Good morning. If I could please have everyone silence their electronic devices So good morning and welcome to our Memorial Day ceremony. I'm Jeff Chung load director of veteran services for the town of Arlington And I'd like to introduce our official party father mark Bishop Mr. Sandy puller town manager Mr. Eric Helmuth chair of the select board state representative Rogers state representative garbly And our keynote speaker major general retired William rep would you please rise Members of the Arlington police department and Arlington fire department will parade the colors Parade the colors the members of the Boston skyline course will now sing our national anthem Father mark Bishop is a Navy veteran and pastor at St. Camelus parish Arlington Belmont and st. Agnes parish in Arlington You will now deliver our invocation Thank you for asking me to pray within my tradition today, and I invite you to pray within your tradition along with me On this Memorial Day, we gather and humble gratitude for all the sacrifices of the men and women of the armed services And that they have endured to ensure our freedoms Those that have died deserve our gratitude and honor Please hold our servicemen and women in your strong arms dear God Cover them with your sheltering grace and your presence as they stand in the gap for our protection We also remember the families of our troops especially Gold star families we ask for your unique blessing to fill their homes, and we pray for your peace provision hope and Strength to fill their lives May the members of our armed services be supplied with the courage to face each day And may they trust in Lord's mighty power to accomplish each task Let our military brothers and sisters Feel our love and support Lord of all nations may we take time to reflect on the great blessings We share as a nation and as a people our blessings have come at a high cost to others May we remember these sacrifices always with deep gratitude We ask that you grant wisdom to the leaders of the armed forces Guide and direct them in their decisions May they be led by your will and your heart as they pursue our nation's freedom We continue to pray for peace in our world Lord, let your kingdom come on earth as it is in heaven. Amen Please be seated This year marks the 50th anniversary of the release of US service members who were in prison during the Vietnam War Members of the armed forces were held as prisoners of war in significant numbers from 1964 to 1973 13 prisons and prison camps were used to house US prisoners in North Vietnam as part of Operation Homecoming on February 12th 1973 The first of 591 US prisoners began to be repatriated and additional flights continued until late March On March 26 1964 the first US service member was in prison Captain Floyd Thompson's plane was brought down by enemy enemy fire and he spent nearly nine years in captivity Making him the longest-held POW in American history US prisoners of war in North Vietnam were subjected to extreme torture and malnutrition during their captivity The aim of the torture was typically not to acquire military information But instead it was to break the will of the prisoners both individually and as a group So they could obtain written or recorded statements from prisoners that criticized America and praised how North Vietnamese treated them During the war only 37 service members escaped captivity Following Operation Homecoming a total of 684 POWs returned home Currently there are still 1582 veterans that remain classified as unaccounted for and there remains a hundred 1056 veterans that have been identified and repatriated at this time. I'd like to call on our town manager Sandy pooler to offer his remarks Thank You Jeff. Thank you all for being here Arlington You're going to hear some remarks from me because I was a history major. So I'm going to talk a little bit about history Arlington has a special connection with this Memorial Day as many of you know the town in 1807 joined with Belmont to become anodomy But in 1867 it changed its name to Arlington in honor of those buried at Arlington National Cemetery. I Brought today this sword, which was my great great-grandfathers he served in The Civil War this sword hangs in my uncle's house. My great-grandfather was a surgeon first in the mass voluntary militia and Then in the National Army he had to perform surgery at Antietam and Spent many many years traveling throughout the South on and the Union Army obviously Serving our troops I've always been very proud of this sword and of the service that he gave to our country to help keep this country together People sometimes talk about who won the Civil War I often think it's the United States that won the Civil War because we stayed together as a unified country Memorial Day started Not really by any official act by a number of citizens simply placing flowers on the graves of those who died in the war and eventually by 1867 by a proclamation by US General asking people to remember those who had died They died to keep our country together they died To give us the freedom that we have today and our ability to come together and honor them And so today I join you in the quiet contemplation that we have for those who have passed before us in the taps that we play for them in our public procession to lay a wreath and Marching together to formally honor those who gave their lives for their country So I join with you who are veterans and thank you for your service I joined with you who have family members who are veterans and I joined with all the families in the United States At one point have lost a family member to a war in this country and we all Honor them and hope for peace in the future I just also want to say and not part of my official remarks and Jeff won't like this But I want to thank Jeff too for all the work that he has done for this town for helping bring together This ceremony for honoring the veterans in this town and for bringing a sense of peace and recognition To to many so thank you Jeff You're right Sandy. I didn't like that At this time I'd like to invite our select board chair mr. Eric helmets offer his remarks Good morning. It's my privilege to be here to represent my colleagues on the select board at this important and somber occasion I'd like to recognize Diane Mahon John Hurd Steve DeCorsi and Len Diggins who are here with me this morning, and I'm proud to see you all here I'm also proud to welcome The veterans who proudly serve in the Arlington police and fire departments. It's good to see you here this morning finally I Extend my appreciation to our guest speaker general wrap who is a four-year member of our town and I turned to him this morning And I said welcome home, sir. It's good to have you finally, I must also thank Jeff Chung low for Reminding us how important today is for organizing this beautiful ceremony and for his service to his country and to our town To respect his wishes, I'm not going to read it But I'm going to hand Jeff a proclamation that the select board voted on at our last meeting Jeff we love you. We appreciate what you've done for this community and you leave it a better place Today we gather to honor the memory of the more than 1 million people who have died in service to their country They left the safety of their hometowns The comfort of their family and their friends to defend our freedom and preserve the peace and not only ours But for countless people all over the globe We also recognize and honor their families spouses children parents siblings aunts uncles whose lives were forever changed To our Gold Star family members here today Thank you We owe you and your loved ones who gave their lives in service a debt of gratitude that can never be repaid We reflect today on the qualities of these women and men who gave their lives for our nation Their love of country and the ideals on which it was founded Their loyalty to the brothers and sisters in arms Their courage which is not the absence of fear but the resolve to do one's duty in the face of it Their selflessness willingly putting putting themselves in harm's way for the sake of others Words are not enough to honor their sacrifice words can never be enough So what can we do? We can remember our fallen heroes with respect and gratitude not just today But every day we can live our lives With the awareness that the freedoms and the opportunities that we enjoy today. We owe to them We can listen to the stories of their family and friends and in so doing Understand the people that they were We can support our veterans and ensure that they get the care respect and opportunities they deserve and Finally we can work for a world that is worthy of their sacrifice a world of compassion understanding tolerance Respect for the rule of law and liberty and we can remain vigilant for the forces of tyranny and oppression That threatens so many corners of the world and even here at home May the memory of those we honor today Inspire us to be the best version of ourselves For Arlington for Massachusetts for our country and for the world God bless our veterans God bless this great country I'd like to invite Representative David Rogers to offer his remarks Well, good morning everybody. It's good to be with you on this Memorial Day, and I too want to thank Jeff Schungal over his amazing service to this town and When I thought about what I might say today, I It occurred to me something that happened in our family my mom was in a terrible car accident and Fortunately the paramedics who showed up on the scene realized she couldn't be taken In an ambulance to the hospital. She needed a chopper. She needed a medevac and or else she wouldn't make it and She was medevac for trauma surgery and it was an intensive care for almost two months and There was a surgeon there Dr. John Pryor and he saved her life and Because she was an intensive care so long I got to know him really well. I was there visiting He would make rounds almost every day and it's just a remarkable human being He was board certified in three different areas of medicine He said he would have been board certified in a fourth area but his wife would have divorced him if he went for the certification so and He was literally a legendary Famous surgeon a brilliant guy and for all of his accomplishment Incredibly down to earth just the nicest guy you'd ever meet Well, it turns out not only was he a brilliant surgeon. He was a reservist in the US Army and He was called and deployed to Iraq Where sadly he he lost his life in an explosion an attack in so It's just a reminder of some of the incredible people that we have in our service that do amazing things for our country and Leave behind the legacy and so today, I'll think of him and the many other veterans we've lost and and God bless America and thank you. I'd like to invite Sean Garbley to offer his remarks Good morning. I Will repeat what everybody has honorably said about Jeff Chung low Jeff. We are going to miss you You have been a safety net for our veterans across this community. We can't thank you Enough would also like to thank our veterans counsel for everything They do and working with Jeff to support our men and women veterans across this town And it is an honor for me to be here with our gold star family members Your commitment and sacrifice cannot be Understated and it is an honor for all of us to be in your presence Into our veterans who are here in the hall and across the town of Arlington Everything you have done for this nation is a sacrifice that we will never be able to repay your incredible service and Supporting and honoring our veterans is something we should be doing Every day as we did last veterans day and we can do that in a number of ways Whether it's around employment, whether it's mental health services, whether it's the most basic principle of housing Supporting our veterans is something that we all should be recommitted to doing every single day but today We honor those who gave everything Those who sacrificed their happiness their families future happiness those who did not come home and It is important for each of us to reflect upon that think of Arlington High or any high school across this incredible country a young person not older than probably half the age of many of the people in this room made the decision to go across seas and into enemy territory and Potentially sacrifice Everything their happiness their families happiness their future their families future for an ideal For the principles and values that we hold so dear in this country many of them Did not come home So today as in the solemn day as We pay tribute to the men of women who did not come home It's not a day to feel guilty It's not a day to feel guilty about Hamburgers or what you do during the weekend with your family. That's not what this day is about. It's not to create a guilt trip But this day is not about hamburgers. It's not about hot dogs. It's not about game seven this day is about coming together as a country and Honoring those who paid the ultimate sacrifice That's what today is All about and that's what we do here as a country as a Commonwealth and as a town to pay tribute To those soldiers who left Arlington High and every high school across this country at far To young of an age and who did not come home Thank you. I was very pleased that general rap Responded to my email about his willingness to participate today as our keynote speaker It's nice to have an official Arlingtonian to offer remarks but William rap is a retired Army major general educator College administrator and proven leader with more than 30 33 years as an active-duty Army officer During his career. He served over five years in Germany one year in Japan and three and a half years in Iraq and Afghanistan He commanded an airborne engineer company in the first Gulf War an engineer bergate brigade in Iraq from 2005 to 2006 and was deputy commander of support for us forces in Afghanistan from 2011 to 2012 He served as the Army senior liaison to the US Congress Commandant of the United States Military Academy and most recently served as a commandant of the US Army War College in Carlisle, Pennsylvania His research interests include civil and military relations and organizational leadership His military career included the following commands commander of Northwestern Division of the Army Corps of Engineers from 2008 to 2009 Commandant of cadets US Military Academy West Point 2009 to 2011 Commanding general national support element us forces Afghanistan 2011 to 2012 Chief of legislative liaison liaison US Army 2012 to 2014 Commodant US Army War College and commanding general Carlisle barracks in 2014 to 2017 He conducts frequent lectures on military affairs and at the Harvard Kennedy School and Harvard University He did a brief stint there for his service a distinguished graduate of West Point in 1984 he also holds a master's degree in Strategic studies from the US Army War College and a master's and PhD in political science from Stanford University It's my pleasure to introduce major general William Rapp Hey, good morning to you all To the veterans who are out there Especially those veterans that served in war zones in our nation's history This day is is special for you. It's not Veterans Day, which comes in November But it is when you especially remember the colleagues who are with you that did not come home It's a day to to remember them Today's the unofficial first day of summer It's a long weekend for most of us and we pull out the grill and we gather friends and family for the warmth of and fun of summertime But a minced amidst all of that sunshine and fun It's fitting for us to take the time today room to remember why they call this Memorial Day This federal holiday was set aside to honor those that gave their lives so that we could live free and celebrate this weekend and so many others with friends and family in late May of 1866 after only after about a year of the since the guns had gone silent at Appomattox General John Logan organized what was then called Decorations Day To place flowers on the graves of the Civil War dead Logan wanted to be sure that the sacrifice of so many was not quickly forgotten After World War One the country changed its name to Memorial Day to recognize the ultimate sacrifice of military men and women from all wars At national cemeteries across this country Small American flags have been placed at the headstones of these brave and trepid souls who stepped forward Served with honor and did not come home My family and I live for a few years at right outside of Arlington National Cemetery And it's a ritual for the old guard soldiers of third u.s. Infantry to place a small American flag at every single headstone there 400,000 honored dead are buried in that one piece of hollowed ground I Recommend someday soon if you haven't done it already to walk over to the intersection of Mass Ave and Broadway and spend some time At the memorial wall there you'll find the names of those in the military and all of our nation's wars up to the late 90s Those marked with a small asterix Gave their lives for this country and Arlington has a long and proud history of Fighting to defend all that is right in this great land Jason Russell Jason Winship Jebez Wyman's names. They're all there from their deaths on the 19th of April 1775 not far from where we are sitting today The American Civil War saw 42 killed from Arlington then called monotony World War One had 24 killed The town had then been renamed to Arlington World War two took a toll on this community 156 were killed in the four terrible years in which the United States and others fought Imperial Japan Nazi Germany and fascist Italy 14 more were killed in Korea and nine were killed in Vietnam There's a single asterix next to a name in the wars after Vietnam to the year 1997 John Patrick Conner a Navy SEAL officer who died heroically on the small airfield of Patia in Western Panama on 20 December 1989 Joan Conner is here with us today. Thank you. We honor your son at the core of our military lie unique themes The selfless desire to serve a strong desire to do what's right and just And the willingness if called upon in the heat of battle to sacrifice oneself for our teammates to defend this nation Those brave men and women were supported now and now grieved by their families and loved ones Those who left behind or those left behind the fallen bear the most difficult of burdens and we honor them as well With their lives these brave soldiers sailors airmen and Marines have bought for us The right to use this day. However, we choose They would not begrudge Your barbecue they might get a bit upset if you overcooked the stakes They would not be offended Unless you failed to take some bit of time today to remember them to keep them in mind these true American heroes fought as they did so that you could live your lives with a sense of peace and Security that allows you to fully enjoy days like this So take that moment to reflect upon their service and sacrifice Lawrence Binion was a British poet who volunteered at age 46 to serve in a field hospital near the killing fields of Verdun in 1915 and He wrote a poem a couple of lines of which about the young men that he watched Marching forward to the front knowing in his heart that many that he saw would not have the chance to grow old He wrote they went with songs to the battle. They were young Straight of limb true of I steady in a glow They were staunch to the end Against odds and countered and they fell with their faces to the foe They shall not grow old as we who are left grow old Aides shall not worry them Nor the years condemn at the going down of the sun and in the morning. We will remember them We must not break faith with those who have died on our behalf We must strive to honor their sacrifice in many small ways every day We must remember them President Harry Truman soon after he took over the presidency in the spring of 1945 The war was still going on with both Germany and Japan and he said the following Our debt to the heroic men and valiant women in the service of our country can never be repaid They have earned our undying gratitude America will never forget their sacrifices because of these sacrifices the dawn of justice and freedom Throughout the world slowly casts its gleam across the horizon The respect and admiration that we give our fallen hopefully not only on this day But in silent or vocal tribute year-round pays tribute to their memory Their stories must not be forgotten by sharing these stories We keep their memories alive and we inspire ourselves to create a better world a stronger nation kinder communities It's up to us to use the gifts that they secured With their ultimate sacrifice to do as much good as possible and to honor that debt that can never be repaid Joan if you allow me to talk just briefly about your son Patrick Connor grew up in Sitchwood in Arlington in a attended WPI in Worcester Upon graduation he joined the United States Navy and volunteered for SEAL training Earning his Trident badge signifying qualification as a US Navy SEAL Lieutenant Connor was privileged to become a leader in golf platoon of SEAL team for which was based out of Virginia Beach and focused on Central and South America on the evening of 20 December 1989 48 members of SEAL team 4 would Were tasked to seize the airfield at Padilla and destroy manual Noriega's Private aircraft so that he could not escape Panamanian forces were an unexpectedly heavy strength at the airfield and John Connor was killed while purposefully drawing fire So that it could take pressure off of the members of his team only age 25 Connor earned the Silver Star and Purple Heart for his Valor that evening He's buried here at Mount Pleasant Cemetery and his name is found on our town's Memorial Wall For me on this day, I remember colleagues that I served with who died in combat as you heard I commanded Bravo Company 27th Airborne engineers and we were working to clear mines on the north side of the runway at Assam and airfield in southern Iraq on February 26 1991 Captain Mario Fajardo who commanded Alpha Company 27th engineers and five of his men First Lieutenant Terry Plunk Sir in first-class Russell Smith Staff Sergeant Michael Harris Sergeant Brian Scott and Corporal Luis Delgado were killed that day clearing mines on the south side of the runway We came from different backgrounds Mario was a Citadel graduate, and I'm a West Pointer But in Iraq we were brothers and I think about him often On Memorial Day, we should reflect upon the experiences of our fallen loved ones And those like John Connor whom we may not have known personally But tomorrow and in the days that follow we must act and ensure that their sacrifice was not in vain We honor them by being the very best versions of ourselves every single day We honor them by being good neighbors To each other and good citizens to this country We honor them as William Wordsworth wrote for our little Nameless unremembered acts of kindness and of love We honor them by fully appreciating that it is a privilege to live in this great country of ours and Finally, we honor them by remembering that the freedom we enjoy is not free. I Join this whole crowd in Recognizing the wonderful support that Jeff has given to the veterans of this community for so many years And I thank you for the opportunity to speak with you this morning Thank you very much People often thank me for my service and I am absolutely Humbled by that, but I will tell you I think I got more than I gave The ability to serve and wear the colors of this nation is something that is a treasure. Thank you for so Massachusetts a couple of years ago created the Medal of Fidelity and It's been established To be presented to the next of kin of a service member or veteran who died as a result of a service-connected disability or illness. I Would like to ask Marilyn Dolan Goldstein to stand Her husband captain Alan Goldstein was a Vietnam veteran who passed away on February 24th as a result of exposure to Agent Orange at this time General Rapp and I will present the award to Mrs. Dolan at this time the skyline course Will perform let freedom ring so the following presentation is a time-honored tradition in the Navy And yes, I have to go Navy When a sailor retires a fold of flag is ceremoniously passed from sailor to sailor and it reflects the different ranks that they have achieved throughout their career The retiree then presents a flag to a spouse or significant other in honor of their support and sacrifices that they provided during their military career and This ceremony is often referred to as old glory. I Would like the members of the Arlington Veterans Council to come forward So I'd like to thank Amory Russo Bill Hainer and Steve Sautel For their support for me While serving as your director of veteran services, but they have done an awesome job and I can't thank them enough I am the flag of the United States of America My name is old glory I fly atop the World's tallest buildings. I stand watch in America's halls of justice. I fly majestically over great Institution of learning. I stand guard with the greatest military power in the world Look up and see me I stand for peace honor truth and justice. I stand for freedom. I Am confident.  I am worshiped. I am saluted. I am respected. I Am revered. I am loved and I am feared. I Have fought in every battle of every war for more than 200 years Gettysburg Shiloh Appomattox San Juan Hill the trenches of France the Argonne Forest Anzio Rome the beaches of Normandy Guam Okinawa Tarawa Korea Vietnam The Persian Gulf and scores of places long forgotten by all except those who were there with me. I Was there. I led my sailors marine airmen and soldiers. I Followed them. I watched over them and they loved me. I was on a small hill on Iwo Jima I was dirty battle-worn and tired But my sailors and Marines cheered me and I was proud I Was at ground zero in New York City on September 11th as cowardly fanatics attacked America. I Was raised from the ashes of once proud buildings by brave firefighters heroes who risk their lives to save others Showing the world that America although bloodied will never be beaten Those who would destroy me cannot win For I am the symbol of freedom of one nation under God Indivisible with liberty and justice for all I Have been soiled burned torn and trampled on the streets of my own country and When it is done by those who I have served with in battle it hurts But I shall overcome for I am strong I have slipped the surly bonds of earth and from my vantage point on the moon I stand watch over the uncharted new frontiers of space. I Have been a silent witness to all of America's finest hours But my finest hour comes when I am torn in strips to be used as bandages for my wounded comrades on the battlefield When I fly at half-mast to honor my soldiers and When I lie in the trembling arms of a grieving mother at the gravesite of her fallen son or daughter, I am proud My name is all glory Long may I wave dear God? Long may I wave? I'm tired of the colors I'd like to once again invite the Boston skyline chorus to perform Hallelujah So I'd like to thank the skyline chorus for your support of our ceremonies over the years So thank you very much. I appreciate everything you do. You do a great job. Thank you Memorial Day honors the service and devotion of our fallen heroes and Serves as a solemn reminder of the extraordinary courage and selflessness. They displayed in the face of danger It's also an important opportunity for us to reflect on the importance of military service For the preservation of our freedom and the safety and security of our nation This Memorial Day It is our responsibility to offer prayers and thanks for these brave men and Individuals and their sacrifices. Let us not forget the remarkable level of selfless service That these men and women provided especially those who paid the ultimate price We can do our part by honoring their memory through acts of reverence respect and remembrance This time I'd like you to please stand and join me in a moment of silence I'd like to invite Father Bishop to offer our benediction in the quiet sanctuaries of our own hearts Let let each of us name and call on the one whose power over us is great and gentle firm and forgiving Holy and healing You create us who sustain us who call us to live in peace hear our prayers this day Hear our prayer for all who have died whose hearts and hopes are known to you alone Hear our prayer for those who put the welfare of others ahead of their own and give us hearts as generous as theirs Hear our prayer for those who gave their lives in the service of others and accept the gift of Their sacrifice Help us to shape and make a world where we will lay down the arms of war and turn our swords into plowshares for a harvest of justice and peace Cover those who grieve the loss of their loved ones and let your healing be the hope in our hearts Hear our prayer this day and in your mercy answer us in the name of all that is holy and good. We pray Amen And shipmate fair winds and following seas Please be seated So this concludes our indoor portion of the ceremony I'd like to thank all of our speakers for their comments today and reflections. Thank you very much So for those of you who would like to go with us to Mount Pleasant Cemetery for our wreath laying ceremonies We'll gather in front of the town hall in a few minutes and proceed to the cemetery For those that would like to go but have issues with mobility. We will have a van outside to take people down so that will be in the front as well and again, I'd like to Acknowledge the police and fire department honor guards and their service to the community. Thank you very much And I'd like to thank everyone that I had the opportunity to work with in this community Arlington is a great community You're all very lucky residents, but a great group of people. I can't say enough about them and We'll just leave it at that. So thank you very much, and we'll see you out front in a few minutes A total of 2.2 million Union soldiers conducted military operations from 1861 to 1865 Over 204,000 soldiers lost their lives during these battles with over 476,000 suffering battle injuries and Additionally, there were over 413,000 non-combat deaths During the Spanish-American War conducted between 1898 and 1901 over 306,000 men and women served our country of this number 362 lost their lives While over 1600 received combat-related injuries, and there were over 2,600 non-combat deaths. This time, Major General Wraps will The monotony Minutemen will render honors. 414.7 million service members served between 1917 and 1918 over 53,500 lost their lives and over 402,000 suffered combat injuries, and there were over 63,000 non-combat deaths During World War II 16.1 million service members served from 1941 to 1945 Over 293,000 lost their lives over 670,000 sustained combat injuries, and there were over 115,000 non-combat deaths During the Korean War, 5.7 million service members served from 1950 to 1953 Over 33,700 lost their lives Over 103,000 received combat injuries, and there were over 10,000 non-combat deaths At this time, Steve Sautel and I will lay a wreath for our World War I veteran The monotony Minutemen will render honors 47,000 for over 150,000 sustained combat injuries and 700 non-hostile deaths Bill Hainer, the Vietnam veteran, will lay a wreath and will render honors Forest and Coast Guard and related reserve and National Guard units have been deployed since 9-Eleven 2001 Over 7,000 military personnel have been killed Over 50,600 military personnel have been wounded Major General Rapp will lay a wreath The field day is a day of remembrance for those who've made the ultimate sacrifice It's a day to honor and respect those who have given their lives in defense of our freedoms and to remember their selfless service commitment to others And to remember the personal sacrifices on behalf of their families It's also an important opportunity for us to reflect on the values of courage service and sacrifice that our fallen heroes embody Remember our Gold Star families who have lost loved ones to illness injury and combat They stood strong through uncertainty and those resilient dedication and commitment to their loved ones Continue to shine and serve as support for others during their darkest hours Remember the men and women currently conducting military operations around the globe and the Blue Star families awaiting their safe return I ask that you remember and honor the missing as well In closing, I'd like to thank Major General Rapp for his participation in our ceremony today I'd like to acknowledge all the veterans present here today our elected officials the Arlington Police Department, the Arlington Fire Department and The Department of Public Works and the crew here at Mount Pleasant Cemetery But most of all, thanks you very much for your support while I've been here And I look forward to many more of these type of services in the future so thank you very much enjoy your day and Keep it in the true spirit of the meaning of the day. Thank you


New work about the topic, and so we say that f is multiplicative, if f of m times n is always f of m times f of n, when the greatest common device of f, m and n is 1, and in this talk I will concentrate on multiplicative functions taking values in minus 1 plus 1. So they are bounded and real valued for purposes of this talk, and I give some examples first of multiplicative functions, so the Mobius function, which is 0 if n has a square factor, and otherwise it's minus 1 if n has odd number of distinct prime factors and it's plus 1 if n has even number of distinct prime factors. And another example is the indicator function of the set n of numbers that can be written as a sum of two squares. So this is 1 if p is 102 mod 4, so it's 1 mod 4 or it's the prime 2, and for primes as a 3 mod 4 it's 1 only if the power of prime is even. So in particular, any prime that is 3 mod 4 is not representable as a sum of two squares, and it's 0 if it has an odd power of a prime mod, which is 3 mod 4. And the third example of the multiplicative function taking funded values is the indicator function of the set of y smooth numbers, and by y smooth I mean a number whose all prime factors are up to y. So this is three sort of fundamental examples of multiplicative functions. And if one looks also at unbounded function, then for instance divisor function is an example, but I won't today talk about those. And I'm interested about averages of multiplicative function, what's the average value, and let's first consider long averages over the interval from 1 to x. And these are well understood. It is known that the mean value of the multiplicative function is 0 if f doesn't pretend to be 1, if the values of primes are not close to 1, and otherwise the mean value is non-zero and can be calculated. So I will give the details. So if f is typically about 1, if the sum of 1 minus fp over b converges, this means that for most primes fp is 1 or very close to 1. In this case, the average of the mean value of the function fn can be calculated, and it's this product here. And it's a converging product, thanks to this condition. And on the other hand, if f is not close to 1 for most primes, if this condition doesn't hold, then it is known that the mean value of f is 0. And if we, for instance, look at the Mobius function at primes, the Mobius function takes the value minus 1, so 1 minus minus 1 is plus 2. So this series definitely diverges. So this is the case of Versing's theorem. One doesn't hold. It's a diverging series. And from that, we know that the average value of the Mobius function is 0. And this is actually equivalent to the primes number theorem, saying that the number of primes up to x is x over log x. And also equivalent to the Priemann set of functions not having zeroes real, but s equal to 1. So this sort of fundamental result in number theory. And these are sort of qualitative results, but there's a theorem of how last that gives one quantitative results. One gets more, depending on how this sort of sum behaves, one gets precise amount of consideration in this sum and so on. But what's my topic today are the short averages. If you look at the average of fn over short interval, then previously Maxim and I sold. It was published in 2016. We showed that if one looks at almost all very short intervals of yh with h tending to infinity with x as low as uvis. If we compare the very short average, we take a very short segment of ns and calculate the average of that segment. Then the average is very close to the average over the long segment. And this is what we know by how last theorem and so on. We know how to evaluate this. So essentially we get the same story for the very short intervals for almost all x. So as soon as h tends to infinity, the error term here tends to zero. And the exceptional set tends to be the low of x. So we got a good result for almost all short intervals. And this has led to numerous applications, developments such as 30,000 resolution of logarithmic average in chococonceptual concerning essentially the correlation of nubius function. And also the resolution of the Earth's discrepancy program. But even though there are still few shortcomings about this theorem, and in this talk I will discuss our new work, which addresses these shortcomings. We get better results in several senses. And the first shortcoming is that the quantitative bounds are pretty weak. Here we win the log h to small power and in the paper we get a bit better result than this, but still at the best we win the small power of log x in the exceptional set. So the bounds are not very good. And the second thing is that if we look at the example of the indicator function of the sums of two squares, then this theorem becomes trivial. The sums of two squares means spacing at log x to half, which means that if we look at this long average, it's already zero. It's of size log x to minus half, which means that it's much smaller than the error that we get here. So this theorem doesn't help us in determining how these sort of vanishing multiplicative functions behave in short intervals. It doesn't tell us anything about sums of two squares, because we already know that almost all intervals don't contain any sums of two squares if the interval length is very small. And for long intervals it doesn't give us anything non-trivial. So that's a bad thing about the theorem. And the third thing is that for many applications one needs result for a complex F. This original result is only for the real value of F. And actually in this form it doesn't hold for complex F, because F might be, say, N to IT. And in this case the solved version doesn't quite pretend to be the long version, but there must be a twist. And in this new work we also address this case of complex F, but in this talk I will concentrate on the case of the real function F. Okay, so let's go into the example of sums of two squares. So recall that N is the set of numbers that can be written as a sum of two squares. And it is well known that the number of sums of two squares is x over log x to half times a constant. And in particular this means that if you got the average gap between numbers that can be represented as a sum of two squares, then the average gap is of such log x to half. And in particular this means that if we look at intervals shorter than the average gap, then typically there are no numbers in N in that interval. So for typical y it's more than h1 the short average is simply zero. But if we look at longer intervals than the average gap, then one would expect that we have regular behavior. If we look at a short interval, it's going to be somewhat longer than this average log x to half, then we expect that we get asymptotic formula for almost all short intervals of this gang. As soon as the length divided by the average gap log x to half tends to infinity. So this is what one would expect to hold. And this is what we saw, that it's actually true. So we saw that even any delta and any h0 if we look at the number of integrals that can be represented as a sum of two squares in an interval of length h0 times log x to half. Then it corresponds to the long density. This is the density of integrals in this interval and this is the density of integrals representable as a sum of two squares in long interval in the interval of length x. And what we get is that this is smaller than the momentum delta times the momentum for all but at most x times h0 to minus c delta to 12 integrals. The point is that this is sort of polynomial exceptional set in h0. And this holds for some constant c which we have explicitly but it's quite small. And node here, the exceptional set saves polynomial in h0 and that we get the error term now taking into account the size of the momentum. Previously we just had log h to some small power here which wasn't helpful at all for this sort of very short intervals. But now we get a non-trivial result as soon as the length of the interval tends to log x to half times something tending to infinity. And we get an exceptional set which tends to little of one and it tends so quite rapidly. And this improves on the result of h0, who saw that for almost all x, one has a lower bound for this of right order of magnitude, but he didn't get asymptotic. So we improve by getting an asymptotic formula for almost all x transplants of the order of magnitude. Kajsa, we have a question from Igor Szparlinski. Igor, could you please ask the question by unmuting your microphone. Thank you. Kajsa, sorry, you say that c is a function of delta, but then what's the meaning of delta to the twelfth next to it? Oh, sorry, so it's not a function of delta. It's an absolute constant. Oh, it's an absolute constant. Yeah, it's an absolute constant. Okay, thank you. You can just see delta here, but then it tends to delta to 12. Yeah, it's an absolute constant. I think it's up to 1000 to minus 12. Thank you. No problem. Okay. So this is that the fp has, then it's known that the average value of fp is like log x alpha minus one. More precisely, it's got this. This, you get the product of fp minus one over p, but if it has the average value of alpha, then it's of this as log x alpha minus one. And now we write h1 for the inverse of this. And because fn has mean value like this, then if it takes values zero and one, the absolute value, then the average in the worth of rent h1 is becoming zero. Because this is again similar to the space of two squares, this is the mean spacing of the values because this is the mean. And then the mean spacing is the. So again, if we look at intervals of shorter than h1, then the typically the average of fn is zero. And we don't have an interesting theorem. But again, as soon as the length of the interval is longer than the mean space, we expect that we get a meaningful result, which is that we get that for almost all x. And the third interval average is of the same size as the interval average. And we have a error term, which is of smaller order of magnitude than the main term. This is the order of the main term, and we get that we do all. So this is what one expects to have. And this is what we prove again.  So if we don't think that the average is exactly something, we just think that if we get it. Kaisa, sorry for interrupting you again. So the audio is a bit uneven. Perhaps, maybe perhaps speak a bit closer to the microphone. Sometimes the volume is a bit uneven. Okay, I will try to do that. Thanks a lot. We have this theorem, which says that if the average value of f at prime is a bit epsilon, so that it's not zero for all the primes in long intervals, then we get the result that we wanted. So h1 is the expected gap between non-zero values of f. And then for any delta and any h0, we get that the short average is the same as long average apart from an error term, which is smaller or smaller size delta times the size of the long average for all. But again, we have a polynomial exceptional set. And now the constant c and kappa depend on the proportion of non-validation. But anyway, we get the same story for all multiplicative functions that don't vanish too much. They have to have a positive average and so get the primes. And then we get the result that in short intervals of the expected length, we can get the same average for f as in the long intervals. And we get the same result for complex valued f, but then we have to have a fist in main term and it doesn't look very nice, but it's a natural fist that we have to take, but I don't talk about the complex case anymore. Okay, so let's get back to the sums of two squares. He called that Huli was able to solve that for almost all x. One gets the correct order of magnitude of sums of two squares in intervals of length h0 times log x12 for any h0 tending to infinity with x. And Huli did this in a different way than us. His main arithmetic information was the solution for the shifted convolution problem for the coefficients of the deducing set of function of the qi. And these are essentially the number of representations of n as sum of two squares. So this is what allowed Huli to solve his result that one has the word. One knows asymptotic formula for this shifted convolution problem. But it's worth pointing out that it is completely out of place if one looks at number field of degree greater than two. If one takes the coefficients of the deducing set of function of any number field that has degree greater than two, then there is no way one can do the shifted convolution problem, which means that if one wants to generalize Huli's result, it doesn't work. It's completely, the approach completely fails because we don't have this crucial arithmetic information. But on the other hand, in our case, we only use multiplicativity of the function. So we have chances to generalize. In particular, we don't need the density of the prime to be half for the case of sum of two squares. The density of the fb at prime is half, but we can have fc1 for the density. So we are allowed to go into higher degree number fields for generalization. And so we will talk about norm forms. The sum of two squares are norm forms in qi. And now I'm going to talk about higher degree extensions. So let's take a number field k. And we say that n is a norm form if it's a norm of an algebraic integer in k. This is a stronger condition that's just requiring that it's norm of an ideal. We make a stronger condition here. And we write ca of n for the indicator function of this case. So in particular, if we look at c of qi, then it's the indicator function of sums of two squares. And there is some work of Odoni on telling these norm forms. And in particular, he thought that the density of norm forms of k is this product, which were quite similar to the products we have had before. So it's the product of all primes for which there doesn't exist an integral idea of norm p1 minus 1 over p. And in particular, if k is a normal extension of degree k, then this is of size of x to minus 1 plus 1 over k. So essentially it means that comparing to the condition we had before, it's that the average times is of size of normal k. And k tends to infinity the average primes n to 0. But there is one issue. When the class number of k is less than 1, then this function tk and is not modificative. So we are not able to directly use our results for this decay of n. Density is not an issue, but the issue is that it's not modificative. But for this, the same work of Odoni helped us to show that tk of n can be written as a linear combination of complex value modificative functions. And then we can apply our results to each function in the linear combination of this theorem in any number field. So let me write this theorem here. So we take k to be any number field over q and we take delta kx to be the density of norm forms in k up to x. And then we get an asymptotic formula for the number of integrals that can be represented as some of norm forms in short intervals as soon as it's possible, I mean in intervals of length than the average gap. And then we get an asymptotic formula for number of norm forms in short intervals for almost all x. And again, we get the polynomial exceptional set. And this k extends who is work, which only works for ui and which didn't have an asymptotic formula. So we get a very wide extension of his work. And who we also studied another question concerning sums of two squares is study the gaps between sums of two squares. So remember that the average gap is log x to half, but then the question is how often do gaps of this length or this length appear. And so writing si for the sequence of numbers that are sums of two squares, it's proved that if we take the gamma movement of the gap function and sum it over the gaps, then we get an asymptotic formula for this. And in other words, this means that for any interval length h, for any interval length h times log x to half, the number of x's such that this interval doesn't contain sums of two squares is at most of all the x times h to minus 2. So this is the equivalent of these two things. So you get a good exceptional set here. Here if one doesn't require an asymptotic formula, but is happy with the existence of sums of two squares in short intervals. And again, who is work is based on the shifted convolution problem for RKN and so it doesn't extend beyond quadratic number fields. If the degree of the number field is greater than 2, who is work doesn't have any chance to generalize the current knowledge about the shift convolution problem. But for us, we again use only multiplicativity. So if we don't anymore want an asymptotic formula before we get an exceptional set for any K, the better exceptional set than we had previously. So this is the result Maxim and I have about gaps. So if K is any number field and delta K is the density of non-forms, then for any interval of h times longer than the average gap, we get that the number of intervals that don't contain the correct order of money to non-forms is of size, the exceptional set is of size h times h to minus half epsilon. So in the previous result, I thought that we had that we got asymptotic formula for this, but we had exceptional set that had a very small power here. Now we have the exceptional set which has size x times h to. Kaisa, I'm not hearing you at the moment. Okay, I'm not sure what I can do about that. Okay, it's better now, I can hear you now. I will try to be closer to the. Okay, thank you. Okay, so if we don't want an asymptotic formula, but just want the correct order of magnitude of non-forms, then we can get a good exceptional set for the number of exceptional intervals. So all but this many intervals contain expected order of magnitude of numbers that are non-forms in K. And if we look at Hooley's problem of looking at the gamma moment of the gaps, then we get the asymptotic formula for gamma up to 3 over 2. And remember that Hooley had this for gamma up to 503, which is a bit larger than our exponent, but worked only for the case of K equal to Q i and our episodic extends to any number fields. And also in the forthcoming work, we are able to do the Q i case for all gamma up to 2 by using some properties of the Deakin set of function for the Q i. So this generalizes who is the result to any number field. And this is not restricted just to non-forms, but this result work for other multiplicative functions. So for instance, we can study gaps between sums of two squares. If we look at the number of intervals from X to X plus 8, which do not contain an extra exceptional number, then the number of them is at most X times H2 minus half plus eta for any eta. And consequently, we get a corresponding result for the gamma powers of the gaps for gamma up to 3 over 2 again. And this improves on the recent result of Hitron, who had this for this gamma moment. He had an upper bound of size X to 1 plus eta. The lower bound that it's at least X is trivial, but he got the upper bound X to 1 plus eta, but now we get the last one. So we can use this also for studying other multiplicative functions and gaps between multiplicative sequences. And let me now move on to some of the ideas in the proofs. I will concentrate on what's new in this work compared to our previous work, but I don't expect that you know how our previous work worked. So for simplicity, let's just concentrate on the case when the average of F is 0. This means that we don't have to worry about the main term. And the starting point for us is Perron's formula, which gets us to express this short average as a Perron in the crowd, as an in the crowd in the complex plane, where we have the corresponding diricate series in the middle and this X plus H and X come from the ending points and starting points of the interval. And this expression here can at least morally be estimated by the mean value theorem to be H times the derivative of X, which is H times X to IT. So we can morally write this sum of Fn over 15.28 and this form we can... And if now we just put absolute values here, we are in trouble, because we always hope in any denominator that we have square root cancellation and we can't hope for more. And if we had square root cancellation in this n-solve, it would be of size X to minus half. And multiplying by H and X over H would mean that the right-hand side with absolute values would be of size X to half, but the trivial bound for the left-hand side is H, which is much smaller. So this is not a good idea to put absolute values here. Instead, we have to somehow... We have to somehow take advantage of the oscillations of X by T. And what one would normally go on to do is to take a square mean. We want to get the result for almost all X, so we can take a square mean of this and if we get a good one for the square mean, then we get that this is more exceptional set. And if we take a square mean of this and square out and do some work, we find out that we get an upper bound of this form, where we get the mean square of the same dirigate polynomial here. And now we don't anymore have this problem that if we have square rule oscillation behind trouble. So that's good. So that's what one would normally do. And recall that we had chosen this H, the length of the interval is H0 times H1. The H1 is the sort of average gap between the numbers. And so if we want to show our desired result that this short average is of size delta times the density of F, then we wanted to get a polynomial exceptional set. Then for this thing here, we would need a bound that is of size this squared times the wanted exceptional set. So what we would like to achieve here would be that the mean square of the dirigate polynomial is of size third time H0 to this power delta kappa times the square of the mean value of F. And if we could show this, it would be done. So this is what we would need to be done. And there is a mean value theorem for dirigate polynomials, which tells us that this mean value here is at most of size the length of the interval plus the length of the sum, as you mean square of the terms. And we can prove a variant of it that works well for the case where Fn has sparse support, like some of those pairs. And in particular, we can show a mean value theorem variant, which has that if we look at the mean value of An, dirigate polynomial, whose coefficients are bounded by a modicative function F, then we get the mean, the square of the density of F. And so we get the same term here and here, but we would need to win this delta square times H0 to minus C delta kappa compared to this sort of trivial mean value theorem bound. And this is exactly the same situation we had in our previous work. We need to save something compared to the mean value theorem bound. The mean value theorem gives us a trivial result. If we had just this, we get that the set of exceptions is of size big O of X, which is completely trivial, because the X numbers, it's not a good thing to have an exceptional set of size big O of X, but we had to win something compared to this mean value theorem bound. And we can do it. We can repeat the same arguments because we have this good mean value theorem now. We have to prove some hollows and giblets that estimates for modicative functions in the sparse setting, where we don't have mean value smaller than zero, but we can repeat all those arguments. That's not a problem, but the problem is that at best this gives us, instead of this power saving H0, this gives us log X2 minus kappa, because hollows can never win more than this and it's an actual problem. There are modicative functions, where you only save this much value. Average is large. And so actually showing what we wanted is in general not possible. There might be some points here, where this did we get polynomial as size log X2 minus kappa and these contribute like log X2 minus kappa to the left-hand side, and we wanted to win the power of H0. So in particular if H0 is larger than the power of log X, we are in trouble here. So we can't quite show this if we want it. But the method would work if we had a good bound for theoretical polynomials, if we had a good bound for theoretical polynomial masterware price in certain industries for say, branch of science object, but if we had good saving in those theoretical polynomials then our previous method could work. We could factor out this theoretical polynomial and then we could use the main value theorem or we could do a similar thing in what we did in our previous method of factoring out different sizes of time even if it would win thanks to this calculation. So if we would have this, we would be happy. But if it was possible for 10 or X, we would need to have that. So we have to FB instead of 1 here. And another thing, we won't have prime factors from some intervals but that's something we can also do. But anyway, the key idea is to handle the exceptional T, the T for which we don't have the desired bound like this before taking the mean square away. So we handle those T for which we don't have this bound before taking the mean square and then once we take the mean square we can assume that we are integrating only over those T for which we have this bound. Okay, so let's get back because of my first formula we had that this thought average and we've written that onto integral and now we split the integration range to two sets T and U where T belongs to T if we have this desired consideration for the theoretical polynomial over primes of sci-tech tools. And by the mean value theorem for theoretical polynomials we can show that for most points T this is actually true. The size of the exceptional set is at most x over h square root. So most of the points we have this and then the next set. And by the previous discussion our order work is basically this of V where we have good consideration in the theoretical polynomial over primes. So for that we get the desired bound and then we are left with handing the integral over U. So we are left for studying this integral over U where the set U is sitting past it's a subset of minus x over h to x over h but it has size only x over h square root. And now we take a new trick. We note that most integrals have at least two prime factors from long interval at least x to epsilon squared to x to epsilon. So we can at least morally we can replace this fn over h we can replace the prior product Diricast polynomial where we have sort of taken out the prime from those intervals. And also we can use Paxi's law theorem which is sort of mean value theorem for Diricast polynomials but over a subset and now we have a subset because we have this one for U and we can use it for those keys for which we have some answer in these polynomials we can again look at the mean square and we can use the point wise point for this. Thanks to this point here and other parts can be taken care of in the mean square thanks to the part set. And on the other hand it's possible to show that this holds for most keys they still are only like x to epsilon exceptions to this. Now we have handed almost all keys they have just like x to epsilon for this one and it's so tiny that we get that this mean here is little of h by using alas mean value theorem for multiplicative functions for this and then some alas-monkometry type large value theorem so like a mean value theorem for Diricast polynomials but over a subset and now we have a very, very subset so we are able to prove a very sparset version that works here but we know we get wanted so this is the theorem we got by looking both moving into the ground formula moving into the alas of Diricast polynomials but working both on L1 and L2 we alternate between looking at the alas over x and not looking at the alas over x to be able to handle this thing here so let's get back to one last thing which is the results about positive proportion lower bound getting the improvement of who is results for gaps so if one only wants for an F a good exceptional set if one only wants to show that in almost all short intervals one has that fn some of average of fn is at least delta times the expected average we don't anymore ask for an asymptotic formula but we just want a positive proportion lower bound whether there exists some sort of spars or whatever we have then it's an easier task because we don't have to get an asymptotic formula but we can hook up in whatever way we wish and only consider those ends and in this case it's a good idea to instead of looking at all possible ends to sum over products of very many numbers we take k to be like 1 over f10 and then we look at product of k things epsilon to 1 over epsilon to 10 things that are small primes we take k minus 1 very small primes and we take m which is also very small body and then we study the product of all these and because fn is multiplicative we can also also split f as a product of k things and now what we want to show is that this has size at least a constant times the right hand side here and this sum over this product is definitely of smaller order of magnitude than this thing here we can get a lower bound like this we might double count some things but at least epsilon to 10 times so we get just a constant depending on epsilon so if we want to count all n we get definitely a lower bound if we just count products of k things in this into us and now the things get easier because if you look at the rocket polynomial for responding to this it can be factored into a product of very short factors now we have like k very short factors and this gives us a lot more flexibility when we use all these here as for the rocket polynomials and this allows us to show these that we want and that's all I wanted to say thank you thank you guys for this excellent talk so you can unmute your microphone so we can thank the speaker altogether and by clapping okay so now we have time for some questions so I will mute all for just a moment so there's time for questions so maybe I'll start with a question you mentioned that let me stop


What we recognize is that this pandemic presents a significant hardship for many, many, many people. And for a lot of us, it does rise to the level of trauma. Our bodies actually release more stressed hormones, and those include things like adrenaline and cortisol. And they can have lots of effects on our brains and bodies. They can increase the risk for things like heart attacks or strokes, or having an experience of depression or anxiety. There's people who are coming in with unexplained symptoms of, my arm feels weird, I'm tired, my chest hurts, I'm short of breath. And they've had extensive work-ups been seen by doctors multiple times. And a lot of it's just this stress that people are experiencing because of the pandemic. Some kids, littler kids especially, may see some developmental regression. Some kids will have a harder time focusing. Some kids will have more angry outbursts. Some kids, they'll have frequent headaches or tummy aches, right? Or their appetite will go way up or way down. And these are some of the things that we can look out for to recognize when a child is experiencing distress. It's a really devastating time for a much larger swath of the society than I think people realize. We estimate that of the 1.5 to 1.7 million restaurant and service workers in California, 1.1 million workers are out of work. The vast majority of our folks have absolutely no access to any kind of healthcare they never did. And now it's a source of incredible stress. Just incredible stress to think, what would I do if I got sick or my child got sick? What would we do? Rossi, Daniel, I want to learn from you more about your own experience of having COVID-19 and what psychological toll that took on you. For me, although I felt sick physically with muscle aches, fevers, not being able to breathe particularly well, I was more affected emotionally because I was so worried about what was going to happen to me. So for me personally, I think the biggest thing to start was all of the anxiety that came with it. I'd love to hear what each of you do to restore common focus at this time. I've tried to continue to do a lot of things that kind of keep me sane on the regular, which is continue to exercise, try to eat some healthy food. I've been talking a lot more on the phone to family and friends, definitely, and trying to be kind of open and honest about how I'm feeling. What's keeping me sane is that I see a moment of opportunity here. Really, I see two pathways for us. Either we can go in a much worse direction because the pandemic has created a lot of serious trauma, as we've all talked about, or we can use this crisis and moment of opportunity to act collectively and demand something totally different and much better than what we've had all along.