The basic processing unit of a neural network i s a linear threshold element. I t has been known that neural networks can be much more powerful than traditional logic circuits, assuming that each threshold element can be built at a cost comparable to that o f AND, OR, Nor logic elements. Whereas any logic circuit o f polynomial size (in n) that computes the product of two n-bit numbers requires...

Rational arithmetic functions are arithmetic functions of the form g1 ∗···∗ gr ∗ h−1 1 ∗ ···∗h−1 s , where gi, hj are completely multiplicative functions and ∗ denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type ide...

Abstract We show that the conjecture of [27] for special value at $s=1$ zeta function an arithmetic surface is equivalent to Birch–Swinnerton–Dyer Jacobian generic fibre.