Structure and properties of ceramic composites based on ZrO2

Monodisperse fine powders of high purity obtained by sol-gel method are used for production of high technical data ceramics. The fiber reinforcement is used for hardening of composite materials. It was of interest to study production possibility of reinforced composite material based on ZrO₂ obtained by sol-gel method with filler from fibers of partially stabilized zirconia. ZrO₂ powders were obtained by precipitation of its hydrated gel from aqueous zirconium oxychloride solution by ammonium hydroxide followed by thermal treatment. For composite reinforcement ceramic partially stabilized (8 mole Y₂O₃) ZrO₂ fibers 0.16–0.67 mm in length and 5–7 m in diameter were used. Content of the fibers in composite was 20 wt.%. From powders and their mixtures with fibers, the samples were pressed as disks, beams and cylinders, and anneal in air at 1100–1600°C temperature range. The investigation has shown that the fibers of partially stabilized zirconia change the composite structure, increase the content of tetragonal modification that promotes its hardening. Treatment temperature of precursor determines physical chemical properties of compositions with fibers. Their high specific surface and reaction ability provides a workability of forming and sintering processes into strong composite material. The ceramics was increased by 2.5–3 times as strength after fibrous filler introduction into ZrO₂ hydrogel matrix.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Presented at the Ninth International conference on the Mechanics of Composite Materials (Riga, October 1995). Institute of General and Inorganic Chemistry of the Academy Sciences of the Belarus. Minsk. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 418–427, May–June. 1996.     

Mechanical properties of irradiated silicone compounds

The change in the mechanical properties of silicone compounds subjected to irradiation with electrons and x rays has been investigated. The compounds contained the fillers ZnO and SiO₂ and, moreover, Be₂O₃ and Fe₂O₃. It is shown that as a result of prolonged irradiation the capacity of the compounds for high elastic deformation is reduced by a factor of 5–6, the true strength values are reduced by a factor of 2–3, and the stiffness of the compounds increases by a factor of 2–3. The compound containing Be₂O₃ proved to be the more radiation-resistant material.Institute of the Chemical Industry, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 610–615, July–August, 1973.     

Structure-sensitive liquid flowable compositions based on polyvinyl alcohol and silicotungstic acid

Conditions of the formation of structure-sensitive liquid flowable media based on 1.5 and 10% aqueous solutions of polyvinyl alcohol containing silicotungstic acid and water-soluble carboxy-methylcellulose in the form of its potassium salt (Na-CMC) are examined. Rheological, optical, and dielectric relaxation methods revealed the formation of several types of interpolymer complexes in the examined liquid flowable media. This leads to the formation of associates and an increase in the molecular mobility of the macromolecules and their fragments. The structure and properties of the complexes depend on the composition of the media as well as the method of introducing polyacids. It was found that it was possible to control the structure of such solutions by applying a mechanical of magnetic field. The compositions obtained can be used for producing anisotropic light-, electric-, and heat-sensitive film materials, as well as sensors of different types.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Institute of General and Inorganic Chemistry, Belarus Academy of Sciences, Minsk, Belarus. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 413–424, May–June, 1998.     

Single machine group scheduling with ordered criteria

The single machine group scheduling problem is considered. Jobs are classified into several groups on the basis of group technology, i.e. jobs of the same group have to be processed jointly. A machine set-up time independent of the group sequence is needed between each two consecutive groups. A schedule specifies the sequence of groups and the sequence of jobs in each group. The quality of a schedule is measured by the criteriaF ₁, ...,F _m ordered by their relative importance. The objective is to minimize the least important criterionF _m subject to the schedule being optimal with respect to the more important criterionF _m–1 which is minimized on the set of schedules minimizing criterionF _m–2 and so on. The most important criterion isF ₁, which is minimized on the set of all feasible schedules. An approach to solve this multicriterion problem in polynomial time is presented if functionsF ₁, ...,F _m have special properties. The total weighted completion time and the total weighted exponential time are the examples of functionsF ₁, ...,F _m–1 and the maximum cost is an example of functionF _m for which our approach can be applied.The research of the authors was partially supported by a KBN Grant No. 3 P 406 003 05, the Fundamental Research Fund of Belarus, Project N 60-242, and the Deutsche Forschungsgemeinschaft, Project Schema, respectively. The paper was completed while the first author was visiting the University of Melbourne.     

Solution of nonstationary boundary layer problems by fastest descent method

We consider the formation of a boundary layer above a semi-infinite plate that moves with velocity V_t8=At ⁿ in a viscous incompressible fluid. The problem is solved by Meksyn's asymptotic method. First, second, and third approximation equations are obtained. A numerical calculation is performed for various n, and especially interesting results are obtained for –0.5 < n < 0. The numerical solution of the third approximation equation is consistent with other known results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 51–55, 1988.     

Bäcklund autotransformation for the equationu xt=eu−e−2u

The system of differential relations that arises in connection with the Bullough-Dodd-Zhiber-Shabat equationu _xt=e^u–e^–2u is considered. The consistency of this system is established, and it is shown that the system realizes a Bäcklund autotransformation for the equationu _xt=e^u–e^–2u. The associated three-dimensional dynamical systems, which are compatible on a two-dimensional invariant submanifold, are investigated, and a construction of their general solution, which gives the explicit form of the three-parameter soliton for the equationu _xt=e^u–e^–2u, is proposed.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 146–159, April, 1993.     

Distribution of integral points on certain two-sheeted hyperboloids

Yu. V. Linnik's investigations [Vestn. Leningr. Univ., No. 2, 3–23; No. 5, 3–32; No. 8, 15–27 (1955)] are refined and generalized to indefinite ternary quadratic forms of a sufficiently general form (to forms contained in the form x₁x₃—x ₂ ² ]. The method of investigation is improved. The presentation is substantially simplified.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 87–141, 1980.     

Regularized Resolvent of Sums of Commuting Operators

For B ₁ and B ₂ commuting linear operators on a Banach space such that B ₁ generates a bounded strongly continuous semigroup and –B ₂ generates an exponentially decaying strongly continuous holomorphic semigroup, it is shown that (B ₁ – B ₂)^–1 B ₂ ^–r and (B ₁ – B ₂)^–1(–B ₁)^–r are bounded and everywhere defined, for any r > 0. Density of domains may also be removed. The results are applied to various abstract Cauchy problems.     

Boundary-value problems for systems of differential equations with operator coefficients

The unique strong solvability is proved for several boundary-value problems for the systems Au_xx–Bu_y=f, A₁u_xx–Bu_yy=f, where A, A₁, B are operators of parabolic and hyperbolic types.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1635–1641, December, 1990.     

Precise distribution properties of the van der Corput sequence and related sequences

The discrepancy is a quantitative measure for the irregularity of distribution of sequences in the unit interval. This article is devoted to the precise study of L_p–discrepancies of a special class of digital (0,1)–sequences containing especially the van der Corput sequence. We show that within this special class of digital (0,1)–sequences over ℤ₂ the van der Corput sequence is the worst distributed sequence with respect to L₂–discrepancy. Further we prove that the L_p–discrepancies of the van der Corput sequence satisfy a central limit theorem and we study the discrepancy function of (0,1)–sequences pointwise.     

Central limit theorems in D[0, 1]

Summary Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {X _n} of independent copies of X, let S _n=X₁++X_n. Conditions which are either necessary or sufficient for the weak convergence of n ^–1/2(S _n–ES_n) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.     

Analysis of the Adhesion and Adhesion Strength of Two-Layer Polymer Films and Coatings

The formation of adhesion bonds during the creation of two-layer polymer films and coatings capable of acid–base interaction is considered. The coatings are obtained from polymer solutions, and the second layer is formed from the polymer solutions and their mixtures on a hard polymer coating. As a result of absorption by the coating of a considerable amount of solvents (to 14%), polymer macrochains are able to approach one another at a distance necessary for the formation of acid–base (hydrogen) bonds and a transition layer. The polar component of the work of adhesion is equal to about 50% of the whole work W _a. There is no interrelation of the thermodynamic work W _a between the films and the work of their separation into layers. The main part of the external work upon the separation is spent (if strong donor–acceptor bonds are formed) for extracting the segments and macrochains of polymers out of the films.     

Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions

This paper provides a new integrable hierarchy. The DP equation: m _t +um _x +3mu _x =0, m=u–u _xx , proposed recently by Degasperis and Procesi, is the first member in the negative order hierarchy while the first equation in the positive order hierarchy is: m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separately discuss the negative order and the positive order hierarchies. For the negative order hierarchy, its 3×3 Lax pairs and corresponding adjoint representations are cast in Liouville-integrable Hamiltonian canonical systems under the Dirac–Poisson bracket defined on a symplectic submanifold of R ^6N . Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of all equations in the negative order hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive order hierarchy, we consider a different constraint and process a procedure similar to the negative case to obtain the parametric solutions of the positive order hierarchy. In a special case, we give the parametric solution of the 5th-order PDE m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . Finally, we discuss the stationary solutions of the 5th-order PDE, which may be included in the parametric solution.     

Structure and Properties of Electrodeposited Composite Metal–Nanodisperse Inorganic Compound Coatings

By using the electrodeposition method, composite Sn-inorganic coatings have been obtained. As the inorganic compounds, ultrafine PbMo₆S₈, NbC_xN_y, Ti_xN_y, and Ti_0.63Nb_0.34C_0.38N_0.58 powders were utilized. The Sn–PbMo₆S₈ coatings containing 28-88 wt.% PbMo₆S₈ were superconductive. The highly electroconductive Sn–NbC_xN_y coatings containing 1-88 wt.% of the disperse phase and the Sn–Ti_xN_y coatings with 42-70 wt.% Ti_xN_y were structurally stable at a low temperature (–50°C), solderable, and corrosion resistant on copper. The formation of the structure of the coatings is determined by the electrolyte composition, the cathode current density during the electrolysis, and the codeposition of the disperse phase. By electrodeposition of the chromium carbonitride Cr₃C_1.6N_0.4, in the presence of H₄P₂O₇ in an CrO₃-based electrolyte, mat composite Cr–Cr₃C_1.6N_0.4 coatings with 6.8 wt.% Cr₃C_1.6N_0.4 were obtained, which had a finely uneaven surface structure. Such surfaces are needed in the cases where lubricants are used.     

Mackey and Frobenius Structures on Odd Dimensional Surgery Obstruction Groups

C. T. C. Wall formulated surgery-obstruction groups L _n (Z[G]) in terms of quadratic modules and automorphisms. C. B. Thomas showed that the Wall-group functors L _n (Z[–],w|_–) are modules over the Hermitian-representation-ring functor G₁(Z, –) if the orientation homomorphism w is trivial. A. Bak generalized the notion of quadratic module by introducing quadratic-form parameters, and obtained various K-groups related to quadratic modules and automorphisms. One of the authors established that some Bak groups W _n (Z[G], w) are equivariant-surgery-obstruction groups and showed in the case of even dimension n that the Bak-group functor W _n (Z)[–], _–; w|_–) is a w-Mackey functor as well as a module over the Grothendieck–Witt-ring functor GW₀(Z, –), where w is possibly nontrivial. In this paper, we prove the same facts in the case of odd dimension n.     

Time-temperature superposition applied to the relaxation properties of a glass-reinforced plastic and its matrix

The relaxation behavior in compression of an unsaturated polyester resin and the corresponding glass-reinforced plastic has been investigated. The time-temperature superposition principle is shown to be applicable to the cross-linked polyester resin and the glass-reinforced plastic. The temperature dependences of the shift functionsa _T of the resin and the glass-reinforced plastic are found to satisfy an Arrhenius equation with the constant equal to 21–25 kcal/mole.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 360–363, March–April, 1972.     

Bayesian testing of nonparametric hypotheses and its application to global optimization

Random distribution functions are the basic tool for solving nonparametric decision-theoretic problems. In 1974, Doksum introduced the family of distributions neutral to the right, that is, distributions such thatF(t ₁),[F(t ₂)–F(t ₁)]/[1 –F(t ₁)],...,[F(t _k)–F(t _{k – 1})]/[1 –F(t _{k – 1})] are independent whenevert ₁ < ... <t _kIn practice, application of distributions neutral to the right has been prevented by the lack of a manageable analytical expression for probabilities of the typeP(F(t)<q) for fixedt andq. A subclass of such distributions can be provided which allows for a close expression of the characteristic function of log[1–F(t)], given the sample. Then, thea posteriori distribution ofF(t) is obtained by numerical evaluation of a Fourier integral. As an application, the global optimization problem is formulated as a problem of inference about the quantiles of the distributionF(y) of the random variableY=f(X), wheref is the objective function andX is a random point in the search domain.The author thanks J. Koronacki and R. Zielinski of the Polish Academy of Sciences for their valuable criticism during the final draft of the paper.     

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Ein Einschliessungsverfahren für Lösungen Fehlerbehafteter Linearer Gleichungen

In a partially ordered space, the method x_n+1 = L⁺x _n ⁺ – N⁺x _n ^- – L^–y⁺ + N^– y _n ^- + r, y_n+1 = N⁺y⁺ – L⁺y _n ^- – N^–x _n ⁺ + L^–x^– + t of successive approximation is developed in order to enclose the solutions of a set of linear fixed point equations monotonously. The method works if only the inequalities x₀ y₀, x₀ x₁, y₁ y₀ related to the starting elements are satisfied. In finite-dimensional spaces suitable starting vectors can be computed if a sufficiently good approximation for the fixed points is known.

     

Some properties of filled polyvinyl chloride

The effect of TiO₂, SiO₂, and carbon fillers on rigid, partly plasticized PVC has been investigated. Interaction in the polymer-filler system improves certain important properties — hardness, stiffness, wear resistance, etc., — the effect of the fillers being highly specific.M. V. Lomonosov Odessa Technological Institute. Translated from Mekhanika Polimerov, No. 1, pp 124–126, January–February, 1970.