Abstract: Lambda lifting is a technique for transforming a functional program with local function definitions, possibly with free variables in the function definitions, into a program consisting only of global function (combinator) definitions which will be used as rewrite rules. Different ways of doing lambda lifting are presented, as well as reasons for rejecting or selecting the method used in our Lazy ML compiler. A functional program implementing the chosen algorithm is given.

Abstract: The theme of this paper is to investigate to what extent approximation, possibly together with randomization, can reduce the complexity of problems in Valiant’s class # P. In general, any function in # P can be approximated to within any constant factor by a function in the class $\Delta _3^p $ of the polynomial-time, hierarchy. Relative to a particular oracle, $\Delta _3^p $ cannot be replaced by $\Delta _2^p $ in this result. Another part of the paper introduces a model of random sampling where the size of a set X is estimated by checking, for various “sample sets” S, whether or not S intersects X For various classes of sample sets, upper and lower bounds on the number of samples required to estimate the size of X are discussed. This type of sampling is motivated by particular problems in # P such as computing the size of a backtrack search tree. In the case of backtrack search trees, a sample amounts to checking whether a certain path exists in the tree. One of the lower bounds suggests that such tests...

Abstract: The univariate Weierstrass-Mandelbrot function is generalized to many variables to model higher dimensional stochastic processes such as undersea topography. Because this topography is difficult to measure at small length scales over the many large regions that affect long-ranged acoustic propagation in the ocean, one needs a stochastic description that can be extrapolated to both large and small features. Fractal surfaces are a convenient framework for such a description. Computer-generated plots for the two-variable case are presented. It is shown that in the continuum limit the multivariate function is equivalent to the multivariate fractional Brownian motion.

Abstract: The sequence with nth term defined by [(n + 1)p] − [np] is an extremal zero-one valued sequence of asymptotic mean p in the following sense (for example): if a fraction p of customers from a point process with iid interarrival times is sent to an exponential server queue according to a prespecified splitting sequence, then the long-term average queue size is minimized when the above sequence is used. The proof involves consideration of the lower convex envelope J (which is a function on Rm) of a function J on Zm. An explicit representation is given for J in terms of J, for J in a broad class of functions, which we call “multimodular.” The expected queue size just before an arrival, considered as a function of the zero-one splitting sequence, is shown to belong to this class.

Abstract: A choice function picks some outcome(s) from every issue (subset of a fixed set A of outcomes). When is this function derived from one preference relation on A (the choice set being then made up of the best preferred outcomes within the issue), or from several preference relations (the choice set being then the Pareto optimal outcome within the issue, or the union of the best preferred outcomes for each preference relation)? A complete and unified treatment of these problems is given based on three functional properties of the choice function. None of the main results is original.

Abstract: Earlier work of Bixby, Cunningham, and Topkis is extended to give a combinatorial algorithm for the problem of minimizing a submodular function, for which the amount of work is bounded by a polynomial in the size of the underlying set and the largest function value (not its length).

Abstract: In the local basis-function approach, a reconstruction is represented as a linear expansion of basis functions, which are arranged on a rectangular grid and possess a local region of support. The basis functions considered here are positive and may overlap. It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms. By employing repetitive basis functions, the computational effort involved in these algorithms can be minimized through the use of tabulated values for the line or strip integrals over a single-basis function. The local nature of the basis functions reduces the difficulties associated with applying local constraints on reconstruction values, such as upper and lower limits. Since a reconstruction is specified everywhere by a set of coefficients, display of a coarsely represented image does not require an arbitrary choice of an interpolation function.

Abstract: It is shown that the modified Emden equation q+α(t)q+qn=0 possesses first integrals for functions α(t) other than kt−1. The function α(t) is obtained explicitly in the case n=3 and parametrically for other n(≠2). The case n=2 is seen to be particularly difficult to solve.

Abstract: In this paper we investigate the following two extremal problems: A) Let F be a continuum in the extended complex plane that does not divide and let f(z) be a function analytic on F By D we denote domains in such that f(z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal condenser capacity B) Let f(z) be a function analytic in a neighborhood of infinity. By D we denote domains in , such that f{z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal logarithmic capacity It is proved that there exist extremal domains D 0 in both problems. In a second part of the paper it will be shown that these domains are uniquely determined up to a boundary set of capacity zero.

Abstract: Motivated by V. B. Mountcastle's organizational principle for neocortical function, and by M. E. Fisher's model of physical spin systems, we introduce a cooperative model of the cortical column incorporating an idealized substructure, the trion, which represents a localized group of neurons. Computer studies reveal that typical networks composed of a small number of trions (with symmetric interactions) exhibit striking behavior--e.g., hundreds to thousands of quasistable, periodic firing patterns, any of which can be selected out and enhanced with only small changes in interaction strengths by using a Hebb-type algorithm.

Abstract: This paper presents an algorithm for optimal data collection in random fields, the so-called variance reduction analysis, which is an extension of kriging. The basis of variance reduction analysis is an information response function (i.e., the amount of information gain at an arbitrary point due to a measurement at another site). The ranking of potential sites is conducted using an information ranking function. The optimal number of new points is then identified by an economic gain function. The selected sequence of sites for further sampling shows a high degree of stability with respect to noisy inputs.

Abstract: In the first part of the paper we show how to extend recent methods for solving a special case of the union-find problem in linear time, to a special case of the eval-link-update problem for computing the minimum function defined on paths of trees. In the cases where our approach is applicable, we give a way to perform m eval, link, and update operations on n elements in O(m + n) time and O(n) space, improved from O(m a(m + n, n) + n) time and O(n) space in the more general case, where a is a functional inverse of Ackermans function. The technique gives similar improvements in the efficiency of algorithms for solving several network optimization problems in the case where all the keys involved are integers in some suitable range. In the second part of the paper we show how to use the new technique for speeding up the fastest known algorithm for finding dominators in flow graphs so that it runs in linear time. We introduce the notions of pseudo and external dominators which are both computable in linear time and make the technique introduced in the first part applicable for finding immediate dominators. We first give an algorithm for a limited class of graphs which include cycle free graphs, and thus can be used to find dominators in reducible flow graphs. We then show how to extend our technique for computing dominators on any flow graph. All the algorithms we describe run on a Random Access Machine.

Abstract: A practical method of computation is described for making periodic signals V(t) which have a given frequency spectrum and which minimize the variance 〈(V2−〈V〉2)2〉 in power as a function of time. These signals can be used to measure the spectral response of a system (the transfer function): They represent a significant improvement over traditional random‐phase and minimum peak value approaches. The signals are also of psychophysical interest.

Abstract: We prove that the optimal cost function of a deterministic impulse control problem is the unique viscosity solution of a first-order Hamilton–Jacobi quasi-variational inequality in $\mathbb{R}^N $.

Abstract: © Foundation Compositio Mathematica, 1985, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: Average uniform flow takes place in a heterogeneous aquifer of infinite extent. The input to the problem is the hydraulic conductivity, which is regarded as a random space function that is lognormal, stationary and statistically isotropic. The output is the water head field, which is a random function satisfying the equation of steady flow. The first-order approximation of the head, in an asymptotic expansion for small log-conductivity variance, is a normal function characterized completely by the head-log-conductivity cross-covariance and the head covariance or variogram. These covariances are proportional to the log-conductivity variance. By using spectral methods, second-order corrections of the head covariances, proportional to the log-conductivity variance squared, are derived explicitly. Detailed calculations are carried out for an exponential log-conductivity covariance. The main finding of the note is that the first-order approximation is very robust and even for a log-conductivity variance equal to unity, the second-order correction of the head variances is smaller than 10% of the first-order approximation.

Abstract: interpretation described in subsection 311, the process of partitioning parameter lists and annotating operators, described in subsection 332, and the proper function specialization process, described in subsection 333 Some further post-transformations are described in subsection 334 that closes this section 331 KnoWDlUnknOWD Abstract InterpretatioD The purpose of this phase is to compute for every function in the subject program a safe description of its parameters, whether they are definitely known or possibly unknown at partial evaluation time Inputs to this phase are 1) the call annotated subject program, and 2) a description of which of the subject program's (ie which of the goal function's) parameters are known and which are unknown at partial evaluation time That is, this phase does not use the actual values of the known input, just a description telling which of theinput parameters are known (Equivalent to providing a value for m in Kleene's S-m-n Theorem) Output is a description, ie a mapping that associates with every function a parameter description, classifying each of its parameters as Known resp Unknown at partial evaluation time Here Known means "definitely known for all possible values of the subject program's known input", and Unknown means "possibly unknown for some (or all) values of the known input" For the following exposition we will assume this L subject program given Fii:ure 2: An L Subject PrQi:ram consisting ofn ~ 1 functions fi each having ki ~ 0 parameters, i=I, ,n Then the program's input is a k I-tuple E okl , where D is the domain of LISP lists Ali:orithm 331 The phase works by an abstract interpretation of the subject program over a domain with two values for expressions, D = {Known, Unknown} During this abstract interpretation, for every function a parameter description is maintained, telling for every parameter of the function whether it can be called with an unknown value (Note that a parameter description may be considered an "abstract environment", associating with every parameter of a function an abstract value) Initially, all parameters except the goal function's are considered Known, and the parameter description for the goal function is the initial description given for the subject program's input parameters The abstract interpretation proceeds as follows: The body of the goal function is evaluated ~ '·I~,lY the parameter description) to see which functions it may call, giving them Unknown oaranlett::r values The parameter descriptions for these functions are modified according to these tln(l111J~s to tell which of their paramete~ may be Unknown Then the bodies ofthese: functions are evaluated using the new parameter descriptions to see which functions they in tum may call with Unknown parameter values and so on Each time a parameter description of a fupction b~ , the "most known" description Noticethat thel~ast upper bound 01 u 02 of any two descriptions 01,02 E Descr exists, and is the tn0st known description safely approximating 01 as well as 02

Abstract: This note is about a simple and algorithmic proof of the striking result of BAUR-STRASSEN [1] showing that the complexity of the evaluation of a rational function of several variables and all its derivatives is bounded above by three times the complexity of the evaluation of the function itself.

Abstract: We consider several generalizations of the exponential ansatz in a rather formal way, giving several new wave functions which we call exponentially generated (EG) wave functions. There are three distinct ways of the exponential‐type generations of the wave functions, two of which are new. They are named ESAC (extended symmetry‐adapted‐cluster) wave function and exponentially generated CI (EGCI) wave function. The ESAC wave function is a simple extension of the SAC wave function and is applicable even when the Hartree–Fock reference configuration is not dominant. The EGCI wave function is a CI wave function constructed in the spirit of the cluster expansion theory. Formally, it has the merits of both the CI theory and the cluster expansion theory; for example, the upper bound nature, size consistency, and the applicability to quasidegenerate states and excited states. We then introduce several new wave functions by a multiple and mixed use of the exponential‐type operators. We call such a class of wave functions multiexponentially generated(MEG)wave functions. There are many possibilities for the MEGwave functions, and the MR‐SAC wave function proposed previously is one of them. When the system involves several classes of electron correlations, the MEGwave function permits an optimal (physically and practically) use of the exponential‐type operators for the distinct classes of electron correlations. We described the method of solution of the EG and MEGwave functions and examined size consistency and some other properties.

Abstract: A perfect hash function PHF is an injection F from a set W of M objects into the set consisting of the first N nonnegative integers where N g M. If N = M, then F is a minimal perfect hash function, MPHF. PHFs are useful for the compact storage and fast retrieval of frequently used objects such as reserved words in a programming language or commonly employed words in a natural language.The mincycle algorithm for finding PHFs executes with an expected time complexity that is polynomial in M and has been used successfully on sets of cardinality up to 512. Given three pseudorandom functions h0, h1, and h2, the mincycle algorithm searches for a function g such that F(w) = (h0(w) + g ° h1(w) + g ° h2(w)) mod N is a PHF.

Abstract: The Fresnel differential approach is used to derive a formula for the modulus of the normal polar Kerr coefficient for a multilayer system containing a single magnetic film. This particular function of the magneto-optical effect is chosen since it invariably appears in expressions for the signal-to-noise ratio associated with first order effects. The resulting formula determines how much superior optimized quadrilayers are than simpler bilayer structures, if at all; and also how the signal-to-noise ratio depends on the fundamental optical constants of the magnetic film.

Abstract: A new approach to the problem of time-variant filtering is presented. This approach is based upon the generation of the mixed time-frequency representation (MTFR) of a signal, multiplication of that representation by a time-frequency function H(ω,t), and obtaining a filtered output by an inverse operation. The resultant filter is linear if the time-frequency representation used is the complex spectrogram. In contrast, the filter is nonlinear if the Wigner distribution function is used. Not every function of two variables is an allowed MTFR of a signal; some conditions must be satisfied. If the function produced by the product of the signal MTFR and the filter function is not an allowed MTFR, an approximation based on projection onto the space of allowed MTFR functions is investigated. This approximation yields a filtered function whose MTFR is as close as possible (in the least-squared sense) to the desired.