When we think of superfood leafy green vegetables, often kale and spinach come to mind. But a more under the radar leafy green that's coming up in the latest health craze is seaweed, so much so it has been fondly dubbed as 'the next kale'.
Because of its mythos, seaweed was considered a luxury food due to its seemingly magical healing properties. Today, these magical healing properties are attributed to the many health benefits it brings. Here's why you should be getting more of this super green food in your diet and see why it belongs way up there with other power leafy greens such as kale and spinach.
Need your fix of iodine to support healthy thyroid function? Eat more seaweed which is rich in this nutrient. Since our bodies can't produce iodine on its own, food sources are a must. Lack of iodine has been linked to goiter, weight gain, mood imbalances such as anger and depression, constipation and fatigue. Eat seaweed to prevent iodine deficiency and ensure that your thyroid is working in good condition.
While many vegetables contain a variety of nutrients, micronutrients can be hard to come by, but you can find it in seaweed. Micronutrients such as folate, selenium, chromium, manganese, and sulphur are found in seaweed, which is important for a variety of metabolic functions. Bonus points that these micronutrients are in a form that's easily absorbed by the body.
Yes seaweed contains good amounts of micronutrients, it is also high in nutrients such as vitamins A, B, C, and E making the nutrient value in seaweed very valuable as it promotes healthy skin, good eyesight, and strong immune function. The dynamic variety of nutrients found in seaweed make it a superfood.
One of the few vegan foods that have EPA and DHA, seaweed is high in omega 3 fatty acids. The importance of getting plenty of omega fatty acids in our diet is integral to good health as it is reduces inflammation, promotes mental health, and prevents heart disease. Consuming seaweed gives you all these benefits.
This veggie from the sea, in comparison to other vegetables is easy to 'farm'. Seaweed grows quickly and abundantly, and is a hardy plant as it doesn't take too many natural resources to grow, needing only rocks or the ocean floor to grow out off. Getting seaweed is a matter of harvesting this fast growing plant, making it a sustainable choice for a leafy green vegetable staple.
Add seaweed to your diet and get the seaweed health benefits by incorporating them into salads, eating roasted seaweed sheets, adding it to soups and sprinkling it on top of rice. It goes without saying too, apart from the health benefits of adding seaweed to your diet, it's delicious as well!
Wellington Water Crackers are made with only the finest quality ingredients by Master Bakers with a heritage and tradition dating back to the 1800s.
Field Day Organic Traditional Water Crackers are the ideal choice for appetizers and entertaining!They have no high fructose corn syrup, no preservatives, and no artificial colors or flavors. What they do have is simply delicious taste that you can feel good about serving each day.


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Do you get 3v at pin 3/U1950?
Check all SPI Bus Termination Resistors and traces for U6100.
Probably knocked off component, or broken trace.
Check some voltages and signals on touchbar connector.
Some time ago you reported machine was working.
Did you try that BIOS file again?
2informaticos replied to 820-2936 Fan turn on and off cycle.
Check if R7402 is OK.
Same thing happens in SMC bypass mode (with RAM)?
Try new BIOS, just in case.
Before OSX (i.e. Boot Menu), trackpad/keyboard work through USB way.
Once logged into OSX, SPI communication is enabled.
Sometime can be soldered even without reballing it; re-using rest of solder on the board pads.
However, I do NOT recommend this method.
Same thing happens with backup and first file I sent you?
So you get stop sign, not restart, or stuck on logo?
Then disconnect all devices, keep only LCD and SSD.
Check continuity from its pads to nearest caps.
Check PWM signal at U9701 (pin 2) when press backlight adjust keys.
DC scale good enough to see the variation.
Remove F7140 and see where the short remains.
Which exact value (ohm scale) has that short?
You can increase injected voltage untill 3.5V and get higher current.
Did you reflash original chip, or used new chip?
Check if board still works with your backup.
Normal value is higher than 10 ohms in CPU VCORE, for this board.
That's good to see where to investigate, after board is clean.
"R8350 again and its 50k."
Onboard measurement is not relevant in this case.


text:
Advice for GOP Presidential Wannabes—C'mon In, The Water's Fine
Posted by Brendan Skwire | Aug 21, 2019 | Elections | 3 |
Here's some fun news.
Joe Walsh, a pugnacious former congressman, is preparing a Republican primary challenge to President Trump that he previewed as a daily "bar fight" with the incumbent over his morality and competency.
Mark Sanford, a former South Carolina governor and congressman, said he is inching closer to a bid of his own by sounding out activists in New Hampshire and other early-voting states about an insurgency focused on the ballooning deficit.
Jeff Flake, a former Arizona senator and Trump antagonist, said he has taken a flurry of recruitment calls in recent days from GOP donors rattled by signs of an economic slowdown and hungry for an alternative to Trump.
And former Ohio governor John Kasich will head to New Hampshire next month to "take a look at things" after experiencing "an increase" in overtures this summer, an adviser said.
The anti-Trump movement inside the Republican Party — long a political wasteland — is feeling new urgency to mount a credible opposition to Trump before it's too late.
Sanford, I think, hits the nail on the head when he calls these challenges—including his own—"preposterous," but there's still value in challenging Trump. As the reporters Bob Costa and Phil Rucker note, "anti-Trump organizers are courting wealthy independents or even liberals to contribute in the GOP primary, if only to bruise the president and help the eventual Democratic nominee in the general election."
So my message to Republican candidates who are disgusted by Trump is pretty simple: if you can't bring yourself to follow Tom Nichols' advice to vote for the Democrats in 2020, feel free to jump into the race and siphon off a few protest votes. It can't hurt!
PreviousSit Down, Shut Up, and Be Civil
NextOur President is an Idiot: Part Infinity
Brendan Skwire
Brendan Skwire is a cultural and media critic. He offers nearly two decades of experience as a journalist, video editor, blogger, and community organizer. Skwire has worked for the Philadelphia Weekly, Scrapple TV, and Raw Story, and is a former member of the News Guild.
Why Are the Republicans So Uptight?
Should Mitt Romney Just Switch Parties
Mainstream Media Once Again Buys Into Republican Fear-Mongering
When Real Right-Wing Fake News Gets Someone Killed

redpajama_set_name: RedPajamaCommonCrawl


text:
Inside Cybersecurity & Privacy Law
Exploring the evolution of cybersecurity and privacy law
Subscribe to Department of the Treasury
OFAC and FinCEN Communicate Ransomware Expectations in New Guidance
By Rajesh De, David A. Simon, Tamer A. Soliman, Ori Lev, Matthew Bisanz & Joshua M. Silverstein on October 5, 2020
Posted in Breaches & Incident Response, Cybersecurity, Ransomware, United States
On October 1, 2020, the US Treasury Department issued important guidance on what victims of ransomware attacks, as well as financial institutions (particularly money services businesses ("MSBs") and other companies that facilitate such payments), should consider when confronted with potential ransomware demands. First, the Office of Foreign Assets Control ("OFAC") issued an advisory that emphasizes…
2019 Outlook: Cybersecurity and Data Privacy
By Rajesh De, Matthew Bisanz, Samantha Machock, Marcus A. Christian, Veronica R. Glick, Sasha Keck, Gabriela Kennedy, Mickey Leibner, Stephen Lilley, Ian McDonald, Cristiane Manzueto, John Nadolenco, Mark A. Prinsley, Linda L. Rhodes, Joshua M. Silverstein, David A. Simon, Jeffrey P. Taft, Evan M. Wooten & Oliver Yaros on January 30, 2019
Posted in Cybersecurity, Privacy / Data Protection
Cybersecurity and data privacy presented some of the most complex legal questions and business risks that multinational companies faced in 2018. Businesses should expect continued growth in cyber and data privacy challenges in 2019.
Cyber attacks became even more sophisticated and severe in 2018, with incidents ranging from exfiltration and extortion schemes, to attacks on…
US Federal Banking Agencies Invite Public Comment on Enhanced Cyber Risk Management Standards for Large and Interconnected Financial-Sector Entities
By Rajesh De, Jeffrey P. Taft, Stephen Lilley & Matthew Bisanz on October 31, 2016
Posted in Breaches & Incident Response, Financial Services, United States
On October 19, 2016, the Board of Governors of the Federal Reserve System, Office of the Comptroller of the Currency and Federal Deposit Insurance Corporation issued an advance notice of proposed rulemaking indicating that they were considering establishing new heightened cyber risk management standards for certain large financial institutions and their service providers.
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An opportunity exists for an Education Support Worker to join our team based in Lawrence for a minimum of four hours per week on a fixed-term basis for six months with potential for this to extend to 12 months. The Education Support Worker will support children's presence, participation and learning in early childhood centres.
Please review the job description attached and if this opportunity interests you and you meet the requirements, please apply now using the 'apply online' button.
Applications close Midnight – Sunday, 24 February 2019.
Please review the job description attached and if this opportunity interests you and you meet the requirements, please apply now using the 'apply online' button. If this sounds like you, please visit our website to apply for the role. For further information contact Colleen at colleen wakefield@education.govt.nz Please review the position description by copy and pasting this link into your browser: https://expressonline.haineslink.co.nz/pdfs/615995_jobdesc.pdf Applications close Midnight - Sunday, 24 February 2019.
Job Categories: Education / Training. Job Types: Fixed-term.


What is the Sports Capital of Texas?
The Sports Capital of Texas is the premier destination for amateur, youth, and recreational sports in the state. Fueled by an economy that includes the headquarters of Dell and Emerson, our dynamic city has invested heavily in the infrastructure needed to support a variety of sports, entertainment, and business events. Over the years, Round Rock has poured tens of millions into its sports facilities, including the recent addition of the Round Rock Sports Center, which adds a world-class indoor showcase facility to the championship outdoor venues that originally earned us our Sports Capital of Texas reputation.
What is Round Rock's claim to the Sports Capital of Texas title?
From soccer tournaments to cycling events, Round Rock has advantages over other cities across the state when attracting youth and adult athletes in amateur and recreational sports to our city. Great facilities, like the Dell Diamond, Round Rock Sports Center and Old Settler's Park; fun activities; plenty of dining and lodging; and a central location within three hours from 90 percent of the state's population make Round Rock's goal of becoming the Sports Capital of Texas a reality.
Is Round Rock the Sports Capital of Texas?
Round Rock is able to own this brand because of the many events currently taking place in the city and those that will be there in the years ahead. Great facilities, fun activities, a central location and welcoming citizens all combine to give Round Rock its competitive advantage.
What does the city hope to accomplish with this campaign?
The Sports Capital of Texas campaign is designed to make Round Rock the top choice of tournament and meeting planners when selecting the location for their next big event. Through this effort, we will bring more tournaments and sporting events to our town, along with the athletes and their families. The net result will be more visitors buying our products and services, staying in our hotels and contributing to our city's sales tax revenue.
What does the Sports Capital of Texas mean to the citizens of Round Rock and why should they care?
Local businesses and citizens play an important role in building the Sports Capital of Texas brand. They are the face of our city for the athletes and families who come to experience all that the ultimate host city for youth, amateur and recreational sports has to offer. And, we encourage them to participate in our activities and continue to be gracious hosts to these visitors. The Sports Capital of Texas campaign adds dollars of sales tax revenue and contributes to the general fund. Ultimately, this means more money to fund the important city services that our citizens rely on; such as parks, libraries and public safety.
How can citizens get involved in this effort?


text:
Mueller tower could become landmark open to public
Classic midcentury design by Fehr & Granger
By Cindy Widner@CurbedAustin Feb 27, 2018, 5:20pm CST
The Austin Ballet Society at the Municipal Airport, photograph, 1962
University of North Texas Libraries, The Portal to Texas History; Austin History Center, Austin Public Library.
The 84-foot former control tower at what was once Austin's frighteningly central Robert Mueller Airport has been recommended for landmark status, Community Impact Newspaper reported Tuesday.
The Austin Historic Landmark Commission took up the matter at its Monday meeting, voting unanimously to recommend the 50-year-old structure for designation as a city landmark.
Designed by local midcentury-modern masters Fehr & Granger in 1961, the tower was the control center for the Robert Mueller Airport, which was built in the 1930s, closed in 1999 upon the opening of the Austin-Bergstrom International Airport.
The former airport site became, of course, the Mueller community—the master-planned, mixed-use development we all know and love today—but the tower (as well as two other buildings original to the airport) was left standing in the transformation and remains a sort of iconic beacon for the area.
The leader in the move toward landmarking the tower, according to Community Impact, is Girard Kinney of Kinney Architects; he told the paper that he will also push for the eventual opening the tower for public.
The two other historic structures on the site, according to CI, are a former hangar called the Browning Terminal (the only wooden bow-trussed hangar in the country) and the airport's former administrative office—now Mueller Central. While the former is in good shape and now hosts farmers markets, a spokesman for Mueller master developer Catellus Development told CI that the latter has probably had too many alterations to be considered for the status.
For commission's recommendation will now go before the city Planning Commission for consideration and, if it gets a thumbs-up there, on to the City Council for a vote.
• Mueller airport control tower recommended for historic zoning, may be opened for public use [Community Impact Newspaper]


Numerous applications refreshed, went marked down, or made declarations as of late, and assembled the most critical ones in our standard round up beneath. The present harvest incorporates a couple of giveaways by engineer Shape.ag, an other option to Apple Maps highlighting Google information, striking iPhone 5 refreshes, value drops for mainstream iOS applications, and so forth. Per common, we will keep on updating this rundown for the duration of the day.
2. Dolphine Browser for iOS | Free Version 6.1 landed today, and it presently incorporates a formerly existing choice to get to tabs on the best or right sidebar. The program likewise features new Tab match up and History synchronize highlights, auto full screen, look inbox, and an enhanced show with iPhone 5 similarity.
3. Adobe Revel for iOS | Free Version 1.6 included a Red-eye expulsion apparatus and the capacity to send out full-determination photographs to Camera Roll.
1. Gmail for iOS | Free Version 1.3.1415 is presently upgraded for the iPhone 5's bigger screen.
1. dJay for iPad | $4.99 Algoriddim's profoundly acclaimed DJ application, which won the Apple Design Award for 2011, is currently $15 off temporarily. The iPhone partner was additionally on special yesterday; see 9to5Toys for points of interest.
Google is commencing its yearly Google I/O engineer meeting today with the occasion's opening keynote introduction and amid had a few declarations for Android as well as iOS and Mac clients.
Among the declarations so far today, Google Assistant is extending to more dialects, notwithstanding coming to iOS with an independent iPhone application.
Google Home is coming to new nations and dialects and furthermore getting a pack of new highlights in front of a normal divulging for Apple's Siri Speaker rival. The Gmail portable applications are getting another "Keen Reply" include with brisk answer proposal catches similar to the Inbox application. Google Photos is getting new highlights including machine learning knowledge. What's more, Google is discharging another VR application for designers today called 'Stare off into space Elements'.
The organization likewise reported some refreshed numbers for its administrations today, including affirming it currently has 2 billion Android dynamic gadgets, 800 million+ Drive clients, and 500 million+ Photos clients.
Refresh: Google affirmed it is "researching reports of an issue with Google Mail. We will give more data without further ado."
Reports have been coming in that Gmail is encountering blackouts in numerous parts of the world.
Google's App Status Dashboard, which offers execution data for Google Apps administrations, is right now guaranteeing that Gmail is down.
Google just changed Gmail to "Administration Disruption" status, however we will fill you in as to whether it is moved up to a full "Administration Outage."
Inbox by Gmail, which Google seems to see as a trade for the prominent email customer Sparrow, has been discreetly refreshed, notes TechCrunch, to incorporate a local form for iPad. The application is as of now just accessible in a few nations (counting the U.S.), and neither the App Store depiction nor What's New section specify the iPad fabricate.
No new usefulness has been included, despite everything you require a welcome to utilize the app– which you can apply for here. The application itself is a free download from iTunes.
It's trusted that the Inbox application has been to a great extent created by the Sparrow group Google procured in 2012. Google evacuated both the iOS and Mac adaptations of the Sparrow email customer a couple of days prior.
Presently your telephone can ensure you always remember to pay your charge card charge. In the wake of being circled as gossipy tidbits in March, the charging update was moved to Google Now.
New element based solicitations are messaged to your Gmail account. Google Now can gather that information and show it right when you require it.
The charging brief is entirely fundamental at the present time, as per Android police, the primary site to find this element. All you'll find in Google Now cards is the name of the charging organization, the sum due, the due date, and the alternative to see the email.
Yet, the Android police said the code for this component likewise has choices to see least installments, past parities, earlier month's solicitations, and even more seasoned bills. Thus, this can be only the beginning of a more element rich approach to monitor your solicitations in Google Now.
Installment updates may not work for the greater part of your solicitations, as it gives the idea that Google must target particular suppliers that are like how the organization incorporates continuous bundle following.
With the choice to consequently pay your bills, your receipt update might be somewhat obsolete nowadays. In any case, for the individuals who can not chance programmed withdrawals before reviving, an update bill will be helpful.
The bill update joins a progression of late improvements to Google Now. Toward the beginning of May, Google included a notice when a thing you hunt down was in a close-by store. Beforehand, Google Now had auto-stopping identification and disconnected mode.
As of late, a flawless Google Now trap has been found to repudiate auto restore participation. Set Google Now suggestions to drop Netflix, for instance, and an update card flies up with an immediate connect to the Netflix undoing page.
There might be more points of interest to Google Now in progress. There are bits of gossip that Google Now can at long last set up a genuine clock rather than a caution when you utilize voice summons to tally down your bubbled eggs.
Need more Google Now tips? See our guide on "5 fast tips to get more out of Google Now" and "How to make Google Now your closest companion."
At last, Google reported iOS bolster for Android Wear on Monday, doing great on an officially existing talk chain since the last real Wear refresh and topped a week ago when Huawei outperformed bolster. iOS on Amazon irregular rundown. IPhone proprietors won't get the full Wear encounter that Android clients get (see points of interest beneath), however their shrewd equipment choice is going to detonate.
On the telephone side, the Android Wear for iOS application chips away at iPhone 5, 5c, 5s, 6 and 6 Plus running iOS 8.2 or later. On the watch confront, bolster is more constrained: The application begins today, however just works with LG's Urbane Watch. Be that as it may, most Android Wear watches later on - including the Huawei, Asus, Motorola and maybe even TAG Heuer - must be perfect with the iPhone.
Why is this critical: For most iPhone clients, Android Wear bolster isn't vital. Customers most inquisitive of Apple's savvy have purchased an Apple Watch or have plans to do as such when the stage is enhanced with watchOS 2. What's more, those iPhone clients who don't fall into these two camps are most likely too smartwatch-irresolute to bring Android Wear a genuine look.
Undoubtedly, we can take a gander at a thin potential change: iPhone clients cherish the Google administrations universe - Gmail, Calendar, and particularly Google Now - and the individuals who need to have a clock. Shrewd, however incline toward the plan, round face Huawei Watch over the Apple Watch capsule.
Be that as it may, this move is an easy decision on Google. On the off chance that it has spent so much riches yet on keen contact focal points and broadband inflatables, for what reason not drop a little advancement cash on a generally straightforward iOS application? The reward in multi-stage gloating rights is justified regardless of the cost alone.
What does Android Wear for iOS do?
Right now, we're just ready to accumulate leap forward pointers of what the iPhone proprietor will have the capacity to do with the Android Wear watch. Lamentably (yet as anyone might expect) iOS clients won't have the capacity to utilize the current outsider Android Wear applications accessible in Google Play. In any case, they will get a basic notice for every one of the iOS applications that have showed up on their iPhone. They will likewise approach more extravagant applications and informing for various Google administrations.
Google Notes underpins rich notices for Gmail, Google Calendar, and Apple Calendar. Clients will likewise get all AndroidWear Google Now cards - the title of the stage - and access to Google-made projects, for example, Google Fit (watch heart rate and information), Weather , Alarm and Translation.
IPhone clients will have the capacity to see warnings for telephone calls and messages, yet not at all like Android clients, they won't have the capacity to answer to messages with voice transcription from their Wear watch. That is a low number, yet in any event Google Now looks set to be safe. Google Digital Assistants are pressed with astonishments and fun highlights, convey movement card alarms, flight updates, climate cautions and other applicable individual things at simply the ideal place. time.
What's more, obviously, you can utilize Google Now to tackle basic math issues, become acquainted with the most recent games scores, and draw an assortment of Googly look tips specifically from your watch. yours.
So despite the fact that Android Wear for iOS does not give a total Wear understanding, it underpins "the vast majority of the Wear highlights we discover our Android clients are utilizing and cherishing," as per sources. I'm at Google. We will know the entire story when we match up an iPhone with LG's Urbane Watch, so keep following up for a hands-on work out.
At the point when Android clients change to Android Lollipop in the coming weeks, Google trusts the Android Gmail application will be the main informing application that most clients will require.
Android Police as of late distributed a video that will report client direction through Gmail 5.0 with regards to Android Lollipop. In that video, Gmail gets not just a material plan outline - it fits well with numerous other Android applications - it additionally can interface with outsider email accounts like Outlook.com and Yahoo. As indicated by the video, Gmail will work with numerous other email benefits past the two noteworthy webmail administrations.
Story behind the story: As Ars Technica called attention to multi year prior, Google has a propensity for supplanting its Android share applications as a feature of its Android Open Source venture with restrictive applications, for example, Message Transfer. Transfer Photos. By expanding the capacities of Gmail, Google might endeavor to execute - or if nothing else sidestep - Android's stock email application, which was repaired in November. In any case, before it can, Gmail for Android should deal with extraordinary conditions like an Exchange server or a typical POP3 and IMAP account.
From the client's perspective, the possibility of using Gmail for all your webmail administrations resembles a truly decent arrangement. Gmail is constantly a standout amongst other applications on Android and is said to be the best portable email understanding.
Adding an outer record to Gmail implies a less demanding email understanding, in addition to you don't need to contend with the stock email customer or the private informing application from Microsoft and Yahoo.
The new Gmail application does not seem all that unique in relation to what we at present have in route and use. To get to the new record, tap the "ground sirloin sandwich" menu symbol in the upper left corner. At that point simply contact or swipe on a level plane over your record name to change from Gmail to Yahoo or another outsider record.
Different changes you may find in the video incorporate moving the alter catch from the highest point of the application to the base (the symbol itself has changed). Furthermore, in the make window, you currently have the connection symbol, which you can press as opposed to covering it under the vertical menu symbol.
By and large, the new Gmail application looks quite alluring. Ideally the spilled video really speaks to what Google expects to convey with Lollipop.


text:
Prehistory3000000 BCE - 5001 BCE
Antiquity5000 BCE - 399 CE
Middle Ages400 CE - 1500 CE
Age of Reason1500 CE - 1879 CE
Modern Times1880 CE - 1980 CE
500000 BCE :
Flints Galore
60000 BCE :
Not Just Cave Dwellers
Rhinos in Samaria
7000 BCE :
Masking Death Prehistoric City
What would you like, Egyptian or Philistine ?
4,000 Year Old Jerusalem Tomb: a Treasure Trove of Decapitated Toads
Where did the Philistines come from?
Is This Ziklag?
800 BCE :
Horses in the rain Ruin of Samaria!
Which Isaiah? How many clerks ?
Temple Off the Mount
Stamped by the Mayor
Signs of Destruction
Who are You, Samaritans?
Esther – the Persian Version
Nehemiah on the Wall
Forgotten Archive
A Brief History of the Hasmoneans
Pools and Palaces
Fighting for Heart and Soul The Youngest Maccabee
Telltale Tremor
Cast a Giant Shadow
A Dig Full of Holes
אוצר ממצולות ים Anonymous Hasmonean
20 BCE :
Mystery of Caesarea's Disappearing Port Jerusalem Potters
Paving the Past
0 BCE :
Nabateans in the Bible Lords of the Desert Pilgrim City
40 CE :
Wanton Destruction on a Calamitous Scale Golden Nostalgia
King's Canopy in Shilo
The Pilgrims' Progress
Don't Call Me Joseph Dead Sea DNA
Romans on the Roofs of Gamla
130 CE :
Backs to the Western Wall
Bar Kokhba in Jerusalem
Bathing Rabbis
Trio in Togas
Washed Out by the Rain
זה השער
Blessed Wine
Shofar – Blasting Away Pilgrims' Riches Playing with Water? Byzantine Cistern in Jerusalem Playground
Georgians in Ashdod
The Fire of Faith
What's a Jewish Menorah doing on early Islamic coins and vessels ?
Whose Head is it Anyway? Potter's Treasure
Cave of Revenge
1096 CE :
Heroes on the Walls of Haifa
Locking Horns at the Battle of Hattin
Z-rated: For Forties Plus
Mystery of the Zohar Zohar Unzipped
Ancient Ring in the Flowerbed
Ready for Elijah
Divinely Plagued
A Widow in Print
Once Bitten, Twice Shy – Portuguese Jewry
Portuguese Messiah at the Stake
Salonika's Mystic Quartet
Who Was David Ha-Reuveni?
Kabbalists in Salonika
The Price of Dissent
Was 'The Jewish Bride' Really Jewish? Messianic Mania
Newton's Fourth Law In the Service of the Czar Haman's Pockets Trying to Belong
Groping for Truth
Napoleon's Jewish Court
Red Rose of Petra
A Soul Divided
Blackface Minstrel Shows
With Thanks from Damascus
Charlotte Rothschild – First Jewish Female Artist
The Angry Convert
Jewish? French? Italian!
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Home / Age of Reason / Salonika's Mystic Quartet
Gateway to the Empire
Mother Salonika
Practical Kabbala
Salonika's Messiah
Heir to a Mission
The Shekhina Speaks
Rising from the Dust
By: Roee Horen
As one of the Ottoman Empire's few Mediterranean ports, Salonika welcomed Jewish refugees from Spain and Portugal. Four great mystics emerged from its shores – each with a different approach to redeeming the Jewish people // Roee Horen
[To hear this article on our podcast please click here]
In 1492, four hundred glorious years of Spanish Jewish literature and culture abruptly ended. The land that had nurtured brilliant Jewish legalists, philosophers, mystics, poets, and grammarians began to heave and tremble beneath the ancient Jewish community. Fearing for their lives and livelihoods, many converted to Christianity, emulating their leader, Rabbi Abraham Senior. But for a multitude of others who clung to their faith, the only option was to leave Spain.
Some settled in nearby Portugal, only to be forcibly baptized in 1497. Some crossed the narrow stretch of sea to North Africa, while others sought out less hostile shores on the Mediterranean, such as those of Christian Italy and Holland. Not all these Jews were welcome, as their contemporary Solomon ibn Verga wrote:
The Spanish exiles' ships reached Italy, and there too, great famine was everywhere, and plague raged onboard. The poor souls knew not what to do […] until finally they disembarked. But the townspeople wouldn't grant them entry. So they went to Genoa, where there was also famine, but they were allowed into the town.
Unable to abide their suffering, the young men went to the house of idolatry to convert just so they would be given a little bread. Many of the uncircumcised walked through the markets with a crucifix in one hand and a morsel of bread in the other, bidding the sons of Israel: "If you bow to this, here is bread." In this fashion, many became apostates and intermingled with the non-Jews. (Solomon ibn Verga, Shevet Yehuda [Scepter of Judah] [Amsterdam, 1654], p. 62a [Hebrew])
Aware of how Jews could enrich his empire. Ottoman emperor Bayezid II
But as one door closed on the Spanish exiles, another unexpectedly opened. Sultan Bayezid II, ruler of the Ottoman Empire, unlocked the gates of his realm, either in anticipation of the benefits these talented merchants could bring to his economy, or in response to pressure from Jewish leaders in Turkey. Regardless of his motives, a firman (royal edict) ordered Ottoman governors to receive all Jews kindly.
The refugees arrived penniless and sometimes – not unlike Holocaust survivors – without a clue as to what had become of their nearest and dearest. They soon viewed the Ottoman Empire as their new homeland. Nevertheless, despite their suffering at the hands of Spain's Christian monarchs, these Jews always considered themselves Sefaradim, Spanish.
Of all their destinations, the port of Salonika (now Thessaloniki), western gateway to the empire, was unique. The cosmopolitan city was a melting pot of Turks and Greeks, Bulgars and Macedonians, Slavs, Russians, and even Frenchmen. Thousands of Jews who'd been there for generations mixed with newcomers from Germany, Italy, and North Africa. Salonika's strategic location on the Mediterranean was also not far from Spain, making it literally a lifesaver for Spanish exiles, who could otherwise have been stranded at sea indefinitely.
A Portuguese poem by Portuguese Jewish author Samuel Usque conveys the overwhelming relief with which those lucky enough to reach Salonikan shores fell into the city's embrace:
Behold, Salonika, you are a mother city of Israel. A faithful orchard of Jewish law and faith, filled with fair flowers and fine trees, famed for its glory among Israel. Its fruits exude beauty and majesty, for it grows alongside springs of charity, and streaming rivers of kindness and mercy water its soil. The brokenhearted and downtrodden who flee there from European lands and all other parts of the world will find healing and cure. It will absorb them lovingly, like a merciful mother, like the mother of Israel, like Jerusalem on its festivals. (Samuel Usque, cited in Y. S. Emanuel, Leaders of Salonika in Chronological Order, part 1, p. 6 [Hebrew])
Salonika, the Ottoman Empire's western gateway, as depicted in Braun and Hogenberg's atlas of the world's cities, Cologne, 1572
Salonika's new inhabitants quickly made it a bustling international trading hub. Jews dealt in lead for artillery, wine, spices, and precious stones. The textile business took off, and silver and salt mines were developed. Like most Jewish centers, Salonika became a printing powerhouse as well. Many Jews grew so rich that, just thirty years after the trauma of the Spanish expulsion, Jewish ethical tracts began criticizing their luxurious lifestyle.
A tremendous spiritual renaissance accompanied the economic boom, with Salonika even becoming known as the Balkan Jerusalem. A rare concentration of kabbalists from the Iberian Peninsula gathered in the city: rabbis Yosef Taitazak, Yosef Karo, Solomon Molkho, and Shlomo Halevi Alkabetz. All four endeavored to redeem the Jewish people through various but interrelated methods. What were these paths, and to what extent did they leave a mark on Jewish history?
The eldest of the group was Rabbi Yosef Taitazak, born circa 1465 in Castile. This Portuguese Jewish leader headed Salonika's largest Talmudic academy. In many ways, Taitazak was the preeminent halakhic authority after the expulsion, certainly as far as the scattered Andalusian communities were concerned. Aside from being a rabbinical judge and preacher, however, he was a philosopher versed in the writings of Thomas Aquinas and many other thinkers. Yosef Karo called him "sage in all matters" (Avkat Rokhel 51).
British Library, London
Practical kabbala as a means of revelation. Sixteenth-century manuscript of The Book of the Responding Entity
Taitazak seems to have been part of a kabbalistic circle that coalesced around the author of the mystical Sefer Ha-meshiv (The Book of the Responding Entity), also known as The Maggid's Book of Visions. Apparently predating the Spanish expulsion, this work discusses the use of specific techniques to summon a divine revelation in the form of an angel.
The secret lies in the fact that the hidden meaning of this number [forty-five, the numerical value of one of the divine names], when uttered in full, will draw down My power and Divinity in the form of an actual man standing before you to answer whatever you would ask in real speech before you, without [causing] terror. (cited in Moshe Idel, "Research on the Method of the Author in The Book of the Responding Entity," Sephunot 17, p. 215 [Hebrew])
The book describes a group of scholars who magically produce just such a divine manifestation. The best-known tale associated with this cabal involves one Rabbi Joseph De-La Reina's efforts to employ these methods to hasten the final redemption. Though the story sounds utterly fantastic, Kabbala scholar Gershom Scholem proved that De-La Reina was in fact a historical figure, having contributed to Sha'arei Ziyyon, a collection of ancient and medieval prayers and liturgical poems attributed to a disciple of the renowned Rabbi Hayyim Vital.
De-La Reina and his circle supposedly used kabbalistic incantations to force Satan to appear as a black dog. While his colleagues fled, De-La Reina bound the dog by means of a ring inscribed with the forty-two-letter name of God (which still appears in the Hasidic Sabbath eve liturgy alongside the eight-line poem Ana Be-khoah [Please, by the Strength]). Then Satan broke free and punished the sage who'd dared to restrain him: De-La Reina reputedly either converted or committed suicide.
Led by the legendary Rabbi Isaac Luria (a.k.a. Arizal, 1534–1572), the kabbalists of Safed subsequently outlawed "practical Kabbala," such as making angels heal or reveal mystical secrets. Rabbi Luria lived in Safed only from 1570 to 1572, but his main pupil, Rabbi Vital (1542–1620), cited De-La Reina's fate as a cautionary tale:
Take proof from Rabbi Joseph De-La Reina and Rabbi Solomon Molkho, who engaged in practical Kabbala and were cast out of the world. […] Even more reason [to refrain] is that [the angels guarding the gates] must be coerced by oath, against their will. Then they entice [the kabbalists], leading them down unworthy paths until they lose their souls. (Rabbi Hayyim Vital, Sha'arei Kedusha [Gates of Holiness], part 3, gate 6)
Evidently, Rabbi Taitazak wasn't spooked by such incidents. One work attributed to him is Kaf Ha-ketoret (The Incense Spoon), written after the expulsion and concerned primarily with estimating when redemption is due and bringing it about using kabbalistic meditations on Psalms.
The second cautionary figure mentioned by Rabbi Vital also appeared in Salonika while Taitazak was active. Like him, Solomon Molkho claimed that a heavenly emissary, or maggid, had appeared to him (see Motti Benmelech, "Salvation at Stake," Segula 33). The similarities don't end there.
Molkho was born Diogo Pires to a converso family in Portugal in 1501. (Despite the forced baptism of Portuguese Jewry in 1497, a strong crypto-Jewish community remained.) In his twenties, Pires served as a senior lawyer in the royal court in Lisbon.
Molkho's messianic pretensions led to his death at the hands of the Spanish Inquisition but also made him a hero. Molkho's standard, inscribed with biblical verses, preserved in the Jewish Museum, Prague
A few hundred kilometers away in Italy, a sensational series of events was under way. Short, swarthy David Ha-Reuveni had miraculously gained access to Pope Clement VII in Rome in 1524. Purporting to be commander in chief of the armies of his brother, ruler of the tribes of Reuben, Gad, and Manasseh in a distant Arabian kingdom, Ha-Reuveni persuaded the pope to assist him in conquering Mecca! The pontiff sent him on to John III of Portugal, recommending that the king assign him additional troops.
Molkho's signature included a flag-shaped flourish
The arrival of this colorful character in Portugal excited the country's crypto-Jews. Diogo Pires, for one, resolved to return to Judaism. Ha-Reuveni refused to circumcise him, however, fearing accusations that his true mission was to reclaim Portugal's New Christians, as the Jews were known. Undeterred, Pires performed the operation himself (though he almost bled to death), changed his name to Solomon Molkho, and fled to either Italy or Salonika in 1525.
In 1529, Molkho published Sefer Ha-mefo'ar (The Magnificent Book), a surprisingly erudite book of mystical sermons. As he wrote in his prologue, the work was intended
"For my dear brothers and friends in Salonika, who asked me to send them some interpretations of the true path, or commentary on the weekly Torah portion, or sayings of our sages, may they rest in peace."
What brought Molkho to Salonika? Having written to Yosef Taitazak, perhaps he'd heard of the rabbi's mystical revelations and hoped to find him a kindred spirit. Alternatively, Taitazak may have summoned Molkho, hoping their encounter would reawaken his own mystic visions after they'd ceased following the Spanish expulsion. Whatever the case, the two had much in common – including a determination to hasten Israel's redemption through kabbalistic means.
Molkho resided in Salonika sometime between 1526 and 1529. There he met Rabbi Shlomo Alkabetz, who was born in the city in 1505 and wrote of scriptural interpretations heard from him. But then, having printed his sermons, Molkho left for Italy.
Like David Ha-Reuveni, Molkho foresaw redemption coming about through world conflict, which he duly tried to set in motion. He regarded himself as the Messiah son of Ephraim, harbinger of the true Messiah, son of David. Molkho joined Ha-Reuveni in the summer of 1532 for a meeting with the holy Roman emperor, Charles V, who also occupied the Spanish throne. They urged him to order the armies of Europe to attack the Ottoman Empire, instigating a war between Christians and Muslims that would constitute the biblical showdown between Gog and Magog prior to the redemption of Israel.
Unlike Pope Clement and John of Portugal, however, Charles promptly handed Ha-Reuveni over to the Spanish Inquisition, and he was never heard of again. Molkho, a lapsed New Christian, was sent to Mantua for trial and burnt at the stake.
The 16th-century figures of Solomon Molkho and David Ha-Reuveni inspired modern-day political Zionists. Engraving of Ha-Reuveni by Arthur Schick
Between 1528 and 1531, a fourth scholar arrived at Rabbi Yosef Taitazak's Talmudic academy. Born in Portugal in 1488, Rabbi Yosef Karo famously authored two classic works of Jewish law, Bet Yosef and Shulhan Arukh (see Tirza Kelman, "Coded for Print,"Segula 44). Yet he came to Greece an unknown, his kabbalistic studies still ahead of him.
Though heavily influenced by Solomon Molkho, Rabbi Yosef Karo chose a different path. 19th-century woodcut portrait of Rabbi Karo
Although there's no clear evidence of direct contact between Rabbi Karo and Solomon Molkho, they overlapped in Salonika in 1529, and Molkho's influence changed Karo's life. The two were also close in age and countrymen. At the very least, Karo's closest friends were profoundly impressed by Molkho.
Rabbi Yosef Karo was devastated by Molkho's execution. Karo idealized him as an outstanding kabbalist, a prophet, and – above all – a martyr for his people. Shortly after Molkho's death, Karo too began envisioning a heavenly voice, or maggid, speaking unbidden from his mouth, sometimes personifying the Mishna itself as he recited its passages. Rabbi Karo seems to have believed that Molkho's soul had entered his own body and that he was to continue his hero's mission. This heavenly messenger promised the rabbi a "bright" future, including the same martyrdom as "my friend Shlomo" (Rabbi Yosef Karo, Maggid Meisharim [Teller of Truths], 1st ed., "Genesis," 27 Heshvan).
Rabbi Yosef Karo's signature
Why did Rabbi Karo wish such a violent death upon himself? Perhaps like the great Rabbi Akiva, he simply sought to fulfill the biblical injunction to "love the Lord, your God, with all your soul" (Berakhot 61b). Alternatively, traumatized by the previous generation's expulsion, he might have concluded that God no longer had place for Jews in this world. Similarly, the maggid ordered Karo to minimize his intake of food and water, thus reducing his bodily existence. This second-generation post-trauma may have reflected the collective subconscious of his entire era.
Though he'd inherited Solomon Molkho's mission, Rabbi Yosef Karo chose an utterly different path to redemption. In 1533, a few months after Molkho's demise, Rabbi Karo and fellow kabbalists (Alkabetz included) studied all night on Shavuot – the holiday marking the revelation at Sinai and hence the wedding of God and Israel – just as described in the Zohar (1, 8a):
Rabbi Shimeon would sit and study Torah all night when the bride was about to be united with her husband [i.e,. on the eve of Shavuot]. And we have learned that the companions of the household in the bride's palace [i.e., the members of Rabbi Shimeon's circle of mystics] are needed on that night when the bride is prepared for her meeting on the morrow with her husband [i.e., God Almighty] under the bridal canopy. They need to be with her all that night and rejoice with her in the preparations with which she is adorned, studying Torah, [progressing] from the Pentateuch to the Prophets, and from the Prophets to the Writings, and then to the midrashic and mystical interpretations of the verses, for these are her adornments and her finery. (Isaiah Tishby,The Wisdom of the Zohar, trans. David Goldstein [Littman Library of Jewish Civilization, 1991], vol. 3, p. 1318)
For probably the first time in history, Rabbi Yosef Karo and his circle sought to reenact this scene. The resulting life-altering revelation was later depicted by Shlomo Alkabetz in a letter reproduced by Rabbi Isaiah Halevi Horowitz in his Shnei Luhot Ha-brit (The Two Tablets of the Covenant), published in 1648 and known by the acronym Shelah. The Shekhina spoke from Rabbi Karo's lips, praising the small group who'd restored the divine presence while Salonika slept:
Happy are those who bore you, My friends, in that, by denying yourselves sleep you have ascended so far on high. Throgh you I have become elevated this night and through the companions in the great city, a mother-city in Israel.
You are not like those who sleep on beds of ivory in sleep which is a sixtieth of death, who stretch themselves out upon their couches. But you cleave to your Creator and He rejoices in you. (Louis Jacobs, Jewish Mystical Testimonies, Schocken Books, 1977, p. 100)
The Shekhina then lamented its own misery:
If you could only imagine one millionth of the anguish which I endure, no joy would ever enter your hearts and no mirth your mouth, for it is because of you that I am cast to the ground. (ibid.)
Finally, the divine presence suggested a course of action:
Happy are you. Return to your studies, not interrupting them for one moment. Go up to the land of Israel, for not all times opportune. There is no hindrance to salvation, be it much or little. Let not your eyes have pity on your worldly goods for you eat of the goodness of the higher land. If you will but hearken, of the goodness of that land will you eat. Make haste, therefore, to go up to the land for I sustain you here and will sustain you there. To you will be peace, and to all that is yours peace. (ibid., p. 101)
The Shekhina called on Karo and his colleagues to abandon the fleshpots of Salonika for the Holy Land, taking the divine presence with them and thus ending its suffering. This path clearly differed not only from that of Rabbi Yosef Taitazak and the circle of the author of The Book of the Responding Entity, but from that of David Ha-Reuveni and Solomon Molkho. The divine voice speaking from Karo's throat intimated that redemption would come not through incantations or political intrigue but only through Jews' physical return to their homeland.
Photo: Yosef Stepansky
Could this be where Rabbi Karo and his colleagues purified themselves before their fateful night of study on Shavuot in 1533? Surviving the fire that raged through Salonika in 1917, this stone bathhouse is known today as the Yahudi Hamam
In contemporary terms, if Molkho and Ha-Reuveni represent political Zionism, marshaling international forces to create a Jewish nation-state, then Rabbi Yosef Karo and the Shekhina that spoke through him represent practical Zionism, such as that of the pioneer movement of the early 19th-century. Rather than using diplomacy to change the world, they sought to make a difference as individuals, believing that the small steps taken by committed pioneers could eventually make a great impact.
Though their Shavuot night vigil left an enormous impression on Rabbi Karo and his companions, not until three years later – after the death of his wife and three children in a plague – did Karo take ship with Shlomo Alkabetz and other rabbis and their families from Constantinople to the Land of Israel. In his farewell sermon in Salonika, Rabbi Alkabetz remarked:
When [the Talmudic sage] Rabbi Meir [lit. "Shining One"] used to look in his Torah scroll, he saw there that Adam, the first man, lost his raiment [of] supernal light. For after he sinned, he was cursed with the burden of physical matters, [represented by the] skin tunics [provided by God to clothe Adam and Eve]. Through the Torah, however, the original, spiritual raiment could be regained […], and [instead of] "skin garments" [as written in the traditional Torah text], read "garments of light." (quoted in Reuven Kimelman, The Mystical Significance of "Lekha Dodi" and Kabbalat Shabbat [Magnes Press, 2003], p. 138 [Hebrew])
Lekha Dodi was quickly adopted by almost all Jewish communities and included in the standard prayer book. Tomb of its author, Rabbi Shlomo Halevi Alkabetz, in Safed
Adam's raiment of skin represented the curse of physical existence, whereas Torah study revealed a spiritual reality – replacing the animal skins with garments of light. For Alkabetz, moving to the Holy Land meant shedding the burdensome skins of physical existence and living a more spiritual, enlightened life. Together with other mystics, including Rabbi Karo, Alkabetz contributed to the kabbalistic renaissance in Safed.
To welcome the Sabbath bride, these kabbalists dressed in white and ventured out to the fields every Friday at sunset. There they attempted to usher in redemption, raising the Shekhina from the dust.
For this ceremony, Rabbi Elkabetz composed the famous liturgical poem Lekha Dodi, whose refrain entreats, "Come, my beloved, to greet the bride." The Hebrew word orekh (your light) marks the exact midpoint of this hymn, with the middle stanza playing on the Hebrew term or (light), beginning with the letter alef, and its homonym, spelled with an ayin and meaning both "skin" and "awaken" (or literally, "shake off one's skin"):
Shake off your skin, shake off your sleep,
For your light has come. Arise, shine!
Awake, awake, declaim a song
The glory of God is revealed in you.
In this poem, light is clothed in garments of slumber. Alkabetz returns to the same metaphor used in his parting speech in Salonika, planting it firmly at the center of the ceremony welcoming the Sabbath. Coming full circle, the sons of Salonika shed the false skin of exile and revealed the vibrant, true colors of the Promised Land, greeting the Sabbath and redeeming the Shekhina.
The messianic cabal surrounding Rabbi Yosef Karo shifted from Salonika to Safed. Engraving of Safed from 1888
Mor Altshuler, "Prophecy and Maggidism in the Life and Writings of Rabbi Joseph Karo," Frankfurter Judaistische Beiträge 33 (2006), pp. 81–110
Haviva Pedaya, "Crisis and Repair, Trauma and Recovery," in Jewish Mysticism and the Spiritual Life, eds. Lawrence Fine, Eitan Fishbane, and Or N. Rose (Woodstock, Vt., 2011), pp. 171–82
Bilha Rubinstein, "A Terrible Fable and Enchanting Fiction: The Story of Joseph De-La Reina and Its Reflections in Two Novels of Yehoshua Bar Yosef," in With Both Feet on the Clouds: Fantasy and Israeli Culture, eds. Danielle Gurevitch, Elana Gomel, and Rani Graff (Academic Studies Press, 2012), pp. 248–262.
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Travel industry giant Sabre said Wednesday an intruder using stolen account credentials for its widely used reservations software had access to payment card details and personal information over a seven-month period. But it declined to say how many people are affected.
Sabre, which is based in Southlake, Texas, disclosed in early May a suspected breach affecting its SynXis Central Reservations system. The software-as-a-service system is used by travel agencies, hotels and booking services for such functions as rate and inventory management (see Sabre Warns Hotels: Card Data Potentially Compromised).
The exposure period started in August 2016 and ran through March. The information at risk includes payment cardholder names, card numbers and expiration dates, Sabre says.
For some reservations, the three-digit security code on the reverse of the card was exposed, but a "large percentage" of bookings were made without the code, the company says. Some bookings were made using virtual payment card numbers, it adds.
Guest names, phone numbers, addresses and other information were at risk, but not Social Security, driver's license or passport numbers, according to Sabre.
"Our investigation did not uncover forensic evidence that the unauthorized party removed any information from the system, but it is a possibility," Sabre says. In May, the company said FireEye's Mandiant investigations unit assisted with the investigation.
Sabre did not give a figure for how many payment cards or individuals were affected. Sabre spokesman Tim Enstice tells Information Security Media Group that "less than 15 percent of the average daily bookings" using the reservation system were viewed.
Enstice declined to answer how many daily bookings, on average, are made. But the SHS reservation system is used at 36,000 locations, from small hotels to large global chains, as well as for property management.
If each location only made one booking a day, the number of transactions would exceed 1 million in a month. At the bare minimum, 15 percent exposure would mean 150,000 transactions a month would be at risk.
Enstice disputed that estimate, saying it was "pure speculation." But Computerworld reported in August 2015 that Sabre's various software systems processes 2 billion transactions per day affecting 1 billion travelers a year.
Sabre says it has contacted travel management companies and travel agencies that do not use SHS reservations software, as well as those that do. "We have engaged Epiq Systems to provide complimentary notice support for those customers that determine they have a notification obligation," Sabre says.
The company also has created a website to notify consumers. It advised consumers to monitor account statements and report suspicious activity to financial institutions.
The breach is at least the second cybersecurity incident for Sabre in as many years.
In an Aug. 4, 2015, filing with the U.S. Securities and Exchange Commission, Sabre said it was investigating a "cybersecurity incident involving several servers managed by a third party."
Bloomberg reported a month later that investigators believed hackers linked with China attacked Sabre as well as American Airlines. The hacking group was suspected to be the same one that struck health insurer Anthem and the U.S. government's personnel office.
In February 2016, Sabre said it concluded its investigation, writing in its annual report that it found "no loss of traveler data, including no unauthorized access to or acquisition of sensitive protected information, such as payment card industry data or personally identifiable information in connection with this incident."
In February 2015, Anthem said the personal information of 78.8 million individuals was stolen, including names, dates of birth, Social Security numbers and healthcare identity numbers. Anthem has agreed to a proposal to settle a related class-action suit for $115 million, which a federal court will review next month (see Analyzing the Anthem Breach Class Action Settlement).
In one of the largest breaches to affect the U.S. government, the details of 4.2 million federal employees and up to 10 million former employers and contractors were stolen from the U.S. Office of Personnel Management (see Millions More Affected by OPM Breach).


The "Dirty Dozen" items - popular fruits and veggies ranked by pesticide contamination - are 100% organic at FCS.
Locally sourced items include milk (Lehigh Valley), which is also 100% antibiotic and hormone-free, eggs and yogurt (Common Market), produce (Ambrogi and Common Market), and grass-fed beef (Common Market).
Friends' Central has also applied for Green Restaurant Association Certification, and a weekly course in our new teaching kitchen has already begun, with Jacob Fogel's 8th graders learning about safe, sustainable cooking from Chef Bob and Chef Michael.
On Monday, October 1, the new Phoenix Café officially opened! The café, featuring after-school snacks and early dinner options like Naan bread pizza, sweet and savory crepes, beverages, and dessert, will open from 4-7 pm a few days a week based on school events. Click here for some highlights of the Phoenix Cafe!
Also, if you're a Middle/Upper School parent and you haven't already downloaded the "Bite" app for your phone, you're missing out! The app not only shows what the options are for the day and the week, at all five food stations, it also shows the nutrition facts for each item offered at both breakfast and lunch. Also, on screens throughout the dining hall, students and faculty are able to see both nutrition facts and where the food is coming from, with bios for the local farmers providing our food.
With these new offerings, Friends' Central is enjoying a truly innovative approach to learning about, and enjoying, healthy and delicious food.


text:
\section{Introduction and statement of results}
Let $\alpha$ be a fixed positive real number, and $\gamma$ run over the imaginary parts of the zeros of the Riemann zeta-function. We are interested in the distribution of the fractional parts $\{\alpha\gamma\}$.
 Rademacher \cite{radema} was the first to consider this problem and he conjectured that, for a certain specific type of $\alpha$,
there should be a "predominance of terms which fulfill $|\{\alpha\gamma\}-1/2|<1/4$".
Since the fractional parts are uniformly distributed modulo $1$, as proved by Hlawka \cite{hla} in 1975, any
discrepancy must be very subtle.
The first and third authors uncovered in \cite{fz} this delicate inequity in the fractional parts and
 not only proved Rademacher correct, but also gave a much more precise measure for this phenomenon.

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the torus. Since the fractional parts $\{\alpha\gamma\}$ are uniformly distributed (mod 1) for any fixed $\alpha$, for all continuous functions $h: \mathbb{T}\rightarrow\mathbb{C}$, we have
$$\sum_{0<\gamma\leq T}h(\alpha\gamma)=N(T)\int_{\mathbb{T}}h(u)du+o(N(T)),$$
as $T\rightarrow\infty$, where $N(T)$ denotes the number of zeros $0<\gamma\leq T$.  We know that (\cite{Titch}, Theorem 9.4)
$$N(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}-\frac{T}{2\pi}+O(\log T),\qquad(T\geq 1).$$

The first and third authors \cite{fz} showed that, for a large class of functions $h: \mathbb{T}\rightarrow\mathbb{C}$, as $T\rightarrow\infty$,
$$\frac{1}{T}\sum_{0<\gamma\leq T}h(\alpha\gamma)-\frac{N(T)}{T}\int_{\mathbb{T}} h(u)du=\int_{\mathbb{T}} h(u)g_{\alpha}(u)du+o(1),$$
where $g_{\alpha}(u)$ is a function depending on the form of $\alpha$. From this result, we can see that the right hand side is close to a constant, and thus the discrepancy of the set $\{h(\alpha\gamma): 0<\gamma\leq T\}$ is of order $O(\frac{1}{\log T})$.
In \cite{fsz}, the first author, Soundararajan, and the third author established connections between the discrepancy of this set, Montgomery's pair correlation function and the distribution of primes in short intervals.

In the present paper, our goal is to generalize the results from \cite{fz} to the case of simultaneous distribution of fractional parts $(\{\alpha_1\gamma\}, \{\alpha_2\gamma\}, \cdots, \{\alpha_n \gamma\})$, where $\alpha_1,\ldots,\alpha_n$ are distinct, positive real numbers.
As we will see below, a new phenomenon involving Diophantine approximation appears in the higher dimensional case $n\geq 2$.

For  $n\geq 2$, consider a function $h(\mathbf{x})$ defined on $\mathbb{T}^n$, which has Fourier expansion
$$h(\mathbf{x})=\sum_{\mathbf{m}\in \mathbb{Z}^n} c_{\mathbf{m}} e^{2\pi i \mathbf{m}\cdot \mathbf{x}},$$
and assume that
\begin{equation}\label{n-dim-condt}
|c_{\mathbf{m}}|\ll \left(\frac{1}{\|\mathbf{m}\|+1}\right)^B,
\end{equation}
for some constant $B>n+2$, and $\|\cdot\|$ is the sup-norm.

Let $\boldsymbol{\alpha}\in \mathbb{R}^n$.   Suppose that there is
a full-rank $r\times n$ ($r\leq n$) matrix $M=(b_{ij})$ with integer entries,
 $\gcd(b_{i, 1}, b_{i, 2}, \cdots, b_{i, n})=1$ for all $1\leq i\leq r$, and
 such that
\begin{equation}\label{n-dim-condition}
M\boldsymbol{\alpha}^{\top}=P,
\end{equation}
where  $P=(\frac{a_1}{q_1}\frac{ \log p_1}{2\pi }, \cdots, \frac{a_r }{q_r}\frac{\log p_r}{2\pi })^{\top}$ for some integers $a_i$, $q_i$ and distinct primes $p_i$ ($i=1, \cdots, r$).  Among all possible such matrices, we take
one with maximal $r$,  in which case the above conditions ensure that $M$
is uniquely determined.  We then define
\begin{eqnarray}\label{n-dim-g1}
g_{\boldsymbol{\alpha}}(\mathbf{x})&=&-\frac{1}{\pi}\sum_{j=1}^r (\log p_j) \Re\left\{ \sum_{k\geq 1} p_j^{-\frac{a_j k}{2}} e^{-2kq_j \pi i(\mathbf{b_j}\cdot \mathbf{x})}  \right\}\nonumber\\
&=&-\frac{1}{\pi}\sum_{j=1}^r \frac{(\log p_j)(p_j^{\frac{a_j}{2}}\cos(2\pi q_j(\mathbf{b_j}\cdot \mathbf{x})-1)}{p_j^{a_j}-2p_j^{\frac{a_j}{2}}\cos(2\pi q_j(\mathbf{b_j}\cdot\mathbf{x})+1},
\end{eqnarray}
where $\mathbf{b_j}=(b_{j, 1}, b_{j, 2}, \cdots, b_{j, n})$ for $1\leq j\leq r$.
If no such matrix exists, we simply set
\begin{equation}\label{n-dim-g2}
g_{\boldsymbol{\alpha}}(\mathbf{x})=0 \qquad (\mathbf{x} \in \mathbb{T}^n).
\end{equation}
Then we have the following.
\begin{theorem}\label{n-dim-aa}
	Let $n\geq 2$. Assume $h(\mathbf{x})$ satisfies condition (\ref{n-dim-condt}). Then,
	if $\boldsymbol{\alpha}\in \mathbb{R}^n$ satisfies
	\begin{equation}\label{n-dim-alpha}
	|\mathbf{m}\cdot\boldsymbol{\alpha}|>\frac{C}{e^{\|\mathbf{m}\|}}
	\end{equation}
	for all $\mathbf{m}\in \mathbb{Z}^n$ and some fixed constant $C$,
	we have
	$$\lim_{T\rightarrow\infty}\frac{1}{T}\left(\sum_{0<\gamma\leq T}h(\gamma\boldsymbol{\alpha})-N(T)\int_{\mathbb{T}^n} h\right)=\int_{\mathbb{T}^n} hg_{\boldsymbol{\alpha}}.$$
\end{theorem}
\noindent\textbf{Remark 1.} By a theorem of Khintchine \cite{khin} and Theorem 2 in \cite{kem}, the set of such $\boldsymbol{\alpha}$ has full Lebesgue measure.

\noindent\textbf{Remark 2.} It is possible to prove a similar conclusion
with a larger class of test functions h, using more complicated Fourier analysis techniques, along the lines of what was done in \cite{fz}, or by assuming the Riemann Hypothesis.

\vspace{1em}
For the special case when $r=n$ and $Rank(M)=n$, \eqref{n-dim-alpha} will
always be satisfied as a consequence of Baker's Theorem \cite{Baker}, and we have
the following.
\begin{coro} \label{n-dim-as}
	If $h(\mathbf{x})$ satisfies condition (\ref{n-dim-condt}) and $\boldsymbol{\alpha}$ satisfies condition (\ref{n-dim-condition}) for $r=n$, we have
	$$\lim_{T\rightarrow\infty}\frac{1}{T}\left(\sum_{0<\gamma\leq T}h(\gamma\boldsymbol{\alpha})-N(T)\int_{\mathbb{T}^n} h\right)=\int_{\mathbb{T}^n} hg_{\boldsymbol{\alpha}}.$$
\end{coro}

In general, the conclusion of Theorem \ref{n-dim-aa} may fail if the
Diophantine condition \eqref{n-dim-alpha} fails; for details see the Remarks at
the conclusion of Section \ref{section-2dim}.
For the case $n=2$, the behavior of $\sum h(\gamma \boldsymbol{\alpha})$ is determined
by the behavior of the convergents  $\frac{p_n}{q_n}$ of the continued fraction of $\xi:=\frac{\alpha_1}{\alpha_2}$.  Even if \eqref{n-dim-alpha} fails, which occurs when some $q_{n+1}$ are very large compared to $q_n$, we show that
 there are ``long'' intervals of $T$-values on which we may draw the conclusion of the
previous theorem.

\begin{theorem} \label{2-dim-range}
	Assume \eqref{n-dim-condt} and fix any $\epsilon>0$.  Let $U_{\boldsymbol{\alpha}}=\bigcup_{n=1}^{\infty}[q_n^{1+\epsilon}, e^{q_n^{B-\epsilon}}]$. Then,
	\begin{equation*}
	\lim_{\substack{T\in U_{\boldsymbol{\alpha}}\\ T\rightarrow\infty}}\frac{1}{T}\left(\sum_{0<\gamma\leq T}h(\alpha_1 \gamma, \alpha_2 \gamma)-N(T)\int_{\mathbb{T}^2} h\right)=\int_{\mathbb{T}^2}  hg_{\boldsymbol{\alpha}}.
	\end{equation*}
\end{theorem}

Finally, we show
 some graphs comparing the limiting density function $g_{\boldsymbol{\alpha}}$ with numerical computations obtained from the first 100 million  zeros of the Riemann zeta-function (which were kindly provided by Tom\'as Oliveira e Silva, see \cite{Si}).
Define
\begin{equation}\
	M(y_1, y_2; T)=M_{\boldsymbol{\alpha}}(y_1, y_2; T):=\frac{1}{T}\sum_{\substack{0<\gamma\leq T\\ \{\alpha_1\gamma\}<y_1, \{\alpha_2\gamma\}<y_2}} 1-y_1 y_2\frac{N(T)}{T}.\nonumber
\end{equation}

Take $T=42653549.761$, then $N(T)=10^8$. Denote $\Delta=\frac{1}{100}$. We partition $[0, 1)\times[0, 1)$ into $\Delta^{-2}$ small rectangles with side length $\Delta$. For some values of $\alpha_1$, $\alpha_2$, we plot the values of $DM:=\frac{1}{\Delta^2}(M(y_1+\Delta, y_2+\Delta)-M(y_1+\Delta, y_2)-M(y_1, y_2+\Delta)+M(y_1, y_2))$ for each small rectangle $[y_1, y_1+\Delta)\times[y_2, y_2+\Delta)$.

\textbf{Example 1.} Let $\alpha_1$ and $\alpha_2$ satisfy
\begin{align*}
\alpha_1+\alpha_2& =\dfrac{\log 2}{2\pi},\\
\alpha_1-\alpha_2& =\dfrac{\log 3}{4\pi}.
  \end{align*}
Figure $\ref{g12}$ shows the graph of $g_{\boldsymbol{\alpha}}$. Figure $\ref{22}$ shows the values of $DM$, the graph of which is a close approximation of  $g_{\boldsymbol{\alpha}}$.

\textbf{Example 2.} Figure $\ref{23}$ and Figure $\ref{24}$ show the analogous graphs for the case when $\alpha_1$ and $\alpha_2$ satisfy
\begin{align*}
  2\alpha_1+\alpha_2&=\dfrac{\log 5}{2\pi},\\
  2\alpha_1+3\alpha_2&=\dfrac{\log 7}{2\pi}.

\end{align*}

\begin{figure}[ht!]
  \centering

  \includegraphics[width=14cm]{f1-cc.jpg}\\
  \caption{Graph of $g_{\boldsymbol{\alpha}}$}\label{g12}
\end{figure}
\FloatBarrier

\begin{figure}[ht!]
  \centering

  \includegraphics[width=14cm]{e1-100o.jpg}\\
  \caption{Values of $DM$}\label{22}
\end{figure}
\FloatBarrier

\begin{figure}[ht!]
  \centering

  \includegraphics[width=14cm]{f2-cc.jpg}\\
  \caption{Graph of $g_{\boldsymbol{\alpha}}$}\label{23}
\end{figure}
\FloatBarrier

\begin{figure}[ht!]
  \centering

  \includegraphics[width=14cm]{e2-100c1-2.jpg}\\
  \caption{Values of $DM$}\label{24}
\end{figure}
\FloatBarrier

\section{Proof of Theorem \ref{n-dim-aa}}

We need Lemma 1 from \cite{fz}, which is an extension of a famous formula of Landau.  Let $\rho$ denote a generic nontrivial zero of the Riemann zeta-function, and denote $\gamma=\Im \rho$.
\begin{lem}\label{lem-fz}
	Let $x, T>1$, and denote by $n_x$ the nearest prime power to $x$. Then
	$$\sum_{0<\gamma\leq T} x^{\rho}=-\frac{\Lambda(n_x)}{2\pi}\frac{e^{iT\log(x/n_x)}-1}{i\log(x/n_x)}+O\left(x\log^2(2xT)
	+\frac{\log 2T}{\log x}\right),$$
where if $x=n_x$ the first term is $-T\frac{\Lambda(n_x)}{2\pi}$.
\end{lem}

Next, we extract from the proof of (3.8) in \cite{fz} the following
estimate for difference between two sums over zeros, one with the
real parts all normalized to be equal to 1/2.

\begin{lem}\label{lem-ford2}
	For $1<x\le \exp\{\frac{\log T}{50\log\log T} \}$, we have
	\begin{equation}\label{fz-relat}
	\sum_{0<\gamma\leq T}(x^{i\gamma}-x^{\rho-1/2})\ll \frac{T\log^2 x}{\log T}+\frac{T}{\log^{10} T}.
	\end{equation}
\end{lem}

\noindent \textbf{\textit{Proof of Theorem \ref{n-dim-aa}.}}

By Theorem 2 in \cite{kem}, we know that the set of $\boldsymbol{\alpha}$'s under the condition (\ref{n-dim-alpha}) has full Lebesgue measure.

Write the Fourier expansion of $h(\mathbf{x})$ as
\begin{equation}\label{hidim-hfunc}
h(\mathbf{x})=\sum_{\mathbf{m}}c_{\mathbf{m}} e^{2\pi i(\mathbf{m}\cdot \boldsymbol{\alpha})}= \sum_{\|\mathbf{m}\|\leq J}c_{\mathbf{m}} e^{2\pi i (\mathbf{m}\cdot \boldsymbol{\alpha})}+O\left(\sum_{\|\mathbf{m}\|>J}|c_{\mathbf{m}}| \right),
\end{equation}
where $J\leq \log T$ is a parameter to be chosen later.

By our assumption (\ref{n-dim-condt}), one can see that the error term above is $O(1/J^{B-n})$. Thus, by (\ref{hidim-hfunc}), we have
\begin{align*}
\sum_{0<\gamma\leq T} h(\gamma\boldsymbol{\alpha})&=\sum_{0<\gamma\leq T} \left( \sum_{\|\mathbf{m}\|\leq J}c_{\mathbf{m}} e^{2\pi i \gamma (\mathbf{m}\cdot \boldsymbol{\alpha})}\right)+O\left(\frac{N(T)}{J^{B-n}}\right)\\
&=N(T)\int h+2\Re \sum_{\substack{0<\|\mathbf{m}\|\leq J\\ \mathbf{m}\cdot \boldsymbol{\alpha}>0}}c_{\mathbf{m}}\sum_{0<\gamma\leq T}e^{2\pi i \gamma (\mathbf{m}\cdot \boldsymbol{\alpha})}+O\left(\frac{N(T)}{J^{B-n}}\right).
\end{align*}
The second equality is a consequence of the identity $c_{-\mathbf{m}}=\overline{c_{\mathbf{m}}}$. Let $x_{\mathbf{m}}=e^{2\pi (\mathbf{m}\cdot \boldsymbol{\alpha})}$. Hence, by \eqref{n-dim-condt} and Lemma \ref{lem-ford2}, we have for $T$ large and $J\ll \frac{\log T}{(\log\log T)^2}$ that
\begin{equation}\label{hidim-h}
\sum_{0<\gamma\leq T} h(\gamma\boldsymbol{\alpha})-N(T)\int h=2\Re \sum_{\substack{0<\|\mathbf{m}\|\leq J\\ \mathbf{m}\cdot \boldsymbol{\alpha}>0}}c_{\mathbf{m}}\sum_{0<\gamma\leq T}x_{\mathbf{m}}^{ \rho-1/2}+O\left(\frac{N(T)}{J^{B-n}}+\frac{T}{\log T}\right).
\end{equation}
Write
$$I=\sum_{\substack{0<\|\mathbf{m}\|\leq J\\ \mathbf{m}\cdot \boldsymbol{\alpha}>0}}c_{\mathbf{m}}\sum_{0<\gamma\leq T}x_{\mathbf{m}}^{ \rho-1/2}.$$
Applying Lemma \ref{lem-fz}, we get
\begin{eqnarray}\label{n-dim-lem}
I&=&-\frac{1}{2\pi}\sum_{\substack{\mathbf{m}\cdot\boldsymbol{\alpha}>0\\ 0<\|\mathbf{m}\|\leq J}}\frac{c_{\mathbf{m}}\Lambda(n_{x_{\mathbf{m}}})}{\sqrt{x_{\mathbf{m}}}}   \frac{\sin(T\log{\frac{x_{\mathbf{m}}}{n_{x_{\mathbf{m}}}}})}{\log{\frac{x_{\mathbf{m}}}{n_{x_{\mathbf{m}}}}}}      \nonumber\\
&& + O\left(\sum_{\substack{\mathbf{m}\cdot\boldsymbol{\alpha}>0\\ 0<\|\mathbf{m}\|\leq J}} |c_{\mathbf{m}}| \left(\sqrt{x_{\mathbf{m}}}\log^2(2x_{\mathbf{m}}T)+\frac{\log 2T}{\sqrt{x_{\mathbf{m}}}\log x_{\mathbf{m}}}  \right) \right)\nonumber\\
&=&-\frac{1}{2\pi}\sum_{\substack{\mathbf{m}\cdot\boldsymbol{\alpha}>0\\ 0<\|\mathbf{m}\|\leq J}}\frac{c_{\mathbf{m}}\Lambda(n_{x_{\mathbf{m}}})}{\sqrt{x_{\mathbf{m}}}}   \frac{\sin(T\log{\frac{x_{\mathbf{m}}}{n_{x_{\mathbf{m}}}}})}{\log{\frac{x_{\mathbf{m}}}{n_{x_{\mathbf{m}}}}}} +O\left(e^{O(J)}\log^2 T\right).
\end{eqnarray}
Here we used our assumption (\ref{n-dim-alpha}) that
$$\log x_{\mathbf{m}}=2\pi (\mathbf{m}\cdot\boldsymbol{\alpha}) \geq \frac{2\pi C}{e^{J}}.$$

Now let $J=\sqrt{\log T}$. By (\ref{hidim-h}), (\ref{n-dim-lem}) and $B>n+2$, we get
\begin{equation}\label{hidim-t}
\sum_{0<\gamma\leq T} h(\gamma\boldsymbol{\alpha})-N(T)\int h =-\frac{T}{\pi}\Re\sum_{\substack{\mathbf{m}\cdot\boldsymbol{\alpha}>0\\ 0<\|\mathbf{m}\|\leq J}}\frac{c_{\mathbf{m}}\Lambda(n_{x_{\mathbf{m}}})}{\sqrt{x_{\mathbf{m}}}}   \frac{\sin(T\log{\frac{x_{\mathbf{m}}}{n_{x_{\mathbf{m}}}}})}{T\log{\frac{x_{\mathbf{m}}}{n_{x_{\mathbf{m}}}}}}+o(T).
\end{equation}
Since $\frac{\Lambda(n_{x_{\mathbf{m}}})}{\sqrt{x_{\mathbf{m}}}}$ is bounded, the first sum is absolutely and uniformly convergent in $T$ under the assumption $(\ref{n-dim-condt})$. And the terms with $n_{x_{\mathbf{m}}}\neq x_{\mathbf{m}}$ tend to zero as $T\rightarrow\infty$. Thus, by (\ref{hidim-t}), as $T\rightarrow\infty$,
\begin{equation}\label{n-dim-conclu}
\frac{1}{T}\left( \sum_{0<\gamma\leq T} h(\gamma\boldsymbol{\alpha})-N(T)\int h\right)
=-\frac{1}{\pi}\Re \left\{\sum_{\substack{\mathbf{m}\cdot\boldsymbol{\alpha}>0\\0<\|\mathbf{m}\|\leq J}}\frac{c_{\mathbf{m}}\Lambda({x_{\mathbf{m}}})}{\sqrt{x_{\mathbf{m}}}}         \right\}+o(1).
\end{equation}
Since $\Lambda({x_{\mathbf{m}}})=0$ unless $x_{\mathbf{m}}$ is prime power, by (\ref{n-dim-g2}), when $g_{\boldsymbol{\alpha}}=0$, the sum in the right hand side of (\ref{n-dim-conclu}) is also $0$. If $\boldsymbol{\alpha}$ satisfies (\ref{n-dim-condition}) with $M$ being maximal (maximal $r$), then the only terms with
$n_{x_{\mathbf{m}}} =  x_{\mathbf{m}}$ (that is, with $x_{\mathbf{m}}$ being a prime power)
are those with $\mathbf{m}$ being a multiple of some row of $M$.
Hence, recalling the definition \eqref{n-dim-g1} of $g_{\boldsymbol{\alpha}}$, as
$T\rightarrow \infty$, the right hand side of (\ref{n-dim-conclu}) becomes
$$-\frac{1}{\pi}\Re\left\{\sum_{j=1}^r \sum_{k\geq 1}(\log p_j) p_j^{-\frac{a_j k}{2}} c_{k{q}_j \boldsymbol{m}}\right\},$$
which equals $\int_{\mathbb{T}^n} hg_{\boldsymbol{\alpha}}$ by (\ref{n-dim-g1}).
So in all cases, we have
$$\lim_{T\rightarrow\infty}\frac{1}{T}\left(\sum_{0<\gamma\leq T}h(\gamma\boldsymbol{\alpha})-N(T)\int_{\mathbb{T}^n} h\right)=\int_{\mathbb{T}^n} hg_{\boldsymbol{\alpha}}.$$

Therefore,  we get the conclusion of this theorem.

\vspace{1em}

\noindent\textbf{\textit{Proof of Corollary \ref{n-dim-as}.}}

For the special form of $\boldsymbol{\alpha}$ when $r=n$,  we can solve the above linear equations for $\boldsymbol{\alpha}$, and find that each $2\pi\alpha_i$ is a linear combination of these $\log p_i$'s with rational coefficients. Hence, by Baker's theorem \cite{Baker}, for such $\boldsymbol{\alpha}$, there exist constants  $D=D(\boldsymbol{\alpha})$  and $\mu$, such that, for all $\mathbf{m}\in \mathbb{Z}^n$,
\begin{equation}\label{n-dim-Baker}
|\mathbf{m}\cdot \boldsymbol{\alpha}|\geq \frac{D}{(\|\mathbf{m}\|+1)^\mu}.
\end{equation}
Thus, such $\boldsymbol{\alpha}$ satisfies condition (\ref{n-dim-alpha}) for some constant $C$.
The conclusion follows from Theorem \ref{n-dim-aa}.

\section{Proof of Theorem \ref{2-dim-range}}\label{section-2dim}
We may assume that $\alpha_1/\alpha_2$ is positive and irrational, else the problem reduces to the $n=1$ case.

Assume a function $h(x,y)$ is defined on the two dimensional torus. By (\ref{hidim-h}), we have
\begin{equation}\label{star-h2}
 \sum_{0<\gamma\leq T}h(\alpha_1 \gamma, \alpha_2 \gamma)-N(T)\int h = 2\Re \sum_{\substack{|m|,|l|\le J \\ m\alpha_1+l\alpha_2>0}} c_{m,l}
 \sum_{0<\gamma\leq T} x_{m,l}^{\rho-1/2} + O\left(\frac{N(T)}{J^{B-2}}+\frac{T}{\log T}\right),
\end{equation}
where $J\leq \log T$ is a parameter to be chosen later.

We now break the double sum into two pieces, over those pairs $(m,l)$ corresponding to ``large'' $x_{m,l}$ and those
pairs $(m,l)$ corresponding to ``small'' $x_{m,l}$.
Fix an arbitrary positive constant $C$ and denote
\begin{align*}
  E_J&:=\left\{(m, l)\in \mathbb{Z}^2:0<\max\{|m|, |l|\}\leq J, m\alpha_1+l\alpha_2>\min\(\frac{C}{e^{\max\{|m|, |l|\}}}, \frac{|\alpha_2|}{2m}, \frac1{4\pi}\)\right\}, \\
  F_J&:=\left\{(m, l)\in \mathbb{Z}^2:0<\max\{|m|, |l|\}\leq J, 0 < m\alpha_1+l\alpha_2 \le \min\(\frac{C}{e^{\max\{|m|, |l|\}}}, \frac{|\alpha_2|}{2m}, \frac1{4\pi}\)\right\}.
\end{align*}

Write
\[
I_E=\sum_{(m,l)\in E_J}c_{m,l}\sum_{0<\gamma\leq T} x_{m,l}^{\rho-1/2}, \qquad
I_F=\sum_{(m,l)\in F_J}c_{m,l}\sum_{0<\gamma\leq T} x_{m,l}^{\rho-1/2}.
\]
Applying Lemma \ref{lem-fz} and using $B>4$,  we get
\begin{eqnarray}\label{est-I1}
  I_E&=&-\frac{1}{2\pi} \sum_{(m,l)\in E_J}\frac{c_{m,l}\Lambda(n_{x_{m,l}})}{\sqrt{x_{m,l}}}\frac{\sin(T\log\frac{x_{m,l}}{n_{x_{m,l}}})}{\log\frac{x_{m,l}}{n_{x_{m,l}}}}\nonumber\\
  &&+O\left(\sum_{(m, l)\in E_J}|c_{m,l}|\left(\sqrt{x_{m,l}}\log^2(2x_{m,l}T)+\frac{\log 2T}{\sqrt{x_{m,l}}\log x_{m,l}}\right)\right)\nonumber\\
  &=&-\frac{1}{2\pi} \sum_{(m,l)\in E_J}\frac{c_{m,l}\Lambda(n_{x_{m,l}})}{\sqrt{x_{m,l}}}\frac{\sin(T\log\frac{x_{m,l}}{n_{x_{m,l}}})}{\log\frac{x_{m,l}}{n_{x_{m,l}}}}
  +O\left(e^{O(J)}\log^2 T\right).
\end{eqnarray}
Here we have used the fact that for $(m,\ell)\in E_J$,
\[
 \log x_{m,\ell} = 2\pi (m\alpha_1+l\alpha_2) \gg e^{-J}.
\]

By standard facts on Diophantine approximation, we know that if $(m,l)\in F_J$ then $$\left| \frac{\alpha_1}{\alpha_2}+\frac{l}{m}\right| < \frac{1}{2m^2},$$
and hence all elements of $F_J$ are of the form $(q_n, -p_n)$, where $p_n/q_n$ is a convergent
of the continued fraction of $\frac{\alpha_1}{\alpha_2}$.
 As $q_n$ grows at least exponentially and $q_n\le J$ for $(q_n,-p_n)\in F_J$,
there are $O(\log J)=O(\log\log T)$ elements in $F_J$.  Let $n^*$ denote the largest $n$
such that $(q_n,-p_n)\in F_J$.

By elementary properties of  continued fractions (Chapter X, \cite{hw}, or Corollary 1.4, \cite{bu}), we have
\begin{equation}\label{cont-fracp}
  \frac{\alpha_2}{q_j+q_{j+1}}<|q_j \alpha_1-p_j \alpha_2|<\frac{\alpha_2}{q_{j+1}}.
\end{equation}
Thus, for all $j$ except possibly $j=n^*$, we have $q_{j+1} \le J$ and
from Lemma \ref{lem-fz} we get (writing $x_j=x_{q_j,-p_j}$)
\begin{align}\label{est-I2}
\sum_{j\ne n^*}c_{q_j,-p_j}\sum_{0<\gamma\leq T} x_j^{\rho-1/2}&=-\frac{1}{2\pi}
\sum_{j\ne n^*}\frac{c_{q_j,-p_j}\Lambda(n_{x_j})}{\sqrt{x_j}}\frac{\sin(T\log\frac{x_j}{n_{x_j}})}{\log\frac{x_j}{n_{x_j}}}
+O(\log^3 T) \nonumber\\
&=O(\log^3 T),
\end{align}
where we have used that $n_{x_j}=2$. We adopt the convention that the term $j=n^*$ is included in the sum above
if $q_{n^*+1} \le J$.
In fact, the error term from Lemma \ref{lem-fz} is acceptable, namely $o(T)$, if $q_{n^*+1}=o(T/\log T)$ as well.

Now let $J=\sqrt{\log T}$ and assume that $(\log T)^{1/B}\log\log T< q_{n^*} \le J$.
We revert to the sum over $x_{q_{n^*},-p_{n^*}}^{i\gamma}$.  By \eqref{fz-relat} and the the trivial
bound for $\sum_{0 <\gamma\le T} x_{q_{n^*},-p_{n^*}}^{i\gamma}$, we have
$$c_{q_{n^*}, -p_{n^*}}\sum_{0<\gamma\leq T} x_{q_{n^*},-p_{n^*}}^{\rho-1/2} \ll
\frac{T\log T}{q_{n^*}^B} =o(T).$$
Combined with \eqref{est-I1} and \eqref{est-I2}, we see that if (a) $F_J$ is empty or if (b)
there is some convergent of $\alpha_1/\alpha_2$ with
$q_n \in ((\log T)^{1/B}\log\log T, o(T/\log T)]$, then
\begin{eqnarray}\label{star-I}
  \sum_{0<\gamma\leq T}h(\alpha_1 \gamma, \alpha_2 \gamma)&=&-\frac{T}{\pi}\Re \sum_{\substack{m\alpha_1+l\alpha_2>0\\0<\max\{|m|, |l|\}\leq J}}\frac{c_{m,l}\Lambda(n_{x_{m,l}})}{\sqrt{x_{m,l}}}\frac{\sin(T\log\frac{x_{m,l}}{n_{x_{m,l}}})}{T\log\frac{x_{m,l}}{n_{x_{m,l}}}}\nonumber\\
  &&+N(T) \int h + o(T).
\end{eqnarray}
Note that (b) is satisfied  if $T$ is large enough and
$T\in \bigcup_{n=1}^{\infty}[q_n^{1+\epsilon}, e^{q_n^{B-\epsilon}}]$.

Then, similar to the proof of (\ref{n-dim-conclu}) in the proof of Theorem \ref{n-dim-aa}, we deduce that
\begin{equation*}
  \lim_{\substack{T\in U_{\boldsymbol{\alpha}}\\ T\rightarrow\infty}}\frac{1}{T}\left(\sum_{0<\gamma\leq T}h(\alpha_1 \gamma, \alpha_2 \gamma)-N(T)\int h\right)=\int hg_{\boldsymbol{\alpha}}.
\end{equation*}
Thus, Theorem \ref{2-dim-range} follows.

\medskip

\textbf{Remarks.}  Suppose that there is a very long gap between convergents,
say $q_{n^*}$ is small and $q_{n^*+1}$ is very large.  Put $x=x_{q_{n^*},-p_{n^*}}$ and assume that
$\log x \ll \frac{\log T}{T}$.  Using the known estimate $N(T)=\frac{T}{2\pi}\log(\frac{T}{2\pi e})+O(\log T)$,
we get
\begin{align*}
 \sum_{0<\gamma\le T} x^{i\gamma} &=\int_0^T x^{it} \log \left(\frac{t}{2\pi}\right)\, dt + O(\log^2 T)\\
 &=T\log(T/2\pi) \frac{e^{iT\log x}-1}{iT\log x} + O(T).
\end{align*}
In the range $T\ll 1/\log x$, there is thus a term in the sum \eqref{star-h2} of order $N(T)|c_{q_{n^*},-p_{n^*}}|$,
which may be larger than $T$.

\vspace{1em}
\textbf{Acknowledgement.} Research of the first author and second author is partially
supported by NSF grants DMS-1201442 and DMS-1501982. The authors would like to thank the helpful comments of the referee.

redpajama_set_name: RedPajamaArXiv


text:
Home / Historical / TODAY IN KIMBERLEY'S HISTORY 10 JUNE
TODAY IN KIMBERLEY'S HISTORY 10 JUNE
Posted by: Steve Lunderstedt in Historical, Today in Kimberley's History June 10, 2020
10 JUNE 1964, Melanie Karam of the Convent awarded the City Council Medal.
10 JUNE 1964, Boys High beat CBC 24-13 at the KAC grounds.
10 JUNE 1975, Fire nearly destroys Mulder's Shoe Bar.
The Mulder's Shoe Bar fire
The fire on Tuesday 10 June 1975 at Mulder's Shoe Bar on Old Main Road caused over R40 000 worth of damage.
The fire, which broke out at 5.30pm destroyed the stock in the women's fashion parlour but left the goods in the adjoining shoe stores undamaged.
Mr WH Mulder, the managing director of the shop believed the fire had been started by an electrical fault or the air conditioner in a back office, his view being supported by Kimberley's fire chief, Mr WS Kruger.
Firemen had to break in through a side door and window to get at the flames, and with the help of the SAP and the Kimberley Commando they had the fire under control within 30 minutes.
Shoes, stockings, handbags and tins of show polish were destroyed by flames, heat or water and duplicate books of the store's records were charred.
Hundreds of people crowded around as smoke poured out the shop and Old Main Road was closed to vehicular traffic.
"It has taken me 11 years to build up this shop and it has been ruined in one night," Mr Mulder stated.
Three short stories.
Firstly, Miss Melanie Karam of the Convent High School, daughter of Mr and Mrs Edward Karam of Douglas, has been awarded the City Council Medal for the highest marks in Kimberley schools, in the Junior Certificate (Standard 8) examination. She would be presented with the medal in the Convent School Hall on 11 June 1964.
Secondly, Boys' High School beat Christian Brothers' College 24-13 at the KAC grounds before a large crowd on 10 June 1964 in a fast and entertaining match.
Though there was not as much of the snappy hand-to-hand passing and backing up as clashes between these two schools usually produce, there was never a dull moment with a lot of hard running, particularly by the High School backs and a lot of excellent tackling, in which Drier, the College centre did well.
For the winners, Derek Sterley at flyhalf overcame an erratic service and Smit ran beautifully on the wing.
Both packs of forwards maintained a terrific pace throughout the game with High School's the stronger combination.
Tries for Boys' High were scored by winger Smit (2), Clements, Gordon, (Unknown) and Reg Billett while Smit converted three of the tries. (A try was 3 points and a conversion 2 points). CBC replied with tries by McNamara and Brokenshire, both of which were converted by Doherty. Doherty also added a penalty.
Thirdly, the fire on Tuesday 10 June 1975 at Mulder's Shoe Bar on Old Main Road caused over R40 000 worth of damage.
Nothing has yet been found for this day in Kimberley's history. Assuredly something did happen….research continues.
Rosemary Weir was born in Kimberley on 22 July 1905, the daughter of George Alfred Lawrence Green (the then editor of the Diamond Fields Advertiser) and his wife Katherine (nee Bell). The family left Kimberley in 1910 when her father was transferred to the Cape as editor of the Cape Times. She was then aged five years. Her brother is the well-known Lawrence Green, famous in South Africa for his series of entertaining travel books.
One of Rosemary Weir's Books
Another one of Rosemary Weir's Books
She was educated privately in South Africa and at schools in England where she trained for the stage, becoming a partner in a touring theatrical company, playing one-night stands in rural districts of England and Wales for three years. Later she turned to farming in North Wales and Devonshire and taught elocution and sewing in a school, ran a furniture renovating business, and worked as a domestic and cook, both privately and in hotels and lodges. During the Second World War she served with the First Aid Nursing Yeomanry before being invalided out.
She did not start writing until she was already over 50 but produced a respectable number of books over the next twenty-five years, later claiming, "I remembered my own childhood very clearly and perhaps my books for children were a way of returning to a happy time in my life. On the whole I have enjoyed writing for younger children, The Smallest Dog in the World being my own favourite." Apart from her writing for children, she also wrote for the BBC, adapting The Secret Journey and penning a series entitled Strange Tales.
Margaret Greaves, in Twentieth Century Children's Writers, discusses a number of her books, and sums up her entry by saying that it is her "books for older readers that show Rosemary Weir as an original and perceptive writer. The characterization is strong and clear, including satisfactory and fruitful relationships with adults as well as children. She stimulates interest by developing an unusual situation, but through it she explores the emotional experiences common to most children. She is concerned with the problems of growing up, but she makes them the material of art and not of social therapy."
She was married to Napier Weir in 1931 (they later separated before his death in 1973), and had one daughter, Alison. She lived in Somerset, England, where she died in 1994, aged 89 years.
Pictured are two of her books.
From Kimberley Calls and Recalls on Facebook By Steve Lunderstedt
Shortlink for this post: https://www.kimberley.org.za/?p=9326
BBC Boys' High School Cape Times Christian Brothers College City Council Medal Convent High School Derek Sterley Edward Karam First Aid Nursing Yeomanry George Alfred Lawrence Green Kimberley Commando Kimberley Streets Margaret Greaves Melanie Karam Mulder's Shoe Bar Napier Weir Reg Billett Rosemary Weir Steve Lunderstedt Strange Tales The Secret Journey WH Mulder World War 2 WS Kruger	2020-06-10
Steve Lunderstedt
Tagged with: BBC Boys' High School Cape Times Christian Brothers College City Council Medal Convent High School Derek Sterley Edward Karam First Aid Nursing Yeomanry George Alfred Lawrence Green Kimberley Commando Kimberley Streets Margaret Greaves Melanie Karam Mulder's Shoe Bar Napier Weir Reg Billett Rosemary Weir Steve Lunderstedt Strange Tales The Secret Journey WH Mulder World War 2 WS Kruger
About Steve Lunderstedt
TODAY IN KIMBERLEY'S HISTORY – 15 JANUARY
UPDATED: 12/01/2021 12 January 1915, The South African forces move in strength ...
UPDATED: 11/01/2021 11 January 1905, Two books signed by Florence Nightingale donated ...


This essay reviews the recent book by Michael Storper et al The Rise and Fall of Urban Economies: Lessons from San Francisco and Los Angeles. The book uses the case studies of San Francisco and Los Angeles to reflect on how economic growth in many cities has diverged over time and seeks to explain why this may be. The same can be seen in a UK context, so understanding what takes place and why in different urban economic contexts has wide relevance and importance, particularly as it provokes questions about theory and method in urban and regional research.
This working paper explores how successful Storper's book is in explaining the phenomena of divergent urban growth, and what further questions it raises. The review was originally published in Regional Studies, vol. 50, no. 9, pp. 1623-1627.