create other notes to complete a final composition.

A transposition of a sequence $x$ of pitch classes by $n$ semitones is the sequence $T^n(x)$ in which each of the pitch classes in $x$ has been increased by $n$ semitones.

So for example if

$x=3 0 8$, which the numbers denote to notes in numbers,

then

$T^4(x)=T^4(3 0 8) = 7 4 0$.

When doing the operation $T^n(x)$, add $n$ to each digit of $x$, and use Modular arithmetic of 12, when the resulting digit is over 12. For instance, add 4 to 8, the result is 12, then $12 \equiv 0 \pmod 12.\,$

Turning to the next operation, inversion $I(x)$ of a sequence $x$ just re-places each pitch class by its negative (in clock arithmetic).

So in the first example above with $x = 3 0 8$, we have

$I(x) = 9 0 4$.

To do the operation $I(x)$, we need to do the opposite way of $T^n(x)$. For instance, if we want to get 12 from 3, we need to add 9. 0 is already 12, so we need to add 0.

References

[2] de Cornulier, Yves. (n.d). Group Presentation. From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/GroupPresentation.html

[4] Milson, Robert and Foregger, Thomas. dihedral group. From PlanetMath.org. June, 12. 2007. Retrieved from http://planetmath.org/encyclopedia/DihedralGroup.html

[5] Gallian, Joseph A. Contemporary Abstract Algebra Seventh Edition. Belmont: Brooks/Cole, Cengage Learning. 2010.

[6] Dahlke, Karl. (n.d). Groups, Dihedral and General Linear Groups. Retrieved from http://www.mathreference.com/grp,dih.html

[7] Sharifi, Yaghoub. Subgroups of dihedral groups (1)&(2). Feb, 17, 2011. Retrieved from http://ysharifi.wordpress.com/2011/02/17/subgroups-of-dihedral-groups-1/

[8] [1]Scott, W. R. Group Theory. New York: Dover, 1987.

[9] [2]Hungerford, Thomas W. Graduate Texts in Mathematics - Algebra. New York: Springer, 1974.

[10] [3]Crans, Alissa S., Fiore, Thomas M. and Satyendra, Ramon. Musical Actions of Dihedral Groups. University of South Florida. Nov 3, 2007. Retrieved from http://myweb.lmu.edu/acrans/MusicalActions.PDF

[11] Benson, Dave J. Music: A Mathematical Offering. Cambridge University Press. Nov 2006. Retrieved from http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf

[12] Rowland, Todd and Weisstein, Eric W. (n.d). Root of Unity. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/RootofUnity.html

2. Add more about Dihedral Groups in 3D. I only talk one property in 3D, but there must be some more.
3. In subgroup part, it is hard to explain only in words, so I use lots of notations, but they are still not very clear, I hope can find a better way to illustrate it.


text:
# Slant Asymptote Calculator

Slant Asymptote Calculator is a free online tool that displays the asymptote value for the given function. BYJU’S online slant asymptote calculator tool makes the calculation faster, and it displays the asymptote value in a fraction of seconds.

## How to Use the Slant Asymptote Calculator?

The procedure to use the slant asymptote calculator is as follows:
Step 1: Enter the function in the input field
Step 2: Now click the button “Calculate Slant Asymptote” to get the result
Step 3: Finally, the asymptotic value and graph will be displayed in the new window

### What is Meant by Slant Asymptote?

In Mathematics, a slant asymptote, also known as an oblique asymptote, occurs when the degree of the numerator polynomial is greater than the degree of the denominator polynomial. The slant asymptote gives the linear function which is neither parallel to x-axis nor parallel to the y-axis. It is easy to calculate the oblique asymptote. It can be found by dividing the numerator polynomial by the denominator polynomial using either synthetic division method or long division method.


text:
### TWOSAMPLEFREQ Statement

Subsections:

TWOSAMPLEFREQ <options> ;

The TWOSAMPLEFREQ statement performs power and sample size analyses for tests of two independent proportions. Pearson’s chi-square, Fisher’s exact, and likelihood ratio chi-square tests are supported.

#### Summary of Options

Table 75.19 summarizes the options available in the TWOSAMPLEFREQ statement.

Table 75.19: TWOSAMPLEFREQ Statement Options

Option

Description

Define analysis

Specifies the statistical analysis

Specify analysis information

Specifies the significance level

Specifies the null odds ratio

Specifies the null proportion difference

Specifies the null relative risk

Specifies the number of sides and the direction of the statistical test or confidence interval

Specify effects

Specifies the two independent proportions, and

Specifies the odds ratio

Specifies the proportion difference

Specifies the reference proportion

Specifies the relative risk

Specify sample size and allocation

Specifies the two group sample sizes

Specifies the sample size allocation weights for the two groups

Enables fractional input and output for sample sizes

Specifies the common sample size per group

Specifies the sample size

Specify power

Specifies the desired power of the test

Control ordering in output

Controls the output order of parameters

Table 75.20 summarizes the valid result parameters for different analyses in the TWOSAMPLEFREQ statement.

Table 75.20: Summary of Result Parameters in the TWOSAMPLEFREQ Statement

Analyses

Solve For

Syntax

TEST=FISHER

Power

Sample size

TEST=LRCHI

Power

Sample size

TEST=PCHI

Power

Sample size

#### Dictionary of Options

ALPHA=number-list

specifies the level of significance of the statistical test. The default is 0.05, corresponding to the usual 0.05 100% = 5% level of significance. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

GROUPPROPORTIONS=grouped-number-list
GPROPORTIONS=grouped-number-list
GROUPPS=grouped-number-list
GPS=grouped-number-list

specifies the two independent proportions, and . For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.

GROUPNS=grouped-number-list
GNS=grouped-number-list

specifies the two group sample sizes. For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.

GROUPWEIGHTS=grouped-number-list
GWEIGHTS=grouped-number-list

specifies the sample size allocation weights for the two groups. This option controls how the total sample size is divided between the two groups. Each pair of values for the two groups represents relative allocation weights. Additionally, if the NFRACTIONAL option is not used, the total sample size is restricted to be equal to a multiple of the sum of the two group weights (so that the resulting design has an integer sample size for each group while adhering exactly to the group allocation weights). Values must be integers unless the NFRACTIONAL option is used. The default value is (1 1), a balanced design with a weight of 1 for each group. For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.

NFRACTIONAL
NFRAC

enables fractional input and output for sample sizes. See the section Sample Size Adjustment Options for information about the ramifications of the presence (and absence) of the NFRACTIONAL option.

NPERGROUP=number-list
NPERG=number-list

specifies the common sample size per group or requests a solution for the common sample size per group with a missing value (NPERGROUP=.). Use of this option implicitly specifies a balanced design. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NTOTAL=number-list

specifies the sample size or requests a solution for the sample size with a missing value (NTOTAL=.). For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NULLODDSRATIO=number-list
NULLOR=number-list

specifies the null odds ratio. The default value is 1. This option can be used only with the ODDSRATIO= option in the TEST=PCHI analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NULLPROPORTIONDIFF=number-list
NULLPDIFF=number-list

specifies the null proportion difference. The default value is 0. This option can be used only with the GROUPPROPORTIONS= or PROPORTIONDIFF= option in the TEST=PCHI analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NULLRELATIVERISK=number-list
NULLRR=number-list

specifies the null relative risk. The default value is 1. This option can be used only with the RELATIVERISK= option in the TEST=PCHI analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

ODDSRATIO=number-list
OR=number-list

specifies the odds ratio . For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

OUTPUTORDER=INTERNAL | REVERSE | SYNTAX

controls how the input and default analysis parameters are ordered in the output. OUTPUTORDER=INTERNAL (the default) arranges the parameters in the output according to the following order of their corresponding options:

The OUTPUTORDER=SYNTAX option arranges the parameters in the output in the same order in which their corresponding options are specified in the TWOSAMPLEFREQ statement. The OUTPUTORDER=REVERSE option arranges the parameters in the output in the reverse of the order in which their corresponding options are specified in the TWOSAMPLEFREQ statement.

POWER=number-list

specifies the desired power of the test or requests a solution for the power with a missing value (POWER=.). The power is expressed as a probability, a number between 0 and 1, rather than as a percentage. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

PROPORTIONDIFF=number-list
PDIFF=number-list

specifies the proportion difference . For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

REFPROPORTION=number-list
REFP=number-list

specifies the reference proportion . For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

RELATIVERISK=number-list
RR=number-list

specifies the relative risk . For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

SIDES=keyword-list

specifies the number of sides (or tails) and the direction of the statistical test or confidence interval. For information about specifying the keyword-list, see the section Specifying Value Lists in Analysis Statements. Valid keywords and their interpretation are as follows:

1

one-sided with alternative hypothesis in same direction as effect

2

two-sided

U

upper one-sided with alternative greater than null value

L

lower one-sided with alternative less than null value

If the effect size is zero, then SIDES=1 is not permitted; instead, specify the direction of the test explicitly in this case with either SIDES=L or SIDES=U. The default value is 2.

TEST=FISHER | LRCHI | PCHI

specifies the statistical analysis. TEST=FISHER specifies Fisher’s exact test. TEST=LRCHI specifies the likelihood ratio chi-square test. TEST=PCHI (the default) specifies Pearson’s chi-square test.

#### Restrictions on Option Combinations

To specify the proportions, choose one of the following parameterizations:

To specify the sample size and allocation, choose one of the following parameterizations:

• sample size per group in a balanced design (by using the NPERGROUP= option)

• total sample size and allocation weights (by using the NTOTAL= and GROUPWEIGHTS= options)

• individual group sample sizes (by using the GROUPNS= option)

#### Option Groups for Common Analyses

This section summarizes the syntax for the common analyses supported in the TWOSAMPLEFREQ statement.

##### Pearson Chi-Square Test for Two Proportions

You can use the NPERGROUP= option in a balanced design and express effects in terms of the individual proportions, as in the following statements. Default values for the SIDES= and ALPHA= options specify a two-sided test with a significance level of 0.05.

proc power;
twosamplefreq test=pchi
groupproportions = (.15 .25)
nullproportiondiff = .03
npergroup = 50
power = .;
run;

You can also specify an unbalanced design by using the NTOTAL= and GROUPWEIGHTS= options and express effects in terms of the odds ratio. The default value of the NULLODDSRATIO= option specifies a test of no effect.

proc power;
twosamplefreq test=pchi
oddsratio = 2.5
refproportion = 0.3
groupweights = (1 2)
ntotal = .
power = 0.8;
run;

You can also specify sample sizes with the GROUPNS= option and express effects in terms of relative risks. The default value of the NULLRELATIVERISK= option specifies a test of no effect.

proc power;
twosamplefreq test=pchi
relativerisk = 1.5
refproportion = 0.2
groupns = 40 | 60
power = .;
run;

You can also express effects in terms of the proportion difference. The default value of the NULLPROPORTIONDIFF= option specifies a test of no effect, and the default value of the GROUPWEIGHTS= option specifies a balanced design.

proc power;
twosamplefreq test=pchi
proportiondiff = 0.15
refproportion = 0.4
ntotal = 100
power = .;
run;

##### Fisher’s Exact Conditional Test for Two Proportions

The following statements demonstrate a power computation for Fisher’s exact conditional test for two proportions. Default values for the SIDES= and ALPHA= options specify a two-sided test with a significance level of 0.05.

proc power;
twosamplefreq test=fisher
groupproportions = (.35 .15)
npergroup = 50
power = .;
run;

##### Likelihood Ratio Chi-Square Test for Two Proportions

The following statements demonstrate a sample size computation for the likelihood ratio chi-square test for two proportions. Default values for the SIDES= and ALPHA= options specify a two-sided test with a significance level of 0.05.

proc power;
twosamplefreq test=lrchi
oddsratio = 2
refproportion = 0.4
npergroup = .
power = 0.9;
run;


text:
Articles written in Proceedings – Mathematical Sciences

• On 𝐴-nilpotent abelian groups

Let 𝐺 be a group and $A = \text{Aut}(G)$ be the group of automorphisms of 𝐺. Then, the element $[g, \alpha] = g^{-1}\alpha(g)$ is an autocommutator of $g \in G$ and $\alpha \in A$. Hence, for any natural number 𝑚 the 𝑚-th autocommutator subgroup of 𝐺 is defined as

$K_{m}(G)=\langle [g,\alpha_{1},\ldots,\alpha_{m}]|g\in G,\alpha_{1},\ldots,\alpha_{m}\in A\rangle$,

where $[g, \alpha_{1}, \alpha_{2},\ldots, \alpha_{m}] = [[g,\alpha_{1},\ldots,\alpha_{m−1}], \alpha_{m}]$. In this paper, we introduce the new notion of 𝐴-nilpotent groups and classify all abelian groups which are 𝐴-nilpotent groups.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# EvolutionarySQS¶

class EvolutionarySQS(alloy, cluster_max_diameters, repetitions=(1, 1, 1), seed_configuration=None, number_of_generations=40, population_size=20, number_to_promote=2, heredity_probability=0.5, permutation_probability=0.5, n_best=10, rng=None)

Generate a special quasi-random structure (SQS) using genetic optimization.

Parameters: alloy (AlloyConfiguration) – An alloy configuration. cluster_max_diameters (dict) – A dictionary that maps the cluster order to the maximum diameter for clusters of that order. They keys should be integers greater than equal to two and the values are positive PhysicalQuantities with units of distance (typically at least as large as the nearest neighbor distance). repetitions (list of type int) – The number of times to repeat the alloy configuration. Default: (1, 1, 1) seed_configuration (BulkConfiguration) – An initial guess of the SQS structure. number_of_generations (int) – The number of times the population should be evolved. population_size (int) – The number of individuals which are part of a population. The population size must be at least 5, but it is recommeded to use a large population size (>= 20) to obtain higher quality results. number_to_promote (int) – The number of most fittest individuals to promote to the next generation. Default: 2 heredity_probability (float) – The unnormalized probabilty of using the heredity operator. Default: 0.5 permutation_probability (float) – The unnormalized probabilty of using the permutation operator. The permutation operator will only be used when there is more than one type of an element in the system. Default: 0.5 n_best (int) – The n best structures the correlation functions of which are closest to the truely random one. rng (numpy.random.RandomState) – The random number generated used to generate the random number stream. Default: A new random number generator (seeded from OS entropy pool)
alloyConfiguration()
Returns: The alloy configuration. This is the same as the alloy configuration that was initially passed in. AlloyConfiguration
bestStructure()
Returns: The SQS structure that best matches the random correlation function. BulkConfiguration
bestStructures()
Returns: A list of the configurations sorted in order of increasing error. list of type BulkConfiguration
evolvePopulation(number_of_generations)

Evolves the current population for number_of_generations generations.

Parameters: number_of_generations (int) – The number of times the genetic algorithm is applied to the population.
fitnesses()

Return the fitnesses of the current population. Sorted by descending fitness.

Returns: a list of fitnesses list
objectiveFunction(configuration)

Evaluate the objective function for a configuration.

Parameters: configuration (BulkConfiguration) – The configuration to evaluate. The SQS objective function. float
population()

Return the current population.

Returns: a population list of individuals
symmetries()

Return the international symbol for each individual of the current population. Sorted by descending fitness.

Returns: a list of international symbols list of str

## Usage Examples¶

Set up an evolutionary SQS optimization for a 64 atom SiGe alloy.

# Set up lattice
vector_a = [5.4306, 0.0, 0.0]*Angstrom
vector_b = [0.0, 5.4306, 0.0]*Angstrom
vector_c = [0.0, 0.0, 21.7224]*Angstrom
lattice = UnitCell(vector_a, vector_b, vector_c)

# Define site types
site_a = AlloySite(Germanium=0.5, Silicon=0.5)

# Define sites
sites = [
site_a, site_a, site_a, site_a, site_a, site_a, site_a, site_a,
site_a, site_a, site_a, site_a, site_a, site_a, site_a, site_a,
site_a, site_a, site_a, site_a, site_a, site_a, site_a, site_a,
site_a, site_a, site_a, site_a, site_a, site_a, site_a, site_a,
]

# Define coordinates
cartesian_coordinates = [[  0.     ,   0.     ,   0.     ],
[  0.     ,   0.     ,   5.4306 ],
[  0.     ,   0.     ,  10.8612 ],
[  0.     ,   0.     ,  16.2918 ],
[  1.35765,   1.35765,   1.35765],
[  1.35765,   1.35765,   6.78825],
[  1.35765,   1.35765,  12.21885],
[  1.35765,   1.35765,  17.64945],
[  2.7153 ,   2.7153 ,   0.     ],
[  2.7153 ,   2.7153 ,   5.4306 ],
[  2.7153 ,   2.7153 ,  10.8612 ],
[  2.7153 ,   2.7153 ,  16.2918 ],
[  4.07295,   4.07295,   1.35765],
[  4.07295,   4.07295,   6.78825],
[  4.07295,   4.07295,  12.21885],
[  4.07295,   4.07295,  17.64945],
[  2.7153 ,   0.     ,   2.7153 ],
[  2.7153 ,   0.     ,   8.1459 ],
[  2.7153 ,   0.     ,  13.5765 ],
[  2.7153 ,   0.     ,  19.0071 ],
[  4.07295,   1.35765,   4.07295],
[  4.07295,   1.35765,   9.50355],
[  4.07295,   1.35765,  14.93415],
[  4.07295,   1.35765,  20.36475],
[  0.     ,   2.7153 ,   2.7153 ],
[  0.     ,   2.7153 ,   8.1459 ],
[  0.     ,   2.7153 ,  13.5765 ],
[  0.     ,   2.7153 ,  19.0071 ],
[  1.35765,   4.07295,   4.07295],
[  1.35765,   4.07295,   9.50355],
[  1.35765,   4.07295,  14.93415],
[  1.35765,   4.07295,  20.36475]]*Angstrom

# Define the maximum diameter of each cluster.
cluster_max_diameters = {
2: 4.9*Angstrom,
3: 4.4*Angstrom,
4: 2.5*Angstrom,
}

# Perform the SQS optimization.
sqs = EvolutionarySQS(
alloy_configuration,
cluster_max_diameters,
number_of_generations=200,
population_size=100,
number_to_promote=10,
heredity_probability=0.5,
permutation_probability=0.5,
)

# Extract and save the best SQS structure after the optimization
best_sqs_configuration = sqs.bestStructure()
nlsave('sqs_sige_optimized.hdf5', best_sqs_configuration)

sige_evolutionary_sqs.py

## Notes¶

EvolutionarySQS can be used to generate an optimized random alloy, based on the special quasi-random structure (SQS) method [vdWTdJ+13].

The desired composition and structure of an alloy can be set up via the AlloySite and AlloyConfiguration classes, which are used as input to EvolutionarySQS.

QuantumATK uses a genetic algorithm (similar to CrystalStructurePrediction) to permute the positions of the atoms in order to obtain correlation functions that match the correlation functions of an ideal random alloy configuration. After the optimization, the bulk configuration with the best correlation functions can be extracted via the bestStructure() method. This configuration can then be used in further calculations.

 [vdWTdJ+13] A. van de Walle, P. Tiwary, M. de Jong, D.L. Olmsted, M. Asta, A. Dick, D. Shin, Y. Wang, L.-Q. Chen, and Z.-K. Liu. Efficient stochastic generation of special quasirandom structures. Calphad, 42:13 – 18, 2013. URL: http://www.sciencedirect.com/science/article/pii/S0364591613000540.


text:
# The Snappit Browser

(Snapified version of the internet)

The browser

Snap! Build Your Own Blocks

Connect to other servers with the Snappit browser.
Just enter the Snappit URL!

Hosting Snappit servers

Snap! Build Your Own Blocks
Host your own server for other people to connect to!
How to use:

• Open the costumes tab inside of the "Server" sprite
• Import an image you would like to display as you site
• Run the project
• Type in a subdomain
Example: Typing joecooldoo would become joecooldoo.snappit.snap, which is the URL other people can connect to.
• Keep the tab open and displayed (not minimized or hidden behind windows)

NOTE: the result cannot be visited in an actual browser. This only works with the Snappit browser.

Post your Snappit URLs below!

@mr_owlssssnap2 here is the project I was making

helicoptur_test.snappit.snap

What's this all about? Just image hosting?

basically
its not working good right now
of course as soon as I open it again it stops working

How can it tell if the connection is secure or insecure?

its all insecure lol
i dont know how to make a recreation of an ssl certificate in snap

its working now yay
i also made it faster

one way or another, i will figure out how to make my project compatible with yours, and vice-versa.
But theres needs to be a way for like text to be transfered

But its cool!

Do you mind if i borrow some code for images with credit to you for my project?

Sure.
Out of curiosity, what are you going to use it for?

you can make your program create the text as an image and send it over that way.

to display images received

oh smart thanks

out of curiosity that stems from absolutely nowhere, how would you like it if someone makes this into like a ds looking thing

why the ds specifically

because

idk

Essentially, it's not functioning properly at the moment. And, invariably, once I reopen it, it ceases to work.

Can you give a more descriptive problem? Like:

• Are you on a school network?
• What icon is at the top left?
• Do you get an error?
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• Is the server you're trying to connect to active and running?
• Did you follow the instructions in the original post?