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Adic Topologies for the Rational Integers
"... Abstract. A topology on Z, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to Q, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on Z, which includes the pad ..."
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Abstract. A topology on Z, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to Q, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on Z, which includes the p
ON THE KTHEORY OF TRUNCATED POLYNOMIAL ALGEBRAS OVER THE RATIONAL INTEGERS
, 2008
"... We show that K2i(Z[x]/(xm), (x)) is finite of order (mi)!(i!) m−2 and that K2i−1(Z[x]/(xm), (x)) is free abelian of rank m − 1. This is accomplished by showing that the equivariant homotopy groups TRn q−λ (Z; p) of the topological Hochschild Tspectrum T(Z) are free abelian, if q is even, and finite ..."
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Cited by 8 (5 self)
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We show that K2i(Z[x]/(xm), (x)) is finite of order (mi)!(i!) m−2 and that K2i−1(Z[x]/(xm), (x)) is free abelian of rank m − 1. This is accomplished by showing that the equivariant homotopy groups TRn q−λ (Z; p) of the topological Hochschild Tspectrum T(Z) are free abelian, if q is even, and finite, if q is odd, and by determining their ranks and orders, respectively.
Modular elliptic curves and Fermat’s Last Theorem
 ANNALS OF MATH
, 1995
"... When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c n ..."
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Cited by 617 (2 self)
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When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c
Integers
"... This paper deals with the identification of effective masses and modal masses from basedriven tests. When performing a basedriven test with an elastomechanical structure. the structural responses can be related to the base ac&rations and a modal identification of the structure can be accompl ..."
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This paper deals with the identification of effective masses and modal masses from basedriven tests. When performing a basedriven test with an elastomechanical structure. the structural responses can be related to the base ac&rations and a modal identification of the structure can
Computing Rational Forms of Integer Matrices
 J. SYMBOLIC COMPUT
, 2000
"... A new algorithm is presented for finding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially on the fastest previously known a ..."
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Cited by 10 (3 self)
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A new algorithm is presented for finding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially on the fastest previously known
Rational Points on Elliptic Curves
, 1992
"... Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set of ..."
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Cited by 125 (1 self)
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containing all the archimedean ones, and denote the ring of Sintegers by OS. Let j be the jinvariant of E. In [Sil6], Silverman proved that if j is integral, then #{P ∈ E(K) : x(P) ∈ OS} can be bounded in terms of the field K, #S, and the rank of E(K). More generally, Silverman proved that if the j
Integer and Rational Arithmetic on MasPar
 In DISCO'96
, 1996
"... . The speed of integer and rational arithmetic increases significantly by systolic implementation on a SIMD architecture. For multiplication of integers one obtains linear speedup (up to 29 times), using a serialparallel scheme. A twodimensional algorithm for multiplication of polynomials gives ..."
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Cited by 2 (1 self)
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. The speed of integer and rational arithmetic increases significantly by systolic implementation on a SIMD architecture. For multiplication of integers one obtains linear speedup (up to 29 times), using a serialparallel scheme. A twodimensional algorithm for multiplication of polynomials gives
RATIONALITY OF THE FOLSOM ONO GRID
"... Abstract. In a recent paper Folsom and Ono constructed a grid of Poincare ́ series of weights 3/2 and 1/2. They conjectured that the coefficients of the holomorphic parts of these series are rational integers. We prove that these coefficients are indeed rational numbers with bounded denominators. 1. ..."
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Cited by 1 (0 self)
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Abstract. In a recent paper Folsom and Ono constructed a grid of Poincare ́ series of weights 3/2 and 1/2. They conjectured that the coefficients of the holomorphic parts of these series are rational integers. We prove that these coefficients are indeed rational numbers with bounded denominators. 1.
RATIONAL APPROXIMATIONS TO ALGEBRAIC FUNCTIONS By
"... the approximability of algebraic numbers by rational numbers. Liouville’s result states that if $\alpha $ is an algebraic number of degree $n\geqq 2 $ then $\alpha\frac{p}{q}\geqq\frac{A}{q^{n}}$ for all rational integers $p$, $q(q>0) $ , where $A $ is a positive constant ..."
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the approximability of algebraic numbers by rational numbers. Liouville’s result states that if $\alpha $ is an algebraic number of degree $n\geqq 2 $ then $\alpha\frac{p}{q}\geqq\frac{A}{q^{n}}$ for all rational integers $p$, $q(q>0) $ , where $A $ is a positive constant
An exact rational mixedinteger programming solver
, 2010
"... We present an exact rational solver for mixedinteger linear programming which avoids the numerical inaccuracies inherent in the floatingpoint computations adopted in existing software. This allows the solver to be used for establishing fundamental theoretical results and in applications where corr ..."
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Cited by 8 (1 self)
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We present an exact rational solver for mixedinteger linear programming which avoids the numerical inaccuracies inherent in the floatingpoint computations adopted in existing software. This allows the solver to be used for establishing fundamental theoretical results and in applications where
Results 1  10
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