On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

Optimal priorities in GI|Gn|1 queue

A polynomial-time algorithm based on reduction to a polyhedron minimization problem is proposed for minimizing a given function F(W₁,...,W_n) that depends on the mean waiting times in the Gl|G_n|l queue.Translated from Kibernetika, No. 2, pp. 80–85, March–April, 1991.     

Radiocoloring in planar graphs: Complexity and approximations

The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters, by exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function such that |Φ(u)-Φ(v)|2, when u,v are neighbors in G, and |Φ(u)-Φ(v)|1 when the distance of u,v in G is two. The number of discrete frequencies and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the radiocoloring problem for general graphs is hard to approximate (unless NP=ZPP) within a factor of n^1/2-ε (for any ), where n is the number of vertices of the graph. However, when restricted to some special cases of graphs, the problem becomes easier. We prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(nΔ) time algorithm (|V|=n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where Δ the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with λ colors, in the case where λ4Δ+50.     

Distributions having bases which generate finite dimensional Lie algebras

Let X¹,…, X^k be real analytic vector fields on an n-dimensional manifold M, k < n, which are linearly independent at a point p ε M and which, together with their Lie products at p, span the tangent space TM_p. Then X¹,…, X^k form a local basis for a real analytic k-dimensional distribution x→D^k(x)=span{X¹(x),…,X^k(x)}. We study the question of when D^k admits a basis which generates a nilpotent, or solvable (or finite dimensional) Lie algebra. If this is the case the study of affine control systems, or partial differential operators, described via X¹,…, X^k can often be greatly simplified.     

Stability and boundedness in difference systems with finite delay

In this paper, stability and boundedness theorems for delay difference systems with the condition

δV(n,x(n))≤-W₂(|x(n)|)+σ

are given, where Δ is the backward difference operator. Some known results are generalized.     

Elements of small norm in Shanks’ cubic extensions of imaginary quadratic fields

Let be an imaginary quadratic number field with ring of integers Z_k and let k(α) be the cubic extension of k generated by the polynomial f_t(x)=x³−(t−1)x²−(t+2)x−1 with tZ_k. In the present paper we characterize all elements γZ_k[α] with norms satisfying |N_k(α)/k|≤|2t+1| for |t|≥14. This generalizes a corresponding result by Lemmermeyer and Pethő for Shanks’ cubic fields over the rationals.     

Homomorphic characterizations of recursively enumerable languages with very small language classes

In this paper, we attempt to characterize the class of recursively enumerable languages with much smaller language classes than that of linear languages. Language classes, and , of (i,j) linear languages and (i,j) minimal linear languages are defined by posing restrictions on the form of production rules and the number of nonterminals. Then the homomorphic characterizations of the class of recursively enumerable languages are obtained using these classes and a class, , of minimal linear languages. That is, for any recursively enumerable language L over Σ, an alphabet Δ, a homomorphism h : Δ^*→Σ^* and two languages L₁ and L₂ over Δ in some classes mentioned above can be found such that L = h(L₁∩L₂). The membership relations of L₁ and L₂ of the main results are as follows:(I) For posing restrictions on the forms of production rules, the following result is obtained:(1) and .This result is the best one and cannot be improved using . However, with posing more restriction on L₂, this result can be improved and the follwing statement is obtained.(2) and .(II) For posing restrictions on the numbers of nonterminals, the follwing result is obtained.(3) and .We believe this result is also the best.     

Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ² for any simple graph with maximum degree Δ≥2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.     

Uniform finite generation of compact Lie groups

Consider a compact connected Lie group G and the corresponding Lie algebra . Let {X₁,…,X_m} be a set of generators for the Lie algebra . We prove that G is uniformly finitely generated by {X₁,…,X_m}. This means that every element KG can be expressed as K=e^Xt₁e^Xt₂···e^Xt_l, where the indeterminates X are in the set {X₁,…,X_m}, , and the number l is uniformly bounded. This extends a previous result by F. Lowenthal in that we do not require the connected one dimensional Lie subgroups corresponding to the X_i, i=1,…,m, to be compact. We link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems.     

Embedding Cartesian Products of Graphs into de Bruijn Graphs

Given a Cartesian productG=G₁× … ×G_m(m≥ 2) of nontrivial connected graphsG_iand then-dimensional baseBde Bruijn graphD=D_B(n), it is investigated whether or notGis a spanning subgraph ofD. Special attention is given to graphsG₁× … ×G_mwhich are relevant for parallel computing, namely, to Cartesian products of paths (grids) or cycles (tori). For 2-dimensional de Bruijn graphsD, we present a theorem stating that certain structural assumptions on the factorsG₁, …,G_mensure thatG₁× … ×G_mis a spanning subgraph ofD. As corollaries, we obtain results improving previous results of Heydemannet al.(J. Parallel Distrib. Comput.23 (1994), 104–111) on embedding grids and tori into de Bruijn graphs. Specifically, we obtain that (i) any gridG=G₁× … ×G_mis a spanning subgraph ofD=D_B(2) provided that |G| = |D|, and (ii) any torusG=G₁× … ×G_mis a spanning subgraph ofD=D_B(2) provided that |G| = |D| and that theG_iare cycles of even length ≥4. We show that these results have consequences for the casen> 2, too: for evenn, we apply our results to obtain embeddings of grids and toriGinto de Bruijn graphsD_B(n) with dilationn/2, where the baseBis a fixed integer ≥2, andnis big enough to ensure |G| ≤ |D_B(n)|. We also contrast our results forn= 2 with nonexistence results forn≥ 3 and briefly describe experimental results in the area of parallel image processing.     

Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions

Given a stable rational transfer function G(s) and weighting function W(s), the problem of finding of MacMillan degree k so as to minimise is considered. This problem is solved for W(s) =(s-β)/(s-α) with no assumptions on the signs of α and β. This gives rise to approximations where can be accurately bounded from above and below in terms of the Hankel singular values of WG (when α, β > 0).     

Fast modular exponentiation of large numbers with large exponents

In many problems, modular exponentiation |x^b|_m is a basic computation, often responsible for the overall time performance, as in some cryptosystems, since its implementation requires a large number of multiplications.It is known that |x^b|_m=|x^|b|_(m)|_m for any x in [1,m−1] if m is prime; in this case the number of multiplications depends on (m) instead of depending on b. It was also stated that previous relation holds in the case m=pq, with p and q prime; this case occurs in the RSA method.In this paper it is proved that such a relation holds in general for any x in [1,m−1] when m is a product of any number n of distinct primes and that it does not hold in the other cases for the whole range [1,m−1].Moreover, a general method is given to compute |x^b|_m without any hypothesis on m, for any x in [1,m−1], with a number of modular multiplications not exceeding those required when m is a product of primes.Next, it is shown that representing x in a residue number system (RNS) with proper moduli m_i allows to compute |x^b|_m by n modular exponentiations |x_i^b|_mi in parallel and, in turn, to replace b by |b|_(mi) in the worst case, thus executing a very low number of multiplications, namely log₂m_i for each residue digit.A general architecture is also proposed and evaluated, as a possible implementation of the proposed method for the modular exponentiation.     

Some counterexamples in robust stability theory

This note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters Δ_i, I = 1,…,n, is robustly stable. It is shown by example that a polynomial in n variables may have a unique real root, and that this observation disposes of several natural conjectures in robust stability theory. In particular, we show that, in a certain sense, there are no ‘edge’ or ‘m-dimensional face’ Kharitonov-like theorems for the general multilinear case. The result holds even when restricted to that subset of multilinear functions which can be written in the form f(Δ₁,…, Δ_n) = det(I + diag(Δ₁,…,Δ_n)M) for some complex matrix M.     

Existence and efficient construction of fast Fourier transforms on supersolvable groups

The linear complexityL _K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L _K(G):= min{L _K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|²log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L _K(G)8.5|G|log|G| for any supersolvable groupG andL _K(G)1.5|G|log|G| for any 2-groupG.     

A class of projection and contraction methods for asymmetric linear variational inequalities and their relations to Fukushima's descent method

For solving asymmetric linear variational inequalities, we present a class of projection and contraction methods under the general G-norm. The search direction of our methods is just a convex combination of two descent directions of Fukushima's merit function. However, we use the direction to reduce the distance function (1/2)u − u*²_G, where μ* is a solution point of the problem. Finally, we report some numerical results for spatial price equilibrium problems by using the presented methods.     

Constructing normalisers in finite soluble groups

This paper describes algorithms for constructing a Hall π-subgroup H of a finite soluble group G and the normaliser N_G(H). If G has composition length n, then H and N_G(H) can be constructed using O(n⁴ log |G|) and O(n⁵ log |G|) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative system normalisers. Computer implementations of these algorithms can compute a Sylow 3-subgroup of a group with n = 84, and its normaliser in 47 seconds and 30 seconds, respectively. Constructing normalisers of arbitrary subgroups of a finite soluble group can be complicated. This is shown by an example where constructing a normaliser is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

Achieving prescribed mean waiting times in a single-server queueing system

A new queueing discipline is proposed, which achieves any prescribed mean waiting time under stationary conditions in the GI|G_n|1 queue. Mean waiting times for customers of each type are obtained for the HM|G_n|1 and GI|HM_n|1 queues. A polynomial-time algorithm is described to determine the parameters of the queueing discipline given the mean waiting times.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 97–101, November–December, 1991.     

Detection of three-axis angles by an optical sensor

This article describes an optical method for measurement of three-axis angles (roll angle Δθ_X, pitch angle Δθ_Y, yaw angle Δθ_Z). The laser autocollimation method is improved from two-axis angle measurement to three-axis angle measurement by employing a diffraction grating instead of a plane mirror as the target reflector. The three-axis angle components can be calculated by three methods of using different diffraction light spots reflected from the diffraction grating. Method 1 uses three beams, which are the 0th-order and the ±1st-order diffraction light spots. Both Method 2 and Method 3 use two light spots. The former uses the ±1st-order diffraction light spots and the latter combines the 0th-order with the +1st-order or the −1st-order diffraction light spots. A prototype three-axis angle sensor is also designed and fabricated to compare the characteristics of the three methods from the viewpoints of sensitivity, linearity and resolution of the sensor output.