Let X ,Y are linear space. In this paper, we prove the generalized Hyers-Ulam stability of the following quartic equation n ∑ k=2 ( k ∑ i1=2 k+1 ∑ i2=i1+1 . . . n ∑ in−k+1=in−k+1 ) f ( n ∑ i=1,i =i1,...,in−k+1 xi − n−k+1 ∑ r=1 xir )

Approved for public release; distribution unlimited. An exact formula is given for the grazing angle of specular reflection from a sphere by displaying the zeros of a certain self-inversive quartic polynomial.

This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n)4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail.

A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.

We prove Manin’s conjecture for a split singular quartic del Pezzo surface with singularity type 2A1 and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics, we prove a result of independent interest on a certain restricted divisor problem for four binary linear forms.

In this paper we analyze and exactly compute the number of affine equivalence classes under permutations for quartic monomial rotation symmetric functions in prime and prime power dimensions.

The quartic gauge boson couplings in the SU(3) C ⊗ SU(3) L ⊗ U(1) N models are presented. We find that the couplings of four differrent gauge bosons may have unusual Lorentz structure and the couplings sastify the tree unitarity requirement at high energy limit.

In this paper we classify the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.

We present three symbolic–numeric techniques for computing the intersection and self–intersection curve(s) of two B ézier surface patches of bidegree (2,2). In particular, we discuss algorithms, implementation, illustrative examples and provide a comparison of the methods.

Let a, b, c, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in P defined by ax + by + cz + dw = 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic