On the Solution of the Monge-Ampere Equation ZxxZyy - Z²xy= f (x, y) with Quadratic Right Side

For the Monge{Ampere equation ZxxZyy-Z²xy = b₂₀x²+b₁₁xy+b₀₂y²+b₀₀ we consider the question on the existence of a solution Z(x, y) in the class of polynomials such that Z = Z(x, y) is a graph of a convex surface. If Z is a polynomial of odd degree, then the solution does not exist. If Z is a polynomial of 4-th degree and 4b₂₀b₀₂ - b₁₁² > 0, then the solution also does not exist. If 4b₂₀b₀₂ - b₁₁² = 0, then we have solutions

Dynamical spacetimes in conformal gravity

The conformal gravity remarkably boosts our prehension of gravity theories. We find a series of dynamical solutions in the $W^2$-conformal gravity, including generalized Schwarzschild-Friedmann-Robertson-Walker (GSFRW), charged generalized Schwarzschild-Friedmann-Robertson-Walker (CGSFRW), especially rotating Friedmann-Robertson-Walker (RFRW), charged rotating Friedmann-Robertson-Walker (CRFRW), and a dynamical cylindrically symmetric solutions. The RFRW, CRFRW and the dynamical cylindrically symmetric solutions are never found in the Einstein gravity and modified gravities. The GSFRW and CGSFRW solutions take different forms from the corresponding solutions in the Einstein gravity.Comment: 13 pages, no fi

Simple Loops on Surfaces and Their Intersection Numbers

Given a compact orientable surface $\Sigma$, let \Cal S(\Sigma) be the set of isotopy classes of essential simple loops on $\Sigma$. We determine a complete set of relations for a function from \Cal S(\Sigma) to $\bold Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in \bold R^{\Cal S(\Sigma)} and P(\bold R^{\Cal S(\Sigma)}) defining Thurston's space of measured laminations and Thurston's compactification of the Teichm\"uller space. These equations are not only piecewise integral linear but also semi-real algebraic.Comment: 42 pages, 29 figure

An alternative method to access diverse N,N′-diquaternised-3,3′-biquinoxalinium “biquinoxen” dications

An alternative synthetic route for the design of N,N′-diquaternised-3,3′-biquinoxalinium “biquinoxen” dications is reported, involving oxidative radical coupling of dithionite reduced quinoxaline quaternary salts. Although the reaction is not regioselective, leading to relatively modest yields (up to 32%), the advantages of this new synthetic protocol lie in a simple potentially gram scale synthesis using inexpensive easily accessible reagents with no metal catalysts and no purification steps. Thus whereas the method reported previously to access the N,N′-dimethyl-3,3′-biquinoxalinium, “methylbiquinoxen” precursor gave higher yield than the new method reported here, this new method avoids the limitation of using scarce oxonium reagents. Overall, the new protocol is a robust synthetic strategy which offers new design possibilities

Studi Daya Dukung Pondasi Dangkal pada Tanah Gambut Menggunakan Kombinasi Perkuatan Anyaman Bambu dan Grid Bambu dengan Variasi Lebar dan Jumlah Lapisan Perkuatan

Pembangunan konstruksi di atas tanah gambut semakin sering dilakukan karena kebutuhan lahan untuk pembangunan yang semakin lama semakin sempit. Kendala yang dihadapi pada pembangunan di tanah gambut diantaranya adalah daya dukung tanah yang rendah. Dalam penelitian ini, anyaman bambu dan grid bambu digunakan sebagai material perkuatan yang diharapkan dapat menjadi alternatif material perkuatan untuk meningkatkan daya dukung tanah gambut yang digunakan sebagai tanah dasar dari pondasi dangkal dengan variasi lebar perkuatan dan jumlah lapis perkuatan. Tujuan dari penelitian ini adalah untuk mengetahui peningkatan daya dukung dari setiapvariasi dengan nilai daya dukung tanpa perkuatan. Metodologi penelitian yang digunakan adalah pengujian dengan skala laboratorium. Data yang didapatkan dari pengujian tersebut kemudian dianalisa dengan membandingkan nilai daya dukung antara tanah tanpa perkuatan dengan menggunakan perkuatan yang dinyatakan dalam Bearing Capacity Ratio (BCR). Dari studi model di laboratorium diperoleh hasil bahwa dengan adanya penambahan dimensi perkuatan dan penambahan jumlah lapis perkuatan akan memberikan angka rasio daya dukung (BCR) yang semakin besar. Setelah diuji variasi lebar perkuatan 2B, 3B, dan 4B dengan jumlah lapisan 1 lapis, 2 lapis dan 3 lapis diperoleh kombinasi yang memberikan nilai daya dukung tertinggi adalah penggunaan 3 lapis perkuatan dengan lebar 4B (B adalah lebar pondasi). Nilai daya dukung tersebut sebesar 23,11 kPa dengan rasio daya dukung (BCR) sebesar 4,272 atau persen peningkatannya sebesar 327,2%

Spin-density-wave instabilities in the organic conductor (TMTSF)_2ClO_4: Role of anion ordering

We study the spin-density-wave instabilities in the quasi-one-dimensional conductor (TMTSF)_2ClO_4. The orientational order of the anions ClO_4 doubles the unit cell and leads to the presence of two electrnic bands at the Fermi level. From the Ginzburg-Landau expansion of the free energy, we determine the low-temperature phase diagram as a function of the strength of the Coulomb potential due to the anions. Upon increasing the anion potential, we first find a SDW phase corresponding to an interband pairing. This SDW phase is rapidly supressed, the metallic phase being then stable down to zero temperature. The SDW instability is restored when the anion potential becomes of the order of the hopping amplitude. The metal-SDW transition corresponds to an intraband pairing which leaves half of the Fermi surface metallic. At lower temperature, a second transition, corresponding to the other intraband pairing, takes place and opens a gap on the whole Fermi surface. We discuss the consequences of our results for the experimental phase diagram of (TMTSF)_2ClO_4 at high magnetic field.Comment: 13 pages, 10 figures, Version 2 with minor correction