Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume

The Law of Large Numbers in a Metric Space with a Convex Combination Operation

We consider a separable complete metric space equipped with a convex combination operation. For such spaces, we identify the corresponding convexification operator and show that the invariant elements for this operator appear naturally as limits in the strong law of large numbers. It is shown how to uplift the suggested construction to work with subsets of the basic space in order to develop a systematic way of proving laws of large numbers for such operations with random set

On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

We mainly discussed pseudointegrals based on a pseudoaddition decomposable measure. Particularly, we give the definition of the pseudointegral for a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function. Furthermore, we got several important properties of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Computing multi-scale organizations built through assembly

The ability to generate and control assembling structures built over many orders of magnitude is an unsolved challenge of engineering and science. Many of the presumed transformational benefits of nanotechnology and robotics are based directly on this capability. There are still significant theoretical difficulties associated with building such systems, though technology is rapidly ensuring that the tools needed are becoming available in chemical, electronic, and robotic domains. In this thesis a simulated, general-purpose computational prototype is developed which is capable of unlimited assembly and controlled by external input, as well as an additional prototype which, in structures, can emulate any other computing device. These devices are entirely finite-state and distributed in operation. Because of these properties and the unique ability to form unlimited size structures of unlimited computational power, the prototypes represent a novel and useful blueprint on which to base scalable assembly in other domains. A new assembling model of Computational Organization and Regulation over Assembly Levels (CORAL) is also introduced, providing the necessary framework for this investigation. The strict constraints of the CORAL model allow only an assembling unit of a single type, distributed control, and ensure that units cannot be reprogrammed - all reprogramming is done via assembly. Multiple units are instead structured into aggregate computational devices using a procedural or developmental approach. Well-defined comparison of computational power between levels of organization is ensured by the structure of the model. By eliminating ambiguity, the CORAL model provides a pragmatic answer to open questions regarding a framework for hierarchical organization. Finally, a comparison between the designed prototypes and units evolved using evolutionary algorithms is presented as a platform for further research into novel scalable assembly. Evolved units are capable of recursive pairing ability under the control of a signal, a primitive form of unlimited assembly, and do so via symmetry-breaking operations at each step. Heuristic evidence for a required minimal threshold of complexity is provided by the results, and challenges and limitations of the approach are identified for future evolutionary studies