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Para Resources Announces That Production at El Limon Has Recommenced
November 21, 2017, Vancouver, B.C. — Para Resources Inc. (the "Company" or "Para") (TSXV – "PBR") (WKN – "A14YF1") (OTC – "PRSRF") is pleased to announce that production at the El Limon Mine in Zaragoza, Colombia, recommenced on November 15th, 2017.
The original start up at El Limon occurred in May/June of 2017, after a US$ 7 million rehabilitation and plant upgrade. That initial planned production ramp up was hampered by a through-put constraint at the filter press. This bottleneck prevented production from ramping up to the design limit of 225 tons per day ("TPD"). While the problem was being identified and remediated, production was limited.
The filter presses have now been replaced with a larger scale belt filter. Sourcing of the necessary equipment and parts has taken four months but the equipment has now been installed and tested. We are pleased to confirm that it is functioning as designed.
Geoff Hampson, Para's CEO states, "The team in Colombia has worked hard to correct the problems that were associated with legacy equipment that should have been, in hindsight, replaced in the original renovation plan. Ian Harris, Para's President and his team in Zaragoza, have systematically worked through the various issues and developed and implemented our remediation strategy. We are now in a position to get back to our original plan of processing 225 TPD.
The grade from the El Limon underground operations continues to be in the predicted range of 7 – 9 gpt Au. As production ramps up El Limon will increase purchases of mineralized rock that has been mined by the local, formalized small miners who operate on the Company's mineral claims."
Para is a junior producing gold mining company. Para owns approximately 80% of the El Limon project, in Colombia, which in addition to its current underground operation is purchasing mineralized rock mined by small artisanal miners working on the Company's property. The El Limon and Otu properties also have exploration and development upside. The Company also owns 88% of the Gold Road Mine in the Oatman District of Arizona. The Company has hired RPM Global as consulting engineers in order to produce a National Instrument 43-101 ("NI 43-101") Technical Report which it expects will establish a current Mineral Resource estimate and anticipates that it will publish a NI 43-101 Preliminary Economic Assessment thereafter. Para will continue to take advantage of current market conditions to acquire and develop additional highly economic, near-term production assets that have strong exploration and development upside.
C. Geoffrey Hampson, Chairman, Chief Executive Officer and Director
For further information, please contact Andrea Laird, telephone: +1-604-259-0302
Statements contained in this news release that are not historical facts are "forward-looking information" or "forward-looking statements" (collectively, "Forward-Looking Information") within the meaning of applicable Canadian securities laws. Forward Looking Information includes, but is not limited to, disclosure regarding possible events, conditions or financial performance that is based on assumptions about future economic conditions and courses of action; the timing and costs of future activities on the Company's properties; success of exploration, development and mill processing activities; and the anticipated results to be achieved from operation of the El Limon and Gold Road Mines. The Company also cautions that there is no assurance that past production at El Limon or Gold Road or production at nearby mines is indicative that the Company will achieve similar results. In certain cases, Forward-Looking Information can be identified by the use of words and phrases such as "plans", "expects", "scheduled", "estimates", "forecasts", "intends", "anticipates" or variations of such words and phrases. Forward-Looking Information involves known and unknown risks, uncertainties and other factors which may cause the actual results, performance or achievements of the Company to be materially different from any future results, performance or achievements expressed or implied by the Forward-Looking Information. There can be no assurance that Forward-Looking Information will prove to be accurate, as actual results and future events could differ materially from those anticipated in such statements. Accordingly, readers should not place undue reliance on Forward-Looking Information. Except as required by law, the Company does not assume any obligation to release publicly any revisions to Forward-Looking Information contained in this news release to reflect events or circumstances after the date hereof or to reflect the occurrence of unanticipated events.


Built in 2009, Sunworld 6 Gulet was completely renovated in 2016 for your comfort. Gulet Sunworld 6 will be a great choice if you want to rent a comfortable, clean, comfortable boat and make a pleasant holiday without spending a high budget. Our gulet yacht can accommodate up to 14 guests. Meet Gulet Sunworld 6, which offers you a perfect holiday in the comfort of your home and take a pleasant journey.
When renting gulet sunworld 6, you can choose the all-inclusive package and bring your own shopping.
There are separate living areas in the gulet for the ship personnel.
On the rear deck there is a seating area with a large cushion, a dining table and a dining area.
There is a separate dining table in the lounge and your meals are prepared in a separate fully equipped kitchen.
Sunworld 6 Gulet has enough sunbathing mats and space for all guests.
2 Triple cabins of 9.00 m2. There is 1.40*2.00 cm double bed and 85*2.00 cm single bed in the cabin. Private bathroom with shower box and electrical toilet.
2 Triple cabins of 10.00 m2. There is 140*200 cm double bed and 85*2.00 cm single bed in the cabin. Private bathroom with shower box and electrical toilet.
2 Double cabins of 5.50 m2. There is 1.40*1.95 cm double bed in the cabin. Private bathroom with shower box and electrical toilet.
1 Double cabin of 6.00 m2.There is 1.55*1.95 cm double bed in the cabin. Private bathroom with shower box and electrical toilet.


LBJ was a notorious pervert.
UPDATE: All day I've been wondering what seemed off about today's comic. Finally found it but when I fixed it, the comic didn't make sense anymore. So, no, you're not imagining things, I did revise the dialogue a bit.


books that I read in the last couple years that frame what I am thinking about lately.
Cal Newport on doing Deep Work and escaping social media, on The Ezra Klein Show.
Tim Ferris on suffering, psychedelics, and spirituality on The Ezra Klein Show.
Jack Kornfield-Finding Freedom, Love and Joy in the Present.


Ben Tre is one of the major provinces of Vietnam, belonging to the Mekong Delta.
About 90km away from Saigon, this is one of the cities that have direct links with Ho Chi Minh City.
Ben Tre is famous for its specialties and famous tourist sites.
Tourism here is very developed because the proximity to Ho Chi Minh City and the tourist destinations is also very special, which attracts many tourists every year.
Along with that is combining with special fruits such as: Coconut, longan, dragon fruit, ..
Lịch sử cảng bến đầm có thực sự đáng tìm hiểu không?


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The 8 Most-Anticipated Short Films of the 2019 Fall Film Festival Circuit
By Tober January 10, 2020 Short Films
Tober
The Venice Film Festival begins in less than a week, with Telluride and the Toronto Intl. Film Festival not too far behind. This means we are officially in the fall film festival season. Oscar hopefuls pit off against indie darlings for the top prizes and, perhaps more importantly, for the biggest buzz.
Just like with feature films, there are certain shorts that come along each year and take the festival world by storm. In today's article, we point you to 8 of the shorts that we see becoming a fest staple throughout 2019.
Anna — Dekel Berenson
Plot: Living in war-torn Eastern Ukraine, Anna is an aging single mother who is desperate for a change. Lured by a radio advertisement, she goes to party with a group of American men who are touring the country, searching for love.
Director's Statement (excerpted): Observing what went on at the nightclubs inspired me to look deeper into this cultural phenomenon and to discover the existence of the "love tours" that are depicted in 'Anna.' I found the subject compelling enough to return to Ukraine six years after my original visit, and capture it on film. In 'Anna' the backdrop is wintry and cold, but regardless of a country's regions everyone ultimately harbors dreams and aspirations. These international dating events, which have been organized since the fall of the Iron Curtain, continue to take place, with groups arriving in Ukraine many times a year. The mass parties in which men are usually outnumbered 1 to 10, now take place almost exclusively in East Ukraine, the industrial and more impoverished part of the country… This is an important story of hope and despair, one which I have attempted to capture as authentically as I could.
Marquee Festival Selection: Cannes Film Festival — Nominee, Palme d'Or
Why The Anticipation: "Anna" is Dekel Berenson's second of three shorts (with the third also having premiered this summer). With each work Berenson is becoming more and more widely received. Typically centered around social and humanitarian issues, Berenson has more than 12 festival wins and nearly 50 nominations under his belt. Biggest among them have been his Academy Award qualifying run with "Ashima" and most recently with the Cannes and TIFF selected "Anna." Believe it or not, he has a second film touring the world at the same time ("The Girls Were Doing Nothing").
Austral Fever — Thomas Woodroffe
Plot: Amanda, a complacent and lonely woman in her mid-forties, lives with Daniel, her sixteen-year-old son, in the mountains of southern Chile. Octavio, a friend of Daniel, is accidentally injured while hunting in the forest. Amanda takes over the healing sessions, developing an intimate bond between them. They discover that the contact with the wound produces an addictive pleasure.
Director's Statement: "Fiebre Austral" is an atmospheric pursuit with which I try to understand what happens in that magical gap that exists between pleasure and pain. It is a film that wonders about the body, about the personal perception of its sensitivity limits. It was inspired by my aunt Carmen's pictorial work, which is violent, seductive, static, sexual, outrageous. For me, "Fiebre Austral" gathers all of that.
Marquee Festival Selection: Venice Film Festival — Official Selection, In-Competition, Orizzonti
Why The Anticipation: Just from the trailer alone, you can see Thomas Woodroffe's long-standing respect for cinema and the tradition. Referencing classics like Citizen Kane in its depth of field techniques and contemporary slow cinema masters in its contemplative camera, "Austral Fever" has a lot on its mind — and a lot to express. This twenty minute fever dream of a film is also playing TIFF, and we imagine many more festivals. This is Woodroffe's third short film.
Blue Boy — Manuel Abramovich
Plot: Young men from all over the world have been meeting at the Blue Boy Bar in Berlin for forty years. According to their website, this is where the city's lonely hearts, business people and tourists go.
Marquee Festival Selection: Berlin Intl. Film Festival — Winner, Silver Berlin Bear
Director's Statement: I wanted to make a series of portraits of sex workers applying the same rules of the sex trade. To place emphasis on the performativity of such dynamics and to create an experiment in which the roles of everyone involved (protagonists, audience, myself) were interchangeable. What would it be like for these young men to take distance from the character they play? How would they react to their own stories?
Why The Anticipation: This masterclass in observational documentary cinema has been on the festival circuit since the very beginning of the year, but it looks to continue rolling with an official selection at the upcoming San Sebastian Film Festival. What's most revelatory about the film lies in its method. Abramovich has his subjects listen to recordings of their experiences, adding a meta-layer to the film's nuances and complicated subject matter. It won't be an easy sit for many, but we think it will prove rewarding for enough festival audiences to be championed further into the year.
Nimic —Yorgos Lanthimos
Plot: A professional cellist has an encounter with a stranger on the subway which has unexpected and far-reaching ramifications on his life.
Marquee Festival Selection: Locarno — World Premiere
Why The Anticipation: This is perhaps the most anticipated short film at any festival it plays in soley based on the star power behind and in front of the screen. Coming off of his Oscar-nominated period piece The Favourite, Yorgos Lanthimos curiously chose to go with an under-the-radar short film piece as his follow-up. But who better to pair him with than the also Oscar-nominated Matt Dillon as star. From what we heard at Locarno, it's a film that's sure to "mess you up", and its continued run at TIFF and NYFF ensure that many more audiences can witness the mind trip on a big screen.
Pearl — Yuchao Feng
Plot: Six-year-old Lin lives in an isolated fishing village with her brother and their widowed mother. When the family is given an opportunity to start a new life, Lin has a startling revelation. A melancholy tale inspired by real events.
Director's Statement: I received a call from my mother one Sunday afternoon while I sat around doing nothing in my shabby Jersey City apartment. It was 3 am in China, where my mom was calling from. She told me that she had just had a nightmare about her mother, a nightmare she had been having since she was six. "I think I am probably never going to be able to forgive her," she told me.
I was shocked. My mother was forty-five, which meant that she'd been having this nightmare for forty years. As the conversation unfolded, a piece of buried memory was unveiled.
Pearl is inspired and loosely based on this memory from my mother's tragic past. I don't intend to recreate her nightmare but I want to put myself in this little girl's shoes and see what she saw, through the art of cinematic storytelling. By examining the event from different perspectives, I hope to better understand this traumatic moment. I believe that the process of writing and reexamining what went on in each character's mind will help me shed some light on the complex fabric of the human condition.
Marquee Festival Selection: Tribeca — Nominee, Student Visionary Award
Why The Anticipation: After playing at NYU's First Run Festival (where Feng was getting his MFA), this visually-arresting tale — based in part on real events — has been surely building up momentum in the festival world. After Tribeca and Short Shorts Asia, its next stop is Raindance, where it will be in great company. Feng has always created some personal and singularly creative work, and garnered attention for it, but we expect this narrative short to make him a known name in short film circles.
She Runs — Qiu Yang
Plot: In an ordinary Chinese winter, a small city junior high student, YU, tries to quit her school aerobic dancing team.
Marquee Festival Selection: Cannes Film Festival — Winner, Canal+ Award
Why The Anticipation: A new Qiu Yang film — what's not to anticipate? This contemporary auteur has been surpassing audience expectations in film after film. Yang's entire filmography has led him to major short film fests around the world. His crowning festival achievement came in 2017, when he won the short film Palme d'Or. Perhaps the most interesting part of his career? He's only made short films so far! We're sure a debut feature can't be far off.
Subject To Review — Theo Anthony
Plot: It charts the rise and development of the instant replay system Hawk-Eye in professional tennis, cleverly relating innovative technology and the imperfections of the human experience to the history of cinema, sports entertainment, and humanity's desire to objectively interpret the world.
Marquee Festival Selection: New York Film Festival — World Premiere
Why The Anticipation: This is Theo Anthony's follow-up to the incredibly successful debut documentary feature, Rat Film. Its exploration of Baltimore haunted and compelled festival juries from SXSW to Cinema Eye Honors. Like Yorgos Lanthimos, he's curiously chosen to go with a short form film on a completely different subject (sports). Since it hasn't come out, we can't be sure to how it will be received — but we have a feeling this documentary essay will have a long fest shelf life.
What Do You Know About The Water And The Moon — Layla Jian Luo
Plot: During an abortion, instead of aborting the fetus, a girl gives birth to a live jellyfish.
Marquee Festival Selection: Palm Springs— Special Mention, Best Student Live-Action Short Over 15 Minutes
Why The Anticipation: Like Yuchao Feng, Layla Jian Luo completed her MFA at NYU and has effectively used this hypnotic short film as a career launch pad. So far, so good, as the film premiered at Clermont-Ferrand and has since played at many major festivals, and continues to have more (like New Orleans Film Festival) coming up. Layla stays busy — when she isn't directing, she guest lectures at the School of Visual Arts. We are so excited for a career that's just beginning!
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redpajama_set_name: RedPajamaCommonCrawl


Best Dating Site For Females. Do we need to mention that women like to know a lot many things about men, before accepting their online date request? When it comes to dating, women want to know some specifics about their date, like who he is? How is his dressing sense? What does he do to earn a living, is he a reliable person? And what we found is that this information about men can be easily found on his Facebook profile. But as a female, do you also reveal information about yourself?
You can read, write, add favorites and show interest at no cost as long as you're a member of our family! But to be completely honest, there can be additionals in the feature only to make your wishes goal faster! We never do phising, scamming or hooking!!! Match by interest, hobby or location? They all don't work! It has been proven that the simplest and the best way to do this is just search someone that interests you.
All 5 of the recommended singles sites have a reasonable male to female ratio.
Numerous studies show that Canadians spend more days in a month and more minutes in a day checking out dating site than users in the U. What this means is that there is a huge market for hook up sites in Canada.
Malaysian Cupid offers thousands of local members, girls and guys, and also claims to be the leading local dating website. Most of the girls, and guys too, are from Kuala Lumpur but there is also of a lot from Penang and the coastal beach cities.


Are the Charges Against You Legally Sufficient?
Criminal Defense Blog 2015 February Are the Charges Against You Legally Sufficient?
Are the DA's Charges Against You Legally Sufficient?
When you are charged with a crime, the DA's Office must file formal charges that states the facts of their case against you. These charges be "facially sufficient." In other words, the facts of the DA's case must be specific and detailed enough to make out the elements of the charges against you. If they are not, then your attorney may be able to file a motion to have the charges dismissed.
The New York County Criminal Court recently issued a decision that is a reminder to good criminal defense attorneys to carefully review the charges in an accusatory instrument for facial sufficiency. In People v. McGinnis, the court dismissed an accusatory instrument as facially insufficient, holding that the facts alleged in it failed to support the charges. 2013 NY Slip Op 23353 (Criminal Ct, New York Co 10/15/2013). In that case, the defendant was charged with Loitering for the Purpose of Prostitution, which is a criminal charge in New York. The DA's Office had attempted to use standard and non-specific language to state the facts of their case and the court found that the language did not make out the elements of the crime.
It stated that there is a "need for an individualized recitation of the facts which set forth the elements of an offense based upon actual observation." Id.
This ruling is a practical reminder about the importance of reviewing the complaint and looking for any possibly way to file a motion to dismiss. At the Law Offices of Lance Fletcher, we carefully review our client's charges and can file a "Motion to Dismiss" when we find mistakes or omissions made by the District Attorney. We have successfully gotten charges dismissed for facial insufficiency on several different occasions, including charges of Driving While Intoxicated, Criminal Possession of a Weapon, Petit Larceny, and others. Contact us today for a Evaluation initial consultation.


Jim Burrows has more than 30 years of professional experience in both the public and private sectors of the financial services industry, including commercial banking, retail banking, and investment banking. Burrows joined EXIM Bank as the Vice President of Small Business in October 2012 and was promoted to the Senior Vice President in 2013. Burrows has also held management positions at both large and regional commercial banks in the U.S.


Saint Paul School of Theology is committed to a safe and secure learning environment for students studying at the Church of the Resurrection or the Oklahoma City University Campus. Our commitment extends to employees and visitors to both campuses, and Avila University's library and housing. This page provides the information required by The Jeanne Clery Act.
Saint Paul School of Theology publishes and distributes an annual campus security report to students, faculty, and staff by October 1st each year. The information contained in the report includes current policies and procedures, as well as statistical data obtained through collaborations among the Campus Security Authorities and local law enforcement offices in Leawood and Overland Park, Kansas and Oklahoma City, Oklahoma.
Security and Safety Reports for Saint Paul School of Theology's Kansas and Oklahoma Campuses: The following reports provide you with the latest statistics on crime on campus, as well as Saint Paul's policies for ensuring a safe and secure learning environment. You may request a printed copy of these reports by contacting the Campus Security Authority at the appropriate campus. Our annual reports (available October 1) contain data related to the Violence Against Women Act (VAWA), as recently specified by the Department of Education.
Reporting Emergencies at Kansas Campus: Students, employees, and visitors at our Kansas Campus should report emergencies, criminal actions, and suspicious behavior to local authorities by dialing 911. You should also notify the Associate Dean of Students Rev. Margaretta Narcisse, of the incident at 913-253-5097.
Reporting Emergencies at Oklahoma Campus: Emergencies, criminal actions, and suspicious behavior at our Oklahoma Campus should be reported to campus police at 405-208-5911. You should also notify the Associate Dean of Students Rev. Margaretta Narcisse, of the incident at 913-253-5097.
Campus Security Authorities (CSAs): The Director of Institutional Effectiveness & Registrar and CSAs are responsible for publishing the Annual Security and Fire Safety Reports by campus. To request a hard copy of the report, please contact the CSA for your campus, listed below. To report an issue of discrimination, please contact the Equity Compliance Officer, Melissa Whalen.


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Three New Online Journals Join the Fertility and Sterility Family; ASRM Expands Opportunities for Readers and Authors
ASRM is pleased to announce that its flagship journal, Fertility and Sterility, will soon be joined by three sister publications: F&S Reports, F&S Science, and F&S Reviews.
All three new journals will begin publishing in the first quarter of 2020 and will be published quarterly online by Elsevier, the publisher of Fertility and Sterility.
Our new open-access journal, F&S Reports, will publish original papers and reviews of clinical, investigative, and translational research in the field of human reproduction. Elsevier is the second largest open-access publisher in the world and F&S Reports will expand ASRM's global reach, using an open-access model meeting the requirements of many European governing bodies. Thus, it may serve as a preferred choice for many international authors and readers. Richard Paulson, M.D., Professor, University of Southern California, will serve as editor of F&S Reports.
F&S Science will publish high-quality, innovative, basic and translational science articles involving research with a strong potential to transform the clinical practice of human reproductive medicine and impact the laboratory practice of assisted human reproduction. William Catherino, M.D., Ph.D., Professor, Uniformed Services, University of the Health Sciences, has been named editor of F&S Science.
F&S Reviews will publish high-quality, timely, in-depth, and highly-citable review manuscripts addressing contemporary issues in reproductive medicine. The journal will provide content for continuing medical education, one of ASRM's key missions. Anne Steiner, M.D., M.P.H., Professor, Duke University, will serve as editor of F&S Reviews.
ASRM President Catherine Racowsky, PhD, said, "The addition of three new journals to the Fertility and Sterility family is an exciting development in ASRM's commitment to publishing and education. The new journals will expand opportunities and options for Society members to publish. To advance this mission further, ASRM and Elsevier are waiving the publication fee of the open access journal, F&S Reports, for the first year. Readers will benefit likewise. The standard of excellence set by our flagship journal, Fertility and Sterility, will carry on in our new journals, with an increased range of content and the open-access option."


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Page Overview: BMW Welt - Collect your BMW - Story
A special kind of rendezvous.
Great expectations — followed by lasting impressions.
When our customer Jennifer picked up her first BMW from her BMW dealer, BMW Welt was still a futuristic construction site. But one could, of course, already spot the building's pioneering architecture from the early construction stages and see that a future visit would inevitably lead to astonishment. If you add the excitement and anticipation of receiving a new BMW to this feeling, then such a day is — without a doubt — going to be a perfect one.
The vehicle handover was planned with so much attention to detail and passion for the BMW brand…
First stop: the post box.
The Collector's Package.
Anticipation is half the fun — despite the fact that all the remaining fun is likely going to fly right by you once you actually get driving. But there are still a few things left to do before Jennifer gets the key to her new BMW. But the anticipation just keeps on growing: not least as a result of our Collector's Package. It contains all the relevant information on the upcoming vehicle delivery. From the arrival to the precise collection procedure — even a bag for the new licence plates is included in the shipment. This allows Jennifer to relax, go over her day at BMW Welt, and start planning all the things to do around Munich or which of the routes would be best for the return journey home.
In the heart of the brand: increased heartbeat.
The excitement keeps growing, the checklist on the Collector's Package is calmly waiting inside the handbag and the anticipation is growing.
Welcome to BMW Welt.
We know how you feel.
When the subject of new vehicle collectors comes up, our experienced employees can't help but smile. Yes, they can be recognised from miles away and can immediately be told apart from other customers. There is a certain smile and a certain joyful excitement that surrounds them — you have to see it for yourself. But our employees know how to quickly eliminate that initial nervousness during check-in. The handful of formalities are taken care of quickly and precisely. Jennifer immediately feels at home, setting the stage for a relaxed day at BMW Welt.
Waiting has never been so tasty.
The buffet at our Premium Lounge doesn't just look delicious. Our exclusive catering partner Feinkost Käfer offers an insight into the sophistication of Bavarian cuisine that goes beyond the well-known clichés. Now it's time to take it all in with all your senses: the spatial experience with the brilliant view of the Premiere, where the vehicle handovers take place, as well as of the surrounding architectural highlights, such as the famed BMW four-cylinder building.
So real — yet virtual.
At the Product Info Center, Jennifer will now get to see her new BMW with all the features that she has ordered for the first time. Using a virtual model, her personal vehicle specialist can explain all the key highlights to her; the functionality of the various driving assistance systems in particular can be explained visually in this way — they appear to, quite literally, be within reach. And yes, there are only a few moments left to go…
The new BMW becomes your new BMW.
Only a few metres now separate Jennifer from her new BMW. Colours, features, technology — everything is as expected. But how will it feel? From the collector's staircase, Jennifer can already see her BMW glistening under the spotlight. With a smile, the vehicle specialist hands her the key. One push of the button and the vehicle's LED lights shine right at Jennifer. She then has all the systems adjusted to her own preferences and connects her smartphone to the vehicle. A smile, a deep, happy breath and Jennifer leaves BMW Welt in her brand new BMW. In the rear-view mirror she sees the building's impressive architecture and reflects on an unforgettable day.
Your day — your BMW.
Picking up your new BMW from BMW Welt is an event that you can already look forward to when ordering your vehicle. Plan this day in comfort and enjoy exactly those programmes that fascinate you the most.
And remember: The joy of driving begins with anticipation.
Packages & Services.
EXCLUSIVE package.
PREMIUM package.
BASIC package.


This policy does not apply to transfer students, defined institutionally as those students who have completed at least 9 semester hours (or equivalent) of college course work after high school graduation and before matriculation at Ithaca College.
Students may apply credits earned at other institutions or granted on the basis of AP/IB/CLEP scores to fulfillment of the following Integrative Core Curriculum (ICC) components: First Year Composition, Quantitative Literacy, Diversity, and Writing Intensive. Such credits may also apply to the 12 Complementary Liberal Arts credits depending on school and department requirements. The degree to which such credits could apply to a specific ICC requirement is dependent on a match between the Ithaca College courses meeting the requirement and the credit granted for courses completed elsewhere and on program-specific requirements for the 12 Complementary Liberal Arts credits.
Students may not apply credits earned at other institutions or granted on the basis of AP/IB/CLEP score or any other standardized examination program from which Ithaca College awards credit toward fulfillment of the Themes and Perspectives or ICC Capstone components of the ICC.
These policies (see the table of advanced placement equivalents) are in effect for students entering Ithaca College in the 2017-2018 academic year. They are subject to change at any time.
For additional information about Ithaca College's advanced placement policies, contact the Office of the Registrar, Peggy Ryan Williams Center, phone 607-274-3127. Ithaca College's ETS school code is 2325. Scores must be sent directly to the Office of the Registrar, Peggy Ryan Williams Center, Ithaca, NY 14850-7013, before credit is granted.
In all schools, scores of 1 or 2 (and in some cases 3) will not be awarded credit. Scores of 3, 4, and 5 are awarded credit as indicated below.
Students can recieve course-specific credit for CHEM 12100, provided they earn a C or better on a department exam. Otherwise, the students should plan on enrolling in CHEM 12300.
Combined with an AB subscore of 3, 4, or 5 only.
The credit for this exam does not include the DV (Diversity) designation.
Students may not receive credit for both the English Literature/Composition and English Language/Composition exams.


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Preston Oshita (Chicago, Illinois, Estados Unidos, 26 de junio de 1993), mejor conocido como Towkio, es un rapero estadounidense de ascendencia japonesa y mexicana. Anteriormente fue conocido como Preston San y Tokyo Shawn. Es integrante del equipo de Savemoney. Su álbum de estudio debut, WWW, fue publicado en 2018.

Discografía

Álbumes de estudio
 2018: WWW

Referencias

Enlaces externos

 Página Web Oficial

Artistas de Republic Records
Músicos de Estados Unidos
Nacidos en Chicago

redpajama_set_name: RedPajamaWikipedia


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MAIN : All : Marmoset Glomerulonephritis & Cardiovascular Disease
Glomerulonephritis and Cardiovascular Disease in the Pygmy Marmoset (Callithrix pygmaea)
Luis R. Padilla1, DVM; James R. Raymond1, DVM; Richard J. Montali1, DVM; Sharda G. Sabnis2, MD
1Department of Pathology, Smithsonian National Zoological Park, Washington, DC, USA; 2Division of Nephropathology, Armed Forces Institute of Pathology, Washington, DC, USA
Renal disease has been a common pathologic finding among the Callitrichidae. This study characterizes a specific type of glomerulonephritis in the pygmy marmoset (Callithrix pygmaea), which can progress to a hypertensive syndrome. A retrospective study of adult pygmy marmosets (n=25) necropsied at the Smithsonian National Zoological Park, Washington, DC revealed a 56% prevalence of glomerulonephritis in this captive population. The disease was determined to be the primary cause of death in 36% of the cases. It was most commonly seen in young adult animals (36–60 months), with no sex predilection. Histologically, glomeruli had a prominent lobular pattern, with varying degrees of mesangial cell proliferation and matrix hyperplasia. Electron microscopy of some affected animals showed minimal increase in basement membrane thickness, but marked increases in mesangial matrix and cells, mixed with some deposits. Immunohistochemistry studies performed on kidney sections of selected animals showed focal mesangial staining to IgG and IgM, but not to IgA. Staining for immunoglobulins was not observed in kidney sections of an age-matched control that died due to traumatic injuries.
Histopathology, electron microscopy and immunohistochemistry were consistent with an immune-mediated membranoproliferative glomerulonephritis (MPGN). Renal vessels in animals with advanced renal disease were often thickened with hyaline arteriosclerosis. Affected animals in the end stages of the disease often showed evidence of hypertension with cardiovascular disease, including left-sided heart failure, chronic passive pulmonary congestion and cavitary effusions. These findings suggest a clinically significant, progressive immune mediated process leading to chronic renal deposition of IgG and IgM in the affected animals and subsequent cardiovascular disease. Although specific antigenic causes initiating glomerulonephritis were not determined in these pygmy marmosets, MPGN in humans can be triggered by chronic infections or parasitism.
12dd69d0-9867-4205-8c3e-0d51f0690833.1675062602
Luis R. Padilla, DVM
Smithsonian National Zoological Park


There will be an increase of 2% to 4% after the implementation of GST on 1st April 2015. While most of the bills in private hospitals would be exempted from GST, the component relating to consultant fees would be subjected to the six per cent tax which will be levied to patients.
Early detection on our health are essential especially when charges are getting higher year by year in private hospitals and clinics. Other than having a good diet and lifestyle, we should have a work balance in life to achieve optimum health. We can check our health using ph urine to check our urine for eg: urine ph showing acidity range means we should take precaution to our health in terms of food we consume, diet, exercise because acidity means we are doing something harmful to our health. If your urine ph shows alkaline means we are in good shape and should keep up the good job in keeping our body in healthy state.
Lots of people neglect their health because they are too busy with their schedule of working and personal life. When down with sickness, pop a pill and everything will be back in order. However, popping a pill for an ill is detrimental to our liver in long term. It's always best to check our urine for ph and change our lifestyle and diet rather than consuming pills for life.
Other than checking for ph urine, you can also check for protein, ketone, blood and others simply by dipping into urine and get instant results in few minutes without paying for expensive bills everytime you need to check.
1 bottle consist of 100 dipsticks and it's way better, cheaper and you can check and repeat checking for few days without having to visit doctor every now and then which also quite inconvenient as we need to take leave and basically no time to visit doctor's office.


text:
Distichophyllum submucronatum är en bladmossart som beskrevs av Fleischer 1908. Distichophyllum submucronatum ingår i släktet Distichophyllum och familjen Hookeriaceae. Inga underarter finns listade i Catalogue of Life.

Källor

Egentliga bladmossor
submucronatum


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text:
\section{Introduction}
In this paper, we investigate the~\citet{Backus1962} average of a stack of anisotropic layers, wherein the tensors are oriented randomly.
In spite of a conceptual relation between randomness and isotropy, herein, the Backus average results in a medium, whose anisotropy, even though weak, is irreducible to isotropy, regardless of increasing randomness.

Each layer is expressed by Hooke's law,
\begin{equation*}
\sigma_{ij}=\sum\limits_{k=1}^3\sum_{\ell=1}^3c_{ijk\ell}\,\varepsilon_{k\ell}
\,,\qquad
i,j=1,2,3\,,
\end{equation*}
where the stress tensor, $\sigma_{ij}$\,, is linearly related to the strain tensor,
\begin{equation*}
\varepsilon_{k\ell}:=\frac{1}{2}\left(\frac{\partial u_k}{\partial x_\ell}+\frac{\partial u_\ell}{\partial x_k}\right),
\qquad
k,\ell=1,2,3\,,
\end{equation*}
where $u$ and $x$ are the displacement and position vectors, respectively, and
\begin{equation*}
c_{ijk\ell}=c_{jik\ell}=c_{k\ell ij}
\end{equation*}
is the elasticity tensor, which has to be positive-definite.
Under the index symmetries, this tensor has twenty-one linearly independent components, and can be written as \citep[e.g.,][expression~(2.1)]{BonaEtAl2008}
\begin{equation}
C
=
\left[\begin{array}{cccccc}
c_{1111} & c_{1122} & c_{1133} & \sqrt{2}\,c_{1123} & \sqrt{2}\,c_{1113} & \sqrt{2}\,c_{1112}\\
c_{1122} & c_{2222} & c_{2233} & \sqrt{2}\,c_{2223} & \sqrt{2}\,c_{2213} & \sqrt{2}\,c_{2212}\\
c_{1133} & c_{2233} & c_{3333} & \sqrt{2}\,c_{3323} & \sqrt{2}\,c_{3313} & \sqrt{2}\,c_{3312}\\
\sqrt{2}\,c_{1123} & \sqrt{2}\,c_{2223} & \sqrt{2}\,c_{3323} & 2\,c_{2323} & 2\,c_{2313} & 2\,c_{2312}\\
\sqrt{2}\,c_{1113} & \sqrt{2}\,c_{2213} & \sqrt{2}\,c_{3313} & 2\,c_{2313} & 2\,c_{1313} & 2\,c_{1312}\\
\sqrt{2}\,c_{1112} & \sqrt{2}\,c_{2212} & \sqrt{2}\,c_{3312} & 2\,c_{2312} & 2\,c_{1312} & 2\,c_{1212}
\end{array}\right]\,.
\label{eq:Chapman}
\end{equation}
Any elasticity tensor of this form is also positive-definite~\citep[e.g.,][]{BonaEtAl2007}.

A rotation of~$c_{ijk\ell}$\,, which is in~${\mathbb{R}}^3$\,, expressed in terms of quaternions, is
\begin{equation*}\label{eq:A(q)}
A=A(q)=
\left[\begin{array}{ccc}
a^{2} + b^{2} - c^{2} - d^{2} & -2\,a\,d + 2\,b\,c & 2\,a\,c + 2\,b\,d \\
2\,a\,d + 2\,b\,c & a^{2} - b^{2} + c^{2} - d^{2} & -2\,a\,b + 2\,c\,d \\
-2\,a\,c + 2\,b\,d & 2\,a\,b + 2\,c\,d & a^{2} - b^{2} - c^{2} + d^{2}
\end{array}\right]
\,,
\end{equation*}
where~$q=[\,a\,,\,b\,,\,c\,,\,d\,]$ is a unit quaternion.
The corresponding rotation of tensor~\eqref{eq:Chapman} is~\citep[diagram~(3.1)]{BonaEtAl2008}
\begin{equation*}
\widetilde{C}=\widetilde{A}\,C\widetilde{A}^{\,T}
\,,
\end{equation*}
where~$\tilde{A}$ is expression~\eqref{eq:At} in Appendix~\ref{app:Atilde}.
\section{The Backus and Gazis et al. averages}
\label{sec:BackusGazis}
To examine the elasticity tensors,~$C\in{\mathbb{R}}^{6\times 6}$\,, which are positive-definite, let us consider the space of all matrices~$\mathcal{M}:= {\mathbb{R}}^{6\times 6}$\,.
Its subspace of isotropic matrices is
\begin{equation*}
\mathcal{M}_{\rm iso }:=\{M\in \mathcal{M}\,:\, \widetilde{Q}\,M\,\widetilde{Q}^{\,T}=M\,,\,\,\forall\, Q\in SO(3)\}.
\end{equation*}

$\mathcal{M}_{\rm iso}$ is a linear space, since, as is easy to verify, if~$M_1,M_2\in\mathcal{M}_{\rm iso}$\,, then~$\alpha M_1+\beta M_2\in \mathcal{M}_{\rm iso}$\,, for all~$\alpha,\beta\in {\mathbb{R}}$\,.
Let us endow $\mathcal{M}$ with an inner product,
\begin{equation*}
\langle M_1,M_2\rangle_F:={\rm tr}(M_1\,M_2^{\,T})=\sum_{i,j=1}^6 (M_1)_{ij}(M_2)_{ij}\,,
\end{equation*}
and the corresponding Frobenius norm,
\begin{equation*}
\|M\|:=\sqrt{\langle M_1,M_2\rangle_F}
\,.
\end{equation*}

In such a context, \citet{GazisEtAl1963} prove the following theorem.

\begin{theorem}\label{thm:Gazis}
The closest element---with respect to the Frobenius norm---to~$M\in \mathcal{M}\subset\mathcal{M}_{\rm iso}$  is uniquely given by
\begin{equation*
M_{\rm iso}
:=
\int\limits_{SO(3)}\widetilde{Q}\,M\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)\,,
\end{equation*}
where~${\rm d}\sigma(Q)$ represents the Haar probability measure on $SO(3)$\,.
\end{theorem}
\begin{proof}
It suffices to prove that
\begin{equation*}
M-M_{\rm iso}\,\,\perp \,\,\mathcal{M}_{\rm iso}\,.
\end{equation*}
To do so, we let~$N\in \mathcal{M}_{\rm iso }$ be arbitrary.
Then, for any~$A\in SO(3)$\,,
\begin{align*}
\langle
M-M_{\rm iso}\,,N
\rangle_F
&=
{\rm tr}\left(\left(M-M_{\rm iso}\right)N^{\,T}\right)
\\
&=
{\rm tr}\left(\left(
M-\int\limits_{SO(3)} \widetilde{Q}\,M\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\right)N^T\right)
\\
&=
{\rm tr}\left(\widetilde{A}\left(\left(
M - \int\limits_{SO(3)}\widetilde{Q}\,M\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\right)N^{\,T}\right)\widetilde{A}^{\,T}\right)
\tag*{{\hbox{(as~$\widetilde{A}$ is orthogonal)}}}
\\
&=
{\rm tr}\left(\widetilde{A}\,M\,N^T\widetilde{A}^{\,T} -\widetilde{A}\left(\,
\int\limits_{SO(3)}\widetilde{Q}\,M\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\right)\widetilde{A}^{\,T}\left(\widetilde{A}\,N^{\,T}\widetilde{A}^{\,T}\right)\right)\,.
\end{align*}
But
\begin{align*}
\widetilde{A}
\left(\,
\int\limits_{SO(3)}\widetilde{Q}\,M\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\right)\widetilde{A}^{\,T}
&=
\int\limits_{SO(3)}\widetilde{A}\left(
\widetilde{Q}\,M\,\widetilde{Q}^{\,T}
\right)\widetilde{A}^{\,T}\,{\rm d}\sigma(Q)
\quad
\hbox{(by linearity)}
\\
&=
\int\limits_{SO(3)}\left(\widetilde{A}\,\widetilde{Q}\right)M \left(\widetilde{A}\,\widetilde{Q}\right)^T{\rm d}\sigma(Q)\\
&=
\int\limits_{SO(3)}\left(\widetilde{A\,Q\,}\right)M\left(\widetilde{A\,Q\,}\right)^T\,{\rm d}\sigma(Q)
\quad
\hbox{(by the properties of the tilde operation)}
\\
&=
\int\limits_{SO(3)}\widetilde{Q}\,M\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\quad
\hbox{(by the invariance of the measure)}
\\
&=
M_{\rm iso}\,.
\end{align*}
Hence,
\begin{align*}
\langle  M-M_{\rm iso}\,,N\rangle_F
&=
{\rm tr}\left(\widetilde{A}\,M\,N^T\widetilde{A}^{\,T}\right)-
M_{\rm iso}\left(\widetilde{A}\,N^T\widetilde{A}^{\,T}\right)\\
&=
{\rm tr}\left(
\left(\widetilde{A}\,M\,\widetilde{A}^{\,T}\right)\left(\widetilde{A}\,N^{\,T}\widetilde{A}^{\,T}\right)-
M_{\rm iso}\left(\widetilde{A}\,N^{\,T}\widetilde{A}^{\,T}\right)
\right)\\
&=
{\rm tr}\left(
\left(\widetilde{A}\,M\,\widetilde{A}^{\,T}\right)N^{\,T} -
M_{\rm iso}\,N^{\,T}
\right)\\
&=
{\rm tr}\left(\left(\widetilde{A}\,M\,\widetilde{A}^{\,T}\right)N^{\,T}\right) -
{\rm tr}\left(M_{\rm iso}\,N^{\,T}\right)
\,,
\end{align*}
as by assumption,~$N\in \mathcal{M}_{\rm iso}$\,.

Finally, integrating over~$A\in SO(3)$\,, we obtain
\begin{align*}
\langle  M-M_{\rm iso}\,,N\rangle_F
&=
{\rm tr}\left(\left(\,\int\limits_{SO(3)}
\widetilde{A}\,M\,\widetilde{A}^T\,{\rm d}\sigma(A)
\right)N^{\,T}\right)-
{\rm tr}(M_{\rm iso}\,N^{\,T})\\
&=
{\rm tr}(M_{\rm iso}\,N^{\,T}) - {\rm tr}(M_{\rm iso}\,N^{\,T})\\
&=0\,,
\end{align*}
as required.
\end{proof}
Since any elasticity tensor, $C\in{\mathbb{R}}^{6\times6}$\,, is positive-definite, it follows that
\begin{equation*}
C_{\rm iso} =
\int\limits_{SO(3)}\widetilde{Q}\,C\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\end{equation*}
is both isotropic and positive-definite, since it is the sum of positive-definite matrices~$\widetilde{Q}\,C\,\widetilde{Q}^{\,T}$\,.
Hence,~$C_{\rm iso}$ is the closest isotropic tensor to~$C$\,, measured in the Frobenius norm.

If $Q_i\in SO(3)$\,,~$i=1\,,\,\ldots\,,\,n$\,, is a sequence of random samples from~$SO(3)$\,, then the sample means converge almost surely to the true mean,
\begin{equation}\label{eq:SampleMeans}
\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n \widetilde{Q_i}\,C\, \widetilde{Q_i}^{\,T}
=
\int\limits_{SO(3)} \widetilde{Q}\,C\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)=C_{\rm iso}\,,
\end{equation}
which---in accordance with Theorem~\ref{thm:Gazis}---is the Gazis et al. average of $C$\,.

This paper relies on replacing the arithmetic average in expression~\eqref{eq:SampleMeans} by the Backus average, which provides a single, homogeneous model that is long-wave-equivalent to a thinly layered medium.
According to~\citet{Backus1962}, the average of the function~$f(x_3)$ of ``width''~$\ell'$ is the moving average given by
\begin{equation*
\overline{f}(x_3):=\int\limits_{-\infty}^\infty w(\zeta-x_3)f(\zeta)\,{\rm d}\zeta
\,,
\end{equation*}
where the weight function,~$w(x_3)$\,, acts like the Dirac delta centred at~$x_3=0$\,, and exhibits the following properties.
\begin{equation*}
w(x_3)\geqslant0\,,\!\!
\quad w(\pm\infty)=0\,,\!\!
\quad
\int\limits_{-\infty}^\infty w(x_3)\,{\rm d}x_3=1\,,\!\!
\quad
\int\limits_{-\infty}^\infty x_3\,w(x_3)\,{\rm d}x_3=0\,,\!\!
\quad
\int\limits_{-\infty}^\infty x_3^2\,w(x_3)\,{\rm d}x_3=(\ell')^2\,.
\end{equation*}
These properties define~$w(x_3)$ as a probability-density function with mean zero and standard deviation~$\ell'$\,, thus explaining the term ``width'' for~$\ell'$\,.
\section{The block structure of $C$ $\to$ $\widetilde{A}\,C\widetilde{A}^{\,T}$}
The action $C\to \widetilde{A}\,C\widetilde{A}^{\,T}$ has a simple block structure that is exploited in Section~\ref{sec:FundamentalSymmetry}.
To see this, we consider $q=[\,a\,,\,0\,,\,0\,,\,d\,]$\,, with~$a:=\cos(\theta/2)$\,,~$d:=\sin(\theta/2)$\,; thus, in accordance with expression~\eqref{eq:A(q)},
\begin{equation}\label{eq:A1}
A=A(q)=\left[\begin{array}{ccc}
\cos\theta & -\sin\theta & 0 \\
\sin\theta & \cos\theta & 0 \\
0 & 0 & 1
\end{array}\right]
\end{equation}
and, in accordance with expression~\eqref{eq:At},
\begin{equation*}
\widetilde{A}=
\left[\def1.25}\begin{array}{cccccc{1.5}\begin{array}{c*{5}{c}}
\cos^{2}\theta & \sin^{2}\theta & 0 & 0 & 0 & -\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) \\
\sin^{2}\theta & \cos^{2}\theta & 0 & 0 & 0 & \tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & \cos\theta & \sin\theta & 0 \\
0 & 0 & 0 & -\sin\theta & \cos\theta & 0 \\
\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) & -\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) & 0 & 0 & 0 & \cos\left(2\,\theta\right)
\end{array}\right]\,.
\end{equation*}
For~$q=[\,0\,,\,b\,,\,c\,,\,0\,]$\,, with~$b:=\cos(\theta/2)$ and~$c:=\sin(\theta/2)$\,,
\begin{equation}\label{eq:A2}
A=A(q)=\left[\begin{array}{ccc}
\cos\theta & \sin\theta & 0 \\
\sin\theta & -\cos\theta & 0 \\
0 & 0 & -1
\end{array}\right]
\end{equation}
and
\begin{equation*}\widetilde{A}=
\left[\def1.25}\begin{array}{cccccc{1.5}\begin{array}{c*{5}{c}}
\cos^{2}\theta & \sin^{2}\theta & 0 & 0 & 0 & \tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) \\
\sin^{2}\theta & \cos^{2}\theta & 0 & 0 & 0 & -\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & \cos\theta & -\sin\theta & 0 \\
0 & 0 & 0 & -\sin\theta & -\cos\theta & 0 \\
\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) & -\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) & 0 & 0 & 0 & -\cos\left(2\,\theta\right)
\end{array}\right]
\,.
\end{equation*}
In both cases permuting the rows and columns to the order~$(\,3\,,\,4\,,\,5\,,\,1\,,\,2\,,\,6\,)$ results in a diagonal block structure for~$\widetilde{A}$\,.
For expression~\eqref{eq:A1}, we have
\begin{equation*}
\widetilde{A}
\to
\left[\begin{array}{cc}
\widetilde{A}_1 & 0 \\
0 & \widetilde{A}_2
\end{array}\right]
\,,
\end{equation*}
where
\begin{equation*}
\widetilde{A}_1
=
\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos\theta & \sin\theta \\
0 & -\sin\theta & \cos\theta
\end{array}\right]
\quad\hbox{and}\quad
\widetilde{A}_2=
\left[\def1.25}\begin{array}{cccccc{1.5}\begin{array}{ccc}
\cos^{2}\theta & \sin^{2} & -\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) \\
\sin^{2}\theta & \cos^{2} & \tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) \\
\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) & -\tfrac{1}{\sqrt{2}}\sin\left(2\,\theta\right) & \cos\left(2\,\theta\right)
\end{array}\right].
\end{equation*}
Both~$\widetilde{A}_1,\widetilde{A}_2\in{\mathbb{R}}^{3\times3}$ are rotation matrices.
Similarly, for expression~\eqref{eq:A2},
\begin{equation*}
\widetilde{A}
\to
\left[\begin{array}{cc}
\widetilde{A}_1 & 0 \\
0 & \widetilde{A}_2
\end{array}\right]
\,;
\end{equation*}
herein, $\widetilde{A}_1,\widetilde{A}_2\in{\mathbb{R}}^{3\times3}$ are reflection matrices.
Thus, in both cases, $\widetilde{A}_1$ and $\widetilde{A}_2$ are orthogonal matrices.

In either case, the following lemma holds.
\begin{lemma}\label{lem:BlockStructure}
Suppose that the rows and columns of~$C$ are permuted to the order~$(\,3\,,\,4\,,\,5\,,\,1\,,\,2\,,\,6\,)$ to have the block structure
\begin{equation*}
C
\to
\left[\begin{array}{cc}
M & B \\
K & J
\end{array}\right]\,,
\end{equation*}
with~$M\,,\,B\,,\,K\,,\,J\in{\mathbb{R}}^{3\times3}$\,, and that the rows and columns of~$\widetilde{A}$ are also so permuted.
Then,
\begin{equation*}
\widetilde{A}\,C\widetilde{A}^{\,T}\,\,\to\,\,
\left[\def1.25}\begin{array}{cccccc{1.25}\begin{array}{cc}
\widetilde{A}_1\,M\,\widetilde{A}_1^{\,T} & \widetilde{A}_1\,B\,\widetilde{A}_2^{\,T} \cr
\widetilde{A}_2\,K\,\widetilde{A}_1^{\,T} & \widetilde{A}_2\,J\,\widetilde{A}_2^{\,T}
\end{array}\right]
\,.
\end{equation*}
\end{lemma}
\begin{proof}
Let~$P\in{\mathbb{R}}^{6\times6}$ be the matrix obtained by permuting the rows of the identity to the order~$(\,3\,,\,4\,,\,5\,,\,1\,,\,2\,,\,6\,)$\,.
Our assumption is that
\begin{equation*}
P\,C\,P^{\,T}
=
\left[\begin{array}{cc}
M & B \\
K & J
\end{array}\right]\,.
\end{equation*}
Then,
\begin{align*}
P\left(\widetilde{A}\,C\,\widetilde{A}^{\,T}\right)P^{\,T}
&=
\left(P\widetilde{A}\,P^{\,T}\right)\left(P\,C\,P^{\,T}\right)\left(P\,\widetilde{A}^{\,T}P^{\,T}\right)\\
&=
\left[\begin{array}{cc}
\widetilde{A}_1 & 0 \\
0 & \widetilde{A}_2
\end{array}\right]
\left[\begin{array}{cc}
M & B \\
K & J
\end{array}\right]
\left[\begin{array}{cc}
\widetilde{A}_1^{\,T} & 0 \\
0 & \widetilde{A}_2^{\,T}
\end{array}\right]\\
&=
\left[\def1.25}\begin{array}{cccccc{1.25}\begin{array}{cc}
\widetilde{A}_1\,M\,\widetilde{A}_1^{\,T} & \widetilde{A}_1\,B\,\widetilde{A}_2^{\,T} \\
\widetilde{A}_2\,K\,\widetilde{A}_1^{\,T} & \widetilde{A}_2\,J\,\widetilde{A}_2^{\,T}
\end{array}\right]\,,
\end{align*}
as required.
\end{proof}
\section{The fundamental symmetry of the Backus average}
\label{sec:FundamentalSymmetry}
Let us examine properties of the Backus average, which---for elasticity tensors, $C_{i}$---we denote by
\begin{equation*}
\overline{\overline{\left(\,C_{1}\,,\,\ldots\,,\,C_{n}\,\right)}}\,.
\end{equation*}
\begin{theorem}\label{thm:SymProp}
For~$A\in{\mathbb{R}}^{3\times 3}$ of the form of expression~\eqref{eq:A1} or \eqref{eq:A2} and any elasticity tensor, $C_{1}\,,\,\dots\,,\,C_{N}$ $\in\mathbb{R}^{6\times6}$\,,
\begin{equation}\label{eq:BAsym}
\widetilde{A}\,\,\overline{\overline{
\left(C_{1}\,,\,\ldots\,,\,C_{n}\right)
}}\,\,\widetilde{A}^{\,T}
=
\overline{\overline{
\left(\widetilde{A}\,C_{1}\,\widetilde{A}^{\,T}\,,\,\dots\,,\widetilde{A}\,C_{n}\,\widetilde{A}^{\,T}\right)
}}
\,,
\end{equation}
which is a symmetry condition.
Conversely, if for an orthogonal matrix, $A\in{\mathbb{R}}^{3\times3}$\,, we have equality~\eqref{eq:BAsym}, for any collection of elasticity tensors, $C_1\,,\,\ldots\,,\,C_n\in{\mathbb{R}}^{6\times6}$\,, then $A$ must be of the form of expression~\eqref{eq:A1} or \eqref{eq:A2}.
\end{theorem}
\begin{proof}
As in Lemma~\ref{lem:BlockStructure}, we permute the rows and columns of~$C\in\mathbb{R}^{6\times6}$ to the order~$(\,3\,,\,4\,,\,5\,,\,1\,,\,2\,,\,6\,)$\,.
Thus, we have the block structure
\begin{equation*}
C
=
\left[\begin{array}{cc}
M & B \\
K & J
\end{array}\right]
\,,\quad
M\,,\,B\,,\,K\,,\,J\in\mathbb{R}^{3\times3};
\end{equation*}
herein, we use the notation of equations (5)--(9) of~\citet{BDaltonSS2017}.
Also,~$\widetilde{A}$ has the block structure of
\begin{equation*}
\widetilde{A}
=
\left[\begin{array}{cc}
\widetilde{A}_{1} & 0 \\
0 & \widetilde{A}_{2}
\end{array}\right]
\,,\quad
\widetilde{A}_{1}\,,\,\widetilde{A}_{2}\in\mathbb{R}^{3\times3}\,,
\end{equation*}
and is orthogonal.

Let
\begin{equation*}
\widetilde{C}
=
\widetilde{A}\,C\widetilde{A}^T
=
\left[\begin{array}{cc}
\widetilde{A}_{1} & 0 \\
0 & \widetilde{A}_{2}
\end{array}\right]
\left[\begin{array}{cc}
M & B \\
K & J
\end{array}\right]
\left[\begin{array}{cc}
\widetilde{A}_{1}{}^T & 0 \\
0 & \widetilde{A}_{2}{}^T
\end{array}\right]
=
\left[\begin{array}{cc}
\widetilde{M} & \widetilde{B} \\
\widetilde{K} & \widetilde{J}
\end{array}\right]\,,
\end{equation*}
where, by Lemma~\ref{lem:BlockStructure},
\begin{equation*}
\widetilde{M} = \widetilde{A}_{1}\,M\,\widetilde{A}_{1}{}^T
\,,\qquad
\widetilde{B} = \widetilde{A}_{1}\,B\,\widetilde{A}_{2}{}^T
\,,\qquad
\widetilde{K} = \widetilde{A}_{2}\,K\,\widetilde{A}_{1}{}^T
\,,\qquad
\widetilde{J} = \widetilde{A}_{2}\,J\,\widetilde{A}_{2}{}^T
\,.
\end{equation*}
In particular,
\begin{align}
\label{eq:tildeMinverse}
\widetilde{M}^{-1}
&=
\left(\widetilde{A}_{1}\,M\,\widetilde{A}_{1}{}^T\right)^{-1} \\
\nonumber
&=
\widetilde{A}_{1}\,M^{-1}\,\widetilde{A}_{1}{}^T
\,.
\end{align}
The Backus-average equations are given by \citep{BDaltonSS2017}
\begin{equation}\label{eq:BAforms}
C_{BA}
=
\left[\begin{array}{cc}
M_{BA} & B_{BA}\\
K_{BA} & J_{BA}
\end{array}\right]
\,,
\end{equation}
where
\begin{align*}
M_{BA} &= \left(\,\overline{M^{-1}}\,\right)^{\!-1}\,, \\
B_{BA} &= \left(\,\overline{M^{-1}}\,\right)^{\!-1}\,\overline{M^{-1}\,B}\,, \\
K_{BA} &= \overline{K\,M^{-1}}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\,,\\
J_{BA}
&=
\left(\overline{J} - \overline{K\,M^{-1}\,B} + \overline{K\,M^{-1}}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\overline{M^{-1}\,B}\right)\,,
\end{align*}
and where~$\overline{\vphantom{\left(\right.}\hspace*{0.5pt}\circ\vphantom{\left.\right)}}$ denotes the arithmetic average of the expression~$\circ$\,; for example,
\begin{equation*}
\overline{M^{-1}}=\frac{1}{n}\sum_{i=1}^n M^{-1}_i\,.
\end{equation*}
Let $M_{BA}$\,, $B_{BA}$\,, $K_{BA}$ and $J_{BA}$ denote the associated sub-blocks of the Backus average of the~$\widetilde{A}\,C_{j}\widetilde{A}^{\,T}$\,.
Then,
\begin{align*}
\widetilde{A}_{1}\,M_{BA}\,\widetilde{A}_{1}{}^T
&=
\widetilde{A}_{1}\left(\,\overline{M^{-1}}\,\right)^{-1}\widetilde{A}_{1}{}^T &\\
&=
\left(\widetilde{A}_{1}\,\overline{M^{-1}}\widetilde{A}_{1}{}^T\right)^{\!-1} &\\
&=
\left(\,\overline{\widetilde{A}_{1}\,M^{-1}\,\widetilde{A}_{1}{}^T}\,\right)^{\!-1}
&
\left(\mathrm{by\,\,linearity}\right) \\
&=
\left(\,\overline{\left(\widetilde{A}_{1}\,M\,\widetilde{A}_{1}{}^T\right)^{-1}}\,\right)^{\!-1}
&
\left({\rm by\,\,equation~\eqref{eq:tildeMinverse}}\right) \\
&=
\left(\,\overline{\left(\widetilde{M}^{-1}\right)}\,\right)^{\!-1} \\
&=
\widetilde{M}_{BA}
\,,
\end{align*}
\begin{align*}
\widetilde{A}_{1}\,B_{BA}\,\widetilde{A}_{2}{}^T
&=
\widetilde{A}_{1}
\left(\,\left(\,\overline{M^{-1}}\,\right)^{-1}\,\overline{M^{-1}B}\,\right)
\widetilde{A}_{2}{}^T
&
\\
&=
\left(\,\widetilde{A}_{1}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\widetilde{A}_{1}{}^{T}\right)
\left(\widetilde{A}_{1}\,\overline{M^{-1}B}\,\widetilde{A}_{2}{}^T\,\right)
&
\\
&=
\widetilde{M}_{BA}\left(\widetilde{A}_{1}\,\overline{M^{-1}B}\,\widetilde{A}_{2}{}^T\right)
&
\left({\rm by~the~previous~result }\right)
\\
&=
\widetilde{M}_{BA}\,\overline{\left(\widetilde{A}_{1}\,M^{-1}B\,\widetilde{A}_{2}{}^T\right)}
&
\left({\rm by\,\,linearity}\right)
\\
&=
\widetilde{M}_{BA}\,\overline{\left(\widetilde{A}_{1}\,M^{-1}\widetilde{A}_{1}{}^T\right)\left(\widetilde{A}_{1}\,B\,\widetilde{A}_{2}{}^T\right)}
\\
&=
\widetilde{M}_{BA}\,\overline{\widetilde{M}^{-1}\,\widetilde{B}\,}
\\
&=
\widetilde{B}_{BA}
\,,
\end{align*}
\begin{align*}
\widetilde{A}_{2}\,K_{BA}\,\widetilde{A}_{1}{}^T
&=
\widetilde{A}_{2}\,\overline{K\,M^{-1}}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\!\widetilde{A}_{1}{}^T
\\
&=
\left(\widetilde{A}_{2}\,\overline{K\,M^{-1}}\,\widetilde{A}_{1}{}^T\right)
\left(\widetilde{A}_{1}\left(\overline{M^{-1}}\,\right)^{\!-1}\!\widetilde{A}_{1}{}^T\right)
\\
&=
\overline{\left(
\widetilde{A}_{2}\,K\,M^{-1}\,\widetilde{A}_{1}{}^T
\right)}\,\widetilde{M}_{BA}
\\
&=
\overline{\widetilde{K}\,\widetilde{M}^{-1}\,}\,\widetilde{M}_{BA}
\\
&=
\widetilde{K}_{BA}
\,,
\end{align*}
and
\begin{align*}
\widetilde{A}_{2}\,J_{BA}\,\widetilde{A}_{2}{}^T
&=
\widetilde{A}_{2}
\left(\overline{J\,} - \overline{K\,M^{-1}\,B} + \overline{K\,M^{-1}}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\overline{M^{-1}\,B}\right)
\widetilde{A}_{2}{}^T
\\
&=
\widetilde{A}_{2}\left(\,\overline{J\,}\,\right)\widetilde{A}_{2}{}^T -
\widetilde{A}_{2}\left(\,\overline{K\,M^{-1}\,B}\,\right)\widetilde{A}_{2}{}^T +
\widetilde{A}_{2}\left(\overline{K\,M^{-1}}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\overline{M^{-1}\,B}\,\right)\widetilde{A}_{2}{}^T
\\
&=
\overline{\widetilde{A}_{2}\left(\,J\,\right)\widetilde{A}_{2}{}^T} -
\overline{\widetilde{A}_{2}\left(\,K\,M^{-1}\,B\,\right)\widetilde{A}_{2}{}^T} +
\widetilde{A}_{2}\,\overline{K\,M^{-1}}\,\widetilde{A}_{1}{}^T\,\,
\widetilde{A}_{1}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\!\widetilde{A}_{1}{}^T\,\,
\widetilde{A}_{1}\,\overline{M^{-1}\,B}\,\widetilde{A}_{2}{}^T
\\
&=
\overline{\widetilde{J}\,\,} -
\overline{
\left(\widetilde{A}_{2}\,K\,\widetilde{A}_{1}{}^T\right)
\left(\widetilde{A}_{1}\,M^{-1}\,\widetilde{A}_{1}{}^T\right)
\left(\widetilde{A}_{1}\,B\,\widetilde{A}_{2}{}^T\right)
} +
\\
&\qquad\quad
\overline{\widetilde{A}_{2}\,K\,M^{-1}\widetilde{A}_{1}{}^T}
\left(\widetilde{A}_{1}\left(\,\overline{M^{-1}}\,\right)^{\!-1}\!\widetilde{A}_{1}{}^T\right)
\overline{\left(
\widetilde{A}_{1}\,M^{-1}\widetilde{A}_{1}{}^T\,\,
\widetilde{A}_{1}\,B\,\widetilde{A}_{2}{}^T
\right)}
\\
&=
\overline{\widetilde{J}\,\,} -
\overline{\widetilde{K}\,\widetilde{M}^{-1}\widetilde{B}\,} +
\overline{\widetilde{K}\,\widetilde{M}^{-1}\,}\,\widetilde{M}_{BA}\,\overline{\widetilde{M}^{-1}\widetilde{B}\,}
\\
&=
\widetilde{J}_{BA}
\,,
\end{align*}
which completes the proof of equality~\eqref{eq:BAsym}.

To show the converse claimed in the statement of Theorem~\ref{thm:SymProp}, let us consider $C_{1}=I$ and $C_{2}=2\,I$\,.
Their Backus average is
\begin{equation}\label{eq:BAofC1andC2}
B
:=
\left[\begin{array}{cccccc}
\tfrac{3}{2} & 0 & 0 & 0 & 0 & 0 \\
0 & \tfrac{3}{2} & 0 & 0 & 0 & 0 \\
0 & 0 & \tfrac{4}{3} & 0 & 0 & 0 \\
0 & 0 & 0 & \tfrac{4}{3} & 0 & 0 \\
0 & 0 & 0 & 0 & \tfrac{4}{3} & 0 \\
0 & 0 & 0 & 0 & 0 & \tfrac{3}{2}
\end{array}\right]\,.
\end{equation}
Following rotation,
\begin{equation}\label{eq:tildeB}
\widetilde{B}
:=
\widetilde{A}\,B\widetilde{A}^{\,T}
\,,
\end{equation}
where $\widetilde{A}$ is given in expression~\eqref{eq:At}.
It can be shown by direct calculation that the $(3,3)$ entry of~$\widetilde{B}$ is
\begin{equation}
\label{eq:33}
\widetilde{B}_{33}
=
\frac{4}{3}\left(
1 +
2\left(b^{2} + c^{2}\right)^{2}\left(a^{2} + d^{2}\right)^{2}
\right)
\,.
\end{equation}
Since $C_{1}$ and $C_{2}$ are multiples of the identity,
\begin{equation*}
\widetilde{A}\,C_{1}\widetilde{A}^{\,T} = C_{1}
\quad{\rm and}\quad
\widetilde{A}\,C_{2}\widetilde{A}^{\,T} = C_{2}
\,,
\end{equation*}
and the Backus average of~$\widetilde{A}\,C_{1}\widetilde{A}^{\,T}$ and~$\widetilde{A}\,C_{2}\widetilde{A}^{\,T}$ equals the Backus average of~$C_{1}$ and~$C_{2}$\,, which is matrix~\eqref{eq:BAofC1andC2}\,.
Hence,
\begin{equation*
\widetilde{A}\,\,\overline{\overline{\left(\,C_{1}\,,\,C_{2}\,\right)}}\,\widetilde{A}^{\,T}
=
\overline{\overline{\left(\,\widetilde{A}\,C_{1}\widetilde{A}^{\,T}\,,\,\widetilde{A}\,C_{2}\widetilde{A}^{\,T}\,\right)}}
\end{equation*}
implies that, for expression~\eqref{eq:33},
\begin{equation*}
\frac{4}{3}\left(
1 +
2\left(b^{2} + c^{2}\right)^{2}\left(a^{2} + d^{2}\right)^{2}
\right)
=
\frac{4}{3}\,,
\end{equation*}
which results in
\begin{equation*}
\frac{2\sqrt{2}}{3}\left(b^{2} + c^{2}\right)\left(a^{2} + d^{2}\right)
=0\,.
\end{equation*}
Thus, either $b=c=0$ or $a=d=0$\,.
This is a necessary condition for symmetry~\eqref{eq:BAsym} to hold, as claimed.
\end{proof}
\begin{remark}\label{rem:DiagMatrices}
Theorem~\ref{thm:SymProp} is formulated for general positive-definite matrices~$C\in{\mathbb{R}}^{6\times6}$\,, not all of which represent elasticity tensors.
However, expression~\eqref{eq:BAsym} is continuous in the~$C_i$ and hence is true in general only if it is also true for~$C_i,$ such as diagonal matrices, which are limits of elasticity tensors.
\end{remark}
\section{The Backus average of randomly oriented tensors}
In this section, we study the Backus average for a random orientations of a given tensor.
As discussed in Section~\ref{sec:BackusGazis}, the arithmetic average of such orientations results in the Gazis et al. average, which is the closest isotropic tensor with respect to the Frobenius norm.
We see that---for the Backus average---the result is, perhaps surprisingly, different.

Given an elasticity tensor, $C\in{\mathbb{R}}^{6\times6}$\,, let us consider a sequence of its random rotations given by
\begin{equation*}
C_j :=
\widetilde{Q}_j\,C\,\widetilde{Q}_j^{\,T},\quad j=1\,,\,\ldots\,,\,n\,,
\end{equation*}
where~$Q_j\in{\mathbb{R}}^{3\times3}$ are random matrices sampled from $SO(3)$\,.

The~$C_j$ are samples from some distribution and, hence, almost surely,
\begin{equation*}
\overline{C}:=\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n} C_{j} =\mu(C)\,,
\end{equation*}
the true mean,
\begin{equation*}
\mu(C)=\int\limits_{SO(3)} \widetilde{Q}\,C\,\widetilde{Q}^{\,T}\,{\rm d}\sigma(Q)
\,,
\end{equation*}
where~${\rm d}\sigma(Q)$ is Haar measure on~$SO(3)$\,.
Note that~$\mu(C)$ is just the Gazis et al. average of~$C$\,.

Similarly, for any expression~$X(C)$ of submatrices of~$C$\,, which appear in the Backus-average formulas,
\begin{equation*}
\overline{X}:=\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n} X_{j} =\mu(X)\,.
\end{equation*}
Hence, almost surely~$\lim_{n\to\infty}\overline{\overline{\left(\,C_{1}\,,\,\ldots\,,\,C_{n}\right)}}$ equals the Backus average formula with each expression~$\overline{X}$ replaced by
\begin{equation*}
\mu(X)=\int\limits_{SO(3)} X\left(\widetilde{Q_j}\,C\,\widetilde{Q_j}{}^{\,T}\right){\rm d}\sigma(Q)\,.
\end{equation*}

\begin{theorem}\label{thm:BTI}
The~$\lim_{n\to\infty}\overline{\overline{\left(\,C_{1}\,,\,\ldots\,,\,C_{n}\right)}}$ exists almost surely, in which case it is transversely isotropic.
It is not, in general, isotropic.
\end{theorem}
\begin{proof}
Let~$A\in{\mathbb{R}}^{3\times3}$ be an orthogonal matrix of type \eqref{eq:A1} or \eqref{eq:A2}.
Then
\begin{align*}
\widetilde{C}_j
&:=\widetilde{A}\,C_j\widetilde{A}^{\,T},\quad j=1\,,\,\ldots\,,\,n\,,\\
&=\widetilde{A}\left(\widetilde{Q_j}\,C_{j}\,\widetilde{Q_j}{}^{\,T}\right)\widetilde{A}^{\,T}\\
&=\left(\widetilde{A}\,\widetilde{Q_j}\right)C_{j}\left(\widetilde{A}\,\widetilde{Q_j}\right)^{\,T}\\
&=\widetilde{\left(AQ_j\right)}\,C_{j}\,\widetilde{\left(AQ_j\right)}^{\,T}
\end{align*}
by the properties of the tilde operation, are also random samples from the same distribution.
Hence, almost surely,
\begin{equation*}
\lim_{n\to\infty}\overline{\overline{
\left(\,\widetilde{C}_1\,,\,\ldots\,,\,\widetilde{C}_n\,\right)
}}
=
\lim_{n\to\infty}\overline{\overline{
\left(\,C_1\,,\,\ldots\,,\,C_n\,\right)
}}
=
B\,,
\quad\hbox{say}\,.
\end{equation*}
But by the symmetry property of the Backus average, Theorem~\ref{thm:SymProp},
\begin{equation*}
\overline{\overline{
\left(\,\widetilde{C}_1\,,\,\ldots\,,\,\widetilde{C}_n\,\right)
}}
=
\widetilde{A}\,\,\overline{\overline{
\left(\,C_1\,,\,\ldots\,,\,C_n\,\right)
}}\,\widetilde{A}^{\,T}.
\end{equation*}
Thus
\begin{equation*}
B=
\widetilde{A}\,B\,\widetilde{A}^{\,T}\,,
\end{equation*}
which means that $B$ is invariant under a rotation of space by~$A$\,.
Consequently, $B$ is a transversely isotropic tensor.

In general, the limit tensor is not isotropic, as illustrated by the following example.
Let
\begin{equation*}
C={\rm diag}\left[\,1\,,\,1\,,\,1\,,\,1\,,\,0\,,\,0\,\right]\,,
\end{equation*}
which, as stated in Remark~\ref{rem:DiagMatrices}, represents a limiting case of an elasticity tensor.
Numerical evidence strongly suggests  that
\begin{equation*}
B
=
\left[\def1.25}\begin{array}{cccccc{1.25}\begin{array}{cccccc}
\tfrac{1}{2} & \tfrac{1}{4} & \tfrac{1}{4} & 0 & 0 & 0 \\
\tfrac{1}{4} & \tfrac{1}{2} & \tfrac{1}{4} & 0 & 0 & 0 \\
\tfrac{1}{4} & \tfrac{1}{4} & \tfrac{1}{2} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \tfrac{1}{4}
\end{array}\right]\,,
\end{equation*}
which is {\it not} isotropic.

Although this is rather an artificial example, it could---with some computational effort---be ``promoted'' to a legal proof.
The conclusion is readily confirmed by the numerical examples presented in Section~\ref{sec:NumericalExamples}.
\end{proof}
In fact, it is easy to identify the limiting matrix~$B$\,; it is just the Backus average expression~\eqref{eq:BAforms}, with an expression~$\overline{X(C)}$ replaced by the true mean
\begin{equation}\label{eq:SectionB}
\mu(X(C))=\int\limits_{SO(3)} X\left(\widetilde{Q}\,C\,\widetilde{Q}^{\,T}\right){\rm d}\sigma(Q)\,.
\end{equation}
This limiting transversely isotropic tensor is of natural interest in its own right.
It plays the role of the Gazis et al. average in the context of the Backus average, and is the subject of a forthcoming work.
\section{Numerical example}
\label{sec:NumericalExamples}
Let us consider the elasticity tensor obtained by~\citet{DewanganGrechka2003}; its components are estimated from seismic measurements in New Mexico,
\begin{equation}\label{eq:GrechkaTensor}
C=
\left[\begin{array}{c*{5}{c}}
7.8195 & 3.4495 & 2.5667 & \sqrt{2}\,(0.1374) & \sqrt{2}\,(0.0558) & \sqrt{2}\,(0.1239)\\
3.4495 & 8.1284 & 2.3589 & \sqrt{2}\,(0.0812) & \sqrt{2}\,(0.0735) & \sqrt{2}\,(0.1692)\\
2.5667 & 2.3589 & 7.0908 & \sqrt{2}\,(-0.0092) & \sqrt{2}\,(0.0286) & \sqrt{2}\,(0.1655)\\
\sqrt{2}\,(0.1374) & \sqrt{2}\,(0.0812) & \sqrt{2}\,(-0.0092) & 2\,(1.6636) & 2\,(-0.0787) & 2\,(0.1053)\\
\sqrt{2}\,(0.0558) & \sqrt{2}\,(0.0735) & \sqrt{2}\,(0.0286) & 2\,(-0.0787) & 2\,(2.0660) & 2\,(-0.1517)\\
\sqrt{2}\,(0.1239) & \sqrt{2}\,(0.1692) & \sqrt{2}\,(0.1655) & 2\,(0.1053) & 2\,(-0.1517) & 2\,(2.4270)
\end{array}\right]
\,.
\end{equation}
Using tensor~\eqref{eq:GrechkaTensor}, let us demonstrate two methods to obtain~$B$ and their mutual convergence in the limit.

The first method to obtain~$B$ requires a stack of layers, whose elasticity tensors are $C$\,.
We rotate each~$C$\,, using a random unit quaternion, and perform the Backus average of the resulting stack of layers.
Using~$10^{7}$ layers, the Backus average is
\begin{equation}\label{eq:Brot}
B_{\equiv}
=
\left[\begin{array}{c*{5}{c}}
7.3008 &
2.9373 &
2.9379 &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right)
\\
2.9373 &
7.3010 &
2.9381 &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right)
\\
2.9379 &
2.9381 &
7.2689 &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(-0.0001\right) &
\sqrt{2}\left(0.0000\right)
\\
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
2\left(2.1710\right) &
2\left(0.0000\right) &
2\left(0.0000\right)
\\
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(-0.0001\right) &
2\left(0.0000\right) &
2\left(2.1710\right) &
2\left(0.0000\right)
\\
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
2\left(0.0000\right) &
2\left(0.0000\right) &
2\left(2.1819\right)
\end{array}\right]
\,.
\end{equation}
For an explicit formulation of the Backus average of generally anisotropic media, see~\citet[expressions~(5)--(9)]{BDaltonSS2017}.

The second method requires integrals in place of arithmetic averages.
Similarly to the first method, we use a random unit quaternion, which is tantamount to a point on a 3-sphere.
We approximate the triple integral using Simpson's and trapezoidal rules.
Effectively, the triple integral is replaced by a weighted sum of the integrand evaluated at discrete points.
The sums that approximate the integrals are accumulated and are used in expressions~\eqref{eq:BAforms}.

Using the Simpson's and trapezoidal rules, with a sufficient number of subintervals, the Backus average is
\begin{equation}\label{eq:Bint}
B_{\!\int\!\!\!\int\!\!\!\int}
=
\left[\begin{array}{c*{5}{c}}
7.3010 &
2.9373 &
2.9380 &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right)
\\
2.9373 &
7.3010 &
2.9380 &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right)
\\
2.9380 &
2.9380 &
7.2687 &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right)
\\
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
2\left(2.1711\right) &
2\left(0.0000\right) &
2\left(0.0000\right)
\\
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(-0.0001\right) &
2\left(0.0000\right) &
2\left(2.1711\right) &
2\left(0.0000\right)
\\
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
\sqrt{2}\left(0.0000\right) &
2\left(0.0000\right) &
2\left(0.0000\right) &
2\left(2.1818\right)
\end{array}\right]
\,.
\end{equation}

In the limit, the components of expressions~\eqref{eq:Brot} and~\eqref{eq:Bint} are the same; their similarity is illustrated in Figure~\ref{fig:Dist}, where the horizontal axis is the number of layers and the vertical axis is the maximum componentwise difference between the two tensors.

\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Dist_BintBrotDiff.pdf}
\caption{
Difference between tensors~\eqref{eq:Brot} and~\eqref{eq:Bint}
}
\label{fig:Dist}
\end{figure}

Expression~\eqref{eq:Bint} is transversely isotropic, as expected from Theorem~\ref{thm:BTI}, and in accordance with \citet[Section~4.3]{BonaEtAl2007}, since its four distinct eigenvalues are
\begin{equation}\label{eq:EigValsBint}
\lambda_{1}=13.1658\,,\,
\lambda_{2}=4.3412\,,\,
\lambda_{3}=4.3636\,,\,
\lambda_{4}=4.3421\,,\,
\end{equation}
with multiplicities of $m_{1}=m_{2}=1$ and $m_{3}=m_{4}=2$\,.
The eigenvalues of expression~\eqref{eq:Brot} are in agreement---up to $10^{-3}$---with eigenvalues~\eqref{eq:EigValsBint} and their multiplicities.
Furthermore, in accordance with Theorem~\ref{thm:BTI}, in the limit, the distance to the closest isotropic tensor for expression~\eqref{eq:Bint} is $0.0326\neq 0$\,; thus the distance does not reduce to zero.

Expressions~\eqref{eq:Brot} and~\eqref{eq:Bint} are transversely isotropic, which is the main conclusion of this work, even though, for numerical modelling, one might view them as isotropic.
This is indicated by~\citet{Thomsen1986} parameters, which for tensor~\eqref{eq:Bint} are
\begin{equation*}
\gamma=2.4768\times10^{-3}\,,\,
\delta=1.5816\times10^{-3}\,,\,
\epsilon=2.2219\times10^{-3}\,;
\end{equation*}
values much less than unity indicate very weak anisotropy.
\section{Conclusions and future work}
\label{sec:FutureWork}
Examining the Backus average of a stack of layers consisting of randomly oriented anisotropic elasticity tensors, we show that---in the limit---this average results in a homogeneous transversely isotropic medium, as stated by Theorems~\ref{thm:SymProp} and~\ref{thm:BTI}.
In other words, the randomness within layers does not result in a medium lacking a directional pattern.
Both the isotropic layers, as shown by~\citet{Backus1962}, and randomly oriented anisotropic layers, as shown herein, result in the average that is transversely isotropic, as a consequence of inhomogeneity among parallel layers.
This property is discussed by~\citet{AdamusEtAl2018}, and herein it is illustrated in Appendix~\ref{app:Alternating}.

In the limit, the transversely isotropic tensor is the Backus counterpart of the Gazis et al. average.
Indeed, the arithmetic average of randomized layers of an elasticity tensor produces the Gazis et al. average and is its closest isotropic tensor, according to the Frobenius norm.
On the other hand, the Backus average of the layers resulting from a randomization of the same tensor produces the transversely isotropic tensor given in expression~\eqref{eq:SectionB}.
This tensor and its properties are the subject of a forthcoming paper.
\section*{Acknowledgments}
We wish to acknowledge discussions with Michael G. Rochester, proofreading of David R. Dalton, as well as the graphic support of Elena Patarini.
This research was performed in the context of The Geomechanics Project supported by Husky Energy.
Also, this research was partially supported by the Natural Sciences and Engineering Research Council of Canada, grant 238416-2013.
\bibliographystyle{apa}

redpajama_set_name: RedPajamaArXiv


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text:
Q: Other semantics associated with the register keyword The register keyword is deprecated and mostly ignored.
But this little part of its documentation on the msdn made me wonder.

The compiler does not accept user requests for register variables; [...]  However, all other semantics associated with the register keyword are honored.

What are those other semantics?

A: For example in C you cannot take the address of an object declared with register specifier.
void foo(void)
{
    register int a = 42;
    &a;  // constraint violation
}

Another example, you cannot use register in a file scope declaration:
register int b = 42;  // constraint violation

int main(void)
{
}

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HomeFeaturesRound-upsSNL round-up: Kestrels add two more, Racers sign import, Tigers appoint Reekie
SNL round-up: Kestrels add two more, Racers sign import, Tigers appoint Reekie
June 21, 2019 Michael Black Round-ups, SNL Comments Off on SNL round-up: Kestrels add two more, Racers sign import, Tigers appoint Reekie
Kirkcaldy Kestrels have further boosted their roster for next season by confirming the return of two promising teenagers.
Great Britain U18 forward Scott Jamieson has agreed a third campaign with the Kestrels, while Lewis Kellock is back for his second year.
A 73-point tally in the Scottish U16 and U20 Leagues in 2016/17 helped Jamieson receive the nod to join the Kestrels in the SNL for the following season.
He has since gone on to grab 76 points in 33 league matches to establish himself as an important part of the team's forward ranks.
Jamieson has also continued to ice at U20 level for Fife Falcons, helping the team continue their dominance in both league and cup competitions.
His efforts were rewarded this year as he was selected to represent GB at the World Championship (Division 1B) tournament in Hungary.
Kellock also progressed through the Fife junior system, before making the step up to the Kestrels last season.
The 18-year-old has previously represented Scotland at junior level and will be eager to make his mark in the senior ranks next term.
Murrayfield Racers have continued to build their roster for the upcoming campaign by securing the services of German-born Alex Christian.
The 25-year-old utility player previously iced in his homeland in the same league as new teammate, Rihards Grigors, before stepping up into Germany3 with Hamburger SV.
He subsequently joined Rodovre SIK in Denmark2, helping the team reach the final of the play-offs in his first season with the club.
Dundee Tigers have confirmed that experienced forward Grant Reekie will be the club's player/coach next season.
The 30-year-old rejoined the Tigers in 2012 after spells with Dundee Stars and Dundee Comets, and has gone on to average almost two points per game in the SNL since then.
(Image permission: Andy Lee)
SNL round-up: Kestrels' forwards back for another year, Racers keep building
Solway Sharks join the Scottish National League
May 19, 2015 Val McLennan News Comments Off on Team Managers
The SIH-UK National Development Program is currently looking for team managers for the coming season. […]
Lightning hoping to reap the rewards from new Pirates link-up
January 12, 2019 Michael Black News Comments Off on Lightning hoping to reap the rewards from new Pirates link-up
Lanarkshire Lightning Junior Ice Hockey Club is hoping a newly-formalised partnership with Paisley Pirates will be the catalyst for continued growth over the next few years. […]
SNL round-up: Boyd bags treble as Murrayfield Racers book place in final
December 31, 2018 Michael Black Round-ups, SNL Comments Off on SNL round-up: Boyd bags treble as Murrayfield Racers book place in final
Murrayfield Racers are through to the final of the Stuart Robertson Cup after recording a 14-3 win at Dundee Tigers. […]