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Sunset Beach was een Amerikaanse soapserie die werd uitgezonden op NBC om 11 uur 's ochtends van 6 januari 1997 tot 31 december 1999. Het werd in 70 landen uitgezonden en was onder meer erg populair in het Verenigd Koninkrijk, Finland, Australië, Zweden, Israël, Duitsland en Nieuw-Zeeland.

Verhaallijnen
Een van de eerste verhaallijnen draaide om Meg, een jongedame uit Kansas die ontdekt dat haar verloofde Tim een affaire heeft met haar bruidsmeisje. Ze laat hem voor het altaar staan en gaat naar Sunset Beach, waar iemand woont die ze via het internet kent. Dit blijkt de rijke weduwnaar en advocaat Ben Evans te zijn. Ze zet haar zinnen op hem en krijgt een hekel aan zijn vriendin Annie. Zij wordt ook jaloers op haar aanwezigheid. Uiteindelijk gaat ook Tim naar Sunset Beach om Meg terug te winnen en krijgt hier een romantische connectie met Annie. Problemen ontstaan als Meg begint te vermoeden dat Ben een moordenaar is.

Meg en Ben worden een van de eerste koppels van de soapserie, samen met juwelendief Cole en Caitlin. Cole bezwangert uiteindelijk zowel Caitlin als diens ongelukkig getrouwde moeder Olivia. Olivia en Gregory plannen om Caitlins baby te stelen en deze te opvoeden als hun eigen, omdat ze haar relatie met Cole afkeuren vanwege zijn criminele verleden. Caitlin verliest uiteindelijk haar kind bij een auto-ongeluk. Als ze achter hun plan komt, huurt ze Annie in om hen uit elkaar te drijven. Annie steelt uiteindelijk Olivia's baby en zegt dat deze dood is geboren. Pas als Olivia achter de waarheid komt, wordt ze herenigd met haar zoon. Andere verhaallijnen gingen over bijvoorbeeld Virginia, die badmeester Michael en Vanessa uit elkaar probeert te drijven omdat ze Michael voor zichzelf wil.

Controverses
Er speelden verscheidene controversiële verhaallijnen, zoals hoe Virginia Vanessa pillen geeft zodat ze vervolgens sperma kan gebruiken van Dr. Tyus Robinson om haar te bezwangeren. Dit was een poging om haar en Michael uit elkaar te drijven. Michael is namelijk onvruchtbaar.

Een gedeelte van de serie heeft de bijnaam Terror Island gekregen. Dit is een verhaallijn die gaat over een aantal van de hoofdpersonages die op een eiland stranden en hier geteisterd worden door een seriemoordenaar. Het is voornamelijk Meg die constant gestalkt wordt. De moordenaar slacht voornamelijk minder voorkomende personages af, met uitzondering op Mark, een hoofdpersonage. De moordenaar blijkt Derek te zijn, Bens kwaadaardige tweelingbroer. Hij ontvoert zijn broer en doet zich maandenlang voor al Ben. Zo gaat hij met Bens vrouw Maria naar bed. Uiteindelijk wordt hij door zijn broer doodgeschoten.

Maria verdwijnt later plotseling en er werd vermoed dat ze is verdronken. Ben trouwt met Meg, waarna Maria terugkeert. Later komt ook Tess ten tonele, een vrouw die beweert het kind Benjy van Ben en Maria te hebben. Op dat moment keert Derek terug. Hij blijkt nog te leven en het wordt ook duidelijk dat hij en Tess de ouders van Benjy zijn.

Op een gegeven moment wordt Sunset Beach door een tsunami/aardbeving getroffen. Verscheidene hoofdpersonages belanden in een val en kunnen geen kant meer op. Ook zijn veel personages op een cruiseschip dat door de tsunami omslaat.

Amerikaanse soapserie

redpajama_set_name: RedPajamaWikipedia


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Democrats Must Control Speech & Firearms to Control Americans
by Roger Katz
Joe Biden with a sign behind him telling a vert uncomfortable truth for the Democratic Presidential Candidate. IMG NRA-ILA
U.S.A. –-(AmmoLand.com)- The Radical Left Democrats, Collectivists all, are sly, deceitful creatures. They erode our fundamental rights under the cloak of morality and pragmatism, hoping that few Americans will notice.
Consider the Biden campaign's war on the fundamental right of free speech. Recently, Biden and other Collectivists argue that free speech ought not to extend to "hate speech." Superficially, that may seem reasonable to some Americans. But is it? What constitutes "hate speech?" Indeed what constitutes "speech" as free expression under the Constitution. Does Flag Burning constitute "speech" protected under the First Amendment? Does display of firearms at rallies constitute "speech" protected under the First Amendment? Is the one but not the other an example of "hate speech?" If so, how does one make that determination?
As one academic writer aptly said: "Hate speech is a vague concept with varying definitions. Generally, it includes speech that is abusive, offensive, or insulting that targets an individual's race, religion, ethnicity, or national origin."*
The author continues, "Regulating hate speech in the United States is problematic because of the value the nation places on free speech. The First Amendment to the U.S. Constitution provides that, 'Congress shall make no law . . . abridging the freedom of speech, or of the press. . . .'" The author adds, "There are three prominent justifications for protecting free speech: (1) it acknowledges human autonomy and dignity, (2) it promotes the marketplace of ideas, and (3) it is an effective tool of democracy."
But the Democrats and other Collectivists of all stripes—Marxists, Communists, Socialists, Globalists, Anarchists, and others—will have none of that.
But assuming that Congress could devise an operational definition of 'hate speech,' would such statute prohibiting such speech still conflict with the First Amendment? Yes! The U.S. Supreme Court has made this point clear, succinct, and categorical, opining, in Snyder V. Phelps, 562 U.S. 443 (2011):
"Such [hate] speech cannot be restricted simply because it is upsetting or arouses contempt. 'If there is a bedrock principle underlying the First Amendment, it is that the government may not prohibit the expression of an idea simply because society finds the idea itself offensive or disagreeable.' Texas v. Johnson, 491 U.S. 397 (1989). Indeed, 'the point of all speech protection . . . is to shield just those choices of content that in someone's eyes are misguided, or even hurtful.' Hurley v. Irish-American Gay, Lesbian and Bisexual Group of Boston, Inc., 515 U.S. 557 (1995).
But Collectivists don't give a damn about the First Amendment's freedom of speech clause; nor do they give a damn about the High Court's interpretation of it. As a prime example of what this means, what this entails, consider the Collectivists' seamy, degenerate attacks on Zuckerberg's social media vehicle, Facebook. The Collectivists' have recently vented their fury on Zuckerberg's Facebook. And, the toady and mentally deficient, presumptive Democrat Party nominee for U.S. President, Joe Biden, in whose name the Collectivists present their aims to the American public, doesn't really have a clue what is going on all around him; how it is the Collectivist puppet masters are playing him for the fool he is and parading him, now and then, before the public.
Of course, Biden's policy planks, marching ever leftward toward a cliff, are and must be coextensive with those of the Collectivists, who are feeding Biden his lines; his messages. They have simply stepped in his shoes, and, in his dim-witted muddled mind, Biden accepts whatever his handlers require of him, understanding nothing, and caring little, if at all, of the clown he has become; his words meaningless jabber, both to him and everyone else.
The website, Reason, says, "After being asked by the Times about previous comments Biden has made regarding Facebook's refusal to remove negative ads targeting his campaign, the Democratic front-runner attacked both the social media platform and its CEO, Mark Zuckerberg.
'I've never been a fan of Facebook,' Biden says. 'I've never been a big Zuckerberg fan, I think he's a real problem.'
Biden and Facebook have been feuding for months, as Reason has previously covered. In an October letter to Facebook, Biden's campaign called on the social media site to reject political ads containing 'previously debunked content'—like a Trump campaign ad linking Biden and his son, Hunter, to corruption in Ukraine. Shortly afterwards, Zuckerberg said the company's policies were 'grounded in Facebook's fundamental belief in free expression, respect for the democratic process, and the belief that, in mature democracies with a free press, political speech is already arguably the most scrutinized speech there is.'
Zuckerberg is correct, but that didn't sit well with Biden. In a CNN town hall event in November, Biden said he would be willing to rewrite the rules for all online platforms in order to force social media companies to 'be more socially conscious.'"
The Collectivists have gone to task on Zuckerberg. On July 9, 2020, as reported in the NY Times, Facebook's "auditors," said, "the prioritization of free expression over all other values such as equality and nondiscrimination is deeply troubling."
Deeply troubling to whom? The Collectivist censors? Apparently, these Facebook auditors aren't familiar with the critical importance of the First Amendment in a free Constitutional Republic.
"Free expression" isn't a mere "value," it's a fundamental, unalienable, immutable, illimitable, natural right, bestowed on man by the Divine Creator, and its meaning is straightforward. The expressions, 'equality,' and 'non-discrimination,' though, are vague concepts and apply to aspirations, not fundamental rights.
In the absence of explication, those expressions, 'equality' and 'non-discrimination that the Facebook auditors mention, do not denote "rights," fundamental or secondary.
People as individuals are decidedly unequal. Some have been blessed with one or more gifts such as intelligence, or beauty, or athletic ability, or business acumen. Others do not have such gifts. In terms of talents, abilities, physical features, and even with respect to motivations and drives, people are decidedly and decisively unequal. Yet, even in physical, mental, and emotional attributes, Collectivists strive to force commonality and uniformity on everyone, destroying that especial aspect of a person that makes people individual souls. This has nothing to do with equal protection under the law as guaranteed under the Fourteenth Amendment to the U.S. Constitution.
But, apropos of "free speech," everyone has an "equal" right to say their mind. If someone's words hurt me, then all the worse for me. If my words hurt another, then all the worse for him. But all the worse for both of us and our Nation if the Collectivist censors determine what either of us can assert verbally or in writing, thereby denigrating and curbing the force of the free speech clause of the First Amendment.
Similarly, people discriminate all the time: in terms of their interests, their proclivities, the people with whom they choose to associate or not, and in terms of their political and social and religious preferences; and, while the law prohibits discrimination, as for example, on the basis of race, color, age, or sex; and as, for another, example, in employment, and in restaurant or hotel accommodations, anti-discrimination laws are statutory constructs, not fundamental rights.
But, Collectivists subsume aspirations to the level of fundamental rights; raise secondary man-made rights, such as 'abortion,' to the level of fundamental rights; and dismiss out-of-hand rights that are natural, fundamental, God-given, such as the right of the people to keep and bear arms as codified in the Second Amendment.
Biden and his handlers have made clear that preservation of the Second Amendment does not factor into their Party plank. While some Collectivists, like retired Associate Justice John Paul Stevens would strike the Second Amendment from the Bill of Rights altogether, Biden and the Democrats are, at the moment at least, circumspect about their intentions, couching the denial of the right in terms of a desire to curb "gun violence." See "The Biden Plan To End Our Gun Violence Epidemic."
And keep in mind how the Biden gun safety plank insinuates the First Amendment into the Second Amendment, and observe how the Destructors of our Nation don't attack the Second Amendment head-on, but obliquely:
"Close the 'hate crime loophole.' Biden will enact legislation prohibiting an individual 'who has been convicted of a misdemeanor hate crime, or received an enhanced sentence for a misdemeanor because of hate or bias in its commission' from purchasing or possessing a firearm."
This so-called "hate crime loophole" would add another criterium to the Federal Penal Code, denying a person the right to possess firearms for "thought" crimes. Eventually, the Collectivists wouldn't even bother to use the excuse of a misdemeanor conviction to deny an American the right to keep and bear arms. If one's speech is construed as "hate speech," that would be enough to deny a person the right to own and possess firearms, expanding the domain of those not permitted to own firearms, exponentially. Would Collectivists argue that merely to desire to own and possess a firearm is tantamount to "hate speech" on its face? Considering how far the Radical Left Collectivists have come since Charlottesville—defacing the monuments of Confederate War Heroes—to arguing for the removal of monuments to the Father of our Nation, George Washington, and to the other Founders, there is no limit to the extravagant outrageous, laws, rules, regulations, ordinances, and executive orders that will come down the pike if the Collectivists take control over all three Branches of Government.
But if Radical Left Marxist control of all thought, deed, and action in our Country is what you fancy, then feel free to give a sawbuck or two to Biden's campaign at "Can you donate to Elect Joe Biden?" I'm sure he would appreciate it.
This entry was posted on Saturday, July 18th, 2020 at 2:10 pm	and is filed under Communists, Conspiracy, Corruption, Free Speech, Government Evil, Gun Rights, New World Order. You can follow any responses to this entry through the RSS 2.0 feed.

redpajama_set_name: RedPajamaCommonCrawl


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Research Shows a (Maybe Not So) Surprising Lack of Black Representation in Award Wins
Cecilia De La Viesca
It's not news that diversity and representation is lacking across many industries and areas. While there have been efforts to improve this over the years, we still have a long way to go – and the first step towards getting better is recognising this.
October is Black History Month. In honour of this important initiative, we wanted to raise awareness about the lack of black representation, particularly when it comes to prestigious awards wins for the Oscars, the Golden Globe Awards and the Nobel Prize.
This year's theme for Black History Month is Time for Change: Action Not Words. We wanted to put a spotlight on the lack of diversity throughout various industries, encourage society to embrace the spirit of inclusivity and recognise the often overlooked accomplishments of black people. We know that we can all do more and we firmly believe that every effort helps, no matter how small!
With all that in mind, we've crunched the numbers and collated a report that illustrates the shockingly low number of awards won by black people across various categories as opposed to their non-black counterparts. Spoiler alert: it's not equal representation.
How Many Black People Have Won an Oscar?
Our research revealed that only 74 black people have ever won an Oscar, an award ceremony that has been hosted since 1929 – despite the fact 3,140 statuettes to date have been awarded.
The shocking data also revealed that no black person has ever won an Oscar in categories like Best Cinematography, Best Director, Best Film Editing and Best International Feature Film. And when it comes to gender representation, only one black woman won an Oscar for Best Actress in Leading Role over the years – yet five black males won Best Actor in a Leading Role.
How Many Black People Have Won a Golden Globe?
When it comes to Golden Globe Awards, only 54 black people won an award since they were established in 1944. According to the study, only two black women have ever won the prestigious award for 'Best Performance by an Actress in a Motion Picture – Drama' and only five black men have won for 'Best Performance by an Actor in a Motion Picture – Drama'.
How Many Black People Have Won the Nobel Prize?
The Nobel Prize has been awarded to 954 laureates between 1901 and 2022, of whom only 17 black people ever received this award – 1 Nobel Prize in Economics, 4 Nobel Prizes in Literature and 12 Nobel Peace Prizes. No black person has ever won a Nobel Prize in Physics, Medicine or Chemistry categories.
There is an increasing acknowledgement that diversity and inequality issues exist but there is still a long way to go until the balance is achieved. More companies and organisations should consider having diversity and inclusion at the forefront of their agendas. Everything starts with awareness and helping to make sure the black voices are heard and their achievements are recognised.
How Passion is Embracing Diversity in Digital Marketing
At Passion, we believe in diversity in the workplace, and that people from all different backgrounds, regardless of their race, ethnicity, gender, sexuality or religion, should be given equal opportunity to succeed in a workplace. We pride ourselves on having employees of different cultural backgrounds, allowing us ample learning opportunities about different countries, histories and more. We believe that Passion is a safe space to encourage these conversations.
We are proud that our company is celebrating Black History Month this year, and we will continue this dialogue all year long – not just throughout October.
To celebrate Black History Month we have come up with an itinerary for the month, including sharing learning and reading material, enjoying African and Caribbean food supplied by one of our partner Charities, Platform Cafe, and inviting guest speakers into Passion HQ to talk about their experience as black women, and share some advice on how companies can become more inclusive. And it doesn't end there, we're hosting a movie night with popcorn and watching a film to end BHM celebrations.
Afsana Begum, Executive Assistant at Passion Digital, says:
"For myself personally, Black History Month has different connotations. On one hand, we get to remember and celebrate prominent black figures who have made significant changes in society, such as Martin Luther King and Rosa Parks and on the other, it is a month of reflection and acknowledging the imbalances in society that have persisted throughout history and still persist today.
I feel proud that Passion is celebrating BHM and that we get to take part in series of events to embrace African and Caribbean culture, but also, we believe that it is important to have conversations around topics that may seem 'difficult' or 'uncomfortable', but in reality are important for our knowledge and understanding, such as slavery and oppression of black people."
If you are interested in working for a company that cares about diversity and inclusion, keep an eye on our recruitment page for career opportunities.


Edit your thesis statement in the box above so that the parts of the thesis flow smoothly, check for proper grammar and standard spelling. In one or two sentences, present your thesis, including a qualification, a reason, and a position. The first time the in-text reference is cited you must include, authors first name, authors last name, title, place of publication, publisher name, year and referenced pages.
In fact, a neurosis is characterized by anxiety (kristen & warb, 2012, p. Apa style encourages the inclusion of page numbers, but it is not mandatory. In the pop-up window, you can print your thesis or save a copy to your computer by going under the file menu.
What is one thing about your topic that you believe to be true, and that you wish to argue? You may already have a position in mind, or you may just have a general topic you know you wish to work with. This immediately demonstrates your interest in accuracy. If the quotation is over 40 words, you must indent the entire quotation and start the quotation on a new line.
Use italics for title of a periodical, a book, a brochure or a report. Alberta is occasionally divided into two regions, northern alberta and southern alberta. Cite the quoted source after the final punctuation mark.
Include all authors last names the first time the citation is used. Dont worry if its not a complete sentence. If your work cited page contains multiple references by the same author, include the authors last name and a full or shortened title of their work and the page number.
When you are satsified with your thesis statement click on the final draft button. Heres a trick begin your qualification with a word like although or it is true that. However you get there, write a short statement describing your position in the space below. In-text citations are required when you use someone elses ideas, theories or research in your paper. In chicago style, footnotes or endnotes are used to reference pieces of work in the text.
Thesis Question Generator When paraphrasing from multiple sources, include all authors name in parentheses in alphabetical order. This was his masters thesis project. Cite the quoted source after the final punctuation mark. These are placed in parentheses at end of sentence before period. Apa style encourages the inclusion of page numbers, but it is not mandatory. Heres a trick begin your qualification with a word like although or it is true that. Notice the placement of the period is before the brackets for long quotations. Check that your document does not fall under any of the other categories (e. Use italics for title of a periodical, The majority of albertas population is located in large urban cities. If the citation has been referenced immediately prior, the note may be shortened even further to ibid with the page number. Use the title in place of author. The first time the in-text reference is cited you must include, authors first name, authors last name, title, place of publication, publisher name, year and referenced pages. Include page or paragraph numbers if it will help reader find the information. The majority of albertas population is located in large urban cities, Alberta is occasionally divided into two regions.
. Heres a trick begin your qualification with a word like although or it is true that. Cite the quoted source after the final punctuation mark. If your work cited page contains multiple references by the same author, include the authors last name and a full or shortened title of their work and the page number. Use italics for title of a periodical, a book, a brochure or a report.
Each superscript then refers to a numbered citation in the footnotes or endnotes. What is one thing about your topic that you believe to be true, and that you wish to argue? You may already have a position in mind, or you may just have a general topic you know you wish to work with. In general why do you believe your position to be correct in spite of your qualification? This is a general statement your specific reasons will follow in the body of your essay. This can be done by inserting a parenthetical reference in your text at the spot where you have used the sources ideas or words. Question write the the question you have been assigned or the prompt you are going to answer with your essay in the box below.
In this context, may refer not only to one or more authors or an institution but also to one or more editors, translators, or compilers. Include page or paragraph numbers if it will help reader find the information. If authors name is in text only use the page number, it is not necessary to repeat the authors name. You should keep parenthetical references as brief and as few as clarity and accuracy permit. When you are satsified with your thesis statement click on the final draft button. In fact, a neurosis is characterized by anxiety (kristen & warb, 2012, p. Spell out the name in full the first time and abbreviate subsequent times only if abbreviation is well known. ). In-text citations are called parenthetical references in mla. The classic, traditional way of combining is to first present your qualification.
Authors last name followed by the page number referenced in work. ). In the pop-up window, you can print your thesis or save a copy to your computer by going under the file menu. The information in the parenthetical references must match the corresponding information in the list of works cited. What do you have an opinion about? what are some things that interest you? Topics are just general categories--school, fishing, parenting, hunting, cars, womens rights, racial issues, the law, etc.
If the citation has already been cited it may be shortened to authors last name, shortened title, and page referenced number. Dont worry if its not a complete sentence. Heres a trick begin your qualification with a word like although or it is true that. The last example shows how one might cite a section of a work that contains no page or section numbers or other numerical signpoststhe case for some electronic documents (see thank you to all those who emailed us with their suggestions for improvements. For each author-date citation in the text, there must be a corresponding entry in the reference list under the same name and date.
Use the title in place of author. If the citation has already been cited it may be shortened to authors last name, shortened title, and page referenced number. Directions this web page explains the different parts to a thesis statement and helps you create your own. The last example shows how one might cite a section of a work that contains no page or section numbers or other numerical signpoststhe case for some electronic documents (see thank you to all those who emailed us with their suggestions for improvements. When a volume as a whole is referred to, without a page number, ).
When you are satsified with your thesis statement click on the final draft button. Include all authors last names the first time the citation is used. In fact, a neurosis is characterized by anxiety (kristen & warb, 2012, p. Abbreviations such as when a specific page, section, equation, or other division of the work is cited, it follows the date, preceded by a comma. In chicago style, footnotes or endnotes are used to reference pieces of work in the text.
Include all authors last names the first time the citation is used. Alberta is canadas most populous province of all three canadian prairie provinces. Use when citing an unpublished web document. Then present your general reason which demonstrates your thinking process, and finally the punch line--your position. Include page or paragraph numbers if it will help reader find the information.
If the quotation is over 4 lines, you must indent the whole quotation and start the quotation on a new line. Authors last name followed by the page number referenced in work. To cite from a source a superscript number is placed after a quote or a paraphrase. An author-date citation in running text or at the end of a block quotation consists of the last (family) name of the author, followed by the year of publication of the work in question. You can click on the example button in each section to see an example of a thesis statement.
Another good way is to begin a free write, im supposed to write an essay about. Edit your thesis statement in the box above so that the parts of the thesis flow smoothly, check for proper grammar and standard spelling. For each author-date citation in the text, there must be a corresponding entry in the reference list under the same name and date. These are placed in parentheses at end of sentence before period. If the quotation is over 4 lines, you must indent the whole quotation and start the quotation on a new line.
If authors name is in text only use the page number, it is not necessary to repeat the authors name. ). Programs example qualification although schools of over a thousand students have flourished in america. If you use the same citation again within the same paragraph, use only the first last name followed by et al. In this context, may refer not only to one or more authors or an institution but also to one or more editors, translators, or compilers.


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"Let me tell you what I really, really do. I put people in prison for 20 years...I know I don't look it," he said, acknowledging his 5'9" height and average build. "But I'm the guy."
The leading cause of death in black men ages 13 to 34 is murder, Tyler said. He has investigated 400 homicides, he said, and "they've got my brown eyes, my tan skin; they remind me of my sons."
Police officers are going to be proactive in neighborhoods, he said. "I have a duty to make sure you grow up," Tyler said.
Trooper Calvin Dye Jr. told the students school administrators had asked the state police for assistance fighting cyber-bullying at the school. Many of the students were clad in school hoodies with "SWAT Stop Walk and Talk" on the back.
"Who can tell me what good character is," he asked. "It's when you do the right thing when an adult isn't watching."
Dye told the students that the murderers, rapists and armed robbers he has talked to have often had one thing in common: starting in fifth or sixth grade, they were bullying other students. He linked it to the cyber world, which the students are more familiar with as many raised their hands to show they have Facebook accounts, and nearly every hand went up to show cell phone ownership.
There are lifelong consequences to the choices they make now, he said, telling of a young teenage girl who sent her boyfriend a picture of herself "in her underclothes."
"How many of you know that at age 10, we can fingerprint you?" he asked. Sending such pictures online is child pornography, he explained, and even a juvenile can be charged with child pornography.
After the presentation, A'Marus Sims, 12, Dejae Phipps, 13, Tamaya Kimble, 11 and Mariah Johnson, 12, agreed that parts of the officers' talk were known to them, but other parts were surprising. One student confided of a bullying incident earlier in the day, that the student had not felt comfortable telling the school administration.
"I watch 'Law & Order,'" Tamaya said, and she already knew about cyber bullying but not about some of the gang information. She said cyber bullying would not be a problem for her, because her parents, aunts and others are all on her social media accounts and keep an eye on her activities.
"I didn't know they can take your fingerprints," A'Marus said.
Tyler has been working with students for about three weeks, but they knew him only as a lieutenant in the U.S. Army, which he is. They've been working on self-esteem skills and anti-bullying, and the students had asked for undercover officers and canine demonstrations, which he arranged.
"I didn't want to come to them as a police officer, kids tend to shut down," he said before the presentation.
Tyler and Dye both talked about how some actions the sixth-graders take now can follow them for the rest of their lives, detailing how extensive the background checks are for law enforcement, military and some other careers.
"Once you're in the system as a gang member, you're in there for life," Tyler said. "Some of you guys are going to be running this country... don't let a little mistake ruin that."
Contact reporter Mary Cooley at mcooley@bnd.com or 618-239-2535. Follow her on Twitter: @MaryCooleyBND.


A Toolkit for those who are exhausted from solving neverending problems, working harder and harder and not arriving at the destination where they truly want to be.
A Map for how to make the journey from head to heart and then integrate the two so that the power of ego is diminished and the Authentic Self can emerge to live and work from the power of the human spirit.
A Reference book you can use for many years to come as the reader meets life's challenges with success that satisfies both the head and the heart.
Journey from Head to Heart is exactly that, integrating logic, reason, emotion, spirituality, recovery, science, and ancient wisdom from a variety of sources to create a recipe for wholeness. The tools and processes are designed for people who are a little wary of "touchy-feely" or "New Age" approaches.
"For beginners on a spiritual voyage, as well as for experienced travelers, Journey From Head To Heart is very powerful. I couldn't put it down. Its stories, told with clarity and simplicity, make it a treasure."
"This book is for people struggling with work/life balance, for entrepreneurs, for those seeking their authentic purpose in life and work. It lays out a plan to get ego out of the way, lead with humility, and communicate so that others are attracted to your cause."
"What separates this book from the majority of self-help manuals is the author's awareness that many people in today's world neglect to acknowledge the importance of personal faith and spirituality in the growth process, and in turn, are unable to integrate their actions with their beliefs."
"Rushing water is the only sound I hear, punctuated by the occasional call of the cardinal atop the tallest tree and the swish of light wind through the tall pines. My eyes follow the sound to the cantilevered boulders and terraced pools that guide the water's path through this place that bridges mountain and valley. Cool droplets glance my shins as I jockey, positioning my body for the best view. But there is no best view, precisely why I feel so safe in this place.
"Here, I know I will not be abused by someone's idea of what I need to do to be the best. My laurels don't matter. Neither do yours. No measuring. No comparing. No pertinent questions: 'Why don't you [do this]? Why can't you [do that]? Have you ever thought about…?' None of that. Just peace.
"No leapfrogging or catapulting here. Every rock has its unique place in the water's glide and fall. Every tree. Every fallen leaf and insect. Every patch of sky and wispy cloud.
"So I come here in silence, in reverence, and in the serenity that knows.
To this point, my greatest handicap in life had been that I appeared attractive, smart, and strong. I looked as if I had it all together, so most people left me alone. But in this place, when I checked in, they discovered I had a hearing problem and had left my hearing aids at home because they needed cleaning. So, for the first class I was respectfully escorted to a seat with earphones wired to the ceiling. Someone gently placed them over my tired ears. What relief! I was so grateful I almost cried. Listening was easy. For the first time, I didn't have to strain to read the teacher's lips.
A few years later, an audiologist explained to me that the type of hearing loss I had, which she called a cookie, was hereditary. I was born with it! The line graph she showed me revealed a normal pattern until the lines suddenly dropped, as if someone had taken a bite out of a cookie. For the first time, I understood why I read lips so well; why I am such an intent, focused listener; why I always sat near the front in my classes; why my parents used to say, Do you hear me!? when I didn't immediately obey.
As I yielded to the imperfection of a hearing impairment, I relaxed. Completely relaxed. Maybe for the first time in my life.
Growing up, what I had learned about love was very confusing. I had taken refuge in school and church. But even in these places I learned that to get approval I had to meet someone else's expectations, which I now realize was conditional love. (i.e. you meet my conditions, and I give you love.) In effect, I made those people and their beliefs my gods.
But in this beautiful, remote place, for 28 days, I lived in an environment in which no one was a favorite. It seemed that everyone was getting what he or she needed. For the first time in my life, the alcoholics were not getting more attention than I was. There was no one I needed to please. It was an amazing difference from what I had experienced out there in the real world. Yet, this new world was also real. I learned that if I didn't follow the rules, I would be corrected gently, with respect. But I would be corrected. I would not be overlooked.
Here, all my books, even my journal, were taken from me and locked up. For 28 days, I was not to exercise my mental faculties. Instead, I was to open more of my emotional and spiritual being. Long hikes were encouraged, and beautiful trails made them enjoyable.
I learned how dependent I had become on books. Some people might even call it an addiction. In a sense, books had also been my gods. If I had a problem, I went to the bookstore or library—not to the chapel to pray. But here I could learn simply by sitting and being quiet. I learned that I didn't have all the answers for other people, even when I thought I did. I learned that I suffer from the halo effect , the notion that I have to be better than everyone else, making a god of my own image of myself. I learned that all addictions grow from four roots: control, security, suffering, and sensation. I learned about process addictions, like addiction to work, exercise, relationships, Internet—even books! I experienced the miracle of witnessing someone (my roommate) finding God for the first time.
Based on watching what areas of the brain activate when stimulated by spiritual language and images, neuroscientists have identified what some call the God spot. It's in the right hemisphere, the same part of the brain that wordlessly appreciates art and creation. As one of my clients put it, Once that God spot lights up, then I can see the tiny lights along the trail leading me forward. Some say we are hard-wired for a relationship with God. Frankly, I don't care about scientific or religious debate that attempts to establish what God is, where God is, who is going to heaven, or who is right. What I care about is living my life as fully as possible. Within me is the capacity, if fleetingly, to leave my ego and be one with God. Those are moments of grace. When they occur, I am at my best. People come. They want me to teach them. As I teach them, I learn and my life continuously improves. This book contains many of their stories. The names I've used are fictitious, to preserve anonymity. As I live out my life, I want to spend more and more moments in a state of grace. That is all.
Contact (1997), a movie adaption of Carl Sagan's 1985 novel of the same name, announced a new frontier for explorers. In the movie, Jodie Foster plays the role of an astronomer named Ellie who receives significant funding to erect huge satellite dishes, like stethoscopes, to tune in to deep space and listen for signs of life. The central question of the movie asks if there is other intelligent life in the universe, and a refrain goes, If it's just us, it seems like an awful waste of space. After an agonizingly long period the sounds begin to come, like a heartbeat, and scientists put the pieces of the puzzle together. In the aggregate, these signals give instructions for building a spacecraft for one person to travel to the heart of the universe. With federal funding the spacecraft is built, and ultimately Ellie is selected as its pilot. Along the way she has fallen in love with Palmer, an author and theologian played by Matthew McConaughey. While she has faith in the pursuit of truth through science, he has faith in the pursuit of truth through God.
At the launch the movie focuses on what happens to Ellie. She is shaken, catapulted, rattled, and hurtled through fantastic light shows with dramatic speed. The compass locket that Palmer has given her is shaken loose and she extricates herself from the chair to retrieve it, symbolically choosing love over her own personal safety. She loses consciousness, and when she awakens she has landed on a calm, perfect beach with white, white sands, lapping blue water, and gentle breezes. Through the plasma-like substance that surrounds her walks her father, as she remembers him from childhood when her love affair with space began as together they tuned into faraway voices on his ham radio. Ever the skeptical scientist, Ellie realizes he isn't real, to which he answers, We thought it would be easier for you this way. When she asks, Are there others? her dad replies, Many others.
Why did you contact us? is her next question.
We didn't contact you; you contacted us, was his reply. His bottom-line message to her: In all our searching, the only thing we find that makes things bearable is each other… This was just a step. When you're ready, you'll take another. Make small moves. The most important thing is to keep searching for your own answers.
When she alights from the spacecraft with only a small facial wound, Ellie finds a dismayed crowd of witnesses who tell her the spacecraft never took off; instead, it crashed into the ocean. Moreover, they said her trip had lasted only a fraction of a second while her estimate of time elapsed was eighteen hours.
Of course, since the project was federally funded there was a Congressional hearing, with Ellie caught in a dilemma. On the one hand, she was a skeptical scientist. On the other hand, I had an experience I can't explain. It was real. It changed me forever. It made me realize how tiny, insignificant, great, and precious we are. We are not alone. I experienced awe, humility, hope. As a scientist, she believed that everything could be explained without the existence of God, but through this experience she had stepped into a spiritual dimension unlike anything she had ever known. She was torn between what she knew from academic, scientific discipline, and what she knew in her heart as true. Thus conflicted, and in the face of no tangible evidence, she is discredited using the argument of Occam's Razor, a medieval scientific principle that affirms that, all things being equal, the simplest explanation has to be the right one. The simplest explanation said this space journey was a hoax.
Palmer, man of faith, is the only person on Capitol Hill who believes her. Her experience aligned with what he knew to be real. He said, As a person of faith, I'm bound by a different covenant, but my goal is the same—the pursuit of truth. At the end of the movie, the two of them drive off together—perhaps a symbolic marriage of science and faith—as a crowd representing the 80-95% of people in the U.S. who believe in God look on.
Bruce Lipton (2005) is another person who found God through science—the science of cellular biology. It startled him to realize that it was the environment, not the genetics, of the cell that controls it.
The nature of the cell's membrane that was 'downloaded' into my awareness'. convinced me that we are immortal, spiritual beings who exist separately from our bodies. I had heard an undeniable inner voice informing me that I was leading a life based not only on the false premise that genes control biology, but also on the false premise that we end when our physical bodies die… At that astounding moment [I] came to realize that the protein 'switches' that control life are primarily turned on and off by signals from the environment… the universe (p. 184).
God is Truth. God is Love. God is Light. God is the Great I AM . The Dalai Lama has affirmed that the new frontier is the journey within, and that assertion is echoed by the enormously successful DVD, The Secret (2006). The chorus for transcendence is growing larger and louder.
Rabbi Michael Lerner (2006) avows that the so-called Red States are not Far Right strongholds; rather, they are the voice of the heartland calling out for meaning, which the Far Left has not given for fear of casting their lot with the right-wingers.
Increasing numbers believe that it may be possible to slip the surly bonds of earth, in the words of John Gillespie Magie, Jr. in his poem High Flight, "…and touch the face of God."
How does one make this journey? How does one know where to start? How to steer? What to take along?
This book is a guide for the journey.
Someday, after we have mastered the winds, the waves, and gravity, mankind will harness for God the energy of love. Then, for the second time in the history of the world, we will have discovered fire.
Grandmother used to give me graham crackers to munch on in church to keep me quiet. In her church—or any other at that time—there wasn't much to interest children. But I was a fairly calm child so I did a lot of listening, especially to the hymns. I remember one that went, I am a poor, wayfaring stranger, traveling through this world of woe. The idea is that here on earth we walk dusty roads as wayfarers, lost from our homes. Around us is sin, woe, and all manner of unpleasant experiences. But if we'll just hang in there eventually we'll get to a different place—heaven presumably, although the word is never mentioned—where we'll see our deceased loved ones and then we'll be …over Jordan… home. When we find that heaven, we will be dead according to this hymn.
Well, I've done my share of wayfaring. That is I've learned what is needed for success in this world, as I define success. First, I learned to walk, feed myself, dress myself, talk, and then read and write. Along the way I learned how to keep myself reasonably safe and get along with other people reasonably well. Also, I made reasonably good grades in school and completed advanced degrees. I've held responsible positions without ever getting fired and have led efforts for change. People familiar with Maslow's Hierarchy of Needs (see p. 87) might say I reached the level of Self Actualization, or becoming all that I could be.
At one time—not too long ago—I read as many as five books a week in a quest to know more. At other times I became very confused about which road to take. In my darkest time, I woke up living a life I hated—violating even my own principles, values, and standards for living. I had gotten lost while wayfaring. I had read the signs incorrectly.
The idea of having to die before I can reach anything resembling a state of joy and peace isn't very attractive to me. I began to feel very strongly that there must be more I could do right here on terra firma to improve my state of mind and find a more meaningful purpose for my life. So I took stock of myself.
On the roadway of life.
I was a performer. If people wanted to get something done, they asked me. Do you know what people do for performers? They applaud them. So I had received much positive reinforcement for doing what other people wanted me to do—so much so that those other people had, in effect, become my gods.
My life was out of balance. According to Loehr and Schwartz (2003), the human system has four sources of energy: physical (including health, food, exercise, rest), mental (where I had overdosed), emotional, and spiritual. I discovered that I had shut down my emotional system many years before in order to avoid pain. I have learned that the emotional center of the brain doesn't have switches for every emotion. Instead, it has one master switch labeled emotion. So when I unwittingly told my brain to turn off pain, it heard, Turn off emotion. I even used to take pride in the fact that people could not hurt my feelings.
Some people say we change only out of pain. This was true for me. As I look back at the poetry that flowed so freely at the time, I remember the pain.
Not even for beauty's sake.
Years after I wrote this poem, when I found myself longing to buy new clothes but unable to find anything in the stores that was different from what I already had, I hired a wardrobe consultant to help me. Her first step was to interview me about what I did in every part of my life—personally as well as professionally. Then we went to my closet to find what was there to fit my needs. After a morning of sorting, I discovered I had very little in my wardrobe that was suitable for relaxation and recreation. Three large plastic lawn bags of excess professional clothing went to Goodwill. That afternoon I returned to my office and began returning telephone calls. In the middle of one call I was overcome with nausea and felt that I might faint. At that moment, I recognized the symptoms of withdrawal and realized that I had overdosed on my addiction to clothing. Unable to find the sensation or high that I had previously reached from a new outfit, I had, just as alcoholics do, bottomed out. I just no longer got a kick out of going shopping as I had before.
How self-deluded can you be?
And get your kicks from migraines.
Pretending you don't need them.
So they don't get close.
Pretending that you need them.
Why don't more people make needed change? Throughout history there have been fewer pioneers than settlers. Shakespeare offers this explanation, transposed from Act III, Scene I of Hamlet: Thus… [what we know] makes cowards of us all—and makes us rather bear the ills we have than to fly to others that we know not. A scientific term for the phenomenon Shakespeare describes is inertia—the tendency of a body to keep on doing what it has been doing.
And no one even knew.
Although I was very active in my church, even my spiritual life wasn't spiritual; it was superficially religious with the same intellectual approach that I had learned so well in school and that served me well in my career.
I began to realize I had to make a change when I drove into a parking space at 7:30 on the first morning of a new project. Early morning is my best time of day and I love new beginnings. Yet, on this morning when I normally would have been full of energy and excitement, I heard an inner voice say, "I am so tired." For the first time this voice got my attention.
The next time I heard my inner voice, I was driving home from work in the late afternoon. This time its message was more disturbing. If you don't get out of here, you're going to die.
Simply stated, I had been living someone else's life. Whose life? The life of the woman I thought I ought to be based on some notion of what was acceptable to other people. But this woman I had concocted was not in alignment with what was in my soul, and some perceptive people saw the symptoms of my pain—allergies, debilitating migraines, workaholism. Increasingly, something inside told me things had to change. Granddaughter of one of the last pioneers who settled the territory that is now called Oklahoma, I felt the strong pull of my ancestry to leave that which was comfortable, secure, and safe to venture into the unknown.
Freight trains are heavy, important, on track.
They're noisy and fearless and prompt.
They carry the freight that runs the world.
You can count on them solid to get the job done.
Gliding toward vast father sky.
From steel rail to white wind?
Pulling me toward important tasks.
But the sail in my soul wants the wind.
Beneath my pain, I was yearning for a better life for myself and those I loved. Love is the most powerful force in the universe. When our yearning grows from love, nothing can stop us. Pain may be an incentive or a byproduct of breaking the barriers—but if we love, we are safe.
In a segment of Family Ties, the television show that catapulted Michael J. Fox to stardom during the 1980s, the character of Alex P. Keaton asked, How do you know if you're good? The clearest answer he got was, You know if you're good from what other people tell you. But this answer is incomplete. Which other people? If people are giving you conflicting information, which ones do you listen to? And what about the danger of becoming dependent on other people to the detriment of listening to your own inner voice?
On a journey of self-discovery how do you know if you're on the right track? Out of all the self-help books, spiritual teachings, and conflicting religious beliefs, how do you know which ones are right for you?
Not through wayfaring, but through wayfinding.


text:
Q: Understanding autocorrelation using the FFT and accelerate framework - iPhone I have been trying to implement an autocorrelation algorithm on the iPhone using the vDSP libraries supplied by Apple (Accelerate framework).
So far i created an audio unit following the auriotouch example of apple, but i want to use the accelerate framework for performing the autocorrelation instead of the older implementation in the example code of auriotouch.
The IORemote audio unit is routed trough my renderCallBack method like so:
{
    AudioGraphController *controller = (AudioGraphController *) inRefCon;

    // Remove DC component
for(UInt32 i = 0; i < ioData->mNumberBuffers; ++i)
    controller.dcFilter[i].InplaceFilter((SInt32*)(ioData->mBuffers[i].mData),         inNumberFrames, 1);

    OSStatus result = AudioUnitRender(controller.inputUnit, ioActionFlags, inTimeStamp, 1,     inNumberFrames, ioData);

    if (result) { printf("InputRenderCallback: error %d\n", (int)result); return result; }

    controller.manager->ProcessAudioData(inNumberFrames, ioData);

    return noErr;
}

The input data from the microphone is sent to a ProcessAudioData method which performs the autocorrelation, according the the c++ snippet in this post: Using the Apple FFT and Accelerate Framework
However i have some trouble understanding the information in the displaydata array.
When i try to access the information all i get is nan, the only time i get an idea of the information is when i cast the displaydata array like so:
SInt16* buf = (SInt16 *)displayData;
The steps to compute the autocorrelation i follow these steps:
- Split the real input (ioData->mBuffers[0].mData) to even and odd inputs.
- Perform a FFT (forward)
- Take the absolute square of the values generated by the FFT.
- Take the IFFT (backward/inverse)
- Convert the complex split to real.
Could someone give me some pointers/advice as how to interpret the information in the displaydata array, also when i do examine display data like this they seem to be all the same values, altho they do vary depending on the mic input.
The input of the microphone is expected to be a signal with some echoes of the original signal, the goal of the autocorrelation is to determine the lag where the autocorrelation peaks, so that i can determine the offset of the echo to the original signal.
Should i create a echoed version of the signal first (with some offset) and use that in the to multiple the values of the FFT?
I appreciate any input, also if you can guide me to information which explains this more clearly, since i fairly new with vDSP techniques especially on the iPhone. I do have mathematical experience with convolution and fourier transforms, but the in place packing of Apple is having me guessing where i can find the information i expect to get from this calculation.

A: Check the data types for each of the routines (vDSP, etc.) you call and make sure you are providing the correct C data type.  Use the debugger to check the input to each subroutine, starting just before you get the NaN result and working backward, to make sure the input data is correct (type and scale, etc.)  You can also copy the data to/from buffers that are formatted more to your liking for debug purposes, and so that you can see before/after results even for in-place calculations.
Also note that inNumberFrames may not be related to the length of the vector you want to autocorrelate, depending on what result information you want.  You may need extra vector buffers.

redpajama_set_name: RedPajamaStackExchange


The 21st celebration of a person is a significant milestone marked by his passage into adulthood. While it is true that time passes by very quickly, I guess it has shifted a little too fast for me. I remembered 9 years ago when I was 12, I said that National Service was a long way ahead. And then after passing through the phases of O Levels, A Levels, and serving the country for two fulfilling years, I have now hit the magic 21. It perhaps marks the end of the childhood I have lived in, but also it signifies a new beginning into this world. A moment of double celebration, this is one night to remember for that very special moment.
Last year for my 20th birthday, I had the pleasure of the company of both my grandmothers with me in celebrating this day at Jade. This year, only my paternal grandmother joined in on the day itself with my family and Godparents. My maternal grandmother was very ill and recently discharged from hospital the day before my birthday, but that didn't stop me from lifting her spirits by having a birthday cake and song to celebrate the occasion with her first at home. The years passed by very quickly, and suddenly it seems like everyone is aging, though finely in the way of shimmering lights and crystals.
For my 21st Birthday this year, I chose to visit The Line Restaurant at the Shangri-La Hotel. Located amidst the elaborate decor of a mystical ambience, with warm wood brown colours and rich lights aplenty, The Line was my place of choice to celebrate my big day with a galore of gourmet varieties and to leave a lasting memory of an experience to remember. I last visited The Line two years ago in December 2009 to celebrate my mum's 49th Birthday and it was through the well thought spread of delectable goodies such as Honey Baked Ham, freshly shucked oysters, oodles of fresh sashimi, a decent cheese spread and sweet tantalizing delights coupled with the good service that brought me back.
The Line is probably the only buffet restaurant I have visited with freshly shucked oysters to order. Just name your desired quantity and be treated to the rough and hardy mollusc being pried open to serve on your plate. The seafood section also features crabs, fresh prawns and mussels with servings of lemon wedges and tabasco sauce.
This Sunday night, Middle Eastern cuisine takes center stage as one of the features being served for guests. The roast rack of lamb looks extermely appetizing.
And if something hot would suit the palate, Middle Eastern Chef Khaled Elelimi treated my table to a spread of Fried Kibbih, Sambousk Jebanh and Sambousk Laham. The Fried Kibbih was exceptionally delicious due to its serving being made of deep fried ground lamb with crushed wheat, onions and pine nuts.
A Mezze spread of Hoummous, Baba Ghannoush, Labneh and Muhammara are available as dips to go with choice Mediterranean breads. I especially adored the Hoummous which was a Puree of Chickpeas with tahina and lemon juice, topped with olive oil as the warm nutty flavour and texture was refreshingly sparkled with an acidic touch of citrus and made smooth and silky with the oil.
And for a more European collection, there were the classics of Fish Pie in Puff Pastry, French Snow Peas, Lobster bisque and Snow Fish with Tomato sauce. Imagine the Snow Fish with Tomato sauce to be extremely smooth and silky matched with the rich umaminess of the tomato and cream.
The Red Wine Braised Lamb Cheek was well marbled in fat with a firm yet melt in the mouth texture. The sauce was also well constructed in savoury flavours in delicate fashion to bring out the beauty of the meat while retaining a robust creamy taste.
And for the classics, there were traditional Roast Beef served with Yorkshire pudding in a smooth savoury brown sauce, roast potatoes, pesto raviolli and pasta with either cream,tomato and beef sauce prepared live.
The Japanese counter was also well presented with fine cuts of Salmon, Tuna, Yellowtail, Octopus and Squid. Smooth, creamy and fresh with a crisp clear taste, this is very good sashimi.
Sushis were decently varied with choices from Inari, Pickled Lettuce, Sweet Prawn, Hamachi, and various makis.
This is one wasabi block that certainly got my attention.
I have always enjoyed the Salad and Greek section at The Line. Witness freshly shaved parma ham being served or even a spread of cheeses to start, or end the meal.
While the spread is good, I found the festive salads like the Pomelo Salad to be a little bitter, and goodness why have I not tried the Pasta and Vegetable salad. Truth to be said, I was too full to try every single item on the menu.
Its hard to go wrong with the Parma Ham with Rockmelon, a classic in salty sweet flavours combined in a refreshing way.
And yes there is the good ol' Chinese cuisine section where dishes of crowd favourites like Cereal Prawns, Coffee Pork Ribs, Curry Chicken and Dim Sum are whipped up for the feast.
The Line is one of the few buffet restaurants in Singapore which features a fixed Indian/Asian Cuisine section. Various curries are up to savour, together with freshly flipped prata and barbecued sticks of satay! And if there is a crave for Roast Duck, Char Siew or even Wanton Mee, there is a Chinese live station catering to hungry tummies. Thumbs up for the variety though the portrayal of dishes needs a louder shout out by the restaurant.
The dessert section this time round is of choice variety with select cakes and tarts but with a few pieces which could do with better interest. The chocolate eclairs were cutely adorned with little flowers, and the Mango Sago Puree with smooth and velvety sweet.
The cakes were very appealing and tempting especially with pieces like the Strawberry Shortcake with layers of strawberry, cream, sponge, cream, strawberry and sponge to top it all off.
Assorted Sponges and Chocolate Classics are a must have to any buffet.
While 7 different varieties of cakes and an assortment of cocktail desserts are available, there is also a fixed Crepe making station, an ice cream booth with toppings of nuts, berries and other dried condiments together with an irresistible chocolate fondue that would make many go chocolate crazy.
The Line at Shangri-La in all has a very decent spread of delicacies and classics served out for its guests. Service, ambience, and quality are still very good, though the restaurant could perhaps do more in bringing out the varieties or spicing up its choices on the menu. The selections could do with that much needed sparkle. This visit to The Line was with the promotion of Shangri-La's 39th Anniversary with 39% discount for reservations made up to 39 days in advance, and it is with good food, excellent company, and a little something for everybody, that makes this birthday dinner worthwhile.
My Birthday Cake! Compliments of The Line. And it was decadently creamy, almost like a chocolate fudge cake made into an ice cream cake.
Middle Eastern Chef Khaled Elelimi and Me!
Next PostNext Kampong Chicken Rice – Soft, Warm, Satisfying. A taste of home made affordable!


Italy has plenty of enchanting places with breathaking views. However, the ones that you can enjoy in Venice are simply unique in their own way. In my opinion, nothing beats the marvelous mixure between the spectacular Italian art and the lovely Byzantine!
Leeney and Willy, our first couple from Indonesia, chose Venice to be their prewedding's scenery, splitting the photo session in three parts.
This young and beautiful fashion designer wore a fantastic pastel pink gown in the morning and an elegant golden attire for the gondola ride. The choice for the last part by night was a classic black evening dress, savouring a glass of wine at Florian Caffe.
Me and my fiance would like to have a prewedding photo shooting in Venice on 21st of May 2018 Monday.
Could you please confirm you availability on that date? Could you please inform us about details and the price.


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text:
\section{Introduction}

For a long time,  the study  of open  quantum systems~\cite{petruccione} remained  confined to maps which are completely positive (CP). However, phenomenological studies of spin relaxation showed that complete positivity was not a necessary requirement~\cite{simmons_completely_1981,raggio_remarks_1982,simmons_another_1982}. Based on the concept of linear assignment maps, a better view on the issue of complete positivity was brought forward in the debate between Pechukas and Alicki~\cite{pechukas_reduced_1994,alicki_comment_1995,pechukas_pechukas_1995}.
With  the  understanding  of  the importance  of  initial  correlations~\cite{jordan_dynamics_2004,carteret_dynamics_2008},
non-Markovian dynamics~~\cite{breuer_colloquium:_2016,li_concepts_2017}
and advances in  control of quantum systems, not-completely-positive (NCP)  maps are now  studied ever
more actively~~\cite{Szarek-2008,cuffaro_debate_2013}. It is also now known that NCP maps arise naturally as intermediate maps of quantum non-Markovian processes~\cite{rivas_entanglement_2010}.

Within the space of positive maps, the relative measure for  CP and NCP maps was addressed for Pauli unital channels in~\cite{jagadish_measure_2019}. The present work revisits the question of measure of NCP maps, but by relaxing the requirement that the map has a full positivity domain on the system of interest, i.e., all states of the reduced system produce valid output states. An example of this type is the map corresponding to the partial-trace operation. But, more generally, we can allow maps, namely NCP maps, whose positivity domain is restricted. An example of this type would be the intermediate (NCP) map for the non-Markovian dephasing channel~\cite{shrikant_non-markovian_2018}, for which, at the singularity in the decoherence rate, only a set of states of zero measure produces a valid output.

In~\cite{jagadish_measure_2019}, we showed that for CP maps, the eigenvalue spectrum is always bounded. Here, our main result is that the NCP maps no longer form a compact set, essentially because the eigenvalues of the corresponding dynamical matrices are not bounded. Hence, for the set of qubit maps, including NCP maps, the volume measure may not be well defined, in general. For simplicity, we restrict ourselves to the case of unital maps. The paper is organized as follows.  In Sec. \ref{positivemaps}, we discuss the preliminaries. Section \ref{motivate} sets the idea presented in the paper through three examples. The main results are then presented  and discussed in Sec. \ref{results}. We then conclude in Sec. \ref{conclusion}.

\section{Preliminaries}
 \label{positivemaps}
\subsection{Positivity domains and valid maps}

  Let  $\mathcal{E}$ be  a positive  map acting  on a
  qubit represented by the density matrix
\begin{equation}
  \label{eq:1q1}
  \rho = \frac{1}{2}(\mathbbm{1} + a_i \sigma_i) = \frac{1}{2} \left( \begin{array}{cc}1 + a_3 & a_1 - \imath a_2\\ a_1 + \imath a_2  & 1-a_3 \end{array} \right),
\end{equation}
where the vector ${\bf  a} = (a_1 \,,\,  a_2 \,,\, a_3)$,
  with
\begin{equation}
 a_1^2 + a_2^2 + a_3^2 \le 1,
 \label{eq:cond1}
\end{equation}  is called the  Bloch vector. All one
qubit states lie on or inside the
``Bloch   ball,''\index{Bloch   sphere}   which  is   the   unit   ball
in the space $\mathbb{R}^3$ parametrized by the axes
$a_1$, $a_2,$ and $a_3$.

The map $\mathcal{E}$ can be represented by a four dimensional Hermitian matrix, usually referred to as the dynamical matrix~\cite{sudarshan_stochastic_1961}. The dynamical matrix is also called the Choi matrix~\cite{choi_completely_1975}. If the dynamical matrix is positive, the map is completely positive (CP). Otherwise, it is not-completely-positive (NCP). The trace of the dynamical matrix acting on a qubit is 2.
Positivity of a map addresses its action on density matrices whereas complete positivity (CP) is a statement on the map itself.

The important point to note is that the definition of a map makes sense only for a valid set of states. For a CP map acting on a qubit, the domain is the entire Bloch ball $\mathcal{B}$. But, for NCP maps, only certain states on the Bloch ball act as a valid domain for their action.

\begin{Definition}[Positivity domain]
\label{Positivity domain}
Given the map $\mathcal{E}$, the set $\mathcal{P}_\mathcal{E} \equiv \{p \in \mathcal{B} | \mathcal{E}(p) \in \mathcal{B}\}$ constitutes the positivity domain of $\mathcal{E}$.
\end{Definition}
In words, $\mathcal{P}_\mathcal{E}$ is the subset of the Bloch ball $\mathcal{B}$ whose image under map $\mathcal{E}$ falls within $\mathcal{B}$.

\begin{Definition}[Valid map]
A map $\mathcal{E}$ acting on a qubit is valid iff its associated positivity domain is nonempty. i.e., $\mathcal{P}_\mathcal{E} \ne \emptyset$.
\end{Definition}
In view of this definition, a map is valid precisely if its positivity domain is nonvanishing. This definition will be motivated in the examples discussed later.

Note that the evolution of a qubit may contain a singularity at some instance, meaning that the decoherence rate in the master equation at that point, and hence the (instantaneous) intermediate map $\mathcal{E}^\ast$, diverges. The map can still be valid if the positivity domain $\mathcal{P}_{\mathcal{E}^\ast}$ is nonvanishing. That this is physically well motivated will be clarified in examples discussed later.
\subsection{Unital maps acting on a qubit}
\label{dynamicalunital}
The general form of a dynamical matrix $\tilde{B}$ acting on a qubit which is unital and trace preserving (TP) can be parametrized as follows:
\begin{equation}
\label{equnitalgeneral}
\tilde{B}= \left(
\begin{array}{cccc}
 a & x & y & z \\
 x* & 1-a & w & -y \\
 y* & w* & 1-a & -x \\
 z* & -y* & -x* & a \\
\end{array}
\right).
\end{equation}This is obtained by considering a four dimensional Hermitian matrix such that $\mathrm{tr}_{1}\tilde{B} = \mathbbm{1} = \mathrm{tr}_{2}\tilde{B}$. This means that the partial trace with respect to each of the two bipartite subsystems is identity. Note that $a$ is real and $x,y,z,$ and $w$ are in general complex. More general maps, allowing damping, can be considered, but the above suffices to prove our main result.

\section{Motivating the concept of the validity of a map}
\label{motivate}
Allowing NCP maps gives considerable freedom in what maps we can consider, even if they are not CP. Our earlier definition of validity is the only restriction we place on what we consider as valid maps. The trivial map that maps every qubit state to the fixed state $\begin{pmatrix}
2 & 0 \\ 0 & -1
\end{pmatrix}$
is invalid (as this is not a bona fide density matrix), whereas a map that takes all states to the maximally mixed state, is indeed valid, according to our definition.

\subsection{A numerical example}
\label{numex}
This numerically simulated example is to show that the eigenvalues of the dynamical matrix representing a NCP map on a qubit can exceed 2 in absolute values, unlike that for CPTP maps.
Consider a NCP map, whose dynamical matrix is given by
\begin{equation}
\label{bnumex}
B_{NCP}=\left(
\begin{array}{cccc}
 0.20 & 0.95 & 0.70 & 0.10 \\
 0.95 & 0.80 & 0.30 & -0.70 \\
 0.70 & 0.30 & 0.80 & -0.95 \\
 0.10 & -0.70 & -0.95 & 0.20 \\
\end{array}
\right).
\end{equation}Clearly, the map is unital and trace preserving. It has eigenvalues 2.32409, -1.12409, 0.669258, 0.130742. As we can see, one eigenvalue is greater than 2, which indicates that the eigenvalues are not bounded by 2, for general NCP maps acting on a qubit.
	Its action on a general qubit density operator
produces an output  with eigenvalues
\begin{eqnarray}
\lambda_1 &\approx& 0.5\, -0.47\sqrt{4.2 a_1^2-1.3 a_1a_3+0.04 a_2^2+2.6 {a_3}^2}, \nonumber\\
\lambda_2 &\approx& 0.5+0.47 \sqrt{4.2 {a_1}^2-1.3 {a_1} {a_3}+0.04 {a_2}^2+2.6 {a_3}^2}.
\nonumber
\end{eqnarray}
from which it follows that any choice of $a_j$ that satisfies Eq. (\ref{eq:cond1}) and
$4.2 {a_1}^2-1.3 {a_1} {a_3}+0.04 {a_2}^2+2.6 {a_3}^2 \ge 0$ (for example, $a_3=0.5, a_1=0.05, a_2=0.1$) is an element of the positivity domain of the map. Since this is a non-vanishing set, it follows from our definition that this constitutes a valid map.
Note that in this case, the set $\mathcal{P}_\mathcal{E}$ has a non zero measure. However, this is not necessary. So long as the positivity domain is non-vanishing, the map is well defined. This is shown to be the case in the next two examples, where the positivity domain is of measure zero.

\subsection{Two successive CNOT gates \label{sec:cnot}}
Consider the action of two successive CNOT gates, $U = U_{CNOT} \circ U_{CNOT},$ on the initial product state $\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)_A \otimes |\phi\rangle_B$ of two qubits  labeled $A$ and $B$.

Let $|\psi' \rangle$ denote the total state of the systems $A$ and $B$ after the first $U_{CNOT}$ has been applied:
\begin{eqnarray}
|\psi' \rangle
&=&  \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B +|1\rangle_A \otimes |1\rangle_B) ;\thinspace \text{for}\thinspace |\phi\rangle = |0\rangle, \nonumber\\
&=&  \frac{1}{\sqrt{2}} (|1\rangle_A \otimes |0\rangle_B +|0\rangle_A \otimes |1\rangle_B) ;\thinspace \text{for}\thinspace |\phi\rangle = |1\rangle.
\end{eqnarray}

Since $U = \mathbbm{1}$, the output state of $B$ is the same
as its input, which could be $|0\rangle_B$ or $|1\rangle_B$. Therefore, to discern the reduced dynamics of $B$ between the application of the two CNOT operations, one would need a map that
takes the identity state as the input, and outputs either $|0\rangle_B$ or $|1\rangle_B$.

Note that this is a one-to-many relation and hence non-linear behavior~\cite{schmid_why_2018} would invalidate it as a map in certain conventional situations. However, this intermediate map corresponding to the application of the second CNOT is indeed valid according to our definition. This is physically well motivated since this intermediate map, only when supplemented to the CP map representing the evolution up to the application of the first CNOT, recreates the full identity map after the application of the second CNOT.

The problem may be understood as an instance of a singularity in the intermediate map, while the full map is well defined. Consider the control qubit to be in the state $\cos\theta\ket{0}+\sin\theta\ket{1}$ and the system to be the target qubit. Let us represent the map acting on the density matrix expressed as a column vector, and call it the $A$ matrix following~\cite{sudarshan_stochastic_1961,Quanta77}. Denote the $A$ matrix corresponding to the first application of CNOT to be $A_1$, and that of the second one to be $A_2$. Then we have $A_1\cdot A_2 = \mathbbm{1}_4$, from which it follows that
\begin{equation}
A_2 = (A_1)^{-1},
\label{eq:a2}
\end{equation}
meaning that $A_2$ corresponds to the inverse of the first, which is seen to be a NCP map as shown below.

The $A$ matrix for the first application of CNOT is
\begin{equation}
A_1= \left(
\begin{array}{cccc}
\cos^2(\theta) & 0 & 0 & \sin^2(\theta) \\
0 & \cos^2(\theta) & \sin^2(\theta)  & 0 \\
0 & \sin^2(\theta)  & \cos^2(\theta) & 0 \\
\sin^2(\theta)  & 0 & 0 & \cos^2(\theta)\\
\end{array}
\right),
\label{eq:a1}
\end{equation}
which corresponds to a bit flip channel.  In view of Eq. (\ref{eq:a2}), it follows from Eq. (\ref{eq:a1}) that
\begin{equation}
A_2 = \sec(2\theta)\left(
\begin{array}{cccc}
\cos ^2(\theta ) & 0 & 0 & -\sin ^2(\theta ) \\
0 & \cos ^2(\theta ) & -\sin ^2(\theta ) & 0 \\
0 & -\sin ^2(\theta ) & \cos ^2(\theta ) & 0 \\
-\sin ^2(\theta ) & 0 & 0 & \cos ^2(\theta ) \\
\end{array}
\right),
\end{equation}
from which it follows that the associated dynamical matrix is
\begin{equation}
\mathfrak{B}_2 = \sec(2\theta)\left(
\begin{array}{cccc}
\cos ^2(\theta ) & 0 & 0 &  \cos ^2(\theta ) \\
0 & -\sin ^2(\theta )& -\sin ^2(\theta ) & 0 \\
0 & -\sin ^2(\theta ) & -\sin ^2(\theta ) & 0 \\
\cos ^2(\theta )  & 0 & 0 & \cos ^2(\theta ) \\
\end{array}
\right),
\label{eq:b2}
\end{equation}
whose non vanishing eigenvalues are seen to be $-2 \sin ^2(\theta )\sec(2\theta)$ and $2\cos ^2(\theta)\sec(2\theta)$.
Exactly one of the two eigenvalues is negative for any choice of $\theta \ne 0, n\pi$ ($n$ an integer), showing that $A_2$ is NCP for such choices.
The parameter $\theta=\frac{\pi}{4}+n\pi$ represents a singularity, where the intermediate map represented by $A_2$ diverges, and corresponds to the nonlinearity mentioned above, which is now seen as a manifestation of the singularity rather than grounds to invalidate the map.

The operator sum representation for the map $\mathfrak{B}_2$ is
\begin{equation}
\rho^\prime = \sqrt{1 + \sec (2 \theta )} \mathbbm{1} \rho \mathbbm{1} + \sqrt{1 - \sec (2 \theta )} \sigma_x \rho \sigma_x.
\end{equation}
Consider the fixed points of $\sigma_x$, i.e., states that are invariant under the action of $\sigma_x$. They will also be invariant under the above map, irrespective of $\theta$. Thus, when $\theta=\frac{\pi}{4}$, only these states, being invariant, will produce a valid output, whereas any other states will lead to divergent outputs. Thus, the set  of these fixed points, constitute the positivity domain.

In particular, these states have the form
\begin{equation}
\rho^X_p \equiv p\ket{+}\bra{+}+(1-p)\ket{-}\bra{-} = \begin{bmatrix}
1 & 2p-1 \\ 2p-1 & 1
\end{bmatrix},
\label{eq:X}
\end{equation}
where $0 \le p \le 1$.
Under the action of the above map, the state (\ref{eq:X}) evolves to
$[1+\sec(2\theta)]\rho^X_p + [1-\sec(2\theta)]\rho^X_p = \rho^X_p$, independently of $\theta$.
At $\theta=\pi/4$, the function $\sec(2\theta)$ and hence the map diverge, but the set $\Sigma^X_\sigma$ of all points of the form $\rho^X_p$ is unaffected by the infinity and thus constitutes our positivity domain. Notice that $\Sigma^X_\sigma$ represents a straight line connecting two extreme points on the Bloch ball, and thus has measure zero.

It is important to stress here that even when the (intermediate) map is singular-- i.e., the absolute values of the eigenvalues of the dynamical matrix increase without bound to infinity (such that their sum is 2)--the set $\mathcal{P}_{\mathcal{E}^\ast}$ is non vanishing, ensuring that the map is well-defined. This is of course validated by the fact that the full map itself corresponds to identity operation and is thus well defined.  The singularity thus gives an extreme instance of the unboundedness of eigenvalues of valid dynamical maps.

\subsection{Intermediate map with non-Markovian dephasing}
Consider an example of a non-Markovian channel based on a familiar process, namely quantum dephasing. This is described by the Lindblad equation,
\begin{equation}
\frac{d\rho}{dq} = \lambda(q) [-\rho(q) + \sigma_z\rho(q)\sigma_z],
\label{eq:lindblad}
\end{equation}
where $\lambda$ is the decoherence rate and the parameter $q$ rises monotonically from 0 to (asymptotically with time) $\frac{1}{2}$. For Markovianity to hold in the sense of CP divisibility, $\lambda$ must  always be positive as a function of  $q$ . Furthermore, for the noise to be described by a quantum dynamical semigroup, $\lambda$ must be a positive constant.

In a non-Markovian dephasing channel such as quantum random telegraph noise (RTN) \cite{daffer_depolarizing_2004}, the negativity of $\lambda$ arises from the presence of terms with (co)sinusoidal dependence on time, but in the  non-Markovian dephasing model proposed in \cite{shrikant_non-markovian_2018}
\begin{equation}
\lambda(q) =
\frac{\frac{1}{2}(\alpha_- + \alpha_+) - q}{(q - \alpha_+)(q - \alpha_-)},
\label{eq:rate}
\end{equation}
where $\alpha_-$ is a parameter that lies in the range $[\frac{1}{2},1]$ and $\alpha_-<\alpha_+$. Thus, the regions for which $q > \alpha_-$ lead to non-Markovianity. Furthermore, $q = \alpha_-$ represents a singularity, which we discuss below.

In Ref. \cite{shrikant_non-markovian_2018}, the authors choose
\begin{equation*}
\alpha_\pm = \frac{1}{2\nu}(\nu + 1 \pm \sqrt{\nu^2+1}),
\end{equation*}
which corresponds to the Kraus operators $K_I = \sqrt{(1-\beta)}\mathbbm{1}$ and $K_Z = \sqrt{\beta} \sigma_z$, with $\beta = [1 + \nu(1 - q)]q$, where the dimensionless quantity $\beta$ can turn negative for sufficiently large value of parameter $\nu$. In this model,  $0 \le \nu \le 1$ and serves as a non-Markovianity parameter.

 The singularity mentioned above occurring at $q = \alpha_-$ can be physically interpreted as follows. At this point, all initial states of the form
\begin{equation*}
\rho(0)\equiv\left( \begin{array}{cc} r & s \\ s^\ast & 1-r \end{array}\right),
\end{equation*}
 are mapped to
 \begin{equation*}
 \rho(\alpha_-) \equiv \left( \begin{array}{cc} r & 0 \\ 0 & 1-r \end{array}\right), \end{equation*}
creating an instance of non invertibility. The family of states $\rho(\alpha_-)$ corresponds to a straight line in the Bloch ball from the north to south pole, having measure zero. (For general consistency conditions pertaining to non-invertible time evolution, see~\cite{andersson_finding_2007,rivas_quantum_2014}.)

However, as in the previous example, the singularity is not pathological, and only momentary. This follows from the fact that the infinitesimal evolution determined by Eq. (\ref{eq:lindblad}),
\begin{equation}
d\rho = \lambda(\alpha_-) [-\rho(q) + \sigma_z\rho(q)\sigma_z]dq,
\label{eq:infint}
\end{equation}
is well-behaved when acting on (only) the elements of the family $\rho(\alpha_-)$.  The intermediate map (\ref{eq:infint}) thus gives us an example of a map whose positivity domain is of zero measure.

We shall now state and prove our main result.

\section{Results and Discussion}
\label{results}
The following result on unboundedness, which is shown for the unital maps, also can be extended to non-unital maps. But, by virtue of applying it to unital qubit maps, it obviously implies the unboundedness of the general set of maps (CP or NCP) on qubits.
\begin{Theorem}
\label{nonbounded}
The set of maps (including NCP maps) acting on a qubit is neither closed nor bounded.
\end{Theorem}
We represent maps in the space of eigenvalues of the dynamical matrix. By the Choi-Jamiolkowski isomorphism~\cite{jamiolkowski-1972}, the set of CP maps is isomorphic to that of two-qubit states, and hence bounded. Thus, we shall be concerned with the (un)boundedness of NCP maps. By definition, the positivity domain corresponding to any valid NCP map acting on a qubit should be non empty. Our proof proceeds by providing an example that evinces the unboundedness of  the eigenvalues of $\tilde{B}$ of Eq. (\ref{equnitalgeneral}). Specifically, consider Eq. (\ref{eq:b2}) which is in the form of Eq. (\ref{equnitalgeneral}) with $a = z = \sec(2\theta)\cos ^2(\theta ), w =  \sec(2\theta)\sin ^2(\theta ),$ and the rest are all zero. Clearly, the eigenvalues $-2 \sin ^2(\theta )\sec(2\theta)$ and $2\cos ^2(\theta)\sec(2\theta)$ diverge to $\infty$ at $\theta = \frac{\pi}{4}$, providing an instance of the unboundedness of eigenvalues of NCP maps.
This unboundedness of eigenvalues arises only for NCP maps, where the eigenvalues can take negative values. That the eigenvalues are non bounded implies that the space of NCP maps is not closed.

For NCP maps that arise as intermediate maps of a CP map, even if they diverge, one can find the positivity domain to be non vanishing, affirming the validity of the NCP map. An instance was discussed in the context of example of Sec. \ref{sec:cnot}. As another instance, consider the action of the map as in Eq. (\ref {equnitalgeneral}) on Eq. (\ref{eq:1q1}). We assume all parameters are real, for simplicity.
The output density matrix has eigenvalues
\begin{equation}
\lambda_{\pm} = \frac{1}{2}\Big(1\pm\sqrt{a_1^2 \left((w+z)^2+4 x^2\right)+4  a_1a_3 ((2 a-1) x+y (w+z))+a_2^2 (w-z)^2+a_3^2 \left((1-2 a)^2+4 y^2\right)}\Big).
\label{eq:ev}
\end{equation}
In this case, setting the positivity condition on the two eigenvalues, one can see that the point $(a_1,a_2,a_3) = (0,0,0)$ corresponding to the maximally mixed state serves as one point in the positivity domain of the NCP map. In general, one can find sets of points $(a_1,a_2,a_3)$ in the Bloch ball such that the eigenvalues $\lambda_{\pm}$ in Eq. (\ref{eq:ev}) are both positive. This set will constitute the positivity domain of the map (for an illustration, see below).
\hfill $\blacksquare$
\bigskip

For the case of Eq. (\ref{bnumex}) in the example discussed in Sec. \ref{numex}, the positivity domain is plotted in Fig. (\ref{fig1}). That this domain is a subset of the Bloch ball implies that the map in question is not positive, and thereby goes beyond the restriction under which a simple measure of NCP maps was found in Ref. \cite{jagadish_measure_2019}.
\begin{figure}[t!]
\resizebox{6 cm}{6 cm}{\includegraphics{Positivity}}
\caption{(Color online) The positivity domain for the NCP map represented by the dynamical matrix in Eq. (\ref{bnumex}). The outer ball (of unit radius) shown in white is the Bloch ball. \label{fig1}}
\end{figure}
This result for the unital map can be extended (albeit in more cumbersome way) to the non unital case, also. However, the present example suffices for the proof.

It follows from the unboundedness of NCP maps, that the set of qubit maps, including NCP maps, do not form a compact set. Now, if the set of divergent maps form a discrete set of singularities, then one might hope that the set of maps obtained by excepting these points could be endowed with a bona fide measure. However, it is easy to see that the divergent maps themselves form a continuous family.

To see this,  note that we can replace the CNOT in the double-CNOT example with any other entangling operation, and we obtain a divergent map corresponding to the intermediate map represented by the second application of the control operation. Indeed, consider any control-$Q(\theta,\phi, \xi)$, where $Q(\theta,\phi, \xi)$ is any unitary, i.e.,
$$
I_2 \otimes I_2 + I_2 \otimes Q(\theta,\phi, \xi),
$$
and the angles $\theta, \phi,$ and $\xi$ may be varied continuously. This strongly suggests that a well-defined measure of maps for this general case does not seem to arise, unless further restrictions are made.

We enumerate a few such interesting cases:
\begin{enumerate}
	\item If we restrict to the case of Pauli channels, subject to the condition of positivity (i.e., having full positivity domain), then the volume measure of NCP maps is twice that of CP maps \cite{jagadish_measure_2019}.
	\item A similar result also holds if the Pauli channels are rotated by a fixed unitary $U$. This corresponds to the simplex whose four vertices are $UPU^\dagger$, where $P$ is a Pauli matrix. This may be considered as a rotated version of the Pauli tetrahedron $\mathcal{T}$, discussed in \cite{jagadish_measure_2019}.
\end{enumerate}
Finally, note that the double-CNOT, or equivalently a double-C-phase gate, provides perhaps the simplest method to practically realize NCP maps, and even  a singularity, conveniently on a quantum computer in a way that is well within the reach of present-day quantum technology. Indeed, the double-control application of any two-qubit gate will be acceptable, based on the argument given above.

\section{Conclusions}
\label{conclusion}

Positivity domains for NCP qubit maps, arising from initial correlations with an environment, were first addressed in~\cite{jordan_dynamics_2004}, where it was shown that they can be strict subsets of the Bloch ball. Here, we show that arbitrary (even divergent) NCP maps, subject only to the trace-preserving condition, can be physically valid in the sense of having a non-vanishing (possibly zero-measure) positivity domain.

The physical significance of this result is that the volume measure of qubit maps, including NCP maps, may in general not be well defined, even though the dynamics is quite regular~\cite{chrusinski2010nonmarkovian}. Specific restrictions may be imposed in certain cases, as in~\cite{jagadish_measure_2019}, to define a measure.

\section{Acknowledgements}
The work  of V.J.  and F.P.  is based upon  research supported  by the
South African Research  Chair Initiative of the  Department of Science
and  Technology and  National Research  Foundation.  R.S.   thanks the
Defense Research  and Development  Organization (DRDO), India  for the
support      provided       through      the       Project No. ERIP/ER/991015511/M/01/1692.

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