We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $(\varphi(N)$? We show that this can be done, under certain size conditions on the prime factors of $N$. The key technique is lattice basis reduction using the LLL algor...

a toroidalization of a dominant morphism $varphi: xto y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $x$ and $y$. we give a proof of toroidalization of locally toroidal morphisms of 3-folds.

We point out that, if quantum field renormalization is taken into account and the counterterms are evaluated at the Hubble-radius crossing time or few e-foldings after it, the predictions of slow-roll inflation for both the scalar and the tensorial power spectrum change significantly. This leads to a change in the consistency condition that relates the tensor-to-scalar amplitude ratio with spec...

Let  $left{a_alpharight}_{alphain I}$ be a bounded net in a Banach algebra $A$ and $varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $|aa_alpha-varphi(a)a_alpha|to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such  as Segal algebras and $L^1$-algebras are resp...

‎generalizing the notion of character amenability for banach‎ ‎algebras‎, ‎we study the concept of $varphi$-connes amenability of‎ ‎a dual banach algebra $mathcal{a}$ with predual $mathcal{a}_*$‎, ‎where $varphi$ is a homomorphism from $mathcal{a}$ onto $bbb c$‎ ‎that lies in $mathcal{a}_*$‎. ‎several characterizations of‎ ‎$varphi$-connes amenability are given‎. ‎we also prove that the‎ ‎follo...

the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

Glucose oxidation to CO(2) in man at the fasted, steady state has been investigated in normal, hypothyroid, patients by monitoring the specific activity of plasma glucose and expired CO(2) after intravenous injection of glucose-1-(14)C, glucose-6-(14)C, and sodium bicarbonate-(24)C in tracer amounts. Making certain stoichiometric assumptions about the oxidation of the C-1 and C-6 carbons of glu...