Illuminating high-dimensional convex sets

A setL of points in thed-spaceE ^d is said toilluminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d if for every setS _i inF and every pointx in the boundary ofS _i there is a pointv inL such thatv illuminatesx, i.e. the line segment joiningv tox intersects the union of the elements ofF in exactly {x}.The problem we treat is the size of a setS needed to illuminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d . We also treat the problem of putting these convex sets in mutually disjoint convex polytopes, each one having at most a certain number of facets.     

Effect of annealing on the viscoelastic and viscoplastic responses of low‐density polyethylene

Two series of tensile tests with constant crosshead speeds (ranging from 5 to 200 mm/min) and tensile relaxation tests (at strains from 0.03 to 0.09) were performed on low‐density polyethylene in the subyield region of deformations at room temperature. Mechanical tests were carried out on nonannealed specimens and on samples annealed for 24 h at the temperatures T = 50, 60, 70, 80, and 100 °C. Constitutive equations were derived for the time‐dependent response of semicrystalline polymers at isothermal deformations with small strains. A polymer is treated as an equivalent heterogeneous network of chains bridged by temporary junctions (entanglements, physical crosslinks, and lamellar blocks). The network is thought of as an ensemble of mesoregions linked with each other. The viscoelastic behavior of a polymer is modeled as a thermally induced rearrangement of strands (separation of active strands from temporary junctions and merging of dangling strands with the network). The viscoplastic response reflects sliding of junctions in the network with respect to their reference positions driven by macrostrains. Stress‐strain relations involve five material constants that were found by fitting the observations. Fair agreement was demonstrated between the experimental data and the results of numerical simulation. This study focuses on the effects of strain rate and annealing temperature on the adjustable parameters in the constitutive equations. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1638–1655, 2003     

Influence of noise on dynamics with fractal Fourier spectra

In deterministic dynamics, the processes with fractal power spectra occupy the intermediate position between ordered and chaotic temporal behavior. We study numerically the influence of different kinds of noise upon these processes. In general, under the action of noise the spectral measure becomes absolutely continuous and the correlations decay. However, during the certain intermediate period the autocorrelation retains the pattern which is typical for dynamics with fractal spectra. Duration of this period is related to the noise intensity by the power law.     

Extending an extension of Grinberg's theorem

Grinberg's well-known necessary condition for a planar graph to be Hamiltonian was extended by Shimamoto to a condition for having a 2-factor. We derive a necessary condition for a planar graph to contain a collection of n disjoint circuits, none of which separates any other two. All is done in terms of the numbers p_k which denote the numbers of k-gonal faces.     

Non-Hamiltonian non-Grinbergian graphs

Settling a question of Tutte and a similar question of Grünbaum and Zaks, we present a 3-valent 3-connected planar graph that has only pentagons and octagons, has 92 (200, respectively) vertices and its longest circuit (path, respectively) contains at most 90 (198, respectively) vertices; moreover, it is shown that the family of all 3-valent 3-connected planar graphs, having n-gons only if n≡ 2 (mod3) (or n≡ 1 (mod3)), has a shortness exponent which is less than one.     

Reduction of several thiolene 1,1-dioxides by lithium in liquid ammonia

Conclusions The isoprene and piperylene thiolene dioxide derivatives undergo regioselective oxidative ring cleavage at the C⁵-S bond upon reduction using lithium in liquid ammonia.The authors thank K. M. Gitis for assistance in carrying out the gas chromatographic analysis.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 12, pp. 2799–2800, December, 1984.     

Non-Hamiltonian simple 3-polytopes having just two types of faces

It is shown that for every value of an integer k, k?11, there exist 3-valent 3-connected planar graphs having just two types of faces, pentagons and k-gons, and which are non- Hamiltonian. Moreover, there exist ?=?(k) > 0, for these values of k, and sequences (G_n)^∞_n=1 of the said graphs for which V(G_n)→∞ and the size of a largest circuit of G_n is at most (1??)V(G_n); similar result for the size of a largest path in such graphs is established for all k, k?11, except possibly for k = 14, 17, 22 and k = 5m+ 5 for all m?2.     

Stochastic quantization and regularization

In this paper the method of stochastic quantization introduced by Parisi and Wu is extended to field theories that include fermions and are supersymmetric. A new non-perturbative regulator based on stochastic quantization is introduced. This regulator preserves all the symmetries of the lagrangian, including gauge, chiral, and supersymmetries, at the expense of introducing non-locality.