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We prove the local logarithmic Brunn-Minkowski inequality for bodies of revolution. Furthermore, we give a generalization for one origin symmetric body of revolution and one body of revolution that does not need to be symmetric and restrict possible equality cases. The proof uses an operator theoretic approach together with the decomposition of spherical functions into isotypical components with respect to rotations around a fixed axis.
The study of quadric surfaces of revolution is a cornerstone of classical Euclidean geometry, but its extension to the three-dimensional sphere $\mathbb{S}^3$ has not been sufficiently explored. This article addresses this important gap by providing a rigorous classification and characterization of non-degenerate quadric surfaces of revolution in $\mathbb{S}^3$, namely spherical ellipsoids, hyperboloids and paraboloids, generated by the rotation of spherical conics around a geodesic axis containing their foci or is orthogonal to them. Using the concept of spherical angular momentum as a prominent geometric invariant, we discover that these surfaces constitute a remarkable class of Weingarten surfaces and prove that they are uniquely characterised by a specific cubic functional relation between their principal curvatures. This result not only provides a unified description of spherical quadric surfaces of revolution, but also highlights a profound geometric universality, reflecting exactly the same cubic Weingarten relations observed in their Euclidean and Lorentzian counterparts.
In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial parametrization, over $\mathbb{C}$, of the surface. Furthermore, we perform the first steps towards the analysis of the existence, and actual computation, of real polynomial parametrizations of surfaces of revolution. As a consequence, we give a complete picture of the real polynomiality of quadrics and we formulate a conjecture for the general case.
Given a compact surface of revolution with Laplace-beltrami operator $Δ$, we consider the spectral projector $P_{λ,δ}$ on a polynomially narrow frequency interval $[λ-δ,λ+ δ]$, which is associated to the self-adjoint operator $\sqrt{-Δ}$. For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the $L^2 \to L^{\infty}$ norm of $P_{λ,δ}$ is of order $λ^{\frac{1}{2}} δ^{\frac{1}{2}}$ up to $δ\geq λ^{-\frac{1}{32}}$. We adapt the microlocal approach introduced by Sogge for the case $δ= 1$, by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdière. This reduces the analysis to a number of estimates of explicit oscillatory integrals, for which we introduce new quantitative tools.This is the first sharp result in the case $δ\ll 1$ beyond the case of locally symmetric surfaces (torus, sphere, arithmetic hyperbolic surfaces).
We introduce the Economic Productivity of Energy (EPE), GDP generated per unit of energy consumed, as a quantitative lens to assess the sustainability of the Artificial Intelligence (AI) revolution. Historical evidence shows that the first industrial revolution, pre-scientific in the sense that technological adoption preceded scientific understanding, initially disrupted this ratio: EPE collapsed as profits outpaced efficiency, with poorly integrated technologies, and recovered only with the rise of scientific knowledge and societal adaptation. Later industrial revolutions, such as electrification and microelectronics, grounded in established scientific theory, did not exhibit comparable declines. Today's AI revolution, highly profitable yet energy-intensive, remains pre-scientific and may follow a similar trajectory in EPE. We combine this conceptual discussion with cross-country EPE data spanning the last three decades. We find that the advanced economies exhibit a consistent linear growth in EPE: those countries are the ones that contribute most to global GDP production and energy consumption, and are expected to be the most affected by the AI transition. Therefore, we advocate
It was known that if the Gaussian curvature function along each meridian on a surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ is decreasing, then the cut locus of each point of $θ^{-1}(0)$ is empty or a subarc of the opposite meridian $θ^{-1}(π).$ Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ is called a generalized von Mangoldt surface of revolution if the cut locus of each point of $θ^{-1}(0)$ is empty or a subarc of the opposite meridian $θ^{-1}(π).$ For example, the surface of revolution $(R^2, dr^2+m_0(r)^2dθ^2),$ where $m_0(x):=x/(1+x^2),$ has the same cut locus structure as above and the cut locus of each point in $r^{-1}( (0, \infty ) )$ is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature $c,$ there exists a generalized von Mangoldt surface of revolution with the same total curvature $c$ such that the Gaussian curvature
How has the credibility revolution shaped political science? We address this question by classifying 91,632 articles published between 2003 and 2023 across 156 political science journals using large language models, focusing on research design, credibility-enhancing practices, and citation patterns. We find that design-based studies -- those leveraging plausibly exogenous variation to justify causal claims -- have become increasingly common and receive a citation premium. In contrast, model-based approaches that rely on strong modeling assumptions have declined. Yet the rise of design-based work is uneven: it is concentrated in top journals and among authors at highly ranked institutions, and it is driven primarily by the growth of survey experiments. Other credibility-enhancing practices that help reduce false positives and false negatives, such as placebo tests and power calculations, remain rare. Taken together, our findings point to substantial but selective change, more consistent with a partial reform than a revolution.
Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider scaled copies of the origin-centered closed unit ball, whereas in the second problem, we search among all affine copies of the unit ball. In this paper, we interpolate between these two classical problems by considering ellipsoids of revolution. More generally, we study pairs of convex bodies $K$ and $L$, and seek a largest-volume affine image of $K$ contained within $L$, subject to certain restrictions on the allowed affine transformations. We derive first-order necessary conditions for optimality, generalizing known conditions from the unrestricted affine setting. Using these conditions, we show that an extremal ellipsoid of revolution exhibits properties analogous to those of either the largest-volume ellipsoid or the largest Euclidean ball, depending on whether the ellipsoid is considered along its axis of revolution or along the orthogonal complement of that axis.
This paper establishes an interesting connection between the family of CMC surfaces of revolution in $\mathbb E_1^3$ and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the $\mathbb E_1^3$, only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in $\mathbb E^3$ and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in $\mathbb E^3$ are spheres and right circular cylinders.
Let $K\subset \mathbb{R}^n$, $n\geq 3$, be a convex body. A point $p$ the interior of $K$ is said to be a Larman point of $K$ if for every hyperplane $Π$ passing through $p$ the section $Π\cap K$ has a $(n-2)$-plane of symmetry. If $p$ is a Larman point of $K$ and, in addition, for every section $Π\cap K$, $p$ is in the corresponding $(n-2)$-plane of symmetry, then we call $p$ a revolution point of $K$. We conjecture that if $K$ contains a Larman point which is not a revolution point, then $K$ is either an ellipsoid or a body of revolution. This generalizes a conjecture of K. Bezdek for convex bodies in $\mathbb{R}^3$ to $n \geq 4$. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if $K \subset \mathbb{R}^n$ is a strictly convex origin symmetric body that contains a revolution point $p$ which is not the origin, then $K$ is a body of revolution. This generalizes the False Axis of Revolution Theorem. We also show that if $p$ is a Larman point of $K \subset \mathbb{R}^3$ and there exists a line $L$ such that $p otin L$ and, for every plane $Π$ passing through $p$, the line of symmetry of the section $Π\cap K$ intersects $L$, then
A major shift from skilled to unskilled workers was one of the many changes caused by the Industrial Revolution, when the switch to machines contributed to decline in the social and economic status of artisans, whose skills were dismembered into discrete actions by factory-line workers. We consider what may be an analogous computing technology: the recent introduction of AI-generated art software. AI art generators such as Dall-E and Midjourney can create fully rendered images based solely on a user's prompt, just at the click of a button. Some artists fear if the cheaper price and conveyor-belt speed that comes with AI-produced images is seen as an improvement to the current system, it may permanently change the way society values/views art and artists. In this article, we consider the implications that AI art generation introduces through a post-industrial revolution historical lens. We then reflect on the analogous issues that appear to arise as a result of the AI art revolution, and we conclude that the problems raised mirror those of industrialization, giving a vital glimpse into what may lie ahead.
The Fourth Industrial Revolution, particularly Artificial Intelligence (AI), has had a profound impact on society, raising concerns about its implications and ethical considerations. The emergence of text generative AI tools like ChatGPT has further intensified concerns regarding ethics, security, privacy, and copyright. This study aims to examine the perceptions of individuals in different information flow categorizations toward AI. The results reveal key themes in participant-supplied definitions of AI and the fourth industrial revolution, emphasizing the replication of human intelligence, machine learning, automation, and the integration of digital technologies. Participants expressed concerns about job replacement, privacy invasion, and inaccurate information provided by AI. However, they also recognized the benefits of AI, such as solving complex problems and increasing convenience. Views on government involvement in shaping the fourth industrial revolution varied, with some advocating for strict regulations and others favoring support and development. The anticipated changes brought by the fourth industrial revolution include automation, potential job impacts, increased socia
Newton's mechanical revolution unifies the motion of planets in the sky and falling of apple on earth. Maxwell's electromagnetic revolution unifies electricity, magnetism, and light. Einstein's relativity revolution unifies space with time, and gravity with space-time distortion. The quantum revolution unifies particle with waves, and energy with frequency. Each of those revolution changes our world view. In this article, we will describe a revolution that is happening now: the second quantum revolution which unifies matter/space with information. In other words, the new world view suggests that elementary particles (the bosonic force particles and fermionic matter particles) all originated from quantum information (qubits): they are collective excitations of an entangled qubit ocean that corresponds to our space. The beautiful geometric Yang-Mills gauge theory and the strange Fermi statistics of matter particles now have a common algebraic quantum informational origin.
Giving explicit parametrizations of discrete constant Gaussian curvature surfaces of revolution that are defined from an integrable systems approach, we study Ricci flow for discrete surfaces, and see how discrete surfaces of revolution have a geometric realization for the Ricci flow that approaches the constant Gaussian curvature surfaces we have parametrized.
Machine learning is expected to enable the next Industrial Revolution. However, lacking standardized and automated assembly networks, ML faces significant challenges to meet ever-growing enterprise demands and empower broad industries. In the Perspective, we argue that ML needs to first complete its own Industrial Revolution, elaborate on how to best achieve its goals, and discuss new opportunities to enable rapid translation from ML's innovation frontier to mass production and utilization.
A surface of revolution is created by taking a curve in the $xy$-plane and rotating it about some axis. We develop a program which automatically generates crochet patterns for surfaces by revolution when they are obtained by rotating about the $x$-axis. In order to accomplish this, we invoke the arclength integral to determine where to take measurements for each row. In addition, a distance measure is created to optimally space increases and decreases. The result is a program that will take a function, $x$-bounds, crochet gauge, and a scale in order to produce a polished crochet pattern.
The classical laws of physics are usually invariant under time reversal. Here, we reveal a novel class of magnetomechanical effects rigorously breaking time-reversal symmetry. The effect is based on the mechanical rotation of a hard magnet around its magnetization axis in the presence of friction and an external magnetic field, which we call spin revolution. The physical reason for time-reversal symmetry breaking is the spin revolution and not the dissipation. The time-reversal symmetry breaking leads to a variety of unexpected effects including upward propulsion on vertical surfaces defying gravity as well as magnetic gyroscopic motion that is perpendicular to the applied force. In contrast to the spin, the angular momentum of spin revolution can be parallel or antiparallel to the equilibrium magnetization. The spin revolution emerges spontaneously, without external rotations, and offers various applications in areas such as magnetism, robotics and energy harvesting.
The digital transformation of the art world has become a revolution for the sector. Cryptoart, based on non-fungible tokens (NFT), is attracting the attention of artists, collectors and enthusiasts for its ability to tokenise any element that can be sold as art in the digital market. That means it is able to become a scarce resource and an economic asset by encapsulating the market value of a piece of digital art, which may or may not have a reference in the real world. This study will delve into the ethical aspects underlying what is known as the NFT Revolution, particularly impacts related to the abuse or destruction of cultural heritage, speculation and the generation of economic bubbles and environmental unsustainability.
In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution.
Revolution dynamics is studied through a minimal Ising model with three main influences (fields): personal conservatism (power-law distributed), inter-personal and group pressure, and a global field incorporating peer-to-peer and mass communications, which is generated bottom-up from the revolutionary faction. A rich phase diagram appears separating possible terminal stages of the revolution, characterizing failure phases by the features of the individuals who had joined the revolution. An exhaustive solution of the model is produced, allowing predictions to be made on the revolution's outcome.