Formal verification of tail distribution bounds in the HOL theorem prover

Tail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher‐order‐logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli(p), Geometric(p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety‐critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well‐known commercially used algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Adsorption Equilibrium, Kinetics and Thermodynamics of α-Amylase on Poly(DVB-VIM)-Cu+2 Magnetic Metal-Chelate Affinity Sorbent

Designing an immobilised metal ion affinity process on large-scale demands that a thorough understanding be developed regarding the adsorption behaviour of proteins on metal-loaded gels and the characteristic adsorption parameters to be evaluated. In view of this requirement, interaction of ??-amylase as a model protein with newly synthesised magnetic-poly(divinylbenzene-1-vinylimidazole) [m-poly(DVB-VIM)] microbeads (average diameter, 53?C212???m) was investigated. The m-poly(DVB-VIM) microbeads were prepared by copolymerising of divinylbenzene (DVB) with 1-vinylimidazole (VIM). The m-poly(DVB-VIM) microbeads were characterised by N₂ adsorption/desorption isotherms, electron spin resonance, elemental analysis, scanning electron microscope and swelling studies. Cu²⁺ ions were chelated on the m-poly(DVB?CVIM) beads and used in adsorption of ??-amylase in a batch system. The maximum ??-amylase adsorption capacity of the m-poly(DVB?CVIM)?CCu²⁺ beads was determined as 10.84?mg/g at pH?6.0, 25?°C. The adsorption data were analyzed using three isotherm models, which are the Langmuir, Freundlich and Dubinin?CRadushkevich isotherm models. The pseudo-first-order, pseudo-second-order, modified Ritchie??s-second-order and intraparticle diffusion models were used to test dynamic experimental data. The study of temperature effect was quantified by calculating various thermodynamic parameters such as Gibbs free energy, enthalpy and entropy changes.     

Kinetics and mechanism of ion exchange of Fe3+, Cd2+ and Na+/H+ on Lewatite S-100 cation exchanger in aqueous and aqueous-detergent media

Sorption mechanism of Fe (III), Cd (II) and Na (I) on cation exchange resins in H-form was investigated from aqueous and aqueous-detergent media by a modified limited batch technique. The cation exchange studies involved the sorption of metallic ions onto a Lewatite S-100 exchanger. Effects of mesh size of the exchanger, temperature and detergent on the exchange rate have been investigated. The mechanisms of cation exchanges have been determined in the temperature range of 25–65°C. In all cases of the reactions ions, diffusion is found to be the rate determining step in the exchange process. The exchange rate in the exchange process was found to increase with a decrease of particle size and an increase of temperature. However, in case of the influence of detergent, the exchange reactions are dependent on the reaction technique. The effective-diffusion coefficients have been evaluated at three different temperatures. The energy barriers (ΔE_a), entropies of activation (ΔS*), (ΔH*) and (ΔG*) for various sorption systems have also been calculated. The results are discussed in terms of size and valences of the counter ions. No change in the internal structure of Lewatite S-100 is inferred due to the sorption of counter ions.     

Bernstein series solutions of pantograph equations using polynomial interpolation

In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of the pantograph-type differential equations with retarded case or advanced case. In addition, the method is presented with error and stability analysis.     

Synthesis of new anthracene derivatives

An efficient synthesis is described for hexabromoanthracenes 3 and 4 by direct bromination of 9,10-dibromoanthrecene 2. Whereas base-induced elimination of hexabromide 3 with t-BuOK gave 2,3,9,10-tetrabromoanthracene 5, the reaction of hexabromide 4 with DBU afforded 1,3,9,10-tetrabromoanthracene 6 as the sole product. Tetrabromide 5 was also obtained by aromatization of 1,4-dinitroxy-2,3,9,10-tetrabromo-1,2,3,4-tetrahydroanthracene 17. Efficient and convenient synthetic routes are described for the preparation of dinotroxy 17, dimethoxy 23, and dihydroxides 18 and 19 with silver-induced substitution of hexabromides 3 and 4. The hydroxy compounds 19 and 18 were converted to diepoxide 20 and monoepoxide 21, respectively, with sodium methoxide. Base-promoted aromatization of dimethoxide 23 afforded dibromomonomethoxides 26 and 27. Bromoanthracenes and isomeric arene oxides constitute valuable precursors for the preparation of functionalized substituted anthracene derivatives that are difficult to prepare by other routes.     

A new approach to the characterization of the nano-surface structure by using factor analysis

The study provides a new approach to explore the existence of a nanostructure on the benzoic acid-modified glassy carbon electrode (BA-GC) system by applying factor analysis with varimax rotation of factors. The infrared spectra (IR) spectra of the bare glassy carbon electrodes (GC) and the BA-GC electrode systems with benzoic acid-nanofilm (BA-FILM) were obtained by using 10 different surface modification experiments. The IR data of the bare GC, BA-GC and BA-FILM were categorized into three series corresponding to the GC, BA-GC and BA-FILM. The factor analysis method was separately applied to each group to extract the factor loadings. By making use of these loadings, the rotated factor loadings corresponding to the first two factors were computed to estimate the relationship among and within groups. The results provided by the factor analysis method indicate the existence of the BA-FILM on the BA modified glassy carbon electrode.     

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.     

A nonnegative extension of the affine demand function and equilibrium analysis for multiproduct price competition

We derive a nonnegative extension of the affine demand function for differentiated substitute products from the optimization problem facing a representative consumer whose utility function is quadratic. We show that the extended demand function reduces to a linear program. The linear program has a simple intuitive interpretation in terms of a shifted price vector. We prove the existence and uniqueness of the Bertrand equilibrium in oligopolies consisting of multiproduct firms under the proposed demand function. The equilibrium, available in closed form, coincides with that obtained when allowing negative demands.     

Synthesis,Crystal Structures and Thermal Analysis of Two New Silver(I)–Dithioether Lamellar Coordination Polymers

Abstract  Two new silver(I)–dithioether coordination polymers based on the self-assembly of the dithioether ligands L ^{n −R} = CH₃–S(CH₂) _n S–CH₃ (n = 1, 3) and AgX silver salts, where X = ClO₄ ⁻ (1) and PF₆ ⁻ (2), have been synthesized and characterized by single crystal X-ray diffraction and TGA. The crystallographic data of those complexes are: (1) Monoclinic P2₁/c; a = 16.5861(3) ?; b = 10.1048(2) ?; c = 15.7843(3) ?; β = 112.570(1)°; V = 2442.83(8) ?³; Z = 4. (2) Monoclinic P2₁/n; a = 8.6500(2) ?; b = 9.4053(2) ?; c = 24.0357(6) ?; β = 92.750(2)°; V = 1953.19(8) ?³; Z = 4. In both complexes, the dithioether building blocks propagated the silver(I) centers to build cationic lamellar coordination polymers, while ClO₄ ⁻ and PF₆ ⁻, which were sandwiched between silver-ligand sheets, counterbalance only the charge of the networks. The thermogravimetric investigation reveals that those complexes decompose in a single step respectively to metallic silver and silver salt. Index Abstract  Two new silver(I)–dithioether coordination polymers, obtained from the self-assembly of dithioether ligands and AgX salts with noncoordinating counteranions (X = ClO₄ ⁻ and PF₆ ⁻), were characterized by X-ray crystallography and thermogravimetric analysis (TGA).      

On a class of bilevel linear mixed-integer programs in adversarial settings

We consider a class of bilevel linear mixed-integer programs (BMIPs), where the follower’s optimization problem is a linear program. A typical assumption in the literature for BMIPs is that the follower responds to the leader optimally, i.e., the lower-level problem is solved to optimality for a given leader’s decision. However, this assumption may be violated in adversarial settings, where the follower may be willing to give up a portion of his/her optimal objective function value, and thus select a suboptimal solution, in order to inflict more damage to the leader. To handle such adversarial settings we consider a modeling approach referred to as \(\alpha \)-pessimistic BMIPs. The proposed method naturally encompasses as its special classes pessimistic BMIPs and max–min (or min–max) problems. Furthermore, we extend this new modeling approach by considering strong-weak bilevel programs, where the leader is not certain if the follower is collaborative or adversarial, and thus attempts to make a decision by taking into account both cases via a convex combination of the corresponding objective function values. We study basic properties of the proposed models and provide numerical examples with a class of the defender–attacker problems to illustrate the derived results. We also consider some related computational complexity issues, in particular, with respect to optimistic and pessimistic bilevel linear programs.