Related papers: A Pointwise Bound for a Holomorphic Function which…

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$…

Let $D$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$ and let $f$ be a positive continuous function on $\partial D$. Under some assumptions on $\varphi$, it is shown that the problem $\Delta u=2\varphi(u)$ in $D$ and $u=f$ on $\partial…

We show an extention of a theorem of Kaczynski to boundary functions in n-dimensional space. Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the $x$-axis. Suppose that $f$ is a function mapping $H$ into some metric…

For a bounded function $\varphi$ on the unit circle $\mathbb T$, let $T_\varphi$ be the associated Toeplitz operator on the Hardy space $H^2$. Assume that the kernel $$K_2(\varphi):=\{f\in H^2:\,T_\varphi f=0\}$$ is nontrivial. Given a…

The singularities of the $\Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$…

Let $H$ be a Hilbert space of entire functions. Let $H'$ be the space of the functions $f(z)/\prod_i(z-z_i)$ where $f$ belongs to $H$ and vanishes at $n$ given complex points $z_i$. We compute a suitable $E$ function for $H'$ when one is…

This paper focuses on a numerical invariant for local rings of characteristic $p$ called $h$-function, that recovers several important invariants, including the Hilbert-Kunz multiplicity, $F$-signature, $F$-threshold, and $F$-signature of…

The Bekenstein bound takes the holographic principle into the realm of flat space, promising new insights on the relation of non-gravitational physics to quantum gravity. This makes it important to obtain a precise formulation of the bound.…

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined…

Suppose that $M$ is a Riemann surface with boundary $\partial M$, $\Lambda$ is its DN-map, and $\mathscr E:M\to\mathbb{C}^{n}$ % $\mathfrak{J}_{M}$ is a holomorphic immersion. Let $M'$ be diffeomorphic to $M$, $\partial M=\partial M'$; let…

In a previous paper we produced a complex iteration of a holomorphic function $\phi$ in the immediate basin of a fixed point whose multiplier is a real number and in between zero and one. We further explore this problem, allowing the…

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper…

Let $(M,\omega)$ be a Kahler manifold. An integrable function on M is called $\omega^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $\omega^q$-plurisubharmonic function is…

Let $K$ be a global function field. We obtain a set of formulas for the densities of the Kodaira types and Tamagawa numbers of elliptic curves over a completion of $K$ that is independent of the field's characteristic. Furthermore, for a…

Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $\mu$ in the sense that $ M \setminus \text{supp}(\pi_* \mu) \neq \emptyset$…

We consider the number of zeros of holomorphic functions in a bounded domain that depend on a small parameter and satisfy an exponential upper bound near the boundary of the domain and similar lower bounds at finitely many points along the…

This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These…

Let $R$ be a real closed field and $K:=R(i)$ its algebraic closure. Let $U\subset K^n$ be an open and definable set in a fixed o-minimal structure. In this note, we study the relationship between definability of a $K$-holomorphic function…

This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of…

For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose…