Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case

We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0}$$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries

Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $

DAN: a Segmentation-free Document Attention Network for Handwritten Document Recognition

Unconstrained handwritten text recognition is a challenging computer vision task. It is traditionally handled by a two-step approach, combining line segmentation followed by text line recognition. For the first time, we propose an end-to-end segmentation-free architecture for the task of handwritten document recognition: the Document Attention Network. In addition to text recognition, the model is trained to label text parts using begin and end tags in an XML-like fashion. This model is made up of an FCN encoder for feature extraction and a stack of transformer decoder layers for a recurrent token-by-token prediction process. It takes whole text documents as input and sequentially outputs characters, as well as logical layout tokens. Contrary to the existing segmentation-based approaches, the model is trained without using any segmentation label. We achieve competitive results on the READ 2016 dataset at page level, as well as double-page level with a CER of 3.43% and 3.70%, respectively. We also provide results for the RIMES 2009 dataset at page level, reaching 4.54% of CER. We provide all source code and pre-trained model weights at https://github.com/FactoDeepLearning/DAN

Measuring connectedness among herds in mixed linear models: From theory to practice in large-sized genetic evaluations

A procedure to measure connectedness among groups in large-sized genetic evaluations is presented. It consists of two steps: (a) computing coefficients of determination (CD) of comparisons among groups of animals; and (b) building sets of connected groups. The CD of comparisons were estimated using a sampling-based method that estimates empirical variances of true and predicted breeding values from a simulated n-sample. A clustering method that may handle a large number of comparisons and build compact clusters of connected groups was developed. An aggregation criterion (Caco) that reflects the level of connectedness of each herd was computed. This procedure was validated using a small beef data set. It was applied to the French genetic evaluation of the beef breed with most records and to the genetic evaluation of goats. Caco was more related to the type of service of sires used in the herds than to herd size. It was very sensitive to the percentage of missing sires. Disconnected herds were reliably identified by low values of Caco. In France, this procedure is the reference method for evaluating connectedness among the herds involved in on-farm genetic evaluation of beef cattle (IBOVAL) since 2002 and for genetic evaluation of goats from 2007 onwards

Experiments and numerical modelling of the performance and wake of a tidal converter

Water Qualit