Integer Square Research Articles

Zonotopes, Braids, and Quantum Groups

Various results about the action of the binomial braids and other braid analogs [2] on some particular higher dimensional representations of the braid groups are presented. These representations are constructed from a fixed integer square matrix A. The common nullspace of the binomial braids is studied in some detail. This space is graded over \( {\Bbb N}^r \), where r is the size of A. Our main results state that the non-trivial components occur only on the lattice points of a certain hypersurface in r-space that is canonically associated to A, and that when A is the symmetrization of a Cartan matrix C of finite type, these lattice points are closely related to the vertices of the zonotope associated to C (the precise relationship is given in Theorem 5.3). The same action is used to construct a quantum group \( U_q^0 (A) \) from an arbitrary integer square matrix A. The simplest choices of A yield the usual polynomial and Eulerian Hopf algebras of Joni and Rota (in the corresponding representations, the binomial braids become the usual binomial or q-binomial coefficients). The other choice we consider is that when A is the symmetrization of a symmetrizable Cartan matrix C. Some of the previous results are used to prove that in this case \( U_q^0 (A) \) coincides with the usual quantum group of Drinfeld and Jimbo. The quantum group is actually defined in a more general setting involving Hopf algebras and crossed bimodules. This paper is a continuation of [2].

Automating Inductive Specification Proofs

We present an automatic method which combines logical proof search and rippling heuristics to prove specifications. The key idea is to instantiate meta-variables in the proof with a simultaneous match based on rippling/reverse rippling heuristic. Underlying our rippling strategy is the rippling distance strategy which introduces a new powerful approach to rippling, as it avoids termination problems of other rippling strategies. Moreover, we are able to synthesize conditional substitutions for meta-variables in the proof. The strength of our approach is illustrated by discussing the specification of the integer square root and automatically synthesizing the corresponding algorithm. The described procedure has been integrated as a tactic into the NUPRL system but it can be combined with other proof methods as well.

Orthogonal Colorings of Graphs

An orthogonal coloring of a graph $G$ is a pair $\{c_1,c_2\}$ of proper colorings of $G$, having the property that if two vertices are colored with the same color in $c_1$, then they must have distinct colors in $c_2$. The notion of orthogonal colorings is strongly related to the notion of orthogonal Latin squares. The orthogonal chromatic number of $G$, denoted by $O\chi(G)$, is the minimum possible number of colors used in an orthogonal coloring of $G$. If $G$ has $n$ vertices, then the definition implies that $\left\lceil \sqrt{n} \, \right\rceil \leq O\chi(G) \leq n$. $G$ is said to have an optimal orthogonal coloring if $O\chi(G) = \left\lceil \sqrt{n} \, \right\rceil$. If, in addition, $n$ is an integer square, then we say that $G$ has a perfect orthogonal coloring, since for any two colors $x$ and $y$, there is exactly one vertex colored by $x$ in $c_1$ and by $y$ in $c_2$. The purpose of this paper is to study the parameter $O\chi(G)$ and supply upper bounds to it which depend on other graph parameters such as the maximum degree and the chromatic number. We also study the structure of graphs having an optimal or perfect orthogonal coloring, and show that several classes of graphs always have an optimal or perfect orthogonal coloring. We also consider the strong version of orthogonal colorings, where no vertex may receive the same color in both colorings.

On the Unicity of Discontinuous Transitions in the Two-Dimensional Potts and Ashkin–Teller Models

We investigate the uniqueness of the discontinuous phase transition of the q-state Potts model and the symmetric q = (s × s) cubic model (generalized Ashkin–Teller model) on the integer square lattice with q ≥ 1. For the ferromagnetic cases, when there is a discontinuous phase transition we show that the discontinuity can only occur at a point of self-duality.

Rules for multidimensional multirate structures

Identifies a comprehensive set of compact rules and efficient algorithms for simplifying and rearranging structures common in multidimensional multirate signal processing. The authors extend the 1D rules reported by Crochiere and Rabiner (1983), especially the many equivalent forms of cascades of upsamplers and downsamplers. They also include rules reported by other authors for completeness. The extension to mD is based primarily on the Smith form decomposition of resampling (nonsingular integer square) matrices. The Smith form converts non-separable multidimensional operations into separable ones by means a shuffling of input samples and a reshuffling of the separable operations. Based on the Smith form, the authors have developed algorithms for 1) computing coset vectors 2) finding greatest common sublattices 3) simplifying cascades of up/downsampling operations. The algorithms and rules are put together in a form that can be implemented efficiently in a symbolic algebra package. The authors have encoded the knowledge in the commercially available Mathematica environment. >

The integer square root of N via a binary search

An algorithm is presented which may be used to find the integer square root of N. The method is intended for use on a binary computer, where only addition, subtraction, multiplication, or division by 2 is required. The problem arose when the author was working on factoring large numbers, where the machine, the Honeywell DPS 8, had double precision integer addition and subtraction, and the simulation of multiplication was easy. The actual factoring of the large number was to be Fermat's Method, requiring only addition and subtraction, but the integer square root is required in order to test for termination. The algorithm is implemented in FORTRAN for ease of reading.Students enjoy the unconventional approach to solving this problem. It isn't long before some of them think of other unusual solutions.

Calculating with procedure calls

Although various proof rules for procedure calls in weakest precondition semantics have been proposed over the years [3, chapter 12] [4] [1], none of these is particularly suitable for calculational program construction. The problem is that they tend to yield a precondition that is tortuously expressed. For instance, let a procedure for finding the integer square root of a parameter y be specified, using a specification variable m , by the precondition

Combinatorial approach to enumeration of doubly stochastic non-negative integer square matrices

Article Combinatorial approach to enumeration of doubly stochastic non-negative integer square matrices was published on January 1, 1993 in the journal Discrete Mathematics and Applications (volume 3, issue 6).

A Modified Babylonian Algorithm

One may infer (Mainzer, p. 44) from nature of their approximations that Babylonians had discovered following algorithm for a sequence x1, x2,... of successively improving approximations to v/;. The sequence is defined inductively as x1 = x and Xk = (xk1 + x/xkl1)/2 for k ? 2. In case of C2, their best approximation was fourth term of this sequence rounded to two sexagesimal places. It was accurate to within 10-6. The choice of x1 is flexible, but for purposes of our exposition x1 = x is best. For Babylonians it was important that Xk was rational if x was, and so could easily be approximated in decimal notation. Modern calculus texts efficiently treat this algorithm as a special case of Newton-Raphson method, but one could object to this treatment on grounds that it obscures fact that convergence of Xk to Cx is an easier consequence of algebra than of calculus, and it does not justify fact that convergence is quadratic (one usually states this to students as the number of significant digits is better than doubled by proceeding from Xk to Xk+1,' but seldom is it proven to freshmen). Finally it does not address issue of non-uniformity of convergence of sequence x1, x2, ... to Cx/. That fact justifies normal strategy of conditioning x by multiplying or dividing it by a suitably chosen perfect integer square to obtain a number larger than 1 and less than or equal to (say) 10, and then applying algorithm to result. We propose following modification of Babylonian algorithm for finding square roots as a good means of illucidating above points.

NP-Completeness of the linear complementarity problem

We consider the linear complementarity problem (q, M) for which the data are the integer column vectorq eR n and the integer square matrixM of ordern. GLCP is the decision problem: Does (q, M) have a solution? We show that GLCP is NP-complete in the strong sense.

The Decomposition of a Rectangle into Rectangles of Minimal Perimeter

We solve the problem of decomposing a rectangle R into p rectangles of equal area so that the maximum rectangle perimeter is as small as possible. This work has applications in areas such as flexible object packing and data allocation. Our solution requires only a constant number of arithmetic operations and integer square roots to characterize the decomposition, and linear time to print the decomposition. The discrete analogue of the problem in which the rectangle R is replaced by a rectangular array of lattice points is also considered, and three heuristic methods of solution are given. All of the heuristic methods operate by finding a discrete approximation to our optimal decomposition of R, but with different tradeoffs between the accuracy of the approximation and running time.

Some NP-complete problems in quadratic and nonlinear programming

In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.

A digit count for integer square roots

In this short paper we discuss the origins of and give a solution to the problem of determining the frequency distribution of particular digits in the integer square roots of a range of integers. After deriving the formulae in the case of the general integer base b(≥ 2), a computer program to evaluate the formulae and examples of its results are given. The generalization to the digit distribution in the case of a qth root is also presented.

The Enumeration of Generalized Double Stochastic Nonnegative Integer Square Matrices

The problem of enumerating generalized double stochastic integer square matrices is considered. The superposition theorem is used in conjunction with Schur functions to obtain the counting series for the $5 \times 5$ and $6 \times 6$ cases.