text:
Blog Life Chances - Going Forward into Employment (GFiE)
Civil Service Commission, Cabinet Office
https://gfie.blog.gov.uk/2022/08/11/breaking-down-walls/
Breaking Down Walls
Posted by: Andrew Ashworth, Posted on: 11 August 2022 - Categories: Uncategorized
Engaging with people is a huge part of what we do as a Going Forward Into Employment team, in fact it's pretty much key to our success.
We know that people leaving prisons or with unspent convictions can face barriers to employment, and many of these are related to what people think or how they feel - whether it is a manager who has misconceptions about people who have been in prison, or someone with lived experience believing that the Civil Service isn't a potential employer for them because of their backgrounds.
Virtual meetings and networking are great, and have helped us to keep conversations going during the pandemic, but nothing beats making new connections in person and giving people the opportunity to find out more about each other. This is particularly important when it comes to prisons, when the people inside are separated from the community and there is a physical barrier in the shape of the walls.
We recently led two visits to HMP Sudbury and HMP Berwyn, bringing together colleagues from across the Civil Service to see the prisons, and meet with staff and prisoners, and all with one shared objective - to break down some walls and start making the links that lead to employment opportunities (and in turn hope) for people on release. Read on if you'd like to find out how we got on!
More about our visits
HMP Sudbury is a Category D prison, which means that it is an open prison and the men there are progressing towards release. Many of these men go out to work each day, with numerous prisoners leaving the grounds to work and returning in the evening each day.
As a Category C prison, HMP Berwyn similarly has a strong focus on employment and resettlement for those nearing the end of their sentence. Opening in 2017, it is the largest prison in the UK, with 20-30 men released from the prison each week
We were joined by colleagues from the Department for Business, Energy and Industrial Strategy, Her Majesty's Revenue and Customs, Food Standards Agency, Cabinet Office, Office of National Statistics, Ministry of Justice and New Futures Network. Many colleagues hadn't been in a working prison before, so part of the visit aims to demystify life behind the walls.
Each visit featured a tour of the prison and its facilities, with both HMP Sudbury and HMP Berwyn showcasing the support that was on offer - HMP Sudbury with an on-site garden centre selling produce from the market garden tended to by the prisoners, and HMP Berwyn with an Resettlement Hub which brought together all the key services such as Housing, Benefits and Employment support. We saw vibrant Education departments, offering courses in a variety of subjects for those who wanted to improve their prospects via qualifications.
We also held information sessions for groups of prisoners who wanted to hear more about the Civil Service opportunities on offer. Many of our visitors said this was the best part of their day, and we enjoyed hearing from the men about what interested them about working for the Civil Service and their hopes for the future. There was a common theme from many about wanting to make a difference and help people, particularly when they had faced difficult experiences in their lives and felt this could be channelled in a positive way.
It was great to see opportunities we had provided displayed in the employment hub and prisoners asking us about the work and roles to find out more - after all, the Civil Service does have a massive range of different departments and roles!
What we heard
We asked people who were involved in the visits to tell us their thoughts on the day, and wanted to share some snippets..
'Overall, the visit helped me to understand the importance, both individually and at a societal level, of positive and secure employment options for prison leavers. Among the group of prisoners we met it was clear that they had a strong desire to work after release.'
'It will help me to challenge people's conceptions of prisoners and their ability to find gainful employment.'
'I have had so many approach me to thank us and let me know how excited they are about the potential for it. You have all started to spread some hope for the guys, which is so important. Word spreads around the Prison so fast, and with it, this hope also.'
'The fact you all came to visit meant so much. To see and listen to you meant that these were not just 'vacancies' on the wall. It shows that they really are genuine opportunities.'
It's important to build on these visits and maintain our engagement, making sure that the hope becomes a reality for those we met. We have plans to go back to provide support to staff and people in prison - so keep an eye out for a future blog to follow us on our journey to what happens next!
In the meantime if you want to know more about GFiE then take a look at our recent series of 'All About GFiE' blogs and follow us on Twitter @CS_GFiE.
If your a departmental representative who would like to come on one of other visits later this year then drop us an email at GFiE@cabinetoffice.gov.uk.
Tags: diversity, Inclusion, levelling up, Life Chance, Prison Leavers, Social Mobility
The Recruitment Internship Supporting Employment (RISE) scheme launches Civil Service wide.
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Going Forward into Employment is a government wide scheme aimed at providing life chances for people through the provision of opportunities in the civil service.
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Breaking Down Walls 11 August, 2022
The Recruitment Internship Supporting Employment (RISE) scheme launches Civil Service wide. 5 July, 2022
A complete guide to GFiE - Care Leavers 5 July, 2022
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A Chinese remote sensingmission, believed to be a military reconnaissance satellite, lifted off from adesert launch pad on a Long March rocket on Wednesday, state media reported.
The Long March2D rocket blasted off at 0842 GMT (3:42 a.m. EST), or during the afternoonat the Jiuquan launching base near the border of northern China's InnerMongolia and Gansu provinces.
The 135-foot-tall booster'stwo stages, fueled by hydrazine and nitrogen tetroxide, guided the Yaogan 7payload into a sun-synchronous orbit about 400 miles high, according to publictracking data.
The state-run Xinhua newsagency reported the satellite will be used for scientific experiments, landresources surveys, crop yield estimates and disaster response applications.
But the craft is likely anelectro-optical spy satellite to be operated by the Chinese military. Observersbelieve the Yaogan series, which began launching in 2006, is a new fleet ofhigh-resolution optical and radar reconaissance satellites. The new satellitewould be the third Yaogan spacecraft fitted with an optical imager.
Yaogan 7 was built by theChina Academy of Space Technology under the China Aerospace Science andTechnology Corp., according to Xinhua.
Wednesday's launch wasannounced less than a day in advance, typical publicity for Chinese militarylaunches.
The flight was the fifthlaunch of a ChineseLong March rocket this year, and the 69th space launch to reach orbitworldwide in 2009.


text:
Q: Data not coming up on database I followed this tutorial on youtube, for making a social network. I had to register my user. But when I register, nothing shows up on the database.. My files are:
index.php
 <?php include ("./inc/header.inc.php");?>
    <?php
    date_default_timezone_set('UTC');
    $reg = @$_POST['reg'];
    //declaring variables
    $fn = "";//First name
    $ln ="";//Last name
    $un = "";//Username
    $em = "";//Email
    $em2 = "";//Email 2
    $pswd = "";//Password
    $pswd2 = "";//Password 2
    $d = "";//Sign Up Date
$u_check = ""; // Check if username exists
//registration form
$fn = strip_tags(@$_POST['fname']);
$ln = strip_tags(@$_POST['lname']);
$un = strip_tags(@$_POST['username']);
$em = strip_tags(@$_POST['email']);
$em2 = strip_tags(@$_POST['email2']);
$pswd = strip_tags(@$_POST['password']);
$pswd2 = strip_tags(@$_POST['password2']);
$d = date("Y-m-d"); // Year - Month - Day

if ($reg) {
if ($em==$em2) {
// Check if user already exists
$u_check = mysqli_query("SELECT username FROM users WHERE username='$un'");
// Count the amount of rows where username = $un
$check = mysqli_num_rows($u_check);
//Check whether Email already exists in the database
$e_check = mysqli_query("SELECT email FROM users WHERE email='$em'");
//Count the number of rows returned
$email_check = mysqli_num_rows($e_check);
if ($check == 0) {
  if ($email_check == 0) {
//check all of the fields have been filed in
if ($fn&&$ln&&$un&&$em&&$em2&&$pswd&&$pswd2) {
// check that passwords match
if ($pswd==$pswd2) {
// check the maximum length of username/first name/last name does not exceed 25 characters
if (strlen($un)>25||strlen($fn)>25||strlen($ln)>25) {
echo "The maximum limit for username/first name/last name is 25 characters!";
}
else
{
// check the maximum length of password does not exceed 25 characters and is not less than 5 characters
if (strlen($pswd)>30||strlen($pswd)<5) {
echo "Your password must be between 5 and 30 characters long!";
}
else
{
//encrypt password and password 2 using md5 before sending to database
$pswd = md5($pswd);
$pswd2 = md5($pswd2);
$query = mysqli_query("INSERT INTO users VALUES ('','$un','$fn','$ln','$em','$pswd','$d','0','Write something about yourself.','','','no')");
die("
Welcome to findFriends
Login to your account to get started ...");
}
}
}
else {
echo "Your passwords don't match!";
}
}
else
{
echo "Please fill in all of the fields";
}
}
else
{
 echo "Sorry, but it looks like someone has already used that email!";
}
}
else
{
echo "Username already taken ...";
}
}
else {
echo "Your E-mails don't match!";
}
}
?>
<?
//Login Script
if (isset($_POST["user_login"]) && isset($_POST["password_login"])) {
    $user_login = preg_replace('#[^A-Za-z0-9]#i', '', $_POST["user_login"]); // filter everything but numbers and letters
    $password_login = preg_replace('#[^A-Za-z0-9]#i', '', $_POST["password_login"]); // filter everything but numbers and letters
    $md5password_login = md5($password_login);
    $sql = mysqli_query("SELECT id FROM users WHERE username='$user_login' AND password='$md5password_login' AND closed='no' LIMIT 1"); // query the person
    //Check for their existance
    $userCount = mysqli_num_rows($sql); //Count the number of rows returned
    if ($userCount == 1) {
        while($row = mysqli_fetch_array($sql)){
             $id = $row["id"];
    }
         $_SESSION["id"] = $id;
         $_SESSION["user_login"] = $user_login;
         $_SESSION["password_login"] = $password_login;
         exit("<meta http-equiv=\"refresh\" content=\"0\">");
        } else {
        echo 'That information is incorrect, try again';
        exit();
    }
}
?>

Already a Memeber? Login below ...

            <form action="index.php" method="post" name="form1" id="form1">
                <input type="text" size="40" name="user_login" id="user_login" class="auto-clear" title="Username ..." />

                <input type="text" size="40" name="password_login" id="password_login" value="Password ..." />

                <input type="submit" name="button" id="button" value="Login to your account">
            </form>

Sign up Below ...

           <form action="#" method="post">
           <input type="text" size="40" name="fname"  class="auto-clear" title="First Name" value="<? echo $fn; ?>">

           <input type="text" size="40" name="lname" class="auto-clear" title="Last Name" value="<? echo $ln; ?>">

           <input type="text" size="40" name="username" class="auto-clear" title="Username" value="<? echo $un; ?>">

           <input type="text" size="40" name="email" class="auto-clear" title="Email" value="<? echo    $em; ?>">

           <input type="text" size="40" name="email2" class="auto-clear" title="Repeat Email" value="        <? echo $em2; ?>">

           <input type="password" size="40" name="password" value="Password ...">

           <input type="password" size="40" name="password2" value="Password ...">

           <input type="submit" name="reg" value="Sign Up!">
           </form>

connect.inc.php:
<?php

$con = mysqli_connect("localhost", "root", "password") or die("Unable to connect");
mysqli_select_db($con, "socialnetworkdatabase") or die("Could not open the db");

mysqli_close($con);

?>

When I open the database on phpmyadmin, It shows : MySQL returned an empty result set (i.e. zero rows). (Query took 0.0000 seconds.)
I edited the code index.php and it looks like this:
<?php include ("./inc/header.inc.php");?>
<?php include("./inc/connect.inc.php");?>
<?php
$con = mysqli_connect("localhost" ,"root" ,"iamanasian", "theworlddatabase" );
date_default_timezone_set('UTC');
if(isset($_POST['reg'])){
$reg = $_POST['reg'];
//declaring variables
$fn = "";//First name
$ln ="";//Last name
$un = "";//Username
$em = "";//Email
$em2 = "";//Email 2
$pswd = "";//Password
$pswd2 = "";//Password 2
$d = "";//Sign Up Date
$u_check = ""; // Check if username exists
//registration form
$fn = stripslashes($_POST['fname']);
$fn = mysqli_real_escape_string($con,$_POST['fname']);
$ln = stripslashes($_POST['lname']);
$ln = mysqli_real_escape_string($con,$_POST['lname']);
$un = stripslashes($_POST['username']);
$un = mysqli_real_escape_string($con,$_POST['username']);
$em = stripslashes($_POST['email']);
$em = mysqli_real_escape_string($con,$_POST['email']);
$em2 = stripslashes($_POST['email2']);
$em2 = mysqli_real_escape_string($con,$_POST['email2']);
$pswd = stripslashes($_POST['password']);
$pswd = mysqli_real_escape_string($con,$_POST['password']);
$pswd2 = stripslashes($_POST['password2']);
$pswd2 = mysqli_real_escape_string($con,$_POST['password2']);
$d = date("Y-m-d"); // Year - Month - Day
if ($reg) {
if ($em==$em2) {
// Check if user already exists
$u_check = mysqli_query($con, "SELECT username FROM users WHERE username='$un'");
// Count the amount of rows where username = $un
$check = mysqli_num_rows($u_check);
//Check whether Email already exists in the database
$e_check = mysqli_query($con,"SELECT email FROM users WHERE email='$em'");
//Count the number of rows returned
$email_check = mysqli_num_rows($e_check);
if ($check == 0) {
  if ($email_check == 0) {
//check all of the fields have been filed in
if ($fn&&$ln&&$un&&$em&&$em2&&$pswd&&$pswd2) {
// check that passwords match
if ($pswd==$pswd2) {
// check the maximum length of username/first name/last name does not exceed 25 characters
if (strlen($un)>25||strlen($fn)>25||strlen($ln)>25) {
echo "The maximum limit for username/first name/last name is 25 characters!";
}
else
{
// check the maximum length of password does not exceed 25 characters and is not less than 5 characters
if (strlen($pswd)>30||strlen($pswd)<5) {
echo "Your password must be between 5 and 30 characters long!";
}
else
{
//encrypt password and password 2 using md5 before sending to database
$pswd = md5($pswd);
$pswd2 = md5($pswd2);
$query = mysqli_query($con,"INSERT INTO users (id, username, first_name, last_name, email, password, sign_up_date, activated ) VALUES ('','$un','$fn','$ln','$em','$pswd','$d','0','Write something about yourself.','','','no')");
die("
Welcome to findFriends
Login to your account to get started ..." ) or die(mysqli_error($con));
}
}
}
else {
echo "Your passwords don't match!";
}
}
else
{
echo "Please fill in all of the fields";
}
}
else
{
 echo "Sorry, but it looks like someone has already used that email!";
}
}
else
{
echo "Username already taken ...";
}
}
else {
echo "Your E-mails don't match!";
}
}
}
?>

Enter the New World Today!

Sign Up Below!

                                <form action="#" method="POST">
                                    <input type="text" name="fname" size="25" placeholder="First Name">

                                    <input type="text" name="lname" size="25" placeholder="Last Name">

                                    <input type="text" name="username" size="25" placeholder="Username">

                                    <input type="text" name="email" size="25" placeholder="Email">

                                    <input type="text" name="email2" size="25" placeholder="Re-enter Email">

                                    <input type="password" name="password" size="25" placeholder="Password">

                                    <input type="password" name="password2" size="25" placeholder="Re-enter Password">

                                    <input type="submit" name="reg" value="Enter The World!">
                                </form>

<?php include ("./inc/footer.inc.php");?>

And it's still not working

A: Firstly, you are not passing the DB connection variable $con to any of your queries, it's required.
$u_check = mysqli_query("SELECT username FROM users WHERE username='$un'");

$e_check = mysqli_query("SELECT email FROM users WHERE email='$em'");

$query = mysqli_query("INSERT INTO users VALUES ('','$un','$fn','$ln','$em','$pswd','$d','0','Write something about yourself.','','','no')");

$sql = mysqli_query("SELECT id FROM users WHERE username='$user_login' AND password='$md5password_login' AND closed='no' LIMIT 1"); // query the person

Use it like this and do the same for the others:
mysqli_query($con, "SELECT ...
mysqli_query($con, "INSERT ...
Sidenote about your INSERT: It is best to include the actual columns when doing an INSERT.
I.e.: INSERT INTO table (column_x, column_y) VALUES ('value_x', 'value_y')
You're also potentially missing session_start(); since you are using sessions, it is required and to be placed at the top of every file using sessions.
You should be using or die(mysqli_error($con)) to mysqli_query() in order to get the errors, if any.
Also, adding this to the top of your files:
error_reporting(E_ALL);
ini_set('display_errors', 1);

Sidenote: Error reporting should only be done in staging, and never production.
Remove the @ symbols from your POST variables; they suppress potential errors.

Plus, instead of strip_tags() which strips out HTML and PHP tags from a string, use mysqli_real_escape_string() and stripslashes()
I.e.:
$fn = stripslashes($_POST['fname']);
$fn = mysqli_real_escape_string($con,$_POST['fname']);

and do the same for the others.

*

*Your present code is open to SQL injection.

Use mysqli with prepared statements, or PDO with prepared statements.
For password storage, use any of the following and do not use MD5, it is old and considered broken.

*

*CRYPT_BLOWFISH or PHP 5.5's password_hash() function.
For PHP < 5.5 use the password_hash() compatibility pack.

Edit:
Place the following and wrap the braces within the code you wish to execute:
// This is related to your named submit button
if(isset($_POST['reg'])){

// code to execute

}

which is why you're getting an Undefined index: reg notice.

Edit #2:
Place the following and wrap the braces within the code you wish to execute:
<?php include ("./inc/header.inc.php");?>
<?php
date_default_timezone_set('UTC');

if(isset($_POST['reg'])){

$reg = $_POST['reg'];

//declaring variables
$fn = "";//First name
$ln ="";//Last name

// put the rest of your code

    }
}

} // closing brace for if(isset($_POST['reg']))
?>

// rest of your code

Parse error: syntax error, unexpected '{

that is most likely caused by short tags not being set.
change
<?
//Login Script

to
<?php
//Login Script

redpajama_set_name: RedPajamaStackExchange


2002-09-10 Assigned to NOKIA CORPORATION reassignment NOKIA CORPORATION ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: LAITINEN, PASI P., TIMONEN, JUHA T.
Disclosed is a method for performing an inter-system handover. The method triggers the handover of a mobile station (100) from a UTRAN system to a GSM system by sending a HANDOVER FROM UTRAN COMMAND message from the UTRAN system. For the case where a HANDOVER COMMAND is sent transparently with the HANDOVER FROM UTRAN COMMAND, where the HANDOVER COMMAND has an information element set for indicating that ciphering is to be started, the method begins data ciphering immediately after handing over to the GSM system such that the first data frame transmitted by the mobile station in the GSM system is a ciphered data frame. For the case where there is a pre-existing UTRAN system ciphered data connection when the HANDOVER FROM UTRAN COMMAND message is received, the method continues without interruption the use of data ciphering with the data connection after handing over to the GSM system.
This patent application claims priority under 35 U.S.C. § 119(e) from Provisional Patent Application No.: 60/336,510, filed Nov. 15, 2001, the content of which is incorporated by reference herein in its entirety. It is noted that references in this patent application to the "current" specification or version of a specification mean the specification as written at the time of filing the Provisional Patent Application No.: 60/336,5 10.
After having switched to the assigned GSM channel received in the HANDOVER FROM UTRAN COMMAND, the GSM MS sends HANDOVER ACCESS in successive layer  1 frames, just as it typically would have done for a conventional GSM handover initiation.
After layer  1 and 2 connections are successfully established, the GSM MS returns the HANDOVER COMPLETE message. The CN/AS is then able to release the UTRAN resources that were used for the UE 1 in the UTRAN Connected Mode. The CN/AS send an IU RELEASE COMMAND to the UTRAN, after which UTRAN can release all network resources from the RLC, MAC and the physical layer. When the release operation is complete, a IU RELEASE COMPLETE message is sent to the CN/AS.
The use of the teachings of this invention does not require any modifications to be made to the presently specified HANDOVER COMMAND, and thus has a minimal impact on already defined messaging and signaling formats and protocols.
FIG. 4 is a logic flow diagram that is descriptive of a method in accordance with this invention.
Referring to FIG. 3, there is illustrated a simplified block diagram of an embodiment of a wireless communications system  5 that is suitable for practicing this invention. The wireless communications system 5 includes at least one mobile station (MS) 100, also referred to herein as User Equipment (UE). FIG. 3 also shows an exemplary network operator having, for example, a Serving GPRS Support Node (SGSN) 30 for connecting to a telecommunications network, such as a Public Packet Data Network or PDN, at least one base station controller (BSC) 40, and a plurality of base transceiver stations (BTS) 50 that transmit in a forward or downlink direction both physical and logical channels to the mobile station 100 in accordance with a predetermined air interface standard. A reverse or uplink communication path also exists from the mobile station 100 to the network operator, which conveys mobile originated access requests and traffic. The air interface standard can may enable both voice and data traffic, such as data traffic enabling Internet 70 access and web page downloads.
Each BTS  50 supports a cell, such as a serving cell that is currently servicing the MS 100, and at least one neighbor cell. In this exemplary system the BSC 40 and BTSs 50 are assumed to be associated with a UTRAN cellular system, while a second BSC 40A and at least one BTS 50A are assumed to be associated with a GSM cellular system. As such, a handover from the BTS 50 and BSC 40 to the BTS 50A and BSC 40A is an inter-system UTRAN to GSM handover.
The network operator may also include a suitable type of Message Center (MC)  60 that receives and forwards messages for the mobile stations 100. Other types of messaging service may include Supplementary Data Services and possibly Multimedia Messaging Service (MMS), wherein image messages, video messages, audio messages, text messages, executables and the like, and combinations thereof, can be transferred between the network and the mobile station 100.
The mobile station  100 may be a handheld radiotelephone, such as a cellular telephone or a personal communicator. The mobile station 100 could also be contained within a card or module that is connected during use to another device. For example, the mobile station 10 could be contained within a PCMCIA or similar type of card or module that is installed during use within a portable data processor, such as a laptop or notebook computer, or even a computer that is wearable by the user.
The user equipment or mobile station  100 typically includes a data processor such as a microcontrol unit (MCU) 120 having an output coupled to an input of a display 140 and an input coupled to an output of a keyboard or keypad 160. The MCU 120 is assumed to include or be coupled to some type of a memory 130, including a read-only memory (ROM) for storing an operating program, as well as a random access memory (RAM) for temporarily storing required data, scratchpad memory, received packet data, packet data to be transmitted, and the like. A separate, removable SIM (not shown) can be provided as well, the SIM storing, for example, a preferred Public Land Mobile Network (PLMN) list and other subscriber-related information. The ROM is assumed, for the purposes of this invention, to store a program enabling the MCU 120 to execute the software routines, layers and protocols required to implement the inter-system UTRAN to GSM handover, and related methods, in accordance with these teachings.
The ROM of the MS  100 also typically stores a program that provides a suitable user interface (U1), via display 140 and keypad 160. Although not shown, a microphone and speaker are typically provided for enabling the user to conduct voice calls in a conventional manner.
The mobile station  100 also contains a wireless section that includes a digital signal processor (DSP) 180, or equivalent high speed processor or logic, as well as a wireless transceiver that includes a transmitter 200 and a receiver 220, both of which are coupled to an antenna 240 for communication with the network operator. At least one local oscillator (LO) 260, such as a frequency synthesizer, is provided for tuning the transceiver. Data, such as digitized voice and packet data, is transmitted and received to through the antenna 240.
The ROM of the MS  100 is assumed to store a program that enables the MS 100 to receive and process a HANDOVER FROM UTRAN COMMAND message, as defined in 3GPP TS 25.331, v4.2.0, chapter 10.2.15. This message includes a number of information elements (IEs), shown in the following Table, and is used for handover of the MS 100 from UMTS to another system, such as to a GSM system. One or several messages from the other system can be included within the Inter-RAT message information element in this message. Note, for example, that one or a plurality of GSM messages can be embedded into this message, as could cdma2000 messages if handing off to a cdma2000 system. The other-system message or messages are structured and coded according to that systems' specification.
In accordance with the teachings of this invention, an exception is added to the currently specified procedure of 3GPP 04.18 to enable the MS  100 to immediately begin ciphering, when entering the GSM connected mode after handing over from a UTRAN system, without first having to receive a CIPHERING MODE COMMAND.
The wireless network is assumed to include a suitable controller or control element(s) for causing the HANDOVER COMMAND message to be sent transparently to the mobile station  100, with the IE set for indicating "start ciphering".
Referring to FIG. 4, at Step A it is assumed that the MS  100 is in the process of executing an inter-system handover. In the preferred embodiment the inter-system handover is a UTRAN to GSM handover, but in other embodiments of this invention other types of system handovers could occur, so long as the system being handed over to is one that does not permit the use of ciphering without the MS 100 receiving a command that expressly instructs the MS 100 to begin the use of ciphering.
The tests made at Steps B and C can be made by the MCU  120 checking the state of appropriate software flags stored in a register or in the memory 130.
The end result is that ciphering may be started whether or not ciphering was in effect when the inter-system handover was initiated. If ciphering was already in effect, as will typically be the case in the UTRAN system, then ciphering continues without interruption in the GSM system when performing the inter-system handover. If ciphering was not in effect in the UTRAN system, then it can be started immediately in the GSM system after the handover by programming the cipher mode setting IE appropriately in the GSM-specific HANDOVER COMMAND message that is sent transparently to the MS  100.
starting the use of ciphering after handing over to the target system in response to receiving a target system-specific handover command that is sent transparently with the current system-specific handover command, where the target system-specific handover command has an information element set for indicating that ciphering is to be started.
2. A method as in claim 1, wherein the target system is a GSM system.
3. A method as in claim 1, wherein the current system is a UTRAN system.
4. A method as in claim 1, wherein the current system-specific handover command is a HANDOVER FROM UTRAN COMMAND, and where the target system-specific handover command is a HANDOVER COMMAND with a cipher mode settings information element that indicates "start ciphering".
5. A mobile station operable for being handed over from a current system to a target system, comprising an RF transceiver coupled to a data processor, said data processor being responsive to a receipt through said RF transceiver of a current system-specific handover command from the current system for triggering the handover of the mobile station from the current system to the target system, where the target system is one that requires the mobile station to initiate the use of ciphering only in response to receiving a specific command from the target system to initiate the use of ciphering, said data processor operating to immediately start the use of ciphering after handing over to the target system in response to receiving a target system-specific handover command that is sent transparently with the current system-specific handover command, where the target system-specific handover command has an information element set for indicating that ciphering is to be started.
6. A mobile station as in claim 5, wherein the target system is a GSM system.
7. A mobile station as in claim 5, wherein the current system is a UTRAN system.
8. A mobile station as in claim 5, wherein the current system-specific handover command is a HANDOVER FROM UTRAN COMMAND, and where the target system-specific handover command is a HANDOVER COMMAND with a cipher mode settings information element that indicates "start ciphering".
for the case where a HANDOVER COMMAND is sent transparently with the HANDOVER FROM UTRAN COMMAND, where the HANDOVER COMMAND has an information element set for indicating that ciphering is to be started, beginning ciphering immediately after handing over to the GSM system such that a first data frame transmitted by the mobile station in the GSM system is a ciphered data frame.
10. A method as in claim 9, where for the case where there is a pre-existing UTRAN system ciphered connection when the HANDOVER FROM UTRAN COMMAND message is received, continuing without interruption the use of ciphering with the connection after handing over to the GSM system.
mobile station means, responsive to a receipt of the predetermined command, for beginning ciphering immediately after handing over to the GSM system.
12. A system as in claim 11, where for the case where there is a pre-existing UTRAN system ciphered connection when the HANDOVER FROM UTRAN COMMAND message is received, continuing without interruption the use of ciphering with the connection after handing over to the GSM system.
BRPI0717727A2 (en) * 2006-10-31 2013-10-29 Qualcomm Inc Procedure handover inter eNodeB.


BBC One's political debate programme, Question Time, is being hosted from Wrexham on Thursday, 29th January 2015.
David Dimbleby chairs the programme from Wrexham in Wales, where Conservative Culture Secretary Sajid Javid MP, former Labour Secretary of State for Wales Peter Hain MP, Plaid Cymru's Economy Spokesman Rhun ap Iorwerth AM, author and critic Germaine Greer and Telegraph.co.uk blogger Kate Maltby, face questions from the audience.
On Sunday 25th January, as part of the BBC's Holocaust memorial coverage, Nicky Campbell presides over a special edition of The Big Questions live from Salford.
Among those taking part are Rabbi Laura Janner-Klausner, senior rabbi to the Movement for Reform Judaism, chief executive of the Holocaust Memorial Day Trust Olivia Marks-Woldman, human rights campaigner Peter Tatchell, historian Professor Tom Lawson from Northumbria University, Holocaust survivor Iby Knill, Dr Paul Darke from Motion Disabled, journalist Angela Epstein, Dr Yaakov Wise from the Centre for Jewish Studies at Manchester University, Prof Nira Yuval-Davis from the University of East London, Mukesh Kapila, professor of global health and humanitarian affairs at Manchester University and Rabbi Benjy Rickman from King David and Yavneh High School, Manchester.


text:
Q: WiX VS Project, after switching to NuGet package Heat is not running I am  using VS2017, there is installed WiX Toolset 3.11.1. I have add HarvestDirectory Item to the project file.
wixproj file:
 <ItemGroup>
    <HarvestDirectory Include="SourceDir\$(Configuration)">
      <PreprocessorVariable>var.HarvestPath</PreprocessorVariable>
      <DirectoryRefId>INSTALLFOLDER</DirectoryRefId>
      <ComponentGroupName>HeatGenerated</ComponentGroupName>
      <SuppressRootDirectory>true</SuppressRootDirectory>
      <SuppressCom>true</SuppressCom>
      <SuppressRegistry>true</SuppressRegistry>
    </HarvestDirectory>
  </ItemGroup>

We decided to include WiX toolset via NuGet package. After installing the NuGet package the project fails to build and the current reason is that Harvest is not running  anymore.
We are using following nuget package.
Install-Package WiX -Version 3.11.1

to reproduce the error run command above on any 3.11.1 WiX projects and try to build it after.
The error was complaining about missing definition of some attribute. do not have access to dev box at the moment will update later.
What else we need to to to make VS build the project the same way as with intalled WiX Toolset?


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text:
Q: Deep link into iPhone app without custom scheme Is there a way to register an iPhone app with a specific URL without a custom scheme? We would like people to be able to post links like this:
https://www.ourcompany.com/OurProduct/DeepLinks?param1=a&&param2=b
So that if the link is clicked on from an iOS device which has our application installed, our application would be opened and have the URL passed to it, and in the case where it was clicked on from a device that doesn't have our application on it, it would simply open the link in a browser.
Ideally we could detect if the user agent is an iOS device and in that case redirect them to the app store.

A: Have you looked at branch.io yet? https://branch.io
It allows you to setup deep links pretty easily (without much of a headache) and covers your requirement as far as I can tell. i.e it features the ability for you to pass parameters with your link and also specify where the user should be directed in the event that they click the link on a device that does/doesn't have the app installed or has been shared on Facebook or somewhere else.

A: Universal Links are what you want. Unfortunately they only work on iOS 9+, and not everywhere even then (see here for a list). I work with Branch.io and our goal is to help solve this problem!
With Universal Links, when a user opens a link from your website (e.g. https://www.ourcompany.com/OurProduct/DeepLinks?param1=a&&param2=b) in most situations (iOS 9+, your app installed, opening the link in an app that supports Universal Links) your app will immediately launch. You can then parse the incoming URL to determine which in-app content to show.
If the user does not have your app installed, they will simply end up on your website and you can send them to the App Store. The problem here is after they download the app and open it, the trail is lost. You have no way to know which product they were looking at.
Branch can help close this loop. When a user without your app lands on your website, they'll click a button to download the app. That button will 'fingerprint' this user and then after installing, Branch will return data you can use to immediately display the correct content. You can use our pre-configured Smart Banner or build your own solution.
Since you already have a live website on your domain, Branch can't automatically configure the apple-app-site-association entitlement file for you. You'll need to handle this yourself. We have a guide for that here.


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Nouvelle Blogger: That's cold, man, cold. . . .
If you are one of the few people who has read my blog, you'll remember how creative my kids can get with the insults. Hey, how many of you have been called "an artichoke?" That lovely was uttered by my son in a fit of pique. Now, his sister has anted up. The other day, the two were (gasp! shock!) bickering and, voila, she shrieked at my son, "You're an ice cube!" Fits of giggles ensued, of course.
Ah, there's a whole world of ingestible insults out there, just waiting to be used. The possibilities are practically limitless. I can't wait for round three.
Dude! I just tracked your site down through LA Moms (i'm a blogger for L.A.M., too) and you're awesome. i actually read most of your postings back to april. that's a staggering accomplishment, considering my a.d.d. leanings. I'm blogrollin' ya.
Thanks Lucia--And may I just say I love you? No, really, I LOOOVE you and your blog. After reading your writing, I suspect we're on the same wavelength.
Thanks for reading, and may I suggest, if you're not feeling too ADD to go back to some of my 2006-07 posts. Frankly, I find many of them better than my more recent ones. . .And, of course, I'm desperate for readership!
Can't wait to meet you one of these days soon.


text:
As-Sab Bijar (arab. السبع بيار) – miejscowość w Syrii w muhafazie Damaszek. W 2004 roku liczyła 395 mieszkańców.

Przypisy

Miejscowości w muhafazie Damaszek

redpajama_set_name: RedPajamaWikipedia


You Couldn't Make it Up !
So have introduced Shakespeare's 'Romeo and Juliet' to Year 9 by watching 'Shakespeare in Love'. At the end Gwyneth Paltrow is shipped off to Virginia by her horrible husband, Colin Firth.
What other people do to you is not your fault.
We concluded a lively discussion talking about how victims of abuse are made to feel 'it is their fault', and any therapy to repair the emotional damage usually involves convincing the victim that 'it was not their fault' – think 'Good WIll Hunting'. Ergo, we concluded, when people are 'good' to you' it is not your 'fault' either. They, it could be argued, are doing 'good' not because you deserve it but because they are good people.
All pretty standard GCSE Literature.
But, and you just couldn't make it up … just as this was being said The Head turned up to my classroom and with him marched in the new Bishop, two priests and four nuns !!!! It was like – and here, Ladies and gentlemen, are a group of 'good people' I made earlier ….
This entry was first posted on January 15 2013.
This entry was posted in You Couldn't Make it Up and tagged 'Romeo and Juliet', Queen Elizabeth I, Shakespeare, teaching GCSE English Literature on April 6, 2013 by themushroomdiaries.
"When nothing goes right, go left"


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text:
\section{Introduction}\label{sec:introduction}
Nematic shells are (mostly undeformable) colloidal particles coated with a thin film of a nematic liquid crystal,
whose rod-like molecules are subject to a \emph{degenerate planar} anchoring exerted by the solid substrate.
This anchoring allows the molecules to rotate freely on the local tangent plane, but makes
it hard for them to flip out of that plane.

We still do not know whether the futuristic dream of the all-optical computer will ever come true.
For sure, Ozin's ``colloidal photonic crystal metropolis'' \cite{arsenault:towards} is a natural
technological evolution of Nelson's ``colloidal chemistry'' \cite{nelson:toward},
which envisioned replacing atoms with nematic decorated colloids.

This may explain the recent surge of experimental studies on nematic shells and their aim to produce different stable configurations with a \emph{tunable} number of \emph{defects}, which in Nelson's paradigm would replace the atomic valence.\footnote{Nelson's original contribution \cite{nelson:toward} showed that a spherical nematic shell can become a \emph{tetravalent} colloid. A similar construction is valid for the wetting layer that surrounds spherical colloids dispersed in a liquid crystal at a temperature above the nematic-isotropic phase transition \cite{huber:tetravalent}.} See, for example, \cite{fernandez-nieves:novel,lopez-leon:frustrated,lopez-leon:nematic-smectic,liang:nematic-smectic,sec:defect,lopez-leon:defect,noh:taming,noh:influence,koning:spherical}, only to mention a few representative experimental works.

A number of simulation studies are also available, which bridge experiment and theory \cite{skacej:controlling,shin:topological,bates:defects,dhakal:nematic,li:topological,li:defect-free,mbanga:simulating,wand:monte}.

Apart from the director elastic theories that will be discussed more closely  below, as they form the background of our contribution, theories from different lineages have also been proposed; they can be broadly classified in two groups: the group of phenomenological theories formulated in terms of an appropriate two-dimensional version of de Gennes' order tensor, especially suitable to the description of defects and their controllability \cite{kralj2011curvature,napoli2013curvature,jesenek:defect,mesarec:effective}, and the group of molecular theories phrased within Onsager's excluded volume density functional formalism, duly extended to surfaces, which are also interesting for the light they shed  on the interplay between defects and density depletion \cite{zhang:onsager,zhang:solution,liang:rigid,ye:nematic}.

Luckily, there are excellent reviews that might safely navigate the reader into the vast literature on nematic shells \cite{lopez-leon:drops,lagerwall:new,mirantsev:defect,serra2016curvature,urbanski2017liquid}.

Our contribution in this paper is placed within the director elastic theories of nematic shells,
at the heart of the debate on what should be the most appropriate form of the energy.
Two groups of theories have been proposed so far: in one group, pioneered by \cite{nelson:fluctuations},
the only molecular distortions that matter are measured within the surface's metric;
in the other group, pioneered by \cite{helfrich:intrinsic},  the way molecules couple
with the curvature of the surface should also be accounted for in the energy balance.
Theories in the former group are conventionally called \emph{intrinsic}, whereas theories
in the latter group are called \emph{extrinsic}. An interesting third way was opened by Selinger
and his co-workers \cite{nguyen:nematic}, who offered reasons to subject to
\emph{unequal} weights these different sources of energy. Much in the spirit of this latter view,
here we start from an elementary, direct way to identify one  component of these energies,
which we shall call the \emph{fossil} energy because it is inextricably interwoven with the
background surface. Not surprisingly perhaps, we find in an appropriate rendition of
Levi-Civita's parallel transport the elementary geometric tool that extracts the
fossil energy out of any proposed phenomenological energy.

This tool is more general than the application of it we make to the analogue of Frank's elastic energy.
When written in the frame of principal directions of curvatures,
the fossil energy reveals itself as a \emph{curvature potential} for the torque acting
on molecules and responsible for their fossil preferred alignment.
The clarity that may be gained with this approach has an interesting consequence:
we predict that at hyperbolic points of the shell, and only there,
the fossil preferred alignment of molecules may fail to be a principal direction of curvature. By symmetry, whenever this applies, the fossil energy becomes \emph{bistable}.
The preferred alignments
depend on the elastic anisotropy, possibly only the anisotropy of the fossil energy,
in the dissociated approach of \cite{nguyen:nematic}.

The structure of the paper is the following. In Sec.~\ref{sec:ground_state},
we present our elementary way to construct the fossil energy, starting from the notion
of parallel transport, which is recalled in a simple, but rigorous language.
A form of elastic energy is considered in Sec.~\ref{sec:energies}, where we derive
formally the associated  fossil energy and represent it so as to make explicit its
nature of curvature potential for the molecular torque. In Sec.~\ref{sec:curvature_potential},
we study the general equilibria for the curvature potential; their multiplicity and stability
is analysed at hyperbolic points of a shell, where these properties become more intriguing.
Sec.~\ref{sec:conclusions} draws the conclusions of this paper.
There we also discuss in some detail where our study stands in the context of many other similar ones.
We also outline problems that in our view could profit from the insights afforded by our approach.
The paper is closed by an appendix, somewhat more technical in character.

\section{Parallel transport and local ground state}\label{sec:ground_state}
In three-dimensional space, the \emph{ground state} of a nematic liquid crystal, that is, the configuration for the director $\bm{n}$ that realises the minimum distortion energy, is easily identified: this is any configuration in which $\bm{n}$ is uniform in space, no matter in which orientation, and so $\nabla\bm{n}\equiv\mathbf{0}$. As is well known from the classical theory of Frank (see, for example \cite[Chap.~3]{virga:variational}), the distortional cost associated with the ground state in three dimensional space is \emph{everywhere} zero.

On a nematic shell, here represented by a smooth surface $\mathscr{S}$ with outer unit normal $\bm{\nu}$,
on which molecules are constrained to lie subject to the only constraint that
\begin{equation}\label{eq:orthogonality_constraint}
\bm{n}\cdot\bm{\nu}\equiv0,
\end{equation}
the ground state is not identified with equal ease. But it is even more important to have a clear concept of
the ground state,
as we shall see that it brings along a distortional cost, not uniformly distributed on $\mathscr{S}$,
reminiscent of a \emph{fossil} energy, inextricable from its geometric background.

We start with the nearly obvious remark that $\nabla_\mathrm{{\!s}}\bm{n}\equiv\bm{0}$, where
$\nabla_\mathrm{{\!s}}$ denotes the surface gradient,
cannot be the appropriate definition for the ground state of nematic shells,
despite the fact that $\nabla_\mathrm{{\!s}}\bm{n}$ appropriately measures the degree of
distortion of molecules interacting in three-dimensional space.
The reason for this follows at once from \eqref{eq:orthogonality_constraint},
under the assumption that both $\bm{n}$ and $\bm{\nu}$ are smooth unit vector
fields on $\mathscr{S}$. Differentiating both sides of \eqref{eq:orthogonality_constraint}, we see that
\begin{eqnarray}\label{eq:why_not}
(\nabla_\mathrm{{\!s}}\bm{n})^\mathsf{T}\bm{\nu}+(\nabla_\mathrm{{\!s}}\bm{\nu})\bm{n}\equiv\bm{0},
\end{eqnarray}
where $\sgrad\normal$ is the (symmetric) curvature tensor of $\mathscr{S}$.
A local distortion state with $\nabla_\mathrm{{\!s}}\bm{n}=\bm{0}$ would also require $(\sgrad\normal)\bm{n}=\bm{0}$,
which in turns wants $\bm{n}$ to be oriented along a direction of zero curvature of $\mathscr{S}$,
which may in general fail to exist. On the other hand, at the points on $\mathscr{S}$ where $\sgrad\n\neq\bm{0}$,
the two-dimensional layer covering $\mathscr{S}$ carries a distortional energy, if seen from the embedding three-dimensional space, because there the molecules with average orientation  $\bm{n}$ necessarily fail to be parallel to one another, disagreeing  with the three-dimensional ground state (as there cannot be any director field $\bm{n}^\ast$ in space that agrees with $\bm{n}$ on $\mathscr{S}$ and has $\nabla\bm{n}^\ast=\bm{0}$).

Thus, we are presented with a clear task: given $\bm{n}$ at a point on $\mathscr{S}$, identify geometrically the least distortion $\sgrad\n^{(0)}$ that can locally be compatible with this assignment (and may well fail to be a gradient). We shall see shortly below that a natural geometric tool to accomplish this task is provided by Levi-Civita's parallel transport. The energy associated with $\sgrad\n^{(0)}$, which by necessity incorporates appropriate local measures of the curvature of $\mathscr{S}$, will be called the \emph{fossil} energy. This is the energy that according to our theory is stored in the \emph{local} ground state, which may be different at different points on $\mathscr{S}$.

\subsection{Parallel transport}\label{sec:parallel_transport}
Little new, if anything, can be said about a subject that has been extensively studied for over a century.
Already Cartan in his memoir \cite{cartan:geometrie} gave a lucid presentation of the role played by
Levi-Civita's notion of parallelism in Riemannian geometry \cite{levi-civita:nozione}; there the distinction
between what are now often called intrinsic and extrinsic properties is admirably illuminated (though in a different,
more descriptive language).
Here we follow closely the kinematic interpretation given in \cite{rosso2012parallel} to Levi-Civita's
parallel transport on surfaces. We build on much earlier work of Persico~\cite{persico:realizzazione},
lately also revived in \cite{pfister:spatial}.

Let two perpendicular unit vectors, $\bm{u}_1$ and $\bm{u}_2:=\bm{\nu}\times\bm{u}_1$,
tangent to $\mathscr{S}$ be prescribed along a smooth curve $\mathscr{C}$ through a given point $p\in\mathscr{S}$.
The triad $(\bm{u}_1,\bm{u}_2,\bm{\nu})$ constitutes a \emph{movable} frame along $\mathscr{C}$ constrained to have
an axis aligned with the surface's normal. The movable frame is parametrized in the arc length $s$ of $\mathscr{C}$
and differentiation with respect to $s$ is denoted by a prime $'$.
Letting $\bm{t}$ be the unit tangent to $\mathscr{C}$, $\bm{\nu}'$ is delivered by
\begin{equation}\label{eq:nu_prime}
\bm{\nu}'=(\sgrad\normal)\bm{t},
\end{equation}
but both $\bm{u}_1'$ and $\bm{u}_2'$ have an extra degree of freedom. Elementary kinematics shows that
\begin{equation}\label{eq:frame_evolution}
\bm{u}_1'=\bm{\omega}\times\bm{u}_1,\quad\bm{u}_2'=\bm{\omega}\times\bm{u}_2,\quad\bm{\nu}'=\bm{\omega}\times\bm{\nu},
\end{equation}
where $\bm{\omega}$ is the \emph{spin vector} defined along $\mathscr{C}$. Only the component $\bm{\omega}_\parallel$ of the spin vector tangent to $\mathscr{S}$ is prescribed by \eqref{eq:nu_prime} and the third of \eqref{eq:frame_evolution}, whereas the component parallel to $\bm{\nu}$ remains completely free. Different choices of $\bm{\omega}\cdot\bm{\nu}$ impart different spins to the evolution of the movable frame along $\mathscr{C}$. Clearly, the evolution corresponding to the least admissible spin is obtained for $\bm{\omega}\cdot\bm{\nu}=0$, which is said to characterize the \emph{parallel transport} of the frame. The least spin vector $\bm{\omega}_\parallel$ is easily obtained from \eqref{eq:frame_evolution} as
\begin{equation}\label{eq:spin_parallel}
\bm{\omega}_\parallel=\bm{\nu}\times\bm{\nu}'.
\end{equation}
Combining \eqref{eq:nu_prime} and \eqref{eq:spin_parallel}, we easily arrive at
\begin{equation}\label{eq:spin_parallel_final}
\bm{\omega}_\parallel=\mathbf{N}(\sgrad\normal)\bm{t},
\end{equation}
where $\mathbf{N}$ is the skew-symmetric tensor associated with the normal $\bm{\nu}$, so that
\begin{equation}\label{eq:skew_normal}
\mathbf{N}\bm{v}=\bm{\nu}\times\bm{v},
\end{equation}
for all vectors $\bm{v}$.
More generally, a vector $\bm{u}$ tangent to $\mathscr{S}$ at $p$ is said to be parallel transported along $\mathscr{C}$ if it is rigidly conveyed by a parallel transported frame, which amounts to prescribe for it the following evolution law,
\begin{equation}\label{eq:parallel_trasport_evolution}
\bm{u}'=\bm{\omega}_\parallel\times\bm{u}.
\end{equation}

Let now $\bm{n}$ be prescribed at $p$. We want to construct a distortion field $\sgrad\n^{(0)}$, if it exists,
that delivers in the vicinity of $p$ on $\mathscr{S}$ the same vector delivered by parallel transporting
$\bm{n}$ along all possible curves emanating from $p$. This requires that the following equation
\begin{equation}\label{eq:minimu_distortion_equation}
(\sgrad\n^{(0)})\bm{t}=\bm{\omega}_\parallel\times\bm{n}=-\bm{n}\times\mathbf{N}(\sgrad\normal)\bm{t}
\end{equation}
be valid for all unit tangent vectors $\bm{t}$. Since, by \eqref{eq:skew_normal},
\begin{equation}\label{eq:minimum_distortion_computation}
\bm{n}\times\mathbf{N}\bm{v}=(\bm{\nu}\otimes\bm{n})\bm{v},
\end{equation}
for all vectors $\bm{v}$, \eqref{eq:minimu_distortion_equation} is satisfied by
\begin{equation}\label{eq:minimum_distortion_solution}
\sgrad\n^{(0)}=-\bm{\nu}\otimes(\sgrad\normal)\bm{n}.
\end{equation}
This is the \emph{minimal} distortion induced locally by the curvature of $\mathscr{S}$ that would agree
with the orientation $\bm{n}$ prescribed at a given point $p\in\mathscr{S}$. The reader
should not be misled by our (perhaps infelicitous) notation for the measure of minimal distortion,
as in general this is \emph{not} the surface gradient of a field.
As recalled in Appendix~\ref{sec:app_developable}, for this to be the case,
the surface $\mathscr{S}$ must possess zero Gaussian curvature $K$, that is,
it must be \emph{developable}. For a non-developable surface,
$\sgrad\n^{(0)}$ in \eqref{eq:minimum_distortion_solution} merely represents a measure of local distortion,
which cannot be globally integrated.

Suppose that $\sgrad\n$ is a \emph{genuine} measure of distortion, that is, the surface gradient of a local field $\bm{n}$, not necessarily in the form \eqref{eq:minimum_distortion_solution}. We define
\begin{equation}\label{eq:covariant_derivative_definition}
\operatorname{D}\!\bm{n}:=\sgrad\n-\sgrad\n^{(0)}=\sgrad\n+\bm{\nu}\otimes(\sgrad\normal)\bm{n}
\end{equation}
and attribute to it the meaning of an \emph{extra} distortion, one that comes on top of the minimal distortion compatible with the value assigned to the field $\bm{n}$ at $p\in\mathscr{S}$. $\operatorname{D}\!\bm{n}$ is better known as the \emph{covariant} gradient of $\bm{n}$ on $\mathscr{S}$, as it can be given an \emph{intrinsic} representation, independent of the way the surface is embedded in space.

To see this, we consider a frame $(\bm{u}_1,\bm{u}_2,\bm{\nu})$ that is locally parallel transported on $\mathscr{S}$. In this frame we represent $\bm{n}$ as
\begin{equation}\label{eq:n_alpha_representation}
\bm{n}=\cos\alpha\,\bm{u}_1+\sin\alpha\,\bm{u}_2,
\end{equation}
so that
\begin{equation}\label{eq:grad_n_alpha_representation}
\sgrad\n=\bm{n}_\perp\otimes\nabla_\mathrm{{\!s}}\alpha+\cos\alpha\nabla_\mathrm{{\!s}}\bm{u}_1+\sin\alpha\nabla_\mathrm{{\!s}}\bm{u}_2,
\end{equation}
where $\bm{n}_\perp:=\mathbf{N}\bm{n}$. By applying \eqref{eq:minimum_distortion_solution} to both $\bm{u}_1$ and $\bm{u}_2$, we easily rewrite \eqref{eq:grad_n_alpha_representation} as
\begin{equation}\label{eq:grad_n_alpha_rewritten}
\sgrad\n=\bm{n}_\perp\!\otimes\nabla_\mathrm{{\!s}}\alpha-\bm{\nu}\otimes(\sgrad\normal)\bm{n},
\end{equation}
which combined with \eqref{eq:covariant_derivative_definition} shows that
\begin{equation}\label{eq:covariant_derivative_representation}
\operatorname{D}\!\bm{n}=\bm{n}_\perp\!\otimes\nabla_\mathrm{{\!s}}\alpha.
\end{equation}
The latter two equations have an interesting consequence: they show that $\operatorname{D}\!\bm{n}$ and $\sgrad\n^{(0)}$ decompose $\sgrad\n$
into two orthogonal components. Thus, $\operatorname{D}\!\bm{n}$ can alternatively be obtained by projecting $\sgrad\n$ on
the plane tangent to $\mathscr{S}$,
\begin{equation}\label{eq:covariant_derivative_projection}
\operatorname{D}\!\bm{n}=(\mathbf{I}-\bm{\nu}\otimes\bm{\nu})\sgrad\n.
\end{equation}

Assuming that the  distortion energy density $W$ is arising from the way molecules interact (and, as it were, ``see'' each other)
in the ambient three-dimensional space, an assumption corroborated by the spin interaction model studied in \cite{selinger:monte},
we are led to consider $W$ as a function of the form $W(\bm{n},\sgrad\n)$. The least distortion in \eqref{eq:minimum_distortion_solution}
then identifies the \emph{local} ground state of the system, solely dictated by the orientation of $\bm{n}$ on $\mathscr{S}$ and the
curvature tensor of $\mathscr{S}$.
Computing $W$ on the local ground state we identify the \emph{fossil} energy density, an energy that
can be attributed to the curvature of the shell and cannot be extracted from it, as long as $\mathscr{S}$
is thought of as undeformable. In general, the fossil energy is given by
\begin{equation}\label{eq:fossil_energy_definition}
W_0(\bm{n},\sgrad\normal):=W(\bm{n},\sgrad\n^{(0)}).
\end{equation}
We shall see in the following section that for the most general $W$ quadratic in $\sgrad\n$ the orthogonal decomposition in \eqref{eq:grad_n_alpha_rewritten}, appropriately extended, will be reflected in splitting $W$ into the sum of $W_0$ and an extra energy depending only on the intrinsic measure of distortion.

\section{Energetics}\label{sec:energies}
We follow \cite{chen:nematic,napoli2012surface} to obtain the two-dimensional energy density $W$ by
a dimension reduction of Frank's classical three-dimensional formula, viewing the shell as a
\emph{thin} layer coating $\mathscr{S}$. Denoting by $K_i$, $i=1,\dots,4$, Frank's elastic constants
(see Chap.~3 of \cite{virga:variational}), and by $h$ the layer's thickness (assumed to be much
smaller than the smallest radius of curvature of $\mathscr{S}$), we easily arrive at
\begin{equation}\label{eq:W_dimension_reduction}
\begin{split}
W(\bm{n},\sgrad\n)&=
\frac{1}{2}k_1    (\diver_\mathrm{{\!s}}\!\vctr{n})^2 +
\frac{1}{2}k_2    (\vctr{n}\cdot\curl_\mathrm{{s}}\!\vctr{n})^2\\
&+
\frac{1}{2}k_3    (\vctr{n}\times\curl_\mathrm{{s}}\!\vctr{n})^2,
\end{split}
\end{equation}
where $k_i=K_ih$, $i=1,2,3$, and no saddle-splay energy is inherited from the parent three-dimensional
energy (see \cite{napoli2012surface} for further details). By Ericksen's inequalities for the constants
$K_i$ \cite{ericksen:inequalities}, all three reduced elastic constants in \eqref{eq:W_dimension_reduction}
must be non-negative, $k_i\geq0$. $W$ represents the elastic energy per unit area of $\mathscr{S}$, and so all
the $k_i$ have the physical dimensions of an energy (whereas all the $K_i$ have the physical dimensions of a force).

\subsection{Fossil energy and curvature potential}\label{sec:fossil_energy}
Here, primarily, we want  to isolate the fossil energy part of $W$.
To this end, we first need to generalize the gradient decomposition in \eqref{eq:grad_n_alpha_rewritten}
to the case where the movable frame $(\bm{u}_1,\bm{u}_2,\bm{\nu})$ used to parametrize $\bm{n}$ in \eqref{eq:n_alpha_representation}
is not locally parallel transported.

This is achieved in a standard way by use of a \emph{spin connection} $\bm{w}$, defined as in \cite{kamien:geometry}
by\footnote{The spin connection $\bm{w}$ is nothing but the \emph{geometric vector potential} introduced by
Nelson and Peliti~\cite{nelson:fluctuations}; see also \cite{vitelli:defect} and \cite{nguyen:nematic}
for the use of this latter name for $\bm{w}$.}
\begin{equation}\label{eq:spin_connector_definition}
\bm{w}:=(\nabla_\mathrm{{\!s}}\bm{u}_2)^\mathsf{T}\bm{u}_1=-(\nabla_\mathrm{{\!s}}\bm{u}_1)^\mathsf{T}\bm{u}_2.
\end{equation}
It follows from using the orthonormality conditions linking the vectors of the movable frame that
\begin{subequations}\label{eq:frame_gradients}
\begin{align}
\nabla_\mathrm{{\!s}}\bm{u}_1&=-\bm{u}_2\otimes\bm{w}-\bm{\nu}\otimes(\sgrad\normal)\bm{u}_1,\\
\nabla_\mathrm{{\!s}}\bm{u}_2&=\bm{u}_1\otimes\bm{w}-\bm{\nu}\otimes(\sgrad\normal)\bm{u}_2,
\end{align}
\end{subequations}
which clearly show the mismatch with the law of parallel transport \eqref{eq:minimum_distortion_solution}.
Combining \eqref{eq:frame_gradients} and \eqref{eq:grad_n_alpha_representation}, we extend \eqref{eq:grad_n_alpha_rewritten} as follows,
\begin{equation}
\sgrad\n=\bm{n}_\perp\otimes(\nabla_\mathrm{{\!s}}\alpha-\bm{w})-\bm{\nu}\otimes(\sgrad\normal)\bm{n},
\end{equation}
from which we readily arrive at
\begin{subequations}\label{eq:div_curl_connector}
\begin{gather}
\diver_\mathrm{{\!s}}\!\bm{n}=\bm{n}_\perp\cdot(\nabla_\mathrm{{\!s}}\alpha-\bm{w}),\\
\curl_\mathrm{{s}}\!\bm{n}=(\nabla_\mathrm{{\!s}}\alpha-\bm{w})\times\bm{n}_\perp+\bm{\nu}\times(\sgrad\normal)\bm{n}.
\end{gather}
\end{subequations}
From the latter equation, which for a unit vector field $\bm{n}$ tangent to $\mathscr{S}$ is an
orthogonal decomposition of $\curl_\mathrm{{s}}\!\bm{n}$, it easily follows that
\begin{subequations}\label{eq:curl_computations}
\begin{gather}
|\bm{n}\cdot\curl_\mathrm{{s}}\!\bm{n}|=|(\sgrad\normal)\bm{n}\times\bm{n}|,\\
|\bm{n}\times\curl_\mathrm{{s}}\!\bm{n}|^2=[(\nabla_\mathrm{{\!s}}\alpha-\bm{w})\cdot\bm{n}]^2+(\bm{n}\cdot(\sgrad\normal)\bm{n})^2.
\end{gather}
\end{subequations}
Use of \eqref{eq:curl_computations} and
\begin{equation}
(\bm{n}\cdot(\sgrad\normal)\bm{n})^2=|(\sgrad\normal)\bm{n}|^2-|(\sgrad\normal)\bm{n}\times\bm{n}|^2
\end{equation}
in \eqref{eq:W_dimension_reduction} delivers the following  general representation formula for $W$,
\begin{align}\label{eq:W_general_representation_formula}
W&=\frac12k_1[\bm{n}_\perp\cdot(\nabla_\mathrm{{\!s}}\alpha-\bm{w})]^2+\frac12(k_2-k_3)|(\sgrad\normal)\bm{n}\times\bm{n}|^2\nonumber\\&+\frac12k_3\{[\bm{n}\cdot(\nabla_\mathrm{{\!s}}\alpha-\bm{w})]^2+|(\sgrad\normal)\bm{n}|^2\},
\end{align}
which neatly reveals the \emph{fossil} energy as
\begin{equation}\label{eq:fossil_energy_revealed}
\begin{split}
W_0(\bm{n},\sgrad\normal)&
=\frac12(k_2-k_3)|(\sgrad\normal)\bm{n}\times\bm{n}|^2\\&
+\frac12k_3|(\sgrad\normal)\bm{n}|^2.
\end{split}
\end{equation}
For definiteness, we shall call the difference
\begin{equation}\label{eq:distortional_energy}
W-W_0=\frac12k_1[\bm{n}_\perp\cdot(\nabla_\mathrm{{\!s}}\alpha-\bm{w})]^2+\frac12k_3[\bm{n}\cdot(\nabla_\mathrm{{\!s}}\alpha-\bm{w})]^2
\end{equation}
the \emph{distortional} energy.

As already remarked in \cite{napoli2012surface}, in contrast with the planar case, a twist contribution arises if $\mathscr{S}$ is generally curved. However, in our theory such a twist energy is always a part of the fossil energy. Moreover, and more importantly, as clearly indicated by \eqref{eq:fossil_energy_revealed}, for $k_2\geq k_3$ the fossil energy $W_0$ is minimized when $\bm{n}$ is oriented along the eigenvector of $\sgrad\normal$ with the least absolute eigenvalue, but for $k_2<k_3$ this is no longer the trivial minimizing choice, and new interesting scenarios may arise, as will be shown below.

Having established \eqref{eq:W_general_representation_formula} for a general spin connection $\bm{w}$, we can now afford representing $\bm{n}$ in a local frame more geometrically telling, which is not necessarily parallel transported on the surface. This is the frame $(\bm{e}_1,\bm{e}_2,\bm{\nu})$, where $(\bm{e}_1,\bm{e}_2)$ are the principal directions of curvature, with $(\kappa_1,\kappa_2)$ the
corresponding principal curvatures. Without loss of generality, we can
assume that $\kappa_1^2\le\kappa_2^2$. So
\begin{subequations}\label{eq:n_parametrization}
\begin{align}
\nabla_\mathrm{{\!s}}\vctr{\nu}&=\kappa_1\vctr{e}_1\otimes\vctr{e}_1+\kappa_2\vctr{e}_2\otimes\vctr{e}_2,\\
\vctr{\nu}&=\vctr{e}_1\times\vctr{e}_2,\\
\vctr{n}&=\cos\theta\vctr{e}_1+\sin\theta\vctr{e}_2,
\end{align}
\end{subequations}
where $\theta$ is the angle that the director $\vctr{n}$ makes with
$\vctr{e}_1$. Making use of \eqref{eq:n_parametrization} in \eqref{eq:fossil_energy_revealed}, we thus convert the fossil energy density $W_0$ into a function of $\theta$, parametrized in the principal curvatures,
\begin{equation}\label{eq:curvature_potential}
\begin{split}
W_0&=\frac18(k_2-k_3)(\kappa_1-\kappa_2)^2\sin^2 2\theta\\
&+\frac{1}{4}k_3\left[
\kappa_1^2+\kappa_2^2+(\kappa_1^2-\kappa_2^2)\cos 2\theta
\right].
\end{split}
\end{equation}
A representation equivalent to \eqref{eq:curvature_potential} was also obtained in \cite{segatti:equilibrium} (see their equation (5), then reiterated in (6.19) of \cite{segatti2016analysis}), though not endowed with the same meaning.

In this form, $W_0$ appears like the potential energy for a torque on the nematic director, a torque that is not imparted by external agents,
such as a field, but which
stems
from the curvature of the shell. Having this interpretation in mind,
we shall refer to $W_0$ in \eqref{eq:curvature_potential} as the \emph{curvature potential}.
Its equilibria are the orientations that make the local fossil energy achieve a critical value
(while the corresponding torque vanishes); its minimizer is the orientation that the local
fossil energy would enforce, \emph{if} it were the only energy at play.

It was first shown in \cite{napoli2012extrinsic} how the fossil energy may be minimized when $\bm{n}$ is aligned
along the direction of minimum absolute principal curvature ($\bm{e}_1$, in our parametrization).
\begin{comment}
and a wide consensus has grown around the opinion (expressed in a different language) that this could be the generic role played by the fossil energy  \cite{serra2016curvature}.
\end{comment}
We shall see  in the following section that this is not generally the case:
at hyperbolic points of a shell, the curvature potential can induce a preferred direction
coincident with neither of the principal directions of curvature.

A new interesting perspective on the energetics of \emph{flexible} nematic shells was recently
opened up by Selinger and his co-workers \cite{nguyen:nematic}. Building upon the decomposition of
$\sgrad\n$ obtained from \eqref{eq:covariant_derivative_definition},
it was proposed in \cite{nguyen:nematic} to write $W(\bm{n},\sgrad\n)$ as the sum of two components,
weighted by \emph{different} elastic constants. One component, the \emph{intrinsic} energy $W(\bm{n},\operatorname{D}\!\bm{n})$,
would be associated with the variations of $\bm{n}$ \emph{inside} the surface, whereas the other component,
our fossil energy $W(\bm{n},\sgrad\n^{(0)})$, would be associated with the variations of $\bm{n}$, as it were,
\emph{outside} the surface. Such a dual treatment of the energy rests on the physical presumption
that different costs should assigned to distortions of  different origins. In our theory,
we have set equal the elastic constants of both energies, for simplicity. Adopting the point of
view of \cite{nguyen:nematic} would simply amount to divorce the elastic constants of the fossil
energy from those of the intrinsic energy, resulting in more freedom of action for the curvature
potential $W_0$ in \eqref{eq:curvature_potential}.

\begin{figure}
\includegraphics[width=0.48\textwidth]{solutionPlane.eps}
\caption{\label{fig:solutionPlane}%
Value of $\xi$ defined by \eqref{eq:xi} as a function of
the ratio $k=k_2/k_3$ of the elastic constants and
the ratio $\kappa=\kappa_1/\kappa_2$ of the principal
curvatures. In the lower part of the $k$-$\kappa$ plane
where $|\xi|\le 1$, a stationary director orientation exists
that is not along a principal direction of curvature.
This orientation is stable in the region under the solid
line where $0<\xi<1$, and it is unstable in the region
under the dashed line where $-1<\xi<0$.}
\end{figure}

\section{Curvature potential equilibria}\label{sec:curvature_potential}
The curvature potential \eqref{eq:curvature_potential} depends on the angle $\theta$ that the director makes with
the principal direction $\vctr{e}_1$ corresponding to the principal
curvature $\kappa_1$ with, as chosen above, $\kappa_1^2\le\kappa_2^2$.
To find the director orientations favoured by the curvature potential, we consider
\begin{equation}\label{eq:dW0dtheta}
\frac{dW_0}{d\theta}=
\frac{1}{2}\sin 2\theta\left[
k_3(\kappa_2^2-\kappa_1^2)+\cos 2\theta(k_2-k_3)(\kappa_1-\kappa_2)^2
\right]
\end{equation}
and
\begin{equation}
\frac{d^2W_0}{d\theta^2}=
\cos 2\theta\,k_3(\kappa_2^2-\kappa_1^2)
+\cos 4\theta(k_2-k_3)(\kappa_1-\kappa_2)^2.
\end{equation}

\subsection{Equilibrium multiplicity and bifurcation}\label{sec:multiplicity_bifurcation}
We start by discussing \eqref{eq:dW0dtheta}. First of all,
at umbilical points, where $\kappa_1=\kappa_2$, the derivative
is identically zero and there is no preferred direction for
the director.

Let us now assume that $\kappa_1\ne\kappa_2$. In the case of
equal elastic constants, $k_2=k_3$, the only zeros
of the derivative are found when $\sin 2\theta=0$,
that is, at $0$, $\pi$, and $\pm\pi/2$, which means
that the director prefers to align along one of the
principal directions. The second derivative is positive
if $\theta$ is zero or $\pi$ and negative when $\theta$
is $\pm\pi/2$.
So the director aligns along the principal direction
corresponding to the  principal curvature with smaller
absolute value. The fossil energy densities for orientation
along the lines of principal curvature are
$\left.W_0\right|_{\theta=0}=\frac12 k_3 \kappa_1^2$
and
$\left.W_0\right|_{\theta=\pi/2}=\frac12 k_3 \kappa_2^2$.

In the case of unequal elastic constants,
$k_2-k_3\ne 0$, there is the possibility that the term in
square brackets on the right-hand side of \eqref{eq:dW0dtheta}
might vanish. This is the case if
\begin{equation}\label{eq:cos2theta0}
\cos 2\theta_0=\xi
\end{equation}
where
\begin{equation}\label{eq:xi}
\xi:= \frac{k_3}{k_3-k_2}\times\frac{\kappa_2+\kappa_1}{\kappa_2-\kappa_1}.
\end{equation}
Provided that $-1\le \xi\le 1$, there are critical
points at $\pm\theta_0$ and $\pi\pm\theta_0$.
The second derivative at the critical points then is
\begin{equation}
\frac{d^2W_0}{d\theta^2}=
(\kappa_2-\kappa_1)^2(k_2-k_3)\times\left\{
\begin{array}{ll}
1-\xi, &\theta=0,\\
1+\xi, &\theta=\pi/2,\\
\xi^2-1, &\theta= \pm\theta_0.
\end{array}\right.
\end{equation}
This shows that if $k_2>k_3$, then both lines of principal curvature
are minimisers. As before, the absolute minimum is attained for alignment
along 0 or $\pi$, but alignment along $\pm\pi/2$ now corresponds to
a local minimum. The directions corresponding to $\cos 2\theta_0=\xi$
yield the maximum of the fossil free energy density.

When $k_2<k_3$, the situation is reversed. Along the lines of
principal curvature, maxima are attained, and the minimum free
energy density is found along the directions corresponding
to $\cos 2\theta_0=\xi$. The fossil energy density for orientation
along $\pm\theta_0$ is
\begin{equation}\label{eq:newEnergy}
\left.W_0\right|_{\theta=\pm\theta_0}= \frac{1}{2}k_3
\left[
\frac{1}{2}(\kappa_1^2+\kappa_2^2)+
\frac{1+\xi^2}{4\xi}(\kappa_1^2-\kappa_2^2)
\right].
\end{equation}

To discuss further the existence and stability of the solution
corresponding to \eqref{eq:cos2theta0}, we define $k:=k_2/k_3$
and $\kappa:=\kappa_1/\kappa_2$, so that we can
write
\begin{equation}\label{eq:xi_simple}
\xi=\frac{1}{1-k}\times\frac{1+\kappa}{1-\kappa}.
\end{equation}
Because both $k_2$ and $k_3$ are positive, we have
$k\in[0,\infty)$, and with our choice of $\kappa_2^2>\kappa_1^2$,
we have $\kappa\in[-1,1]$. The term
$\frac{1+\kappa}{1-\kappa}$ is always positive. It is less
than one if and only if $\kappa$ is negative. The
term $\frac{1}{1-k}$ is greater than one if $k<1$, that is
if $k_2<k_3$. It is negative if $k>1$, that is if $k_2>k_3$,
and its modulus can become arbitrarily small.

Figure \ref{fig:solutionPlane} shows the regions
in the $k$-$\kappa$-plane where this solution exists.
The solid line marks $\xi=1$ ($\theta_0=0$), and the dashed line
marks $\xi=-1$ ($\theta_0=\pi/2$).
The solution is stable for $k<1$ and unstable for
$k>1$. While the unstable solution exists for all admissible ratios
of the principal curvatures, if only $k$ is large enough, the stable
solution can only be found when $\kappa<0$, that is at hyperbolic
points on the surface.
When $\kappa=-1$, as long as $k\ne 1$, we
have $\xi=0$ ($\theta_0=\pi/4$).

It should be noted that in the region of the $k$-$\kappa$-plane delimited by the solid line in Fig.~\ref{fig:solutionPlane}, the fossil energy is minimised along \emph{two} directions, symmetrically oriented with respect to the principal direction of curvature $\bm{e}_1$ with minimum absolute principal curvature $\kappa_1$. The region $\mathscr{S}_0$ on $\mathscr{S}$ where this is the case, if existing, is delimited by the curve where
\begin{equation}\label{eq:bistable_region_delimiter}
\kappa=-\frac{k}{2-k},\quad k<1,
\end{equation}
as easily follows from \eqref{eq:xi_simple}, upon setting $\xi=1$. $\mathscr{S}_0$, if not empty, is the \emph{bistable} region for the fossil energy $W_0$.

\begin{figure}
\includegraphics[width=0.48\textwidth]{saddle.eps}\\[10pt]
\includegraphics[width=0.48\textwidth]{saddleFlat.eps}
\caption{\label{fig:principal}Principal lines of curvature on the hyperbolic paraboloid.
Both panels of the figure show the same part of the manifold, the view in the second panel
is along the $z$-axis, showing the projection onto the $x$-$y$-plane.
The solid lines correspond to the principal curvature with smaller absolute value. If $k_2=k_3$, the
curvature potential would always prompt the director to align along these lines.}
\end{figure}

\subsection{Curvature along lines of preferred directions}
Suppose that the director is at an angle $\theta_0$ corresponding
to \eqref{eq:cos2theta0},
\begin{equation}
\vctr{n}=\cos\theta_0\vctr{e}_1+\sin\theta_0\vctr{e}_2.
\end{equation}
The curvature $\kappa_n$ in this direction is
\begin{equation}
\kappa_n=\vctr{n}\cdot (\nabla_\mathrm{{\!s}}\vctr{\nu})\vctr{n}=
\kappa_1\cos^2\theta_0+\kappa_2\sin^2\theta_0.
\end{equation}
As $\cos 2\theta_0=\xi$, we have $\cos^2\theta_0=\frac12(1+\xi)$
and $\sin^2\theta_0=\frac12(1-\xi)$, so
\begin{equation}\label{eq:curvature_n}
\kappa_n=\frac12[\kappa_1+\kappa_2+\xi(\kappa_1-\kappa_2)].
\end{equation}
In the limiting cases of existence for this solution, it lines up with the
principal directions of curvature: when $\xi=1$, then $\kappa_n=\kappa_1$, and when $\xi=-1$, then
$\kappa_n=\kappa_2$. In the extreme case when $k_2=0$,
we have $\xi=\frac{\kappa_2+\kappa_1}{\kappa_2-\kappa_1}$, and it then follows that $\kappa_n=0$,
independent of the actual manifold: at hyperbolic points the director prefers to align along directions
of zero curvature.

In general, using \eqref{eq:xi} in \eqref{eq:curvature_n} shows that
\begin{equation}\label{eq:curvature_n_2}
\kappa_n=H\frac{k_2}{k_2-k_3},
\end{equation}
where $H=\frac12(\kappa_1+\kappa_2)$ is the mean curvature.
So $\kappa_n$ will normally neither be zero nor one of the principal
curvatures.

\begin{figure}[h]
\includegraphics[width=0.48\textwidth]{newLines.eps}\\[10pt]
\includegraphics[width=0.48\textwidth]{newLinesFlat.eps}
\caption{When $k_2=0$, the director would like to align along directions of zero curvature.
In the case of the hyperbolic paraboloid, the projections of these lines onto the
$x$-$y$-plane are lines with slope $\pm 1$.\label{fig:newLines}}
\end{figure}

\subsection{Example: hyperbolic paraboloid}\label{sec:example}
To illustrate our results, we consider as an example
the hyperbolic paraboloid, a two-dimensional manifold which has negative
Gaussian curvature everywhere. It can be represented in the form
\begin{equation}
f(x,y)=(x,y,h(x,y))
\end{equation}
with the height of the manifold above the $x$-$y$-plane given by
$h(x,y)=x^2-y^2$. All geometric properties of the hyperbolic paraboloid, such
as the principal curvatures and principal lines of curvature, can be expressed
in terms of the height function $h$ and its derivatives~\cite[p. 137]{spivak:diffferential3}.

The principal lines of curvature are shown in
Fig. \ref{fig:principal}, with the solid lines corresponding to the
lower fossil energy density.

In terms of polar coordinates $(r,\phi)$, the Gaussian curvature $K$ and the mean curvature $H$ are
\begin{equation}\label{eq:H_and_K}
K=-\frac{4}{(1+4r^2)^2}\quad\text{and}\quad
H=-\frac{4r^2\cos2\phi}{(1+4r^2)^{3/2}}.
\end{equation}
As $K<0$, all points are hyperbolic, so depending on the ratio between $k_2$ and
$k_3$ the orientation corresponding to \eqref{eq:cos2theta0} can be the one favoured
by the curvature potential. We consider the two cases $k_2=0$ and $k_2=\frac12 k_3$.

When $k_2=0$, we have $\xi=\frac{\kappa_2+\kappa_1}{\kappa_2-\kappa_1}$, and
from \eqref{eq:xi_simple} it is evident that $0<\xi<1$ everywhere. Therefore,
the solution corresponding to \eqref{eq:cos2theta0} minimises everywhere the curvature potential
and by \eqref{eq:curvature_n_2} the director thus would align along lines with vanishing
curvature. It is known that the hyperbolic paraboloid is a
doubly-ruled surface~\cite[p. 155]{spivak:diffferential3},
and those rules are precisely the lines along which the curvature potential would
like to align the director, see Fig.~\ref{fig:newLines}.
Note that at every point of the manifold there are two preferred directions with
equal fossil energy density; the bistable region $\mathscr{S}_0$ is the whole of $\mathscr{S}$.

\begin{figure}
\includegraphics[width=0.48\textwidth]{preferred.eps}\\[10pt]
\includegraphics[width=0.48\textwidth]{preferredFlat.eps}
\caption{Directions with minimal fossil energy density for $k_2=\frac12 k_3$.
Near the origin and the quadrant bisectors $0\le\xi<1$.
This is the bistable region for the fossil energy, there the two
preferred directions are depicted in grey. Where $\xi>1$, the director has only one
preferred orientation, shown in black: along the principal direction with
curvature with smaller absolute value.
The dashed lines, along which $\xi=1$, delimit the bistable region,
according to \eqref{eq:bistable_region_delimiter_example}.
\label{fig:preferred}}
\end{figure}

When $k_2\ne0$, the orientation corresponding to \eqref{eq:cos2theta0}
only exists near the origin and near the quadrant-bisectors where $0\le\xi <1$.
In this region, there are two preferred directions, depicted in grey
in Fig.~\ref{fig:preferred}
for the case where $k_2=\frac12 k_3$,
and so $\xi=2\frac{\kappa_2+\kappa_1}{\kappa_2-\kappa_1}$. In the
same figure, the single preferred orientation in regions where $\xi>1$
is shown in black: it is along the principal direction with
curvature with smaller absolute value. In the same parametrization as in \eqref{eq:H_and_K},
\begin{equation}
\xi=\left|\frac{4r^2\cos2\phi}{\sqrt{1+4r^2+4r^4\cos^22\phi}}\right|,
\end{equation}
so that the bistable region $\mathscr{S}_0$ is now characterized by
\begin{equation}\label{eq:bistable_region_delimiter_example}
r^2<\frac{2+\sqrt{4+12\cos^22\phi}}{12\cos^22\phi}.
\end{equation}

\section{Conclusions and Discussion}\label{sec:conclusions}
This paper is mainly concerned with the notion of a fossil energy that arises in nematic shells upon properly defining
the local ground state of their distortion by use of Levi-Civita's parallel transport on surfaces. From the perspective
afforded  by this notion, it is easily possible to distinguish and categorize the different contributions to the
elastic energy of nematic shells.

Some studies, pioneered by \cite{nelson:fluctuations}  and \cite{lubensky:orientational},\footnote{Occasionally,
\cite{straley:liquid} is also  indicated as a remote, inspiring antecedent for these studies.} simply  proposed
to disregard the fossil energy and to penalize only the distortions built upon the ground state, which are
measured by the covariant gradient $\operatorname{D}\!\bm{n}$ of the nematic director field $\bm{n}$. These are the \emph{intrinsic}
theories of nematic shells. Other studies, of which presumably the oldest is \cite{helfrich:intrinsic},
followed by \cite{kamien:extrinsic}, \cite{mbanga:frustrated}, and
\cite{napoli2012extrinsic},%
\footnote{Contributions such as \cite{biscari:nematic,santangelo:geometric,jiang:vesicle,frank:defects} should also be mentioned,
for completeness, even if some are originating from the world of flexible vesicles, rather than from that of rigid nematic shells.}
have duly incorporated the fossil energy into the energetics of nematic shells, mostly advocating, however, a different
interpretation of its effects in terms of certain \emph{extrinsic} geometric features revealed by the flux lines of $\bm{n}$.

From the latest studies, especially on applications to generalized cylinders \cite{napoli2013curvature},
a widespread opinion has emerged, elevated to the dignity of \emph{heuristic principle} in \cite{segatti2016analysis},
which in its proponents' own words reads: ``the nematic elastic energy promotes the alignment of the flux lines
of the nematic director towards geodesics and/or lines of curvature of the surface'' \cite{napoli2012extrinsic}.
The clear-cut distinction between distortional and fossil energies, which we have based on an elementary argument,
has unambiguously identified the antagonistic players striving for a minimum.
Clarity breeds novelty: the
fossil energy acts as a curvature potential that in general does \emph{not} orient $\bm{n}$ along the
direction with minimum absolute principal curvature, as is the case for cylinders.
Actually, we prove in Apppendix \ref{sec:app_developable} that cylinders are  very special, as they are the only surfaces on which
the fossil energy can be uniformly minimized. \emph{Two} equivalent, but distinct fossil preferred orientations, \emph{none} of which along a principal direction
of curvature, may arise at \emph{hyperbolic points} of the shell, where the Gaussian curvature is negative.
A noticeable example is offered by a quadratic saddle surface, whose points are all hyperbolic.
\begin{comment}
We have shown that such a preferred orientation indeed manifests itself in a whole region on the saddle
surrounding the lines with zero mean curvature.
\end{comment}

It would be interesting to see whether inclusion of the fossil energy in the shape functional studied by
Giomi~\cite{giomi:hyperbolic} would affect the stability analysis (and the related shape transformation)
for the soft hyperbolic interfaces he considered. We know already from \cite{mbanga:frustrated} that
catenoids endowed with a phenomenological  \emph{ad hoc} extrinsic energy, not inherited from the fossil energy,
induce on the nematic director a generic preference to be oriented in the direction of local maximum or minimum curvature,
a preference more pronounced in regions with higher curvature anisotropy. We should ask whether
the preference for a skew fossil orientation would result in a different energy landscape.

These are problems yet to be solved. One already tackled is the equilibrium problem for the elastic energy density
in \eqref{eq:W_general_representation_formula} on a circular toroidal shell
\cite{li:defect-free,segatti:equilibrium,segatti2016analysis} (see also \cite{bowick:curvature-induced,evans:phase}
for parallel treatments with only intrinsic, distortional energies, and \cite{ye:nematic} for an
interesting application to toroidal shells of a promising Onsager excluded volume theory for wormlike polymer
chains elaborated by Chen~\cite{chen:theory}). But, unfortunately, this problem has only been solved for
the singular case where $k_2=k_3$,
which by \eqref{eq:fossil_energy_revealed} implies that the fossil preferred direction coincides with
the direction of minimal absolute curvature (that is, with either parallles or meridians).
Since all around the hole of the torus the Gaussian curvature is negative, for $k_2<k_3$ the fossil preferred directions
are different from both parallels and meridians. The equilibrium bifurcation pattern should arguably be richer
than anticipated for the case $k_2=k_3$, where for sufficiently fat tori the equilibrium director was found
to bend towards the meridians while swinging through the hole.

As our explicit example indicated, there may be two equivalent, but distinct  fossil preferred directions on the shell.
This should not come as a surprise, since in general the fossil energy cannot be uniformly minimized by the same orientation
in the principal curvature frame, save on generalized cylinders. One might think that bistability of the fossil energy
should herald  defect formation, in analogy to the criterion established by some intrinsic
theories according to which defects (of whatever topological charge) are attracted to regions of negative Gaussian
curvature (and repelled from regions of positive Gaussian curvature); see, in particular,
\cite{vitelli:anomalous,vitelli:defect} for learning more about this specific claim, and \cite{bowick:two-dimensional}
for a broader, enlightening review. Though the connection between fossil energy and defect formation is worth exploring,
it may turn out to be more complicated than expected. In general, distortional  and fossil energies compete one against
the other, just because a single fossil preferred direction cannot be uniformly enforced; defects will arise wherever such a
competition cannot be smoothly resolved and the system would rather prefer paying the extra cost that in a
doctored\footnote{The divergence of the elastic energy at a point defect in a two-dimensional setting is usually tamed
by carving out a small core about the defect and attributing to it a standard melting cost \cite{lubensky:orientational}.}
two-dimensional director theory is due by a defect for the total energy to remain finite.
\begin{acknowledgments}
	E.G.V. acknowledges the kind hospitality of the Oxford Centre for
	Nonlinear PDE, where part of this work was done while he was visiting the
	Mathematical Institute at the University of Oxford.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Iran General NewsBlair refuses to back Iran strike
Iran General News
Blair refuses to back Iran strike
Scotland on Sunday: Tony Blair has told George Bush that Britain cannot offer military support to any strike on Iran, regardless of whether the move wins the backing of the international community, government sources claimed yesterday.
Scotland on Sunday
BRIAN BRADY
WESTMINSTER EDITOR
TONY Blair has told George Bush that Britain cannot offer military support to any strike on Iran, regardless of whether the move wins the backing of the international community, government sources claimed yesterday.
Amid increasing tension over Tehran's attempts to develop a military nuclear capacity, the Prime Minister has laid bare the limits of his support for President Bush, who is believed to be considering an assault on Iran, Foreign Office sources revealed.
US Secretary of State Condoleezza Rice is calling on the United Nations to consider new sanctions against Tehran when the Security Council meets next week to discuss the developing crisis. Blair is expected to support the call for a "Chapter 7" resolution, which could effectively isolate Iran from the international community.
But, in the midst of international opposition to a pre-emptive strike on Tehran, and Britain's military commitments around the world, the government maintains it cannot contribute to a military assault. "We will support the diplomatic moves, at best," a Foreign Office source told Scotland on Sunday. "But we cannot commit our own resources to a military strike."
Meanwhile, a new report on the Iran crisis has warned that neo-conservatives in the Bush administration are on "collision course" with Tehran.
The Foreign Policy Centre (FPC), often referred to as Blair's "favourite think-tank", will appeal for a greater effort to find a diplomatic solution in a report to be published later this week. FPC director Stephen Twigg, formerly a Labour minister, explained: "It is essential UK policy on Iran is well informed… We want to engage with the various reformist elements in Iran, both inside and outside the structures of power.
"There is potential for political dialogue, economic ties and cultural contacts to act as catalysts for the strengthening of civil society in Iran."
While the sense of crisis over Iran has been escalated by the fiery rhetoric between Tehran and the West – particularly Washington – many within the British government are now convinced that the impasse can be resolved by repeating the same sort of painstaking diplomatic activity that returned Libya to the international fold.
The approach contrasts sharply with the strategy employed during the run-up to the war in Iraq, when ministers repeatedly issued grim warnings to Saddam Hussein over the consequences of not falling in line with their demands.
"The only long-term solution to Iran's problems is democracy," said Alex Bigham, co-author of the FPC report. "But it cannot be dictated, Iraq-style, or it will backfire. Iran may seem superficially like Iraq but we need to treat Iran more like Libya. Diplomatic engagement must be allowed to run its course. There need to be bigger carrots as well as bigger sticks."
However, the conciliatory language was not reflected in the approach from Washington, where senior figures in the Bush administration remain keen to stress the danger of Tehran's intentions.
In a declaration aimed at America's allies as much as Iran, Rice claimed the Security Council's handling of the Iranian nuclear issue would be a test of the international community's credibility. "If the UN Security Council says: 'You must do these things and we'll assess in 30 days,' and Iran has not only not done those things, but has taken steps that are exactly the opposite of those that are demanded, then the Security Council is going to have to act."
Rice dismissed Iran's declaration that it is only interested in enriching uranium for use in civil nuclear power facilities, saying the international community must remain focused on the potential military applications of this technology.
"The world community does not want them to have that nuclear know-how and that's why nobody wants them to be able to enrich and reprocess on their territory, getting to the place that they can produce what we call a full-scale nuclear plant to be able to do this," she said.
Rice reiterated that President Bush has not taken any option off the table, including a military response, if Iran fails to comply with the demands of the international community.
Previous articleThe frightening truth of why Iran wants a bomb
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redpajama_set_name: RedPajamaCommonCrawl


text:
Raven's advanced progressive matrices within a South African context
Electronic Theses and Dissertations (ETD)
ETD Collection
dc.contributor.author Israel, Nicole
dc.date.issued 2006-11-14T12:26:28Z
dc.description faculty of Humanities School of Human and Community Development 9807986h IsrealN@umthombo.wits.ac.za en
dc.description.abstract The issue of bias, whether a psychological test measures what it claims to measure similarly across different groups, remains a fundamental concern within the field of psychometrics, particularly within South Africa, where a history riddled with oppression, discrimination and malpractice in the area has led to suspicion, mistrust and legislation banning the use of many psychological tests as invalid and unfair (Foxcroft & Roodt, 2001; Murphy & Davidshofer, 2001; Nell, 1999). There is thus clearly a need for additional and more detailed investigations of the way specific individual tests function in the South African context. This study attempted to creatively examine systematic differences in performance on one specific test, the Raven's Advanced Progressive Matrices (RAPM), on the basis of home language and gender, factors seldom investigated in the literature. A sample of one hundred Psychology first-year students completed a demographic questionnaire, the RAPM, the Similarities sub-test of the South African Wechsler Adult Intelligence Scales (SAWAIS) and an adapted version of the Reading Comprehension subtest of the Stanford Diagnostic Reading Test (SDRT). The data gathered was then utilized to explore four main research questions. Firstly, in order to establish construct comparability, the relationship between the RAPM and a more verbally-oriented measure of g, the Similarities sub-test of the SAWAIS, was assessed. Results indicated a relatively strong positive relationship between the two measures (r = 0.66), and no significant differences between the correlations on the basis of either gender or home language. Secondly, in order to explore the relationship between the RAPM and English comprehension, the study assessed the relationship between overall, literal and inferential scores on an adapted version of the Reading Comprehension sub-test of the SDRT and the RAPM. Results indicated only a moderate relationship between the two tests (r = 0.65), no difference in the relationship between RAPM performance and literal comprehension or inferential comprehension, and no difference in the relationship between the two tests on the basis of either gender or home language. Thirdly, in order to establish whether items were found to be difficult in a similar way across the different gender and home language groups, p-values and regression lines were calculated. These indicated that significant differences in level of item difficulty were experienced between English and African language speakers, although no differences were apparent in item difficulty on the basis of gender. 5 Lastly, in order to establish whether qualitative differences in performance on the basis of ability (as estimated by performance on the RAPM), gender or language existed, a discrimination analysis examining the types of errors made by each group was performed. Repeated measures ANOVAs and multiple comparison post-hoc analyses revealed significant differences in the types of errors made on the basis of ability and home language, but not gender. The post-hoc analyses suggested that those of higher ability or first language English speakers were more likely to make incomplete correlate errors, while those of lower ability or speaking African first languages were more likely to make confluence of ideas errors. In general, the findings of the study seemed to suggest that the RAPM, while not biased on the basis of gender, might contain a deep-seated language bias despite their non-verbal presentation. en
dc.format.extent 635620 bytes
dc.format.extent 20494 bytes
dc.subject Errors en
dc.subject South Africa en
dc.subject Reasoning en
dc.subject Intelligence en
dc.subject differences en
dc.title Raven's advanced progressive matrices within a South African context en
dc.type Thesis en
Name: AbstractFinal2006 ...
Name: RAPMResRepFinal20 ...