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.

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.

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.

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.

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.

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.

In this paper, we construct a simple chemostat-based variable yield model of competition between two bacterial strains, one of which is capable of wall growth [14]. In this model we prove the boundedness of solutions, analyze the local stability of equilibria and establish the global stability for the locally stable extinction equilibrium. Furthermore, when the extinction equilibrium becomes unstable, we prove the existence and uniqueness of the positive equilibrium and the uniform persistence of the system.

In the present paper, we deal with the global regularity estimates for the $p$-Laplace equations with data in divergence form \[ \operatorname{div}(|\nabla u|^{p-2} \nabla u) = \operatorname{div}(|F|^{p-2} F) \quad \textrm{in $\Omega$}, \] in Morrey spaces with natural data $F \in L^p(\Omega;\mathbb{R}^n)$ and nonhomogeneous boundary data belongs to $W^{1,p}(\Omega)$. Motivated by the work of [M.-P. Tran, T.-N. Nguyen, New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data, J. Differential Equations 268 (2020), no. 4, 1427-1462], this paper extends that of global Lorentz--Morrey gradient estimates in which the ‘good-$\lambda$’ technique was undertaken for a class of more general equations, and further, global regularity of weak solutions will be given in terms of fractional maximal operators.

We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of Matsumoto zeta-functions $\varphi_h(s)$ and a collection of periodic Hurwitz zeta-functions $\zeta(s,\alpha;\mathfrak{B})$ under the condition that the common difference of arithmetical progression $h \gt 0$ is such that $\exp \big\{ \frac{2\pi}{h} \big\}$ is a rational number and parameter $\alpha$ is a transcendental number.

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

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).

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincaré polynomial of the compactified space.

We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.

We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.

In this paper, we provide an effective method to compute the topological entropies of $G$-subshifts of finite type ($G$-SFTs) with $G = F_{d}$ and $S_{d}$, the free group and free semigroup with $d$ generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of $S_{2}$-SFTs, namely the types $\mathbf{E}$, $\mathbf{D}$, $\mathbf{C}$ and $\mathbf{O}$, are introduced, and we could compute their entropies explicitly. This enables us to give the complete characterization of $S_{2}$-SFTs on two symbols. That is, the set of entropies of $S_{2}$-SFTs on two symbols is equal to $\mathbf{E} \cup \mathbf{D} \cup \mathbf{C} \cup \mathbf{O}$. The methods developed in $S_{d}$-SFTs will also apply to the study of the entropy theory of $F_{d}$-SFTs. The entropy formulae of $S_{d}$-, $F_{d}$-golden mean shifts and $k$-colored chessboards are also presented herein.

This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

For a fixed nonnegative integer $m$, an analytic map $\varphi$ and an analytic function $\psi$, the generalized integration operator $I^{(m)}_{\varphi,\psi}$ is defined by \[ I^{(m)}_{\varphi,\psi} f(z) = \int_0^z f^{(m)}(\varphi(\zeta)) \psi(\zeta) \, d\zeta \] for $X$-valued analytic function $f$, where $X$ is a Banach space. Some estimates for the norm of the operator $I^{(m)}_{\varphi,\psi} \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ are obtained. In particular, it is shown that the Volterra operator $J_b \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ is bounded if and only if $J_b \colon A^2_{\alpha} \to A^2_{\alpha}$ is in the Schatten class $S_p(A^2_{\alpha})$ for $2 \leq p \lt \infty$ and $\alpha \gt -1$. Some corresponding results are established for $X$-valued Hardy spaces and $X$-valued Fock spaces.

We study independence of symplectic embeddings of the theory of Ekedahl--Oort stratifications on Shimura varieties of Hodge type, by comparing two different embeddings with a third one. The main results are as follows.

1. The Ekedahl--Oort stratification is independent of the choices of symplectic embeddings.

2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl--Oort stratifications with respect to morphisms of Shimura varieties.

The purpose of this paper is to work out the theoretical aspects of the vibration problem of rubber wiper blade on convex windshield. Over the past 20 years, some $2$-dimensional spring-mass models were presented in engineering science to simulate the vibration of rubber wiper blade on windshield. In this paper, we will consider the elasticity perspective on this $3$-dimensional vibration problem. Our theoretical analysis suggests that there should exist two classes of vibration frequencies corresponding to “$\ast$-exact deformations (Class I)” and “$\ast$-closed deformations (Class II)”. We prove mathematical theorems on the characterization of deformations of Class I. We also explain how elementary deformations of Class II can be constructed. We then deduce two mathematical formulas, for the vibration problem of rubber wiper blade on convex windshield, from our theoretical analysis. Our theoretical predictions are in almost perfect agreement with experimental data. One of the crucial steps of our analysis is a decomposition theorem motivated by the de Rham Cohomology and the Hodge Theory.

We prove that for each nonseparable and reflexive Banach space $(X,\|\cdot\|_X)$ with the nonstrict Opial and Kadec--Klee properties, there exists an equivalent norm $\|\cdot\|_0$ such that the Banach space $(X,\|\cdot\|_0)$ is LUR and contains a diametrically complete set with empty interior.

In this paper, we study the existence of semiclassical solutions of the fractional Schrödinger--Poisson system with critical exponent in $\mathbb{R}^{3}$. Under suitable assumptions, we show that the problem has at least one nontrivial solution provided that $\epsilon \leq \varepsilon_{\delta}$ where $\varepsilon_{\delta}$ is sufficiently small positive number.

Using the method of moving planes, we establish the radial symmetry of positive solutions to the fractional system \[ \begin{cases} (-\Delta)^s_p u = f(u,v), \\ (-\Delta)^t_q v = g(u,v) \end{cases} \] in the entire Euclidean space $\mathbb{R}^n$ and in the unit ball, where $0 \lt s,t \lt 1$ and $p,q \geq 2$. In particular, our result can be applied to the nonlinearities $f(u,v) \equiv u^a v^b$ and $g(u,v) \equiv u^c v^d$, where $a,d \in \mathbb{R}$ and $b,c \gt 0$.

In this paper, we consider the condition number theory for a linear function of the solution to the constrained and weighted least squares problem. We first present two explicit expressions without Kronecker product of normwise condition number using the classical method for condition numbers. Then, we derive the explicit expression of mixed and componentwise condition numbers by the dual techniques. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method and devise three algorithms. Numerical experiments are provided to illustrate the obtained results.

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. The set of leaves of $T$ is denoted by $L(T)$ and the set of branch vertices of $T$ is denoted by $B(T)$. For two distinct vertices $u$, $v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v$. Let $T$ be a tree with $B(T) \neq \emptyset$, for each vertex $x \in L(T)$, set $y_x \in B(T)$ such that $(V(P_T[x,y_x]) \setminus \{y_x\}) \cap B(T) = \emptyset$. We delete $V(P_T[x,y_x]) \setminus \{y_x\}$ from $T$ for all $x \in L(T)$. The resulting graph is a subtree of $T$ and is denoted by $\operatorname{R\_Stem}(T)$. It is called the reducible stem of $T$. A leaf of $\operatorname{R\_Stem}(T)$ is called a peripheral branch vertex of $T$. In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph $G$ to have a spanning tree with few peripheral branch vertices.

In the paper a second order differential system with a periodic boundary data is considered. Using some variational methods sufficient conditions for the continuous dependence of trajectories on controls are proved. The obtained results are applied then to the optimal control problem governed by the above system and the cost functional of a Bolza-type. In the end of the paper, an example of periodic optimal control problem demonstrating the applicability of the results is presented.

This paper is concerned with the biholomorphism of two equidimensional Hartogs type domains over irreducible bounded symmetric domains $H_{\Omega}(p)$ (see (1.1)) which is a Hua construction of Cartan--Hartogs domain. We develop a new simple methods to give an sufficient and necessary condition for the two Hua domains to be biholomorphic equivalent by using the function $\mathcal{L}_{\Omega}(z,\omega)$ (see (1.2)). Furthermore as an application, we can also give an equivalent description for the automorphism of Hua domain.

Let $H \xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H \xrightarrow{s} G \}$, and the degree bipartite Ramsey number $br_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H \xrightarrow{s} G \textrm{ and } \chi(H) = 2 \}$. In this note, we show that $r_{\Delta}(K_{m,n};s)$ is linear on $n$ with fixed $m$. We also evaluate $br_{\Delta}(G;s)$ for paths and other trees.

Given a field $k$, a finite-dimensional $k$-algebra $A$, and a finite acyclic quiver $Q$, let $AQ$ be the path algebra of $Q$ over $A$. Then the category of representations of $Q$ over $A$ is equivalent to the category of $AQ$-modules. The main result of this paper explicitly describes the strongly Gorenstein-projective $AQ$-modules via the separated monic representations with a local strongly Gorenstein-property. As an application, a necessary and sufficient condition is given on when each Gorenstein-projective $AQ$-module is strongly Gorenstein-projective. As a direct result, for an integer $t \geq 2$, let $A = k[x]/\langle x^t \rangle$, each Gorenstein-projective $AQ$-module is strongly Gorenstein-projective if and only if $A = k[x]/\langle x^2 \rangle$.

The influence of the geometry of the domain on the behavior of generalized solutions of Dirichlet problems for elliptic partial differential equations has been an important subject for over a century. We investigate the boundary behavior of variational solutions $f$ of Dirichlet problems for prescribed mean curvature equations in a domain $\Omega \subset \mathbb{R}^{2}$ near a point $\mathcal{O} \in \partial \Omega$ under different assumptions about the curvature of $\partial \Omega$ on each side of $\mathcal{O}$. We prove that the radial limits at $\mathcal{O}$ of $f$ exist under different assumptions about the Dirichlet boundary data $\phi$, depending on the curvature properties of $\partial \Omega$ near $\mathcal{O}$.

In this paper, we mainly discuss the nonuniform average sampling and reconstruction in multiply generated shift-invariant subspaces \[ V_{p,q}(\Phi_r) = \bigg\{ \sum_{k_{1} \in \mathbf{Z}} \sum_{k_{2} \in \mathbf{Z}^{d}} c^T(k_{1},k_{2}) \Phi_r(\,\cdot-k_{1},\,\cdot-k_{2}): (c(k_{1},k_{2}))_{(k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}} \in \big( \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d) \big)^r \bigg\} \] of mixed Lebesgue spaces $L^{p,q}(\mathbf{R} \times \mathbf{R}^{d})$, $1 \leq p,q \leq \infty$, where $\Phi_r = (\varphi_1, \varphi_2, \ldots, \varphi_r)^T$ with $\varphi_i \in L^{p,q}(\mathbf{R} \times \mathbf{R}^d)$ and $c = (c_1,c_2,\ldots,c_r)^T$ with $c_i \in \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d)$, $i = 1,2,\ldots,r$, under the assumption that the family $\{ \varphi_{i}(x-k_{1},y-k_{2}): (k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}, 1 \leq i \leq r \}$ constitutes a $(p,q)$-frame of $V_{p,q}(\Phi_r)$. First, iterative approximation projection algorithms for two kinds of average sampling functionals are established. Then, we estimate the convergence rates of the corresponding algorithms.

In this paper, a diffusive tumour-immune model is presented. By comparing the effect of Neumann boundary conditions and Dirichlet boundary conditions on the stability of trivial equilibrium, we derive that the former can provide more mechanisms for spatial pattern formation of the model. By taking the diffusion rate of tumour cells as a parameter, we first give the local and global steady-state bifurcations emitting from the positive equilibrium of the model. Then the stability of the bifurcation solution is discussed by computing the second derivative of an appropriate function, which is different from the general case. Furthermore, numerical simulations provide an indication of the wealth of patterns that the system can exhibit. In particular, periodic oscillation and spot-like patterns can be observed in one-dimensional and two-dimensional simulations, respectively. All results obtained reveal the mechanism of interaction between tumour cells and immune system, which have profound significance for the development of tumour immunotherapy.