We establish upper bounds for the first two nonzero Steklov eigenvalues of bounded domains in Euclidean spaces of dimension $d \geq 3$, under a natural normalization involving volume and boundary measure, and show that these bounds are sharp for $d \geq 7$.
We obtain the simplicity of the first Neumann eigenvalue of convex thin domain with boundary in $R^n$ and compact thin manifolds with non-negative Ricci curvature. For convex thin domain in $R^2$, we get the simplicity of the first k Neumann eigenvalues. The number k depends on the ratio of the corresponding width over the diameter of the domain. For convex thin domain in $R^n$, we obtain the eigenvalue comparison with collapsing segment.
Corry Corazon Marzuki, Aminah Utami, Mona Elviyenti
et al.
his study aims to determine the total vertex irregularity strength on a series parallel graph for and . Total labeling is said to be vertex irregular, if the weights for each vertices are different. Determination of the total vertex irregularity of series parallel graph is done by obtaining the largest lower bound and the smallest upper bound. The lower bound is obtained by analyzing the structure of the graph to obtain the largest minimum label of k and the upper bound is analyzed by labeling the vertices and edges of the graph, where the largest label is k and the values for each vertices weight is different. The result obtained for the total vertex irregularity strength of a series parallel graph is .
We prove eigenvalue bounds for Schrödinger operator $-Δ_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the round sphere and more general Zoll manifolds. These bounds are natural analogues of Frank's \cite{MR2820160} results in the Euclidean case.
Dan-Hua Li, Rexiding Abuduaini, Meng-Xuan Du
et al.
Non-human primates harbour diverse microbiomes in their guts. As a part of the China Microbiome Initiatives, we cultivated and characterized the gut microbiome of cynomolgus monkeys (Macaca fascicularis). In this report, we communicate the characterization and taxonomy of eight bacterial strains that were obtained from faecal samples of captive cynomolgus monkeys. The results revealed that they represented eight novel bacterial species. The proposed names of the eight novel species are Alkaliphilus flagellatus (type strain MSJ-5T=CGMCC 1.45007T=KCTC 15974T), Butyricicoccus intestinisimiae MSJd-7T (MSJd-7T=CGMCC 1.45013T=KCTC 25112T), Clostridium mobile (MSJ-11T=CGMCC 1.45009T=KCTC 25065T), Clostridium simiarum (MSJ-4T=CGMCC 1.45006T=KCTC 15975T), Dysosmobacter acutus (MSJ-2T=CGMCC 1.32896T=KCTC 15976T), Paenibacillus brevis MSJ-6T (MSJ-6T=CGMCC 1.45008T=KCTC 15973T), Peptoniphilus ovalis (MSJ-1T=CGMCC 1.31770T=KCTC 15977T) and Tissierella simiarum (MSJ-40T=CGMCC 1.45012T=KCTC 25071T).
The spectral properties of the singular Schrödinger operator with complex-valued potential which takes values in a wider region than the half-plane, have been little studied. In general case, the operator is non-sectorial, and the numerical range coincides with the entire complex plane. In this situation we propose sufficient conditions for discreteness of the spectrum and compactness of the resolvent.
Denote by $N_{\cal N} (Ω,λ)$ the counting function of the spectrum of the Neumann problem in the domain $Ω$ on the plane. G. Pólya conjectured that $N_{\cal N} (Ω,λ) \ge (4π)^{-1} |Ω| λ$. We prove that for convex domains $N_{\cal N} (Ω,λ) \ge (2 \sqrt 3 \,j_0^2)^{-1} |Ω| λ$. Here $j_0$ is the first zero of the Bessel function $J_0$.
We prove that the Weyl function of the one-dimensional Dirac operator on the half-line $\mathbb{R}_+$ with exponentially decaying entropy extends meromorphically into the horizontal strip $\{0\ge \mbox{Im}\,z > -δ\}$ for some $δ> 0$ depending on the rate of decay. If the entropy decreases very rapidly then the corresponding Weyl function turns out to be meromorphic in the whole complex plane. In this situation we show that poles of the Weyl function (scattering resonances) uniquely determine the operator.
We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a connection between asymptotic properties of solutions to the eigenvalue equations and continuity properties of the spectral measure with respect to the Lebesgue measure. We also use this theory in order to derive results regarding the multiplicity of the singular spectrum.
We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse eigenvalue problem for specific finite Jacobi matrices and theory of polynomials. We determine forbidden domains for resonances and maximal possible multiplicities of real and complex resonances.
A new class of isospectral graphs is presented. These graphs are isospectral with respect to both the normalised Laplacian on the discrete graph and the standard differential Laplacian on the corresponding metric graph. The new class of graphs is obtained by gluing together subgraphs with the Steklov maps possessing special properties. It turns out that isospectrality is related to the degeneracy of the Steklov eigenvalues.
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
We study location of eigenvalues of one-dimensional discrete Schrödinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two different spectral bounds are obtained. The method relies on the Birman-Schwinger principle and various techniques for estimations of the norm of the Birman-Schwinger operator.
We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty$.
We show that any second eigenfunction of the Laplacian with standard vertex conditions on a metric tree graph attains its extremal values only at degree one vertices, and give an example where these vertices do not realise the diameter of the graph.
Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also discuss the exponential types of the transfer matrices.
The Stretched Sierpinski Gasket (or Hanoi attractor) was subject of several prior works. In this work we use this idea of stretching self-similar sets to obtain non-self-similar ones. We are able to do this for a subset of the connected p.c.f. self-similar sets that fulfill a certain connectivity condition. We construct Dirichlet forms and study the associated self-adjoint operators by calculating the Hausdorff dimension w.r.t. the resistance metric as well as the leading term of the eigenvalue counting function.