In this study we produced a new method for solving regular differential equations with step size h and Taylor series. This method analyzes a regular differential equation with initial values and step size h. this types of equations include quadratic and cubic homogenous equations with constant coeffcients and cubic and second-level equations.

3/— Abstract. To calculate R, the regulator of a pure cubic field Q(\Jd), a complete period of Voronoi's continued fraction algorithm over Q(\JD) is usually generated. In this paper it is shown how, in certain pure cubic fields, R can be determined by generating only about one third of this period. These results were used on a computer to find R and 3 — then the class number for all pure cubic ...

Both the superelastic and the shape memory properties of the cubic NiTi alloy known as Nitinol arise from two-phase transitions; from a simple cubic austenite phase to a trigonal R-phase and to a lower symmetry martensite phase. In this article, we speculate that two competing forces drive these transitions. One is “mechanical” in nature and it takes the cubic austenite structure of NiTi to a d...

We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic graphs. The APX-...

We report a preparation of high-quality cubic PbTe nanocrystals and their assembly into both square-array, two-dimensional patterns and three-dimensional simple cubic super crystals. The influence of oleylamine in the nanocrystal synthesis and core-shell formation through an anion-exchange mechanism was also studied. The simple cubic super crystals together with two-dimensional assembly pattern...

We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an ( , √ )-approximately local minimum within Õ(n/ ) second-order oracl...

For k = 2, 3 and a cubic graph G let νk(G) denote the size of a maximum k-edge-colorable subgraph of G. Mkrtchyan, Petrosyan and Vardanyan proved that ν2(G) ≥ 4 5 · |V (G)|, ν3(G) ≥ 7 6 · |V (G)| [13]. They were also able to show that ν2(G) + ν3(G) ≥ 2 · |V (G)| [3] and ν2(G) ≤ |V (G)|+2·ν3(G) 4 [13]. In the present work, we show that the last two inequalities imply the first two of them. Moreo...

The morphology of hexagonal phase NaYF4:Er(3+) nanorods synthesized by hydrothermal method changed greatly after a continuing calcination, along with a phase transformation to cubic phase. Photoluminescence (PL) spectra indicated that mid-infrared (MIR) emission was obtained in both hexagonal and cubic phase NaYF4:Er(3+) nanocrystals for the first time. And the MIR emission of NaYF4:Er(3+) nano...

A Ti-rich oxide, (Ti0.50Zr0.26Mg0.14Cr0.10) 1⁄4 1.0O1.81, was synthesized at 8.8GPa and 1600 1C using a multi-anvil apparatus. Its crystal structure at ambient conditions and compressibility up to 10.58GPa were determined with single-crystal X-ray diffraction. This high-pressure phase is isomorphous with cubic zirconia (fluorite-type) with space group Fm3̄m and unit-cell parameters a 1⁄4 4.8830(...

+ S&iX2(Xi — X2)} = 0. Clearly, the quartic degenerates into the #-line of the pencil of cubics, and into a cubic having (si, s2) sz) as the $-point. Hence the THEOREM: Any cubic of the net with a given S-point may be generated by any pencil of cubics within the net, not containing the given cubic, and the projective pencil of lines joining the Spoint of the given cubic to the S-points of the c...