The diffusive limit of Carleman-type models in the range of very fast diffusion equations

We improve the existing results on the limiting behavior of the Cauchy problem for a class of Carleman-like models with power-type interaction rate in the diffusive scaling with data in the spaces L ^p , 1 ≤ p ≤ ∞. The convergence result, which has been carefully established before for exponents of the interaction rate α ≤ 1, is extended here to the range of exponents 1 < α < 4/3. In addition, we discuss the problem of establishing a good theory in the still remaining range , by introducing a modified kinetic system which admits an explicit self-similar solution. The analysis of this solution clarifies the role of the exponent .     

Polyethylene with improved environmental stress cracking resistance

The stress cracking resistance of low-pressure polyethylene can be improved by introducing finely dispersed mineral additives (TiO₂, SiO₂, Al₂O₃ · SiO₂) and surface-active agents (anthranilic, adipic, and sebacic acids; mannitol). The amounts needed to effectively increase the cracking resistance without detriment to the principal physicomechanical and other characteristics of the material have been determined.Mekhanika Polimerov, Vol. 4, No. 1, pp. 67–71, 1968Communication V of the series "Modification of the Structure and Properties of Polyolefins."     

Semilinear dirichlet problem with nearly critical exponent, asymptotic location of hot spots

We study asymptotic properties of the positive solutions of

Mechanical properties of polypropylene modified by decomposition products of alkali sulfate lignin

During processing of polypropylene and alkali sulfate lignin at a temperature of 180° C, a reaction occurs resulting in the formation of a polymer which appears to have a three-dimensional structure. The observed cross-linking is accompanied by an increase in reversible deformation, a decrease in irreversible deformation, a transition from solubility to limited swelling in solvents, an increase in tensile strength, and an increase in freeze resistance. This is achieved by also adding a plasticizer to the polymer.Mekhanika Polimerov, Vol. 1, No. 6, pp. 3–9, 1965     

Biomagnetic fluid flow over a stretching sheet with non linear temperature dependent magnetization

The flow of a heated ferrofluid over a linearly stretching sheet is studied in the pres- ence of an applied magnetic field due to a magnetic dipole. It is assumed that the applied magnetic field is sufficiently strong to saturate the ferrofluid and the variation of magnetization with temperature can be approximated by a non linear function of temperature difference. By introducing appropriate non dimensional variables the problem is described by a coupled and non linear system of ordinary differential equations with its boundary conditions which is solved numerically by applying an efficient numerical technique based on the common finite difference method. The obtained results are presented graphically for different values of the parameters entering into the problem under consideration and the dependence of the flow field from these parameters is discussed. A comparative study, with a similar problem which has already been solved and documented in literature, is also made wherever necessary, emphasizing the impor- tance of the non-linear variation of magnetization with temperature. Emphasis is also given in the obtained results for Prandtl number equal to 21 and critical exponent = 0.368 which are important and interesting in Biomagnetic Fluid Dynamics.     

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.     

Biomagnetic fluid flow over a stretching sheet with non linear temperature dependent magnetization

The flow of a heated ferrofluid over a linearly stretching sheet is studied in the pres- ence of an applied magnetic field due to a magnetic dipole. It is assumed that the applied magnetic field is sufficiently strong to saturate the ferrofluid and the variation of magnetization with temperature can be approximated by a non linear function of temperature difference. By introducing appropriate non dimensional variables the problem is described by a coupled and non linear system of ordinary differential equations with its boundary conditions which is solved numerically by applying an efficient numerical technique based on the common finite difference method. The obtained results are presented graphically for different values of the parameters entering into the problem under consideration and the dependence of the flow field from these parameters is discussed. A comparative study, with a similar problem which has already been solved and documented in literature, is also made wherever necessary, emphasizing the impor- tance of the non-linear variation of magnetization with temperature. Emphasis is also given in the obtained results for Prandtl number equal to 21 and critical exponent = 0.368 which are important and interesting in Biomagnetic Fluid Dynamics.     

The flux of primary cosmic ray nuclei of atomic numberZ⩾2 at geomagnetic latitude 30°

The method of determining the primary flux of nuclei with atomic number Z?3 developed by Bradt and Peters, has been extended to include all nuclei of charge Z?2. It is shown that the method permits the identification of primary helium nuclei with an efficiencyν?90%. The primary α-particle flux obtained in this way is in very good agreement with that obtained by other methods and the relative flux values for nuclei of atomic number 3?Z?5 and 6?Z?9 agree with previous determinations.     

Compactness in Categories of S-Net Spaces

The $S$ -net spaces studied are convergence structures whose convergences are expressed by using generalized nets, the so called $S$ -nets, which are obtained from the usual nets by replacing the category of directed sets and cofinal maps with an arbitrary construct $S$ . We investigate compactness in categories of $S$ -net spaces defined by introducing continuous maps in a natural way and imposing some usual convergence axioms.     

Quasi-stationary stefan problem with values on the front depending on its geometry

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5–8, 10, 12, 13].     

Automatic Completely-Simple Semigroups

The notion of automaticity has been widely studied in groups and some progress has been made in understanding the notion in the wider context of semigroups. The purpose of this paper is to study automatic completely-simple semigroups. We show that, if is a completely-simple semigroup (with and finite), then is automatic if and only if the group is automatic. As a consequence, we deduce that automatic completely-simple semigroups are finitely presented. We also show that automatic completely-simple semigroups are characterized by the fellow traveller property and also that the existence of an automatic structure is independent of the choice of generating set.     

An analytic algorithm for time fractional nonlinear reaction–diffusion equation based on a new iterative method

A new analytic algorithm for highly nonlinear time fractional reaction–diffusion equations is proposed in this paper. The proposed method is an amalgamation of variational iteration method (VIM), Adomian decomposition method (ADM) and further refined by introducing a new correction functional. This new correction functional is obtained from the standard correction functional of VIM by introducing an auxiliary parameter γ and an auxiliary function H(x) in it. Further, a sequence G_n(x, t), with suitably chosen support, is also introduced in the new correction functional. The algorithm is easy to implement and only four to six iterations are sufficient for fairly accurate solutions. The algorithm is tested on Fitzhugh – Nagumo and generalized Fisher equations with nonlinearity ranging from 2 to 5.     

Design of orthotropic parts of complex shape by the method of nonplanar sections

Procedures based on the hypothesis of nonplanar sections are used to derive equations for determining the normal stresses at an arbitrary point of a body of complex shape in tension and bending. The anisotropy of the material properties is taken into account by introducing a variable modulus of elasticity. The theoretical results have been checked experimentally.Gor'kii Zhadanov Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 351–354, March–April, 1969.     

Some characteristics of the flow of polyisobutylene

The flow of polyisobutylene under a uniaxial constant tensile load has been studied over a broad interval of temperatures and stresses. In general, the isothermal flow curve (deformation-time) is divided by molecular rearrangement and crystallization into three sections, the first corresponding to flow in the amorphous state and the third, to flow in the crystalline state. The Kargin-Sogolova flow law describes well the first section and almost the third, but not the second section. The results obtained, together with those of an electron-microscope investigation, indicate that supermolecular structures play an important part in elastomer flow. The exhaustion of the lifetime of these structures also leads to a rapid development of deformation after a certain time t*_d (i.e., to the appearance of the second section of the flow curve). A relation analogous to the Zhurkov lifetime equation has been found among t*_d, the temperature T, and the true stress .Mekhanika Polimerov, Vol. 3, No. 4, pp. 659–666, 1967     

Effect of temperature on the fracture surfaces of filled systems based on SKS-85 butadiene-styrene copolymer

The tear fracture surfaces of mixtures of SKS-85 butadiene — styrene copolymer (85% styrene) with ordinary (carbon black, chalk) and polymeric (Kapron and cellophane powder) fillers have been investigated on the interval from –60 to +40°C. As the temperature varies within the limits of the glassy state (T_g SKS-85=+24°C) of the filled polymer, the nature of the fracture surface of specimens of filled mixtures, like that of the unfilled polymer, changes; in the region of the temperature transition of the copolymer associated with the mobility of the phenyl groups (–10±5°C) there is a slowing of the fracture process. At temperatures below the T_g of the copolymer the tear fracture surfaces of specimens of mixtures containing ordinary and polymeric fillers differ sharply. The introduction of fillers (20 vol. %) with a coefficient of thermal expansion different from that of the filled polymer considerably reduces the resistance of the material to fracture and leads to a sharp increase in the rate of crack propagation; the introduction of polymeric fillers with coefficients of thermal expansion similar to that of the filled polymer leads to an increase in the resistance of the material to fracture and to a decrease in the rate of crack propagation.Moscow Technological Institute of the Meat and Dairy Industry; State Institute of Polymer Adhesives, Kirovakan. Translated from Mekhanika Polimerov, No. 5, pp. 819–826, September–October, 1969.     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数,得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时,吸引子的维数减小．特别地,得到了零维吸引子存在的参数条件．     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数，得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时，吸引子的维数减小．特别地，得到了零维吸引子存在的参数条件．     

A new proof of the Ramanujan congruences for the partition function

Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class of finite sets. We describe a combinatorial quantity that is of the same order as We then give a related result allowing to compute the supremum of the empirical process on a class of sets. Work partially supported by an NSF grant.     

Explicit formula for scalar non-linear conservation laws with boundary condition

We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form , with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ? ? . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.