Interactions Between Tsetse Endosymbionts and Glossina pallidipes Salivary Gland Hypertrophy Virus in Glossina Hosts

Tsetse flies are the sole cyclic vector for trypanosomosis, the causative agent for human African trypanosomosis or sleeping sickness and African animal trypanosomosis or nagana. Tsetse population control is the most efficient strategy for animal trypanosomosis control. Among all tsetse control methods, the Sterile Insect Technique (SIT) is one of the most powerful control tactics to suppress or eradicate tsetse flies. However, one of the challenges for the implementation of SIT is the mass production of target species. Tsetse flies have a highly regulated and defined microbial fauna composed of three bacterial symbionts (Wigglesworthia, Sodalis and Wolbachia) and a pathogenic Glossina pallidipes Salivary Gland Hypertrophy Virus (GpSGHV) which causes reproduction alterations such as testicular degeneration and ovarian abnormalities with reduced fertility and fecundity. Interactions between symbionts and GpSGHV might affect the performance of the insect host. In the present study, we assessed the possible impact of GpSGHV on the prevalence of tsetse endosymbionts under laboratory conditions to decipher the bidirectional interactions on six Glossina laboratory species. The results indicate that tsetse symbiont densities increased over time in tsetse colonies with no clear impact of the GpSGHV infection on symbionts density. However, a positive correlation between the GpSGHV and Sodalis density was observed in Glossina fuscipes species. In contrast, a negative correlation between the GpSGHV density and symbionts density was observed in the other taxa. It is worth noting that the lowest Wigglesworthia density was observed in G. pallidipes, the species which suffers most from GpSGHV infection. In conclusion, the interactions between GpSGHV infection and tsetse symbiont infections seems complicated and affected by the host and the infection density of the GpSGHV and tsetse symbionts.

Frame-cyclic operators and their properties

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

CONSTRUCTION OF A MEASURE THAT MAPS EVERY CARLESON SET TO ZERO

According to an important result by J. Shapiro (see \cite{Shapiro}), the construction of such measure provides a corresponding singular inner cyclic vector for the shift operator on the Bergman space $L_a^t([0,1])$.

Monic representations of finite higher-rank graphs

In this paper, we define the notion of monic representation for the$C^{\ast }$-algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative$C^{\ast }$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the$\unicode[STIX]{x1D6EC}$-semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs.J. Math. Anal. Appl. 434 (2015), 241–270) and also provide a universal representation model for non-negative monic representations.

COVARIANT GNS REPRESENTATION FOR OPEN C*-DYNAMICAL SYSTEMS

An open C*-dynamical systems is a triple {𝔄, Φ, φ} where 𝔄 is a C* algebra with unit, φ as its state, Φ : 𝔄 → 𝔄 a unital completely positive map with φ ◦ Φ = φ. Such system is called purely implementable if there exists a representation π of 𝔄 in the bounded operators on Hilbert spaces [Formula: see text], an isometry [Formula: see text] and a V-invariant vector [Formula: see text] such that π(Φ(a)) = V*π(a)V and φ(a) = 〈Ω, π(a)Ω〉 for all a ∈ 𝔄, with Ω cyclic vector for the algebra generated by π(𝔄) and V. The quadruple [Formula: see text] is said be a covariant GNS representation of the open C*-dynamical system. We prove that every open C*-dynamical system is purely implementable. In the case when the dynamics Φ is a *-homomorphism this result was obtained by Niculescu Ströh and Zsidó. Moreover we prove that, in the case of *-homomorphisms, the above-mentioned construction provides an alternative construction of a minimal dilation in the sense of Kümmerer in which the dilation of the dynamics Φ has the same ergodic properties as Φ.