The purpose of this article is to explore the solutions of the following nonlinear partial differential equations:

\begin{equation*}\mathcal{P}_1(u)^2+\mathcal{P}_2(u)^2=e^{g}\end{equation*}

and

\begin{equation*}\mathcal{P}_1(u)^2+2\alpha \mathcal{P}_1(u)\mathcal{P}_2(u)+\mathcal{P}_2(u)^2=e^{g},\end{equation*}

where $\alpha^2\in \mathbb{C}\setminus\{0,1\}$, g(z) is a polynomial, $a_j,b_j,c_j(j=1,2)$ are constants in $\mathbb{C}$, and

\begin{equation*}\mathcal{P}_1(u)=a_1 u+b_1 u_{z_1}+c_1u_{z_2}\quad \text{and} \quad \mathcal {P}_2(u)=a_2 u+b_2u_{z_1}+c_2u_{z_2}.\end{equation*}

The description of the existence conditions and the forms of the solutions for the above partial differential equations demonstrate that our results improve and generalise the previous results given by Saleeby, Cao and Xu. Moreover, some of our examples corresponding to every case in our theorems reveal the significant difference in the order of solutions for equations from a single variable to several variables.

We show that the set of Julia limiting directions of a transcendental-type K-quasiregular mapping $f:\mathbb {R}^n\to \mathbb {R}^n$ must contain a component of a certain size, depending on the dimension n, the maximal dilatation K, and the order of growth of f. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. We also show that if every component of the set of Julia limiting directions is a point, then f has infinite order. The main tool in proving these results is a new version of a Phragmén–Lindelöf principle for sub-F-extremals in sectors, where we allow for boundary growth of the form $O( \log |x| )$ instead of the previously considered $O(1)$ bound.

This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by

\begin{equation*}L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0.\end{equation*}

$ L_k[f] $

$ f^n+g^n=1 $

Macroscopically, a Darcian unsaturated moisture flow in the top soil is usually represented by an one-dimensional volume scale of evaporation from a static water table. On the microscale, simple pore-level models posit bundles of small-radius capillary tubes of a constant circular cross-section, fully occupied by mobile water moving in the Hagen–Poiseuille (HP) regime, while large-diameter pores are occupied by stagnant air. In our paper, cross-sections of cylindrical pores are polygonal. Steady, laminar, fully developed two-dimensional flows of Newtonian water in prismatic conduits, driven by a constant pressure gradient along a pore gradient, are more complex than the HP formula; this is based on the fact that the pores are only partially occupied by water and immobile air. The Poisson equation in a circular tetragon, with no-slip or mixed (no-shear-stress) boundary conditions on the two adjacent pore walls and two menisci, is solved by the methods of complex analysis. The velocity distribution is obtained via the Keldysh–Sedov type of singular integrals, and the flow rate is evaluated for several sets of meniscus radii by integrating the velocity over the corresponding tetragons.

Let $\mu $ be a finite positive Borel measure on $[0,1)$ and $f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} \in H(\mathbb {D})$. For $0<\alpha <\infty $, the generalized Cesàro-like operator $\mathcal {C}_{\mu ,\alpha }$ is defined by

$$ \begin{align*}\mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, \end{align*} $$

where, for $n\geq 0$, $\mu _n$ denotes the nth moment of the measure $\mu $, that is, $\mu _n=\int _{0}^{1} t^{n}d\mu (t)$.

For $s>1$, let X be a Banach subspace of $H(\mathbb {D})$ with $\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$. In this article, for $1\leq p <\infty $, we characterize the measure $\mu $ for which $\mathcal {C}_{\mu ,\alpha }$ is bounded (resp. compact) from X into the analytic Besov space $B_{p}$.

In this article, we describe meromorphic solutions of certain partial differential equations, which are originated from the algebraic equation $P(f,g)=0$, where P is a polynomial on $\mathbb {C}^2$. As an application, with the theorem of Coman–Poletsky, we give a proof of the classic theorem: Every meromorphic solution $u(s)$ on $\mathbb {C}$ of $P(u,u')=0$ belongs to W, which is the class of meromorphic functions on $\mathbb {C}$ that consists of elliptic functions, rational functions and functions of the form $R(e^{a s})$, where R is rational and $a\in \mathbb {C}$. In addition, we consider the factorization of meromorphic solutions on $\mathbb {C}^n$ of some well-known PDEs, such as Inviscid Burgers’ equation, Riccati equation, Malmquist–Yosida equation, PDEs of Fermat type.

In this note, we give a new necessary condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterize the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak {D}_{\vec {a}}(\mathbb {D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb {D}^2$ of the form $\Phi (z_1,z_2)=(\phi _1(z_1),\phi _2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\Phi }:\mathfrak {D}(\mathbb {D}^2)\to A^2(\mathbb {D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the Dirichlet space of the bidisc.

In sharp contrast to the Hardy space case, the algebraic properties of Toeplitz operators on the Bergman space are quite different and abnormally complicated. In this paper, we study the finite-rank problem for a class of operators consisting of all finite linear combinations of Toeplitz products with monomial symbols on the Bergman space of the unit disk. It turns out that such a problem is equivalent to the problem of when the corresponding finite linear combination of rational functions is zero. As an application, we consider the finite-rank problem for the commutator and semi-commutator of Toeplitz operators whose symbols are finite linear combinations of monomials. In particular, we construct many motivating examples in the theory of algebraic properties of Toeplitz operators.

In the early 1900s, Maillet [Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions (Gauthier–Villars, Paris, 1906)] proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimensions turns out to fail. In fact, using a result by Kleinbock and Margulis [‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148(1) (1998), 339–360], we show that among analytic matrix functions in dimension $n\ge 2$, Maillet’s invariance property is only true for Möbius transformations with special coefficients. This implies that the analogue in higher dimensions of an open question of Mahler on the existence of transcendental entire functions with Maillet’s property has a negative answer. However, extending a topological argument of Erdős [‘Representations of real numbers as sums and products of Liouville numbers’, Michigan Math. J. 9 (1962), 59–60], we prove that for any injective continuous self-mapping on the space of rectangular matrices, many Liouville matrices are mapped to Liouville matrices. Dropping injectivity, we consider setups similar to Alniaçik and Saias [‘Une remarque sur les $G_{\delta }$-denses’, Arch. Math. (Basel) 62(5) (1994), 425–426], and show that the situation depends on the matrix dimensions $m,n$. Finally, we discuss extensions of a related result by Burger [‘Diophantine inequalities and irrationality measures for certain transcendental numbers’, Indian J. Pure Appl. Math. 32 (2001), 1591–1599] to quadratic matrices. We state several open problems along the way.

We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax–Schanuel and André–Oort-type theorems in genus zero. As a consequence, one obtains several equivalent characterizations of bi-algebraic varieties. It follows that bi-algebraic varieties in genus zero are foliated by affine-linear varieties. Furthermore, bi-algebraic varieties with constant residues in strata with only simple poles are affine-linear. In addition, we produce infinitely many new linear varieties in strata of genus zero, including infinitely many new examples of meromorphic Teichmüller curves.

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the logarithmic derivative $df/f$ of the polynomial. We determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by the framing of the punctured disk induced by the horizontal foliation on the translation surface.

We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splitting with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the geometrization conjecture by Perelman. In particular, we give a proof of Maher’s result that random 3-manifolds are hyperbolic that bypasses geometrization.

The article deals with isometric dilation and commutant lifting for a class of n-tuples $(n\ge 3)$ of commuting contractions. We show that operator tuples in the class dilate to tuples of commuting isometries of BCL type. As a consequence of such an explicit dilation, we show that their von Neumann inequality holds on a one-dimensional variety of the closed unit polydisc. On the basis of such a dilation, we prove a commutant lifting theorem of Sarason’s type by establishing that every commutant can be lifted to the dilation space in a commuting and norm-preserving manner. This further leads us to find yet another class of n-tuples $(n\ge 3)$ of commuting contractions each of which possesses isometric dilation.

In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface X of genus g is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of X. In this survey, we give a detailed account of Mirzakhani’s proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.

M. Heins demonstrated that any finite Blaschke product defined on the open unit disc, provided it has at least one finite pole, possesses a nonzero residue. In this work, we extend Heins’ result by generalizing the class of functions under consideration. Specifically, we prove that a broader class of rational functions, defined on certain star-shaped domains in the complex plane, also exhibits this nonzero residue property. This class includes, as a special case, the family of finite Blaschke products. Our findings contribute to a better understanding of the analytic behavior of rational functions on more complex domains, opening new avenues for exploration in this area.

Let $\mathcal {H}$ be the class of all analytic self-maps of the open unit disk $\mathbb {D}$. Denote by $H^n f(z)$ the nth-order hyperbolic derivative of $f\in \mathcal H$ at $z\in \mathbb {D}$. We develop a method allowing us to calculate higher-order hyperbolic derivatives in an expeditious manner. We also generalise certain classical results for variability regions of the nth derivative of bounded analytic functions. For $z_0\in \mathbb {D}$ and $\gamma = (\gamma _0, \gamma _1 , \ldots , \gamma _{n-1}) \in {\mathbb D}^{n}$, let ${\mathcal H} (\gamma ) = \{f \in {\mathcal H} : f (z_0) = \gamma _0,H^1f (z_0) = \gamma _1,\ldots ,H^{n-1}f (z_0) = \gamma _{n-1} \}$. We determine the variability region $\{ f^{(n)}(z_0) : f \in {\mathcal H} (\gamma ) \}$ to prove a Schwarz–Pick lemma for the nth derivative. We apply this result to establish an nth-order Dieudonné lemma, which provides an explicit description of the variability region $\{h^{(n)}(z_0): h\in \mathcal {H}, h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,\ldots , h^{(n-1)}(z_0)=w_{n-1}\}$ for given $z_0$, $w_0$, $w_1,\ldots ,w_{n-1}$. Moreover, we determine the form of all extremal functions.

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim toward applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into $\mathbb {R}$-trees have Lipschitz free spaces isomorphic to $L^1$-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into $\mathbb {R}$-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to $\ell ^1$, (2) the Lipschitz free space over hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-space and (3) every infinite, finitely generated hyperbolic group stochastically embeds into an $\mathbb {R}$-tree, has Lipschitz free space isomorphic to $\ell ^1$, and admits a proper, uniformly Lipschitz affine action on $\ell ^1$.

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion. We consider mappings of domains of the Euclidean space that satisfy weighted Poletsky inequality. Assume that, the definition domain is finitely connected on its boundary and, in addition, on the set of all points which are pre-images of the cluster set of this boundary. Then the specified mappings have a continuous boundary extension provided that the majorant in the Poletsky inequality satisfies some integral divergence condition, or has a finite mean oscillation at every boundary point.