A new representation of the many body wave function and its application as a post Hartree-Fock energy variation method

In this paper, we introduce a new representation of many body electron wave function and a few calculation results of the ground state energies of many body systems using that representation, which is systematically better than the Hartree-Fock approximation.Comment: 9 pages, no figur

On the Feynman path integral for the magnetic Schroedinger equation with a polynomially growing electromagnetic potential

The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials $(V,A_{1},A_{2},...,A_{d})$ such that $V(t,x) = |x|^{2(l+1)} +$`` a polynomial of degree $(2l + 1)$ in $x$ " ($l = 0,1,2,...$) and $A_{j}(t,x)$ are polynomials of degree $l$ in $x$. The Feynman path integrals are defined as $L^2$-valued continuous functions with respect to the time variable.Comment: to appear in Review Mathematical Physics (2020

Mathematical Remarks on the Feynman Path Integral for Nonrelativistic Quantum Electrodynamics

The Feynman path integral for nonrelativistic quantum electrodynamics is studied mathematically of a standard model in physics, where the electromagnetic potential is assumed to be periodic with respect to a large box and quantized thorough its Fourier coefficients. In physics, the Feynman path integral for nonrelativistic quantum electrodynamics is defined very formally. For example, as is often seen, even independent variables are not so clear. First, the Feynman path integral is defined rigorously under the constraints familiar in physics. Secondly, the Feynman path integral is also defined rigorously without the constraints, which is stated in Feynman and Hibbs' book without any comments. So, our definition may be completely new. Thirdly, the vacuum and the state of photons of momentums and polarization states are expressed by means of concrete functions of variables consisting of the Fourier coefficients of the electromagnetic potential. Our results above have many applications as is seen in Feynman and Hibbs' book, though the applications are not rigorous so far. It is also proved rigorously by means of the distribution theory that the Coulomb potentials between charged particles naturally appear in the Feynman path integral above. As is well known, this shows that photons give the Coulomb forth

On the finiteness of solutions for polynomial-factorial Diophantine equations

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exists only finitely many $l$ such that $l!$ is represented {by} $N_A(x)$, where $N_A$ is a norm form constructed from the field norm of a field extension $K/\mathbf Q$. We also deal with the equation $N_A(x)=l!_S$, where $l!_S$ is the Bhargava factorial. In this paper, we also show that the Oesterl\'e-Masser conjecture implies that for any infinite subset $S$ of $\mathbf Z$ and for any polynomial $P(x)\in\mathbf Z[x]$ of degree $2$ or more the equation $P(x)=l!_S$ has only finitely many solutions $(x,l)$. For some special infinite subsets $S$ of $\mathbf Z$, we can show the finiteness of solutions for the equation $P(x)=l!_S$ unconditionally.Comment: 21 page

A higher dimensional generalization of Lichtenbaum duality in terms of the Albanese map

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the N\'eron-Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparison result of cohomological- and Azumaya-Brauer groups. We will also mention the local-global problem of the Albanese-cokernel; the abelian group on the "local side" turns out to be a finite group.Comment: 20 pages, to appear in Compositio Mathematic

Poisson algebras of curves on bordered surfaces and skein quantization

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quotient of the vector space spanned by the regular homotopy classes of curves on the bordered surface by generalizing the Goldman bracket and the Turaev cobracket. Moreover, we define a Poisson algebra of unoriented curves on a bordered surface and show that a quantization of the Poisson algebra coincides with the skein algebra of the bordered surface defined by Muller.Comment: 23 pages, 9 figures; v2: references added; v3: references added, minor change

Uniform upper bounds of the distribution of relatively r-prime lattice points

We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under assuming the Lindel\"of hypothesis. And we also showed unconditional results for abelian extensions with a degree less than or equal to $6$. In this paper we remove all assumption about number fields and improve uniform upper bounds. Throughout this paper we consider estimates for distribution of ideals of the ring of integer $\mathcal{O}_K$ and obtain uniform upper bounds. And when $K$ runs through cubic extension fields we show better uniform upper bounds than that under the Lindel\" of Hypothesis.Comment: 20 page

Continuous Varieties of Metric and Quantitative Algebras

A metric algebra is a metric variant of the notion of $\Sigma$-algebra, first introduced in universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. In this paper, we showed metric versions of the variety theorem, which characterizes strict varieties (classes of metric algebras defined by metric equations) and continuous varieties (classes defined by a continuous family of basic quantitative inferences) by means of closure properties. To this aim, we introduce the notion of congruential pseudometric on a metric algebra, which corresponds to congruence in classical universal algebra, and we investigate its structure

The APM/Matched-Filter Cluster Catalog

A catalog of nearby clusters in the 5800 deg$^2$ area in the southern Galactic cap is constructed by applying a matched-filter cluster-finding algorithm to the sample of 3.3 million galaxies from the APM Galaxy Survey. I have preliminarily detected more than 4000 cluster candidates with estimated redshift of less than 0.2 and with richness similar to those of ACO clusters. Generally, a good correspondence is found between the nearest cluster candidates in our catalog and the ACO clusters which have measured redshift. While the ACO catalog becomes incomplete at z>0.08, the completeness limit of our cluster catalog reaches z=0.15.Comment: 5 pages LaTeX, 6 PostScript figures, uses newpasp.sty and epsf.sty (included), to appear in the Proceedings of IGRAP99 International Conference, Marseille, 29/06/1999-02/07/199

A2A_2 Skein Representations of Pure Braid Groups

We define a family of representations $\{\rho_n\}_{n\geq 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its boundary. We also introduce a triangle-free basis of such an $A_2$ web space and calculate matrix representations of $\rho_n$ about the standard generators of $P_{2k}$.Comment: 13 pages, many TikZ picture