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.     

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.     

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.     

Direct electrochemistry of cytochrome c and biosensing for hydrogen peroxide on polyaniline grafted multi-walled carbon nanotube electrode

Cytochrome c (cyt c) was immobilized into a matrix consisting of polyaniline (PANI) and multi-walled carbon nanotubes (MWNT) by a new strategy. First, PANI chains were grafted onto MWNT through electropolymerization. Second, the amine groups in PANI chains were oxidized at an applied potential of +0.80 V to acquire positive charges that would effectively immobilize negatively charged cyt C. The ITO/MWNT-g-PANI(O)/cyt c electrode exhibited a pair of redox peaks with a peak potential separation (anodic to cathodic) of 0.25 V (vs Ag/AgCl) in 0.1 M phosphate buffer (pH 7.0). The results demonstrated that ITO/MWNT-g-PANI(O)/cyt c promoted direct electron transfer between cyt c and electrode with a high electron transfer rate constant (17 s⁻¹). The ITO/MWNT-g-PANI(O)/cyt c electrode catalyzes the reduction of H₂O₂. The ITO/MWNT-g-PANI(O)/cyt c biosensor displays an amperometric response to H₂O₂ with a linear concentration range from 0.5 μM to 1.5 mM (r = 0.99, n = 12), a high sensitivity (32.2 μAm M⁻¹) and fast response (9 s) and detection limit of 0.3 μM (S/N = 3).     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

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.     

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.     

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 (*).     

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.     

One-step t-fault diagnosis for hypermesh optical interconnection multiprocessor systems

To maintain high reliability and availability, system-level diagnosis should be considered for the multiprocessor systems. The self-diagnosis problem of hypermesh, emerging potential optical interconnection networks for multiprocessor systems, is solved in this paper. We derive that the precise one-step diagnosability of kⁿ-hypermesh is n(k − 1). Based on the principle of cycle decomposition, a one-step t-fault diagnosis algorithm for kⁿ-hypermesh which runs in O(kⁿn(k − 1)) time also is described.     

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 sensitive nonenzymatic hydrogen peroxide sensor based on DNA–Cu complex electrodeposition onto glassy carbon electrode

In this paper, DNA–Cu²⁺ complex was electrodeposited onto the surface of glassy carbon (GC) electrode, which fabricated a DNA–Cu²⁺/GC electrode sensor to detect H₂O₂ with nonenzyme. Cyclic voltammogram of DNA–Cu²⁺/GC electrode showed a pair of well-defined redox peaks for Cu²⁺/Cu⁺. Moreover, the electrodeposited DNA–Cu²⁺ complex exhibited excellent electrocatalytic behavior and good stability for the detection of H₂O₂. The effects of Cu²⁺ concentration, electrodeposition time and determination conditions such as pH value, applied potential on the current response of the DNA–Cu²⁺/GC electrode toward H₂O₂ were optimized to obtain the maximal sensitivity. The linear range for the detection of H₂O₂ is 8.0 × 10⁻⁷ M to 4.5 × 10⁻³ M with a high sensitivity of −40.25 μA mM⁻¹, a low detection limit of 2.5 × 10⁻⁷ M and a fast response time of within 4 s. In addition, the sensor has good reproducibility and long-term stability and is interference free.     

Direct amperometric determination of l-ascorbic acid (Vitamin C) at octacyanomolybdate-doped-poly(4-vinylpyridine) modified electrode in fruit juice and pharmaceuticals

The glassy carbon electrode (GCE) modified with Mo(CN)₈⁴⁻-incorporated-poly(4-vinylpyridine) (PVP/Mo(CN)₈⁴⁻), which has been recently shown to possess several attractive attributes as an efficient electrocatalytic electrode for l-ascorbic acid oxidation and its estimation, is used for l-ascorbic acid estimation directly in orange fruit juice and Celin tablet in a 0.1 M H₂SO₄ acid solution without any special treatment. Constant potential amperometry at 570 mV (saturated calomel electrode, SCE) in stirred solutions is used for this purpose. A good correlation is attained with the official titrametric method. To understand the possible electrocatalytic reaction mechanism for the electro-oxidation of l-ascorbic acid, calibration graphs over the range 1 × 10⁻⁵ to 1 × 10⁻² mol dm⁻³ l-ascorbic acid are compared for the three electrodes, ca. PVP/Mo(CN)₈⁴⁻, undoped PVP, and GCE; the curvature at high ascorbic acid concentration for the PVP/Mo(CN)₈⁴⁻ electrode is explained in terms of Michaelis–Menten (MM) saturation kinetics. The apparent MM constant (K_M), the maximum catalytic current (i_M), the complex decomposition rate constant (k_c), and the heterogeneous modified electrode rate constant (k^′_ME) are calculated from three different approaches. A reasonably high value of ≈1 × 10⁻² cm s⁻¹ is obtained for k^′_ME, indicating efficient l-ascorbic acid mediation at the PVP/Mo(CN)₈⁴⁻ electrode, thus accounting for quite a high sensitivity of this modified film electrode compared to several other modified electrodes.     

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.     

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}.     

Single quantum dot-micelles coated with gemini surfactant for selective recognition of a cation and an anion in aqueous solutions

The synthesis of quantum dot coated with cetyltrimethyl ammonium bromide (CTAB) and gemini surfactant [C₁₂H₂₅N⁺(CH₃)₂(CH₂)₄(CH₃)₂N⁺C₁₂H₂₅]·2Br⁻ (C_12-4-12) in aqueous solution have been described. It is characterized by photoluminescent spectroscopy, UV–vis spectroscopy and transmission electron microscopy (TEM), etc. In comparison with CTAB-coated QDs, the QDs coated with C_12-4-12 respond selectively to both transition metal ion copper and fluoride ion in aqueous media. When Cu²⁺ is bound to C_12-4-12-coated QD micelles, the fluorescence intensity is quenched. Linear relationships are found between the relative fluorescence intensity and the concentration of Cu²⁺ in the range 0–500 μM, which is best described by a Stern–Volmer-type equation. Meanwhile, it is found that F⁻ enhanced the luminescence of the C_12-4-12-coated QD micelles in a concentration dependence that is described by a Langmuir binding isotherm equation in the range 0–300 μM. The limits of detection of Cu²⁺ and F⁻ are 1.1 and 0.68 μM, respectively. The possible mechanism is discussed.     

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.     

Semigroup generation and stabilization by A-bounded feedback perturbations

Let A be a generator of a strongly continuous semigroup of operators, and assume that C and H are operators such that A + CH generates a strongly continuous semigroup S_H(t) on X. Let λ₀ be a real number in the resolvent set of A, and let ε [−1, 1]. Then there are some fairly unrestrictive conditions under which A+(λ₀ − A)CH(λ₀ − A)⁻ also generates a strongly continuous semigroup S_K(t) on X which has the same exponential growth rate as S_H(t). Given an input operator B, we can use this to identify a class of feedback perturbations K such that A + BK generates a strongly continuous semigroup. We can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system. For example, we show that if the control on a cantilever beam in the state space H₀²[0, 1] × L²[0, 1] is a moment force on the free end, then we cannot stabilize the beam with an A^−1/2-bounded feedback, but we can find an A^−1/4-bounded feedback, for any > 0, which does stabilize the beam.     

Concurrent Open Shop Scheduling to Minimize the Weighted Number of Tardy Jobs

We consider a relaxed version of the open shop scheduling problem–the concurrent open shop scheduling problem, in which any two operations of the same job on distinct machines are allowed to be processed concurrently. The completion time of a job is the maximum completion time of its operations. The objective is to schedule the jobs so as to minimize the weighted number of tardy jobs, with 0–1 operation processing times and a common due date d. We show that, even when the weights are identical, the problem has no (1 – )ln m-approximation algorithm for any > 0 if NP is not a subset of DTIME(n ^loglogn ), and has no c·ln m-approximation algorithm for some constant c > 0 if P NP, where m is the number of machines. This also implies that the problem is strongly NP-hard. We also give a (1 + d)-approximation algorithm for the problem.