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E. Oja, T. Viil, and D. Werner showed, in [Totally smooth renormings, Archiv der Mathematik, 112, 3, (2019), 269--281] that a weakly compactly generated Banach space $(X,\|\cdot \|)$ with the property that every linear functional on $X$ has a unique Hahn--Banach extension to the bidual $X^{**}$ (the so-called Phelps' property U in $X^{**}$, also known as the Hahn--Banach smoothness property) can be renormed to have the stronger property that for every subspace $Y$ of $X$, every linear functional on $Y$ has a unique Hahn--Banach extension to $X^{**}$ (the so-called total smoothness property of the space). We mention here that this result holds in full generality -- without any restriction on the space -- and in a stronger form, thanks to a result of M. Raja, [On dual locally uniformly rotund norms, Israel Journal of Mathematics 129 (2002), 77--91].

We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ containing $c_0$ is an $r$-Lipschitz retract for $r < 2$, which implies in particular that $\mathcal{F}(\ell_\infty)$ is not $r$-Plichko for $r < 2$., Comment: 16 pages

We provide the explicit formula for the numerical index of any $2$-dimensional Lipschitz-free space, also giving the construction of operators attaining this value as its numerical radius. As a consequence, the numerical index of $2$-dimensional Lipschitz-free spaces can take any value of the interval $[\frac{1}{2},1]$, and this whole range of numerical indices can be attained by taking $2$-dimensional subspaces of any Lipschitz-free space of the form $\mathcal{F}(A)$, where $A\subset {\mathbb{R}}^n$ with $n\geq 2$ is any set with non-empty interior.

The growing popularity of Machine Learning (ML) has led to its deployment in various sensitive domains, which has resulted in significant research focused on ML security and privacy. However, in some applications, such as Augmented/Virtual Reality, integrity verification of the outsourced ML tasks is more critical--a facet that has not received much attention. Existing solutions, such as multi-party computation and proof-based systems, impose significant computation overhead, which makes them unfit for real-time applications. We propose Fides, a novel framework for real-time integrity validation of ML-as-a-Service (MLaaS) inference. Fides features a novel and efficient distillation technique--Greedy Distillation Transfer Learning--that dynamically distills and fine-tunes a space and compute-efficient verification model for verifying the corresponding service model while running inside a trusted execution environment. Fides features a client-side attack detection model that uses statistical analysis and divergence measurements to identify, with a high likelihood, if the service model is under attack. Fides also offers a re-classification functionality that predicts the original class whenever an attack is identified. We devised a generative adversarial network framework for training the attack detection and re-classification models. The evaluation shows that Fides achieves an accuracy of up to 98% for attack detection and 94% for re-classification., Comment: 15 pages, 12 figures

A subspace $X$ of a Banach space $Y$ has $\textit{Property U}$ whenever every continuous linear functional on $X$ has a unique norm-preserving (i.e., Hahn$-$Banach) extension to $Y$ (Phelps, 1960). Throughout this document we introduce and develop a systematic study of the existence of $\textit{U-embeddings}$ between Banach spaces $X$ and $Y$, that is, isometric embeddings of $X$ into $Y$ whose ranges have property U. In particular, we are interested in the case that $Y=C(K)$, where $K$ is a compact Hausdorff topological space. We provide results for general Banach spaces and for some specific set-ups, such as $X$ being a finite-dimensional space or a $C(K)$-space.

Parkinson's disease (PD) is the second most frequent neurodegenerative disease with prevalence among general population reaching 0.1-1 %, and an annual incidence between 1.3-2.0/10000 inhabitants. The mean age at diagnosis of PD is 55 and most patients are between 50 and 80 years old. The most obvious symptoms are movement-related; these include tremor, rigidity, slowness of movement and walking difficulties. Frequently these are the symptoms that lead to the PD diagnoses. Later, thinking and behavioral problems may arise, and other symptoms include cognitive impairment and sensory, sleep and emotional problems. In this paper we will present an ongoing project that will evaluate if voice and handwriting analysis can be reliable predictors/indicators of swallowing and balance impairments in PD. An important advantage of voice and handwritten analysis is its low intrusiveness and easy implementation in clinical practice. Thus, if a significant correlation between these simple analyses and the gold standard video-fluoroscopic analysis will imply simpler and less stressing diagnostic test for the patients as well as the use of cheaper analysis systems., Comment: 10 pages

With the rise of knowledge management and knowledge economy, the knowledge elements that directly link and embody the knowledge system have become the research focus and hotspot in certain areas. The existing knowledge element representation methods are limited in functions to deal with the formality, logic and reasoning. Based on description logic ALC and the common knowledge element model, in order to describe the knowledge element, the description logic ALC is expanded. The concept is extended to two diferent ones (that is, the object knowledge element concept and the attribute knowledge element concept). The relationship is extended to three (that is, relationship between object knowledge element concept and attribute knowledge element concept, relationship among object knowledge element concepts, relationship among attribute knowledge element concepts), and the inverse relationship constructor is added to propose a description logic KEDL system. By demonstrating, the relevant properties, such as completeness, reliability,of the described logic system KEDL are obtained. Finally, it is verified by the example that the description logic KEDL system has strong knowledge element description ability.

In Gao's previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if $b=f(x)=g(x)=a$ for all vertices $x$ in $G$. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference $\Delta$ between $g(x)$ and $f(x)$ for every vertex $x$ in $G$. These obtained new degree conditions reformulate Gao's previous conclusions, and show how $\Delta$ acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting $f(x)=g(x)+\Delta$.

We study the class of Banach spaces $X$ such that the locally convex space $(X,\mu(X,Y))$ is complete for every norming and norm-closed subspace $Y \subset X^*$, where $\mu(X,Y)$ denotes the Mackey topology on $X$ associated to the dual pair $\langle X,Y\rangle$. Such Banach spaces are called fully Mackey complete. We show that fully Mackey completeness is implied by Efremov's property ($\mathcal{E}$) and, on the other hand, it prevents the existence of subspaces isomorphic to $\ell_1(\omega_1)$. This extends previous results by Guirao, Montesinos and Zizler [J. Math. Anal. Appl. 445 (2017), 944-952] and Bonet and Cascales [Bull. Aust. Math. Soc. 81 (2010), 409-413]. Further examples of Banach spaces which are not fully Mackey complete are exhibited, like $C[0,\omega_1]$ and the long James space $J(\omega_1)$. Finally, by assuming the Continuum Hypothesis, we construct a Banach space with $w^*$-sequential dual unit ball which is not fully Mackey complete. A key role in our discussion is played by the (at least formally) smaller class of Banach spaces $X$ such that $(Y,w^*)$ has the Mazur property for every norming and norm-closed subspace $Y \subset X^*$.

Mechanical strain and stress play a major role in biological processes such as wound healing or morphogenesis. To assess this role quantitatively, fixed or live images of tissues are acquired at a cellular precision in large fields of views. To exploit these data, large numbers of cells have to be analyzed to extract cell shape anisotropy and cell size. Most frequently, this is performed through detailed individual cell contour determination, using so-called segmentation computer programs, complemented if necessary by manual detection and error corrections. However, a coarse grained and faster technique can be recommended in at least three situations. First, when detailed information on individual cell contours is not required, for instance in studies which require only coarse-grained average information on cell anisotropy. Second, as an exploratory step to determine whether full segmentation can be potentially useful. Third, when segmentation is too difficult, for instance due to poor image quality or too large a cell number. We developed a user-friendly, Fourier transform-based image analysis pipeline. It is fast (typically $10^4$ cells per minute with a current laptop computer) and suitable for time, space or ensemble averages. We validate it on one set of artificial images and on two sets of fully segmented images, one from a Drosophila pupa and the other from a chicken embryo; the pipeline results are robust. Perspectives include \textit{in vitro} tissues, non-biological cellular patterns such as foams, and $xyz$ stacks., Comment: 13 pages; 9 figures

We review some of the most recent approaches to colorize gray-scale images using deep learning methods. Inspired by these, we propose a model which combines a deep Convolutional Neural Network trained from scratch with high-level features extracted from the Inception-ResNet-v2 pre-trained model. Thanks to its fully convolutional architecture, our encoder-decoder model can process images of any size and aspect ratio. Other than presenting the training results, we assess the "public acceptance" of the generated images by means of a user study. Finally, we present a carousel of applications on different types of images, such as historical photographs., Comment: 12 pages

We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in $X$ such that the triangle inequality $d(p,q)\leq d(p,r)+d(q,r)$ is uniformly strict for $r$ away from $p,q$. For compact $X$, this condition reduces to the triangle inequality being strict. This result gives an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points., Comment: 15 pages. Results have been generalized to the general (i.e. non-compact) case. Comments are welcome

The Bishop-Phelps-Bollob\'{a}s property deals with simultaneous approximation of an operator $T$ and a vector $x$ at which $T$ nearly attains its norm by an operator $T_0$ and a vector $x_0$, respectively, such that $T_0$ attains its norm at $x_0$. In this note we extend the already known results about {the} Bishop-Phelps-Bollob\'{a}s property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: $\Gamma$-flat operators and Banach spaces with ACK$_\rho$ structure. In particular, we prove a general BPB-type theorem for $\Gamma$-flat operators acting to a space with ACK$_\rho$ structure and show that uniform algebras and spaces with the property $\beta$ have ACK$_\rho$ structure. We also study the stability of the ACK$_\rho$ structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces $Y$ such that the Bishop-Phelps-Bollob\'{a}s property for Asplund operators is valid for all pairs of the form ($X,Y$)., Comment: 24 pages