Search results for "Hypersurface"
showing 8 items of 48 documents
An Ab Initio Study on the Mechanism of the Atmospheric Reaction NH2+O3→H2NO+O2
2003
The atmospheric reaction NH 2 +0 3 →H 2 NO+O 2 has been investigated theoretically by using MP2, QCISD, QCISD(T), CCSD(T), CASSCF, and CASPT2 methods with various basis sets. At the MP2 level or theory, the hypersurface of the potential energy (HPES) shows a two step reaction mechanism. Therefore, the mechanism proceeds along two transition states (TS1 and TS2), seperated by an intermediate disignated as Int. However, when the single-reference higler correlated QCISD and the multiconfigurational CASSCF methodologies have been employed, the minimum structure Int and TS2 are not found on the HPES, which thus confirms a direct reaction mechanism. Single-reference high correlated and multiconfi…
Ab initiostudy of the mechanism of the atmospheric reaction: NO2+ O3→ NO3+ O2
2003
The atmospheric reaction NO2 + O3 --> NO3 + O2 (1) has been investigated theoretically by using the MP2, G2, G2Q, QCISD, QCISD(T), CCSD(T), CASSCF, and CASPT2 methods with various basis sets. The results show that the reaction pathway can be divided in two different parts at the MP2 level of theory. At this level, the mechanism proceeds along two transition states (TS1 and TS2) separated by an intermediate, designated as A. However, when the single-reference higher correlated QCISD methodology has been employed, the minimum A and the transition state TS2 are not found on the hypersurface of potential energy, which confirms a direct reaction mechanism. Single-reference high correlated and mu…
A Computational Study of Two-State Conformational Changes in 16-Electron [CpW(NO)(L)] Complexes (L=PH3, CO, CH2, HCCH, H2CCH2)
1999
International audience; High-spin and low-spin [CpW(NO) (L)] complexes are calculated to be remarkably close in energy. Several critical conformational changes in the singlet compounds are predicted to proceed more readily by spin crossover to the triplet hypersurface. The relationships between spin state, π bonding, ligand orientation, and geometry at W are explored.
Three-qutrit entanglement and simple singularities
2016
In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety $X$ of separable three-qutrit states within the projective Hilbert space $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|\varphi\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_\varphi$-hypersuface by cutting $X$ with a hyperplane $H_\varphi$ defined by the linear form $\langle\varphi|$ (the $X_\varphi$-hypersurface of $X$ is $X\cap H_\varphi \subset X$). We prove that when $|\varphi\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_\varphi$-hypersuface with isolated singularities, has…
Humbert surfaces and the Kummer plane
2003
A Humbert surface is a hypersurface of the moduli space A 2 \mathcal A_2 of principally polarized abelian surfaces defined by an equation of the form a z 1 + b z 2 + c z 3 + d ( z 2 2 − z 1 z 3 ) + e = 0 az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0 with integers a , … , e a,\ldots ,e . We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.
The F-pure threshold of quasi-homogeneous polynomials
2018
Abstract Inspired by the work of Bhatt and Singh [3] we compute the F-pure threshold of quasi-homogeneous polynomials. We first consider the case of a curve given by a quasi-homogeneous polynomial f in three variables x , y , z of degree equal to the degree of xyz and then we proceed with the general case of a Calabi–Yau hypersurface, i.e. a hypersurface given by a quasi-homogeneous polynomial f in n + 1 variables x 0 , … , x n of degree equal to the degree of x 0 ⋯ x n .
Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
2020
This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with…
Embeddings of Danielewski hypersurfaces
2008
In this thesis, we study a class of hypersurfaces in $\mathbb{C}^3$, called \emph{Danielewski hypersurfaces}. This means hypersurfaces $X_{Q,n}$ defined by an equation of the form $x^ny=Q(x,z)$ with $n\in\mathbb{N}_{\geq1}$ and $\deg_z(Q(x,z))\geq2$. We give their complete classification, up to isomorphism, and up to equivalence via an automorphism of $\mathbb{C}^3$. In order to do that, we introduce the notion of standard form and show that every Danielewski hypersurface is isomorphic (by an algorithmic procedure) to a Danielewski hypersurface in standard form. This terminology is relevant since every isomorphism between two standard forms can be extended to an automorphism of the ambiant …