0.3. For the singular Q-Fanos one can try to use the same approach. The first difference is that, if X has a non-Gorenstein singularity, then locally in a neighborhood of such a point a general element of I-Kxl should have Du Val singularity. The second observation is that -Kx, restricted on S, is not a Cartier divisor any more but only a Weil divisor such that its multiple is an ample Cartier ...

The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized greatest common divisor are presented. Applications to finite simple continued fraction expansion of rational numbers and Smith normal form of integral matrices with an integer parameter are also given.

Abstract In previous experimental study with three-way-reversal and juggling sequence rotation algorithms, using 20,000,000 elements for type LONG in Java, the average execution times have been shown to be 49.66761ms 246.4394ms, respectively. These results revealed appreciable low performance algorithm despite its proven optimality. However, has also exhibited efficiency some offset ranges. Due...

This paper defends the claim that a part of human mathematical cognition can be realized in a computational cognitive agent (thereby rejecting the claim that the human embodiment has a privileged status with respect to mathematical cognition). The defense of this position rests on two points: (1) The embodied activities that provide a semantics for arithmetic are performable by computational ag...

Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid’s algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems...

We discuss computation of approximate Gröbner bases at high but finite precision. We show how this can be used to deduce exact results for various applications. Examples include implicitizing surfaces, finding multivariate polynomial greatest common divisors and factorizations over the rational and complex number fields. This is an extended version of a paper for SYNASC 2010, titled úPolynomial...

An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for non-realizable conditions. The improvement lies in the direct use of ‘sign lists’, obtained from the discriminant sequence, rather than ‘revised sign lists’. It is show...

We present a cgal-based univariate algebraic kernel, which provides certi ed real-root isolation of univariate polynomials with integer coe cients and standard functionalities such as basic arithmetic operations, greatest common divisor (gcd) and square-free factorization, as well as comparison and sign evaluations of real algebraic numbers. We compare our kernel with other comparable kernels, ...

1. Let (a, b) and [a, b] be the greatest common divisor and the least common multiple of a and b, respectively . p„ denotes the n'th prime ; p, q, q1 , q2 , . . . are prime numbers. A sum Z and a product f f denote a summation and a multiplication, respectively, over primes indicated . The symbol # { . . .} denotes the number of elements indicated in the bracket 11. Pμ is the product of the fir...