We consider constant depth circuits augmented with few modular, or more generally, arbitrary symmetric gates. We prove that circuits augmented with o(log n) symmetric gates must have size n n) to compute a certain (complicated) function in ACC. This function is also hard on the average for circuits of size n logn augmented with o(log n) symmetric gates, and as a consequence we can get a pseudor...

Abstract Let k be a finite field of characteristic p , and G compact -adic analytic group. Write kG for the completed group ring over . In this paper, we describe structure / P where is minimal prime ideal We give an explicit isomorphism between matrix with coefficients in ${(k'G')_\alpha }$ $k'/k$ extension, $G'$ large subquotient no normal subgroups, (–) α “twisting” operation that preserves ...

Let A be a subset of an abelian group G with |G| = n. We say that A is sum-free if there do not exist x, y, z ∈ A with x+ y = z. We determine, for any G, the maximal density μ(G) of a sum-free subset of G. This was previously known only for certain G. We prove that the number of sum-free subsets of G is 2, which is tight up to the o-term. For certain groups, those with a small prime factor of t...

Let p be a prime number, and F p the nite eld with p elements. Let S(m) be the \smoothness" function that for integers m is deened as the largest prime divisor of m. In this note, we prove the following theorem. Theorem. There is a deterministic algorithm for factoring polynomials over F p , which on poly-nomials over F p of degree n runs in time S(p ? 1) 1=2 (n log p) O(1) under the assumption...

we prove the generalized hyers--ulam stability  of n--th order linear differential equation of the form $y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x)$, with condition that there exists a non--zero solution of corresponding homogeneous equation. our main results extend and improve the corresponding results obtained by many authors.