Let $\Omega \subset {ℂ}^n$ be a bounded, simply connected $ℂ$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^n$ and let $f$ be a function on $\Omega$ which is separately $C^{2{\alpha }_j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2{\alpha }_j-1}$-smooth with respect to $\mathrm{Re}{z}_j$, $\mathrm{Im}{z}_j$). If $f$ is $\alpha$-analytic on $\Omega \setminus f^{-1}\left(0\right)$, then $f$ is $\alpha$-analytic on $\Omega$. The result is well-known for the case ${\alpha }_i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any...

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s ≥ 3. Also let ${\Omega }_{\pi }$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f:\partial {\Omega }_{\pi }\to ℝ$ that can be extended to a real-valued pluriharmonic function in ${\Omega }_{\pi }$.

Let ${𝒫}_n$ denote the class of analytic functions $p\left(z\right)$ of the form $p\left(z\right)=1+c_n{z}^n+c_{n+1}{z}^{n+1}+\cdots$ in the open unit disc $𝕌$. Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p\left(z\right)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨$\alpha u\left(0\right)-\beta u^{\text{'}}\left(0\right)={\sum }_{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)$ ⎩$\gamma u\left(1\right)+\delta u^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\xi }_i\right)$, where ${\xi }_i\in (0,1)$, $a_i,b_i\in (0,\infty )$ (for i ∈ 1,…,m-2) are given constants satisfying $\varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)>0$, $\varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)>0$ and $\Delta :=\left|\begin{array}{cc}-{\sum }_{i=1}^{m-2}{a}_i\psi \left({\xi }_i\right)& \varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)\\ \varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)& -{\sum }_{i=1}^{m-2}{b}_i\varphi \left({\xi }_i\right)\end{array}\right|<0$. We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and...

We consider functionals of a potential energy ${\mathbb{P}}_{\psi }\left(u\right)$ corresponding to $\mathrm{𝑎𝑛}\mathrm{𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}-\mathrm{𝑣𝑎𝑙𝑢𝑒}\mathrm{𝑝𝑟𝑜𝑏𝑙𝑒𝑚}$. We are dealing with $𝑎\mathrm{𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛}\mathrm{𝑜𝑓}𝑎\mathrm{𝑡ℎ𝑖𝑛}\mathrm{𝑎𝑛𝑛𝑢𝑙𝑎𝑟}\mathrm{𝑝𝑙𝑎𝑡𝑒}$ with $\mathrm{𝑁𝑒𝑢𝑚𝑎𝑛𝑛}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}\mathrm{𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠}$. Various types of the subsoil of the plate are described by various types of the $\mathrm{𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒}$ nonlinear term $\psi \left(u\right)$. The aim of the paper is to find a suitable computational algorithm.

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺. ...

The 3x+k function $T_k\left(n\right)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_k\left(·\right)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_k\left(·\right)$. We consider the generating functions $f_{k,m}\left(z\right)={\sum }_{n>0,n\to m}{z}^n$, which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_{k,m}\left(z\right)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold...

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega$ whose boundary $\partial \Omega$ is formed by two circles ${\Gamma }_1$, ${\Gamma }_2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho$, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in W_0^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^n$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^n$ with smooth boundary, and $ℝ\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

We consider linear elliptic equations $-\Delta u+q\left(x\right)u=\lambda u+f$ in bounded Lipschitz domains $D\subset {ℝ}^N$ with mixed boundary conditions $\partial u/\partial n=\sigma \left(x\right)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma \left(x\right)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and ${\int }_{D}q\left(x\right)dx>0$ then there exist two eigenvalues ${\lambda }_{-1}$, ${\lambda }_1$ such the positivity...

Per ogni soluzione della (1) nel dominio limitato $\mathrm{\Omega }$,, appartenente a $H_0^2(\mathrm{\Omega })$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^0$ del contorno; si consente a $\partial \mathrm{\Omega }$ di essere singolare in $x^0$.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

We study questions related to exceptional sets of pluri-Green potentials $V_{\mu }$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{\mu }$ are defined by $V_{\mu }\left(z\right)={\int }_{B}log(1/|{\varphi }_z\left(w\right)\left|\right)d\mu \left(w\right)$, where for a fixed z ∈ B, ${\varphi }_z$ denotes the holomorphic automorphism of B satisfying ${\varphi }_z\left(0\right)=z$, ${\varphi }_z\left(z\right)=0$ and $\left({\varphi }_z\circ {\varphi }_z\right)\left(w\right)=w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.