Exact boundary null controllability for a coupled system of plate equations with variable coefficients

<p style='text-indent:20px;'>This paper studies a coupled system of plate equations with variable coefficients, subject to the clamped boundary conditions. By the Riemannian geometry approach, the duality method, the multiplier technique and a compact perturbation method, we establish exact boundary null controllability of the system under verifiable assumptions.</p>

Perturbation of the spectra of complex symmetric operators

An operator T on a complex Hilbert space H is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for H. This paper focuses on the perturbation theory for the spectra of complex symmetric operators. We prove that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and having the single-valued extension property. Also it is proved that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and satisfying generalized Weyl?s theorem.

Electromagnetic Steklov eigenvalues: approximation analysis

We continue the work of [Camano, Lackner, Monk, SIAM J.\ Math.\ Anal., Vol.\ 49, No.\ 6, pp.\ 4376-4401 (2017)] on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues due not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Steklov eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function $A(\cdot)$. We construct a ``test function operator function'' $T(\cdot)$ so that $A(\lambda)$ is weakly $T(\lambda)$-coercive for all suitable $\lambda$, i.e.\ $T(\lambda)^*A(\lambda)$ is a compact perturbation of a coercive operator. The construction of $T(\cdot)$ relies on a suitable decomposition of the function space into subspaces and an apt sign change on each subspace. For the approximation analysis, we apply the framework of T-compatible Galerkin approximations. For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence convergence. For the original problem, we require the projection operators to satisfy an additional commutator property involving the tangential trace. The existence and construction of such projection operators remain open questions.

Remarks on Surjectivity of Gradient Operators

Let X be a real Banach space with dual X∗ and suppose that F:X→X∗. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent result of ours, we prove that any gradient map that has this property and is additionally bounded, coercive and continuous is surjective. As before, the main tool for the proof is the Ekeland Variational Principle. Comparison with known surjectivity results is made; finally, as an application, we discuss a Dirichlet boundary-value problem for the p-Laplacian (1<p<∞), completing our previous result which was limited to the case p≥2.

Otopy classification of gradient compact perturbations of identity in Hilbert space

We prove that the inclusion of the space of gradient local maps into the space of all local maps from Hilbert space to itself induces a bijection between the sets of the respective otopy classes of these maps, where by a local map we mean a compact perturbation of identity with a compact preimage of zero.

Branched Sidechain bPNA + Reports on RNA-Loop Tertiary Interactions via Triplex Hybridization Stem Replacement at Internal Domains

We report herein the synthesis and characterization of bPNA+, a new variant of bifacial peptide nucleic acid (bPNA) that binds oligo T/U nucleic acids to form triplex hybrids with good affinity but half the molecular footprint of bPNA. This is accomplished via display of two melamine (M) bases melamine per lysine sidechain on bPNA+ rather than one as previously reported. Lysine derivatives bearing two bases were prepared by double reductive alkylation with melamine acetaldehyde, resulting in a tertiary amine branch point. Importantly, this amino sidechain fosters oligo T/U binding through both base-triple formation and electrostatic interactions, while maintaining selectivity and peptide solubility. The bPNA+ binding site roughly the size of a 6 base-pair stem; this relatively compact perturbation can be genetically encoded at virtually any position within an RNA transcript, and may replace existing stem elements. Subsequent triplex hybridization with fluorophore-labeled bPNA+ triplex hybrids thus accomplishes site-specific labeling of internal RNA locations without the need for chemical modification. We demonstrate herein the use of this strategy for reporting on intermolecular RNA-RNA kissing loop interactions, RNA-protein binding as well as intramolecular RNA tetraloop-tetraloop receptor binding. We anticipate that bPNA+ will have utility as structural probes for dynamic tertiary interactions in long noncoding RNAs.

Essential Spectra of General Second-Order Differential Operators

In this chapter, the operators considered are those m-sectorial operators discussed in Chapter VII, and the essential spectra are the sets defined in Chapter IX that remain invariant under compact perturbation. A generalization of a result of Persson is used to determine the least point of the essential spectrum. Davies’ mean distance function is introduced and consequences investigated.