A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators

A striking result by Nordgren, Rosenthal and Wintrobe states that the Invariant Subspace Problem is equivalent to the fact that any minimal invariant subspace for a composition operator Cφ induced by a hyperbolic automorphism φ of the unit disc D acting on the classical Hardy space H² is one dimensional. We provide a completely different proof of Nordgren, Rosenthal and Wintrobe’s Theorem based on analytic Toeplitz operators

A hyperbolic universal operator commuting with a compact operator

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a non-trivial, quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a `hyperbolic' composition operator

Finite Blaschke Products as Compositions of Other Finite Blaschke Products

These notes answer the question "When can a finite Blaschke product $B$ be written as a composition of two finite Blaschke products $B_1$ and $B_2$, that is, $B=B_1\circ B_2$, in a non-trivial way, that is, where the order of each is greater than 1." It is shown that a group can be computed from $B$ and its local inverses, and that compositional factorizations correspond to normal subgroups of this group. This manuscript was written in 1974 but not published because it was pointed out to the author that this was primarily a reconstruction of work of Ritt from 1922 and 1923, who reported on work on polynomials. It is being made public now because of recent interest in this subject by several mathematicians interested in different aspects of the problem and interested in applying these ideas to complex analysis and operator theory.Comment: 22 page

Constructing Invariant Subspaces as Kernels of Commuting Matrices

Given an n by n matrix A over the complex numbers and an invariant subspace L, this paper gives a straightforward formula to construct an n by n matrix N that commutes with A and has L equal to the kernel of N. For Q a matrix putting A into Jordan canonical form J = RAQ with R the inverse of Q, we get N = RM$ where the kernel of M is an invariant subspace for J with M commuting with J. In the formula M = P ZVW with V the inverse of a constructed matrix T and W the transpose of P, the matrices Z and T are m by m and P is an n by m row selection matrix. If L is a marked subspace, m = n and Z is an n by n block diagonal matrix, and if L is not a marked subspace, then m > n and Z is an m by m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a finite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A.Comment: 12 pages with two illustrations of invariant subspace lattice diagram

Constructing invariant subspaces as kernels of commuting matrices

Given an n n matrix A over C and an invariant subspace N, a straightforward formula constructs an n n matrix N that commutes with A and has N = kerN. For Q a matrix putting A into Jordan canonical form, J = Q1AQ, we get N = Q1M where M= ker(M) is an invariant subspace for J with M commuting with J. In the formula J = PZT1Pt, the matrices Z and T are m m and P is an n m row selection matrix. If N is a marked subspace, m = n and Z is an n n block diagonal matrix, and if N is not a marked subspace, then m > n and Z is an m m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a nite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A

No evidence for differential gene expression in major depressive disorder PBMCs, but robust evidence of elevated biological ageing.

The increasingly compelling data supporting the involvement of immunobiological mechanisms in Major Depressive Disorder (MDD) might provide some explanation forthe variance in this heterogeneous condition. Peripheral blood measures of cytokines and chemokines constitute the bulk of evidence, with consistent meta-analytic data implicating raised proinflammatory cytokines such as IL6, IL1β and TNF. Among the potential mechanisms linking immunobiological changes to affective neurobiology is the accelerated biological ageing seen in MDD, particularly via the senescence associated secretory phenotype (SASP). However, the cellular source of immunobiological markers remains unclear. Pre-clinical evidence suggests a role for peripheral blood mononuclear cells (PBMC), thus here we aimed to explore the transcriptomic profile using RNA sequencing in PBMCs in a clinical sample of people with various levels of depression and treatment response comparing it with that in healthy controls (HCs). There were three groups with major depressive disorder (MDD): treatment-resistant (n = 94), treatment-responsive (n = 47) and untreated (n = 46). Healthy controls numbered 44. Using PBMCs gene expression analysis was conducted using RNAseq to a depth of 54.5 million reads. Differential gene expression analysis was performed using DESeq2. The data showed no robust signal differentiating MDD and HCs. There was, however, significant evidence of elevated biological ageing in MDD vs HC. Biological ageing was evident in these data as a transcriptional signature of 888 age-associated genes (adjusted p  0.6) that also correlated strongly with chronological age (spearman correlation coefficient of 0.72). Future work should expand clinical sample sizes and reduce clinical heterogeneity. Exploration of RNA-seq signatures in other leukocyte populations and single cell RNA sequencing may help uncover more subtle differences. However, currently the subtlety of any PBMC signature mitigates against its convincing use as a diagnostic or predictive biomarker

In Situ Observations of the Deformation Behavior and Fracture Mechanisms of Ti-45Al-2Nb-2Mn+0.8 vol pct TiB₂

The deformation and fracture mechanisms of a nearly lamellar Ti-45Al-2Nb-2Mn (at. pct) + 0.8 vol pct TiB₂ intermetallic, processed into an actual low-pressure turbine blade, were examined by means of in situ tensile and tensile-creep experiments performed inside a scanning electron microscope (SEM). Low elongation-to-failure and brittle fracture were observed at room temperature, while the larger elongations-to-failure at high temperature facilitated the observation of the onset and propagation of damage. It was found that the dominant damage mechanisms at high temperature depended on the applied stress level. Interlamellar cracking was observed only above 390 MPa, which suggests that there is a threshold below which this mechanism is inhibited. Failure during creep tests at 250 MPa was controlled by intercolony cracking. The in situ observations demonstrated that the colony boundaries are damage nucleation and propagation sites during tensile creep, and they seem to be the weakest link in the microstructure for the tertiary creep stage. Therefore, it is proposed that interlamellar areas are critical zones for fracture at higher stresses, whereas lower stress, high-temperature creep conditions lead to intercolony cracking and fracture.The authors are grateful to Industria de Turbo Propulsores, S.A. for supplying the intermetallic blades. Funding from the Spanish Ministry of Science and Innovation through projects MAT2009-14547-C02-01 and MAT2009-14547-C02-02 is acknowledged. The Madrid Regional Government supported this project partially through the ESTRUMAT grant P2009/MAT-1585. C.J.B. acknowledges the support from Grant SAB2009-0045 from the Spanish Ministry of Education for his sabbatical stage in Madrid.Publicad

AI is a viable alternative to high throughput screening: a 318-target study

: High throughput screening (HTS) is routinely used to identify bioactive small molecules. This requires physical compounds, which limits coverage of accessible chemical space. Computational approaches combined with vast on-demand chemical libraries can access far greater chemical space, provided that the predictive accuracy is sufficient to identify useful molecules. Through the largest and most diverse virtual HTS campaign reported to date, comprising 318 individual projects, we demonstrate that our AtomNet® convolutional neural network successfully finds novel hits across every major therapeutic area and protein class. We address historical limitations of computational screening by demonstrating success for target proteins without known binders, high-quality X-ray crystal structures, or manual cherry-picking of compounds. We show that the molecules selected by the AtomNet® model are novel drug-like scaffolds rather than minor modifications to known bioactive compounds. Our empirical results suggest that computational methods can substantially replace HTS as the first step of small-molecule drug discovery