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The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at t=0 even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any \delta >0, we find a solution to the heat equation on the whole plane, with exponential growth of order 2+\delta, which is nowhere analytic in time. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]

Characterization of simultaneous similarity for commuting m- tuples of operators is an open problem even in finite-dimensional spaces; known as "A wild problem in linear algebra". In this paper we offer a criterion for simultaneous similarity of m-tuples of k-cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m-tuples of commuting square matrices. [ABSTRACT FROM AUTHOR]

In this paper we characterize the nonlinear superposition operators mapping the space \mathcal {Q}_s into the Dirichlet-type space \mathcal {D}^p_\alpha and answer basically the remaining open cases provided by Galanopoulos, Girela, and Auxiliadora Márquez [J. Math. Anal. Appl. 463 (2018), pp. 659–680]. Meanwhile, we give some sufficient-necessary conditions for the superposition operators between \mathcal {Q}_s spaces in all cases. [ABSTRACT FROM AUTHOR]

In this paper, we study some geometric properties of a class of analytic functions which is defined from the J-fraction expansion of the ratio {zf'(z)}/{f(z)}. We find the disk domain which is mapped into a starlike domain by these functions. Moreover, we study similar results for two different normalized forms of regular Coulomb wave functions and a normalized Bessel function of the first kind by using continued fractions expansions. [ABSTRACT FROM AUTHOR]

This paper deals with the dynamics of a system governed by a multi-dimensional linear hyperbolic PDE. The dynamical behaviors of linear PDEs extremely depend on the selection of space, and a conventional way is to define a infinite-dimensional space with a tuning parameter. Thereby, the linear PDEs can exhibit chaos or stability in the different range of tuning parameter. In this work, the chaos of the C_0-semigroup corresponding to the system is established on the Banach space of multivariate analytic functions when the tuning parameter exceeds some given positive number. Based on this, both Devaney and distributional chaos of the system are further obtained. Meanwhile, the C_0-semigroup is proved to be uniformly exponentially stable when the tuning parameter is less than a certain positive number, which contributes to showing the global stability of the system. Finally, two examples are given to illustrate effectiveness of our results. [ABSTRACT FROM AUTHOR]

Let \Omega (n) and \omega (n) be the number of all prime factors and different prime factors of n, respectively. An elegant result of Alladi showed that the probability of (n,\Omega (n))=1 or (n,\omega (n))=1 is 6/\pi ^2. In the other aspect, denoting by d_k the density of the numbers n with \Omega (n)-\omega (n)=k, then another elegant formula of Rényi states that d_k\sim c_0/2^k as k tends to infinity, where c_0=(1/4)\prod _{p>2}(1-(p-1)^{-2})^{-1}. In this paper, we investigate the numbers n which are relatively prime to \Omega (n)-\omega (n). Let \mathscr {D} denote the set of such numbers. It is proved that there exists a positive constant c_1 such that \mathscr {D}(x)=c_1x+O_{\varepsilon }(\sqrt {x}\log ^{1+\varepsilon }x) for any \varepsilon >0. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a version of multidimensional analogues of the refined Bohr's inequality. Then we establish two versions of multidimensional analogues of improved Bohr's inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr's inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the study of the Bohr phenomenon for a quasisubordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for K-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the structure of the infinitesimal spaces of generic quasiconformal mappings. One of the main results is that for any countable set in the complex plane, the infinitesimal spaces of generic K-quasiconformal mappings at any point in the set is the totality of K-quasiconformal mappings. [ABSTRACT FROM AUTHOR]

In this paper, we correct some errors that appeared in ''The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero'', Proc. Amer. Math. Soc. 139 (2011), no. 1, 1-9. [ABSTRACT FROM AUTHOR]

The logarithmic coefficients γn of an analytic and univalent function f in the unit disk D = {z ∈ C : |z| < 1} with the normalization f(0) = 0 = f'(0) - 1 are defined by log .... Recently, D. K. Thomas [Proc. Amer. Math. Soc. 144 (2016), 1681-1687] proved that |γ3| ≤ 7/12 for functions in a subclass of close-to-convex functions (with argument 0) and claimed that the estimate is sharp by providing a form of an extremal function. In the present paper, we point out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument 0). We also determine a sharp upper bound of |γ3| for close-to-convex functions (with argument 0) with respect to the Koebe function. [ABSTRACT FROM AUTHOR]

Let Γ be a Fuchsian group. Any [μ] in the Teichmüller space T(Γ)determines a quasi-circle fμ(∂D). In this paper, we prove that, for any Fuchsian group Γ of the second kind, the Hausdorff dimension ... is not a real analytic function in T(Γ). [ABSTRACT FROM AUTHOR]

Given a sequence of points in the unit disk, a well known result due to Carleson states that if given any point of the sequence it is possible to interpolate the value one in that point and zero in all the other points of the sequence, with uniform control of the norm in the Hardy space of bounded analytic functions on the disk, then the sequence is an interpolating sequence (i.e. every bounded sequence of values can be interpolated by functions in the Hardy space). It turns out that such a result holds in other spaces. In this short paper we would like to show that for a given sequence it is sufficient to find just one function suitably interpolating zeros as well as ones to deduce interpolation in the Hardy space. The result has an interesting interpretation in the context of model spaces. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to extend in Rn a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in Rn a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues λj, j ϵ N, and a corresponding sequence of eigenfunctions uj, j ϵ N, forming an orthonormal basis of L²(Rn). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S½½ spaces, are characterized through expansions Σj ajuj and the estimates of coefficients aj by the power function, resp. exponential function of λj . [ABSTRACT FROM AUTHOR]

In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three standard methods for studying moduli spaces of abelian varieties. [ABSTRACT FROM AUTHOR]

Let $ f$

Let $mathcal {C}_n$ be a local quasi-analytic subring of the ring of germs of $C^infty $ functions on $mathbb {R}^n$, and let $mathcal {C}={ mathcal {C}_n , nin mathbb {N}}$. We suppose that $mathcal {C}$ is closed under composition. Consider a map $varphi : (mathbb {R}^n, 0)rightarrow (mathbb {R}^k, 0)$ vanishing at zero, where $varphi $ is a $k$-tuple $ (varphi _1,ldots ,varphi _k)$ and $varphi _1,ldots ,varphi _k$ are in $mathcal {C}_n$. Then $varphi $ defines uniquely a map $phi : mathcal {C}_k rightarrow mathcal {C}_n$ by composition, and $phi $ induces a morphism $hat {phi }: hat {mathcal {C}_k }rightarrow hat {mathcal {C}_n}$ between completions. We let $phi _* : frac {hat {mathcal {C}_k} }{mathcal {C}_k }rightarrow frac {hat {mathcal {C}_n }}{mathcal {C}_n }$ be the homomorphism of groups induced by $phi $ and $hat {phi }$ in the obvious manner. In the analytic case, i.e. when each $mathcal {C}_n$ is the ring of germs of real analytic functions, M. Eakin and A. Harris give a condition under which $phi _*$ is injective. In this paper we prove that the same statement does not hold for a quasianalytic system unless this system is analytic. [ABSTRACT FROM AUTHOR]

In this paper, we extend the complex-valued Ray-Singer torsion, introduced by Burghelea-Haller, to compact connected Riemannian manifolds with boundary. We also compare it with the refined analytic torsion. [ABSTRACT FROM AUTHOR]