Abstract: Three randomized response models for quantitative data are considered, the extension of the unrelated question model first considered by Greenberg, and two models which involve the addition or multiplication of a random number from a known distribution to the sensitive value. Estimation is considered in all three models and efficiencies compared. The addition and multiplication models appear to have potential but a practical comparison of the three models is needed.

Abstract: A graph-theoretic model is introduced for bilinear algorithms. This facilitates in particular the investigation of the additive complexity of matrix multiplication. The number of additions/subtractions required for each of the problems defined by symmetric permutations on the dimensions of the matrices are shown to differ conversely as the size of each product matrix. It is noted that this result holds for any system of dual problems, not only dual matrix multiplication problems. This additive symmetry is employed to obtain various results, including the fact that 15 additive operations are necessary and sufficient to multiply two $2 \times 2$ matrices by a bilinear algorithm using at most 7 multiplication operations.

Abstract: Results are presented for both theoretical and experimental analyses of low-level avalanche multiplication in an insulated gate field-effect transistor (IGFET). The theoretical model is derived from the ionization integral using a linear field approximation for the electric field at the drain. Experimental multiplication factors are determined by measuring channel and substrate currents. The model is shown to lead to reasonable agreement with data in the range of multiplication factors defined by (M n - 1) less than unity.

Abstract: In this paper we give a new algorithm for matrix multiplication which for n large uses $n^2 + o(n^2 )$ multiplications to multiply $n \times p$ matrices by $p \times n$ matrices provided $p \leqq \log _2 n$. Multiplication and division by 2 is necessary in this algorithm. This is to be compared with $pn^2 $ for the standard algorithm and $ \simeq p^{.58} n^2 + o(n^2 )$ for an algorithm of Hopcroft and Kerr [1] which, however, requires no multiplication and division by 2.

Abstract: Let E be an arithmetic expression involving n variables, each of which appears just once, and the possible operations of addition, multiplication, and division. Although other cases are considered, when these three operations take unit time the restructuring algorithms presented in this paper yield evaluation times no greater than 2.88 log2n + 1 and 2.08 log2n for general expressions and division-free expressions, respectively. The coefficients are precisely given by 2/log2a a 2.88 and 1/log2b a 2.08, where a and b are the positive real roots of the equations z2 = z + 1 and z4 = 2z + 1, respectively. While these times were known to be of order log2n, the best previously known coefficients were 4 and 2.15 for the two cases. The authors conjecture that the present coefficients are the best possible, since they have exhibited expressions which seem to require these times within an additive constant. The paper also gives upper bounds to the restructuring time of a given expression E and to the number of processors required for its parallel evaluation. It is shown that at most O(n1.44) and O(n1.82) operations are needed for restructuring general expressions and division-free expression, respectively. It is pointed out that, since the order of the compiling time is greater than n log n, the numbers of required processors exhibit the same rate of growth in n as the corresponding compiling times.

Abstract: A comparison of the limitations of the eigenvalues for the effective neutron multiplication factor per neutron generation, k, the multiplication factor per collision, ..gamma.., and the fundamental multiplication rate, ..cap alpha.., is presented as they concern the neutron spectrum and the spectral properties of the integral transport operators. Numerical examples of analyses of fast-neutron plutonium systems are given. Advantages of the rarely used ..gamma.. eigenvalue are discussed, leading to the conclusion that it should be used more often.

Abstract: A recursive formula for number conversion from one radix representation to another radix representation is presented. This formula differs from the existing ones in two major aspects. First, it utilizes a digit shift technique which provides faster accumulation of higher significant digits in the final result. Second, it is suitable for parallel computation, so that the length of time necessary for number conversion can be shortened. Thus the longer the digit number, the more appreciation in conversion time-saving will result. Applications of the recursive formula are studied in multiplication and division for negative radix numbers as well as for positive radix numbers. The multiplication and division presented here are especially useful for computations of η word precision, since successive bit-carrying propagation to the most significant digit hardly ever occurs.

Abstract: Two numbers are multiplied together without first changing either of them, if negative, to a positive number, thereby minimizing the time required in the multiplication process. In the multiplication, depending upon the sign of the multiplier and the sign of a bit in a predetermined bit location of the multiplier as shifted in a shift register, the multiplier and the multiplicand are operated on by either a shift operation or operated on by a shift and add operation.

Abstract: A non coherent optical system has been proposed for performing transforms equivalent to the multiplication of three matrices. The possibility of using such a system has been shown for a spectral analysis of images according to an arbitrary basis with separated variables and for a multichannel processing of signals. For the optical system characteristic parameters the size of the matrices under multiplication is of the order of 100 × 100.

Abstract: This correspondence presents a new multiplication algorithm for Atrubin's one-dimensional real-time iterative multiplier such that all the cells including the first cell in the array are identical in all respects, for the no-delay case.

Abstract: This paper describes the scheme (as well as circuit details) of frequency multipliers that can be used for multiplying frequencies of square waves by factors of 2 or 3. By a coupling of these units in sequence, it should be possible to achieve frequency multiplication by a power of 2 or 3, within the limitations mentioned in the text.

Abstract: The present invention relates to a simplified apparatus for converting a ary fraction input into a Natural Binary Coded Decimal (8421 code) and subsequently into a decimal output with the proper sign. The invention provides means for selectively shifting and summing the binary fraction to effect a multiplication by ten. An integer portion to the left of a fixed binal point and a remaining fraction portion to the right are produced. The integer portion is extracted as a BCD character which is converted via a nixie display into decimal form. Shifting and summing successive remaining fractions and displaying successive integer portions produces a final decimal number which corresponds to the binary fraction input.

Abstract: Summary The application of exponential weighting to homomorphic deconvolution has been treated extensively by Schafer. This paper considers complex exponential weighting, i.e. multiplication by a" ei6n. At first glance it appears that the phase factor will have no significant effect (just a phase shift) on the complex cepstrum. However, it is shown that a slight modification of the weighting procedure yields a useful technique for the determination of delay times in the cepstrum.

Abstract: PURPOSE:To effect drop-out with respect to an aural signal when both the aural signal and a picture signal, which are recorded by frequency multiplication, are regenerated.

Abstract: The present note gives an improvement on [8]. There it was shown that the liar's antinomy can be reconstructed in any recursively enumerable arithmetical theory in which all elementary functions are definable, if the theory is assumed to be complete. Here the same construction is done by requiring only definability of addition and multiplication. This constitutes a natural and therefore straight-forward proof of a strong version of the incompleteness theorem for arithmetical theories. The improvement on [8] consists in the fact that addition and multiplication are obviously fewer and less complex than all elementary functions: the former belong respectively to the first and to the second class of Grzegorczyk's hierarchy [5], whereas the latter constitute its third class. The present result can be considered optimal in so far as it is impossible to obtain the same result for the less complex function used because definability of addition alone allows completeness, see [l].

Abstract: PURPOSE:To increase the S/N ratio by giving a/b-times multiplication to the video signal read every cycle, giving c/d-times multiplication to the 2nd signal applied to the picture display part, and by inputting the sum of both signals featuring a relation of a/b+c/d=1.

Abstract: PURPOSE:A console electronic computer, that is unnecessary to depress an arithmetic Key at the time of continuous calculation for addition or multiplication alone.

Abstract: In the previous paper [I] with the same title we discussed the admissible multiplications in cr-coefficient cohomology theories and we gave a suthcient condition for existence of admissible multiplication in the case crEzr+le-i(Sr) satisfies lcAa== (sri)crr(sr+le'tz) and t'W(skev)=o. This paper is the continuations of [I] and is devoted to the discussion of commutativity and associativity of admissible multiplication pt. which is given by [I]. In gl and g2, we discuss the associativity and commutativity of pt. in the case lcAa=O. For the case lcAcr;O, we discuss in g3 to g5. We use all notations and notions defined in [I].