Daniel C. Fielder

Simplified algebra for the bilinear and related transformations

An integer matrix method for implementation of the bilinear transformation is discussed and extended to a more general case that is useful in the design of digital filters.

Pascal’s Triangle: Top Gun or Just One of the Gang?

Pascal’s triangle can appear as a member of classes of triangular arrays where presumably no class member should be ranked in importance over any other. Two such cases which came to mind were the multinomial triangles [6] and the Hoggatt triangles [2]. No doubt there are others. We selected the multinomial triangles. Was Pascal’s triangle only a binomial triangle in a sea of trinomial, quadrinomial, pentanomial, etc., triangles, or might it exhibit a significant influence on the makeup of the other multinomial triangles? We admit a certain prejudice in our choice. Computer experimentation with partition counting, large multinomial expansions, and generating functions using computer algebra systems (muMath, Derive, Mathematica) hinted at a definite Pascal influence. A few years ago, such experimentation would have been virtually impossible.

Number representation effects in truth-table look-up processing: 8-bit addition example

The parallelism and interconnectivity of optical systems may provide important advantages for these systems in massively parallel processing applications. Electronic systems, however, retain all the advantages of a highly developed technology that has been widely applied with excellent success. In both of these technologies, the methods of direct truth-table look-up processing are becoming increasingly important as the need grows for increased speed and throughput. A major issue in truth-table look-up processing is the number representation used for data. In this paper, the effects of number representation are investigated for the important case of 8-bit addition as a specific example. The inputs are two 8-bit binary numbers together with an input carry. The output is a full precision 9-bit binary sum. For the intermediate processing three number representations are treated: binary, residue, and modified signed-digit. The numbers in all three representations are in binary-coded form throughout the processing. The critically important steps of encoding the numbers into the residue and modified signed-digit systems and then decoding the results back into direct binary are also performed using truth-table look-up methods. For the direct binary representation, a total of 2545 gates (2519 holograms) are required. For the residue representation, a total of 1764 gates (1686 holograms) are required. For the modified signed-digit representation, a total of 4142 gates (4052 holograms) are required.

Observations from Computer Experiments on An Integer Equation

Computer algebra programs such as Mathematica [7], Maple, Macsyma, etc., with their exact integer and infinite (almost) precision capabilities, have opened the way to meaningful experimental solution of Diophantine and other integer equations.

An Investigation of Sequences Derived from Hoggatt Sums and Hoggatt Triangles

In a recent note [1], the authors discuss derivations of integer sequences called Hoggatt Sums and associated triangular arrays called Hoggatt Triangles. The nomenclature was proposed as a tribute to the late Verner Hoggatt, Jr. since the investigation and extension of an unpublicized conjecture of Hoggatt ultimately resulted in the above sums and triangles. In personal correspondence [2], Hoggatt conjectured that the third (counting as 0, 1, 2, 3, …) right diagonal of Pascal’s triangle could be used to determine the sequence of integers, S0, S1, S2, … S m ,…, which are identically the Baxter permutation counts [3] of indices 0, 1, 2, …, m, … Hoggatt based his calculation algorithm for S m on sums of products between third diagonal terms from Pascal’s triangle and appropriately corresponding terms from a completed S m -1. The authors’ note [1] supplied the missing proof of Hoggatt’s conjecture. Hoggatt’s conjecture was then extended to include all right Pascal triangle diagonals indexed as 0, 1, 2, 3, …, d, …. For each d, the set of S m ’s became Hoggatt sums of order d, and the individual integers which sum to a particular S m became row members of a triangular array called a Hoggatt triangle of order d. With the inclusion of d as a variable parameter, the numerical results of [1] can be interpreted as sequences of (S d ) m ’s with Fixed index d and variable index m. For example, the Baxter permutation count values are Hoggatt sums of order three whose general sequence term is (S3) m . Sequences of Hoggatt sums follow a linear recursion which is index-variant in m, i.e., the calculation of (S d ) m for d fixed depends not only on previous members of the sequence but also depends on the value of m. Difference equations for this type of recursion are known to be difficult, if not impossible, to obtain by operational methods [4].

On Shannon's Almost Uniform Distribution

This report describes a new digital differential analyzer (DDA) technique for computing equations of the form f(x)=c/x (c=constant), which was developed to solve a problem posed by the Systems Laboratory. The major advantages of this new method over conventional DDA methods are: a much smaller truncation error because only one integration is performed, and reduced size of the equipment, since it requires only one Y register and one time-shared adder.

More Applications of a Partition Driven Symmetric Table

For many years mathematicians and scientists have been intrigued with the algebraic, symmetric, and partition properties associated with operations on polynomials such as a 0xn + a1xn-1 + a2xn-2 +…+ an-1x+an. A typical operation is the summation of the kth powers of the roots (without first finding the roots). In the early 1960’s, Fielder [7], [8] developed a tabular approach which generalizes such operations including the above. Known existing examples were systematized, and several additional examples were presented. Undoubtedly there are many more just begging to be discovered.

Investigating Special Binary Sequences with Some Computer Help

In this note, we study exclusively n-length binary sequences where at least one 1 is adjacent to (or touching) another 1. For brevity, we refer to them as “binary sequences with some touching 1’s.” They are not to be confused with n-length binary sequences where every 1 is adjacent to at least one other 1 [1]. We let C n denote the collection with unrestricted content, and we let D n denote the subset of C n with even content. (Content is defined as the number of 1’s in a binary sequence.)

Contributions from Cascaded Combinations to the Naming of Special Permutations

In the sub-title of his text, “Mathematics of Choice,” Niven [7] characterized the study of combinatorics as “How to count without counting,” to highlight the role of enumeration in combinatorics. While the importance of knowing “how many” cannot be denied, there are many instances where the unique names of members of an enumeration can serve as codes to perform external control tasks or even adaptively influence the future course of the enumeration. This is particularly true when the names are represented as ordered collections of integers. To form a next name from a present name, application of a suitable algorithm is the adaptive influence needed. To cite one example which we will expand in detail later, Even [1, page 32] introduces a position shifting algorithm for successively generating unique names of combinations C(n, m). In recent work on counting and naming conversations on crossbar systems [2–4], we developed an algorithm for finding the successive names of a structure we call cascaded combinations. (Our concept of cascaded combinations is fully described in a subsequent section.) Additionally, we needed a simple algorithm for naming multikind permutations of n. These are the familiar permutations of n with r 1 of the first kind, r 2 of the second kind, etc., up through r k of the kth (and last) kind. To our surprise, an extensive library search and inquiries of colleagues did not yield a statement of what should be a commonplace algorithm. We temporarily abandoned the search when we discovered we could use our algorithm for generating cascaded combinations as a very simple “guide” algorithm for generating the permutation names.

Counting and Naming Connection Islands on a Grid of Conductors

Consider a grid of T horizontal and T vertical information conductors. At each crossing point it is possible to either connect a horizontal to a vertical conductor or just let the connection be open. When connected at a crossing point, both conductors share the same information state throught their lengths.