Let the sign of a standard Young tableau (SYT) be the sign of the permutation obtained by reading the entries row by row from left to right, starting from the top row. By a sign-reversing involution on the tableaux in terms of lattice paths, we obtain the sign-balance of $n$-cell SYTs of all shapes with at most three rows. For skew shapes, we obtain partial results on the sign-balance enumeration of $2n$-cell skew SYTs of all shapes with at most three rows, under a parity condition of the skew part.

We consider a twisted Laplacian $\Delta_{\nu,\mu}$ on the $n$-complex space associated with the sub-Laplacian of the Heisenberg group $\mathbb{C} \times_{\omega} \mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. The main results to which is aimed this paper concern the spectral theory of $\Delta_{\nu,\mu}^{\Gamma}$ when acting on some $L^2$ space of $\Gamma$-automorphic functions of biweight $(\nu,\mu)$ associated to given cocompat discrete subgroup $\Gamma$ of the additive group $\mathbb{C}^n$. We describe its spectrum proving a stability theorem. Using the Selberg's approach, we give the explicit dimension formula for the corresponding $L^2$-eigenspaces. We also construct a concrete basis of such $L^2$-eigenspaces.

Foliations on the space of $p$-divisible groups were studied by Oort in 2004. In his theory, special leaves called central streams play an important role. In this paper, we give a complete classification of the boundary components of the central streams for an arbitrary Newton polygon in the unpolarized case. Hopefully this classification would help us to know the boundaries of other leaves and more detailed structure of the boundaries of central streams.

This paper extends many conclusions based on phantom envelopes and Ext-phantom covers of modules, and we find that many important properties still hold after replacing phantom and Ext-phantom with $n$-phantom and $n$-Ext-phantom respectively. In addition, we also obtain some extra results. Specifically, we give a characterization of the weak dimensions of rings in terms of $n$-phantom envelopes and $n$-Ext-phantom covers of modules with the unique mapping property respectively. We show that $\operatorname{wD}(R) \leq 2n$ whenever every right $R$-module has an $n$-phantom envelope with the unique mapping property or every left $R$-module has an $n$-Ext-phantom cover with the unique mapping property over left coherent rings.

Time distributed order two-sided space differential equations of arbitrary order offer a robust approach to modelling complex dynamical systems. In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations. The numerical discretization scheme is a hybrid scheme, comprising a Newton–Cotes quadrature formula and a spectral collocation method. The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination of shifted Chebyshev polynomials in all variables. Convergence analysis of the numerical scheme is presented. Some one- and two-dimensional time distributed order two-sided space fractional differential equations, such as the fractional advection-dispersion and diffusion equations, are presented to demonstrate the accuracy and computational efficiency of the numerical scheme, and numerical solutions are compared with the exact solutions, where these are available.

This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in $\mathbb{R}^N$ involving the fractional pseudo-differential operators defined as the generalizations of the $p(x)$-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory. The obtained results of this note significantly contribute to the study of Kirchhoff-type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.

This paper is devoted to studying Sobolev regularity properties of commutators of Hardy–Littlewood maximal operator and its fractional case with Lipschitz symbols, both in the global and local case. Some new pointwise estimates for the weak gradients of the above commutators will be established. As applications, some bounds for the above commutators on the Sobolev spaces will be obtained.

We consider inverse subinterval-nodal problems for the Sturm–Liouville operator on a star graph with mixed boundary conditions in pendant vertices and the standard matching conditions at the interior vertex. Can subinterval nodes recognize star-shaped quantum graphs? In this paper we give a positive answer. It is shown that the data of subinterval-nodes near the interior vertex can uniquely determine the potential on a graph and boundary conditions.

Let $(\Sigma,g)$ be a closed Riemannian surface, $\mathbf{G} = \{ \sigma_1,\ldots,\sigma_N \}$ be an isometric group acting on it. Denote a positive integer $\ell = \min_{x \in \Sigma} I(x)$, where $I(x)$ is the number of all distinct points of the set $\{ \sigma_1(x),\ldots,\sigma_N(x) \}$. By a method of flow due to Castéras (Pacific J. Math. 2015), we prove that the solution to the mean field equation \[ -\Delta_g u = 8\pi \ell \left( \frac{he^u}{\int_{\Sigma} he^u \, dv_g} - \frac{1}{\operatorname{Vol}_g(\Sigma)} \right) \] exists under given conditions. This gives a new proof of Yang and Zhu's result in (Internat. J. Math. 2020). The case $\ell = 1$ was studied by Li and Zhu (Calc. Var. Partial Differential Equations 2019).

In this paper, we give optimality conditions for the quadratic programming problems with constraints defined by finitely many convex quadratic constraints in Hilbert spaces. As special cases, we obtain optimality conditions for the quadratic programming problems under linear constraints in Hilbert spaces.