We have used the He I discontinuities at 3421A to determine the electron temperatures, designated Te(He I), for a sample of five Galactic planetary nebulae (PNe). We compared Te(He I) with the electron temperatures derived from the hydrogen Balmer jump at 3646A, designated Te(H I), and found that Te(He I) are generally lower than Te(H I). There are two possible interpretations, a) the presence of substantial He+2 zone, or b) the presence of hydrogen-deficient cold clumps within diffuse nebulae. A series of photoionization models were constructed to test the two scenarios. We found that the observed Te(He I)/Te(H I) discrepancies are beyond the predictions of chemically homogeneous models. Our modelling shows that the presence of a small amount of hydrogen-deficient inclusions seems to be able to reproduce the observed intensities of He I discontinuities. We stress the value of He I discontinuities in investigating nebular physical conditions. Albeit with some observational and technical limitations, He I discontinuities should be considered in future modelling work.

Y. Zhang H. -B. Yuan C. -T. Hua X. -W. Liu J. Nakashima S. Kwok

We present a microscopic theory of superfluid $^4$He, formulated using the overall experimental observations as input information. With the theory of a consistent basis, we answer all of the essential questions regarding He II.

J. X. Zheng-Johansson B. Johansson P-I. Johansson

Let X and X' be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles, in the sense of Henning Meyer's graduate thesis, between the tropicalization of the intersection product of X and X' and the stable intersection of trop(X) and trop(X'), when restricted to (the inverse image under the tropicalization map of) a connected component C of the intersection of trop(X) and trop(X'). This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in loc.cit.. As a result we get a numerical equivalence (after a compactification and restricting to C) between the intersection product of X and X' and the stable intersection of trop(X) and trop(X') via the compactified stable intersection. In particular, when X and X' have complementary codimensions, this equivalence generalizes the work of Osserman and Rabinoff, in the sense that the intersection of X and X' is allowed to be of positive dimension. Moreover, if the intersection of the closures of X and X' has finitely many points which tropicalize to the closure of C, we prove a similar equation as in Theorem 6.4 of the paper of Osserman and Rabinoff when the ambient space is a reduced closed subscheme of T (instead of T itself).

Xiang He

We derive the structure factor for superfluid He II about the energy dispersion minimum 1.93 1/A.

J. X. Zheng-Johansson

In this paper, we investigate the Diophantine equation \[ (2^k - 1)(3^k - 1) = x^n \] and prove that it has no solutions in positive integers $k, x, n > 2$.

Bo He Chang Liu

Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem: "Assume that F(0;0) = 0. Then for every non-zero integer d, the equation F(X;Y) = 0 has only finitely many solutions in integers (X;Y) with gcd(X;Y) = d". Skolem method allows one to bound the solutions explicitly in terms of the coefficients of the polynomial F and the integer d. In 2008, Abouzaid gave a far-going generalization of Skolem theorem. He extended it in two directions: first, he studied solutions not only in rational integers, but in arbitrary algebraic numbers. Second, he not only bounded the solution in terms of the logarithmic gcd, but obtained a sort of asymptotic relation between the heights of the coordinates and their logarithmic gcd. Unfortunately, Abouzaid assumption is slightly more restrictive than Skolem: he assumes not only that the point (0;0) belongs to the plane curve F(X;Y) = 0, but that (0;0) is a non-singular point on this curve. The purpose of the present article is to get rid of this non singularity hypothesis.

Boris Bartolome

Potential and dipole moment curves for the CrH(X$^6\Sigma^+$) molecule were obtained at the internally-contracted multi-reference configuration interaction with single and double excitations and Davidson correction (ic-MRCISD+Q) level using large basis set augmented with additional diffused functions and using Douglass-Kroll Hamiltonian for scalar relativistic effects. Also bound states, average positions and rotational constants calculated on the CrH(X$^6\Sigma^+$) potential are reported. The He-CrH(X$^6\Sigma^+$) potential energy surface was calculated with the coupled cluster singles, doubles, and noniterative triples [RCCSD(T)] method. The global minimum was found for the collinear He---Cr-H geometry with the well depth of 1143.84 cm$^{-1}$ at $R_e=4.15$ a$_0$. An insight in the character of the complex was gained by means of symmetry-adapted perturbation theory (SAPT) based on DFT description of the monomers. The presence of the so called "exchange cavity" was observed. Finally, bound states of the He-CrH complex for J = 0 are presented.

Jacek Kłos Michał Hapka Grzegorz Chałasiński

$^3$He spin-dependent structure functions, $g^3_1(x)$ and $g^3_2(x)$, which parametrize the hadronic tensor in polarized deep-inelastic scattering, were evaluated within the Poincar\'e covariant light-front framework. The Bakamjian-Thomas construction of the Poincar\'e generators allows us to make use of a realistic $^3$He wave function, obtained from refined nuclear phenomenological potentials. The same approach was already successfully applied to the $^3$He and $^4$He unpolarized deep-inelastic scattering. To investigate the neutron polarized structure functions, $g^n_1$ and $g^n_2$, a readily implementable procedure, aimed at extracting the neutron spin structure functions from those of $^3$He, is shown to hold. Moreover, the first moment of $g^3_1(x)$ was evaluated, aiming at providing a valuable check of the Bjorken sum rule. The present analysis is relevant for experiments nvolving polarized beams planned at the future facilities, like the Electron Ion Colliders.

Eleonora Proietti Filippo Fornetti Emanuele Pace Matteo Rinaldi Giovanni Salmè Sergio Scopetta

The notion of a pseudo cluster tilting subcategory $\mathcal X$ in an extriangulated category $\mathcal C$ is defined in this article. We prove that the quotient category $\mathcal C/\mathcal X$, obtained by factoring an extriangulated category by a pseudo cluster tilting subcategory, is a semi-abelian category. Furthermore, we also show that the quotient category $\mathcal C/\mathcal X$ is an abelian category if and only if certain self-orthogonal conditions are satisfied. As an application, these results generalize the work of Xu and Zheng in the exact category.

Jian He Jing He

Let $\mathcal{L}=-\Delta+\mathit{V}(x)$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse H\"{o}lder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schr\"{o}dinger operators, which include Riesz transforms, standard Calder\'{o}n-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.

Qianjun He Pengtao Li