Last time we have seen that the proof of Fermat’s Last Theorem for the exponent 4 provides us with two elliptic curves (y = x + x and y = x − 4x) in the guise of the quartic equations (X + Y 4 = Z and X − 4Y 4 = Z). Moreover, in the actual proof we used connections between these quartic curves and the ‘underlying conic’, namely the quadratic curves x+ y = z and x− 4y = z, which actually both re...

We consider negabent Boolean functions that have Trace representation. We completely characterize quadratic negabent monomial functions. We show the relation between negabent functions and bent functions via a quadratic function. Using this characterization, we give infinite classes of bent-negabent Boolean functions over the finite field F2n , with the maximum possible degree, n 2 . These are ...

We give a set of inference rules for E-uniication, similar to the inference rules for Syntactic Mutation. If the E is nitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore, E-uniication is decidable in NP, as is also the case for Basic Narrowing. However, if we further restrict E, then our algorithm runs in quadratic time, whereas Basic Narrowing...

We present a technique for speeding up inference of structured variables using a prioritydriven search algorithm rather than the more conventional dynamic programing. A priority-driven search algorithm is guaranteed to return the optimal answer if the priority function is an underestimate of the true cost function. We introduce the notion of a probable approximate underestimate, and show that i...

The paper tackles the problem of executing high-dimensional fuzzy inferences following zerothand "rst-order Takagi}Sugeno inference models. The choice of a proper AND operator, which is compatible with all the semantic requirements for a conjunctive aggregation, results in an input}output relationship which is piecewise a$ne or quadratic. The proposed inference procedures exploit this to perfor...

Switch and medial are two inference rules that play a central role in many deep inference proof systems. In specific proof systems, the mix rule may also be present. In this paper we show that the maximal length of a derivation using only the inference rules for switch, medial, and mix, modulo associativity and commutativity of the two binary connectives involved, is quadratic in the size of th...

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indelinite quadratic constraints. Under a suitable simultaneous diagonalization assumption {which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem ...

In this paper we consider the problem of minimizing a quadratic function with a quadratic constraint. We point out some new properties of the problem. In particular, in the rst part of the paper, we show that (i) the number of values of the objective function at KKT points is bounded by 3n + 1 where n is the dimension of the problem; (ii) given a KKT point that is not a global minimizer, it is ...