Matrix equations over rings

The main purpose of this work is to establish necessary conditions and sufficient conditions for the existence of a solution of matrix equations whose coefficient matrices have elements belonging to the ring $\text{R=}\text{C}\text{[z}{}_1\text{,z}{}_2\text{,…z}{}_{\mathrm{n}}\text{]}$ of polynomials in n variables with complex coefficients and the ring $\text{R=}\text{R}\text{[z}{}_1\text{,z}{}_2\text{,…z}{}_{\mathrm{n}}\text{]}{}^{\mathrm{n}}$ of rational functions a(z₁,z₂,…z_n)b(z₁,z₂,…,z_n)^?1 with real coefficients and b(z₁,z₂,…,z_n)≠0 for all (z₁,z₂,…,z_n) in Rⁿ. Results obtained are useful in multidimensional systems theory and elsewhere.     

On n-Dimensional Sequences. I

Let R be a commutative ring and let n ≥ 1. We study Γ(s), the generating function and Ann(s), the ideal of characteristic polynomials of s, an n-dimensional sequence over R .We express f(X₁,…,X_n) · Γ(s)(X^-1₁,…,X^-1_n) as a partitioned sum. That is, we give (i) a 2ⁿ-fold "border" partition (ii) an explicit expression for the product as a 2ⁿ-fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is βo(f, s), the "border polynomial" of f and s, which is divisible by X₁ … X_n.We say that s is eventually rectilinear if the elimination ideals Ann(s)∩R[X_i] contain an f_i (X_i) for 1 ≤ i ≤ n. In this case, we show that Ann(s) is the ideal quotient ($\text{∑}{}^{\mathrm{n}}{}_{\mathrm{i}}\text{=1(fi) : βo(f, s)/(X}{}_1\text{… X}{}_{\mathrm{n}}\text{)}$).When R and R[[X₁, X₂ ,…, X_n]] are factorial domains (e.g. R a principal ideal domain or F [X₁,…, X_n]), we compute the monic generator γ_i of Ann(s) ∩ R[X_i] from known f_i ϵ Ann(s) ∩ R[X_i] or from a finite number of 1-dimensional linear recurring sequences over R. Over a field F this gives an O($\text{∏}{}_{\mathrm{n}}{}^{\mathrm{i}}\text{=1 δγ}{}^3{}_{\mathrm{i}}$) algorithm to compute an F-basis for Ann(s).     

Some results on addition/subtraction chains

Let ?(n) (respectively $\text{?}$(n)) be the length of the shortest addition chain respectively addition/subtraction chain for n. We shall present several results on $\text{?}$(n). In particular, we determine $\text{?}$(n) for all n satisfying $\text{s}\text{(}\text{n}\text{) ? 3}$ and show $\text{?}\text{log n}\text{? + 2 ?}\text{?}\text{(}\text{n}\text{)}$ for all n satisfying $\text{s}\text{(}\text{n}\text{) ? 3}$, where $\text{s}\text{(}\text{n}\text{)}$ is the extended sum of digits of n. These results are based on analogous results for ?(n) and on the following two inequalities: |n| ? 2^d?1F_f+3 < 2^k?b and $\text{f + b}\text{?}\text{log}\text{s}\text{(n)}$ for a chain of length k = d + f + b with d doublings, f short steps, and b back steps for n. Moreover, we show that the difference $\text{?(}\text{n}\text{)?}\text{?}\text{(}\text{n}\text{)}$ (respectively $\text{?}\text{(}\text{n}\text{)??}\text{log n}\text{?}$) can be made arbbitrarily large. In addition, we prove that $\text{?}\text{(}\text{m}\text{) ?}\text{?}\text{(?}\text{m}\text{) ?}\text{?}\text{(}\text{m}\text{) + 1}\text{for m}\text{> 0}$ and characterize the case $\text{?}\text{(?}\text{m}\text{) =}\text{?}\text{(}\text{m}\text{)}$. Finally, we determine $\text{?}{}_{\mathrm{<mtext>k</mtext>}}\text{(}\text{n}{}_1\text{,…,}\text{n}{}_{\mathrm{k}}\text{)}$, the k-dimensional generalization of $\text{?}$, with the help of $\text{?}\text{(}\text{n}{}_1\text{,…,}\text{n}{}_{\mathrm{k}}\text{)}$, the k-element generalization of $\text{?}$.     

computer aided intuition in abstract algebra

Two examples are given in which the computer was used to supplement intuition in abstract algebra. In the first example, the computer was used to search Cayley tables of 4 element groupoids to find those which are 5-associative but not 4-associative. (n-associative means that the product of any n elements is independent of the way the factors are grouped by parentheses.) The computer generated examples suggested the existence of n element groupoids which are (2^n?2+1)-associative but not (2^n-2)-associative, for each integer n≧4.In the second example, the computer counted the numbers g₂(m) of invertible 2×2 matrices with entries chosen from the ring Z_mof integers, for m = 2, 3, 4,…, 18. The insight gained from these results led to a proof that there are

invertible n×n matrices over Z_m.Some applications to graduate and undergraduate instruction are indicated.     

Kleene chain completeness and fixedpoint properties

Let L denote the nonscalar complexity in k(x₁,…, x_n). We prove $\text{L(?,??/?x}{}_1\text{,…,??/?x}{}_{\mathrm{n}}\text{)?3L(?)}$. Using this we determine the complexity of single power sums, single elementary symmetric functions, the resultant and the discriminant as root functions, up to order of magnitude. Also we linearly reduce matrix inversion to computing the determinant.     

Mathematical foundations of Bézier's technique

Bezier's method is one of the most famous in computational geometry. In his book Numerical control Bezier gives excellent expositions of the mathematical foundations of this method. In this paper a new expression of the functions {f_n,i(u)}

$\text{f}{}_{\mathrm{n,i}}\text{(u)=1?}\text{Σ}\text{p=0}\text{i?1}\text{C}{}^{\mathrm{p}}{}_{\mathrm{n}}\text{u}{}^{\mathrm{p}}\text{(1?u)}{}^{\mathrm{n?p}}\text{(i=1,2,…,n)}$

is obtained.Using this formula, we have not only derived some properties of the functions {f_n,i(u)} (for instance $\text{f}{}_{\mathrm{n,n}}\text{(u) < f}{}_{\mathrm{n,n?1}}\text{(u)<...<f}{}_{\mathrm{n,1}}\text{(u)}\text{u}\text{? [0, 1]}$ and functions {f_n,i(u)} increase strictly at [0, 1] etc) but also simplified systematically all the mathematical discussions about Bezier's method.Finally we have proved the plotting theorem completely by matrix calculation.     

Uniform computational complexity of the derivatives of C∞-functions

We discuss the uniform computational complexity of the derivatives of C^∞-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function g∈C^∞[−1,1] such that the sequence $\text{{|g}{}^{\mathrm{(n)}}\text{(0)|}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is not bounded by any recursive function. On the other hand, we show that if f∈C^∞[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f⁽ⁿ⁾(x)| is bounded by 2^p(n) for all x∈[−1,1] for some polynomial p, then the sequence $\text{{f}{}^{\mathrm{(n)}}\text{}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is uniformly polynomial time computable.     

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on ƒ involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem $\text{y}{}_{\mathrm{<mtext>k</mtext><mtext>+1</mtext>}}\text{− 2y}{}_{\mathrm{k}}\text{+ y}{}_{\mathrm{<mtext>k</mtext><mtext>-1</mtext>}}\text{+ ƒ(k,y}{}_{\mathrm{k}}\text{,v}{}_{\mathrm{k}}\text{) = 0, for k = 1,…, n − 1, y}{}_0\text{= 0 = y}{}_{\mathrm{n}}$, where ƒ is continuous and v_k = y_k − y_k−1, for k = 1,…,n. In the special case $\text{ƒ(k,t,p) = ƒ(t) ≥ 0}$, we give growth conditions on ƒ and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue.     

Improving loops and vanishing variables in long transportation problems

In this paper the following two results are presented: (1)A method which determines the optimal values of certain variables during the iterative solution process. The closer the current primal feasible solution is to the optimal solution, the greater the number of variables which may be determined. (2) For each current feasible solution (X_ij) of the given m × n transportation problem A, a feasible solution ($\text{X}\text{?}{}_{\mathrm{ij}}$) of an auxiliary m × m(m ?1) transportation problem $\text{A}\text{?}$ is constructed. Problem $\text{A}\text{?}$ is shown to be equivalent to an m(m ? 1) × m(m ? 1) assignment problem with two admissible cells per column. The optimally of (X_ij) is shown to imply the optimality of ($\text{X}\text{?}{}_{\mathrm{ij}}$) and conversely. The best “improving loops” (including the improving loops used in MODI) of $\text{A}\text{?}$ are shown to be the best “improving loops” of A as well.     

Normal forms for nonautonomous difference equations

We extend Henry Poincaré's normal form theory for autonomous difference equations χ_{k + 1} = f(χ_k) to nonautonomous difference equations χ_{k + 1} = f_k(χ_k). Poincaré's nonresonance condition $\text{α}{}_{\mathrm{j}}\text{-П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{=1α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for eigenvalues is generalized to the new nonresonance condition λ_j∪$\text{α}{}_{\mathrm{j}}\text{⊔П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for spectral intervals.     

Point pattern matching by relaxation

Let $\text{P}\text{= P}{}_1\text{, …, P}{}_{\mathrm{m}}$ and $\text{Q}\text{= Q}{}_1\text{, …, Q}{}_{\mathrm{n}}$ be two patterns of points. Each pairing (P_i, Q_j) of a point of $\text{P}$ with a point of $\text{Q}$ defines a relative displacement δ_ij of the two patterns. We can define a figure of merit for δ_ij according to how closely other point pairs coincide under δ_ij. If there exists a displacement δ₀ for which $\text{P}$ and $\text{Q}$ match reasonably well, the pairings for which δ_ij ? δ₀ will have high merit scores, while other pairings will not. The scores can then be recomputed, giving weights to the other point pairs based on their own scores; and this process can be iterated. When this is done, the scores of pairs that correspond under δ₀ remain relatively high, while those of other pairs become low. Examples of this method of point pattern matching are given, and its possible advantages relative to other methods are discussed.     

Unbounded solutions of quasi-linear difference equations

We study positive increasing solutions of the nonlinear difference equation $\text{δ(a}{}_{\mathrm{n}}\text{φ}{}_{\mathrm{p}}\text{(δχ}{}_{\mathrm{n}}\text{))=b}{}_{\mathrm{n}}\text{f(χ}{}_{\mathrm{n+1}}\text{,}\text{φ}{}_{\mathrm{p}}\text{(u)=|u|}{}^{\mathrm{p-2}}\text{u,}\text{p>1}$ where {a_n}, {b_n} are positive real sequences for n ≥ 1, fR → R is continuous with uf(u) > 0 for u ≠ 0. A full characterization of limit behavior of all these solutions in terms of a_n, b_n is established. Examples, showing the essential role of used hypotheses, are also included. The tools used are the Schauder fixed-point theorem and a comparison method based on the reciprocity principle.     

Determination of a general economic location constant by monte carlo simulation

The problem of estimating the number of markets (or plants) to serve a set of sources in a given geographical area was considered. Markets were located so as to minimize total assembly cost which was considered a linear function of the weighted Euclidean distances between sources and markets. The following predictive function C_m was proposed for estimating the minimum total assembly cost for a given number of markets: $\text{C}{}_{\mathrm{m}}\text{= C}{}_1\text{?(}\text{m?1}\text{m}\text{)}{}^{\mathrm{k}}\text{(}\text{M}\text{M ? 1}\text{)}{}^{\mathrm{k}}\text{(C}{}_1\text{? C}{}_{\mathrm{M}}\text{),m = 1, 2, 3, …, M}$ where m = number of markets being located. M = maximum number of potential market sites. C₁ = minimum assembly cost when only one market is located. C_M = minimum assembly cost when all M markets are located. k = an undetermined constant which specifies the shape of the function.The validity of the C_m function and the range of the k constant were determined by computer Monte Carlo experimentation. The constant k, to a sufficient degree of approximation and ordinary use, was found independent of the number of sources and their distribution. A general economic location co     

On the form equivalence of L-forms

In [5] the notion of an L form F defining a family $\text{L}$_d(F) of languages by means of X-interpretations has been introduced. Here X is one of a number of possible variations of the notion of interpretation originally used in [1] for grammar forms. In this paper it is shown that the questions whether $\text{L}{}_{\mathrm{d}}\text{(F) =}\text{L}{}_{\mathrm{d}}\text{(F}{}_1\text{)}$ for L forms F and F₁ is decidable, if deterministic interpretations of PDOL systems are considered, where L(F) and L(F₁) contain at most one word of length n for any n ? 0, and it is shown that same question is undecidable, if full or uniform interpretations are chosen. In contrast to this, no such results are known for grammar forms at this point.