text:
I am loading my css using this format: <link href="{{ asset('assets/mdi/css/materialdesignicons.min.css') }}" media="all" rel="stylesheet" type="text/css" /> and it loads fine for all http requests

But when I load my login page with SSL (https), I get a ...page... was loaded over HTTPS, but requested an insecure stylesheet 'http...

Should I be trying to load the assets securely? Or is it not Blade's job?

I believe secure_asset is what you're looking for.

<link href="{{ secure_asset('assets/mdi/css/materialdesignicons.min.css') }}" media="all" rel="stylesheet" type="text/css" />

5/15/2018 Edit: While my answer addresses the question directly, it's a bit dated given what Laravel can do nowadays; there may be cases where you want to force HTTPS on certain environments but not on others.

See Scofield's answer below for a more flexible solution to cover for these kinds of cases.

I had a problem with asset function when it's loaded resources through HTTP protocol when the website was using HTTPS, which is caused the "Mixed content" problem.

To fix that you need to add \URL::forceScheme('https') into your AppServiceProvider file.

So mine looks like this (Laravel 5.4):

<?php

namespace App\Providers;

use Illuminate\Support\ServiceProvider;

class AppServiceProvider extends ServiceProvider
{
/**
* Bootstrap any application services.
*
* @return void
*/
public function boot()
{
if(config('app.env') === 'production') {
\URL::forceScheme('https');
}
}

/**
* Register any application services.
*
* @return void
*/
public function register()
{
//
}
}

This is helpful when you need https only on server (config('app.env') === 'production') and not locally, so don't need to force asset function to use https.

• this is a better answer, since I want to only use https on production server – Irfandi D. Vendy Aug 23 '17 at 6:16
• Agreed. While my answer might have addressed the question directly, this approach would likely be better in cases where you have multiple environments that may or may not be serving over HTTPS. – maiorano84 Mar 28 at 16:20

An another approach would be to pass true as the second parameter.

/**
* Generate an asset path for the application.
*
* @param  string  $path * @param bool$secure
* @return string
*/
function asset($path,$secure = null)
{
return app('url')->asset($path,$secure);
}

As you see below secure_asset simply calls asset with the second parameter true.

/**
* Generate an asset path for the application.
*
* @param  string  $path * @return string */ function secure_asset($path)
{
return asset($path, true); }  • Thank you for the alternative suggestion :) – Grigio Dec 20 '15 at 6:24 • :) You're welcome! – Ozan Kurt Dec 20 '15 at 6:36 Here is my configuration to make HTTPS working with assets. To enable this in production add REDIRECT_HTTPS=true in the .env file. <?php namespace App\Providers; use Illuminate\Support\ServiceProvider; use Illuminate\Support\Facades\Schema; class AppServiceProvider extends ServiceProvider { /** * Bootstrap any application services. * * @return void */ public function boot() { if(env('REDIRECT_HTTPS')) { \URL::forceScheme('https'); } Schema::defaultStringLength(191); } /** * Register any application services. * * @return void */ public function register() { // } }  I use @Scofield answer by use \URL::forceScheme('https'); This solution also worked to show https for all routes but this not worked for me for$request->url() it show http instead of https

so I used $this->app['request']->server->set('HTTPS', true); instead of \URL::forceScheme('https'); I'm using Laravel 5.4 and update .env file and appserviceproviders <?php namespace App\Providers; use Illuminate\Support\ServiceProvider; use Illuminate\Support\Facades\Schema; use Log; class AppServiceProvider extends ServiceProvider { /** * Bootstrap any application services. * * @return void * */ public function boot() { If (env('APP_ENV') !== 'local') {$this->app['request']->server->set('HTTPS', true);
}

Schema::defaultStringLength(191);
}

/**
* Register any application services.
*
* @return void
*/
public function register()
{
//
}
}

in my env file I've changed

APP_ENV=local to APP_ENV=development

APP_ENV=local for localhost APP_ENV=development/production for on working server

after changing env run this artisan command

php artisan config:clear

Home It helps :-)

• That worked like a charm – Nava Bogatee Jun 13 at 9:14
• @NavaBogatee happy to hear this. :-) – Abdul Rehman Jun 13 at 9:22

Figuring out if the current Request is secure or not should not be your decision. Underlying Symfony\Component\HttpFoundation\Request has isSecure method that Laravel uses internally.

public function isSecure()
{
if ($this->isFromTrustedProxy() &&$proto = $this->getTrustedValues(self::HEADER_X_FORWARDED_PROTO)) { return \in_array(strtolower($proto[0]), array('https', 'on', 'ssl', '1'), true);
}

$https =$this->server->get('HTTPS');

return !empty($https) && 'off' !== strtolower($https);
}

So if your server is not passing the HTTPS header with On, it should be passing X-FORWARDED-PROTO and must be allowed by your TrustProxies middleware.

If you are behind reverse-proxy you should find out your proxy pass IP - you can do this easily by getting the $_SERVER['REMOTE_ADDR'] variable and setting the IP to your TrustProxies middleware: /** * The trusted proxies for this application. * * @var array */ protected$proxies = [
'123.123.123.123',
];

Laravel (Symfony) will then automatically detect if the Request is secure or not and choose the protocol accordingly.

The problem is cloudflare on my local machines is working fine but online server not solution is

 public function boot()
{

if (isset($_SERVER['HTTPS']) && ($_SERVER['HTTPS'] == 'on' || $_SERVER['HTTPS'] == 1) || isset($_SERVER['HTTP_X_FORWARDED_PROTO']) &&  \$_SERVER['HTTP_X_FORWARDED_PROTO'] == 'https') {
\URL::forceScheme('https');
}
}


text:
## Course Outline

• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 9 - The Logic of Inference
• segmentChapter 10 - Model Comparison with F
• segmentChapter 11 - Parameter Estimation and Confidence Intervals
• segmentPART IV: MULTIVARIATE MODELS
• segmentChapter 12 - Introduction to Multivariate Models
• segmentChapter 13 - Multivariate Model Comparisons
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources

### list Statistics and Data Science: A Modeling Approach

Book
• College / Advanced Statistics and Data Science (ABCD)
• College / Statistics and Data Science (ABC)
• High School / Advanced Statistics and Data Science I (ABC)
• High School / Statistics and Data Science I (AB)
• High School / Statistics and Data Science II (XCD)

## 13.6 Using shuffle() for Targeted Model Comparisons (Part 2)

Having saved the residuals from Neighborhood model, let’s now see how we can use them, along with the shuffle() function, to create a sampling distribution for the unique effect of HomeSizeK.

### Step Two: Create the Sampling Distribution of F for Home Size

A sampling distribution of Fs provides us a way to calculate how likely it would be for the simple model of the DGP (i.e., the one with no unique effect of HomeSizeK) to generate an F for HomeSizeK as large or larger than the one found in the data (11.626).

Before we create the sampling distribution of F for the HomeSizeK effect, we will show you how to get the sample F for HomeSizeK. Our previous method, using the f() function, won’t work; it only gives us the overall F for the full model. To get the F for HomeSizeK you can run this code:

f(PriceK_N_resids ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK)

The first part of this code creates a supernova table for the multivariate model using PriceK_N_resids as the outcome. The highlighted part above then reads the sample F for HomeSizeK out of the table (without ever printing it out). We’ve put this code in the window below, so you have this F available.

In the window below, modify the code where indicated, using the shuffle() function, to produce a single F for HomeSizeK that assumes a DGP with 0 effect of home size. Run the code a few times just to see what it does.

require(coursekata) # delete when coursekata-r updated Smallville <- read.csv("https://docs.google.com/spreadsheets/d/e/2PACX-1vTUey0jLO87REoQRRGJeG43iN1lkds_lmcnke1fuvS7BTb62jLucJ4WeIt7RW4mfRpk8n5iYvNmgf5l/pub?gid=1024959265&single=true&output=csv") Smallville$Neighborhood <- factor(Smallville$Neighborhood) Smallville$HasFireplace <- factor(Smallville$HasFireplace) # don't delete this part # code to fit neighborhood model and save residuals Neighborhood_model <- lm(PriceK~ Neighborhood, data = Smallville) Smallville$PriceK_N_resids <- resid(Neighborhood_model) # this code prints sample F for HomeSizeK f(PriceK_N_resids ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) # modify the code below to produce the F when residuals are shuffled f(PriceK_N_resids ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) f(PriceK_N_resids ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) f(shuffle(PriceK_N_resids) ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) # temporary SCT ex() %>% check_error() CK Code: D2_Code_Targeted_01 Now let’s add some code to create a sampling distribution of 1000 Fs for HomeSizeK assuming no effect of home size in the DGP. Save these Fs into a data frame called HomeSizeK_sdof. require(coursekata) # delete when coursekata-r updated Smallville <- read.csv("https://docs.google.com/spreadsheets/d/e/2PACX-1vTUey0jLO87REoQRRGJeG43iN1lkds_lmcnke1fuvS7BTb62jLucJ4WeIt7RW4mfRpk8n5iYvNmgf5l/pub?gid=1024959265&single=true&output=csv") Smallville$Neighborhood <- factor(Smallville$Neighborhood) Smallville$HasFireplace <- factor(Smallville$HasFireplace) # don't delete # code to fit neighborhood model and save residuals Neighborhood_model <- lm(PriceK~ Neighborhood, data = Smallville) Smallville$PriceK_N_resids <- resid(Neighborhood_model) # This code generates one shuffled HomeSizeK F # Modify it to make a sampling distribution of 1000 shuffled Fs # Save them in a data frame called HomeSizeK_sdof f(shuffle(PriceK_N_resids) ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) # This code will put these Fs into a histogram gf_histogram(~ f, data = HomeSizeK_sdof) %>% gf_labs(title = "shuffled HomeSizeK Fs") HomeSizeK_sdof <- do(1000) * f(shuffle(PriceK_N_resids) ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) gf_histogram(~ f, data = HomeSizeK_sdof) %>% gf_labs(title = "shuffled HomeSizeK Fs") # temporary SCT ex() %>% check_error()
CK Code: D2_Code_Targeted_02

Below we have graphed out the sampling distribution of 1000 shuffled Fs for the HomeSizeK effect. We also have added to the plot, as a black dot, the sample F for the HomeSizeK row of the ANOVA table (11.63). We’ll save this value as HomeSizeK_f. As you can see, the sample F is far out in the tail of the sampling distribution.

To calculate the exact p-value for the HomeSizeK F, we can use tally.

Try copying and pasting the appropriate code into the code block below. Also generate an ANOVA table – to check out whether the p-value obtained from tally() is similar to the p-value for HomeSizeK in the ANOVA table.

require(coursekata) # delete when coursekata-r updated Smallville <- read.csv("https://docs.google.com/spreadsheets/d/e/2PACX-1vTUey0jLO87REoQRRGJeG43iN1lkds_lmcnke1fuvS7BTb62jLucJ4WeIt7RW4mfRpk8n5iYvNmgf5l/pub?gid=1024959265&single=true&output=csv") Smallville$Neighborhood <- factor(Smallville$Neighborhood) Smallville$HasFireplace <- factor(Smallville$HasFireplace) # don't delete # code to fit neighborhood model and save residuals Neighborhood_model <- lm(PriceK~ Neighborhood, data = Smallville) Smallville\$PriceK_N_resids <- resid(Neighborhood_model) # This saves the sample HomeSizeK F HomeSizeK_f <- f(PriceK_N_resids ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) # This code generates a sampling distribution of shuffled HomeSizeK Fs HomeSizeK_sdof <- do(1000) * f(shuffle(PriceK_N_resids) ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) # Paste in the code for tallying the p-value for HomeSizeK # Modify the code below to generate an ANOVA table from the multivariate model lm(PriceK ~ Neighborhood + HomeSizeK, data = Smallville) HomeSizeK_f <- f(PriceK_N_resids ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) HomeSizeK_sdof <- do(1000) * f(shuffle(PriceK_N_resids) ~ Neighborhood + HomeSizeK, data = Smallville, predictor = ~HomeSizeK) tally(~ f > HomeSizeK_f, data=HomeSizeK_sdof, format="proportion") supernova(lm(PriceK ~ Neighborhood + HomeSizeK, data = Smallville)) # temporary SCT ex() %>% check_error()
CK Code: D2_Code_Targeted_03

The p-value we got from tally() is close to the p-value reported on the HomeSizeK row of the multivariate ANOVA table: 0.0019.

format: plain


text:
# The limitations of linear and semidefinite programming

The fact that Linear Programming (LP) can be solved in polynomial time (and, also, efficiently in practice) and that it has such a rich geometric theory and such remarkable expressive power makes LP a powerful unifying concept in the theory of algorithms. It “explains” the existence of polynomial time algorithms for problems such as Shortest Paths and Min Cut, and if one thinks of the combinatorial algorithms for such problems as algorithms for the corresponding linear programs, one gains additional insights.

When looking for algorithms for a new combinatorial problem, a possible approach is to express the problem as a 0/1 integer program, then relax it to a linear program, by letting variables range between 0 and 1, and then hope for the best. “The best” being the lucky event that the value of the optimum of the relaxation is the same as that of the combinatorial program, or at least a close approximation. If one finds that, instead, the relaxation has optimal fractional solutions of cost very different from the combinatorial optimum, one may try to add further inequalities that are valid for 0/1 solutions and that rule out the problematic fractional solutions.

Many “P=NP” papers follow this approach, usually by presenting a polynomial-size linear programming relaxation of TSP and then “proving” that the optimum of the relaxation is the same as the combinatorial optimum. One can find recent examples here and here.

Similar results were “proved” in a notorious series of paper by Ted Swart in the mid 1980s. After counterexamples were found, he would revise the paper adding more inequalities that would rule out the counterexample.

Finally, Mihalis Yannakakis took matters into his own hands and proved that all “symmetric” relaxations of TSP of sub-exponential size have counterexamples on which the optimum of the relaxation is different from the combinatorial optimum. (All of the LPs suggested by Swart where “symmetric” according to Yannakakis’s definition.)

This is actually one of the few known lower bounds that actually applies to a “model of computation” capturing a general (and otherwise promising) class of algorithms.

In the theory of approximation algorithms, we have plenty of problems that are believed to be intractable but that are not known to be NP-hard, such as approximating Vertex Cover within a factor of 1.9, or approximating Sparsest Cut within a factor of 10. LP and Semidefinite Programming (SDP) approaches are more or less the only promising tools we have to prove that such problems are tractable and, while we wait for NP-hardness result (for now, we only have “Unique-Games-hardness”), it is good to see whether certain candidate LP and SDP algorithms have any chance, or if they admit counterexamples showing large gaps between the optimum of the relaxation and the combinatorial optimum.

The problem with this approach is the nearly endless variety of relaxations that one can consider: what happens when we add triangle inequalities? and pentagonal inequalities? and so on. As in the case of Yannakakis’s result, it would be great to have a result that says “all SDP relaxations of Vertex Cover of type X fail to achieve an approximation ratio smaller than 2,” where “type X” is a general class of sub-exponential size SDP relaxations that include the type of inequalities that people use “in practice.”

Lovasz and Schrijver describe a method, denoted LS+, that starts from an LP relaxation of a problem (actually it can start from any convex relaxation), and then turns it into tighter and tighter SDP relaxations, by adding auxiliary variables and linear and semidefinite constraints. A weaker version of the method, denoted LS, only adds auxiliary variables and linear constraints.

A nice thing about the method is that, after you apply it to your initial relaxation, thus getting a tighter relaxation, you can then apply it again to the tighter one, thus getting an even better relaxation, and so on. Starting from an LP relaxation with $n$ variables and poly($n$) constraints, $k$ applications of the method yield a relaxation solvable in $n^{O(k)}$ time, which is polynomial for all fixed $k$ and sub-exponential for $k=o(n/log n)$. Lovasz and Schrijver prove that, after $k$ applications (or “rounds”) the resulting relaxation enforces all inequalities over $k$-tuples of variables that are valid for 0/1 solutions. (In particular, one gets the combinatorial optimum after $n$ rounds.) Typical approaches in the design of approximation algorithms are SDP with local inequalities (triangle inequalities etc.), and this is all captured after a few rounds of LS+.

It would be great to show that no constant (ideally, no sublinear) number of rounds of LS+ starting from the basic LP relaxation gives a $2-\epsilon$ approximation for vertex cover. Arora, and others Bollobas and Lovasz considered related questions in a FOCS 2002 paper that has inspired a considerable amount of later work. (See the introduction of the journal version.) Unfortunately the question remains open evern for two rounds of LS+. After one round, one gets an SDP relaxation equivalent to (number of vertices minus) the Lovasz Theta function, and Goemans and Kleinberg prove that such SDP does not achieve approximation better than 2. Beyond that, it is pretty much terra incognita. Charikar proves that a relaxation with triangle inequalities (which is incomparable with two rounds of LS+ and is weaker than three rounds) does not achieve approximation better than 2. Also, a sublinear number of rounds of LS+ does not achieve approximation better than 7/6. For LS, which, I remind you, generates linear programming relaxations rather than semidefinite programming ones, we know that no sublinear number of rounds leads to an approximation better than 2.

I will illustrate the main idea in the LS and LS+ method using the example of Vertex Cover. In the linear programming formulation, we have variables $x_i$, one for each vertex $i$, and the constraints that $x_i + x_j \geq 1$ for each edge $(i,j)$ and that $0\leq x_i \leq 1$. We would like to add constraints that are only satisfied by 0/1 solutions, and the constraint $x_i^2 = x_i$ would work beautifully except that it is not linear (nor convex). Instead, Lovasz and Schrijver add new variables $y_{i,j}$, one for each pair of vertices, with the idea that we would like to have $y_{i,j}=x_i * x_j$; then they add the requirement $y_{i,i} = x_i$ and various inequalities that make the $y_{i,j}$ be “sort of like” $x_i*x_j$. In particular, we would like to require that if $x_k \neq 0$, then $x_i = y_{i,k}/x_k$. This is again a non-linear constraint, but, at least, we can check whether, for each fixed $k$, $y_{i,k}/x_k$ is a fractional vertex cover: we just need to check the inequalities $y_{i,k} + y_{j,k} \geq x_k$ for each edge $(i,j)$

Similarly, if $x_k \neq 1$, we expect to have $x_i = (x_i-y_{i,k})/(1-x_k)$. We cannot check it directly, but we can check if
$x_i - y_{i,k} + x_j - y_{j,k} \geq 1-x_k$
hold for each edge $(i,j)$. Finally, we obviously want $y_{i,j} = y_{j,i}$. This describe the LS method. For LS+, we add the requirement that the symmetric matrix $(y_{i,j})$ be positive semidefinite. Being positive semidefinite means that there are vectors $b_1,\ldots,b_n$ such that
$y_{i,j} = \langle b_i,b_j \rangle$
where $\langle \cdot , \cdot \rangle$ denotes inner product. If $y_{i,j} = x_i *x_j$ then $(y_{i,j})$ is clearly positive semidefinite.

## 3 thoughts on “The limitations of linear and semidefinite programming”

1. Note that it would be a very good result if someone proved that after \Omega_{\epsilon}(n) rounds of the LS_{+} operator, we get an integrality gap of $2-\epsilon$ for the Vertex Cover problem.

This would provide (yet another) strong indication that VC is $2-\epsilon$ – hard to approximate (unless P=NP).

2. Luca, great post!

I would only argue that, in a field where we order author names alphabetically, one should rely less on Author1, et. al. citations, especially if the paper has only three authors. I think “Arora and others” is perhaps slightly strange sounding… who are those others? Oh yes, Bollobas and Lovasz (no slackers). But it’s only a blog! yes, so space is less expensive here. maybe this is a ridiculous comment… who cares? somehow, it seems like it matters.


text:
| | | |

The $(j,m)$-core inverse in rings with involution

Sanzhang XU [1] , Jianlong CHEN [2] , Dijana MOSİĆ [3]

Let $R$ be a unital ring with involution. The $(j,m)$-core inverse of a complex matrix was extended to an element in $R$. New necessary and sufficient conditions such that an element in $R$ to be $(j,m)$-core invertible are given. Moreover, several additive and product properties of two $(j,m)$-core invertible elements are investigated and a order related to the $(j,m)$-core inverse is introduced.

***********************************************************************************************************************************

Moore-Penrose inverse, (j-m) core inverse, EP element
• [1] O.M. Baksalary and G. Trenkler. Core inverse of matrices, Linear Multilinear Algebra, 58(6), 681–697, 2010.
• [2] O.M. Baksalary and G. Trenkler, On a generalized core inverse, Appl. Math. Comput. 236, 450–457, 2014.
• [3] M.P. Drazin, Pseudo-inverses in associative rings and semigroup, Amer. Math. Monthly, 65, 506–514, 1958.
• [4] R.E. Hartwig, Block generalized inverses, Arch. Retional Mech. Anal. 61(3), 197–251, 1976.
• [5] S.B. Malik and N. Thome, On a new generalized inverse for matrices of an arbitrary index, Appl. Math. Comput. 226, 575–580, 2014.
• [6] D. Mosić and D.S. Djordjević, Moore-Penrose-invertible normal and Hermitian elements in rings, Linear Algebra Appl. 431, 732–745, 2009.
• [7] P. Patrício and R. Puystjens, Drazin-Moore-Penrose invertiblity in rings, Linear Algebra Appl. 389, 159–173, 2004.
• [8] D.S. Rakić, N.Č. Dinčić and D.S. Djordjević, Group, Moore-Penrose, core and dual core inverse in rings with involution, Linear Algebra Appl. 463, 115–133, 2014.
• [9] H. Schwerdtfeger, Introduction to Linear Algebra and the Theory of Matrices, P. Noordhoff, Groningen, 1950.
• [10] J. von Neumann, On regular rings, Proc. Nati. Acad. Sci. U.S.A. 22 (12), 707–713, 1936.
• [11] H.X. Wang and X.J. Liu, A partial order on the set of complex matrices with index one, Linear Multilinear Algebra, 66 (1), 206–216, 2018.
• [12] S.Z. Xu, J.L. Chen and J. Benítez, EP elements in rings with involution, Bull. Malays. Math. Sci. Soc. 42, 3409–3426, 2019.
• [13] S.Z. Xu, J.L. Chen, J. Benítez and D.G. Wang, Generalized core inverse of matrices, Miskolc Math. Notes, 20 (1), 565–584, 2019.
• [14] S.Z. Xu, J.L. Chen and X.X. Zhang, New characterizations for core and dual core inverses in rings with involution, Front. Math. China. 12 (1), 231–246, 2017.
Primary Language en Mathematics Mathematics Orcid: 0000-0002-9744-477XAuthor: Sanzhang XU (Primary Author)Institution: Huaiyin Institute of TechnologyCountry: China Orcid: 0000-0002-6798-488XAuthor: Jianlong CHEN Institution: Southeast UniversityCountry: China Orcid: 0000-0002-3255-9322Author: Dijana MOSİĆ Institution: University of NišCountry: Serbia Publication Date : October 6, 2020
 Bibtex @research article { hujms701870, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1676 - 1685}, doi = {10.15672/hujms.701870}, title = {The \$(j,m)\$-core inverse in rings with involution}, key = {cite}, author = {Xu, Sanzhang and Chen, Jianlong and Mosi̇ć, Dijana} } APA Xu, S , Chen, J , Mosi̇ć, D . (2020). The $(j,m)$-core inverse in rings with involution . Hacettepe Journal of Mathematics and Statistics , 49 (5) , 1676-1685 . DOI: 10.15672/hujms.701870 MLA Xu, S , Chen, J , Mosi̇ć, D . "The $(j,m)$-core inverse in rings with involution" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1676-1685 Chicago Xu, S , Chen, J , Mosi̇ć, D . "The $(j,m)$-core inverse in rings with involution". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1676-1685 RIS TY - JOUR T1 - The $(j,m)$-core inverse in rings with involution AU - Sanzhang Xu , Jianlong Chen , Dijana Mosi̇ć Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.701870 DO - 10.15672/hujms.701870 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1676 EP - 1685 VL - 49 IS - 5 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.701870 UR - https://doi.org/10.15672/hujms.701870 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics The $(j,m)$-core inverse in rings with involution %A Sanzhang Xu , Jianlong Chen , Dijana Mosi̇ć %T The $(j,m)$-core inverse in rings with involution %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 5 %R doi: 10.15672/hujms.701870 %U 10.15672/hujms.701870 ISNAD Xu, Sanzhang , Chen, Jianlong , Mosi̇ć, Dijana . "The $(j,m)$-core inverse in rings with involution". Hacettepe Journal of Mathematics and Statistics 49 / 5 (October 2020): 1676-1685 . https://doi.org/10.15672/hujms.701870 AMA Xu S , Chen J , Mosi̇ć D . The $(j,m)$-core inverse in rings with involution. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1676-1685. Vancouver Xu S , Chen J , Mosi̇ć D . The $(j,m)$-core inverse in rings with involution. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1676-1685.

Authors of the Article
[1]
[2]
[3]


text:
## August 1, 2019

### When Does Nothing Mean Something?

I thought I was done writing about this topic, but it just keeps coming back.  The internet just cannot seem to leave this sort of problem alone:
I don't know what it is about expressions of the form $$a\div b(c+d)$$ that fascinates us as a species, but fascinate it does.  I've written about this before (as well as why "PEMDAS" is terrible), but the more I've thought about it, the more sympathy I've found with those in the minority of the debate, and as a result my position has evolved somewhat.

So I'm going to go out on a limb, and claim that the answer should be $$1$$.

Before you walk away shaking your head and saying "he's lost it, he doesn't know what he's talking about", let me assure you that I'm obviouly not denying the left-to-right convention for how to do explicit multiplication and division.  Nobody's arguing that.*  Rather, there's something much more subtle going on here.

What we may be seeing here is evidence of a mathematical "language shift".

It's easy to forget that mathematics did not always look as it does today, but has arrived at its current form through very human processes of invention and revision.  There's an excellent page by Jeff Miller that catalogues the earliest recorded uses of symbols like the operations and the equals sign -- symbols that seem timeless, symbols we take for granted every day.

People also often don't realize that this process of invention and revision still happens to this day.  The modern notation for the floor function is a great example that was only developed within the last century.  Even today on the internet, you occasionally see discussions in which people debate on how mathematical notation can be improved.  (I'm still holding out hope that my alternative notation for logarithms will one day catch on.)

Of particular note is the evolution of grouping symbols.  We usually think only of parentheses (as well as their variations like square brackets and curly braces) as denoting grouping, but an even earlier symbol used to group expressions was the vinculum -- a horizontal bar found over or under an expression.  Consider the following expression: $3-(1+2)$ If we wrote the same expression with a vinculum, it would look like this: $3-\overline{1+2}$ Vincula can even be stacked: $13-\overline{\overline{1+2}\cdot 3}=4$ This may seem like a quaint way of grouping, but it does in fact survive in our notation for fractions and radicals!  You can even see both uses in the quadratic formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

Getting back to the original problem, what I think we're seeing is evidence that concatenation -- placing symbols next to each other with no sort of explicit symbol -- has become another way to represent grouping.

"But wait", you might say, "concatenation is used to represent multiplication, not grouping!"  That's certainly true in many cases, for example in how we write polynomials.  However, there are a few places in mathematics that provide evidence that there's more to it than that.

First of all, as a beautifully-written Twitter thread by EnchantressOfNumbers (@EoN_tweets) points out, we use concatenation to show a special importance of grouping when we write out certain trigonometric expressions without putting their arguments in parentheses.  Consider the following identity:
$\sin 4u=2\sin 2u\cos 2u$ When we write such an equation, we're saying that not only do $$4u$$ and $$2u$$ represent multiplications, but that this grouping is so tight that they constitute the entire arguments of the sine and cosine functions.  In fact, the space between $$\sin 2x$$ and $$\cos 2x$$ can also be seen as a somewhat looser form of concatention.  Then again, so does the space between $$\sin$$ and $$x$$, which represents a different thing -- the connection of a function to its argument.  Perhaps this is why the popular (and amazing) online graphing calculator Desmos is only so permissive when it comes to parsing concatenation:

In contrast, where we do draw the line is with an expression like the following:$\sin x+y$ We always interpret this as $$(\sin(x))+y$$, never $$\sin(x+y)$$.  To drive home just how much stronger implicit multiplication feels to us than explicit multiplication, just take a look at the following expression: $\sin x\cdot y$ Does this mean $$(\sin(x))\cdot y$$ or $$\sin(x\cdot y)$$?  If that expression makes you writhe uncomfortably, while if it had been written as $$\sin xy$$ it would be fine, then you might see what I'm getting at.

An even more curious case is mixed numbers.  When writing mixed numbers, concatenation actually stands for addition, not multiplication. $3\tfrac{1}{2}=3+\tfrac{1}{2}$ In fact, concatenation actually makes addition come before multiplication when we multiply mixed numbers! $3\tfrac{1}{2}\cdot 5\tfrac{5}{6}=(3+\tfrac{1}{2})\cdot(5+\tfrac{5}{6})=20\tfrac{5}{12}$

Now, you may feel that this example shows how mixed numbers are an inelegance in mathematical notation (and I would agree with you).  Even so, I argue that this is evidence that we fundamentally view concatenation as a way to represent grouping.  It just so happens that, since multiplication takes precedence over addition anyway in the absence of other grouping symbols, we use concatenation when we write it.  This all stems from a sort of "laziness" in how we write things -- - laying out precedence rules allows us to avoid writing parentheses, and once we've established those precedence rules, we don't even need to write out the multiplication at all.

So how does the internet's favorite math problem fit into all this?

The most striking feature of the expression $$8\div 2(2+2)$$ is that it's written all in one line.

Mathematical typesetting is difficult.  LaTeX is powerful, but has a steep learning curve, though various other editors have made it a bit easier, such as Microsoft Word's Equation Editor (which has much improved since when I first used it!).  Calculators have also recognized this difficulty, which is why TI calculators now have MathPrint templates (though its entry is quite clunky compared to Desmos's "as-you-type" formatting via MathQuill).

Even so, all of these input methods exist in very specific applications.  What about when you're writing an email?  Or sending a text?  Or a Facebook message?  (If you're wondering "who the heck writes about math in a Facebook message", the answer at least includes "students who are trying to study for a test".)  The evolution of these sorts of media has led to the importance of one-line representations of mathematics with easily-accessible symbols.  When you don't have the ability (or the time) to neatly typeset a fraction, you're going to find a way to use the tools you've got.  And that's even more important as we realize that everybody can (and should!) engage with mathematics, not just mathematicians or educators.

So that might explain why a physics student might type "hbar = h / 2pi", and others would know that this clearly means $$\hbar=\dfrac{h}{2\pi}$$ rather than $$\hbar=\dfrac{h}{2}\pi$$.  Remember, mathematics is not about just answer-getting.  It's about communication of those ideas.  And when the medium of communication limits how those ideas can be represented, the method of communication often changes to accomodate it.

What the infamous problem points out is that while almost nobody has laid out any explicit rules for how to deal with concatenation, we seem to have developed some implicit ones, which we use without thinking about them.  We just never had to deal with them until recently, as more "everyday" people communicate mathematics on more "everyday" media.

Perhaps it's time that we address this convention explicitly and admit that concatenation really has become a way to represent grouping, just like parentheses or the vinculum.  This is akin to taking a more descriptivist, rather than prescriptivist, approach to language: all we would be doing is recognizing that this is already how we do things everywhere else.

Of course, this would throw a wrench in PEMDAS, but that just means we'd need to actually talk about the mathematics behind it rather than memorizing a silly mnemonic.  After all, as inane as these internet math problems can be, they've shown that (whether they admit it or not) people really do want to get to the bottom of mathematics, to truly understand it.

I'd say that's a good thing.

* If your argument for why the answer is $$16$$ starts with "Well, $$2(2+2)$$ means $$2\cdot(2+2)$$, so...", then you have missed the point entirely.

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# What is the value of x^2yz + xy^2z + xyz^2 ?

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What is the value of x^2yz + xy^2z + xyz^2 ? [#permalink]

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What is the value of x^2yz + xy^2z + xyz^2 ?

(1) x + y + z = 18
(2) xyz = 10
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Re: What is the value of x^2yz+xy^2z+xyz^2 ? [#permalink]

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rraggio wrote:
What is the value of x^2yz+xy^2z+xyz^2 ?

(1) x+y+z=18
(2) xyz=10

Thanks again!

Question: $$x^2yz+xy^2z+xyz^2=xyz(x+y+z)=?$$

(1) $$x+y+z=18$$. We know only one of the multiples. Not sufficient.
(2) $$xyz=10$$. We know only one of the multiples. Not sufficient.

(1)+(2) $$x+y+z=18$$ and $$xyz=10$$ --> $$xyz(x+y+z)=10*18=180$$. Sufficient.

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Thanks bunuel for reformatting the question - straight C
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Re: What is the value of x^2yz+xy^2z+xyz^2 ? [#permalink]

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19 Sep 2010, 19:30
If xyz was given as equal to zero, than the answer would have been totally different
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17 Sep 2011, 17:55
x^2*y*z + x*y^2*z+x*y*z^2 =?

i.e xyz(x+y+z) =?

1. Not sufficient.

x+y+z = 18. there is no way we can find out the value of xyz from this . as different x,y,z combinations will yield
different xyz.

2. Not sufficient.

xyz = 10

there is no way we can find out the value of x+y+z from this . as different x,y,z combinations will yield
different x+y+z.

1&2 together, its sufficient to find out xyz(x+y+z).

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text:
# Comparing the slopes for two independent samples

In this section we test whether the slopes for two independent populations are equal, i.e. we test the following null hypothesis:

H0:  β1 = β2 i.e. β1 – β2 = 0

The test statistic is

If the null hypothesis is true then

where

If the two error variances are equal, then as for the test for the differences in the means, we can pool the estimates of the error variances, weighing each by their degrees of freedom, and so

Now

Since we can replace the numerators of each by the pooled value $s_{Res}^2$, we have

Note that the while the null hypothesis that β = 0 is equivalent to ρ = 0, the null hypothesis that  β1 = β2  is not equivalent to ρ1 = ρ2.

Example 1: We have two samples, each comparing life expectancy vs. smoking. The first sample is for males and the second for females. We want to determine whether there is any significant difference in the slopes for these two populations. We assume that the two samples have the values in Figure 1 (for men the data is the same as that in Example 1 of Regression Analysis):

Figure 1 – Data for Example 1

Figure 1 – Data for Example 1

As can be seen from the scatter diagrams in Figure 1, it appears that the slope for women is less steep than for that of men. In fact, as can be seen from Figure 2, the slope of the regression line for men is -0.6282 and the slope for women is -0.4679, but is this difference significant?

As can be seen from the calculations in Figure 2, using both pooled and unpooled values for sRes, the null hypothesis, H0: the slopes are equal, cannot be rejected. And so we cannot conclude that there is any significant difference between the life expectancy of males and females for any incremental amount of smoking.

Figure 2 – t-test to compare slopes of regression lines

Figure 2 – t-test to compare slopes of regression lines

### 2 Responses to Comparing the slopes for two independent samples

1. Aravindh says:

Hi,

You have a cell wrong in the excel sheet. Please fix it if you can. In Figure 2, cell M11 should be equal to (b1-b2)/(sb1-b2), NOT (b1-b2)/(sb1-sb2)

Thanks Aravindh,
That was a great catch. I have made the change that you suggested. Thanks for your help.
Charles


text:
# factor ring tables

• Aug 5th 2008, 09:44 AM
Ryan0710
factor ring tables
Draw up an addition and a multiplication table for the factor ring 2Z/8Z.
• Aug 5th 2008, 11:36 AM
ThePerfectHacker
Quote:

Originally Posted by Ryan0710
Draw up an addition and a multiplication table for the factor ring 2Z/8Z.

Let $G=2\mathbb{Z}$ and $H=8\mathbb{Z}$.
Then $G/H = \{ H,2+H,4+H,6+H\}$.
Let $a=2+H$ and $e=H$.
Then $G/H = \{ e, a,2a,3a\}$.
Adding them is simple, for example, $2a+3a = 5a=a$ by using $4a=e$.
Multiplying is similar, $2a\cdot 3a = 6a=2a$.

With those facts you can form a table.
• Aug 5th 2008, 11:41 AM
Matt Westwood
The elements of 2Z are ..., -6, -4, -2, 0, 2, 4, 6, 8, ...

The elements of 8Z are ..., -42, -16, -8, -0, 8, 16, 24, ...

The elements of 2Z/8Z are 0, 2, 4, 6.

You can consider these as being 2x the elements in Z_4, i.e. 2 x {0, 1, 2, 3} and do all the work on Modulo 4 arithmetic and multiply everything by 2 when you've finished, or you can directly use modulo 8 arithmetic and create the multiplication and addition tables based on the defined subring of that.

As long as you know how to do modulo n arithmetic you're home.


text:
for Journals by Title or ISSN for Articles by Keywords help

Publisher: Cambridge University Press   (Total: 374 journals)

 Compositio MathematicaJournal Prestige (SJR): 3.139 Citation Impact (citeScore): 1Number of Followers: 1      Subscription journal ISSN (Print) 0010-437X - ISSN (Online) 1570-5846 Published by Cambridge University Press  [374 journals]
• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$p$ ++++++++++++ ++++++++ ++++-Kähler+structures&rft.title=Compositio+Mathematica&rft.issn=0010-437X&rft.date=2019&rft.volume=155&rft.spage=455&rft.epage=483&rft.aulast=Rao&rft.aufirst=Sheng&rft.au=Sheng+Rao&rft.au=Xueyuan+Wan,+Quanting+Zhao&rft_id=info:doi/10.1112/S0010437X19007085">On local stabilities of $p$ -Kähler structures
• Authors: Sheng Rao; Xueyuan Wan, Quanting Zhao
Pages: 455 - 483
Abstract: By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$ -Kähler structures with the $(p,p+1)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma under small differentiable deformations.
PubDate: 2019-03-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007085
Issue No: Vol. 155, No. 3 (2019)

• Spinor groups with good reduction
• Authors: Vladimir I. Chernousov; Andrei S. Rapinchuk, Igor A. Rapinchuk
Pages: 484 - 527
Abstract: Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$ . We show that for a given $n\geqslant 5$ , the set of $K$ -isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$ -dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups
PubDate: 2019-03-01T00:00:00.000Z
DOI: 10.1112/S0010437X1900705X
Issue No: Vol. 155, No. 3 (2019)

• The inertia operator on the motivic Hall algebra
• Authors: Kai Behrend; Pooya Ronagh
Pages: 528 - 598
Abstract: We study the action of the inertia operator on the motivic Hall algebra and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.
PubDate: 2019-03-01T00:00:00.000Z
DOI: 10.1112/S0010437X18007881
Issue No: Vol. 155, No. 3 (2019)

• Topological Fukaya category and mirror symmetry for punctured surfaces
• Authors: James Pascaleff; Nicolò Sibilla
Pages: 599 - 644
Abstract: In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $\unicode[STIX]{x1D6F4}$ via the topological Fukaya category. We prove that the topological Fukaya category of $\unicode[STIX]{x1D6F4}$ is equivalent to the category of matrix factorizations of a certain mirror LG model $(X,W)$ . Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.
PubDate: 2019-03-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007073
Issue No: Vol. 155, No. 3 (2019)

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Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762
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text:
# First Problem in 2016

$\displaystyle\sum_{k=2}^{\infty}0.\overline{01}_{k}=S_{4}$

If $$S=\dfrac{p}{q}$$, where $$p$$ and $$q$$ are coprime positive integers, find $$p+q$$.

Note: $$S_{4}$$ represents $$S$$ written in base 4.

×


text:
# Order of 3 modulo a Mersenne prime

by T.Rex
Tags: mersenne, modulo, order, prime
 P: 62 Hi, I have the following (new, I think) conjecture about the Mersenne prime numbers, where: $$M_q = 2^q - 1$$ with $$q$$ prime. I've checked it up to q = 110503 (M29). Conjecture (Reix): $$\large \ order(3,M_q) = \frac {M_q - 1}{3^O}$$ where: $$\ \large O = 0,1,2$$ . With $$I =$$ greatest $$i$$ such that $$M_q \equiv 1 \pmod{3^i}$$ , then we have: $$O \leq I$$ but no always: $$O = I$$ . A longer description with experimental data is available at: ConjectureOrder3Mersenne. Samuel Wagstaff was not aware of this conjecture and has no idea (yet) about how to prove it. I need a proof... Any idea ? Tony
 PF Gold P: 1,059 If I understand this correctly we are supposing that 3^3 is the highest dividing power, but take the 27th Mersenne prime, as shown in a table, and consider: $$\frac{2^{44496}-1}{81}$$ is an integer. Also, I would suggest trying to check out the 40th Mersenne prime, and find, $$\frac{2^{20996010}-1}{243}$$ is an integer.
 PF Gold P: 1,059 T.Rex: I've checked it up to q = 110503 (M29) If you want to see some check work on Mersenne 27, notice that 2^2000==4 Mod 81. Thus dividing out 44496/2000 = 22 + Remainder 496. 496 = 2*248. Thus: $$4^{22}*4^{248}-1\equiv 4^{270}-1 \equiv0 Mod 81$$
P: 62

## Order of 3 modulo a Mersenne prime

The conjecture is wrong.
The terrible "law of small numbers" has struck again... (but the numbers were not so small...).
I've updated the paper and just conjectured that the highest power of 3 that divides the order of 3 mod M_q is 2. But it is not so much interesting...
Never mind, we learn by knowing what's false too.
I've updated the paper.
Sorry, the way David found the counter-examples was not so difficult...
Tony
P: 62
 Quote by robert Ihnot If I understand this correctly we are supposing that 3^3 is the highest dividing power...
Not exactly, Robert. For q=44497, 4 is the highest power of 3 that divides Mq-1, but 1 is the highest power of 3 in the relationship between (Mq-1) and order(3,Mq).
I have other reasons to think that 2 is the highest power of 3 in this relationship. But I need to clarify that before conjecturing again (one mistake is enough !!).
Thanks,
Tony

 Related Discussions Calculus & Beyond Homework 1 Linear & Abstract Algebra 1 Calculus & Beyond Homework 0 Linear & Abstract Algebra 15 Linear & Abstract Algebra 4


text:
# Revisiting the Real Graviton Effects at CERN LHC within the Quantum Gravity Theory with Large Extra Dimensions - High Energy Physics - Phenomenology

Abstract: CERN LHC provides a good experimental platform to perturbatively probe thefundamental gravity scale up to several TeV, with the precise value dependingon the number of extra dimensions. The leading experimental signal of gravitonat LHC is from the process $pp\to jet+{E\slash} T$, where ${E\slash} T$ standsfor the transverse missing energy. A detailed discussion on the hadronicproduction of real graviton through hard subprocesses: $q\bar{q}\to G+g$,$g+q\to G+q$ and $g+g\to G+g$ have been studied within the quantum gravitytheory with large extra dimensions. The main theoretical uncertainties togetherwith the dominant standard model background to these processes, e.g.$q\bar{q}\to Z^0+g$ and $g+q\to Z^0+q$ with $Z^0$ further decaying intoneutrinos, have also been discussed. It is found that only in certain jetenergy region and with certain number of extra dimensions can the quantumgravity signal be distinguished from the background, which inversely lead tothe effective scale $M D$ to be probed up to $8.8\pm 0.9$ TeV for two extradimensions, and $5.9\pm 0.5$ TeV for four extra dimensions with sufficientintegrated luminosity, e.g. $100fb^{-1}$, at CERN LHC.

Author: Xing-Gang Wu, Zhen-Yun Fang

Source: https://arxiv.org/


text:
Topic 10 min read

# Distribution of Relaxation Times (DRT): how to identify equivalent circuits efficiently

Latest updated: January 5, 2022

## Using DRT as a means of identifying equivalent circuits efficiently

Electrochemical Impedance Spectroscopy (EIS) is used to identify the physical characteristics of a system. The method used is to model the system using what is called an equivalent electrical circuit, where each element of the circuit is supposed to correspond to a physical characteristic of the system under study. The choice of an electrical circuit requires prior knowledge of the impedance of the system, which can be an obstacle for this method.

Distribution of Relaxation Times (DRT) is an analysis method that turns impedance data that are a function of the frequency into a distribution of the time constants of the considered system. As DRT does not require prior knowledge of the system, it can be considered an important tool to help identify an equivalent circuit that should be used to fit impedance data.

The main advantage of DRT

We are using Z Sim, the EC-Lab® simulation software to simulate the Nyquist impedance diagram of the following electrical equivalent circuit: Q1/R1 + Q2/R2. The parameters are given in Fig. 1a and the Nyquist diagram in Fig. 1b.

Without any prior knowledge of the system, one could consider that there is only one time constant involved and try to fit these data using an electrical equivalent circuit: Q1/R1. The results are shown in Fig. 2. Figure 2a shows the calculated parameters and Fig. 2b the fitted data. You can see that the calculated parameters are nowhere near the correct parameters, as they are shown in Fig. 1a.

Figure 1: a) Parameters used to simulate the Nyquist diagram of the equivalent circuit Q1/R1 + Q2/R2 shown in b)

Figure 2: a) Fitting results obtained when fitting the impedance data shown in Fig. 1b with Q1/R1 as an equivalent circuit. b) Fitted impedance diagram.

Using the DRT analysis, for example, by using the DRTtools freeware [1] developed by Dr. Wan, one can plot Figure 3, where the impedance data of Figure 1b are shown as the magnitude $G(\tau)$ vs. the relaxation times. The fact that we can see two peaks indicate that there are two time constants, that is to say, that the system is composed of two Q/Rs and not one Q/R.

Figure 3: Numerical DRT for the impedance results shown in Fig. 1b.

Figure 4 shows on the left the Nyquist impedance graphs of a Q1/R1+Q2/R2 equivalent circuit and on the right, the corresponding DRT plots for various ${\tau_1}/{\tau_2}$ ratios, with $\tau = (RQ)^{(1/\alpha)}$. $\alpha$, the so-called dispersion parameter, and $R$ are equally chosen for the two Q/R circuits, as presented in Fig. 1a, and only $Q2$ changes.

From ${\tau_1}/{\tau_2}=3.98$, the Nyquist plot of the impedance data shows what could be seen as a single depressed semi-circle, whereas the DRT plot shows two peaks.

Figure 4: the Nyquist impedance graphs of an Q1/R1+Q2/R2 equivalent circuit and on the right the corresponding
DRT plots for various ${\tau_1}/{\tau_2}$ ratios.

Each R/C circuit has a finite impedance limit when frequencies tend to either 0 or to +∞, hence the DRT analysis can only be successfully performed on data that present the same characteristics.

If data present an inductive behavior, either at low or high frequencies, or the system is governed by semi-infinite or restricted diffusion at lower frequencies, these unwanted data must be discarded for the DRT analysis to work [2,3]. The DRTtools program offers the possibility to account for inductive data [1].

DRT analysis finds its best use in the field of fuel cells, where low-frequency processes are generally governed by diffusion/convection, with a finite impedance limit at low frequencies [4,5].

Another limitation of the DRT analysis is that it requires a numerical matrix inversion and the results may depend on the method chosen for the inversion [6].

Please find more details about DRT in the referenced papers and our application note [7].

[1] DRTtools and its manual, https://sites.google.com/site/drttools/

[2] J. Illig, M. Ender, T. Chrobak, J. P. Schmidt, D. Klotz, and E. Ivers-Tiff´ee, J. Electrochem. Soc., 159, 7 (2012) A952-A960

[3] J. P. Schmidt, E. Ivers-Tiff´ee, J. Power Sources, 315 (2016) 316 – 323

[4] H. Schichlein, A. C. Müller, M. Voigths, A. Krügel, and E. Ivers-Tiff´ee, J. Applied Electrochem., 32 (2002) 875- 882

[5] C. M. Rangel V. V. Lopes and A. Q. Novais. IV Iberian Symposium on Hydrogen, Fuel Cells and Advanced Batteries, Estoril, Portugal, 2013

[6] B. A. Boukamp, J. Phys. Energy 2 (2020) 1

DRT DRT Distribution of Relaxation Times EIS DRT Distribution of Relaxation Times EIS Electrochemical Impedance Spectroscopy Equivalent Electrical Circuit Modelling

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# is calciuma metal nonmetal or metalloid in denmark

#### Metals and Nonmetals - CHISD

METAL, NONMETAL, METALLOID Hydrogen Magnesium Calcium Arsenic Carbon Helium Lead Aluminum NONMETAL METAL METALLOID * * * Fe, Au, Cu, and Al all solid at room temp- who can think of one metal not solid at room temp?- Mercury! * Chlorine is a greenish-yellow gas with a poisonous odor * * * * * Fe, Au, Cu, and Al all solid at room temp- who can

#### is chlorine a metal nonmetal or metalloid

These metals are very reactive. share electrons to produce neutral molecules. ATP 5-19 - Risk Management? 1 Answer anor277 Dec 14, 2016 Chlorine is a non-metal. a typical property of a non-metal… g) Metal. The metals which follow the transition metals (towards the nonmetals, or Steel is an alloy of iron and carbon,

#### Solved: 14. Calcium: A. Group Nuer: B. Syol: C. Valenc

Answer to 14. Calcium: a. Group Nuer: b. Syol: C. Valence Electrons: d. Metal, Metalloid, or Nonmetal? 15. Arsenic: a. Group N

#### Is calcium a non metal? - Answers

Dec 18, 2012· Calcium Chloride is an ionic compound. You see, for a substance to be classified as either a non-metal or a metal, it has to be in elemental form. Calcium Chloride …

#### METALLOID | meaning in the Caridge English Dictionary

metalloid definition: 1. a chemical element with some of the properties of a metal and some of a non-metal, for example…. Learn more.

#### Is argon a metal, nonmetal, or metalloid? - Quora

Argon is a nonmetal — a noble gas. I know that because it’s on the far right of the table in the 18th column (family?). Metallic character decreases from left to right on a diagonal: I read some interesting things on argon recently.. * The first s

#### is calcium a metal or nonmetal or metalloid?

Is calcium a metal or nonmetal or metalloid? - 16668652

#### Is argon a metal, nonmetal, or metalloid? - Quora

Argon is a nonmetal — a noble gas. I know that because it’s on the far right of the table in the 18th column (family?). Metallic character decreases from left to right on a diagonal: I read some interesting things on argon recently.. * The first s

#### Metals Nonmetals metalloids

Oct 29, 2013· Metals Nonmetals metalloids 1. Use the periodic table in your agenda to fill in the chart. Element Neon (Ne) Sodium (Na) Chlorine (Cl) Silicon (Si) Calcium (Ca) Gold (Au) Arsenic (As) Metal, Nonmetal, or Metalloid? Nonmetal Metal Nonmetal Metalloid Metal Metal Metalloid 2. SCIENTIST DIVIDE ALL ELEMENTS INTO THREE SECTIONS 3.

#### Is krypton a metal nonmetal or metalloid?

May 17, 2020· Classifiion: Krypton is a noble gas and a no. SA Metal Group is South Africa''s oldest and largest metal recycling company. For the last century, the Group has purchased, collected, processed and recycled all forms of ferrous metals (iron and steel) and non-ferrous metals (aluminium, copper, zinc, stainless steel, lead, nickel, brass, tin, bronze and others).

#### Is Phosphorus a metal nonmetal or metalloid?

Feb 08, 2020· Similarly, it is asked, is Phosphorus is a metal? Phosphorus is a typical non-metal element that is loed in the 15th group of the periodic table of chemical elements. In nature, phosphorus is present in the composition of the minerals called Apatite. Our body contains 1% of phosphorus by mass, mostly as part of the bones. Red phosphorus burns well in air.

#### is calcium a metal nonmetal or metalloid

Question: Is calcium a metal metalloid or nonmetal. Todd Helmenstine Worksheet Answers . Determine whether the element Calcium is a metal or a nonmetal without looking at a periodic table. Physical Properties of nonmetals: Physical State: Most of the non-metals exist in two of the three states of matter at room temperature: gases (oxygen) and

#### Is argon a metal, nonmetal, or metalloid? - Quora

Argon is a nonmetal — a noble gas. I know that because it’s on the far right of the table in the 18th column (family?). Metallic character decreases from left to right on a diagonal: I read some interesting things on argon recently.. * The first s

#### 5fdceed4421873887b8b456e.docx - Course Hero

Classify each of the following elements as a metal, non-metal, or metalloid a. Calcium metal b. Silicon metalloid c. Chlorine nonmetal d. Potassium metal e. Titanium metal f. Sulfur nonmetal g. Antimony metalloid 2. Determine if the listed pairs of elements have similar or different properties a. Potassium & Rubidium similar b.

#### Chapter 4 Review Flashcards

Is lead a metal, nonmetal,or metalloid? Term. metal: Definition. Is potassium a metal, nonmetal, or metalloid? Term. metal: Definition. Is calcium a metal, nonmetal, or metalloid? Term. metalloid: Definition. Is silicon a metal, nonmetal, or metalloid? Supporting users have an ad free experience! Tweet . My Flashcards. My Sets;

#### Chemistry Chapter 5 Flashcards | Quizlet

Is calcium a metal nonmetal or metalloid. metal. What element has 28 protons. Nickel. What happens to the nuer of valence electrons as you go down a group. Stays the same. What happens to the nuer of valence electrons as you go across a period. Increases. When an electron is …

#### is calcium a metal or nonmetal or metalloid?

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#### Elements - Metals, Nonmetals and Metalloids

Aug 18, 2019· In this activity, students will investigate several properties of the given elements and decide whether each element is a metal, non-metal or a metalloid. They will examine the appearance of the given samples and note the color, luster and form. Using a …

#### Is Carbon a metal, nonmetal or metalloid - A Plus Topper

Dec 03, 2020· Is Carbon a metal, nonmetal or metalloid. The food you eat, the clothes you wear, the cosmetics you use, the fuels you use to run automobiles are all the compounds of carbon. Carbon was discovered in prehistory and it was known to the ancients. They used to manufacture charcoal by burning organic material.

#### is calcium a metal nonmetal or metalloid

Nonmetal without looking at a periodic table 2/3 of density of air ( d ) silicon:. Whether the element calcium is a metal, nonmetal, or metalloid, calcium and! Calcium, and metalloids on the periodic table metal: metalloid is calcium a metal nonmetal or metalloid nonmetal copper metal oxygen boron.

#### is chlorine a metal nonmetal or metalloid

These metals are very reactive. share electrons to produce neutral molecules. ATP 5-19 - Risk Management? 1 Answer anor277 Dec 14, 2016 Chlorine is a non-metal. a typical property of a non-metal… g) Metal. The metals which follow the transition metals (towards the nonmetals, or Steel is an alloy of iron and carbon,

#### Metal, Nonmetal, or Metalloid

Metal, Nonmetal, or Metalloid? Objective: Explore the physical and chemical properties of eight elements with the goal of classifying them as metals, nonmetals, or metalloids. Background: Physical Properties: – properties that can be observed without changing the identity of a substance

#### 7.6: Metals, Nonmetals, and Metalloids - Chemistry LibreTexts

Mar 19, 2021· Chemical Properties of Nonmetals. Non-metals have a tendency to gain or share electrons with other atoms. They are electronegative in character. Nonmetals, when reacting with metals, tend to gain electrons (typically attaining noble gas electron configuration) and become anions: $\ce{3Br2(l) + 2Al(s) \rightarrow 2AlBr3(s)} \nonuer$

#### Solved: Identity Each Of The Following Elements As A Metal

Identity each of the following elements as a metal, a nonmetal, or a metalloid Drag the appropriate items to their respective bins. Reset Help calcium a shiny element

#### Is Oxygen (O) a Metal, Non-Metal, or Metalloid? -

Sep 29, 2020· To determine whether Oxygen (O) is a metal, nonmetal, or metalloid we look at its position on the Periodic Table. Elements on the left-hand side (with the e

#### What do you think about hydrogen, is it a metal, a

Jun 17, 2018· In the metalloid got in the lattice-space as interstitial structure of the titanium-metal, hydrogen does not influence on the strength and hardness, but on the impact.In the case of …

#### Is Oxygen (O) a Metal, Non-Metal, or Metalloid? -

Sep 29, 2020· To determine whether Oxygen (O) is a metal, nonmetal, or metalloid we look at its position on the Periodic Table. Elements on the left-hand side (with the e


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# How do you sketch the graph of f(x) = (x^2 - 3x + 9)/(x - 3)?

Apr 4, 2015

$f \left(x\right) = \frac{{x}^{2} - 3 x + 9}{x - 3} = \frac{{x}^{2} - 3 x}{x - 3} + \frac{9}{x - 3}$

$= \frac{x \left(x - 3\right)}{x - 3} + \frac{9}{x - 3} = x + \frac{9}{x - 3}$

So you actually wanna sketch $f \left(x\right) = x + \frac{9}{x - 3}$

Domain:
The domain of this function is all Real x except 3
because $\frac{9}{x - 3}$ would become unreal!

This brings us to Asymptotes:
They're two Asymptotes:

No 1: $y = x$ because as $x \to \infty$ $f \left(x\right) \to x$

No 2: $x = 3$ is a vertical asymptote because as $f \left(x\right) \to \infty$ $f \left(x\right) \to 3$

Next, turning points:
To find turning points (maxima and minima),
We let$y = f \left(x\right) \implies y = x + \frac{9}{x - 3}$

We make $x$ the subject of the equation,
$\implies y x - 3 y = {x}^{2} - 3 x + 9$
$\implies {x}^{2} - \left(3 + y\right) x + 9 + 3 y = 0$

Since all the values of $x$ are real, ${b}^{2} - 4 a c \ge 0$

$\implies {\left(3 + y\right)}^{2} - 4 \left(9 + 3 y\right) \ge 0$

$\implies {y}^{2} - 6 y - 27 \ge 0 \implies \left(y - 9\right) \left(y + 3\right) \ge 0$

Hence, $y \le - 3 \cup y \ge 9$

Putting the values of $y$ in $f \left(x\right)$ to get $x$
we have the points $\left(0 , - 3\right)$ maximum and $\left(6 , 9\right)$ as minimum

To find the intercepts :
$y$- intercept is when $x = 0$ we have $\left(0 , - 3\right)$
$x$- intercept is when $y = 0$ but $y$ cannot be $0$(see the range above)

Table of variation:

So now we're all set to sketch that graph;
Here's what i got:


text:
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# Four real roots (Tomato subjective 67)

problem: Describe the set of all real numbers x which satisfy 2 $${\log_{{2x+3}^x}}$$ <1.

solution: 2 $${\log_{{2x+3}^x}}$$ < 1

Now x< 0 is not in the domain of logarithm.

So x> 0.

Now as x>0           2x+3 > 1.


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# owner’s equity and the Accounting equation e5A. Daiichi Company’s total assets and liabilities at...

owner’s equity and the Accounting equation

e5A. Daiichi Company’s total assets and liabilities at the beginning and end of the year follow.

 Beginning of the year Assets $175,000 Liabilities$  68,750 End of the year 275,000 162,500

Determine Daiichi’s net income or loss for the year under each of the following alternatives:

1.    The owner made no investments in or withdrawals from the business during the year.

2.    The owner made no investments in the business but withdrew $27,500 during the year. 3. The owner invested$16,250 in the business but made no withdrawals during the year.

4.    The owner invested $12,500 in the business and withdrew$27,500 during the year.

Answer : We know that Capital = Assets - liabilities Capital at the Beginning of the Year = 175000 -68750 = 106250 Capital at the end of the years = 275000-162500 = 112500 Answer for alternative 1. Profit or loss Profit/loss = Capital at the end of the years - Capital at the Beginning of the year = 112500-106250 Profit = $6250 Answer for alternative 2 Profit or loss when owner withdraw$ 27, 500. Profit/loss = Capital at the end of the years - Capital at the Beginning of the year+ with drawal = 112500-106250+27500 Profit = \$ 33750 Answer for alternative 3 Profit or loss when...


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# Orders of magnitude (data)

An order of magnitude is a factor of ten. A quantity growing by four orders of magnitude implies it has grown by a factor of 10,000 or 104.

This article presents a list of multiples, sorted by orders of magnitude, for digital information storage measured in bits.

The byte is a common unit of measurement of information (kilobyte, kibibyte, megabyte, mebibyte, gigabyte, gibibyte, terabyte, tebibyte, etc.); for the purpose of this article, a byte is a group of 8 bits (octet), a nibble is a group of four bits. Historically, both assumptions have not always been true.

The decimal SI prefixes kilo, mega, giga, tera, etc., are powers of 103 = 1000. The binary prefixes kibi, mebi, gibi, tebi, etc. respectively refer to the corresponding power of 210 = 1024. In casual usage, when 1024 is a close enough approximation of 1000, the two corresponding prefixes are equivalent.

Orders of magnitude (data)
Binary
[bits]
Decimal Item
Factor Term Factor Term
2−3 10−3 millibit
2−2 10−2 centibit
2−1 10−1 decibit 0.415 bits (${\displaystyle \log _{2}4/3}$) – The amount of information needed to eliminate one option out of four.
0.6 - 1.3 bits - Approximate information per letter of English text.[1]
20 bit 100 bit 1 bit – 0 or 1, false or true, Low or High (aka unibit)
1.442695 bits (${\displaystyle \log _{2}e}$) – approximate size of a nat (a unit of information based on natural logarithms)
1.5849625 bits (${\displaystyle \log _{2}3}$) – approximate size of a trit (a base-3 digit)
21 2 bits – a crumb (aka dibit) enough to uniquely identify one base pair of DNA
3 bits – a triad(e), (aka tribit) the size of an octal digit
22 nibble 4 bits – (aka tetrad(e), nibble, quadbit, semioctet, or halfbyte) the size of a hexadecimal digit; decimal digits in binary-coded decimal form
5 bits – the size of code points in the Baudot code, used in telex communication (aka pentad)
6 bits – the size of code points in Univac Fieldata, in IBM "BCD" format, and in Braille. Enough to uniquely identify one codon of genetic code. The size of code points in Base64; thus, often the entropy per character in a randomly-generated password.
7 bits – the size of code points in the ASCII character set

– minimum length to store 2 decimal digits

23 byte 8 bits – (aka octet or octad(e)) on many computer architectures.

– Equivalent to 1 "word" on 8-bit computers (Apple II, Atari 800, Commodore 64, et al.).
– the "word size" (instruction length) for 8-bit console systems including: Atari 2600, Nintendo Entertainment System

101 decabit 10 bits

– minimum bit length to store a single byte with error-correcting computer memory
– minimum frame length to transmit a single byte with asynchronous serial protocols

12 bits – wordlength of the PDP-8 of Digital Equipment Corporation (built from 1965–1990)
24 16 bits

– the Basic Multilingual Plane of Unicode, containing character codings for almost all modern languages, and a large number of symbols
– the basic unit in UTF-16; the full Universal Character Set (Unicode) can be encoded in one or two of these
– commonly used in many programming languages, the size of an integer capable of holding 65,536 different values
– Equivalent to 1 "word" on 16-bit computers (IBM PC, Commodore Amiga)
– the "word size" (instruction length) for 16-bit console systems including: Sega Genesis, Super Nintendo, Mattel Intellivision

25 32 bits (4 bytes)

– size of an integer capable of holding 4,294,967,296 different values
– size of an IEEE 754 single-precision floating point number
– size of addresses in IPv4, the current Internet protocol
– Equivalent to 1 "word" on 32-bit computers (Apple Macintosh, Pentium-based PC).
– the "word size" (instruction length) for various console systems including: PlayStation, Nintendo GameCube, Xbox, Wii

36 bits – size of word on Univac 1100-series computers and Digital Equipment Corporation's PDP-10
56 bits (7 bytes) – cipher strength of the DES encryption standard
26 64 bits (8 bytes)

– size of an integer capable of holding 18,446,744,073,709,551,616 different values
– size of an IEEE 754 double-precision floating point number
– Equivalent to 1 "word" on 64-bit computers (Power, PA-Risc, Alpha, Itanium, Sparc, x86-64 PCs and Macintoshes).
– the "word size" (instruction length) for 64-bit console systems including: Nintendo 64, PlayStation 2, PlayStation 3, Xbox 360

80 bits (10 bytes) – size of an extended precision floating point number, for intermediate calculations that can be performed in floating point units of most processors of the x86 family.
102 hectobit 100 bits
27 128 bits (16 bytes)

– size of addresses in IPv6, the successor protocol of IPv4
– minimum cipher strength of the Rijndael and AES encryption standards, and of the widely used MD5 cryptographic message digest algorithm
– size of an SSE vector register, included as part of the x86-64 standard

160 bits – maximum key length of the SHA-1, standard Tiger (hash), and Tiger2 cryptographic message digest algorithms
28 256 bits (32 bytes)

– minimum key length for the recommended strong cryptographic message digests as of 2004
– size of an AVX2 vector register, present on newer x86-64 CPUs

29 512 bits (64 bytes)

– maximum key length for the standard strong cryptographic message digests in 2004
– size of an AVX-512 vector register, present on some x86-64 CPUs

103 kilobit 1,000 bits
210 kibibit 1,024 bits (128 bytes) - RAM capacity of the Atari 2600
1,288 bits – approximate maximum capacity of a standard magnetic stripe card
211 2,048 bits (256 bytes) – RAM capacity of the stock Altair 8800
212 4,096 bits (512 bytes)

– typical sector size, and minimum space allocation unit on computer storage volumes, with most file systems
– approximate amount of information on a sheet of single-spaced typewritten paper (without formatting)

4,704 bits (588 bytes) – uncompressed single-channel frame length in standard MPEG audio (75 frames per second and per channel), with medium quality 8-bit sampling at 44,100 Hz (or 16-bit sampling at 22,050 Hz)
kilobyte 8,000 bits (1,000 bytes)
213 kibibyte 8,192 bits (1,024 bytes) – RAM capacity of a Sinclair ZX81.
9,408 bits (1,176 bytes) – uncompressed single-channel frame length in standard MPEG audio (75 frames per second and per channel), with standard 16-bit sampling at 44,100 Hz
104 15,360 bits – one screen of data displayed on an 8-bit monochrome text console (80x24)
214 16,384 bits (2 kibibytes) – one page of typed text,[2] RAM capacity of Nintendo Entertainment System
215 32,768 bits (4 kibibytes)
216 65,536 bits (8 kibibytes)
105 100,000 bits
217 131,072 bits (16 kibibytes) – RAM capacity of the smallest Sinclair ZX Spectrum.
218 262,144 bits (32 kibibytes) - RAM capacity of Matra Alice 90
393,216 bits (48 kibibytes) - RAM capacity of 48K Sinclair ZX Spectrum
506 kilobits – approximate size of this article as of 20 May 2019
219 524,288 bits (64 kibibytes) – RAM capacity of a lot of popular 8-bit computers like the C-64, Amstrad CPC etc.
106 megabit 1,000,000 bits
220 mebibit 1,048,576 bits (128 kibibytes) – RAM capacity of popular 8-bit computers like the C-128, Amstrad CPC etc. Or a 1024 x 768 pixel jpeg image.
1,978,560 bits – a one-page, standard-resolution black-and-white fax (1728 × 1145 pixels)
221 2,097,152 bits (256 kibibytes)
4,147,200 bits – one frame of uncompressed NTSC DVD video (720 × 480 × 12 bpp Y'CbCr)
222 4,194,304 bits (512 kibibytes)
4,976,640 bits – one frame of uncompressed PAL DVD video (720 × 576 × 12 bpp Y'CbCr)
5,000,000 bits – Typical English book volume in plain text format of 500 pages × 2000 characters per page and 5-bits per character.
5,242,880 bits (640 kibibytes) – the maximum addressable memory of the original IBM PC architecture
megabyte 8,000,000 bits (1,000 kilobytes) – the preferred definition of megabyte
8,343,400 bits – one "typical" sized photograph with reasonably good quality (1024 × 768 pixels).
223 mebibyte 8,388,608 bits (1,024 kibibytes), one of a few traditional meanings of megabyte
107 11,520,000 bits – capacity of a lower-resolution computer monitor (as of 2006), 800 × 600 pixels, 24 bpp
11,796,480 bits – capacity of a 3.5 in floppy disk, colloquially known as 1.44 megabyte but actually 1.44 × 1000 × 1024 bytes
224
16,777,216 bits (2 mebibytes)
25,000,000 bits – amount of data in a typical color slide
30,000,000 bits – The first commercial harddisk IBM 350 in 1956 could store 3.75 MiB for a cost of 50,000 USD, equivalent to 460771.11 USD in 2013.[3]
225 33,554,432 bits (4 mebibytes) – RAM capacity of stock Nintendo 64 and average size of a music track in MP3 format.
41,943,040 bits (5 mebibytes) – approximate size of the Complete Works of Shakespeare[2]
80,000,000 bits – In 1985 a 10 MB harddisk cost 710 USD, equivalent to 1653.96 USD in 2013.[3]
98,304,000 bits – capacity of a high-resolution computer monitor as of 2011, 2560 × 1600 pixels, 24 bpp
50 – 100 megabits – amount of information in a typical phone book
226 108 67,108,864 bit (8 mebibytes)
227 134,217,728 bits (16 mebibytes)
150 megabits – amount of data in a large foldout map
228 268,435,456 bits (32 mebibytes)
144,000,000 bits: In 1980 an 18 MB harddisk cost 4,199 USD, equivalent to 12768.28 USD in 2013.[3]
423,360,000 bits: a five-minute audio recording, in CDDA quality
229 536,870,912 bits (64 mebibytes)
109 gigabit 1,000,000,000 bits
230 gibibit 1,073,741,824 bits (128 mebibytes)
231 2,147,483,648 bits (256 mebibytes)
232 4,294,967,296 bits (512 mebibytes)
5.45×109 bits (650 mebibytes) – capacity of a regular compact disc (CD)
5.89×109 bits (702 mebibytes) – capacity of a large regular compact disc
6.4×109 bits – capacity of the human genome (assuming 2 bits for each base pair)
6,710,886,400 bits – average size of a movie in Divx format in 2002.[4]
gigabyte 8,000,000,000 bits (1,000 megabytes) – In 1995 a 1 GB harddisk cost 849 USD, equivalent to 1395.96 USD in 2013.[3]
233 gibibyte 8,589,934,592 bits (1,024 mebibytes) – The maximum disk capacity using the 21-bit LBA SCSI standard introduced in 1979.
1010 10,000,000,000 bits
234 17,179,869,184 bits (2 gibibytes). The storage limit of IDE standard for harddisks in 1986, also the volume size limit for the FAT16B file system (with 32 KiB clusters) released in 1987 as well as the maximum file size (2 GiB-1) in DOS operating systems prior to the introduction of large file support in DOS 7.10 (1997).
235 34,359,738,368 bits (4 gibibytes) – maximum addressable memory for the Motorola 68020 (1984) and Intel 80386 (1985), also the volume size limit for the FAT16B file system (with 64 KiB clusters) as well as the maximum file size (4 GiB-1) in MS-DOS 7.1-8.0.
3.76×1010 bits (4.7 gigabytes) – capacity of a single-layer, single-sided DVD
236 68,719,476,736 bits (8 gibibytes)
79,215,880,888 bits9.2 GiB size of Wikipedia article text compressed with bzip2 on 2013-06-05
1011 100,000,000,000 bits
237 137,438,953,472 bits (16 gibibytes).
1.46×1011 bits (17 gigabytes) – capacity of a double-sided, dual-layered DVD
2.15×1011 bits (25 gigabytes) – capacity of a single-sided, single-layered 12-cm Blu-ray disc
238 274,877,906,944 bits (32 gibibytes)
239 549,755,813,888 bits (64 gibibytes)
1012 terabit 1,000,000,000,000 bits
240 tebibit 1,099,511,627,776 bits (128 gibibytes) – estimated capacity of the Polychaos dubium genome, the largest known genome. The storage limit for ATA-1 compliant disks introduced in 1994.
1.6×1012 bits (200 gigabytes) – capacity of a hard disk that would be considered average as of 2008. In 2005 a 200 GB harddisk cost 100 USD, equivalent to 128.28 USD in 2013.[3] As of April 2015, this is the maximum capacity of a fingernail-sized microSD card.
241 2,199,023,255,552 bits (256 gibibytes) – As of 2017, this is the maximum capacity of a fingernail-sized microSD card
242 4,398,046,511,104 bits (512 gibibytes)
terabyte 8,000,000,000,000 bits (1,000 gigabytes) – In 2010 a 1 TB harddisk cost 80 USD, equivalent to 91.92 USD in 2013.[3]
243 tebibyte 8,796,093,022,208 bits (1,024 gibibytes)
(approximately) 8.97×1012 bits – as of 2010, data of π to the largest number of decimal digits ever calculated (2.7×1012)
1013 10,000,000,000,000 bits (1.25 terabytes) – capacity of a human being's functional memory, according to Raymond Kurzweil in The Singularity Is Near, p. 126
16,435,678,019,584 bits (1.9 terabytes) – Size of all multimedia files used in English wikipedia on May 2012
244 17,592,186,044,416 bits (2 tebibytes) – Maximum size of MBR partitions used in PCs introduced in 1983, also the maximum disk capacity using the 32-bit LBA SCSI introduced in 1987
245 35,184,372,088,832 bits (4 tebibytes)
246 70,368,744,177,664 bits (8 tebibytes)
1014 100,000,000,000,000 bits
247 140,737,488,355,328 bits (16 tebibytes). NTFS volume capacity in Windows 7, Windows Server 2008 R2 or earlier implementation.[5]
1.5×1014 bits (18.75 terabytes)
248 281,474,976,710,656 bits (32 tebibytes)
249 562,949,953,421,312 bits (64 tebibytes)
1015 petabit 1,000,000,000,000,000 bits
250 pebibit 1,125,899,906,842,624 bits (128 tebibytes)
251 2,251,799,813,685,248 bits (256 tebibytes)
252 4,503,599,627,370,496 bits (512 tebibytes)
petabyte 8,000,000,000,000,000 bits (1,000 terabytes)
253 pebibyte 9,007,199,254,740,992 bits (1,024 tebibytes)
1016 10,000,000,000,000,000 bits
254 18,014,398,509,481,984 bits (2 pebibytes)
255 36,028,797,018,963,968 bits (4 pebibytes) – theoretical maximum of addressable physical memory in the AMD64 architecture[citation needed]
4.5×1016 bits (5.625 petabytes) – estimated hard drive space in Google's server farm as of 2004[citation needed]
256 72,057,594,037,927,936 bits (8 pebibytes)
10 petabytes (1016 bytes) – estimated approximate size of the Library of Congress's collection, including non-book materials, as of 2005.[6] Size of the Internet Archive topped 10 PB in October 2013[7]
1017 100,000,000,000,000,000 bits
257 144,115,188,075,855,872 bits (16 pebibytes)
2×1017 bits (25 petabytes) – Storage space of Megaupload file-hosting service at the time it was shut down in 2012[8]
258 288,230,376,151,711,744 bits (32 pebibytes)
259 576,460,752,303,423,488 bits (64 pebibytes)
8 ×1017, the storage capacity of the fictional Star Trek character Data
1018 exabit 1,000,000,000,000,000,000 bits
260 exbibit 1,152,921,504,606,846,976 bits (128 pebibytes). The storage limit using the 48-bit LBA ATA-6 standard introduced in 2002.
1.6×1018 bits (200 petabytes) – total amount of printed material in the world[citation needed]
2×1018 bits (250 petabytes) – storage space at Facebook data warehouse as of June 2013,[9] growing at a rate of 15 PB/month.[10]
261 2,305,843,009,213,693,952 bits (256 pebibytes)
2.4×1018 bits (300 petabytes) – storage space at Facebook data warehouse as of April 2014, growing at a rate of 0.6 PB/day.[11]
262 4,611,686,018,427,387,904 bits (512 pebibytes)
exabyte 8,000,000,000,000,000,000 bits (1,000 petabytes)
263 exbibyte 9,223,372,036,854,775,808 bits (1,024 pebibytes)
1019 10,000,000,000,000,000,000 bits
264 18,446,744,073,709,551,616 bits (2 exbibytes).
265 36,893,488,147,419,103,232 bits (4 exbibytes)
50,000,000,000,000,000,000 bits (50 exabit)
266 73,786,976,294,838,206,464 bits (8 exbibytes)
1020 100,000,000,000,000,000,000 bits
1.2×1020 bits (15 exabytes) – estimated storage space at Google data warehouse as of 2013[12]
267 147,573,952,589,676,412,928 bits (16 exbibytes) – maximum addressable memory using 64-bit addresses without segmentation.[13] Maximum volume and file size for ZFS filesystem.
268 295,147,905,179,352,825,856 bits (32 exbibytes)
3.5 × 1020 bits – increase in information capacity when of energy is added to a heat-bath at 300 K (27 °C)[14]
269 590,295,810,358,705,651,712 bits (64 exbibytes)
1021 zettabit 1,000,000,000,000,000,000,000 bits
270 zebibit 1,180,591,620,717,411,303,424 bits (128 exbibytes)
271 2,361,183,241,434,822,606,848 bits (256 exbibytes)
3.4×1021 bits (0.36 zettabytes) – amount of information that can be stored in 1 gram of DNA[15]
4.7×1021 bits (0.50 zettabytes) – amount of digitally stored information in the world as of May 2009[16]
4.8×1021 bits (0.61 zettabytes) – total hard drive capacity shipped in 2016[17]
272 4,722,366,482,869,645,213,696 bits (512 exbibytes)
zettabyte 8,000,000,000,000,000,000,000 bits (1,000 exabytes)
273 zebibyte 9,444,732,965,739,290,427,392 bits (1,024 exbibytes)
1022 10,000,000,000,000,000,000,000 bits
276 276 bits – Maximum volume and file size in the Unix File System (UFS) and maximum disk capacity using the 64-bit LBA SCSI standard introduced in 2000 using 512-byte blocks.[18]
1023 1.0×1023 bits – increase in information capacity when 1 Joule of energy is added to a heat-bath at 1 K (−272.15 °C)[19]
277 6.0×1023 bits – information content of 1 mole (12.01 g) of graphite at 25 °C; equivalent to an average of 0.996 bits per atom.[20]
1024 yottabit 1,000,000,000,000,000,000,000,000 bits
7.3×1024 bits – information content of 1 mole (18.02 g) of liquid water at 25 °C; equivalent to an average of 12.14 bits per molecule.[21]
280 yobibit 1,208,925,819,614,629,174,706,176 bits (128 zebibytes)
yottabyte 8,000,000,000,000,000,000,000,000 bits (1,000 zettabytes)
283 yobibyte 9,671,406,556,917,033,397,649,408 bits (1,024 zebibytes)
1025 1.1×1025 bits – entropy increase of 1 mole (18.02 g) of water, on vaporizing at 100 °C at standard pressure; equivalent to an average of 18.90 bits per molecule.[22]
1.5×1025 bits – information content of 1 mole (20.18 g) of neon gas at 25 °C and 1 atm; equivalent to an average of 25.39 bits per atom.[23]
Beyond standardized SI / IEC (binary) prefixes
2150 N/A 1045 N/A ~ 1045 bits – the number of bits required to perfectly recreate the natural matter of the average-sized U.S. adult male human being down to the quantum level on a computer is about 2×1045 bits of information (see Bekenstein bound for the basis for this calculation).
2193 1058 ~ 1058 bits – thermodynamic entropy of the sun[24] (about 30 bits per proton, plus 10 bits per electron).
2230 1069 ~ 1069 bits – thermodynamic entropy of the Milky Way Galaxy (counting only the stars, not the black holes within the galaxy)[citation needed]
2255 1077 1.5×1077 bits – information content of a one-solar-mass black hole.[25]
2305 1090 The information capacity of the observable universe, according to Seth Lloyd. (not including gravitation)[26]

1. Entropy (information theory), such as the amount of information that can be stored in DNA
2. Entropy (thermodynamics), such as entropy increase of 1 mole of water

These two definitions are not entirely equivalent, see Entropy in thermodynamics and information theory.

For comparison, the Avogadro constant is 6.02214179(3)×1023 entities per mole, based upon the number of atoms in 12 grams of carbon-12 isotope.

In 2012, some hard disks used ~984,573 atoms to store each bit. In January 2012, IBM researchers announced they compressed 1 bit in 12 atoms using antiferromagnetism and a scanning tunneling microscope with iron and copper atoms. This could mean a practical jump from a 1 TB disk to a 100 TB disk.[3][27]

## References

1. ^ Mark Nelson (24 August 2006). "The Hutter Prize". Retrieved 27 November 2008.
2. ^ a b "A special report on managing information: All too much". The Economist. 25 February 2010. Retrieved 4 March 2010.
3. "Cost of Hard Drive Space". 11 May 2013. Retrieved 23 June 2013.
4. ^ "How much does DivX shrink a file?". 18 April 2002. Retrieved 24 June 2013.
5. ^ Microsoft TechNet (28 March 2003). "How NTFS Works". Windows Server 2003 Technical Reference. Retrieved 12 September 2011.
6. ^ Hickey, Thom (OCLC Chief Scientist) (21 June 2005). "Entire Library of Congress". Outgoing. Retrieved 5 May 2010.
7. ^ The Internet Archive Has Now Saved a Whopping 10,000,000,000,000,000 Bytes of Data, retrieved October 2nd 2013
8. ^ 25 petabyte on Megaupload. Archived 1 August 2012 at Archive.today Retrieved 16 February 2012
9. ^ [1]
10. ^ [2]
11. ^ [3]
12. ^ Estimated storage space at Google
13. ^ "A brief history of virtual storage and 64-bit addressability". Retrieved 17 February 2007.
14. ^ ${\displaystyle \scriptstyle {\frac {1}{300}}}$[citation needed] J K−1
15. ^ http://www.tmrfindia.org/ijcsa/V2I29.pdf
16. ^ "Internet data heads for 500bn gigabytes", The Guardian, 18 May 2009. Retrieved on 23 April 2010.
17. ^ 424 Million HDDs Shipped in 2016 – Trendfocus
18. ^ "Working Draft T10, American National Standard Project 1417-D, Revision 4, 28 July 2001" (PDF). o3one.org. 8 January 2002. p. 72. Retrieved 23 June 2013.
19. ^ 1 J K−1. Equivalent to 1/(k ln 2) bits, where k is Boltzmann's constant
20. ^ Equivalent to 5.74 J K−1. Standard molar entropy of graphite.
21. ^ Equivalent to 69.95 J K−1. Standard molar entropy of water.
22. ^ Equivalent to 108.9 J K−1
23. ^ Equivalent to 146.33 J K−1. Standard molar entropy of neon. An experimental value, see [4] for a theoretical calculation.
24. ^ Given as 1042 erg K−1 in Bekenstein (1973), Black Holes and Entropy, Physical Review D 7 2338
25. ^ Entropy = ${\displaystyle \scriptstyle Ac^{3}/4G\hbar }$ in nats, with ${\displaystyle A=16\pi G^{2}M^{2}/c^{4}}$ for a Schwarzschild black hole. 1 nat = 1/ln(2) bits. See Jacob D. Bekenstein (2008), Bekenstein-Hawking entropy, Scholarpedia.
26. ^ Lloyd, Seth (24 May 2002). "Computational Capacity of the Universe" (PDF). Physical Review Letters. 88 (23). arXiv:quant-ph/0110141. Bibcode:2002PhRvL..88w7901L. doi:10.1103/PhysRevLett.88.237901. Archived (PDF) from the original on 11 November 2017.
27. ^ "IBM Smashes Moore's Law, Cuts Bit Size to 12 Atoms". 12 January 2012. Retrieved 23 June 2013.


text:
What is the condensed structural formula for toluene with bromine?

• $\text{C"_6"H"_5"CH"_2"Br}$ - if bromine is on the methyl group
• $\text{C"_6"H"_4"CH"_3"Br}$ - if bromine is directly on the ring; note that this is the ortho isomer, and the meta and para isomers exist as well.
Their molecular formulas are identical ($\text{C"_7"H"_7"Br}$), so it wouldn't be productive to use that in this case.


text:
# Homework Help: Identify Type of Graph

1. Apr 17, 2010

### realism877

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

4x2 - 8x + 3y2 + 18y - 23 = 0

2. Relevant equations

3. The attempt at a solution

4(x-4)^+3(y+9)^=120

What did I do wrong?

2. Apr 17, 2010

### Mathnomalous

3. Apr 17, 2010

### realism877

Can I get assistance to break it down from a to z?

4. Apr 17, 2010

### Mathnomalous

Well, sure. We can both learn as we go since I'm only familiar with parabolas and circles. Let's both work on our ends to find the solution.

To start, re-check your equation and make sure you typed it as it appears on your notebook or textbook, please. In the meantime, we are looking for something of the form:

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

or

(y-k)^2/a^2 + (x-k)^2/b^2 = 1

That should look very familiar to the circle equation, no?

5. Apr 18, 2010

### Staff: Mentor

Mathnomalous,
This is NOT the equation of a circle, if that's what you were driving at. The equation you gave is similar to (not familiar to) the equation of a circle, but the equation is of an ellipse.
4x2 - 8x + 3y2 + 18y - 23 = 0

Realism877,
You started off on the right track, but you complete the square in the x and y terms incorrectly.

4x2 - 8x + 3y2 + 18y - 23 = 0
4(x2 - 2x) + 3(y2 + 6y) = 23
Now figure out what you need to add in the first group to complete the square in the x terms, and what you need to add in the second group to complete the square in the y terms. Be sure to add both numbers to the right side.

When you have done that, divide both sides by whatever number you have on the right side. You should end up with something that looks like this:
$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$