Una clase especial de Hipersuperficies parametrizadas por líneas de curvatura en R4

In this paper we study hypersurfaces in R4 parametrized by lines of curvature with three distinct principal curvatures and with Laplace invariants mji = mki = 0; mjik 6= 0 for i; j; k distinct fixed indices. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable, this family includes a classe of Dupin hypersurfaces. Moreover, weshow that these vector valued functions are invariant under inversions and homotheties.En este artículo estudiamos hipersuperficies en R4 parametrizadas por líneas de curvatura con tres curvaturas principales distintas y con invariantes de Laplace mji = mki = 0; mjik 6= 0 para índices fijos i; j; k distintos. Caracterizamos localmente una familia genérica de tales hipersuperficies en términos de las curvaturas principales y tres funciones vectoriales de una variable, esta familia incluye una clase de hipersuperficies de Dupin. Ademas, mostramos que estas funciones vectoriales son invariantes por inversiones y dilataciones

Quasipinning and selection rules for excitations in atoms and molecules

Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for fermionic natural occupation numbers {ni }. A recent analysis of the pure N-representability problem provides a wide set of inequalities for the {ni}, leading to constraints on these numbers. This has a strong potential impact on reduced density matrix functional theory as we know it. In this work we continue our study of the nature of these inequalities for some atomic and molecular systems. The results indicate that (quasi)saturation of some of them leads to selection rules for the dominant configurations in configuration interaction expansions, in favorable cases providing means for significantly reducing their computational requirements

Ecuación de Helmholtz generalizada

In this paper we introduce the generalized Helmholtz equation and present explicit solutions to this generalized Helmholtz equation, these solutions depend on three holomorphic functions. As an application we present explicit solutions to the Helmholtz equation. We note that these solutions are not necessarily limited to certain domains of the complex plane C.En este artículo introducimos la ecuación de Helmholtz generalizada y presentamos soluciones explícitas para esta ecuación de Helmholtz generalizada, estas soluciones dependen de tres funciones holomorfas. Como aplicación presentamos soluciones explícitas para la ecuación de Helmholtz. Observamos que estas soluciones nonecesariamente estan limitadas a ciertos dominios del plano complejo C

Superfícies bi-harmonicas de tipo gráfico em R3

In this work, we study biharmonic surfaces that are parameterized by biharmonic coordinate functions. We study a class of biharmonic surfaces called graph-type biharmonic surfaces. Also, we define a class of surfaces associated to two harmonic functions (FH2A-surfaces), these surfaces satisfy a relation between the Gaussian curvature, the projection of the Gauss map on a fixed plane and two harmonic functions. We show that a particular class of graph-type biharmonic surfaces are FH2A-surfaces. Finally, we classify the FH2A-surfaces of rotation.Neste artigo, estudamos superfícies bi-harmonicas que sao parametrizadas por funcoes coordenadas biharmonicas. Estudamos uma classe de superfícies bi-harmonicas, chamadas superficies bi-harmonicas de tipo gráfico. Tambem, definimos uma classe de superfícies associadas a duas funcoes harmonicas (FH2Asurfaces), essas superfícies satisfazem uma relacao entre a curvatura Gaussiana, a projecao da aplicacao de Gauss sobre um plano fixo e duas funcoes harmonicas. Mostramos que uma classe particular de superfícies bi-harmonicas de tipo gráfico sao FH2A-surfaces. Finalmente, classificamos as FH2A-surfaces de rotacao

Physical Wigner functions

In spite of their potential usefulness, the characterizations of Wigner functions for Bose and Fermi statistics given by O'Connell and Wigner himself almost thirty years ago has drawn little attention. With an eye towards applications in quantum chemistry, we revisit and reformulate them in a more convenient way.Comment: Latex, 10 page

Machine learning the derivative discontinuity of density-functional theory

Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (a) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (b) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation

Testing one-body density functionals on a solvable model

There are several physically motivated density matrix functionals in the literature, built from the knowledge of the natural orbitals and the occupation numbers of the one-body reduced density matrix. With the help of the equivalent phase-space formalism, we thoroughly test some of the most popular of those functionals on a completely solvable model.Comment: Latex, 16 pages, 4 figure

Congruencias de esferas geodésicas em H3 e S3

In [2], was obtained a characterization of the surfaces in R3 which are envelopes of a sphere congruence in R3, in which the other envelope is in R2. In this paper, we characterize the surfaces of H3 and S3 which are envelopes of a congruence of geodesic spheres in H3 and S3, respectively, in which the other envelope is contained in H2 H3and S2 S3. We show that this characterization allows locally to obtain a parameterization of the surfaces contained in H3 and S3, this characterization extends the result obtained in [2]. Moreover, we provide sufficient conditions for these surfaces to be locally associated by a transformation of Ribaucour. Also, we present families of surfaces parameterized by lines of curvature in H3 and S3, which depend on a function of two variables which is solution of a differential equation. Finally, we characterize the surfaces of the spherical type in H3 and S3, as the surfaces where its radius function is the solution of the Helmholtz equation. Em [2], foi obtida uma caracterização das superfícies em R3 que são envelopes de uma congruência de esferas em R3, na qual o outro envelope está em R2. Neste artigo, caracterizamos as superfícies de H3 e S3 que são envelopes de uma congruência de esferas geodésicas em H3 e S3, respectivamente, na qual o outro envelope está contido em H2 H3 e S2 S3. Mostramos que esta caracterização permite obter localmente uma parametrização das superfícies contidas em H3 e S3, esta caracterização estende o resultado obtido em [2]. Além disso, damos condições suficientes para que estas superficies estejam associadas localmente por uma transformação de Ribaucour. Também, apresentamos famílias de superfícies parametrizadas por linhas de curvatura H3 e S3, que dependem unicamente de uma função de duas variavéis, a qual é solução de uma equação diferencial. Finalmente, caracterizamos as superfícies de tipo esférico em H3 e S3, como as superfícies onde sua função raio é solução da equação de Helmholtz

Congruencias de esferas geodésicas em H3 e S3

In [2], was obtained a characterization of the surfaces in R3 which are envelopes of a sphere congruence in R3, in which the other envelope is in R2. In this paper, we characterize the surfaces of H3 and S3 which are envelopes of a congruence of geodesic spheres in H3 and S3, respectively, in which the other envelope is contained in H2⊂H3 and S2⊂S3. We show that this characterization allows locally to obtain a parameterization of the surfaces contained in H3 and S3, this characterization extends the result obtained in [2]. Moreover, we provide sufficient conditions for these surfaces to be locally associated by a transformation of Ribaucour. Also, we present families of surfaces parameterized by lines of curvature in H3 and S3, which depend on a function of two variables which is solution of a differential equation. Finally, we characterize the surfaces of the spherical type ∑ in H3 and S3, as the surfaces where its radius function is the solution of the Helmholtz equation