Computing Homomorphisms Between HolonomicD-modules

Let K C be a subfield of the complex numbers, and let D be the ring of K -linear differential operators on R = K [ x₁, . . . , x_n]. If M and N are holonomic left D -modules we present an algorithm that computes explicit generators for the finite dimensional vector space _D(M, N). This enables us to answer algorithmically whether two given holonomic modules are isomorphic. More generally, our algorithm can be used to get explicit generators for _Dⁱ(M, N) for any i in the sense of Yoneda.     

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.     

(J, J′)-lossless factorization based on conjugation

This paper is concerned with a derivation of the state-space form of the (J, J′)-lossless factorization which contains both the inner-outer factorization and the spectral factorization of positive matrices as special cases. Also, the (J, J′)-lossless factorization gives a unified framework of H^∞ control theory. We use the method of conjugation which makes the derivation much simpler than the previous literature, most of which used the technique of (J, J′)-spectral factorization. A necessary and sufficient condition is represented in terms of two Riccati equations one of which is degenerated.     

An estimate on the convergence of MKZ–Bézier operators

The pointwise approximation properties of the MKZ–Bézier operators M_n,α(f,x) for α≥1 have been studied in [X.M. Zeng, Rates of approximation of bounded variation functions by two generalized Meyer–König–Zeller type operators, Comput. Math. Appl. 39 (2000) 1–13]. The aim of this paper is to study the pointwise approximation of the operators M_n,α(f,x) for the other case 0<α<1. By means of some new estimate techniques and a result of Guo and Qi [S. Guo, Q. Qi, The moments for the Meyer–König and Zeller operators, Appl. Math. Lett. 20 (2007) 719–722], we establish an estimate formula on the rate of convergence of the operators M_n,α(f,x) for the case 0<α<1.     

On Some Schützenberger Conjectures

In this paper we consider factorizing codes C over A, i.e., codes verifying the factorization conjecture by Schützenberger. Let n be the positive integer such that aⁿC, we show how we can construct C starting with factorizing codes C′ with a^n′C′ and n′ < n, under the hypothesis that all words aⁱza^j in C, with z(A\a)A*(A\a) (A\a), satisfy i, j, > n. The operation involved, already introduced by Anselmo, is also used to show that all maximal codes C=P(A−1)S+1 with P, SZA and P or S in Za can be constructed by means of this operation starting with prefix and suffix codes. Old conjectures by Schützenberger have been revised.     

Uq(sl(2)) Satisfies a Bernstein Duality

It is well known that in some cases the functorExt_R^μ( − , R) defines a duality between module categories. In earlier papers we studied when this duality can be represented by a bimodule and have characterized when this happens. In this paper, using some computational methods of noncommutative Gröbner bases in the construction of projective resolutions of irreducible finite-dimensional representations, we show new examples of algebras satisfying this property.     

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.     

On the non-existence of stationary diffusions which satisfy the Bene condition

The one-dimensional diffusion x_t satisfying dx_t = f(x_t)dt + dw_t, where w_t is a standard Brownian motion and f(x) satisfies the Bene condition f′(x) + f²(x) = ax² + bx + c for all real x, is considered. It is shown that this diffusion does not admit a stationary probability measure except for the linear case f(x) = αx + β, α < 0.     

Lower Bounds for Merging Networks

A lower bound theorem is established for the number of comparators in a merging network. Let M(m, n) be the least number of comparators required in the (m, n)-merging networks, and let C(m, n) be the number of comparators in Batcher's (m, n)-merging network, respectively. We prove for n≥1 that M(4, n)=C(4, n) for n≡0, 1, 3 mod 4, M(4, n)≥C(4, n)−1 for n≡2 mod 4, and M(5, n)=C(5, n) for n≡0, 1, 5 mod 8. Furthermore Batcher's (6, 8k+6)-, (7, 8k+7)-, and (8, 8k+8)-merging networks are optimal for k≥0. Our lower bound for (m, n)-merging networks, m≤n, has the same terms as C(m, n) has as far as n is concerned. Thus Batcher's (m, n)-merging network is optimal up to a constant number of comparators, where the constant depends only on m. An open problem posed by Yao and Yao (Lower bounds on merging networks, J. Assoc. Comput. Mach.23, 566–571) is solved: lim_n→∞M(m, n)/n=log m/2+m/2^{log m}.     

Sample complexity of worst-case H-identification

We show that the sample complexity of qorst-case H^∞-identification is of order n², by proving that the minimal length of a fractional H^∞-cover for C_n, regarded as the linear space of complex-valued sequences of length n, is of order n². A unit vector u in ^∞ is a fractional H^∞-cover for C_n if for some

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for all rh ε C_n, where is the z-transform of h. We also give similar results for real-valued sequences.     

The complete inclusion structure of leaf power classes

Let k≥2 be an integer and G=(V,E) be a finite simple graph. A tree T is a k-leaf root of G, if V is the set of leaves of T and, for any two distinct x,yV, the distance between x and y in T is at most k if and only if xyE. We say that G is a k-leaf power if there is a k-leaf root of G. The main result of this paper is that, for all 2≤k<k^′, the classes of k- and k^′-leaf powers are inclusion-incomparable, if and only if k^′≤2k−3 and k^′−k is an odd number. With this result, an open problem from the literature about the inclusion structure of these graph classes is solved completely. In addition, the intersection of the smallest pair of inclusion-incomparable classes is studied.     

Optimizing a damped system – a case study

We consider a second-order damped-vibrational system described by the equation M ?+C(v) [xdot]+K x=0, where M, C(v), K are real, symmetric matrices of order n. We assume that the undamped eigenfrequencies (eigenvalues of (λ² M+K) x=0) ω₁, ω₂, …, ω _n , are multiple in the sense that ω₁=ω₂, ω₃=ω₄, …, ω _n?1=ω _n , or are given in close pairs ω₁ ≈ ω₂, ω₃ ≈ ω₄, …, ω _n?1 ≈ ω _n . We present a formula which gives the solution of the corresponding phase space Lyapunov equation, which then allows us to calculate the first and second derivatives of the trace of the solution, with no extra cost. It can serve for the efficient trace minimization.     

On the rate of convergence of 2-term recursions in ℝ d

LetG be a compact set in ℝ ^d d≥1,M=G×G andϕ:M →G a map inC ₃(M). Suppose thatϕ has a fixed pointξ, i.e.ϕ(ξ, ξ)=ξ. We investigate the rate of convergence of the iterationx ⁿ⁺²=φ(x ⁿ⁺¹,x ⁿ) withx ⁰,x ¹∈G andx ⁿ→ξ. Iff _n=Q‖x ⁿ−ξ‖ with a suitable norm and a constantQ depending onξ, an exact representation forf _n is derived. The error terms satisfyf _2m+1≍(ƒ_2m)^γ,f _2m+2≍(ƒ_2m+1)^2γ,m≥0, with 1<gg<2, andγ=γ(x ¹,x ⁰). According to our main result we have lim_n→∞{‖x ⁿ⁺²‖/(‖x ⁿ‖)²}=Q, 0<Q<∞. This paper constitutes an extension of a part of the author’s doctoral thesis realized under the direction of Prof. E. Wirsing and Prof. A. Peyerimhoff, University of Ulm (Germany).     

New gas sensitive MIS structures Pt/Al2O3(M = Pt, Rh)/Si with a granular dielectric layer

New gas sensitive MIS structures Pt/Al₂O₃(M)/p-Si, where M = Pt, Rh, with granular dielectric Al₂O₃ layers doped with noble metals were obtained by an aerosol pyrolysis method. Surface morphology and composition of the structures were studied by TEM, AFM and EPMA. Sensor properties of the MIS structures were studied towards reducing gases (1000 ppm H₂, 300 ppm CO, 1000 ppm CH₄ in air) at 100 and 200 °C. The Pt/Al₂O₃(M = Pt, Rh)/Si structures showed a very high sensor response to reducing gases. A shift of C–V characteristics was up to 2.5 V under CO, 2.2 V under hydrogen and 0.7 V under methane. High values of shift of C–V curves can be related with cooperative influence of a change of surface state density in dielectric layer, reduction of platinum electrode and dipole layer formation.     

Optimal strategies against a liar

We consider the following scenario: There are two individuals, say Q (Questioner) and R (Responder), involved in a search game. Player R chooses a number, say x, from the set S={1,…,M}. Player Q has to find out x by asking questions of type: “which one of the sets A₁,A₂,…,A_q, does x belong to?”, where the sets A₁,…,A_q constitute a partition of S. Player R answers “i” to indicate that the number x belongs to A_i. We are interested in the least number of questions player Q has to ask in order to be always able to correctly guess the number x, provided that R can lie at most e times. The case e=0 obviously reduces to the classical q-ary search, and the necessary number of questions is [log_qM]. The case q=2 and e1 has been widely studied, and it is generally referred to as Ulam's game. In this paper we consider the general case of arbitrary q2. Under the assumption that player R is allowed to lie at most twice throughout the game, we determine the minimum number of questions Q needs to ask in order to successfully search for x in a set of cardinality M=qⁱ, for any i1. As a corollary, we obtain a counterexample to a recently proposed conjecture of Aigner, for the case of an arbitrary number of lies. We also exactly solve the problem when player R is allowed to lie a fixed but otherwise arbitrary number of times e, and M=qⁱ, with i not too large with respect to q. For the general case of arbitrary M, we give fairly tight upper and lower bounds on the number of the necessary questions.     

A note on a problem on ω-limit sets of N–dimensional skew-product maps

In 2003, Balibrea et al. stated the problem of finding a skew-product map G on 𝕀³ holding ω _G ={0}×𝕀²=ω _G (x, y, z) for any (x, y, z)∈𝕀³, x≠0. We present a method for constructing skew-product maps F on 𝕀 ⁿ⁺¹ holding ω _F ={0}×𝕀 ⁿ =ω _F (x ₁, x ₂, …, x _n+1), (x ₁, x ₂, …, x _n+1)∈𝕀 ⁿ⁺¹, x ₁≠0.     

Inversion Problems in theq-Hahn Tableau

If { P_n(x;q)}_nis a family of polynomials belonging to the q -Hahn tableau then each polynomial of this family can be written as P_n(x;q) = ∑_{m = 0}ⁿD_m(n)_m(x) where _m(x) stands for (x;q)_mor x^m. In this paper we solve the corresponding inversion problem, i.e. we find the explicit expression for the coefficients I_m(n) in the expansion _n(x) = ∑_{m = 0}ⁿI_m(n)P_m(x;q).     

Finding the Extrema of a Distributed Multiset

We consider the problem of finding the extrema of a distributed multiset in a ring, that is, of determining the minimum and the maximum values,x_minandx_max, of a multisetX= {x₀,x₂, ...,x_n−1} whose elements are drawn from a totally ordered universeUand stored at thenentities of a ring network. This problem is unsolvable if the ring size is not known to the entities, and it has complexity Θ(n²) in the case of asynchronous rings of known size. We show that, in synchronous rings of known size, this problem can always be solved inO((c+ logn) ·n) bits andO(n·c·x^1/c) time for any integerc> 0, wherex= Max{|x_min|, |x_max|}. The previous solutions requiredO(n²) bits and the same amount of time. Based on these results, we also present a bit-optimal solution to the problem of finding the multiplicity of the extrema.     

Analytical and numerical results onM-variable generalized Bessel functions

Recently, some multivariable special functions have been obtained by generalizing functions of Bessel type. Here, we continue the treatment of these functions starting fromJ _n (x, y; i), which is of noticeable practical interest. Finally, we consider the cases of functionsJ _n(x₁, x₂,..., x_M) and the related modified version,I _n(x₁, x₂,..., x_M), with two significant physical applications. Calculations of multivariable generalized Bessel functions are discussed and numerical results are given forJ _n(x₁,x₂;i), withn=0, 1, in a region of interest.