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Ukraine's new president has demanded that fighting between pro-Russian insurgents and government troops in the east of the country come to a halt by the end of the week.

The deadly violence has intensified in recent weeks, with Ukraine stepping up its "anti-terrorism" operation after Petro Poroshenko won the country's presidential elections on May 25.

Poroshenko said Sunday, one day after being sworn in as leader, that there must be a cease-fire this week. But his calls for peace have been rejected in the past by rebel leaders who see them as one-sided demands for the insurgents to lay down their arms.

"We must cease-fire this week," Poroshenko told a meeting Sunday, according to his official website. "Each day when people die, when Ukraine pays such a high price, is inadmissible for me."

Ukraine's President Petro Poroshenko walks in front of the guard of honor during a ceremony of his inauguration in Kiev on June 7, 2014. Petro Poroshenko was sworn in as Ukraine's fifth post-Soviet president today, vowing to maintain the unity of his country amid a continuing crisis with Russia. Poroshenko, a 48-year-old billionaire who won the presidential election on May 25 with 54.7 percent of the vote, took the oath in the Ukrainian parliament in Kiev.SERGEI SUPINSKY / AFP - Getty Images

"That is why we must first of all restore the work of the state border of Ukraine in order to guarantee the security of all Ukrainian citizens living in the Donbas [east Ukraine] irrespective of their political views." he added.

Ukrainian troops and separatists have been mired in fighting largely in the Donetsk region, one of two areas to declare themselves independent from the capital Kiev and ask to be annexed by Russia.

- Alexander Smith