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Q: Convergence of a series in a Hilbert space Let $\{y_k:k=1,2,\dots\}$ be an orthonormal set in a Hilbert space $\mathcal{H}$. Give and prove the necessary and sufficient conditions for the scalar sequence $\{\lambda_k\}$ such that  the series $$\sum_{k=1}^\infty\lambda_k y_k$$
converges. What is the norm of the sum of the series then?
This is my preparatory task before an exam, but I'm struggling with the first step, could you explain in detail what I'm supposed to do and what theorems to make use of?

A: Hint: $\sum \lambda_k y_k$ converges $\iff$ $\sum |\lambda_k|^2$ converges.
Sketch of proof: since a Hilbert space is complete, the two series converge iff their partial sums are Cauchy. Since $(y_n)_{n\in \mathbb N}$ is orthonormal, by Pythagoras' Theorem the Cauchy conditions for the two partial sums are equivalent.
Edit: let
$$S_n:=\sum_{k=1}^n \lambda_k y_k \qquad \sigma_n:=\sum_{k=1}^n|\lambda_k|^2.$$
Then, as a conseguence of Pythagoras' theorem,
$$\begin{align}
 ||S_{n+p}-S_n||^2=&
||\sum_{k=n}^{n+p}\lambda_k y_k||^2 \\=
& \sum_{k=n}^{n+p}|\lambda_k|^2||y_k||^2 \\=
& \sum_{k=n}^{n+p}|\lambda_k|^2 \\=
&\sigma_{n+p}-\sigma_n
\end{align}
$$
From here it is clear that $(S_n)_{n\in \mathbb N}$ is Cauchy iff $(\sigma_n)_{n\in \mathbb N}$ is Cauchy, which concludes the proof.

A: Hint: for $n \in \mathbb N$ we have
$||\sum_{k=1}^n\lambda_k y_k||^2=\sum_{k=1}^n |\lambda_k|^2 $.
Its now your turn to show:
$\sum_{k=1}^\infty\lambda_k y_k$ converges $ \iff (\lambda_k) \in l^2$.


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What Happens to AIDS Funding in Russia After Policy Changes?
September 25, 2013 / Sue-Lynn Moses
The Clinton Foundation and the Bill & Melinda Gates Foundation are only two of the many well-known U.S. non-profit organizations entrenched in a long battle against HIV and AIDS. For both organizations, the fight rages on in all parts of the world, including the Russian Federation. That fight may all but end in Russia — at least for the Gates and Clinton foundations. Earlier this year, as Vladimir Putin was beginning his latest six-year term as president, he and his house announced the requirement that all nonprofits receiving funding from organizations outside of the Russian Federation must identify themselves as "foreign agents."
This means that Russian nonprofits receiving money from any organization outside of Russia will be considered to be engaging in political activities and carrying out the interests of a foreign country. Say what now?
To say that politics doesn't have a place in non-profit work is a bit misleading. To say that politics shouldn't have a place in non-profit work is more accurate. Politics has its place, whether we like it or not, but that's an issue to be discussed at a different time, on a different platform.
Fact is, counts of the number of people living with AIDS in the Russian Federation vary wildly, from 300,000 officially registered cases to more than 1 million when considering unregistered cases. The U.N. has reported that annual AIDS cases have grown by 251% in Eastern Europe — with 90% of those new cases in Russia.
The Russian government has already attacked USAID and its estimated $50 million worth of programs, including health initiatives focused on HIV and AIDS. USAID had no choice but to comply and shuttered its Russian doors in October 2012. Both the Clinton and Gates foundations head up global health initiatives toward the education, prevention, and treatment of HIV and AIDS. The Clinton and Gates foundations are powerhouses, to say the least, with combined net assets of nearly $37.8 billion. For Russia's government to put the kibosh on access to that money borders on unconscionable.
Related: Gates Foundation: Grants for Global Health
This isn't about politics. This is about people. Sure, anyone could say that the Russian nonprofits could simply accept funds from non-Russian foundations and register themselves as foreign agents. Aside from the mind-boggling amount of bureaucratic red tape it takes to register, once an agency chooses to do so, it opens itself up to governmental control when it comes to the issues for which it advocates.
If the agency chooses to accept foreign money, it risks criminal penalties and financial penalties up to three million Rubles ($100,000). If an agency chooses to accept foreign money and not register as a foreign agent, it is choosing to break the law.
Russian nonprofits have found themselves steeped in a difficult if not impossible situation. It's definitely going to take some time to see how this new legislation plays out. Unfortunately, time is not something people suffering from diseases such as AIDS tend to have a lot of.
September 25, 2013 / Sue-Lynn Moses/
Gates Foundation: Grants for HIV/AIDS
Sue-Lynn Moses
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redpajama_set_name: RedPajamaCommonCrawl


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Joey Hulbert
Denny like this.
Discovering Plant Destroyers in South Africa with Citizen Science
Forestry and Agriculture Biotechnology Institute (FABI), University of Pretoria
Phytophthora, the Greek term for "plant destroyer", is a group of plant killing microbes that threaten forests and agriculture around the world. More than 100 species have been discovered and between 100-600 species are estimated to remain unknown. Discovering new species improves research and regulations to protect our forest and agriculture systems. The native forests of South Africa are remarkably diverse and have not been surveyed for Phytophthora species. We want to survey the native forests for Phytophthora species with the help of the public!
Help us engage South African communities in science to discover new species! The purpose of this campaign is to raise support for a citizen science program that provides opportunities for South Africans to learn and participate in the scientific process! By backing this research you are supporting awesome research and helping hundreds of individuals participate!
The Smiles We've Inspired—Thanks to you!
Crowdfunding part 2: a small ask for a HUGE achievement
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Exploring the effects of captive environmental stimuli on Ceratophrys ornata (Pacman Frog)
The goal is to examine the effects of various environmental stimuli on Ceratophrys ornata, commonly known...
Gregory Ryder
Baxter Academy for Science and Technology
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A biology project funded by 71 people


As they get older, many people discover they have grown a second chin. A double chin can make a person look much older, and can also make a person look much heavier, neither of which is particularly desirable. A surgical liposuction and chin lift procedure can restore the chin to its normal state, but surgery is expensive and painful. Kybella injections offer a minimally invasive, affordable alternative.
Kybella is a minimally invasive FDA-approved procedure to treat double chins. It consists of a solution of deoxycholic acid that is injected into the double chin. It is a natural component of the human body that acts to break down fatty tissue. Once injected into the double chin, it destroys the fatty tissue, allowing the sagging skin to retract into a more normal position.
What About Weight Loss Instead?
The old theory was that a double chin was simply due to excessive body fat, and therefore diet and exercise could effectively get rid of one. However, double chins are not primarily caused by being overweight. Many slim, athletic persons develop double chins and many overweight people do not develop them. They are highly resistant to diet and exercise programs. The appearance of a double chin is caused by abnormal accumulation of fat in the chin area accompanied by an excessively lax under-chin skin, leading to sagging and bulging. The development of this condition appears to be primarily genetic. If your relatives have double chins, it's likely you will develop one too. In some families, double chins occur during middle age, but in others they can occur as early as the mid-20s. In a minority of cases, a double chin may be due to hypothyroidism, so if you have other typical symptoms (e.g, fatigue, feeling cold) you may wish to consult your doctor.
Each treatment session takes less than 30 minutes. After numbing the area with a topical agent, the doctor will inject Kybella into multiple sites under the chin. Most patients need two to four, sometimes up to six, treatment sessions to completely eliminate their double chin. Treatments are administered once every 4 to 8 weeks.
After each treatment session, there will be some swelling, redness, bruising, and pain in the chin area. These side effects can subside after 2 days or, rarely, last for up to 3 weeks. Older individuals and those with very large double chins tend to experience a longer period of swelling. Ice pack application can help reduce the swelling. Patients can, if they desire, return to work immediately after treatment. A scarf or turtleneck can be used to hide the swelling. A more practical approach is to be treated on Friday and allow the swelling to subside over the weekend before returning to work. Due to the swelling and the fact that Kybella takes time to work, a reduction in the size of the double chin will not be immediately apparent.
How Effective is Kybella? Does Kybella Work?
In clinical trials, eight percent of patients experienced an improvement after only one treatment. After four treatments, 55 percent of patients experienced an improvement, and after six sessions, 77 percent had improved. Around 16 percent of patients experienced a dramatic improvement (grade 2) after six treatments. The vast majority (79 percent) of patients reported being very satisfied with their appearance after treatment.
Kybella permanently destroys the fatty tissue in the skin. Retreatments are not necessary, and it is very unlikely the double chin will re-occur. In patients with large double chins, some type of non-invasive skin-tightening procedure after treatment for the double chin can help improve the patient's profile even more.
The safety of Kybella was established in clinical trials that treated 1,600 patients for double chins. It is quite safe. The most common side effects of Kybella include temporary redness, bruising, swelling, and pain at the sites of injection. Very rarely, damage to a facial nerve may occur, which can lead to a lopsided smile, facial muscle weakness, or difficulty swallowing.
How to reduce double chin without Kybella? As mentioned above, diet and exercise will not eliminate or reduce a double chin. Plastic surgery, such as liposuction with or without a chin lift, is the only effective alternative to Kybella. However, these procedures are invasive, painful, and have long recovery times. They can also be much more expensive than Kybella. After a chin liposuction procedure, patients usually need to rest at home for 3 to 4 days and cannot return to work for a week. If a chin lift is also performed, patients usually cannot return to work for at least 2 weeks. Even after returning to work, patients need to avoid strenuous activity for several weeks, and may need to wear a special chin strap during this time period. Risks of liposuction and neck lift procedures include reactions to anesthesia, infections, and scarring.
Find out if you're a candidate for Kybella. Make your appointment today!


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Solving India's mental health issues during COVID-19: these 5 startups show the way
By Shreya Pothula
Innerhour, Trijog, ePsyclinic, YourDOST, and Wysa, have launched various initiatives and are tackling deep-rooted stigma associated with mental health.
When it comes to the prevalence of mental health issues in Indian society, there are many things linked to it that make it slightly more difficult for it to be addressed. One is the intangibility of such issues, and another is the stigma associated with it.
For long, identifying a mental illness in one's self was popularly synonymous to being weak or "taking things too seriously." In fact, even today, the idea of seeing a psychiatrist or therapist is far-fetched and only serves as a last resort in many Indian households.
According to the Indian Psychiatry Society, the number of mental health cases – including anxiety and depression – has risen 20 percent since the lockdown was first announced on March 24, mostly attributed to the decline in social activity and restricted movement.
Get connected to Wysa
Working hard towards the welfare of mental health of people in the country in these trying times are these five startups:
InnerHour
This startup was founded by Dr Shefali Batra and Dr Amit Mailk in 2016. The app launched by them is a wholesome guide that has been curated by mental health professionals with the right attitude, values, and skill set. The app, which can be used by anyone over the age of 18, is a tool for customised self-help, and provides digital therapy - a bonus, especially in the times of a pandemic.
Mind control: Here's how to take care of your mental health during the COVID-19 lockdown
Based in Mumbai, Inner Hour empowers people to overcome mental and emotional hardships. Their goal is to ensure that all users find themselves happier and healthier. The team comprises psychologists, product designers, and developers.
Trijog
Founded by mother-daughter duo Anureet and Arushi Seth in 2014, Trijog is a Mumbai-based startup. Anureet has been a practicing psychologist for over three decades. It was Arushi's final-year assignment that led to the creation of Trijog. It offers various mental health wellness solutions for individual clients, corporates, and even organisations. It is also affiliated with the Rehabilitation Council of India.
Teenagers and mental health during COVID 19
Specifically designing a way to keep Indians mentally healthy in times of the COVID-19 crisis, Trijog is currently providing discounted solutions for people facing increased mental difficulties. It has the provision of webinars and online counselling too.
Get news and updates from the world of social entrepreneurship
ePsyclinic
This startup offers 24x7 free online chat support sessions and call sessions to aid the mental wellbeing of everyone whose lives have come to a standstill in these unprecedented times.
Based in Gurgaon and founded by Shipra Dawar in 2015, this mental health startup has now curated specialised digital counselling sessions for frontline health workers, senior citizens, women, and working professionals. Their blog also features a wide range of resources which shed light on the current scenario of interpersonal and intrapersonal relationships.
This non-profit is breaking stereotypes surrounding mental health by providing free counselling
2017 was when the team at ePsyclinic launched an app called IWill, which has more than 45 in-house therapists and counsellors.
YourDOST
This platform is a way to connect those who are suffering from mental illnesses with mental health experts. It has been offering counselling in relationships, self-improvement, well-being, education, and career, ever since its inception in 2014. With the option to stay anonymous during chat, phone, or video call sessions, there is an escalated level of comfort that individuals have when seeking help.
YourDOST reaches SAIF-er shores, raises funding
Founded by Richa Chadha, YourDOST startup recently took up the responsibility of helping those who have been hit by the COVID-19 crisis. In addition, they have a list of resources to read through which provide relatable struggles subject to these times and tools to deal with those struggles.
Wysa
Wysa is essentially an AI (artificial intelligence) chatbot created to address mental health concerns. This innovative idea to enable mental health welfare in the country came from Jo Aggarwal in 2015. The chatbots are designed by coaches, users, AI experts and therapists to ensure an effective and credible experience.
10 ways pregnant women can manage mental health during COVID-19 pandemic
In the current times, Wysa is providing support to those who have been subject to anxiety and vulnerability. So far, Wysa has been anonymously used by residents of over 30 countries and benefitted more than than 1.2 million people. In fact, Wysa has been chosen as the best app for COVID-19 stress and anxiety by ORCHA, the world's leading health app evaluation and adviser organisation.
Edited by Javed Gaihlot
Inner Hour
InnerHour is a Mumbai-based startup that has a team of trained professionals and psychologists to provide discreet counselling.
1. Shefali Batra
2. Amit Malik
3. Neha Kirpal
HealthTech / MedTech / Healthcare
Trijog is India's trusted organization for mental health care and holistic wellness. They offer a range of counseling solutions for Individuals, Corporates and Institutions globally. They have catered to 33,500+ individuals & 23+ corporates.
1. Anureet Sethi
2. Arushi Sethi
3. Sahej Sethi
4. Sameer Arora
YourDOST is an online-based counselling platform which helps to connect customers with psychologists & counsellors for guidance in regards to personal, professional and academic matters.
1. Satyajeet Nandekar
2. Richa Singh
3. Puneet Manuja
4. Prakhar Verma
Wysa is a Bengaluru based artificial intelligence based chatbot for managing mental health.
1. Ramakant Vempati
2. Jo Aggarwal
mental health in India
artificialintelligence
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Introduction: Shimano will be unveiling a number of new spinning reels this ICAST, mostly a revamp of the middle of their mainstream range, which includes the Symetre and Sahara. Positioned right below the iconic Stradic is the Saros Series and the new FG model is probably the most intriguing spinning reel update for the company this season.
The newly updated Saros FA will debut at ICAST and is designed to offer a lot of what makes Shimano spinning reels so popular at an aggressive $139.99 price point. While the current Saros features an aluminum construction this new reel follows the Stradic CI4 and makes use of a new graphite composite material. This new graphite is called �XGT7� and is designed to be lightweight but also extremely durable. How durable? Shimano says the XGT7 material is 200% stronger than the material used in the CI4.
This new graphite material is utilized in the frame, sideplate and rotor and the reel is finished with an aluminum spool. The Saros FA also makes use of a reinforced stamped bail for reliable line pickup. The Saros features the S and SR Concept designs and the Propulsion Line Management System, and internally the reel also benefits from the company�s Paladin gear durability enhancements which improve gear strength and endurance.
The aluminum and brass guts combined with the corrosion resistant exterior of the new Saros make this a good reel for not just freshwater applications but inshore as well, and Shimano will be introducing sizes ranging from 1000 to 4000 to tackle everything from Panfish to small Tuna. To stand up to these fish Shimano has now incorporated the X-Ship designs which was previously only found on higher end reels, and should help deliver much improved cranking power and feel over the previous Saros.
When there is a fish on the line the Saros will also benefit from Shimano�s Rapid Fire Drag design which was first introduced on the Sustain FG. This design delivers a much more aggressive drag curve so that rotating the drag knob just half way delivers the same pressure change as up to two full rotations on traditional reels. Anglers fishing light tackle will find this feature delivers much quicker adjustment when needing to make changes to keep pressure on fish and away from snag ridden structure.
Overall the Saros FA features compact styling and a much darker finish which helps makes the reel look even smaller and also ties it in with the Stradic CI4 and Sustain FG. Weight-wise the Saros FG is the same or slightly lighter than the previous aluminum based version depending on model. The Saros FA will come with a 6.0:1 retrieve ratio except the 4000 size which will feature a slower more powerful 5.8:1 gear ratio. Each reel comes with 5A-RB + 1 A-RB roller bearing configuration.
These reels will be available for order from retailers and e-tailers like Tackle Warehouse on or around July 11th, and will retail for $139.99 each, except the larger 4000 size which will be just 10 dollars more. In typical Shimano style some of the features that were once reserved for the company�s higher end offerings has now trickled down the spectrum and made their way into the Saros FA, but the story is not just how many features Shimano has managed to cram in here for $139.99, but also the new extra durable XGT7 graphite material which makes this reel possible.


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Which DISC Personality Style Does Each Avenger Have?
By Shanna Bennell on December 2, 2016 3:25:55 PM EST
With the next iteration of "The Avengers" series beginning production in January of 2017, the popularity of these characters doesn't seem to be declining anytime soon.
These characters are iconic- classic superhero archet ypes with admirable and imitable traits. The Avengers are strong, tenacious, talented, and intelligent. And they all have distinct personalities.
So what DISC personality is best attributed to each Avenger?
C- Black Widow and Hawkeye:
Hawkeye's name is even about accuracy. He excels at precision and is the definition of detail oriented. He knows precisely where and when to strike to achieve his objectives- and he never misses. He is there to get the job done, and is very task oriented. He has always done his research before he goes on a mission, he isn't winging it, he knows what needs to happen for the mission to succeed. He has also trained rigorously to achieve this level of technical prowess.
Black Widow, or Agent Natasha Romanov, is also a C style personality. She persuades using facts and information; her attacks are precise. Everything she does is in service of getting the job done correctly. She only expresses emotions when they are used to manipulate information out of people, for example, when she feigns heartbreak to draw Loki's intentions out of him.
I- Ironman (Tony Stark):
Ironman is the epitome of an I style personality. Social, fast paced, and prone to showboating, Tony Stark is not one to hide from the limelight. Ironman is always interrupting everyone and talking over them. He always has to be the center of attention. He overrides Black Widows PA system to play his own intro music and asks Agent Romanov, "You miss me?" He is inspiring, persuasive, impulsive, charismatic, quick thinking, and optimistic. His "Look at me look at me!" style even permeates his superhero identity. While most superheroes keep their real identity secret to protect themselves and their loved ones, Stark does the opposite.There is no separation between Tony Stark and Ironman. He didn't keep it a secret because he wants the attention. He made a suit that is practically impermeable and indestructible, yet it has a faceplate that comes up so you can see Tony inside. Like any I under stress, Tony always makes a joke to lighten the mood. Aspirational to some, he is the pinnacle of many people's beliefs about what a rich man's life would be like, with all the toys, girls and attention. Stark prefers to inspire others by leading, not following. At one point he even describes himself as "billionaire playboy philanthropist.". He's even got a giant building with his name on it. Sure, Ironman is really a combination of an I and a D (which spells Id- Freud joke!), but it is his persuasive and inspiring traits that make him such an I style.
S- Captain America (Steve Rogers)
Captain America literally hides behind a shield that looks like the American flag. He literally shields himself from harm by hiding behind the symbol for his country. He believes that systems and loyalty will protect us, and that these concepts are worth fighting for. Symbolically, he not only subscribes to these systems and seeks stability, he fights for them. Steve Rogers is an S style personality- steady, predictable, loyal, team and people oriented. Captain America is staunchly loyal to his country. His only form of rebellion is against people who are not in his herd (people who believe in Democracy.) In the 2012 film, Rogers tells Loki about Hitler and America's part in the fight against fascism. He uses this as an analogy for the battle he will rage against Loki's dictatorial tendencies. Sure, S Styles are notoriously peaceful. But this peacekeeping tendency only goes so far. S's are peaceful because they want to maintain stability within a group. When something external threatens the group, they will fight for it to return to homeostasis. Captain America strives to create solidarity between all Americans against what he perceives as an evil outsider threatening the world order.
Everything that he says and does supports the idea that he is an S style personality. When Captain America is talking to Ironman about Thor's attempt to purloin Loki, he says "We can't just attack, we need a plan!" He is both saying that they are a team, not just a bunch of individual heroes; and that you have to plan as a team before acting. You don't act impulsively or quickly.
Tony Stark's response to C.A.'s prompting him to plan is "I have a plan- attack." This also serves to exemplify his I style personality: impulsive, fast acting, and change oriented.
Additionally, Romanov exhibits her C style personality in this exchange. She says "you cannot just attack them, they are basically gods." This shows she believes that the facts need to be taken into consideration, and procedures must be followed. She also indicates that she is pessimistic about the probable outcome. And then Captain America says "there is only one god, and I'm pretty sure he doesn't dress like that." This shows that he exhibits a belief in organized monotheistic religion, he believes in appealing to the group, loyalty, belief in hierarchy, and a group mentality. This drive for security and stability and loyalty puts him at ends with Ironman.
Stark's style conflicts directly with Captain America's. Rogers tells Ironman he should follow orders; Ironman says following isn't his style. C.A. asks questions, Ironman thinks he knows all the answers. Captain America says "threatening the safety of everyone on this ship isn't funny" when Ironman pokes at the Hulk. He chastises Ironman for not being willing to take one for the team, like he is.
D- Hulk (Bruce Banner)
Dominant, brute force, his secret is that he is always angry. This is the D style personality's superpower, too. D's don't wait for permission, they act. They aren't concerned with consensus or teamwork; they are concerned with achieving their objectives. Banner once warns Romanov to not try to forcibly convince him, or he will get angry.This is also true of a D style personality. D styles hate to feel like they are being taken advantage of. Loki calls Hulk " a mindless beast that makes play he's a man," but Banner describes it as always having a raw nerve. His anger and his power to act on it keeps him from paying attention to details and plans when he is "the Other Guy." Ironman indicates at one point that the amount of gamma radiation that Banner has been exposed to should have killed him, so the other guy saved him. In many ways, this is true for all D styles; they are the ones you turn to in an emergency. Hulk (not Banner) is good in a crisis; he is not afraid of confrontation or danger. Loki says "I have an army" Ironman says "We have a Hulk, which is pretty much the same thing. " This encourages the idea that one D style can accomplish more than a number of other people combined. Even his catchphrase indicates that he is fast paced, active, and task oriented. "Hulk smash!"
Many other superheroes fit these DISC archetypes, and we will continue to show you how DISC theory can be applied to any personality, even fictitious ones.
About the Author: Shanna Bennell


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Pan-Thessalonikian Athletic Club of Constantinopolitans F.C.
Pan-Thessalonikian Athletic Club of Constantinopolitans F.C. (Greece)
all competitionsChampions League Women
Europe: Champions League Women - Qualification
Semi-finals Valerenga W PAOK W 2:0 (0:0, 2:0) details 21.08.2021
Quarter-finals PAOK W Anenii Noi W 6:0 (3:0, 3:0) details 18.08.2021
Semi-finals PAOK W SL Benfica W 1:3 (0:2, 1:1) details 04.11.2020
Group 8 - 3. Round PAOK W LSK Kvinner W 0:1 (0:0, 0:1) details 13.08.2019
Group 8 - 2. Round PAOK W Linfield W 2:3 (2:2, 0:1) details 10.08.2019
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