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Concept# Sphere

Summary

A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.
Geometrically, a sphere can be formed by rotating a ci

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We prove that every Schwartz function in Euclidean space can be completely recovered given only its restrictions and the restrictions of its Fourier transform to all origin-centered spheres whose radii are square roots of integers. In particular, the only Schwartz function which, together with its Fourier transform, vanishes on these spheres, is the zero function. We show that this remains true if we replace the spheres by surfaces or discrete sets of points which are sufficiently small perturbations of these spheres. In a complementary, opposite direction, we construct infinite dimensional spaces of Fourier eigenfunctions vanishing on on certain discrete subsets of those spheres. The proofs combine harmonic analysis, the theory of modular forms and algebraic number theory.

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radii are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an applica-tion, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Via-zovska [1].(c) 2022 Published by Elsevier Inc.

Innovation is crucial and decisive for the long-term development of a metropolitan region in the global economy. While firms certainly play a significant role in the development of specific innovation, the environment that nur-tures innovative firms and disseminates innovations in the economy involves a complex web of interactions among a range of actors, including firms, universities, and local governments. The performance of these actors, but also their interactions, have significant impacts on the overall innovation performance of the region. The great complexity of these interactions makes it challenging for many regions to strengthen the innovation per-formance of their areas. To address this challenge, the main innovation actors â ranging from firms and universi-ties to governments â have made great efforts to learn from the famous example of Silicon Valley.
However, despite some self-promotion, almost all such efforts of these regions have failed. The root of this failure lies in the lack of holistic and historical view in the learning process and the lack of proper coordination among actors in the implementation process. The existing learning about the Silicon Valley model mostly focuses on the period since the 1970s when Silicon Valleyâs success has been widely acknowledged; and it provides static analysis by listing the key features of an already successful metropolitan innovation system. While such an approach cer-tainly helps with understanding, it could be misleading, since the actors often implement the lessons learned from the advanced stage that do not fit the current stage of the region. Hence, this thesis studies the 170 years of Silicon Valleyâs development to shed light on how the actors in the three institutional spheres interact and coordi-nate in the development process. Provided that the changed global environment and the different local cultures invite the additional issue of whether the learnings are applicable to the new contexts, three case studies have been conducted in this thesis â about Beijing, Shenzhen, and Hangzhou in China â over a time span of 70 years (from 1950 to 2020). These cases are then analyzed with the framework and theory grounded from the case of Silicon Valley.
Synthesizing the analysis of the four selected cases, the development path of the Metropolitan Innovation System (MIS) in Beijing most closely resembles the Silicon Valley model, since both moved from a âlaissez-faireâ model to a balanced âTriple Helixâ model (where industry, university, and government have intensive interactions and play interwoven roles). The MIS in Shenzhen reveals a path that starts from the âstatistâ model, while the one in Hangzhou started with a semi-balanced Triple Helix model. The three paths converge in the sense of heading towards the balanced Triple Helix model and all three improve the interaction and coordination between the ac-tors in the three spheres in a process that ultimately creates an open and conducive environment. This includes the ability to attract high-quality talent and cognitive capital to the region.
By providing thick-descriptions of the four cases and the tentative theory about the development of MIS, this the-sis contributes to the literature of innovation system, the Triple Helix model, and the New Institutional Economics, and contributes to practice by providing 20 recommendations for the actors in the three spheres.