scholarly journals Some improvements to Turner's algorithm for bracket abstraction

1990 ◽  
Vol 55 (2) ◽  
pp. 656-669 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Finite Number ◽  
Translation Process ◽  
Special Cases ◽  
Infinite Set

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.There are several ways in which these problems can be alleviated:(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.(2) The use of an extra clause (c), appropriate to βη reduction.(3) The introduction of a finite number of extra combinators.The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

How to Program an Infinite Abacus

1961 ◽  
Vol 4 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Finite Number ◽  
Recursive Function ◽  
Computable Function ◽  
Unlimited Supply ◽  
Countably Infinite ◽  
Infinite Set

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

scholarly journals Exact distributions of order statistics from ln,p-symmetric sample distributions

2017 ◽  
Vol 5 (1) ◽  
pp. 221-245 ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Distribution Function ◽  
Finite Number ◽  
Order Statistics ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Symmetric Distributions ◽  
Sample Distribution ◽  
Special Cases ◽  
Exact Distributions ◽  
Gaussian Sample

Abstract We derive the exact distributions of order statistics from a finite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end,we first review the special cases of order statistics fromspherical aswell as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure.We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ {1, . . . , n − 1}.

Singularity confinement for a class of m -th order difference equations of combinatorics

2007 ◽  
Vol 366 (1867) ◽  
pp. 877-922 ◽  
Author(s):  
Mark Adler ◽  
Pierre van Moerbeke ◽  
Pol Vanhaecke
Keyword(s):  
Finite Number ◽  
Difference Equations ◽  
Random Permutations ◽  
Order Difference ◽  
Painlevé Property ◽  
Special Cases ◽  
Singularity Confinement ◽  
Time Differential ◽  
Free Parameters ◽  
The Difference

In a recent publication, it was shown that a large class of integrals over the unitary group U ( n ) satisfy nonlinear, non-autonomous difference equations over n , involving a finite number of steps; special cases are generating functions appearing in questions of the longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painlevé property ; roughly speaking, this means that after a finite number of steps the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again (‘ singularity confinement ’). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the Painlevé property for the discrete relations is inherited from the Painlevé property of the (continuous) Toeplitz lattice.

scholarly journals Angle Sums of Schläfli Orthoschemes

2021 ◽  
Author(s):  
Thomas Godland ◽  
Zakhar Kabluchko
Keyword(s):  
Finite Number ◽  
Random Walks ◽  
Stirling Numbers ◽  
Convex Hulls ◽  
Tangent Cones ◽  
Expected Number ◽  
Intrinsic Volumes ◽  
Special Cases ◽  
Minkowski Sums ◽  
Finite Products

AbstractWe consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

scholarly journals Large Deviations Principle for Occupancy Problems with Colored Balls

2007 ◽  
Vol 44 (01) ◽  
pp. 115-141
Author(s):  
Paul Dupuis ◽  
Carl Nuzman ◽  
Phil Whiting
Keyword(s):  
Large Deviations ◽  
Finite Number ◽  
Scale Parameter ◽  
Large Deviations Principle ◽  
Special Cases ◽  
Boltzmann Statistics ◽  
Occupancy Problems

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are r balls thrown into n urns with the probability of a ball entering a given urn being 1/n (i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter, n, tending to infinity and r increasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.

Dimension theory and homogeneity for elementary extensions of a model

1982 ◽  
Vol 47 (1) ◽  
pp. 147-160 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Large Class ◽  
Countable Model ◽  
Dimension Theory ◽  
Elementary Embedding ◽  
Minimal Extension ◽  
Infinite Set ◽  
Countable Models

We take a fixed countable model M0, and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M0). Assuming that S(M0) is countable, we prove that if N is a weakly minimal extension of , and if then there is an elementary embedding of N into M over M0), then N is homogeneous over M0. Moreover the condition that ∣S(M0)∣ = ℵ0 cannot be removed. Under the hypothesis that M0 contains no infinite set of tuples ordered by a formula, we prove that M0 has infinitely many countable elementary extensions up to isomorphism over M0. A preliminary result is that all types over M0 are definable, and moreover is definable over M0 if and only if is definable over M0 (forking symmetry). We also introduce a notion of relative homogeneity, and show that a large class of elementary extensions of M0 are relatively homogeneous over M0 (under the assumptions that M0 has no order and S(M0) is countable).I will now discuss the background to and motivation behind the results in this paper, and also the place of this paper relative to other conjectures and investigations. To simplify notation let T denote the complete diagram of M0. First, our result that if M0 has no order then T has infinitely many countable models is related to the following conjecture: any theory with a finite number (more than one) of countable models is unstable.

On the generation of viscous toroidal eddies in a cylinder

1979 ◽  
Vol 95 (2) ◽  
pp. 209-222 ◽  
Author(s):  
J. R. Blake
Keyword(s):  
Finite Number ◽  
Stream Function ◽  
Radial Direction ◽  
Finite Cylinder ◽  
Infinite Cylinder ◽  
Cylinder Length ◽  
Infinite Set

The streamlines due to a stokeslet on the axis in a finite, semi-infinite and infinite cylinder are obtained together with the case of a Stokes-doublet and source-doublet in an infinite cylinder. In the infinite and semi-infinite cylinder examples an infinite set of toroidal eddies are obtained. The eddies alternate in sign and the magnitude of the stream function decays exponentially with distance from the driving singularity. In the finite cylinder a primary interior eddy adjacent to the singularity is always obtained and, depending on location of the singularity within the cylinder and the ratio of cylinder length to radius, a finite number of secondary interior eddies. In the case of long cylinders, the eddies are generated along the axis, whereas, for squat cylinders, secondary eddies occur in the radial direction. The interior eddies emerge from the corner as the length of the cylinder is increased. Moffatt corner eddies exist but they are very much smaller than the interior eddies.

Bifurcations of meromorphic vector fields on the Riemann sphere

1995 ◽  
Vol 15 (6) ◽  
pp. 1211-1222 ◽  
Author(s):  
Jesús Muciño-Raymundo ◽  
Carlos Valero-Valdés
Keyword(s):  
Vector Field ◽  
Finite Number ◽  
Vector Fields ◽  
Riemann Sphere ◽  
Fixed Degree ◽  
Multiplicity One ◽  
Accumulation Points ◽  
Dense Sets ◽  
Infinite Set ◽  
Zeros And Poles

AbstractLet {Xθ} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {θ}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.

Multigranulation roughness based on soft relations

2021 ◽  
pp. 1-16
Author(s):  
Muhammad Shabir ◽  
Jamalud Din ◽  
Irfan Ahmad Ganie
Keyword(s):  
Finite Number ◽  
Rough Set ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Soft Sets ◽  
Upper Approximation ◽  
Special Cases ◽  
Multigranulation Rough Set ◽  
Empty Subset ◽  
Lower And Upper Approximations

The original rough set model, developed by Pawlak depends on a single equivalence relation. Qian et al, extended this model and defined multigranulation rough sets by using finite number of equivalence relations. This model provide new direction to the research. Recently, Shabir et al. proposed a rough set model which depends on a soft relation from an universe V to an universe W . In this paper we are present multigranulation roughness based on soft relations. Firstly we approximate a non-empty subset with respect to aftersets and foresets of finite number of soft binary relations. In this way we get two sets of soft sets called the lower approximation and upper approximation with respect to aftersets and with respect to foresets. Then we investigate some properties of lower and upper approximations of the new multigranulation rough set model. It can be found that the Pawlak rough set model, Qian et al. multigranulation rough set model, Shabir et al. rough set model are special cases of this new multigranulation rough set model. Finally, we added two examples to illustrate this multigranulation rough set model.

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