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'SNL': Why Lin-Manuel Miranda's 'Diego Calls Home' Was the Best Sketch Of the Night
By Dustin Rowles | Saturday Night Live | October 9, 2016 |
Many of those who watched Lin-Manuel Miranda's "Diego Calls Home" short last night on SNL may have initially misread it, expecting a funny short film on a comedy show. That it was largely in Spanish may have made it even more hard to decipher, but watch it again. Watch it with an understanding of what it was about, an understanding of Miranda was trying to say.
Taking place in a phone booth in the middle a corn field — an image that evokes both the loneliness and the vulnerability of the immigrant experience — Miranda plays a Latino immigrant who uses a calling card to phone home to talk to his mother. He speaks in Spanish, except when referring to the huge, bizarre wonders of North Dakota — the marshmallow salad, Little Debbie snacks, big mounds of yellow and orange foods — and he tells his mom about the friend he made working as a dishwasher. Preston. He's the QB of his high school football team, and Preston took him to watch the fireworks in the bed of his pick-up truck. He has dinner with Preston's family. They're full of American platitudes, and they make him feel at home.
Miranda speaks of what a great country America is — big, wonderful, strange. It's not a funny short (it's more akin to the melancholy "Sad Mouse") but it's not meant to be, either. It captures the vulnerability of those who enter this country earnest and hopeful, who miss their families but who relish those new connections they make in America, connections that remind them of their long-lost connections to home.
And this is the guy that Donald Trump and his supporters want to prevent from coming into our country by building a wall. This is a guy Donald Trump and his supporters call a rapist.
This guy isn't a criminal. He's a human being. He's a human being who misses his mother, but wants her to know that he's doing well in America. This is the American immigrant.
It's an amazingly sweet, poignant short that left my wife in tears. It also recalls Under the Same Moon (La Misma Luna), a beautiful, touching movie that no one talks about but that everyone should watch (it's on Netflix).
← Kate McKinnon Steals the 'SNL' Cold Open From Alec Baldwin's Donald Trump | Highlights From 'SNL' Hosted by Lin-Manuel Miranda →

redpajama_set_name: RedPajamaCommonCrawl


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Home » Archive » 352 » John de St Jorre
20th Century, D H Lawrence, France, Paris, Publishing
John de St Jorre
Chariots of Flesh
Obelisk: A History of Jack Kahane and the Obelisk Press
By Neil Pearson
Liverpool University Press 494pp £25 order from our bookshop
In 1930, Jack Kahane, a tall gangling Englishman who favoured Savile Row suits, a monocle and cane, launched an unlikely publishing venture in Paris. The early signs were not propitious. He had no publishing experience; he suffered from poor health; he was hobbled by a disadvantageous deal with an unreliable printer-partner; and he found himself operating in a bleak economic environment following the stock market meltdown a year earlier.
Yet over the next decade, Kahane's Obelisk Press not only kept afloat, but published works by Henry Miller (Tropic of Cancer, Tropic of Capricorn and three other novels), Anaïs Nin (House of Incest and Winter of Artifice), Lawrence Durrell (The Black Book), James Joyce (a fragment of Haveth Childers Everywhere and Pomes Penyeach), Cyril Connolly (The Rock Pool), Radclyffe Hall (The Well of Loneliness), and D H Lawrence (Lady Chatterley's Lover).
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Judge Overturns New York City Ban On Big Sugary Sodas
By Maria Godoy,
Published March 11, 2013 at 1:34 PM PDT
Mario Tama
A customer fills a 21-ounce cup with soda at a New York City McDonald's.
A New York state judge has knocked down New York City's landmark new ban on big, sugary drinks, just one day before it was set to take effect.
Calling them "arbitrary and capricious," state Supreme Court Justice Milton Tingling on Monday invalidated regulations that would have banned New York City restaurants, movie theaters and other food service establishments from serving sugary drinks in sizes bigger than 16 ounces. The ban would have covered not just sodas but a wide array of other sugar-sweetened drinks, from smoothies to coffee. (Here's a PDF of the ruling.)
Mayor Michael Bloomberg unveiled the regulations last May, part of an effort to curb runaway obesity rates by encouraging people to drink less sugar-laden beverages. The New York City Board of Health approved the measure last September.
But Tingling found that the board of health had overstepped the authority it was granted to fulfill its mission: protecting against and preventing diseases. That authority, the judge said, does not include the power to "limit or ban a legal item under the guise of 'controlling chronic disease.' "
What's more, Tingling noted, the regulations wouldn't have applied equally across eating establishments. For example, sugary milk products would have been exempt, as would 7-Eleven and other convenience stores, and supermarkets.
Those gaps in the law would have seriously limited its effectiveness, says David Just of Cornell University's Center for Behavioral Economics in Child Nutrition Programs. He says the ban would not likely have made much of an impact on overall calorie intake.
"The overwhelming majority of soda is bought in grocery stores," he told The Salt earlier Monday. He says the ban would have targeted the occasional consumer rather than the regular consumer.
And some advocates of tougher policies say the judge's ruling opens the door to new strategies — such as, perhaps, a sugar-sweetened beverage tax.
"The biggest problem with this measure was that it could have jeopardized the ability to introduce more comprehensive — and potentially more effective — obesity prevention policies," Gary Bennett, an expert on obesity, weight loss and physical activity at Duke University, said in a statement.
Industry groups that sued to have the ban overturned expressed relief.
"The court ruling provides a sigh of relief to New Yorkers and thousands of small businesses in New York City that would have been harmed by this arbitrary and unpopular ban," a spokesperson for the American Beverage Association tells The Salt in an email.
"With this ruling behind us, we look forward to collaborating with city leaders on solutions that will have a meaningful and lasting impact on the people of New York City."
City officials say they plan to appeal the ruling, as our colleagues at WNYC report.
"This measure is part of the city's multipronged effort to combat the growing obesity epidemic," Michael Cardozo, corporate counsel for the city's law department, said in a statement. "We believe the Board of Health has the legal authority — and responsibility — to tackle its leading causes."
Meanwhile, in Mississippi, lawmakers are taking things one step further than Judge Tingling did in New York. A bill on the governor's desk would keep counties and towns from enacting rules governing the size of big sodas and other restrictions on food makers.
The bill — nicknamed the "Anti-Bloomberg Bill" — is popular with state lawmakers, despite the fact that one out of every three Mississippians is obese.
Maria Godoy
Maria Godoy is a senior science and health editor and correspondent with NPR News. Her reporting can be heard across NPR's news shows and podcasts. She is also one of the hosts of NPR's Life Kit.
See stories by Maria Godoy
Allison Aubrey is a correspondent for NPR News, where her stories can be heard on Morning Edition and All Things Considered. She's also a contributor to the PBS NewsHour and is one of the hosts of NPR's Life Kit.


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Engineering at Scale
Retail and Logistics
Oct 16, 2020 by Andrea Lorenzo Soler & Ronald Roccioletti
Still Playing Catch-Up with Core Banking Technology?
A growing number of digital commentators seem to believe that the word "bank" is synonymous with "IT museum."
Harsh? Not surprising though, when you consider that the financial sector (in keeping with large governmental departments, insurers, healthcare providers and most Fortune 500 companies) depends on legacy systems which can be as much as 40 years old. After all, these core banking systems were built on COBOL, a near-forgotten language which predates Coke cans, computer mice and The Beatles.
That said, 220 billion lines of COBOL are still in use today. And because of the scarcity of people versed in the language, veteran engineers have been coaxed out of retirement to fix ancient banking system software at handsome hourly rates. Which is why there's a gathering movement to train a new generation of COBOL engineers to replace the former employees who built and maintained these core banking systems the first time around.
It's a Costly Business
Typically, banks spend around 80% of their IT budgets on maintaining legacy systems. And with something like $3 trillion flowing through them daily, replacing a core banking system is more of a must-do than a might-do. But where do you start?
What are the best practices and pitfalls?
Make Sure You Know What You're Dealing with
Familiarize yourself with the new banking software's capabilities and, possibly, exchange legacy satellite systems where possible, weighing-up the pros and cons of monolithic and distributed systems (see our other blog on this topic). Very often, new standard banking software automatically enables functionalities which were located in those old satellite systems. It's all too easy to find yourself duplicating functionalities in a new system.
Make sure you clean up the data in your legacy systems. Migrating bad data to a new system will generate poor results, however advanced the banking software and technology are – garbage in, garbage out. Check if the data in the new system differs from the original data. You could find that the functionality is also different; something you might only realize at the end of the project and that can be expensive to put right before the new system goes live.
Testing is really important but remember, when you test functionalities, keep to the same transaction each time because, otherwise, you may not catch all the bugs in the system.
Business Wise
Standard core banking software is widely used, but all too often, brokers, custodians and counterparty banks are set up incorrectly. Not enough emphasis is placed on counterparty alignment and, invariably, that's another problem only recognized at the end of a project.
Don't forget to involve all stakeholders as early as possible. Get them on board and committed to easing your journey to the project vision. However, if they're encouraging you to cram functionalities into the last couple of days before go-live, refuse politely and change the subject. And make sure the business doesn't use the old system too much in the days running up to go-live, in case you snag in-flight transactions.
Core banking implementation usually triggers massive organizational change. Whether you're building a new business unit or a completely new business, or implementing a major regulatory update, you need to have a sound strategy for communicating the changes. You'll get a ton of questions like "How are you organizing the bank with the new software?", "What are the new processes?" and "What is the new target operating model?", because everyone needs to feel part of the new system and be aware of what's coming.
Often, you'll need to share post-implementation concerns as well. For instance, someone will flag up that some checks are missing in particular processes – "we're sending reports to the client without anybody keeping an eye on them" – and this proactive move will initiate a remedy.
People Need People
People are a key element in the success of the core banking implementation, so you need to make sure everyone is properly on board and no one is left behind. Today's managers are tomorrow's leaders and plans to facilitate that should be locked in. Promising young talent should get similar treatment. However, while the current project leader and project manager are focusing on the success of the project and future obstacles and opportunities, issues that have been left behind in the run-the-bank (RTB) area can easily be overlooked instead of overseen.
HR and KPIs
It should come as no surprise to find out that decision-makers want it both ways: to keep the business running, as well as being involved in the change project to ensure the new software capabilities match business requirements. In spite of that, the bank is often reluctant to commit adequate resources to the project. One way around this is to ask HR to make involvement part of business executive KPIs. For instance, if they take part in testing, it will have a positive impact on their KPI and encourage them to loosen the purse strings a little.
It sounds obvious, but involving people who actually understand how both the systems and the business elements work is another smart move. Business people need to have an affinity with IT and vice versa – the stronger the better. It's particularly important for positions like stream leader, people writing business specifications, user acceptance test cases, training and so on. Over the years this has proved to be an integral part of successful core banking implementation and change acceptance. Often, the consulting companies have limited access to these key resources. Therefore, it's imperative to address this point at a very early stage and even before starting a new project.
10 Things to Avoid
As well as employing the positive implementation actions, here are some negatives you'd do well to avoid:
A temporary solution becoming permanent
Not spending enough time designing new business processes for the new banking software
Being too ambitious – allowing scope creep, or providing Hollywood solutions to problems with a small number of transactions
Not maintaining standard functionality
Allowing the number of change requests to rise
Allowing change requests even if not regulatory or requested by bank policy
Implementing weak change management
Establishing unrealistic timelines guaranteed to disappoint
Not accounting for the time it takes to adopt a new core banking system and processes, planning project phases like "stabilization phase," followed by "enhancement and new or second priority features phase"
Inadequate defect management (e.g., bugs, functional/change requests, performance issues, etc.). Not focusing on go-live elements or deprioritizing/moving to second phase
Sharpening the Competitive Edge
The plain fact is that while a bank is weighed down by monolithic legacy infrastructure, it cannot hope to compete against a bustling horde of agile, innovative and funky digital competitors. However, as technology advances reduce the cost of upgrading/replacing systems, more and more former naysayers will feel able to go with the flow.
This new clarity of vision will help banks recognize the futility of patching threadbare technology with rolls of $100 bills, and the brilliance of implementing a new core banking system.
Andrea Lorenzo Soler
Vendor Solutions Core Banking DACH Delivery Head, Luxoft
Andrea Lorenzo Solèr gained an M.A. in Business Administration from the University of Zurich, Switzerland. Since 2006, he has been working in the financial sector in various Swiss and international projects as management consultant / business analyst. He has significant cross-functional expertise in consulting, banking, advisory, project management and change management. Andrea is fluent in English, German and Italian.
Ronald Roccioletti
Global Head Core Banking, Luxoft
Ronald Roccioletti studied Electrical Engineering at the Federal Institute of Technology in Zurich (Dipl. El. Ing. ETH), later completing a Master in Advanced Studies in Management, Technology and Economics (MAS MTEC). Subsequently, he chose to work in the financial industry, gaining valuable experience as a consultant, a team lead and a project lead. He speaks German, Italian, English, French and Spanish fluently.
2019-03-28 blog post
LIBOR Transition: Trade documentation management solutions
The progressive transition away from legacy benchmark rates such as Libor is likely to be a key area of CTB expenditure for many Capital Markets organizations in the next 3 years. Benchmark transition...
How to prepare for Brexit, the new normal
Brexit is delivering a wave of change for UK-based finance institutions: for many, it's move certain functions or risk detrimental losses. Brexit's impact on business, particularly financial services ...
Reallocating resources to SFTR – the challenge facing compliance functions, and how to make it an opportunity
The more things change, the more they stay the same.
With the still-growing regulatory and reporting requirements of MiFID II to the challenges of Brexit IT migrations, resources in IT and compli...
UNAFORTIS Joins Luxoft – Bringing the future of financial services to life
Imagine how banking could change in just a few years. Will we be greeted by robots when we walk through the door? Will banking establishments even still exist? Or will we be doing everything from our ...
Think. Create. Accelerate.
with Luxoft
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LA CAÑADA — Flintridge Prep boys' swimming captain Bruce Park admitted his team wasn't expecting a victory Thursday when it hosted South Pasadena High, a CIF Southern Section Division I opponent.
Six races into the dual, the Rebels, who dropped to Division III this year, not only had hope for an upset, but an 18-point lead. The Tigers pulled as close as eight and 10 with one race to go, but were held off as Prep picked up a signature 93-77 win.
It was in the 400 free that the Rebels (3-0) not only took first — thanks to the relay team of Park, Marco Flores, Carlo Sivilotti and Gary Lee — but third to secure the win.
South Pasadena proved far too dominant on the girls' side in a 119-32 victory. The Tigers (2-0) took three of the top-four spots in all eight of the individual events and swept first to third on five occasions.
The Rebels' (0-3) lone victory and top-two individual finishes came in the 200 individual medley from Jennifer Langen, who finished in 2:20.3.
Flintridge Prep's boys came out strong, as Park and Flores teamed with David Shin and Ethan Vandeventer for an opening victory in the 200-medley relay in 1:47.9. Park (1:53) and Gabriel Fernandez (1:58.14) then went first and third, respectively, in the 200 freestyle.
Prep went on to take two of the top three spots in the next three races to build a 56-38 lead.
Sivilotti (22.8) and Lee (23.7) took first and second, respectively, in the 50 free; Park (56.7) and William Newman (1:05) were then No. 1 and 3 to wall in the 100 butterfly before Sivilotti (52.4) and Kevin Chang (54.0) took the top two spots in the 100 free.
"This was a good chance to gauge where we're at because we went down a division and today we were up against a big school," Sivilotti said.
South Pasadena (1-1) then charged back into contention. It took second to fourth in the 500 free, which Fernandez won in 5:28.50; second and third in the 200-freestyle relay; first and second in the 100 back and first and third in the 100 breaststroke.
It was good enough to bring the Tigers, who trailed 76-64 after the backstroke, within 10, 83-73, going into the 400 relay.
"I don't think people knew it would be this close, I don't think people felt we were going to win, so it put a little pressure on us but I think it was the little push that the guys needed," Goto said of the Tigers' comeback attempt.
The Rebels also won the 200-freestyle relay in 1:35.80, thanks to Sivilotti, Lee, Chang and Vandeventer. Flores also took second in the 100 breast (1:03.80) and Fernandez took third in the 100 back (1:07).


The DOJ is considering filing suit against the Agency 5 publishers and Apple. If that goes through, Amazon may be able to control and drive down the price of eBooks again. Hopefully something will be worked out so independents can compete and sell eBooks profitably.


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GE acquires University MRI technology
The University will pocket about one-third of the gained revenue.
by Kate Raddatz
A new MRI technique developed by University of Minnesota researchers was acquired by General Electric Co. earlier this month.
GE Health Care announced in early April it plans to manufacture devices that use technology developed by Steady State Imaging, a group of University researchers. SSI researches and creates MRI technologies including the SWIFT method.
The SWIFT method has been in the MRI research spotlight for its potential to capture images of the body where normal MRIs donâÄôt typically function, like near the lungs. GE Health Care devices using the SWIFT method will be better able to image ligaments, tendons and other body tissues.
SWIFT was developed by a team of researchers led by Michael Garwood at the UniversityâÄôs Center for Magnetic Resonance Research.
GE Health Care has not yet disclosed the financial terms of the deal, but the University will receive royalties and other compensation.
Vice President for Research at the University Tim Mulcahy said the University will pocket about one-third of the revenue, which will be used to reinvest in graduate research. Mulcahy said the agreement would help boost the UniversityâÄôs brand, both in the Legislature and in the medical equipment field.
âÄúIt will be a very strong asset for the University,âÄù Mulcahy said. âÄúOn a reputational standpoint, folks in the MRI field will recognize the product immediately.âÄù
Mulcahy added that while the University will provide the technology and research for GE, it is not selling a product.
âÄúThe University makes the breakthrough discoveries which have the potential to benefit the product of life, but we donâÄôt make products âÄî GE will make those,âÄù he said. âÄúItâÄôs great validation of the quality of work we do to see the full promise of
technology.âÄù
Vice president of GE Health CareâÄôs Global MR business, Jim Davis said in a statement that the acquisition of SSI would help increase access and decrease costs for MRIs.
âÄúWe are excited at the prospect of implementing SSIâÄôs technology,âÄù Davis said. âÄú[WeâÄôll] continue to invest in innovations that will have an impact on our customers and their patients.âÄù
SSI was licensed through the Office of Technology Commercialization in 2006, five years after its creation.
A majority of funding for SSI comes from Minnesota investors, many of whom have taken advantage of the recent Minnesota Angel Tax Credit Program, which provides incentives for investors to put money into a startup company focused on new
technologies.
The CMRR laboratory has been doing biomedical research at the University since it opened in 1991.
Mulcahy called the partnership with GE âÄúfantasticâÄù and said it will help bring University research to a public platform.
âÄúOverall it reflects what we should be about as a research University,âÄù he said. âÄúWe want to ultimately benefit the quality of life and the long-term process of this acquisition is just a part of that.âÄù


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//
//  Copyright 2012 Christopher Dudley
//
//  Licensed under the Apache License, Version 2.0 (the "License");
//  you may not use this file except in compliance with the License.
//  You may obtain a copy of the License at
//
//      http://www.apache.org/licenses/LICENSE-2.0
//
//  Unless required by applicable law or agreed to in writing, software
//  distributed under the License is distributed on an "AS IS" BASIS,
//  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
//  See the License for the specific language governing permissions and
//  limitations under the License.
//
#ifndef _DP_RANDOM_H_
#define _DP_RANDOM_H_

#include "Config.h"

/*
 *  This header defines functions used for thread-safe random number generation
 *  that will not lock when different threads attempt to generate random
 *  numbers.
 */

#ifdef _HAS_STD_RANDOM

// Use STL pseudo-random number generators with thread-local storage.

#include <random>

unsigned long RandInt(void);

double RandDouble(void);

#else // _HAS_STD_RANDOM

// Use rand48 class of functions with thread-local storage for the state.
#include <cstdlib>
#include <cstdio>

unsigned long int RandInt(void);

double RandDouble(void);

#endif // _HAS_STD_RANDOM

#endif // _DP_RANDOM_H_

redpajama_set_name: RedPajamaGithub


Lately all people talk about when mattresses come up is the latex mattress. It is the hottest product used right now, it is used as a topper, it replaces the interspersing mattresses, and it has been selling more and more each day. Why is this so?
There are a lot of reasons why people are buying latex mattresses these days. One of them is the green materials that mattress producers use to make the latex mattresses. The mattresses labeled as "green" are made from all natural materials, sustainable materials and they do not release any unnatural materials. So, what is a latex mattress?
Latex offers something that the other materials can't and that is greater elasticity and buoyant characteristics. It does not het much firmer like cotton mattresses or saggy like wool mattresses.
Another reason why people choose latex is its hypo-allergenic, anti-dust and anti-bacterial qualities.
However, the most important reason why latex has become the number one mattress material is the comfortable sleeping surface that it offers. So if you have been having problems sleeping at night, try buying yourself a latex mattress because it unbelieving pain issues, and it offers unparalleled support and comfort as well.
But when you have decided to buy 100% latex mattress make sure that you know what you are buying. A mattress can be made of synthetic, natural or a mix of synthetic and natural latex and is still considered latex. This is important for you to keep in mind because if you are interested in the "green" properties of latex you should look for mattresses made of 100% organic latex. And even though natural latex and synthetic latex may look identical to scientists, they do not feel the same and they do not last the same as well.
If you want comfort and a mattress without any chemical materials then go for natural latex. But if you have made up your mind to finding a mattress made of pure natural latex you have to keep in mind the different manufacture processes with which latex can be made.
The first one is the Dunlop method. This method is old, traditional and it is used to make natural latex.
The second one is the Talalay method which is a newer method used to make synthetic latex.
There are a lot of advantages which can help you decide whether to buy a latex mattress.
Latex doesn't absorb your body temperature so if you have trouble being warm when sleeping this is the right choice. It does not collect dust or any other microbial organisms because of his organic characteristics. Natural latex is highly durable; it resists body down and adjusts to the the curves of your body. Because of it elasticity you can easily roll from side to belly or back without worrying if you are going to disturb your partner.
It is hard to find a 100% latex mattress because of cost. Manufacturers do not produce so many natural latex mattresses because they are expensive and people do not buy them. That is why they prefer using synthetic or blended latex because it is cheaper. But if you are looking for good quality, without money being an issue than you should look for a 100% pure natural latex mattress.


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Posted byRenée Bess August 24, 2021 Posted inUncategorized
"The world is too much with us; late and soon…"
Amen, Mr. Wordsworth!
But if you thought your world was "too much with [you,]" consider this deck of cards:
A Pandemic, hundreds of thousands of Americans refusing to be vaccinated as the Pandemic's variant is in full bloom, an earthquake in the Western Hemisphere's most destitute island nation, people washed into oblivion by hurricane-induced floods, various school board officials fending off death threats when they propose to keep children safe and healthy, off the charts murders and shootings in Chicago, Philadelphia, Boston and elsewhere, the ever-menacing presence of the Big Lie/Liar, Afghanistan citizens trampled to death in their effort to reach Kabul Airport…
Have you ever felt so overwhelmed by the here and now, that you've found yourself a tad confused as to where "here" really is, not to mention if "now" is the present, past or future?
My "here" is where it's been for the past year and a half.. .at home. My "now" lacks clarity. If I see my neighbor backing into her driveway, I know it's Thursday morning. She always markets early on Thursdays. If I see her in the passenger seat while her husband is behind the wheel, I know which day and time it is. Every Saturday at half past four they leave home to attend the five o'clock Mass. Bereft of those two visual clues, I'm forced to look at the calendar to figure out "now's" identity.
I used to associate sports with the seasons. Not any more. Baseball, football, pro soccer, and ice hockey seem to blur their seasonal boundries like over cooked strands of vermicelli. The sports are meshed, mixed together in the same three or four stadiums. Thanks to the ever-growing number of sports channels at my TV's disposal, I can watch Alpine skiing and luge competitions in August, and the World Series in February.
I've learned to hone in on "now" by watching nature's signals. Nature however, sends jumbled, unreliable signals these days. Tornados used to touch down in Indiana or Kansas. In the last three weeks my area, just outside of Philadelphia, has had six tornados crash down, sucking cars and buildings into their funnels. One more of these unwanted visits and we'll hire a crew to move furniture into our basement. Nothing elaborate, a bed, a table, some chairs, maybe a sleep sofa, and two camping lanterns.
Photo by Ralph W. lambrecht on Pexels.com
Back-to-school sales began in June, just as kids were leaving their virtual learning screens to begin the summer break. Many of the city kids wondered if their community swimming pools would once again hold virtual water and staff virtual lifeguards.
As of last week the local supermarkets began selling dozens of pumpkin flavored products, including dish detergent. The CVS I tend to patronize has begun hawking Halloween candy. I'm pretty sure there are boxes of candy canes in the store's back room. Did fall come and go when I wasn't looking?
Photo by RODNAE Productions on Pexels.com
I believe my loss of "here and now" began in 2020 as I searched online for N95 masks and settled for the KN95 models. Or, did it start when my yearly wellness check was conducted online? Certainly the loss was in progress as get-togethers with friends revealed only our faces. I do recall caring about the "here and now" during the run up to the 2020 Presidential election, even if I didn't feel I could risk my health by working at the Polls.
Joe Biden and Kamala Harris were elected! Pandemic or not, we were headed out of Bizarro World and back to Planet Normalcy. Or so I thought…for the first forty-eight hours after Election Day. It's been downhill since then.
The specious court challenges to the election results, the willingness of so many people to buy into the Big Lie, the proliferation of in-your-face racist animus, and Covid-19's Delta variant have plunged my mind and spirit into the nether world of existential limbo.
And let's face it, limbo is best appreciated as a dance one does while balancing a fruity adult beverage in one hand, and using the other one to stay balanced. Limbo is not a place where one should live.
As I free-fall into the valley of adult onset ADHD, I've decided to lose myself in old movies whose plots and characters never related to my reality. I've stopped holding on to every word spoken by the CNN and MSNBC reporters. (Although I do worry about the women and young girls who won't be able to leave Afghanistan.) I've lowered the volume of my rage against the hypocrisy of the Fox News talking heads who publicly scorn those of us who got our vaccinations and privately received theirs without having to stand in long lines or struggle to get appointments. I avoid reading incomprehensible comments on social media sites, especially those remarks that favor sending mask-less children into petri dish classrooms. I DON'T UNDERSTAND risking children's health.
I defend the social media's existence, but there are times when I believe it's a weapon of mass destruction.
Anyhoo… my sense of "here and now" tells me it's late August. Gardening season, with all of its work and rewards is still upon us. I know this because my perpetually peaceful spouse is still at war with the weeds that get in her way and the garden pests that try her patience. Our First Aid supplies are still in the laundry room closet, close by just in case cuts, scratches, bee stings, and smashed fingers persist.
Patient First, a nearby doc-in-the-box facility, has affixed Viv's name to a chair in their waiting room. And rumor has it that a medical school of fine repute might endow a chair in Viv's name for a visiting instructor in their newest department, "Emergency Medicine for Yard and Garden Mishaps."
Bring on September! Did we celebrate the fourth of July? What about New Year's Eve?
Renée Bess is the author of five novels, the co-curator of an award winning anthology, OUR HAPPY HOURS, LGBT VOICES FROM THE GAY BARS, and BETWEEN A ROCK AND A SOFT PLACE, COLLECTED WORKS, her latest book. She blogs when she cannot help herself from doing so. Here's more info, http://www.reneebess.com
hoto by Artem on Pexels.com
SOMEWHERE BETWEEN A WATERFALL AND A TRICKLE
2 thoughts on "PROPS TO WILLIAM WORDSWORTH"
Lee Pierce says:
Thank you, Renee, for your description of these times in which we live. What's helping me through the madness you described so well is a growing consciousness that — "I am not this Body and I'm not even the Mind"; and that there is a Consciousness each of us have within — THIS CONSCIOUSNESS is Who we are. (Our bodies will return to the earth — evidenced by the fact that food we eat turns into our bodies which the Earth reclaims when we transition ; and, our Minds are simply an accumulation of all memories — like the software in a computer.) It is our Consciousness that is part of the infinite COSMOS — eternal Universe —
A Yogi Mystic named Sadhguru explains these ideas in videos produced by his ISHA Foundation, available on Y'Tube. Isha's website is at — isha.sadhguru.org — Ten months ago I participated in Isha's INNER ENGINEERING sessions (a 3 day, on-line webinar); and, I'm glad that I did — the lessons learned are helping me make "sense" out of the "sense-less" confusion of the times and Era in which we live. Stay safe
jaiebaxter21 says:
Thank you for reading my post and responding to it. You've given me something about which I'll have to think. Stay safe, my good friend.


text:
Tori Spelling's Family Album With Dean McDermott Through the Years
Growing up with famous parents, Tori Spelling has always lived her life in the public eye. The daughter of Aaron Spelling and Candy Spelling, who later developed a successful career of her own, has never shied away from putting herself and her family in the spotlight.
Her love life has also played out publicly, most notably her marriage to husband Dean McDermott. The couple, who tied the knot in 2006, have since welcomed five children together: son Liam in March 2007, daughter Stella in June 2008, daughter Hattie in October 2011, son Finn in August 2012, and son Beau in March 2017. The Scary Movie 2 actress is also stepmom to McDermott's son Jack, whom he shares with ex Mary Jo Eustace.
The tight-knit family isn't afraid of giving their fans a close look at their lives at home — even the more challenging moments. Tori and her husband have been honest and open about the difficulties they've faced in their marriage, including infidelity, health scares and financial troubles. Despite their ups and downs, the BH90210 alum told Us Weekly in February 2019 that she and the former Chopped Canada host "don't have relationship problems."
"We communicate now, I have a voice," she explained at the time. "He probably hears too much of it but he [is so] good at listening. Women like to talk and over-explain."
The longtime couple has been dedicated to working through their past troubles for the sake of their children, but McDermott admitted to having a hard time reconciling with the concept of monogamy — especially when his relationship has been put "under a microscope."
"Men, we're like, we need to spread our seed everywhere. It's [been] in our DNA for millions and millions of years," he told Us in October 2019. "There's that, but then there's also, 'Oh, God. This relationship is work.' And it is. Every relationship is work and when you get to that point, you can't bail. You've gotta knuckle under and work it out."
Tori made headlines in June 2021 when she revealed that she and McDermott were sleeping separately. "Right now, my kids and dogs sleep in my bed," she explained at the time. "Since he left [and] he was gone for six months filming in another country, they all stayed with me. So I currently still have four in the bedroom with me who have yet to go back to their rooms, yes."
The pair were experiencing "major issues" at the time, a source exclusively told Us. "Tori knows her kids see what she says online, so for her to say something publicly means the end could be very near."
Four months later, another source exclusively told Us that Spelling and McDermott were staying together "for the kids," explaining that the actress wanted "a divorce but [felt] trapped."
Scroll down to see the most adorable photos of the actress — and her brood!
Everything Kourtney Kardashian Has Said About Having Another Baby
Vanderpump Rules' Scheana Shay and Brock Davies' Daughter Summer's Album: Baby Photos


text:
//! Bindings to the Windows SChannel APIs.
#![cfg(windows)]
#![warn(missing_docs)]
#![allow(non_upper_case_globals)]

use std::ffi::c_void;
use std::ptr;

use windows_sys::Win32::Security::Authentication::Identity;

macro_rules! inner {
    ($t:path, $raw:ty) => {
        impl crate::Inner<$raw> for $t {
            unsafe fn from_inner(t: $raw) -> Self {
                $t(t)
            }

            fn as_inner(&self) -> $raw {
                self.0
            }

            fn get_mut(&mut self) -> &mut $raw {
                &mut self.0
            }
        }

        impl crate::RawPointer for $t {
            unsafe fn from_ptr(t: *mut ::std::os::raw::c_void) -> $t {
                $t(t as _)
            }

            unsafe fn as_ptr(&self) -> *mut ::std::os::raw::c_void {
                self.0 as *mut _
            }
        }
    };
}

macro_rules! null_terminate {
    ($input:expr) => {{
        const OUTPUT: [u8; $input.as_bytes().len() + 1] = {
            let mut output = [0u8; $input.as_bytes().len() + 1];

            let input = $input.as_bytes();

            // The output is 1 byte longer, so the last byte stays initialized to 0
            let mut i = 0usize;
            while i < input.len() {
                output[i] = input[i];
                i += 1;
            }

            output
        };

        &OUTPUT
    }};
}

/// Allows access to the underlying schannel API representation of a wrapped data type
///
/// Performing actions with internal handles might lead to the violation of internal assumptions
/// and therefore is inherently unsafe.
pub trait RawPointer {
    /// Constructs an instance of this type from its handle / pointer.
    /// # Safety
    /// This function is unsafe
    unsafe fn from_ptr(t: *mut ::std::os::raw::c_void) -> Self;

    /// Get a raw pointer from the underlying handle / pointer.
    /// # Safety
    /// This function is unsafe
    unsafe fn as_ptr(&self) -> *mut ::std::os::raw::c_void;
}

pub mod cert_chain;
pub mod cert_context;
pub mod cert_store;
pub mod crypt_key;
pub mod crypt_prov;
/* pub */ mod ctl_context;
pub mod key_handle;
pub mod ncrypt_key;
pub mod schannel_cred;
pub mod tls_stream;

mod alpn_list;
mod context_buffer;
mod security_context;

#[cfg(test)]
mod test;

const ACCEPT_REQUESTS: u32 = Identity::ASC_REQ_ALLOCATE_MEMORY
    | Identity::ASC_REQ_CONFIDENTIALITY
    | Identity::ASC_REQ_SEQUENCE_DETECT
    | Identity::ASC_REQ_STREAM
    | Identity::ASC_REQ_REPLAY_DETECT;

const INIT_REQUESTS: u32 = Identity::ISC_REQ_CONFIDENTIALITY
    | Identity::ISC_REQ_INTEGRITY
    | Identity::ISC_REQ_REPLAY_DETECT
    | Identity::ISC_REQ_SEQUENCE_DETECT
    | Identity::ISC_REQ_MANUAL_CRED_VALIDATION
    | Identity::ISC_REQ_ALLOCATE_MEMORY
    | Identity::ISC_REQ_STREAM
    | Identity::ISC_REQ_USE_SUPPLIED_CREDS;

trait Inner<T> {
    unsafe fn from_inner(t: T) -> Self;

    fn as_inner(&self) -> T;

    fn get_mut(&mut self) -> &mut T;
}

unsafe fn secbuf(buftype: u32, bytes: Option<&mut [u8]>) -> Identity::SecBuffer {
    let (ptr, len) = match bytes {
        Some(bytes) => (bytes.as_mut_ptr(), bytes.len() as u32),
        None => (ptr::null_mut(), 0),
    };
    Identity::SecBuffer {
        BufferType: buftype,
        cbBuffer: len,
        pvBuffer: ptr as *mut c_void,
    }
}

unsafe fn secbuf_desc(bufs: &mut [Identity::SecBuffer]) -> Identity::SecBufferDesc {
    Identity::SecBufferDesc {
        ulVersion: Identity::SECBUFFER_VERSION,
        cBuffers: bufs.len() as u32,
        pBuffers: bufs.as_mut_ptr(),
    }
}


Lots of ritual and Mustahab acts for "Hajj" and "Umrah" has been mentioned in the words of great Faqihs (Allah's heaven be upon them) and we have brought a considerable part of them here.
But it is better to perform them with the intention of "Rija'", which means hoping for the goodness of the act in the eyes of religion and having benefits and blessings, because some of them do not have enough proof (and we do not consider the principle of tolerance correct in the proof of traditions and Mustahab acts).
Another important point is that performing some of those Mustahab acts today, with crowd congestion, is impossible for many people. Therefore they can perform those actions which is possible, and if the Hajji wants to perform an act which is difficult to perform but have an intention of performing that, Allah will give him/her the reward according to his/her intention as it has been mentioned in hadeeth of Ma'soomin (a.s.).
اللَّهُمَّ إنّی أتَقَرَّبُ إلَیْکَ بهِنَّ، فَاجْعَلْهُنَّ خَلیفَتی فی أهْلِی وَ مَالی .
مَعىَ وَ سَلِّمْنی وَ سَلِّم مَا مَعِىَ وَ بَلِّغْنی وَ بَلِّغْ مَا مَعِىَ ببَلاغِکَ الحَسَنِ الجَمیلِ.


Two words that can put the fear into many people and students, are "essay" and "research".
Apparently Einstein, the Great Genius, said: "If we knew what it was we were doing, it would not be called research, would it?" I love this sentence because of the mystery of research. One starts with a vague question, or something that you wonder about. And then you begin to find many interesting things.
Before you know it, you have many more questions that in the beginning. You even wonder what your original question was, and why you tried to get an answer! But if one sticks to the plan for essay-writing and research, it can be as easy as pie.
Many people need to do research. You need to do research to get answers. Maybe you need to develop a product or design something or advertise a brand or find out what readers of a newspaper or magazine like. Or maybe you need to write an essay for school. I will show you how to make a mind-map after making sure you understand the question well. The main goal is to focus!
Where do I start with research for an essay?
As an academic and a writer, I can help you to understand research better.
I can also help you write a great essay or report based on the research that you have done.
Why don't you give me a ring if you need help with your essay, project or research?


text:
Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection
José L. Mate, Aurelio Ariza, Ana Muñoz, José L. Molinero, Dolores López, José J. Navas-Palacios
Department of Morphological Sciences
With the goal of facilitating viral reproduction, cytomegalovirus (CMV) induces changes in the host cell replication machinery. Very little information is available, however, on the effects brought about by CMV on proliferating cell nuclear antigen (PCNA) and Ki-67 expression in infected cells. Fifty-five paraffin-embedded tissue samples (43 gastrointestinal, 10 skin, and 2 kidney biopsies) with both histological and immunohistochemical evidence of CMV infection were investigated for PCNA and Ki-67 expression by the avidinbiofin-peroxidase method. Of the 55 cases studied, 47 were positive for PCNA and 46 for Ki-67. PCNA and Ki-67 immunostaining was more striking in CMV-immunoreactive, inclusion-free cell nuclei, whereas cell nuclei exhibiting well-developed CMV inclusions either showed a weak peripheral signal for both proliferation markers, or were completely negative. Enhanced PCNA and Ki-67 expression appears to be among the changes induced by CMV infection in host cells. Moreover, this induction seems to reach its peak during the earlier phases of CMV infection and abate as the infection proceeds to its inclusion-forming phases, when a sufficiently high viral load would have been attained.
Journal of Pathology
https://doi.org/10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4
PCNA
10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4
Fingerprint Dive into the research topics of 'Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection'. Together they form a unique fingerprint.
Cytomegalovirus Infections Medicine & Life Sciences
Proliferating Cell Nuclear Antigen Medicine & Life Sciences
Cell Nucleus Medicine & Life Sciences
Viral Load Medicine & Life Sciences
Reproduction Medicine & Life Sciences
Peroxidase Medicine & Life Sciences
Paraffin Medicine & Life Sciences
Mate, J. L., Ariza, A., Muñoz, A., Molinero, J. L., López, D., & Navas-Palacios, J. J. (1998). Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection. Journal of Pathology, 184(3), 279-282. https://doi.org/10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4
Mate, José L. ; Ariza, Aurelio ; Muñoz, Ana ; Molinero, José L. ; López, Dolores ; Navas-Palacios, José J. / Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection. In: Journal of Pathology. 1998 ; Vol. 184, No. 3. pp. 279-282.
@article{8d202c9441884dce8fabdd4c08a24062,
title = "Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection",
abstract = "With the goal of facilitating viral reproduction, cytomegalovirus (CMV) induces changes in the host cell replication machinery. Very little information is available, however, on the effects brought about by CMV on proliferating cell nuclear antigen (PCNA) and Ki-67 expression in infected cells. Fifty-five paraffin-embedded tissue samples (43 gastrointestinal, 10 skin, and 2 kidney biopsies) with both histological and immunohistochemical evidence of CMV infection were investigated for PCNA and Ki-67 expression by the avidinbiofin-peroxidase method. Of the 55 cases studied, 47 were positive for PCNA and 46 for Ki-67. PCNA and Ki-67 immunostaining was more striking in CMV-immunoreactive, inclusion-free cell nuclei, whereas cell nuclei exhibiting well-developed CMV inclusions either showed a weak peripheral signal for both proliferation markers, or were completely negative. Enhanced PCNA and Ki-67 expression appears to be among the changes induced by CMV infection in host cells. Moreover, this induction seems to reach its peak during the earlier phases of CMV infection and abate as the infection proceeds to its inclusion-forming phases, when a sufficiently high viral load would have been attained.",
keywords = "Cell cycle, Cytomegalovirus, Ki-67, PCNA",
author = "Mate, {Jos{\'e} L.} and Aurelio Ariza and Ana Mu{\~n}oz and Molinero, {Jos{\'e} L.} and Dolores L{\'o}pez and Navas-Palacios, {Jos{\'e} J.}",
doi = "10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4",
journal = "Journal of Pathology",
Mate, JL, Ariza, A, Muñoz, A, Molinero, JL, López, D & Navas-Palacios, JJ 1998, 'Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection', Journal of Pathology, vol. 184, no. 3, pp. 279-282. https://doi.org/10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4
Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection. / Mate, José L.; Ariza, Aurelio; Muñoz, Ana; Molinero, José L.; López, Dolores; Navas-Palacios, José J.
In: Journal of Pathology, Vol. 184, No. 3, 01.03.1998, p. 279-282.
T1 - Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection
AU - Mate, José L.
AU - Ariza, Aurelio
AU - Muñoz, Ana
AU - Molinero, José L.
AU - López, Dolores
AU - Navas-Palacios, José J.
N2 - With the goal of facilitating viral reproduction, cytomegalovirus (CMV) induces changes in the host cell replication machinery. Very little information is available, however, on the effects brought about by CMV on proliferating cell nuclear antigen (PCNA) and Ki-67 expression in infected cells. Fifty-five paraffin-embedded tissue samples (43 gastrointestinal, 10 skin, and 2 kidney biopsies) with both histological and immunohistochemical evidence of CMV infection were investigated for PCNA and Ki-67 expression by the avidinbiofin-peroxidase method. Of the 55 cases studied, 47 were positive for PCNA and 46 for Ki-67. PCNA and Ki-67 immunostaining was more striking in CMV-immunoreactive, inclusion-free cell nuclei, whereas cell nuclei exhibiting well-developed CMV inclusions either showed a weak peripheral signal for both proliferation markers, or were completely negative. Enhanced PCNA and Ki-67 expression appears to be among the changes induced by CMV infection in host cells. Moreover, this induction seems to reach its peak during the earlier phases of CMV infection and abate as the infection proceeds to its inclusion-forming phases, when a sufficiently high viral load would have been attained.
AB - With the goal of facilitating viral reproduction, cytomegalovirus (CMV) induces changes in the host cell replication machinery. Very little information is available, however, on the effects brought about by CMV on proliferating cell nuclear antigen (PCNA) and Ki-67 expression in infected cells. Fifty-five paraffin-embedded tissue samples (43 gastrointestinal, 10 skin, and 2 kidney biopsies) with both histological and immunohistochemical evidence of CMV infection were investigated for PCNA and Ki-67 expression by the avidinbiofin-peroxidase method. Of the 55 cases studied, 47 were positive for PCNA and 46 for Ki-67. PCNA and Ki-67 immunostaining was more striking in CMV-immunoreactive, inclusion-free cell nuclei, whereas cell nuclei exhibiting well-developed CMV inclusions either showed a weak peripheral signal for both proliferation markers, or were completely negative. Enhanced PCNA and Ki-67 expression appears to be among the changes induced by CMV infection in host cells. Moreover, this induction seems to reach its peak during the earlier phases of CMV infection and abate as the infection proceeds to its inclusion-forming phases, when a sufficiently high viral load would have been attained.
KW - Cell cycle
KW - Cytomegalovirus
KW - Ki-67
KW - PCNA
U2 - 10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4
DO - 10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4
JO - Journal of Pathology
JF - Journal of Pathology
Mate JL, Ariza A, Muñoz A, Molinero JL, López D, Navas-Palacios JJ. Induction of proliferating cell nuclear antigen and Ki-67 expression by cytomegalovirus infection. Journal of Pathology. 1998 Mar 1;184(3):279-282. https://doi.org/10.1002/(SICI)1096-9896(199803)184:3<279::AID-PATH7>3.0.CO;2-4


text:
Run `meteor` from the `api` directory to start the Meteor server.

## Bundling Meteor client

`meteor-client-bundler` is a module bundler which will take a bunch of Atmosphere package and put them into a single module, so we can load Meteor's client scripts regardless of what framework we're using to run our server.

Run `./node_modules/.bin/meteor-client bundle -s api`.

## Development server

Then run `npm run start` for a dev server. Navigate to `http://localhost:4200/`. The app will automatically reload if you change any of the source files.

## Build

Run `npm run build` to build the project. The build artifacts will be stored in the `dist/` directory. Use `npm run build-prod` for a production build with AOT.

## Running unit tests

Run `npm run test` to execute the unit tests via [Karma](https://karma-runner.github.io).

## Running end-to-end tests

Run `npm run e2e` to execute the end-to-end tests via [Protractor](http://www.protractortest.org/).

## The process I've done to recreate this

```
ng new angular-cli-meteor --style scss
cd angular-cli-meteor
```

```
meteor create api
yarn add babel-runtime
yarn add meteor-node-stubs
yarn add meteor-rxjs
yarn add meteor-client-bundler --dev
yarn add @types/meteor --dev
rm -rf api/node_modules
rm -rf api/client
mv api/server/main.js api/server/main.ts
rm api/package.json api/yarn.lock
cd api;
ln -s ../package.json
ln -s ../yarn.lock
ln -s ../tsconfig.json
ln -s ../node_modules
meteor add barbatus:typescript;
cd ..;
```

Create `meteor-client.config.json`.
Add `"generateNodeModules": true` to `meteor-client.config.json`.
Add `node_modules/meteor-client.js` to `scripts` section in `angular.json`.
Add `meteor` to `types` section in `src/tsconfig.app.json`


Looking for a barber or a hairstylist specialised in taking care of your hair in RIVER CHAPEL?
CREATIVE STYLES, welcomes you for a unique and unforgettable salon experience.
CREATIVE STYLES offers solutions and the latest trends in haircuts, colour, hair care and styling for a total look.


Students and staff members can print black-and-white prints free of charge to the Printing Point devices in A4 or A3 sizes.
First you have to print your document to the virtual printer SecureprintPS, and then you can release the prints from any of the devices at Aalto University by logging in with your Aalto IT credentials. Once logged in you can register your HSL travel, Frank student card, or Canon card to the service, and so you don't need to do type in the credentials in future.
The other way is to send the file to the email address mobileprint@aalto.fi instead of printing to the virtual printer.
Learning Hub Atrium: Room 1156 (corridor).
Library of Mathematics & Systems Analysis: Student hub.
Non-Aalto customers can print black-and-white A4 to the customer service printer U91000-ps13 which costs 0,10 €/page. Payment only with credit and debit cards.
use Print in City service.
Please see the tab Printing in colours for details.
Only A4 or A3 printing is possible at us. Please contact Unigrafia in need of printing on other sizes.


text:
\section{Introduction}
Since the work of Bennett \cite{benn} and Pettitt \cite{pett} on the
proportional odds (PO) model in the survival analysis context, it
has been used by many researchers. The assumption of constant hazard
ratio is unreasonable in many practical cases as discussed by
Bennett \cite{benn}, Kirmani and Gupta \cite{kirmani} and Rossini
and Tsiatis \cite{ross}. The PO model has been used by Bennett
\cite{benn} to demonstrate the effectiveness of a cure, when the
mortality rate of a group having some disease approaches that of a
(disease-free) control group as time progresses. After Bennet's
\cite{benn} work, the PO model has found many practical
applications, see, for instance, Collett \cite{coll}, Dinse and
Lagakos \cite{dinse}, Pettitt \cite{pett} and Rossini and Tsiatis
\cite{ross}. Let $X$ and $Y$ be two random variables with
distribution functions $F(\cdot)$, $G(\cdot)$, survival functions
$\bar{F}(\cdot)$, $\bar{G}(\cdot)$, probability density functions
$f(\cdot)$, $g(\cdot)$ and hazard rate functions
$r_{X}(\cdot)=f(\cdot)/\bar{F}(\cdot)$,
$r_{Y}(\cdot)=g(\cdot)/\bar{G}(\cdot)$. Let the odds functions of
$X$ and $Y$ be denoted respectively by
$\theta_{X}(t)=\bar{F}(t)/F(t)$ and $\theta_{Y}(t)=\bar{G}(t)/G(t)$.
The random variables $X$ and $Y$ are said to satisfy PO model with
proportionality constant $\alpha$ if $\theta_{Y}(t)=\alpha
\theta_{X}(t)$. For more discussion on PO models one may refer to
Kirmani and Gupta \cite{kirmani}. It is observed that, in terms of
survival functions, the PO model can be represented as
\begin{equation}\label{poalt}\bar{G}(t)=\frac{\alpha
\bar{F}(t)}{1-\bar{\alpha}\bar{F}(t)},\end{equation} where
$\bar{\alpha}=1-\alpha$. From the above representation we have
$$\frac{r_{Y}(t)}{r_{X}(t)}=\frac{1}{1-\bar{\alpha}\bar{F}(t)}=\frac{G(t)}{F(t)},$$
so that the hazard ratio is increasing (resp. decreasing) for
$\alpha>1$ (resp. $\alpha<1$) and it convergence to 1 as $t$ tends
to $\infty$. This property of hazard functions makes the PO model
reasonable in many practical applications. This is in contrast to
the proportional hazards model where the ratio of the hazard rates
remains constant with time.\par The model (\ref{poalt}), with
$0<\alpha <\infty$; gives a method of introducing new parameter
$\alpha$ to a family of distributions for obtaining more flexible
new family of distributions as discussed by Marshall and Olkin
\cite{marsh1}. The family of distributions so obtained is known as
Marshall-Olkin family of distributions or Marshall-Olkin extended
distributions (for details see \cite{marsh1,marsh2}). For more
discussion and applications of Marshall-Olkin family of
distributions one can see \cite{caron,cord1,cord,ghit2}. The
parameter $\alpha$ is called `tilt parameter'. This is because the
hazard rate of the new family is shifted below or above the hazard
rate of the underlying (baseline) distribution for $\alpha\geq 1$
and $0<\alpha \leq 1$ respectively. Thus, Marshall-Olkin family of
distributions has implications both in terms of PO model as well as
in generating a new family of flexible distributions, and hence it
is worth to investigate this family of distributions.\par Kirmani
and Gupta \cite{kirmani} studied some ageing properties of the PO
model with fixed tilt parameter $\alpha$. Nanda and Das
\cite{nanda1} studied different ageing classes of Marshall-Olkin
family of distributions taking the tilt parameter as random
variable. Ghitany and Kotz \cite{ghit} studied the reliability
properties by taking $\bar{F}$ as the reliability function of the
linear failure-rate distribution. Gupta et al. \cite{guptar}
compared the Marshall-Olkin extended distribution and the original
distribution with respect to some stochastic orderings. Nanda and
Das \cite{nanda2} compared this family of distributions with respect
to different stochastic orderings by taking the tilt parameter
random. All the studies mentioned above consider the original
(baseline) distribution to be continuous. However, not much work is
available in the literature for discrete case. In this paper, we
study different ageing properties, as well as different stochastic
orderings of this family of distributions with discrete baseline
distribution and with fixed tilt parameter.
\section{Preliminaries}
Here we discuss the survival function, the hazard (failure) rate
function, and the mean residual life of a discrete random variable
$X$ with support $\mathbb{N}=\{1,2,...\}$. Let the probability mass
function (pmf) of $X$ be given by $f(k)=P\{X=k\}$, and the
distribution function be $F(\cdot)$ so that the reliability
(survival) function $\bar{F}(\cdot)$ of X becomes
$$\bar{F}(k)=P\{X>k\}=\sum_{j=k+1}^\infty f(j),~k=1,2,...,$$ with $\bar{F}(0)=1$. The
failure rate function $r(\cdot)$ (Shaked et al. \cite{shaked}) is
given by $$r(k)=P\{X=k|X\geq k\}=\frac{P\{X=k\}}{P\{X\geq
k\}}=\frac{f(k)}{\bar{F}(k-1)},$$ and the reversed hazard rate
function is given by $\tilde{r}(k)=f(k)/F(k)$. Below we give the
definitions of different discrete ageing classes.
\begin{definition}
A discrete random variable $X$ is said to be
\begin{enumerate}[label=(\roman*)]
\item ILR (DLR) i.e. increasing (decreasing) in likelihood ratio if $f(k)$ is log-concave (log-convex),
i.e. if $f(k+2)f(k)\leq(\geq)~f^2(k+1)$, $k\in \mathbb{N}$ (Dewan
and Sudheesh \cite{dewan});
\item IFR (DFR) i.e. increasing (decreasing) failure rate if $r(k)$ is increasing (decreasing) in
$k\in \mathbb{N}$. This is equivalent to the fact that $\bar{F}(k +
1)/\bar{F}(k)$ is decreasing (increasing) in $k\in \mathbb{N}$
(Salvia and Bollinger \cite{salvia}, Shaked et al. \cite{shaked},
Gupta et al. \cite{gupta});
\item IFRA (DFRA) i.e. increasing (decreasing) in failure rate average if $[\bar{F}(k)]^{1/k}$ is decreasing (increasing) in $k$, i.e., $[\bar{F}(k)]^{1/k}\geq(\leq)~[\bar{F}(k+1)]^{1/(k+1)}$, $k\in \mathbb{N}$
(Esary et al. \cite{esary}, Shaked et al. \cite{shaked});
\item NBU (NWU) i.e. new better (worse) than used if
$\bar{F}(j + k)\leq(\geq)~\bar{F}(j)\bar{F}(k),~j,k\in \mathbb{N}$
(Esary et al. \cite{esary}, Shaked et al. \cite{shaked});
\item DRHR (decreasing reversed hazard rate) if $\tilde{r}(k)$ is decreasing in
$k$, i.e. if $F(k)$ is log-concave, i.e. if $[F(k+1)]^2\geq F(k)
F(k+2)$, $k\in \mathbb{N}$ (Nanda and Sengupta \cite{nanda}, Li and
Xu \cite{lix}).
\item NBAFR (new better than used in failure rate average) if
$[\bar{F}(k)]^{1/k}\leq \bar{F}(1)$, $k\in \mathbb{N}$ (Fagiuoli and
Pellerey \cite{fagi}).
\end{enumerate}
\end{definition}
\subsection{Proportional odds family of discrete distributions} Let $X$
be a discrete random variable with support $\mathbb{N}=\{1,2,...\}$
having pmf $f(\cdot)$, distribution function $F(\cdot)$, survival
function $\bar{F}(\cdot)$, hazard rate function $r_X(\cdot)$, and
reversed hazard rate function $\tilde{r}_X(\cdot)$. Starting with
the survival function $\bar{F}$, the survival function of the
proportional odds family (also known as Marshall-Olkin family) of
discrete distribution is given by
\begin{equation}\label{eqmos}\bar{G}(k;\alpha)=\frac{\alpha \bar{F}(k)}{1-\bar{\alpha}\bar{F}(k)},~k=1,2,...,~0<\alpha <\infty,~\bar{\alpha}=1-\alpha,\end{equation}
with $\bar{G}(0;\alpha)=1$. Let the corresponding random variable be
denoted by $Y$. Now the distribution function of $Y$ is given by
\begin{equation}\label{eqmod}G(k;\alpha)=1-\bar{G}(k;\alpha)=\frac{F(k)}{1-\bar{\alpha}\bar{F}(k)},\end{equation}
whereas the pmf is given by
\begin{equation}\label{eqmom}g(k;\alpha)=\bar{G}(k-1;\alpha)-\bar{G}(k;\alpha)=\frac{\alpha f(k)}{[1-\bar{\alpha}\bar{F}(k-1)][1-\bar{\alpha}\bar{F}(k)]}.\end{equation}
The corresponding hazard rate and the reversed hazard rate functions
are given by
\begin{equation}\label{eqmoh}r_{Y}(k;\alpha)=\frac{r_X(k)}{1-\bar{\alpha}\bar{F}(k)},\end{equation}
\begin{equation}\label{eqmorh}\tilde{r}_{Y}(k;\alpha)=\frac{\alpha \tilde{r}_X(k)}{1-\bar{\alpha}\bar{F}(k-1)}.\end{equation}
It is to be mentioned here that different properties of
(\ref{eqmos}) have been studied by D\'{e}niz and Sarabia
\cite{deniz} by taking $F$ as the cdf of Poisson random variable.
\section{Stochastic Ageing properties}
In this section we study how different ageing properties of $X$ are
transmitted to the random variable $Y$.\par With the following two
counterexamples, one with $\alpha>1$ and the other with $\alpha<1$,
we show that if $X$ is ILR, then $Y$ is neither ILR nor DLR.
\begin{counterexample}
Consider the random variable $X$ with the mass function given by
$$f(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            0.1, & \hbox{if $k=2$;} \\
                                            0.25, & \hbox{if $k=3$;} \\
                                            0.35, & \hbox{if $k=4$;} \\
                                            0.3, & \hbox{if $k=5$.}
                                          \end{array}\right.$$
Clearly $X$ is ILR. For $\alpha=5$, we have the mass function of $Y$
as
$$g(k;5)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            \frac{1}{46}, & \hbox{if $k=2$;} \\
                                            \frac{125}{1656}, & \hbox{if $k=3$;} \\
                                            \frac{175}{792}, & \hbox{if $k=4$;} \\
                                            \frac{15}{22}, & \hbox{if $k=5$.}
                                          \end{array}
                                        \right.$$ It is observed
that $Y$ is neither ILR nor DLR.
\end{counterexample}
\begin{counterexample}
Consider the random variable $X$ with mass function given by
$$f(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            0.3, & \hbox{if $k=2$;} \\
                                            0.34, & \hbox{if $k=3$;} \\
                                            0.26, & \hbox{if $k=4$;} \\
                                            0.1, & \hbox{if $k=5$.}
                                          \end{array}\right.$$ Here
$X$ is ILR. For $\alpha=0.2$, we have the mass function of $Y$ as
$$g(k;0.2)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            \frac{15}{22}, & \hbox{if $k=2$;} \\
                                            \frac{425}{1958}, & \hbox{if $k=3$;} \\
                                            \frac{325}{4094}, & \hbox{if $k=4$;} \\
                                            \frac{1}{46}, & \hbox{if $k=5$.}
                                          \end{array}
                                        \right.$$ It is observed
that $Y$ is neither ILR nor DLR.$\hfill\Box$
\end{counterexample}
\par With the following two counterexamples, one for $\alpha>1$ and
the other for $\alpha<1$, we show that if $X$ is DLR, then $Y$ is
neither DLR nor ILR.
\begin{counterexample}
Consider the random variable $X$ with mass function given by
$$f(k)=\left\{
                                          \begin{array}{ll}
                                            0.36, & \hbox{if $k=1$;} \\
                                            0.26, & \hbox{if $k=2$;} \\
                                            0.21, & \hbox{if $k=3$;} \\
                                            0.17, & \hbox{if $k=4$.}
                                          \end{array}\right.$$ Here
$X$ is DLR. For $\alpha=2$, we have the mass function of $Y$ as
$$g(k;2)=\left\{
                                          \begin{array}{ll}
                                            \frac{9}{41}, & \hbox{if $k=1$;} \\
                                            \frac{650}{2829}, & \hbox{if $k=2$;} \\
                                            \frac{700}{2691}, & \hbox{if $k=3$;} \\
                                            \frac{34}{117}, & \hbox{if $k=4$.}
                                          \end{array}
                                        \right.$$ It is observed
that $Y$ is neither DLR nor ILR.
\end{counterexample}
\begin{counterexample}
Consider the random variable $X$ with mass function given by
$$f(k)=\left\{
                                          \begin{array}{ll}
                                            0.26, & \hbox{if $k=1$;} \\
                                            0.18, & \hbox{if $k=2$;} \\
                                            0.24, & \hbox{if $k=3$;} \\
                                            0.32, & \hbox{if $k=4$.}
                                          \end{array}\right.$$ Here
$X$ is DLR. For $\alpha=0.4$, we have the mass function of $Y$ as
$$g(k;0.4)=\left\{
                                          \begin{array}{ll}
                                            \frac{65}{139}, & \hbox{if $k=1$;} \\
                                            \frac{2250}{11537}, & \hbox{if $k=2$;} \\
                                            \frac{1500}{8383}, & \hbox{if $k=3$;} \\
                                            \frac{16}{101}, & \hbox{if $k=4$.}
                                          \end{array}
                                        \right.$$ It is observed
that $Y$ is neither DLR nor ILR.$\hfill\Box$
\end{counterexample}
\par The following theorem gives the condition under which IFR/DFR
property of $X$ is transmitted to the random variable $Y$. The proof
follows from the fact that $1/\left(1-\bar{\alpha}\bar{F}(k)\right)$
is increasing (resp. decreasing) in $k$ for $\alpha\geq (resp. \leq)
1$.
\begin{thm}If $X$ is IFR (resp. DFR) and $\alpha\geq (resp. \leq)~1$, then $Y$ is IFR (resp. DFR).$\hfill\Box$\end{thm}
\par The following counterexample shows that if $\alpha<1$, then the
IFR property of $X$ may not be transmitted to the random variable
$Y$.
\begin{counterexample}\label{exdifr}
Consider the random variable $X$ following discrete IFR distribution
(cf. Salvia and Bollinger \cite{salvia}) with
$$f(k)=(k-c)c^{k-1}/k!,~k\in \mathbb{N},~0<c\leq 1.$$ Here
$\bar{F}(k)=c^{k}/k!$ and $r_X(k)=1-c/k$ so that $X$ is IFR. Now
$$r_{Y}(k;\alpha)=\frac{1-c/k}{1-\bar{\alpha}c^{k}/k!}.$$
It is observed that, for $c=0.8$ and $\alpha=0.2$, we have
$r_{Y}(2;0.2)=0.8064516$, $r_{Y}(3;0.2)=0.7870635$, and
$r_{Y}(4;0.2)=0.8110739$. This shows that $Y$ is neither IFR nor
DFR. $\hfill\Box$\end{counterexample} \par The following
counterexample shows that the DFR property of $X$ may not be
transmitted to the random variable $Y$ when $\alpha>1$.
\begin{counterexample}\label{ext1wd}
Let $X$ follow the Type I discrete Weibull distribution (cf.
Nakagawa and Osaki \cite{nakag}) with pmf given by
$$f(k)=q^{(k-1)^\beta}-q^{k^\beta},~k\in \mathbb{N},~q\in(0,1),~\beta>0.$$ Then the corresponding survival function is given by
$\bar{F}(k)=q^{k^\beta}$, and the hazard rate function is given by
$r_X(k)=1-q^{k^\beta -(k-1)^\beta}$. Here $X$ is DFR for
$0<\beta<1$. Note that
$$r_{Y}(k;\alpha)=\frac{1-q^{k^\beta -(k-1)^\beta}}{1-\bar{\alpha}q^{k^\beta}}.$$
It is observed that, for $\beta=0.8$, $\alpha=5$ and $q=0.5$, we
have $r_{Y}(7;5)=0.2759209$, $r_{Y}(10;5)=0.2834942$, and
$r_{Y}(13;5)=0.2793229$. This shows that $Y$ is neither DFR nor
IFR.$\hfill\Box$
\end{counterexample}
\par Below we see that, for $\alpha\geq (resp. \leq)~1$, the NBU
(resp. NWU) property of $X$ is transmitted to the random variable
$Y$.
\begin{thm}
If $X$ is NBU (resp. NWU) and $\alpha\geq (resp. \leq)~1$, then $Y$
is NBU (resp. NWU).\end{thm} \textbf{Proof:} Let $X$ be NBU (resp.
NWU). Then $Y$ will be  NBU (resp. NWU) if and only if
$$\frac{1-\bar{\alpha} \bar{F}(j+k)}{\bar{F}(j+k)}\geq (resp. \leq)
\frac{(1-\bar{\alpha}\bar{F}(j))(1-\bar{\alpha}\bar{F}(k))}{\alpha
\bar{F}(j)\bar{F}(k)}.$$ This is equivalent to the fact that
$$\frac{1}{\bar{F}(j+k)} \geq (resp. \leq)
\frac{1-\bar{\alpha}\left(\bar{F}(j)+\bar{F}(k)\right)+\bar{\alpha}\bar{F}(k)\bar{F}(j)}{\alpha
\bar{F}(j)\bar{F}(k)},$$ which holds if $$\frac{1}{\bar{F}(j+k)}
\geq (resp. \leq) \frac{1}{\bar{F}(j)\bar{F}(k)}.$$ The last
inequality follows from the fact that, for $\alpha\geq (resp.
\leq)~1$,
$$1-\bar{\alpha}\left(\bar{F}(j)+\bar{F}(k)\right)+\bar{\alpha}\bar{F}(k)\bar{F}(j)\leq (resp. \geq)~
\alpha.$$ Hence the theorem follows.$\hfill\Box$
\par The following counterexamples show that, for $\alpha< (resp.
>)~1$, the NBU (resp. NWU) property of $X$ may not be transmitted to
the random variable $Y$.
\begin{counterexample}\label{notnbu}
Consider $X$ following the discrete S-distribution (cf. Bracuemond
and Gaudoin \cite{brac}) with pmf given by
$$f(k)=p(1-a^k)\prod_{i=1}^{k-1}(1-p+p a^i),~k\in \mathbb{N},~
0<p\leq 1,~0<a<1.$$ This gives the survival function as
$\bar{F}(k)=\prod_{i=1}^{k}(1-p+p a^i)$, and hazard rate function as
$r_X(k)=p(1-a^k)$. Here $X$ is NBU. Now
$$\bar{G}(k;\alpha)=\frac{\alpha \prod_{i=1}^{k}(1-p+p a^i)}{1-\bar{\alpha}\prod_{i=1}^{k}(1-p+p a^i)}.$$
For $j=2$, $k=3$, $p=0.3$, $a=0.6$, $\alpha=0.2$, we have
$\bar{G}(j+k;\alpha)=0.075737$ and
$\bar{G}(j;\alpha)\bar{G}(k;\alpha)=0.063494$. This shows that $Y$
is not NBU.
\end{counterexample}
\begin{counterexample}
Let $X$ follow the distribution as given in Counterexample
\ref{ext1wd}. Then clearly $X$ is NWU for $\beta\in(0,1)$. Now, for
$j=2$, $k=3$, $\alpha=5$, $q=0.5$, we have
$\bar{G}(j+k;\alpha)=0.3062174$ and
$\bar{G}(j;\alpha)\bar{G}(k;\alpha)=0.3657684$. This shows that $Y$
is not NWU.$\hfill\Box$
\end{counterexample}
\par Kirmani and Gupta \cite{kirmani} have observed that if $X$ is
IFRA (DFRA), then $Y$ is IFRA (DFRA) for $\alpha>(<)~1$. Below we
show that if $X$ is IFRA, then $Y$ may not be IFRA or DFRA when
$\alpha<1$.
\begin{counterexample}
Let $X$ follow the distribution as given in Counterexample
\ref{notnbu}. Here $X$ is IFRA. Now, for $\alpha=0.2$, $p=0.5$, and
$a=0.6$, we have
$$[\bar{G}(k;0.2)]^{1/k}=\left\{
  \begin{array}{ll}
    0.44444, & \hbox{for $k=1$;} \\
    0.438901, & \hbox{for $k=2$;} \\
    0.457806, & \hbox{for $k=4$.}
  \end{array}
\right.$$ This shows that $Y$ is neither IFRA nor DFRA.$\hfill\Box$
\end{counterexample}
\par Below we show that if $X$ is DFRA, then, for $\alpha>1$, $Y$ may
not be DFRA or IFRA.
\begin{counterexample}
Let $X$ follow the discrete Pareto distribution with survival
function $$\bar{F}(k)=\left(\frac{d}{k+d}\right)^c,~k\in
\mathbb{N},~c,d>0,$$ which is DFR and hence DFRA. Now
$$\bar{G}(k;\alpha)=\frac{\alpha \left(\frac{d}{k+d}\right)^c}{1-\bar{\alpha}\left(\frac{d}{k+d}\right)^c}.$$
For $\alpha=6$, $d=2$, $c=3$, we have
$$[\bar{G}(k;6)]^{1/k}=\left\{
    \begin{array}{ll}
      0.7164179, & \hbox{for $k=1$;} \\
      0.658037, & \hbox{for $k=4$;} \\
      0.68081, & \hbox{for $k=8$,}
    \end{array}
  \right.$$
which is neither increasing nor decreasing in $k$, i.e. $Y$ is
neither DFRA nor IFRA.$\hfill\Box$
\end{counterexample}
\par Following theorem shows that, for $\alpha\leq 1$, the DRHR
property of $X$ is transmitted to the random variable $Y$. The proof
follows from the fact that
$1/\left(1-\bar{\alpha}\bar{F}(k-1)\right)$ is decreasing in $k$,
for $\alpha\leq 1$.
\begin{thm}If $X$ is DRHR, then $Y$ is DRHR for $\alpha\leq 1$.$\hfill\Box$\end{thm}
\par The following counterexample shows that, for $\alpha>1$, DRHR
property of $X$ may not be transmitted to the random variable $Y$.
\begin{counterexample}
Consider the random variable $X$ having distribution function given
by
$$F(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $1\leq k<2$;} \\
                                            \frac{4}{25}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{2}{5}, & \hbox{if $3\leq k<4$;} \\
                                            \frac{2}{3}, & \hbox{if $4\leq k<5$;} \\
                                            1, & \hbox{if $k\geq 5$.}
                                          \end{array}
                                        \right.$$ Clearly $X$ is DRHR. For $\alpha=4$, the distribution function of $Y$ is given by
$$G(k;4)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $1\leq k<2$;} \\
                                            \frac{1}{22}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{1}{7}, & \hbox{if $3\leq k<4$;} \\
                                            \frac{1}{3}, & \hbox{if $4\leq k<5$;} \\
                                            1, & \hbox{if $k\geq 5$,}
                                          \end{array}
                                        \right.$$ which is not DRHR.$\hfill\Box$
\end{counterexample}
\par The following counterexample shows that, for $\alpha<1$, NBAFR
property of $X$ may not be transmitted to the random variable $Y$.
\begin{counterexample}
Consider the random variable $X$ with reliability function given by
$$\bar{F}(k)=\left\{
                                          \begin{array}{ll}
                                             \frac{4}{5}, & \hbox{if $1\leq k<2$;} \\
                                            \frac{8}{13}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{1}{2}, & \hbox{if $3\leq k<4$;} \\
                                            0, & \hbox{if $k\geq 4$.}
                                          \end{array}
                                        \right.$$
Here $X$ is NBAFR. For $\alpha=0.4$, we have the reliability
function of $Y$ as
$$\bar{G}(k;0.4)=\left\{
                                          \begin{array}{ll}
                                            \frac{8}{13}, & \hbox{if $1\leq k<2$;} \\
                                            \frac{16}{41}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{2}{7}, & \hbox{if $3\leq k<4$;} \\
                                            0, & \hbox{if $k\geq 4$.}
                                          \end{array}
                                        \right.$$
It is clear that $Y$ is not NBAFR.$\hfill\Box$\end{counterexample}
We summarize the above findings in Table 1.
\begin{table}[h]
\begin{center} Table 1: Preservation of ageing classes\\
\begin{tabular}{|l|l|l|}
\hline
Ageing properties  & \multicolumn{1}{c|}{$\alpha<1$} &
\multicolumn{1}{c|}{$\alpha>1$}
\\\hline ILR & Not Preserved & Not Preserved
 \\
DLR & Not Preserved & Not Preserved
 \\
IFR & Not Preserved & Preserved
 \\
DFR & Preserved & Not Preserved
 \\
NBU & Not Preserved & Preserved
 \\
NWU & Preserved & Not Preserved
 \\
IFRA & Not Preserved & Preserved
 \\
DFRA & Preserved & Not Preserved
 \\
DRHR & Preserved & Not Preserved
 \\
NBAFR & Not Preserved &
 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Stochastic Orderings}
Let $X_{1}$ and $X_{2}$ be two discrete random variables with
support $\mathbb{N}=\{1,2,...\}$ having respective pmf
$f_{1}(\cdot)$, $f_{2}(\cdot)$, distribution function
$F_{1}(\cdot)$, $F_{2}(\cdot)$, and survival function
$\bar{F}_{1}(\cdot)$, $\bar{F}_{2}(\cdot)$. Let the survival
function of the Marshall-Olkin family of discrete distributions be
given by $$\bar{G}_{i}(k;\alpha)=\frac{\alpha
\bar{F}_{i}(k)}{1-\bar{\alpha} \bar{F}_{i}(k)},~0<\alpha<\infty,~
\bar{\alpha}=1-\alpha,$$ and let the corresponding random variable
be $Y_{i}$, $i=1,2$. The following theorem shows that the usual
stochastic order between $X_{1}$ and $X_{2}$ and that of $Y_{1}$ and
$Y_{2}$ are equivalent.
\begin{thm}\label{thst}
$Y_1\leq_{st} Y_2$ if and only if $X_1\leq_{st} X_2$.
\end{thm}
\textbf{Proof:} Note that $Y_1\leq_{st} Y_2$ if, and only if
\begin{eqnarray*}
\frac{\alpha\bar{F}_{1}(k)}{1-\bar{\alpha}\bar{F}_{1}(k)}&\leq&
\frac{\alpha\bar{F}_{2}(k)}{1-\bar{\alpha}\bar{F}_{2}(k)},
\end{eqnarray*} which is equivalent to the fact that $\bar{F}_{1}(k)\leq
\bar{F}_{2}(k)$. Hence the theorem follows.$\hfill\Box$ \par The
following theorem gives condition on $\alpha$, under which hazard
rate order between $X_1$ and $X_2$ is transmitted to that between
$Y_1$ and $Y_1$.
\begin{thm}\label{thhr}
If $X_1\leq_{hr} X_2$, then $Y_1\leq_{hr} Y_2$, provided $\alpha\geq
1$.
\end{thm}
\textbf{Proof:} Since hazard rate order is stronger than usual
stochastic order, we have, for $\alpha\geq 1$,
$$\frac{1}{1-\bar{\alpha}\bar{F}_{1}(k)}\geq\frac{1}{1-\bar{\alpha}\bar{F}_{2}(k)}.$$
Now, using the hypothesis we have, from (\ref{eqmoh}),
$$r_{Y_{1}}(k;\alpha)=\frac{r_{X_{1}}(k)}{1-\bar{\alpha}
\bar{F}_{1}(k)}\geq \frac{r_{X_{2}}(k)}{1-\bar{\alpha}
\bar{F}_{2}(k)}=r_{Y_{2}}(k;\alpha).$$ Hence the theorem follows.$\hfill\Box$\\
\par The following counterexample shows that the above theorem does
not hold if $\alpha<1$.
\begin{counterexample}
Consider the random variables $X_1$ and $X_2$ with respective
reliability function $$\bar{F}_{1}(k)=\left\{
                                          \begin{array}{ll}
                                            1, & \hbox{if $1\leq k<2$;} \\
                                            \frac{1}{2}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{2}{5}, & \hbox{if $3\leq k<4$;} \\
                                            0, & \hbox{if $k\geq 4$,}
                                          \end{array}
                                        \right.$$
and
$$\bar{F}_{2}(k)=\left\{
                                          \begin{array}{ll}
                                            1, & \hbox{if $1\leq k<2$;} \\
                                            \frac{5}{8}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{11}{20}, & \hbox{if $3\leq k<4$;} \\
                                            0, & \hbox{if $k\geq 4$.}
                                          \end{array}
                                        \right.$$
This shows that $X_1\leq_{hr} X_2$. For $\alpha=0.2$, we have the
respective reliability function of $Y_1$ and $Y_2$ as
$$\bar{G}_{1}(k;\alpha)=\left\{
                                          \begin{array}{ll}
                                            1, & \hbox{if $1\leq k<2$;} \\
                                            \frac{1}{6}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{2}{17}, & \hbox{if $3\leq k<4$;} \\
                                            0, & \hbox{if $k\geq 4$,}
                                          \end{array}
                                        \right.$$
and
$$\bar{G}_{2}(k;\alpha)=\left\{
                                          \begin{array}{ll}
                                            1, & \hbox{if $1\leq k<2$;} \\
                                            \frac{1}{4}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{11}{56}, & \hbox{if $3\leq k<4$;} \\
                                            0, & \hbox{if $k\geq 4$.}
                                          \end{array}
                                        \right.$$ This shows that $Y_1\nleq_{hr} Y_2$.$\hfill\Box$
\end{counterexample}
\par The following theorem gives the condition on $\alpha$ such that
reversed hazard rate order between  $X_1$ and $X_2$ is transmitted
to that between $Y_1$ and $Y_2$.
\begin{thm}\label{thrhr}
If $X_1\leq_{rhr} X_2$, then $Y_1\leq_{rhr} Y_2$, for $0<\alpha\leq
1$.
\end{thm}
\textbf{Proof:} Since reversed hazard rate order is stronger than
usual stochastic order, we have, for $\alpha\leq 1$,
$$\frac{1}{1-\bar{\alpha}\bar{F}_{1}(k-1)}\leq\frac{1}{1-\bar{\alpha}\bar{F}_{2}(k-1)}.$$
Now, using the hypothesis we have, from (\ref{eqmorh}),
$$\tilde{r}_{Y_1}(k;\alpha)=\frac{\alpha
\tilde{r}_{X_{1}}(k)}{1-\bar{\alpha}\bar{F_1}(k-1)}\leq \frac{\alpha
\tilde{r}_{X_{2}}(k)}{1-\bar{\alpha}\bar{F_2}(k-1)}=\tilde{r}_{Y_2}(k;\alpha).$$
Hence the theorem follows.$\hfill\Box$\par That the above theorem
does not hold in case of $\alpha>1$ is shown in the following
counterexample.
\begin{counterexample}
Consider the random variables $X_1$ and $X_2$ with respective
distribution function $$F_{1}(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $1\leq k<2$;} \\
                                            \frac{5}{24}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{1}{2}, & \hbox{if $3\leq k<4$;} \\
                                            \frac{3}{4}, & \hbox{if $4\leq k<5$;} \\
                                            1, & \hbox{if $k\geq 5$,}
                                          \end{array}
                                        \right.$$ and $$F_{2}(k)=\left\{
                                          \begin{array}{ll}
                                           0, & \hbox{if $1\leq k<2$;} \\
                                            \frac{1}{6}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{5}{12}, & \hbox{if $3\leq k<4$;} \\
                                            \frac{2}{3}, & \hbox{if $4\leq k<5$;} \\
                                            1, & \hbox{if $k\geq 5$.}
                                          \end{array}
                                        \right.$$
Clearly $X_1\leq_{rhr} X_2$. For $\alpha=4$, we have the
distribution functions of $Y_1$ and $Y_2$ respectively as
$$G_{1}(k;4)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $1\leq k<2$;} \\
                                            \frac{5}{81}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{1}{5}, & \hbox{if $3\leq k<4$;} \\
                                            \frac{3}{7}, & \hbox{if $4\leq k<5$;} \\
                                            1, & \hbox{if $k\geq 5$,}
                                          \end{array}
                                        \right.$$
and
$$G_{2}(k;4)=\left\{
                                          \begin{array}{ll}
                                           0, & \hbox{if $1\leq k<2$;} \\
                                            \frac{1}{21}, & \hbox{if $2\leq k<3$;} \\
                                            \frac{5}{33}, & \hbox{if $3\leq k<4$;} \\
                                            \frac{1}{3}, & \hbox{if $4\leq k<5$;} \\
                                            1, & \hbox{if $k\geq 5$.}
                                          \end{array}
                                        \right.$$ This shows
that $Y_1\nleq_{rhr} Y_2$.$\hfill\Box$
\end{counterexample}
\par Following two counterexamples show that the likelihood ratio
order between $X_1$ and $X_2$ is not necessarily transmitted to that
between $Y_1$ and $Y_2$.
\begin{counterexample}
Let $X_1$ and $X_2$ have the respective probability mass function
$$f_{1}(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            0.3, & \hbox{if $k=2$;} \\
                                            0.4, & \hbox{if $k=3$;} \\
                                            0.2, & \hbox{if $k=4$;} \\
                                            0.1, & \hbox{if $k=5$,}
                                          \end{array}
                                        \right.$$
and
$$f_{2}(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            0.2, & \hbox{if $k=2$;} \\
                                            0.3, & \hbox{if $k=3$;} \\
                                            0.2, & \hbox{if $k=4$;} \\
                                            0.3, & \hbox{if $k=5$.}
                                          \end{array}
                                        \right.$$
Clearly $X_1\leq_{lr} X_2$. For $\alpha=5$, we have the mass
functions of $Y_1$ and $Y_2$ respectively as
$$g_{1}(k;5)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            \frac{3}{38}, & \hbox{if $k=2$;} \\
                                            \frac{50}{209}, & \hbox{if $k=3$;} \\
                                            \frac{25}{77}, & \hbox{if $k=4$;} \\
                                            \frac{5}{14}, & \hbox{if $k=5$,}
                                          \end{array}
                                        \right.$$
and
$$g_{2}(k;5)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            \frac{1}{21}, & \hbox{if $k=2$;} \\
                                            \frac{5}{42}, & \hbox{if $k=3$;} \\
                                            \frac{5}{33}, & \hbox{if $k=4$;} \\
                                            \frac{15}{22}, & \hbox{if $k=5$.}
                                          \end{array}
                                        \right.$$ This shows
that $Y_1\nleq_{lr} Y_2$.$\hfill\Box$
\end{counterexample}
\begin{counterexample}
Take the random variables $X_1$ and $X_2$ having respective mass
functions
$$f_{1}(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            0.3, & \hbox{if $k=2$;} \\
                                            0.3, & \hbox{if $k=3$;} \\
                                            0.2, & \hbox{if $k=4$;} \\
                                            0.2, & \hbox{if $k=5$,}
                                          \end{array}
                                        \right.$$
and
$$f_{2}(k)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            0.2, & \hbox{if $k=2$;} \\
                                            0.3, & \hbox{if $k=3$;} \\
                                            0.24, & \hbox{if $k=4$;} \\
                                            0.26, & \hbox{if $k=5$.}
                                          \end{array}
                                        \right.$$
Clearly $X_1\leq_{lr} X_2$. For $\alpha=0.2$, we have the mass
functions of $Y_1$ and $Y_2$ as
$$g_{1}(k;0.2)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            \frac{15}{22}, & \hbox{if $k=2$;} \\
                                            \frac{75}{374}, & \hbox{if $k=3$;} \\
                                            \frac{25}{357}, & \hbox{if $k=4$;} \\
                                            \frac{1}{21}, & \hbox{if $k=5$,}
                                          \end{array}
                                        \right.$$
and
$$g_{2}(k;0.2)=\left\{
                                          \begin{array}{ll}
                                            0, & \hbox{if $k=1$;} \\
                                            \frac{5}{9}, & \hbox{if $k=2$;} \\
                                            \frac{5}{18}, & \hbox{if $k=3$;} \\
                                            \frac{10}{99}, & \hbox{if $k=4$;} \\
                                            \frac{13}{198}, & \hbox{if $k=5$.}
                                          \end{array}
                                        \right.$$ This shows
that $Y_1\nleq_{lr} Y_2$.$\hfill\Box$
\end{counterexample}
We summarize the above findings in Table 2.
\begin{table}[h]
\begin{center} Table 2: Preservation of stochastic orderings\\
\begin{tabular}{|l|l|l|}
\hline
Stochastic orders between & \multicolumn{1}{c|}{$\alpha<1$} &
\multicolumn{1}{c|}{$\alpha>1$}\\baseline distributions   &
 &
\\\hline Usual stochastic order & Preserved & Preserved
 \\
Hazard rate order & Not Preserved & Preserved
 \\
Reversed hazard rate order & Preserved & Not Preserved
 \\
Likelihood ratio order & Not Preserved & Not Preserved
 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}Marshall and Olkin \cite{marsh1} introduced a method of
adding a new parameter, called tilt parameter, to a family of
distributions for obtaining more flexible new families of
distributions. In the literature, some reliability properties of
this family of distributions are studied with continuous baseline
distributions. However, not much study is done in the literature, to
the best of our knowledge, for discrete baseline distributions. This
paper discusses various stochastic ageing properties, as well as
different stochastic orderings of this family with discrete baseline
distributions.
\subsection*{Acknowledgements:} The support received from IISER
Kolkata to carry out this research work is gratefully acknowledged
by Pradip Kundu. The financial support from NBHM, Govt. of India
(vide Ref. No. 2/48(25)/2014/NBHM(R.P.)/R\&D II/1393 dt. Feb. 3,
2015) is duly acknowledged by Asok K. Nanda.

redpajama_set_name: RedPajamaArXiv


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