Singular integrals in quantum Euclidean spaces

We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce L p L_p -boundedness and Sobolev p p -estimates for regular, exotic and forbidden symbols in the expected ranks. In the L 2 L_2 level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove L p L_p -regularity of solutions for elliptic PDEs.

A Bibliometric Analysis for Lebesgue Measure Integration in Optimization

In solving mathematical problems so far, Riemann's integral theory is quite adequate for solving pure mathematics and applications problems. But not all problems can be solved using this integration, such as a discontinuous function that is not Riemann's integration. Lebesgue integral is an integration concept based on measure and can solve finite and unlimited function problems and be solved in a more general set domain. One of the bases of this integration is the Lebesgues measure includes the set of real numbers, where the length of the interval is the endpoints. The alternative use of this integral is widely used in various studies such as partial differential equations, quantum mechanics, and probabilistic analysis, requiring the integration of arbitrary set functions. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to Lebesgue measure in integration. Search results are obtained 832 papers in the google scholar database and 997 papers using Lebesgue measure integration in optimization. It can also be seen that the research have 4 clusters and 3 clusters respectively with scattered keywords for each cluster. Finally, using bibliographic data can be obtained Lebesgues measure in integration and optimization supports many of the research and provides productive citations to citing the study.

Ken Wilber’s Integral Spirituality

The book of Ken Wilber, a philosopher, psychologist and the founder of integral theory, is devoted to the role of spirituality, faith and religion nowadays and in the future. The author systematically applies the holist integral theory to many aspects of spirituality, considering modern methodologies and philosophical and religious traditions. He analyzes various practical approaches to the states and the stages of consciousness, views parallels and finds common ground between the Western and the Eastern, scientific and meditative branches of various spiritual systems, and complements them to build his own project – integral postmetaphysics – an area that, according to K. Wilber, can withstand the criticism of postmodernism. Although the author sometimes presents the ideas in a rather partial manner, one should note his erudition and his broad outlook on world religions, phenomenology, behaviorism, structuralism, yoga, meditation, philosophy of mind, psychophysiology and many other spheres. Certainly, the integral theory is a sphere of philosophic analysis that tends to be ambiguous and not universal in application. As it is noted by K. Wilber: “An integral map is just a map. This is not a territory. It would be an obvious mistake to confuse them” [Wilber, 2006: 2]. Nevertheless, it seems that this book can be useful for those interested in the development of holism and modern approaches to spirituality.