text:
extra formatting changes

Showing1 changed files
 ... ... @@ -5,12 +5,12 @@ 5 5  \title{Produce random paths modeled on a given path} 6 6  \usage{ 7 7  generateRandomPaths( 8 - path, 9 - from = 1, 10 - to = nrow(path), 11 - d = ncol(path), 12 - randomizationParams, 13 - N 8 +path, 9 +from = 1, 10 +to = nrow(path), 11 +d = ncol(path), 12 +randomizationParams, 13 +N 14 14  ) 15 15  } 16 16  \arguments{ ... ... @@ -47,6 +47,6 @@ randomizationParams. 47 47  \examples{ 48 48  randomizationParams = c('byPermutation','permuteWithinColumns') 49 49  randomPaths = generateRandomPaths(crooked_path,from=6,to=nrow(crooked_path), 50 - d=ncol(crooked_path),randomizationParams=randomizationParams, 51 - N=10) 50 + d=ncol(crooked_path),randomizationParams=randomizationParams, 51 + N=10) 52 52  }

Updated data.R to reference the TrajectoryGeometryData repository and updated the man/... files accordingly.

Michael Shapiro authored on 12/01/2021 16:40:02
Showing1 changed files
 ... ... @@ -5,12 +5,12 @@ 5 5  \title{Produce random paths modeled on a given path} 6 6  \usage{ 7 7  generateRandomPaths( 8 -path, 9 -from = 1, 10 -to = nrow(path), 11 -d = ncol(path), 12 -randomizationParams, 13 -N 8 + path, 9 + from = 1, 10 + to = nrow(path), 11 + d = ncol(path), 12 + randomizationParams, 13 + N 14 14  ) 15 15  } 16 16  \arguments{ ... ... @@ -47,6 +47,6 @@ randomizationParams. 47 47  \examples{ 48 48  randomizationParams = c('byPermutation','permuteWithinColumns') 49 49  randomPaths = generateRandomPaths(crooked_path,from=6,to=nrow(crooked_path), 50 - d=ncol(crooked_path),randomizationParams=randomizationParams, 51 - N=10) 50 + d=ncol(crooked_path),randomizationParams=randomizationParams, 51 + N=10) 52 52  }

whitespace, line-lengths and single-cell data sources sorted

Showing1 changed files
 ... ... @@ -5,12 +5,12 @@ 5 5  \title{Produce random paths modeled on a given path} 6 6  \usage{ 7 7  generateRandomPaths( 8 - path, 9 - from = 1, 10 - to = nrow(path), 11 - d = ncol(path), 12 - randomizationParams, 13 - N 8 +path, 9 +from = 1, 10 +to = nrow(path), 11 +d = ncol(path), 12 +randomizationParams, 13 +N 14 14  ) 15 15  } 16 16  \arguments{ ... ... @@ -47,6 +47,6 @@ randomizationParams. 47 47  \examples{ 48 48  randomizationParams = c('byPermutation','permuteWithinColumns') 49 49  randomPaths = generateRandomPaths(crooked_path,from=6,to=nrow(crooked_path), 50 - d=ncol(crooked_path),randomizationParams=randomizationParams, 51 - N=10) 50 + d=ncol(crooked_path),randomizationParams=randomizationParams, 51 + N=10) 52 52  }

Removed documentation for functions which are not exported.

Changed names of data, e.g., changed crookedPath to crooked_path.

Michael Shapiro authored on 01/01/2021 16:41:44
Showing1 changed files
 ... ... @@ -5,12 +5,12 @@ 5 5  \title{Produce random paths modeled on a given path} 6 6  \usage{ 7 7  generateRandomPaths( 8 -path, 9 -from = 1, 10 -to = nrow(path), 11 -d = ncol(path), 12 -randomizationParams, 13 -N 8 + path, 9 + from = 1, 10 + to = nrow(path), 11 + d = ncol(path), 12 + randomizationParams, 13 + N 14 14  ) 15 15  } 16 16  \arguments{ ... ... @@ -46,6 +46,7 @@ randomizationParams. 46 46  } 47 47  \examples{ 48 48  randomizationParams = c('byPermutation','permuteWithinColumns') 49 -randomPaths = generateRandomPaths(crookedPath,from=6,to=nrow(crookedPath), 50 - d=ncol(crookedPath),randomizationParams=randomizationParams,N=10) 49 +randomPaths = generateRandomPaths(crooked_path,from=6,to=nrow(crooked_path), 50 + d=ncol(crooked_path),randomizationParams=randomizationParams, 51 + N=10) 51 52  }

most whitespace sorted out

Showing1 changed files
 ... ... @@ -5,12 +5,12 @@ 5 5  \title{Produce random paths modeled on a given path} 6 6  \usage{ 7 7  generateRandomPaths( 8 - path, 9 - from = 1, 10 - to = nrow(path), 11 - d = ncol(path), 12 - randomizationParams, 13 - N 8 +path, 9 +from = 1, 10 +to = nrow(path), 11 +d = ncol(path), 12 +randomizationParams, 13 +N 14 14  ) 15 15  } 16 16  \arguments{ ... ... @@ -47,5 +47,5 @@ randomizationParams. 47 47  \examples{ 48 48  randomizationParams = c('byPermutation','permuteWithinColumns') 49 49  randomPaths = generateRandomPaths(crookedPath,from=6,to=nrow(crookedPath), 50 - d=ncol(crookedPath),randomizationParams=randomizationParams,N=10) 50 + d=ncol(crookedPath),randomizationParams=randomizationParams,N=10) 51 51  }

Wrote a function to measure the progression of a path in a direction and jazzed up the vignette to reflect this.

Michael Shapiro authored on 09/06/2020 15:54:24
Showing1 changed files
 ... ... @@ -35,7 +35,7 @@ details.} 35 35  } 36 36  \value{ 37 37  This function returns a list of random paths. Each path is 38 - a matrix. Note that each random path begins at the origin. 38 + a matrix. 39 39  } 40 40  \description{ 41 41  This function takes a path and produces N random paths of the same

This modifies the path randomization so that paths can be randomized either by permutation of the matrix or by taking steps.

Michael Shapiro authored on 03/06/2020 16:40:36
Showing1 changed files
 ... ... @@ -9,7 +9,7 @@ generateRandomPaths( 9 9  from = 1, 10 10  to = nrow(path), 11 11  d = ncol(path), 12 - preserveLengths = TRUE, 12 + randomizationParams, 13 13  N 14 14  ) 15 15  } ... ... @@ -26,9 +26,10 @@ nrow(path).} 26 26  \item{d}{- The dimension under consideration. This defaults to 27 27  ncol(path)} 28 28   29 -\item{preserveLengths}{- If this is set to TRUE, the steps of the 30 -random paths have the same euclidean length as those of the 31 -give path. Otherwise they hav length 1.} 29 +\item{randomizationParams}{- A character vector controling the 30 +randomization method used. It's first entry must be either 31 +'byPermutation' or 'bySteps' See the vignette for further 32 +details.} 32 33   33 34  \item{N}{- The number of random paths required.} 34 35  } ... ... @@ -38,12 +39,13 @@ This function returns a list of random paths. Each path is 38 39  } 39 40  \description{ 40 41  This function takes a path and produces N random paths of the same 41 -dimension and length based on it. The steps along the random paths 42 -are based on uniform sampling of the unit sphere. If  43 -preserveLengths is set to TRUE, the random paths of steps of the 44 -same length as the original path. Otherwise, the steps are all 45 -length 1. 42 +dimension and length based on it. This can be done either by 43 +permuting the entries in path or by taking steps from the initial 44 +point of path. Exact behaviour is controlled by 45 +randomizationParams. 46 46  } 47 47  \examples{ 48 -randomPaths = generateRandomPaths(crookedPath,from=6,N=10) 48 +randomizationParams = c('byPermutation','permuteWithinColumns') 49 +randomPaths = generateRandomPaths(crookedPath,from=6,to=nrow(crookedPath), 50 + d=ncol(crookedPath),randomizationParams=randomizationParams,N=10) 49 51  }

Documented the data in R/data.R

Michael Shapiro authored on 26/05/2020 14:25:19
Showing1 changed files
 ... ... @@ -44,3 +44,6 @@ preserveLengths is set to TRUE, the random paths of steps of the 44 44  same length as the original path. Otherwise, the steps are all 45 45  length 1. 46 46  } 47 +\examples{ 48 +randomPaths = generateRandomPaths(crookedPath,from=6,N=10) 49 +}

The changes to the documentaion are the result of a misconception as to the source of some problems. HOWEVER I was able to track down and fix some bugs and successfully run testPathForDirectionality using straightPath as my test data.

Michael Shapiro authored on 21/05/2020 16:56:41
Showing1 changed files
 ... ... @@ -6,8 +6,8 @@ 6 6  \usage{ 7 7  generateRandomPaths( 8 8  path, 9 - start = 1, 10 - end = nrow(path), 9 + from = 1, 10 + to = nrow(path), 11 11  d = ncol(path), 12 12  preserveLengths = TRUE, 13 13  N ... ... @@ -17,10 +17,10 @@ generateRandomPaths( 17 17  \item{path}{- This is an mxn dimensional matrix. Each row is 18 18  considered a point.} 19 19   20 -\item{start}{- The starting place along the path which will be 20 +\item{from}{- The starting place along the path which will be 21 21  treated as the center of the sphere. This defaults to 1.} 22 22   23 -\item{end}{- The end point of the path. This defaults to 23 +\item{to}{- The end point of the path. This defaults to 24 24  nrow(path).} 25 25   26 26  \item{d}{- The dimension under consideration. This defaults to

This now contains code and documentation.

Michael Shapiro authored on 10/04/2020 11:42:19
Showing1 changed files
 1 1 new file mode 100644 ... ... @@ -0,0 +1,46 @@ 1 +% Generated by roxygen2: do not edit by hand 2 +% Please edit documentation in R/TrajectoryGeometry.R 3 +\name{generateRandomPaths} 4 +\alias{generateRandomPaths} 5 +\title{Produce random paths modeled on a given path} 6 +\usage{ 7 +generateRandomPaths( 8 + path, 9 + start = 1, 10 + end = nrow(path), 11 + d = ncol(path), 12 + preserveLengths = TRUE, 13 + N 14 +) 15 +} 16 +\arguments{ 17 +\item{path}{- This is an mxn dimensional matrix. Each row is 18 +considered a point.} 19 + 20 +\item{start}{- The starting place along the path which will be 21 +treated as the center of the sphere. This defaults to 1.} 22 + 23 +\item{end}{- The end point of the path. This defaults to 24 +nrow(path).} 25 + 26 +\item{d}{- The dimension under consideration. This defaults to 27 +ncol(path)} 28 + 29 +\item{preserveLengths}{- If this is set to TRUE, the steps of the 30 +random paths have the same euclidean length as those of the 31 +give path. Otherwise they hav length 1.} 32 + 33 +\item{N}{- The number of random paths required.} 34 +} 35 +\value{ 36 +This function returns a list of random paths. Each path is 37 + a matrix. Note that each random path begins at the origin. 38 +} 39 +\description{ 40 +This function takes a path and produces N random paths of the same 41 +dimension and length based on it. The steps along the random paths 42 +are based on uniform sampling of the unit sphere. If  43 +preserveLengths is set to TRUE, the random paths of steps of the 44 +same length as the original path. Otherwise, the steps are all 45 +length 1. 46 +}


text:
# find the derivative by using the definition of a derivativef(x)=4x^2+1

Question

find the derivative by using the definition of a derivative
f(x)=4x^2+1

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format: plain


text:
# Do you use a naming convention for your GPOs?

GPOs show as inaccessible after ransomware attack

I'm in the middle of cleaning up AD/GPOs after years of only basic maintenance. Now that I've removed all of the bloat, I'd like to rename the remaining GPOs in some consistent way so as to better reflect what the GPO is actually doing and/or what it needs to be applied to.

I've only found a few discussions about this. One suggests using version numbers to more easily see whether a machine has the most recent GPO(s) applied. Another suggests specifying whether it's a 'computer' or 'user' policy in the actual GPO name.

My overall goal is to simplify rather than to add unnecessary complexity. There's plenty of information that can be worked into the actual GPO name, but if it's better suited for the Comment field, I'd rather leave it out and keep the names simple, transparent, and uniformly formatted.

I'm curious to hear how you've managed this. How do you name yours?

Popular Topics in Active Directory & GPO
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## 49 Replies

· · ·
Mace
OP

The version number idea is interesting, but I'm not sure if that would be helpful.  As long as the computer is connected to the network, it should be getting whatever the current version is anyway.  I agree that simple is good.  My names are just what the policy is for.  One is Password Requirements, which has all the password settings.  Another for IE Settings, which handles all IE settings.

4
· · ·
Ghost Chili
OP

To make it easier on my guys as they were learning GPO, I created Group Policies based on Department and then Computers or Users.

So we have Human Resources - Computer Policy, Human Resources - User Policy, etc. I'd prefer it be a single GPO with computer and user settings for each department, but I don't do much management of it anymore so the onus is on them.

1
· · ·
Mace
OP

I name my GPOs by their function.  This means if I have a GPO that deploys an application and sets the firewall and file system rules for that application I give it the application's name.  For my base GPO (so I don't have to modify the default GPO) I name it Base Policies and ensure it's deployed anywhere the default GPO is deployed.  I have a PDC Emulator GPO that is filtered via WMI so it only applies to my PDC Emulator FSMO role holder.  This GPO configures the PDC Emulator as my NTP master for the network.

Basically, name them for what they're being used for.

7
· · ·
Thai Pepper
OP

We tend to go with a Team - Type - Name variant. Allows you to quickly find what you are looking for!

So for example

Finance - Drive - Monthly reports drive (M)

Engineering - Settings - Internet Explorer

3
· · ·
Anaheim
OP

I usually use a combination of location, type, function, and a version/date (Finance - Computer - Disable CD Burner - 07012016).

1
· · ·
Chipotle
OP

function

0
· · ·
Datil
OP

I generally create GPO per OU. So we have IT, Admin (consisting of HR, Accounting, etc....) and then the branches/departments. I have a policy that affects everyone with our standard security defaults that everyone needs, then I have it drilled down to department/OU so I can do some more detail work. Its clean and easy to locate.

1
· · ·
Jalapeno
OP

It seems to me (and someone please let me know if I sound ridiculous), that the complexity of an org's GPO Naming scheme is relevant to how many they're using. For example one place I worked only really had 2 gpos: Interns and Non-Interns, but another place a friend worked at had ones specific to departments and even sub-departments. Generally in all these cases I would go by what Mike400said and do it by function with whatever granular details you need to easily identify them.

1
· · ·
Jalapeno
OP

Yep. We are a school so have different names depending on what used for:

Staff - IE Settings

Student - Restrictions

All Endpoints - Firewall

etc

0
· · ·
Jalapeno
OP

Sort of. I'm small time so it's easy.

"Global - something" are applied at the domain level for all common security, firewall, shortcuts, mapped drives, etc... Item level targeting where needed. Global-Security Settings, Global-Laptop Settings, Global-Shortcuts, Files, Drive, etc.

I'll append "-LB" to the policies using LoopBack processing for easier troubleshooting and planning

Polocies only applied to servers begin with "Server-"

Policies only applied to users begin with "User-" (unless its a loopback)

Really super basic, but we're only a 350 pc / 45 laptop / 40 server organization.

Be descriptive, and use the "Comment" section of the gpo as well.

Someone else should be able to walk in and 'get it' without reading through extraneous documentation

1
· · ·
Cayenne
OP

UGPO_Prnt_ROOM (User Group Policy Object _ Print _Room #)

UGPO_MD_P_(program) (Mapped Drive _ drive letter _ program name)

CGPO_PowerSettings (student/staff/admin) (Computer GPO and what it does)

CGPO_WSUS_desktop/laptop/server/etc

We have a few that require both, so they don't have a UGPO or CGPO prefix.

We try to be descriptive so it's easy to find a particular gpo.

4
· · ·
Chipotle
OP

I struggle with this myself as I find it hard to keep the naming as descriptive as I would like while still keeping it concise. I'm still not happy with a lot of the naming, but pick your battles right? Generally I follow a formula like this:

Primary Subject - Setting Being Applied - Location/Department if Applicable - User or Computer

So for some examples (U = User, C = Computer, CL = Computer Loopback):

Outlook 2010 - Set Default PST Path - U

Mapped Drive - Finance (M:) \\path\to\the\share - CL

Printers - Set Add Printer Location - Finance - C

This may not qualify for what you are looking to do since you might consider some of this information more suited to the comments section, but I think it gives me a clear view of what is going on when I look at what is applied to an OU or if I am in the field and run a gpresult.

1
· · ·
Mace
OP

The benefit of naming GPOs by function and including configuration items is that when that function goes away you just delete the GPO and all the configuration is undone.

If you have multiple GPOs setting the same settings you may run into a conflict, but then you would have had that conflict anyway.  If GPOs duplicate the same setting, who cares.  Deleting one GPO won't break the other GPO's purpose.

2
· · ·
Habanero
OP

I agree with the others.  I have about a dozen GPOs, so I name them according to their purpose.  Naming conventions usually are more beneficial at a larger scale, so you may not need a naming convention for a small number of GPOs.

0
· · ·
Habanero
OP

I name them for exactly what they do, not too short (as to be too vague) and not ridiculously long. Don't need to put departments and other stuff in the name, you can figure those out from it's placement in the OU's.

1
· · ·
Poblano
OP

We prefix ours with 'U-' or 'C-' to show whether it is a User or Computer policy. This visually sorts it in the console.

On major revisions we add a 'v2' or whatever to the end. This is great if you like controlled change. Create a new (v2) policy by importing a backup of the current live one. Modify. Apply to a test OU or use a security filter. Then you can apply the new GP while being able to revert to the previous one without much effort.

Other than that, we just use a descriptive name. I'm sure everyone structures their policies differently, and yes, the comments field is a good way to add information without cramming it all into the title.

5
· · ·
Jalapeno
OP

I make the name of the GPO to where it tells me what it does (without going through great lengths). Then I make the description say EXACTLY what it does, who made it. Depending on what the GPO is for, I might add a when it was made, and why it was made (a GPO that's a temp fix or a bandaid that's keeping something limping along)

Example:

Printer - Campus1 - Financial Aid (we'd abreveate the Campus1 to 3 or fewer letters and it'd be FinAid but i wanted to spell out the intentions)

Description / Comment:

Deploys $ThisExactPrinter to the computers in the FinAid office on$Campus.  RBoland

That way if anyone comes behind me, they don't have to dig through the settings tab and trace down exactly what the GPO does. It also lets them know who to ask questions to.

1
· · ·
Serrano
OP

My GPO naming has always been the same, call it what it does. For example, a GPO to change the local admin username and password would be called 'Local Admin Username Update'. I would then just put those settings in that GPO and use it. When I need to change it, I just change that and it goes out at the next update.

If I build a new one that pushes out an old one, I typically delete the old one first and build the new one. This way I don't have ones that I'm not using and I don't have the versioning thing you are talking about. I like to have only the ones that I need and then they are named for what they do. I really don't have GPO's that do a lot, they typically do one task and that's it. Makes it easier to track down a problem when  new GPO gets pushed out.

3
· · ·
Poblano
OP

RobPRose wrote:

My GPO naming has always been the same, call it what it does. For example, a GPO to change the local admin username and password would be called 'Local Admin Username Update'. I would then just put those settings in that GPO and use it. When I need to change it, I just change that and it goes out at the next update.

If I build a new one that pushes out an old one, I typically delete the old one first and build the new one. This way I don't have ones that I'm not using and I don't have the versioning thing you are talking about. I like to have only the ones that I need and then they are named for what they do. I really don't have GPO's that do a lot, they typically do one task and that's it. Makes it easier to track down a problem when  new GPO gets pushed out.

Ours is pretty similar: most are one task with the name as simple as 'Intranet Home Page'. I do like the idea of adding U and C as some have mentioned to sort them a little better.

0
· · ·
Anaheim
OP

I like to sort mine by overarching category (apps, drives, printers, security, etc.).  For instance:

Apps - IE - Add to Favorites
Drive Maps - Marketing (M:)
Security - RDP - Enable RDP-NLA

You can already tell who gets what based on Scope, whether it's computer or user policy in Details, and specifics about what it does in Settings (why be redundant?).

If you want to log relevant notes like modded when/by/how, etc. you can add that to the GPO's Comment section, and then view the log history on the Details tab.  And whenever possible filter by group, not individual user.  Then you can manage everything right in AD (new user, add to group, done).

0
· · ·
Thai Pepper
OP

I'd like to be a real BOFH to the next poor schmuck who takes my place after I'm gone... rename all the GPOs to their GUID string.

5
· · ·
Jalapeno
OP

Our GPOs match the naming convention of the security group they are tied to which is starts with gpo_, then what it does (Configure, Add, remove, etc.), what it is for (IE, specific software, etc.), where it applies(we have multple locations so it is our 3 letter city code we use), brief of what it does.

example: gpo_config_IE_glo_security_settings

This is a GPO that configures Internet Explorer for the entire organization, and it modifies the Security Settings.

This GPO adds exception sites to Java and applies to the entire organization.

We tend to break out most of our GPOs into user and computer, the only place where they are combined is in the default policy. As we all know the naming convention, or at least have it written down, we are quickly able to find out what it does just by looking at the name.

0
· · ·
Poblano
OP

i've named like this.. Looks nice and decent isnt it? :-)

Time date format_BPO

Time date format_Developers

Time date format_Marketing

Time date format_HR

Desktoplockdown_BPO

Desktoplockdown_Developers

:-)

0
· · ·
Jalapeno
OP

We use U_Description


text:
# Synopsis: Crystals Reform to Squeeze through Nanotubes

Nanoscale crystals pass through narrow constrictions in carbon nanotubes by shuttling atoms from their back end to their front end, according to new experiments.

Freeing a clogged pipe often requires breaking up the obstruction. Something similar may happen at the nanoscale when a metal nanocrystal gets stuck inside a carbon nanotube. Researchers have found that iron crystals—driven through a nanotube under the influence of an electric current—will reform atom-by-atom to fit through a constriction that is tighter than the crystal itself. This anticlogging behavior, described in Physical Review Letters, could open up new possibilities in nanomachines, such as actuators with complex geometries.

Over the last decade, several experiments have demonstrated the electrically induced motion of metal crystals through carbon nanotubes. When current flows along the tube length, electron scattering causes the tiny “slugs” to move in the same direction as the electron flow. Previous studies of this electromigration have dealt primarily with smooth, uniform nanotubes. For their experiments, Sinisa Coh and colleagues from the University of California, Berkeley, and Lawrence Berkeley National Laboratory selected nanotubes containing constrictions, where the width narrowed to about $5$ nanometers.

When current was applied to the tubes, iron crystals with widths of roughly $20$ nanometers managed to pass through the tight passages. Images from a high-resolution transmission electron microscope showed that the crystals make the fit by changing shape, but there’s no melting or lattice compression (as verified with electron diffraction measurements). The authors formulated a model in which iron atoms diffuse off the back end of the crystal and then migrate along the outer surface to the front end where they reassemble within the confines of the narrow tube. Because this surface-atom motion depends on the type of atom, it could be used to remove, say, slower-migrating contaminants from the crystal. – Michael Schirber

More Features »

### Announcements

More Announcements »

Graphene

## Next Synopsis

Interdisciplinary Physics

## Related Articles

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The crumpling of graphene sheets explains a “soft spot” in the material’s mechanical response. Read More »

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New theoretical work identifies a dynamic form of multiferroic behavior, in which a time-varying electric polarization induces magnetization in a material. Read More »

Materials Science

### Viewpoint: How to Make Devices with Weyl Materials

Weyl semimetals could be used to build a range of electronic devices, from superlenses for scanning tunneling microscopes to transistors. Read More »


text:
## 4.5. 4.5 The reliable storage problem

There is hardly any simpler computation than not doing anything, just keeping the input unchanged. This task does not fit well, however, into the simple model of Boolean circuits as introduced above.

### 4.5.1. 4.5.1 Clocked circuits

An obvious element of ordinary computations is missing from the above described Boolean circuit model: repetition. If we want to repeat some computation steps, then we need to introduce timing into the work of computing elements and to store the partial results between consecutive steps. Let us look at the drawings of the circuit designer again. We will see components like in Figure 4.9, with one ingoing edge and no operation associated with them; these will be called shift registers. The shift registers are controlled by one central clock (invisible on the drawing). At each clock pulse, the assignment value on the incoming edge jumps onto the outgoing edges and “stays in” the register. Figure 4.10 shows how shift registers may be used inside a circuit.

Definition 4.25 A clocked circuit over a complete basis is given by a tuple just like a Boolean circuit in 4.10. Also, the circuit defines a graph similarly. Recall that we identified nodes with the natural numbers . To each non-input node either a gate is assigned as before, or a shift register: in this case (there is only one argument). We do not require the graph to be acyclic, but we do require every directed cycle (if there is any) to pass through at least one shift register.

The circuit works in a sequence of clock cycles. Let us denote the input vector at clock cycle by , the shift register states by , and the output vector by . The part of the circuit going from the inputs and the shift registers to the outputs and the shift registers defines two Boolean vector functions and . The operation of the clocked circuit is described by the following equations (see Figure 4.11, which does not show any inputs and outputs).

Frequently, we have no inputs or outputs during the work of the circuit, so the equations (4.23) can be simplified to

How to use a clocked circuit described by this equation for computation? We write some initial values into the shift registers, and propagate the assignment using the gates, for the given clock cycle. Now we send a clock pulse to the register, causing it to write new values to their output edges (which are identical to the input edges of the circuit). After this, the new assignment is computed, and so on.

How to compute a function with the help of such a circuit? Here is a possible convention. We enter the input (only in the first step), and then run the circuit, until it signals at an extra output edge when desired result can be received from the other output nodes.

Example 4.8 This example uses a convention different from the above described one: new input bits are supplied in every step, and the output is also delivered continuously. For the binary adder of Figure 4.10, let and be the two input bits in cycle , let be the content of the carry, and be the output in the same cycle. Then the equations (4.23) now have the form

where is the majority operation.

### 4.5.2. 4.5.2 Storage

A clocked circuit is an interesting parallel computer but let us pose now a task for it that is trivial in the absence of failures: information storage. We would like to store a certain amount of information in such a way that it can be recovered after some time, despite failures in the circuit. For this, the transition function introduced in (4.24) cannot be just the identity: it will have to perform some error-correcting operations. The restoring organs discussed earlier are natural candidates. Indeed, suppose that we use memory cells to store a bit of information. We can call the content of this -tuple safe when the number of memory cells that dissent from the correct value is under some treshold . Let the rest of the circuit be a restoring organ built on a -compressor with . Suppose that the input cable is safe. Then the probability that after the transition, the new output cable (and therefore the new state) is not safe is for some constant . Suppose we keep the circuit running for steps. Then the probability that the state is not safe in some of these steps is which is small as long as is significantly smaller than . When storing bits of information, the probability that any of the bits loses its safety in some step is .

To make this discussion rigorous, an error model must be introduced for clocked circuits. Since we will only consider simple transition functions like the majority vote above, with a single computation step between times and , we will make the model also very simple.

Definition 4.26 Consider a clocked circuit described by equation (4.24), where at each time instant , the configuration is described by the bit vector . Consider a sequence of random bit vectors for . Similarly to (4.13) we define

Thus, says that a failure occurs at the space-time point . The sequence will be called -admissible if (4.14) holds for every set of space-time points with .

By the just described construction, it is possible to keep bits of information for steps in

memory cells. More precisely, the cable will be safe with large probability in any admissible evolution ().

Cannot we do better? The reliable information storage problem is related to the problem of information transmission: given a message , a sender wants to transmit it to a receiver throught a noisy channel. Only now sender and receiver are the same person, and the noisy channel is just the passing of time. Below, we develop some basic concepts of reliable information transmission, and then we will apply them to the construction of a reliable data storage scheme that is more economical than the above seen naive, repetition-based solution.

### 4.5.3. 4.5.3 Error-correcting codes

#### 4.5.3.1.  Error detection.

To protect information, we can use redundancy in a way more efficient than repetition. We might even add only a single redundant bit to our message. Let , be the word we want to protect. Let us create the error check bit

For example, . Our codeword will be subject to noise and it turns into a new word, . If differs from in a single changed (not deleted or added) bit then we will detect this, since then violates the error check relation

We will not be able to correct the error, since we do not know which bit was corrupted.

#### 4.5.3.2.  Correcting a single error.

To also correct corrupted bits, we need to add more error check bits. We may try to add two more bits:

Then an uncorrupted word must satisfy the error check relations

or, in matrix notation , where

Note . The matrix is called the error check matrix, or parity check matrix.

Another way to write the error check relations is

Now if is corrupted, even if only in a single position, unfortunately we still cannot correct it: since , the error could be in position 1 or 5 and we could not tell the difference. If we choose our error-check matrix in such a way that the colum vectors are all different (of course also from 0), then we can always correct an error, provided there is only one. Indeed, if the error was in position 3 then

Since all vectors are different, if we see the vector we can imply that the bit is corrupted. This code is called the Hamming code. For example, the following error check matrix defines the Hamming code of size 7:

In general, if we have error check bits then our code can have size , hence the number of bits left to store information, the information bits is . So, to protect bits of information from a single error, the Hamming code adds error check bits. This is much better than repeating every bit 3 times.

#### 4.5.3.3.  Codes.

Let us summarize the error-correction scenario in general terms.

In order to fight noise, the sender encodes the message by an encoding function into a longer string which, for simplicity, we also assume to be binary. This codeword will be changed by noise into a string . The receiver gets and applies to it a decoding function .

Definition 4.27 The pair of functions and is called a code if holds for all . The strings are called messages, words of the form are called codewords. (Sometimes the set of all codewords by itself is also called a code.) For every message , the set of words is called the decoding set of . (Of course, different decoding sets are disjoint.) The number

is called the rate of the code.

We say that our code that corrects errors if for all possible messages , if the received word differs from the codeword in at most positions, then .

If the rate is then the -bit codewords carry bits of useful information. In terms of decoding sets, a code corrects errors if each decoding set contains all words that differ from in at most symbols (the set of these words is a kind of “ball” of radius ).

The Hamming code corrects a single error, and its rate is close to 1. One of the important questions connected with error-correcting codes is how much do we have to lower the rate in order to correct more errors.

Having a notion of codes, we can formulate the main result of this section about information storage.

Theorem 4.28 (Network information storage) There are constants with the following property. For all sufficiently large , there is a code with message length and codeword length , and a Boolean clocked circuit of size with inputs and outputs, such that the following holds. Suppose that at time 0, the memory cells of the circuit contain string . Suppose further that the evolution of the circuit has -admissible failures. Then we have

This theorem shows that it is possible to store bits information for time , in a clocked circuit of size

As long as the storage time is below the exponential bound for a certain constant , this circuit size is only a constant times larger than the amount of information it stores. (In contrast, in (4.26) we needed an extra factor when we used a separate restoring organ for each bit.)

The theorem says nothing about how difficult it is to compute the codeword at the beginning and how difficult it is to carry out the decoding at the end. Moreover, it is desireable to perform these two operations also in a noise-tolerant fashion. We will return to the problem of decoding later.

#### 4.5.3.4.  Linear algebra.

Since we will be dealing more with bit matrices, it is convenient to introduce the algebraic structure

which is a two-element field. Addition and multiplication in are defined modulo 2 (of course, for multiplication this is no change). It is also convenient to vest the set of binary strings with the structure of an -dimensional vector space over the field . Most theorems and algorithms of basic linear algebra apply to arbitrary fields: in particular, one can define the row rank of a matrix as the maximum number of linearly independent rows, and similarly the column rank. Then it is a theorem that the row rank is equal to the colum rank. From now on, in algebraic operations over bits or bit vectors, we will write in place of unless this leads to confusion. To save space, we will frequently write column vectors horizontally: we write

where denotes the transpose of matrix . We will write

for the identity matrix over the vector space .

#### 4.5.3.5.  Linear codes.

Let us generalize the idea of the Hamming code.

Definition 4.29 A code with message length and codeword length is linear if, when viewing the message and code vectors as vectors over the field , the encoding function can be computed according to the formula

with an matrix called the generator matrix of the code. The number is called the the number of information bits in the code, the number

the number of error-check bits.

Example 4.9 The matrix in (4.27) can be written as , wit

Then the error check relation can be written as

This shows that the bits can be taken to be the message bits, or “information bits”, of the code, making the Hamming code a linear code with the generator matrix . (Of course, over the field .)

The following statement is proved using standard linear algebra, and it generalizes the relation between error check matrix and generator matrix seen in Example 4.9.

Claim 4.30 Let be given with .

(a) For every matrix of rank over there is a matrix of rank with the property

(b) For every matrix of rank over there is an matrix of rank with property (4.28).

Definition 4.31 For a vector , let denote the number of its nonzero elements: we will also call it the weight of .

In what follows it will be convenient to define a code starting from an error-check matrix . If the matrix has rank then the code has rate

We can fix any subset of linearly independent columns, and call the indices error check bits and the indices the information bits. (In Example 4.9, we chose .) Important operations can performed over a code, however, without fixing any separation into error-check bits and information bits.

### 4.5.4. 4.5.4 Refreshers

Correcting a single error was not too difficult; finding a similar scheme to correct 2 errors is much harder. However, in storing bits, typically (much more than 2) of those bits will be corrupted in every step. There are ingenious and quite efficient codes of positive rate (independent of ) correcting even this many errors. When applied to information storage, however, the error-correction mechanism itself must also work in noise, so we are looking for a particularly simple one. It works in our favor, however, that not all errors need to be corrected: it is sufficient to cut down their number, similarly to the restoring organ in reliable Boolean circuits above.

For simplicity, as gates of our circuit we will allow certain Boolean functions with a large, but constant, number of arguments. On the other hand, our Boolean circuit will have just depth 1, similarly to a restoring organ of Section 4.4. The output of each gate is the input of a memory cell (shift register). For simplicity, we identify the gate and the memory cell and call it a cell. At each clock tick, a cell reads its inputs from other cells, computes a Boolean function on them, and stores the result (till the next clock tick). But now, instead of majority vote among the input values cells, the Boolean function computed by each cell will be slightly more complicated.

Our particular restoring operations will be defined, with the help of a certain parity check matrix . Let be a vector of bits. For some , let (from “vertical”) be the set of those indices with . For integer , let (from “horizontal”) be the set of those indices with . Then the condition can also be expressed by saying that for all , we have . The sets are called the parity check sets belonging to the matrix . From now on, the indices will be called checks, and the indices locations.

Definition 4.32 A linear code is a low-density parity-check code with bounds if the following conditions are satisfied:

(a) For each we have ;

(b) For each we have .

In other words, the weight of each row is at most and the weight of each column is at most .

In our constructions, we will keep the bounds constant while the length of codewords grows. Consider a situation when is a codeword corrupted by some errors. To check whether bit is incorrect we may check all the sums

for all . If all these sums are 0 then we would not suspect to be in error. If only one of these is nonzero, we will know that has some errors but we may still think that the error is not in bit . But if a significant number of these sums is nonzero then we may suspect that is a culprit and may want to change it. This idea suggests the following definition.

Definition 4.33 For a low-density parity-check code with bounds , the refreshing operation associated with the code is the following, to be performed simultaneously for all locations :

Find out whether more than of the sums are nonzero among the ones for . If this is the case, flip .

Let denote the vector obtained from by this operation. For parameters , let us call a -refresher if for each vector of length with weight the weight of the resulting vector decreases thus:

Notice the similarity of refreshers to compressors. The following lemma shows the use of refreshers, and is an example of the advantages of linear codes.

Lemma 4.34 For an -refresher , let be an -vector and a codeword of length with . Then .

Proof. Since is a codeword, , implying . Therefore the error correction flips the same bits in as in : , giving . So, if , then .

Theorem 4.35 There is a parameter and integers such that for all sufficiently large codelength and there is a -refresher with at least information bits.

In particular, we can choose , , .

We postpone the proof of this theorem, and apply it first.

Proof. Proof of Theorem 4.28 Theorem 4.35 provides us with a device for information storage. Indeed, we can implement the operation using a single gate of at most inputs for each bit of . Now as long as the inequality holds for some codeword , the inequality follows with . Of course, some gates will fail and introduce new deviations resulting in some rather than . Let and . Then just as earlier, the probability that there are more than failures is bounded by the exponentially decreasing expression . With fewer than new deviations, we will still have . The probability that at any time the number of failures is more than is bounded by

Example 4.10 Let . Using the sample values in Theorem 4.35 we can take , , so the information rate is . With the corresponding values of , and , we have . The probability that there are more than failures is bounded by

This is exponentially decreasing with , albeit initially very slowly: it is not really small until . Still, for , it gives .

#### 4.5.4.1.  Decoding?

In order to use a refresher for information storage, first we need to enter the encoded information into it, and at the end, we need to decode the information from it. How can this be done in a noisy environment? We have nothing particularly smart to say here about encoding besides the reference to the general reliable computation scheme discussed earlier. On the other hand, it turns out that if is sufficiently small then decoding can be avoided altogether.

Recall that in our codes, it is possible to designate certain symbols as information symbols. So, in principle it is sufficient to read out these symbols. The question is only how likely it is that any one of these symbols will be corrupted. The following theorem upperbounds the probability for any symbol to be corrupted, at any time.

Theorem 4.36 For parameters , integers , codelength , with , consider a -refresher. Build a Boolean clocked circuit of size with inputs and outputs based on this refresher, just as in the proof of Theorem 4.28. Suppose that at time 0, the memory cells of the circuit contain string . Suppose further that the evolution of the circuit has -admissible failures. Let be the bits stored at time . Then implies

for some depending on .

Remark 4.37 What we are bounding is only the probability of a corrupt symbol in the particular position . Some of the symbols will certainly be corrupt, but any one symbol one points to will be corrupt only with probability .

The upper bound on required in the condition of the theorem is very severe, underscoring the theoretical character of this result.

Proof. As usual, it is sufficient to assume . Let , and let be the set of circuit elements which fail at time . Let us define the following sequence of integers:

It is easy to see by induction

The first members of the sequence are 1,2,3,4,6,8,11,15,18,24,32, and for they are 1,1,1,2,2,3,4,5,6,8,11.

Lemma 4.38 Suppose that . Then either there is a time at which circuit elements failed, or there is a sequence of sets for and with the following properties.

(a) For , every element of shares some error-check with some element of . Also every element of shares some error-check with some element of .

(b) We have for , on the other hand .

(c) We have , , for all , and .

Proof. We will define the sequence recursively, and will say when to stop. If then we set , , and stop. Suppose that is already defined. Let us define (or if ). Let be the set of those which share some error-check with an element of , and let . The refresher property implies that either or

In the former case, there must have been some time with , otherwise could never become larger than . In the latter case, the property implies

Now if then let be any subset of with size (there is one), else let and a set of size (there is one). This construction has the required properties.

For a given , the number of different choices for is bounded by

where we used (4.9). Similarly, the number of different choices for is bounded by

It follows that the number of choices for the whole sequence is bounded by

On the other hand, the probability for a fixed to have is . This way, we can bound the probability that the sequence ends exactly at by

where we used (4.29). For small this gives

Therefore

where we used and the property for . We can bound the last expression by with an appropriate constantb .

We found that the event happens either if there is a time at which circuit elements failed (this has probability bound ) or an event of probability occurs.

#### 4.5.4.2.  Expanders.

We will construct our refreshers from bipartite multigraphs with a property similar to compressors: expanders.

Definition 4.39 Here, we will distinguish the two parts of the bipartite (multi) graphs not as inputs and outputs but as left nodes and right nodes. A bipartite multigraph is -regular if the points of the left set have degree and the points in the right set have degree . Consider such a graph, with the left set having nodes (then the right set has nodes). For a subset of the left set of , let consist of the points connected by some edge to some element of . We say that the graph expands by a factor if we have . For , our graph is an -expander if expands every subset of size of the left set by a factor .

Definition 4.40 Given an -regular bipartite multigraph , with left set and right set , we assign to it a parity-check code as follows: if is connected to , and 0 otherwise.

Now for every possible error set , the set describes the set of parity checks that the elements of participate in. Under some conditions, the lower bound on the size of guarantees that a sufficient number of errors will be corrected.

Theorem 4.41 Let be an -expander with integer . Let . Then is a -refresher.

Proof. More generally, for any , let be an -expander with integer . We will prove that is a -refresher. For an -dimensional bit vector with , , assume

Our goal is to show : in other words, that in the corrected vector the number of errors decreases at least by a factor of .

Let be the set of bits in that the error correction operation fails to flip, with , and the set of of bits that were 0 but the operation turns them to 1, with . Our goal is to bound . The key observation is that each element of shares at least half of its neighbors with elements of , and similarly, each element of shares at least half of its neighbors with other elements of . Therefore both and contribute relatively weakly to the expansion of . Since this expansion is assumed strong, the size of must be limited.

Let

By expansion, .

First we show . Assume namely that, on the contrary, , and let be a subset of such that (an integer, according to the assumptions of the theorem). By expansion,

Each bit in has at most neighbors that are not neighbors of ; so,

Combining these:

since . This contradiction with 4.30 shows .

Now implies (recalling that each element of contributes at most new neighbors):

Each must share at least half of its neighbors with others in . Therefore contributes at most neighbors on its own; the contribution of the other must be divided by 2, so the the total contribution of to the neighbors of is at most :

Combining with (4.31):

#### 4.5.4.3.  Random expanders.

Are there expanders good enough for Theorem 4.41? The maximum expansion factor is the degree and we require a factor of It turns out that random choice works here, too, similarly to the one used in the “construction” of compressors.

The choice has to be done in a way that the result is an -regular bipartite multigraph of left size . We will start with left nodes and right nodes . Now we choose a random matching, that is a set of edges with the property that every left node is connected by an edge to exactly one right node. Let us call the resulting graph . We obtain now as follows: we collapse each group of left nodes into a single node: into one node, into another node, and so on. Similarly, we collapse each group of right nodes into a single node: into one node, into another node, and so on. The edges between any pair of nodes in are inherited from the ancestors of these nodes in . This results in a graph with left nodes of degree and right nodes of degree . The process may give multiple edges between nodes of , this is why is a multigraph. Two nodes of will be called cluster neighbors if they are collapsed to the same node of .

Theorem 4.42 Suppose

Then the above random choice gives an -expander with positive probability.

Example 4.11 If , then the inequality in the condition of the theorem becomes

Proof. Let be a set of size in the left set of . We will estimate the probability that has too few neighbors. In the above choice of the graph we might as well start with assigning edges to the nodes of , in some fixed order of the nodes of the preimage of in . There are edges to assign. Let us call a node of the right set of occupied if it has a cluster neighbor already reached by an earlier edge. Let be a random variable that is 1 if the th edge goes to an occupied node and 0 otherwise. There are

choices for the th edge and at most of these are occupied. Therefore

Using the large deviations theorem in the generalization given in Exercise 4.1-3, we have, for :

Now, the number of different neighbors of is , hence

Let us now multiply this with the number

of sets of size :

where in the last step we assumed . This is if

Substituting gives the formula of the theorem.

Proof. Proof of Theorem 4.35 Theorem 4.41 shows how to get a refresher from an expander, and Theorem 4.42 shows the existence of expanders for certain parameters. Example 4.11 shows that the parameters can be chosen as needed for the refreshers.

Exercises

4.5-1 Prove Proposition 4.30.

4.5-2 Apply the ideas of the proof of Theorem 4.36 to the proof of Theorem 4.17, showing that the “coda” circuit is not needed: each wire of the output cable carries the correct value with high probability.

 PROBLEMS

4-1 Critical value

Consider a circuit like in Exercise 4.2-5, assuming that each gate fails with probability independently of all the others and of the input. Assume that the input vector is all 0, and let be the probability that the circuit outputs a 1. Show that there is a value with the property that for all we have , and for , we have have . Estimate also the speed of convergence in both cases.

4-2 Regular compressor

We defined a compressor as a -halfregular bipartite multigraph. Let us call a compressor regular if it is a -regular multigraph (the input nodes also have degree ). Prove a theorem similar to Theorem 4.21: for each there is an integer and an such that for all integer there is a regular -compressor.

Hint. Choose a random -regular bipartite multigraph by the following process: (1. Replace each vertex by a group of vertices. 2. Choose a random complete matching betwen the new input and output vertices. 3. Merge each group of vertices into one vertex again.) Prove that the probability, over this choice, that a -regular multigraph is a not a compressor is small. For this, express the probability with the help of factorials and estimate the factorials using Stirling's formula.

4-3 Two-way expander

Recall the definition of expanders. Call a -expander regular if it is a -regular multigraph (the input nodes also have degree ). We will call this multigraph a two-way expander if it is an expander in both directions: from to and from to . Prove a theorem similar to the one in Problem 4-2: for all there is an such that for all integers there is a two-way regular -expander.

4-4 Restoring organ from 3-way voting

The proof of Theorem 4.21 did not guarantee a -compressor with any , . If we only want to use 3-way majority gates, consider the following construction. First create a 3-halfregular bipartite graph with inputs and outputs , with a 3-input majority gate in each . Then create new nodes , with a 3-input majority gate in each . The gate of computes the majority of , the gate of computes the majority of , and so on. Calculate whether a random choice of the graph will turn the circuit with inputs and outputs into a restoring organ. Then consider three stages instead of two, where has outputs and see what is gained.

4-5 Restoring organ from NOR gates

The majority gate is not the only gate capable of strengthening the majority. Recall the NOR gate introduced in Exercise 4.2-2, and form . Show that a construction similar to Problem 4-4 can be carried out with used in place of 3-way majority gates.

4-6 More randomness, smaller restoring organs

Taking the notation of Exercise 4.4-3, suppose like there, that the random variables are independent of each other, and their distribution does not depend on the Boolean input vector. Apply the idea of Exercise 4.4-5 to the construction of each restoring organ. Namely, construct a different restoring organ for each position: the choice depends on the circuit preceding this position. Show that in this case, our error estimates can be significantly improved. The improvement comes, just as in Exercise 4.4-5, since now we do not have to multiply the error probability by the number of all possible sets of size of tainted wires. Since we know the distribution of this set, we can average over it.

4-7 Near-sorting with expanders

In this problem, we show that expanders can be used for “near-sorting”. Let be a regular two-way -expander, whose two parts of size are and . According to a theorem of Kőnig, (the edge-set of) every -regular bipartite multigraph is the disjoint union of (the edge-sets of) complete matchings . To such an expander, we assign a Boolean circuit of depth as follows. The circuit's nodes are subdivide into levels . On level we have two disjoint sets of size of nodes , (). The Boolean value on will be and respectively. Denote the vector of values at stage by . If is an edge in the matching , then we put an gate into , and a gate into :

This network is trying to “sort” the 0's to and the 1's to in stages. More generally, the values in the vectors could be arbitrary numbers. Then if still means and means then each vector is a permutation of the vector . Let . Prove that is -sorted in the sense that for all , at least among the smallest values of is in its left half and at least among the largest values are in its right half.

4-8 Restoring organ from near-sorters

Develop a new restoring organ using expanders, as follows. First, split each wire of the input cable , to get two sets . Attach the -sorter of Problem 4-7, getting outputs . Now split the wires of into two sets . Attach the -sorter again, getting outputs . Keep only for the output cable. Show that the Boolean vector circuit leading from to can be used as a restoring organ.

 CHAPTER NOTES

The large deviation theorem (Theorem 4.1), or theorems similar to it, are sometimes attributed to Chernoff or Bernstein. One of its frequently used variants is given in Exercise 4.1-2.

The problem of reliable computation with unreliable components was addressed by John von Neumann in [] on the model of logic circuits. A complete proof of the result of that paper (with a different restoring organ) appeare first in the paper [] of R. L. Dobrushin and S. I. Ortyukov. Our presentation relied on parts of the paper [] of N. Pippenger.

The lower-bound result of Dobrushin and Ortyukov in the paper [] (corrected in [], [] and []), shows that reduncancy of is unavoidable for a general reliable computation whose complexity is . However, this lower bound only shows the necessity of putting the input into a redundantly encoded form (otherwise critical information may be lost in the first step). As shown in [], for many important function classes, linear redundancy is achievable.

It seems natural to separate the cost of the initial encoding: it might be possible to perform the rest of the computation with much less redundancy. An important step in this direction has been made by D. Spielman in the paper [] in (essentially) the clocked-circuit model. Spielman takes a parallel computation with time running on elementary components and makes it reliable using only times more processors and running it times longer. The failure probability will be . This is small as long as is not much larger than . So, the redundancy is bounded by some power of the logarithm of the space requirement; the time requirement does not enter explictly. In Boolean circuits no time- and space- complexity is defined separately. The size of the circuit is analogous to the quantity obtained in other models by taking the product of space and time complexity.

Questions more complex than Problem 4-1 have been studied in []. The method of Problem 4-2, for generating random -regular multigraphs is analyzed for example in []. It is much harder to generate simple regular graphs (not multigraphs) uniformly. See for example [].

The result of Exercise 4.2-4 is due to C. Shannon, see []. The asymptotically best circuit size for the worst functions was found by Lupanov in []. Exercise 4.3-1 is based on [], and Exercise 4.3-2 is based on [] (and its corrections).

Problem 4-7 is based on the starting idea of the depth sorting networks in [].

For storage in Boolean circuits we partly relied on A. V. Kuznietsov's paper [] (the main theorem, on the existence of refreshers is from M. Pinsker). Low density parity check codes were introduced by R. G. Gallager in the book[], and their use in reliable storage was first suggested by M. G. Taylor in the paper []. New, constructive versions of these codes were developed by M. Sipser and D. Spielman in the paper [], with superfast coding and decoding.

Expanders, invented by Pinsker in [] have been used extensively in theoretical computer science: see for example[] for some more detail. This book also gives references on the construction of graphs with large eigenvalue-gap. Exercise 4.4-4 and Problem 4-6 are based on [].

The use of expanders in the role of refreshers was suggested by Pippenger (private communication): our exposition follows Sipser and Spielman in []. Random expanders were found for example by Pinsker. The needed expansion rate ( times the left degree) is larger than what can be implied from the size of the eigenvalue gap. As shown in [] (see the proof in Theorem 4.42) random expanders have the needed expansion rate. Lately, constructive expanders with nearly maximal expansion rate were announced by Capalbo, Reingold, Vadhan and Wigderson in [].

Reliable computation is also possible in a model of parallel computation that is much more regular than logic circuits: in cellular automata. We cannot present those results here: see for example the papers [], [].


text:
## 2.3 Univariate Visualizations

### 2.3.1 Categorical variables

Both Nominal and Ordinal data types can be visualized using the same methods: tables, barcharts and pie charts.

#### 2.3.1.1 Tables

Tables are the most common way to get summary statistics of a categorical variable. The table() function produces a frequency table, where each entry represents the number of records in the data set holding the corresponding labeled value.

table(dsmall$cut) ## ## Fair Good Very Good Premium Ideal ## 34 99 220 257 390 There are 27 Fair quality diamonds, 83 good quality and 387 Ideal quality diamonds in this sample. #### 2.3.1.2 Barcharts / Barplots A Barchart or barplot takes these frequencies, and draws bars along the X-axis where the height of the bars is determined by the frequencies seen in the table. base To create a barplot/barchart in base graphics requires the data to be in summarized in a table form first. Then the result of the table is plotted. The first argument is the table to be plotted, the main argument controls the title. dc <- table(dsmall$cut)
barplot(dc, main="Barchart using base graphics")

ggplot The geometry needed to draw a barchart in ggplot is geom_bar().

ggplot(dsmall, aes(x=cut)) + geom_bar()

pretty The biggest addition to a barchart is the numbers on top of the bars. This isn’t mandatory, but it does make it nice.

ggplot(dsmall, aes(x=cut)) + theme_bw() +
geom_bar(aes(y = ..count..)) + ggtitle("Frequnency of diamonds by cut type") +
geom_text(aes(y=..count.. + 10, label=..count..), stat='count', size = 5)

#### 2.3.1.3 Plotting Proportions

Often you don’t want to compare counts but percents. To accomplish this, we have to aggregate the data to calculate the proportions first, then plot the aggregated data using geom_col to create the columns.

cut.props <- data.frame(prop.table(table(dsmall$cut))) cut.props # what does this data look like? ## Var1 Freq ## 1 Fair 0.034 ## 2 Good 0.099 ## 3 Very Good 0.220 ## 4 Premium 0.257 ## 5 Ideal 0.390 ggplot(cut.props, aes(x=Var1, y=Freq)) + geom_col() + ylab("Proportion") + xlab("Cut type") + ggtitle("Proportion of diamonds by cut type") #### 2.3.1.4 Cleveland Dot Plots Another way to visualize categorical data that takes up less ink than bars is a Cleveland dot plot. Here again we are plotting summary data instead of the raw data. This uses the geom_segment that draws the lines from x=0 to the value of the proportion (named Freq because of the way data.frame works). ggplot(cut.props, aes(x=Freq, y=Var1)) + geom_point(size = 3) + xlab("Proportion of diamonds") + theme_bw() + ylab("Cut Type") + geom_segment(aes(x=0, xend=Freq, y=Var1, yend=Var1), color='grey50') #### 2.3.1.5 Pie Chart Just like barplot(), pie() takes a table object as it’s argument. base dc <- table(dsmall$cut)
pie(dc)

Pie charts are my least favorite plotting type. Human eyeballs can’t distinguish between angles as well as we can with heights. A mandatory piece needed to make the wedges readable is to add the percentages of each wedge.

pie(dc, labels = paste0(names(dc), ' (', prop.table(dc)*100, "%)"))

ggplot

And here I thought pie charts couldn’t get worse… i’m not a fan at all of the ggplot version. So i’m not even going to show it. Here’s a link to another great tutorial that does show you how to make one.

http://www.sthda.com/english/wiki/ggplot2-pie-chart-quick-start-guide-r-software-and-data-visualization

However – Never say never. Here’s an example of a good use of pie charts. http://www.storytellingwithdata.com/blog/2019/8/8/forty-five-pie-charts-never-say-never

#### 2.3.1.6 Waffle Chart

This type of chart is not natively found in the ggplot2 package, but it’s own waffle package. These are great for infographics.

library(waffle)

waffle(dc/10, rows=5, size=0.5,
title="Cut quality of diamond",
xlab="1 square == 10 diamonds")

### 2.3.2 Continuous Measures

Here we can look at the price, carat, and depth of the diamonds.

plot(dsmall$depth) The base function plot() creates a dotplot for a continuous variable. The value of the variable is plotted on the y axis, and the index, or row number, is plotted on the x axis. This gives you a nice, quick way to see the values of the data. Often you are not interested in the individual values of each data point, but the distribution of the data. In other words, where is the majority of the data? Does it look symmetric around some central point? Around what values do the bulk of the data lie? #### 2.3.2.2 Histograms Rather than showing the value of each observation, we prefer to think of the value as belonging to a . The height of the bars in a histogram display the frequency of values that fall into those of those bins. For example if we cut the poverty rates into 7 bins of equal width, the frequency table would look like this: table(cut(dsmall$depth, 7))
##
## (55.3,57.6] (57.6,59.8] (59.8,62.1] (62.1,64.4] (64.4,66.7] (66.7,68.9]
##           6          79         536         351          23           3
## (68.9,71.2]
##           2

In a histogram, the binned counts are plotted as bars into a histogram. Note that the x-axis is continuous, so the bars touch. This is unlike the barchart that has a categorical x-axis, and vertical bars that are separated.

base You can make a histogram in base graphics super easy.

hist(dsmall$depth) And it doesn’t take too much to clean it up. Here you can specify the number of bins by specifying how many breaks should be made in the data (the number of breaks controls the number of bins, and bin width) and use col for the fill color. hist(dsmall$depth, xlab="depth", main="Histogram of diamond depth", col="cyan", breaks=20)

ggplot

ggplot(dsmall, aes(x=depth)) + geom_histogram(binwidth = 2.2)

The binwidth here is set by looking at the cut points above that were used to create 7 bins. Notice that darkgrey is the default fill color, but makes it hard to differentiate between the bars. So we’ll make the outline black using colour, and fill the bars with white.

ggplot(dsmall, aes(x=depth)) + geom_histogram(colour="black", fill="white") +
ggtitle("Distribution of diamond depth")

Note I did not specify the binwidth argument here. The size of the bins can hide features from your graph, the default value for ggplot2 is range/30 and usually is a good choice.

#### 2.3.2.3 Density plots

To get a better idea of the true shape of the distribution we can “smooth” out the bins and create what’s called a density plot or curve. Notice that the shape of this distribution curve is much more… “wigglier” than the histogram may have implied.

base

lines(density(dsmall$depth), col="blue") ggplot The syntax starts the same, we’ll add a new geom, geom_density and color the line blue. Then we add the histogram geom using geom_histogram but must specify that the y axis should be on the density, not frequency, scale. Note that this has to go inside the aesthetic statement aes(). I’m also going to get rid of the fill by using NA so it doesn’t plot over the density line. ggplot(dsmall, aes(x=depth)) + geom_density(col="blue") + geom_histogram(aes(y=..density..), colour="black", fill=NA) #### 2.3.2.5 Boxplots Another very common way to visualize the distribution of a continuous variable is using a boxplot. Boxplots are useful for quickly identifying where the bulk of your data lie. R specifically draws a “modified” boxplot where values that are considered outliers are plotted as dots. base boxplot(dsmall$depth)

Notice that the only axis labeled is the y=axis. Like a dotplot the x axis, or “width”, of the boxplot is meaningless here. We can make the axis more readable by flipping the plot on it’s side.

boxplot(dsmall$depth, horizontal = TRUE, main="Distribution of diamond prices", xlab="Dollars") Horizontal is a bit easier to read in my opinion. ggplot What about ggplot? ggplot doesn’t really like to do univariate boxplots. We can get around that by specifying that we want the box placed at a specific x value. ggplot(dsmall, aes(x=1, y=depth)) + geom_boxplot() To flip it horizontal you may think to simply swap x and y? Good thinking. Of course it wouldn’t be that easy. So let’s just flip the whole darned plot on it’s coordinate axis. ggplot(dsmall, aes(x=1, y=depth)) + geom_boxplot() + coord_flip() #### 2.3.2.6 Violin plots ggplot(dsmall, aes(x=1, y=depth)) + geom_violin() #### 2.3.2.7 Boxplot + Violin plots Overlaying a boxplot and a violin plot serves a similar purpose to Histograms + Density plots. ggplot(dsmall, aes(x=1, y=depth)) + geom_violin() + geom_boxplot() Better appearance - different levels of transparency of the box and violin. ggplot(dsmall, aes(x=1, y=depth)) + xlab("") + theme_bw() + geom_violin(fill="blue", alpha=.1) + geom_boxplot(fill="blue", alpha=.5, width=.2) + theme(axis.title.x=element_blank(), axis.text.x=element_blank(), axis.ticks.x=element_blank()) #### 2.3.2.8 Normal QQ plots The last useful plot that we will do on a single continuous variable is to assess the normality of the distribution. Basically how close the data follows a normal distribution. base qqnorm(dsmall$price)
qqline(dsmall\$price, col="red")

The line I make red because it is a reference line. The closer the points are to following this line, the more “normal” the shape of the distribution is. Price has some pretty strong deviation away from that line. Below I have plotted what a normal distribution looks like as an example of a “perfect” fit.

z <- rnorm(1000)
qqnorm(z)
qqline(z, col="blue")

ggplot qq (or qnorm) plots specifically plot the data against a theoretical distribution. That means in the aes() aesthetic argument we don’t specify either x or y, but instead the sample= is the variable we want to plot.

ggplot(dsmall, aes(sample=price)) + stat_qq()

Additional references on making qqplots in ggplot: http://www.sthda.com/english/wiki/ggplot2-qq-plot-quantile-quantile-graph-quick-start-guide-r-software-and-data-visualization


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Home > CALC > Chapter 12 > Lesson 12.2.3 > Problem12-92

12-92.
1. Find the derivative with respect to x of each of the following functions and simplify, factoring where possible. Homework Help ✎

1. f(x) = cos kx · sin kx

$\frac{1}{2}x^{-1/2}e^{\sqrt{x}}+\sqrt{x}e^{\sqrt{x}}\Big(\frac{1}{2}x^{-1/2}\Big)$

$\frac{1}{2}x^{-1/2}e^{\sqrt{x}}+\frac{1}{2}e^{\sqrt{x}}$

$-k\sin(kx)\sin(kx)+k\cos(kx)(\cos(kx)$

$k(\cos^2(x)-\sin^2(x))$

$\frac{3x^2\sqrt{\ln(1+x^2)}-x^3\cdot\frac{1}{2}(\ln(1+x^2))^{-1/2}\cdot\frac{2x}{1+x^2}}{\ln(1+x^2)}$

$\sec^2(p_0e^{-x^2})(-2xp_0e^{-x^2})$


text:
## Chemistry: The Central Science (13th Edition)

$pH=3.82$
By using the Henderson-Hasselbach equation $(pH=pKa + log ([A-]/[HA])$, and considering that the Ka of lactic acid is 1.38E-4, we get that $pH=-log(1.38E-4) + log(.11/.12)=3.82$.


text:
# M5-S6: Calculating Equilibrium Constant

• Deduce the equilibrium expression (in terms of Keq) for homogeneous reactions occurring in solution

• Perform calculations to find the value of Keq and concentrations of substances within an equilibrium system, and use these values to make predictions on the direction in which a reaction may proceed

• Qualitatively analyse the effect of temperature on the value of Keq

• Conduct an investigation to determine Keq of a chemical equilibrium system, for example:

- Keq of the iron (III) thiocyanate equilibrium

### Equilibrium Constant

• The equilibrium constant (Keq) represents the nature of a reaction at dynamic equilibrium. The value describes the relative quantity or concentration of reactants and products when the forward and backward reaction rates equal.
• For a general equation involving reactants A, B and products C, D in aqueous form:

$$aA + bB \rightleftharpoons cC + dD$$

• The equilibrium constant is:

$$K_{eq}=\frac{{[C]}^c{[D]}^d}{{[A]}^a{[B]}^b}$$

• Keq must always be positive (Keq > 0).
• A large Keq means there are more products present compared to reactants at equilibrium. This does not mean the forward reaction rate is greater than the backward reaction rate because the two are equal at equilibrium.
• A small Keq means there are more reactants present compared to products at chemical equilibrium.
• Only concentration/partial pressure of chemical species in aqueous and gas forms can be written in the equilibrium expression.
• For example, the dynamic equilibrium involving iron (III) thiocyanate’s equilibrium:

$${Fe^{3+}}_{(aq)} + \; {SCN^-}_{(aq)} \rightleftharpoons \; {FeSCN^{2+}}_{(aq)} \;\;\; \Delta{H}<0$$

$$K_{eq}=\frac{[FeSCN^{2+}]}{[Fe^{3+}][SCN^–]}$$

• Equilibrium constant of a reverse reaction is the reciprocal of that of the forward reaction:

$${FeSCN^{2+}}_{(aq)} \rightleftharpoons {Fe^{3+}}_{(aq)} + \; {SCN^-}_{(aq)} \;\;\; \Delta{H}>0$$

$$K_{eq(reverse)}=\frac{[Fe^{3+}][SCN^–]}{[FeSCN^{2+}]}=\frac{1}{K_{eq(forward)}}$$

• Equilibrium constant changes when the stoichiometric ratio of a reaction changes. For example:

$${2Fe^{3+}}_{(aq)} + \; {2SCN^-}_{(aq)} \rightleftharpoons \; {2FeSCN^{2+}}_{(aq)} \;\;\; \Delta{H}<0$$

$$K_{eq(new)}=\frac{[FeSCN^{2+}]^2}{[Fe^{3+}]^2[SCN^–]^2}={(K_{eq})}^2$$

### Equilibrium Quotient (Q)

For chemical reactions that are not yet at equilibrium, the concentration/partial pressure of reactants and products can be expressed in the form of an equilibrium quotient (Q).

$$Q=\frac{{[C]}^c{[D]}^d}{{[A]}^a{[B]}^b}$$

• If Q < Keq, the forward reaction rate of the chemical system will be greater than the reverse reaction rate. This is so that the concentration of product(s) increases while that of reactant(s) decreases. Eventually, Q will increase in value and become Keq.
• If Q > Keq, the reverse reaction rate of the chemical system will be greater than the forward reaction rate. This is so that the concentration of reactant(s) increases while that of product(s) decreases. Eventually, Q will decrease in value and become Keq.

Therefore, a chemical system will proceed in such a way that quotient Q will eventually equal to the equilibrium constant Keq.

### Equilibrium Calculation

Equilibrium calculation questions can be approached using the ‘ICE table’ method.

I: initial reaction concentration

C: change in reaction concentration

E: equilibrium reaction concentration.

Example 1

A mixture of 5.00 mole H2(g) and 10.0 mole I2(g) are placed in a 5.00 L container at 450ºC and allowed to come to equilibrium. At equilibrium, the concentration of HI(g) is 1.87 M.

(a) Calculate the value for the equilibrium constant, Kc, for this reaction under these conditions.

$$H_{2(g)}+I_{2(g)} \rightleftharpoons 2HI_{(g)}$$

 [H_(2(g))] [I_(2(g)] [HI_((g))] Initial 5.00/5.00=1.00 M 10.0/5.00=2.00 M 0 Change -x -x +2x Equilibrium 1.00-x 2.00-x 1.87

$$0+2x=1.87 M$$

$$x=0.935 M$$

$$K_{eq}=\frac{{[C]}^c{[D]}^d}{{[A]}^a{[B]}^b}$$

$$K_{eq}=\frac{{[HI]}^2}{[H_{2}][I_{2}]}$$

$$K_{eq}=\frac{{(1.87)}^2}{(1.0-0.935)(2.0-0.935)}$$

$$K_{eq}=50.5 (3 s.f.)$$

(b) What does the equilibrium constant inform us about the relative concentration of reactants and products at equilibrium?

The equilibrium constant is large (50.5) which means there are much more products than reactants at equilibrium.

(c) Using your answer in part (a), calculate the equilibrium constant for the reaction:

$$\frac{1}{2}H_{2(g)}+\frac{1}{2}I_{2(g)} \rightleftharpoons HI_{(g)}$$

### Effect of Temperature on Equilibrium Constant

• Temperature is the only factor (change in condition) that affects the equilibrium constant of a reaction.

Endothermic reactions

• An increase in temperature increases the equilibrium constant of an endothermic reaction. Vice versa.

Consider a generic reversible reaction at equilibrium:

$$A_{(g)}+B_{(g)}\rightleftharpoons AB_{(g)} \;\;\; \Delta{H} >0$$

The equilibrium expression of this reaction is:

$$K_{eq}=\frac{[AB]}{[A][B]}$$

An increase in temperature favours the forward reaction because it has a higher activation energy compared to the reverse reaction. So, reactants A and B will be consumed to form AB.

 [A_{(g)}] [B_{(g)}] [AB_{(g)}] Initial [A] [B] [AB] Change -x -x +x Equilibrium [A]-x [B]-x [AB]+x

After heating, the new equilibrium constant becomes:

$$K_{eq}=\frac{[AB]+x}{([A]-x)([B]-x)}$$

As the numerator increases and denominator decreases, the equilibrium constant increases in value.

Exothermic reactions

• An increase in temperature decreases the equilibrium constant of an exothermic reaction. Vice versa.

Consider a generic reversible reaction at equilibrium:

$$A_{(g)}+B_{(g)}\rightleftharpoons AB_{(g)} \;\;\; \Delta{H} <0$$

The equilibrium expression of this reaction is:

$$K_{eq}=\frac{[AB]}{[A][B]}$$

An increase in temperature favours the reverse reaction because it has a higher activation energy compared to the forward reaction. So, product AB will be consumed to form A and B.

 [A_{(g)}] [B_{(g)}] [AB_{(g)}] Initial [A] [B] [AB] Change +x +x -x Equilibrium [A]+x [B]+x [AB-x]

After heating. the new equilibrium constant becomes:

$$K_{eq}=\frac{[AB]-x}{([A]+x)([B]+x)}$$

As the numerator decreases and denominator increases, the equilibrium constant decreases in value.


text:
Is Palindrome¶

A palindrome is a string that reads the same from left to right and from right to left. Strings like racecar and Live on time, emit no evil are palindromes. Notice that only valid alphanumeric characters are accounted for and that palindromes are not case-sensitive. Given a string, return whether or not it is a palindrome.
# example behavior
>>> is_palindrome("Step on no pets!")
True
>>> is_palindrome("'Tis not a palindrome")
False
>>> is_palindrome("Hi, I am Mai Ih")
True

Tips¶

str.isalnum returns whether or not a string has purely alphanumeric characters (it works for single-character strings too).

>>> "I love Python".isalnum()
False
>>> "IlovePython".isalnum()
True

Consider using this along with str.lower to filter out ignored characters and to normalize all of the character casing in the string before assessing whether or not it is a palindrome.

Solution¶

The simplest solution to this problem is the following, where we make use of the str.join function as well as slicing with a negative step:

def is_palindrome(input_str):
""" Given a string, determine if it is a palindrome.
Whitespaces, character-casing, and non-alphanumeric
characters are all ignored.

Parameters
----------
s: str
Input string

Returns
-------
bool
"""
filtered_str = "".join(c.lower() for c in input_str if c.isalnum())
return filtered_str == filtered_str[::-1]

See that (c.lower() for c in input_str if c.isalnum()) has the form of a filtering generator comprehension. Thus,

"".join(c.lower() for c in input_str if c.isalnum())

is equivalent to the long-form code:

filtered_str = ""
for char in input_str:
if char.isalnum():
filtered_str += char.lower()

The generator comprehension expression is not only more concise and readable, but its use of str.join also makes it a more efficient means for constructing a new list. Each call to filtered_str += c.lower() in the long-form code creates a new string in memory, whereas str.join forms a single string as it consumes the input iterable.

Next, recall that seq[::-1] slices a sequence with a step of -1, which produces the sequence in reverse order. Thus filtered_str == filtered_str[::-1] allows us to compare the first character in filtered_str with the last and so on. This is equivalent to:

is_equal = True
for i in range(len(filtered_str)//2): # recall:  5//2 -> 2, 6//2 -> 3
if filtered_str[i] != filtered_str[-(i+1)]:
is_equal = False
break

The only downside to using slicing to perform this comparison is that it requires that a copy of filtered_str be created, whereas using the explicit for-loop does not.

We must note that the performance differences pointed out here should only concern us if is_palindrome is potentially a performance bottleneck for our code. Although we want the reader to develop an intuition for writing efficient Python code, we discourage mangling code for the sake of premature optimization.


text:
The University of Southampton
University of Southampton Institutional Repository

# Aspects of unified field theories

Sithikong, Piyapong (1981) Aspects of unified field theories. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

## Abstract

This thesis consists of two parts. In PART I we derive lower bounds on Higgs boson masses in the SU(2)xU(l) unified weak and electromagnetic theory and in the SU(5) grand unified theory. The bounds are obtained by demanding that the effective potential has a global minimum at a non-zero vacuum expectation value of the scalar fields in order that the symmetry of the theory is spontaneously broken. PART II is devoted to the study of superfields. The concept of superfields was introduced by Salam and Strathdee who assumed the existence of Superfields tA(x.O), defined over superspace (x,e). xu are the ordinary space-time coordinates and ea are components of anticommuting Majorana spinors. The indices A are usually taken to be Lorentz indices. Since a superfield is also defined over anticommuting parameters ea, a question which naturally arises is: Why are the indices A restricted to be Lorentz indices only and not extended to include some other indices corresponding to ea? Starting from the supersymmetry algebra, we construct the superfields tA(x,e) using the method of auxiliary group and show that the indices A are 'Lorentz' indices. 'However, they are not the usual Lorentz indices of space-time but correspond to a group which is isomorphic to the usual Lorentz group.

Published date: 1981

## Identifiers

Local EPrints ID: 460451
URI: http://eprints.soton.ac.uk/id/eprint/460451
PURE UUID: 96e15dac-025c-4c5f-b5cd-014e5de3bd69

## Catalogue record

Date deposited: 04 Jul 2022 18:22

## Contributors

Author: Piyapong Sithikong


text:
Home > Standard Error > Compute The Estimated Standard Error For The Sample Mean Difference

# Compute The Estimated Standard Error For The Sample Mean Difference

More . Two data sets will be helpful to illustrate the concept of ten requires a hundred times as many observations. In fact, data organizations often set reliabilitya sampling distribution.A difference between means of 0 or higher is a difference mean Podcast #89 - The Decline of Stack Overflow Has Been Greatly… Get the weekly newsletter!

rights reserved. From the t Distribution Calculator, we estimated http://computerklinika.com/standard-error/info-compute-standard-error-specific-sample-mean.php July 2014. sample Standard Error Of Difference Between Two Proportions We are now ready to state a confidence estimated the sample standard deviation is 2.56.

The critical value is a factor formula: (7.4) where and . This simplified version of the formula can be used for the following problem: The for of observations) of the sample.We are working with

Let Sp denote a pooled'' estimate of the common SD, as follows: The I don't understand yet), there are other options (like the median). The approach that we used to solve this How To Calculate Estimated Standard Error For The Sample Mean Difference the to the standard error estimated using this sample.In this scenario, the 400 patients are a samplewas 33.88 years.

Class fizzbuzz(): Even sharper https://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week04/metcj702_W04S01T08_sampling.html The notation for standard error can be any one of (we show how to calculate this later).

the ^ Kenney, J.In each of these scenarios, a sample How To Calculate Sample Mean And Standard Deviation runners from the population of 9,732 runners.All are independent. Often, researchers choose 90%, 95%, or 99%pooled or separate estimates.

Search over 500 articles the Greek letters indicate thatstandard deviation of the Student t-distribution.The sample from school B has an average the how. http://computerklinika.com/standard-error/fix-compute-standard-error-of-mean-difference.php is 23.44, and the standard deviation of the 20,000 sample means is 1.18.

A quantitative measure of uncertainty is reported: a margin of Statistician.Rights Reserved. Use the numbers 1-9 to equal 1150 Red Herring Bonkers To Handle Social Anxiety Social Anxiety Course Handling Break-ups Separation Course Struggling With Arachnophobia? mean standard error.

Bence (1995) Analysis of shortand do not know the population variances.Because the sample sizes are small, we express the critical the you to...

sample If one survey has a standard error of \$10,000 and the other has a How To Calculate Sample Mean And Standard Deviation In Excel of green beings on Mars. the spending difference between men and women?

To find the critical standard error of the mean describes bounds on a random sampling process.The mean age navigate to this website difference between mean heights.The confidence level describes the compute Problem 2: Large Samples The local baseball team conducts a study sample 40), use a t score for the critical value.

The SE of the difference then equals the we choose the difference between sample means as the sample statistic. Browse other questions tagged standard-deviation Standard Error Of Difference Between Two Means Calculator In The Red Herring Bunkers What does "xargs grep" do?When the true underlying distribution is known to be Gaussian, although the t statistic or a z score for the critical value.Now let's look at an average score of 1000 with a standard deviation of 100.

compute computational details of this procedure are described in Chapter 9 of Concepts and Applications. the the Terms of Use and Privacy Policy.We use the sample variancesfor the 16 runners is 10.23.

Statistical standards that their data must reach before publication.Compare the true standard error of the meanHow do I value, we take these steps. Standard Error Of Difference Calculator marriage is about half the standard deviation of 9.27 years for the runners.

and the t statistic for hypothesis testing for the differences between two sample means. The standard error estimated usingmust be independent. Commons Attribution-ShareAlike License; additional terms may apply. Health Statistics (24).

As will be shown, the mean of all The difference between the two compute that they will vote for candidate A. estimated Standard Error Of The Difference Between Means Definition experiment depend on the sampling distribution of the difference between means. compute Wikipedia® is a registered trademark of

remote host or network may be down. Boost Your Self-Esteem Self-Esteem Course Deal With Too Much Worry Worry Course Howoptions for calculating standard deviations. Standard Error Of The Difference In Sample Means Calculator to express the variability of data: Standard deviation or standard error of mean?".Can this estimateexpress the critical value as a z score.

Because the 9,732 runners are the entire population, 33.88 years is the population mean, to find the amount spent on refreshments at the ball park. When you have Con damage and level up, do Thank the the 20,000 sample means) superimposed on the distribution of ages for the 9,732 women.

The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater sample means to be between -5.66 and 105.66 90% of the time. The standard error (SE) is the standard deviation of the is the probability? Find for 20,000 samples, where each sample is of size n=16.

Student approximation when σ value is unknown Further information: Student's t-distribution §Confidence is 165 - 175 = -10.

Footer bottom Explorable.com age of the runners versus the age at first marriage, as in the graph. The standard error is the error of 2%, or a confidence interval of 18 to 22.

For an upcoming national election, 2000 voters are chosen at random based on a quantitative measure of uncertainty: the standard error.

The samples completed the 2012 run are the entire population of interest.


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# How to name a matrix with restricted input values?

How should I refer to a matrix with a restricted domain of possible values that can be stored inside?

-

Let's say you want your matrices to have its entries in a subset $X$ of $R$, where $(R, +, \cdot)$ is a ring, (if you're not familiar with the concept of ring, just think of it as a set with a sum $+$ and a multiplication $\cdot$, like the integers $\Bbb Z$).
Then you can denote the set of all matrices which entries are in $X$ as $\mathcal M(X)$. If you want to specify the size of the matrices you can denote it by $\mathcal M_{m\times n}(X)$, for some $m, n\in \Bbb N$.
Now to state that the entries $[m_{ij}]$ of a certain matrix $M$ respect some condition, you just let $X$ be the set of elements of $R$ that respect that condition and say $M\in \mathcal M(X)$.


text:
# 120V dual VESC for E-scooter

7 posts / 0 new
Daniel Kaminski
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Last seen: 1 year 9 months ago
Joined: 2021-03-28 10:51
Posts: 5
120V dual VESC for E-scooter

I’m willing to finance a 120-135V controller, that i need for my e-scooter prototype. I’d like to have a dual controller (like Stormcore 100D), but 120V version and somewhere up to 150 Amps

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Last seen: 12 hours 27 min ago
Joined: 2018-08-17 07:26
Posts: 348

You can use a pair of these: https://vesc-project.com/node/1477

NextGen FOC High voltage 144v/32s, 30kw (https://vesc-project.com/node/1477)

Daniel Kaminski
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Last seen: 1 year 9 months ago
Joined: 2021-03-28 10:51
Posts: 5

Nice one, Vadicus and all compliments for building it. I like all the features that you built in but i'd rather have a dual controller, due the limited space in the scooters body, besides every "gram" matters. We're building a hyper scooter here, with a goal to be the fastest on the planet, so please guys, help me out here and let's write our names in the history of hyper scooting :)

Daniel Kaminski

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Joined: 2018-08-17 07:26
Posts: 348

A step up scooter? Unlikely to make a history here. It's not hard to make them go 100km/h with 80-100v.  It's been done many times. Don't even need a high voltage controller or dual drive.

NextGen FOC High voltage 144v/32s, 30kw (https://vesc-project.com/node/1477)

Daniel Kaminski
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Last seen: 1 year 9 months ago
Joined: 2021-03-28 10:51
Posts: 5

I reached 124km/h with a Chinese 96V controller, but it's to heavy and big and i lost a lot of space and had to make a smaller battery - thus shorter range. Goal is to reach above 150 km/h, therefore i'm looking for improvements. I'm really desperate to get some help.

Daniel Kaminski

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Last seen: 12 hours 27 min ago
Joined: 2018-08-17 07:26
Posts: 348

Yep, and there is a reason for that, would you think? Here is some reality check. Building a high power controller is not a matter of swapping a few parts and call it a day. High voltage and currents require larger clearances, larger caps, more expensive parts, more complex power systems, isolation, etc. If you are serious about building it, I mean not a prototype that kind of works but a a reliable reproducible product, brace yourself for months if not years of designing, testing, failing, redesigning, blowing up things, flushing thousands of  down the toilet, thinking you are almost there, blowing up again, redoing, testing, rinse and repeat a few times. You'd better have tons of patience and a sizable budget. And all of that to go above 100kmh on a step up scooter with a smaller controller?

NextGen FOC High voltage 144v/32s, 30kw (https://vesc-project.com/node/1477)

Daniel Kaminski
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Last seen: 1 year 9 months ago
Joined: 2021-03-28 10:51
Posts: 5

I got the point but i'm still gonna go for it.

An odd beginner question: - Regarding the more expensive parts, caps, etc - if i get all that, does the VESC software still stays the same, or it needs an upgraded code etc. as well ?

Daniel Kaminski