A multiparameter fractional Laplace problem with semipositone nonlinearity

<p style='text-indent:20px;'>In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u&amp; = &amp; \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&amp;&gt;&amp;0 \text{ in } \Omega\\ u&amp;\equiv &amp;0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>when the positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> belong to certain range. Here <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is assumed to be a bounded open set with smooth boundary, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0, 1), N&gt; 2s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;q&lt;1&lt;r\leq \frac{N+2s}{N- 2s}. $\end{document}</tex-math></inline-formula> First we consider <inline-formula><tex-math id="M6">\begin{document}$ (P_ \lambda^\mu) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula> and prove that there exists <inline-formula><tex-math id="M8">\begin{document}$ \lambda_0\in(0, \infty) $\end{document}</tex-math></inline-formula> such that for all <inline-formula><tex-math id="M9">\begin{document}$ \lambda&gt; \lambda_0 $\end{document}</tex-math></inline-formula> the problem <inline-formula><tex-math id="M10">\begin{document}$ (P_ \lambda^0) $\end{document}</tex-math></inline-formula> admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of <inline-formula><tex-math id="M11">\begin{document}$ (P_\lambda^0) $\end{document}</tex-math></inline-formula> emanating from infinity. Next for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda&gt;\lambda_0 $\end{document}</tex-math></inline-formula> and for all <inline-formula><tex-math id="M13">\begin{document}$ 0&lt;\mu&lt;\mu_{\lambda} $\end{document}</tex-math></inline-formula> we establish the existence of at least one positive solution of <inline-formula><tex-math id="M14">\begin{document}$ (P_\lambda^\mu) $\end{document}</tex-math></inline-formula> using variational method. Also in the sub critical case, i.e., for <inline-formula><tex-math id="M15">\begin{document}$ 1&lt;r&lt;\frac{N+2s}{N-2s} $\end{document}</tex-math></inline-formula>, we show the existence of second positive solution via mountain pass argument.</p>

Stability of rotation relations in -algebras

Let $\Theta =(\theta _{j,k})_{3\times 3}$ be a nondegenerate real skew-symmetric $3\times 3$ matrix, where $\theta _{j,k}\in [0,1).$ For any $\varepsilon>0$ , we prove that there exists $\delta>0$ satisfying the following: if $v_1,v_2,v_3$ are three unitaries in any unital simple separable $C^*$ -algebra A with tracial rank at most one, such that $\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$ for all $\tau \in T(A)$ and $j,k=1,2,3,$ where $\log _{\theta }$ is a continuous branch of logarithm (see Definition 4.13) for some real number $\theta \in [0, 1)$ , then there exists a triple of unitaries $\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$ such that $\begin{align*}\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.\end{align*}$ The same conclusion holds if $\Theta $ is rational or nondegenerate and A is a nuclear purely infinite simple $C^*$ -algebra (where the trace condition is vacuous). If $\Theta $ is degenerate and A has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity conditions to get the above conclusion.

Numerical Verification of 10000-dimensional Linear Systems 10000x Faster

Tool Presentation: We evaluate an improved reachability algorithm for linear (and affine) systems implemented in the continuous branch of the Hylaa tool. While Hylaa’s earlier approach required n simulations to verify an n-dimensional system, the new method takes advantage of additional problem structure to produce the same verification result in significantly less time. If the initial states can be defined in i dimensions, and the output variables related to the property being checked are o-dimensional, the new approach needs only min(i,o) simulations to verify the system, or produce a counter-example. In addition to reducing the number of simulations, a second improvement speeds up individual simulations when the dynamics is sparse by using Krylov subspace methods.At ARCH 2017, we used the original approach to verify nine large linear benchmarks taken from model order reduction. Here, we run the new algorithm on the same set of benchmarks, and get an identical verification result in a fraction of the time. None of the benchmarks need more than tens of seconds to complete. The largest system with 10922 dimensions, which took over 24 hours using last year’s method, is verified in 3.4 seconds.

Inheritance of branching in sunflower (Helianthus annuus L.)

Aim. The purpose of our research was to study genetic diversity and establish the inheritance of the branching trait in the collection of sunflower lines of the Institute of Oilseeds of the National Academy of Sciences. Methods. 47 lines were used as a material for studying the genetics of the branching feature. Methods of crossing with pre-castration, self-pollination and analysis of offspring were used. Results. In 25 lines, a monogenic recessive control of the trait of the upper branching to the continuous branch was established. In 9 lines of the collection, the sign of continuous branching is due to the dominant allele of the gene. In two lines, the presence of two genes of the dominant alleles of which cause the sign of continuous branching is established. In 1 lines, the trait of continuous branching is controlled by the dominant alleles of the three genes. In 5 lines, the sign of the basal branch is due to the recessive homozygote of one gene b2. In 4 lines, the trait of the basal branch is due to the recessive homozygote for the two genes b3 and b4. Conclusions. In total, four genes are found in our collection, recessive alleles of which cause branching and three genes whose dominant alleles cause branching.Keywords: genetics, sunflower, branching, gene, inheritance.

Exact and approximate limit behaviour of the Yule tree’s cophenetic index

In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure–birth tree conditioned onncontemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and inL2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction–type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure–birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time–consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.

REPRESENTATIONS OF N=1 EXTENDED SUPERCONFORMAL ALGEBRAS WITH TWO GENERATORS

In this paper we consider the representation theory of N=1 Super-W-algebras with two generators for conformal dimension of the additional superprimary field between two and six. In the superminimal case our results coincide with the expectation from the ADE-classification. For the parabolic algebras we find a finite number of highest weight representations and an effective central charge [Formula: see text]. Furthermore we show that most of the exceptional algebras lead to new rational models with [Formula: see text]. The remaining exceptional cases show a new ‘mixed’ structure. Besides a continuous branch of representations discrete values of the highest weight also exist.

Leader growth and the architecture of three North American hemlocks

Height growth in hemlock (Tsuga canadensis (L.) Carr., T. heterophylla (Raf.) Sarg., T. mertensiana (Bong.) Carr.) is by rhythmic growth of a monopodial axis with continuous branch production throughout the growing season. Leader growth is plagiotropic and leader erection is a process lasting several years. Two types of events disrupt the basically monopodial nature of the axis. (1) Frequent (43%) apical meristem death shifts dominance to a nearby lateral branch in T. canadensis. (2) Weak apical control allows occasional shifts in dominance from the leader to a branch without meristem death (13 and 24% in T. heterophylla and T. canadensis, respectively, but none in T. mertensiana). These growth patterns contain elements of several tree architectural models but fit none well.