Results 1  10
of
53
Applying interval arithmetic to real, integer and Boolean constraints
 JOURNAL OF LOGIC PROGRAMMING
, 1997
"... We present in this paper a general narrowing algorithm, based on relational interval arithmetic, which applies to any nary relation on!. The main idea is to define, for every such relation ae, a narrowing function \Gamma! ae based on the approximation of ae by a block which is the cartesian product ..."
Abstract

Cited by 186 (20 self)
 Add to MetaCart
We present in this paper a general narrowing algorithm, based on relational interval arithmetic, which applies to any nary relation on!. The main idea is to define, for every such relation ae, a narrowing function \Gamma! ae based on the approximation of ae by a block which is the cartesian product of intervals. We then show how, under certain conditions, one can compute the narrowing function of relations defined in terms of unions and intersections of simpler relations. We apply the use of the narrowing algorithm, which is the core of the CLP language BNRProlog, to integer and disequality constraints, to boolean constraints and to relations mixing numerical and boolean values. The result is a language, called CLP(BNR), where constraints are expressed in a unique structure, allowing the mixing of real numbers, integers and booleans. We end by the presentation of several examples showing the advantages of such approach from the point of view of the expressiveness, and give some computational results from a first prototype
Apollonian Circle Packings: Geometry and Group Theory II. SuperApollonian Group and Integral Packings
, 2006
"... Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
(Show Context)
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group. It is observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the superApollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a superpacking. The circles in a superpacking never cross each other but are nested
Proving the IEEE Correctness of Iterative FloatingPoint Square Root, Divide, and Remainder Algorithms
"... The work presented in this paper was initiated as part of a study on software alternatives to the hardware implementations of floatingpoint operations such as divide and square root. The results of the study proved the viability of software implementations, and showed that certain proposed algorith ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
The work presented in this paper was initiated as part of a study on software alternatives to the hardware implementations of floatingpoint operations such as divide and square root. The results of the study proved the viability of software implementations, and showed that certain proposed algorithms are comparable in performance to current hardware implementations. This paper discusses two components of that study: (1) A methodology for proving the IEEE correctness of the result of iterative algorithms that implement the floatingpoint square root, divide, or remainder operation. (2) Identification of operands for the floatingpoint divide and square root operations that lead to results representing difficult cases for IEEE rounding. Some general properties of floatingpoint computations are presented first. The IEEE correctness of the floatingpoint square root operation is discussed next. We show how operands for the floatingpoint square root that lead to difficult cases for rounding can be generated, and how to use this knowledge in proving the IEEE correctness of the result of iterative algorithms that calculate the square root of a floatingpoint number. Similar aspects are analyzed for the floatingpoint divide operation, and we present a method for generating difficult cases for rounding. In the case of the floatingpoint divide operation, however, it is more difficult to use this information in proving the IEEE correctness of the result of an iterative algorithm than it is for the floatingpoint square root operation. We examine the restrictions on the method used for square root. Finally, we present possible limitations due to the finite exponent range.
The geometry of continued fractions and the topology of surface singularities
, 2005
"... We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a cano ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ> 1 and λ/λ−1.
On Upsampling, Downsampling And Rational Sampling Rate Filter Banks
 COMPUTATIONAL MATHEMATICS LABORATORY, RICE UNIVERSITY
, 1992
"... Recently, solutions to the problem of design of rational sampling rate filter banks in one dimension has been proposed. The ability to interchange the operations of upsampling, downsampling, and filtering plays an important role in these solutions. This paper develops a complete theory for the analy ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
Recently, solutions to the problem of design of rational sampling rate filter banks in one dimension has been proposed. The ability to interchange the operations of upsampling, downsampling, and filtering plays an important role in these solutions. This paper develops a complete theory for the analysis of arbitrary combinations of upsamplers, downsamplers and filters in multiple dimensions. Though some of the simpler results are well known, the more difficult results concerning swapping upsamplers and downsamplers and variations thereof are new. As an application of this theory, we obtain algebraic reductions of the general multidimensional rational sampling rate problem to a multidimensional uniform filter bank problem. However, issues concerning the design of the filters themselves are not addressed. In multiple dimensions, upsampling and downsampling operators are determined by integer matrices (as opposed to scalars in one dimension), and the noncommutativity of matrices, makes th...
Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions
"... Abstract. This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2, Z) with respect to the standard fundamental domain F = {z = x+iy: −1 1 2 ≤ x ≤ ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Abstract. This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2, Z) with respect to the standard fundamental domain F = {z = x+iy: −1 1 2 ≤ x ≤
The Number of Real Quadratic Fields Having Units of Negative Norm
, 1993
"... this paper, although formulated and treated in terms of real quadratic number fields, is a very old problem that does not need anything in its formulation beyond ordinary integers. More precisely, we will be concerned with the solvability of the negative Pell equation ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
this paper, although formulated and treated in terms of real quadratic number fields, is a very old problem that does not need anything in its formulation beyond ordinary integers. More precisely, we will be concerned with the solvability of the negative Pell equation
Correctly rounded multiplication by arbitrary precision constants
 IEEE Symposium on Computer Arithmetic, Research Report, n o 5354, INRIA
, 2005
"... Abstract—We introduce an algorithm for multiplying a floatingpoint number x by a constant C that is not exactly representable in floatingpoint arithmetic. Our algorithm uses a multiplication and a fused multiply and add instruction. Such instructions are available in some modern processors such as ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
Abstract—We introduce an algorithm for multiplying a floatingpoint number x by a constant C that is not exactly representable in floatingpoint arithmetic. Our algorithm uses a multiplication and a fused multiply and add instruction. Such instructions are available in some modern processors such as the IBM Power PC and the Intel/HP Itanium. We give three methods for checking whether, for a given value of C and a given floatingpoint format, our algorithm returns a correctly rounded result for any x. When it does not, some of our methods return all of the values x for which the algorithm fails. The three methods are complementary: The first two do not always allow one to conclude, yet they are simple enough to be used at compile time, while the third one always either proves that our algorithm returns a correctly rounded result for any x or gives all of the counterexamples. We generalize our study to the case where a wider internal format is used for the intermediate calculations, which gives a fourth method. Our programs and some additional information (such as the case where an arbitrary nonbinary even radix is used), as well as examples of runs of our programs, can be downloaded from
The Role of Openended Problems in Mathematics Education
 J. Math. Behavior
, 1994
"... this article. For reasons of authenticity, I have purposely transcribed these problems essentially verbatim from the original sources, including the use of double question marks and the proliferation of capital letters. Here is the first one: ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
this article. For reasons of authenticity, I have purposely transcribed these problems essentially verbatim from the original sources, including the use of double question marks and the proliferation of capital letters. Here is the first one:
Finding Factors of Factor Rings over the Gaussian
 Integers”, the American Mathematical Monthly
, 2005
"... set Z[i] = {a + bi: a, b ∈ Z, i = √−1}. These sit inside the complex numbers C and thus obey the usual rules of addition and multiplication; indeed, despite the presence of the imaginary i, they are quite similar to the “traditional ” integers. In fact, in the set Z[i] one can define (Gaussian inte ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
set Z[i] = {a + bi: a, b ∈ Z, i = √−1}. These sit inside the complex numbers C and thus obey the usual rules of addition and multiplication; indeed, despite the presence of the imaginary i, they are quite similar to the “traditional ” integers. In fact, in the set Z[i] one can define (Gaussian integer) primes, construct analogues of the Euclidean