Let G = (V,E) be a connected finite graph and Δ be the usual graph Laplacian. In this paper, the authors consider a generalized self-dual Chern-Simons equation on the graph G

where

$\tilde F(u)$ satisfies $u=1+\tilde F(u)-\rm{{e}}^{\tilde F(u)}$, λ > 0, M is any fixed positive integer, $\delta_{p_{j}}$ is the Dirac delta mass at the vertex p_j, and p₁, p₂, …, p_M are arbitrarily chosen distinct vertices on the graph. They first prove that there is a critical value λ_c such that if λ ≥ λ_c, then the generalized self-dual Chern-Simons equation has a solution u_λ. Applying the existence result, they develop a new method to construct a solution of (0.1) which is monotonic with respect to λ when λ ≥ λ_c. Then they establish that there exist at least two solutions of the equation for λ > λ_c via the variational method. Furthermore, they give a fine estimate of the monotone solution, which can be applied to other related problems.

This study explores a broader variety of single-index multiplicative models (SIMM for short) with an unknown, discontinuous link function. Relaxing the continuity assumption in nonparametric functions enhances the applicability to positive data. However, the authors find an issue with the existing least product relative error (LPRE for short) technique at jump points, posing a challenge in estimating the link function accurately in SIMMs. They propose an automated method that combines the LPRE technique with jump-preserving methods to simultaneously estimate the unknown parameter vector and the discontinuous link function. Their approach is flexible and practical, not requiring prior knowledge of jump point details. Furthermore, they establish the asymptotic properties of the estimators for the parametric vector and the discontinuous function components under reasonable conditions. To validate their approach, they conduct numerical simulations evaluating the performance with finite samples. Additionally, they demonstrate the effectiveness of the approach through real data analysis.

As a generalization of Ahlfors’s results for analytic functions, by using the pre-Schwarzian derivative of harmonic mappings, the authors obtain a criterion of univalence and quasiconformal extension for harmonic functions. As applications, they give a lower bound of the inner radius of univalency by means of pre-Schwarzian derivative of harmonic mappings for a planar domain.

In this paper, the authors investigate the double-indexed version of the strong law of large numbers under some general conditions in a sub-linear expectation space. The weighted version of the Marcinkiewicz-Zygmund type strong law of large numbers is also established. These results extend or improve some existing ones in the classical probability space or a sub-linear expectation space. As an application, they further study the nonparametric regression model under the sub-linear expectation framework. Some numerical simulations are also presented.

The arboricity of a graph is the minimum number of forests needed to cover all edges of the graph. Let G be a connected graph embedded in a surface. This paper shows that the arboricity of G is at most $\left\lceil {{{1003} \over {460}} + \sqrt {3g + {1 \over {16}}} } \right\rceil$ (or $\left\lceil {{{487} \over {244}} + \sqrt {{3 \over 2}h + {1 \over {16}}} } \right\rceil$) if the orientable genus (or the nonorientable genus) of G is g (or h), and that these bounds are tight. As a conclusion of the results, an upper bound for the outerthickness of a connected graph embedded in an orientable surface (or a non-orientable surface) is obtained. The paper proves that if the orientable genus (or the non-orientable genus) of G is g(≥ 1) (or h), then G can be decomposed into $\lceil {\sqrt {3g} } \rceil + 3$ (or $\left\lceil {\sqrt {{3 \over 2}h} } \right\rceil + 3$) forests in which one has maximum degree at most $\left\lfloor {{1 \over 3}\lceil {\sqrt {3g} }\rceil } \right\rfloor + 1$ (or $\left\lfloor {{1 \over 3}\left\lceil {\sqrt {{3 \over 2}h} } \right\rceil } \right\rfloor + 1$).

A finite non-abelian group G is called an ${\cal{M}}{\cal{C}}$-group if all non-abelian subgroups H of G have minimum centralizers (i.e., C_G(H) = Z(G)). In this paper, the authors give some characterizations of ${\cal{M}}{\cal{C}}$-groups, and it is proved that ${\cal{M}}{\cal{C}}$-groups are just the finite groups with modular centralizer lattice of length 2 depicted by Schmidt, which leads to a classification of ${\cal{M}}{\cal{C}}$-groups. However, Schmidt’s depiction said nothing for ${\cal{M}}{\cal{C}}$-p-groups. They give a characterization of ${\cal{M}}{\cal{C}}$-p-groups. In particular, they characterize special ${\cal{M}}{\cal{C}}$-p-groups by means of the commutator matrices, and provide a method to determine or classify special ${\cal{M}}{\cal{C}}$-p-groups. As applications, some examples are given, and special ${\cal{M}}{\cal{C}}$-p-groups with an abelian maximal subgroup are classified up to isoclinism.

The author proves that braided fusion categories of Frobenius-Perron dimension p^mqⁿd or p²q²r² are weakly group-theoretical, where p, q, r are distinct prime numbers, d is a square-free natural number such that (pq, d) = 1. As an application, the author obtains that weakly integral braided fusion categories of Frobenius-Perron dimension less than 1800 are weakly group-theoretical, and weakly integral braided fusion categories of odd dimension less than 33075 are solvable. For the general case, the author proves that fusion categories (not necessarily braided) of Frobenius-Perron dimension 84 and 90 are either solvable or group-theoretical. Together with the results in the literature, this shows that every weakly integral fusion category of Frobenius-Perron dimension less than 120 is either solvable or group-theoretical. Thus the author completes the classification of all these fusion categories in terms of Morita equivalence.

Over an algebraically closed field of characteristic p > 2, the 0-dimensional and 1-dimensional cohomology of the queer Lie superalgebra ${\frak{q}}(2)$ with coefficients in all baby Verma modules and all simple modules are determined.