On the existence and smoothness of radially symmetric solutions of a BVP for a class of nonlinear, non-Lipschitz perturbations of the Laplace equation

The existence of radially symmetric solutions $u(x;a)$ to the Dirichlet problems $$\Delta u(x)+f(|x|,u(x),|\nabla u(x)|)=0\qquad x\in B,\ u|_\Gamma=a\in{\mathbb{R}}\ (\Gamma:=\partial B)$$ is proved, where $B$ is the unit ball in ${\mathbb{R}}^n$ centered at the origin $(n\ge2)$, $a$ is arbitrary $(a>a_0\ge-\infty);f$ is positive, continuous and bounded. It is shown that these solutions belong to $C^2(\overline{B})$. Moreover, in the case $f\in C^1$ a sufficient condition (near necessary) for the smoothness property $u(x;a)\in C^3(\overline{B})\quad\forall a>a_0$ is also obtained

Molecular analysis of S-haplotypes in peach, a self-compatible Prunus species

The most commercially grown peach [Prunus persica (L.) Batsch.] cultivars do not require cross-pollination for reasonable fruit set; however, self-incompatibility is a well-known feature within the Prunoideae subfamily. Isoelectric focusing and native polyacrylamide gel electrophoresis of S-ribonucleases; PCR analyses of S-RNase and S-haplotype-specific F-box genes as well as DNA sequencing were carried out to survey the self- (in)compatibility allele pool and to uncover the nature of self-compatibility in peach. From 25 cultivars and hybrids with considerable diversity in phenotype and origin, only two S-haplotypes were detected. Allele identity could be checked by exact length determination of the PCR-amplified fragments and/or partial sequencing of the peach S-1-, S-2-, and Prunus davidiana (Carr.) Franch. S-1 RNases. S-RNases of peach were detected to possess ribonuclease activity, and a single nucleotide polymorphism in the S,-RNase was shown, which represents a synonymous substitution and does not change the amino acid present at the position in the protein. A 700-bp fragment of the peach SFB gene was PCR-amplified, which is similar to the fragment size of functional Prunus L. SFBs. All data obtained in this study may support the contribution of genes outside the S-locus to the self-compatible phenotype of peaches

Observability of string vibrations

Transversal vibrations $u=u(x,t)$ of a string of length $l$ under three essential boundary conditions are studied, where $u$ is governed by the Klein--Gordon equation: $$u_{tt}(x,t) = a^2u_{xx}(x,t) - cu(x,t), (x,t) \in [0,l]\times \mathbb{R}; \ 0 < a, c \in \mathbb{R}.$$ Sufficient conditions are obtained that guarantee the unique solvability of a general observation problem with the given state functions $f, g \in D^s(0,l), s \in \mathbb{R}$ at two distinct instants of time $-\infty < t_1 < t_2 < \infty$: \begin{array} \displaystyle A_1 u|_{t=t_1} + B_1 u_t|_{t=t_1} = f, & |A_1|+|B_1| > 0, \ A_1 B_1\geq 0, \\ \displaystyle A_2 u|_{t=t_2} + B_2 u_t|_{t=t_2} = g, & |A_2|+|B_2| > 0, \ A_2 B_2\leq 0. \end{array} Here $s$ is arbitrary, the space $D^s(0,l)$ (see [2] and [13]) is some subspace of the Sobolev space $H^s(0,l)$. The essential condition of the solvability is that $(t_2-t_1) a/l$ is a rational number. In fact, this result is a consequence of a general observability result related to the vibration $u = u(x,t)$ governed by the equation $$u_{tt} = (p(x) u_{x})_x - q(x) u, (x,t)\in [0,l]\times \mathbb{R}, 0 < p, q \in C^{\infty}([0,l]),$$ subject to some initial data and linear boundary conditions (see in Proposition 1 below). This time the main restrictions are some Diophantine conditions and asymptotic properties of the eigenfrequencies $\omega_n$ as $n \to \infty$. Some other results without these restrictions are also presented

Observation problems posed for the Klein-Gordon equation

Transversal vibrations $u=u(x,t)$ of a string of length $l$ with fixed ends are considered, where $u$ is governed by the Klein-Gordon equation $$u_{tt}(x,t) = a^2u_{xx}(x,t)+cu(x,t), \qquad (x,t) \in [0,l] \times \mathbb{R}, \quad a>0, \ c<0.$$ Sufficient conditions are obtained that guarantee the solvability of each of four observation problems with given state functions $f, \ g$ at two distinct time instants $-\infty<t_1<t_2 < \infty$. The essential conditions are the following: smoothness of $f, \ g$ as elements of a corresponding subspace $D^{s+i}(0,l)$ (introduced in [2]) of a Sobolev space $H^{s+i} (0,l)$, where $i=1,2$ depending on the type of the observation problem, and the representability of $t_2-t_1$ as a rational multiple of $\frac{2l}{a}$. The reconstruction of the unknown initial data $(u(x,0), u_t(x,0))$ as the elements of $D^{s+1}(0,l) \times D^s(0,l)$ are given by means of the method of Fourier expansions

Preliminary characterization of the self-incompatibility genotypes of European plum (Prunus domestica L.) cultivars

European plum is an important fruit crop with complex, hexaploid genome of unknown origin. The characterization of the selfincompatibility (S) locus of 16 European plum cultivars was carried out using the PaConsI-F primer in combination with the EM-PC1consRD primer for the first intron and the EM-PC2consFD and EM-PC3consRD primers for the second intron amplification. Altogether, 18 different alleles were scored indicating high genetic diversity. These alleles were labelled using alphabetical codes from SA to SS. We&nbsp; identified 5 different alleles in 9 cultivars, 4 alleles in 5 cultivars, while 3 alleles were shown in two of the assayed cultivars. A total of 16 different S-genotypes were assigned, and discrimination of all plum cultivars was successful based on their unique S-genotypes. However, further research is required to reliably identify the S-alleles based on their DNA sequence and clarify complete S-genotypes