Related papers: Liouville theorem with parameters: asymptotics of …
We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear…
Let $K$ be a field of characteristic $p>0$ and let $f(t_1,...,t_d)$ be a power series in $d$ variables with coefficients in $K$ that is algebraic over the field of multivariate rational functions $K(t_1,...,t_d)$. We prove a generalization…
Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in…
In this work, we introduce the notion of Gevrey asymptotic expansion and we show how the classical concept of a convergent power series can be generalized to include the case in which the radius of convergence is zero. This technique can be…
Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…
We construct Euclidean Liouville conformal field theories in odd number of dimensions. The theories are nonlocal and non-unitary with a log-correlated Liouville field, a ${\cal Q}$-curvature background, and an exponential Liouville-type…
We consider a three-dimensional Fourier integral in which the exponent in the exponential factor is the product of some phase function and a large parameter. The asymptotics of this integral is sought when the large parameter tends to…
In this paper, under certain restrictions on linear factors of the denominator of a rational function of two variables, the leading term of the asymptotic expansion of the coefficients is found.
We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with $2\pi$-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation…
We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an $L^{p}$ Liouville type theorem which is a quantitative integral $L^{p}$ estimate of harmonic functions analogous to Karp's…
A model of a relativistic particle moving in the Liouville field is investigated. Symmetry group of the system is $SL(2,R)/Z_2$. The corresponding dynamical integrals describe full set of classical trajectories. Dynamical integrals are used…
We consider the asymptotic expansion of the functional series \[S_{\mu}^\pm(a;\lambda)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-\lambda n}}{(n^2+a^2)^\mu}\] for $\lambda>0$ and $\mu\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<\pi/2$. The…
In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u(y)=\intpr \frac{…
We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey asymptotics with respect to a parameter. The unique implicitly defined solution admits a Gevrey asymptotic expansion and furthermore it is the Borel resummation of…
We establish a quantitative approximation formula of the Lyapunov exponent of a rational function of degree more than one over an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and possibly…
In this paper, we study the equation $\mathcal{L} u=0$ in $\mathbb{R}^N$, where $\mathcal{L}$ belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this…
In this paper we study elementary extensions of differential fields in prime characteristic. In particular, we show that, in contrast to Liouville's result in characteristic zero, all elements of an elementary extension admit an…
We present an asymptotic evaluation unitary formula for large argument values existing for defined class of functions. The asymptotic evaluation is obtained using only power series expansion coefficients of a function, what is a new result…
We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the…
According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension…