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25
Elementary algebraic specifications of the rational complex numbers
 of Lecture Notes in Computer Science
, 2006
"... From the range of techniques available for algebraic specifications we select a core set of features which we define to the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary equational spe ..."
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Cited by 8 (7 self)
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From the range of techniques available for algebraic specifications we select a core set of features which we define to the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary equational specification of the field operations and conjugation operator on the rational complex numbers Q(i) and discuss some open problems. For Joseph Goguen 1
Simple Equational Specifications of Rational Arithmetic
 Discrete Mathematics and Theoretical Computer Science
"... We exhibit an initial specification of the rational numbers equipped with addition, subtraction, multiplication, greatest integer function, and absolute value. Our specification uses only the sort of rational numbers. It uses one hidden function; that function is unary. But it does not use an error ..."
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Cited by 7 (0 self)
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We exhibit an initial specification of the rational numbers equipped with addition, subtraction, multiplication, greatest integer function, and absolute value. Our specification uses only the sort of rational numbers. It uses one hidden function; that function is unary. But it does not use an error constant, or extra (hidden) sorts, or conditional equations. All of our work is elementary and selfcontained.
THE MOMENTS OF MINKOWSKI ?(x) FUNCTION: DYADIC PERIOD FUNCTIONS
, 2007
"... We examine the generating function of moments of the Minkowski question mark function?(x), which describes the distribution of rationals according to their continued fraction expansion. It appears that the generating function possesses certain modular properties and is defined in C \ R≥1. The expo ..."
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Cited by 6 (3 self)
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We examine the generating function of moments of the Minkowski question mark function?(x), which describes the distribution of rationals according to their continued fraction expansion. It appears that the generating function possesses certain modular properties and is defined in C \ R≥1. The exponential generating function satisfies the integral equation, with kernel being the Bessel function of the first kind. Finally, the solution of this integral equation leads to the definition of dyadic period functions of weight 2 and index λ. Such a form is defined and is holomorphic in the domain C \ R≥1, it satisfies the semimodular functional equation, and it exists for every λ, which is the eigenvalue of the properly defined HilbertSchmidt integral operator.
Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function
, 2008
"... In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, Lseries, distributions, spectral properties. These objects can b ..."
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Cited by 6 (4 self)
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In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, Lseries, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Various previous investigations of ?(x) are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects.
Regularity Properties of the Stern Enumeration of the Rationals
"... s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1 ..."
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Cited by 6 (1 self)
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s(n) + s(n + 1). Stern showed in 1858 that gcd(s(n), s(n + 1)) = 1, and that every positive rational number a s(n) b occurs exactly once in the form s(n+1) for some n ≥ 1. We show that in a strong sense, the average value of these fractions is 3 2. We also show that for d ≥ 2, the pair (s(n), s(n+1)) is uniformly distributed among all feasible pairs of congruence classes modulo d. More precise results are presented for d = 2 and 3. 1
Enumerating the Rationals
"... We present a series of lazy functional programs for enumerating the rational numbers without duplication, drawing on some elegant results of Neil Calkin, Herbert Wilf and Moshe Newman. ..."
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Cited by 5 (0 self)
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We present a series of lazy functional programs for enumerating the rational numbers without duplication, drawing on some elegant results of Neil Calkin, Herbert Wilf and Moshe Newman.
An asymptotic formula for the moments of Minkowski question mark function
 in the interval
"... Abstract. In this paper we prove the asymptotic formula for the moments of Minkowski question mark function in the interval [0, 1]. The main idea is to demonstrate that certain variation of Laplace method (a forerunner of saddlepoint approximation) is applicable in this problem, hence the task redu ..."
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Cited by 5 (4 self)
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Abstract. In this paper we prove the asymptotic formula for the moments of Minkowski question mark function in the interval [0, 1]. The main idea is to demonstrate that certain variation of Laplace method (a forerunner of saddlepoint approximation) is applicable in this problem, hence the task reduces to a number of technical calculations.
Minkowski question mark function and its generalizations, associated with pcontinued fractions: fractals, explicit series for the dyadic period function and moments (submitted); arXiv:0805.1717
"... Abstract. Previously, several natural integral transforms of Minkowski question mark function F(x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F(x). One of them the dyad ..."
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Cited by 4 (2 self)
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Abstract. Previously, several natural integral transforms of Minkowski question mark function F(x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F(x). One of them the dyadic period function G(z) was defined via certain transcendental integral. In this paper we introduce a family of “distributions ” Fp(x) for ℜ p ≥ 1, such that F1(x) is the question mark function and F2(x) is a discrete distribution with support on x = 1. Thus, all the aforementioned integral transforms are calculated for such p. As a consequence, the generating function of moments of F p(x) satisfies the three term functional equation. This has an independent interest, though our main concern is the information it provides about F(x). This approach yields certain explicit series for G(z). This also solves the problem in expressing the moments of F(x) in closed form.
On Euclid’s algorithm and elementary number theory
, 2009
"... 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 v ..."
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Cited by 3 (2 self)
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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). The theorems that we verify are wellknown and most of them are included in standard numbertheory books. The new results concern distributivity properties of the greatest common divisor and a new algorithm for efficiently enumerating the positive rationals in two different ways. One way is known and is due to Moshe Newman. The second is new and corresponds to a deforestation of the SternBrocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In this way, we construct a SternBrocot enumeration algorithm with the same time and space complexity as Newman’s algorithm. A short review of the original papers by Stern and Brocot is also included.