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• Optica (IF 9.778) Pub Date : 2020-09-28
Zheng Gong, Xianwen Liu, Yuntao Xu, and Hong X. Tang

Thin film lithium niobate (LN) is bringing renewed potential to the realm of integrated photonics. Its strong quadratic and cubic nonlinearities as well as wide transparency window are ideal for realizing on-chip self-referenced soliton microcombs. However, broadband Kerr cavity soliton generation in LN thin films remains challenging due to limited dispersion control and competition from strong stimulated

更新日期：2020-09-25
• Optica (IF 9.778) Pub Date : 2020-09-28
Yanwen Hu, Shenhe Fu, Hao Yin, Zhen Li, Zhaohui Li, and Zhenqiang Chen

Nondiffracting light beams have been attracting considerable attention for their various applications in both classical and quantum optics. Whereas substantial investigations on generation of the nondiffracting beams were made, their lateral dimension is much larger than optical wavelength. Here we present both theoretically and experimentally a study of the generation of nondiffracting light beams

更新日期：2020-09-25
• Optica (IF 9.778) Pub Date : 2020-09-28
John Steeves, J. Kent Wallace, Christian Kettenbeil, and Jeffrey Jewell

Precision wavefront sensing and interferometry are essential in many fields of industry and fundamental research. Characterization of semiconductor devices, optics in lithography systems, and biologic features of living cells all require measurement resolution at the nanometer level. The field of high-contrast imaging in space-based astronomy has pushed wavefront sensing requirements to a new regime

更新日期：2020-09-25
• Optica (IF 9.778) Pub Date : 2020-09-21
Erika Janitz, Mihir K. Bhaskar, and Lilian Childress

Coherent interfaces between optical photons and long-lived matter qubits form a key resource for a broad range of quantum technologies. Cavity quantum electrodynamics (cQED) offers a route to achieve such an interface by enhancing interactions between cavity-confined photons and individual emitters. Over the last two decades, a promising new class of emitters based on defect centers in diamond has

更新日期：2020-09-22
• Optica (IF 9.778) Pub Date : 2020-09-21
Thomas Bauer, Svetlana N. Khonina, Ilya Golub, Gerd Leuchs, and Peter Banzer

Topological properties of light attract tremendous attention in the optics communities and beyond. For instance, light beams gain robustness against certain deformations when carrying topological features, enabling intriguing applications. We report on the observation of a topological structure contained in an optical beam, i.e., a twisted ribbon formed by the electric field vector per se, in stark

更新日期：2020-09-22
• Optica (IF 9.778) Pub Date : 2020-09-21
Istvan Gyongy, Sam W. Hutchings, Abderrahim Halimi, Max Tyler, Susan Chan, Feng Zhu, Stephen McLaughlin, Robert K. Henderson, and Jonathan Leach

Imaging systems with temporal resolution play a vital role in a diverse range of scientific, industrial, and consumer applications, e.g., fluorescent lifetime imaging in microscopy and time-of-flight (ToF) depth sensing in autonomous vehicles. In recent years, single-photon avalanche diode (SPAD) arrays with picosecond timing capabilities have emerged as a key technology driving these systems forward

更新日期：2020-09-22
• Optica (IF 9.778) Pub Date : 2020-09-04
Pengcheng Huo, Maowen Song, Wenqi Zhu, Cheng Zhang, Lu Chen, Henri J. Lezec, Yanqing Lu, Amit Agrawal, and Ting Xu

We realize a dielectric metasurface that enables full-color generation and ultrasmooth brightness variation. The reproduced artwork “girl with a pearl earring” features photorealistic color representation and stereoscopic image impression, mimicking the texture of an oil-painting.

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-04
Zhiquan Yuan, Heming Wang, Lue Wu, Maodong Gao, and Kerry Vahala

The linewidth of regenerative oscillators is enhanced by amplitude–phase coupling of the oscillator field [Phys. Rev. 160, 290 (1967)]. In laser oscillators, this effect is well known for its impact on semiconductor laser performance. Here, this coupling is studied in Brillouin lasers. Because their gain is parametric, the coupling and linewidth enhancement are shown to originate from phase mismatch

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-25
Norihiko Nishizawa and Masahito Yamanaka

Using nonlinear optical effects, we can control the optical spectra in various ways. Here, we discovered a novel phenomenon in ultrafast nonlinear fiber optics, namely, periodical spectral peaking. A narrow spectral dip on a pulse turns into a sharp, intense peak through a nonlinear effect. This phenomenon shows periodical evolution behavior in the soliton regime. If we use absorption spectra with

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-03
Melissa A. Guidry, Ki Youl Yang, Daniil M. Lukin, Ashot Markosyan, Joshua Yang, Martin M. Fejer, and Jelena Vučković

Silicon carbide (SiC) is rapidly emerging as a leading platform for the implementation of nonlinear and quantum photonics. Here, we find that commercial SiC, which hosts a variety of spin qubits, possesses low optical absorption that can enable SiC integrated photonics with quality factors exceeding ${10^7}$. We fabricate multimode microring resonators with quality factors as high as 1.1 million, and

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-28
Dongin Jeong, Dohyeon Kwon, Igju Jeon, In Hwan Do, Jungwon Kim, and Hansuek Lee

Silica microcombs have a high potential for generating tens of gigahertz of optical pulse trains with ultralow timing jitter, which is highly suitable for higher speed and higher bandwidth information systems. So far, the accurate characterization of timing jitter in microcombs has been limited by the measurement methods—although theoretically predicted to be ${\gt}{20 {\rm dB}}$ better performance

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-25
Enrico Ridente, Matthew Weidman, Mikhail Mamaikin, Clemens Jakubeit, Ferenc Krausz, and Nicholas Karpowicz

Simple and compact laser systems facilitate the stable and reproducible generation of high-power few-cycle laser pulses. We demonstrate the amplification of 15 fs pulses at 2.1 µm, employing a hybrid phase-matching scheme for optical parametric chirped pulse amplification. A combination of two BBO crystals with type-I and type-II phase-matching placed in close vicinity is utilized as a single amplification

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-21
Ruth R. Sims, Sohaib Abdul Rehman, Martin O. Lenz, Sarah I. Benaissa, Ezra Bruggeman, Adam Clark, Edward W. Sanders, Aleks Ponjavic, Leila Muresan, Steven F. Lee, and Kevin O’Holleran

We introduce single molecule light field microscopy (SMLFM), a new class of three-dimensional (3D) single molecule localization microscopy. By segmenting the back focal plane of a microscope objective with an array of microlenses to generate multiple 2D perspective views, the same single fluorophore can be imaged from different angles. These views, in combination with a bespoke fitting algorithm, enable

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-24
Baurzhan Muminov and Luat T. Vuong

Deep learning convolutional neural networks generally involve multiple-layer, forward-backward propagation machine-learning algorithms that are computationally costly. In this work, we demonstrate an alternative scheme to convolutional neural nets that reconstructs an original image from its optically preprocessed, Fourier-encoded pattern. The scheme is much less computationally demanding and more

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-04
Martin Schnell, Shravan Gupta, Tomasz P. Wrobel, Michael G. Drage, Rohit Bhargava, and P. Scott Carney

Label-free imaging approaches seek to simplify and augment histopathologic assessment by replacing the current practice of staining by dyes to visualize tissue morphology with quantitative optical measurements. Quantitative phase imaging (QPI) operates with visible/UV light and thus provides a resolution matched to current practice. Here we introduce and demonstrate confocal QPI for label-free imaging

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-04
Chen Zhu, Liangjun Lu, Wensheng Shan, Weihan Xu, Gangqiang Zhou, Linjie Zhou, and Jianping Chen

Optical beam-forming networks (OBFNs) based on optical true-time delay lines (OTTDLs) are well known as the promising candidate for solving the bandwidth limitation of traditional electronic phased array antennas (PAAs) due to beam squinting. Here we report, to the best of our knowledge, the first monolithic $1 \times 8$ microwave photonic beamformer, based on switchable OTTDLs on the silicon-on-insulator

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-08
Yaowen Hu, Christian Reimer, Amirhassan Shams-Ansari, Mian Zhang, and Marko Loncar

Crystals are ubiquitous in nature and are at the heart of material research, solid-state science, and quantum physics. Unfortunately, the controllability of solid-state crystals is limited by the complexity of many-body dynamics and the presence of defects. In contrast, synthetic crystal structures, realized by, e.g., optical lattices, have recently enabled the investigation of various physical processes

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-21
Petr Zapletal, Bogdan Galilo, and Andreas Nunnenkamp

Combining topologically protected chiral light transport and laser amplification gives rise to topological lasers, whose operation is immune to fabrication imperfections and defects, uncovering the role of topology in a novel nonlinear non-Hermitian regime. We study a topological laser based on the photonic Haldane model with selective pumping of chiral edge modes described by saturable gain. We investigate

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-03
Reshma A. Mohandas, Lalitha Ponnampalam, Lianhe Li, Paul Dean, Alwyn J. Seeds, Edmund H. Linfield, A. Giles Davies, and Joshua R. Freeman

The accuracy of high-resolution spectroscopy depends critically on the stability, frequency control, and traceability available from laser sources. In this work, we report exact tunable frequency synthesis and phase control of a terahertz laser. The terahertz laser is locked by a terahertz injection phase lock loop for the first time, with the terahertz signal generated by heterodyning selected lines

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-31
Yang Cao, Kathirvel Nallappan, Hichem Guerboukha, Guofu Xu, and Maksim Skorobogatiy

While in most existing terahertz communications systems, the THz carrier wave is transmitted via free-space channels, the THz waveguide-based integrated solutions can be of great utility at both the transmitter and receiver ends, thus simplifying the miniaturization and mass production of cost-effective THz communications systems. Here we present a new type of modular THz integrated circuits based

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-03
Jun Wang, Marco Clementi, Momchil Minkov, Andrea Barone, Jean-François Carlin, Nicolas Grandjean, Dario Gerace, Shanhui Fan, Matteo Galli, and Romuald Houdré

Second-harmonic generation in nonlinear materials can be greatly enhanced by realizing doubly resonant cavities with high quality factors. However, fulfilling such doubly resonant condition in photonic crystal (PhC) slab cavities is a long-standing challenge, because of the difficulty in engineering photonic bandgaps around both frequencies. Here, by implementing a second-harmonic bound state in the

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-03
Cheng Guo, Haiwen Wang, and Shanhui Fan

There has been substantial interest in miniaturizing optical systems by flat optics. However, one essential optical component, free space, fundamentally cannot be substituted with conventional local flat optics with space-dependent transfer functions, since the transfer function of free space is momentum-dependent instead. Overcoming this difficulty is important to achieve the utmost miniaturization

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-08
Zongda Li, Yiqing Xu, Stéphane Coen, Stuart G. Murdoch, and Miro Erkintalo

Driven Kerr nonlinear optical resonators can sustain localized structures known as dissipative Kerr cavity solitons, which have recently attracted significant attention as the temporal counterparts of microresonator optical frequency combs. While conventional wisdom asserts that bright cavity solitons can only exist when driving in the region of anomalous dispersion, recent theoretical studies have

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-24
Sushree S. Dash, Pinaki Mukherjee, Daniel Haskel, Richard A. Rosenberg, and Miguel Levy

Bi-substituted iron garnet films are extensively used in the fabrication of nonreciprocal devices in optical telecommunications. The miniaturization of these devices for on-chip integration requires the development of more efficient magneto-optic materials than presently available. Recent evidence has emerged of large near-surface enhancements in the magneto-optic response in these materials. However

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-21
Daodang Wang, Ping Xu, Zhendong Wu, Xiangyu Fu, Rengmao Wu, Ming Kong, Jian Liang, Bo Zhang, and Rongguang Liang

Freeform optics, due to the more general surface geometry that offers high degrees of design freedom to control light propagation, has already been widely used in both nonimaging optics and imaging optics. With the recent advances in design and fabrication of freeform optics, one of the remaining challenges is how to accurately measure freeform optical surfaces, especially those included in freeform

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-09
Z. D. Reed, D. A. Long, H. Fleurbaey, and J. T. Hodges

Accurate and spectroscopic measurements of molecular transition frequencies are increasingly being employed in a variety of rigorous tests of physics, including the validity of quantum electrodynamics, the proton–electron mass ratio, and the dipole moment of the electron. Near-infrared molecular transitions may also underpin secondary frequency standards for length metrology and provide convenient

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-08
Abdullah Rasmita, Chongyun Jiang, Hui Ma, Zhurun Ji, Ritesh Agarwal, and Wei-bo Gao

Utilizing the spin or valley degree of freedom is a promising approach for realizing more energy-efficient information processing devices. Circularly polarized light can be used to generate spin/valley current in monolayer 2D transition metal dichalcogenides. We observe a geometrically dependent photocurrent in heterostructure ${{\rm MoS}_2}/{{\rm WSe}_2}$, where light with a different circular polarization

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-21
Yong-Nan Sun, Zhao-Di Liu, Joseph Bowles, Geng Chen, Xiao-Ye Xu, Jian-Shun Tang, Chuan-Feng Li, and Guang-Can Guo

Dimensionality plays an important role in various information-theoretic tasks and can be seen as a key resource for quantum information processing. In this work, for the first time to the best of our knowledge, we report an experimental test of classical and quantum dimensions in a prepare-and-measure scenario through a nonlinear dimension witness (NDW), where the preparer and the measurer have no

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-08
Hairun Guo, Wenle Weng, Junqiu Liu, Fan Yang, Wolfgang Hänsel, Camille Sophie Brès, Luc Thévenaz, Ronald Holzwarth, and Tobias J. Kippenberg

High resolution and fast detection of molecular vibrational absorption is important for organic synthesis, pharmaceutical processes, and environmental monitoring, and is enabled by mid-infrared (mid-IR) laser frequency combs via dual-comb spectroscopy. Here, we demonstrate a novel and highly simplified approach to broadband mid-IR dual-comb spectroscopy via supercontinuum generation, achieved using

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-04
Xinhao Li, Zheng Jie Tan, and Nicholas X. Fang

Flat optics for spatially resolved amplitude and phase modulation usually rely on 2D patterning of layered structures with spatial thickness variation. For example, Fabry–Perot-type multilayer structures have been applied widely as spectral filter arrays. However, it is challenging to efficiently fabricate large-scale multilayer structures with spatially variable thicknesses. Conventional photo/eB

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-09-11
Marie-Christine Zdora, Pierre Thibault, Willy Kuo, Vincent Fernandez, Hans Deyhle, Joan Vila-Comamala, Margie P. Olbinado, Alexander Rack, Peter M. Lackie, Orestis L. Katsamenis, Matthew J. Lawson, Vartan Kurtcuoglu, Christoph Rau, Franz Pfeiffer, and Irene Zanette

High-contrast, high-resolution imaging of biomedical specimens is indispensable for studying organ function and pathologies. Conventional histology, the gold standard for soft-tissue visualization, is limited by its anisotropic spatial resolution, elaborate sample preparation, and lack of quantitative image information. X-ray absorption or phase tomography have been identified as promising alternatives

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-27
Sander A. Mann, Dimitrios L. Sounas, and Andrea Alù

We reply to the comment on our recent article [Optica 6, 104 (2019) [CrossRef] ], in which we demonstrated that nonreciprocal cavities comply with the resonance time-bandwidth limit. We fully stand by our original claims and further elucidate how breaking of reciprocity is not required to achieve the large field enhancements and time-bandwidth products observed in the comment and our article. We also

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-27
Kosmas L. Tsakmakidis, Yun You, Tomasz Stefański, and Linfang Shen

In their paper in Optica 6, 104 (2019) [CrossRef] , Mann et al. claim that linear, time-invariant nonreciprocal structures cannot overcome the time-bandwidth limit and do not exhibit an advantage over their reciprocal counterparts, specifically with regard to their time-bandwidth performance. In this Comment, we argue that these conclusions are unfounded. On the basis of both rigorous full-wave simulations

更新日期：2020-09-20
• Optica (IF 9.778) Pub Date : 2020-08-07
Tianyu Wang and Chris Xu

Three-photon microscopy has been increasingly adopted for probing neural activities beyond the typical two-photon imaging depth. In this review, we outline the unique properties that differentiate three-photon microscopy from two-photon microscopy for in vivo imaging in biological samples, especially in the mouse brain. We present a systematic summary of the optimization of three-photon imaging parameters

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-07
Bowen Li, Jian Xing, Dohyeon Kwon, Yijun Xie, Neeraj Prakash, Jungwon Kim, and Shu-Wei Huang

The bidirectional mode-locked oscillator is an emerging light source architecture suitable for dual-comb applications. Therein, bidirectional mode-locked fiber lasers (MLFLs) are particularly promising for their cost effectiveness, system compactness, and environmental robustness. However, the pulse energy has been limited to tens of picojoules, restricting practical dual-comb applications, especially

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-14
Yuta Ueda, Takahiko Shindo, Shigeru Kanazawa, Naoki Fujiwara, and Mitsuteru Ishikawa

The huge amount of data traffic behind the rapid growth of cloud computing is putting pressure on the operation of mobile fronthauls and data center networks so there is a need to improve their power consumption and latency. We developed an electro-optically tunable laser diode employing a tunable filter that is practically tuned even with small refractive index change of the electro-optic effect.

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-23
Viacheslav Mazlin, Kristina Irsch, Michel Paques, Jose-Alain Sahel, Mathias Fink, and Claude A. Boccara

Cell-resolution optical imaging methods, such as confocal microscopy and full-field optical coherence tomography, capture flat optical sections of the sample. If the sample is curved, the optical field sections through several sample layers, and the view of each layer is reduced. Here we present curved-field optical coherence tomography, capable of capturing optical sections of arbitrary curvature

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-14
Florian Döring, Benedikt Rösner, Manuel Langer, Adam Kubec, Armin Kleibert, Jörg Raabe, Carlos A. F. Vaz, Maxime Lebugle, and Christian David

X-ray free-electron lasers (XFELs) are paving the way towards new experiments in many scientific fields, such as ultrafast science, nonlinear spectroscopy, and coherent imaging. However, the strong intensity fluctuations inherent to the lasing process in these sources often lead to problems in signal normalization. In order to address this challenge, we designed, fabricated, and characterized diffractive

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-03
Xiong Dun, Hayato Ikoma, Gordon Wetzstein, Zhanshan Wang, Xinbin Cheng, and Yifan Peng

Diffractive achromats (DAs) promise ultra-thin and light-weight form factors for full-color computational imaging systems. However, designing DAs with the optimal optical transfer function (OTF) distribution suitable for image reconstruction algorithms has been a difficult challenge. Emerging end-to-end optimization paradigms of diffractive optics and processing algorithms have achieved impressive

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-10
Taeseong Woo, Su Hyun Jung, Cheolwoo Ahn, Byungjae Hwang, Hyunggee Kim, Joo H. Kang, and Jung-Hoon Park

To date, imaging systems have generally been designed to provide an even spatiotemporal resolution across the field of view (FOV). However, this becomes a fundamental limitation when we aim to simultaneously observe varying dynamics at different parts of the FOV. In conventional imaging systems, to capture fast dynamics occurring at only a small portion of the FOV, the entire imaging system’s sampling

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-28
Md Borhan Mia, Syed Z. Ahmed, Ishtiaque Ahmed, Yun Jo Lee, Minghao Qi, and Sangsik Kim

Electromagnetic coupling is ubiquitous in photonic systems and transfers optical signals from one device to the other, creating crosstalk between devices. While this allows the functionality of some photonic components such as couplers, it limits the integration density of photonic chips, and many approaches have been proposed to reduce the crosstalk. However, due to the wave nature of light, complete

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-10
Po Dong, Argishti Melikyan, Kwangwoong Kim, Noriaki Kaneda, Brian Stern, and Yves Baeyens

In the last decade, coherent optical transmission technology has dominated the provision of high-capacity communications in metro and long-haul networks and is expected to expand to short-reach networks such as data center and passive optical networks. Capacities of more than 1.5 Tb/s have been demonstrated for a single wavelength. In-phase/quadrature (IQ) modulation allows information to be encoded

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-05
Tianran Liu and Andrea Fiore

On-chip spectrometers with tailored spectral range and compact footprint have been pursued widely in the last decade. Splitting different frequencies typically requires a propagation length that scales inversely with the frequency resolution, which leads to a trade-off between resolution and size. Scattering media in the diffusive regime provide a long light path and multipath interference in a compact

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-05
Yiyin Zhou, Yuanhao Miao, Solomon Ojo, Huong Tran, Grey Abernathy, Joshua M. Grant, Sylvester Amoah, Gregory Salamo, Wei Du, Jifeng Liu, Joe Margetis, John Tolle, Yong-hang Zhang, Greg Sun, Richard A. Soref, Baohua Li, and Shui-Qing Yu

Monolithic lasers on Si have long been anticipated as an enabler of full photonic integration, and significant progress in GeSn material development shows promise for such laser devices. While there are many reports focused on optically pumped lasers, in this work, we demonstrate electrically injected GeSn lasers on Si. We grew a GeSn/SiGeSn heterostructure diode on a Si substrate in a ridge waveguide

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-07
Yi Zhou, Yu-Xuan Ren, Jiawei Shi, Huade Mao, and Kenneth K. Y. Wong

Solitons can self-assemble into stable bound states, also denoted as soliton molecules that exhibit molecule-like dynamics. Soliton molecules have been predominantly investigated in the anomalous dispersion mode-locked fiber lasers. However, the soliton molecule dynamic evolution is still largely unexplored in the normal dispersion regime. We reveal here that, in the normal dispersion regime, the buildup

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-03
A. T. M. Anishur Rahman and P. F. Barker

The ability to create dynamic, tailored optical potentials has become important across fields ranging from biology to quantum science. We demonstrate a method for the creation of arbitrary optical tweezer potentials using the broadband spectral profile of a superluminescent diode combined with the chromatic aberration of a lens. A tunable filter, typically used for ultrafast laser pulse shaping, allows

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-28
Sepehr Benis, Claudiu M. Cirloganu, Nicholas Cox, Trenton Ensley, Honghua Hu, Gero Nootz, Peter D. Olszak, Lazaro A. Padilha, Davorin Peceli, Matthew Reichert, Scott Webster, Milton Woodall, David J. Hagan, and Eric W. Van Stryland

This paper presents three-photon absorption (3PA) measurement results for nine direct-gap semiconductors, including full 3PA spectra for ZnSe, ZnS, and GaAs. These results, along with our theory of 3PA using an eight-band Kane model (four bands with double spin degeneracy), help to explain the significant disagreements between experiments and theory in the literature to date. 3PA in the eight-band

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-07
Yiqing Xu, Yi Lin, Alexander Nielsen, Ian Hendry, Stéphane Coen, Miro Erkintalo, Huilin Ma, and Stuart G. Murdoch

With demonstrated applications ranging from metrology to telecommunications, soliton microresonator frequency combs have emerged over the past decade as a remarkable new technology. However, standard implementations allow only for the generation of combs whose repetition rate is tied closely to the fundamental resonator free-spectral range (FSR), offering little or no dynamic control over the comb

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-11
Declan J. Armstrong, Timo A. Nieminen, Alexander B. Stilgoe, Anatolii V. Kashchuk, Isaac C. D. Lenton, and Halina Rubinsztein-Dunlop

We demonstrate how optical tweezers combined with a three-dimensional force detection system and high-speed camera are used to study the swimming force and behavior of trapped micro-organisms. By utilizing position sensitive detection, we measure the motility force of trapped particles, regardless of orientation. This has the advantage of not requiring complex beam shaping or microfluidic controls

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-20
Cecilia Clivati, Roberto Aiello, Giuseppe Bianco, Claudio Bortolotti, Paolo De Natale, Valentina Di Sarno, Pasquale Maddaloni, Giuseppe Maccaferri, Alberto Mura, Monia Negusini, Filippo Levi, Federico Perini, Roberto Ricci, Mauro Roma, Luigi Santamaria Amato, Mario Siciliani de Cumis, Matteo Stagni, Alberto Tuozzi, and Davide Calonico

Among the most powerful techniques for the exploration of the Universe is very long baseline interferometry (VLBI), which is based on the simultaneous observation of radio sources in the sky with arrays of distant ground-based antennas. One of the effects currently limiting its ultimate sensitivity is the phase-instability of the reference clocks adopted at each antenna. This term can be made negligible

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-23
Hadas Frostig, Itamar Vidal, Robert Fischer, Hanan Herzig Sheinfux, and Yaron Silberberg

Efforts to understand the physics of rogue waves have motivated the study of mechanisms that produce rare, extreme events, often through analogous optical setups. As many studies have reported nonlinear generation mechanisms, recent work has explored whether optical rogue events can be produced in linear systems. Here we report the observation of linear rogue events with tunable height, generated from

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-31
Alex Turpin, Gabriella Musarra, Valentin Kapitany, Francesco Tonolini, Ashley Lyons, Ilya Starshynov, Federica Villa, Enrico Conca, Francesco Fioranelli, Roderick Murray-Smith, and Daniele Faccio

Traditional paradigms for imaging rely on the use of a spatial structure, either in the detector (pixels arrays) or in the illumination (patterned light). Removal of the spatial structure in the detector or illumination, i.e., imaging with just a single-point sensor, would require solving a very strongly ill-posed inverse retrieval problem that to date has not been solved. Here, we demonstrate a data-driven

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-14
Jacob B. Khurgin

We show that concept of parity-time (PT) symmetry can be expanded to include mixed photon-exciton modes by demonstrating that eigenmodes of active (pumped) strongly coupled cavity polaritons with population inversion exhibit characteristics that are remarkably akin to those of coupled photonic structures with parity-time symmetry. The exceptional point occurs when the Rabi splitting of polariton branches

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-17
Sergey Lepeshov and Alex Krasnok

Progress in light scattering engineering made it feasible to develop optical tweezers allowing capture, hold, and controllable displacement of submicrometer-size particles and biological structures. However, the momentum conservation law imposes a fundamental restriction on the optical pressure to be repulsive in paraxial fields, which severely limits the capabilities of optomechanical control, e.g

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-22
Kuniaki Konishi, Daisuke Akai, Yoshio Mita, Makoto Ishida, Junji Yumoto, and Makoto Kuwata-Gonokami

Circularly polarized light in the vacuum ultraviolet (VUV) region is important for probing the structural and electronic properties of matter. Moreover, a circularly polarized VUV coherent light enables one to observe the dynamics of biomolecules and electron spins in solids. The development of a table-top technology to directly generate circularly polarized VUV coherent light is of great value, owing

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-10
Vyacheslav E. Leshchenko, Bradford K. Talbert, Yu Hang Lai, Sha Li, Yaguo Tang, Stephen J. Hageman, Greg Smith, Pierre Agostini, Louis F. DiMauro, and Cosmin I. Blaga

The development of mid-infrared, high-energy pulses of extremely short pulse duration is of great importance to the strong-field physics community, as it enables extension of the high-harmonic cutoff towards the keV range of photon energies. Here we demonstrate a ${\rm{C}}{{\rm{r}}^{2 +}}:{\rm ZnSe}$ laser amplifier delivering 7 mJ, 100 fs pulses at 1 kHz repetition rate and 2.4 µm central wavelength

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-11
Helder M. Crespo, Tobias Witting, Miguel Canhota, Miguel Miranda, and John W. G. Tisch

In laser–matter interaction experiments, it is of paramount importance to characterize the laser pulse on target (in situ) and at full power. This allows pulse optimization and meaningful comparison with theory, and it can shed fundamental new light on pulse distortions occurring in or on the target. Here we introduce and demonstrate a new technique based on dispersion-scan using the concurrent third

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-08-04
Michael Chen, David Ren, Hsiou-Yuan Liu, Shwetadwip Chowdhury, and Laura Waller

This publisher’s note reports typographical corrections to Optica 7, 394 (2020) [CrossRef] .

更新日期：2020-08-20
• Optica (IF 9.778) Pub Date : 2020-07-01
Quandong Huang and Kin Seng Chiang

We propose a polarization-insensitive ultra-broadband mode filter based on a 3D graphene structure buried in a few-mode optical waveguide. Our experimental device fabricated by burying an L-shaped graphene structure in a three-mode polymer waveguide provides polarization-insensitive filtering of the fundamental mode, while keeping the higher-order modes, with a mode extinction ratio of ${\sim}{{17}}\;{\rm{dB}}$

更新日期：2020-07-21
• Optica (IF 9.778) Pub Date : 2020-07-13
Jian Wang,Xuanwen Hua,Changliang Guo,Wenhao Liu,Shu Jia

We introduce Airy-beam tomographic microscopy (ATM) for high-resolution, volumetric, inertia-free imaging of biological specimens. The work exploits the highly adjustable Airy trajectories in the 3D space, transforming the conventional telecentric wide-field imaging scheme that requires sample or focal-plane scanning to acquire 3D information. The results present a consistent near-diffraction-limited

更新日期：2020-07-21
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Nuclear Fusion Physics and Technology/Algebra summary

Definition: Cartesian multiplicationEdit

Cartesian multiplication of two sets A, B with |a>,|b> elements is defined as
$A \times B = \{ (|a>,|b>): |a> \in A \wedge |b> \in B \}$

Definition: ProjectionEdit

Projection f from set A to set B is AxB subset defined as
$f = \{ (|a>,|b>) \in A \times B : \forall |a> \in A \exists_1 |b> \in B \}$
and notation $f(|a>) = |b>$ is used.

Definition: Body of numbersEdit

Body of numbers T is defined as
$T = \{ c \in \mathbb{C}: (\exists c_1, c_2)(c_1 \neq c_2) \wedge (\forall c_1, c_2)(\exists c_3 = c_1 + c_2 \wedge \exists c_4 = c_1 . c_2 \wedge \exists c_5 = -c_1 \wedge \exists 0 \neq c_6 = c^{-1}_1) \}$

Definition: Vector spaceEdit

Vector space V is defined as
$\mathbb{V} = \{ (V, T, f_1, f_2): (\forall |a>,|b> \in V)(f_1(|a>,|b>) = f_1(|b>,|a>))(...) \}$

Definition: FunctionEdit

Function is a projection $f: \mathbb{C} \rightarrow \mathbb{C}$, which meets
$(\forall x \in D_f )(\exists_1 y \in H_f) (f(x) = y)$

Definition: FunctionalEdit

Functional is a projection $f: \mathbb{V} \rightarrow \mathbb{C}$, which meets
$(\forall x \in D_f )(\exists_1 y \in H_f) (f(x) = y)$

Definition: OperatorEdit

Operator is a projection $f: \mathbb{V} \rightarrow \mathbb{V}$, which meets
$(\forall x \in D_f )(\exists_1 y \in H_f) (f(x) = y)$

format: plain


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• ### PandaX-III: Searching for Neutrinoless Double Beta Decay with High Pressure $^{136}$Xe Gas Time Projection Chambers(1610.08883)

Oct. 28, 2016 hep-ex, nucl-ex, physics.ins-det
Searching for the Neutrinoless Double Beta Decay (NLDBD) is now regarded as the topmost promising technique to explore the nature of neutrinos after the discovery of neutrino masses in oscillation experiments. PandaX-III (Particle And Astrophysical Xenon Experiment III) will search for the NLDBD of $^{136}$Xe at the China Jin Ping underground Laboratory (CJPL). In the first phase of the experiment, a high pressure gas Time Projection Chamber (TPC) will contain 200 kg, 90% $^{136}$Xe enriched gas operated at 10 bar. Fine pitch micro-pattern gas detector (Microbulk Micromegas) will be used at both ends of the TPC for the charge readout with a cathode in the middle. Charge signals can be used to reconstruct tracks of NLDBD events and provide good energy and spatial resolution. The detector will be immersed in a large water tank to ensure $\sim$5 m of water shielding in all directions. The second phase, a ton-scale experiment, will consist of five TPCs in the same water tank, with improved energy resolution and better control over backgrounds.
• ### Fast Timing for High-Rate Environments with Micromegas(1601.00123)

Jan. 12, 2016 hep-ex, physics.ins-det
The current state of the art in fast timing resolution for existing experiments is of the order of 100 ps on the time of arrival of both charged particles and electromagnetic showers. Current R&D on charged particle timing is approaching the level of 10 ps but is not primarily directed at sustained performance at high rates and under high radiation (as would be needed for HL-LHC pileup mitigation). We demonstrate a Micromegas based solution to reach this level of performance. The Micromegas acts as a photomultiplier coupled to a Cerenkov-radiator front window, which produces sufficient UV photons to convert the ~100 ps single-photoelectron jitter into a timing response of the order of 10-20 ps per incident charged particle. A prototype has been built in order to demonstrate this performance. The first laboratory tests with a pico-second laser have shown a time resolution of the order of 27 ps for ~50 primary photoelectrons, using a bulk Micromegas readout.


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# Acting for a covariant derivative on charged spinor [closed]

For field, theory what i know $i.e$,complex scalar QED \begin{align} D_\mu \phi = \partial_{\mu} \phi - i Q A_{\mu} \phi \end{align} and \begin{align} D_\mu \phi^{\dagger} = \partial_{\mu} \phi^{\dagger} + i Q A_{\mu} \phi^{\dagger} \end{align}

In Geometry, by adapting vielbein formulation for spinor $\psi$, \begin{align} D_\nu \psi = \partial_\nu \psi + \frac{1}{4} w_{\nu ab} \gamma^{[a} \gamma^{b]} \psi, \quad D_\nu \bar{\psi} = \partial_\nu \bar{\psi} - \frac{1}{4} w_{\nu ab} \bar{\psi} \gamma^{[a}\gamma^{b]} \end{align}

Consider charged spinor, then the covariant derivatives

\begin{align} D_a \psi= e^{\mu}{}_a (\partial_\mu - i eQ A_\mu + \frac{1}{2} \sigma^{bc} e_b{}^\nu \partial_\mu e_{c\nu})\psi, \quad \sigma^{ab} = \frac{\gamma^a \gamma^b - \gamma^b \gamma^a}{4} \end{align} Then how about $D_a \bar{\psi}$?

My guess is \begin{align} D_a \bar{\psi}= e^{\mu}{}_a (\partial_\mu\bar{\psi} + i eQ A_\mu \bar{\psi} - \frac{1}{2} \bar{\psi}\sigma^{bc} e_b{}^\nu \partial_\mu e_{c\nu}), \quad \sigma^{ab} = \frac{\gamma^a \gamma^b - \gamma^b \gamma^a}{4} \end{align} Is this right?

## closed as off-topic by ACuriousMind♦, user36790, honeste_vivere, CuriousOne, knzhouJul 7 '16 at 7:27

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, Community, honeste_vivere, CuriousOne, knzhou
If this question can be reworded to fit the rules in the help center, please edit the question.

• Use the fact that $\bar \psi \psi$ is a scalar, apply the derivative and the Leibniz rule, it should be obvious enough – Slereah Jul 6 '16 at 9:40
• – Qmechanic Jul 7 '16 at 18:36


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# You can also size point together with your thumb otherwise hand

You can also size point together with your thumb otherwise hand

Exactly how, this new digit occupies throughout the $10$ level of examine whenever kept straight out. So, pacing off in reverse up until the little finger completely occludes the tree tend to provide the distance of one’s adjoining edge of a right triangle. If it range is $30$ paces what is the height of your forest? Well, we need specific facts. Guess the rate was $3$ feet. Then the adjoining size is $90$ legs. The fresh multiplier ‘s the tangent regarding $10$ values, or:

And therefore getting benefit out of thoughts we’ll say is $1/6$ (an excellent $5$ percent mistake). To make sure that answer is more or less $15$ feet:

Furthermore, you should use your own thumb rather than your first. To make use of your first you might proliferate because of the $1/6$ the brand new surrounding front side, to make use of the thumb throughout the $1/30$ because this approximates the fresh tangent of $2$ degrees:

This is reversed. If you know the newest peak out of one thing a radius aside you to definitely is included by the flash otherwise little finger, then you definitely would multiply one to peak from the appropriate add up to look for the point.

## Very first attributes

The sine function is scheduled for everyone actual $\theta$ and contains various $[-step 1,1]$ . Clearly because the $\theta$ gusts of wind in the $x$ -axis, the position of $y$ coordinate actually starts to repeat in itself. We say new sine mode is actually unexpected having months $2\pi$ . A chart often instruct:

The fresh chart shows a couple of periods. The fresh new wavy facet of the graph ‘s the reason it setting is always model unexpected actions, like the amount of sunshine in one day, and/or alternating current powering a pc.

Using this chart – otherwise considering if the $y$ accentuate try $0$ – we see the sine setting has actually zeros at any integer several off $\pi$ , or $k\pi$ , $k$ inside $\dots,-dos,-step one, 0, step 1, 2, \dots$ .

The brand new cosine means is similar, where it offers an equivalent website name and variety, but is “out-of phase” to your sine bend. A graph out-of both reveals the two was relevant:

The brand new cosine form is simply a shift of one’s sine setting (otherwise the other way around). We see your zeros of one’s cosine form happens on facts of form $\pi/dos + k\pi$ , $k$ in the $\dots,-dos,-step one, 0, step 1, dos, \dots$ .

The latest tangent form doesn’t have most of the $\theta$ because of its website name, rather those individuals issues in which department of the $0$ takes place http://datingranking.net/it/incontri-video/ are omitted. Such can be found if cosine was $0$ , otherwise once again at $\pi/dos + k\pi$ , $k$ from inside the $\dots,-2,-1, 0, step 1, 2, \dots$ . All of the the newest tangent form will be all the genuine $y$ .

New tangent setting is even unexpected, although not with period $2\pi$ , but rather just $\pi$ . A graph will teach this. Here i avoid the vertical asymptotes by continuing to keep him or her off the new spot domain and you will layering multiple plots.

$r\theta = l$ , where $r$ is the radius out-of a group and you can $l$ the duration of the fresh arc molded of the perspective $\theta$ .

Both try related, while the a group from $2\pi$ radians and you may 360 level. So to alter out-of levels with the radians it will take multiplying from the $2\pi/360$ and to transfer out-of radians to amount it will take multiplying from the $360/(2\pi)$ . The newest deg2rad and you can rad2deg functions are available for this.

Inside the Julia , the new features sind , cosd , tand , cscd , secd , and you will cotd are around for explain the job regarding writing the two operations (that’s sin(deg2rad(x)) is equivalent to sind(x) ).

## The sum of-and-distinction formulas

Take into account the point-on these devices network $(x,y) = (\cos(\theta), \sin(\theta))$ . With regards to $(x,y)$ (otherwise $\theta$ ) is there an approach to portray brand new angle located by the spinning a supplementary $\theta$ , that is what are $(\cos(2\theta), \sin(2\theta))$ ?

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# Progressions - 1

Algebra Level 2

The Harmonic Mean of two numbers $$a,b$$ is 4. It is also given that $$2A + G^2 = 27$$.

• $$A$$ represents Arithmetic Mean of $$a,b$$ and $$G$$ represents Geometric Mean of $$a,b$$.

Find the numbers $$a, \space b$$ .

Are you weak in maths ? Do you want to practice more maths for competitive exams ? Then try my set Mathematics Done Right

×


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# Should Passive LPF/HPF Component Selection Depend on Ability of Input to Source Current?

When choosing R and C for a passive LPF or HPF, should you take into account the input's ability to source current? I'm currently working on a passive HPF for an input that can supply up to 10 mA @ 3.3 V. The input will be PWM. Should this be a factor in choosing R, for example?

Side note: The output of the HPF will go through an op amp buffer.

Not really. It's the load impedance as a whole that matters. For example:

Network 1 has a DC input impedance of 10.1k and network2 has a DC input impedance of 11k. For your 3.3v input that makes a difference of $$\i=\frac{3.3v}{10.1k} = 326uA\$$ for network 1 and $$\i=\frac{3.3v}{11k} = 300uA\$$ for network 2.

So if your load impedance is sized correctly, then the LPF impedance will have minimal impact on current draw.

• The load will be high impedance as the output of the filter will go to an op amp buffer. – HumbleOne Nov 27 '19 at 17:04
• @HumbleOne That's even better, as an opamp input impedance can be set pretty high. – Aaron Nov 27 '19 at 17:20

Within the pass-band, the filter will not have much effect on how much current the source has to provide. It will essentially just connect the source to the load, and the load will determine the current.

In the stop-band(s) it might matter. If your high-pass filter provides a DC short to ground, for example, that could alter the operating point of the source, even if it doesn't affect the current required at the operating point. Generally it's not hard to design your filter to avoid this problem, but you must be aware of it and design for it.

• Why would there be a DC short to ground? – HumbleOne Nov 27 '19 at 17:07
• For example if you connected an inductor from the signal line to ground. – The Photon Nov 27 '19 at 17:26


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# Sigma-awgebra

In madematicaw anawysis and in probabiwity deory, a σ-awgebra (awso σ-fiewd) on a set X is a cowwection Σ of subsets of X dat incwudes X itsewf, is cwosed under compwement, and is cwosed under countabwe unions.

The definition impwies dat it awso incwudes de empty subset and dat it is cwosed under countabwe intersections.

The pair (X, Σ) is cawwed a measurabwe space or Borew space.

A σ-awgebra is a type of awgebra of sets. An awgebra of sets needs onwy to be cwosed under de union or intersection of finitewy many subsets, which is a weaker condition, uh-hah-hah-hah.[1]

The main use of σ-awgebras is in de definition of measures; specificawwy, de cowwection of dose subsets for which a given measure is defined is necessariwy a σ-awgebra. This concept is important in madematicaw anawysis as de foundation for Lebesgue integration, and in probabiwity deory, where it is interpreted as de cowwection of events which can be assigned probabiwities. Awso, in probabiwity, σ-awgebras are pivotaw in de definition of conditionaw expectation.

In statistics, (sub) σ-awgebras are needed for de formaw madematicaw definition of a sufficient statistic,[2] particuwarwy when de statistic is a function or a random process and de notion of conditionaw density is not appwicabwe.

If X = {a, b, c, d}, one possibwe σ-awgebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is de empty set. In generaw, a finite awgebra is awways a σ-awgebra.

If {A1, A2, A3, …} is a countabwe partition of X den de cowwection of aww unions of sets in de partition (incwuding de empty set) is a σ-awgebra.

A more usefuw exampwe is de set of subsets of de reaw wine formed by starting wif aww open intervaws and adding in aww countabwe unions, countabwe intersections, and rewative compwements and continuing dis process (by transfinite iteration drough aww countabwe ordinaws) untiw de rewevant cwosure properties are achieved - de σ-awgebra produced by dis process is known as de Borew awgebra on de reaw wine, and can awso be conceived as de smawwest (i.e. "coarsest") σ-awgebra containing aww de open sets, or eqwivawentwy containing aww de cwosed sets. It is foundationaw to measure deory, and derefore modern probabiwity deory, and a rewated construction known as de Borew hierarchy is of rewevance to descriptive set deory.

## Motivation

There are at weast dree key motivators for σ-awgebras: defining measures, manipuwating wimits of sets, and managing partiaw information characterized by sets.

### Measure

A measure on X is a function dat assigns a non-negative reaw number to subsets of X; dis can be dought of as making precise a notion of "size" or "vowume" for sets. We want de size of de union of disjoint sets to be de sum of deir individuaw sizes, even for an infinite seqwence of disjoint sets.

One wouwd wike to assign a size to every subset of X, but in many naturaw settings, dis is not possibwe. For exampwe, de axiom of choice impwies dat, when de size under consideration is de ordinary notion of wengf for subsets of de reaw wine, den dere exist sets for which no size exists, for exampwe, de Vitawi sets. For dis reason, one considers instead a smawwer cowwection of priviweged subsets of X. These subsets wiww be cawwed de measurabwe sets. They are cwosed under operations dat one wouwd expect for measurabwe sets; dat is, de compwement of a measurabwe set is a measurabwe set and de countabwe union of measurabwe sets is a measurabwe set. Non-empty cowwections of sets wif dese properties are cawwed σ-awgebras.

### Limits of sets

Many uses of measure, such as de probabiwity concept of awmost sure convergence, invowve wimits of seqwences of sets. For dis, cwosure under countabwe unions and intersections is paramount. Set wimits are defined as fowwows on σ-awgebras.

• The wimit supremum of a seqwence A1, A2, A3, ..., each of which is a subset of X, is
${\dispwaystywe \wimsup _{n\to \infty }A_{n}=\bigcap _{n=1}^{\infty }\bigcup _{m=n}^{\infty }A_{m}.}$
• The wimit infimum of a seqwence A1, A2, A3, ..., each of which is a subset of X, is
${\dispwaystywe \wiminf _{n\to \infty }A_{n}=\bigcup _{n=1}^{\infty }\bigcap _{m=n}^{\infty }A_{m}.}$
• If, in fact,
${\dispwaystywe \wiminf _{n\to \infty }A_{n}=\wimsup _{n\to \infty }A_{n},}$
den de ${\dispwaystywe \wim _{n\to \infty }A_{n}}$ exists as dat common set.

### Sub σ-awgebras

In much of probabiwity, especiawwy when conditionaw expectation is invowved, one is concerned wif sets dat represent onwy part of aww de possibwe information dat can be observed. This partiaw information can be characterized wif a smawwer σ-awgebra which is a subset of de principaw σ-awgebra; it consists of de cowwection of subsets rewevant onwy to and determined onwy by de partiaw information, uh-hah-hah-hah. A simpwe exampwe suffices to iwwustrate dis idea.

Imagine you and anoder person are betting on a game dat invowves fwipping a coin repeatedwy and observing wheder it comes up Heads (H) or Taiws (T). Since you and your opponent are each infinitewy weawdy, dere is no wimit to how wong de game can wast. This means de sampwe space Ω must consist of aww possibwe infinite seqwences of H or T:

${\dispwaystywe \Omega =\{H,T\}^{\infty }=\{(x_{1},x_{2},x_{3},\dots ):x_{i}\in \{H,T\},i\geq 1\}.}$

However, after n fwips of de coin, you may want to determine or revise your betting strategy in advance of de next fwip. The observed information at dat point can be described in terms of de 2n possibiwities for de first n fwips. Formawwy, since you need to use subsets of Ω, dis is codified as de σ-awgebra

${\dispwaystywe {\madcaw {G}}_{n}=\{A\times \{H,T\}^{\infty }:A\subset \{H,T\}^{n}\}.}$

Observe dat den

${\dispwaystywe {\madcaw {G}}_{1}\subset {\madcaw {G}}_{2}\subset {\madcaw {G}}_{3}\subset \cdots \subset {\madcaw {G}}_{\infty },}$

where ${\dispwaystywe {\madcaw {G}}_{\infty }}$ is de smawwest σ-awgebra containing aww de oders.

## Definition and properties

### Definition

Let X be some set, and wet ${\dispwaystywe {\madcaw {P}}(X)}$ represent its power set. Then a subset ${\dispwaystywe \Sigma \subseteq {\madcaw {P}}(X)}$ is cawwed a σ-awgebra if it satisfies de fowwowing dree properties:[3]

1. X is in Σ, and X is considered to be de universaw set in de fowwowing context.
2. Σ is cwosed under compwementation: If A is in Σ, den so is its compwement, X \ A.
3. Σ is cwosed under countabwe unions: If A1, A2, A3, ... are in Σ, den so is A = A1A2A3 ∪ … .

From dese properties, it fowwows dat de σ-awgebra is awso cwosed under countabwe intersections (by appwying De Morgan's waws).

It awso fowwows dat de empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts dat its compwement, de empty set, is awso in Σ. Moreover, since {X, ∅} satisfies condition (3) as weww, it fowwows dat {X, ∅} is de smawwest possibwe σ-awgebra on X. The wargest possibwe σ-awgebra on X is 2X:=${\dispwaystywe {\madcaw {P}}(X)}$.

Ewements of de σ-awgebra are cawwed measurabwe sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-awgebra over X, is cawwed a measurabwe space. A function between two measurabwe spaces is cawwed a measurabwe function if de preimage of every measurabwe set is measurabwe. The cowwection of measurabwe spaces forms a category, wif de measurabwe functions as morphisms. Measures are defined as certain types of functions from a σ-awgebra to [0, ∞].

A σ-awgebra is bof a π-system and a Dynkin system (λ-system). The converse is true as weww, by Dynkin's deorem (bewow).

### Dynkin's π-λ deorem

This deorem (or de rewated monotone cwass deorem) is an essentiaw toow for proving many resuwts about properties of specific σ-awgebras. It capitawizes on de nature of two simpwer cwasses of sets, namewy de fowwowing.

A π-system P is a cowwection of subsets of X dat is cwosed under finitewy many intersections, and
a Dynkin system (or λ-system) D is a cowwection of subsets of X dat contains X and is cwosed under compwement and under countabwe unions of disjoint subsets.

Dynkin's π-λ deorem says, if P is a π-system and D is a Dynkin system dat contains P den de σ-awgebra σ(P) generated by P is contained in D. Since certain π-systems are rewativewy simpwe cwasses, it may not be hard to verify dat aww sets in P enjoy de property under consideration whiwe, on de oder hand, showing dat de cowwection D of aww subsets wif de property is a Dynkin system can awso be straightforward. Dynkin's π-λ Theorem den impwies dat aww sets in σ(P) enjoy de property, avoiding de task of checking it for an arbitrary set in σ(P).

One of de most fundamentaw uses of de π-λ deorem is to show eqwivawence of separatewy defined measures or integraws. For exampwe, it is used to eqwate a probabiwity for a random variabwe X wif de Lebesgue-Stiewtjes integraw typicawwy associated wif computing de probabiwity:

${\dispwaystywe \madbb {P} (X\in A)=\int _{A}\,F(dx)}$ for aww A in de Borew σ-awgebra on R,

where F(x) is de cumuwative distribution function for X, defined on R, whiwe ${\dispwaystywe \madbb {P} }$ is a probabiwity measure, defined on a σ-awgebra Σ of subsets of some sampwe space Ω.

### Combining σ-awgebras

Suppose ${\dispwaystywe \textstywe \{\Sigma _{\awpha }:\awpha \in {\madcaw {A}}\}}$ is a cowwection of σ-awgebras on a space X.

• The intersection of a cowwection of σ-awgebras is a σ-awgebra. To emphasize its character as a σ-awgebra, it often is denoted by:
${\dispwaystywe \bigwedge _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }.}$
Sketch of Proof: Let Σ denote de intersection, uh-hah-hah-hah. Since X is in every Σα, Σ is not empty. Cwosure under compwement and countabwe unions for every Σα impwies de same must be true for Σ. Therefore, Σ is a σ-awgebra.
• The union of a cowwection of σ-awgebras is not generawwy a σ-awgebra, or even an awgebra, but it generates a σ-awgebra known as de join which typicawwy is denoted
${\dispwaystywe \bigvee _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }=\sigma \weft(\bigcup _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }\right).}$
A π-system dat generates de join is
${\dispwaystywe {\madcaw {P}}=\weft\{\bigcap _{i=1}^{n}A_{i}:A_{i}\in \Sigma _{\awpha _{i}},\awpha _{i}\in {\madcaw {A}},\ n\geq 1\right\}.}$
Sketch of Proof: By de case n = 1, it is seen dat each ${\dispwaystywe \Sigma _{\awpha }\subset {\madcaw {P}}}$, so
${\dispwaystywe \bigcup _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }\subset {\madcaw {P}}.}$
This impwies
${\dispwaystywe \sigma \weft(\bigcup _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }\right)\subset \sigma ({\madcaw {P}})}$
by de definition of a σ-awgebra generated by a cowwection of subsets. On de oder hand,
${\dispwaystywe {\madcaw {P}}\subset \sigma \weft(\bigcup _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }\right)}$
which, by Dynkin's π-λ deorem, impwies
${\dispwaystywe \sigma ({\madcaw {P}})\subset \sigma \weft(\bigcup _{\awpha \in {\madcaw {A}}}\Sigma _{\awpha }\right).}$

### σ-awgebras for subspaces

Suppose Y is a subset of X and wet (X, Σ) be a measurabwe space.

• The cowwection {YB: B ∈ Σ} is a σ-awgebra of subsets of Y.
• Suppose (Y, Λ) is a measurabwe space. The cowwection {AX : AY ∈ Λ} is a σ-awgebra of subsets of X.

### Rewation to σ-ring

A σ-awgebra Σ is just a σ-ring dat contains de universaw set X.[4] A σ-ring need not be a σ-awgebra, as for exampwe measurabwe subsets of zero Lebesgue measure in de reaw wine are a σ-ring, but not a σ-awgebra since de reaw wine has infinite measure and dus cannot be obtained by deir countabwe union, uh-hah-hah-hah. If, instead of zero measure, one takes measurabwe subsets of finite Lebesgue measure, dose are a ring but not a σ-ring, since de reaw wine can be obtained by deir countabwe union yet its measure is not finite.

### Typographic note

σ-awgebras are sometimes denoted using cawwigraphic capitaw wetters, or de Fraktur typeface. Thus (X, Σ) may be denoted as ${\dispwaystywe \scriptstywe (X,\,{\madcaw {F}})}$ or ${\dispwaystywe \scriptstywe (X,\,{\madfrak {F}})}$.

## Particuwar cases and exampwes

### Separabwe σ-awgebras

A separabwe σ-awgebra (or separabwe σ-fiewd) is a σ-awgebra ${\dispwaystywe {\madcaw {F}}}$ dat is a separabwe space when considered as a metric space wif metric ${\dispwaystywe \rho (A,B)=\mu (A{\madbin {\triangwe }}B)}$ for ${\dispwaystywe A,B\in {\madcaw {F}}}$ and a given measure ${\dispwaystywe \mu }$ (and wif ${\dispwaystywe \triangwe }$ being de symmetric difference operator).[5] Note dat any σ-awgebra generated by a countabwe cowwection of sets is separabwe, but de converse need not howd. For exampwe, de Lebesgue σ-awgebra is separabwe (since every Lebesgue measurabwe set is eqwivawent to some Borew set) but not countabwy generated (since its cardinawity is higher dan continuum).

A separabwe measure space has a naturaw pseudometric dat renders it separabwe as a pseudometric space. The distance between two sets is defined as de measure of de symmetric difference of de two sets. Note dat de symmetric difference of two distinct sets can have measure zero; hence de pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a singwe eqwivawence cwass, de resuwting qwotient set can be properwy metrized by de induced metric. If de measure space is separabwe, it can be shown dat de corresponding metric space is, too.

### Simpwe set-based exampwes

Let X be any set.

• The famiwy consisting onwy of de empty set and de set X, cawwed de minimaw or triviaw σ-awgebra over X.
• The power set of X, cawwed de discrete σ-awgebra.
• The cowwection {∅, A, Ac, X} is a simpwe σ-awgebra generated by de subset A.
• The cowwection of subsets of X which are countabwe or whose compwements are countabwe is a σ-awgebra (which is distinct from de power set of X if and onwy if X is uncountabwe). This is de σ-awgebra generated by de singwetons of X. Note: "countabwe" incwudes finite or empty.
• The cowwection of aww unions of sets in a countabwe partition of X is a σ-awgebra.

### Stopping time sigma-awgebras

A stopping time ${\dispwaystywe \tau }$ can define a ${\dispwaystywe \sigma }$-awgebra ${\dispwaystywe {\madcaw {F}}_{\tau }}$, de so-cawwed ${\dispwaystywe \sigma }$-Awgebra of τ-past, which in a fiwtered probabiwity space describes de information up to de random time ${\dispwaystywe \tau }$ in de sense dat, if de fiwtered probabiwity space is interpreted as a random experiment, de maximum information dat can be found out about de experiment from arbitrariwy often repeating it untiw de time ${\dispwaystywe \tau }$ is ${\dispwaystywe {\madcaw {F}}_{\tau }}$.[6]

## σ-awgebras generated by famiwies of sets

### σ-awgebra generated by an arbitrary famiwy

Let F be an arbitrary famiwy of subsets of X. Then dere exists a uniqwe smawwest σ-awgebra which contains every set in F (even dough F may or may not itsewf be a σ-awgebra). It is, in fact, de intersection of aww σ-awgebras containing F. (See intersections of σ-awgebras above.) This σ-awgebra is denoted σ(F) and is cawwed de σ-awgebra generated by F.

If F is empty, den σ(F)={X, ∅}. Oderwise σ(F) consists of aww de subsets of X dat can be made from ewements of F by a countabwe number of compwement, union and intersection operations.

For a simpwe exampwe, consider de set X = {1, 2, 3}. Then de σ-awgebra generated by de singwe subset {1} is σ({{1}}) = {∅, {1}, {2, 3}, {1, 2, 3}}. By an abuse of notation, when a cowwection of subsets contains onwy one ewement, A, one may write σ(A) instead of σ({A}); in de prior exampwe σ({1}) instead of σ({{1}}). Indeed, using σ(A1, A2, ...) to mean σ({A1, A2, ...}) is awso qwite common, uh-hah-hah-hah.

There are many famiwies of subsets dat generate usefuw σ-awgebras. Some of dese are presented here.

### σ-awgebra generated by a function

If f is a function from a set X to a set Y and B is a σ-awgebra of subsets of Y, den de σ-awgebra generated by de function f, denoted by σ(f), is de cowwection of aww inverse images f -1(S) of de sets S in B. i.e.

${\dispwaystywe \sigma (f)=\{f^{-1}(S)\,|\,S\in B\}.}$

A function f from a set X to a set Y is measurabwe wif respect to a σ-awgebra Σ of subsets of X if and onwy if σ(f) is a subset of Σ.

One common situation, and understood by defauwt if B is not specified expwicitwy, is when Y is a metric or topowogicaw space and B is de cowwection of Borew sets on Y.

If f is a function from X to Rn den σ(f) is generated by de famiwy of subsets which are inverse images of intervaws/rectangwes in Rn:

${\dispwaystywe \sigma (f)=\sigma \weft(\{f^{-1}((a_{1},b_{1}]\times \cdots \times (a_{n},b_{n}]):a_{i},b_{i}\in \madbb {R} \}\right).}$

A usefuw property is de fowwowing. Assume f is a measurabwe map from (X, ΣX) to (S, ΣS) and g is a measurabwe map from (X, ΣX) to (T, ΣT). If dere exists a measurabwe map h from (T, ΣT) to (S, ΣS) such dat f(x) = h(g(x)) for aww x, den σ(f) ⊂ σ(g). If S is finite or countabwy infinite or, more generawwy, (S, ΣS) is a standard Borew space (e.g., a separabwe compwete metric space wif its associated Borew sets), den de converse is awso true.[7] Exampwes of standard Borew spaces incwude Rn wif its Borew sets and R wif de cywinder σ-awgebra described bewow.

### Borew and Lebesgue σ-awgebras

An important exampwe is de Borew awgebra over any topowogicaw space: de σ-awgebra generated by de open sets (or, eqwivawentwy, by de cwosed sets). Note dat dis σ-awgebra is not, in generaw, de whowe power set. For a non-triviaw exampwe dat is not a Borew set, see de Vitawi set or Non-Borew sets.

On de Eucwidean space Rn, anoder σ-awgebra is of importance: dat of aww Lebesgue measurabwe sets. This σ-awgebra contains more sets dan de Borew σ-awgebra on Rn and is preferred in integration deory, as it gives a compwete measure space.

### Product σ-awgebra

Let ${\dispwaystywe (X_{1},\Sigma _{1})}$ and ${\dispwaystywe (X_{2},\Sigma _{2})}$ be two measurabwe spaces. The σ-awgebra for de corresponding product space ${\dispwaystywe X_{1}\times X_{2}}$ is cawwed de product σ-awgebra and is defined by

${\dispwaystywe \Sigma _{1}\times \Sigma _{2}=\sigma (\{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\}).}$

Observe dat ${\dispwaystywe \{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\}}$ is a π-system.

The Borew σ-awgebra for Rn is generated by hawf-infinite rectangwes and by finite rectangwes. For exampwe,

${\dispwaystywe {\madcaw {B}}(\madbb {R} ^{n})=\sigma \weft(\weft\{(-\infty ,b_{1}]\times \cdots \times (-\infty ,b_{n}]:b_{i}\in \madbb {R} \right\}\right)=\sigma \weft(\weft\{(a_{1},b_{1}]\times \cdots \times (a_{n},b_{n}]:a_{i},b_{i}\in \madbb {R} \right\}\right).}$

For each of dese two exampwes, de generating famiwy is a π-system.

### σ-awgebra generated by cywinder sets

Suppose

${\dispwaystywe X\subset \madbb {R} ^{\madbb {T} }=\{f:f{\text{ is a function from }}\madbb {T} {\text{ to }}\madbb {R} \}}$

is a set of reaw-vawued functions on ${\dispwaystywe \madbb {T} }$. Let ${\dispwaystywe {\madcaw {B}}(\madbb {R} )}$ denote de Borew subsets of R. A cywinder subset of X is a finitewy restricted set defined as

${\dispwaystywe C_{t_{1},\dots ,t_{n}}(B_{1},\dots ,B_{n})=\{f\in X:f(t_{i})\in B_{i},1\weq i\weq n\}.}$

Each

${\dispwaystywe \{C_{t_{1},\dots ,t_{n}}(B_{1},\dots ,B_{n}):B_{i}\in {\madcaw {B}}(\madbb {R} ),1\weq i\weq n\}}$

is a π-system dat generates a σ-awgebra ${\dispwaystywe \textstywe \Sigma _{t_{1},\dots ,t_{n}}}$. Then de famiwy of subsets

${\dispwaystywe {\madcaw {F}}_{X}=\bigcup _{n=1}^{\infty }\bigcup _{t_{i}\in \madbb {T} ,i\weq n}\Sigma _{t_{1},\dots ,t_{n}}}$

is an awgebra dat generates de cywinder σ-awgebra for X. This σ-awgebra is a subawgebra of de Borew σ-awgebra determined by de product topowogy of ${\dispwaystywe \madbb {R} ^{\madbb {T} }}$ restricted to X.

An important speciaw case is when ${\dispwaystywe \madbb {T} }$ is de set of naturaw numbers and X is a set of reaw-vawued seqwences. In dis case, it suffices to consider de cywinder sets

${\dispwaystywe C_{n}(B_{1},\dots ,B_{n})=(B_{1}\times \cdots \times B_{n}\times \madbb {R} ^{\infty })\cap X=\{(x_{1},x_{2},\dots ,x_{n},x_{n+1},\dots )\in X:x_{i}\in B_{i},1\weq i\weq n\},}$

for which

${\dispwaystywe \Sigma _{n}=\sigma (\{C_{n}(B_{1},\dots ,B_{n}):B_{i}\in {\madcaw {B}}(\madbb {R} ),1\weq i\weq n\})}$

is a non-decreasing seqwence of σ-awgebras.

### σ-awgebra generated by random variabwe or vector

Suppose ${\dispwaystywe (\Omega ,\Sigma ,\madbb {P} )}$ is a probabiwity space. If ${\dispwaystywe \textstywe Y:\Omega \to \madbb {R} ^{n}}$ is measurabwe wif respect to de Borew σ-awgebra on Rn den Y is cawwed a random variabwe (n = 1) or random vector (n > 1). The σ-awgebra generated by Y is

${\dispwaystywe \sigma (Y)=\{Y^{-1}(A):A\in {\madcaw {B}}(\madbb {R} ^{n})\}.}$

### σ-awgebra generated by a stochastic process

Suppose ${\dispwaystywe (\Omega ,\Sigma ,\madbb {P} )}$ is a probabiwity space and ${\dispwaystywe \madbb {R} ^{\madbb {T} }}$ is de set of reaw-vawued functions on ${\dispwaystywe \madbb {T} }$. If ${\dispwaystywe \textstywe Y:\Omega \to X\subset \madbb {R} ^{\madbb {T} }}$ is measurabwe wif respect to de cywinder σ-awgebra ${\dispwaystywe \sigma ({\madcaw {F}}_{X})}$ (see above) for X, den Y is cawwed a stochastic process or random process. The σ-awgebra generated by Y is

${\dispwaystywe \sigma (Y)=\weft\{Y^{-1}(A):A\in \sigma ({\madcaw {F}}_{X})\right\}=\sigma (\{Y^{-1}(A):A\in {\madcaw {F}}_{X}\}),}$

de σ-awgebra generated by de inverse images of cywinder sets.

## References

1. ^ "Probabiwity, Madematicaw Statistics, Stochastic Processes". Random. University of Awabama in Huntsviwwe, Department of Madematicaw Sciences. Retrieved 30 March 2016.
2. ^ Biwwingswey, Patrick (2012). Probabiwity and Measure (Anniversary ed.). Wiwey. ISBN 978-1-118-12237-2.
3. ^ Rudin, Wawter (1987). Reaw & Compwex Anawysis. McGraw-Hiww. ISBN 978-0-07-054234-1.
4. ^ Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiwey & Sons. p. 12. ISBN 978-0-470-31795-2.
5. ^ Džamonja, Mirna; Kunen, Kennef (1995). "Properties of de cwass of measure separabwe compact spaces" (PDF). Fundamenta Madematicae: 262. If ${\dispwaystywe \mu }$ is a Borew measure on ${\dispwaystywe X}$, de measure awgebra of ${\dispwaystywe (X,\mu )}$ is de Boowean awgebra of aww Borew sets moduwo ${\dispwaystywe \mu }$-nuww sets. If ${\dispwaystywe \mu }$ is finite, den such a measure awgebra is awso a metric space, wif de distance between de two sets being de measure of deir symmetric difference. Then, we say dat ${\dispwaystywe \mu }$ is separabwe iff dis metric space is separabwe as a topowogicaw space.
6. ^ Fischer, Tom (2013). "On simpwe representations of stopping times and stopping time sigma-awgebras". Statistics and Probabiwity Letters. 83 (1): 345–349. arXiv:1112.1603. doi:10.1016/j.spw.2012.09.024.
7. ^ Kawwenberg, Owav (2001). Foundations of Modern Probabiwity (2nd ed.). Springer. p. 7. ISBN 978-0-387-95313-7.


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# Scale of prior for non-linear beta regression

I have a pretty basic question regarding how to specify priors in a model with a logit link function. I stumbled upon this trying to fit the unimodal Sharpe-Schoolfield equation to food consumption rate of fishes as a function of temperature using brms:

b(T) = \frac{b_0(T_c)e^{E(\frac{1}{kT_c} - \frac{1}{kT})}} {1+e^{E_h(\frac{1}{kT_h} - \frac{1}{kT})}}

where b(T) is the rate at temperature, T, in Kelvin (K), b_0(T_c) is the rate at a common temperature, E (eV) describes the thermal sensitivity of the biological rate, k is Boltzmann’s constant (8.62 × 10−5 eV K−1), Eh (eV) characterises the decline in the rate past the optimum temperature and Th (K) is the temperature at which half the rate is reduced due to high temperatures. This is the parameterization in Padfield et al., (2020).

The data I have compiled consist of many species, and their feeding rate is expressed in as fraction of max. Hence I want to model this with species-varying parameter b_0. I further rescaled the feeding rates to not have 1’s in order to fit a Beta model (can also consider using a zero_one_inflated_beta once I figure out my little issue). Temperature has also been centred so that 0 is the peak temperature within species. The data can be found here:

d <- readr::read_csv("https://raw.githubusercontent.com/maxlindmark/stan_data/main/dat.csv")

Given how the data have been centred, I think these priors make sense (I’m using uniform priors here to make sure they don’t wander off… sometimes I get biomodal posteriors in non-linear models unless at least one parameter has a uniform prior):

prior <-
prior("uniform(0, 1)", nlpar = "b0", lb = 0, ub = 1) + # consumption at the reference temperature, 0 (note all y are bounded in [0,1])
prior("uniform(0, 3)", nlpar = "E", lb = 0, ub = 3) + # activation energy
prior("uniform(0, 4)", nlpar = "Eh", lb = 0, ub = 4) + # de-activation energy
prior("uniform(0, 10)", nlpar = "Th", lb = 0, ub = 10) # temperature at half rate after decline - hence it should be above 0 which is the peak temperature in the data

With that, we can fit the model using an identity link first:

library(readr)
library(brms)
library(dplyr)
library(ggplot2)
library(modelr)
library(tidybayes)

# Fit a Beta model with an identity link
beta_identity <-
brm(bf(con_percent_scaled ~ (b0 * exp(E*((1/(k*(T_c + 273.15))) - 1/(k*(temp_ct + 273.15))))) / (1 + exp(Eh*((1/(k*(Th + 273.15))) - (1/(k*(temp_ct + 273.15)))))),
b0 ~ 1 + (1|species_ab), E ~ 1, Eh ~ 1, Th ~ 1, nl = TRUE),
data = d, family = Beta(link = "identity"), prior = prior, iter = 4000, cores = 3, chains = 3)

d %>%
data_grid(temp_ct = seq_range(temp_ct, n = 50)) %>%
mutate(k = 8.62*10^-5,
T_c = 0) %>%
add_predicted_draws(beta_identity, re_formula = NA, scale = "response") %>%
ggplot(aes(x = temp_ct, y = con_percent_scaled)) +
stat_lineribbon(aes(y = .prediction), .width = c(.95), alpha = 0.4, fill = "grey") +
geom_point(data = d, aes(temp_ct, con_percent_scaled, color = species_ab), alpha = 1)

This looks OK. Next I wanted to try using a logit link function, to make the mean of the Beta-distribution bounded [0,1]:

beta_logit <-
brm(bf(con_percent_scaled ~ (b0 * exp(E*((1/(k*(T_c + 273.15))) - 1/(k*(temp_ct + 273.15))))) / (1 + exp(Eh*((1/(k*(Th + 273.15))) - (1/(k*(temp_ct + 273.15)))))),
b0 ~ 1 + (1|species_ab), E ~ 1, Eh ~ 1, Th ~ 1, nl = TRUE),
data = d, family = Beta(link = "logit", link_phi = "log"), prior = prior, iter = 4000, cores = 3, chains = 3)

d %>%
data_grid(temp_ct = seq_range(temp_ct, n = 50)) %>%
mutate(k = 8.62*10^-5,
T_c = 0) %>%
add_predicted_draws(beta_logit, re_formula = NA, scale = "response") %>%
ggplot(aes(x = temp_ct, y = con_percent_scaled)) +
stat_lineribbon(aes(y = .prediction), .width = c(.95), alpha = 0.4, fill = "grey") +
geom_point(data = d, aes(temp_ct, con_percent_scaled, color = species_ab), alpha = 1)

Ok, here’s where I get stuck: should the priors in the model with an logit link function be on the logit scale? Because when I extract the stancode, mu[n] is the inv_logit of the predictors: ... mu[n] = inv_logit((nlp_b0[n] .... And if that’s the case, my second problem is I don’t really have a feeling for how to “convert” the priors that make sense on the scale of the reponse to the logit scale.

• Operating System: MacOS Catalina
• brms Version: brms_2.13.5

edited: response is a fraction, not %

Anyone with some advice for me on how to deal with priors when changing from a Gaussian to a Beta model? :)

At least in brms, which goes through Stan, the priors you specify for your mean beta values are not specified a between 0 and 1, but as you suggest on the logit scale from - to + infinity. You likely want to constrain the values with a prior because even quite low numbers on the logit scale already correspond to quite extreme values on the beta scale e.g., 4.6 is about 0.99. This graph may be of some use in thinking about what values you expect on the beta scale, and the corresponding values on a logit scale. The mean and SDs on the graph correspond to a prior on the logit scale with a mean of 0, and an SD of 1.25.

Thank you for the graph! It’s very helpful.

Right, so if I would have a model of the mean (no covariates) that could take any value between 0 and 1 (call it \mu), I could write:

prior("uniform(-10, 10)", nlpar = "mu", lb = -10, ub = 10), saying all values for the mean of the Beta model are equally likely (instead of narrowing it down between 0.5 and 0.7310586 as my Gaussian model would do, i.e. the "prior(uniform(-0, 1))", ... code I had above).

But how do I find good priors when \mu is modelled as a combination of non linear parameters and variables (for example, the Sharpe-Schoolfield model)? I.e., it’s not clear to me how to set priors on specific parameters on the Sharpe-Schoolfield parameters, when the whole equation is within the inv_logit function. Sorry if I’m missing something obvious here…

Hi Max,
I certainly wouldn’t model it with the basic prior on mu as a uniform distribution from -10 to 10. I’m not sure if that is part of what you would want to do or if it is just an example, but you need to think about the conversion from that logit scale to the beta scale you care about. If you plot that uniform prior in its transformed form between 0 and 1, you will see that it places huge amount of weight well into the extremes of the distribution at either end. I presume you don’t think that most of the probability on the beta scale should be less than .01 or greater than .99? Hence, I would suggest a normal distribution or something similar, centered around 0, with not too high a standard deviation - or again, the weight will be placed in the extremes again. You can see what the implications of the priors are on this single parameter by actually simulating with runif and asking for several thousand observations, then convert them with plogis or inv_logit, and plot the numbers as a density plot. It will be really peaked at the extremes, which I imagine is not what you want? Then compare it with rnorm, mean0 and sd 1.25, transformed with plogis or inv_logit - the probability mass tapers off around the extremes.

Things indeed get more complicated and difficult to imagine when you are combining multiple parameters. You can specify that you want to really see the priors in brms by putting this into the model specification:

sample_prior = ‘only’

This will run a prior predictive simulation, and when you see or plot the results, it is showing combinations of the different priors you have put in, without updating them with any data - i.e., your results are just the priors.

The graph above should however help with thinking about what plausible values might be, and you’d have to use some of your domain knowledge for what is plausible. Let’s say you think your average or intercept value should be about .5 probability (0 on the logit scale). Then think, what size effect could one of your variables have - you can sort of eyeball effects on the different scales by looking at the graph. Perhaps the variable in question is likely to influence the average by + or - .15 on the beta scale. That effect would be easily captured (and then some) by a prior on that parameter centered around 0, with an sd of .75. These are just very rough examples, but that’s how I think of it. I can’t comment regarding the exact model you have because I don’t know anything about it, and am also not quite that good at Bayesian analysis to give you more than pointers I’m afraid!

1 Like

Yes, I only meant that as an example. I.e., if my model had been an intercept-only model, even I would have (probably) been able to convert between the scales! (sorry if that was more confusing than helpful…).

But now I have non-linear model with 4 parameters. I have a good feeling for what these parameters should be on the same scale as the response, as in the identity link example above, and here’s a figure showing the figure the code above produces with the identity-link model:

and here for the posteriors:

But for the logit link… I do like your approach to it and I think it makes sense. Especially if I had a linear model with standardized predictors variables! But I’m not sure how to apply it to this case. For instance, the parameter E describes how fast the response variable increases (exponentially) with X at X<0, where X=0 is where the function reaches its maximum, and Eh describes the rate of the fall (also exponential) for X>0. X here is temp_ct. So I struggle with how to think about it in terms of how much the average (intercept) changes with 1 unit or sd in predictor x1 as if it had been a more straightforward linear model… Thanks though! It’s much appreciated!

That is certainly a bit tricky to wrap your head around. Perhaps you might need to test out some different parameter values for these variables: plug them into your regression equation, and calculate predicted outcome scores (just using addition/multiplication etc to work out the regression outcomes, not using brms). That way you can get a better sense of what different values imply about the outcome. It is certainly difficult to think about just in your head. This could at least help you note points at which outcomes start to become totally weird and implausible.

1 Like

That sounds like a good idea! But before I dig into that, maybe I can add one more question to the thread. I read in Douma & Weedon, 2019 that it’s recommended to test alternative link functions, including even an “identity” link, “albeit with care”.

In my case it seems fine to me to use an identity link (figure in my last post), and the model for sure becomes easier to interpret and fit… I guess this is a question more fore the authors, but would this be an example of where this is done with care…? Or what are the pitfalls I’m missing here?

I would have to read the paper and properly understand your model to answer, I think. I’m not sure if the question of modeling with different link functions is the same as saying you could just treat the data as normally distributed or something, and just model it directly on the ‘probability scale’? That would be possible to do, I think it just depends on how well the model then fits the data. I work with psychological responses and if I do something like that, I typically end up with regression models that predict data points well under or over the 0-1 bounds, and the beta distribution can recreate the look of the data much better.

Yes, I understand!

I’m also not sure about that. Here’s the same plot as two posts above, but with a Gaussian model and an identity link. Here the prediction interval (not the prediction) goes below 0 (and just slightly above 1),

but with the Beta distribution and identity link the prediction interval is bound between 0 and 1 (quoted figure).

So, in my view, it seems Beta(link = "identity") does what I hoped the Beta(link = "logit")would do (not predicting outside possible data) if I had gotten that to work.

Nice, I somehow haven’t managed to make the beta thing run with “identity” as the link. Maybe I should try again.

I think the interval spreading that far over 0 in the gaussian model could be considered problematic, as with a Bayesian model we really want to pay attention to the uncertainty in the estimate, and there is no single ‘prediction’, but a range of plausible predictions. Of course, it all depends on your degree of tolerance for these things and possibly other aspects of the model that you might find more appropriate or useful. But nice that it looks like you can try to just set priors on the identity scale now, and get the benefits of the beta model without the confusion.

Thanks for your input JimBob, it’s much appreciated! I can update this thread once I land on something final, but for now I might stick to the Beta(link = "identity") approach.

Cool. It would be interesting to see, though perhaps not possible, what your regression ends up looking like and how you interpret it. As I use beta models a lot at the moment, I’m interested to see what other people do with it. Good luck!


text:
## Algebra and Trigonometry 10th Edition

The determinant of a $2 \times 2$ matrix can be computed by using the formula $det =ps-qr$ where $det =\begin{bmatrix}p & q \\r & s\end{bmatrix}$ Considering the left side, we have: $det=zw-xy$ Now, we have $det=-(yx-zw)=zw-xy$ So, it can be seen that both sides are the same.


text:
$$n^\text{th}$$ derivative?

Calculus Level 3

If $$n$$ is a non-negative integer, find the $$n^\text{th}$$ derivative of $$\dfrac{\ln x}x$$.

×


text:
# Newlfm class and indentation

I am trying to write cover letter using Latex, I found out about newlfm class, I almost got what I wanted, but I can't make him indent any paragraphs.

What I want is to get first line of every paragraph indented. Is it possible to achieve something like that using this class?

-
How about \setlength\parindent{5pt} (Place this in the preamble.)? You can change the paragraph indentation with this for the entire document. –  azetina Jun 7 '12 at 21:30
Easiest solutions are the best it seems :). Thanks –  Andna Jun 7 '12 at 21:33

You need to modify the paragraph indentation command \parindent with \setlength\parindent{length}. For example:

\documentclass{newlfm}
\usepackage{lipsum}
\setlength\parindent{1cm}

\begin{document}
\lipsum[2]
\lipsum[3]
\end{document}

See result:

-


text:
# Volumes by Cylindrical Shells Method

Let’s consider the problem of finding the volume of the solid obtained by rotating about the $x$-axis or parallel to $x$-axis of the region, where the core idea of cylindrical shells method for finding volumes. We get a cylinder if we slice perpendicular to the $y$-axis. But to compute the washer’s inner and outer radius, we would have to solve the cubic equation for $x$ in terms of $y$. The method of cylindrical shells is used to find the volume in this case, which is easier to use in such a case. We can see a cylindrical shell with an inner radius, outer radius, and height. Its volume is calculated by subtracting the inner cylinder’s volume from the outer cylinder’s volume.

### Worked Example of Finding a Volume by Cylindrical Shells Method

The diagram shows the graph of $\displaystyle f(x) = \frac{x}{1 + x^2}$.

The area bounded by $y = f(x)$, the line $x = 1$ and the $x$-axis is rotated about the line $x=1$ to form a solid volume. Use the cylindrical shells method to find the volume of the solid.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \delta V &= \pi \Big[(1-x + \delta x)^2-(1-x)^2 \Big] \times y \\ &= \pi \Big[(1-x)^2 + 2(1-x) \delta x +(\delta x)^2-(1-x)^2 \Big] \times y \\ &= \pi \Big[2(1-x) \delta x \Big] \times \frac{x}{1+x^2} &\color{red} (\delta x)^2 \approx 0 \\ &= 2 \pi \Big[ \frac{x-x^2}{1 + x^2} \Big] \delta x \\ &= 2 \pi \Big[ \frac{1-(1+x^2) + x}{1 + x^2} \Big] \delta x \\ &= 2 \pi \Big[ \frac{1}{1+x^2}-1 + \frac{x}{1+x^2} \Big] \delta x \\ V &= 2 \pi \int_{0}^{1} \Big[ \frac{1}{1+x^2}-1 + \frac{x}{1+x^2} \Big] dx \\ &= 2 \pi \Big[\tan^{-1} x-x + \frac{1}{2}\log_e (1 + x^2) \Big]_{0}^{1} \\ &= 2 \pi \Big(\frac{\pi}{4}-1 + \frac{1}{2} \log_e{2} \Big) \end{aligned}


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## How to calculate the coordinate by depth， resolution and perspective angle

Typically: "How do I... ", "How can I... " questions
Zhuang
Posts: 28
Joined: 02 Aug 2017, 14:43

### How to calculate the coordinate by depth， resolution and perspective angle

Hi， coppelia
Recently, I try to use depth sensor to calculate a world position of a pixel. I see the sensor is different from real camera. I am not sure if the method that is used to calculate a position by perspective angel and depth is right.

Code: Select all

    camXHalfAngle = camXAngleInDegrees * 0.5 * math.pi/180
camYHalfAngle = math.atan(math.tan(camXAngleInDegrees*0.5* camYResolution/camXResolution))*math.pi/180
nearClippingPlane = 0.01
depthAmplitude = 3.5
xAngle = ((320-u+0.5)/320)*camXHalfAngle
yAngle = ((240-v+0.5)/240)*camYHalfAngle
depthValue = image_depth[v-1][u-1]  # u and v are the coordinate of the pixel. because the depth is turned so [v-1]#[u-1]
zCoord = nearClippingPlane+depthAmplitude * depthValue
xCoord = math.tan(xAngle) * zCoord
yCoord = math.tan(yAngle) * zCoord
pixels_coordinare_kinetic = np.mat([[xCoord],[yCoord],[zCoord],[1]], np.float)

cheers,
zhuang

coppelia


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# Integrate $\int_C(\frac{1}{z-4}+\frac{i}{z-i})dz$

Integrate $$\int_C\left(\frac{1}{z-4}+\frac{i}{z-i}\right)dz$$ where $C$ is the straight line starting from $2 + 2i$ to $2 − 2i$

As far as I understand, branch cut is $$z\leq4\quad\text{and}\quad z\leq i$$ But $2 + 2i$ and $2 - 2i$ are both in these intervals. How would I have to change the argument then?

• This integral is integrated over the curve C which is a straight line starting from 2 + 2i to 2 − 2i – JMK Mar 9 '17 at 16:33
• Just write the line $C$ in function of $z$, after make a change of variable to make it a real line and integrate as if it was over some real interval. – Masacroso Mar 9 '17 at 17:28

Hint: Let $C: z=2-2ti$ where $t\in[-1,1]$ then set it to your integral with $dz=-2i\,dt$.


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# Resistors for preventing short circuiting

Discussion in 'Homework Help' started by wuz, Jul 13, 2011.

1. ### wuz Thread Starter New Member

Oct 4, 2010
8
0
This may be a very stupid question and I apologise in advance.
I find the following schematics (source:http://www.circuitdb.com). 220R are used between the micro pins RA0 - RA5 & the connector, supposedly to prevent accidental short circuiting.
- I'm wondering when do you actually need these resistors? My guess is that these resistors are only used when the pins may be exposed to voltage higher than the micro rating (e.g. connectors that can be plugged into the wrong V)?
- How do you decide the values of the resistors? Is it R = (Vmaximum possible) - V[micro rating]) / I[micro rating]? So to protect a 5V micro with 10mA single pin current rating may be connected to a 12V rail, R required = (12 - 5) / 0.01 = 700 Ohms?

Thanks!

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2. ### Jony130 AAC Fanatic!

Feb 17, 2009
3,993
1,116
The main job for this 220 Ohm resistor is to limit the base current for the BJT.
And in the same time provide enough base current to ensure the BJT will be in saturation.
You can not connect the BJT base directly to the output PIC output, without limiting the base current otherwise you'll burn the PIC and the BJT.

wuz likes this.
3. ### Georacer Moderator

Nov 25, 2009
5,151
1,266
What Jony said. You need an upper limit for your voltage. The value you are looking for is
$R=\frac{V_{out\-PIC}}{I_{base\-max}}$

You don't want to put anything much higher though.

wuz likes this.
4. ### ErnieM AAC Fanatic!

Apr 24, 2011
7,442
1,628
The 220R resistors going from the RA port are as you suspected a "belt and suspender" item, you don't absolutely need them, they may interfere with something else connected there, but they do provide a slight amount of protection from whatever unknown device you add onto that connector.

I would suspect their value is based on using the same value as other resistors on the board.

Generally, when you connect a 5V pin to a 12V source you are no longer looking at the pins drive (here, sink) capability, but the internal ESD protection diode typically found on most (not all) micro I/O pins. The current into that diode is a separate spec from the drive current spec.

When the pin is either an input or outputting a high then you have (12-5 + diode drop) V on your resistor as you are driving the internal diode.

When the output is being driven low you have the full 12V on it, and the current is the output sink current as you are driving the (low) output pin.

wuz likes this.
5. ### praondevou AAC Fanatic!

Jul 9, 2011
2,936
489
Your RA ports go to a connector "for limit switches, sensors etc." . This means these RA ports are configured as inputs referenced to ground. It doesn't look like if you are supposed to use sensors with higher voltages on this port (which you then could accidentally pass to the MC).

In the PICs datasheet there is indicated 25mA absolute maximum current source or sink on a port pin.
If for any reason, maybe at a reset/startup or something like this, the pin is configured as an output and you have the corresponding pin connected to ground via a NO switch for example, current will be limited to 23mA...

wuz likes this.
6. ### wuz Thread Starter New Member

Oct 4, 2010
8
0
Thanks for all the replies

Jony130 & Georacer: I was referring to the 220R (connected to the connectors) on the RA ports not the RB ports.

praondevou: I think I now understand the usage of 220R after reading your comment. I'm probably just repeating you here, but just want to confirm I fully understand the issue.
These are meant for protecting the directly short between the PIC pins (when configured as a low impedance output pin) & the 5V/0V voltage rails. And the value of R needs to be > 5V / I [maximum sourcing / sinking current of the PIC]. So I guess it's good practice to have current limiting resistors whenever the I/O pins are directly routed to a connector going to a different board?

7. ### praondevou AAC Fanatic!

Jul 9, 2011
2,936
489
Actually it depends on what you have on the other board, how far it is away, if it has to be protected against noise pick-up etc. The only thing I could imagine in your case was the one I described: Accidentally set a portA pin configured as output and having something (like a switch) connecting it to ground.

wuz likes this.
8. ### wuz Thread Starter New Member

Oct 4, 2010
8
0
Thanks very much for your help


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### 10.   Eigenvector Composition of Levels

The wave functions of levels are often expressed as eigenvectors that are linear combinations of basis states in one of the standard coupling schemes. Thus, the wave function Ψ(α J) for a level labeled α J might be expressed in terms of normalized LS coupling basis states Φ(γSLJ):

 (4) The c(γSLJ) are expansion coefficients, and (5)
The squared expansion coefficients for the various γSL terms in the composition of the α J level are conveniently expressed as percentages, whose sum is 100 %. Thus the percentage contributed by the pure Russell-Saunders state γSLJ is equal to 100  × |cSLJ)|2. The notation for Russell-Saunders basis states has been used only for concreteness; the eigenvectors may be expressed in any coupling scheme, and the coupling schemes may be different for different configurations included in a single calculation (with configuration interaction). "Intermediate coupling" conditions for a configuration are such that calculations in both LS and jj coupling yield some eigenvectors representing significant mixtures of basis states.

The largest percentage in the composition of a level is called the purity of the level in that coupling scheme. The coupling scheme (or combination of coupling schemes if more than one configuration is involved) that results in the largest average purity for all the levels in a calculation is usually best for naming the levels. With regard to any particular calculation, one does well to remember that, as with other calculated quantities, the resulting eigenvectors depend on a specific theoretical model and are subject to the inaccuracies of whatever approximations the model involves.

Theoretical calculations of experimental energy level structures have yielded many eigenvectors having significantly less than 50 % purity in any coupling scheme. Since many of the corresponding levels have nevertheless been assigned names by spectroscopists, some caution is advisable in the acceptance of level designations found in the literature.

### 11.   Ground Levels and Ionization Energies for the Neutral Atoms

Table of Ground Levels and Ionization Energies for the Neutral Atoms
When using the above table, be sure to use your "Back" button to return to this document.

The ground-state electron configurations of elements heavier than neon are shortened in the table by using rare-gas element symbols in brackets to represent the corresponding electrons. The ground levels of all neutral atoms have reasonably meaningful LS-coupling names, the corresponding eigenvector percentages lying in the range from ~55 % to 100 %. These names are listed in the table, except for Pa, U, and Np; the lowest few ground-configuration levels of these atoms comprise better 5N(L1S1J1), 6dj7s2 (J1 j) terms than LS-coupling terms. The relatively large spin-orbit interaction of the 6p electrons produces jj-coupling structures for the (6p21/2)0, (6p21/26p3/2)o3/2, and (6p21/26p23/2)2 ground levels of the 6p2, 6p3, and 6p4 configurations of neutral Pb, Bi, and Po, respectively. As noted in the section jj Coupling of Equivalent Electrons, the jj-coupling names are more appropriate for these atoms than the alternative LS-coupling designations in the table.

The ionization energies in the table are from recent compilations [13, 14]. The uncertainties are mainly in the range from less than one to several units in the last decimal place, but a few of the values may be in error by 20 or more units in the final place; i.e., the error of some of the two place values could be greater than 0.2 eV. Although no more than four decimal places are given here, values for both the neutral and singly-ionized atoms are given to their full accuracies in [14].


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# Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds

I have two questions which are somewhat related:

(a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that this does not happen for any $\mathbb R^n$, $n\neq 4$. In other words, every smooth structure on $\mathbb R^n$, $n\neq 4$, is diffeomorphic to the standard structure. (Whose result is it?!)

MY QUESTION: What about analytic structures on $\mathbb R^n$? or complex-analytic structures on $\mathbb C^n$? Have these questions been answered?

(b) Let $(M,O_M)$ be a smooth (respectively, analytic) supermanifold (in the sense of Berezin, Kostant, Leites, Manin, etc.), that is, $M$ is a manifold and $O_M$ is a sheaf of $\mathbb Z_2$-graded algebras which is locally isomorphic to $C^\infty(\mathbb R^m)\otimes\Lambda \mathbb R^n$ (respectively, $C^\omega(\mathbb R^m)\otimes\Lambda \mathbb R^n$).

MY QUESTION: In his book "Gauge Field Theory and Complex Geometry", Manin defines the sheaf of ideals $J_M\subseteq O_M$ by $J_M=O_{M,1}^2+O_{M,1}$ and calls it the the "ideal generated by odd elements" (see $\S4.1.3$, page 182). Then he claims that for supermanifolds, $J_M$ is equal to the sheaf of ideals of nilpotent elements. I can prove (using partition of unity) that this statement is correct for smooth supermanifolds, but not for analytic supermanifolds. I have seen this statement in various places, but without a proper explanation.

I am not even sure why $O_{M,1}^2+O_{M,1}$ is a sheaf! Of course one can consider its sheafification, but I am not convinced that this is what they are doing, and in any case I don't have a counterexample.

• Michael Freedman found the first exotic $\Bbb R^4$ in 1982. As far as I know Milnor never did any work on exotic $\Bbb R^4$'s. – Henry T. Horton Oct 25 '13 at 1:15
• Thanks. Corrected. Sorry, I am by no means an expert in differential topology. – Valerie Oct 25 '13 at 2:00
• You might split this into 2 questions, since (a) and (b) are not very closely related. – Ben McKay Oct 25 '13 at 6:25
• (b): Since it is a sheaf, you only have to work locally, looking at nilpotent elements defined in an open set, so you can work with a ball in Euclidean space, if you like. This should give global supermanifold coordinates. – Ben McKay Oct 25 '13 at 6:27