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Prime Minister Malcolm Turnbull visits SMEC's Cooma office.
On Wednesday 28 June, Prime Minister Malcolm Turnbull, Deputy Premier John Barilaro and Parliamentary Secretary Bronnie Taylor visited SMEC's Cooma office.
On Wednesday 28 June, Prime Minister Malcolm Turnbull, Deputy Premier John Barilaro and Parliamentary Secretary Bronnie Taylor visited SMEC's Cooma office ahead of investigative drilling commencing for Snowy 2.0.
The Prime Minister, Deputy Premier and Parliamentary Secretary's visit was also an opportunity for Snowy Hydro and SMEC to bring together their local project teams to celebrate the reopening of the SMEC office in Cooma.
SMEC's CEO Andy Goodwin said the reopening of their Cooma office and work on the Snowy 2.0 feasibility study will continue to generate economic activity in the region. "It's great to be working on such an iconic hydropower project with Snowy Hydro that reflects on our namesake and history."
The Snowy 2.0 Feasibility Study will be completed by the end of this year
Full media release here.


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        <meta property="og:title" content="{% block og_title %}Open States{% endblock %}">
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        <link rel="apple-touch-icon" sizes="180x180" href="{% static 'images/favicon/apple-touch-icon.png?v=20190106' %}">
        <link rel="icon" type="image/png" sizes="32x32" href="{% static 'images/favicon/favicon-32x32.png?v=20190106' %}">
        <link rel="icon" type="image/png" sizes="16x16" href="{% static 'images/favicon/favicon-16x16.png?v=20190106' %}">
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        <link rel="mask-icon" href="{% static 'images/favicon/safari-pinned-tab.svg?v=20190106' %}" color="#233240">
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        <meta name="msapplication-TileColor" content="#ffffff">
        <meta name="msapplication-config" content="{% static 'images/favicon/browserconfig.xml?v=20190106' %}">
        <meta name="theme-color" content="#ffffff">

        {% block styles %}
        {% endblock %}

        {% header %}

                    {% block content %}{% endblock %}

        {% include "./footer.html" %}

        {% block scripts %}
        {% endblock %}

redpajama_set_name: RedPajamaGithub


text:
T-Mobile data breach compromises data of millions. Cyberattack at PayPal highlights the importance of password security.
T-Mobile data breach compromises data of millions.
Cyberattack at PayPal highlights the importance of password security.
Yesterday wireless giant T-Mobile disclosed it suffered a data breach that exposed the data of approximately 37 million of its current postpaid and prepaid customer accounts. Bloomberg reports that the compromised data include names, billing addresses, email addresses, phone numbers, and dates of birth, as well as account numbers and plan details, but no payment info or passwords were accessed. T-Mobile said in a statement, "Some basic customer information (nearly all of which is the type widely available in marketing databases or directories) was obtained." According to the company's filing with US Securities and Exchange Commission, T-Mobile detected the incident on January 5 and secured its networks within a day, but it's believed the hacker had access to the network starting on November 25.
Security Week explains that the attackers appeared to access the customer data by abusing a single API entry point, but T-Mobile says the intruders did not breach the company's systems or network. "Based on our investigation to date, customer accounts and finances were not put at risk directly by this event," T-Mobile said in the filing. The Wall Street Journal notes that this is the second major breach the company has experienced in the past two years. In 2021 a hacker stole the data of over 13 million active and 40 million prospective T-Mobile customers, and last year the company settled a class-action suit related to that breach for $500 million. T-Mobile says it's likely the newest breach will be costly as well. A spokesperson from the US Federal Communications Commission stated, "This incident is the latest in a string of data breaches at the company, and the FCC is investigating."
Online payment platform PayPal has begun notifying nearly 35,000 users that their data were exposed in a credential stuffing attack. Bleeping Computer reports that the incident occurred between December 6 and December 8, and PayPal worked swiftly to mitigate it and launch an investigation. By December 20 the company confirmed that unauthorized third parties had used stolen credentials to log into accounts, but there was no evidence of a breach of the company's systems and it was determined the credentials had been acquired by some other means. The potentially compromised data include account holders' full names, dates of birth, postal addresses, Social Security numbers, tax IDs, and payment info. The company informed the impacted users, "We reset the passwords of the affected PayPal accounts and implemented enhanced security controls that will require you to establish a new password the next time you log in to your account."
An attack of such scale and visibility has elicited commentary from security experts. Baber Amin, the COO of Veridium, told PCWorld that companies like PayPal need to do more to protect user data. "As trusted vendors, PayPal and others need to set a higher bar here." Amin suggests companies should rigorously monitor their systems for suspicious behavior (like a high volume of failed login attempts, a telltale sign of a credential stuffing attack), actively encourage users to use two-factor authentication, and "eliminate passwords from their user-facing systems by fast tracking FIDO Passkey adoption." Dr. Ilia Kolochenko, founder of ImmuniWeb and a member of the Europol Data Protection Experts Network, told Forbes he questions "why MFA authentication is not enforced by default for such a sensitive service as PayPal." Jasson Casey, chief technology officer at Beyond Identity, says perhaps it's time to do away with passwords entirely, claiming that "passwords - whether unique or complex - are fundamentally flawed." Casey suggests organizations should be moving to phishing-resistant credentials like FIDO Alliance standard blueprints.
Update On LastPass Security Breach: Hackers Stole Encrypted Password Vaults (Business2Community) Password manager LastPass encountered a security breach in August this year, which led to the loss of confidential technical data and a part of the company's source code. Hackers later used the stolen information to hack LastPass again at the end of November.
LastPass faces mounting criticism over recent breach | TechTarget (Security) LastPass disclosed last month that a recent breach involved the theft of significant, personally identifiable customer information.
More than 19,000 records released in B.C. school district data breach (Prince George Citizen) After 19,126 records were released by a school district in Maple Ridge, B.C., staff warn of potential phishing scams.
QUT confirms personal data of thousands of staff compromised in cyber attack (ABC) Thousands of staff and 67 students at one of Queensland's largest universities have had their personal data compromised in a cyber attack last month.
Suspected Chinese Threat Actors Exploiting FortiOS Vulnerability (CVE-2022-42475) (Mandiant) A suspected China-nexus campaign exploiting a vulnerability in Fortinet's FortiOS.
T-Mobile Says Hackers Stole Data on About 37 Million Customers (Wall Street Journal) The carrier says addresses, birth dates and other personal records were exposed, but not financial records.
T-Mobile Says Hackers Used API to Steal Data on 37 Million Accounts (SecurityWeek) T-Mobile fessed up to another massive data breach affecting approximately 37 million current postpaid and prepaid customer accounts.
Nearly 35,000 PayPal users had SSNs, tax info leaked during December cyberattack (The Record from Recorded Future News) PayPal is sending out breach notification letters to 34,942 customers after a December 6 credential stuffing attack exposed SSNs and more.


text:
<!DOCTYPE html>
<html>
  <head>
    <meta charset="utf-8">
    <meta name="viewport" content="initial-scale=1, maximum-scale=1, user-scalable=no, width=device-width">
    <title></title>

    <link href="lib/ionic/css/ionic.css" rel="stylesheet">
    <link href="css/style.css" rel="stylesheet">

    <!-- IF using Sass (run gulp sass first), then uncomment below and remove the CSS includes above
    <link href="css/ionic.app.css" rel="stylesheet">
    -->

    <!-- ionic/angularjs js -->
    <script src="lib/ionic/js/ionic.bundle.js"></script>
    <script src="lib/angular-cookies/angular-cookies.min.js"></script>
    <script src="lib/query-string/query-string.js"></script>
    <script src="lib/angular-oauth2/dist/angular-oauth2.min.js"></script>
    <script src="lib/angular-ui-router/release/angular-ui-router.min.js"></script>

    <!-- cordova script (this will be a 404 during development) -->
    <script src="cordova.js"></script>

    <!-- your app's js -->
    <script src="js/app.js"></script>
    <script src="js/controllers.js"></script>
    <script src="js/login_controller.js"></script>
    <script src="js/product_controller.js"></script>
    <script src="js/order_controller.js"></script>
    <script src="js/ptype_controller.js"></script>
  </head>
  <body ng-app="starter">
		<ion-nav-view></ion-nav-view>

        <script src="//ajax.googleapis.com/ajax/libs/jquery/1.10.1/jquery.min.js"></script>
  </body>
</html>


The Villa dos Corais Pousada has a privileged location right in between the third and fourth beaches, with access to both and is right in front of the natural pools. The hotel offers 40 comfortable and spacious rooms, each equipped with air conditioning, mini fridge, cable TV, super king size bed, telephone, individual safe, balcony with hammock and an ocean view. There is a swimming pool, lounge and restaurant for guests to take pleasure in.


text:
// Originally from UpdatedComponentTests/AcceptProtocols/Rest/Rest_Get_XMLRequest_UnsupportedAcceptProtocol.xls;
package com.betfair.cougar.tests.updatedcomponenttests.acceptprotocols.rest;

import com.betfair.testing.utils.cougar.misc.XMLHelpers;
import com.betfair.testing.utils.cougar.assertions.AssertionUtils;
import com.betfair.testing.utils.cougar.beans.HttpCallBean;
import com.betfair.testing.utils.cougar.beans.HttpResponseBean;
import com.betfair.testing.utils.cougar.enums.CougarMessageProtocolRequestTypeEnum;
import com.betfair.testing.utils.cougar.manager.AccessLogRequirement;
import com.betfair.testing.utils.cougar.manager.CougarManager;

import org.testng.annotations.Test;
import org.w3c.dom.Document;

import java.sql.Timestamp;
import java.util.HashMap;
import java.util.Map;

/**
 * Ensure that when a Rest XML Get is performed on Cougar, specifying an unsupported response protocol, the correct error response is generated
 */
public class RestGetXMLRequestUnsupportedAcceptProtocolTest {
    @Test
    public void doTest() throws Exception {
        // Create the HttpCallBean
        CougarManager cougarManager1 = CougarManager.getInstance();
        HttpCallBean httpCallBeanBaseline = cougarManager1.getNewHttpCallBean();
        CougarManager cougarManagerBaseline = cougarManager1;
        // Get the cougar logging attribute for getting log entries later
        // Point the created HttpCallBean at the correct service
        httpCallBeanBaseline.setServiceName("baseline", "cougarBaseline");

        httpCallBeanBaseline.setVersion("v2");
        // Set up the Http Call Bean to make the request
        CougarManager cougarManager2 = CougarManager.getInstance();
        HttpCallBean getNewHttpCallBean2 = cougarManager2.getNewHttpCallBean("87.248.113.14");
        cougarManager2 = cougarManager2;

        cougarManager2.setCougarFaultControllerJMXMBeanAttrbiute("DetailedFaults", "false");

        getNewHttpCallBean2.setOperationName("testSimpleGet", "simple");

        getNewHttpCallBean2.setServiceName("baseline", "cougarBaseline");

        getNewHttpCallBean2.setVersion("v2");

        Map map3 = new HashMap();
        map3.put("message","foo");
        getNewHttpCallBean2.setQueryParams(map3);
        // Set the response protocols (with an unsupported protocol ranked highest)
        Map map4 = new HashMap();
        map4.put("application/pdf","q=70");
        getNewHttpCallBean2.setAcceptProtocols(map4);
        // Get current time for getting log entries later

        Timestamp getTimeAsTimeStamp10 = new Timestamp(System.currentTimeMillis());
        // Make the XML call to the operation
        cougarManager2.makeRestCougarHTTPCall(getNewHttpCallBean2, com.betfair.testing.utils.cougar.enums.CougarMessageProtocolRequestTypeEnum.RESTXML);
        // Create the expected response as an XML document
        XMLHelpers xMLHelpers6 = new XMLHelpers();
        Document createAsDocument12 = xMLHelpers6.getXMLObjectFromString("<fault><faultcode>Client</faultcode><faultstring>DSC-0013</faultstring><detail/></fault>");
        // Convert the expected response to JSON
        Map<CougarMessageProtocolRequestTypeEnum, Object> convertResponseToRestTypes13 = cougarManager2.convertResponseToRestTypes(createAsDocument12, getNewHttpCallBean2);
        // Check the response is as expected (fault)
        HttpResponseBean getResponseObjectsByEnum14 = getNewHttpCallBean2.getResponseObjectsByEnum(com.betfair.testing.utils.cougar.enums.CougarMessageProtocolResponseTypeEnum.REST);
        AssertionUtils.multiAssertEquals(convertResponseToRestTypes13.get(CougarMessageProtocolRequestTypeEnum.RESTXML), getResponseObjectsByEnum14.getResponseObject());
        AssertionUtils.multiAssertEquals((int) 406, getResponseObjectsByEnum14.getHttpStatusCode());
        AssertionUtils.multiAssertEquals("Not Acceptable", getResponseObjectsByEnum14.getHttpStatusText());

        Map<String, String> map8 = getResponseObjectsByEnum14.getFlattenedResponseHeaders();
        AssertionUtils.multiAssertEquals("application/xml", map8.get("Content-Type"));
        // Check the log entries are as expected

        cougarManagerBaseline.verifyAccessLogEntriesAfterDate(getTimeAsTimeStamp10, new AccessLogRequirement("87.248.113.14", "/cougarBaseline/v2/simple", "MediaTypeNotAcceptable") );
    }

}


You receive a list of Pre-Qualified Leads built specifically for your needs and immediately execute your campaign!
demandDrive and our Data Science Team works with you + your team to define your top priority lead, and help establish (or optimize) an Ideal Customer Profile (ICP). Using these parameters along with our Predictive Modeling capabilities, we design the Qualification Criteria necessary for Pre-Qualified Leads to be both valid, and deliver the best value.
Your team specifies its requirements and works side by side with our specialists to design the right campaign. Whether involving data validation, content syndication, survey execution, or your specific data functions, we create a custom outreach plan based on your campaign specifics which will result in the Pre-Qualified Leads you need.
This is where you come in. After the physical handover of your Pre-Qualified Leads, you deal with them as you wish. The data is guaranteed to be 90+% accurate, and it's all yours! Run these leads through your next campaign and watch the profits blossom.
Need some tweaks? Kick the data back to us for further optimization. After analyzing your initial results it might become clear that some standards need adjustment. By creating a feedback loop we can further refine your qualification criteria and better define your ICP to grant you access to superior Pre-Qualified Leads with each successive campaign.
Interested in Trialing a Pre-Qualified Leads Program?


text:
Should Men Who Identify as Women Compete in Women's Sports?
Regardless of what postmodernists say, there are real, objective, measurable biological differences between men and women. That's why sports are segregated by sex.
Men with Slightly Longer Legs Are More Attractive to Women
It's hardly a secret that men find women with long legs attractive. What's less obvious is that the reverse also appears to be true. Even after controlling for height, women find men with slightly longer legs than average to be more attractive.
Stop Mommy-Shaming Pregnant Women Who Have a Glass of Wine
When it comes to alcohol, the United States is incredibly puritanical. Our society has promoted the view that even a single drop of alcohol is harmful to developing babies. However, the totality of evidence does not support that belief at all.
Top 10 Causes of Death Aren't Same for Men and Women
The weekly report from the Centers for Disease Control and Prevention on the dead and dying is morbidly fascinating. In both men and women, heart disease and cancer are #1 and #2 killers, respectively. However, everything changes after that.
Women Have Always Lived Longer, Study Finds
By Stephanie Bucklin, Live Science Contributor Men still aren't living as long as women — and that holds true for humans' primate cousins as well, a new study shows. In the study, researchers looked at data from six populations of humans from both modern and historical times, in different countries. The investigators found that, "in spite of the huge gains in human longevity over the past century, the male-female difference has not shrunk," said Susan Alberts, a professor of biology at Duke University and a co-author of the new study.
Kathleen Gyllenhaal: Health meets Hollywood Q&A
Welcome to our three part series discussing the science behind director Kathleen Gyllenhaal's latest documentary, IN UTERO. "To Avoid Adult Dysfunction Start 'IN UTERO'" took a deeper look at the health ramifications of toxic stress in prenatal life and its role in contributing to adult disease. Part I of our Q&A with Gyllenhaal emphasizes health, Hollywood, how a story gets told and the parallels between funding for film and scientific research. Welcome to our three part series discussing the science behind director Kathleen Gyllenhaal's latest documentary, IN UTERO. In Part II of the Q&A, we explore her insights into motherhood and the impact of her recent film.
Woman's Heart Attack Differs from a Man's
A comprehensive study of heart attacks in women by the American Heart Association reveals many important differences between those that afflict men and women: symptoms, risk factors, diagnosis and treatment. Racial disparities also come into play.
Women even (especially?) young women should protect their hearts
If you asked a representative sampling of American women which health threat most concerned them, it s likely that a goodly proportion would say cancer, especially breast cancer. But they re not on target because,
Exercise reduces AFib risk in older and obese women
Atrial fibrillation (an irregular and often rapid heart rate resulting in poor blood flow) is the most common abnormal heart rhythm. According to the American Heart Association, AFib can lead to blood clots, stroke, heart failure, and other heart complications. Currently, an estimated 2.7 million Americans are affected by AFib. AFib can affect anyone, but typically people
Depression just as common in men, but it may look different than you think
Depression has often been associated with symptoms such as sadness, apathy and anxiety. But researchers are coming to understand that depression can manifest itself differently in men than it does in women


text:
from zeit.cms.checkout.helper import checked_out
from mechanize import LinkNotFoundError
import zeit.cms.testing

class DAVPropertiesListingTest(zeit.cms.testing.ZeitCmsBrowserTestCase):

    login_as = 'producer:producerpw'

    def test_not_shown_in_workingcopy(self):
        b = self.browser
        b.open(
            'http://localhost/++skin++vivi/repository/testcontent/@@checkout')
        with self.assertRaises(LinkNotFoundError):
            b.getLink('DAV Properties')

    def test_shows_properties_via_columns(self):
        b = self.browser
        b.open('http://localhost/++skin++vivi/repository/testcontent')
        b.getLink('DAV Properties').click()
        self.assertEllipsis(
            """... http://namespaces.zeit.de/CMS/meta
             type 	  testcontenttype
             type...testcontenttype
            	...""", b.contents)

    def test_handles_non_ascii(self):
        with checked_out(self.repository['testcontent']) as co:
            co.ressort = u'Zeit für die Schule'
        b = self.browser
        b.open('http://localhost/++skin++vivi/repository/testcontent')
        b.getLink('DAV Properties').click()
        self.assertEllipsis('...Zeit für die Schule...', b.contents)


text:
\section{Introduction}
\label{introduction} The search for a better and easy to use tool
for the solution of nonlinear equations illuminating the nonlinear
phenomena of our life keeps continuing.

A variety of methods therefore were proposed to find approximate
solutions. One of the most recent popular technique is the
homotopy analysis method, which is a combination of the classical
perturbation technique and homotopy concept as used in topology.
In the homotopy analysis method, which requires neither a small
parameter nor a linear term in a differential equation, a homotopy
with an embedding parameter $p \in [0, 1]$ is constructed. In
\cite{Liao92a} a basic idea of homotopy analysis method for
solving nonlinear differential equations was presented. In this
method, the solution is considered as the sum of an infinite
series, which converges rapidly to accurate solutions. A numerous
nonlinear problems in science, finance and engineering were
recently treated by the method, see at least
\cite{Abbasbandy2006}, \cite{Zhu2006}, \cite{Hayat2008},
\cite{VanGorder2008}, \cite{Akyildiz2008}, \cite{Wu2009},
\cite{Liang2009},\cite{Molabahrami2009} and \cite{Zhao2009}.
Particularly, a few new solutions of some nonlinear problems were
found by means of the method \cite{Liao2007a}, which were
neglected by other analytic methods and even by numerical
techniques. All of these show the potential of the homotopy
analysis method for strongly nonlinear problems. Recently an
optimal homotopy analysis approach for strongly nonlinear
differential equations were proposed in \cite{Liao2010}. Even
though a great deal of equations were solved using the homotopy
analysis method, the question of convergence of the method is yet
to be answered.

We in the present paper investigate the homotopy analysis
technique from a mathematical point of view. The aim is to analyze
the method and to show that under certain circumstances the
homotopy analysis method converges to the exact solution desired,
without a prior knowledge of the exact solution. An optimal value
of the convergence control parameter is defined through the square
residual error concept. Our another emphasis is to address the
error estimate of the approximate solution. The given theorem is
justified exemplifying it by basic examples from ordinary and
partial differential equations from the literature. The presented
theory not only gives the convergence, but it also provides the
information about the interval of convergence of the homotopy
series.

In the rest of the paper, \S \ref{Method} lays the basis of
homotopy analysis method. A theory is outlined in \S \ref{Results}
for the convergence and error estimate. Illustrative examples in
\S \ref{Examp} are followed by the conclusions in \S
\ref{conclusions}.

\section{The Homotopy Analysis Method}
\label{Method} Liao \cite{Liao92a} described the early form of the
homotopy analysis method in 1992. The essential idea of this
method is to introduce a homotopy parameter, say $p$, which varies
from 0 to 1 and a nonzero auxiliary parameter so-called the
convergence control parameter $h$. At $p=0$, the system of
equations usually has been reduced to a simplified form which
normally admits a rather simple solution. As $p$ gradually
increases continuously toward 1, the system goes through a
sequence of deformations, and the solution at each stage is close
to that at the previous stage of the deformation. Eventually at
$p=1$, the system takes the original form of the equation and the
final stage of the deformation gives the desired solution. To
illustrate the basic ideas of this method, consider the nonlinear
boundary value problem
\begin{eqnarray}\label{eq1}
   N(u(r))=0;\;r\in \Omega, \quad B(u(r),{du\over dn}) = 0;\;r\in
   \Gamma
\end{eqnarray}
where $u(r)$ defined over the region $\Omega$ is the function to
be solved under the boundary constraints in $B$ defined over the
boundary $\Gamma$ of $\Omega$. The homotopy analysis technique
defines a homotopy $u(r,p): R\times [0,1]\rightarrow R$ so that
\begin{eqnarray}\label{eq2}
  H(u,p) &=& (1-p)[L(u)-L(u_0)]+p h N(u),
\end{eqnarray}where $L$ is a suitable auxiliary linear operator,
$u_0$ is an initial approximation of equation (\ref{eq1})
satisfying exactly the boundary conditions. It is obvious from
equation (\ref{eq2}) that
\begin{eqnarray}\label{eq3}
  H(u,0) &=& L(u)-L(u_0), \quad H(u,1)\; =\; N(u).
\end{eqnarray}As $p$ moves from 0 to 1, $u(r,p)$ moves from
$u_0(r)$ to $u(r)$. In topology, this called a deformation and
$L(u)-L(u_0)$ and $N(u)$ are said to be homotopic. Our basic
assumption is that the solution of equation (\ref{eq2}) when
equated to zero can be expressed as a power series in $p$
\begin{eqnarray}\label{eq4}
   u(r,p) &=& u_0(r)+pu_1(r)+p^2u_2(r)+\cdots =
   \sum_{k=0}^\infty u_k(r)p^k.
\end{eqnarray}
The appropriate solutions of the coefficients $u_k(r)$ in
(\ref{eq4}) can be found from the homotopy deformation equations,
see \cite{Liao2003book}. Hence, the approximate solution of
equation (\ref{eq1}) can be readily obtained as
\begin{eqnarray}\label{eq5}
   u(r)=\lim_{p\to 1}u(r,p) = \sum_{k=0}^\infty u_k(r).
\end{eqnarray}It was found that the auxiliary parameter $h$ can
adjust and control the convergence region and rate of homotopy
series solutions (\ref{eq4}). Whenever the series (\ref{eq4}) is
known to be convergent, then (\ref{eq5}) represents the exact
solution of (\ref{eq1}), as proved in \cite{Liao2003book}.

\section{A convergence Theorem and error estimate}\label{Results}
Using the methodology underlined above, the number of problems
treated by the homotopy analysis method approaches a couple of
hundreds now. However, the very basic question of why the series
obtained by setting $p=1$ in (\ref{eq4}) should be convergent
remains unanswered till today. To remedy this issue up to a point,
we provide the subsequent theorems here. It should be noted that
how to find a proper convergence control parameter $h$, or even
better, to get a faster convergent one, to be used in Theorem 1
will be clarified later in this section.

\vspace{0.2in}{\bf Theorem 1.} Suppose that $A\subset R$ be a
Banach space donated with a suitable norm $\|\|$ (depending on the
physical problem considered), over which the sequence $u_k(t)$ of
(\ref{eq4}) is defined for a prescribed value of $h$. Assume also
that the initial approximation $u_0(t)$ remains inside the ball of
the solution $u(t)$. Taking $r\in R$ be a constant, the following
statements hold true:

{\bf (i)} If $\|v_{k+1}(t)\|\le r \|v_k(t)\|$ for all $k$, given
some $0<r<1$, then the series solution $u(t,p) = \sum_{k=0}^\infty
u_k(t)p^k$, defined in (\ref{eq4}) converges absolutely at $p=1$
to (\ref{eq5}) over the domain of definition of $t$,

{\bf (ii)} If $\|v_{k+1}(t)\|\ge r \|v_k(t)\|$ for all $k$, given
some $r>1$, then the series solution $u(t,p) = \sum_{k=0}^\infty
u_k(t)p^k$, defined in (\ref{eq4}) diverges at $p=1$ over the
domain of definition of $t$.

\vspace{0.2in}{\bf Proof.} {\bf (i)} In compliance with the ratio
test for the power series in $p$, the proof is clear. However, in
order to give an estimate to the truncation error of homotopy
analysis method, we shortly give the whole proof here. If $S_n(t)$
denote the sequence of partial sum of the series (\ref{eq5}), we
need to show that $S_n(t)$ is a Cauchy sequence in $A$. For this
purpose, consider,
\begin{eqnarray}\label{eq6}
  \|S_{n+1}(t)-S_n(t)\| = \|u_{n+1}(t)\|\le r \|u_n(t)
  \|\le r^2 \|u_{n-1}(t)\|\le \cdots \le r^{n+1}\|u_0(t)\|.
\end{eqnarray}
It should be remarked that owing to (\ref{eq6}), all the
approximations produced by the homotopy method (\ref{eq2}) will
lie within the ball of $u(t)$. For every $m,n \in N$, $n\ge m$,
making use of (\ref{eq6}) and the triangle inequality
successively, we have,
\begin{eqnarray}\label{eq7}
  \|S_{n}(t)-S_m(t)\| = \|(S_n(t)-S_{n-1}(t))+(S_{n-1}(t)-S_{n-2}(t))+
  \cdots+(S_{m+1}(t)-S_m(t))\|\\\nonumber \le {1-r^{n-m}\over
  1-r}r^{m+1}\|u_0(t)\|.
\end{eqnarray}
Since $0<r<1$, we get from (\ref{eq7})
\begin{eqnarray}\label{eq8}
  \lim_{n,m\to\infty}\|S_{n}(t)-S_m(t)\| &=& 0.
\end{eqnarray}
Therefore, $S_n(t)$ is a Cauchy sequence in the Banach space $A$,
and this implies that the series solution (\ref{eq5}) is
convergent. This completes the proof {\bf (i)}. $\diamond$

The proof of {\bf (ii)} follows from the fact that under the
hypothesis supplied in {\bf (ii)}, there exist a number $l$,
$l>r>1$, so that the interval of convergence of the power series
(\ref{eq4}) is $|p|<1/l<1$, which obviously excludes the case of
$p=1$. $\diamond$

\vspace{0.2in}{\bf Remark 1.} Since the finite number of terms
does not affect the convergence, Theorem 1 is equally valid if the
inequalities stated in {\bf (i-ii)} are true for sufficiently
large $k'$s. Thus, it is sufficient to keep track of magnitudes of
the ratios $rat_k$ defined by
\begin{eqnarray}\label{rat}
  rat_k &=& {\|v_{k+1}(t)\|\over \|v_k(t)\|},
\end{eqnarray}and whether they remain less than unity.

\vspace{0.2in}{\bf Remark 2.} By enforcing the ratio in {\bf (i)}
to hold true in the infinite limit, the validity region of $t$ for
the series solution can also be constructed, see Example 1 below.

\vspace{0.2in}{\bf Theorem 2.} If the series solution defined in
(\ref{eq5}) is convergent, then it converges to an exact solution
of the nonlinear problem (\ref{eq1}).

\vspace{0.2in}{\bf Proof.} The proof can be found in
\cite{Liao2003book}.

\vspace{0.2in}{\bf Theorem 3.} Assume that the series solution
$\sum_{n=0}^\infty u_n(t)$, defined in (\ref{eq5}) is convergent
to the solution $u(t)$ for a prescribed value of $h$. If the
truncated series $\sum_{n=0}^Mu_n(t)$ is used as an approximation
to the solution $u(t)$ of problem (\ref{eq1}), then an upper bound
for the error, $E_M(t)$, is estimated as
\begin{eqnarray}\label{eq10}
  E_M(t) \le {r^{M+1}\over 1-r}\|u_0(t)\|.
\end{eqnarray}

\vspace{0.2in}{\bf Proof.} Making use of the inequality
(\ref{eq7}) of Theorem 1, we immediately obtain
\begin{eqnarray}\label{eq11}
  \|u(t)-S_M(t)\| \le {1-r^{n-M}\over
  1-r}r^{M+1}\|u_0(t)\|,
\end{eqnarray}
and taking into account $(1-r^{n-M})<1$, (\ref{eq11}) leads to the
desired formula (\ref{eq10}). This completes the proof. $\diamond$

\vspace{0.2in}{\bf Remark 3.} An optimal value of the convergence
control parameter $h$ can be found by means of the exact square
residual error integrated in the whole region of interest
$\Gamma$, at the order of approximation $M$, that is,
\begin{eqnarray}\label{eqres1}
  Res(h) &=& \int_\Gamma [N(\sum_{k=0}^Mu_k(r))]^2dr.
\end{eqnarray}
Obviously, the more quickly $Res(h)$ in (\ref{eqres1}) decreases
to zero, the faster the corresponding homotopy series solution
(\ref{eq5}) converges. So, at the given order of approximation
$M$, the corresponding optimal value of the convergence control
parameter $h$ is given by the minimum of $Res(h)$, corresponding
to a nonlinear algebraic equation of the form
\begin{eqnarray}\label{eqres2}
  {d Res \over d h} &=& 0.
\end{eqnarray}

Therefore, the convergence control parameter obtained via
(\ref{eqres1}-\ref{eqres2}) can be supplied into the Theorem 1.
However, it is unfortunate that the exact square residual error
$Res(h)$ defined by (\ref{eqres1}) needs too much CPU time to
calculate even if the order of approximation is not very high, and
thus is often useless in practice. To avoid the time-consuming
computation, Liao in \cite{Liao2003book} suggested to investigate
the convergence of some special quantities, which often have
important physical meanings. For example, one can consider the
convergence of $u'(0)$ and $u''(0)$ of a nonlinear differential
equation (\ref{eq1}), if they are unknown. It is found by the
homotopy analysis researchers that there often exists such a
region that certain values of $h$ give a convergent series
solution of such kind of quantities. Besides, such a region can be
found, although approximately, by plotting the curves of these
unknown quantities versus $h$. These curves are called {\em
$h$-curves} or {\em curves for convergence-control parameter},
which have been successfully applied in many nonlinear problems as
cited herein. This approach constitutes another way of finding
proper values of $h$ for the Theorem 1.

\vspace{0.2in}{\bf Remark 4.} Similar to the constant $h-$curves
idea of Liao, an approximate interval of convergence for $h$ can
be determined by application of Theorem 1 to some certain physical
quantities, see Example 3 below.

\section{Illustrative Examples}\label{Examp}
To illustrate the validity of the Theorems outlined, we take into
account the following examples taken from the homotopy analysis
studies in the literature.

\vspace{0.2in}{\bf Example 1.} Consider the first-order
nonlinear differential equation \cite{Liao2003book}
\begin{eqnarray}\label{eq12}
   u'+u^2 &=& 1,\quad u(0) = 0,
\end{eqnarray}that governs the steady free convection flow over
a vertical semi-infinite flat plate which is embedded in a fluid
saturated porous medium of ambient temperature \cite{Crane70} and
also the steady-state boundary-layer flows over a permeable
stretching sheet \cite{Keller2000}. In accordance with Theorem 1,
choosing $u_0(t)=1-e^{-2t}$ and $L={d\over dt}+2$, the homotopy
series solution via the homotopy approach (\ref{eq2}) can be
straightforwardly constructed. Employing the $L^2$ norm in $R$,
then Theorem 1 assures the convergency of (\ref{eq12}) provided
that
\begin{eqnarray}\label{eq13}
  {\|u_{n+1}(t)\| \over \| u_n(t)\|} <1.
\end{eqnarray}
Table \ref{Table 1} presents evolution of the ratio (\ref{eq13})
for a variety of the convergence control parameter $h$ for the
problem (\ref{eq12}). Additionally, Table \ref{Table 1} gives the
error, see the last column, defined by
\begin{eqnarray}\label{eq14}
  err &=& \int_0^\infty|u_e(t)-u(t)|dt,
\end{eqnarray}where $u_e(t)$ is the solution of (\ref{eq12})
calculated numerically. Data displayed in Table \ref{Table 1}
clearly explains why the homotopy analysis method generates
completely convergent series solution to the problem (\ref{eq12})
for the chosen parameters $h$.
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
  \hline $h$ &$M=1$    & $M=10$  & $M=20$  & $M=30$  & $M=40$  & $M=50$  & $err$\\\hline
  2.0        & 0.44721 & 0.87228 & 0.93102 & 0.95275 & 0.96407 & 0.97044 & $9.609\times 10^{-5}$ \\
  1.5        & 0.15811 & 0.43509 & 0.46523 & 0.47625 & 0.48196 & 0.48517 & $3.768\times 10^{-20}$\\
  1.3        & 0.15811 & 0.31877 & 0.33354 & 0.33881 & 0.34152 & 0.34304 & $8.261\times 10^{-26}$\\
  1.0        & 0.31623 & 0.45743 & 0.47701 & 0.48425 & 0.48802 & 0.49015 & $8.384\times 10^{-18}$\\
  0.5        & 0.65192 & 0.69906 & 0.71905 & 0.72798 & 0.73295 & 0.73583 & $1.492\times 10^{-8}$ \\\hline
\end{tabular}
\end{center}
\caption{The evolution of the ratio (\ref{eq13}) for the equation
(\ref{eq12}). Last column corresponds to the error defined by
(\ref{eq14}).}\label{Table 1}
\end{table}

The optimal value of convergence control parameter $h$ is further
calculated from equations (\ref{eqres1}-\ref{eqres2}) only for
$M=7$ as $h=1.30405$, which strongly indicates the reason of the
faster converge of the homotopy series near this value, see Table
\ref{Table 1}. Moreover, the uniform validity region of the
homotopy series solution (\ref{eq5}) can also be analytically
evaluated for the special value of $h=1$. In this particular case,
the terms in the homotopy series (\ref{eq5}) satisfy the ratio
\begin{eqnarray}\label{eq15}
  u_{n+1}(t)\over u_n(t) &=& {1\over 2}(1-e^{-2t}).
\end{eqnarray}Hence, regarding (\ref{eq15}), Theorem 1 assures
the convergence of the corresponding homotopy series (\ref{eq5})
for all values of $t$ valid in the interval $t>-{\ln 3\over 2}$.

\vspace{0.2in}{\bf Example 2.} Consider now the second-order
nonlinear differential equation
\begin{eqnarray}\label{eq16}
   2u''+u-u^2 &=& 0,\quad u(0) = 0,\quad u(\infty)=1,
\end{eqnarray}that governs the steady mixed convection flow past a plane
of arbitrary shape under the boundary layer and Darcy-Boussinesq
approximations \cite{Keller2001}. With the choices of $L={d^2\over
dt^2}-1$ and $u_0(t)=1-e^{-t}$, the homotopy (\ref{eq2}) generates
a homotopy series (\ref{eq5}) whose ratios of successive terms are
tabulated in Table \ref{Table 2} at several order of
approximations for some selected values of $h$. As observed from
the Table, $h=1.2$ generates divergent homotopy solution, and also
the convergence of the homotopy series to the exact solution takes
place at a considerably slow rate near $h=1$ as compared to the
smaller values of $h$.
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
  \hline $h$ &$M=1$    & $M=10$  & $M=20$  & $M=30$  & $M=40$  & $M=50$  & $err$\\\hline
  1.2        & 0.42070 & 1.24479 & 1.31373 & 1.34407 & 1.35657 & 1.36525 & $8828.16$              \\
  1.0        & 0.24684 & 0.88919 & 0.94119 & 0.96001 & 0.96970 & 0.97513 & $3.444\times 10^{-4}$\\
  0.8        & 0.21484 & 0.63937 & 0.68278 & 0.69872 & 0.70700 & 0.71167 & $1.332\times 10^{-9}$\\
  0.6        & 0.36425 & 0.69932 & 0.74524 & 0.76240 & 0.77137 & 0.77643 & $1.753\times 10^{-7}$\\\hline
\end{tabular}
\end{center}
\caption{The ratio of successive terms in the homotopy series
corresponding to equation (\ref{eq16}). Last column corresponds to
the error defined by (\ref{eq14}).}\label{Table 2}
\end{table}

The optimal value of convergence control parameter $h$ is
calculated from equations (\ref{eqres1}-\ref{eqres2}) only for
$M=7$ as $h=0.73258$, which explains why the homotopy series
(\ref{eq5}) should converge faster near this value to the solution
of (\ref{eq16}), see Table \ref{Table 2}.

\vspace{0.2in}{\bf Example 3.} Consider now the nonlinear partial
differential Burger's equation
\begin{eqnarray}\label{eq17}
   u_t+uu_x &=& u_{xx},\quad u(x,0) = 2x,
\end{eqnarray}
that has been found to describe various kind of phenomena, such as
a mathematical model of turbulence and the approximate theory of
the flow through a shock wave traveling in a viscous fluid
\cite{Benton66a}. Equation (\ref{eq17}) admits an exact solution
given by
\begin{eqnarray}\label{eq18}
   u(x,t) &=& \frac{2x}{1+2t}.
\end{eqnarray}

To approximate the exact solution (\ref{eq18}), we choose the
auxiliary parameters as $u_0(x,t)=2x$ and $L={\partial \over
\partial t}$. Then, the homotopy (\ref{eq2}) turns out to be
\begin{eqnarray}\label{eq19}
   \left(1+\frac{1-p}{h p}\right)u_t(x,t,p)+u(x,t,p)u_x(x,t,p)-
   u_{xx}(x,t,p) = 0,\; u(x,0,p)=2x.
\end{eqnarray}
Equation (\ref{eq19}) produces the below homotopy series for the
solution of (\ref{eq17})
\begin{eqnarray}\label{eq19a}
   u(x,t) =
   2 x - 4 h t x + 4 h t (-1 + h + 2 h t) x -
 4 h t (-1 + h + 2 h t)^2 x \\\nonumber + 4 h t (-1 + h + 2 h t)^3 x -
 4 h t (-1 + h + 2 h t)^4 x+\cdots,
\end{eqnarray}whose convergence to the exact solution (\ref{eq18})
takes place for the values satisfying
$\lim_{n\to\infty}{|u_{n+1}|\over |u_n|}=|1 - h (1 + 2 t)|<1$.
Thus, in accordance with Theorem 1, this holds exactly for the
values of $-\frac{1}{2}<t<\frac{2-h}{2 h}$ together with $0<h<2$.
It is easy to demonstrate that the case $h=1$ corresponds to the
traditional Taylor series expansion of the solution (\ref{eq18}),
which is only valid in the region $-\frac{1}{2}<t<\frac{1}{2}$.

\vspace{0.2in}{\bf Example 4.} Consider now the following
fourth-order parabolic partial differential equation arising in
the study of the transverse vibrations of a uniform flexible beam
\cite{Gorman75}
\begin{eqnarray}\label{eq20}
   u_{tt}+({y+z\over 2\cos x}-1)u_{xxxx}+({z+x\over 2\cos y}-1)u_{yyyy}
   +({x+y\over 2\cos z}-1)u_{zzzz}&=& 0,\\\nonumber
   u(x,y,z,0) =-u_t(x,y,z,0)= x+y+z-(\cos x+\cos y+\cos z),
\end{eqnarray}
whose exact solution is given by
\begin{eqnarray}\label{eq21}
   u(x,t) &=& (x+y+z-\cos x-\cos y-\cos z)e^{-t}.
\end{eqnarray}
To approximate the exact solution (\ref{eq21}), if we choose the
auxiliary parameters respectively, $u_0(x,t)=(x+y+z-\cos x-\cos
y-\cos z)(1-t)$ and $L={\partial^2 \over \partial t^2}$, the
homotopy (\ref{eq2}) for $h=1$ then generates the subsequent
homotopy series
\begin{eqnarray}\label{eq22}
  u(x,t) &=& \sum_{n=0}^\infty u_n=\sum_{n=0}^\infty(x+y+z-\cos x-\cos y-\cos z)
  \left(\frac{t^{2n}}{(2n)!}-\frac{t^{2n+1}}{(2n+1)!}\right)
\end{eqnarray}which leads to $\lim_{n\to \infty} {|u_{n+1}|\over |u_n|}=0$.
So, the homotopy series (\ref{eq22}) converges to the exact
solution of (\ref{eq20}) for all $t$.

\vspace{0.2in}{\bf Example 5.} Let us consider the linear partial
differential equation
\begin{eqnarray}\label{eq23}
   u_t+u_x-2u_{xxt}&=&0,\quad u(x,0) =e^{-x},
\end{eqnarray}
whose exact separable solution is given by
\begin{eqnarray}\label{eq24}
   u(x,t) &=& e^{-x-t}.
\end{eqnarray}
To approximate the exact solution (\ref{eq24}), if we choose the
auxiliary parameters $u_0(x,t)=e^{-x}$ and $L={\partial \over
\partial t}$ respectively, then the homotopy (\ref{eq2}) turns out to be
\begin{eqnarray}\label{eq25}
   (1-p)u_t(x,t,p)+ p h(u_t(x,t,p)+u_x(x,t,p)-2
   u_{xxt}(x,t,p)) = 0,\; u(x,0,p)=e^{-x}.
\end{eqnarray}
When $h=1$, it appears that the homotopy series solution is
convergent only for $t\leq 0$, that is out of physical interest.
On the other hand, for $h=-1$, the successive ratio yields
$$|{u_{n+1}(t)\over u_n(t)}|={|t|\over n+1},$$ whose limit gives
rise to zero. This explains why the homotopy series solution
(\ref{eq5}) in this case represents the real physical solution
over $t\geq 0$ for $h=-1$, as also explained in \cite{Liang2009}.

\vspace{0.2in}{\bf Example 6.} As a final example, we consider the
classical Blasius flat-plate flow problem of fluid mechanics
governed by the nonlinear initial-value of third-order
\cite{Liao99a}
\begin{eqnarray}\label{eq26}
   y'''+{yy''\over 2} &=& 0,\quad y(0)=y'(0)=y'(\infty)-1=0,
   \quad \eta \in [0,\infty).
\end{eqnarray}With the transformations $$y={u\over \lambda},\quad t=\lambda \eta,$$
where $\lambda$ is a scaling parameter taken here as 4, system
(\ref{eq26}) is converted into
\begin{eqnarray}\label{eq27}
   u'''+{uu''\over 2\lambda^2} &=& 0,\quad u(0)=u'(0)=u'(\infty)-1=0,
   \quad t\in [0,\infty).
\end{eqnarray}
Taking $L={d^3\over dt^3}+{d^2\over dt^2}$ with
$u_0(t)=-1+t+e^{-t}$, the values for the convergence control
parameters $h$ from the homotopy (\ref{eq2}) are shown in Table
\ref{Table 3} along with the corresponding residual errors. It is
seen that the optimal value for the physical problem considered is
$h=-3/2$. Figures \ref{fig1}(a-d) demonstrate the ratios
(\ref{rat}) in Theorem 1 which were computed from the homotopy
series (\ref{eq5}) for the values of $h$ given in Table \ref{Table
3}. The convergence is assured further by the values of the ratios
less than unity, suggesting that limit tends to 0.83, 0.77, 0.76
and 0.78, for the values of $h$ shown.
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
  \hline $h$ & $M=1$    & $M=10$                & $M=20$               & $M=30$               \\\hline
  1.6        & 1.258621 & $1.201\times 10^{-3}$  & $2.463\times 10^{-7}$ & $1.032\times 10^{-6}$ \\
  1.5        & 1.027450 & $1.372\times 10^{-4}$  & $3.602\times 10^{-7}$ & $2.573\times 10^{-9}$ \\
  1.4        & 0.820771 & $1.399\times 10^{-4}$  & $7.621\times 10^{-7}$ & $5.454\times 10^{-9}$ \\
  1.0        & 0.238796 & $7.621\times 10^{-4}$  & $1.511\times 10^{-5}$ & $4.540\times 10^{-7}$ \\\hline
\end{tabular}
\end{center}
\caption{The square residual errors $Res(h)$ for the homotopy
series solutions corresponding to equation (\ref{eq27}) at several
values of convergence control parameters $h$.}\label{Table 3}
\end{table}

\begin{figure}[!htb]
\begin{center}
\mbox{ \subfigure[]{
\includegraphics[width=6cm,height=5cm]{graph1.eps}}
\subfigure[]{
\includegraphics[width=6cm,height=5cm]{graph2.eps}}}
\mbox{ \subfigure[]{
\includegraphics[width=6cm,height=5cm]{graph3.eps}}
\subfigure[]{
\includegraphics[width=6cm,height=5cm]{graph4.eps}}}
\caption[Basic flow]{A list plot of the ratios $rat$ from the
theorem to reveal the convergence of the HAM solutions for the
Blasius equation (\ref{eq27}) for different choices of $h$. (a)
$h=-1$, (b) $h=-1.4$, (c) $h=-1.5$ and (d) $h=-1.6$.} \label{fig1}
\end{center}
\end{figure}

\section{Concluding remarks}
\label{conclusions} In this paper, the homotopy analysis method
has been analyzed with an aim to investigate the conditions which
result in the convergence of the generated homotopy solutions of
the nonlinear ordinary and partial differential equations. The
theorems outlined in the paper have proved that if specific values
are assigned to the auxiliary parameters in the homotopy analysis
method, then the approximate homotopy results successfully
converge to the exact solution. An optimal value approach for the
convergence control parameter has also been given. Examples have
been provided to verify the theory. Via the theorems provided
here, not only the question of the convergence of the homotopy
series is answered, but also the region of validity of the space
variable ensuring the convergence is determined. The traditional
flat-plate boundary layer flow problem has been finally treated by
the convergence theorem.
\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


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