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Triangle centers are usually studied individually or through special geometric relationships, but little attention has been given to global structure among them. In this paper we introduce several natural ways to order triangle centers, including the isosceles order, vertex order, side order, and trace order. These partial orders compare centers by their relative positions in families of triangles, such as acute triangles with a fixed shortest side. Using barycentric coordinates and symbolic computation, we determine ordering relations among many of the first 100 triangle centers listed in Kimberling's Encyclopedia of Triangle Centers. The results reveal surprising structural patterns and suggest new ways to organize and study triangle centers. For example, in an acute triangle $ABC$, with shortest side $BC$, the Gergonne point is always closer to side $BC$ than the nine-point center.
Modern large-scale data centers are known for their engineering complexity, cooling, and oversubscription challenges. To mitigate these issues, this article proposes the implementation of community data centers that are closer to consumers as part of the data center ecosystem. Having a community data center can reduce latency, minimize network burden on Internet Service Providers (ISPs), utilize full computing capability, available during disaster events, and simplify the engineering complexity associated with traditional data centers. In addition to that, this article explores one technical design for such a community data center and the business strategy for operating community data centers.
The nitrogen-vacancy (NV) center in diamond is a solid-state spin defect that has been widely adopted for quantum sensing and quantum information processing applications. Typically, experiments are performed either with a single isolated NV center or with an unresolved ensemble of many NV centers, resulting in a trade-off between measurement speed and spatial resolution or control over individual defects. In this work, we introduce an experimental platform that bypasses this trade-off by addressing multiple optically resolved NV centers in parallel. We perform charge- and spin-state manipulations selectively on multiple NV centers from within a larger set, and we manipulate and measure the electronic spin states of over 100 NV centers in parallel. We show that the high signal-to-noise ratio of the measurements enables the detection of shot-to-shot pairwise correlations between the spin states of 108 NV centers, corresponding to the simultaneous measurement of 5,778 unique correlation coefficients. We discuss how our platform can be scaled to parallel experiments with thousands of individually resolved NV centers. These results enable parallelized high-throughput sensing experiments
Among the variety of quantum emitters in hexagonal boron nitride (hBN), blue-emitting color centers, or B centers, have gathered a particular interest owing to their excellent quantum optical properties. Moreover, the fact that they can be locally activated by an electron beam makes them suitable for top-down integration in photonic devices. However, in the absence of a real-time monitoring technique sensitive to individual emitters, the activation process is stochastic in the number of emitters, and its mechanism is under debate. Here, we implement an in-situ cathodoluminescence monitoring setup capable of detecting individual quantum emitters in the blue and ultraviolet (UV) range. We demonstrate that the activation of individual B centers is spatially and temporally correlated with the deactivation of individual UV centers emitting at 4.1 eV, which are ubiquitous in hBN. We then make use of the ability to detect individual B center activation events to demonstrate the controlled creation of an array with only one emitter per irradiation site. Additionally, we demonstrate a symmetric technique for heralded selective deactivation of individual emitters. Our results provide insight
In this work, we investigate the conditions that guarantee the existence of centers on the center manifold, arising from Hopf points, in the new three-dimensional quadratic chaotic system introduced by B. Khaled et al. in 2024 in the Int. J. Data Netw. Sci. For some of the Hopf points of the system, we solve the center-focus problem on the center manifold, analyzing both its isochronicity and cyclicity. Our results significantly improve the previously known lower bound on the number of limit cycles bifurcating from Hopf points in this system, as established by B. M. Mohammed in 2025 in the Int. J. Bifurc. Chaos Appl. Sci. Eng.
We introduce a mixed characteristic analog of log canonical centers in characteristic $0$ and centers of $F$-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure of) normality of the ring. We also show the existence of a special center of perfectoid purity that detects the perfectoid purity of $R$, analogously to the splitting prime of Aberbach and Enescu, and investigate its behavior under étale morphisms.
We introduce the left, right, and full center of an algebra in a monoidal 2-category and prove their Morita invariance. This categorifies Davydov's theory of centers of algebras in monoidal categories, and specializes to give a uniform, structural account of the Drinfeld center of a fusion category, the crossed braided Drinfeld center of a fusion category graded by a finite group, and the center of a central module monoidal category. Along the way, we develop a theory of 2-adjunctions between monoidal 2-categories and a base-change construction for module 2-categories.
A center of a differential system in the plane $\mathbb{R}^2$ is an equilibrium point $p$ having a neighborhood $U$ such that $U\setminus \{p\}$ is filled of periodic orbits. A center $p$ is global when $\mathbb{R}^2\setminus \{p\}$ is filled of periodic orbits. In general is a difficult problem to distinguish the centers from the foci for a given class of differential systems, and also it is difficult to distinguish the global centers inside the centers. The goal of this paper is to classify the centers and the global centers of the following class of quintic polynomial differential systems $$ \dot{x}= y,\quad \dot{y}=-x+a_{05}\,y^5+a_{14}\,x\,y^4+a_{23}\,x^2\,y^3+a_{32}\,x^3\,y^2+a_{41}\,x^4\,y+a_{50}\,x^5, $$ in the plane $\mathbb{R}^2$.
Muonium (Mu) centers formed upon implantation of $μ^+$ in a solid have long been investigated as an experimentally accessible model of isolated hydrogen impurities. Recent discoveries of hydridic centers formed at oxygen vacancies have stimulated a renewed interest in H$^-$ and corresponding Mu$^-$ centers in oxides. However, the two diamagnetic centers, Mu$^+$ and Mu$^-$, are difficult to separate spectroscopically. In this review article, we summarize established and developing methodologies for identifying Mu$^-$ centers in solids, and review recent Mu studies on said centers supposedly formed in mayenite, oxygen-deficient SrTiO$_{3-x}$, and BaTiO$_{3-x}$H$_x$ oxyhydride.
The universal enveloping algebra $U(\mathfrak{g} )$ of a current (super)algebra or loop (super)algebra $\mathfrak{g} $ is considered over an algebraically closed field $\mathbb{K} $ with characteristic $p\ge 0$. This paper focuses on the structure of the center $Z(\mathfrak{g} )$ of $U(\mathfrak{g} )$. In the case of zero characteristic, $Z(\mathfrak{g} )$ is generated by the centers of $\mathfrak{g} $. In the case of prime characteristic, $Z(\mathfrak{g} )$ is generated by the centers of $\mathfrak{g} $ and the $p$-centers of $U(\mathfrak{g} )$. We also study the structure of $Z(\mathfrak{g} )$ in the semisimple Lie (super)algebra.
With growing use of internet and exponential growth in amount of data to be stored and processed (known as 'big data'), the size of data centers has greatly increased. This, however, has resulted in significant increase in the power consumption of the data centers. For this reason, managing power consumption of data centers has become essential. In this paper, we highlight the need of achieving energy efficiency in data centers and survey several recent architectural techniques designed for power management of data centers. We also present a classification of these techniques based on their characteristics. This paper aims to provide insights into the techniques for improving energy efficiency of data centers and encourage the designers to invent novel solutions for managing the large power dissipation of data centers.
The symmetric Kullback-Leibler centroid also called the Jeffreys centroid of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks including information retrieval, information fusion, and clustering in image, video and sound processing. However, the Jeffreys centroid is not available in closed-form for sets of categorical or normal distributions, two widely used statistical models, and thus need to be approximated numerically in practice. In this paper, we first propose the new Jeffreys-Fisher-Rao center defined as the Fisher-Rao midpoint of the sided Kullback-Leibler centroids as a plug-in replacement of the Jeffreys centroid. This Jeffreys-Fisher-Rao center admits a generic formula for uni-parameter exponential family distributions, and closed-form formula for categorical and normal distributions, matches exactly the Jeffreys centroid for same-mean normal distributions, and is experimentally observed in practice to be close to the Jeffreys centroid. Second, we define a new type of inductive centers generalizing the principle of Gauss arithmetic-geometric double sequence mean for p
In his seminal paper on triangle centers, Clark Kimberling made a number of conjectures concerning the distances between triangle centers. For example, if $D(i; j)$ denotes the distance between triangle centers $X_i$ and $X_j$ , Kimberling conjectured that $D(6; 1) \leq D(6; 3)$ for all triangles. We use symbolic mathematics techniques to prove these conjectures. In addition, we prove stronger results, using best-possible constants, such as $D(6; 1) \leq (2 -\sqrt3)D(6; 3)$.
Among the wealth of single fluorescent defects recently detected in silicon, the G center catches interest for its telecom single-photon emission that could be coupled to a metastable electron spin triplet. The G center is a unique defect where the standard Born-Oppenheimer approximation breaks down as one of its atoms can move between 6 lattice sites under optical excitation. The impact of this atomic reconfiguration on the photoluminescence properties of G centers is still largely unknown, especially in silicon-on-insulator (SOI) samples. Here, we investigate the displacement of the center-of-mass of the G center in silicon. We show that single G defects in SOI exhibit a multipolar emission and zero-phonon line fine structures with splittings up to $\sim1$ meV, both indicating a motion of the defect central atom over time. Combining polarization and spectral analysis at the single-photon level, we evidence that the reconfiguration dynamics are drastically different from the one of the unperturbed G center in bulk silicon. The SOI structure freezes the delocalization of the G defect center-of-mass and as a result, enables to isolate linearly polarized optical lines. Under above-ba
This paper presents a method for determination of the size distribution for diamond nanocrystals containing luminescent nitrogen-vacancy (NV) centers using the luminescence intensity only. We also revise the basic photo physical properties of NV centers and conclude that the luminescence quantum yield of such centers is significantly smaller than the frequently stated 100\%. The yield can be as low as 5\% for centers embedded in nanocrystals and depends on their shape and the refractive index of the surrounding medium. The paper also addresses the value of the absorption cross-section of NV centers.
We present some results on the existence and nonexistence of centers for polynomial first order ordinary differential equations with complex coefficients. In particular, we show that binomial differential equations without linear terms do not complex centers. Classes of polynomial differential equations, with more than two terms, are presented that do not have complex centers. We also study the relation between complex centers and the Pugh problem. An algorithm is described to solve the Pugh problem for equations without complex centers. The method of proof involves phase plane analysis of the polar equations an a local study of periodic solutions.
The conversion of neutral nitrogen-vacancy centers to negatively charged nitrogen-vacancy centers is demonstrated for centers created by ion implantation and annealing in high-purity diamond. Conversion occurs with surface exposure to an oxygen atmosphere at 465 C. The spectral properties of the charge-converted centers are investigated. Charge state control of nitrogen-vacancy centers close to the diamond surface is an important step toward the integration of these centers into devices for quantum information and magnetic sensing applications.
Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a Bousfield-Kan type spectral sequence that computes the homotopy groups. Nontrivial classes of examples are given as illustration throughout.
In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the centers of the Meusnier spheres of the curve is an evolute curve. Then, we give the relations between this evolute curve and the focal curve. Also, we give some relations between helices, slant helices and the locus of the centers of the Meusnier spheres of the curve.
We begin from the quantization algebras and constraint for analyzing the choice of centers in the first-order formulation without losing generality. Then we calculate the entanglement entropy in the non-interacting $p$-form theory in $2p+2$ dimensional Euclidean flat background with an $S^{2p}$ entangling surface. The universal term of the entanglement entropy in the non-interacting $p$-form theory is determined in terms of the universal terms of the non-interacting zero-form theory. We also prove the strong subadditivity in the non-interacting theory with the non-trivial centers. Finally, we calculate the mutual information with centers in two-dimensional conformal field theory. The result shows that the mutual information is independent of the choice of centers.