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.     

Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

The psi-cube: a bus-based cube-type clustering network for high-performance on-chip systems

This paper proposes a bus-based cube-type network, called psi-cube, that alleviates the two problems, long wires and a limited number of I/O pins, against the on-chip systems through a small diameter and dynamic clusters, respectively. The 2ⁿ-node psi-cube is organized on the sets of node-partitions produced with an extended n-bit Hamming code ψ(n, k) [M. Takesue, Ψ-Cubes: recursive bused fat-hypercubes for multilevel snoopy caches, in: Proceedings of the International Symposium on Parallel Architectures, Algorithms, and Networks, IEEE CS Press, 1999, pp. 62–67] if we connect the nodes in each partition to the bus owned by the leader of the partition. Owing to the routing between the leaders separated by the distance of 1–3, the diameter equals n/2 if n≠2^p − 1 or n/2 otherwise. The maximum bus length is O(2^p−1) or O(2^k−1) when the psi-cube is mapped onto an array. We dynamically produce separate sets of clusters for different off-chip targets such as memory blocks, so the traffic to the leaders of clusters is much smaller than in static clusters fixed in hardware. From simulation results, the psi-cube outperforms over the mesh if the bus delay is less than 4 times the mesh link’s, and the dynamic clusters increase the psi-cube bandwidth by over 60%.     

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.     

Connectivity properties of spaces of partial realizations

This paper addresses the problem of determining the number of connected components of the space of real scalar M×N Hankel matrices of rank n. For n<min{M,N} it is shown that the space has n+1 components for M+N even, and is connected for M+N odd. As an application, the number of components of sets of minimal partial realizations of McMillan degree nτ of length τ sequences is determined.     

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.     

Fast Parallel Algorithms for Solving Triangular Systems of Linear Equations on the Hypercube

This paper presents efficient hypercube algorithms for solving triangular systems of linear equations by using various matrix partitioning and mapping schemes. Recently, several parallel algorithms have been developed for this problem. In these algorithms, the triangular solver is treated as the second stage of Gauss elimination. Thus, the triangular matrix is distributed by columns (or rows) in a wrap fashion since it is likely that the matrix is distributed this way after an LU decomposition has been done on the matrix. However, the efficiency of the algorithms is low. Our motivation is to develop various data partitioning and mapping schemes for hypercube algorithms by treating the triangular solver as an independent problem. Theoretically, the computation time of our best algorithm is ((12p + 1)n² + 36p³ − 28p²)/(24p²), and an upper bound on the communication time is 2αp log p (log n − log p) + 2α(log n − log p − 1) log p + (cn/p − 2c)(2 log p − 1) + log p(cn − c − α), where α is the (communication startup time)/(one entry scanning time), c is a constant, n is the order of the triangular system and p is the number of nodes in the hypercube. Experimental results show that the algorithm is efficient. The efficiency of the algorithm is 0.945 when p = 2, n = 513, and 0.93 when p = 8, n = 1025.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Characterizations of distributions of exponential or geometric type by the integrated lack of memory property and record values

Let be a sequence of i.i.d. random variables, the sequence of its upper record values (i.e. L(0) = 1, L(n) = inf{jX_j>X_L(n−1} for n≥1). Without any assumptions to the support of P^X₁ the equidistribution of X₁ and a record increment X_L(n − X_L(n−1), n ≥ 1 yields X₁ to be either exponentially or geometrically distributed according to whether the additive subgroup generated by the support of P^X₁ is dense or a lattice in . The integrated lack of memory property can easily be reduced to the above problem for the case n = 1. Similarly the independence of X_L(n−1) and X_L(n) − X_L(n−1) for some n>1 characterizes X₁ to have e exponential or a geometric tail provided that the support of P^X₁ is bounded to the left and its right extremity no atom. Hence, if also its left extremity is no atom the independence of X_L(n−1) and X_L(n−1) − X_L(n−1) characterizes X₁ to be exponentially distributed.     

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 (*).     

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.     

Electrochemical DNA biosensor for the detection of interaction between di[azino-di(5,6-azafluorene)-κ-NN′]dichlormanganous and DNA

A new Mn(II) complex of MnL₂Cl₂ (L = azino-di(5,6-azafluorene)-κ²-NN′) was synthesized and utilized as an electrochemical indicator for the determination of hepatitis B virus (HBV) based on its interaction with MnL₂Cl₂. The electrochemical behavior of interaction of MnL₂Cl₂ with salmon sperm DNA was investigated on glassy carbon electrode (GCE). In the presence of salmon sperm DNA, the peak current of [MnL₂]²⁺ was decreased and the peak potential was shifted positively without appearance of new peaks. The binding ratio between [MnL₂]²⁺ and salmon sperm DNA was calculated to be 2:1 and the binding constant was 3.72 × 10⁸ mol² L⁻². The extent of hybridization was evaluated on the basis of the difference between signals of [MnL₂]²⁺ with probe DNA before and after hybridization with complementary sequence. Control experiments performed with non-complementary and mismatch sequence demonstrated the good selectivity of the biosensor. With this approach, a sequence of the HBV could be quantified over the range from 1.76 × 10⁻⁸ to 1.07 × 10⁻⁶ mol L⁻¹, with a linear correlation of r = 0.9904 and a detection limit of 6.80 × 10⁻⁹ mol L⁻¹. Additionally, the binding mechanism was preliminarily discussed. The mode of interaction between MnL₂Cl₂ and DNA was found to be primary intercalation binding.     

On Unbounded Tolerable Solution Sets

We prove that the nonempty tolerable solution set of the interval linear system Ax = b is unbounded if and only if the interval matrix of the system has linearly dependent degenerate columns. Also, we prove that the tolerable solution set may be represented as a sum of the linear subspace {c ε ℝⁿ ∨ Ac = 0} and a bounded convex polyhedron and propose a way for suitable estimation of unbounded tolerable solution sets.     

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.     

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 .     

Further oscillation criteria for delay partial difference equations

This paper is concerned with the partial difference equation A_m+1,n+A_m,n+1−A_m,n+p_m,nA_m−k,n−l=0,m,n=0,1,2,…,, where k and l are two positive integers, {p_m,n} is a real double sequence. Some new oscillation criteria for this equation are obtained.     

Application of carbon ionic liquid electrode for the electrooxidative determination of catechol

A carbon ionic liquid electrode (CILE) was constructed using graphite powder mixed with N-butylpyridinium hexafluorophosphate (BPPF₆) in place of paraffin as the binder, which showed strong electrocatalytic activity to the direct oxidation of catechol. In pH 3.0 phosphate buffer solution (PBS) a pair of redox peaks appeared on the CILE with the anodic and the cathodic peak potential located at 387 and 330 mV (vs. SCE), respectively. The electrochemical behaviors of catechol on the CILE were carefully investigated, and the electrochemical parameters were calculated with the results of the electrode reaction standard rate constant k_s as 1.27 s⁻¹, the charge-transfer coefficient α as 0.58 and the electron transferred number n as 2. Under the selected conditions, the anodic peak current increased linearly with the catechol concentration over the range from 1.0 × 10⁻⁶ to 8.0 × 10⁻⁴ mol L⁻¹ by cyclic voltammetry at the scan rate of 100 mV s⁻¹. The detection limit was calculated as 6.0 × 10⁻⁷ mol L⁻¹ (3σ). The CILE showed good ability to separate the electrochemical responses of catechol and ascorbic acid (AA) with the anodic peak potential separation as 252 mV (vs. SCE). The proposed method was further applied to the synthetic samples determination with satisfactory results.     

Molecular valve consisting of poly(acrylic acid) gel

A molecular valve, consisting of poly(acrylic acid) gel-coated Au mesh, was developed based on volume change of the gel in response to cation concentration. The valve closed when concentration of cations such as H⁺, Na⁺, K⁺, Ca²⁺, Cu²⁺, or Al³⁺ was low, whereas opened upon increase in its concentration. The valve re-closed when water was flowed. The concentration where the valve opens was found to increase in the order of Al³⁺, Ca²⁺, and Na⁺ (2 × 10⁻⁴, 5 × 10⁻⁴, and 6 × 10⁻³ M, respectively). The response to Cu²⁺ ion showed similar behaviour, but the opening concentration was ca. 2 × 10⁻⁴ M, which is lower than that of Ca²⁺ ion. The valve appeared to close over the pH range from 3 to 12, whereas to open below and above it. The fastest response time to open the valve (less than 1 min) was obtained for a solution of pH 1–2. The valve showed repeatability at least 25 cycles upon successive loading of a solution of pH 2 and water. Effects of anions and pressure were also studied.     

Electrochemical sensor for hydroperoxides determination based on Prussian blue film modified electrode

An electrochemical sensor for hydroperoxides determination was investigated. The sensor was based on the electrocatalytic reduction of hydroperoxides on Prussian blue (PB)-modified glassy carbon electrode. The modified electrode possesses a high electrocatalytic effect towards all studied peroxides with the highest effect obtained with H₂O₂ followed by tert-butyl hydroperoxide (TBH), cumene hydroperoxide (CH) and linoleic acid hydroperoxide (LAH). In addition, the modified electrode showed a good stability and a fast response time (<20 s). The lower detection limits of H₂O₂, TBH, CH and LAH were found to be 10⁻⁷ mol L⁻¹, 2 × 10⁻⁷ mol L⁻¹, 3.5 × 10⁻⁷ mol L⁻¹ and 4 × 10⁻⁷ mol L⁻¹, respectively. The electrochemical sensor was then applied for amperometric determination of peroxide value (PV) in edible oil at an applied potential of 50 mV (vs. Ag/AgCl (1 M KCl)). A good linearity has been found in the range 0.02–1.0 mequiv. O₂/kg, with a detection limit (S/N = 3) of 0.001 mequiv. O₂/kg. The precision of the method (R.S.D., n = 9) for within and between-days is better than 1.9% and 2.7%, respectively at 0.1 mequiv. O₂/kg. The method was successfully applied to the determination of PV in real edible oil samples with an excellent agreement with results obtained with the official standard procedure. The proposed method is accurate, simple, cheap and could be used to control edible oil rancidity with a high sample throughputs (more than 120 samples/h).     

The asymptotic self-similar behavior for the quasilinear heat equation with nonlinear boundary condition

In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function

v(y,t)=(T−t)^{1/(2p+α−2)}u((T−t)^{(p−1)/(2p+α−2)}y,t),