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In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

[Display omitted] • LN-SDCTC for DR detection. • Classifying color fundus images, SDCTC algorithm. • Multinomial Regression Classification Function. Diabetic retinopathy (DR) is a common eye disease and a notable starting point of blindness in diabetic patients. Detecting the existence of a microaneurysm in the fundus images and the identification of DR in the preliminary stage has been a considerable question for decades. Systematic screening and interference are the most efficient mechanisms for disease management. The sizeable populations of diabetic patients and their enormous screening requirements have given rise to the computer-aided and automatic diagnosis of DR. The utilization of Deep Neural Networks in DR diagnosis has also attracted much attention and considerable advancement has been made. Diabetic retinopathy (DR) includes sensitivity and specificity that are particular to the Diabetic tested (see section on probabilistic reasoning). Even if a test is performed correctly, there is a chance for a false positive or false negative result. However, despite the several advancements that have been made, there remains room for improvement in the sensitivity and specificity of the DR diagnosis. In this work, a novel method called the Luminosity Normalized Symmetric Deep Convolute Tubular Classifier (LN-SDCTC) for DR detection is proposed. The LN-SDCTC method is split into two parts. Initially, with the retinal color fundus images as input, the Luminosity Normalized Retinal Color Fundus Preprocessing model applies to produce a noise-minimized enhanced contrast image. Second, we provide the get-processed image as input to the Symmetric Deep Convolute network. Here, with the aid of the convolutional layer (i.e., the Tubular Neighborhood Window), the average pooling layer (i.e., average magnitude value of tubular neighbors), and the max-pooling layer (i.e., maximum contrast orientation), relevant features are selected. Finally, with the extracted features as input and with the aid of the Multinomial Regression Classification function, the severity of the DR disease is determined. Extensive experimental results in terms of peak signal-to-noise ratio, disease detection time, sensitivity, and specificity reveal that the proposed method of DR detection greatly facilitates the deep learning model and yields better results than various state-of-art methods. [ABSTRACT FROM AUTHOR]

We investigate the maximal open domain E (M) on which the orthogonal projection map p onto a subset M ⊆ R d can be defined and study essential properties of p. We prove that if M is a C 1 submanifold of R d satisfying a Lipschitz condition on the tangent spaces, then E (M) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C 2 or if the topological skeleton of M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C k -submanifold M with k ≥ 2 , the projection map is C k - 1 on E (M) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion M ⊆ E (M) is that M is a C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with M ⊆ E (M) , then M must be C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between E (M) and the topological skeleton of M c . [ABSTRACT FROM AUTHOR]

We show that a closed piecewise flat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdroff topology, there is a Lipschitz Gromov-Hausdorff approximation. [ABSTRACT FROM AUTHOR]

This paper is focused on a cooperative control design which guarantees robust connectivity and invariance of a multiagent network inside a bounded domain, under the presence of additional bounded input terms in each agent's dynamics. In particular, under the assumptions that the domain is convex and has a smooth boundary, we can design a repulsion vector field near its boundary, which ensures invariance of the agents' trajectories and does not affect the robustness properties of the control part that is exploited for connectivity maintenance. [ABSTRACT FROM AUTHOR]