How to plot multicolor phase daigram with three parameters?

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I am trying to plot like the diagram given above. I have two nonlinear equations that I have separated in imaginary and complex forms (u1,v1, and x1,y1). As shown in the figure in the x-axis I want ‘p0’ which varies from (0,2.5*10^3) and in the y-axis, I want ‘del0’ which varies from (0,3) and inside the plot where rectangular color region box it is ‘NL=u1^2+v1^2’which varies from (0,6*10^7). I am trying this but unable to plot this type of diagram and gives me an error: “solve was unable to solve the system with inexact coefficients. The \ the answer was obtained by solving a corresponding exact system and \ numericizing the result. >>” Should I give another command to get this type of plot or anything else? If anyone can short it out will be appreciated.

A1 = 1; B1 = 0; delta = -1.5;(*Subscript[Δ, L] ; Laser-Lower polariton \ detuning*) g0 = .315; (* Subscript[g, 0] ; vacuum optomechanical strength *) ome = 3.014; (* Ω ; Rabi splitting *) ome1 = .125; (* Subscript[Ω, m] ; mechanical \ resonator's frequency *) kexc = 0.002;  (* Subscript[κ, exc  ;  ]exciton decay rate*) kk = .2;   (* κ; cavity decay rate *) gma = 0.00001;  (* Γ ; phonon dcay rate *) ka = (kk + kexc)/2 - del0*(kk - kexc)/(2*ome); Sol = NSolve[{-delta*v1 - ka*u1/2 - (kk - kexc)*B1*g0*x1*u1/ome == 0,      delta*u1 + g0*(1 - del0/ome)*A1*x1*u1 +        p0*(1 - del0/(2*ome))/Sqrt[2] + p0*g0*B1*x1/(Sqrt[2]*ome) -        ka*v1/2 - (kk - kexc)*g0*x1*B1*v1/ome == 0,      ome1*y1 - gma*x1/2 ==       0, -ome1*x1 + g0/2*(1 - del0/ome)*A1*(u1^2 + v1^2) +        p0*g0*u1*B1/(Sqrt[2]*ome) - gma*y1/2 == 0}, {u1, v1, x1, y1},     del0]; ContourPlot[{Evaluate[(u1^2 + v1^2)] /. Sol}, {p0, 0, 5}, {del0, 0,    3}, PlotLegends -> BarLegend[{"LakeColors", {0, 6}}]]  After running this code I am getting the plot. Which is not the same as above. I don't know what the problem with it?    

Need to retrieve IDs from $wpdb but have multiple parameters

What i am trying to achieve is to hyperlink or profile link the alternative contacts that people have written on their profiles.

I have written this code so far that successfully retrieves the alternative contact of the user. The result of $ names is “Joe Bloggs, John Smith” for example as it is stored as a metakey value in one cell on the database.

add_shortcode( 'Alternative_Contact_Links', function () { global $  current_user; global $  current_user_manager; global $  wpdb; global $  details;  $  contacts = ''; $  contactsnames = ''; $  dummyurl = "https://altranet-test/user/"; $  displaycontacts = '';  $  urldata = parse_url($  _SERVER['REQUEST_URI'], PHP_URL_PATH ); $  urldata = preg_replace('/\D/', '', $  urldata);  $  current_user = wp_get_current_user(); $  current_user_manager =get_metadata( 'user', $  urldata, 'user_id', true );  $  sqlquery = "SELECT DISTINCT wp_usermeta.meta_value, wp_usermeta.user_id FROM wp_usermeta WHERE  wp_usermeta.meta_key = 'Alternative_Contact' AND wp_usermeta.user_id = '$  urldata'";  $  usernames = array(); $  nameArray = array(); $  usernames = $  wpdb->get_results($  sqlquery);  foreach ($  usernames as $  details) { $  names = $  details->meta_value; echo $  names; }  $  sqlquery2 = "SELECT wp_users.ID FROM wp_users WHERE wp_users.display_name IN ('$  usernames')"; echo "<br>". $  sqlquery2; }); 

The problem I am having is that I now need to retrieve the user IDs of Joe and John. However, How do i pass their names into the sqlquery2 each time? Do i need some sort of loop? this is where I am lost.

Any help would be great!

Thanks

Right use of canonical when using URL parameters?

On certain pages of my website I use URL parameters on links:

<a href="/example/?parameter=value">Click here</a> 

The target URL is always exactly the same (i.e. /example/). With exactly the same content.

The URL parameter only serves the purpose of being used as a hidden form field in a form on that page. So that I can identify the page the user has visited before actually submitting that form.

My question: How do I correctly set the canonical to prevent the target page to be indexed multiple times?

Only on the actual target page in the head like this?

<link rel="canonical" href="https://www.example.com/example/" /> 

Or should I also do it on the actual original link? How would I ideally solve this?

Currently, the tool ahrefs.com claims those pages as “Duplicate pages without canonical” although I have already set the self-referencing canonical on the target page like this <link rel="canonical" href="https://www.example.com/example/" />.

Hreflang and canonical problem on session parameters

We have a oxid onlineshop with different domains/subdomains depending on currency and language.

Now we have a problem with hreflang tags, because of parameters

1) the session of the basket between domains is set by ?force_sid=(random string for session id)

2) for different views in categories like ?ldtype=grid&_artperpage=100&pgNr=0&cl=alist&searchparam=&cnid=3ae4a2e1dd7501139.35363255

if the url is accessed without the parameters then the canonical and hreflang tags are correct.

If the parameters are set then the canonical and hreflang tags are wrong.

What are the correct tags for example: www.example.de/category-name/?force_sid=e9k6p7d5dbpcu3s41p22tbll71 ?

We have:

<link rel="canonical" href="https://www.example.de/category-name/"> <link rel="alternate" hreflang="x-default" href="https://www.example.de/category-name/"> <link rel="alternate" hreflang="de" href="https://www.example.de/category-name/"> <link rel="alternate" hreflang="de-CH" href="https://www.example.ch/category-name/"> <link rel="alternate" hreflang="fr-CH" href="https://fr.example.ch/category-name/"> <link rel="alternate" hreflang="de-AT" href="https://www.example.at/category-name/"> <link rel="alternate" hreflang="fr" href="https://www.example.fr/category-name/"> <link rel="alternate" hreflang="en" href="https://www.example.com/category-name/"> <link rel="alternate" hreflang="es" href="https://www.example.es/category-name/"> 

How to tackle Big O proofs that involve multiple parameters

I am getting more and more familiar with the whole concept of time complexity but I have never encountered an example where more than one parameter is involved. Therefore, is it possible(well, I am sure it is :”)) and how to prove

a^n = Θ(logn)

or any other, similar-looking expression?

c1 * logn ≤ a^n ≤ c2 * logn

where, e.g. c1 = 1 and c2 = 2,

logn ≤ a^n ≤2 * logn.

Can I go one step further and set n, to be equal e.g. 2? This way I will get

log(2) ≤ a^2 ≤ log(4)

Which is surely true(for a between ~ 0.55 and 0.77)…

…but isn’t that too specific and interfere with the inequality too much? Sorry if the answer is trivial but Google is not helping and I have nobody to ask for explanation.

How to use Mathematica to prove that isotropic materials have only two independent parameters

Posts on related issues can be found from here or here.


Index symmetries:

A stiffness tensor $ C$ is a fourth-order tensor with components $ c_{ijkl}$ which maps symmetric second-order tensors into symmetric second-order tensors, i.e., $ \sigma_{ij} = c_{ijkl} \varepsilon_{kl}$ (linear elastic law), $ \sigma$ (stress) and $ \varepsilon$ (strain) being arbitrary symmetric second-order tensors. Due to the symmetry of the second-order tensors, $ C$ is allowed to be minor symmetric, i.e., $ c_{ijkl} = c_{jikl} = c_{ijlk}$ . The not minor symmetric part of $ C$ is irrelevant for the elastic law and is dropped. If the stress $ \sigma$ is to be related to an elastic energy potential $ W$ (referred to as hyperelastic behavior), i.e., $ \sigma = \partial W / \partial \varepsilon$ , then, due to Schwarz’s theorem, the stiffness tensor $ c_{ijkl} = \partial^2 W / \partial \varepsilon_{ij} \partial \varepsilon_{kl}$ has to possess the major symmetry, i.e., $ c_{ijkl} = c_{klij}$ .

Material symmetry:

A material with stiffness $ C$ is said to possess the material symmetry group $ G$ (e.g., triclinic, orthotropic, transversally isotropic, …) if

\begin{equation} C = Q \star C \qquad Q \in G \end{equation}

holds, where $ Q$ are second-order tensors, referred to as symmetry transformations of $ C$ . The product $ \hat{C} = Q \star C$ (referred to here as Rayleigh product) is defined in components as

\begin{equation} \hat{c}_{ijkl} = Q_{im}Q_{jn}Q_{ko}Q_{lp}c_{mnop} \end{equation}

For solids, $ G$ is a subset of the orthogonal group. In solid mechanics, if suffices to consider rotation matrices $ Q$ from the rotational group $ SO(3)$ . If $ G = \{I\}$ , $ I$ being the identity matrix, then $ C$ is said to triclinic. If $ G$ possesses more than the identity transformation, then different material classes can be defined (different anisotropy types). If $ G = SO(3)$ , the $ C$ is said to be isotropic (no direction dependency).


I want to use Mathematica to get the number of independent parameters needed for the fourth-order tensor to make $ C = Q \star C (Q \in SO(3))$ under the rotation of group $ SO(3)$ .

At present, I can only get 30 independent variables using the following method:

SymmetrizedIndependentComponents[{3, 3, 3, 3},    Symmetric[{1, 2, 3}]] // Length 

However, I still can’t use the rotation of group $ SO(3)$ to further reduce the number of independent variables. What should I do?

Impact of query parameters on SEO for a single page application

Might crawlers visit a page if there is no link referencing this page anywhere but a URL to this page is generated client side with JavaScript ?

Context:

Let’s say I have a SPA with Server Side Rendering. Some pages show a list of items and offer a filtering facility. When the user selects some options or fills in some input field to filter the list, I’d like to embed this information in the URL (eg. /items?sort=price&order=desc&q=something) via the history API (client side routing). Behind the scene, an API call is made to get the results.

Since I do SSR, the server will also be able to understand these URLs and render these pages (hence the user can bookmark the page or share it). But nowhere in the HTML pages these URLs will appear, there are only generated client side in response to user events.

In this context, I think crawlers won’t know these pages exist, and so, they should have no impact on SEO. Even if crawlers are now able to run JavaScript, they don’t use it to simulate user events.

Am I wrong ?

(I guess if someone shares publicly that kind of URL, it could suffice to make this page crawled ? In any case, what I’m worried about is the cost on the crawling budget if all these pages are visited, but I’m ok with a few pages being crawled, they could be marked as “noindex” for instance).

Will irrational parameters make a problem not well-defined on complexity

Given a set $ 𝑁=\{𝑎_1,⋯,𝑎_𝑛\}$ where all $ 𝑎_𝑖$ s are rational positive numbers and $ \sum_{i\in N}a_i=1$ , find a subset 𝑆⊆𝑁 such that $ (\sqrt{2\sum_{i\in S}a_i}-1)^2$ is minimized. Does the appearance of √ make the problem ill-defined with regrading to complexity? If well-defined, it is NP-hard, right?