Showing papers on "Function (mathematics) published in 1982"
TL;DR: A key development in the economic theory of index numbers has been the demonstration that many index number formulas can be explicitly derived from particular aggregator functions as mentioned in this paper, which provides a powerful new basis for selecting an index number procedure.
Abstract: Early in this century economists began to give serious attention to making comparisons using index number techniques. There was extensive debate as to which index number formulas were the most appropriate for carrying out comparisons.1 The debate was extensive in no small part due to the lack of agreement as to criteria for preferring one formula over another. In recent decades there has been a resurgence of interest in index numbers, resulting from discoveries that the properties of index numbers can be directly related to the properties of the underlying aggregator functions that they represent. The underlying functions - production functions, utility functions, etc. - are the building blocks of economic theory, and the study of relationships between these functions and index number formulas has been referred to by Samuelson and Swamy (I974) as the economic theory of index numbers.2 A key development in the economic theory of index numbers has been the demonstration that numerous index number formulas can be explicitly derived from particular aggregator functions. This development provides a powerful new basis for selecting an index number procedure. Rather than starting the selection process with a number of plausible index number formulas, one can specify an aggregator function with desirable properties and derive the corresponding index number procedure. The resulting index is termed exact for that particular aggregator function. Diewert (I976) makes a strong case for limiting the consideration of aggregator functions to those which are flexible, i.e. those which can provide a second order approximation to an arbitrary aggregator function. He has termed index numbers that are exact for flexible aggregator functions 'superlative '. There are two superlative index numbers that are of particular interest - the Fisher Ideal index and the Tornqvist-Theil-translog index. Fisher (I 922) dubbed the following index Ideal since it best satisfied his several criteria for choosing among index numbers:
1,822 citations
TL;DR: Automatic phase-picking algorithms are designed to detect a seismic signal on a single trace and to time the arrival of the signal precisely as mentioned in this paper, but they are inherently less sensitive than one designed only to detect the presence of a signal, but still can approach the performance of a skilled analyst.
Abstract: Automatic phase-picking algorithms are designed to detect a seismic signal on a single trace and to time the arrival of the signal precisely. Because of the requirement for precise timing, a phase-picking algorithm is inherently less sensitive than one designed only to detect the presence of a signal, but still can approach the performance of a skilled analyst. A typical algorithm filters the input data and then generates a function characterizing the seismic time series. This function may be as simple as the absolute value of the series, or it may be quite complex. Event detection is accomplished by comparing the function or its short-term average (STA ) with a threshold value (THR), which is commonly some multiple of a long-term average (LTA) of a characteristic function. If the STA exceeds THR, a trigger is declared. If the event passes simple criteria, it is reported. Sensitivity, expected timing error, false-trigger rate, and false-report rate are interrelated measures of performance controlled by choice of the characteristic function and several operating parameters. At present, computational power limits most systems to one-pass, time-domain algorithms. Rapidly advancing semi-conductor technology, however, will make possible much more powerful multi-pass approaches incorporating frequency-domain detection and pseudo-offline timing.
642 citations
TL;DR: It is shown, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini.
592 citations
TL;DR: In this article, an extended empirical orthogonal function analysis technique is described which expands a data set in terms of functions which are the best representation of that data set for a sequence of time points.
Abstract: An extended empirical orthogonal function analysis technique is described which expands a data set in terms of functions which are the “best” representation of that data set for a sequence of time points. The method takes advantage of the fact that geophysical fields are often significantly correlated in both space and time. Two examples of applications of this technique are given which suggest it may be a highly useful tool for diagnosing the modes of variation of dominant sequences of events. In the first, an analysis of 300 mb relative vorticity, fairly regular advection of the major features of the spatial patterns is evident. Westward speeds of between 0.3 and 0.4 m s−1 are inferred. The second example illustrates extended functions of tropical Pacific Ocean surface temperatures. The dominant function, which is associated with El Nino, shows a high degree of persistence over a six-month sequence. The second most important function suggests opposing variations in the influences of the North a...
414 citations
TL;DR: An alternative method for fitting the gravity model is suggested, in which the interaction variable is treated as the outcome of a discrete probability process, whose mean is a function of the size and distance variables.
Abstract: In this paper [the authors] suggest an alternative method for fitting the gravity model. In this method the interaction variable is treated as the outcome of a discrete probability process whose mean is a function of the size and distance variables. This treatment seems appropriate when the dependent variable represents a count of the number of items (people vehicles shipments) moving from one place to another. It would seem to have special advantages where there are some pairs of places between which few items move. The argument will be illustrated with reference to data on the numbers of migrants moving in 1970-1971 between pairs of the 126 labor market areas defined for Great Britain.... (EXCERPT)
389 citations
TL;DR: In this paper, an algorithm and corresponding computer program for solution of the scattered data interpolation problem is described, which is based on a weighted sum of locally defined thin plate splines, and yields an interpolation function which is differentiable.
Abstract: An algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described. Given points ( x k , y k , f k ), k = 1,…, N a locally defined function F ( x , y ) which has the property F ( x k , y k ) = f k , k = 1,…, N is constructed. The algorithm is based on a weighted sum of locally defined thin plate splines, and yields an interpolation function which is differentiable. The program is available from the author.
360 citations
Patent•
14 Sep 1982
TL;DR: In this article, the GF(2 m ) elements are represented by a vector of m binary digits in such a way that multiplication can be performed by using the same logic function to compute each binary component of the product of two elements, and addition can be formed by logic circuitry that forms the modulo-two sum of the corresponding components of the two vectors representing the elements to be summed.
Abstract: Elements of the finite field GF(2 m ) are represented by a vector of m binary digits in such a way that multiplication can be performed by using the same logic function to compute each binary component of the product of two elements, squaring can be performed by logic circuitry that rotates the vector representing the element to be squared, and addition can be performed by logic circuitry that forms the modulo-two sum of the corresponding components of the two vectors representing the elements to be summed.
335 citations
DSM1
TL;DR: In this article, Spence showed that the modulus and compliance functions are analytic in the lower half of the complex frequency plane, they are limited if the frequency tends to infinity, and the real and imaginary parts are even and odd functions, respectively, of the frequencyω.
Abstract: On the basis of some very plausible assumptions about the response of physical systems to stimuli, such as Boltzmann's superposition principle and the causality principle, Spence showed that the following characteristics obtain for the modulus and compliance functions: (i) They are analytic in the lower half of the complex frequency plane, (ii) they are limited if the frequency tends to infinity, and (iii) the real and imaginary parts are even and odd functions, respectively, of the frequencyω. It can generally be demonstrated that the real and imaginary parts of every function satisfying these three requirements and (iv) without singularities on the real frequency axis, are interrelated by Kramers-Kronig transforms. Similar relations hold between the logarithm of the modulus and the argument of the function.
237 citations
TL;DR: It is shown first that for each nonnegative integer k there is a language L k in NP that does not have O( n k )-size uniform circuits, and it is noted that existence of “small circuits≓ is in suitable contexts equivalent to being reducible to sparse sets.
Abstract: As remarked in Cook (“Towards a Complexity Theory of Synchronous Parallel Computation,≓ Univ. of Toronto, 1980), a nonlinear lower bound on the circuit-size of a language in P or even in NP is not known. The best known published lower bound seems to be due to Paul (“Proceedings, 7th ACM Symposium on Theory of Computing,≓ 1975). In this paper it is shown first that for each nonnegative integer k there is a language L k in σ 2 ⌢ π 2 (of the Meyer and Stockmeyer (“Proceedings, 13th IEEE Symposium on Switching and Automata Theory,≓ 1972) hierarchy) which does not have O( n k )-size circuits. Using the same techniques, one is able to prove several similar results. For example, it is shown that for each nonnegative integer k , there is a language L k in NP that does not have O( n k )-size uniform circuits. This follows as a corollary of a stronger result shown in the paper. This result like the others to follow is not provable by direct diagonalization. It thus points to the most interesting feature of the techniques used hereby using the polynomial-time hierarchy, they are able to prove results about NP that cannot seem to proved by direct diagonalization. Finally, it is noted that existence of “small circuits≓ is in suitable contexts equivalent to being reducible to sparse sets. Using this, one is able to prove, for example, that for any time-constructible superpolynomial function f ( n ), NTIME( f ( n )) contains a language which is not many-to-one p -time reducible to any sparse set.
210 citations
TL;DR: In this article, a simple mathematical inverse method is used to correlate two time series D(x) and R(t), where these two signals are related to each other by the mapping function x(t) The mapping function describes stretching and squeezing of one signal with respect to the other.
Abstract: A simple mathematical inverse method is used to correlate two time series D(x) and R(t), where these two signals are related to each other by the mapping function x(t) The mapping function describes stretching and squeezing of one signal with respect to the other The method assumes that D(x) and R(t) are known, while x(t) is not The mapping function is parameterized in terms of a sum of simple functions of unknown coefficients a i These coefficients are estimated from the time series with the assumption that the best coefficients are those which maximize the coherence between R(t) and D[x(t)] The maximization is performed iteratively, beginning with some initial estimate of ai The standard error in the estimates of a and the uncertainty implied in x(t) is calculated The effect of noise in the signals and sharp slope changes in the mapping function are assessed by empirical testing Results suggest that mapping functions containing features with periods down to 6% of the signal length and as sharp as hiatuses can be recovered even if the signals contain significant noise The method is applied to determining differential sedimentation rates from O 8 profiles measured in two cores and to measuring differential spreading rate about a mid-ocean ridge by using marine magnetic anomalies
175 citations
IBM1
TL;DR: This paper develops a relation between the partitioning properties of computer logic and the distribution of connection lengths and finds that an exponential partitioning function leads to an inverse power law length distribution.
Abstract: This paper develops a relation between the partitioning properties of computer logic and the distribution of connection lengths. The computation of length distributions is important for wirability analysis and delay estimation. The principal result is that an exponential partitioning function leads to an inverse power law length distribution.
TL;DR: In this paper, it was shown that the first-order Edgeworth expansion of a functionals with skew-symmetric influence curve is asymptotically minimax.
Abstract: Let $X_1, X_2, \cdots, X_n$ be i.i.d random variables with d.f. $F$. Suppose the $\{\hat{T}_n = \hat{T}_n(X_1, X_2, \cdots, X_n); n \geq 1\}$ are real-valued statistics and the $\{T_n(F); n \geq 1\}$ are centering functionals such that the asymptotic distribution of $n^{1/2}\{\hat{T}_n - T_n(F)\}$ is normal with mean zero. Let $H_n(x, F)$ be the exact d.f. of $n^{1/2}\{\hat{T}_n - T_n(F)\}$. The problem is to estimate $H_n(x, F)$ or functionals of $H_n(x, F)$. Under regularity assumptions, it is shown that the bootstrap estimate $H_n(x, \hat{F}_n)$, where $\hat{F}_n$ is the sample d.f., is asymptotically minimax; the loss function is any bounded monotone increasing function of a certain norm on the scaled difference $n^{1/2}\{H_n(x, \hat{F}_n) - H_n(x, F)\}$. The estimated first-order Edgeworth expansion of $H_n(x, F)$ is also asymptotically minimax and is equivalent to $H_n(x, \hat{F}_n)$ up to terms of order $n^{- 1/2}$. On the other hand, the straightforward normal approximation with estimated variance is usually not asymptotically minimax, because of bias. The results for estimating functionals of $H_n(x, F)$ are similar, with one notable difference: the analysis for functionals with skew-symmetric influence curve, such as the mean of $H_n(x, F)$, involves second-order Edgeworth expansions and rate of convergence $n^{-1}$.
TL;DR: It is argued that insensitivity to marginal variables undermines not only the specific hypothesis of reinforcement-rate maximization but also the more general theories of value maximization developed by Rachlin, Staddon, .
Abstract: A theory of hyperbolic feedback functions for schedules of reinforcement is developed, followed by an analysis of matching and maximizing behavior in an environment characterized by such feedback functions. The hyperbolic function classifies schedules along two dimensions: one that measures the time and one • that measures the responses that are needed to collect a unit of reinforcement. Among other results it is shown that (a) both response rules predict pairwise linearity, a condition which states that absolute rates of response on any two schedules are mutually constrained by a linear function, (b) matching and maximizing rules predict identical behavior if and only if the predictions of either one are consistent with Luce's choice axiom, and (c) the hyperbolic feedback function is preserved under aggregation of response classes. The evidence collected from single and concurrent schedules of reinforcement strongly favors the matching interpretation of equilibrium behavior, as subjects do not seem to be influenced by the marginal trade-offs that define the maximizing behavior distribution. It is argued that insensitivity to marginal variables undermines not only the specific hypothesis of reinforcement-rate maximization but also the more general theories of value maximization developed by Rachlin, Staddon, . and others.
Journal Article•
TL;DR: In this article, a general principle is formulated according to which the various pseudo-density functions of f should be concentrated around the curve (t, phi prime) and a more detailed qualitative analysis of the behavior of the Wigner distribution of f around this curve is included.
TL;DR: A source coding problem is considered for the Wyner-Ziv type system where the decoder is required to estimate the value of some function of the encoder input and the side information.
Abstract: A source coding problem is considered for the Wyner-Ziv type system where the decoder is required to estimate the value of some function of the encoder input and the side information. The rate-distortion function is established for this system, and for some binary cases parametric expressions are obtained to enable numerical calculations.
TL;DR: An important deduction from this model is that size-selection in tentaculate deposit-feeders need have no morphological correlates ; the lower the adhesive strength of the mucus, the greater will be the animal's selectivity for smaller, lighter particles.
1 Jan 1982
TL;DR: In this paper, Constraint qualifications are revisited, once again reminiscent of transversality theory, and they are used as a useful tool for computing tangent cones, by the means of generalized inverse function theorems.
Abstract: Constraint qualifications are revisited, once again. These conditions are shown to be reminiscent of transversality theory. They are used as a useful tool for computing tangent cones, by the means of generalized inverse function theorems. The finite dimensional case is given a special treatment as the results are nicer and simpler in this case. Some remarks on the nondifferentiable case are also presented.
TL;DR: In this article, a coupled-cluster method which permits the use of multiconfiguration reference states has been developed in this laboratory and applied to several states of H2(1Σg+), Li(2S), HeH2( 1A1), and CH2(3B1,1A1) which include both open and closed shells.
Abstract: A coupled‐cluster method which permits the use of multiconfiguration reference states has recently been developed in this laboratory. In the present work, it is applied to several states of H2 (1Σg+), Li(2S), HeH2(1A1), and CH2(3B1,1A1), which include both open and closed shells. These applications are made within an approximation in which the cluster operator (T) is truncated at T2, T≃T1+T2 and the expansion of e−THeT is truncated at the double‐commutator level. For cases where a single configuration function ceases to be a good starting point, it is found that a single configuration based truncated coupled‐cluster procedure may exhibit serious difficulties. In such cases we find it possible to choose a multiconfigurational reference state for which our coupled‐cluster procedure converges reasonably rapidly. This paper contains several illustrations of such convergence characteristics.
Patent•
IBM1
TL;DR: In this paper, an improved logic simulation machine is presented, in which non-unitary delays of logic functions being simulated are permitted and in which the delay time can be made different for low-to-high and high-tolow transitions.
Abstract: An improved logic simulation machine in which non-unitary delays of logic functions being simulated are permitted and in which the delay time can be made different for low-to-high and high-to-low transitions. A plurality of basic processors are interconnected with a control processor through an inter-processor switch. The logic functions being simulated are divided among the various basic processors. The control processor provides primary input data and communicates the results computed by the basic processors with other ones of the basic processors as needed. All of the basic processors and the control processor operate in variable length work cycles. The length of a work cycle is determined by a minimum work space value among all of the logic functions to be simulated, that is, a minimum time to a next successive transition in a simulated output among all of the simulated logic functions. Further, the presence of glitches in the simulated output is detected. The detected glitches are suppressed if their duration is less than the delay time of the logic function being simulated for a particular transition it is predicted to undergo.
TL;DR: A new algorithm is given for solving the d.c. piecewise-linear equations of non-linear electronic circuits, which depends crucially on a recent development which allows a multi-dimensional piece wise-linear function to be represented in a closed canonical form.
Abstract: A new algorithm is given for solving the d.c. piecewise-linear equations of non-linear electronic circuits. The device models are assumed, with little loss of generality, to be made of 2-terminal piecewise-linear resistors and linear controlled sources. Unlike other methods, this algorithm guarantees that all solutions will be found in a finite number of steps. The method depends crucially on a recent development which allows a multi-dimensional piecewise-linear function to be represented in a closed canonical form. This highly compact representation requires only a minimum amount of computer storage and is responsible for the efficiency of the algorithm.
1 Apr 1982
TL;DR: In this paper, a class of problems described in a somewhat imprecise way is considered, where a linear operator of the form L + V(x), where L is the generator of a Markov process x sub t and the potential v(x) is some real-valued function on the state space sigma of X sub t, is considered.
Abstract: : We are concerned with a class of problems described in a somewhat imprecise way as follows. Consider a linear operator of the form L + V(x), where L is the generator of a Markov process x sub t and the potential V(x) is some real-valued function on the state space sigma of x sub t. We are interested in probabilistic representations for solutions phi(s,x) to a certain backward equation with data phi(T,x) = phi(x) at a final time T.
TL;DR: In this paper, the authors provide an axiomatic approach to marginal cost (MC) pricing and point out its similarity with Aumann-Shapley (A-S) pricing.
Abstract: THE MAIN PURPOSE of this paper is to provide an axiomatic approach to marginal cost (MC) pricing and to point out its similarity with Aumann-Shapley (A-S) pricing. The latter is a cost-sharing price mechanism discussed in [3 and 6] that is derived from a set of five natural axioms. In this paper we consider models in which there is one producer with a given technology who faces fixed input prices and produces a finite number of consumption goods. Thus, we can uniquely derive the cost function that describes the minimal cost of producing a given vector of consumption goods. By a price mechanism P(., ) we mean a rule or a function that associates with each cost function F and vector a of quantities, a vector of prices:
TL;DR: The meromorphic continuation of a function is characterized in terms of the asymptotic behavior of the poles of its local rational approximations (rows of the Pade table).
Abstract: The meromorphic continuation of a function is characterized in terms of the asymptotic behavior of the poles of its local rational approximations (rows of the Pade table).Bibliography: 15 titles.
TL;DR: This paper shows that n(1 + \Theta (1/\sqrt M )) steps are both necessary and sufficient, if M memory cells are available to store values of the function, and explicitly considers the performance of the algorithm as a function of the amount of memory available and the relative cost of evaluating f and comparing sequence elements for equality.
Abstract: Given a function f over a finite domain D and an arbitrary starting point x, the sequence $f^0 (x),f^1 (x),f^2 (x), \cdots $ is ultimately periodic. Such sequences are typically the output of random number generators. The cycle problem is to determine the first repeated element $f^n (x)$ in the sequence. Previous algorithms for this problem have required $3n + O(1)$ operations. In this paper we show that $n(1 + \Theta (1/\sqrt M ))$ steps are both necessary and sufficient, if M memory cells are available to store values of the function. We explicitly consider the performance of the algorithm as a function of the amount of memory available and the relative cost of evaluating f and comparing sequence elements for equality.
TL;DR: In this paper, the dipole autocorrelation function for spectral line broadening is treated in a quantum theory which rigorously satisfies the fluctuation-dissipation theorem on a microscopic level.
Abstract: The dipole autocorrelation function for spectral line broadening is treated in a quantum theory which rigorously satisfies the fluctuation-dissipation theorem on a microscopic level. The basic approximation in the theory is the binary-collision approximation. In the present paper, the two-body interaction is decomposed into one part which commutes with the internal coordinates and another part which does not. The theory, as developed, is appropriate for broadening mechanisms for which the noncommuting term may be treated within the framework of perturbation theory, while the commuting term is to be treated exactly. The theory gives, at long times, a result for the dipole autocorrelation function consistent with the well-known impact approximation. At short times, an autocorrelation function of Gaussian form, with renormalization of the initial-state occupancy is obtained. It is found that the qualitative features discussed above are unaltered in higher-order perturbation theory. The results are consistent with the requirement that all time derivatives of the autocorrelation function at $t=0$ exist. This further satisfies the requirement that all moments of the line-shape function in the frequency domain exist, hence that the line-shape function decays "exponentially" sufficiently far in the wings.
TL;DR: A Newton-like method is presented for minimizing a function ofn variables and is a variant of the discrete Newton algorithm that uses only function and gradient values and requires fewer operations than the standard method whenn > 39.
Abstract: A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn > 39, and storage is proportional ton rather thann2.