text:
229 South Green Street Henderson, KY 42420
Business Equipment is more than office equipment. It's more than office furniture and supplies. It's consulting, training, education, and service. It's commitment to service beyond the sale!
Business Equipment is with you at every phase of your business operations -- and that extends beyond serving your daily operational needs.
Our trained professionals work with you and your personnel to match your specific needs with the highest quality products and service -- from general office supplies and furniture to state of the art digital copying systems.
We take pride in the fact that we're an independent, locally owned and operated office equipment dealer. We've been in the business of serving businesses for over 40 years -- and we're proud of the reputation of quality that we've earned over the years.
Let Business Equipment show you that we can serve your total business needs!
Having difficulties finding an item?
Call our knowledgeable staff and we can help!
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The Hon Company

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Text trimming with column definition with star in wpf In my second grid where I have a column definition at Width="800" I would like to have something responsive like 1* and my text would trim accordingly to the size of actual width of the grid column.
For example:
I have an image 40x40 pixels in the first column with my text. My text would be long, let's say 1000 pixels
if my overall grid width is 1000 pixels.
I would have 200 pixels in the second column and 800 pixels in the first, that would make my text trim at 800 - 40 pixel = 760 pixel
if my overall grid width is 500 pixels.
I would have 100 pixels in the second column and 400 pixels in the first, that would make my text trim at 400 - 40 pixel = 360 pixel,
etc.
     <Grid>
            <Grid.ColumnDefinitions>
                <ColumnDefinition Width="4*" />
                <ColumnDefinition Width="1*" />
            </Grid.ColumnDefinitions>
            <Grid>
                <Grid.ColumnDefinitions>
                    <ColumnDefinition Width="Auto" />
                    <ColumnDefinition Width="800" /> <!--Instead of 800 I want to put 1*-->
                </Grid.ColumnDefinitions>
                <Image
                    Height="40"
                    Margin="7"
                    Source="{Binding FileIcon}"/>
                <TextBlock
                    Grid.Column="1"
                    HorizontalAlignment="Left"
                    VerticalAlignment="Center"
                    Text="{Binding SomeLongText}"
                    TextTrimming="CharacterEllipsis" />
            </Grid>
            <!--<Grid Grid.Column="1"> Other stuff here responsive </Grid>-->
    </Grid>

Can you help?
Thanks,
Olivier

redpajama_set_name: RedPajamaStackExchange


text:
Home › Writing › Putting real people in stories
Putting real people in stories
In writing workshops, the question often comes up: "What about putting people I know in stories? Is that ok?" It's an interesting question, touching on issues of decency, law, human psychology, and the nature of narratives.
I don't consciously import people from my life into stories. A character may be inspired by some quality of someone I know, but the push and pull of narrative structure, plot, and theme will start shaping the character far away from its model. I think that's good. It's how narratives work. Most successful fictive characters are amalgamations of characteristics you've seen and invented. If you want to be faithful to how Uncle Max "really was," memoir is probably a better vehicle than fiction.
Which is not to say that people you know won't see themselves in a character, especially a reasonably sympathetic one. I had an ex-pat female protagonist in a story set in Italy and lost count of the number of female ex-pat friends who assured me that I'd actually modeled my "Helen" character after them. Hum, news to me, but it made them happy. I'm sure every writer has had this experience.
But I do think it's a problem if a writer starts playing the "I'm a writer" card to even a score with family members, ex-partners, or friends. As in "Robert did me wrong. I'll write a story that shows everyone what a disgusting scumbag he is with a character called . . . hum . . . Rupert. After all, I'm a writer and I'm sharing my experience. So there."
Without getting too far into the legal issue of libel, I think a fair moral test would be whether you'd give a free pass on public ridicule to someone on the grounds that s/he was a plumber or physician. Probably not. Being a writer doesn't give you the right to be a creep.
Naturally, we are shaped by our experience, and the characters we create come from the world we've seen and imagined, but I do think that story telling is its own thing. It's not about using your talent and craft to even a score. That's what diaries and journals are for. You can always dis your ex over drinks with a friend—in private. Anyway, a story that exists merely to show that a scumbag gets what's coming to him/her is usually pretty lame anyway.
‹ Novel writing & bridge building
Dangers of over-describing ›
Tagged with: Narratives, Real people in fiction, Real people in stories


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Swiss condemn massacre of Charlie Hebdo journalists
Emergency workers at the scene of the shooting attack at the Charlie Hebdo magazine headquarters in Paris AFP
This content was published on January 7, 2015 - 19:00 January 7, 2015 - 19:00
swissinfo.ch and agencies
Swiss President Simonetta Sommaruga has condemned the terrorist attack on French satirical magazine Charlie Hebdo. On Wednesday, gunmen stormed the publication's Paris office – killing at least 12 people.
"The Swiss government is deeply shocked by this terrible attack. We offer our condolences to all the relatives and friends of the victims. This attack is an assault on democracy, human rights, the freedom of the press and the freedom of expression," Sommaruga told Swiss public television, SRF.
She says there are no indications of a related terrorist attack planned in Switzerland at the moment, but that the Swiss authorities are in close contact with their French peers and that the government is monitoring the security situation attentively.
"It is a difficult situation and probably a difficult phase, once again, in the fight against terrorism which continues or has only just begun," Swiss Foreign Minister Didier Burkhalter told Swiss public radio, SRF.
Charlie Hebdo is a weekly satirical magazine that is known for mocking radical Islam. One of the hooded gunman was filmed shouting "Allahu Akbar! (God is greatest!)".
Will the United Nations soon be obsolete?
Is the international order capable of meeting the enormous challenges facing states and societies around the world?
Of the 12 people killed, two were police officers. Another 11 people were injured, four of them critically. Cartoonist Corinne Rey, alias Coco, managed to hide under her desk. She also works for Vigousse, a Swiss satirical magazine.
Wednesday's edition of Charlie Hebdo AFP
A black day
Médias Suisses, a publishers' association from the French-speaking part of Switzerland, has called it a "black day" for freedom of the press. Journalists' association Impressum said it was deeply shaken by the incident.
"We condemn this attack on freedom of the press and freedom of expression, and call on the authorities to hold those responsible for the massacre, and their supporters, accountable," said Impressum in a statement.
Hanspeter Lebrument, head of the Swiss Publishers' Association, asked whether Swiss editorial departments were safe enough. He also noted that it was getting difficult for journalists to describe the relationship between locals and Muslim immigrants.
Zeid Ra'ad Al Hussein, the UN High Commissioner for Human Rights, said he was concerned that the attack could worsen relations between different groups.
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"Those who try to divide communities based on religion, ethnicity or other reasons must not be allowed to succeed," he said, calling the attack "appalling" and "barbaric".
"With xenophobia and anti-immigrant sentiments already on the rise in Europe, I am very concerned that this terrible act could be exploited by extremists on all sides," he concluded.
Eye for an eye
Jacques Baud, a Swiss anti-terrorism expert, says the attack is a typical example of radical Islam.
"It's an eye for an eye attack. Even if it's been ten years since Charlie Hebdo published its caricatures of the prophet Mohamed, the terrorists have delivered their bloody blow. They considered the caricature to be an attack on the prophet," Baud told swissinfo.ch.
Baud, who has published several books on Islamic terrorism, says he hopes France does not go for the bait laid by the terrorists.
"The famous Mohamed caricature must not be published again, not in the name of the freedom of press. This would only be a new victory for the jihadists and those who oppose Islam in Europe, such as the movement in Germany. The latest assault is the consequence of a binary logic which came after the 9/11 attacks: those who are not with us are against us."
Georgia elections in US showcase the nuts and bolts of democracy building
This content was published on Jan 11, 2021 Jan 11, 2021 Zurich-based American law lecturer Alexandra Dufresne sees both positive signs and cautionary tales in recent US political developments.
But Baud has little hope of a change in attitude. He says the governments still have no strategy other than military or police intervention against radical Islam.
Some 500 people gathered in Geneva on Wednesday evening to pay tribute to the victims. Many held candles, pens or signs reading, "I'm Charlie".
In Bern, about 200 media professionals participated in a rally organized by a media trade union.
This article was automatically imported from our old content management system. If you see any display errors, please let us know: community-feedback@swissinfo.ch
"No topic is unworthy of caricature"
This content was published on Jan 16, 2013 Jan 16, 2013 For caricature there are no taboos and even sensitive topics are admissible, according to Anette Gehrig, director of the Basel Cartoon Museum. On...
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We extend a warm invitation to all members and guests to come out and join us to celebrate the past year's activities, future plans and for the election of officers. This is your club and we appreciate your support and involvement!
This year is the year to elect our new president for the next 2 year term and lots of movement will happen with current board members.
We have an opening in the board this year for a new treasurer and we would love to discuss with anyone who is interested in getting involved how they could play a role in this amazing organization. If you are looking for board experience this is a great opportunity!
As per Article IX of the club's bylaws, nominations will also be accepted from the floor at the AGM.
BPW Greater Moncton is required to provide the venue with the number of attendees in advance. Your cooperation is greatly appreciated. Please register early to ensure your spot.
A reminder that we need quorum for this AGM to take place, so please do your best to attend.
We are women in business and represent the professions from diverse industries. In addition to developing the professional and leadership potential of women through education, awareness, advocacy and mentoring within a supportive network, we are known by our motto as "women working for working women" and we seek to improve the economic, employment, political and social conditions of all working women.


text:
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redpajama_set_name: RedPajamaGithub


text:
New Mexico Wild Wilderness Ranger Program
Category: Wilderness Ranger Program
Published: Friday, 16 June 2017 13:56
2019 Wilderness Rangers
RANGER BIOS
Zack Bumgarner
Santa Fe - West
Zack's love for nature and the outdoors began on family camping trips as a young child and grew through monthly outings with the Boy Scouts. This culminated with attending college in the mountains of western North Carolina, graduating with a degree in Geography. After years in the GIS department of local government, Zack began volunteering with wilderness stewardship groups in North Carolina, building upon his knowledge and skills; something he has continued to do since moving to New Mexico. As a Wilderness Ranger for New Mexico Wild, Zack is now using these skills to help protect Wilderness and public lands here in the Land of Enchantment.
Doug Campbell
Santa Fe - Pecos
Doug, a native Santa Fean, went on his first New Mexico backcountry adventure when he was 10 years old—an overnight backpacking trip with his father into the Pecos Wilderness. Ever since, wilderness exploration and preservation have been his life's passion, so much so that after a career in social work, he made his passion his profession. Since 2008 he has been a professional backpacking guide (for The Wildland Trekking Company) and a wilderness skills instructor (NOLS, United World College-USA, Albuquerque Academy). He has guided over 180 multi-day treks into the Grand Canyon and coordinates Wildland Trekking's conservation-focused philanthropy program. He is a life member of NM Wild, attended the Wilderness Act's 50th Anniversary Conference, and in 2014-15 conducted roadless-area inventories for NM Wild. He is a graduate of NOLS, Outward Bound Canada, and Penn State, where he earned a B.A. in history. The biggest wilderness adventure of his life was in 2018—a 24-day, 350-mile canoeing expedition down Alaska's Noatak River. Last but not least, he knows lots of funny a-guy-walked-into-a-bar jokes.
Brennan Davis
Cibola
Brennan is a proud New Mexico native who grew up exploring the state with his father. From the Gila Wilderness to the vast expanses of the Chihuahuan Desert around Las Cruces, the natural beauty of New Mexico inspired a strong conservation ethic that guides Brennan to this day. He obtained a B.S. degree in Biology from UNM and began pursuing a career in natural resource conservation. Working with the BLM in Carlsbad, Brennan began his professional journey contributing to the management of sensitive wildlife species and cave environments. Since then, he has worked towards conservation projects across the state, from native seed collection to forest health monitoring in the Rio Grande watershed. Brennan is excited to bring his passion for the natural world to NM Wild, to contribute to the management of New Mexico's incredible Wilderness areas.
Becky Fitzpatrick
Becky fell in love with the landscapes and people of the southwestern United States when she studied Natural History and Ecology at Prescott College in Arizona. After attaining her bachelors degree, Becky worked as a wildlife technician on a bear nutritional study in Yellowstone National Park, a tropic cascade study in Waterton Lakes Park, and a bison management study in Montana. She then began teaching wildlife science programs to high school students in the Yellowstone area, which grew into a love of teaching and place-based outdoor education. Becky returned to the southwest and worked for a conservation corps building trail and working on invasive species projects. She currently spends her time working seasonally as an outdoor educator with Outward Bound and Prescott College. Her love for wild places began young, and has continued to grow the more time she spends mentoring and teaching students within it. She has found no better teacher.
Jesse Furr
Jesse is a lifelong lover of wilderness and the outdoors. As a community-minded Northern New Mexican, he has an interest in protecting and stewarding wildlands because of the unrivaled value they offer for wildlife, vegetation, soils, watershed health, and human enjoyment. Jesse is an artist and avid wilderness enthusiast who is excited to use his skills as a New Mexico Wild ranger to maintain and advocate for wilderness character through education and improved access in the Lincoln National Forest.
James Lautzenheiser
Hi, I'm James! After earning a Recreation Management degree from Appalachian State University and spending six seasons filling different Wilderness stewardship roles with Southern Appalachian Wilderness Stewards, I took a side year to explore other parts of the country and fields of work. I am excited to come back to the Wilderness fold in New Mexico!
Tobias Nickel
Hailing from Germany, Tobias Nickel first discovered his love for backpacking and the American West while attending college in California. He graduated from the University of San Diego with a degree in environmental science, philosophy, and political science. Working as a guide for the university's outdoor adventure center, Tobias led trips to wilderness destinations across the Southwest. Since graduation, Tobias has worked for Grand Canyon National Park, the Bureau of Land Management, California Wolf Center, and the Catalina Island Conservancy. Tobias' experiences on public lands in the American West have profoundly impacted his life in positive and transformative ways. Having had the privilege of experiencing many wilderness areas, Tobias strives to share his love for the outdoors with others and help them discover their own land ethic. As a Wilderness Ranger on the Lincoln National Forest, Tobias is excited to support stewardship of New Mexico's wild treasures and to inspire others in the process.
Irene Owsley
Irene Owsley, who has worked as a professional photographer for more than 20 years (following a career in museums), comes to her position as a Wilderness Ranger, primarily through her specialty in outdoors and travel imagery and the stories she has covered in public lands, including several designated wildernesses and national parks. Her work has appeared in such magazines as Canoe & Kayak, Sierra, National Parks, Earthwatch, and Natural History and in the publications of several conservation organizations. Since 2011, she has been selected for five artist-in-residence programs: Bandelier National Monument, New Mexico (where she also volunteered for trail work), and four through Voices of the Wilderness in Alaska (Nellie Juan-College Fjord Wilderness Study Area, Alaska Maritime National Wildlife Refuge and two in Tracy Arm-Ford's Terror Wilderness). A panorama from the latter residency was included in an exhibition at the Smithsonian's Museum of Natural History celebrating the 50th anniversary of The Wilderness Act. She has exhibited her work in venues including the Anchorage Museum, Smithsonian's Museum of Natural History, and the National Institutes of Health. A long-time resident of Washington DC until 2014, Owsley found the "wild" in this metropolitan area by creating a body of work called "Wild Washington," capturing images of many untamed areas within a twenty mile radius of the White House. She now makes her home in Santa Fe.
Andrew Reville
Andrew grew up in Cleveland, Ohio, far from any semblance of Wilderness as a concept or reality. Still, he has valued natural places and the solitude found within them since childhood. A backpacking trip in the Dolly Sods Wilderness of West Virginia as a teen gave him his first taste of these values on a larger scale. He is grateful to be able to protect these values as they exist in the wild lands of New Mexico for a living. He is returning for his second season as a Cibola National Forest Wilderness Ranger. Previously, he has worked for the Student Conservation Association in wilderness areas of Idaho and has a Bachelor's of Science in Biology.
This summer, nine Wilderness Rangers will be afoot across wilderness areas on the Cibola, Lincoln, and Santa Fe National Forests through a public-private partnership between the Forest Service and New Mexico Wild. Ranger pairs will be working in the Sandia Mountain, Manzano Mountains, Apache Kid and Withington wilderness areas on the Cibola National Forest; the White Mountain and Capitan wilderness areas on the Lincoln National Forest; and the Chama River Canyon, San Pedro Parks, and Pecos wilderness areas on the Santa Fe National Forest.
This partnership is designed to increase stewardship in wilderness areas across northern New Mexico. Rangers will be providing a range of important services including wilderness character monitoring, trails assessment, trail clearing, campsite rehabilitation, public outreach and wilderness education. Their work provides valuable information on current wilderness conditions while enhancing public safety and contributing to positive wilderness experiences for wilderness visitors.
Volunteer engagement is an integral aspect of this partnership. Providing the public opportunities to be involved helps leverage scarce public dollars, helps address the backlog of maintenance and improvement needs and instills a sense of pride and stewardship for New Mexico's wilderness areas. Most volunteer projects will focus on monitoring visitor use (see "solitude monitoring" section below), but could also include campsite and trail rehabilitation, signage, trash clean-up and other types of monitoring efforts.
If you would like to volunteer in one of the wilderness areas listed above, please use the information below to contact the rangers working in that wilderness. And, if you see these rangers on the trail, please take a moment to say HELLO!
SOLITUDE MONITORING
Do you like to hike in wilderness and want to give something back? Sign up to help the Cibola, Santa Fe and Carson National Forests conduct wilderness solitude monitoring! We are looking for avid hikers and wilderness lovers to count wilderness encounters across several wilderness areas in northern New Mexico. This fun and easy volunteer project can be done without supervision (after a ½ day of training) in pairs or as a group. This summer, monitoring will take place in the Sandia Mountain, Manzano Mountain, Apache Kid, Withington, Chama River Canyon, Columbine-Hondo, Latir Peak and Wheeler Peak Wilderness areas. Contact the individual rangers assigned to each wilderness area for more information or to sign up!
HIKE WITH A RANGER
Ever wonder what a day in the life of a wilderness ranger is like? Find out by joining our rangers for a day in the field. You'll learn backcountry travel skills, wilderness data collection basics, how to make effective and positive public contacts and more all while spending a fun day with great people in a beautiful place. Contact a ranger in your area of interest to learn more and schedule a day on the trail.
CONTACT THE RANGERS
SANTA FE NATIONAL FOREST
San Pedro Parks Wilderness, Chama River Canyon Wilderness, Pecos Wilderness
Zack Bumgarner, This email address is being protected from spambots. You need JavaScript enabled to view it.
James Lautzenheiser, This email address is being protected from spambots. You need JavaScript enabled to view it.
Doug Campbell, This email address is being protected from spambots. You need JavaScript enabled to view it.
Becky Fitzpatrick, This email address is being protected from spambots. You need JavaScript enabled to view it.
Irene Owsley, This email address is being protected from spambots. You need JavaScript enabled to view it.
CIBOLA NATIONAL FOREST
Sandia Mountain Wilderness, Manzano Mountains Wilderness, Apache Kid Wilderness, Withington Wilderness
Brennan Davis, This email address is being protected from spambots. You need JavaScript enabled to view it.
Andrew Reville, This email address is being protected from spambots. You need JavaScript enabled to view it.
LINCOLN NATIONAL FOREST
White Mountain Wilderness and Capitan Wilderness
Jesse Furr, This email address is being protected from spambots. You need JavaScript enabled to view it.
Tobias Nickel, This email address is being protected from spambots. You need JavaScript enabled to view it.
Read their bios on Facebook>>
Santa Fe National Forest David Greene
Santa Fe National Forest Atieno Ouma
LIncoln National Forest Ezra Sage
Santa Fe National Forest April Fuentes
Lincoln National Forest Rhett Spencer
Carson National Forest Ben Mortensen
Carson National Forest
Will Ribbans
Cibola National Forest Andrew Reville
NM Wilderness Alliance Stewardship Project
Wilderness Education Classroom Pilot
Wilderness Outdoor Connection Leadership Program
Bosque School Partnership
Sembrando Semillas
Wilderness Stewardship Challenge
Working with New Mexico Community Land Grants


text:
Kaiser Permanente to Find Solution to Deal with Colossal Number of Patient-Centric Healthcare Apps
There is a copious amount of patient-centric healthcare mobile applications available in the market. With every new year, new mobile application launches and some of them make a permanent impact on patients that often enter in the health system with a somewhat confused, stressed mind. However, this huge volume of digital tool that patients use has dumfounded doctors as they simply don't have time to go through the App Store to use every digital tool available and find the perfect tool for their patients. According to Allied Market Research, the market of global patient-centric healthcare app market is expected to grow owing to rise in smartphone penetration and increase in collaboration among healthcare providers, phone vendors, app developers, and patients.
In order to tackle this issue, Kaiser Permanente, an Oakland-based healthcare company, launched a project that is aimed to address about 25% of patients with subclinical behavioral health symptoms. The company's Chamai team interviewed about 38 member and dozens of healthcare professionals. The team spent years to develop an ecosystem of useful mobile applications that clinicians suggest to patients to improve their normal care.
Trina Histon, a senior principal consultant at Kaiser Permanente, stated, "Many therapists prescribe their patients some app that offers a various health-related solution, however, it becomes hard to figure out how to get them into a workflow. Therefore, we went deep into our electronic health records (EHR) to make it easy for clinicians and therapists to refer to the most appropriate app that could match with their patients' requirements."
Today, the team's ecosystem support about six mental health apps and among those six, three apps emphasis on mindful meditation and the remaining are based on CBT. Additionally, some apps are video-based, which includes a coach and a navigator. The team has developed a platform that could help non-English-speaking users to use the selection of apps. However, such options are yet to come.
According to a service designer at Kaiser Permanente Design Consultancy, Aubrey Kraft, this kind range on hand has become necessary to confirm that the most suitable tool that has offered to every patient. However, he believes that the staggering digital offerings available in the market and requirement of the patient make the selection a tricky thing.
Kraft said, "If some patient with these subclinical behavioral health symptoms appears, we try to suggest the best suitable option and what kind of content they must use or which app and articles they should read. However, we are still not sure how to do that, but for now, we're trying to make the system as easy as possible for the healthcare professionals." The team knows that it is best that the clinicians themselves select the best suitable app for their patients, however, we should recognize their time constraints.
Despite many hurdles, the feedback of patients as well as clinicians is the most important tool they have to determine whether the mobile application is appropriate to refer it to the patients, even though the team has taken the opinions of a handful of "champion clinicians" while testing and reviewing the ecosystem.
The team asked few participants to use some of the apps and monitored their opinions as it was the best way to inspect these apps with different perspectives and decide whether the tool would help in future or not. Apart from offering a user experience, the team included various clinicians and other members of the healthcare professionals to review the process. The team asked for their professional advice as they could teach and offer their valuable guidance regarding which apps to use to educate patients and what kind of language is suitable while prescribing an application to a patient. They monitored the comments such as "I should prescribe this to you" or "I think you should do this". This way, they can convince other clinicians that some therapists thought that app is useful, providing them some convincing source of review while pitching for the tool. It is observed that clinicians tend to trust the opinions of physicians more than the comments of the app designers.
Apart from user feedback and experience, the team consulted literature borrowed from professional corporations to effectively integrate digital therapeutics in care. The focus was on apps that offer some proof of enhanced outcome among patients and clinicians. However, the lack of reliable data relating to explicit digital tools and products. Moreover, the team believes that the issues would be easy to deal with if the developers of digital therapeutics and digital health industry raise the bar to increase efficacy.
Alex Hillsberg
Chief editor of review team at FinancesOnline
Chief editor of review team at FinancesOnline Alex Hillsberg is an expert in the area of B2B and SaaS related products. He has worked for several B2B startups in the past and gathered a lot of first hand knowledge about the industry during that time.


Please note, we reserve the right to revise these dates in the event of excessive orders.
Europe - 8th of December.
Rest of world - 1st December.
If an item needs to go through customs please check the service in your area is operating over the Christmas period.
You are welcome to order after these dates and items will be made and dispatched as soon as possible. However, we won't be working between the 15th and 17th December (boring hospital stuff).


text:
Soloway: Challenges are opportunities
By Stan Soloway,
By Stan Soloway
GSA must seize the moment to rebuild relations with federal agency customers and Congress.
CIOs or CXOs
The resignation of an agency head always produces its share of drama. But those departures, regardless of the circumstances, are common and generally less important than the agency's overall condition and the leadership that remains.
In the case of Lurita Doan's departure from the General Services Administration, the agency remains under the microscope and faces an array of external problems. At the same time, GSA has moved forward under Doan. The agency has improved its management and financial controls, received a clean audit opinion — one of few agencies to do so — and been named by employees as one of the best places to work in government.
Moreover, GSA has some of the federal government's top senior career leaders, including the highly regarded David Bibb, deputy administrator, and Jim Williams, commissioner of GSA's Federal Acquisition Service. Say what you will about Doan's style or her choice of battles, the truth is GSA is better off today than when she arrived 22 months ago. There is reason for optimism in the near term and through the presidential transition.
That said, GSA has four immediate priorities:
1. Get back to the customer. GSA's customer relations are the core of its business, yet those relationships have become strained in too many important areas. Given the credibility of the agency's remaining leaders, GSA has an opportunity to reassert its value and critical role in federal contracting. However, the growing practice of agencies awarding their own contracts has had a palpable effect on GSA's market share, and some of that business will not be returning.
That trend presents challenges, but it also offers opportunities. GSA's first step must be to rebuild customer relations at the highest levels and, where necessary, continue to adapt its business model to meet the realities of the marketplace.
2. Get back to Congress. GSA's congressional relations are not what they need to be. Others can argue about why, but that is less important than moving quickly to restore those relations. GSA has a good story to tell and great people in positions to tell it. They need to be turned loose to do so.
3. Get back to basics. No organization can function at an optimal level unless its parts are aligned and functioning well together. There are signs that those alignments at GSA are shaky, at best. The last thing GSA needs is internal divisions that imperil its ability to move forward.
In addition, too many disparate business practices and policies remain across GSA's components. That, too, must change.
4. GSA must be unafraid to show some backbone. The agency must be willing to assert itself when circumstances dictate. It would be an enormous mistake for the agency and its leaders to take Doan's departure as a signal to keep their heads down and sit it out. GSA cannot afford to do that. Now is the time for the agency to remain focused, visible and engaged. Simply treading water won't suffice.
Soloway is president and chief executive officer of the Professional Services Council.
NEXT STORY: Fox: Doan picked a good fight


text:
LEstadi Adérito Sena és un camp de futbol situat a l'Avinguda 12 de Setembre de la ciutat de Mindelo a l'illa de São Vicente de Cap Verd. És un estadi certificat per la FIFA per jugar partits internacionals. En aquest camp es juguen els partits de futbol dels diferents campionats organitzats per l'associació regional de futbol de São Vicente

Referències

Instal·lacions esportives de Cap Verd
Futbol a Cap Verd
Estadis de futbol de l'Àfrica
Illa de São Vicente

redpajama_set_name: RedPajamaWikipedia


By now (with the aid of my dictionary to fill in some of your knowledge of vocabulary) you should be able to put sentences together with a degree of competency. This lesson aims to show you how Sindarin adjectives are constructed, and much like the Abstract Suffix in the previous lesson, show you how you might use existing rules to create new words.
This is just listed by Tolkien as 'an adjectival suffix' but it may mean 'full', e.g. Aglar = Glory, Aglareb = Glorious (full glory). It is a much less commonly used suffix than -ui, and I wouldn't advise that you commonly use this for reconstructions.
This is known as the 'augmentative suffix', and is used to make a noun into a bigger version. (Do not confuse this with the abstract suffix, this is only for making things physically larger). This suffix is quite limited in its usage.
This suffix means 'Ornamentation, highlighting, with the intimation of being fair, used especially on colours and shapes'. This isn't a suffix that exists in English and Tolkien left no attested examples; just a paragraph describing how to use it.
There are no rules of any sort attached to the information (as there are no words listed using this), but we can make some fairly easy inferences, based on how other suffixes attach to words (such as -rim and -ren).
Sigil vorn adh valenred = A black necklace (or dagger) with yellow ornamentation/yellow detail (highlights).
This is a suffix that means 'of' or 'like' and is very commonly used. You can use this for reconstructions very easily. In older Noldorin words this appears as -rin (e.g. Doriathrin, Palathrin). For normal reconstructions, always use -ren!
This suffix can be used on any noun or verb, and is the equivalent of -LY or -Y on the end of an English word.
Remove the vowel (e.g. Lost instead of Losta-) and add the suffix as above.
This suffix is attested to work with both I stem verbs and nouns. It is the equivalent of the English -able or -ible on the end of a word, or to say 'of xyz' (Breakable, Horrible). It is the most common adjectival suffix in Sindarin.


text:
October - 2007 - issue	> Feature
Telling Tales The Corporate Way
Jaya Smitha Menon
The stories need not start with 'once upon a time', but they can dramatically improve the odds and culminate into a happy ending. The age old practice of storytelling has redefined its scope and has a larger audience now, especially in the corporate world. In the corporate world ruled by analysis and figures, storytelling is increasingly getting accepted as an effective tool in achieving management goals. "When it comes to igniting the minds, analysis will not be as effective as anecdotes," say Srinivasan Kumar, General Manager, Amazon. In the times of stiff competition, the corporate survival would require leadership to inspire people to act in unfamiliar situations, often in unwelcome manners. Neither the ever growing balance sheet figures nor the powerpoint slides packed with vision statements will be able to spark action in people with energy and enthusiasm.
Kumar remembers an interesting incident on storytelling. When his team was implementing an important product, the team was worried about the risk and had a tinge of pessimism and diffidence. To help motivate the team in the right direction, he narrated a real story about his friend who escaped from Vietnam in a wooden boat. He survived in South China Sea for 14 days without any food or water or high tech navigation tools before they reached land safe. This story was a perfect example to help his team understand the meaning of risk and how you mitigate it. There's risk in everything we do but Kumar believes that the team needs to understand the importance of determination, perseverance, and optimism. "Philosophy may be boring. But when you relate it to practical life it resonates with people and motivate them", explains Kumar.
The award winning story A Peacock in the Land of Penguins is a good example to illustrate the power of storytelling. The story is today used by most corporates to effectively address the diversity issues and to encourage an inclusive work culture. In the story when Perry the Peacock and his exotic feathered friends arrive in the homogenous Land of Penguins they have trouble fitting in, despite their talent and intelligence. However, when Perry and his friends save the island from an attacking pack of wolves, the penguins realize that in their ever-changing world, all birds would have to appreciate each other's skills and contributions. The story highlights the importance of appreciating those whose backgrounds or personalities are different from our own. Learn to recognize the strengths inherent in a diverse work group and see the dangers that lies in seeking conformity.
Many a time, in a session, all the participants know what is being talked about, but from their own perspective. In order to deliver the key concept and add the punch required to drive the idea, stories are employed. "Anecdotes can be of various types depending upon the circumstances and the audience," says Anshul Sharma, Project Manager, Samsung. True incidents stemming from past experiences are used more often, he adds. He narrates his experience while working on new technology. His team was working on a new audio compression technology named spectral band replication. It was an entirely new technology and researchers were just getting familiar with the concepts. Hence they were a bit apprehensive about going forward with the work. When he researched about the history of this technology, he learned that the original inventor of the technology Lars Liljeryd was a person who used to design hearing aid equipments and over a period of time he had discovered a unique way to compress audio information in around half the bandwidth of that used in the existing compression systems. The inventor of the technology was a dedicated audiophile who lives in a houseboat in Sweden and has an astonishing range of diverse interests. "I told this story to the team so that it could inspire them to think creatively and expand their technical bandwidth without conforming to limitations," says Sharma. Upon the impact, some people in his team got really interested in this domain and even today continue with their research in the domain, claims Sharma.
Apart from all this, stories can be an effective tool in transmitting values within the organization. History of the company, past experiences, and experiments in the company and other parables can ensure that the audience understands where the company stands and also how things are done in the company. Stories can be effective in articulating and focusing a vision. It makes information more tangible and memorable. Stories are now used as a medium of communication, both internally within an organization and externally to customers, investors, and others. Increasingly, many companies are realizing the potential of stories as a market research tool, for communicating important institutional knowledge about effective business practices and adapting to innovation. Stories provide a tool for conceptualizing and identifying challenges and opportunities.
Kapil Kanbarkar, Engineering Manager at Razor Sight employs the story telling technique to build a high performance team. As his team embarks on a novel project or entering a crucial phase of delivery, he motivates them with customer stories. "Customer stories can contain customer experience , like narrate how a crucial phase was successfully completed earlier," says Kanbarkar. This helps them clearly understand the gravity of the situation and work proactively to tide over the difficult phase. "But stories should be honest and truthful to have the desired effect," says Kanbarkar. "It is also important to tell the right story at the right time," he explains. Even stories from epics and puranas are cited by leaders to connect to a broad base of people in an easy manner. Speaking in a truly Indian context, we don't have to go into Vedas to enlighten ourselves, whereas turn with ease to Puranas for inspiration.
Those 'once upon a time' stories from our grandmothers which we used to hear as children during bedtime and in leisure often capture our imagination and we are lulled to sleep with the characters looming large in our head and hearts. We never know when and how the values get ingrained in us. But the narrative delight offered by the new era of storyteller in the corporates motivates us to act and achieve results.
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text:
\section{Introduction}

We consider  the Cauchy problem for the continuity
equation in $d$ space dimensions:
\begin{equation}\label{e:continuity}
\left\{ \begin{array}{l}
\partial_t \rho + \div (u \rho) = 0 \\
\rho(0,\cdot) = \bar \rho \,,
\end{array}\right.
\end{equation}
where  the velocity field $u = u(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}^d$ and the
datum ${\bar \rho = \bar \rho(x) : \mathbb{R}^d \to \mathbb{R}}$ are given, and  $\rho =
\rho(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$. Such an equation arises in different contexts and it models conservation of the scalar quantity  $\rho$ under the dynamics generated by the velocity field $u$. Often, as in the case of incompressible fluids, the (distributional) spatial divergence $\div u$ vanishes, and it is well known that, among other factors,  the behavior
of solutions of  \eqref{e:continuity} depends on controlling the growth of the divergence of $u$. In this work we restrict to velocity fields that are divergence free,  in which case the continuity equation coincides with the transport
equation
\begin{equation}\label{e:transport}
\partial_t \rho + u \cdot \nabla \rho = 0.
\end{equation}
Further assumptions on the velocity field $u$ will be specified  later in the paper.

The focus of this work is to investigate the regularity of the solution $\rho$ at time $t$, given a certain regularity on the initial datum $\bar\rho$ and conditions on the velocity, such a uniform-in-time bound on norms of $u$ and its derivatives. In particular, we are interested in determining what type of loss of regularity, if any, occurs in this setting.

The question whether initial regularity is propagated for solutions of
equation \eqref{e:continuity} has a classical and simple answer when the
velocity field is
regular enough, namely Lipschitz continuous with respect to space uniformly with
respect to time:
\begin{equation}
\label{e:lipschitz}
| u(t,x) - u(t,y) | \leq L |x-y|,
\end{equation}
for any $t \geq 0$ and $x$, $y \in \mathbb{R}^d$, for some constant $L>0$ depending on
$u$, but not on $t,\,x,\,y$. The infimum over all $L$ satisfying the above
estimate is called the Lipschitz constant of $u$ and denoted by Lip$(u)$.

With this regularity on the velocity field,  the classical Cauchy-Lipschitz theory
applies, and  the Cauchy problem for the continuity
equation~\eqref{e:continuity} has a unique solution, which is moreover
transported by the unique flow of the velocity field $u$, that is,
\begin{equation}
\label{e:transported}
\rho (t,x) = \bar \rho \big( X(t,\cdot)^{-1} (x) \big) \,,
\end{equation}
where the flow $X = X(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}^d$ is the solution of
\begin{equation}
\label{e:ode}
\begin{cases}
\dot X(t,x) = u(t,X(t,x)) \\
X(0,x) = x \,.
\end{cases}
\end{equation}
Therefore, the solution $\rho$ at time $t$ has the same regularity as the initial datum $\bar\rho$ provided the flow $X$ is regular enough. In particular, Gr\"onwall's  estimate ensures that the flow and its inverse are Lipschitz
continuous with respect to space with a bound on their  Lipschitz constant that
is exponential in time and in~$L$, where $L$ is as the
constant appearing in~\eqref{e:lipschitz}:
\begin{equation}
\label{e:gronwall}
{\rm Lip}\, \big(X(t,\cdot)\big) \leq \exp(tL) \,, \qquad
{\rm Lip}\, \big(X(t,\cdot)^{-1}\big) \leq \exp(tL).
\end{equation}
 In particular, from~\eqref{e:transported} it follows that
\begin{equation}
\label{e:propagated}
{\rm Lip}\, \big( \rho(t,\cdot) \big) \leq {\rm Lip}\, (\bar \rho) \, \exp(tL)
\,,
\end{equation}
and an analogous result holds for the H\"older regularity of the solution.
Alternatively, the exponential bound in~\eqref{e:propagated} can be proved by
performing simple energy
estimates directly on   the continuity equation~\eqref{e:continuity}.

In many situations, for example when $u$ is a solution of the Navier-Stokes and Euler equations of fluid mechanics,  the velocity field is not Lipschitz continuous, but still retains some regularity in terms of weak or distributional derivatives.
In this setting, a theory of well posedness in Lebesgue spaces $L^p$, $1\leq p \leq \infty$, for the continuity
equation~\eqref{e:continuity} and the ordinary differential
equation~\eqref{e:ode} has been developed, starting with the seminal works by
DiPerna-Lions~\cite{DL} and Ambrosio~\cite{A}. In these two papers, the velocity
field is assumed to have, respectively, Sobolev or $BV$ regularity with respect
to the spatial variables, with an integral dependence on time.
(By $BV$ regularity we mean that the velocity is of bounded variation, and by Sobolev regularity we mean that derivatives of $u$, of possibly fractional order, have a certain integrability in space.)
Then one can prove existence, uniqueness, and stability of weak
solutions of the partial differential equation (PDE)~\eqref{e:continuity}, and of suitable flows
(the so-called {\em regular Lagrangian flows\/}), solutions of the ordinary
differential equation (ODE)~\eqref{e:ode}. (See~\cite{notes} for a
recent survey on research in this area.)

This well-posedness result makes it rather natural to ask whether any propagation
of regularity for solutions of~\eqref{e:continuity} holds when the advecting
field has only the Sobolev or $BV$ regularity assumed in~\cite{A,DL}. A positive answer to
this question would be very relevant in particular for applications to
non-linear problems, as propagation of regularity would imply strong
compactness of solutions, allowing the use of approximation schemes for
non-linear problems that include continuity equations. This is  a rather delicate question
because the flow map may be discontinuous.

To the best of our knowledge, the literature addressing propagation of regularity in the framework described above  is rather
limited. Several results of classical flavor for ``almost Lipschitz'' velocity fields (i.e., with a
modulus of continuity satisfying the Osgood condition) are collected for
instance in~\cite{chemin}. Whether or not any propagation of regularity holds for less regular fields again seems  a rather subtle question that, in particular, will depend on dimension. For instance, in one space dimension it is not difficult  to see that $BV$ regularity of the initial datum is propagated in time for solutions
of the transport equation with a Sobolev velocity field, without additional
conditions on its divergence. By contrast,  in~\cite{colrausch} the authors show that, in two space dimensions, neither
$BV$ regularity nor continuity of the data are preserved under the flow,
if the  velocity field lies in the intersection of a Sobolev and a H\"older space.
Their construction is based on the lack of pointwise uniqueness for solutions
of~\eqref{e:ode}, which can be used to generate trajectories that collapse quickly  by  time reversion.
See also~\cite{Figalli} for a related analysis concerning $H^{3/2}$ velocity
fields on the circle.

The main result of this work implies that,  in two or more space dimensions, the Sobolev regularity of initial datum is not necessarily preserved in time when the velocity is not Lipschitz continuous. The loss of regularity is particularly
striking when the velocity field has only one derivative in $L^p$, for $1\leq p<\infty$.
In this case, it is possible for smooth  initial data to loose all Sobolev
regularity, even of fractional order,  instantaneously at $t=0_+$. One should contrast this result with the fact that, within the Ambrosio-DiPerna-Lions theory, the Lebesgue $L^p$ norms of the initial datum are all exactly conserved in time, if $u$ is divergence free. The divergence-free constraint on the velocity field is a strong constraint as it implies the flow of $u$ preserves the Lebesgue measure, so blow up of the norm by concentration is more difficult or impossible to achieve. We do not expect that allowing $u$ to have a non-zero, but bounded, divergence would significantly simplify our construction. Allowing the divergence to become singular would likely lead to simpler examples of loss of regularity, but would also bring the result outside of the Ambrosio-DiPerna-Lions theory for solutions to \eqref{e:continuity}.

This negative result on propagation of regularity in this work complements a few  positive results that have become available recently in the literature. There, (low) regularity is measured in a non-standard way, while we choose to work with traditional Sobolev spaces. The  quantitative
estimates for regular Lagrangian flows in~\cite{estimates} show that any regular
Lagrangian flow can be approximated (in the Lusin sense) with Lipschitz maps,
with a quantitative control on the Lipschitz constant of the approximation that depends
exponentially on the size of the neglected set. This
approximation result implies the propagation of a suitable ``Lipexp regularity''
of the initial datum (see~\cite{estimates}).  More recently, in~\cite{jabin2}  preservation  of ``a logarithm of a derivative'' for the solution of the continuity equation is established  and used  for a new theory of
existence of solutions to the compressible Navier-Stokes equations. This result
is obtained by means of a delicate argument, which utilizes the propagation of a
weighted norm of the solution, with a suitable choice of weights, via an adjoint
problem, and does not require the divergence of the velocity to be bounded.
See also~\cite{leger} for a related approach in the context
of incompressible velocity fields, described in terms of Fourier
multipliers and proved with the use of multilinear harmonic analysis~\cite{SSS},
and the subsequent  sharp results in~\cite{hung}.

Our main result is contained in the following theorem, where $d$ again denotes the spatial dimension.

\begin{theorem}\label{t:main}
Let $d\geq 2$. There exist a bounded velocity field $v:\,[0,\infty)\times \mathbb{R}^d\to \mathbb{R}^d$ and a bounded
solution $\theta:\, \,[0,\infty)\times \mathbb{R}^d\to \mathbb{R}$
of the continuity equation \eqref{e:continuity} with velocity $v$,
such that:
\begin{enumerate}[label=(\roman*),ref=(\roman*)]
\item for any $1 \leq p<\infty$ the velocity field $v$ is bounded in
$\dot{W}^{1,p}(\mathbb{R}^d)$ uniformly in time;
\item the initial datum $\bar \theta:=\theta(0,\cdot)$ belongs to
$C^\infty_c(\mathbb{R}^d)$;
\item the solution $\theta(t,\cdot)$ does not belong to
$\dot{H}^s(\mathbb{R}^d)$ for any $s>0$ and $t>0$.
\end{enumerate}
Moreover both $v$ and $\theta$ have support in a fixed compact subset of $\mathbb{R}^d$ for all $t\geq 0$, and
can be taken to be smooth on the complement of a point in $\mathbb{R}^d$.
\end{theorem}

The main strategy in the proof of Theorem \ref{t:main} above is to obtain sustained (exponential) growth of Sobolev norms with positive exponent from decay to zero of Sobolev norms with negative exponent by means of an interpolation inequality and conservation of the $L^p$ norms for solutions of \eqref{e:continuity}. The decay of negative Sobolev norms is, in turn, a consequence of results on optimal mixing, obtained by the authors \cite{ACM2}. The other main ingredient in the proof is a speed up of the growth of the Sobolev norm that eventually leads to instantaneous blow up, achieved via a suitable rescaling argument. We discuss  these ideas in more details later in the Introduction.

Theorem \ref{t:main} is stated in term of homogeneous Sobolev
spaces $\dot{W}^{s,p}$ and $\dot{H}^s\equiv\dot{W}^{s,2}$ (defined below in Appendix~\ref{s:prelim}), given that we employ rescaling in the proof  and  homogeneous Sobolev norm have the sharpest behavior under rescaling as opposed to non-homogeneous spaces. In our results, it is equivalent to work with homogeneous or inhomogeneous norms, since in
the example we construct both the velocity field and the solution  are uniformly bounded.
Hence the inhomogeneous norm will be bounded if and only if the homogeneous
norm is bounded. Additionally, we state the loss of regularity for the solution
only in terms of $L^2$-based Sobolev spaces $H^s$, $s\in \mathbb{R}_+$. While these
norms have physical meaning, using $L^2$-based Sobolev norms is a technical
restriction, which allows us to use equivalent Gagliardo-type norms for the
fractional spaces $H^s$, $0<s<1$. These norms in turn imply an almost
orthogonality property (see Lemma \ref{l:orto}) in these spaces. We expect that
a statement analogous to that of Theorem \ref{t:main} can be made concerning
blow-up of $L^p$-Sobolev norms, $1<p<\infty$.

\begin{remark} \label{r:ODE}
We work directly with the continuity equation, rather than
with the flow induced by the advecting velocity, that is, we employ directly
the PDE and we do not explicitly refer to the underlying ODE for the flow.  We can, however, draw the following conclusions regarding the flow of the velocity $u$.
The dramatic loss of regularity in Theorem \ref{t:main} indicates a severe lack of
continuity of the
solution map in Sobolev spaces. Indeed, the inverse flow map $X^{-1}(t,\cdot)$
cannot be in any $\dot H^s (\mathbb{R}^d)$, with $s>0$, for $t>0$, otherwise the solution would retain
Sobolev regularity from formula \eqref{e:transported} as the initial data is
smooth. The representation formula \eqref{e:transported} is still valid, though
not pointwise, for the unique weak and renormalized solution to
\eqref{e:continuity}. In a recent paper \cite{jabin1}, Jabin gives in fact an
example of a
velocity field in the Sobolev space $W^{1,p}(\mathbb{R}^d)$, for $1\leq p < \infty$, with respect
to the space variables, for which the associated regular
Lagrangian flow is not in $W^{1,\tilde p}(\mathbb{R}^d)$ for any $\tilde p$. The proof exploits a randomization
argument in the choice of the rotations of certain basic blocks. In fact,
Jabin observed that his construct gives  the same lack of
regularity for the flow in case of velocity fields in $W^{r,p}(\mathbb{R}^d)$, provided $p<\frac{d}{r-1}$,
where $r$ is even allow to be larger than $1$ and $p$ to be smaller than $1$ (cf. Theorem~\ref{t:small} below).
\end{remark}

\begin{remark}\label{r:hung}
Recently, it was shown in~\cite{hung} that adapting the proof of Theorem~\ref{t:main} and exploiting  suitable
logarithmic interpolation inequalities  leads to
examples of loss of logarithmic regularity for the solution of
the continuity equation.
\end{remark}

The proof of the main result in this work is based on a previous construction~\cite{ACM2} (see also~\cite{ACM1}) of optimal mixers in  the context of two-dimensional incompressible flows under Sobolev bounds on
the velocity. Informally, mixing of $\rho$ amounts to
creation of stretching and filamentation that accounts for
a small negative Sobolev norm of the solution. Since the $L^2$ norm of the solution
is preserved under the flow, by interpolation positive Sobolev norms have to
be large. The strategy of proof has two main steps:
\begin{enumerate} [leftmargin=*,label={\bf Step \arabic*.},ref={Step
\arabic*}]
\item Construction of a basic element that saturates the Gr\"onwall
estimate~\eqref{e:gronwall}--\eqref{e:propagated} in integral sense: positive Sobolev norms
of the solution grow exponentially in time. This construction is essentially
taken from our previous paper~\cite{ACM2}. \label{i:step1}
\item Iteration and scaling, which turn the exponential growth into an
instantaneous blow up for the positive Sobolev norms of the solution,
while still keeping a control on the regularity of the velocity field and the
initial datum. \label{i:step2}
\end{enumerate}
We notice that it is straightforward to saturate the Gr\"onwall estimate for \ref{i:step1} above in pointwise
sense. The relevance of our example is, instead, to saturate it in
integral sense.

As a matter of fact, one can prove a complementary result to that in Theorem
\ref{t:main} in the setting of velocity fields with higher regularity, showing
loss of some regularity on the solution. By higher regularity, we mean here  that the velocity field belongs uniformly in time to
a Sobolev space $W^{r,p}$ with $r>1$, as long as this space does not embed in the
Lipschitz class.

\begin{theorem}\label{t:small}
Assume $d\geq 2$. Let $r > 1$ and $1 \leq p < \infty$ be such that $p< \frac{d}{r-1}$.
Let $T>0$ and $\sigma>0$ be fixed.
Then there exist a bounded velocity field $v$ and a solution $\theta$
of the continuity equation \eqref{e:continuity} with velocity $v$, both defined
for $0 \leq t \leq T$,  such
that:
\begin{enumerate}[label=(\roman*),ref=(\roman*)]
\item the velocity field $v$ is bounded in $\dot{W}^{r,p}(\mathbb{R}^d)$ uniformly
in time;
\item the initial datum $\bar \theta:=\theta(0,\cdot)$ belongs to
$\dot{H}^\sigma(\mathbb{R}^d)$;
\item the solution $\theta(t,\cdot)$ does not belong to
$\dot{H}^\sigma(\mathbb{R}^d)$ for any $t>0$;
\item both $v$ and $\theta$ are compactly supported in space, and
can be taken smooth on the complement of a point in $\mathbb{R}^d$.
\end{enumerate}
Furthermore, there is $\bar\mu = \bar\mu(r,p,d)<1$ such that
\begin{enumerate}[label=(\roman*),ref=(\roman*),start=4]
\item[(v)] for any $\mu>\bar \mu$ the solution $\theta(T,\cdot)$ does not belong to
$\dot{H}^{\mu\sigma}(\mathbb{R}^d)$ at the final time $T$.
\end{enumerate}
\end{theorem}

We make a few comments elucidating the similarities and differences between the two theorems:
\begin{enumerate}[leftmargin=*,label=(\arabic*), ref=(\arabic*)]
\item The loss of regularity in Theorem~\ref{t:small}
is much weaker than that in Theorem~\ref{t:main}. This appears as an artifact of our strategy for the proof. In fact,
the examples of optimal mixers in~\cite{ACM2}, the building blocks in the current proof,  give exponential decay of negative norms (we refer to this situation as ``exponential mixing''), if a uniform  $L^p$-bound in time is imposed only on the gradient of the velocity. If the bound is on higher derivatives, then the mixing is polynomial in time, which is not sufficient to show instantaneous blow up of the positive norms by rescaling. One can attempt to remedy this problem   in two ways and obtain a stronger version  of Theorem~\ref{t:small}, by constructing exponential mixers with uniform bounds on higher derivatives or by constructing flows that give directly growth of positive norms for the solution. Very recently, Elgindi and Zlatos have constructed more general exponential mixers under uniform bounds in $W^{s,p}$ for $1<s<2$ and $p$ close to $2$ \cite{ElgindiZlatos2018}. (See Theorem~\ref{t:mixing} for more details
on the examples of optimal mixers from~\cite{ACM2}.)

\item  If $r<1$ and we assume the velocity field is in
$W^{r,p}$, then in general solutions are not unique. However, we can prove the
existence of one solution as in Theorem~\ref{t:main}.

\item  The solution in Theorem~\ref{t:small} is unique in $L^2$  if $p
\geq 2$. If $\sigma>d/2$,  the solution can be taken to be bounded, and is the
unique one for any $p \geq 1$.

\item In the proof of both theorems we do not discuss the case $p=1$ for the velocity. This
choice is by convenience, since we define Sobolev
spaces  using Fourier multipliers and this definition is not equivalent to the
definition using weak derivatives and interpolation for $p=1$ and $p=\infty$.
Clearly, the result holds also for $p=1$ as stated by embedding, as $W^{r,p}\subset
W^{r,1}$ for any $p>1$.
\end{enumerate}

\begin{remark}\label{r:theorem3}
Further variants of Theorem \ref{t:main} are possible. For example, by
modifying the proof of Theorem \ref{t:small}, one can prove instantaneous loss
of some regularity, but with further constraints on the regularity of the
velocity field and of the initial data. Informally, the regularity index $r>1$
must be sufficiently close to $1$, $\sigma$ must be sufficiently large, and $s$
sufficiently close to $\sigma$. Furthermore, these conditions are coupled
together.  For sake of clarity and to keep the article contained, we opt not to
include a precise statement and a proof of these variants.
\end{remark}

The paper is organized as follows. We discuss the example of optimal mixing flows from \cite{ACM2} that constitute the building blocks for the constructive proofs of our main results in Section \ref{s:mixing}. Section \ref{s:proofs} contains the proof of Theorems \ref{t:main} and \ref{t:small}. In Appendix \ref{s:prelim}, we briefly recall main facts about fractional Sobolev spaces and present an almost orthogonality result for functions in these spaces with well separated supports, which yields a key lemma for the iteration step in the proof of the main theorem.

\subsection*{Acknowledgments}
The authors thank the anonymous referees for a careful reading of the manuscript and insightful comments.
This work was started during a visit of the first
and third authors at the University of Basel.
Their stay has been partially supported by the
Swiss National Science Foundation grants 140232 and 156112.
The visits of the second author to Pisa have been supported by the University of Pisa PRA
project ``Metodi variazionali per problemi geometrici [Variational Methods
for Geometric Problems]''.
The second author was partially supported by the
ERC Starting Grant 676675 FLIRT and the third author by the
US~National Science Foundation grants DMS-1312727 and DMS-1615457.

\section{An example of exponential mixing} \label{s:mixing}

As outlined in the Introduction, the first step in proving our main result,
Theorem \ref{t:main}, is the construction of a regular velocity field $u$ and
of a regular solution $\rho$ to the continuity equation with velocity $u$ such
that $u$ is Lipschitz uniformly in time and the positive Sobolev norms of
$\rho$ grow exponentially in time.

Given that all Lebesgue norms of $\rho$ are constant in time,  one way to
achieve the exponential growth in time of the positive Sobolev norms is to
ensure an exponential decay of the negative norms in view of the interpolation
inequality \eqref{e:interpolation}. Negative Sobolev norms will decay in time
if the flow of $u$ is sufficiently ``mixing'', which informally means that the
action of the flow generates small scales in the solution. It was recently
shown \cite{ikx,seis} that, under a $W^{1,p}$ bound on $u$ that is uniform in time,
where $1<p\leq \infty$, the rate of decay of negative Sobolev norms for
the solution $\rho$  is indeed at most exponential in time.

In \cite{ACM2}, we constructed examples of velocity fields satisfying the
required $W^{1,p}$ bound and solutions to the associated continuity equation that
saturate the exponential rate of decay of the negative norms (see
\cite{ElgindiZlatos2018,zlatos} for related examples). We will refer to these examples as
``optimal mixers''. We also consider velocity fields with higher regularity,
see~\eqref{e:expfield} and \eqref{e:expmix} below.
Our examples are based on a quasi self-similar geometric construction.

\begin{theorem}[\protect\cite{ACM2}]\label{t:mixing}
Assume $d\geq 2$ and let ${\mathcal{Q}} \subset \mathbb{R}^d$ be the open cube with unit side
centered at the origin of $\mathbb{R}^d$. There exist a velocity field $u \in C^\infty ( [0,+\infty[ \times \mathbb{R}^d)$ and
a corresponding (non trivial) solution $\rho \in C^\infty ( [0,+\infty[ \times
\mathbb{R}^d)$ of the continuity equation \eqref{e:continuity} such that:
\begin{enumerate}[label=(\roman*),ref=(\roman*)]
\item $u(t,\cdot)$ is bounded, divergence-free, and compactly supported in
${\mathcal{Q}}$ for any $t$;
\item $\rho(t,\cdot)$ has zero average and is bounded and compactly
supported in ${\mathcal{Q}}$ for any $t$;
\item $u(t,\cdot)$ belongs to ${\rm Lip}\,(\mathbb{R}^d)$ uniformly in time;
\item for any $r \geq 0$ and $1 \leq p \leq \infty$, there exist constants
$b>0$ and $B_r > 0$ such that
\begin{equation}\label{e:expfield}
\| u(t,\cdot) \|_{\dot{W}^{r,p}(\mathbb{R}^d)} \leq B_r \exp \big( (r-1) \, b t \big)\,;
\end{equation}
\item for any $0 < s < 2$, there exist constants $c>0$ and $\hat{C}_s >0$
such that
\begin{equation}\label{e:expmix}
\| \rho(t,\cdot) \|_{\dot{H}^{-s}(\mathbb{R}^d)} \leq \hat{C}_s \exp(- s c t)
\qquad t \geq 0 \,.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{remark}\label{r:gronwall}
By the interpolation formula \eqref{e:interpolation} with $s_1 = -s$, $s_2=s$,
and $\vartheta = 1/2$, which reads
$$
\| \rho(t,\cdot) \|_{L^2(\mathbb{R}^d)}
\leq
\| \rho(t,\cdot) \|^{1/2}_{\dot{H}^{-s}(\mathbb{R}^d)} \| \rho(t,\cdot)
\|^{1/2}_{\dot{H}^{s}(\mathbb{R}^d)} \,,
$$
together with the fact that the $L^2$ norm of the solution $\rho$ is conserved
in time, estimate \eqref{e:expmix} implies the following lower bound:
\begin{equation}\label{e:expsol}
\begin{aligned}
\| \rho(t,\cdot) \|_{\dot{H}^{s}(\mathbb{R}^d)}
& \geq
\| \rho(t,\cdot) \|^2_{L^2(\mathbb{R}^d)} \| \rho(t,\cdot) \|^{-1}_{\dot{H}^{-s}(\mathbb{R}^d)}
\geq
\| \rho(0,\cdot) \|_{L^2(\mathbb{R}^d)}^2 \hat{C}_s^{-1} \exp(s c  t) \\
& = C_s \exp(s c  t)
\quad \text{ for every $0<s<2$.}
\end{aligned}
\end{equation}
This bound can be seen as saturating  Gr\"onwall
inequality~\eqref{e:gronwall}--\eqref{e:propagated} and will be used in the
proof of Theorems~\ref{t:main} and~\ref{t:small}.

Estimate \eqref{e:expsol} is  weaker than  \eqref{e:expmix}, in the sense that growth of the positive norms does not imply decay of the negative norms, yet it is still enough to establish the loss of regularity. In fact, what is needed is a lower bound that implies growth of positive norms faster than polynomial, but not necessarily exponential in time.  Furthermore, blow up is a {\em local} phenomenon, while mixing is inherently a {\em non-local} property. One can then ask whether it is possible to use a simpler flow as building block for the construction in Section \ref{s:proofs}. Intuitively, a  circular-type flow, where the angular velocity is suitably chosen, could suffice. However, a preliminary analysis indicates that making this construction rigorous is rather delicate and we choose instead to exploit the work we have already done for the optimal mixers. We reserve to investigate this point further in future work.

\end{remark}

\begin{remark} The example in~\cite{ACM2} was constructed in two space
dimensions. However, it is not difficult to modify such construction to make it
$d$ dimensional. One can argue as follows. Call $\tilde u(t,x_1,x_2)$ and
$\tilde \rho(t,x_1,x_2)$ the velocity field and solution constructed
in~\cite{ACM2}. Fix $\eta = \eta(x_3,\ldots,x_d)$ smooth, with compact support
in $B(0,1/4) \subset \mathbb{R}^{d-2}$ and $\eta=1$ on $B(0,1/8)$. Moreover, fix
$\bar\eta = \bar\eta(x_3,\ldots,x_d)$ smooth, with compact support in $B(0,1/2)
\subset \mathbb{R}^{d-2}$ and $\bar\eta=1$ on $B(0,1/4)$. Then, setting
$$
u(t,x_1,\ldots,x_d) = \tilde u(t,x_1,x_2) \bar\eta(x_3,\ldots,x_d) \,, \quad
\rho(t,x_1,\ldots,x_d) = \tilde\rho(t,x_1,x_2) \eta(x_3,\ldots,x_d),
$$
yields a velocity field $u$ and solution $\rho$ defined on $\mathbb{R}^d$ that satisfy
the conclusions of Theorem~\ref{t:mixing}.
\end{remark}

\section{Proof of Theorems~\ref{t:main} and \ref{t:small}} \label{s:proofs}

The main idea in the proof of both Theorems~\ref{t:main} and \ref{t:small} consists in
constructing the velocity field $v$ and the associated solution $\theta$ by
patching together a countable number of velocity fields $u_n$ and solutions
$\rho_n$ saturating Gr\"onwall's inequality, the
existence of which follows from Theorem~\ref{t:mixing} (see also
Remark~\ref{r:gronwall}),  after rescaling in both space and time, and in size
as well for the solution, in a suitable manner. We then translate them so as
to be supported in cubes that accumulate towards a point in space. The pairs
$u_n$, $\rho_n$ serve as building blocks in our construction.

The choice of the rescaling parameters is the key point of the proof and is done
in such a way that the regularity of the velocity field and that of
the initial datum for the solution can be controlled, while at the same time
any regularity of the solution at later times is destroyed.

In the proof of Theorem~\ref{t:main}, we choose the parameters for the
rescaling to ensure that the velocity field has Sobolev regularity
uniformly in time, while we exploit that the rescaling in time can be chosen so
that the rate of exponential growth of the Sobolev
norms of the solution grows enough at each step to become in the
limit an instantaneous blow up.

When $r>1$, in view of estimate \eqref{e:expfield}, the
Sobolev norm of order $r$ of the velocity field is not uniformly bounded in
time, rather it grows exponentially in time. One then needs to balance
this exponential growth with the exponential growth for the Sobolev norms of
the solution, which accounts for the weaker result in Theorem~\ref{t:small}.

\subsection{A geometric construction.}
We fix a sequence $\{ \lambda_n \}$, with $\lambda_n>0$ and $\lambda_n \downarrow 0$, to be determined later.
For every $n\in \mathbb{N}$, we consider an open cube $Q_n$ in $\mathbb{R}^d$ with side of
length $3\lambda_n$.
The first condition we impose on $\{ \lambda_n \}$ is that
\begin{equation}\label{c:A}
\tag{${\rm A}$}
\left.\begin{array}{c}
\text{$\{ Q_n\}$ can be chosen to be contained} \\
\text{in a compact set and convergent to a point}
\end{array}\right\}
\qquad \Longleftarrow \qquad
\sum_n \lambda_n < \infty\,.
\end{equation}
If \eqref{c:A} is satisfied, we can choose the centers of the cubes so that
all cubes are contained in a compact set in $\mathbb{R}^d$ and accumulate to a point,
i.e., every open ball centered at such a point contains all the cubes
but finitely many. By saying that condition \eqref{c:A} is satisfied, we mean
that the condition on the right hand side, which implies the one on the left,
is satisfied. The same will be true for conditions \eqref{c:B}--\eqref{c:D}
(and variants) below.

Given $u$ and $\rho$ as in Theorem~\ref{t:mixing}, we fix two more sequences
$\{ \tau_n \}$ and $\{ \gamma_n \}$ with $\tau_n>0$ and $\gamma_n>0$
also to be  determined later, where $\tau_n \downarrow 0$.
Up to a translation,  in each cube $Q_n$ we set
\begin{equation}\label{e:pieces}
u_n (t,x) = \frac{\lambda_n}{\tau_n} \, u \left( \frac{t}{\tau_n} , \frac{x}{\lambda_n} \right) \,,
\qquad
\rho_n(t,x) = \gamma_n \, \rho \left( \frac{t}{\tau_n} , \frac{x}{\lambda_n} \right) \,.
\end{equation}
It is immediate to check that $\rho_n$ is a solution of the continuity equation
with velocity field $u_n$, that is,
\begin{equation}\label{e:eqn}
\partial_t \rho_n + \div ( u_n \rho_n) = 0
\qquad \text{on $\mathbb{R}^d$.}
\end{equation}
Moreover, we observe that for every $n$ we have
$$
{\rm dist}\, (\text{Supp}\, \rho_n \,,\, Q_n^c) \geq \lambda_n \,,
$$
since the support of $\rho_n$ is contained in a square of side $\lambda_n$  with the same center as
$Q_n$, the side of which has length $3\lambda_n$.

We set
\begin{equation}\label{e:sums}
v := \sum_n u_n \,,
\qquad
\theta := \sum_n \rho_n \,,
\end{equation}
where convergence of both series is taken pointwise almost everywhere.
We will prove later that, for a suitable choice of the three sequences of
parameters, $\lambda_n$, $\tau_n$, $\gamma_n$, the function
$\theta$ is a (weak) solution of the continuity equation
with velocity field $v$:
\begin{equation}\label{e:sumsol}
\partial_t \theta + \div (v \theta) = 0 \,.
\end{equation}

\subsection{Regularity of the velocity field}
We estimate the $\dot{W}^{r,p}(\mathbb{R}^d)$ norm of the velocity
field $v$ at a fixed time $t$:
$$
\begin{aligned}
\| v(t,\cdot) \|_{\dot{W}^{r,p}(\mathbb{R}^d)}
& \stackrel{\eqref{e:sums}}{\leq}
\sum_n \| u_n(t,\cdot) \|_{\dot{W}^{r,p}(\mathbb{R}^d)}
\stackrel{\eqref{e:pieces}}{=}
\sum_n \frac{\lambda_n}{\tau_n}
\left\| u \left( \frac{t}{\tau_n} , \frac{\cdot}{\lambda_n} \right) \right\|_{\dot{W}^{r,p}(\mathbb{R}^d)} \\
& \stackrel{\eqref{e:scaleW}}{=}
\sum_n \frac{\lambda_n}{\tau_n} \lambda_n^{\frac{d}{p}-r}
\left\| u \left( \frac{t}{\tau_n} , \cdot \right) \right\|_{\dot{W}^{r,p}(\mathbb{R}^d)} \\
& \stackrel{\eqref{e:expfield}}{\leq}
\sum_n \frac{\lambda_n^{1-r+\frac{d}{p}}}{\tau_n} B_r
\exp \left( \frac{ (r-1)bt}{\tau_n} \right) \,,
\end{aligned}
$$
where we have explicitly referenced which property we are using for each
inequality. Therefore, the fact that $v(t,\cdot)$ belongs to $\dot{W}^{r,p}(\mathbb{R}^d)$ is
implied by the following condition:
\begin{equation}\label{c:B}
\tag{${\rm B}$}
v(t,\cdot) \in \dot{W}^{r,p}(\mathbb{R}^d)
\qquad \Longleftarrow \qquad
\sum_n \frac{\lambda_n^{1-r+\frac{d}{p}}}{\tau_n}
\exp \left( \frac{ (r-1)bt}{\tau_n} \right) < \infty \,.
\end{equation}
Moreover, the definition of $u_n$, which is given in \eqref{e:pieces}, and the
fact that the
cubes $\{ Q_n \}$ have been chosen to be pairwise disjoint, gives the following
implication:
\begin{equation}\label{c:B'}
\tag{$\tilde{\rm B}$}
v(t,\cdot) \in L^\infty(\mathbb{R}^d) \text{ uniformly in $t$}
\qquad \Longleftarrow \qquad
\left\{ \frac{\lambda_n}{\tau_n} \right\} \text{ bounded.}
\end{equation}

\subsection{Regularity of the initial datum}
For a given $\sigma \geq 0$, we estimate the $\dot{H}^\sigma(\mathbb{R}^d)$ norm of the
initial
datum $\Bar\theta=\theta(0,\cdot)$ as follows:
$$
\begin{aligned}
\| \Bar \theta(\cdot) \|_{\dot{H}^\sigma(\mathbb{R}^d)}
& \stackrel{\eqref{e:sums}}{\leq}
\sum_n \| \rho_n(0,\cdot) \|_{\dot{H}^\sigma(\mathbb{R}^d)}
\stackrel{\eqref{e:pieces}}{=}
\sum_n \gamma_n \left\| \rho \left( 0,\frac{\cdot}{\lambda_n} \right) \right\|_{\dot{H}^\sigma(\mathbb{R}^d)} \\
& \stackrel{\eqref{e:scaleH}}{=}
\sum_n \gamma_n \lambda_n^{\frac{d}{2} - \sigma} \| \rho \|_{\dot{H}^\sigma(\mathbb{R}^d)} \,.
\end{aligned}
$$
Hence, we obtain the further condition
\begin{equation}\label{c:C}
\tag{${\rm C}$}
\Bar\theta \in \dot{H}^\sigma(\mathbb{R}^d)
\qquad \Longleftarrow \qquad
\sum_n \gamma_n \lambda_n^{\frac{d}{2} - \sigma} < \infty \,.
\end{equation}

In Theorem~\ref{t:main}, we require that the solution $\theta$ is bounded,
which follows if $\Bar\theta$ is bounded. Using
the definition of $\rho_n$ in \eqref{e:pieces} and again the fact that the
cubes $\{ Q_n \}$ have been chosen to be pairwise disjoint gives the following
implication:
\begin{equation}\label{c:C'}
\tag{$\tilde{\rm C}$}
\theta(t,\cdot) \in L^\infty(\mathbb{R}^d) \text{ uniformly in $t$}
\qquad \Longleftarrow \qquad
\{\gamma_n\} \text{ bounded.}
\end{equation}
In Theorem~\ref{t:small}, we only require that $\theta$ belongs to
$L^2(\mathbb{R}^d)$ uniformly in time, which again follows if $\bar\theta \in
L^2(\mathbb{R}^d)$. This requirement corresponds to condition
\eqref{c:C} with $\sigma=0$, that is,
\begin{equation}\label{c:Chat}
\tag{$\hat{\rm C}$}
\theta(t,\cdot) \in L^2(\mathbb{R}^d) \text{ uniformly in $t$}
\qquad \Longleftarrow \qquad
\sum_n \gamma_n \lambda_n^{\frac{d}{2}} < \infty \,.
\end{equation}

\subsection{Loss of regularity for the solution}
To show loss of regularity for $\theta$ at any $t>0$, it is enough to show
that the solution does not belong to $\dot H^s(\mathbb{R}^d)$, for  $0<s<1$, since
$\theta$ is at least in $L^2$ (by condition \eqref{c:C'} or \eqref{c:Chat} and
the compact support above) and $H^s (\mathbb{R}^d)= \dot H^s (\mathbb{R}^d) \cap L^2 (\mathbb{R}^d)$ forms a scale of
spaces ($H^s(\mathbb{R}^d)\subset H^r(\mathbb{R}^d)$, $s>r$).
We then fix an arbitrary $0<s<1$, and estimate the norm of $\theta$ in $\Dot
H^s(\mathbb{R}^d)$ from below as follows:
$$
\begin{aligned}
\| \theta(t,\cdot) \|^2_{\dot{H}^s(\mathbb{R}^d)}
& \stackrel{\eqref{e:nonlocal}}{\geq}
\limsup_{N\to\infty} \sum_{n=1}^N \left( \| \rho_n(t,\cdot) \|^2_{\dot{H}^s(\mathbb{R}^d)}
-\frac{C_d}{s} \frac{1}{\lambda_n^{2s}} \| \rho_n(t,\cdot) \|^2_{L^2(\mathbb{R}^d)}
\right) \\
& \stackrel{\eqref{e:pieces}}{=}
\limsup_{N\to\infty} \sum_{n=1}^N  \left( \gamma_n^2 \left\| \rho \left( \frac{t}{\tau_n} ,
\frac{\cdot}{\lambda_n} \right) \right\|^2_{\dot{H}^s(\mathbb{R}^d)}
 -\frac{C_d}{s} \frac{1}{\lambda_n^{2s}} \gamma_n^2
 \left\| \rho \left( \frac{t}{\tau_n} , \frac{\cdot}{\lambda_n} \right)
\right\|^2_{L^2(\mathbb{R}^d)} \right)
\\
& \stackrel{\eqref{e:scaleH}}{=}
\limsup_{N\to\infty} \sum_{n=1}^N \gamma_n^2 \lambda_n^{d-2s}
\Bigg[
\left\| \rho \left( \frac{t}{\tau_n} , \cdot \right) \right\|^2_{\dot{H}^s(\mathbb{R}^d)}
-\frac{C_d}{s} \left\| \rho \left( \frac{t}{\tau_n} , \cdot \right) \right\|^2_{L^2(\mathbb{R}^d)}
\Bigg] \\
& \stackrel{\eqref{e:expsol}}{\geq}
\limsup_{N\to\infty} \sum_{n=1}^N \gamma_n^2 \lambda_n^{d-2s}
\Bigg[
C_s^2 \exp \left( \frac{2s c t}{\tau_n} \right) - \frac{C_d \hat{C}_0}{s}
\Bigg] \,,
\end{aligned}
$$
where again we have referenced the property used at each step. Here, we exploit an almost orthogonality property in $\dot{H}^s$ for functions with disjoint support, established in Lemma \ref{l:orto} in Appendix \ref{s:prelim}.
In the last inequality,  we have denoted
the $L^2(\mathbb{R}^d)$ norm of $\rho$, which is conserved in time,  by $\hat{C}_0$.
Therefore, since we require $\tau_n \downarrow 0$, for any given $t>0$ we have
\begin{equation}\label{c:D}
\tag{${\rm D}$}
\theta(t,\cdot) \not \in \dot{H}^s(\mathbb{R}^d)
\qquad \Longleftarrow \qquad
\sum_n \gamma_n^2 \lambda_n^{d-2s}
\exp \left( \frac{2s c t}{\tau_n} \right)
= \infty \,.
\end{equation}

\subsection{Verify that $\theta$ is a solution of the continuity equation with
velocity $v$}
We assume that the sequences $\{\lambda_n\}$, $\{\tau_n\}$, and $\{\gamma_n\}$
have been chosen
so that conditions \eqref{c:B}, \eqref{c:B'}, and \eqref{c:C'} (or
\eqref{c:Chat}) are satisfied. Then, the
series in \eqref{e:sums} converge strongly in $L^2_{{\rm loc}}(\mathbb{R}^d)$ on any
finite time interval. This fact, together
with the validity of \eqref{e:eqn} and the fact that the cubes $\{ Q_n \}$ have been chosen
to be pairwise disjoint, implies the validity of \eqref{e:sumsol}.

\subsection{Proof of Theorem~\ref{t:main}} We are now in the position to
complete the proof of our main result.

In the case $r=1$, condition~\eqref{c:B} can be restated as
\begin{equation}\label{c:Bhat}
\tag{$\hat{\rm B}$}
v \in L^\infty((0,\infty);\dot{W}^{1,p}(\mathbb{R}^d))
\qquad \Longleftarrow \qquad
\sum_n \frac{\lambda_n^{\frac{d}{p}}}{\tau_n} < \infty \,.
\end{equation}
Our task is to find sequences $\{ \lambda_n\}$, $\{ \tau_n \}$, and $\{ \gamma_n \}$ such that conditions \eqref{c:A}, \eqref{c:Bhat}, \eqref{c:B'}, \eqref{c:C}, \eqref{c:C'}, and \eqref{c:D} are satisfied, for any $1<p<\infty$, $\sigma>0$, $s>0$, and $t>0$. Choosing
\begin{equation}
\label{e:choice1}
\tau_n = \frac{1}{n^3} \qquad \text{ and } \qquad \lambda_n = e^{-n}\,,
\end{equation}
we immediately see that~\eqref{c:A}, \eqref{c:Bhat}, and~\eqref{c:B'} are satisfied. If we take
\begin{equation}
\label{e:choice2}
\gamma_n = e^{-n^2},
\end{equation}
we see that~\eqref{c:C} and~\eqref{c:C'} are satisfied. We are left with
having to check condition~\eqref{c:D}. With the choices in~\eqref{e:choice1}
and~\eqref{e:choice2} for the parameters, the series appearing in condition
\eqref{c:D} reads:
$$
\sum_n \exp(-2n^2) \,
\exp \big( -(d-2s) n \big) \,
\exp (2sct n^3),
$$
which clearly diverges since $s c  t >0$. This concludes the proof of
Theorem~\ref{t:main}. \hfill \qed

\subsection{Proof of Theorem~\ref{t:small}} The proof is a variation of the previous one.
We fix $T>0$ and choose
$$
\tau_n = \frac{1}{n}\,.
$$
We want to check \eqref{c:B} for $0 \leq t \leq T$. Such a condition now reads
\begin{equation}\label{e:BT}
\sum_n n \lambda_n^{1-r+\frac{d}{p}}
\exp \big( n (r-1) b T \big) < \infty \,.
\end{equation}
By the assumption $p< \frac{d}{r-1}$ (expressing the fact that we consider a
Sobolev space that does not embed in the Lipschitz class),
we have that
$$
\beta = 1-r+\frac{d}{p} > 0 \,.
$$
For $\alpha>0$ to be determined, we choose
$$
\lambda_n = \exp ( - \alpha T n) \,.
$$
Then, condition \eqref{e:BT} becomes
$$
\sum_n n \exp ( - \alpha \beta T n )
\exp \big( n (r-1)bT \big) < \infty \,.
$$
This series indeed converges provided that
\begin{equation}
\label{e:choicealpha}
- \alpha \beta T + (r-1)bT < 0 \,, \quad \text{ i.e. }\quad
\alpha > \frac{(r-1)b}{\beta} \,,
\end{equation}
which is admissible. We remark that the parameters $\alpha$ and $\beta$ can be chosen independent of $T$, since $b$ is independent of $T$.  Under this condition, \eqref{c:B} holds and
also \eqref{c:A} and \eqref{c:B'} follow.

With this choice of $\lambda_n$, condition \eqref{c:C} becomes
$$
\sum_n \gamma_n \exp \left( \alpha \left( \sigma - \frac{d}{2} \right) T n
\right)< \infty \,.
$$
We can guarantee that this condition holds if
$$
\gamma_n = \frac{1}{n^2} \exp \left( \alpha \left( \frac{d}{2} -\sigma \right) T
n \right).
$$
Moreover, condition \eqref{c:Chat} becomes
$$
\sum_n \frac{1}{n^2} \exp \left( \alpha \left( \frac{d}{2} -\sigma \right) T n \right)
\Bigg[ \exp ( - \alpha T n ) \Bigg]^{\frac{d}{2}}
=
\sum_n \frac{1}{n^2} \exp \left( - \alpha  \sigma  T n \right)
< \infty \,,
$$
and we can verify that it is satisfied. We note in passing that, in the case
$\sigma>d/2$,
the stronger condition \eqref{c:C'} in fact holds.

We now substitute all the above expressions into \eqref{c:D} and obtain that
this condition is equivalent to having:
$$
\begin{aligned}
 \sum_n \left( \gamma_n \lambda_n^{\frac{d}{2} - \sigma} \right)^2 \lambda_n^{2(\sigma-s)}
 \exp \left( \frac{2s c t}{\tau_n} \right)
= & \sum_n \frac{1}{n^4}  \exp \big( - 2 (\sigma - s) \alpha T n \big)
\exp \big( 2s c t n \big) \\
= & \sum_n \frac{1}{n^4}  \exp \Big( 2 n \big(-  (\sigma - s) \alpha T  + s c t \big)  \Big)
= \infty \,.
\end{aligned}
$$
The above series diverges if
$$
-  (\sigma - s) \alpha T  + s c t > 0\,,
$$
which is (trivially) the case for any $t \geq 0$ when $s>\sigma$. In the relevant case $0<s \leq \sigma$,
we see
that the norm of the solution in $\dot{H}^s(\mathbb{R}^d)$ becomes infinity only after
a time $(\sigma - s) \alpha T / s c $.
Recalling that we control the Sobolev norm of the velocity field up to time $T$ only,
only those regularity indices $s$ that satisfy
\begin{equation}
\label{e:percent}
\frac{(\sigma - s) \alpha T}{s c }
< T
\qquad \Longleftrightarrow \qquad
(\sigma - s) \alpha
< s c  \,,
\end{equation}
lead to blow up of the norm.
Recalling~\eqref{e:choicealpha}, we can rewrite~\eqref{e:percent} as
$$
\frac{(\sigma-s)(r-1)b}{s c  \beta} < 1
\qquad \Longleftrightarrow \qquad
\frac{s}{\sigma} > 1- \frac{c\beta}{c\beta + (r-1)b} =: \bar\mu \,.
$$
We observe that, with this definition, $\bar\mu$ is independent of $T$, as again $\beta$ and $b$ are, and it also satisfies
$0<\bar{\mu}<1$. Therefore, the amount of regularity for the solution that
can be lost is $\bar\mu\sigma$. Theorem \ref{t:small} is now proved. \hfill \qed

\begin{remark}
 Although $\bar\mu$ is independent of $T$ and $T$ is chosen arbitrarily, we cannot conclude from our construction that such regularity is lost instantaneously by choosing a sequence of times $T=T_n\to 0$, as both the velocity field and the initial data, hence the entire solution, depend on $T$ in a fundamental way.
\end{remark}

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