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We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy kernel with the unknown. Even though the initial datum is bounded and compactly supported, the velocity field may have unbounded divergence. The proof is based on the compactness property of quasiconformal mappings.

We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma$, where $\mathbf{D}$ is the unit disk and $|\Sigma|=\pi R^2$, then $|F_x(z)|(1-|z|^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine disk, and $F$ is linear up to a M\"obius transformation of the unit disk., Comment: 6 pages

Examples by Poletsky and the author and by Zwonek show the existence nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions and contains multiple sheets. We will explain this phenomenon by fine analytic continuation of the function over part of a Cantor-type set involved. This gives more information on the hull, and allows for weakening and effectiveness of the conditions in the original examples., Comment: 12 pages. In version 2 quite some annoying typo's and mistakes were corrected. In version 3 a serious gap has been filled

The goal of this note is to extend the result bounding from bellow the minimal possible growth of frequently oscillating subharmonic functions to a larger class of functions that carry similar properties. We refine and find further applications for the technique presented by Jones and Makarov in their celebrated paper, Density properties of harmonic measure., arXiv admin note: text overlap with arXiv:2004.06148

A subset ${\mathcal D}$ of a domain $\Omega \subset {\mathbb C}^d$ is determining for an analytic function $f:\Omega \to \overline {{\mathbb D}}$ if whenever an analytic function $g:\Omega \rightarrow \overline {{\mathbb D}}$ coincides with f on ${\mathcal D}$ , equals to f on whole $\Omega $ . This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any $N\geq 1$ , a set consisting of $N^2-N+1$ many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.

In a previous paper, we introduced the Collatz polynomials $P_N(z)$, whose coefficients are the terms of the Collatz sequence of the positive integer $N$. Our work in this paper expands on our previous results, using the Enestr\"om-Kakeya Theorem to tighten our old bounds of the roots of $P_N(z)$ and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $\{z\in\mathbb{C}: |z| = 2\}$ are rare: the set of $N$ such that $P_N(z)$ has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study., Comment: 8 pages, 1 figure

We show that the singularities of the twisted Kähler--Einstein metric arising as the long-time solution of the Kähler--Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics is intimately related to the holomorphic sectional curvature of the reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross--Tosatti--Zhang with explicit constants appearing in the divisorial pole., 16 pages, correction of mistakes in the previous version, comments are welcome

The scattering transform network (STN), which has a similar structure as that of a popular convolutional neural network except its use of predefined convolution filters and a small number of layers, can generates a robust representation of an input signal relative to small deformations. We propose a novel Monogenic Wavelet Scattering Network (MWSN) for 2D texture image classification through a cascade of monogenic wavelet filtering with nonlinear modulus and averaging operators by replacing the 2D Morlet wavelet filtering in the standard STN. Our MWSN can extract useful hierarchical and directional features with interpretable coefficients, which can be further compressed by PCA and fed into a classifier. Using the CUReT texture image database, we demonstrate the superior performance of our MWSN over the standard STN. This performance improvement can be explained by the natural extension of 1D analyticity to 2D monogenicity.

Given a compact Sasaki manifold, we endow the space of the Sasaki potentials with an analogue of Mabuchi metric. We show that its metric completion is negatively curved in the sense of Alexandrov., 20 pages, Publicacions Matem\`atiques

Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $\Omega$ with prescribed behaviour near the weighted set $A$. In particular we prove uniform continuity of the exponential Green function in both variables $(z,\mathcal A)$ in the metric space $\bar \Omega \times \mathcal F$, where $\mathcal F$ is a suitable family of sets of weighted poles in $\Omega \times ]0,+ \infty[$ endowed with the Hausdorff distance. Moreover we give a precise estimates on its modulus of continuity. Our results generalize and improve previous results concerning the pluricomplex Green function du to P. Lelong., Comment: 31 pages

We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension 1 (i.e. a double fixed point) under conjugacy. These generic unfolding depend on one real parameter. The classification is done by assigning to each such germ a weak and a strong modulus, which are unfoldings of the modulus assigned to the antiholomorphic parabolic point. The weak and the strong moduli are unfoldings of the \'Ecalle-Voronin modulus of the second iterate of the germ which is a real unfolding of a holomorphic parabolic point. A preparation of the unfolding allows to identify one real analytic canonical parameter and any conjugacy between two prepared generic unfoldings preserves the canonical parameter. We also solve the realisation problem by giving necessary and sufficient conditions for a strong modulus to be realized. This is done simultaneously with solving the probem of the existence of an antiholomorphic square root to a germ of generic analytic unfolding of a holomorphic parabolic germ. As a second application we establish the condition for the existence of a real analytic invariant curve., Comment: 43 pages, 18 figures

We study the Monge-Amp\` ere operator within the framework of Dinh-Sibony's intersection theory defined via density currents. We show that if $u$ is a plurisubharmonic function belonging to the Blocki-Cegrell class, then the Dinh-Sibony $n$-fold self-product of $\text{dd}^c u$ exists and coincides with the classically defined Monge-Amp\`ere measure $(\text{dd}^c u)^n$., Comment: Improved results. Revised presentation following referee's remarks. To appear in Indiana University Mathematics Journal

We find the constant $b_{\infty}$ ($b_{\infty} \approx 4.81058280$) such that if a complex polynomial or entire function $f(z) = \sum_{k=0}^ \omega a_k z^k, $ $\omega \in \{2, 3, 4, \ldots \} \cup \{\infty\},$ with nonzero coefficients satisfy the conditions $\left|\frac{a_k^2}{a_{k-1} a_{k+1}}\right| >b_{\infty} $ for all $k =1, 2, \ldots, \omega-1,$ then all the zeros of $f$ are simple. We show that the constant $b_{\infty}$ in the statement above is the smallest possible. We also obtain an analog of Hutchinson's theorem for polynomials or entire functions with real nonzero coefficients.

We consider an embedding of planar maps into an equilateral triangle $\Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov's proof of Cardy's formula. Under the Cardy embedding, the planar map induces a metric and an area measure on $\Delta$ and a boundary measure on $\partial \Delta$. We prove that for uniformly sampled triangulations, the metric and the measures converge jointly in the scaling limit to the Brownian disk conformally embedded into $\Delta$ (i.e., to the $\sqrt{8/3}$-Liouville quantum gravity disk). As part of our proof, we prove scaling limit results for critical site percolation on the uniform triangulations, in a quenched sense. In particular, we establish the scaling limit of the percolation crossing probability for a uniformly sampled triangulation with four boundary marked points., Comment: 66 pages, 13 figures. Revised according to referee report. Accepted for publication in Acta Mathematica

We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities., Revision based on referee's comments

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=\mu_j(z,\omega) \partial F_j(z),$ where the random dilatation satisfies $|\mu_j|\leq k, Comment: 56 pages

Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 < \rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms $\Xi_{\rho,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles., Comment: 24 pages, no figures, comments are very welcome! v3: several minor corrections, the main application is now stated for strongly positive forms

We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a $d$-dimensional version of a question posed in \cite{BPS22} in the affirmative., 10pp

We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of the same topology. Moreover, we show that this is also true for surfaces as above with non-empty boundary and that the corresponding map can be chosen in a canonical way. Our proof is based on a local argument involving the existence of quasisymmetric parametrizations for metric discs as shown in a paper of Lytchak and Wenger.

Building on H. Tran's study of Loewner hulls generated by complex-valued driving functions, which showed the existence of a phase transition, we answer the question of whether the phase transition for complex-driven hulls matches the phase transition for real-driven hulls. This is accomplished through a detailed study of the Loewner hulls generated by driving functions $c\sqrt{1-t}$ and $c\sqrt{\tau + t}$ for $c \in \mathbb{C}$ and $\tau \geq 0$. This family also provides examples of new geometric behavior that is possible for complex-driven hulls but prohibited for real-driven hulls.

Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in $\{-1,1\}$). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $\text{ADF}(f)$, is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $\text{CDF}(f,g)$, is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of $\text{CDF}(f,g)$ and the geometric mean of $\text{ADF}(f)$ and $\text{ADF}{(g)}$ must be at least $1$. For randomly selected pairs of long binary sequences, this quantity is typically around $2$. In this paper, we show that Pursley and Sarwate's bound is met for binary sequences precisely when $(f,g)$ is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families., 37 pages

We provide a fine classification of rigid hyperelliptic manifolds in dimension four up to biholomorphism and diffeomorphism. These manifolds are explicitly described as finite \'etale quotients of a product of four Fermat elliptic curves., Comment: 22 pages, MAGMA code on personal homepage

It is well-known that for any inner function $\theta$ defined in the unit disk $D$ the following two conditons: $(i)$ there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim_{n \to \infty} \theta(z) p_n(z) = 1$ for all $z \in D$, and $(ii)$ $\sup_n \| \theta p_n \|_\infty < \infty$, are incompatible, i.e., cannot be satisfied simultaneously. In this note we discuss and apply a consequence of a result by Thomas Ransford, which shows that if we relax the second condition to allow for arbitrarily slow growth of the sequence $\{ \theta(z) p_n(z)\}_n$ as $|z| \to 1$, then condition $(i)$ can be met. In other words, every growth class of analytic functions contains cyclic singular inner functions. We apply this observation to properties of decay of Taylor coefficients and moduli of continuity of functions in model spaces $K_\theta$. In particular, we establish a variant of a result of Khavinson and Dyakonov on non-existence of functions with certain smoothness properties in $K_\theta$, and we show that the classical Aleksandrov theorem on density of continuous functions in $K_\theta$, and its generalization to de Branges-Rovnyak spaces $\mathcal{H}(b)$, is essentially sharp., Comment: This is a first version of the paper. Some updates are likely in near future. Any comments are highly appreciated

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models., 29 pages

We extend Quine's bound on the number of self-intersection of curves with polynomial parameterization to the case of Laurent polynomials. As an application, we show that circle embeddings are dense among all maps from a circle to a plane with respect to an integral norm., Comment: 10 pages

The generalized Bohr radius $R_{p, q}(X), p, q\in[1, \infty)$ for a complex Banach space $X$ was introduced by Blasco in 2010. In this article, we determine the exact value of $R_{p, q}(\mathbb{C})$ for the cases (i) $p, q\in[1, 2]$, (ii) $p\in (2, \infty), q\in [1, 2]$ and (iii) $p, q\in [2, \infty)$. Moreover, we consider an $n$-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal{H})$ for an infinite dimensional complex Hilbert space $\mathcal{H}$, (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to\infty$ for finite dimensional $X$. We also study the multidimensional analogue of a related concept called the $p$-Bohr radius, introduced by Djakov and Ramanujan in 2000. In particular, we obtain the asymptotic value of the $n$-dimensional $p$-Bohr radius for bounded complex-valued functions, and in the vector-valued case we provide a lower estimate for the same, which is independent of $n$. In a similar vein, we investigate in detail the multidimensional $p$-Bohr radius problem for functions with positive real part. Towards the end of this article, we pose one more generalization $R_{p, q}(Y, X)$ of $R_{p, q}(X)$-considering functions that map the open unit ball of another complex Banach space $Y$ inside the unit ball of $X$, and show that the existence of nonzero $R_{p, q}(Y, X)$ is governed by the geometry of $X$ alone., Comment: 18 pages

We establish analogues of Liouville's theorem in the complex function theory, with the differential operator replaced by various difference operators. This is done generally by the extraction of (formal) Taylor coefficients using a residue map which measures the obstruction having local "anti-derivative". The residue map is based on a Weyl algebra or $q$-Weyl algebra structure satisfied by each corresponding operator. This explains the different senses of "boundedness" required by the respective analogues of Liouville's theorem in this article., 17 pages

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.

The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painleve-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement., Comment: 138 pages, 45 figures

The main goal of the paper is to parametrize the Weyl matrix balls associated with an arbitrary matricial truncated Hamburger moment problem. For the special case of a non-degenerate matricial truncated Hamburger moment problem the corresponding Weyl matrix balls were computed by I. V. Kovalishina in the framework of V. P. Potapov's method of "Fundamental matrix inequalities".

In this paper, the class of (complex) quasi-Herglotz functions is introduced as the complex vector space generated by the convex cone of ordinary Herglotz functions. We prove characterization theorems, in particular, an analytic characterization. The subclasses of quasi-Herglotz functions that are identically zero in one half-plane as well as rational quasi-Herglotz functions are investigated in detail. Moreover, we relate to other areas such as weighted Hardy spaces, definitizable functions, the Cauchy transform on the unit circle and sum-rule identities., Comment: 35 pages, 4 figures

The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,b} \colon z \mapsto ae^z+be^{-z}$ for $a,b\in \mathbb{C}^\ast$, consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and \textit{split} at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$. We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$, and conclude that no two of its dynamic rays land together., 33 pages, 5 figures. V2: Author accepted manuscript. To appear in Math. Proc. Camb. Philos. Soc

Given two Riemann surfaces with boundary and a homotopy class of topological embeddings between them, there is a conformal embedding in the homotopy class if and only if the extremal length of every simple multi-curve is decreased under the embedding. Furthermore, the homotopy class has a conformal embedding that misses an open disk if and only if extremal lengths are decreased by a definite ratio. This ratio remains bounded away from one under covers., 32 pages, 6 figures; v3: New Section 3.4, improved Example 4.4, other improvements throughout

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in $\mathbb{C}$ and tubular neighborhoods of compact totally real submanifolds in general in $\mathbb{C}^n$, $n \geq 2$, are given. The contractibility of a hyperbolic domain with contracting holomorphic mapping is explained., Comment: 18 pages, 1 diagram

The main purpose of this article is to present a localization of Forelli's theorem for the functions holomorphic along a standard suspension of linear discs. This generalizes one of the main results of \cite{CK21} and the original Forelli's theorem., 10 pages, 0 figure. The article is to appear in Complex Variables and Elliptic Equations

We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular, we show that under quite general conditions a semigroup of rational functions is left amenable if and only if it is a subsemigroup of the centralizer of some rational function., Comment: The extended and polished version

We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by "simple" arithmetic conditions. We explore some examples and give an upper bound for the dimension of this subspace. As an application, we find explicit expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable., 26 pages. Notations have been improved. Final version to appear in Journal of Number Theory

The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by\[ \dfrac{\mathrm{d}^2u}{\mathrm{d}x^2}=\dfrac{1}{u}\left( \dfrac{\mathrm{d}u}{\mathrm{d}x} \right)^2-\dfrac{1}{x} \dfrac{\mathrm{d}u}{\mathrm{d}x}+\dfrac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \quad \alpha,\beta \in \mathbb{C}.\] Starting from a generic initial solution $u_0(x)$ corresponding to parameters $\alpha, \beta$, denoted as the triple $(u_0(x),\alpha,\beta)$, we apply an explicit B\"acklund transformation to generate a family of solutions $(u_n(x),\alpha+4n,\beta+4n)$ indexed by $n\in \mathbb{N}$. We study the large $n$ behavior of the solutions $(u_n(x),\alpha+4n,\beta+4n)$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate Painlev\'e-III equation, known as Painlev\'e-III($D_8$), \[\dfrac{\mathrm{d}^2U}{\mathrm{d}z^2}=\dfrac{1}{U}\left(\dfrac{\mathrm{d}U}{\mathrm{d}z}\right)^2-\dfrac{1}{z} \dfrac{\mathrm{d}U}{\mathrm{d}z}+\dfrac{4U^2 + 4}{z}.\] A notable application of our result is to rational solutions of Painlev\'e-III($D_6$), which are constructed using the seed solution $(1,4m,-4m)$ where $m\in \mathbb{C} \setminus (\mathbb{Z} + \frac{1}{2})$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well-defined, and by its monodromy data in general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlev\'e-III, both $D_6$ and $D_8$ at $z = 0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z = 0$., Comment: 69 pages, 12 figures

We study the central limit distribution $\mu$ for V-monotone independence. Using its Cauchy--Stieltjes transform, we prove that $\mu$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$ and we give its density $\rho$ in an implicit form. We present a computer generated graph of $\rho$.

Burgers' equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers' equation shows an infinite stream of simple poles born at t = 0^+, emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for t > 0. We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations., 30 pages, 6 figures

Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|0$ for $z\in\mathbb{D}$. In the present article, we determine the sharp estimate of the pre-Schwarzian and Schwarzian norms for functions in the class $\mathcal{S}_{\alpha}$.

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) \Delta_cf(z) + b [\Delta_cf(z)]^2 = e^{g(z)}, \end{align*} where $ a, \omega, b\in\mathbb{C} $, $ g $ is a polynomial in $ \mathbb{C}^n $ and $ \Delta_cf(z)=f(z+c)-f(z) $. The main results of the paper improve several existence results in $ \mathbb{C}^n $ for integer $ n\geq 2 $ and $ 1\leq i, Comment: 20. arXiv admin note: text overlap with arXiv:2307.05549

Let $Z_1,\, Z_2,\dots$ be independent and identically distributed complex random variables with common distribution $\mu$ and set $$ P_n(z) := (z - Z_1)\cdots (z - Z_n)\,. $$ Recently, Angst, Malicet and Poly proved that the critical points of $P_n$ converge in an almost-sure sense to the measure $\mu$ as $n$ tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed $k\in \mathbb{N}$, the empirical measure of zeros of the $k$th derivative of $P_n$ converges to $\mu$ in the almost sure sense, as conjectured by Angst-Malicet-Poly., Comment: 8 pages

We introduce an analog of the Kobayashi-Royden metric on a Riemannian manifold and study its basic properties., 11 pp

Given a proper holomorphic surjective morphism $f:X\rightarrow Y$ from a compact K\"ahler manifold to a compact K\"ahler manifold, and a Nakano semipositive holomorphic vector bundle $E$ on $X$, we prove Koll\'ar type vanishing theorems on cohomologies with coefficients in $R^qf_\ast(\omega_X(E))\otimes F$, where $F$ is a $k$-positive vector bundle on $Y$. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane-Takayama, and an $L^2$-Dolbeault resolution of the higher direct image sheaf $R^qf_\ast(\omega_X(E))$, which is of interest in itself., Comment: 11 pages. Comments are welcome

In this paper, we start by introducing the modified Bergman-Dirichlet space $\mathcal D_m^2(\mathbb D_R,\mu^R_{\alpha,\beta})$ and then we study its asymptotic behavior when the parameter $\alpha$ goes to infinity and to $(-1)$ to obtain respectively the modified Bargmann-Dirichlet and the modified Hardy-Dirichlet spaces with their reproducing kernels. Finally, we give some examples of Segal-Bargmann transforms of those spaces., Comment: 19 pages

We extend representation formulas that generalize the similarity principle of solutions to the Vekua equation to certain classes of meta-analytic functions. Also, we solve a generalization of the higher-order Schwarz boundary value problem in the context of meta-analytic functions with boundary conditions that are boundary values in the sense of distributions., Submitted

A Sasakian manifold is a Riemannian manifold whose metric cone admits a certain K\"ahler structure which behaves well under homotheties. We show that the product of two compact Sasakian manifolds admits a family of complex structures indexed by a complex nonreal parameter, none of whose members admits any compatible locally conformally K\"ahler metrics if both Sasakian manifolds are of dimension greater than $1$. We compare this family with another family of complex structures which has been studied in the literature. We compute the Dolbeault cohomology groups of these products of compact Sasakian manifolds.

In this paper, the author solves the long term open problem of Kerzman on sup-norm estimate for Cauchy-Riemann equation on polydisc in $n$-dimensional complex space. The problem has been open since 1971. He also extends and solves the problem on a bounded product domain $\Omega^n$, where $\Omega$ is any bounded domain in $\mathbb{C}$ with $C^{1,\alpha}$ boundary for some $\alpha>0$., Comment: arXiv admin note: substantial text overlap with arXiv:2211.01507