Pattern Analysis of Multidimensional Digital Spaces, Homotopy Theory Approach

Finite topological spaces and their dimensions have many applications in computer science, e.g., in digital topology, computer graphics and the analysis and synthesis of digital images. Georgiou et. al. [11] provided a polynomial algorithm for computing the covering dimension dim( X; 𝒯) of a finite topological space (X; 𝒯). In addition, they asked whether algorithms of the same complexity for computing the small inductive dimension ind( X; 𝒯) and the large inductive dimension Ind( X; 𝒯) can be developed. The first problem was solved in a previous paper [4]. Using results of the that paper, we also solve the second problem in this paper. We present a polynomial algorithm for Ind( X; 𝒯), so that there are now efficient algorithms for the three most important notions of a dimension in topology. Our solution reduces the computation of Ind( X; 𝒯), where the specialisation pre-order of ( X; 𝒯) is taken as input, to the computation of the maximal height of a specific class of directed binary trees within the partially ordered set. For the latter an efficient algorithm is presented that is based on order- and graph-theoretic ideas. Also refinements and variants of the algorithm are discussed.

Achieving total information flow security is feasible through the use of formal mathematical models. A fundamental theorem of a topological theory of Discretionary Access Controls is presented: Let U be a set of UNIX users. The association from each data owner p to its data owners group F(p) defines a generalized topological space (U, F) and a binary relation B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> . Thus, a DAC-system is information-flow-secure (via file movements): if and only if (U, F) is a finite topological space; if and only if (U, B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> ) is a reflexive and transitive graph. An important consequence is the following proved secure system in current UNIX: The information flow (via file movements) of UNIX generated by Read-access is information-flow-secure: if (U, F) is a finite topological space;if (U, B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> ) is a reflexive and transitive graph. This model is based on a modified security principle: Every data owner p has the discretionary authority (Individual Data Owner p's authority, or IDO(p)-authority) over who can access the data. The precise security policy is the integration of IDO(p)-security for All data owner p in All possible data states. Once granted access, information propagation can still be controlled by the topological structure.

Homotopy in digital spaces

Intersection number and topology preservation within digital surfaces

Construct validity of pixel-by-pixel DCE-MRI: Correlation with conventional MRI scores in juvenile idiopathic arthritis

The problem of the robust stability of large-scale dynamical systems including delayed states and parameter perturbations in interconnections is considered. By using algebraic Riccati equations and some analytical methods, some sufficient conditions on linear decentralized state feedback controllers are derived so that the systems remain stable in the presence of delayed states and parameter perturbations. Such conditions give some bounds on the perturbations of interconnections with delayed states and uncertain parameters, and result in a quantitative measures of robustness for large-scale dynamical systems including delayed states and uncertain parameters in interconnections. The results obtained in this paper are applicable not only to large-scale systems with multiple time-varying delays, but also to large-scale systems without exact knowledge of the delays, i.e., large-scale systems with uncertain delays.

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

DC/DC converter and DC distributed power system (DC-DPS) have been widely used in many industrial fields. This paper systematically studies the rule and method of deriving the basic topologies of them. Starting with the overall structure of basic non-isolated DC/DC converters, it can be divided into three cascaded units. Each unit can further be divided into voltage-type (V-type) and current-type (C-type). Then, according to the basic circuit theory, four categories of basic non-isolated DC/DC converter generalized structures and twelve specific non-isolated DC/DC converter topologies are all derived out. Based on the same topology construction principle above, the basic DC V-type and Ctype distributed power system generalized structures are both established, which can be named DCV-DPS and DCC-DPS. On this basis, two practical DC-DPS topology structures are also given out as an application example. Finally, combined with high-power LED driving applications, a 90W prototype with three channel outputs based on the DCC-DPS topology structure is built and the experimental results are presented. It is proved that the proposed topology construction rule and analysis method are valid.

Several recent papers have adapted notions of geometric topology to the emerging field of `digital topology.' An important notion is that of digital homotopy. In this paper, we study a variety of digitally-continuous functions that preserve homotopy types or homotopy-related properties such as the digital fundamental group.

The paper [9] introduces the important tool of the digital covering space for studying the digital fundamental group. From a classical construction of algebraic topology [16,17,19], we show the existence of digital universal covering spaces and their significance for the study of the digital fundamental group.

Backward shift operator and finite dimensional de Branges Rovnyak spaces in the ball

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

ObjectivesLung ultrasound (LUS) has sparked significant interest during COVID‐19. LUS is based on the detection and analysis of imaging patterns. Vertical artifacts and consolidations are some of the recognized patterns in COVID‐19. However, the interrater reliability (IRR) of these findings has not been yet thoroughly investigated. The goal of this study is to assess IRR in LUS COVID‐19 data and determine how many LUS videos and operators are required to obtain a reliable result.MethodsA total of 1035 LUS videos from 59 COVID‐19 patients were included. Videos were randomly selected from a dataset of 1807 videos and scored by six human operators (HOs). The videos were also analyzed by artificial intelligence (AI) algorithms. Fleiss' kappa coefficient results are presented, evaluated at both the video and prognostic levels.ResultsFindings show a stable agreement when evaluating a minimum of 500 videos. The statistical analysis illustrates that, at a video level, a Fleiss' kappa coefficient of 0.464 (95% confidence interval [CI] = 0.455–0.473) and 0.404 (95% CI = 0.396–0.412) is obtained for pairs of HOs and for AI versus HOs, respectively. At prognostic level, a Fleiss' kappa coefficient of 0.505 (95% CI = 0.448–0.562) and 0.506 (95% CI = 0.458–0.555) is obtained for pairs of HOs and for AI versus HOs, respectively.ConclusionsTo examine IRR and obtain a reliable evaluation, a minimum of 500 videos are recommended. Moreover, the employed AI algorithms achieve results that are comparable with HOs. This research further provides a methodology that can be useful to benchmark future LUS studies.

Introduction: Triple immunohistochemical (IHC) stains including antibodies specific for alpha-methylacyl-CoA-racemase and basal cell markers have been a valuable aid in accurate identification of prostate carcinoma. However, accurate quantification of minuscule areas of prostate carcinoma in biopsy specimens can often be a challenge. Here we assessed the diagnostic value and quantitative use of automated digital image analysis on triple IHC stained prostate needle biopsies. Methods: Twelve cases of prostate needle biopsy material including 75 needle cores were stained with triple-antibody cocktail (P504S + 34βE12 + p63). Slides were digitally scanned with the APERIO digital image analyzer and evaluated with the GENIE pattern and color recognition digital image analysis that we developed. A slide with known areas of adenocarcinoma, high grade prostatic intraepithelial neoplasia (PIN), benign glands and stroma was used as a training set for the automated digital image analysis platform. Results: Among 75 needle biopsy cores, 19 (25.33%) contained adenocarcinoma by histology. Digital image analysis recognized adenocarcinoma in 95% of these needle biopsies. The average area of the needle biopsy was 7.63 mm2 and overall the average area of tumor was 0.196 mm2. The smallest area of tumor recognized by the program was 0.0022 mm2 (0.0363% of the core) and the largest was 0.62 mm2 (8.17% of the core) among needle core biopsies. False positives resulted from areas of high grade PIN with patchy basal cells. The false negative was caused by uneven AMACR staining in one area of adenocarcinoma. Digital recognition of areas of interest was improved by three successive image analysis training which increased the sensitivity and specificity from 83% and 89% to 90% and 93%, respectively. Conclusions: Digital image analysis in concert with IHC triple staining may be useful for accurate detection and quantitative analysis of small foci of prostatic adenocarcinoma. Defining methods to increase the sensitivity and specificity of quantitative automated digital image analysis will likely evolve as an area of investigation. Future automated digital scanning and innovative pattern and color recognition technologies may open avenues for classifying a variety of prostate lesions.

Rational homotopy theory and differential graded category