Abstract: This paper describes a single unified algorithm for the calculation of elementary functions including multiplication, division, sin, cos, tan, arctan, sinh, cosh, tanh, arctanh, In, exp and square-root The basis for the algorithm is coordinate rotation in a linear, circular, or hyperbolic coordinate system depending on which function is to be calculated The only operations required are shifting, adding, subtracting and the recall of prestored constants The limited domain of convergence of the algorithm is calculated, leading to a discussion of the modifications required to extend the domain for floating point calculations

Abstract: This chapter is a self-contained development of abstract harmonic analysis applied to a single-output combinational logic functions; linear algebra and elementary group theory are the only mathematical prerequisites. New analysis and synthesis techniques are developed, and the groundwork is laid for future extensions to multiple output combinational logic, sequential machines, and real- or complex-valued functions of binary arguments. Harmonic analysis adds novel conceptual insights and unifying principles, improved computational techniques, and new measures of complexity to the traditional approach to switching theory. The first section is a summary of the chapter. Section II surveys classical Fourier transform properties and introduces the canonical expansion of a switching function as an n-dimensional abstract Fourier transform over the finite two-element field. The two most important transform properties are the convolution theorem, which leads to tests for prime implicants and disjunctive decompositions, and spectrum invariance which is basic to further theoretical developments and to a new synthesis technique called encoded input logic. Section III develops a new algorithm which concurrently extracts prime implicants and detects disjunctive decompositions of a switching function. Implicants of both the function and its complement are detected simultaneously, and “core” implicants can be identified. The algorithm which is not sensitive to functional complexity has been programmed for a commercial time-sharing system. Section IV introduces the restricted affine group (RAG) whose elements, called prototype transformations, encode the arguments and outputs of combinational logic functions. This group partitions the space F of all two-valued functions on Zn into 3, 8, and 48 equivalence classes respectively for n=3, 4, and 5. Unique representatives are identified for each class when n=3 and 4 and for 46 of the 48 classes of 5-argument functions. Section V applies the tools of abstract harmonic analysis to the synthesis problem for large truth tables (many-input combinational logic). A general multilevel synthesis approach, called encoded input logic, is introduced which is compatible with large-scale-integrated circuit technology. Both the conventional macrocellular and the newer microcellular array approach are included as special cases. Prototype encoding transformations are used to reduce the complexity of an imbedded normal form realization. Practical synthesis algorithms are based on Fourier analysis. A realistic 6-argument example is treated in detail. Section V concludes with a list of fundamental problems whose solutions would extend the research presented herein.

Abstract: When a local minimum of a function of several variables has been found by use of an algorithm for finding such minima numerically, one often runs the same algorithm many times with different starting values in the hopes of finding a lower minimum. Here, under the assumption that a local minimum is known, a process with analytical criteria is described which sometimes finds smaller local minima in an algorithmic manner. Methods of descent are useful for minimizing functions of several variables. Generally, one can always obtain points (if such exist) for which the gradient vanishes, and moreover, points which are local minima. At saddle points. one can continue descent with second derivative information. A point which is a local minimum for a function may or may not be a global minimum. At this juncture one resorts to search techniques to attempt to further decrease the function. The process to be described sometimes finds smaller local minima in an algorithmic manner with analytical criteria. One has no general test, of course, for a global minimum. Consider first the problem of finding the global minimum for a 2nth degree polynomial P1(x) in one variable. The coefficient of x24 will be positive. Let xl be a local minimizer of P1. Then one may write

Abstract: A method of calculating lower bounds, quadratic in the arguments of the function, for the complexity of P-schemes.

Abstract: 1 Introduction.- 2 Language and Logic.- 3 Equality.- 4 Classes.- 5 The Elementary Properties of Classes.- 6 Functions and Relations.- 7 Ordinal Numbers.- 8 Ordinal Arithmetic.- 9 Relational Closure and the Rank Function.- 10 Cardinal Numbers.- 11 The Axiom of Choice, the Generalized Continuum Hypothesis and Cardinal Arithmetic.- 12 Models.- 13 Absoluteness.- 14 The Fundamental Operations.- 15 The Godel Model.- 16 The Arithmetization of Model Theory.- 17 Cohen's Method.- 18 Forcing.- 19 Languages, Structures, and Models.- Problem List.- Index of Symbols.

Abstract: Illustrations are given of information carried in the waveforms and frequency spectra of acoustic emissions. The effects of multiple reflections and resonances are discussed. A model of the acoustic‐emission source wave is developed, and arguments are given why this wave should be a pulselike function, rather than an oscillatory function of stress. Further use of this model may allow more quantitative treatment of emission amplitudes, energies, and spectra. Experimental results show the use of two instruments for the evaluation of emission spectra. The feasibility of using frequency analysis to obtain information about source events is demonstrated.

Abstract: The form of the function giving the variation of the induced anisotropic polarizability with interatomic separation in a pair of interacting atoms is obtained by the method of moments. The spectrum of depolarized light in gaseous Ar, Kr, and Xe and the second Kerr virial coefficient of Ar, used to convert relative to absolute scattered light intensities, are used in this analysis. The coefficients in the model for the induced anisotropy, β (x)=(6A2σ−3) [x−3+(y/6) x−p], where A is the polarizability of a single isolated atom, σ is the atomic diameter, and x=r/σ, are for Xe, for example, p=9.6 and y/6=−0.957. These results clearly demonstrate that short‐range interactions have an important effect on the induced anisotropy.

Abstract: The equation of motion for the averaged Green's function in an alloy couples the latter to the Green's function for which the average is restricted so that the composition of one atom is held fixed. The average Green's function may be regarded as the Green's function for a zero-atom cluster, and it is coupled to the Green's function for a one-atom cluster. There is thus an infinite hierarchy of equations of motion in which the $n$-atom functions are coupled to the ($n+1$) atom functions. The coherent potential approximation (CPA) of Soven corresponds to truncation in the equation of motion of the one-atom function. We have generalized the coherent potential theory to a theory of $n$-atom functions with truncation in the equation of motion of the ($n+1$) atom function ($\mathrm{CP}\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$). The formalism is developed, and specific formal results are reported. In particular, the existence of localized states in the band tails can be demonstrated, but the transition region from localized to extended states is beyond the reach of a cluster theory. The theory provides a systematic basis for quantitative improvement over the CPA, and allows for a discussion of the effects of randomness in the off-diagonal elements of the Hamiltonian. The cluster hierarchy is formally solved to provide a multiple-scattering expansion of the average Green's function, where terms involving one, two, etc., atom scattering are grouped together. This expansion can be used to generate recently proposed generalizations of the CPA, but when used in conjunction with the self-consistent $n$-atom functions of the $\mathrm{CP}\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$, it provides the best approximate averaged Green's function for which the lowest-order corrections involve the scattering from compact ($n+1$) atom clusters.

Abstract: This paper is concerned with the nature of speedups. Let f be any recursive func- tion. We show that there is no effective procedure for going from an algorithm forf to another algorithm for f that is significantly faster on all but a finite number of inputs. On the other hand, for a large class of functions f, one can go effectively from any algorithm for f to one that is faster on at least infinitely many integers. Finally, if one has an algorithm for a given function f, and if there is an algorithm which is faster on all but a finite number of inputs, then even though one cannot get this faster algorithm effectively, one can still obtain a pseudo- speedup: This is a very fast algorithm which computes a variant of the function, one which differs from the original function on a finite number of inputs.

Abstract: Kerr (J. Fish. Res. Bd. Canada 28: 809–814, 1971) described an equation system based upon metabolic constraints and foraging limitations that predicts level of rations and growth efficiency as a function of the availability of the prey resource. An algorithm that extends the equation system to a growth model is described here. With reference to increasing predator size, and to the mean size and abundance of prey, the model computes successive growth increments while holding the equations to the configuration that maximizes the ratio of growth efficiency to rations. When given information on the mean size and density of prey organisms, the model accurately predicts observed patterns of lake trout growth.

Abstract: A derivation of test sets S 0 and S 1 for irredundant unate logical circuits is presented. It is shown that these sets (S 0 and S 1 , respectively) detect all stuck-at-0 and stuck-at-1 faults in all realizations with no internal inverters of a given unate function. They can be obtained easily from the minimum sum and minimum product forms, from a Karnaugh map, or from a Hasse diagram of the function. These sets are minimum in the sense that there is no set with a smaller number of elements that detects all faults in the class of realizations of a logical function. In particular, it is found that a two-level AND–OR (OR–AND) network needs all the tests in S 0 (S 1 ).

Abstract: A procedure is given for generating the reliability function directly from the Boolean algebra transmission function. This method is easily programmed on the computer so that it can be utilized both in the derivation of the reliability function and in the evaluation of the reliability. The easily obtained transmission function completely defines the relationship between elements in a configuration. Thus, by utilizing this technique, a large variety of complicated configurations can be easily evaluated and compared.

Abstract: A relation