Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

A generalization of concavity for finite differences

The concept of concavity is generalized to discrete functions, u, satisfying the n^th-order difference inequality, (−1)^n−kΔⁿu(m) ≥ 0, M = 0, 1,..., N and the homogeneous boundary conditions, u(0) = … = u(k−1) = 0, u(N + k + 1) = … = u(N + n) = 0 for some k “1, …, n − 1”. A piecewise polynomial is constructed which bounds u below. The piecewise polynomial is employed to obtain a positive lower bound on u(m) for m = k, …, N + k, where the lower bound is proportional to the supremum of u. An analogous bound is obtained for a related Green's function.     

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.     

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 .     

A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form

(*)

y_n+1 − y_n + p_ny_n−k = 0, N = 0, 1, 2, …,

(*)where p_n ≥ 0 and k is a positive integer. We prove by using a new technique that

guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σⁿ⁻¹_i=n−kp_i ≤ k^k+1/(k + 1)^k+1. In addition, we present a nonoscillation sufficient condition for equation (*).     

Rational Interpolation from Stochastic Data: A New Froissart's Phenomenon

The analytic structure of Rational Interpolants (R.I.) f (z) built from randomly perturbed data is explored; the interpolation nodes x _j , j = 1,...,M, are real points where the function f reaches these prescribed data . It is assumed that the data are randomly perturbed values of a rational function _(n) ^(m) (m / n is the degree of the numerator/denominator). Much attention is paid to the R.I. familyf _(n+1) ^(m–1), in the small stochasticity régime. The main result is that the additional zero and pole are located nearby the root of the same random polynomial, called the Froissart Polynomial (F.P.). With gaussian hypothesis on the noise, the random real root of F.P. is distributed according to a Cauchy-Lorentz law, with parameters such that the integrated probability over the interpolation interval x ₁, x _M is always larger than 1/2; in two cases studied in detail, it reaches 2/3 in one case and almost 3/4 in the other. For the families f _(n+k) ^(m+k), numerical explorations point to similar phenomena; inspection shows that, in the mean, the localization occurs in the complex and/or real vicinity of the interpolation interval.     

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.     

Monoid-labeled transition systems

Given a -complete (semi)lattice , we consider -labeled transition systems as coalgebras of a functor ⁽⁻⁾, associating with a set X the set ^X of all -fuzzy subsets. We describe simulations and bisimulations of -coalgebras to show that L⁽⁻⁾ weakly preserves nonempty kernel pairs iff it weakly preserves nonempty pullbacks iff L is join infinitely distributive (JID).Exchanging for a commutative monoid , we consider the functor ⁽⁻⁾_ω which associates with a set X all finite multisets containing elements of X with multiplicities m M. The corresponding functor weakly preserves nonempty pullbacks along injectives iff 0 is the only invertible element of , and it preserves nonempty kernel pairs iff is refinable, in the sense that two sum representations of the same value, r₁ + … + r_m = c₁ + … + c_n, have a common refinement matrix (m_{(i, j)}) whose k-th row sums to r_k and whose l-th column sums to c_l for any 1≤ k ≤ m and 1 ≤ l ≤ n.     

QFT template generation for time-delay plants based on zero-inclusion test

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f(T,Q)={f(τ,q): τT[τ⁻,τ⁺], qQ∏_k=0^m−1[q_k⁻,q_k⁺]}, where f(τ,q)=g(q)+h(q)e^−jτω*, g(q) and h(q) are both complex-valued affine functions of the m-dimensional real vector q, and ω^* is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2^m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f(T,Q). This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.     

The query complexity of order-finding

We consider the problem where π is an unknown permutation on {0,1,…,2ⁿ−1}, y₀{0,1,…,2ⁿ−1}, and the goal is to determine the minimum r>0 such that π^r(y₀)=1. Information about π is available only via queries that yield π^x(y) from any x{0,1,…,2^m−1} and y{0,1,…,2ⁿ−1} (where m is polynomial in n). The main resource under consideration is the number of these queries. We show that the number of queries necessary to solve the problem in the classical probabilistic bounded-error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.     

Conjunctions of Unate DNF Formulas: Learning and Structure

A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class ^kconsisting of conjunctions ofkunate DNF formulas. This class generalizes the class ofk-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that ²can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P=NP. Thus the barrier to properly learning ²with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellersteinet al.(1997,J. Assoc. Comput. Mach.43(5), 840–862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to ^kand pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ^k. We construct, for any fixedk2, a class of functionsfthat cannot be represented by any formula in ^k, but which cannot be “easily” shown to have this property. More precisely, for any functionfonnvariables in the class, the value offon any polynomial-size set of points in its domain is not a witness thatfcannot be represented by a formula in ^k. Our construction is based on BCH codes.     

On the extension of solutions of linear constant-coefficient PDE

We solve the following over-determined boundary value problem (the “extension problem”): Let R(∂) be a matrix whose entries are linear partial differential operators, with constant coefficients. Let Ω be a non-empty, open, bounded, convex set. We consider the homogeneous system R(∂)f=0 in a neighborhood of , subject to the boundary condition f=g in a neighborhood of ∂Ω. For a given g, we give a criterion for the (unique) existence of a smooth solution f to this problem. There are two obvious necessary conditions: g is smooth and R(∂)g=0 in a neighborhood of ∂Ω. We characterize the class of differential operators R(∂) for which the problem is solvable for any g satisfying the necessary conditions. Finally, in the case where the solution is non-unique, we consider the possibility of obtaining uniqueness by fixing several components of the desired solution.     

Complexity of Computing the Local Dimension of a Semialgebraic Set

The paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the formf  ≥  0 with f  R [ X₁, ,X_n ], deg(f)  < d , andx   V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd)^O(n). Let x belong to a stratum of codimension l_xin V with respect to a smooth stratification ofV . Another algorithm computes the local dimension dim_x(V) with the complexity (k(l_x +  1)d)^O(l_x2n). Ifl  = max_x  Vl_x, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l +  1)d)^O(l2n). If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less thankd^O(l2n) , and the algorithm also finds the dimension of the tangent space to V at x in time kd^O(n). Whenl is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd)^O(n)andkd^O(n) respectively. A third algorithm finds the singular locus ofV in time (kd)^O(n2).     

Upper signed k-domination in a general graph

Let k be a positive integer, and let G=(V,E) be a graph with minimum degree at least k−1. A function f:V→{−1,1} is said to be a signed k-dominating function (SkDF) if _∑u∈N[v]f(u)?k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)?f(v) for every v∈V. The maximum of the values of _∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination numberΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.     

Large deviations for distributions of sums of random variables: Markov chain method

Let {ξ _k } _k=0^∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} , n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| ^p , p > 0, and exponential random variables with g(x) = x for x ≥ 0.     

Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.     

Robustness of star graph network under link failure

The star graph is an attractive underlying topology for distributed systems. Robustness of the star graph under link failure model is addressed. Specifically, the minimum number of faulty links, f(n, k), that make every (n − k)-dimensional substar S_n−k faulty in an n-dimensional star network S_n, is studied. It is shown that f(n,1)=n+2. Furthermore, an upper bound is given for f(n, 2) with complexity of O(n³) which is an improvement over the straightforward upper bound of O(n⁴) derived in this paper.     

Symmetric Rank Codes

As is well known, a finite field _n = GF(q ⁿ ) can be described in terms of n × n matrices A over the field = GF(q) such that their powers A ⁱ , i = 1, 2, ..., q ⁿ – 1, correspond to all nonzero elements of the field. It is proved that, for fields _n of characteristic 2, such a matrix A can be chosen to be symmetric. Several constructions of field-representing symmetric matrices are given. These matrices A ⁱ together with the all-zero matrix can be considered as a _n -linear matrix code in the rank metric with maximum rank distance d = n and maximum possible cardinality q ⁿ . These codes are called symmetric rank codes. In the vector representation, such codes are maximum rank distance (MRD) linear [n, 1, n] codes, which allows one to use known rank-error-correcting algorithms. For symmetric codes, an algorithm of erasure symmetrization is proposed, which considerably reduces the decoding complexity as compared with standard algorithms. It is also shown that a linear [n, k, d = n – k + 1] MRD code _k containing the above-mentioned one-dimensional symmetric code as a subcode has the following property: the corresponding transposed code is also _n -linear. Such codes have an extended capability of correcting symmetric errors and erasures.     

Improving bounds on link failure tolerance of the star graph

Determination of the minimum number of faulty links, f(n,k), that make every n-k-dimensional sub-star graph S_n-k faulty in an n-dimensional star network S_n, has been the subject of several studies. Bounds on f(n,k) have already been derived, and it is known that f(n,1)=n+2. Here, we improve the bounds on f(n,k). Specifically, it is shown that f(n,k)?(k+1)F(n,k), where F(n,k) is the minimum number of faulty nodes that make every S_n-k faulty in S_n. The complexity of f(n,k) is shown to be O(n²k) which is an improvement over the previously known upper bound of O(n³); this result in a special case leads to f(n,2)=O(n^↑2), settling a conjecture introduced in an earlier paper. A systematic method to derive the labels of the faulty links in case of f(n,1) is also introduced.