projected to increase by up to 16 percent because of the war in Ukraine and the pandemic, which made ingredients, packaging and supply chains more costly, according to UNICEF.

displaced by the drought this year. As many as three million Somalis have also been displaced by tribal and political conflicts and the ever-growing threat from the terrorist group Al Shabab.

cyclones, rising temperatures, a locust infestation that destroyed crops, and, now, four consecutive failed rainy seasons.

spend 60 to 80 percent of their income on food. The loss of wheat from Ukraine, supply-chain delays and soaring inflation have led to sharp rises in the prices of cooking oil and staples like rice and sorghum.

At a market in the border town of Doolow, more than two dozen tables were abandoned because vendors could no longer afford to stock produce from local farms. The remaining retailers sold paltry supplies of cherry tomatoes, dried lemons and unripe bananas to the few customers trickling in.

perished since mid-2021, according to monitoring agencies.

The drought is also straining the social support systems that Somalis depend on during crises.

As thousands of hungry and homeless people flooded the capital, the women at the Hiil-Haween Cooperative sought ways to support them. But faced with their own soaring bills, many of the women said they had little to share. They collected clothes and food for about 70 displaced people.

“We had to reach deep into our community to find anything,” said Hadiya Hassan, who leads the cooperative.

likely fail, pushing the drought into 2023. The predictions are worrying analysts, who say the deteriorating conditions and the delayed scale-up in funding could mirror the severe 2011 drought that killed about 260,000 Somalis.

Famine in Somalia.”

For now, the merciless drought is forcing some families to make hard choices.

Back at the Benadir hospital in Mogadishu, Amina Abdullahi gazed at her severely malnourished 3-month-old daughter, Fatuma Yusuf. Clenching her fists and gasping for air, the baby let out a feeble cry, drawing smiles from the doctors who were happy to hear her make any noise at all.

“She was as still as the dead when we brought her here,” Ms. Abdullahi said. But even though the baby had gained more than a pound in the hospital, she was still less than five pounds in all — not even half what she should be. Doctors said it would be a while before she was discharged.

This pained Ms. Abdullahi. She had left six other children behind in Beledweyne, about 200 miles away, on a small, desiccated farm with her goats dying.

“The suffering back home is indescribable,” she said. “I want to go back to my children.”

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Eid Under the Taliban Shows a Changed Afghanistan

KABUL, Afghanistan — Thousands of Afghans had piled into buses and set out down the country’s once perilous highways bound for relatives they had not seen in years. Afghanistan’s only national park was filled with tourists who had only dreamed of traveling to its intensely blue lakes and jagged mountains when fighting raged across the country.

And Zulhijjah Mirzadah, a mother of five, packed a small picnic of dried fruit, gathered her family in a minibus and wove for two hours through the congested streets of the capital, Kabul, to a bustling amusement park.

From the entrance, she could hear the low whoosh of a roller coaster and the chorus of joyous screams from Afghans inside celebrating Eid al-Fitr, the holiday marking the end of the holy month of Ramadan. But she could not go further. Women, she was told at the gate, were barred by the Taliban from entering the park on Eid.

“We’re facing economic problems, things are expensive, we can’t find work, our daughters can’t go to school — but we hoped to have a picnic in the park today,” said Ms. Mirzadah, 25.

country’s economic collapse since the Taliban toppled the Western-backed government, the freedom of travel and luxury of celebratory outings remained out of reach.

City Park, the amusement park in Kabul, and the city’s zoo, had less than half of the number of visitors that typically come each Eid, according to park managers. The low turnout was a reflection of both the country’s economic downturn and the Taliban’s edict barring women from visiting on Eid — the latest in a growing roster of restrictions on women in public spaces.

In a modest house tucked into one of Kabul’s many hillsides, Zhilla, 18, gathered with relatives at her aunt’s house on the second day of Eid. Her young cousins and siblings chased each other in the small courtyard. Inside, Zhilla marveled over her new cousin, just six days old, sleeping peacefully in her mother’s lap.

“The baby knows we’ve been through a lot, she needs to behave for us,” Zhilla joked.

The previous year, she and her relatives had gathered by the city’s Qargha reservoir for a picnic by the river, as boys and girls rode bicycles along its banks and took boats out on the water — a memory that feels like a lifetime ago, she said.

“This Eid is the same as any other day — we cannot go out, we cannot be free,” she said.

Najim Rahim contributed reporting from Houston.

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Live Updates: Russian Forces Battle in Ukraine’s East to Feed Putin’s Hunger for a Victory

DONETSK REGION, Ukraine — Fighting raged on Thursday across eastern Ukraine, from the Kharkiv area in the north where Ukrainian forces regained ground, to Mariupol in the south, where Russians breached the last Ukrainian redoubt in a steel plant, as Moscow’s forces battled to present President Vladimir V. Putin with something he can call victory.

Some of the most ferocious combat took place between those two poles, in or near the north of the Donetsk region, where the earth heaved with constant artillery bombardment. Russian forces approached from the east, north and south, vainly trying to trap and destroy Ukrainian units in and around the cities of Kramatorsk and Sloviansk, and the towns of Lyman and Barvinkove.

At a busy medical field hospital in that cauldron, where the smoke of battle dulled the spring sunlight, a Ukrainian soldier with a concussion lay curled into a fetal position, while another, his face half torn away, lay dead in a black body bag. In Kramatorsk, now largely abandoned, three Russian airstrikes gutted a large apartment complex and a store selling bras and underwear, injuring 26 people.

Credit…Lynsey Addario for The New York Times

The Kremlin is determined to reach some kind of milestone, Western officials and analysts say, by May 9, the day Russia commemorates the Soviet Union’s triumph over Nazi Germany with a military parade full of bombast and martial spirit that Mr. Putin has turned into something close to a religious holiday. After more than two months of his vaunted military’s halting performance and heavy losses in Ukraine, they say, Russia’s autocratic leader needs something to show for the war’s massive cost in lives and treasure.

But it is difficult to evaluate how the actual fighting is going. The Russian advance appears to have been sluggish, with forces taking a few villages each day in one location, while losing just as many in another. Ukrainian forces are mounting a highly mobile defense, maneuvering in small units around the larger masses of Russian forces, ensuring that lines remain fluid and unpredictable.

“The front is swinging this way and that,” said a tattooed 24-year-old army paramedic named Zhenya who was resting at the field hospital. “At first they weren’t hitting nearby here, now shells are coming in over the fence.”

In Mariupol, perhaps the city most devastated by the Russian invasion that began on Feb. 24, furious close-quarters combat shook the sprawling Azovstal steel plant, as Russian forces finally began to penetrate the complex where the last Ukrainians have held out for two months in a warren of underground bunkers. The number of Ukrainian fighters remaining is unclear, but Ukrainian officials said that even after a recent trickle of evacuations, about 200 civilians are still trapped there.

“Heavy, bloody battles are raging,” Lt. Col. Denys Prokopenko, a Ukrainian commander at Azovstal, said in a video posted Wednesday night. On Thursday, Petro Andriushchenko, an adviser to the city government, said that with nonstop shelling and fighting, the plant had been “turned into hell.”

Credit…Associated Press

In its latest assessment, the Institute for the Study of War, a research organization in Washington, said that Moscow wanted “to claim complete control of Mariupol by May 9, with Russian propagandists recently arriving in the city to set conditions for further claims of a Russian victory.”

With Russian efforts now concentrated farther south, Ukrainian forces have been pushing the Russians back in the Kharkiv area, recapturing towns and villages, and in some cases forcing Russian units beyond artillery range of the battered city.

The Kremlin had a muted response on Thursday to The New York Times’s report that the United States had supplied intelligence to Ukrainian forces that had helped them locate and kill Russian generals. Russia was already “well aware” that NATO and its member countries were sharing intelligence with Ukraine, said Dmitri S. Peskov, Mr. Putin’s spokesman, who added that Western aid only lengthens the war and “cannot prevent the fulfillment” of Russia’s goals.

The Pentagon spokesman, John F. Kirby, declined to comment directly, but said the United States did not specifically provide intelligence on the locations of Russian officers, “or participate in the targeting decisions of the Ukrainian military.”

After Russia’s initial drive in the north failed to take Kyiv, Ukraine’s capital, its forces withdrew and began to focus on capturing territory in the eastern Donetsk and Luhansk regions, but their progress has been slow and costly.

In a striking moment of candor, Mr. Putin’s closest foreign ally, Aleksandr G. Lukashenko, the ruler of neighboring Belarus, called the fighting a war — a term forbidden in Russia — and acknowledged that it was not going well for Russia. “I feel like this operation has dragged on,” he said in an interview with The Associated Press.

Credit…Finbarr O’Reilly for The New York Times

In the north of Donetsk, the dead and wounded flowed into the field hospital at a regular clip as Russian artillery pounded the rolling, wooded hills where Ukrainian troops were mounting their defense.

On a visit on Thursday, ordnance whizzed, thumped and boomed in all directions. Military paramedics brought wounded soldiers to the field hospital to stabilize them before sending them by ambulance to a military hospital farther from the front.

Ukrainian military officials asked that the precise location of the field hospital, about a 25-minute drive from Kramatorsk, be withheld to prevent the Russians from targeting it. Even so, Russian artillery shells landed nearby.

The toll on Ukrainian forces could be measured by the columns of ambulances racing away from the front lines, even as trucks and armored vehicles carrying troops and equipment headed in the opposite direction.

“We’re not making any kind of prognoses,” said Valeria Skorik, a press officer for the 81st brigade, among the units fighting in the northern part of the Donetsk region. “I’ve been asked by journalists about what kind of event we might have on May 9, but I’ve just decided not to answer.”

Credit…Lynsey Addario for The New York Times

Western officials and analysts say that Mr. Putin could be planning to make a dramatic announcement on Victory Day, when he traditionally reviews the parade from an elevated platform in Red Square and delivers a speech surrounded by aged World War II veterans. He often has other heads of government with him, too, but the war has left Russia largely isolated, and the Kremlin says no foreign leaders were invited this year.

Speculation has centered on a possible claim of victory by Mr. Putin or, more ominously, an acknowledgment that Russia is at war and the announcement of a mass mobilization with expanded conscription, a move that would be unpopular.

Ukrainian forces in and around northern Donetsk appear to be holding the line for now, offering poor prospects for a Russian achievement there, despite Russia’s incessant hammering at Ukrainian military positions and towns.

The airstrike on Kramatorsk left a large crater and generated a shock wave so powerful that it blew out the interior walls of a row of apartments about 75 feet away and ripped steel doors off hinges. Touring the damage, Pavel Kirilenko, chief of the Donetsk region’s military administration, said that remarkably, no one had been killed.

Credit…Lynsey Addario for The New York Times

“This is yet more confirmation that everyone needs to leave the city,” Mr. Kirilenko said. “The enemy is exclusively targeting elements of civilian infrastructure in order to spread panic — and not only spread panic, but to destroy the civilian population.”

In anticipation of a potential assault, officials have urged anyone who is able to leave the city as soon as possible. Many have done so: The streets of Kramatorsk, an industrial and administrative center with a prewar population of about 150,000, are largely empty. Most businesses are shuttered. Each day, buses leave the city center, evacuating residents to points west.

But not everyone has heeded the calls to leave. Inside the destroyed apartment building on Thursday was a woman in a bathrobe, cradling a small dog. She gave only her first name, Viktoria.

The explosion, at about 4:30 a.m., blew her balcony and the entire front wall of her apartment onto her and her husband as they slept. Her husband suffered a large head wound; drops of blood stained the mattress and floor. Her 24-year-old daughter was left with a broad cluster of bloody cuts from flying glass.

She said local officials had urged her to shelter in a school, at least for the night. But she said she just wanted to seal the front of her apartment in plastic to keep out the elements, and stay there for the night.

“There is shelling everywhere,” she said. “So where are we supposed to go?”

Credit…Lynsey Addario for The New York Times

For the last defenders of Mariupol, long cut off from outside aid with their numbers and supplies dwindling, the situation was even more dire.

Russian forces managed to find their way into the four-square-mile Azovstal complex where they have been sheltering with the help of a former worker familiar with its layout, according to Anton Gerashchenko, an adviser to the Ministry of Internal Affairs. Mr. Gerashchenko, on social media and speaking to reporters, said that an electrician who had worked at the steel plant showed the Russians tunnels to enter the complex.

He said the Russian desire to declare victory on May 9 explained why Russian state television hosts, who are some of Mr. Putin’s leading cheerleaders — including Vladimir Solovyov, under U.S. and European sanctions for promoting Kremlin disinformation — have traveled to Mariupol.

Communications from Azovstal briefly went dark on Wednesday, but on Thursday morning, fighters in the bunkers were again sending messages via social media platforms, promising not to surrender.

“It has been three days since Russian troops broke into the territory of Azovstal,” said Capt. Svyatoslav Palamar, the deputy commander of the Azov regiment at the plant. “Heavy fighting continues to take a bloody toll.”

Reporting was contributed by Richard Pérez-Peña from New York, Cora Engelbrecht and Marc Santora from Krakow, Poland, and Anton Troianovski from Istanbul.

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Ukraine Live Updates: Seeing a Stalled Russia, West Adds Support and Arms for Ukraine

ZAPORIZHZHIA, Ukraine — Some flashed bright smiles and others bent over in heaving sobs, sharing the end of their hellish subterranean ordeal. Here, at last, were ordinary things they thought they might not live to see again: sunlight, enough food and escape from incessant Russian shelling.

On a fleet of city buses, flanked by white United Nations and Red Cross vehicles, nearly 130 women, children and elderly people on Tuesday reached the relative safety of Ukrainian-controlled territory, after weeks huddled in the belly of Mariupol’s sprawling steelworks.

They had sheltered in the near-darkness of underground bunkers, with little food or water as explosives of all shapes and sizes rained down day and night, slowly chipping away the steel and concrete overhead that was their only protection.

“For some reason I remember Easter, Easter Day,” said Inna Papush, who spent 58 days underground with her daughter, Dasha, 17. “We thought it would be a holy day and they would take a break,” she said of the Russian forces.

“But the shelling became even heavier,” Dasha said, completing her mother’s thought.

Leaders of the United States and Europe pressed harder on Tuesday to arm Ukraine, hinder the Kremlin and strengthen the NATO alliance — President Biden visited a factory that makes antitank missiles that have been vital to the Ukrainian cause — even as President Vladimir V. Putin of Russia warned that they were only making matters worse.

And in the parking lot of the Epicenter shopping complex in Zaporizhzhia, in southeastern Ukraine, evacuees from Mariupol stepped from buses, blinking in the sunshine. They were greeted by a parade of aid workers offering tea and snacks and a less-than-quiet place to rest in a large white tent buzzing with journalists, psychologists and the occasional politician. Children were given candy, while an air raid siren sounded briefly, ignored by all.

Their evacuation was a rare but limited victory for diplomacy, and an unusual concession to human dignity by Russian forces who have inflicted death and misery upon civilian populations across a broad swath of Ukraine since the war began on Feb. 24.

Negotiators from the United Nations and the International Committee for the Red Cross brokered a deal with the Russians that allowed for the civilians to escape the Azovstal steel plant, the sprawling complex that had been their refuge. But it came only after more than two months of intense attacks that have turned Mariupol, once a vivacious port city, into a ruin of bomb-blasted buildings and corpse-strewn streets. In addition to 127 evacuees who fled to Zaporizhzhia, about 30 escaped the plant but chose to remain in Mariupol, according to The Associated Press.

In the days leading up to a cease-fire that allowed the civilians to escape, Russian forces escalated their attacks on the plant, causing cave-ins that hampered rescue efforts and killing and injuring unknown numbers, according to Ukrainian officials and troops who are still there.

“I was in Azovstal for two and a half months and they slammed us from all sides,” said Olga Savina, an elderly woman, as she emerged from a white city bus provided by the Zaporizhzhia authorities for the evacuation.

As she spoke, she repeatedly cast her gaze down to the pavement, explaining that the sun burned her eyes after so many days underground.

From the evacuees a picture began to emerge of life in Azovstal. The steel mill was like a small city, with roads and buildings dating to the post-World War II era, when any big Soviet construction project included reinforced bomb shelters equipped with everything needed for long-term survival.

Evacuees described bunkers, most housing 30 to 50 people, with kitchens, bathhouses and sleeping areas. The shelters were spread out around the grounds of the complex, so there was little contact between groups hiding in different places.

There in the dark, a semblance of day-to-day life took shape.

“We got used to it being very dark. We had to economize food,” said Dasha Papush. “The soldiers brought us what they could: water, food, oatmeal.”

“We didn’t eat like we did at home,” she added.

Many of the evacuees had been underground since the earliest days of the war. For a woman named Anna, 29, who placated her young son, Ivan, with a lollipop, it was 57 days. While there, she was separated from her husband, a fighter in the National Guard, by a brisk, 15-minute walk through the factory ruins, though visits were rare because of the shelling and constant fighting.

Leaving the safety of the underground shelter was treacherous, but necessary for survival.

“The guys who are with us went out under fire and tried to find us a generator and fuel, so that we had electricity to charge our flashlights,” she said. “We of course had to search for water.”

For Sergei Tsybulchenko, 60, the reason to emerge was firewood. Scattered around the grounds of the factory were shipping palettes that he and a few men would collect and break up to fuel the cooking fire he and his fellow inmates had made in a part of their bunker. He and the 50 or so others crammed into his bunker would gather to prepare and share one meal a day, he explained — usually a mix of macaroni, oatmeal and canned meat, cooked all together in a large pot.

Mr. Tsybulchenko said the fire had to be kept low, for fear that it could be detected by thermal sensors on Russian jets.

“It was just always, boom, boom, boom, boom,” he said. “It was a real strain on the brain.”

Under constant bombardment, he said, the shelter began to disintegrate, with a portion of it collapsing.

Over the weekend, for the first time in weeks, it stopped.

In Mr. Tsybulchenko’s shelter, three soldiers with the Azov regiment, a Ukrainian military unit whose soldiers make up the bulk of those fighting at Azovstal, asked for anyone suffering from any illnesses to come forward. Mr. Tsybulchenko’s wife, Nelya, who has asthma, raised her hand. The couple walked out of the shelter into the sunlight with their daughter, her husband and a small dog.

Only 11 people from their bunker were chosen to leave, leaving some 40 others behind. Those who remained included a mother with her two children, who Mr. Tsybulchenko said was scared to leave because her husband was a high-ranking officer fighting at the plant.

“She was worried that if they found out, she would end up in a prison camp together with the children,” he said.

The mayor of Mariupol, Vadym Boichenko, said in a televised interview on Tuesday that more than 200 civilians were still hiding at the plant, and that more than 100,000 people remained scattered about the city. Inside Azovstal, supplies of food, water and medicines have dwindled to critical levels.

The Russians resumed shelling the plant almost immediately after international negotiators departed with evacuees, according to soldiers there. On Tuesday, Russian forces attempted to storm the complex after pummeling it with planes, tanks and artillery, Capt. Svyatoslav Palamar, the deputy commander of the Azov regiment at the plant, said in a statement on Telegram. The regiment released video showing the bodies of two women, who it said were killed in the renewed attack.

“We will do everything possible to repel this assault,” Captain Palamar said. “However, we call for immediate action to evacuate civilians from the plant’s grounds.”

It took Mr. Tsybulchenko and his family nearly two hours just to make it out of the complex. An elderly man who was with them had to be carried over twisted equipment, through massive craters and around unexploded ordinance.

Once outside, the evacuees were handed over to Russian troops and eventually put on buses for what would become a three-day, roundabout journey through dozens of checkpoints, where Russian soldiers fingerprinted and photographed them and interrogated them about the locations of Ukrainian fighters still at the plant.

At one point on the journey, Mr. Tsybulchenko looked off in the distance and saw the remains of Mariupol, the city of his birth. The apartment that his grandfather had received from the Soviet authorities in the 1960s and where he had lived since he was 3 years old was gone. On the horizon, he could make out the jagged shapes of the steel factory.

“A black smoke hung over Azovstal,” he said.

Cora Engelbrecht contributed reporting from Krakow, Poland.

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Ukraine Live Updates: Russia’s Top Officer Visited the East’s Front Line

BRUSSELS — House Speaker Nancy Pelosi traveled to Ukraine’s capital over the weekend, leading the second senior American delegation to meet with President Volodymyr Zelensky in a week and declare support for his country’s fight to beat back the Russian invasion.

With each visit — the secretaries of state and defense traveled to Kyiv last weekend — the promise of American commitment to a Ukrainian victory appears to grow, even as how the United States defines victory has remained uncertain.

On Sunday, a day after her visit to Ukraine, Ms. Pelosi told a news conference in Poland: “America stands with Ukraine. We stand with Ukraine until victory is won. And we stand with NATO.”

Ms. Pelosi, the second in line to succeed President Biden, is the highest-ranking American official to visit Kyiv since the war began, and her words carry weight, seeming to underscore an expanded view of American and allied war aims.

Her visit, with a congressional delegation, followed a joint visit to Kyiv by Secretary of State Antony J. Blinken and Defense Secretary Lloyd J. Austin III only last Sunday. Mr. Austin caused some controversy and debate afterward when he appeared to shift the goal of the war from defending Ukraine’s independence and territorial sovereignty to weakening Russia.

“We want to see Russia weakened to the degree that it can’t do the kinds of things that it has done in invading Ukraine,” Mr. Austin said, implying that the United States wanted to erode Russian military power for years to come — presumably so long as Vladimir V. Putin, president of Russia, remains in power.

Credit…Omar Marques/Getty Images

In one positive development on Sunday, the United Nations and the International Committee of the Red Cross helped organize what was described as an “ongoing” evacuation of civilians from the Azovstal steel works in Mariupol, where they have been taking shelter with a dwindling number of Ukrainian soldiers who have refused to surrender to the Russians. Between 80 and 100 civilians arrived in a convoy of buses at a temporary accommodation center 18 miles east of the city, in the village of Bezimenne.

The evacuation appeared to be the fruit of a visit to both Mr. Putin in Moscow and Mr. Zelensky in Kyiv last week by António Guterres, the U.N. secretary general, who called the war in Ukraine “an absurdity.” Mr. Guterres and the Red Cross have been working to get humanitarian aid and supplies of food and water to civilians trapped by the fighting; any serious peace negotiations still appear far off.

In a Twitter message, Mr. Zelensky applauded the evacuation of what he said was a “first group of about 100 people,” and said that “tomorrow we’ll meet them in Zaporizhzhia.”

The U.N. Office for the Coordination of Humanitarian Affairs, or OCHA, said in a statement that it would not provide details of the effort while it was continuing; further evacuations are expected to resume on Monday.

Russian forces have not yet been able to finally take the last slice of Mariupol, which no longer matters militarily but which has been an inspiring symbol of Ukrainian bravery, morale and resistance that is bound to go down in Ukrainian history.

But if there is a new allied consensus about supplying Ukraine with heavier and more sophisticated weapons for the latest stage of the war in eastern Ukraine, there is no allied consensus about switching the war aim from Ukraine to Russia.

Credit…Lynsey Addario for The New York Times

There is a sense in Europe that “the U.S. is dragging everyone into a different war,” said François Heisbourg, a French defense analyst, citing similar comments by President Biden about “the butcher of Moscow” and how “Putin must go.”

Some wonder what Washington is trying to say — or do.

“To help Ukraine prevail is not about waging war against Russia for reasons related to its governance,” Mr. Heisbourg said. “Regime change may be a vision, but not a war aim.”

He and others said that such talk from Washington plays perfectly into Mr. Putin’s narrative that NATO is waging war against Russia, and that Russia is fighting a defensive war for its survival in Ukraine. That may give Mr. Putin the excuse on May 9, the annual celebration of the Soviet victory over Nazi Germany, to declare this “special military operation” a war, which would allow him, if he chooses, to mobilize the population and use conscripts widely in the battle.

Talk of victory over Russia “gives easy ammunition to the other side and creates the fear that the West may go further, and it’s not what we want,” said Ulrich Speck, a German analyst. “We don’t want to cut Russia into pieces.”

Gérard Araud, a former French ambassador to the United States and the United Nations, commented on Twitter: “The support to Ukraine in its modalities and its objectives should be agreed at a political level between allies. Right now, we are sleepwalking to nobody knows where.”

In response, Moscow has raised the tone of its own rhetoric.

On Wednesday, Mr. Putin said that any countries who “create a strategic threat to Russia” during this war in Ukraine can expect “retaliatory strikes” that would be “lightning-fast.” Days before, the Russian foreign minister, Sergey V. Lavrov, said in an interview that “NATO is essentially going to war with Russia through a proxy and arming that proxy.”

Mr. Putin’s military, having lost what Britain estimates to have been at least 15,000 killed in action — that is more than in the Soviet Union’s entire war in Afghanistan — has been struggling to cut supply lines of Western arms, munitions and heavy weapons to Ukrainian forces in eastern Ukraine.

Credit…Daniel Berehulak for The New York Times

On Sunday, the Russians said they had bombed a runway and a munitions dump at a military airfield near Odesa that was storing Western arms, and Russia has been attempting to attack roads and especially railway terminals, since most heavy weapons are traveling east by rail. The Russian aim is to slowly cut off or encircle the bulk of Ukraine’s army east of the Dnipro River and starve it of new supplies.

But that grinding effort is going slowly, with fierce artillery battles and high casualties on both sides.

It is not just Ukraine’s military that is being starved of supplies. There is now a shortage of gasoline and diesel, at least for civilian use, stemming from Russia’s blockade of Ukrainian ports and attacks on refineries and fuel depots. Long lines for gasoline have been seen even in cities like Lviv, and there are concerns about the impact of the shortages on agriculture, even in fields untouched by the war.

A report by the U.N. Food and Agriculture Organization said that only a fifth of almost 1,300 large agribusinesses surveyed by the government in mid-March had enough fuel to operate the farm equipment needed to plant corn, barley and other crops this spring, which is already causing rising food prices in countries far from Ukraine.

In a possible indication of flagging Russian morale, the chief of staff of the Russian armed forces, Gen. Valery V. Gerasimov, the country’s top uniformed officer, made a visit to a dangerous frontline position in eastern Ukraine this weekend in an effort to “change the course” of Russia’s offensive there, according to a senior Ukrainian official with knowledge of the visit.

Ukrainian forces launched an attack on a Russian headquarters in Izium on Saturday evening, but General Gerasimov had already left to return to Russia, the official said. Still, some 200 soldiers, including at least one general, were killed, the Ukrainian official said, speaking on the condition of anonymity to discuss a sensitive military operation. A U.S. official, speaking on the condition of anonymity, confirmed that General Gerasimov had been in eastern Ukraine but did not confirm the rest of the Ukrainian account.

Fighting has intensified around the large eastern city of Kharkiv in recent days as Ukrainian forces have attempted to push away Russian units. Though the gains have been small, they are emblematic of both the Ukrainian and Russian forces’ strategy as the war drags into its third month, one that focuses on a village at a time and leverages concentrated artillery fire to dislodge one another.

Ukraine’s military said in a statement on Saturday that it had been able to retake four villages around Kharkiv: Verkhnya Rohanka, Ruska Lozova, Slobidske and Prilesne. The claims have been hard to verify since much of those areas are currently closed to the media; on Sunday, Ukraine announced that it had rebuffed Russian advances toward villages in the Donbas, but that, too, could not be confirmed.

Credit…Tyler Hicks/The New York Times

Ukrainian forces were also suspected of another attack over the border near the Russian city of Belgorod, a staging area for Russian forces, where a fire broke out in a defense ministry facility, the regional governor said.

The Russian forces in control of the southern Ukrainian city of Kherson and its surrounding province started to enforce a transition to the Russian ruble from Ukrainian currency on Sunday, a move that Ukrainian officials have described as part of an attempt to scrub a part of the country clean of its national identity and embed it in Moscow’s sphere of influence.

At the same time, the Ukrainians reported on Sunday that nearly all cellular and internet service in the area was down. The Ukrainian Ministry of the Interior accused Russian forces of cutting service, saying it was an attempt to keep Ukrainians from seeing truthful information about the war.

The Hollywood actor Angelina Jolie, who has been a special representative of the U.N. High Commissioner for Refugees since 2011, made her own surprise visit to Ukraine over the weekend, visiting the western city of Lviv to meet displaced Ukrainians from the east who have found refuge there, including children undergoing treatment for injuries sustained in Russia’s missile strike on the Kramatorsk railway station in early April.

Ms. Pelosi was accompanied by legislators whose comments largely echoed her own.

“This is a struggle of freedom against tyranny,” said Representative Adam Schiff, a Democrat from California. “And in that struggle, Ukraine is on the front lines.”

Credit…Finbarr O’Reilly for The New York Times

Representative Jason Crow, a Democrat from Colorado, a veteran and a member of the House intelligence and armed services committee, said his focus was on the supply of weapons. “We have to make sure the Ukrainians have what they need to win,” he said. Praising Ukrainian bravery, he said, “The United States of America is in this to win, and we will stand with Ukraine until victory is won.”

But as ever, what is meant by “victory,” whether it involves pushing Russia entirely out of Ukraine or just blocking its advance until its offensive runs out of steam and negotiations ensue, remains an open question. So does the equally central question of what Mr. Putin decides is victory enough for his own war of choice.

Steven Erlanger reported from Brussels, Jane Arraf from Lviv and Marc Santora from Krakow, Poland. Michael Schwirtz contributed reporting from Zaporizhzhia, Ukraine; and Thomas Gibbons-Neff from Kharkiv.

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How Tech Companies Are Trying to Woo Employees Returning to Work

When Google employees returned to their mostly empty offices this month, they were told to relax. Office time should be “not only productive but also fun.” Explore the place a little. Don’t book back-to-back meetings.

Also, don’t forget to attend the private show by Lizzo, one of the hottest pop stars in the country. If that’s not enough, the company is also planning “pop-up events” that will feature “every Googler’s favorite duo: food and swag.”

But Google employees in Boulder, Colo., were still reminded of what they were giving up when the company gave them mouse pads with the image of a sad-eyed cat. Underneath the pet was a plea: “You’re not going to RTO, right?”

R.T.O., for return to office, is an abbreviation born of the pandemic. It is a recognition of how Covid-19 forced many companies to abandon office buildings and empty cubicles. The pandemic proved that being in the office does not necessarily equal greater productivity, and some firms continued to thrive without meeting in person.

a happy hour with its chief executive, Cristiano Amon, at its San Diego offices for several thousand employees with free food, drink and T-shirts. The company also started offering weekly events such as pop-up snack stands on “Take a Break Tuesday” and group fitness classes for “Wellness Wednesday.”

the surveys, is that employees want to see colleagues in person.

After a number of postponements, Google kicked off its hybrid work schedule on April 4, requiring most employees to show up at U.S. offices a few days a week. Apple started easing staff back to the office on Monday, with workers expected to check in at the office once a week at first.

reimburse $49 monthly leases for an electric scooter as part of its transportation options for staff. Google also plans to also start experimenting with different office designs to adapt to changing work styles.

When Microsoft employees returned to their offices in February as part of a hybrid work schedule, they were greeted with “appreciation events” and lawn games such as cornhole and life-size chess. There were classes for spring basket making and canvas painting. The campus pub transformed into a beer, wine and “mocktail” garden.

And, of course, there was free food and drink: pizzas, sandwiches and specialty coffees. Microsoft paid for food trucks with offerings including fried chicken, tacos, gyros, Korean food and barbecue.

Unlike other technology companies, Microsoft expects employees to pay for their own food at the office. One employee marveled at how big a draw the free food was.

signed a letter urging management to be more open to flexible work arrangements. It was a rare show of dissent from the company’s rank-and-file, who historically have been less willing to openly challenge executives on workplace matters.

But as tech companies grapple with offering employees greater work flexibility, the firms are also scaling back some office perks.

cutting back or eliminating free services like laundry and dry cleaning. Google, like some other companies, has said it approved requests from thousands of employees to work remotely or transfer to a different office. But if employees move to a less expensive location, Google is cutting pay, arguing that it has always factored in where a person was hired in setting compensation.

Clio, a legal software company in Burnaby, British Columbia, won’t force its employees back to the office. But last week, it gave a party at its offices.

There was upbeat music. There was an asymmetrical balloon sculpture in Clio’s signature bright blue, dark blue, coral and white — perfect for selfies. One of Clio’s best-known workers donned a safari costume to give tours of the facility. At 2 p.m., the company held a cupcake social.

To make its work spaces feel more like home, the company moved desks to the perimeter, allowing Clions — what the company calls its employees — to gaze out at the office complex’s cherry blossoms while banging out emails. A foosball table was upgraded to a workstation with chairs on either end, “so you could have a meeting while playing foosball with your laptop on it,” said Natalie Archibald, Clio’s vice president of people.

Clio’s Burnaby office, which employs 350, is open at only half capacity. Spaced-out desks must be reserved, and employees got red, yellow and green lanyards to convey their comfort levels with handshakes.

Only around 60 people came in that Monday. “To be able to have an IRL laugh rather than an emoji response,” Ms. Archibald said. “People are just excited for that.”

Karen Weise contributed reporting.

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Live Updates: Missile Strike on Train Station Kills 50 People Seeking to Escape Fighting

One moment, they were packed onto the platforms at the Kramatorsk train station, hundreds of women, children and old people, heeding the pleas of Ukrainian officials imploring them to flee ahead of a feared Russian onslaught.

The next moment, death rained from the air.

At least 50 people were killed and many more wounded in a missile assault on Friday morning that left bodies and luggage scattered on the ground and turned the Kramatorsk station into the site of another atrocity in the six-week-old war.

“There are just children!” one woman cried in a video from the aftermath.

The missile struck as officials in Kramatorsk and other cities in eastern Ukraine had been warning civilians to leave before Russian forces mount what is expected to be a major push into the region, where their troops have been regrouping after withdrawing from areas around Kyiv, the capital.

President Volodymyr Zelensky of Ukraine said that Russia had hit the station with what he identified as a Tochka-U short-range ballistic missile as “thousands of peaceful Ukrainians were waiting to be evacuated.”

Credit…Fadel Senna/Agence France-Presse — Getty Images

“Lacking the strength and courage to stand up to us on the battlefield, they are cynically destroying the civilian population,” Mr. Zelensky said. “This is an evil that has no limits. And if it is not punished, it will never stop.”

Russian officials, denying responsibility, said a Ukrainian battalion had fired the missile in what they called a “provocation.” The Russian Defense Ministry said that Tochka-U missiles are only used by the Ukrainian armed forces and that Russian troops had not made any strikes against Kramatorsk on Friday.

A senior Pentagon official said the United States believed Russian forces had fired the missile. “They originally claimed a successful strike and then only retracted it when there were reports of civilian casualties,” said the official, who spoke on condition of anonymity to discuss a confidential intelligence assessment.

The train station was hit as a top European Union delegation was visiting Mr. Zelensky’s government, and the images of yet another mass killing provoked new Western outrage.

Whether one or more missiles struck the station was not immediately clear, and there was no way to independently verify the origin of the attack. Several parked cars were also hit, catching fire and turning into charred hulks. The waiting area was strewn with bodies and belongings.

After the strike, the Ukrainian police inspected the remains of a large rocket next to the train station with the words “for our children” written on it in Russian. It was unclear who had written the message and where the rocket had come from.

The mayor of Kramatorsk, Oleksandr Honcharenko, said 4,000 people had been at the station when it was attacked, the vast majority of them women, children and elderly people. At least two children were among the dead, he said.

The head of the military administration in the region, Pavlo Kyrylenko, said 50 people had been killed, including 12 who died in the hospital. Another 98 were wounded, including 16 children, he said.

After the attack, Kramatorsk officials said they were trying to find cars and buses to evacuate civilians to western areas presumed to be less vulnerable to Russian attacks.

Credit…Andriy Andriyenko/Associated Press

Ukraine’s railway service said that evacuations would proceed from nearby Sloviansk, where shelters and hospitals have been stocked with food and medicine in anticipation of an imminent Russian offensive.

Western countries, which have been shipping arms to Ukraine and tightening sanctions on Russia to punish President Vladimir V. Putin for the invasion, saw the Kramatorsk slaughter as new justification to intensify their efforts.

“The attack on a Ukrainian train station is yet another horrific atrocity committed by Russia, striking civilians who were trying to evacuate and reach safety,” President Biden said on Twitter. He vowed to send more weapons to Ukraine and to work with allies to investigate the attack “as we document Russia’s actions and hold them accountable.”

President Emmanuel Macron of France called the strike “abominable.”

“Ukrainian civilians are fleeing to escape the worst,” he wrote on Twitter. “Their weapons? Strollers, stuffed animals, luggage.”

The station was hit as the Slovak president, Eduard Heger, and the president of the European Commission, Ursula von der Leyen, were traveling to Kyiv in a show of support for Mr. Zelensky and his country’s bid for European Union membership.

Mr. Heger announced that Slovakia had given Ukraine an S-300 air defense system to help defend against Russian missiles and airstrikes.

Credit…Andriy Andriyenko/Associated Press

To make the transfer possible, the Pentagon said it would reposition one Patriot missile system, operated by U.S. service members, to Slovakia. It was the latest buildup in arms and troops along NATO’s eastern flank, as the alliance seeks to deter any Russian incursion.

“Now is no time for complacency,” Mr. Biden said in a statement announcing the Patriot repositioning. “As the Russian military repositions for the next phase of this war, I have directed my administration to continue to spare no effort to identify and provide to the Ukrainian military the advanced weapons capabilities it needs to defend its country.”

The attack on the railway station came after Russian forces had spent weeks shelling schools, hospitals and apartment buildings in an apparent attempt to pound Ukraine into submission by indiscriminately targeting civilian infrastructure, ignoring Geneva Convention protections that can make such actions war crimes.

Last month, an estimated 300 people were killed in an attack on a theater where hundreds had been sheltering in the battered port of Mariupol, Ukrainian officials said. In recent days, growing evidence has pointed to atrocities in the devastated suburbs of Kyiv, where Ukrainian troops found bodies bound and shot in the head after Russian forces had retreated.

Ms. von der Leyen visited one of those suburbs, Bucha, on Friday before meeting with Mr. Zelensky.

“It was important to start my visit in Bucha,” she wrote on Twitter. “Because in Bucha our humanity was shattered.”

Russia has said its troops have been falsely accused and that the evidence against them is fake.

The repercussions of the fighting are spreading far beyond Europe. The United Nations reported on Friday that world food prices rose sharply last month to their highest levels ever, as the invasion sent shock waves through global grain and vegetable oil markets. Russia and Ukraine are important suppliers of the world’s wheat and other grains.

Credit…Andriy Andriyenko/Associated Press

The report of rising prices came as the British government said Russia was heading for its “deepest recession since the collapse of the Soviet Union,” estimating that the economy could shrink by as much as 15 percent this year.

On Friday, the European Union formally approved its fifth round of sanctions against Moscow, which included a ban on Russian coal and restrictions on Russian banks, oligarchs and Kremlin officials. The coal ban, which will cost Russia about $8.7 billion in annual revenue, takes effect immediately for new contracts. At Germany’s insistence, however, existing contracts were given four months to wind down, softening the blow to Russia and Germany alike.

Nevertheless, Prime Minister Boris Johnson of Britain, meeting with the German chancellor, Olaf Scholz, in London on Friday, applauded what Mr. Johnson called the “seismic decision” by Germany to turn away from Russian fuel. Britain has pushed for a total ban on Russian energy, a move that Germany, which heats half its homes with Russian gas, has resisted.

Mr. Johnson acknowledged the obstacles to transforming Germany’s energy system “overnight,” but said “we know that Russia’s war in Ukraine will not end overnight.” Mr. Scholz said Mr. Putin had tried to divide European powers, but “he will continue to experience our unity.”

On Friday, Russia retaliated for some of the punishments from the West, declaring 45 Polish Embassy and Consulate staff “persona non grata,” and ordering them to leave Russia. Poland had expelled the same number of Russian diplomats.

Russia’s Justice Ministry also said it had revoked the registration of several prominent human rights groups in the country, including Human Rights Watch and Amnesty International, which have accused Russian troops of committing war crimes in Ukraine. The ministry accused the groups of violating an unspecified Russian law. The decision means the organizations are no longer allowed to operate in Russia.

Human Rights Watch said that forcing its office to close would not change its determination to call out Russia’s turn to authoritarianism. The group said it had been monitoring abuses in Russia since the Soviet era.

“We found ways of documenting human rights abuses then, and we will do so in the future,” it said.

Credit…Anatolii Stepanov/Agence France-Presse — Getty Images

Megan Specia reported from Krakow, Poland, and Michael Levenson from New York. Reporting was contributed by Jane Arraf from Lviv, Ukraine, Aurelien Breeden from Paris, Ivan Nechepurenko from Istanbul, Matina Stevis-Gridneff from Brussels, Michael D. Shear and Eric Schmitt from Washington, and Mark Landler and Chris Stanford from London.

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Ukraine Live Updates: 3,000 Civilians Escape Mariupol

The most ambitious effort yet to evacuate desperate civilians from Ukraine’s devastated port of Mariupol, besieged by Russian forces for weeks, was upended by disruptions Friday, with thousands of residents managing to flee but many more still stuck after the Red Cross judged the exodus too dangerous.

The suspended Red Cross evacuation in Mariupol, a city that has come to symbolize the horrors of the war in Ukraine, was among several developments painting a mixed picture on Friday as one of the biggest armed conflicts to convulse Europe in decades rumbled into its sixth week.

New signs emerged that Russian forces, stymied by their own botched planning and fierce Ukrainian resistance, were retreating from areas outside of Kyiv, the capital, and moving north. Ukrainians asserted that they had retaken control of more than two dozen suburban towns and hamlets.

Ukrainian helicopter gunships struck an oil terminal inside Russia, Russian officials said — which, if confirmed, would be the first known Ukrainian airstrike in Russian territory since the Feb. 24 invasion.

Such an attack would be both embarrassing and potentially provocative to President Vladimir V. Putin of Russia in his troubled military campaign to subjugate Ukraine. Ukrainian officials gave conflicting accounts on whether Ukraine was responsible.

Credit…Alexander Ermochenko/Reuters

And in Chernobyl, the toxic defunct nuclear site in northwest Ukraine, which Russia seized in the war’s early days, Russian vehicles apparently stirred up radiation dust as they departed, the United Nations’ top nuclear official said. Whether Russian soldiers or others there suffered radiation poisoning remained unclear.

There had been some early optimism that an organized large-scale evacuation of Mariupol — a thriving port of 450,000 that has been obliterated by Russian shelling and bombs — could be undertaken Friday under the auspices of the Red Cross, after Russia’s Defense Ministry approved a temporary cease-fire.

Many thousands of civilians have been trapped in the city for weeks under constant Russian bombardment with limited access to food, water and electricity. Previous attempts at humanitarian pauses in fighting have repeatedly collapsed.

By some estimates, three-quarters of Mariupol’s population has fled and roughly 100,000 people remain.

A team from the Red Cross that had been en route to Mariupol on Friday to escort out buses and cars carrying civilians had to turn back because it was not guaranteed conditions to ensure safe passage, the organization said in a statement. It said the team, made up of three vehicles and nine personnel, would try again on Saturday.

“For the operation to succeed, it is critical that the parties respect the agreements and provide the necessary conditions and security guarantees,” the Red Cross statement said.

Credit…Alexander Ermochenko/Reuters

The Red Cross had expected about 54 buses, along with an unknown number of private vehicles, to take part in an evacuation convoy carrying thousands of people.

While the larger convoy failed on Friday, smaller groups of people have been able to leave the city in cars, according to local officials. On Friday afternoon, Iryna Vereshchuk, the deputy prime minister, confirmed in a statement on her Telegram page that a corridor had opened from Mariupol to the city of Zaporizhzhia by private transport.

Around noon on Friday, Pyotr Andryuschenko, the Mariupol mayor’s adviser, said that some buses had left for nearby Berdyansk.

By day’s end, it remained unclear exactly how many people from Mariupol had been able to leave. But Kyrylo Tymoshenko, a top aide to President Volodymyr Zelensky of Ukraine, said on his Telegram account that roughly 3,000 people had managed to escape on Friday, and that more than 3,000 had been evacuated from other cities.

The Russians signaled a week ago that they might be pulling forces back from Kyiv and other areas in northern Ukraine and recalibrating their aims in the war, as it became increasingly clear that their military was performing poorly and Kremlin expectations of a quick victory were wrong.

Credit…Daniel Berehulak for The New York Times

Western officials and analysts were initially skeptical, suspecting the Russians were simply repositioning and resupplying for new attacks. While that may still be true, the Russian pullback from the Kyiv area after more than a week of Ukrainian counterattacks appeared to be real, these officials and analysts said, based on Ukrainian military accounts of retaken towns and other signs, including social media videos and satellite images, pointing to a Russian retreat.

“The counterattacks probably prompted the Russian decision to give up on Kyiv,” said Frederick W. Kagan, a military expert with the American Enterprise Institute in Washington. “The counterattacks demonstrated that the Russians were actually not going to be able to hold the positions they occupied anyway.”

The early-morning helicopter assault on Russian territory took place in Belgorod, part of a staging area for the Russian invasion about 20 miles from the eastern Ukrainian border. Ukraine’s military previously had managed to hit Russian territory only with ground-launched missiles, and Russia had boasted that the Ukrainian Air Force had been “practically destroyed” in Russian assaults.

Video posted to VKontakte, a Russian social media site, and verified by The New York Times showed two helicopters firing at the oil depot on the eastern edge of the city. Although it was not possible to determine the nationality of the aircraft, the footage confirmed that an airstrike caused a fire at the site. Other video of the aftermath showed the depot burning well into the daylight hours.

Credit…Russian Emergencies Ministry, via Agence France-Presse — Getty Images

Ukrainian officials were initially evasive about whether Ukraine’s forces had carried out the assault, but a top security aide, Oleksiy Danilov, issued what amounted to a denial by saying, “This does not correspond with reality.”

Whether or how Russia intended to respond remained unclear late Friday, but the attack did not appear to bode well for diplomacy to halt the war. Russia’s deputy permanent representative at the United Nations, Dmitry Polyanskiy, told reporters that such attacks “reflect the real intentions of the Ukrainian side and real intentions toward peace talks.”

Concerns about possible radiation exposure from the Russian seizure of Ukraine’s Chernobyl nuclear plant, where a 1986 meltdown caused the worst radiation accident in history, surfaced again on Friday in remarks by Rafael Mariano Grossi, the head of the International Atomic Energy Agency, the U.N. nuclear monitor. The Russians took control of the Chernobyl area last month and withdrew this week.

While Mr. Grossi told a news conference at the agency’s headquarters in Vienna that radiation levels had not changed at the plant, he said heavy military vehicles had stirred up contaminated ground when Russian forces first invaded the area, “and apparently this might have been the case again on the way out.”

Mr. Grossi said that he was aware of reports that some Russian military personnel had been poisoned by radiation while they held the Chernobyl plant but that the subject had not been discussed during talks he held in Russia with nuclear officials.

“We heard about the possibility of some personnel being contaminated, but we do not have any confirmation,” he said.

Credit…Bulent Kilic/Agence France-Presse — Getty Images

Outside Ukraine, nations that have sought to penalize Russia by banning purchases of Russian oil took further steps Friday to help insulate themselves from the economic shock of higher oil prices caused by the reduced supply.

The International Energy Agency, a 31-member group of oil-consuming nations, said they had agreed to a new release of emergency oil reserves in what is turning into a historic, wide-reaching effort to calm global markets.

The move came a day after the Biden administration announced a 180-million-barrel release over six months from the strategic reserve held by the United States.

Megan Specia reported from Krakow, Poland, Anton Troianovski from Istanbul, Matthew Mpoke Bigg from London and Julian E. Barnes from Washington. Reporting was contributed by Carlotta Gall from Kyiv, Ukraine; Ivan Nechepurenko from Istanbul; Josh Holder and Stanley Reed from London; Lazaro Gamio from Washington; and Denise Lu from New York.

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Live Updates: Russian Pledge to Ease Attack Is Greeted Skeptically by West

ISTANBUL — The first signs of significant progress in peace talks between Russia and Ukraine emerged on Tuesday, but there was no hint of an imminent end to the suffering, with Russia appearing determined to capture more territory in eastern Ukraine and officials predicting that weeks of further negotiation were needed.

After three hours of talks in Istanbul, Ukrainian officials said their country was ready to declare itself permanently neutral — forsaking the prospect of joining NATO, a key Russian demand — and discuss Russian territorial claims in exchange for “security guarantees” from a group of other nations. An aide to Ukraine’s president called the Russian delegation “constructive,” while Russia said it would “drastically” scale back its military activity around Kyiv to “increase mutual trust.”

Russia’s statement that it will de-escalate the fighting around Kyiv — even as it keeps pounding other parts of Ukraine — may be little more than putting a positive gloss on its military being stymied in its attempts to seize or encircle the capital. In recent days, Ukrainian counteroffensives around the city have forced back Russian forces in some of the fiercest street battles of the war, though they remain within striking distance of Kyiv.

Now, Russian officials said, the goal will be to take more territory in Ukraine’s eastern Donbas region, where Russia has installed two separatist statelets that President Vladimir V. Putin recognized last month as independent, but that no other nation has formally acknowledged.

Western officials and security analysts cautioned against taking at face value Russia’s statements about its aims in Kyiv or elsewhere.

Credit…Ivor Prickett for The New York Times

President Biden said he would not draw any conclusions about Russia’s intentions “until I see what their actions are.” Speaking after a White House meeting with the prime minister of Singapore, Mr. Biden added, “We’ll see if they follow through with what they are suggesting.”

“In the meantime,” Mr. Biden said, “we’re going to continue to keep strong the sanctions and will continue to provide the Ukrainian military with the capacity to defend themselves.”

Secretary of State Antony J. Blinken, on a diplomatic trip to Morocco, told reporters, “There is what Russia says and there’s what Russia does,” adding, “and what Russia is doing is the continued brutalization of Ukraine and its people.”

President Volodymyr Zelensky of Ukraine has worked hard to present Mr. Putin with a negotiated way to end the war, accompanied by Ukrainian concessions that would not go so far as to make his country a Russian satellite.

The offer to declare a permanent neutral status, Ukrainian officials in Istanbul said, means it would neither join the NATO alliance nor host foreign troops — a scenario that Mr. Putin used as one of the justifications for his invasion.

Ukrainian officials envision an arrangement in which a diverse group of countries — potentially including the United States, Germany, Turkey and China — would commit, if Ukraine were attacked, to providing it with military assistance and to imposing a no-fly zone if necessary. It was not clear that any of those countries had signed on to such guarantees.

Credit…Nacho Doce/Reuters

“This is probably the best outcome that could’ve been hoped for today,” said Samuel Charap, who studies Russian foreign policy at the RAND Corporation. “The Ukrainians have at least come up with something concrete to address the core Russian demand for Ukraine’s neutrality.”

Ukraine also signaled readiness to consider concessions related to the parts of its territory already occupied by Russia. It proposed a 15-year negotiating process for Crimea, the Ukrainian peninsula seized by Russia in 2014, and said it was ready to rule out trying to retake it by force. Questions surrounding the eastern Donbas region, Ukrainian officials said, could be discussed at a possible meeting between Mr. Putin and Mr. Zelensky.

Russia, for its part, said it was prepared to accelerate planning for such a summit meeting — something that Mr. Zelensky had long sought and that Moscow had resisted, casting Ukraine’s government as a mere puppet of Washington. But now, the Kremlin appears to be increasingly prepared to deal with Mr. Zelensky — in part because Ukraine’s resistance on the battlefield is leaving it no other choice.

Vladimir Medinsky, the head of Russia’s delegation, said that he viewed Ukraine’s proposals as “a constructive step in the search for a compromise.”

“If the treaty is worked out quickly and the required compromise is found, the possibility of making peace will be much closer,” Mr. Medinsky said.

Ukraine and Russia would need at least two more weeks for talks with Russia and potential guarantor countries, Oleksandr Chalyi, a member of the Ukrainian delegation, told reporters after Tuesday’s session. Turkey said it was prepared to host a meeting of the Russian and Ukrainian foreign ministers to flesh out Tuesday’s talks, followed by a possible meeting between the two presidents.

“You have shouldered a historic responsibility,” President Recep Tayyip Erdogan of Turkey, who hosted Tuesday’s talks at a 19th-century Ottoman palace on the banks of the Bosporus, told the Russian and Ukrainian delegates. “All the world is expecting good news from you.”

Credit…Murat Cetin Muhurdar/Turkish Presidential Press Service, via AFP — Getty Images

Turkey has emerged as a pivotal intermediary in the talks, one of the few countries to maintain close ties to both Russia and Ukraine after the invasion. Though it is a NATO member, Turkey has refused to put in place sanctions against Russia, even as it offers military support to Ukraine. To underscore the latter, a member of the Ukrainian delegation, Mykhailo Podolyak, posted on Twitter on Tuesday a selfie with Haluk Bayraktar, a close associate of Mr. Erdogan and the head of a Turkish company that makes armed drones used by Ukraine.

But despite the positive signals, myriad diplomatic pitfalls remained. For example, Ukraine said that the international security guarantees would not apply to the disputed Donbas, but it is not clear how that area would be defined; the separatists claim far more territory than they controlled before the war.

“The Russian delegation is constructive,” Mr. Podolyak, who is an aide to Mr. Zelensky, said. “This doesn’t mean that the negotiations are easy. They are difficult.”

There was also no reprieve on the ground in Ukraine. On Tuesday morning, a Russian cruise missile strike destroyed part of the main regional government office building in the southern city of Mikolaiv, killing at least nine people and injuring at least 28, Ukrainian officials said.

It was one of many attacks since the Feb. 24 invasion that appeared aimed at disabling government operations; the regional administrator, Vitaly Kim, said his own office was destroyed, but he was not in the building at the time. The Russian advance along the Black Sea coast west of Crimea stalled outside Mikolaiv, in an area that has seen intense combat.

Another Russian missile struck an oil depot on Monday night in the Rivne region in northwestern Ukraine, the second to be destroyed there, according to the regional administrator.

“The scale of the challenges has not diminished,” Mr. Zelensky said in a statement. “The Russian army still has significant potential to continue attacks against our state. They still have a lot of equipment and enough people completely deprived of rights whom they can send to the cauldron of war. Therefore, we stay alert and do not reduce our defense efforts.”

The U.S. and British governments confirmed Ukrainian gains in towns around Kyiv — notably Irpin, scene of some of the fiercest street fighting — but advised skepticism about claims that Russia’s stance had changed.

Credit…Daniel Berehulak for The New York Times

“Has there been some movement by some Russian units away from Kyiv in the last day or so? Yeah, we think so, small numbers,” said John F. Kirby, the Pentagon spokesman. But, he added, “We believe that this is a repositioning, not a real withdrawal, and that we should be prepared to watch for a major offensive against other areas in Ukraine.”

The British Defense Ministry warned that “Russia still poses a significant threat to the city through their strike capability.”

Russia has signaled that it was narrowing its war aims to focus on taking more territory in the Donbas. Sergei K. Shoigu, the Russian defense minister, on Tuesday offered a possible justification for winding down the war effort elsewhere in the country by declaring, in televised remarks in Moscow, that Russia’s initial mission was accomplished.

“In general, the main goals of the first stage of the special operation have been completed,” Mr. Shoigu said. “The combat potential of the Ukrainian Armed Forces has been significantly reduced, which makes it possible to focus the main attention and main efforts on achieving the main goal — the liberation of Donbas.”

Russian forces have taken control of a land bridge along Ukraine’s southeastern coast linking Crimea to the Donbas. The lone holdout in that strip is the center of the devastated port city of Mariupol, where, the Institute for the Study of War reported on Tuesday, the Russians are still gaining ground, slowly tightening the noose around the fighters and civilians who remain.

Thousand of civilians have been killed in Mariupol, according to local officials — a claim that cannot be independently verified — and much of the city has been flattened. It remains under “continuous heavy shelling” by Russian forces, the British Defense Ministry said on Tuesday.

Credit…Mauricio Lima for The New York Times

In the nearby port of Berdyansk, captured by the Russians, an evacuation convoy took 34 buses full of civilians out of the city, headed into territory held by Ukraine. In anther occupied city in the region, Melitopol, the mayor said the schools chief had been detained after she and local teachers refused orders by the occupying forces to change what was taught and to teach in Russian, not Ukrainian.

The war has cost Ukraine $564.9 billion in damage and lost economic activity — roughly three times its prewar gross domestic product — Yulia Svyrydenko, the economy minister, said in a Facebook post on Monday.

There are as yet no reliable estimates of civilian casualties, and some four million people have fled the country, along with about six million who are internally displaced, according to the United Nations.

Reporting was contributed by Ivan Nechepurenko and Safak Timur from Istanbul; Megan Specia from Krakow, Poland; Michael D. Shear from Washington; and Lara Jakes from Rabat, Morocco.

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